Properties

Label 4608.2.a.h.1.1
Level $4608$
Weight $2$
Character 4608.1
Self dual yes
Analytic conductor $36.795$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4608,2,Mod(1,4608)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4608, 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("4608.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4608 = 2^{9} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4608.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: \(36.7950652514\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1536)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 4608.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.41421 q^{5} +1.41421 q^{7} +O(q^{10})\) \(q-1.41421 q^{5} +1.41421 q^{7} -2.00000 q^{11} -2.00000 q^{17} +4.00000 q^{19} -2.82843 q^{23} -3.00000 q^{25} +9.89949 q^{29} -7.07107 q^{31} -2.00000 q^{35} +8.48528 q^{37} -6.00000 q^{41} +8.00000 q^{43} -2.82843 q^{47} -5.00000 q^{49} -1.41421 q^{53} +2.82843 q^{55} -12.0000 q^{59} -14.1421 q^{61} +8.00000 q^{67} +14.1421 q^{71} -8.00000 q^{73} -2.82843 q^{77} -4.24264 q^{79} -6.00000 q^{83} +2.82843 q^{85} -2.00000 q^{89} -5.65685 q^{95} -14.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{11} - 4 q^{17} + 8 q^{19} - 6 q^{25} - 4 q^{35} - 12 q^{41} + 16 q^{43} - 10 q^{49} - 24 q^{59} + 16 q^{67} - 16 q^{73} - 12 q^{83} - 4 q^{89} - 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −1.41421 −0.632456 −0.316228 0.948683i \(-0.602416\pi\)
−0.316228 + 0.948683i \(0.602416\pi\)
\(6\) 0 0
\(7\) 1.41421 0.534522 0.267261 0.963624i \(-0.413881\pi\)
0.267261 + 0.963624i \(0.413881\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.82843 −0.589768 −0.294884 0.955533i \(-0.595281\pi\)
−0.294884 + 0.955533i \(0.595281\pi\)
\(24\) 0 0
\(25\) −3.00000 −0.600000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 9.89949 1.83829 0.919145 0.393919i \(-0.128881\pi\)
0.919145 + 0.393919i \(0.128881\pi\)
\(30\) 0 0
\(31\) −7.07107 −1.27000 −0.635001 0.772512i \(-0.719000\pi\)
−0.635001 + 0.772512i \(0.719000\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.00000 −0.338062
\(36\) 0 0
\(37\) 8.48528 1.39497 0.697486 0.716599i \(-0.254302\pi\)
0.697486 + 0.716599i \(0.254302\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.82843 −0.412568 −0.206284 0.978492i \(-0.566137\pi\)
−0.206284 + 0.978492i \(0.566137\pi\)
\(48\) 0 0
\(49\) −5.00000 −0.714286
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.41421 −0.194257 −0.0971286 0.995272i \(-0.530966\pi\)
−0.0971286 + 0.995272i \(0.530966\pi\)
\(54\) 0 0
\(55\) 2.82843 0.381385
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) −14.1421 −1.81071 −0.905357 0.424650i \(-0.860397\pi\)
−0.905357 + 0.424650i \(0.860397\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 14.1421 1.67836 0.839181 0.543852i \(-0.183035\pi\)
0.839181 + 0.543852i \(0.183035\pi\)
\(72\) 0 0
\(73\) −8.00000 −0.936329 −0.468165 0.883641i \(-0.655085\pi\)
−0.468165 + 0.883641i \(0.655085\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.82843 −0.322329
\(78\) 0 0
\(79\) −4.24264 −0.477334 −0.238667 0.971101i \(-0.576710\pi\)
−0.238667 + 0.971101i \(0.576710\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) 2.82843 0.306786
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −5.65685 −0.580381
\(96\) 0 0
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −9.89949 −0.985037 −0.492518 0.870302i \(-0.663924\pi\)
−0.492518 + 0.870302i \(0.663924\pi\)
\(102\) 0 0
\(103\) 12.7279 1.25412 0.627060 0.778971i \(-0.284258\pi\)
