Properties

Label 768.2.a.i.1.2
Level $768$
Weight $2$
Character 768.1
Self dual yes
Analytic conductor $6.133$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [768,2,Mod(1,768)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(768, 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("768.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 768.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: \(6.13251087523\)
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: \( 2 \)
Twist minimal: no (minimal twist has level 384)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 768.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +2.82843 q^{5} -2.82843 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +2.82843 q^{5} -2.82843 q^{7} +1.00000 q^{9} -4.00000 q^{11} -5.65685 q^{13} -2.82843 q^{15} -2.00000 q^{17} -4.00000 q^{19} +2.82843 q^{21} +5.65685 q^{23} +3.00000 q^{25} -1.00000 q^{27} -2.82843 q^{29} +8.48528 q^{31} +4.00000 q^{33} -8.00000 q^{35} +5.65685 q^{39} -10.0000 q^{41} -12.0000 q^{43} +2.82843 q^{45} -5.65685 q^{47} +1.00000 q^{49} +2.00000 q^{51} -2.82843 q^{53} -11.3137 q^{55} +4.00000 q^{57} +4.00000 q^{59} +11.3137 q^{61} -2.82843 q^{63} -16.0000 q^{65} -4.00000 q^{67} -5.65685 q^{69} -5.65685 q^{71} +2.00000 q^{73} -3.00000 q^{75} +11.3137 q^{77} +8.48528 q^{79} +1.00000 q^{81} +4.00000 q^{83} -5.65685 q^{85} +2.82843 q^{87} -6.00000 q^{89} +16.0000 q^{91} -8.48528 q^{93} -11.3137 q^{95} +14.0000 q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{9} - 8 q^{11} - 4 q^{17} - 8 q^{19} + 6 q^{25} - 2 q^{27} + 8 q^{33} - 16 q^{35} - 20 q^{41} - 24 q^{43} + 2 q^{49} + 4 q^{51} + 8 q^{57} + 8 q^{59} - 32 q^{65} - 8 q^{67} + 4 q^{73} - 6 q^{75} + 2 q^{81} + 8 q^{83} - 12 q^{89} + 32 q^{91} + 28 q^{97} - 8 q^{99}+O(q^{100}) \) 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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 2.82843 1.26491 0.632456 0.774597i \(-0.282047\pi\)
0.632456 + 0.774597i \(0.282047\pi\)
\(6\) 0 0
\(7\) −2.82843 −1.06904 −0.534522 0.845154i \(-0.679509\pi\)
−0.534522 + 0.845154i \(0.679509\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) −5.65685 −1.56893 −0.784465 0.620174i \(-0.787062\pi\)
−0.784465 + 0.620174i \(0.787062\pi\)
\(14\) 0 0
\(15\) −2.82843 −0.730297
\(16\) 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 \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 2.82843 0.617213
\(22\) 0 0
\(23\) 5.65685 1.17954 0.589768 0.807573i \(-0.299219\pi\)
0.589768 + 0.807573i \(0.299219\pi\)
\(24\) 0 0
\(25\) 3.00000 0.600000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −2.82843 −0.525226 −0.262613 0.964901i \(-0.584584\pi\)
−0.262613 + 0.964901i \(0.584584\pi\)
\(30\) 0 0
\(31\) 8.48528 1.52400 0.762001 0.647576i \(-0.224217\pi\)
0.762001 + 0.647576i \(0.224217\pi\)
\(32\) 0 0
\(33\) 4.00000 0.696311
\(34\) 0 0
\(35\) −8.00000 −1.35225
\(36\) 0 0
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 5.65685 0.905822
\(40\) 0 0
\(41\) −10.0000 −1.56174 −0.780869 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) −12.0000 −1.82998 −0.914991 0.403473i \(-0.867803\pi\)
−0.914991 + 0.403473i \(0.867803\pi\)
\(44\) 0 0
\(45\) 2.82843 0.421637
\(46\) 0 0
\(47\) −5.65685 −0.825137 −0.412568 0.910927i \(-0.635368\pi\)
−0.412568 + 0.910927i \(0.635368\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) 0 0
\(53\) −2.82843 −0.388514 −0.194257 0.980951i \(-0.562230\pi\)
−0.194257 + 0.980951i \(0.562230\pi\)
\(54\) 0 0
\(55\) −11.3137 −1.52554
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) 0 0
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) 11.3137 1.44857 0.724286 0.689500i \(-0.242170\pi\)
0.724286 + 0.689500i \(0.242170\pi\)
\(62\) 0 0
\(63\) −2.82843 −0.356348
\(64\) 0 0
\(65\) −16.0000 −1.98456
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) −5.65685 −0.681005
