Properties

Label 416.2.b.a.209.1
Level $416$
Weight $2$
Character 416.209
Analytic conductor $3.322$
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] = [416,2,Mod(209,416)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(416, 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("416.209");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 416 = 2^{5} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 416.b (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: \(3.32177672409\)
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 104)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 209.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 416.209
Dual form 416.2.b.a.209.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} -3.00000i q^{5} -3.00000 q^{7} +2.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} -3.00000i q^{5} -3.00000 q^{7} +2.00000 q^{9} -1.00000i q^{13} -3.00000 q^{15} -7.00000 q^{17} -4.00000i q^{19} +3.00000i q^{21} -4.00000 q^{23} -4.00000 q^{25} -5.00000i q^{27} +4.00000i q^{29} +8.00000 q^{31} +9.00000i q^{35} -7.00000i q^{37} -1.00000 q^{39} +2.00000 q^{41} -1.00000i q^{43} -6.00000i q^{45} +7.00000 q^{47} +2.00000 q^{49} +7.00000i q^{51} -4.00000i q^{53} -4.00000 q^{57} -14.0000i q^{59} +10.0000i q^{61} -6.00000 q^{63} -3.00000 q^{65} +2.00000i q^{67} +4.00000i q^{69} +3.00000 q^{71} +14.0000 q^{73} +4.00000i q^{75} +10.0000 q^{79} +1.00000 q^{81} +14.0000i q^{83} +21.0000i q^{85} +4.00000 q^{87} +3.00000i q^{91} -8.00000i q^{93} -12.0000 q^{95} +8.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{7} + 4 q^{9} - 6 q^{15} - 14 q^{17} - 8 q^{23} - 8 q^{25} + 16 q^{31} - 2 q^{39} + 4 q^{41} + 14 q^{47} + 4 q^{49} - 8 q^{57} - 12 q^{63} - 6 q^{65} + 6 q^{71} + 28 q^{73} + 20 q^{79} + 2 q^{81} + 8 q^{87} - 24 q^{95} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/416\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(287\) \(353\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(-0.906785\pi\)
0.957427 0.288675i \(-0.0932147\pi\)
\(4\) 0 0
\(5\) − 3.00000i − 1.34164i −0.741620 0.670820i \(-0.765942\pi\)
0.741620 0.670820i \(-0.234058\pi\)
\(6\) 0 0
\(7\) −3.00000 −1.13389 −0.566947 0.823754i \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) 0 0
\(9\) 2.00000 0.666667
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) − 1.00000i − 0.277350i
\(14\) 0 0
\(15\) −3.00000 −0.774597
\(16\) 0 0
\(17\) −7.00000 −1.69775 −0.848875 0.528594i \(-0.822719\pi\)
−0.848875 + 0.528594i \(0.822719\pi\)
\(18\) 0 0
\(19\) − 4.00000i − 0.917663i −0.888523 0.458831i \(-0.848268\pi\)
0.888523 0.458831i \(-0.151732\pi\)
\(20\) 0 0
\(21\) 3.00000i 0.654654i
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) − 5.00000i − 0.962250i
\(28\) 0 0
\(29\) 4.00000i 0.742781i 0.928477 + 0.371391i \(0.121119\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 9.00000i 1.52128i
\(36\) 0 0
\(37\) − 7.00000i − 1.15079i −0.817875 0.575396i \(-0.804848\pi\)
0.817875 0.575396i \(-0.195152\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) − 1.00000i − 0.152499i −0.997089 0.0762493i \(-0.975706\pi\)
0.997089 0.0762493i \(-0.0242945\pi\)
\(44\) 0 0
\(45\) − 6.00000i − 0.894427i
\(46\) 0 0
\(47\) 7.00000 1.02105 0.510527 0.859861i \(-0.329450\pi\)
0.510527 + 0.859861i \(0.329450\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) 7.00000i 0.980196i
\(52\) 0 0
\(53\) − 4.00000i − 0.549442i −0.961524 0.274721i \(-0.911414\pi\)
0.961524 0.274721i \(-0.0885855\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) − 14.0000i − 1.82264i −0.411693 0.911322i \(-0.635063\pi\)