0.627060 + 0.778971i \(0.284258\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 0 0
\(109\) 11.3137 1.08366 0.541828 0.840489i \(-0.317732\pi\)
0.541828 + 0.840489i \(0.317732\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 4.00000 0.373002
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −2.82843 −0.259281
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.3137 1.01193
\(126\) 0 0
\(127\) −4.24264 −0.376473 −0.188237 0.982124i \(-0.560277\pi\)
−0.188237 + 0.982124i \(0.560277\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 5.65685 0.490511
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −14.0000 −1.16264
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 4.24264 0.347571 0.173785 0.984784i \(-0.444400\pi\)
0.173785 + 0.984784i \(0.444400\pi\)
\(150\) 0 0
\(151\) 4.24264 0.345261 0.172631 0.984987i \(-0.444773\pi\)
0.172631 + 0.984987i \(0.444773\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 10.0000 0.803219
\(156\) 0 0
\(157\) −14.1421 −1.12867 −0.564333 0.825547i \(-0.690866\pi\)
−0.564333 + 0.825547i \(0.690866\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −4.00000 −0.315244
\(162\) 0 0
\(163\) −16.0000 −1.25322 −0.626608 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 11.3137 0.875481 0.437741 0.899101i \(-0.355779\pi\)
0.437741 + 0.899101i \(0.355779\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −15.5563 −1.18273 −0.591364 0.806405i \(-0.701410\pi\)
−0.591364 + 0.806405i \(0.701410\pi\)
\(174\) 0 0
\(175\) −4.24264 −0.320713
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) −11.3137 −0.840941 −0.420471 0.907306i \(-0.638135\pi\)
−0.420471 + 0.907306i \(0.638135\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −12.0000 −0.882258
\(186\) 0 0
\(187\) 4.00000 0.292509
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −22.6274 −1.63726 −0.818631 0.574320i \(-0.805267\pi\)
−0.818631 + 0.574320i \(0.805267\pi\)
\(192\) 0 0
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −7.07107 −0.503793 −0.251896 0.967754i \(-0.581054\pi\)
−0.251896 + 0.967754i \(0.581054\pi\)
\(198\) 0 0
\(199\) 24.0416 1.70427 0.852133 0.523325i \(-0.175309\pi\)
0.852133 + 0.523325i \(0.175309\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 14.0000 0.982607
\(204\) 0 0
\(205\) 8.48528 0.592638
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −8.00000 −0.553372
\(210\) 0 0
\(211\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −11.3137 −0.771589
\(216\) 0 0
\(217\) −10.0000 −0.678844
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −12.7279 −0.852325 −0.426162 0.904647i \(-0.640135\pi\)
−0.426162 + 0.904647i \(0.640135\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 22.0000 1.46019 0.730096 0.683345i \(-0.239475\pi\)
0.730096 + 0.683345i \(0.239475\pi\)
\(228\) 0 0
\(229\) −16.9706 −1.12145 −0.560723 0.828003i \(-0.689477\pi\)
−0.560723 + 0.828003i \(0.689477\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −22.0000 −1.44127 −0.720634 0.693316i \(-0.756149\pi\)
−0.720634 + 0.693316i \(0.756149\pi\)
\(234\) 0 0
\(235\) 4.00000 0.260931
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 22.6274 1.46365 0.731823 0.681495i \(-0.238670\pi\)
0.731823 + 0.681495i \(0.238670\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 7.07107 0.451754
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 6.00000 0.378717 0.189358 0.981908i \(-0.439359\pi\)
0.189358 + 0.981908i \(0.439359\pi\)
\(252\) 0 0
\(253\) 5.65685 0.355643
