Properties

Label 1600.2.c.j.449.2
Level $1600$
Weight $2$
Character 1600.449
Analytic conductor $12.776$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1600,2,Mod(449,1600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1600.449");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.c (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(12.7760643234\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 800)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1600.449
Dual form 1600.2.c.j.449.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{3} -2.00000i q^{7} +2.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} -2.00000i q^{7} +2.00000 q^{9} +5.00000 q^{11} -5.00000i q^{17} -5.00000 q^{19} +2.00000 q^{21} -6.00000i q^{23} +5.00000i q^{27} +4.00000 q^{29} -10.0000 q^{31} +5.00000i q^{33} -10.0000i q^{37} +5.00000 q^{41} +4.00000i q^{43} -8.00000i q^{47} +3.00000 q^{49} +5.00000 q^{51} +10.0000i q^{53} -5.00000i q^{57} +10.0000 q^{61} -4.00000i q^{63} -3.00000i q^{67} +6.00000 q^{69} -5.00000i q^{73} -10.0000i q^{77} +10.0000 q^{79} +1.00000 q^{81} -1.00000i q^{83} +4.00000i q^{87} +9.00000 q^{89} -10.0000i q^{93} -10.0000i q^{97} +10.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{9} + 10 q^{11} - 10 q^{19} + 4 q^{21} + 8 q^{29} - 20 q^{31} + 10 q^{41} + 6 q^{49} + 10 q^{51} + 20 q^{61} + 12 q^{69} + 20 q^{79} + 2 q^{81} + 18 q^{89} + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) 1.00000i 0.577350i 0.957427 + 0.288675i \(0.0932147\pi\)
−0.957427 + 0.288675i \(0.906785\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 2.00000i − 0.755929i −0.925820 0.377964i \(-0.876624\pi\)
0.925820 0.377964i \(-0.123376\pi\)
\(8\) 0 0
\(9\) 2.00000 0.666667
\(10\) 0 0
\(11\) 5.00000 1.50756 0.753778 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 5.00000i − 1.21268i −0.795206 0.606339i \(-0.792637\pi\)
0.795206 0.606339i \(-0.207363\pi\)
\(18\) 0 0
\(19\) −5.00000 −1.14708 −0.573539 0.819178i \(-0.694430\pi\)
−0.573539 + 0.819178i \(0.694430\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) − 6.00000i − 1.25109i −0.780189 0.625543i \(-0.784877\pi\)
0.780189 0.625543i \(-0.215123\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.00000i 0.962250i
\(28\) 0 0
\(29\) 4.00000 0.742781 0.371391 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) −10.0000 −1.79605 −0.898027 0.439941i \(-0.854999\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) 0 0
\(33\) 5.00000i 0.870388i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 10.0000i − 1.64399i −0.569495 0.821995i \(-0.692861\pi\)
0.569495 0.821995i \(-0.307139\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5.00000 0.780869 0.390434 0.920631i \(-0.372325\pi\)
0.390434 + 0.920631i \(0.372325\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 8.00000i − 1.16692i −0.812142 0.583460i \(-0.801699\pi\)
0.812142 0.583460i \(-0.198301\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 5.00000 0.700140
\(52\) 0 0
\(53\) 10.0000i 1.37361i 0.726844 + 0.686803i \(0.240986\pi\)
−0.726844 + 0.686803i \(0.759014\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 5.00000i − 0.662266i
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 0 0
\(63\) − 4.00000i − 0.503953i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 3.00000i − 0.366508i −0.983066 0.183254i \(-0.941337\pi\)
0.983066 0.183254i \(-0.0586631\pi\)
\(68\) 0 0
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) − 5.00000i − 0.585206i −0.956234 0.292603i \(-0.905479\pi\)
0.956234 0.292603i \(-0.0945214\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 10.0000i − 1.13961i
\(78\) 0 0
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 1.00000i − 0.109764i −0.998493 0.0548821i \(-0.982522\pi\)
0.998493 0.0548821i \(-0.0174783\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 4.00000i 0.428845i
\(88\) 0 0
\(89\) 9.00000 0.953998 0.476999 0.878904i \(-0.341725\pi\)