\(70\) 0 0
\(71\) −5.65685 −0.671345 −0.335673 0.941979i \(-0.608964\pi\)
−0.335673 + 0.941979i \(0.608964\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 0 0
\(75\) −3.00000 −0.346410
\(76\) 0 0
\(77\) 11.3137 1.28932
\(78\) 0 0
\(79\) 8.48528 0.954669 0.477334 0.878722i \(-0.341603\pi\)
0.477334 + 0.878722i \(0.341603\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) −5.65685 −0.613572
\(86\) 0 0
\(87\) 2.82843 0.303239
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 16.0000 1.67726
\(92\) 0 0
\(93\) −8.48528 −0.879883
\(94\) 0 0
\(95\) −11.3137 −1.16076
\(96\) 0 0
\(97\) 14.0000 1.42148 0.710742 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 0 0
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) −14.1421 −1.40720 −0.703598 0.710599i \(-0.748424\pi\)
−0.703598 + 0.710599i \(0.748424\pi\)
\(102\) 0 0
\(103\) 2.82843 0.278693 0.139347 0.990244i \(-0.455500\pi\)
0.139347 + 0.990244i \(0.455500\pi\)
\(104\) 0 0
\(105\) 8.00000 0.780720
\(106\) 0 0
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) 5.65685 0.541828 0.270914 0.962604i \(-0.412674\pi\)
0.270914 + 0.962604i \(0.412674\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 0 0
\(115\) 16.0000 1.49201
\(116\) 0 0
\(117\) −5.65685 −0.522976
\(118\) 0 0
\(119\) 5.65685 0.518563
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) 10.0000 0.901670
\(124\) 0 0
\(125\) −5.65685 −0.505964
\(126\) 0 0
\(127\) −8.48528 −0.752947 −0.376473 0.926427i \(-0.622863\pi\)
−0.376473 + 0.926427i \(0.622863\pi\)
\(128\) 0 0
\(129\) 12.0000 1.05654
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) 11.3137 0.981023
\(134\) 0 0
\(135\) −2.82843 −0.243432
\(136\) 0 0
\(137\) −10.0000 −0.854358 −0.427179 0.904167i \(-0.640493\pi\)
−0.427179 + 0.904167i \(0.640493\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 5.65685 0.476393
\(142\) 0 0
\(143\) 22.6274 1.89220
\(144\) 0 0
\(145\) −8.00000 −0.664364
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) 14.1421 1.15857 0.579284 0.815125i \(-0.303332\pi\)
0.579284 + 0.815125i \(0.303332\pi\)
\(150\) 0 0
\(151\) 8.48528 0.690522 0.345261 0.938507i \(-0.387790\pi\)
0.345261 + 0.938507i \(0.387790\pi\)
\(152\) 0 0
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) 24.0000 1.92773
\(156\) 0 0
\(157\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) 0 0
\(159\) 2.82843 0.224309
\(160\) 0 0
\(161\) −16.0000 −1.26098
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 0 0
\(165\) 11.3137 0.880771
\(166\) 0 0
\(167\) 11.3137 0.875481 0.437741 0.899101i \(-0.355779\pi\)
0.437741 + 0.899101i \(0.355779\pi\)
\(168\) 0 0
\(169\) 19.0000 1.46154
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 0 0
\(173\) −19.7990 −1.50529 −0.752645 0.658427i \(-0.771222\pi\)
−0.752645 + 0.658427i \(0.771222\pi\)
\(174\) 0 0
\(175\) −8.48528 −0.641427
\(176\) 0 0
\(177\) −4.00000 −0.300658
\(178\) 0 0
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) 5.65685 0.420471 0.210235 0.977651i \(-0.432577\pi\)
0.210235 + 0.977651i \(0.432577\pi\)
\(182\) 0 0
\(183\) −11.3137 −0.836333
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 8.00000 0.585018
\(188\) 0 0
\(189\) 2.82843 0.205738
\(190\) 0 0
\(191\) 11.3137 0.818631 0.409316 0.912393i \(-0.365768\pi\)
0.409316 + 0.912393i \(0.365768\pi\)
\(192\) 0 0
\(193\) 6.00000 0.431889 0.215945 0.976406i \(-0.430717\pi\)
0.215945 + 0.976406i \(0.430717\pi\)
\(194\) 0 0
\(195\) 16.0000 1.14578
\(196\) 0 0
\(197\) 8.48528 0.604551 0.302276 0.953221i \(-0.402254\pi\)
0.302276 + 0.953221i \(0.402254\pi\)
\(198\) 0 0
\(199\) 2.82843 0.200502 0.100251 0.994962i \(-0.468035\pi\)
0.100251 + 0.994962i \(0.468035\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 0 0
\(203\) 8.00000 0.561490
\(204\) 0 0
\(205\) −28.2843 −1.97546
\(206\) 0 0
\(207\) 5.65685 0.393179
\(208\) 0 0
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 0 0