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) 12.0000 0.745644
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 28.2843 1.74408 0.872041 0.489432i \(-0.162796\pi\)
0.872041 + 0.489432i \(0.162796\pi\)
\(264\) 0 0
\(265\) 2.00000 0.122859
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −21.2132 −1.29339 −0.646696 0.762748i \(-0.723850\pi\)
−0.646696 + 0.762748i \(0.723850\pi\)
\(270\) 0 0
\(271\) −24.0416 −1.46043 −0.730213 0.683220i \(-0.760579\pi\)
−0.730213 + 0.683220i \(0.760579\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 6.00000 0.361814
\(276\) 0 0
\(277\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 2.00000 0.119310 0.0596550 0.998219i \(-0.481000\pi\)
0.0596550 + 0.998219i \(0.481000\pi\)
\(282\) 0 0
\(283\) −20.0000 −1.18888 −0.594438 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −8.48528 −0.500870
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 9.89949 0.578335 0.289167 0.957279i \(-0.406622\pi\)
0.289167 + 0.957279i \(0.406622\pi\)
\(294\) 0 0
\(295\) 16.9706 0.988064
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 11.3137 0.652111
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 20.0000 1.14520
\(306\) 0 0
\(307\) −32.0000 −1.82634 −0.913168 0.407583i \(-0.866372\pi\)
−0.913168 + 0.407583i \(0.866372\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −22.6274 −1.28308 −0.641542 0.767088i \(-0.721705\pi\)
−0.641542 + 0.767088i \(0.721705\pi\)
\(312\) 0 0
\(313\) −4.00000 −0.226093 −0.113047 0.993590i \(-0.536061\pi\)
−0.113047 + 0.993590i \(0.536061\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −9.89949 −0.556011 −0.278006 0.960579i \(-0.589673\pi\)
−0.278006 + 0.960579i \(0.589673\pi\)
\(318\) 0 0
\(319\) −19.7990 −1.10853
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −8.00000 −0.445132
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −4.00000 −0.220527
\(330\) 0 0
\(331\) −32.0000 −1.75888 −0.879440 0.476011i \(-0.842082\pi\)
−0.879440 + 0.476011i \(0.842082\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −11.3137 −0.618134
\(336\) 0 0
\(337\) −8.00000 −0.435788 −0.217894 0.975972i \(-0.569919\pi\)
−0.217894 + 0.975972i \(0.569919\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 14.1421 0.765840
\(342\) 0 0
\(343\) −16.9706 −0.916324
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 26.0000 1.39575 0.697877 0.716218i \(-0.254128\pi\)
0.697877 + 0.716218i \(0.254128\pi\)
\(348\) 0 0
\(349\) 25.4558 1.36262 0.681310 0.731995i \(-0.261411\pi\)
0.681310 + 0.731995i \(0.261411\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) −20.0000 −1.06149
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 19.7990 1.04495 0.522475 0.852654i \(-0.325009\pi\)
0.522475 + 0.852654i \(0.325009\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 11.3137 0.592187
\(366\) 0 0
\(367\) 4.24264 0.221464 0.110732 0.993850i \(-0.464680\pi\)
0.110732 + 0.993850i \(0.464680\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −2.00000 −0.103835
\(372\) 0 0
\(373\) −19.7990 −1.02515 −0.512576 0.858642i \(-0.671309\pi\)
−0.512576 + 0.858642i \(0.671309\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 5.65685 0.289052 0.144526 0.989501i \(-0.453834\pi\)
0.144526 + 0.989501i \(0.453834\pi\)
\(384\) 0 0
\(385\) 4.00000 0.203859
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 24.0416 1.21896 0.609480 0.792802i \(-0.291378\pi\)
0.609480 + 0.792802i \(0.291378\pi\)
\(390\) 0 0
\(391\) 5.65685 0.286079
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 6.00000 0.301893
