Properties

Label 3200.2.a.bk.1.2
Level $3200$
Weight $2$
Character 3200.1
Self dual yes
Analytic conductor $25.552$
Analytic rank $0$
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] = [3200,2,Mod(1,3200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("3200.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3200 = 2^{7} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3200.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: \(25.5521286468\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) 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 640)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 3200.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.23607 q^{3} +1.23607 q^{7} +7.47214 q^{9} +O(q^{10})\) \(q+3.23607 q^{3} +1.23607 q^{7} +7.47214 q^{9} -2.00000 q^{11} +4.47214 q^{13} +4.47214 q^{17} +4.47214 q^{19} +4.00000 q^{21} -9.23607 q^{23} +14.4721 q^{27} +2.00000 q^{29} +2.47214 q^{31} -6.47214 q^{33} -10.9443 q^{37} +14.4721 q^{39} +3.52786 q^{41} -5.70820 q^{43} -2.76393 q^{47} -5.47214 q^{49} +14.4721 q^{51} -8.47214 q^{53} +14.4721 q^{57} +0.472136 q^{59} +6.00000 q^{61} +9.23607 q^{63} +5.70820 q^{67} -29.8885 q^{69} -6.47214 q^{71} +4.47214 q^{73} -2.47214 q^{77} +4.94427 q^{79} +24.4164 q^{81} -9.70820 q^{83} +6.47214 q^{87} +2.94427 q^{89} +5.52786 q^{91} +8.00000 q^{93} +7.52786 q^{97} -14.9443 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 2 q^{7} + 6 q^{9} - 4 q^{11} + 8 q^{21} - 14 q^{23} + 20 q^{27} + 4 q^{29} - 4 q^{31} - 4 q^{33} - 4 q^{37} + 20 q^{39} + 16 q^{41} + 2 q^{43} - 10 q^{47} - 2 q^{49} + 20 q^{51} - 8 q^{53} + 20 q^{57} - 8 q^{59} + 12 q^{61} + 14 q^{63} - 2 q^{67} - 24 q^{69} - 4 q^{71} + 4 q^{77} - 8 q^{79} + 22 q^{81} - 6 q^{83} + 4 q^{87} - 12 q^{89} + 20 q^{91} + 16 q^{93} + 24 q^{97} - 12 q^{99}+O(q^{100}) \) 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
\(3\) 3.23607 1.86834 0.934172 0.356822i \(-0.116140\pi\)
0.934172 + 0.356822i \(0.116140\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.23607 0.467190 0.233595 0.972334i \(-0.424951\pi\)
0.233595 + 0.972334i \(0.424951\pi\)
\(8\) 0 0
\(9\) 7.47214 2.49071
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 4.47214 1.24035 0.620174 0.784465i \(-0.287062\pi\)
0.620174 + 0.784465i \(0.287062\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.47214 1.08465 0.542326 0.840168i \(-0.317544\pi\)
0.542326 + 0.840168i \(0.317544\pi\)
\(18\) 0 0
\(19\) 4.47214 1.02598 0.512989 0.858395i \(-0.328538\pi\)
0.512989 + 0.858395i \(0.328538\pi\)
\(20\) 0 0
\(21\) 4.00000 0.872872
\(22\) 0 0
\(23\) −9.23607 −1.92585 −0.962927 0.269763i \(-0.913055\pi\)
−0.962927 + 0.269763i \(0.913055\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 14.4721 2.78516
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) 2.47214 0.444009 0.222004 0.975046i \(-0.428740\pi\)
0.222004 + 0.975046i \(0.428740\pi\)
\(32\) 0 0
\(33\) −6.47214 −1.12665
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −10.9443 −1.79923 −0.899614 0.436687i \(-0.856152\pi\)
−0.899614 + 0.436687i \(0.856152\pi\)
\(38\) 0 0
\(39\) 14.4721 2.31740
\(40\) 0 0
\(41\) 3.52786 0.550960 0.275480 0.961307i \(-0.411163\pi\)
0.275480 + 0.961307i \(0.411163\pi\)
\(42\) 0 0
\(43\) −5.70820 −0.870493 −0.435246 0.900311i \(-0.643339\pi\)
−0.435246 + 0.900311i \(0.643339\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.76393 −0.403161 −0.201580 0.979472i \(-0.564608\pi\)
−0.201580 + 0.979472i \(0.564608\pi\)
\(48\) 0 0
\(49\) −5.47214 −0.781734
\(50\) 0 0
\(51\) 14.4721 2.02650
\(52\) 0 0
\(53\) −8.47214 −1.16374 −0.581869 0.813283i \(-0.697678\pi\)
−0.581869 + 0.813283i \(0.697678\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 14.4721 1.91688
\(58\) 0 0
\(59\) 0.472136 0.0614669 0.0307334 0.999528i \(-0.490216\pi\)
0.0307334 + 0.999528i \(0.490216\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 0 0
\(63\) 9.23607 1.16364
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 5.70820 0.697368 0.348684 0.937240i \(-0.386629\pi\)
0.348684 + 0.937240i \(0.386629\pi\)
\(68\) 0 0
\(69\) −29.8885 −3.59816
\(70\) 0 0
\(71\) −6.47214 −0.768101 −0.384051 0.923312i \(-0.625471\pi\)
−0.384051 + 0.923312i \(0.625471\pi\)