0.411693 0.911322i \(-0.364937\pi\)
\(60\) 0 0
\(61\) 10.0000i 1.28037i 0.768221 + 0.640184i \(0.221142\pi\)
−0.768221 + 0.640184i \(0.778858\pi\)
\(62\) 0 0
\(63\) −6.00000 −0.755929
\(64\) 0 0
\(65\) −3.00000 −0.372104
\(66\) 0 0
\(67\) 2.00000i 0.244339i 0.992509 + 0.122169i \(0.0389851\pi\)
−0.992509 + 0.122169i \(0.961015\pi\)
\(68\) 0 0
\(69\) 4.00000i 0.481543i
\(70\) 0 0
\(71\) 3.00000 0.356034 0.178017 0.984027i \(-0.443032\pi\)
0.178017 + 0.984027i \(0.443032\pi\)
\(72\) 0 0
\(73\) 14.0000 1.63858 0.819288 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) 0 0
\(75\) 4.00000i 0.461880i
\(76\) 0 0
\(77\) 0 0
\(78\) 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
\(83\) 14.0000i 1.53670i 0.640030 + 0.768350i \(0.278922\pi\)
−0.640030 + 0.768350i \(0.721078\pi\)
\(84\) 0 0
\(85\) 21.0000i 2.27777i
\(86\) 0 0
\(87\) 4.00000 0.428845
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 3.00000i 0.314485i
\(92\) 0 0
\(93\) − 8.00000i − 0.829561i
\(94\) 0 0
\(95\) −12.0000 −1.23117
\(96\) 0 0
\(97\) 8.00000 0.812277 0.406138 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 6.00000 0.591198 0.295599 0.955312i \(-0.404481\pi\)
0.295599 + 0.955312i \(0.404481\pi\)
\(104\) 0 0
\(105\) 9.00000 0.878310
\(106\) 0 0
\(107\) − 8.00000i − 0.773389i −0.922208 0.386695i \(-0.873617\pi\)
0.922208 0.386695i \(-0.126383\pi\)
\(108\) 0 0
\(109\) − 1.00000i − 0.0957826i −0.998853 0.0478913i \(-0.984750\pi\)
0.998853 0.0478913i \(-0.0152501\pi\)
\(110\) 0 0
\(111\) −7.00000 −0.664411
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 12.0000i 1.11901i
\(116\) 0 0
\(117\) − 2.00000i − 0.184900i
\(118\) 0 0
\(119\) 21.0000 1.92507
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) 0 0
\(123\) − 2.00000i − 0.180334i
\(124\) 0 0
\(125\) − 3.00000i − 0.268328i
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 0 0
\(129\) −1.00000 −0.0880451
\(130\) 0 0
\(131\) − 15.0000i − 1.31056i −0.755388 0.655278i \(-0.772551\pi\)
0.755388 0.655278i \(-0.227449\pi\)
\(132\) 0 0
\(133\) 12.0000i 1.04053i
\(134\) 0 0
\(135\) −15.0000 −1.29099
\(136\) 0 0
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 0 0
\(139\) 11.0000i 0.933008i 0.884519 + 0.466504i \(0.154487\pi\)
−0.884519 + 0.466504i \(0.845513\pi\)
\(140\) 0 0
\(141\) − 7.00000i − 0.589506i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 12.0000 0.996546
\(146\) 0 0
\(147\) − 2.00000i − 0.164957i
\(148\) 0 0
\(149\) − 6.00000i − 0.491539i −0.969328 0.245770i \(-0.920959\pi\)
0.969328 0.245770i \(-0.0790407\pi\)
\(150\) 0 0
\(151\) −17.0000 −1.38344 −0.691720 0.722166i \(-0.743147\pi\)
−0.691720 + 0.722166i \(0.743147\pi\)
\(152\) 0 0
\(153\) −14.0000 −1.13183
\(154\) 0 0
\(155\) − 24.0000i − 1.92773i
\(156\) 0 0
\(157\) − 2.00000i − 0.159617i −0.996810 0.0798087i \(-0.974569\pi\)
0.996810 0.0798087i \(-0.0254309\pi\)
\(158\) 0 0
\(159\) −4.00000 −0.317221
\(160\) 0 0
\(161\) 12.0000 0.945732
\(162\) 0 0
\(163\) 24.0000i 1.87983i 0.341415 + 0.939913i \(0.389094\pi\)
−0.341415 + 0.939913i \(0.610906\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) − 8.00000i − 0.611775i
\(172\) 0 0
\(173\) 6.00000i 0.456172i 0.973641 + 0.228086i \(0.0732467\pi\)
−0.973641 + 0.228086i \(0.926753\pi\)
\(174\) 0 0
\(175\) 12.0000 0.907115
\(176\) 0 0
\(177\) −14.0000 −1.05230
\(178\) 0 0
\(179\) 21.0000i 1.56961i 0.619740 + 0.784807i \(0.287238\pi\)
−0.619740 + 0.784807i \(0.712762\pi\)
\(180\) 0 0
\(181\) − 10.0000i − 0.743294i −0.928374 0.371647i \(-0.878793\pi\)
0.928374 0.371647i \(-0.121207\pi\)
\(182\) 0 0
\(183\) 10.0000 0.739221