\(396\) 0 0
\(397\) −2.82843 −0.141955 −0.0709773 0.997478i \(-0.522612\pi\)
−0.0709773 + 0.997478i \(0.522612\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −16.9706 −0.841200
\(408\) 0 0
\(409\) −26.0000 −1.28562 −0.642809 0.766027i \(-0.722231\pi\)
−0.642809 + 0.766027i \(0.722231\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −16.9706 −0.835067
\(414\) 0 0
\(415\) 8.48528 0.416526
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 30.0000 1.46560 0.732798 0.680446i \(-0.238214\pi\)
0.732798 + 0.680446i \(0.238214\pi\)
\(420\) 0 0
\(421\) −5.65685 −0.275698 −0.137849 0.990453i \(-0.544019\pi\)
−0.137849 + 0.990453i \(0.544019\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 6.00000 0.291043
\(426\) 0 0
\(427\) −20.0000 −0.967868
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −2.82843 −0.136241 −0.0681203 0.997677i \(-0.521700\pi\)
−0.0681203 + 0.997677i \(0.521700\pi\)
\(432\) 0 0
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −11.3137 −0.541208
\(438\) 0 0
\(439\) −4.24264 −0.202490 −0.101245 0.994862i \(-0.532283\pi\)
−0.101245 + 0.994862i \(0.532283\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −30.0000 −1.42534 −0.712672 0.701498i \(-0.752515\pi\)
−0.712672 + 0.701498i \(0.752515\pi\)
\(444\) 0 0
\(445\) 2.82843 0.134080
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 22.0000 1.03824 0.519122 0.854700i \(-0.326259\pi\)
0.519122 + 0.854700i \(0.326259\pi\)
\(450\) 0 0
\(451\) 12.0000 0.565058
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −6.00000 −0.280668 −0.140334 0.990104i \(-0.544818\pi\)
−0.140334 + 0.990104i \(0.544818\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 12.7279 0.592798 0.296399 0.955064i \(-0.404214\pi\)
0.296399 + 0.955064i \(0.404214\pi\)
\(462\) 0 0
\(463\) 35.3553 1.64310 0.821551 0.570135i \(-0.193109\pi\)
0.821551 + 0.570135i \(0.193109\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 22.0000 1.01804 0.509019 0.860755i \(-0.330008\pi\)
0.509019 + 0.860755i \(0.330008\pi\)
\(468\) 0 0
\(469\) 11.3137 0.522419
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −16.0000 −0.735681
\(474\) 0 0
\(475\) −12.0000 −0.550598
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −25.4558 −1.16311 −0.581554 0.813508i \(-0.697555\pi\)
−0.581554 + 0.813508i \(0.697555\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 19.7990 0.899026
\(486\) 0 0
\(487\) −4.24264 −0.192252 −0.0961262 0.995369i \(-0.530645\pi\)
−0.0961262 + 0.995369i \(0.530645\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −28.0000 −1.26362 −0.631811 0.775122i \(-0.717688\pi\)
−0.631811 + 0.775122i \(0.717688\pi\)
\(492\) 0 0
\(493\) −19.7990 −0.891702
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 20.0000 0.897123
\(498\) 0 0
\(499\) 20.0000 0.895323 0.447661 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −2.82843 −0.126113 −0.0630567 0.998010i \(-0.520085\pi\)
−0.0630567 + 0.998010i \(0.520085\pi\)
\(504\) 0 0
\(505\) 14.0000 0.622992
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −24.0416 −1.06563 −0.532813 0.846233i \(-0.678865\pi\)
−0.532813 + 0.846233i \(0.678865\pi\)
\(510\) 0 0
\(511\) −11.3137 −0.500489
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −18.0000 −0.793175
\(516\) 0 0
\(517\) 5.65685 0.248788
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 38.0000 1.66481 0.832405 0.554168i \(-0.186963\pi\)
0.832405 + 0.554168i \(0.186963\pi\)
\(522\) 0 0
\(523\) 4.00000 0.174908 0.0874539 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 14.1421 0.616041
