Properties

Label 1900.2.c.b.1749.1
Level $1900$
Weight $2$
Character 1900.1749
Analytic conductor $15.172$
Analytic rank $0$
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] = [1900,2,Mod(1749,1900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1900, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1900.1749");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1900.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: \(15.1715763840\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 76)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1749.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1900.1749
Dual form 1900.2.c.b.1749.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000i q^{3} -3.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-2.00000i q^{3} -3.00000i q^{7} -1.00000 q^{9} +5.00000 q^{11} +4.00000i q^{13} -3.00000i q^{17} +1.00000 q^{19} -6.00000 q^{21} -8.00000i q^{23} -4.00000i q^{27} +2.00000 q^{29} +4.00000 q^{31} -10.0000i q^{33} +10.0000i q^{37} +8.00000 q^{39} +10.0000 q^{41} -1.00000i q^{43} -1.00000i q^{47} -2.00000 q^{49} -6.00000 q^{51} +4.00000i q^{53} -2.00000i q^{57} -6.00000 q^{59} -13.0000 q^{61} +3.00000i q^{63} -12.0000i q^{67} -16.0000 q^{69} +2.00000 q^{71} -9.00000i q^{73} -15.0000i q^{77} -8.00000 q^{79} -11.0000 q^{81} +12.0000i q^{83} -4.00000i q^{87} -12.0000 q^{89} +12.0000 q^{91} -8.00000i q^{93} -8.00000i q^{97} -5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{9} + 10 q^{11} + 2 q^{19} - 12 q^{21} + 4 q^{29} + 8 q^{31} + 16 q^{39} + 20 q^{41} - 4 q^{49} - 12 q^{51} - 12 q^{59} - 26 q^{61} - 32 q^{69} + 4 q^{71} - 16 q^{79} - 22 q^{81} - 24 q^{89} + 24 q^{91} - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\) \(951\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) − 2.00000i − 1.15470i −0.816497 0.577350i \(-0.804087\pi\)
0.816497 0.577350i \(-0.195913\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 3.00000i − 1.13389i −0.823754 0.566947i \(-0.808125\pi\)
0.823754 0.566947i \(-0.191875\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 5.00000 1.50756 0.753778 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) 0 0
\(13\) 4.00000i 1.10940i 0.832050 + 0.554700i \(0.187167\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 3.00000i − 0.727607i −0.931476 0.363803i \(-0.881478\pi\)
0.931476 0.363803i \(-0.118522\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −6.00000 −1.30931
\(22\) 0 0
\(23\) − 8.00000i − 1.66812i −0.551677 0.834058i \(-0.686012\pi\)
0.551677 0.834058i \(-0.313988\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 4.00000i − 0.769800i
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) − 10.0000i − 1.74078i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 10.0000i 1.64399i 0.569495 + 0.821995i \(0.307139\pi\)
−0.569495 + 0.821995i \(0.692861\pi\)
\(38\) 0 0
\(39\) 8.00000 1.28103
\(40\) 0 0
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) − 1.00000i − 0.152499i −0.997089 0.0762493i \(-0.975706\pi\)
0.997089 0.0762493i \(-0.0242945\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 1.00000i − 0.145865i −0.997337 0.0729325i \(-0.976764\pi\)
0.997337 0.0729325i \(-0.0232358\pi\)
\(48\) 0 0
\(49\) −2.00000 −0.285714
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) 0 0
\(53\) 4.00000i 0.549442i 0.961524 + 0.274721i \(0.0885855\pi\)
−0.961524 + 0.274721i \(0.911414\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 2.00000i − 0.264906i
\(58\) 0 0
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) −13.0000 −1.66448 −0.832240 0.554416i \(-0.812942\pi\)
−0.832240 + 0.554416i \(0.812942\pi\)
\(62\) 0 0
\(63\) 3.00000i 0.377964i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 12.0000i − 1.46603i −0.680211 0.733017i \(-0.738112\pi\)
0.680211 0.733017i \(-0.261888\pi\)
\(68\) 0 0
\(69\) −16.0000 −1.92617
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 0 0
\(73\) − 9.00000i − 1.05337i −0.850060 0.526685i \(-0.823435\pi\)
0.850060 0.526685i \(-0.176565\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 15.0000i − 1.70941i