\(184\) 0 0
\(185\) −21.0000 −1.54395
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 15.0000i 1.09109i
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 0 0
\(193\) −6.00000 −0.431889 −0.215945 0.976406i \(-0.569283\pi\)
−0.215945 + 0.976406i \(0.569283\pi\)
\(194\) 0 0
\(195\) 3.00000i 0.214834i
\(196\) 0 0
\(197\) − 17.0000i − 1.21120i −0.795769 0.605600i \(-0.792933\pi\)
0.795769 0.605600i \(-0.207067\pi\)
\(198\) 0 0
\(199\) 10.0000 0.708881 0.354441 0.935079i \(-0.384671\pi\)
0.354441 + 0.935079i \(0.384671\pi\)
\(200\) 0 0
\(201\) 2.00000 0.141069
\(202\) 0 0
\(203\) − 12.0000i − 0.842235i
\(204\) 0 0
\(205\) − 6.00000i − 0.419058i
\(206\) 0 0
\(207\) −8.00000 −0.556038
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) − 15.0000i − 1.03264i −0.856395 0.516321i \(-0.827301\pi\)
0.856395 0.516321i \(-0.172699\pi\)
\(212\) 0 0
\(213\) − 3.00000i − 0.205557i
\(214\) 0 0
\(215\) −3.00000 −0.204598
\(216\) 0 0
\(217\) −24.0000 −1.62923
\(218\) 0 0
\(219\) − 14.0000i − 0.946032i
\(220\) 0 0
\(221\) 7.00000i 0.470871i
\(222\) 0 0
\(223\) 1.00000 0.0669650 0.0334825 0.999439i \(-0.489340\pi\)
0.0334825 + 0.999439i \(0.489340\pi\)
\(224\) 0 0
\(225\) −8.00000 −0.533333
\(226\) 0 0
\(227\) − 18.0000i − 1.19470i −0.801980 0.597351i \(-0.796220\pi\)
0.801980 0.597351i \(-0.203780\pi\)
\(228\) 0 0
\(229\) − 21.0000i − 1.38772i −0.720110 0.693860i \(-0.755909\pi\)
0.720110 0.693860i \(-0.244091\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 19.0000 1.24473 0.622366 0.782727i \(-0.286172\pi\)
0.622366 + 0.782727i \(0.286172\pi\)
\(234\) 0 0
\(235\) − 21.0000i − 1.36989i
\(236\) 0 0
\(237\) − 10.0000i − 0.649570i
\(238\) 0 0
\(239\) −5.00000 −0.323423 −0.161712 0.986838i \(-0.551701\pi\)
−0.161712 + 0.986838i \(0.551701\pi\)
\(240\) 0 0
\(241\) −8.00000 −0.515325 −0.257663 0.966235i \(-0.582952\pi\)
−0.257663 + 0.966235i \(0.582952\pi\)
\(242\) 0 0
\(243\) − 16.0000i − 1.02640i
\(244\) 0 0
\(245\) − 6.00000i − 0.383326i
\(246\) 0 0
\(247\) −4.00000 −0.254514
\(248\) 0 0
\(249\) 14.0000 0.887214
\(250\) 0 0
\(251\) 20.0000i 1.26239i 0.775625 + 0.631194i \(0.217435\pi\)
−0.775625 + 0.631194i \(0.782565\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 21.0000 1.31507
\(256\) 0 0
\(257\) −7.00000 −0.436648 −0.218324 0.975876i \(-0.570059\pi\)
−0.218324 + 0.975876i \(0.570059\pi\)
\(258\) 0 0
\(259\) 21.0000i 1.30488i
\(260\) 0 0
\(261\) 8.00000i 0.495188i
\(262\) 0 0
\(263\) −14.0000 −0.863277 −0.431638 0.902047i \(-0.642064\pi\)
−0.431638 + 0.902047i \(0.642064\pi\)
\(264\) 0 0
\(265\) −12.0000 −0.737154
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 6.00000i − 0.365826i −0.983129 0.182913i \(-0.941447\pi\)
0.983129 0.182913i \(-0.0585527\pi\)
\(270\) 0 0
\(271\) 3.00000 0.182237 0.0911185 0.995840i \(-0.470956\pi\)
0.0911185 + 0.995840i \(0.470956\pi\)
\(272\) 0 0
\(273\) 3.00000 0.181568
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 18.0000i 1.08152i 0.841178 + 0.540758i \(0.181862\pi\)
−0.841178 + 0.540758i \(0.818138\pi\)
\(278\) 0 0
\(279\) 16.0000 0.957895
\(280\) 0 0
\(281\) −8.00000 −0.477240 −0.238620 0.971113i \(-0.576695\pi\)
−0.238620 + 0.971113i \(0.576695\pi\)
\(282\) 0 0
\(283\) 4.00000i 0.237775i 0.992908 + 0.118888i \(0.0379328\pi\)
−0.992908 + 0.118888i \(0.962067\pi\)
\(284\) 0 0
\(285\) 12.0000i 0.710819i
\(286\) 0 0
\(287\) −6.00000 −0.354169
\(288\) 0 0
\(289\) 32.0000 1.88235
\(290\) 0 0
\(291\) − 8.00000i − 0.468968i
\(292\) 0 0
\(293\) 1.00000i 0.0584206i 0.999573 + 0.0292103i \(0.00929925\pi\)
−0.999573 + 0.0292103i \(0.990701\pi\)