\(213\) 5.65685 0.387601
\(214\) 0 0
\(215\) −33.9411 −2.31477
\(216\) 0 0
\(217\) −24.0000 −1.62923
\(218\) 0 0
\(219\) −2.00000 −0.135147
\(220\) 0 0
\(221\) 11.3137 0.761042
\(222\) 0 0
\(223\) −19.7990 −1.32584 −0.662919 0.748691i \(-0.730683\pi\)
−0.662919 + 0.748691i \(0.730683\pi\)
\(224\) 0 0
\(225\) 3.00000 0.200000
\(226\) 0 0
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 0 0
\(229\) −5.65685 −0.373815 −0.186908 0.982377i \(-0.559847\pi\)
−0.186908 + 0.982377i \(0.559847\pi\)
\(230\) 0 0
\(231\) −11.3137 −0.744387
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) −16.0000 −1.04372
\(236\) 0 0
\(237\) −8.48528 −0.551178
\(238\) 0 0
\(239\) −22.6274 −1.46365 −0.731823 0.681495i \(-0.761330\pi\)
−0.731823 + 0.681495i \(0.761330\pi\)
\(240\) 0 0
\(241\) 6.00000 0.386494 0.193247 0.981150i \(-0.438098\pi\)
0.193247 + 0.981150i \(0.438098\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 2.82843 0.180702
\(246\) 0 0
\(247\) 22.6274 1.43975
\(248\) 0 0
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) −22.6274 −1.42257
\(254\) 0 0
\(255\) 5.65685 0.354246
\(256\) 0 0
\(257\) −30.0000 −1.87135 −0.935674 0.352865i \(-0.885208\pi\)
−0.935674 + 0.352865i \(0.885208\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −2.82843 −0.175075
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) −8.00000 −0.491436
\(266\) 0 0
\(267\) 6.00000 0.367194
\(268\) 0 0
\(269\) 8.48528 0.517357 0.258678 0.965964i \(-0.416713\pi\)
0.258678 + 0.965964i \(0.416713\pi\)
\(270\) 0 0
\(271\) −19.7990 −1.20270 −0.601351 0.798985i \(-0.705371\pi\)
−0.601351 + 0.798985i \(0.705371\pi\)
\(272\) 0 0
\(273\) −16.0000 −0.968364
\(274\) 0 0
\(275\) −12.0000 −0.723627
\(276\) 0 0
\(277\) −16.9706 −1.01966 −0.509831 0.860274i \(-0.670292\pi\)
−0.509831 + 0.860274i \(0.670292\pi\)
\(278\) 0 0
\(279\) 8.48528 0.508001
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 0 0
\(285\) 11.3137 0.670166
\(286\) 0 0
\(287\) 28.2843 1.66957
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −14.0000 −0.820695
\(292\) 0 0
\(293\) 19.7990 1.15667 0.578335 0.815800i \(-0.303703\pi\)
0.578335 + 0.815800i \(0.303703\pi\)
\(294\) 0 0
\(295\) 11.3137 0.658710
\(296\) 0 0
\(297\) 4.00000 0.232104
\(298\) 0 0
\(299\) −32.0000 −1.85061
\(300\) 0 0
\(301\) 33.9411 1.95633
\(302\) 0 0
\(303\) 14.1421 0.812444
\(304\) 0 0
\(305\) 32.0000 1.83231
\(306\) 0 0
\(307\) −4.00000 −0.228292 −0.114146 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(308\) 0 0
\(309\) −2.82843 −0.160904
\(310\) 0 0
\(311\) 11.3137 0.641542 0.320771 0.947157i \(-0.396058\pi\)
0.320771 + 0.947157i \(0.396058\pi\)
\(312\) 0 0
\(313\) 14.0000 0.791327 0.395663 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) 0 0
\(315\) −8.00000 −0.450749
\(316\) 0 0
\(317\) 8.48528 0.476581 0.238290 0.971194i \(-0.423413\pi\)
0.238290 + 0.971194i \(0.423413\pi\)
\(318\) 0 0
\(319\) 11.3137 0.633446
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) 8.00000 0.445132
\(324\) 0 0
\(325\) −16.9706 −0.941357
\(326\) 0 0
\(327\) −5.65685 −0.312825
\(328\) 0 0
\(329\) 16.0000 0.882109
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −11.3137 −0.618134
\(336\) 0 0
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) 0 0
\(339\) −2.00000 −0.108625
\(340\) 0 0
\(341\) −33.9411 −1.83801
\(342\) 0 0
\(343\) 16.9706 0.916324
\(344\) 0 0
\(345\) −16.0000 −0.861411
\(346\) 0 0
\(347\) −36.0000 −1.93258 −0.966291 0.257454i \(-0.917117\pi\)
−0.966291 + 0.257454i \(0.917117\pi\)
\(348\) 0 0
\(349\) 11.3137 0.605609 0.302804 0.953053i \(-0.402077\pi\)
0.302804 + 0.953053i \(0.402077\pi\)
\(350\) 0 0
\(351\) 5.65685 0.301941
\(352\) 0 0
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 0 0
\(355\) −16.0000 −0.849192
\(356\) 0 0
\(357\) −5.65685 −0.299392
\(358\) 0 0
\(359\) −28.2843 −1.49279 −0.746393 0.665505i \(-0.768216\pi\)