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) 12.0000i 1.31717i 0.752506 + 0.658586i \(0.228845\pi\)
−0.752506 + 0.658586i \(0.771155\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 4.00000i − 0.428845i
\(88\) 0 0
\(89\) −12.0000 −1.27200 −0.635999 0.771690i \(-0.719412\pi\)
−0.635999 + 0.771690i \(0.719412\pi\)
\(90\) 0 0
\(91\) 12.0000 1.25794
\(92\) 0 0
\(93\) − 8.00000i − 0.829561i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 8.00000i − 0.812277i −0.913812 0.406138i \(-0.866875\pi\)
0.913812 0.406138i \(-0.133125\pi\)
\(98\) 0 0
\(99\) −5.00000 −0.502519
\(100\) 0 0
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 0 0
\(103\) 6.00000i 0.591198i 0.955312 + 0.295599i \(0.0955191\pi\)
−0.955312 + 0.295599i \(0.904481\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.00000i 0.193347i 0.995316 + 0.0966736i \(0.0308203\pi\)
−0.995316 + 0.0966736i \(0.969180\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) 20.0000 1.89832
\(112\) 0 0
\(113\) 10.0000i 0.940721i 0.882474 + 0.470360i \(0.155876\pi\)
−0.882474 + 0.470360i \(0.844124\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 4.00000i − 0.369800i
\(118\) 0 0
\(119\) −9.00000 −0.825029
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 0 0
\(123\) − 20.0000i − 1.80334i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 6.00000i 0.532414i 0.963916 + 0.266207i \(0.0857705\pi\)
−0.963916 + 0.266207i \(0.914230\pi\)
\(128\) 0 0
\(129\) −2.00000 −0.176090
\(130\) 0 0
\(131\) −9.00000 −0.786334 −0.393167 0.919467i \(-0.628621\pi\)
−0.393167 + 0.919467i \(0.628621\pi\)
\(132\) 0 0
\(133\) − 3.00000i − 0.260133i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 11.0000i − 0.939793i −0.882721 0.469897i \(-0.844291\pi\)
0.882721 0.469897i \(-0.155709\pi\)
\(138\) 0 0
\(139\) 3.00000 0.254457 0.127228 0.991873i \(-0.459392\pi\)
0.127228 + 0.991873i \(0.459392\pi\)
\(140\) 0 0
\(141\) −2.00000 −0.168430
\(142\) 0 0
\(143\) 20.0000i 1.67248i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 4.00000i 0.329914i
\(148\) 0 0
\(149\) 15.0000 1.22885 0.614424 0.788976i \(-0.289388\pi\)
0.614424 + 0.788976i \(0.289388\pi\)
\(150\) 0 0
\(151\) 2.00000 0.162758 0.0813788 0.996683i \(-0.474068\pi\)
0.0813788 + 0.996683i \(0.474068\pi\)
\(152\) 0 0
\(153\) 3.00000i 0.242536i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 2.00000i − 0.159617i −0.996810 0.0798087i \(-0.974569\pi\)
0.996810 0.0798087i \(-0.0254309\pi\)
\(158\) 0 0
\(159\) 8.00000 0.634441
\(160\) 0 0
\(161\) −24.0000 −1.89146
\(162\) 0 0
\(163\) 4.00000i 0.313304i 0.987654 + 0.156652i \(0.0500701\pi\)
−0.987654 + 0.156652i \(0.949930\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 6.00000i − 0.464294i −0.972681 0.232147i \(-0.925425\pi\)
0.972681 0.232147i \(-0.0745750\pi\)
\(168\) 0 0
\(169\) −3.00000 −0.230769
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 0 0
\(173\) − 6.00000i − 0.456172i −0.973641 0.228086i \(-0.926753\pi\)
0.973641 0.228086i \(-0.0732467\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 12.0000i 0.901975i
\(178\) 0 0
\(179\) −18.0000 −1.34538 −0.672692 0.739923i \(-0.734862\pi\)
−0.672692 + 0.739923i \(0.734862\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 0 0
\(183\) 26.0000i 1.92198i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 15.0000i − 1.09691i
\(188\) 0 0
\(189\) −12.0000 −0.872872
\(190\) 0 0
\(191\) 25.0000 1.80894 0.904468 0.426541i \(-0.140268\pi\)
0.904468 + 0.426541i \(0.140268\pi\)
\(192\) 0 0
\(193\) − 12.0000i − 0.863779i −0.901927 0.431889i \(-0.857847\pi\)
0.901927 0.431889i \(-0.142153\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2.00000i 0.142494i 0.997459 + 0.0712470i \(0.0226979\pi\)
−0.997459 + 0.0712470i \(0.977302\pi\)
\(198\) 0 0
\(199\) 7.00000 0.496217 0.248108 0.968732i \(-0.420191\pi\)
0.248108 + 0.968732i \(0.420191\pi\)
\(200\) 0 0
\(201\) −24.0000 −1.69283
\(202\) 0 0
\(203\) − 6.00000i − 0.421117i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 8.00000i 0.556038i
\(208\) 0 0
\(209\) 5.00000 0.345857
\(210\) 0 0