\(72\) 0 0
\(73\) 4.47214 0.523424 0.261712 0.965146i \(-0.415713\pi\)
0.261712 + 0.965146i \(0.415713\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.47214 −0.281726
\(78\) 0 0
\(79\) 4.94427 0.556274 0.278137 0.960541i \(-0.410283\pi\)
0.278137 + 0.960541i \(0.410283\pi\)
\(80\) 0 0
\(81\) 24.4164 2.71293
\(82\) 0 0
\(83\) −9.70820 −1.06561 −0.532807 0.846237i \(-0.678863\pi\)
−0.532807 + 0.846237i \(0.678863\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 6.47214 0.693886
\(88\) 0 0
\(89\) 2.94427 0.312092 0.156046 0.987750i \(-0.450125\pi\)
0.156046 + 0.987750i \(0.450125\pi\)
\(90\) 0 0
\(91\) 5.52786 0.579478
\(92\) 0 0
\(93\) 8.00000 0.829561
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 7.52786 0.764339 0.382169 0.924092i \(-0.375177\pi\)
0.382169 + 0.924092i \(0.375177\pi\)
\(98\) 0 0
\(99\) −14.9443 −1.50196
\(100\) 0 0
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 0 0
\(103\) −1.23607 −0.121793 −0.0608967 0.998144i \(-0.519396\pi\)
−0.0608967 + 0.998144i \(0.519396\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.7639 1.23394 0.616968 0.786988i \(-0.288361\pi\)
0.616968 + 0.786988i \(0.288361\pi\)
\(108\) 0 0
\(109\) 2.94427 0.282010 0.141005 0.990009i \(-0.454967\pi\)
0.141005 + 0.990009i \(0.454967\pi\)
\(110\) 0 0
\(111\) −35.4164 −3.36158
\(112\) 0 0
\(113\) −14.9443 −1.40584 −0.702919 0.711269i \(-0.748121\pi\)
−0.702919 + 0.711269i \(0.748121\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 33.4164 3.08935
\(118\) 0 0
\(119\) 5.52786 0.506738
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 11.4164 1.02938
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 2.18034 0.193474 0.0967369 0.995310i \(-0.469159\pi\)
0.0967369 + 0.995310i \(0.469159\pi\)
\(128\) 0 0
\(129\) −18.4721 −1.62638
\(130\) 0 0
\(131\) −2.94427 −0.257242 −0.128621 0.991694i \(-0.541055\pi\)
−0.128621 + 0.991694i \(0.541055\pi\)
\(132\) 0 0
\(133\) 5.52786 0.479327
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 0 0
\(139\) 11.5279 0.977781 0.488890 0.872345i \(-0.337402\pi\)
0.488890 + 0.872345i \(0.337402\pi\)
\(140\) 0 0
\(141\) −8.94427 −0.753244
\(142\) 0 0
\(143\) −8.94427 −0.747958
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −17.7082 −1.46055
\(148\) 0 0
\(149\) 10.9443 0.896590 0.448295 0.893886i \(-0.352031\pi\)
0.448295 + 0.893886i \(0.352031\pi\)
\(150\) 0 0
\(151\) 3.41641 0.278023 0.139012 0.990291i \(-0.455607\pi\)
0.139012 + 0.990291i \(0.455607\pi\)
\(152\) 0 0
\(153\) 33.4164 2.70156
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 22.9443 1.83115 0.915576 0.402145i \(-0.131735\pi\)
0.915576 + 0.402145i \(0.131735\pi\)
\(158\) 0 0
\(159\) −27.4164 −2.17426
\(160\) 0 0
\(161\) −11.4164 −0.899739
\(162\) 0 0
\(163\) −1.70820 −0.133797 −0.0668984 0.997760i \(-0.521310\pi\)
−0.0668984 + 0.997760i \(0.521310\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −16.6525 −1.28861 −0.644304 0.764770i \(-0.722853\pi\)
−0.644304 + 0.764770i \(0.722853\pi\)
\(168\) 0 0
\(169\) 7.00000 0.538462
\(170\) 0 0
\(171\) 33.4164 2.55542
\(172\) 0 0
\(173\) 6.94427 0.527963 0.263982 0.964528i \(-0.414964\pi\)
0.263982 + 0.964528i \(0.414964\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1.52786 0.114841
\(178\) 0 0
\(179\) −11.5279 −0.861633 −0.430817 0.902440i \(-0.641774\pi\)
−0.430817 + 0.902440i \(0.641774\pi\)
\(180\) 0 0
\(181\) 6.94427 0.516164 0.258082 0.966123i \(-0.416910\pi\)
0.258082 + 0.966123i \(0.416910\pi\)
\(182\) 0 0
\(183\) 19.4164 1.43530
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −8.94427 −0.654070
\(188\) 0 0
\(189\) 17.8885 1.30120
\(190\) 0 0
\(191\) 15.4164 1.11549 0.557746 0.830012i \(-0.311666\pi\)
0.557746 + 0.830012i \(0.311666\pi\)
\(192\) 0 0
\(193\) 20.4721 1.47362 0.736808 0.676102i \(-0.236332\pi\)
0.736808 + 0.676102i \(0.236332\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −8.47214 −0.603615 −0.301807 0.953369i \(-0.597590\pi\)
−0.301807 + 0.953369i \(0.597590\pi\)
\(198\) 0 0
\(199\) −16.9443 −1.20115 −0.600574 0.799569i \(-0.705061\pi\)
−0.600574 + 0.799569i \(0.705061\pi\)
\(200\) 0 0
\(201\) 18.4721 1.30292
\(202\) 0 0
\(203\) 2.47214 0.173510
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −69.0132 −4.79675