−0.746393 + 0.665505i \(0.768216\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) −5.00000 −0.262432
\(364\) 0 0
\(365\) 5.65685 0.296093
\(366\) 0 0
\(367\) 8.48528 0.442928 0.221464 0.975169i \(-0.428916\pi\)
0.221464 + 0.975169i \(0.428916\pi\)
\(368\) 0 0
\(369\) −10.0000 −0.520579
\(370\) 0 0
\(371\) 8.00000 0.415339
\(372\) 0 0
\(373\) 33.9411 1.75740 0.878702 0.477370i \(-0.158410\pi\)
0.878702 + 0.477370i \(0.158410\pi\)
\(374\) 0 0
\(375\) 5.65685 0.292119
\(376\) 0 0
\(377\) 16.0000 0.824042
\(378\) 0 0
\(379\) −28.0000 −1.43826 −0.719132 0.694874i \(-0.755460\pi\)
−0.719132 + 0.694874i \(0.755460\pi\)
\(380\) 0 0
\(381\) 8.48528 0.434714
\(382\) 0 0
\(383\) −11.3137 −0.578103 −0.289052 0.957313i \(-0.593340\pi\)
−0.289052 + 0.957313i \(0.593340\pi\)
\(384\) 0 0
\(385\) 32.0000 1.63087
\(386\) 0 0
\(387\) −12.0000 −0.609994
\(388\) 0 0
\(389\) −31.1127 −1.57748 −0.788738 0.614729i \(-0.789265\pi\)
−0.788738 + 0.614729i \(0.789265\pi\)
\(390\) 0 0
\(391\) −11.3137 −0.572159
\(392\) 0 0
\(393\) −12.0000 −0.605320
\(394\) 0 0
\(395\) 24.0000 1.20757
\(396\) 0 0
\(397\) 33.9411 1.70346 0.851728 0.523984i \(-0.175555\pi\)
0.851728 + 0.523984i \(0.175555\pi\)
\(398\) 0 0
\(399\) −11.3137 −0.566394
\(400\) 0 0
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) 0 0
\(403\) −48.0000 −2.39105
\(404\) 0 0
\(405\) 2.82843 0.140546
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) 10.0000 0.493264
\(412\) 0 0
\(413\) −11.3137 −0.556711
\(414\) 0 0
\(415\) 11.3137 0.555368
\(416\) 0 0
\(417\) −4.00000 −0.195881
\(418\) 0 0
\(419\) −28.0000 −1.36789 −0.683945 0.729534i \(-0.739737\pi\)
−0.683945 + 0.729534i \(0.739737\pi\)
\(420\) 0 0
\(421\) −16.9706 −0.827095 −0.413547 0.910483i \(-0.635710\pi\)
−0.413547 + 0.910483i \(0.635710\pi\)
\(422\) 0 0
\(423\) −5.65685 −0.275046
\(424\) 0 0
\(425\) −6.00000 −0.291043
\(426\) 0 0
\(427\) −32.0000 −1.54859
\(428\) 0 0
\(429\) −22.6274 −1.09246
\(430\) 0 0
\(431\) 28.2843 1.36241 0.681203 0.732095i \(-0.261457\pi\)
0.681203 + 0.732095i \(0.261457\pi\)
\(432\) 0 0
\(433\) −34.0000 −1.63394 −0.816968 0.576683i \(-0.804347\pi\)
−0.816968 + 0.576683i \(0.804347\pi\)
\(434\) 0 0
\(435\) 8.00000 0.383571
\(436\) 0 0
\(437\) −22.6274 −1.08242
\(438\) 0 0
\(439\) −31.1127 −1.48493 −0.742464 0.669886i \(-0.766343\pi\)
−0.742464 + 0.669886i \(0.766343\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −20.0000 −0.950229 −0.475114 0.879924i \(-0.657593\pi\)
−0.475114 + 0.879924i \(0.657593\pi\)
\(444\) 0 0
\(445\) −16.9706 −0.804482
\(446\) 0 0
\(447\) −14.1421 −0.668900
\(448\) 0 0
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) 40.0000 1.88353
\(452\) 0 0
\(453\) −8.48528 −0.398673
\(454\) 0 0
\(455\) 45.2548 2.12158
\(456\) 0 0
\(457\) 6.00000 0.280668 0.140334 0.990104i \(-0.455182\pi\)
0.140334 + 0.990104i \(0.455182\pi\)
\(458\) 0 0
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) −8.48528 −0.395199 −0.197599 0.980283i \(-0.563315\pi\)
−0.197599 + 0.980283i \(0.563315\pi\)
\(462\) 0 0
\(463\) −19.7990 −0.920137 −0.460069 0.887883i \(-0.652175\pi\)
−0.460069 + 0.887883i \(0.652175\pi\)
\(464\) 0 0
\(465\) −24.0000 −1.11297
\(466\) 0 0
\(467\) −28.0000 −1.29569 −0.647843 0.761774i \(-0.724329\pi\)
−0.647843 + 0.761774i \(0.724329\pi\)
\(468\) 0 0
\(469\) 11.3137 0.522419
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 48.0000 2.20704
\(474\) 0 0
\(475\) −12.0000 −0.550598
\(476\) 0 0
\(477\) −2.82843 −0.129505
\(478\) 0 0
\(479\) −28.2843 −1.29234 −0.646171 0.763193i \(-0.723631\pi\)
−0.646171 + 0.763193i \(0.723631\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 16.0000 0.728025
\(484\) 0 0
\(485\) 39.5980 1.79805
\(486\) 0 0
\(487\) −14.1421 −0.640841 −0.320421 0.947275i \(-0.603824\pi\)
−0.320421 + 0.947275i \(0.603824\pi\)
\(488\) 0 0