\(211\) 18.0000 1.23917 0.619586 0.784929i \(-0.287301\pi\)
0.619586 + 0.784929i \(0.287301\pi\)
\(212\) 0 0
\(213\) − 4.00000i − 0.274075i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 12.0000i − 0.814613i
\(218\) 0 0
\(219\) −18.0000 −1.21633
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) 0 0
\(223\) − 2.00000i − 0.133930i −0.997755 0.0669650i \(-0.978668\pi\)
0.997755 0.0669650i \(-0.0213316\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 4.00000i 0.265489i 0.991150 + 0.132745i \(0.0423790\pi\)
−0.991150 + 0.132745i \(0.957621\pi\)
\(228\) 0 0
\(229\) −17.0000 −1.12339 −0.561696 0.827344i \(-0.689851\pi\)
−0.561696 + 0.827344i \(0.689851\pi\)
\(230\) 0 0
\(231\) −30.0000 −1.97386
\(232\) 0 0
\(233\) − 3.00000i − 0.196537i −0.995160 0.0982683i \(-0.968670\pi\)
0.995160 0.0982683i \(-0.0313303\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 16.0000i 1.03931i
\(238\) 0 0
\(239\) −21.0000 −1.35838 −0.679189 0.733964i \(-0.737668\pi\)
−0.679189 + 0.733964i \(0.737668\pi\)
\(240\) 0 0
\(241\) −26.0000 −1.67481 −0.837404 0.546585i \(-0.815928\pi\)
−0.837404 + 0.546585i \(0.815928\pi\)
\(242\) 0 0
\(243\) 10.0000i 0.641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 4.00000i 0.254514i
\(248\) 0 0
\(249\) 24.0000 1.52094
\(250\) 0 0
\(251\) 11.0000 0.694314 0.347157 0.937807i \(-0.387147\pi\)
0.347157 + 0.937807i \(0.387147\pi\)
\(252\) 0 0
\(253\) − 40.0000i − 2.51478i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 32.0000i 1.99611i 0.0623783 + 0.998053i \(0.480131\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) 0 0
\(259\) 30.0000 1.86411
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) 0 0
\(263\) 21.0000i 1.29492i 0.762101 + 0.647458i \(0.224168\pi\)
−0.762101 + 0.647458i \(0.775832\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 24.0000i 1.46878i
\(268\) 0 0
\(269\) 24.0000 1.46331 0.731653 0.681677i \(-0.238749\pi\)
0.731653 + 0.681677i \(0.238749\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 0 0
\(273\) − 24.0000i − 1.45255i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 1.00000i 0.0600842i 0.999549 + 0.0300421i \(0.00956413\pi\)
−0.999549 + 0.0300421i \(0.990436\pi\)
\(278\) 0 0
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) 22.0000 1.31241 0.656205 0.754583i \(-0.272161\pi\)
0.656205 + 0.754583i \(0.272161\pi\)
\(282\) 0 0
\(283\) 3.00000i 0.178331i 0.996017 + 0.0891657i \(0.0284201\pi\)
−0.996017 + 0.0891657i \(0.971580\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 30.0000i − 1.77084i
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) −16.0000 −0.937937
\(292\) 0 0
\(293\) 12.0000i 0.701047i 0.936554 + 0.350524i \(0.113996\pi\)
−0.936554 + 0.350524i \(0.886004\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 20.0000i − 1.16052i
\(298\) 0 0
\(299\) 32.0000 1.85061
\(300\) 0 0
\(301\) −3.00000 −0.172917
\(302\) 0 0
\(303\) 20.0000i 1.14897i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 12.0000i 0.684876i 0.939540 + 0.342438i \(0.111253\pi\)
−0.939540 + 0.342438i \(0.888747\pi\)
\(308\) 0 0
\(309\) 12.0000 0.682656
\(310\) 0 0
\(311\) 7.00000 0.396934 0.198467 0.980108i \(-0.436404\pi\)
0.198467 + 0.980108i \(0.436404\pi\)
\(312\) 0 0
\(313\) 10.0000i 0.565233i 0.959233 + 0.282617i \(0.0912024\pi\)
−0.959233 + 0.282617i \(0.908798\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 30.0000i 1.68497i 0.538721 + 0.842484i \(0.318908\pi\)
−0.538721 + 0.842484i \(0.681092\pi\)
\(318\) 0 0
\(319\) 10.0000 0.559893
\(320\) 0 0
\(321\) 4.00000 0.223258
\(322\) 0 0
\(323\) − 3.00000i − 0.166924i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −3.00000 −0.165395
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 0 0
\(333\) − 10.0000i − 0.547997i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 32.0000i − 1.74315i −0.490261 0.871576i \(-0.663099\pi\)
0.490261 0.871576i \(-0.336901\pi\)
\(338\) 0 0
\(339\) 20.0000 1.08625
\(340\) 0 0
\(341\) 20.0000 1.08306
\(342\) 0 0
\(343\) − 15.0000i − 0.809924i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 19.0000i 1.01997i 0.860182 + 0.509987i \(0.170350\pi\)