\(528\) 0 0
\(529\) −15.0000 −0.652174
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 5.65685 0.244567
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 10.0000 0.430730
\(540\) 0 0
\(541\) 28.2843 1.21604 0.608018 0.793923i \(-0.291965\pi\)
0.608018 + 0.793923i \(0.291965\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −16.0000 −0.685365
\(546\) 0 0
\(547\) −16.0000 −0.684111 −0.342055 0.939680i \(-0.611123\pi\)
−0.342055 + 0.939680i \(0.611123\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 39.5980 1.68693
\(552\) 0 0
\(553\) −6.00000 −0.255146
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 38.1838 1.61790 0.808949 0.587879i \(-0.200037\pi\)
0.808949 + 0.587879i \(0.200037\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 6.00000 0.252870 0.126435 0.991975i \(-0.459647\pi\)
0.126435 + 0.991975i \(0.459647\pi\)
\(564\) 0 0
\(565\) 8.48528 0.356978
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 34.0000 1.42535 0.712677 0.701492i \(-0.247483\pi\)
0.712677 + 0.701492i \(0.247483\pi\)
\(570\) 0 0
\(571\) 28.0000 1.17176 0.585882 0.810397i \(-0.300748\pi\)
0.585882 + 0.810397i \(0.300748\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 8.48528 0.353861
\(576\) 0 0
\(577\) −4.00000 −0.166522 −0.0832611 0.996528i \(-0.526534\pi\)
−0.0832611 + 0.996528i \(0.526534\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −8.48528 −0.352029
\(582\) 0 0
\(583\) 2.82843 0.117141
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 28.0000 1.15568 0.577842 0.816149i \(-0.303895\pi\)
0.577842 + 0.816149i \(0.303895\pi\)
\(588\) 0 0
\(589\) −28.2843 −1.16543
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −30.0000 −1.23195 −0.615976 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 0 0
\(595\) 4.00000 0.163984
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −36.7696 −1.50236 −0.751182 0.660096i \(-0.770516\pi\)
−0.751182 + 0.660096i \(0.770516\pi\)
\(600\) 0 0
\(601\) 44.0000 1.79480 0.897399 0.441221i \(-0.145454\pi\)
0.897399 + 0.441221i \(0.145454\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 9.89949 0.402472
\(606\) 0 0
\(607\) −4.24264 −0.172203 −0.0861017 0.996286i \(-0.527441\pi\)
−0.0861017 + 0.996286i \(0.527441\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 8.48528 0.342717 0.171359 0.985209i \(-0.445184\pi\)
0.171359 + 0.985209i \(0.445184\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −46.0000 −1.85189 −0.925945 0.377658i \(-0.876729\pi\)
−0.925945 + 0.377658i \(0.876729\pi\)
\(618\) 0 0
\(619\) 36.0000 1.44696 0.723481 0.690344i \(-0.242541\pi\)
0.723481 + 0.690344i \(0.242541\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −2.82843 −0.113319
\(624\) 0 0
\(625\) −1.00000 −0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −16.9706 −0.676661
\(630\) 0 0
\(631\) −15.5563 −0.619288 −0.309644 0.950852i \(-0.600210\pi\)
−0.309644 + 0.950852i \(0.600210\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 6.00000 0.238103
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 42.0000 1.65890 0.829450 0.558581i \(-0.188654\pi\)
0.829450 + 0.558581i \(0.188654\pi\)
\(642\) 0 0
\(643\) 40.0000 1.57745 0.788723 0.614749i \(-0.210743\pi\)
0.788723 + 0.614749i \(0.210743\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 14.1421 0.555985 0.277992 0.960583i \(-0.410331\pi\)
0.277992 + 0.960583i \(0.410331\pi\)
\(648\) 0 0
\(649\) 24.0000 0.942082
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 29.6985 1.16219 0.581096 0.813835i \(-0.302624\pi\)