\(208\) 0 0
\(209\) −8.94427 −0.618688
\(210\) 0 0
\(211\) −18.9443 −1.30418 −0.652089 0.758143i \(-0.726107\pi\)
−0.652089 + 0.758143i \(0.726107\pi\)
\(212\) 0 0
\(213\) −20.9443 −1.43508
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 3.05573 0.207436
\(218\) 0 0
\(219\) 14.4721 0.977936
\(220\) 0 0
\(221\) 20.0000 1.34535
\(222\) 0 0
\(223\) −13.2361 −0.886353 −0.443176 0.896434i \(-0.646148\pi\)
−0.443176 + 0.896434i \(0.646148\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −12.1803 −0.808438 −0.404219 0.914662i \(-0.632457\pi\)
−0.404219 + 0.914662i \(0.632457\pi\)
\(228\) 0 0
\(229\) 14.9443 0.987545 0.493773 0.869591i \(-0.335618\pi\)
0.493773 + 0.869591i \(0.335618\pi\)
\(230\) 0 0
\(231\) −8.00000 −0.526361
\(232\) 0 0
\(233\) −24.4721 −1.60322 −0.801611 0.597845i \(-0.796024\pi\)
−0.801611 + 0.597845i \(0.796024\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 16.0000 1.03931
\(238\) 0 0
\(239\) −4.94427 −0.319818 −0.159909 0.987132i \(-0.551120\pi\)
−0.159909 + 0.987132i \(0.551120\pi\)
\(240\) 0 0
\(241\) −18.3607 −1.18272 −0.591358 0.806409i \(-0.701408\pi\)
−0.591358 + 0.806409i \(0.701408\pi\)
\(242\) 0 0
\(243\) 35.5967 2.28353
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 20.0000 1.27257
\(248\) 0 0
\(249\) −31.4164 −1.99093
\(250\) 0 0
\(251\) −2.00000 −0.126239 −0.0631194 0.998006i \(-0.520105\pi\)
−0.0631194 + 0.998006i \(0.520105\pi\)
\(252\) 0 0
\(253\) 18.4721 1.16133
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −14.9443 −0.932198 −0.466099 0.884733i \(-0.654341\pi\)
−0.466099 + 0.884733i \(0.654341\pi\)
\(258\) 0 0
\(259\) −13.5279 −0.840581
\(260\) 0 0
\(261\) 14.9443 0.925027
\(262\) 0 0
\(263\) −4.29180 −0.264643 −0.132322 0.991207i \(-0.542243\pi\)
−0.132322 + 0.991207i \(0.542243\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 9.52786 0.583096
\(268\) 0 0
\(269\) −11.8885 −0.724857 −0.362429 0.932012i \(-0.618052\pi\)
−0.362429 + 0.932012i \(0.618052\pi\)
\(270\) 0 0
\(271\) −20.3607 −1.23682 −0.618412 0.785854i \(-0.712224\pi\)
−0.618412 + 0.785854i \(0.712224\pi\)
\(272\) 0 0
\(273\) 17.8885 1.08266
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 6.94427 0.417241 0.208620 0.977997i \(-0.433103\pi\)
0.208620 + 0.977997i \(0.433103\pi\)
\(278\) 0 0
\(279\) 18.4721 1.10590
\(280\) 0 0
\(281\) 12.4721 0.744025 0.372013 0.928228i \(-0.378668\pi\)
0.372013 + 0.928228i \(0.378668\pi\)
\(282\) 0 0
\(283\) −21.7082 −1.29042 −0.645209 0.764006i \(-0.723230\pi\)
−0.645209 + 0.764006i \(0.723230\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 4.36068 0.257403
\(288\) 0 0
\(289\) 3.00000 0.176471
\(290\) 0 0
\(291\) 24.3607 1.42805
\(292\) 0 0
\(293\) 11.8885 0.694536 0.347268 0.937766i \(-0.387109\pi\)
0.347268 + 0.937766i \(0.387109\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −28.9443 −1.67952
\(298\) 0 0
\(299\) −41.3050 −2.38873
\(300\) 0 0
\(301\) −7.05573 −0.406685
\(302\) 0 0
\(303\) 32.3607 1.85907
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 13.7082 0.782369 0.391184 0.920312i \(-0.372066\pi\)
0.391184 + 0.920312i \(0.372066\pi\)
\(308\) 0 0
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) −11.4164 −0.647365 −0.323683 0.946166i \(-0.604921\pi\)
−0.323683 + 0.946166i \(0.604921\pi\)
\(312\) 0 0
\(313\) −24.8328 −1.40363 −0.701817 0.712357i \(-0.747628\pi\)
−0.701817 + 0.712357i \(0.747628\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −11.5279 −0.647469 −0.323735 0.946148i \(-0.604939\pi\)
−0.323735 + 0.946148i \(0.604939\pi\)
\(318\) 0 0
\(319\) −4.00000 −0.223957
\(320\) 0 0
\(321\) 41.3050 2.30542
\(322\) 0 0
\(323\) 20.0000 1.11283
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 9.52786 0.526892
\(328\) 0 0
\(329\) −3.41641 −0.188353
\(330\) 0 0
\(331\) −19.8885 −1.09317 −0.546587 0.837403i \(-0.684073\pi\)
−0.546587 + 0.837403i \(0.684073\pi\)
\(332\) 0 0
\(333\) −81.7771 −4.48136
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −22.9443 −1.24985 −0.624927 0.780683i \(-0.714871\pi\)
−0.624927 + 0.780683i \(0.714871\pi\)
\(338\) 0 0
\(339\) −48.3607 −2.62659
\(340\) 0 0
\(341\) −4.94427 −0.267747
\(342\) 0 0
\(343\) −15.4164 −0.832408