−0.860182 + 0.509987i \(0.829650\pi\)
\(348\) 0 0
\(349\) 11.0000 0.588817 0.294408 0.955680i \(-0.404877\pi\)
0.294408 + 0.955680i \(0.404877\pi\)
\(350\) 0 0
\(351\) 16.0000 0.854017
\(352\) 0 0
\(353\) 6.00000i 0.319348i 0.987170 + 0.159674i \(0.0510443\pi\)
−0.987170 + 0.159674i \(0.948956\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 18.0000i 0.952661i
\(358\) 0 0
\(359\) −21.0000 −1.10834 −0.554169 0.832404i \(-0.686964\pi\)
−0.554169 + 0.832404i \(0.686964\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) − 28.0000i − 1.46962i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 16.0000i 0.835193i 0.908633 + 0.417597i \(0.137127\pi\)
−0.908633 + 0.417597i \(0.862873\pi\)
\(368\) 0 0
\(369\) −10.0000 −0.520579
\(370\) 0 0
\(371\) 12.0000 0.623009
\(372\) 0 0
\(373\) 4.00000i 0.207112i 0.994624 + 0.103556i \(0.0330221\pi\)
−0.994624 + 0.103556i \(0.966978\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 8.00000i 0.412021i
\(378\) 0 0
\(379\) 30.0000 1.54100 0.770498 0.637442i \(-0.220007\pi\)
0.770498 + 0.637442i \(0.220007\pi\)
\(380\) 0 0
\(381\) 12.0000 0.614779
\(382\) 0 0
\(383\) − 4.00000i − 0.204390i −0.994764 0.102195i \(-0.967413\pi\)
0.994764 0.102195i \(-0.0325866\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1.00000i 0.0508329i
\(388\) 0 0
\(389\) 21.0000 1.06474 0.532371 0.846511i \(-0.321301\pi\)
0.532371 + 0.846511i \(0.321301\pi\)
\(390\) 0 0
\(391\) −24.0000 −1.21373
\(392\) 0 0
\(393\) 18.0000i 0.907980i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 5.00000i 0.250943i 0.992097 + 0.125471i \(0.0400443\pi\)
−0.992097 + 0.125471i \(0.959956\pi\)
\(398\) 0 0
\(399\) −6.00000 −0.300376
\(400\) 0 0
\(401\) 28.0000 1.39825 0.699127 0.714998i \(-0.253572\pi\)
0.699127 + 0.714998i \(0.253572\pi\)
\(402\) 0 0
\(403\) 16.0000i 0.797017i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 50.0000i 2.47841i
\(408\) 0 0
\(409\) 20.0000 0.988936 0.494468 0.869196i \(-0.335363\pi\)
0.494468 + 0.869196i \(0.335363\pi\)
\(410\) 0 0
\(411\) −22.0000 −1.08518
\(412\) 0 0
\(413\) 18.0000i 0.885722i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 6.00000i − 0.293821i
\(418\) 0 0
\(419\) 36.0000 1.75872 0.879358 0.476162i \(-0.157972\pi\)
0.879358 + 0.476162i \(0.157972\pi\)
\(420\) 0 0
\(421\) −40.0000 −1.94948 −0.974740 0.223341i \(-0.928304\pi\)
−0.974740 + 0.223341i \(0.928304\pi\)
\(422\) 0 0
\(423\) 1.00000i 0.0486217i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 39.0000i 1.88734i
\(428\) 0 0
\(429\) 40.0000 1.93122
\(430\) 0 0
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 0 0
\(433\) − 2.00000i − 0.0961139i −0.998845 0.0480569i \(-0.984697\pi\)
0.998845 0.0480569i \(-0.0153029\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 8.00000i − 0.382692i
\(438\) 0 0
\(439\) −2.00000 −0.0954548 −0.0477274 0.998860i \(-0.515198\pi\)
−0.0477274 + 0.998860i \(0.515198\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) 0 0
\(443\) 5.00000i 0.237557i 0.992921 + 0.118779i \(0.0378979\pi\)
−0.992921 + 0.118779i \(0.962102\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 30.0000i − 1.41895i
\(448\) 0 0
\(449\) −16.0000 −0.755087 −0.377543 0.925992i \(-0.623231\pi\)
−0.377543 + 0.925992i \(0.623231\pi\)
\(450\) 0 0
\(451\) 50.0000 2.35441
\(452\) 0 0
\(453\) − 4.00000i − 0.187936i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 13.0000i − 0.608114i −0.952654 0.304057i \(-0.901659\pi\)
0.952654 0.304057i \(-0.0983414\pi\)
\(458\) 0 0
\(459\) −12.0000 −0.560112
\(460\) 0 0
\(461\) −19.0000 −0.884918 −0.442459 0.896789i \(-0.645894\pi\)
−0.442459 + 0.896789i \(0.645894\pi\)
\(462\) 0 0
\(463\) − 19.0000i − 0.883005i −0.897260 0.441502i \(-0.854446\pi\)
0.897260 0.441502i \(-0.145554\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 5.00000i − 0.231372i −0.993286 0.115686i \(-0.963093\pi\)
0.993286 0.115686i \(-0.0369067\pi\)
\(468\) 0 0
\(469\) −36.0000 −1.66233
\(470\) 0 0
\(471\) −4.00000 −0.184310
\(472\) 0 0
\(473\) − 5.00000i − 0.229900i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 4.00000i − 0.183147i