0.581096 + 0.813835i \(0.302624\pi\)
\(654\) 0 0
\(655\) 16.9706 0.663095
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −44.0000 −1.71400 −0.856998 0.515319i \(-0.827673\pi\)
−0.856998 + 0.515319i \(0.827673\pi\)
\(660\) 0 0
\(661\) 19.7990 0.770091 0.385046 0.922897i \(-0.374186\pi\)
0.385046 + 0.922897i \(0.374186\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −8.00000 −0.310227
\(666\) 0 0
\(667\) −28.0000 −1.08416
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 28.2843 1.09190
\(672\) 0 0
\(673\) 22.0000 0.848038 0.424019 0.905653i \(-0.360619\pi\)
0.424019 + 0.905653i \(0.360619\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −24.0416 −0.923995 −0.461997 0.886881i \(-0.652867\pi\)
−0.461997 + 0.886881i \(0.652867\pi\)
\(678\) 0 0
\(679\) −19.7990 −0.759815
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −26.0000 −0.994862 −0.497431 0.867503i \(-0.665723\pi\)
−0.497431 + 0.867503i \(0.665723\pi\)
\(684\) 0 0
\(685\) −14.1421 −0.540343
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −8.00000 −0.304334 −0.152167 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 12.0000 0.454532
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −4.24264 −0.160242 −0.0801212 0.996785i \(-0.525531\pi\)
−0.0801212 + 0.996785i \(0.525531\pi\)
\(702\) 0 0
\(703\) 33.9411 1.28011
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −14.0000 −0.526524
\(708\) 0 0
\(709\) −33.9411 −1.27469 −0.637343 0.770580i \(-0.719966\pi\)
−0.637343 + 0.770580i \(0.719966\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 20.0000 0.749006
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 25.4558 0.949343 0.474671 0.880163i \(-0.342567\pi\)
0.474671 + 0.880163i \(0.342567\pi\)
\(720\) 0 0
\(721\) 18.0000 0.670355
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −29.6985 −1.10297
\(726\) 0 0
\(727\) −24.0416 −0.891655 −0.445827 0.895119i \(-0.647090\pi\)
−0.445827 + 0.895119i \(0.647090\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −16.0000 −0.591781
\(732\) 0 0
\(733\) 11.3137 0.417881 0.208941 0.977928i \(-0.432998\pi\)
0.208941 + 0.977928i \(0.432998\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −16.0000 −0.589368
\(738\) 0 0
\(739\) 12.0000 0.441427 0.220714 0.975339i \(-0.429161\pi\)
0.220714 + 0.975339i \(0.429161\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −28.2843 −1.03765 −0.518825 0.854881i \(-0.673630\pi\)
−0.518825 + 0.854881i \(0.673630\pi\)
\(744\) 0 0
\(745\) −6.00000 −0.219823
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −5.65685 −0.206697
\(750\) 0 0
\(751\) 12.7279 0.464448 0.232224 0.972662i \(-0.425400\pi\)
0.232224 + 0.972662i \(0.425400\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −6.00000 −0.218362
\(756\) 0 0
\(757\) 50.9117 1.85042 0.925208 0.379459i \(-0.123890\pi\)
0.925208 + 0.379459i \(0.123890\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −10.0000 −0.362500 −0.181250 0.983437i \(-0.558014\pi\)
−0.181250 + 0.983437i \(0.558014\pi\)
\(762\) 0 0
\(763\) 16.0000 0.579239
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 20.0000 0.721218 0.360609 0.932717i \(-0.382569\pi\)
0.360609 + 0.932717i \(0.382569\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −15.5563 −0.559523 −0.279761 0.960070i \(-0.590255\pi\)
−0.279761 + 0.960070i \(0.590255\pi\)
\(774\) 0 0
\(775\) 21.2132 0.762001
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −24.0000 −0.859889
\(780\) 0 0
\(781\) −28.2843 −1.01209