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) 20.0000 0.902587 0.451294 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) 0 0
\(493\) 5.65685 0.254772
\(494\) 0 0
\(495\) −11.3137 −0.508513
\(496\) 0 0
\(497\) 16.0000 0.717698
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 0 0
\(501\) −11.3137 −0.505459
\(502\) 0 0
\(503\) −5.65685 −0.252227 −0.126113 0.992016i \(-0.540250\pi\)
−0.126113 + 0.992016i \(0.540250\pi\)
\(504\) 0 0
\(505\) −40.0000 −1.77998
\(506\) 0 0
\(507\) −19.0000 −0.843820
\(508\) 0 0
\(509\) −2.82843 −0.125368 −0.0626839 0.998033i \(-0.519966\pi\)
−0.0626839 + 0.998033i \(0.519966\pi\)
\(510\) 0 0
\(511\) −5.65685 −0.250244
\(512\) 0 0
\(513\) 4.00000 0.176604
\(514\) 0 0
\(515\) 8.00000 0.352522
\(516\) 0 0
\(517\) 22.6274 0.995153
\(518\) 0 0
\(519\) 19.7990 0.869079
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 0 0
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 0 0
\(525\) 8.48528 0.370328
\(526\) 0 0
\(527\) −16.9706 −0.739249
\(528\) 0 0
\(529\) 9.00000 0.391304
\(530\) 0 0
\(531\) 4.00000 0.173585
\(532\) 0 0
\(533\) 56.5685 2.45026
\(534\) 0 0
\(535\) −33.9411 −1.46740
\(536\) 0 0
\(537\) −12.0000 −0.517838
\(538\) 0 0
\(539\) −4.00000 −0.172292
\(540\) 0 0
\(541\) −16.9706 −0.729621 −0.364811 0.931082i \(-0.618866\pi\)
−0.364811 + 0.931082i \(0.618866\pi\)
\(542\) 0 0
\(543\) −5.65685 −0.242759
\(544\) 0 0
\(545\) 16.0000 0.685365
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) 0 0
\(549\) 11.3137 0.482857
\(550\) 0 0
\(551\) 11.3137 0.481980
\(552\) 0 0
\(553\) −24.0000 −1.02058
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −8.48528 −0.359533 −0.179766 0.983709i \(-0.557534\pi\)
−0.179766 + 0.983709i \(0.557534\pi\)
\(558\) 0 0
\(559\) 67.8823 2.87111
\(560\) 0 0
\(561\) −8.00000 −0.337760
\(562\) 0 0
\(563\) 4.00000 0.168580 0.0842900 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) 0 0
\(565\) 5.65685 0.237986
\(566\) 0 0
\(567\) −2.82843 −0.118783
\(568\) 0 0
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 0 0
\(573\) −11.3137 −0.472637
\(574\) 0 0
\(575\) 16.9706 0.707721
\(576\) 0 0
\(577\) −22.0000 −0.915872 −0.457936 0.888985i \(-0.651411\pi\)
−0.457936 + 0.888985i \(0.651411\pi\)
\(578\) 0 0
\(579\) −6.00000 −0.249351
\(580\) 0 0
\(581\) −11.3137 −0.469372
\(582\) 0 0
\(583\) 11.3137 0.468566
\(584\) 0 0
\(585\) −16.0000 −0.661519
\(586\) 0 0
\(587\) 36.0000 1.48588 0.742940 0.669359i \(-0.233431\pi\)
0.742940 + 0.669359i \(0.233431\pi\)
\(588\) 0 0
\(589\) −33.9411 −1.39852
\(590\) 0 0
\(591\) −8.48528 −0.349038
\(592\) 0 0
\(593\) 34.0000 1.39621 0.698106 0.715994i \(-0.254026\pi\)
0.698106 + 0.715994i \(0.254026\pi\)
\(594\) 0 0
\(595\) 16.0000 0.655936
\(596\) 0 0
\(597\) −2.82843 −0.115760
\(598\) 0 0
\(599\) 39.5980 1.61793 0.808965 0.587857i \(-0.200028\pi\)
0.808965 + 0.587857i \(0.200028\pi\)
\(600\) 0 0
\(601\) 34.0000 1.38689 0.693444 0.720510i \(-0.256092\pi\)
0.693444 + 0.720510i \(0.256092\pi\)
\(602\) 0 0
\(603\) −4.00000 −0.162893
\(604\) 0 0
\(605\) 14.1421 0.574960
\(606\) 0 0
\(607\) 14.1421 0.574012 0.287006 0.957929i \(-0.407340\pi\)
0.287006 + 0.957929i \(0.407340\pi\)
\(608\) 0 0
\(609\) −8.00000 −0.324176
\(610\) 0 0
\(611\) 32.0000 1.29458
\(612\) 0 0
\(613\) 11.3137 0.456956 0.228478 0.973549i \(-0.426625\pi\)
0.228478 + 0.973549i \(0.426625\pi\)
\(614\) 0 0
\(615\) 28.2843 1.14053
\(616\) 0 0
\(617\) −38.0000 −1.52982 −0.764911 0.644136i \(-0.777217\pi\)
−0.764911 + 0.644136i \(0.777217\pi\)
\(618\) 0 0
\(619\) 36.0000 1.44696 0.723481 0.690344i \(-0.242541\pi\)
0.723481 + 0.690344i \(0.242541\pi\)
\(620\) 0 0
\(621\) −5.65685 −0.227002
\(622\) 0 0
\(623\) 16.9706 0.679911
\(624\) 0 0
\(625\) −31.0000 −1.24000
\(626\) 0 0
\(627\) −16.0000 −0.638978
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 8.48528 0.337794 0.168897 0.985634i \(-0.445980\pi\)
0.168897 + 0.985634i \(0.445980\pi\)
\(632\) 0 0