0.476999 + 0.878904i \(0.341725\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) − 10.0000i − 1.03695i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 10.0000i − 1.01535i −0.861550 0.507673i \(-0.830506\pi\)
0.861550 0.507673i \(-0.169494\pi\)
\(98\) 0 0
\(99\) 10.0000 1.00504
\(100\) 0 0
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) 0 0
\(103\) 16.0000i 1.57653i 0.615338 + 0.788263i \(0.289020\pi\)
−0.615338 + 0.788263i \(0.710980\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 3.00000i 0.290021i 0.989430 + 0.145010i \(0.0463216\pi\)
−0.989430 + 0.145010i \(0.953678\pi\)
\(108\) 0 0
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) 10.0000 0.949158
\(112\) 0 0
\(113\) 15.0000i 1.41108i 0.708669 + 0.705541i \(0.249296\pi\)
−0.708669 + 0.705541i \(0.750704\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −10.0000 −0.916698
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 0 0
\(123\) 5.00000i 0.450835i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 18.0000i 1.59724i 0.601834 + 0.798621i \(0.294437\pi\)
−0.601834 + 0.798621i \(0.705563\pi\)
\(128\) 0 0
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) −20.0000 −1.74741 −0.873704 0.486458i \(-0.838289\pi\)
−0.873704 + 0.486458i \(0.838289\pi\)
\(132\) 0 0
\(133\) 10.0000i 0.867110i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 5.00000i − 0.427179i −0.976924 0.213589i \(-0.931485\pi\)
0.976924 0.213589i \(-0.0685155\pi\)
\(138\) 0 0
\(139\) −5.00000 −0.424094 −0.212047 0.977259i \(-0.568013\pi\)
−0.212047 + 0.977259i \(0.568013\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 3.00000i 0.247436i
\(148\) 0 0
\(149\) −20.0000 −1.63846 −0.819232 0.573462i \(-0.805600\pi\)
−0.819232 + 0.573462i \(0.805600\pi\)
\(150\) 0 0
\(151\) 10.0000 0.813788 0.406894 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) 0 0
\(153\) − 10.0000i − 0.808452i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 10.0000i 0.798087i 0.916932 + 0.399043i \(0.130658\pi\)
−0.916932 + 0.399043i \(0.869342\pi\)
\(158\) 0 0
\(159\) −10.0000 −0.793052
\(160\) 0 0
\(161\) −12.0000 −0.945732
\(162\) 0 0
\(163\) − 21.0000i − 1.64485i −0.568876 0.822423i \(-0.692621\pi\)
0.568876 0.822423i \(-0.307379\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 8.00000i 0.619059i 0.950890 + 0.309529i \(0.100171\pi\)
−0.950890 + 0.309529i \(0.899829\pi\)
\(168\) 0 0
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) −10.0000 −0.764719
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 15.0000 1.12115 0.560576 0.828103i \(-0.310580\pi\)
0.560576 + 0.828103i \(0.310580\pi\)
\(180\) 0 0
\(181\) −18.0000 −1.33793 −0.668965 0.743294i \(-0.733262\pi\)
−0.668965 + 0.743294i \(0.733262\pi\)
\(182\) 0 0
\(183\) 10.0000i 0.739221i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 25.0000i − 1.82818i
\(188\) 0 0
\(189\) 10.0000 0.727393
\(190\) 0 0
\(191\) 10.0000 0.723575 0.361787 0.932261i \(-0.382167\pi\)
0.361787 + 0.932261i \(0.382167\pi\)
\(192\) 0 0
\(193\) 5.00000i 0.359908i 0.983675 + 0.179954i \(0.0575949\pi\)
−0.983675 + 0.179954i \(0.942405\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 10.0000i − 0.712470i −0.934396 0.356235i \(-0.884060\pi\)
0.934396 0.356235i \(-0.115940\pi\)
\(198\) 0 0
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 0 0
\(201\) 3.00000 0.211604
\(202\) 0 0
\(203\) − 8.00000i − 0.561490i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 12.0000i − 0.834058i
\(208\) 0 0
\(209\) −25.0000 −1.72929
\(210\) 0 0
\(211\) −5.00000 −0.344214 −0.172107 0.985078i \(-0.555058\pi\)
−0.172107 + 0.985078i \(0.555058\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 20.0000i 1.35769i
\(218\) 0 0
\(219\) 5.00000 0.337869
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 4.00000i 0.267860i 0.990991 + 0.133930i \(0.0427597\pi\)
−0.990991 + 0.133930i \(0.957240\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 12.0000i − 0.796468i −0.917284 0.398234i \(-0.869623\pi\)