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 12.7639 0.685204 0.342602 0.939481i \(-0.388692\pi\)
0.342602 + 0.939481i \(0.388692\pi\)
\(348\) 0 0
\(349\) −2.94427 −0.157603 −0.0788016 0.996890i \(-0.525109\pi\)
−0.0788016 + 0.996890i \(0.525109\pi\)
\(350\) 0 0
\(351\) 64.7214 3.45457
\(352\) 0 0
\(353\) −22.9443 −1.22120 −0.610600 0.791939i \(-0.709072\pi\)
−0.610600 + 0.791939i \(0.709072\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 17.8885 0.946762
\(358\) 0 0
\(359\) 32.9443 1.73873 0.869366 0.494169i \(-0.164527\pi\)
0.869366 + 0.494169i \(0.164527\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −22.6525 −1.18895
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −31.7082 −1.65515 −0.827577 0.561352i \(-0.810282\pi\)
−0.827577 + 0.561352i \(0.810282\pi\)
\(368\) 0 0
\(369\) 26.3607 1.37228
\(370\) 0 0
\(371\) −10.4721 −0.543686
\(372\) 0 0
\(373\) 22.9443 1.18801 0.594005 0.804462i \(-0.297546\pi\)
0.594005 + 0.804462i \(0.297546\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 8.94427 0.460653
\(378\) 0 0
\(379\) 24.4721 1.25705 0.628525 0.777790i \(-0.283659\pi\)
0.628525 + 0.777790i \(0.283659\pi\)
\(380\) 0 0
\(381\) 7.05573 0.361476
\(382\) 0 0
\(383\) 23.7082 1.21143 0.605716 0.795681i \(-0.292887\pi\)
0.605716 + 0.795681i \(0.292887\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −42.6525 −2.16815
\(388\) 0 0
\(389\) 23.8885 1.21120 0.605599 0.795770i \(-0.292934\pi\)
0.605599 + 0.795770i \(0.292934\pi\)
\(390\) 0 0
\(391\) −41.3050 −2.08888
\(392\) 0 0
\(393\) −9.52786 −0.480617
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −3.52786 −0.177058 −0.0885292 0.996074i \(-0.528217\pi\)
−0.0885292 + 0.996074i \(0.528217\pi\)
\(398\) 0 0
\(399\) 17.8885 0.895547
\(400\) 0 0
\(401\) −2.00000 −0.0998752 −0.0499376 0.998752i \(-0.515902\pi\)
−0.0499376 + 0.998752i \(0.515902\pi\)
\(402\) 0 0
\(403\) 11.0557 0.550725
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 21.8885 1.08497
\(408\) 0 0
\(409\) −9.41641 −0.465611 −0.232806 0.972523i \(-0.574791\pi\)
−0.232806 + 0.972523i \(0.574791\pi\)
\(410\) 0 0
\(411\) −6.47214 −0.319247
\(412\) 0 0
\(413\) 0.583592 0.0287167
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 37.3050 1.82683
\(418\) 0 0
\(419\) 4.47214 0.218478 0.109239 0.994016i \(-0.465159\pi\)
0.109239 + 0.994016i \(0.465159\pi\)
\(420\) 0 0
\(421\) −24.8328 −1.21028 −0.605139 0.796120i \(-0.706882\pi\)
−0.605139 + 0.796120i \(0.706882\pi\)
\(422\) 0 0
\(423\) −20.6525 −1.00416
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 7.41641 0.358905
\(428\) 0 0
\(429\) −28.9443 −1.39744
\(430\) 0 0
\(431\) −20.3607 −0.980739 −0.490370 0.871515i \(-0.663138\pi\)
−0.490370 + 0.871515i \(0.663138\pi\)
\(432\) 0 0
\(433\) −8.47214 −0.407145 −0.203572 0.979060i \(-0.565255\pi\)
−0.203572 + 0.979060i \(0.565255\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −41.3050 −1.97588
\(438\) 0 0
\(439\) −32.9443 −1.57234 −0.786172 0.618008i \(-0.787940\pi\)
−0.786172 + 0.618008i \(0.787940\pi\)
\(440\) 0 0
\(441\) −40.8885 −1.94707
\(442\) 0 0
\(443\) −26.6525 −1.26630 −0.633149 0.774030i \(-0.718238\pi\)
−0.633149 + 0.774030i \(0.718238\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 35.4164 1.67514
\(448\) 0 0
\(449\) 25.4164 1.19947 0.599737 0.800197i \(-0.295272\pi\)
0.599737 + 0.800197i \(0.295272\pi\)
\(450\) 0 0
\(451\) −7.05573 −0.332241
\(452\) 0 0
\(453\) 11.0557 0.519443
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 31.8885 1.49168 0.745842 0.666123i \(-0.232048\pi\)
0.745842 + 0.666123i \(0.232048\pi\)
\(458\) 0 0
\(459\) 64.7214 3.02093
\(460\) 0 0
\(461\) 29.7771 1.38686 0.693429 0.720525i \(-0.256099\pi\)
0.693429 + 0.720525i \(0.256099\pi\)
\(462\) 0 0
\(463\) −39.1246 −1.81827 −0.909137 0.416496i \(-0.863258\pi\)
−0.909137 + 0.416496i \(0.863258\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 7.59675 0.351536 0.175768 0.984432i \(-0.443759\pi\)
0.175768 + 0.984432i \(0.443759\pi\)
\(468\) 0 0
\(469\) 7.05573 0.325803
\(470\) 0 0
\(471\) 74.2492 3.42122
\(472\) 0 0
\(473\) 11.4164 0.524927
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −63.3050 −2.89853
\(478\) 0 0
\(479\) 4.94427 0.225910 0.112955 0.993600i \(-0.463968\pi\)