\(633\) −12.0000 −0.476957
\(634\) 0 0
\(635\) −24.0000 −0.952411
\(636\) 0 0
\(637\) −5.65685 −0.224133
\(638\) 0 0
\(639\) −5.65685 −0.223782
\(640\) 0 0
\(641\) −34.0000 −1.34292 −0.671460 0.741041i \(-0.734332\pi\)
−0.671460 + 0.741041i \(0.734332\pi\)
\(642\) 0 0
\(643\) 44.0000 1.73519 0.867595 0.497271i \(-0.165665\pi\)
0.867595 + 0.497271i \(0.165665\pi\)
\(644\) 0 0
\(645\) 33.9411 1.33643
\(646\) 0 0
\(647\) −39.5980 −1.55676 −0.778379 0.627795i \(-0.783958\pi\)
−0.778379 + 0.627795i \(0.783958\pi\)
\(648\) 0 0
\(649\) −16.0000 −0.628055
\(650\) 0 0
\(651\) 24.0000 0.940634
\(652\) 0 0
\(653\) −42.4264 −1.66027 −0.830137 0.557560i \(-0.811738\pi\)
−0.830137 + 0.557560i \(0.811738\pi\)
\(654\) 0 0
\(655\) 33.9411 1.32619
\(656\) 0 0
\(657\) 2.00000 0.0780274
\(658\) 0 0
\(659\) −4.00000 −0.155818 −0.0779089 0.996960i \(-0.524824\pi\)
−0.0779089 + 0.996960i \(0.524824\pi\)
\(660\) 0 0
\(661\) −45.2548 −1.76021 −0.880105 0.474780i \(-0.842528\pi\)
−0.880105 + 0.474780i \(0.842528\pi\)
\(662\) 0 0
\(663\) −11.3137 −0.439388
\(664\) 0 0
\(665\) 32.0000 1.24091
\(666\) 0 0
\(667\) −16.0000 −0.619522
\(668\) 0 0
\(669\) 19.7990 0.765473
\(670\) 0 0
\(671\) −45.2548 −1.74704
\(672\) 0 0
\(673\) 14.0000 0.539660 0.269830 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) 0 0
\(675\) −3.00000 −0.115470
\(676\) 0 0
\(677\) −19.7990 −0.760937 −0.380468 0.924794i \(-0.624237\pi\)
−0.380468 + 0.924794i \(0.624237\pi\)
\(678\) 0 0
\(679\) −39.5980 −1.51963
\(680\) 0 0
\(681\) 12.0000 0.459841
\(682\) 0 0
\(683\) −20.0000 −0.765279 −0.382639 0.923898i \(-0.624985\pi\)
−0.382639 + 0.923898i \(0.624985\pi\)
\(684\) 0 0
\(685\) −28.2843 −1.08069
\(686\) 0 0
\(687\) 5.65685 0.215822
\(688\) 0 0
\(689\) 16.0000 0.609551
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) 0 0
\(693\) 11.3137 0.429772
\(694\) 0 0
\(695\) 11.3137 0.429153
\(696\) 0 0
\(697\) 20.0000 0.757554
\(698\) 0 0
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) 19.7990 0.747798 0.373899 0.927470i \(-0.378021\pi\)
0.373899 + 0.927470i \(0.378021\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 16.0000 0.602595
\(706\) 0 0
\(707\) 40.0000 1.50435
\(708\) 0 0
\(709\) 16.9706 0.637343 0.318671 0.947865i \(-0.396763\pi\)
0.318671 + 0.947865i \(0.396763\pi\)
\(710\) 0 0
\(711\) 8.48528 0.318223
\(712\) 0 0
\(713\) 48.0000 1.79761
\(714\) 0 0
\(715\) 64.0000 2.39346
\(716\) 0 0
\(717\) 22.6274 0.845036
\(718\) 0 0
\(719\) −16.9706 −0.632895 −0.316448 0.948610i \(-0.602490\pi\)
−0.316448 + 0.948610i \(0.602490\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) 0 0
\(723\) −6.00000 −0.223142
\(724\) 0 0
\(725\) −8.48528 −0.315135
\(726\) 0 0
\(727\) −25.4558 −0.944105 −0.472052 0.881570i \(-0.656487\pi\)
−0.472052 + 0.881570i \(0.656487\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 24.0000 0.887672
\(732\) 0 0
\(733\) 16.9706 0.626822 0.313411 0.949618i \(-0.398528\pi\)
0.313411 + 0.949618i \(0.398528\pi\)
\(734\) 0 0
\(735\) −2.82843 −0.104328
\(736\) 0 0
\(737\) 16.0000 0.589368
\(738\) 0 0
\(739\) −36.0000 −1.32428 −0.662141 0.749380i \(-0.730352\pi\)
−0.662141 + 0.749380i \(0.730352\pi\)
\(740\) 0 0
\(741\) −22.6274 −0.831239
\(742\) 0 0
\(743\) −22.6274 −0.830119 −0.415060 0.909794i \(-0.636239\pi\)
−0.415060 + 0.909794i \(0.636239\pi\)
\(744\) 0 0
\(745\) 40.0000 1.46549
\(746\) 0 0
\(747\) 4.00000 0.146352
\(748\) 0 0
\(749\) 33.9411 1.24018
\(750\) 0 0
\(751\) −48.0833 −1.75458 −0.877292 0.479958i \(-0.840652\pi\)
−0.877292 + 0.479958i \(0.840652\pi\)
\(752\) 0 0
\(753\) −12.0000 −0.437304
\(754\) 0 0
\(755\) 24.0000 0.873449
\(756\) 0 0
\(757\) −28.2843 −1.02801 −0.514005 0.857787i \(-0.671839\pi\)
−0.514005 + 0.857787i \(0.671839\pi\)
\(758\) 0 0
\(759\) 22.6274 0.821323
\(760\) 0 0
\(761\) −42.0000 −1.52250 −0.761249 0.648459i \(-0.775414\pi\)
−0.761249 + 0.648459i \(0.775414\pi\)