0.917284 0.398234i \(-0.130377\pi\)
\(228\) 0 0
\(229\) −4.00000 −0.264327 −0.132164 0.991228i \(-0.542192\pi\)
−0.132164 + 0.991228i \(0.542192\pi\)
\(230\) 0 0
\(231\) 10.0000 0.657952
\(232\) 0 0
\(233\) − 10.0000i − 0.655122i −0.944830 0.327561i \(-0.893773\pi\)
0.944830 0.327561i \(-0.106227\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 10.0000i 0.649570i
\(238\) 0 0
\(239\) −20.0000 −1.29369 −0.646846 0.762620i \(-0.723912\pi\)
−0.646846 + 0.762620i \(0.723912\pi\)
\(240\) 0 0
\(241\) −15.0000 −0.966235 −0.483117 0.875556i \(-0.660496\pi\)
−0.483117 + 0.875556i \(0.660496\pi\)
\(242\) 0 0
\(243\) 16.0000i 1.02640i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 1.00000 0.0633724
\(250\) 0 0
\(251\) −5.00000 −0.315597 −0.157799 0.987471i \(-0.550440\pi\)
−0.157799 + 0.987471i \(0.550440\pi\)
\(252\) 0 0
\(253\) − 30.0000i − 1.88608i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 10.0000i 0.623783i 0.950118 + 0.311891i \(0.100963\pi\)
−0.950118 + 0.311891i \(0.899037\pi\)
\(258\) 0 0
\(259\) −20.0000 −1.24274
\(260\) 0 0
\(261\) 8.00000 0.495188
\(262\) 0 0
\(263\) − 14.0000i − 0.863277i −0.902047 0.431638i \(-0.857936\pi\)
0.902047 0.431638i \(-0.142064\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 9.00000i 0.550791i
\(268\) 0 0
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) −10.0000 −0.607457 −0.303728 0.952759i \(-0.598232\pi\)
−0.303728 + 0.952759i \(0.598232\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 20.0000i 1.20168i 0.799368 + 0.600842i \(0.205168\pi\)
−0.799368 + 0.600842i \(0.794832\pi\)
\(278\) 0 0
\(279\) −20.0000 −1.19737
\(280\) 0 0
\(281\) −10.0000 −0.596550 −0.298275 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) 0 0
\(283\) − 29.0000i − 1.72387i −0.507018 0.861936i \(-0.669252\pi\)
0.507018 0.861936i \(-0.330748\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 10.0000i − 0.590281i
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 10.0000 0.586210
\(292\) 0 0
\(293\) 30.0000i 1.75262i 0.481749 + 0.876309i \(0.340002\pi\)
−0.481749 + 0.876309i \(0.659998\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 25.0000i 1.45065i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 8.00000 0.461112
\(302\) 0 0
\(303\) − 2.00000i − 0.114897i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 7.00000i 0.399511i 0.979846 + 0.199756i \(0.0640148\pi\)
−0.979846 + 0.199756i \(0.935985\pi\)
\(308\) 0 0
\(309\) −16.0000 −0.910208
\(310\) 0 0
\(311\) −10.0000 −0.567048 −0.283524 0.958965i \(-0.591504\pi\)
−0.283524 + 0.958965i \(0.591504\pi\)
\(312\) 0 0
\(313\) − 30.0000i − 1.69570i −0.530236 0.847850i \(-0.677897\pi\)
0.530236 0.847850i \(-0.322103\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 20.0000i 1.12331i 0.827371 + 0.561656i \(0.189836\pi\)
−0.827371 + 0.561656i \(0.810164\pi\)
\(318\) 0 0
\(319\) 20.0000 1.11979
\(320\) 0 0
\(321\) −3.00000 −0.167444
\(322\) 0 0
\(323\) 25.0000i 1.39104i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 10.0000i 0.553001i
\(328\) 0 0
\(329\) −16.0000 −0.882109
\(330\) 0 0
\(331\) 25.0000 1.37412 0.687062 0.726599i \(-0.258900\pi\)
0.687062 + 0.726599i \(0.258900\pi\)
\(332\) 0 0
\(333\) − 20.0000i − 1.09599i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 5.00000i 0.272367i 0.990684 + 0.136184i \(0.0434837\pi\)
−0.990684 + 0.136184i \(0.956516\pi\)
\(338\) 0 0
\(339\) −15.0000 −0.814688
\(340\) 0 0
\(341\) −50.0000 −2.70765
\(342\) 0 0
\(343\) − 20.0000i − 1.07990i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 27.0000i 1.44944i 0.689046 + 0.724718i \(0.258030\pi\)
−0.689046 + 0.724718i \(0.741970\pi\)
\(348\) 0 0
\(349\) −14.0000 −0.749403 −0.374701 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 30.0000i 1.59674i 0.602168 + 0.798369i \(0.294304\pi\)
−0.602168 + 0.798369i \(0.705696\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 10.0000i − 0.529256i