0.112955 + 0.993600i \(0.463968\pi\)
\(480\) 0 0
\(481\) −48.9443 −2.23167
\(482\) 0 0
\(483\) −36.9443 −1.68102
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 17.2361 0.781041 0.390520 0.920594i \(-0.372295\pi\)
0.390520 + 0.920594i \(0.372295\pi\)
\(488\) 0 0
\(489\) −5.52786 −0.249979
\(490\) 0 0
\(491\) −38.9443 −1.75753 −0.878765 0.477254i \(-0.841632\pi\)
−0.878765 + 0.477254i \(0.841632\pi\)
\(492\) 0 0
\(493\) 8.94427 0.402830
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −8.00000 −0.358849
\(498\) 0 0
\(499\) 41.4164 1.85405 0.927027 0.374996i \(-0.122356\pi\)
0.927027 + 0.374996i \(0.122356\pi\)
\(500\) 0 0
\(501\) −53.8885 −2.40756
\(502\) 0 0
\(503\) −35.1246 −1.56613 −0.783065 0.621940i \(-0.786345\pi\)
−0.783065 + 0.621940i \(0.786345\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 22.6525 1.00603
\(508\) 0 0
\(509\) −36.8328 −1.63259 −0.816293 0.577638i \(-0.803974\pi\)
−0.816293 + 0.577638i \(0.803974\pi\)
\(510\) 0 0
\(511\) 5.52786 0.244538
\(512\) 0 0
\(513\) 64.7214 2.85752
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 5.52786 0.243115
\(518\) 0 0
\(519\) 22.4721 0.986417
\(520\) 0 0
\(521\) 19.8885 0.871333 0.435666 0.900108i \(-0.356513\pi\)
0.435666 + 0.900108i \(0.356513\pi\)
\(522\) 0 0
\(523\) 38.0689 1.66464 0.832318 0.554298i \(-0.187013\pi\)
0.832318 + 0.554298i \(0.187013\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 11.0557 0.481595
\(528\) 0 0
\(529\) 62.3050 2.70891
\(530\) 0 0
\(531\) 3.52786 0.153096
\(532\) 0 0
\(533\) 15.7771 0.683382
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −37.3050 −1.60983
\(538\) 0 0
\(539\) 10.9443 0.471403
\(540\) 0 0
\(541\) 11.8885 0.511128 0.255564 0.966792i \(-0.417739\pi\)
0.255564 + 0.966792i \(0.417739\pi\)
\(542\) 0 0
\(543\) 22.4721 0.964372
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 10.6525 0.455467 0.227733 0.973724i \(-0.426869\pi\)
0.227733 + 0.973724i \(0.426869\pi\)
\(548\) 0 0
\(549\) 44.8328 1.91342
\(550\) 0 0
\(551\) 8.94427 0.381039
\(552\) 0 0
\(553\) 6.11146 0.259886
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 30.9443 1.31115 0.655575 0.755130i \(-0.272426\pi\)
0.655575 + 0.755130i \(0.272426\pi\)
\(558\) 0 0
\(559\) −25.5279 −1.07971
\(560\) 0 0
\(561\) −28.9443 −1.22203
\(562\) 0 0
\(563\) −17.7082 −0.746312 −0.373156 0.927769i \(-0.621724\pi\)
−0.373156 + 0.927769i \(0.621724\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 30.1803 1.26746
\(568\) 0 0
\(569\) 10.3607 0.434342 0.217171 0.976134i \(-0.430317\pi\)
0.217171 + 0.976134i \(0.430317\pi\)
\(570\) 0 0
\(571\) 36.8328 1.54141 0.770703 0.637195i \(-0.219905\pi\)
0.770703 + 0.637195i \(0.219905\pi\)
\(572\) 0 0
\(573\) 49.8885 2.08412
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 15.8885 0.661449 0.330724 0.943727i \(-0.392707\pi\)
0.330724 + 0.943727i \(0.392707\pi\)
\(578\) 0 0
\(579\) 66.2492 2.75322
\(580\) 0 0
\(581\) −12.0000 −0.497844
\(582\) 0 0
\(583\) 16.9443 0.701760
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 22.6525 0.934968 0.467484 0.884002i \(-0.345161\pi\)
0.467484 + 0.884002i \(0.345161\pi\)
\(588\) 0 0
\(589\) 11.0557 0.455543
\(590\) 0 0
\(591\) −27.4164 −1.12776
\(592\) 0 0
\(593\) 14.0000 0.574911 0.287456 0.957794i \(-0.407191\pi\)
0.287456 + 0.957794i \(0.407191\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −54.8328 −2.24416
\(598\) 0 0
\(599\) −13.8885 −0.567471 −0.283735 0.958903i \(-0.591574\pi\)
−0.283735 + 0.958903i \(0.591574\pi\)
\(600\) 0 0
\(601\) 22.3607 0.912111 0.456056 0.889951i \(-0.349262\pi\)
0.456056 + 0.889951i \(0.349262\pi\)
\(602\) 0 0
\(603\) 42.6525 1.73694
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 47.1246 1.91273 0.956364 0.292176i \(-0.0943794\pi\)
0.956364 + 0.292176i \(0.0943794\pi\)
\(608\) 0 0
\(609\) 8.00000 0.324176
\(610\) 0 0
\(611\) −12.3607 −0.500060
\(612\) 0 0
\(613\) 28.4721 1.14998 0.574989 0.818161i \(-0.305006\pi\)
0.574989 + 0.818161i \(0.305006\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.41641 0.0570224 0.0285112 0.999593i \(-0.490923\pi\)
0.0285112 + 0.999593i \(0.490923\pi\)
\(618\) 0 0
\(619\) 11.5279 0.463344 0.231672 0.972794i \(-0.425580\pi\)
0.231672 + 0.972794i \(0.425580\pi\)