\(478\) 0 0
\(479\) −4.00000 −0.182765 −0.0913823 0.995816i \(-0.529129\pi\)
−0.0913823 + 0.995816i \(0.529129\pi\)
\(480\) 0 0
\(481\) −40.0000 −1.82384
\(482\) 0 0
\(483\) 48.0000i 2.18408i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 26.0000i − 1.17817i −0.808070 0.589086i \(-0.799488\pi\)
0.808070 0.589086i \(-0.200512\pi\)
\(488\) 0 0
\(489\) 8.00000 0.361773
\(490\) 0 0
\(491\) −28.0000 −1.26362 −0.631811 0.775122i \(-0.717688\pi\)
−0.631811 + 0.775122i \(0.717688\pi\)
\(492\) 0 0
\(493\) − 6.00000i − 0.270226i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 6.00000i − 0.269137i
\(498\) 0 0
\(499\) −19.0000 −0.850557 −0.425278 0.905063i \(-0.639824\pi\)
−0.425278 + 0.905063i \(0.639824\pi\)
\(500\) 0 0
\(501\) −12.0000 −0.536120
\(502\) 0 0
\(503\) 36.0000i 1.60516i 0.596544 + 0.802580i \(0.296540\pi\)
−0.596544 + 0.802580i \(0.703460\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 6.00000i 0.266469i
\(508\) 0 0
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) −27.0000 −1.19441
\(512\) 0 0
\(513\) − 4.00000i − 0.176604i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 5.00000i − 0.219900i
\(518\) 0 0
\(519\) −12.0000 −0.526742
\(520\) 0 0
\(521\) 32.0000 1.40195 0.700973 0.713188i \(-0.252749\pi\)
0.700973 + 0.713188i \(0.252749\pi\)
\(522\) 0 0
\(523\) − 26.0000i − 1.13690i −0.822718 0.568450i \(-0.807543\pi\)
0.822718 0.568450i \(-0.192457\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 12.0000i − 0.522728i
\(528\) 0 0
\(529\) −41.0000 −1.78261
\(530\) 0 0
\(531\) 6.00000 0.260378
\(532\) 0 0
\(533\) 40.0000i 1.73259i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 36.0000i 1.55351i
\(538\) 0 0
\(539\) −10.0000 −0.430730
\(540\) 0 0
\(541\) 11.0000 0.472927 0.236463 0.971640i \(-0.424012\pi\)
0.236463 + 0.971640i \(0.424012\pi\)
\(542\) 0 0
\(543\) − 20.0000i − 0.858282i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 28.0000i 1.19719i 0.801050 + 0.598597i \(0.204275\pi\)
−0.801050 + 0.598597i \(0.795725\pi\)
\(548\) 0 0
\(549\) 13.0000 0.554826
\(550\) 0 0
\(551\) 2.00000 0.0852029
\(552\) 0 0
\(553\) 24.0000i 1.02058i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.00000i 0.0423714i 0.999776 + 0.0211857i \(0.00674412\pi\)
−0.999776 + 0.0211857i \(0.993256\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) −30.0000 −1.26660
\(562\) 0 0
\(563\) − 42.0000i − 1.77009i −0.465506 0.885044i \(-0.654128\pi\)
0.465506 0.885044i \(-0.345872\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 33.0000i 1.38587i
\(568\) 0 0
\(569\) 8.00000 0.335377 0.167689 0.985840i \(-0.446370\pi\)
0.167689 + 0.985840i \(0.446370\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 0 0
\(573\) − 50.0000i − 2.08878i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 13.0000i − 0.541197i −0.962692 0.270599i \(-0.912778\pi\)
0.962692 0.270599i \(-0.0872216\pi\)
\(578\) 0 0
\(579\) −24.0000 −0.997406
\(580\) 0 0
\(581\) 36.0000 1.49353
\(582\) 0 0
\(583\) 20.0000i 0.828315i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 3.00000i 0.123823i 0.998082 + 0.0619116i \(0.0197197\pi\)
−0.998082 + 0.0619116i \(0.980280\pi\)
\(588\) 0 0
\(589\) 4.00000 0.164817
\(590\) 0 0
\(591\) 4.00000 0.164538
\(592\) 0 0
\(593\) − 22.0000i − 0.903432i −0.892162 0.451716i \(-0.850812\pi\)
0.892162 0.451716i \(-0.149188\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 14.0000i − 0.572982i
\(598\) 0 0
\(599\) 12.0000 0.490307 0.245153 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) 0 0
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 0 0
\(603\) 12.0000i 0.488678i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 32.0000i 1.29884i 0.760430 + 0.649420i \(0.224988\pi\)
−0.760430 + 0.649420i \(0.775012\pi\)
\(608\) 0 0
\(609\) −12.0000 −0.486265
\(610\) 0 0
\(611\) 4.00000 0.161823
\(612\) 0 0
\(613\) 47.0000i 1.89831i 0.314806 + 0.949156i \(0.398061\pi\)
−0.314806 + 0.949156i \(0.601939\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 9.00000i 0.362326i 0.983453 + 0.181163i \(0.0579862\pi\)