\(358\) 0 0
\(359\) 10.0000 0.527780 0.263890 0.964553i \(-0.414994\pi\)
0.263890 + 0.964553i \(0.414994\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) 0 0
\(363\) 14.0000i 0.734809i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 12.0000i 0.626395i 0.949688 + 0.313197i \(0.101400\pi\)
−0.949688 + 0.313197i \(0.898600\pi\)
\(368\) 0 0
\(369\) 10.0000 0.520579
\(370\) 0 0
\(371\) 20.0000 1.03835
\(372\) 0 0
\(373\) − 10.0000i − 0.517780i −0.965907 0.258890i \(-0.916643\pi\)
0.965907 0.258890i \(-0.0833568\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −15.0000 −0.770498 −0.385249 0.922813i \(-0.625884\pi\)
−0.385249 + 0.922813i \(0.625884\pi\)
\(380\) 0 0
\(381\) −18.0000 −0.922168
\(382\) 0 0
\(383\) 14.0000i 0.715367i 0.933843 + 0.357683i \(0.116433\pi\)
−0.933843 + 0.357683i \(0.883567\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 8.00000i 0.406663i
\(388\) 0 0
\(389\) −10.0000 −0.507020 −0.253510 0.967333i \(-0.581585\pi\)
−0.253510 + 0.967333i \(0.581585\pi\)
\(390\) 0 0
\(391\) −30.0000 −1.51717
\(392\) 0 0
\(393\) − 20.0000i − 1.00887i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 20.0000i 1.00377i 0.864934 + 0.501886i \(0.167360\pi\)
−0.864934 + 0.501886i \(0.832640\pi\)
\(398\) 0 0
\(399\) −10.0000 −0.500626
\(400\) 0 0
\(401\) −27.0000 −1.34832 −0.674158 0.738587i \(-0.735493\pi\)
−0.674158 + 0.738587i \(0.735493\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 50.0000i − 2.47841i
\(408\) 0 0
\(409\) −5.00000 −0.247234 −0.123617 0.992330i \(-0.539449\pi\)
−0.123617 + 0.992330i \(0.539449\pi\)
\(410\) 0 0
\(411\) 5.00000 0.246632
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 5.00000i − 0.244851i
\(418\) 0 0
\(419\) 15.0000 0.732798 0.366399 0.930458i \(-0.380591\pi\)
0.366399 + 0.930458i \(0.380591\pi\)
\(420\) 0 0
\(421\) 20.0000 0.974740 0.487370 0.873195i \(-0.337956\pi\)
0.487370 + 0.873195i \(0.337956\pi\)
\(422\) 0 0
\(423\) − 16.0000i − 0.777947i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 20.0000i − 0.967868i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −10.0000 −0.481683 −0.240842 0.970564i \(-0.577423\pi\)
−0.240842 + 0.970564i \(0.577423\pi\)
\(432\) 0 0
\(433\) 5.00000i 0.240285i 0.992757 + 0.120142i \(0.0383351\pi\)
−0.992757 + 0.120142i \(0.961665\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 30.0000i 1.43509i
\(438\) 0 0
\(439\) 20.0000 0.954548 0.477274 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 0 0
\(441\) 6.00000 0.285714
\(442\) 0 0
\(443\) 39.0000i 1.85295i 0.376361 + 0.926473i \(0.377175\pi\)
−0.376361 + 0.926473i \(0.622825\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 20.0000i − 0.945968i
\(448\) 0 0
\(449\) −25.0000 −1.17982 −0.589911 0.807468i \(-0.700837\pi\)
−0.589911 + 0.807468i \(0.700837\pi\)
\(450\) 0 0
\(451\) 25.0000 1.17720
\(452\) 0 0
\(453\) 10.0000i 0.469841i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 15.0000i 0.701670i 0.936437 + 0.350835i \(0.114102\pi\)
−0.936437 + 0.350835i \(0.885898\pi\)
\(458\) 0 0
\(459\) 25.0000 1.16690
\(460\) 0 0
\(461\) 12.0000 0.558896 0.279448 0.960161i \(-0.409849\pi\)
0.279448 + 0.960161i \(0.409849\pi\)
\(462\) 0 0
\(463\) − 4.00000i − 0.185896i −0.995671 0.0929479i \(-0.970371\pi\)
0.995671 0.0929479i \(-0.0296290\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 12.0000i 0.555294i 0.960683 + 0.277647i \(0.0895545\pi\)
−0.960683 + 0.277647i \(0.910445\pi\)
\(468\) 0 0
\(469\) −6.00000 −0.277054
\(470\) 0 0
\(471\) −10.0000 −0.460776
\(472\) 0 0
\(473\) 20.0000i 0.919601i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 20.0000i 0.915737i
\(478\) 0 0
\(479\) 30.0000 1.37073 0.685367 0.728197i \(-0.259642\pi\)
0.685367 + 0.728197i \(0.259642\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) − 12.0000i − 0.546019i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 2.00000i 0.0906287i 0.998973 + 0.0453143i \(0.0144289\pi\)