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 20.0000 0.713831
\(786\) 0 0
\(787\) 4.00000 0.142585 0.0712923 0.997455i \(-0.477288\pi\)
0.0712923 + 0.997455i \(0.477288\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −8.48528 −0.301702
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1.41421 −0.0500940 −0.0250470 0.999686i \(-0.507974\pi\)
−0.0250470 + 0.999686i \(0.507974\pi\)
\(798\) 0 0
\(799\) 5.65685 0.200125
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 16.0000 0.564628
\(804\) 0 0
\(805\) 5.65685 0.199378
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 2.00000 0.0703163 0.0351581 0.999382i \(-0.488807\pi\)
0.0351581 + 0.999382i \(0.488807\pi\)
\(810\) 0 0
\(811\) 28.0000 0.983213 0.491606 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 22.6274 0.792604
\(816\) 0 0
\(817\) 32.0000 1.11954
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −9.89949 −0.345495 −0.172747 0.984966i \(-0.555264\pi\)
−0.172747 + 0.984966i \(0.555264\pi\)
\(822\) 0 0
\(823\) −21.2132 −0.739446 −0.369723 0.929142i \(-0.620547\pi\)
−0.369723 + 0.929142i \(0.620547\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −28.0000 −0.973655 −0.486828 0.873498i \(-0.661846\pi\)
−0.486828 + 0.873498i \(0.661846\pi\)
\(828\) 0 0
\(829\) −11.3137 −0.392941 −0.196471 0.980510i \(-0.562948\pi\)
−0.196471 + 0.980510i \(0.562948\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 10.0000 0.346479
\(834\) 0 0
\(835\) −16.0000 −0.553703
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −14.1421 −0.488241 −0.244120 0.969745i \(-0.578499\pi\)
−0.244120 + 0.969745i \(0.578499\pi\)
\(840\) 0 0
\(841\) 69.0000 2.37931
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 18.3848 0.632456
\(846\) 0 0
\(847\) −9.89949 −0.340151
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −24.0000 −0.822709
\(852\) 0 0
\(853\) −14.1421 −0.484218 −0.242109 0.970249i \(-0.577839\pi\)
−0.242109 + 0.970249i \(0.577839\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 26.0000 0.888143 0.444072 0.895991i \(-0.353534\pi\)
0.444072 + 0.895991i \(0.353534\pi\)
\(858\) 0 0
\(859\) 8.00000 0.272956 0.136478 0.990643i \(-0.456422\pi\)
0.136478 + 0.990643i \(0.456422\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −22.6274 −0.770246 −0.385123 0.922865i \(-0.625841\pi\)
−0.385123 + 0.922865i \(0.625841\pi\)
\(864\) 0 0
\(865\) 22.0000 0.748022
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 8.48528 0.287843
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 16.0000 0.540899
\(876\) 0 0
\(877\) −42.4264 −1.43264 −0.716319 0.697773i \(-0.754174\pi\)
−0.716319 + 0.697773i \(0.754174\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 22.0000 0.741199 0.370599 0.928793i \(-0.379152\pi\)
0.370599 + 0.928793i \(0.379152\pi\)
\(882\) 0 0
\(883\) 48.0000 1.61533 0.807664 0.589643i \(-0.200731\pi\)
0.807664 + 0.589643i \(0.200731\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −50.9117 −1.70945 −0.854724 0.519083i \(-0.826273\pi\)
−0.854724 + 0.519083i \(0.826273\pi\)
\(888\) 0 0
\(889\) −6.00000 −0.201234
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −11.3137 −0.378599
\(894\) 0 0
\(895\) −5.65685 −0.189088
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −70.0000 −2.33463
\(900\) 0 0
\(901\) 2.82843 0.0942286
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 16.0000 0.531858
\(906\) 0 0
\(907\) 8.00000 0.265636 0.132818 0.991140i \(-0.457597\pi\)
0.132818 + 0.991140i \(0.457597\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −5.65685 −0.187420 −0.0937100 0.995600i \(-0.529873\pi\)