−0.983453 + 0.181163i \(0.942014\pi\)
\(618\) 0 0
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) 0 0
\(621\) −32.0000 −1.28412
\(622\) 0 0
\(623\) 36.0000i 1.44231i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 10.0000i − 0.399362i
\(628\) 0 0
\(629\) 30.0000 1.19618
\(630\) 0 0
\(631\) −7.00000 −0.278666 −0.139333 0.990246i \(-0.544496\pi\)
−0.139333 + 0.990246i \(0.544496\pi\)
\(632\) 0 0
\(633\) − 36.0000i − 1.43087i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 8.00000i − 0.316972i
\(638\) 0 0
\(639\) −2.00000 −0.0791188
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 11.0000i 0.433798i 0.976194 + 0.216899i \(0.0695942\pi\)
−0.976194 + 0.216899i \(0.930406\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 7.00000i − 0.275198i −0.990488 0.137599i \(-0.956061\pi\)
0.990488 0.137599i \(-0.0439386\pi\)
\(648\) 0 0
\(649\) −30.0000 −1.17760
\(650\) 0 0
\(651\) −24.0000 −0.940634
\(652\) 0 0
\(653\) 27.0000i 1.05659i 0.849060 + 0.528296i \(0.177169\pi\)
−0.849060 + 0.528296i \(0.822831\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 9.00000i 0.351123i
\(658\) 0 0
\(659\) 34.0000 1.32445 0.662226 0.749304i \(-0.269612\pi\)
0.662226 + 0.749304i \(0.269612\pi\)
\(660\) 0 0
\(661\) 16.0000 0.622328 0.311164 0.950356i \(-0.399281\pi\)
0.311164 + 0.950356i \(0.399281\pi\)
\(662\) 0 0
\(663\) − 24.0000i − 0.932083i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 16.0000i − 0.619522i
\(668\) 0 0
\(669\) −4.00000 −0.154649
\(670\) 0 0
\(671\) −65.0000 −2.50930
\(672\) 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
\(677\) − 2.00000i − 0.0768662i −0.999261 0.0384331i \(-0.987763\pi\)
0.999261 0.0384331i \(-0.0122367\pi\)
\(678\) 0 0
\(679\) −24.0000 −0.921035
\(680\) 0 0
\(681\) 8.00000 0.306561
\(682\) 0 0
\(683\) 20.0000i 0.765279i 0.923898 + 0.382639i \(0.124985\pi\)
−0.923898 + 0.382639i \(0.875015\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 34.0000i 1.29718i
\(688\) 0 0
\(689\) −16.0000 −0.609551
\(690\) 0 0
\(691\) −33.0000 −1.25538 −0.627690 0.778464i \(-0.715999\pi\)
−0.627690 + 0.778464i \(0.715999\pi\)
\(692\) 0 0
\(693\) 15.0000i 0.569803i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 30.0000i − 1.13633i
\(698\) 0 0
\(699\) −6.00000 −0.226941
\(700\) 0 0
\(701\) −10.0000 −0.377695 −0.188847 0.982006i \(-0.560475\pi\)
−0.188847 + 0.982006i \(0.560475\pi\)
\(702\) 0 0
\(703\) 10.0000i 0.377157i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 30.0000i 1.12827i
\(708\) 0 0
\(709\) 6.00000 0.225335 0.112667 0.993633i \(-0.464061\pi\)
0.112667 + 0.993633i \(0.464061\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 0 0
\(713\) − 32.0000i − 1.19841i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 42.0000i 1.56852i
\(718\) 0 0
\(719\) −13.0000 −0.484818 −0.242409 0.970174i \(-0.577938\pi\)
−0.242409 + 0.970174i \(0.577938\pi\)
\(720\) 0 0
\(721\) 18.0000 0.670355
\(722\) 0 0
\(723\) 52.0000i 1.93390i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 7.00000i 0.259616i 0.991539 + 0.129808i \(0.0414360\pi\)
−0.991539 + 0.129808i \(0.958564\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) −3.00000 −0.110959
\(732\) 0 0
\(733\) 6.00000i 0.221615i 0.993842 + 0.110808i \(0.0353437\pi\)
−0.993842 + 0.110808i \(0.964656\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 60.0000i − 2.21013i
\(738\) 0 0
\(739\) 43.0000 1.58178 0.790890 0.611958i \(-0.209618\pi\)
0.790890 + 0.611958i \(0.209618\pi\)
\(740\) 0 0
\(741\) 8.00000 0.293887
\(742\) 0 0
\(743\) 40.0000i 1.46746i 0.679442 + 0.733729i \(0.262222\pi\)
−0.679442 + 0.733729i \(0.737778\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 12.0000i − 0.439057i
\(748\) 0 0
\(749\) 6.00000 0.219235
\(750\) 0 0
\(751\) −16.0000 −0.583848 −0.291924 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) 0 0
\(753\) − 22.0000i − 0.801725i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 5.00000i − 0.181728i −0.995863 0.0908640i \(-0.971037\pi\)
0.995863 0.0908640i \(-0.0289629\pi\)
\(758\) 0 0
\(759\) −80.0000 −2.90382
\(760\) 0 0