−0.998973 + 0.0453143i \(0.985571\pi\)
\(488\) 0 0
\(489\) 21.0000 0.949653
\(490\) 0 0
\(491\) −20.0000 −0.902587 −0.451294 0.892375i \(-0.649037\pi\)
−0.451294 + 0.892375i \(0.649037\pi\)
\(492\) 0 0
\(493\) − 20.0000i − 0.900755i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −20.0000 −0.895323 −0.447661 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) 0 0
\(501\) −8.00000 −0.357414
\(502\) 0 0
\(503\) − 4.00000i − 0.178351i −0.996016 0.0891756i \(-0.971577\pi\)
0.996016 0.0891756i \(-0.0284232\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 13.0000i 0.577350i
\(508\) 0 0
\(509\) −6.00000 −0.265945 −0.132973 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 0 0
\(511\) −10.0000 −0.442374
\(512\) 0 0
\(513\) − 25.0000i − 1.10378i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 40.0000i − 1.75920i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −3.00000 −0.131432 −0.0657162 0.997838i \(-0.520933\pi\)
−0.0657162 + 0.997838i \(0.520933\pi\)
\(522\) 0 0
\(523\) 1.00000i 0.0437269i 0.999761 + 0.0218635i \(0.00695991\pi\)
−0.999761 + 0.0218635i \(0.993040\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 50.0000i 2.17803i
\(528\) 0 0
\(529\) −13.0000 −0.565217
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 15.0000i 0.647298i
\(538\) 0 0
\(539\) 15.0000 0.646096
\(540\) 0 0
\(541\) −8.00000 −0.343947 −0.171973 0.985102i \(-0.555014\pi\)
−0.171973 + 0.985102i \(0.555014\pi\)
\(542\) 0 0
\(543\) − 18.0000i − 0.772454i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 23.0000i 0.983409i 0.870762 + 0.491704i \(0.163626\pi\)
−0.870762 + 0.491704i \(0.836374\pi\)
\(548\) 0 0
\(549\) 20.0000 0.853579
\(550\) 0 0
\(551\) −20.0000 −0.852029
\(552\) 0 0
\(553\) − 20.0000i − 0.850487i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 40.0000i − 1.69485i −0.530912 0.847427i \(-0.678150\pi\)
0.530912 0.847427i \(-0.321850\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 25.0000 1.05550
\(562\) 0 0
\(563\) − 24.0000i − 1.01148i −0.862686 0.505740i \(-0.831220\pi\)
0.862686 0.505740i \(-0.168780\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 2.00000i − 0.0839921i
\(568\) 0 0
\(569\) 15.0000 0.628833 0.314416 0.949285i \(-0.398191\pi\)
0.314416 + 0.949285i \(0.398191\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 0 0
\(573\) 10.0000i 0.417756i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 5.00000i 0.208153i 0.994569 + 0.104076i \(0.0331886\pi\)
−0.994569 + 0.104076i \(0.966811\pi\)
\(578\) 0 0
\(579\) −5.00000 −0.207793
\(580\) 0 0
\(581\) −2.00000 −0.0829740
\(582\) 0 0
\(583\) 50.0000i 2.07079i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 7.00000i − 0.288921i −0.989511 0.144460i \(-0.953855\pi\)
0.989511 0.144460i \(-0.0461446\pi\)
\(588\) 0 0
\(589\) 50.0000 2.06021
\(590\) 0 0
\(591\) 10.0000 0.411345
\(592\) 0 0
\(593\) 15.0000i 0.615976i 0.951390 + 0.307988i \(0.0996557\pi\)
−0.951390 + 0.307988i \(0.900344\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 20.0000i 0.818546i
\(598\) 0 0
\(599\) −30.0000 −1.22577 −0.612883 0.790173i \(-0.709990\pi\)
−0.612883 + 0.790173i \(0.709990\pi\)
\(600\) 0 0
\(601\) −5.00000 −0.203954 −0.101977 0.994787i \(-0.532517\pi\)
−0.101977 + 0.994787i \(0.532517\pi\)
\(602\) 0 0
\(603\) − 6.00000i − 0.244339i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 8.00000i − 0.324710i −0.986732 0.162355i \(-0.948091\pi\)
0.986732 0.162355i \(-0.0519090\pi\)
\(608\) 0 0
\(609\) 8.00000 0.324176
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 10.0000i 0.403896i 0.979396 + 0.201948i \(0.0647272\pi\)
−0.979396 + 0.201948i \(0.935273\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 30.0000i − 1.20775i −0.797077 0.603877i \(-0.793622\pi\)
0.797077 0.603877i \(-0.206378\pi\)
\(618\) 0 0
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) 30.0000 1.20386
\(622\) 0 0