\(620\) 0 0
\(621\) −133.666 −5.36382
\(622\) 0 0
\(623\) 3.63932 0.145806
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −28.9443 −1.15592
\(628\) 0 0
\(629\) −48.9443 −1.95154
\(630\) 0 0
\(631\) 25.5279 1.01625 0.508124 0.861284i \(-0.330339\pi\)
0.508124 + 0.861284i \(0.330339\pi\)
\(632\) 0 0
\(633\) −61.3050 −2.43665
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −24.4721 −0.969621
\(638\) 0 0
\(639\) −48.3607 −1.91312
\(640\) 0 0
\(641\) −8.47214 −0.334629 −0.167315 0.985904i \(-0.553510\pi\)
−0.167315 + 0.985904i \(0.553510\pi\)
\(642\) 0 0
\(643\) 0.180340 0.00711191 0.00355596 0.999994i \(-0.498868\pi\)
0.00355596 + 0.999994i \(0.498868\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 17.2361 0.677620 0.338810 0.940855i \(-0.389976\pi\)
0.338810 + 0.940855i \(0.389976\pi\)
\(648\) 0 0
\(649\) −0.944272 −0.0370659
\(650\) 0 0
\(651\) 9.88854 0.387563
\(652\) 0 0
\(653\) −22.5836 −0.883764 −0.441882 0.897073i \(-0.645689\pi\)
−0.441882 + 0.897073i \(0.645689\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 33.4164 1.30370
\(658\) 0 0
\(659\) −5.41641 −0.210993 −0.105497 0.994420i \(-0.533643\pi\)
−0.105497 + 0.994420i \(0.533643\pi\)
\(660\) 0 0
\(661\) −0.111456 −0.00433514 −0.00216757 0.999998i \(-0.500690\pi\)
−0.00216757 + 0.999998i \(0.500690\pi\)
\(662\) 0 0
\(663\) 64.7214 2.51357
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −18.4721 −0.715244
\(668\) 0 0
\(669\) −42.8328 −1.65601
\(670\) 0 0
\(671\) −12.0000 −0.463255
\(672\) 0 0
\(673\) −39.3050 −1.51509 −0.757547 0.652780i \(-0.773602\pi\)
−0.757547 + 0.652780i \(0.773602\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 14.3607 0.551926 0.275963 0.961168i \(-0.411003\pi\)
0.275963 + 0.961168i \(0.411003\pi\)
\(678\) 0 0
\(679\) 9.30495 0.357091
\(680\) 0 0
\(681\) −39.4164 −1.51044
\(682\) 0 0
\(683\) −37.7082 −1.44286 −0.721432 0.692485i \(-0.756516\pi\)
−0.721432 + 0.692485i \(0.756516\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 48.3607 1.84508
\(688\) 0 0
\(689\) −37.8885 −1.44344
\(690\) 0 0
\(691\) −30.0000 −1.14125 −0.570627 0.821209i \(-0.693300\pi\)
−0.570627 + 0.821209i \(0.693300\pi\)
\(692\) 0 0
\(693\) −18.4721 −0.701698
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 15.7771 0.597600
\(698\) 0 0
\(699\) −79.1935 −2.99537
\(700\) 0 0
\(701\) 26.9443 1.01767 0.508836 0.860864i \(-0.330076\pi\)
0.508836 + 0.860864i \(0.330076\pi\)
\(702\) 0 0
\(703\) −48.9443 −1.84597
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 12.3607 0.464871
\(708\) 0 0
\(709\) 18.0000 0.676004 0.338002 0.941145i \(-0.390249\pi\)
0.338002 + 0.941145i \(0.390249\pi\)
\(710\) 0 0
\(711\) 36.9443 1.38552
\(712\) 0 0
\(713\) −22.8328 −0.855096
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −16.0000 −0.597531
\(718\) 0 0
\(719\) 30.8328 1.14987 0.574935 0.818199i \(-0.305027\pi\)
0.574935 + 0.818199i \(0.305027\pi\)
\(720\) 0 0
\(721\) −1.52786 −0.0569006
\(722\) 0 0
\(723\) −59.4164 −2.20972
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −11.7082 −0.434233 −0.217117 0.976146i \(-0.569665\pi\)
−0.217117 + 0.976146i \(0.569665\pi\)
\(728\) 0 0
\(729\) 41.9443 1.55349
\(730\) 0 0
\(731\) −25.5279 −0.944182
\(732\) 0 0
\(733\) 16.1115 0.595090 0.297545 0.954708i \(-0.403832\pi\)
0.297545 + 0.954708i \(0.403832\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −11.4164 −0.420529
\(738\) 0 0
\(739\) −49.1935 −1.80961 −0.904806 0.425824i \(-0.859984\pi\)
−0.904806 + 0.425824i \(0.859984\pi\)
\(740\) 0 0
\(741\) 64.7214 2.37760
\(742\) 0 0
\(743\) 0.652476 0.0239370 0.0119685 0.999928i \(-0.496190\pi\)
0.0119685 + 0.999928i \(0.496190\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −72.5410 −2.65414
\(748\) 0 0
\(749\) 15.7771 0.576482
\(750\) 0 0
\(751\) 28.3607 1.03490 0.517448 0.855715i \(-0.326882\pi\)
0.517448 + 0.855715i \(0.326882\pi\)
\(752\) 0 0
\(753\) −6.47214 −0.235858
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −15.8885 −0.577479 −0.288739 0.957408i \(-0.593236\pi\)
−0.288739 + 0.957408i \(0.593236\pi\)
\(758\) 0 0
\(759\) 59.7771 2.16977
\(760\) 0 0
\(761\) −31.8885 −1.15596 −0.577979 0.816051i \(-0.696159\pi\)
−0.577979 + 0.816051i \(0.696159\pi\)
\(762\) 0 0