\(762\) 0 0
\(763\) −16.0000 −0.579239
\(764\) 0 0
\(765\) −5.65685 −0.204524
\(766\) 0 0
\(767\) −22.6274 −0.817029
\(768\) 0 0
\(769\) 38.0000 1.37032 0.685158 0.728395i \(-0.259733\pi\)
0.685158 + 0.728395i \(0.259733\pi\)
\(770\) 0 0
\(771\) 30.0000 1.08042
\(772\) 0 0
\(773\) −2.82843 −0.101731 −0.0508657 0.998706i \(-0.516198\pi\)
−0.0508657 + 0.998706i \(0.516198\pi\)
\(774\) 0 0
\(775\) 25.4558 0.914401
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 40.0000 1.43315
\(780\) 0 0
\(781\) 22.6274 0.809673
\(782\) 0 0
\(783\) 2.82843 0.101080
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −20.0000 −0.712923 −0.356462 0.934310i \(-0.616017\pi\)
−0.356462 + 0.934310i \(0.616017\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −5.65685 −0.201135
\(792\) 0 0
\(793\) −64.0000 −2.27271
\(794\) 0 0
\(795\) 8.00000 0.283731
\(796\) 0 0
\(797\) −19.7990 −0.701316 −0.350658 0.936504i \(-0.614042\pi\)
−0.350658 + 0.936504i \(0.614042\pi\)
\(798\) 0 0
\(799\) 11.3137 0.400250
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 0 0
\(803\) −8.00000 −0.282314
\(804\) 0 0
\(805\) −45.2548 −1.59502
\(806\) 0 0
\(807\) −8.48528 −0.298696
\(808\) 0 0
\(809\) −10.0000 −0.351581 −0.175791 0.984428i \(-0.556248\pi\)
−0.175791 + 0.984428i \(0.556248\pi\)
\(810\) 0 0
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 0 0
\(813\) 19.7990 0.694381
\(814\) 0 0
\(815\) −11.3137 −0.396302
\(816\) 0 0
\(817\) 48.0000 1.67931
\(818\) 0 0
\(819\) 16.0000 0.559085
\(820\) 0 0
\(821\) 31.1127 1.08584 0.542920 0.839784i \(-0.317319\pi\)
0.542920 + 0.839784i \(0.317319\pi\)
\(822\) 0 0
\(823\) 19.7990 0.690149 0.345075 0.938575i \(-0.387854\pi\)
0.345075 + 0.938575i \(0.387854\pi\)
\(824\) 0 0
\(825\) 12.0000 0.417786
\(826\) 0 0
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 0 0
\(829\) 50.9117 1.76824 0.884118 0.467264i \(-0.154760\pi\)
0.884118 + 0.467264i \(0.154760\pi\)
\(830\) 0 0
\(831\) 16.9706 0.588702
\(832\) 0 0
\(833\) −2.00000 −0.0692959
\(834\) 0 0
\(835\) 32.0000 1.10741
\(836\) 0 0
\(837\) −8.48528 −0.293294
\(838\) 0 0
\(839\) 5.65685 0.195296 0.0976481 0.995221i \(-0.468868\pi\)
0.0976481 + 0.995221i \(0.468868\pi\)
\(840\) 0 0
\(841\) −21.0000 −0.724138
\(842\) 0 0
\(843\) −10.0000 −0.344418
\(844\) 0 0
\(845\) 53.7401 1.84872
\(846\) 0 0
\(847\) −14.1421 −0.485930
\(848\) 0 0
\(849\) −4.00000 −0.137280
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 22.6274 0.774748 0.387374 0.921923i \(-0.373382\pi\)
0.387374 + 0.921923i \(0.373382\pi\)
\(854\) 0 0
\(855\) −11.3137 −0.386921
\(856\) 0 0
\(857\) 6.00000 0.204956 0.102478 0.994735i \(-0.467323\pi\)
0.102478 + 0.994735i \(0.467323\pi\)
\(858\) 0 0
\(859\) −44.0000 −1.50126 −0.750630 0.660722i \(-0.770250\pi\)
−0.750630 + 0.660722i \(0.770250\pi\)
\(860\) 0 0
\(861\) −28.2843 −0.963925
\(862\) 0 0
\(863\) 56.5685 1.92562 0.962808 0.270187i \(-0.0870856\pi\)
0.962808 + 0.270187i \(0.0870856\pi\)
\(864\) 0 0
\(865\) −56.0000 −1.90406
\(866\) 0 0
\(867\) 13.0000 0.441503
\(868\) 0 0
\(869\) −33.9411 −1.15137
\(870\) 0 0
\(871\) 22.6274 0.766701
\(872\) 0 0
\(873\) 14.0000 0.473828
\(874\) 0 0
\(875\) 16.0000 0.540899
\(876\) 0 0
\(877\) −11.3137 −0.382037 −0.191018 0.981586i \(-0.561179\pi\)
−0.191018 + 0.981586i \(0.561179\pi\)
\(878\) 0 0
\(879\) −19.7990 −0.667803
\(880\) 0 0
\(881\) −46.0000 −1.54978 −0.774890 0.632096i \(-0.782195\pi\)
−0.774890 + 0.632096i \(0.782195\pi\)
\(882\) 0 0
\(883\) 12.0000 0.403832 0.201916 0.979403i \(-0.435283\pi\)
0.201916 + 0.979403i \(0.435283\pi\)
\(884\) 0 0
\(885\) −11.3137 −0.380306
\(886\) 0 0
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 24.0000 0.804934
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) 0 0
\(893\) 22.6274 0.757198
\(894\) 0 0
\(895\) 33.9411 1.13453
\(896\) 0 0
\(897\) 32.0000 1.06845
\(898\) 0 0
\(899\) −24.0000 −0.800445