\(623\) − 18.0000i − 0.721155i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 25.0000i − 0.998404i
\(628\) 0 0
\(629\) −50.0000 −1.99363
\(630\) 0 0
\(631\) 30.0000 1.19428 0.597141 0.802137i \(-0.296303\pi\)
0.597141 + 0.802137i \(0.296303\pi\)
\(632\) 0 0
\(633\) − 5.00000i − 0.198732i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 0 0
\(643\) 36.0000i 1.41970i 0.704352 + 0.709851i \(0.251238\pi\)
−0.704352 + 0.709851i \(0.748762\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 12.0000i 0.471769i 0.971781 + 0.235884i \(0.0757987\pi\)
−0.971781 + 0.235884i \(0.924201\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −20.0000 −0.783862
\(652\) 0 0
\(653\) 30.0000i 1.17399i 0.809590 + 0.586995i \(0.199689\pi\)
−0.809590 + 0.586995i \(0.800311\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 10.0000i − 0.390137i
\(658\) 0 0
\(659\) 25.0000 0.973862 0.486931 0.873441i \(-0.338116\pi\)
0.486931 + 0.873441i \(0.338116\pi\)
\(660\) 0 0
\(661\) 20.0000 0.777910 0.388955 0.921257i \(-0.372836\pi\)
0.388955 + 0.921257i \(0.372836\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 24.0000i − 0.929284i
\(668\) 0 0
\(669\) −4.00000 −0.154649
\(670\) 0 0
\(671\) 50.0000 1.93023
\(672\) 0 0
\(673\) 10.0000i 0.385472i 0.981251 + 0.192736i \(0.0617360\pi\)
−0.981251 + 0.192736i \(0.938264\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 40.0000i − 1.53732i −0.639655 0.768662i \(-0.720923\pi\)
0.639655 0.768662i \(-0.279077\pi\)
\(678\) 0 0
\(679\) −20.0000 −0.767530
\(680\) 0 0
\(681\) 12.0000 0.459841
\(682\) 0 0
\(683\) − 31.0000i − 1.18618i −0.805135 0.593091i \(-0.797907\pi\)
0.805135 0.593091i \(-0.202093\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 4.00000i − 0.152610i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −15.0000 −0.570627 −0.285313 0.958434i \(-0.592098\pi\)
−0.285313 + 0.958434i \(0.592098\pi\)
\(692\) 0 0
\(693\) − 20.0000i − 0.759737i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 25.0000i − 0.946943i
\(698\) 0 0
\(699\) 10.0000 0.378235
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 50.0000i 1.88579i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 4.00000i 0.150435i
\(708\) 0 0
\(709\) 44.0000 1.65245 0.826227 0.563337i \(-0.190483\pi\)
0.826227 + 0.563337i \(0.190483\pi\)
\(710\) 0 0
\(711\) 20.0000 0.750059
\(712\) 0 0
\(713\) 60.0000i 2.24702i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 20.0000i − 0.746914i
\(718\) 0 0
\(719\) −10.0000 −0.372937 −0.186469 0.982461i \(-0.559704\pi\)
−0.186469 + 0.982461i \(0.559704\pi\)
\(720\) 0 0
\(721\) 32.0000 1.19174
\(722\) 0 0
\(723\) − 15.0000i − 0.557856i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 28.0000i 1.03846i 0.854634 + 0.519231i \(0.173782\pi\)
−0.854634 + 0.519231i \(0.826218\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) 20.0000 0.739727
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 15.0000i − 0.552532i
\(738\) 0 0
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 6.00000i − 0.220119i −0.993925 0.110059i \(-0.964896\pi\)
0.993925 0.110059i \(-0.0351041\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 2.00000i − 0.0731762i
\(748\) 0 0
\(749\) 6.00000 0.219235
\(750\) 0 0
\(751\) 20.0000 0.729810 0.364905 0.931045i \(-0.381101\pi\)
0.364905 + 0.931045i \(0.381101\pi\)
\(752\) 0 0
\(753\) − 5.00000i − 0.182210i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 40.0000i 1.45382i 0.686730 + 0.726912i \(0.259045\pi\)
−0.686730 + 0.726912i \(0.740955\pi\)
\(758\) 0 0
\(759\) 30.0000 1.08893
\(760\) 0 0
\(761\) 37.0000 1.34125 0.670624 0.741797i \(-0.266026\pi\)
0.670624 + 0.741797i \(0.266026\pi\)
\(762\) 0 0
\(763\) − 20.0000i − 0.724049i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 51.0000 1.83911 0.919554 0.392965i \(-0.128551\pi\)
0.919554 + 0.392965i \(0.128551\pi\)
\(770\) 0 0
\(771\) −10.0000 −0.360141
\(772\) 0 0
\(773\) 20.0000i 0.719350i 0.933078 + 0.359675i \(0.117112\pi\)