\(761\) 9.00000 0.326250 0.163125 0.986605i \(-0.447843\pi\)
0.163125 + 0.986605i \(0.447843\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 24.0000i − 0.866590i
\(768\) 0 0
\(769\) −31.0000 −1.11789 −0.558944 0.829205i \(-0.688793\pi\)
−0.558944 + 0.829205i \(0.688793\pi\)
\(770\) 0 0
\(771\) 64.0000 2.30490
\(772\) 0 0
\(773\) − 18.0000i − 0.647415i −0.946157 0.323708i \(-0.895071\pi\)
0.946157 0.323708i \(-0.104929\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 60.0000i − 2.15249i
\(778\) 0 0
\(779\) 10.0000 0.358287
\(780\) 0 0
\(781\) 10.0000 0.357828
\(782\) 0 0
\(783\) − 8.00000i − 0.285897i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 4.00000i 0.142585i 0.997455 + 0.0712923i \(0.0227123\pi\)
−0.997455 + 0.0712923i \(0.977288\pi\)
\(788\) 0 0
\(789\) 42.0000 1.49524
\(790\) 0 0
\(791\) 30.0000 1.06668
\(792\) 0 0
\(793\) − 52.0000i − 1.84657i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 12.0000i − 0.425062i −0.977154 0.212531i \(-0.931829\pi\)
0.977154 0.212531i \(-0.0681706\pi\)
\(798\) 0 0
\(799\) −3.00000 −0.106132
\(800\) 0 0
\(801\) 12.0000 0.423999
\(802\) 0 0
\(803\) − 45.0000i − 1.58802i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 48.0000i − 1.68968i
\(808\) 0 0
\(809\) −39.0000 −1.37117 −0.685583 0.727994i \(-0.740453\pi\)
−0.685583 + 0.727994i \(0.740453\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 0 0
\(813\) 16.0000i 0.561144i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 1.00000i − 0.0349856i
\(818\) 0 0
\(819\) −12.0000 −0.419314
\(820\) 0 0
\(821\) 45.0000 1.57051 0.785255 0.619172i \(-0.212532\pi\)
0.785255 + 0.619172i \(0.212532\pi\)
\(822\) 0 0
\(823\) − 53.0000i − 1.84746i −0.383040 0.923732i \(-0.625123\pi\)
0.383040 0.923732i \(-0.374877\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 28.0000i − 0.973655i −0.873498 0.486828i \(-0.838154\pi\)
0.873498 0.486828i \(-0.161846\pi\)
\(828\) 0 0
\(829\) 48.0000 1.66711 0.833554 0.552437i \(-0.186302\pi\)
0.833554 + 0.552437i \(0.186302\pi\)
\(830\) 0 0
\(831\) 2.00000 0.0693792
\(832\) 0 0
\(833\) 6.00000i 0.207888i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 16.0000i − 0.553041i
\(838\) 0 0
\(839\) 10.0000 0.345238 0.172619 0.984989i \(-0.444777\pi\)
0.172619 + 0.984989i \(0.444777\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) − 44.0000i − 1.51544i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 42.0000i − 1.44314i
\(848\) 0 0
\(849\) 6.00000 0.205919
\(850\) 0 0
\(851\) 80.0000 2.74236
\(852\) 0 0
\(853\) − 42.0000i − 1.43805i −0.694983 0.719026i \(-0.744588\pi\)
0.694983 0.719026i \(-0.255412\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 18.0000i 0.614868i 0.951569 + 0.307434i \(0.0994704\pi\)
−0.951569 + 0.307434i \(0.900530\pi\)
\(858\) 0 0
\(859\) −1.00000 −0.0341196 −0.0170598 0.999854i \(-0.505431\pi\)
−0.0170598 + 0.999854i \(0.505431\pi\)
\(860\) 0 0
\(861\) −60.0000 −2.04479
\(862\) 0 0
\(863\) − 6.00000i − 0.204242i −0.994772 0.102121i \(-0.967437\pi\)
0.994772 0.102121i \(-0.0325630\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 16.0000i − 0.543388i
\(868\) 0 0
\(869\) −40.0000 −1.35691
\(870\) 0 0
\(871\) 48.0000 1.62642
\(872\) 0 0
\(873\) 8.00000i 0.270759i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 34.0000i 1.14810i 0.818821 + 0.574049i \(0.194628\pi\)
−0.818821 + 0.574049i \(0.805372\pi\)
\(878\) 0 0
\(879\) 24.0000 0.809500
\(880\) 0 0
\(881\) −51.0000 −1.71823 −0.859117 0.511780i \(-0.828986\pi\)
−0.859117 + 0.511780i \(0.828986\pi\)
\(882\) 0 0
\(883\) − 1.00000i − 0.0336527i −0.999858 0.0168263i \(-0.994644\pi\)
0.999858 0.0168263i \(-0.00535624\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 22.0000i 0.738688i 0.929293 + 0.369344i \(0.120418\pi\)
−0.929293 + 0.369344i \(0.879582\pi\)
\(888\) 0 0
\(889\) 18.0000 0.603701
\(890\) 0 0
\(891\) −55.0000 −1.84257
\(892\) 0 0
\(893\) − 1.00000i − 0.0334637i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 64.0000i − 2.13690i
\(898\) 0 0
\(899\) 8.00000 0.266815
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) 0 0