\(294\) 0 0
\(295\) −42.0000 −2.44533
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 4.00000i 0.231326i
\(300\) 0 0
\(301\) 3.00000i 0.172917i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 30.0000 1.71780
\(306\) 0 0
\(307\) 2.00000i 0.114146i 0.998370 + 0.0570730i \(0.0181768\pi\)
−0.998370 + 0.0570730i \(0.981823\pi\)
\(308\) 0 0
\(309\) − 6.00000i − 0.341328i
\(310\) 0 0
\(311\) −32.0000 −1.81455 −0.907277 0.420534i \(-0.861843\pi\)
−0.907277 + 0.420534i \(0.861843\pi\)
\(312\) 0 0
\(313\) 29.0000 1.63918 0.819588 0.572953i \(-0.194202\pi\)
0.819588 + 0.572953i \(0.194202\pi\)
\(314\) 0 0
\(315\) 18.0000i 1.01419i
\(316\) 0 0
\(317\) 18.0000i 1.01098i 0.862832 + 0.505490i \(0.168688\pi\)
−0.862832 + 0.505490i \(0.831312\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −8.00000 −0.446516
\(322\) 0 0
\(323\) 28.0000i 1.55796i
\(324\) 0 0
\(325\) 4.00000i 0.221880i
\(326\) 0 0
\(327\) −1.00000 −0.0553001
\(328\) 0 0
\(329\) −21.0000 −1.15777
\(330\) 0 0
\(331\) 10.0000i 0.549650i 0.961494 + 0.274825i \(0.0886199\pi\)
−0.961494 + 0.274825i \(0.911380\pi\)
\(332\) 0 0
\(333\) − 14.0000i − 0.767195i
\(334\) 0 0
\(335\) 6.00000 0.327815
\(336\) 0 0
\(337\) −17.0000 −0.926049 −0.463025 0.886345i \(-0.653236\pi\)
−0.463025 + 0.886345i \(0.653236\pi\)
\(338\) 0 0
\(339\) 6.00000i 0.325875i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 15.0000 0.809924
\(344\) 0 0
\(345\) 12.0000 0.646058
\(346\) 0 0
\(347\) 7.00000i 0.375780i 0.982190 + 0.187890i \(0.0601648\pi\)
−0.982190 + 0.187890i \(0.939835\pi\)
\(348\) 0 0
\(349\) 19.0000i 1.01705i 0.861048 + 0.508523i \(0.169808\pi\)
−0.861048 + 0.508523i \(0.830192\pi\)
\(350\) 0 0
\(351\) −5.00000 −0.266880
\(352\) 0 0
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) − 9.00000i − 0.477670i
\(356\) 0 0
\(357\) − 21.0000i − 1.11144i
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 3.00000 0.157895
\(362\) 0 0
\(363\) − 11.0000i − 0.577350i
\(364\) 0 0
\(365\) − 42.0000i − 2.19838i
\(366\) 0 0
\(367\) 22.0000 1.14839 0.574195 0.818718i \(-0.305315\pi\)
0.574195 + 0.818718i \(0.305315\pi\)
\(368\) 0 0
\(369\) 4.00000 0.208232
\(370\) 0 0
\(371\) 12.0000i 0.623009i
\(372\) 0 0
\(373\) 36.0000i 1.86401i 0.362446 + 0.932005i \(0.381942\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(374\) 0 0
\(375\) −3.00000 −0.154919
\(376\) 0 0
\(377\) 4.00000 0.206010
\(378\) 0 0
\(379\) − 4.00000i − 0.205466i −0.994709 0.102733i \(-0.967241\pi\)
0.994709 0.102733i \(-0.0327588\pi\)
\(380\) 0 0
\(381\) 8.00000i 0.409852i
\(382\) 0 0
\(383\) 1.00000 0.0510976 0.0255488 0.999674i \(-0.491867\pi\)
0.0255488 + 0.999674i \(0.491867\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 2.00000i − 0.101666i
\(388\) 0 0
\(389\) 34.0000i 1.72387i 0.507020 + 0.861934i \(0.330747\pi\)
−0.507020 + 0.861934i \(0.669253\pi\)
\(390\) 0 0
\(391\) 28.0000 1.41602
\(392\) 0 0
\(393\) −15.0000 −0.756650
\(394\) 0 0
\(395\) − 30.0000i − 1.50946i
\(396\) 0 0
\(397\) − 22.0000i − 1.10415i −0.833795 0.552074i \(-0.813837\pi\)
0.833795 0.552074i \(-0.186163\pi\)
\(398\) 0 0
\(399\) 12.0000 0.600751
\(400\) 0 0
\(401\) 32.0000 1.59800 0.799002 0.601329i \(-0.205362\pi\)
0.799002 + 0.601329i \(0.205362\pi\)
\(402\) 0 0
\(403\) − 8.00000i − 0.398508i
\(404\) 0 0
\(405\) − 3.00000i − 0.149071i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) 0 0
\(411\) 12.0000i 0.591916i
\(412\) 0 0
\(413\) 42.0000i 2.06668i
\(414\) 0 0
\(415\) 42.0000 2.06170
\(416\) 0 0
\(417\) 11.0000 0.538672
\(418\) 0 0
\(419\) 11.0000i 0.537385i 0.963226 + 0.268693i \(0.0865916\pi\)