−0.0937100 + 0.995600i \(0.529873\pi\)
\(912\) 0 0
\(913\) 12.0000 0.397142
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −16.9706 −0.560417
\(918\) 0 0
\(919\) −21.2132 −0.699759 −0.349880 0.936795i \(-0.613777\pi\)
−0.349880 + 0.936795i \(0.613777\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −25.4558 −0.836983
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 6.00000 0.196854 0.0984268 0.995144i \(-0.468619\pi\)
0.0984268 + 0.995144i \(0.468619\pi\)
\(930\) 0 0
\(931\) −20.0000 −0.655474
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −5.65685 −0.184999
\(936\) 0 0
\(937\) −18.0000 −0.588034 −0.294017 0.955800i \(-0.594992\pi\)
−0.294017 + 0.955800i \(0.594992\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 1.41421 0.0461020 0.0230510 0.999734i \(-0.492662\pi\)
0.0230510 + 0.999734i \(0.492662\pi\)
\(942\) 0 0
\(943\) 16.9706 0.552638
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 28.0000 0.909878 0.454939 0.890523i \(-0.349661\pi\)
0.454939 + 0.890523i \(0.349661\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 30.0000 0.971795 0.485898 0.874016i \(-0.338493\pi\)
0.485898 + 0.874016i \(0.338493\pi\)
\(954\) 0 0
\(955\) 32.0000 1.03550
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 14.1421 0.456673
\(960\) 0 0
\(961\) 19.0000 0.612903
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 5.65685 0.182101
\(966\) 0 0
\(967\) −52.3259 −1.68269 −0.841344 0.540500i \(-0.818235\pi\)
−0.841344 + 0.540500i \(0.818235\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 22.0000 0.706014 0.353007 0.935621i \(-0.385159\pi\)
0.353007 + 0.935621i \(0.385159\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −50.0000 −1.59964 −0.799821 0.600239i \(-0.795072\pi\)
−0.799821 + 0.600239i \(0.795072\pi\)
\(978\) 0 0
\(979\) 4.00000 0.127841
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 50.9117 1.62383 0.811915 0.583775i \(-0.198425\pi\)
0.811915 + 0.583775i \(0.198425\pi\)
\(984\) 0 0
\(985\) 10.0000 0.318626
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −22.6274 −0.719510
\(990\) 0 0
\(991\) 46.6690 1.48249 0.741246 0.671234i \(-0.234235\pi\)
0.741246 + 0.671234i \(0.234235\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −34.0000 −1.07787
\(996\) 0 0
\(997\) 31.1127 0.985349 0.492675 0.870214i \(-0.336019\pi\)
0.492675 + 0.870214i \(0.336019\pi\)
\(998\) 0 0
\(999\) 0 0
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 4608.2.a.h.1.1 2
3.2 odd 2 1536.2.a.j.1.2 yes 2
4.3 odd 2 4608.2.a.j.1.1 2
8.3 odd 2 inner 4608.2.a.h.1.2 2
8.5 even 2 4608.2.a.j.1.2 2
12.11 even 2 1536.2.a.c.1.2 yes 2
16.3 odd 4 4608.2.d.g.2305.4 4
16.5 even 4 4608.2.d.g.2305.1 4
16.11 odd 4 4608.2.d.g.2305.2 4
16.13 even 4 4608.2.d.g.2305.3 4
24.5 odd 2 1536.2.a.c.1.1 2
24.11 even 2 1536.2.a.j.1.1 yes 2
48.5 odd 4 1536.2.d.d.769.2 4
48.11 even 4 1536.2.d.d.769.4 4
48.29 odd 4 1536.2.d.d.769.3 4
48.35 even 4 1536.2.d.d.769.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1536.2.a.c.1.1 2 24.5 odd 2
1536.2.a.c.1.2 yes 2 12.11 even 2
1536.2.a.j.1.1 yes 2 24.11 even 2
1536.2.a.j.1.2 yes 2 3.2 odd 2
1536.2.d.d.769.1 4 48.35 even 4
1536.2.d.d.769.2 4 48.5 odd 4
1536.2.d.d.769.3 4 48.29 odd 4
1536.2.d.d.769.4 4 48.11 even 4
4608.2.a.h.1.1 2 1.1 even 1 trivial
4608.2.a.h.1.2 2 8.3 odd 2 inner
4608.2.a.j.1.1 2 4.3 odd 2
4608.2.a.j.1.2 2 8.5 even 2
4608.2.d.g.2305.1 4 16.5 even 4
4608.2.d.g.2305.2 4 16.11 odd 4
4608.2.d.g.2305.3 4 16.13 even 4
4608.2.d.g.2305.4 4 16.3 odd 4