\(903\) 6.00000i 0.199667i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 40.0000i − 1.32818i −0.747653 0.664089i \(-0.768820\pi\)
0.747653 0.664089i \(-0.231180\pi\)
\(908\) 0 0
\(909\) 10.0000 0.331679
\(910\) 0 0
\(911\) −18.0000 −0.596367 −0.298183 0.954509i \(-0.596381\pi\)
−0.298183 + 0.954509i \(0.596381\pi\)
\(912\) 0 0
\(913\) 60.0000i 1.98571i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 27.0000i 0.891619i
\(918\) 0 0
\(919\) 28.0000 0.923635 0.461817 0.886975i \(-0.347198\pi\)
0.461817 + 0.886975i \(0.347198\pi\)
\(920\) 0 0
\(921\) 24.0000 0.790827
\(922\) 0 0
\(923\) 8.00000i 0.263323i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 6.00000i − 0.197066i
\(928\) 0 0
\(929\) −14.0000 −0.459325 −0.229663 0.973270i \(-0.573762\pi\)
−0.229663 + 0.973270i \(0.573762\pi\)
\(930\) 0 0
\(931\) −2.00000 −0.0655474
\(932\) 0 0
\(933\) − 14.0000i − 0.458339i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 31.0000i − 1.01273i −0.862320 0.506363i \(-0.830990\pi\)
0.862320 0.506363i \(-0.169010\pi\)
\(938\) 0 0
\(939\) 20.0000 0.652675
\(940\) 0 0
\(941\) −26.0000 −0.847576 −0.423788 0.905761i \(-0.639300\pi\)
−0.423788 + 0.905761i \(0.639300\pi\)
\(942\) 0 0
\(943\) − 80.0000i − 2.60516i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 12.0000i − 0.389948i −0.980808 0.194974i \(-0.937538\pi\)
0.980808 0.194974i \(-0.0624622\pi\)
\(948\) 0 0
\(949\) 36.0000 1.16861
\(950\) 0 0
\(951\) 60.0000 1.94563
\(952\) 0 0
\(953\) 16.0000i 0.518291i 0.965838 + 0.259145i \(0.0834409\pi\)
−0.965838 + 0.259145i \(0.916559\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 20.0000i − 0.646508i
\(958\) 0 0
\(959\) −33.0000 −1.06563
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) − 2.00000i − 0.0644491i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 48.0000i 1.54358i 0.635880 + 0.771788i \(0.280637\pi\)
−0.635880 + 0.771788i \(0.719363\pi\)
\(968\) 0 0
\(969\) −6.00000 −0.192748
\(970\) 0 0
\(971\) 20.0000 0.641831 0.320915 0.947108i \(-0.396010\pi\)
0.320915 + 0.947108i \(0.396010\pi\)
\(972\) 0 0
\(973\) − 9.00000i − 0.288527i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 56.0000i 1.79160i 0.444459 + 0.895799i \(0.353396\pi\)
−0.444459 + 0.895799i \(0.646604\pi\)
\(978\) 0 0
\(979\) −60.0000 −1.91761
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 12.0000i 0.382741i 0.981518 + 0.191370i \(0.0612931\pi\)
−0.981518 + 0.191370i \(0.938707\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 6.00000i 0.190982i
\(988\) 0 0
\(989\) −8.00000 −0.254385
\(990\) 0 0
\(991\) −38.0000 −1.20711 −0.603555 0.797321i \(-0.706250\pi\)
−0.603555 + 0.797321i \(0.706250\pi\)
\(992\) 0 0
\(993\) 8.00000i 0.253872i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 37.0000i 1.17180i 0.810383 + 0.585901i \(0.199259\pi\)
−0.810383 + 0.585901i \(0.800741\pi\)
\(998\) 0 0
\(999\) 40.0000 1.26554
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 1900.2.c.b.1749.1 2
5.2 odd 4 1900.2.a.b.1.1 1
5.3 odd 4 76.2.a.a.1.1 1
5.4 even 2 inner 1900.2.c.b.1749.2 2
15.8 even 4 684.2.a.b.1.1 1
20.3 even 4 304.2.a.a.1.1 1
20.7 even 4 7600.2.a.p.1.1 1
35.13 even 4 3724.2.a.a.1.1 1
40.3 even 4 1216.2.a.q.1.1 1
40.13 odd 4 1216.2.a.c.1.1 1
55.43 even 4 9196.2.a.f.1.1 1
60.23 odd 4 2736.2.a.q.1.1 1
95.8 even 12 1444.2.e.c.653.1 2
95.18 even 4 1444.2.a.a.1.1 1
95.68 odd 12 1444.2.e.a.653.1 2
95.83 odd 12 1444.2.e.a.429.1 2
95.88 even 12 1444.2.e.c.429.1 2
380.303 odd 4 5776.2.a.p.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
76.2.a.a.1.1 1 5.3 odd 4
304.2.a.a.1.1 1 20.3 even 4
684.2.a.b.1.1 1 15.8 even 4
1216.2.a.c.1.1 1 40.13 odd 4
1216.2.a.q.1.1 1 40.3 even 4
1444.2.a.a.1.1 1 95.18 even 4
1444.2.e.a.429.1 2 95.83 odd 12
1444.2.e.a.653.1 2 95.68 odd 12
1444.2.e.c.429.1 2 95.88 even 12
1444.2.e.c.653.1 2 95.8 even 12
1900.2.a.b.1.1 1 5.2 odd 4
1900.2.c.b.1749.1 2 1.1 even 1 trivial
1900.2.c.b.1749.2 2 5.4 even 2 inner
2736.2.a.q.1.1 1 60.23 odd 4
3724.2.a.a.1.1 1 35.13 even 4
5776.2.a.p.1.1 1 380.303 odd 4
7600.2.a.p.1.1 1 20.7 even 4
9196.2.a.f.1.1 1 55.43 even 4