−0.963226 + 0.268693i \(0.913408\pi\)
\(420\) 0 0
\(421\) − 15.0000i − 0.731055i −0.930800 0.365528i \(-0.880889\pi\)
0.930800 0.365528i \(-0.119111\pi\)
\(422\) 0 0
\(423\) 14.0000 0.680703
\(424\) 0 0
\(425\) 28.0000 1.35820
\(426\) 0 0
\(427\) − 30.0000i − 1.45180i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 3.00000 0.144505 0.0722525 0.997386i \(-0.476981\pi\)
0.0722525 + 0.997386i \(0.476981\pi\)
\(432\) 0 0
\(433\) −1.00000 −0.0480569 −0.0240285 0.999711i \(-0.507649\pi\)
−0.0240285 + 0.999711i \(0.507649\pi\)
\(434\) 0 0
\(435\) − 12.0000i − 0.575356i
\(436\) 0 0
\(437\) 16.0000i 0.765384i
\(438\) 0 0
\(439\) 10.0000 0.477274 0.238637 0.971109i \(-0.423299\pi\)
0.238637 + 0.971109i \(0.423299\pi\)
\(440\) 0 0
\(441\) 4.00000 0.190476
\(442\) 0 0
\(443\) 9.00000i 0.427603i 0.976877 + 0.213801i \(0.0685846\pi\)
−0.976877 + 0.213801i \(0.931415\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −6.00000 −0.283790
\(448\) 0 0
\(449\) −20.0000 −0.943858 −0.471929 0.881636i \(-0.656442\pi\)
−0.471929 + 0.881636i \(0.656442\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 17.0000i 0.798730i
\(454\) 0 0
\(455\) 9.00000 0.421927
\(456\) 0 0
\(457\) −32.0000 −1.49690 −0.748448 0.663193i \(-0.769201\pi\)
−0.748448 + 0.663193i \(0.769201\pi\)
\(458\) 0 0
\(459\) 35.0000i 1.63366i
\(460\) 0 0
\(461\) − 5.00000i − 0.232873i −0.993198 0.116437i \(-0.962853\pi\)
0.993198 0.116437i \(-0.0371472\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) 0 0
\(465\) −24.0000 −1.11297
\(466\) 0 0
\(467\) 12.0000i 0.555294i 0.960683 + 0.277647i \(0.0895545\pi\)
−0.960683 + 0.277647i \(0.910445\pi\)
\(468\) 0 0
\(469\) − 6.00000i − 0.277054i
\(470\) 0 0
\(471\) −2.00000 −0.0921551
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 16.0000i 0.734130i
\(476\) 0 0
\(477\) − 8.00000i − 0.366295i
\(478\) 0 0
\(479\) −5.00000 −0.228456 −0.114228 0.993455i \(-0.536439\pi\)
−0.114228 + 0.993455i \(0.536439\pi\)
\(480\) 0 0
\(481\) −7.00000 −0.319173
\(482\) 0 0
\(483\) − 12.0000i − 0.546019i
\(484\) 0 0
\(485\) − 24.0000i − 1.08978i
\(486\) 0 0
\(487\) −8.00000 −0.362515 −0.181257 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) 0 0
\(489\) 24.0000 1.08532
\(490\) 0 0
\(491\) 15.0000i 0.676941i 0.940977 + 0.338470i \(0.109909\pi\)
−0.940977 + 0.338470i \(0.890091\pi\)
\(492\) 0 0
\(493\) − 28.0000i − 1.26106i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −9.00000 −0.403705
\(498\) 0 0
\(499\) 6.00000i 0.268597i 0.990941 + 0.134298i \(0.0428781\pi\)
−0.990941 + 0.134298i \(0.957122\pi\)
\(500\) 0 0
\(501\) − 12.0000i − 0.536120i
\(502\) 0 0
\(503\) −4.00000 −0.178351 −0.0891756 0.996016i \(-0.528423\pi\)
−0.0891756 + 0.996016i \(0.528423\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 1.00000i 0.0444116i
\(508\) 0 0
\(509\) 14.0000i 0.620539i 0.950649 + 0.310270i \(0.100419\pi\)
−0.950649 + 0.310270i \(0.899581\pi\)
\(510\) 0 0
\(511\) −42.0000 −1.85797
\(512\) 0 0
\(513\) −20.0000 −0.883022
\(514\) 0 0
\(515\) − 18.0000i − 0.793175i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) −3.00000 −0.131432 −0.0657162 0.997838i \(-0.520933\pi\)
−0.0657162 + 0.997838i \(0.520933\pi\)
\(522\) 0 0
\(523\) 44.0000i 1.92399i 0.273075 + 0.961993i \(0.411959\pi\)
−0.273075 + 0.961993i \(0.588041\pi\)
\(524\) 0 0
\(525\) − 12.0000i − 0.523723i
\(526\) 0 0
\(527\) −56.0000 −2.43940
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) − 28.0000i − 1.21510i
\(532\) 0 0
\(533\) − 2.00000i − 0.0866296i
\(534\) 0 0
\(535\) −24.0000 −1.03761
\(536\) 0 0
\(537\) 21.0000 0.906217