−0.933078 + 0.359675i \(0.882888\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 20.0000i − 0.717496i
\(778\) 0 0
\(779\) −25.0000 −0.895718
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 20.0000i 0.714742i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 52.0000i − 1.85360i −0.375555 0.926800i \(-0.622548\pi\)
0.375555 0.926800i \(-0.377452\pi\)
\(788\) 0 0
\(789\) 14.0000 0.498413
\(790\) 0 0
\(791\) 30.0000 1.06668
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 30.0000i − 1.06265i −0.847167 0.531327i \(-0.821693\pi\)
0.847167 0.531327i \(-0.178307\pi\)
\(798\) 0 0
\(799\) −40.0000 −1.41510
\(800\) 0 0
\(801\) 18.0000 0.635999
\(802\) 0 0
\(803\) − 25.0000i − 0.882231i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) −20.0000 −0.702295 −0.351147 0.936320i \(-0.614208\pi\)
−0.351147 + 0.936320i \(0.614208\pi\)
\(812\) 0 0
\(813\) − 10.0000i − 0.350715i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 20.0000i − 0.699711i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −10.0000 −0.349002 −0.174501 0.984657i \(-0.555831\pi\)
−0.174501 + 0.984657i \(0.555831\pi\)
\(822\) 0 0
\(823\) − 36.0000i − 1.25488i −0.778664 0.627441i \(-0.784103\pi\)
0.778664 0.627441i \(-0.215897\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 23.0000i − 0.799788i −0.916561 0.399894i \(-0.869047\pi\)
0.916561 0.399894i \(-0.130953\pi\)
\(828\) 0 0
\(829\) −40.0000 −1.38926 −0.694629 0.719368i \(-0.744431\pi\)
−0.694629 + 0.719368i \(0.744431\pi\)
\(830\) 0 0
\(831\) −20.0000 −0.693792
\(832\) 0 0
\(833\) − 15.0000i − 0.519719i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 50.0000i − 1.72825i
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 0 0
\(843\) − 10.0000i − 0.344418i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 28.0000i − 0.962091i
\(848\) 0 0
\(849\) 29.0000 0.995277
\(850\) 0 0
\(851\) −60.0000 −2.05677
\(852\) 0 0
\(853\) − 10.0000i − 0.342393i −0.985237 0.171197i \(-0.945237\pi\)
0.985237 0.171197i \(-0.0547634\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 45.0000i − 1.53717i −0.639747 0.768585i \(-0.720961\pi\)
0.639747 0.768585i \(-0.279039\pi\)
\(858\) 0 0
\(859\) −5.00000 −0.170598 −0.0852989 0.996355i \(-0.527185\pi\)
−0.0852989 + 0.996355i \(0.527185\pi\)
\(860\) 0 0
\(861\) 10.0000 0.340799
\(862\) 0 0
\(863\) 4.00000i 0.136162i 0.997680 + 0.0680808i \(0.0216876\pi\)
−0.997680 + 0.0680808i \(0.978312\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 8.00000i − 0.271694i
\(868\) 0 0
\(869\) 50.0000 1.69613
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) − 20.0000i − 0.676897i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 20.0000i − 0.675352i −0.941262 0.337676i \(-0.890359\pi\)
0.941262 0.337676i \(-0.109641\pi\)
\(878\) 0 0
\(879\) −30.0000 −1.01187
\(880\) 0 0
\(881\) −10.0000 −0.336909 −0.168454 0.985709i \(-0.553878\pi\)
−0.168454 + 0.985709i \(0.553878\pi\)
\(882\) 0 0
\(883\) 1.00000i 0.0336527i 0.999858 + 0.0168263i \(0.00535624\pi\)
−0.999858 + 0.0168263i \(0.994644\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 28.0000i 0.940148i 0.882627 + 0.470074i \(0.155773\pi\)
−0.882627 + 0.470074i \(0.844227\pi\)
\(888\) 0 0
\(889\) 36.0000 1.20740
\(890\) 0 0
\(891\) 5.00000 0.167506
\(892\) 0 0
\(893\) 40.0000i 1.33855i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −40.0000 −1.33407
\(900\) 0 0
\(901\) 50.0000 1.66574
\(902\) 0 0
\(903\) 8.00000i 0.266223i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 52.0000i 1.72663i 0.504664 + 0.863316i \(0.331616\pi\)
−0.504664 + 0.863316i \(0.668384\pi\)
\(908\) 0 0
\(909\) −4.00000 −0.132672
\(910\) 0 0
\(911\) −20.0000 −0.662630 −0.331315 0.943520i \(-0.607492\pi\)
−0.331315 + 0.943520i \(0.607492\pi\)
\(912\) 0 0
\(913\) − 5.00000i − 0.165476i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 40.0000i 1.32092i
\(918\) 0 0
\(919\) −10.0000 −0.329870 −0.164935 0.986304i \(-0.552741\pi\)