\(763\) 3.63932 0.131752
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 2.11146 0.0762403
\(768\) 0 0
\(769\) 2.94427 0.106173 0.0530866 0.998590i \(-0.483094\pi\)
0.0530866 + 0.998590i \(0.483094\pi\)
\(770\) 0 0
\(771\) −48.3607 −1.74167
\(772\) 0 0
\(773\) −26.3607 −0.948128 −0.474064 0.880490i \(-0.657213\pi\)
−0.474064 + 0.880490i \(0.657213\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −43.7771 −1.57049
\(778\) 0 0
\(779\) 15.7771 0.565273
\(780\) 0 0
\(781\) 12.9443 0.463182
\(782\) 0 0
\(783\) 28.9443 1.03438
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −14.0689 −0.501502 −0.250751 0.968052i \(-0.580677\pi\)
−0.250751 + 0.968052i \(0.580677\pi\)
\(788\) 0 0
\(789\) −13.8885 −0.494445
\(790\) 0 0
\(791\) −18.4721 −0.656794
\(792\) 0 0
\(793\) 26.8328 0.952861
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1.41641 0.0501717 0.0250859 0.999685i \(-0.492014\pi\)
0.0250859 + 0.999685i \(0.492014\pi\)
\(798\) 0 0
\(799\) −12.3607 −0.437289
\(800\) 0 0
\(801\) 22.0000 0.777332
\(802\) 0 0
\(803\) −8.94427 −0.315637
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −38.4721 −1.35428
\(808\) 0 0
\(809\) −47.8885 −1.68367 −0.841836 0.539734i \(-0.818525\pi\)
−0.841836 + 0.539734i \(0.818525\pi\)
\(810\) 0 0
\(811\) −2.00000 −0.0702295 −0.0351147 0.999383i \(-0.511180\pi\)
−0.0351147 + 0.999383i \(0.511180\pi\)
\(812\) 0 0
\(813\) −65.8885 −2.31081
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −25.5279 −0.893107
\(818\) 0 0
\(819\) 41.3050 1.44331
\(820\) 0 0
\(821\) −38.9443 −1.35916 −0.679582 0.733599i \(-0.737839\pi\)
−0.679582 + 0.733599i \(0.737839\pi\)
\(822\) 0 0
\(823\) 11.7082 0.408122 0.204061 0.978958i \(-0.434586\pi\)
0.204061 + 0.978958i \(0.434586\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −16.1803 −0.562646 −0.281323 0.959613i \(-0.590773\pi\)
−0.281323 + 0.959613i \(0.590773\pi\)
\(828\) 0 0
\(829\) 38.0000 1.31979 0.659897 0.751356i \(-0.270600\pi\)
0.659897 + 0.751356i \(0.270600\pi\)
\(830\) 0 0
\(831\) 22.4721 0.779550
\(832\) 0 0
\(833\) −24.4721 −0.847909
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 35.7771 1.23664
\(838\) 0 0
\(839\) −4.00000 −0.138095 −0.0690477 0.997613i \(-0.521996\pi\)
−0.0690477 + 0.997613i \(0.521996\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) 40.3607 1.39010
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −8.65248 −0.297303
\(848\) 0 0
\(849\) −70.2492 −2.41095
\(850\) 0 0
\(851\) 101.082 3.46505
\(852\) 0 0
\(853\) −16.4721 −0.563995 −0.281998 0.959415i \(-0.590997\pi\)
−0.281998 + 0.959415i \(0.590997\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 28.8328 0.984910 0.492455 0.870338i \(-0.336100\pi\)
0.492455 + 0.870338i \(0.336100\pi\)
\(858\) 0 0
\(859\) 22.5836 0.770542 0.385271 0.922803i \(-0.374108\pi\)
0.385271 + 0.922803i \(0.374108\pi\)
\(860\) 0 0
\(861\) 14.1115 0.480917
\(862\) 0 0
\(863\) 31.7082 1.07936 0.539680 0.841870i \(-0.318545\pi\)
0.539680 + 0.841870i \(0.318545\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 9.70820 0.329708
\(868\) 0 0
\(869\) −9.88854 −0.335446
\(870\) 0 0
\(871\) 25.5279 0.864979
\(872\) 0 0
\(873\) 56.2492 1.90375
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 16.1115 0.544045 0.272023 0.962291i \(-0.412307\pi\)
0.272023 + 0.962291i \(0.412307\pi\)
\(878\) 0 0
\(879\) 38.4721 1.29763
\(880\) 0 0
\(881\) −15.5279 −0.523147 −0.261574 0.965184i \(-0.584241\pi\)
−0.261574 + 0.965184i \(0.584241\pi\)
\(882\) 0 0
\(883\) 14.2918 0.480957 0.240479 0.970654i \(-0.422696\pi\)
0.240479 + 0.970654i \(0.422696\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −37.5967 −1.26238 −0.631188 0.775630i \(-0.717432\pi\)
−0.631188 + 0.775630i \(0.717432\pi\)
\(888\) 0 0
\(889\) 2.69505 0.0903890
\(890\) 0 0
\(891\) −48.8328 −1.63596
\(892\) 0 0
\(893\) −12.3607 −0.413634
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −133.666 −4.46297
\(898\) 0 0
\(899\) 4.94427 0.164901
\(900\) 0 0
\(901\) −37.8885 −1.26225
\(902\) 0 0
\(903\) −22.8328 −0.759829
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 4.76393 0.158184 0.0790919 0.996867i \(-0.474798\pi\)
0.0790919 + 0.996867i \(0.474798\pi\)
\(908\) 0 0
\(909\) 74.7214 2.47835
\(910\) 0 0