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 25.0000i 1.07483i 0.843317 + 0.537417i \(0.180600\pi\)
−0.843317 + 0.537417i \(0.819400\pi\)
\(542\) 0 0
\(543\) −10.0000 −0.429141
\(544\) 0 0
\(545\) −3.00000 −0.128506
\(546\) 0 0
\(547\) − 13.0000i − 0.555840i −0.960604 0.277920i \(-0.910355\pi\)
0.960604 0.277920i \(-0.0896450\pi\)
\(548\) 0 0
\(549\) 20.0000i 0.853579i
\(550\) 0 0
\(551\) 16.0000 0.681623
\(552\) 0 0
\(553\) −30.0000 −1.27573
\(554\) 0 0
\(555\) 21.0000i 0.891400i
\(556\) 0 0
\(557\) 13.0000i 0.550828i 0.961326 + 0.275414i \(0.0888149\pi\)
−0.961326 + 0.275414i \(0.911185\pi\)
\(558\) 0 0
\(559\) −1.00000 −0.0422955
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 31.0000i − 1.30649i −0.757145 0.653247i \(-0.773406\pi\)
0.757145 0.653247i \(-0.226594\pi\)
\(564\) 0 0
\(565\) 18.0000i 0.757266i
\(566\) 0 0
\(567\) −3.00000 −0.125988
\(568\) 0 0
\(569\) 25.0000 1.04805 0.524027 0.851701i \(-0.324429\pi\)
0.524027 + 0.851701i \(0.324429\pi\)
\(570\) 0 0
\(571\) 5.00000i 0.209243i 0.994512 + 0.104622i \(0.0333632\pi\)
−0.994512 + 0.104622i \(0.966637\pi\)
\(572\) 0 0
\(573\) 12.0000i 0.501307i
\(574\) 0 0
\(575\) 16.0000 0.667246
\(576\) 0 0
\(577\) −2.00000 −0.0832611 −0.0416305 0.999133i \(-0.513255\pi\)
−0.0416305 + 0.999133i \(0.513255\pi\)
\(578\) 0 0
\(579\) 6.00000i 0.249351i
\(580\) 0 0
\(581\) − 42.0000i − 1.74245i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −6.00000 −0.248069
\(586\) 0 0
\(587\) − 28.0000i − 1.15568i −0.816149 0.577842i \(-0.803895\pi\)
0.816149 0.577842i \(-0.196105\pi\)
\(588\) 0 0
\(589\) − 32.0000i − 1.31854i
\(590\) 0 0
\(591\) −17.0000 −0.699287
\(592\) 0 0
\(593\) 24.0000 0.985562 0.492781 0.870153i \(-0.335980\pi\)
0.492781 + 0.870153i \(0.335980\pi\)
\(594\) 0 0
\(595\) − 63.0000i − 2.58275i
\(596\) 0 0
\(597\) − 10.0000i − 0.409273i
\(598\) 0 0
\(599\) −10.0000 −0.408589 −0.204294 0.978909i \(-0.565490\pi\)
−0.204294 + 0.978909i \(0.565490\pi\)
\(600\) 0 0
\(601\) 27.0000 1.10135 0.550676 0.834719i \(-0.314370\pi\)
0.550676 + 0.834719i \(0.314370\pi\)
\(602\) 0 0
\(603\) 4.00000i 0.162893i
\(604\) 0 0
\(605\) − 33.0000i − 1.34164i
\(606\) 0 0
\(607\) 22.0000 0.892952 0.446476 0.894795i \(-0.352679\pi\)
0.446476 + 0.894795i \(0.352679\pi\)
\(608\) 0 0
\(609\) −12.0000 −0.486265
\(610\) 0 0
\(611\) − 7.00000i − 0.283190i
\(612\) 0 0
\(613\) − 14.0000i − 0.565455i −0.959200 0.282727i \(-0.908761\pi\)
0.959200 0.282727i \(-0.0912392\pi\)
\(614\) 0 0
\(615\) −6.00000 −0.241943
\(616\) 0 0
\(617\) −12.0000 −0.483102 −0.241551 0.970388i \(-0.577656\pi\)
−0.241551 + 0.970388i \(0.577656\pi\)
\(618\) 0 0
\(619\) 6.00000i 0.241160i 0.992704 + 0.120580i \(0.0384755\pi\)
−0.992704 + 0.120580i \(0.961525\pi\)
\(620\) 0 0
\(621\) 20.0000i 0.802572i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 49.0000i 1.95376i
\(630\) 0 0
\(631\) 23.0000 0.915616 0.457808 0.889051i \(-0.348635\pi\)
0.457808 + 0.889051i \(0.348635\pi\)
\(632\) 0 0
\(633\) −15.0000 −0.596196
\(634\) 0 0
\(635\) 24.0000i 0.952411i
\(636\) 0 0
\(637\) − 2.00000i − 0.0792429i
\(638\) 0 0
\(639\) 6.00000 0.237356
\(640\) 0 0
\(641\) 2.00000 0.0789953 0.0394976 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) 0 0
\(643\) − 26.0000i − 1.02534i −0.858586 0.512670i \(-0.828656\pi\)
0.858586 0.512670i \(-0.171344\pi\)
\(644\) 0 0
\(645\) 3.00000i 0.118125i
\(646\) 0 0
\(647\) −18.0000 −0.707653 −0.353827 0.935311i \(-0.615120\pi\)
−0.353827 + 0.935311i \(0.615120\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 24.0000i 0.940634i
\(652\) 0 0