\(900\) 0 0
\(901\) 5.65685 0.188457
\(902\) 0 0
\(903\) −33.9411 −1.12949
\(904\) 0 0
\(905\) 16.0000 0.531858
\(906\) 0 0
\(907\) −44.0000 −1.46100 −0.730498 0.682915i \(-0.760712\pi\)
−0.730498 + 0.682915i \(0.760712\pi\)
\(908\) 0 0
\(909\) −14.1421 −0.469065
\(910\) 0 0
\(911\) 22.6274 0.749680 0.374840 0.927090i \(-0.377698\pi\)
0.374840 + 0.927090i \(0.377698\pi\)
\(912\) 0 0
\(913\) −16.0000 −0.529523
\(914\) 0 0
\(915\) −32.0000 −1.05789
\(916\) 0 0
\(917\) −33.9411 −1.12083
\(918\) 0 0
\(919\) −2.82843 −0.0933012 −0.0466506 0.998911i \(-0.514855\pi\)
−0.0466506 + 0.998911i \(0.514855\pi\)
\(920\) 0 0
\(921\) 4.00000 0.131804
\(922\) 0 0
\(923\) 32.0000 1.05329
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 2.82843 0.0928977
\(928\) 0 0
\(929\) −18.0000 −0.590561 −0.295280 0.955411i \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) 0 0
\(931\) −4.00000 −0.131095
\(932\) 0 0
\(933\) −11.3137 −0.370394
\(934\) 0 0
\(935\) 22.6274 0.739996
\(936\) 0 0
\(937\) 42.0000 1.37208 0.686040 0.727564i \(-0.259347\pi\)
0.686040 + 0.727564i \(0.259347\pi\)
\(938\) 0 0
\(939\) −14.0000 −0.456873
\(940\) 0 0
\(941\) 14.1421 0.461020 0.230510 0.973070i \(-0.425960\pi\)
0.230510 + 0.973070i \(0.425960\pi\)
\(942\) 0 0
\(943\) −56.5685 −1.84213
\(944\) 0 0
\(945\) 8.00000 0.260240
\(946\) 0 0
\(947\) −36.0000 −1.16984 −0.584921 0.811090i \(-0.698875\pi\)
−0.584921 + 0.811090i \(0.698875\pi\)
\(948\) 0 0
\(949\) −11.3137 −0.367259
\(950\) 0 0
\(951\) −8.48528 −0.275154
\(952\) 0 0
\(953\) −42.0000 −1.36051 −0.680257 0.732974i \(-0.738132\pi\)
−0.680257 + 0.732974i \(0.738132\pi\)
\(954\) 0 0
\(955\) 32.0000 1.03550
\(956\) 0 0
\(957\) −11.3137 −0.365720
\(958\) 0 0
\(959\) 28.2843 0.913347
\(960\) 0 0
\(961\) 41.0000 1.32258
\(962\) 0 0
\(963\) −12.0000 −0.386695
\(964\) 0 0
\(965\) 16.9706 0.546302
\(966\) 0 0
\(967\) 14.1421 0.454780 0.227390 0.973804i \(-0.426981\pi\)
0.227390 + 0.973804i \(0.426981\pi\)
\(968\) 0 0
\(969\) −8.00000 −0.256997
\(970\) 0 0
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 0 0
\(973\) −11.3137 −0.362701
\(974\) 0 0
\(975\) 16.9706 0.543493
\(976\) 0 0
\(977\) −2.00000 −0.0639857 −0.0319928 0.999488i \(-0.510185\pi\)
−0.0319928 + 0.999488i \(0.510185\pi\)
\(978\) 0 0
\(979\) 24.0000 0.767043
\(980\) 0 0
\(981\) 5.65685 0.180609
\(982\) 0 0
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 24.0000 0.764704
\(986\) 0 0
\(987\) −16.0000 −0.509286
\(988\) 0 0
\(989\) −67.8823 −2.15853
\(990\) 0 0
\(991\) −25.4558 −0.808632 −0.404316 0.914619i \(-0.632490\pi\)
−0.404316 + 0.914619i \(0.632490\pi\)
\(992\) 0 0
\(993\) −20.0000 −0.634681
\(994\) 0 0
\(995\) 8.00000 0.253617
\(996\) 0 0
\(997\) 33.9411 1.07493 0.537463 0.843287i \(-0.319383\pi\)
0.537463 + 0.843287i \(0.319383\pi\)
\(998\) 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 768.2.a.i.1.2 2
3.2 odd 2 2304.2.a.x.1.1 2
4.3 odd 2 768.2.a.l.1.2 2
8.3 odd 2 inner 768.2.a.i.1.1 2
8.5 even 2 768.2.a.l.1.1 2
12.11 even 2 2304.2.a.r.1.1 2
16.3 odd 4 384.2.d.c.193.3 yes 4
16.5 even 4 384.2.d.c.193.4 yes 4
16.11 odd 4 384.2.d.c.193.2 yes 4
16.13 even 4 384.2.d.c.193.1 4
24.5 odd 2 2304.2.a.r.1.2 2
24.11 even 2 2304.2.a.x.1.2 2
48.5 odd 4 1152.2.d.h.577.2 4
48.11 even 4 1152.2.d.h.577.1 4
48.29 odd 4 1152.2.d.h.577.4 4
48.35 even 4 1152.2.d.h.577.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.2.d.c.193.1 4 16.13 even 4
384.2.d.c.193.2 yes 4 16.11 odd 4
384.2.d.c.193.3 yes 4 16.3 odd 4
384.2.d.c.193.4 yes 4 16.5 even 4
768.2.a.i.1.1 2 8.3 odd 2 inner
768.2.a.i.1.2 2 1.1 even 1 trivial
768.2.a.l.1.1 2 8.5 even 2
768.2.a.l.1.2 2 4.3 odd 2
1152.2.d.h.577.1 4 48.11 even 4
1152.2.d.h.577.2 4 48.5 odd 4
1152.2.d.h.577.3 4 48.35 even 4
1152.2.d.h.577.4 4 48.29 odd 4
2304.2.a.r.1.1 2 12.11 even 2
2304.2.a.r.1.2 2 24.5 odd 2
2304.2.a.x.1.1 2 3.2 odd 2
2304.2.a.x.1.2 2 24.11 even 2