\(911\) −9.30495 −0.308287 −0.154143 0.988048i \(-0.549262\pi\)
−0.154143 + 0.988048i \(0.549262\pi\)
\(912\) 0 0
\(913\) 19.4164 0.642589
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −3.63932 −0.120181
\(918\) 0 0
\(919\) −29.8885 −0.985932 −0.492966 0.870049i \(-0.664087\pi\)
−0.492966 + 0.870049i \(0.664087\pi\)
\(920\) 0 0
\(921\) 44.3607 1.46173
\(922\) 0 0
\(923\) −28.9443 −0.952712
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −9.23607 −0.303352
\(928\) 0 0
\(929\) 15.5279 0.509453 0.254726 0.967013i \(-0.418015\pi\)
0.254726 + 0.967013i \(0.418015\pi\)
\(930\) 0 0
\(931\) −24.4721 −0.802042
\(932\) 0 0
\(933\) −36.9443 −1.20950
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 12.4721 0.407447 0.203723 0.979028i \(-0.434696\pi\)
0.203723 + 0.979028i \(0.434696\pi\)
\(938\) 0 0
\(939\) −80.3607 −2.62247
\(940\) 0 0
\(941\) 8.83282 0.287942 0.143971 0.989582i \(-0.454013\pi\)
0.143971 + 0.989582i \(0.454013\pi\)
\(942\) 0 0
\(943\) −32.5836 −1.06107
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −44.1803 −1.43567 −0.717834 0.696214i \(-0.754866\pi\)
−0.717834 + 0.696214i \(0.754866\pi\)
\(948\) 0 0
\(949\) 20.0000 0.649227
\(950\) 0 0
\(951\) −37.3050 −1.20970
\(952\) 0 0
\(953\) 25.7771 0.835002 0.417501 0.908677i \(-0.362906\pi\)
0.417501 + 0.908677i \(0.362906\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −12.9443 −0.418429
\(958\) 0 0
\(959\) −2.47214 −0.0798294
\(960\) 0 0
\(961\) −24.8885 −0.802856
\(962\) 0 0
\(963\) 95.3738 3.07338
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −32.6525 −1.05003 −0.525016 0.851092i \(-0.675941\pi\)
−0.525016 + 0.851092i \(0.675941\pi\)
\(968\) 0 0
\(969\) 64.7214 2.07915
\(970\) 0 0
\(971\) −11.8885 −0.381522 −0.190761 0.981637i \(-0.561095\pi\)
−0.190761 + 0.981637i \(0.561095\pi\)
\(972\) 0 0
\(973\) 14.2492 0.456809
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −23.3050 −0.745591 −0.372796 0.927913i \(-0.621601\pi\)
−0.372796 + 0.927913i \(0.621601\pi\)
\(978\) 0 0
\(979\) −5.88854 −0.188199
\(980\) 0 0
\(981\) 22.0000 0.702406
\(982\) 0 0
\(983\) −37.0132 −1.18054 −0.590268 0.807207i \(-0.700978\pi\)
−0.590268 + 0.807207i \(0.700978\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −11.0557 −0.351908
\(988\) 0 0
\(989\) 52.7214 1.67644
\(990\) 0 0
\(991\) 26.4721 0.840915 0.420458 0.907312i \(-0.361870\pi\)
0.420458 + 0.907312i \(0.361870\pi\)
\(992\) 0 0
\(993\) −64.3607 −2.04242
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −61.4164 −1.94508 −0.972539 0.232742i \(-0.925230\pi\)
−0.972539 + 0.232742i \(0.925230\pi\)
\(998\) 0 0
\(999\) −158.387 −5.01114
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 3200.2.a.bk.1.2 2
4.3 odd 2 3200.2.a.bf.1.1 2
5.2 odd 4 3200.2.c.v.2049.1 4
5.3 odd 4 3200.2.c.v.2049.4 4
5.4 even 2 640.2.a.j.1.1 yes 2
8.3 odd 2 3200.2.a.bl.1.2 2
8.5 even 2 3200.2.a.be.1.1 2
15.14 odd 2 5760.2.a.cd.1.1 2
20.3 even 4 3200.2.c.x.2049.1 4
20.7 even 4 3200.2.c.x.2049.4 4
20.19 odd 2 640.2.a.l.1.2 yes 2
40.3 even 4 3200.2.c.u.2049.4 4
40.13 odd 4 3200.2.c.w.2049.1 4
40.19 odd 2 640.2.a.i.1.1 2
40.27 even 4 3200.2.c.u.2049.1 4
40.29 even 2 640.2.a.k.1.2 yes 2
40.37 odd 4 3200.2.c.w.2049.4 4
60.59 even 2 5760.2.a.bw.1.2 2
80.19 odd 4 1280.2.d.m.641.4 4
80.29 even 4 1280.2.d.k.641.1 4
80.59 odd 4 1280.2.d.m.641.1 4
80.69 even 4 1280.2.d.k.641.4 4
120.29 odd 2 5760.2.a.ci.1.1 2
120.59 even 2 5760.2.a.ch.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
640.2.a.i.1.1 2 40.19 odd 2
640.2.a.j.1.1 yes 2 5.4 even 2
640.2.a.k.1.2 yes 2 40.29 even 2
640.2.a.l.1.2 yes 2 20.19 odd 2
1280.2.d.k.641.1 4 80.29 even 4
1280.2.d.k.641.4 4 80.69 even 4
1280.2.d.m.641.1 4 80.59 odd 4
1280.2.d.m.641.4 4 80.19 odd 4
3200.2.a.be.1.1 2 8.5 even 2
3200.2.a.bf.1.1 2 4.3 odd 2
3200.2.a.bk.1.2 2 1.1 even 1 trivial
3200.2.a.bl.1.2 2 8.3 odd 2
3200.2.c.u.2049.1 4 40.27 even 4
3200.2.c.u.2049.4 4 40.3 even 4
3200.2.c.v.2049.1 4 5.2 odd 4
3200.2.c.v.2049.4 4 5.3 odd 4
3200.2.c.w.2049.1 4 40.13 odd 4
3200.2.c.w.2049.4 4 40.37 odd 4
3200.2.c.x.2049.1 4 20.3 even 4
3200.2.c.x.2049.4 4 20.7 even 4
5760.2.a.bw.1.2 2 60.59 even 2
5760.2.a.cd.1.1 2 15.14 odd 2
5760.2.a.ch.1.2 2 120.59 even 2
5760.2.a.ci.1.1 2 120.29 odd 2