\(653\) − 44.0000i − 1.72185i −0.508729 0.860927i \(-0.669885\pi\)
0.508729 0.860927i \(-0.330115\pi\)
\(654\) 0 0
\(655\) −45.0000 −1.75830
\(656\) 0 0
\(657\) 28.0000 1.09238
\(658\) 0 0
\(659\) − 4.00000i − 0.155818i −0.996960 0.0779089i \(-0.975176\pi\)
0.996960 0.0779089i \(-0.0248243\pi\)
\(660\) 0 0
\(661\) − 30.0000i − 1.16686i −0.812162 0.583432i \(-0.801709\pi\)
0.812162 0.583432i \(-0.198291\pi\)
\(662\) 0 0
\(663\) 7.00000 0.271857
\(664\) 0 0
\(665\) 36.0000 1.39602
\(666\) 0 0
\(667\) − 16.0000i − 0.619522i
\(668\) 0 0
\(669\) − 1.00000i − 0.0386622i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 29.0000 1.11787 0.558934 0.829212i \(-0.311211\pi\)
0.558934 + 0.829212i \(0.311211\pi\)
\(674\) 0 0
\(675\) 20.0000i 0.769800i
\(676\) 0 0
\(677\) − 22.0000i − 0.845529i −0.906240 0.422764i \(-0.861060\pi\)
0.906240 0.422764i \(-0.138940\pi\)
\(678\) 0 0
\(679\) −24.0000 −0.921035
\(680\) 0 0
\(681\) −18.0000 −0.689761
\(682\) 0 0
\(683\) − 26.0000i − 0.994862i −0.867503 0.497431i \(-0.834277\pi\)
0.867503 0.497431i \(-0.165723\pi\)
\(684\) 0 0
\(685\) 36.0000i 1.37549i
\(686\) 0 0
\(687\) −21.0000 −0.801200
\(688\) 0 0
\(689\) −4.00000 −0.152388
\(690\) 0 0
\(691\) − 20.0000i − 0.760836i −0.924815 0.380418i \(-0.875780\pi\)
0.924815 0.380418i \(-0.124220\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 33.0000 1.25176
\(696\) 0 0
\(697\) −14.0000 −0.530288
\(698\) 0 0
\(699\) − 19.0000i − 0.718646i
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) −28.0000 −1.05604
\(704\) 0 0
\(705\) −21.0000 −0.790906
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) − 6.00000i − 0.225335i −0.993633 0.112667i \(-0.964061\pi\)
0.993633 0.112667i \(-0.0359394\pi\)
\(710\) 0 0
\(711\) 20.0000 0.750059
\(712\) 0 0
\(713\) −32.0000 −1.19841
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 5.00000i 0.186728i
\(718\) 0 0
\(719\) 30.0000 1.11881 0.559406 0.828894i \(-0.311029\pi\)
0.559406 + 0.828894i \(0.311029\pi\)
\(720\) 0 0
\(721\) −18.0000 −0.670355
\(722\) 0 0
\(723\) 8.00000i 0.297523i
\(724\) 0 0
\(725\) − 16.0000i − 0.594225i
\(726\) 0 0
\(727\) −8.00000 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) 7.00000i 0.258904i
\(732\) 0 0
\(733\) 1.00000i 0.0369358i 0.999829 + 0.0184679i \(0.00587886\pi\)
−0.999829 + 0.0184679i \(0.994121\pi\)
\(734\) 0 0
\(735\) −6.00000 −0.221313
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) − 34.0000i − 1.25071i −0.780340 0.625355i \(-0.784954\pi\)
0.780340 0.625355i \(-0.215046\pi\)
\(740\) 0 0
\(741\) 4.00000i 0.146944i
\(742\) 0 0
\(743\) 21.0000 0.770415 0.385208 0.922830i \(-0.374130\pi\)
0.385208 + 0.922830i \(0.374130\pi\)
\(744\) 0 0
\(745\) −18.0000 −0.659469
\(746\) 0 0
\(747\) 28.0000i 1.02447i
\(748\) 0 0
\(749\) 24.0000i 0.876941i
\(750\) 0 0
\(751\) −2.00000 −0.0729810 −0.0364905 0.999334i \(-0.511618\pi\)
−0.0364905 + 0.999334i \(0.511618\pi\)
\(752\) 0 0
\(753\) 20.0000 0.728841
\(754\) 0 0
\(755\) 51.0000i 1.85608i
\(756\) 0 0
\(757\) − 32.0000i − 1.16306i −0.813525 0.581530i \(-0.802454\pi\)
0.813525 0.581530i \(-0.197546\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 22.0000 0.797499 0.398750 0.917060i \(-0.369444\pi\)
0.398750 + 0.917060i \(0.369444\pi\)
\(762\) 0 0
\(763\) 3.00000i 0.108607i
\(764\) 0 0
\(765\) 42.0000i 1.51851i
\(766\) 0 0
\(767\) −14.0000 −0.505511
\(768\) 0 0
\(769\) −20.0000 −0.721218 −0.360609 0.932717i \(-0.617431\pi\)
−0.360609 + 0.932717i \(0.617431\pi\)
\(770\) 0 0
\(771\) 7.00000i 0.252099i
\(772\) 0 0
\(773\) 21.0000i 0.755318i 0.925945 + 0.377659i \(0.123271\pi\)
−0.925945 + 0.377659i \(0.876729\pi\)