−0.164935 + 0.986304i \(0.552741\pi\)
\(920\) 0 0
\(921\) −7.00000 −0.230658
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 32.0000i 1.05102i
\(928\) 0 0
\(929\) −30.0000 −0.984268 −0.492134 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 0 0
\(931\) −15.0000 −0.491605
\(932\) 0 0
\(933\) − 10.0000i − 0.327385i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 5.00000i 0.163343i 0.996659 + 0.0816714i \(0.0260258\pi\)
−0.996659 + 0.0816714i \(0.973974\pi\)
\(938\) 0 0
\(939\) 30.0000 0.979013
\(940\) 0 0
\(941\) −8.00000 −0.260793 −0.130396 0.991462i \(-0.541625\pi\)
−0.130396 + 0.991462i \(0.541625\pi\)
\(942\) 0 0
\(943\) − 30.0000i − 0.976934i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 32.0000i 1.03986i 0.854209 + 0.519930i \(0.174042\pi\)
−0.854209 + 0.519930i \(0.825958\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −20.0000 −0.648544
\(952\) 0 0
\(953\) 25.0000i 0.809829i 0.914354 + 0.404915i \(0.132699\pi\)
−0.914354 + 0.404915i \(0.867301\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 20.0000i 0.646508i
\(958\) 0 0
\(959\) −10.0000 −0.322917
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) 0 0
\(963\) 6.00000i 0.193347i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 8.00000i − 0.257263i −0.991692 0.128631i \(-0.958942\pi\)
0.991692 0.128631i \(-0.0410584\pi\)
\(968\) 0 0
\(969\) −25.0000 −0.803116
\(970\) 0 0
\(971\) 45.0000 1.44412 0.722059 0.691831i \(-0.243196\pi\)
0.722059 + 0.691831i \(0.243196\pi\)
\(972\) 0 0
\(973\) 10.0000i 0.320585i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 15.0000i 0.479893i 0.970786 + 0.239946i \(0.0771298\pi\)
−0.970786 + 0.239946i \(0.922870\pi\)
\(978\) 0 0
\(979\) 45.0000 1.43821
\(980\) 0 0
\(981\) 20.0000 0.638551
\(982\) 0 0
\(983\) 46.0000i 1.46717i 0.679597 + 0.733586i \(0.262155\pi\)
−0.679597 + 0.733586i \(0.737845\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 16.0000i − 0.509286i
\(988\) 0 0
\(989\) 24.0000 0.763156
\(990\) 0 0
\(991\) −10.0000 −0.317660 −0.158830 0.987306i \(-0.550772\pi\)
−0.158830 + 0.987306i \(0.550772\pi\)
\(992\) 0 0
\(993\) 25.0000i 0.793351i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 60.0000i − 1.90022i −0.311916 0.950110i \(-0.600971\pi\)
0.311916 0.950110i \(-0.399029\pi\)
\(998\) 0 0
\(999\) 50.0000 1.58193
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 1600.2.c.j.449.2 2
4.3 odd 2 1600.2.c.g.449.1 2
5.2 odd 4 1600.2.a.s.1.1 1
5.3 odd 4 1600.2.a.h.1.1 1
5.4 even 2 inner 1600.2.c.j.449.1 2
8.3 odd 2 800.2.c.d.449.2 2
8.5 even 2 800.2.c.c.449.1 2
20.3 even 4 1600.2.a.r.1.1 1
20.7 even 4 1600.2.a.g.1.1 1
20.19 odd 2 1600.2.c.g.449.2 2
24.5 odd 2 7200.2.f.bc.6049.1 2
24.11 even 2 7200.2.f.a.6049.2 2
40.3 even 4 800.2.a.c.1.1 yes 1
40.13 odd 4 800.2.a.g.1.1 yes 1
40.19 odd 2 800.2.c.d.449.1 2
40.27 even 4 800.2.a.h.1.1 yes 1
40.29 even 2 800.2.c.c.449.2 2
40.37 odd 4 800.2.a.b.1.1 1
120.29 odd 2 7200.2.f.bc.6049.2 2
120.53 even 4 7200.2.a.o.1.1 1
120.59 even 2 7200.2.f.a.6049.1 2
120.77 even 4 7200.2.a.bq.1.1 1
120.83 odd 4 7200.2.a.bm.1.1 1
120.107 odd 4 7200.2.a.k.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
800.2.a.b.1.1 1 40.37 odd 4
800.2.a.c.1.1 yes 1 40.3 even 4
800.2.a.g.1.1 yes 1 40.13 odd 4
800.2.a.h.1.1 yes 1 40.27 even 4
800.2.c.c.449.1 2 8.5 even 2
800.2.c.c.449.2 2 40.29 even 2
800.2.c.d.449.1 2 40.19 odd 2
800.2.c.d.449.2 2 8.3 odd 2
1600.2.a.g.1.1 1 20.7 even 4
1600.2.a.h.1.1 1 5.3 odd 4
1600.2.a.r.1.1 1 20.3 even 4
1600.2.a.s.1.1 1 5.2 odd 4
1600.2.c.g.449.1 2 4.3 odd 2
1600.2.c.g.449.2 2 20.19 odd 2
1600.2.c.j.449.1 2 5.4 even 2 inner
1600.2.c.j.449.2 2 1.1 even 1 trivial
7200.2.a.k.1.1 1 120.107 odd 4
7200.2.a.o.1.1 1 120.53 even 4
7200.2.a.bm.1.1 1 120.83 odd 4
7200.2.a.bq.1.1 1 120.77 even 4
7200.2.f.a.6049.1 2 120.59 even 2
7200.2.f.a.6049.2 2 24.11 even 2
7200.2.f.bc.6049.1 2 24.5 odd 2
7200.2.f.bc.6049.2 2 120.29 odd 2