\(774\) 0 0
\(775\) −32.0000 −1.14947
\(776\) 0 0
\(777\) 21.0000 0.753371
\(778\) 0 0
\(779\) − 8.00000i − 0.286630i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 20.0000 0.714742
\(784\) 0 0
\(785\) −6.00000 −0.214149
\(786\) 0 0
\(787\) 22.0000i 0.784215i 0.919919 + 0.392108i \(0.128254\pi\)
−0.919919 + 0.392108i \(0.871746\pi\)
\(788\) 0 0
\(789\) 14.0000i 0.498413i
\(790\) 0 0
\(791\) 18.0000 0.640006
\(792\) 0 0
\(793\) 10.0000 0.355110
\(794\) 0 0
\(795\) 12.0000i 0.425596i
\(796\) 0 0
\(797\) − 12.0000i − 0.425062i −0.977154 0.212531i \(-0.931829\pi\)
0.977154 0.212531i \(-0.0681706\pi\)
\(798\) 0 0
\(799\) −49.0000 −1.73350
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) − 36.0000i − 1.26883i
\(806\) 0 0
\(807\) −6.00000 −0.211210
\(808\) 0 0
\(809\) 25.0000 0.878953 0.439477 0.898254i \(-0.355164\pi\)
0.439477 + 0.898254i \(0.355164\pi\)
\(810\) 0 0
\(811\) 40.0000i 1.40459i 0.711886 + 0.702295i \(0.247841\pi\)
−0.711886 + 0.702295i \(0.752159\pi\)
\(812\) 0 0
\(813\) − 3.00000i − 0.105215i
\(814\) 0 0
\(815\) 72.0000 2.52205
\(816\) 0 0
\(817\) −4.00000 −0.139942
\(818\) 0 0
\(819\) 6.00000i 0.209657i
\(820\) 0 0
\(821\) 45.0000i 1.57051i 0.619172 + 0.785255i \(0.287468\pi\)
−0.619172 + 0.785255i \(0.712532\pi\)
\(822\) 0 0
\(823\) 46.0000 1.60346 0.801730 0.597687i \(-0.203913\pi\)
0.801730 + 0.597687i \(0.203913\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 8.00000i − 0.278187i −0.990279 0.139094i \(-0.955581\pi\)
0.990279 0.139094i \(-0.0444189\pi\)
\(828\) 0 0
\(829\) 14.0000i 0.486240i 0.969996 + 0.243120i \(0.0781709\pi\)
−0.969996 + 0.243120i \(0.921829\pi\)
\(830\) 0 0
\(831\) 18.0000 0.624413
\(832\) 0 0
\(833\) −14.0000 −0.485071
\(834\) 0 0
\(835\) − 36.0000i − 1.24583i
\(836\) 0 0
\(837\) − 40.0000i − 1.38260i
\(838\) 0 0
\(839\) 40.0000 1.38095 0.690477 0.723355i \(-0.257401\pi\)
0.690477 + 0.723355i \(0.257401\pi\)
\(840\) 0 0
\(841\) 13.0000 0.448276
\(842\) 0 0
\(843\) 8.00000i 0.275535i
\(844\) 0 0
\(845\) 3.00000i 0.103203i
\(846\) 0 0
\(847\) −33.0000 −1.13389
\(848\) 0 0
\(849\) 4.00000 0.137280
\(850\) 0 0
\(851\) 28.0000i 0.959828i
\(852\) 0 0
\(853\) − 9.00000i − 0.308154i −0.988059 0.154077i \(-0.950760\pi\)
0.988059 0.154077i \(-0.0492404\pi\)
\(854\) 0 0
\(855\) −24.0000 −0.820783
\(856\) 0 0
\(857\) −42.0000 −1.43469 −0.717346 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) 0 0
\(859\) 36.0000i 1.22830i 0.789188 + 0.614152i \(0.210502\pi\)
−0.789188 + 0.614152i \(0.789498\pi\)
\(860\) 0 0
\(861\) 6.00000i 0.204479i
\(862\) 0 0
\(863\) 51.0000 1.73606 0.868030 0.496512i \(-0.165386\pi\)
0.868030 + 0.496512i \(0.165386\pi\)
\(864\) 0 0
\(865\) 18.0000 0.612018
\(866\) 0 0
\(867\) − 32.0000i − 1.08678i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 2.00000 0.0677674
\(872\) 0 0
\(873\) 16.0000 0.541518
\(874\) 0 0
\(875\) 9.00000i 0.304256i
\(876\) 0 0
\(877\) − 7.00000i − 0.236373i −0.992991 0.118187i \(-0.962292\pi\)
0.992991 0.118187i \(-0.0377081\pi\)
\(878\) 0 0
\(879\) 1.00000 0.0337292
\(880\) 0 0
\(881\) −3.00000 −0.101073 −0.0505363 0.998722i \(-0.516093\pi\)
−0.0505363 + 0.998722i \(0.516093\pi\)
\(882\) 0 0
\(883\) − 41.0000i − 1.37976i −0.723924 0.689880i \(-0.757663\pi\)
0.723924 0.689880i \(-0.242337\pi\)
\(884\) 0 0
\(885\) 42.0000i 1.41181i
\(886\) 0 0
\(887\) −18.0000 −0.604381 −0.302190 0.953248i \(-0.597718\pi\)
−0.302190 + 0.953248i \(0.597718\pi\)
\(888\) 0 0
\(889\) 24.0000 0.804934
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 28.0000i − 0.936984i
\(894\) 0 0
\(895\) 63.0000 2.10586
\(896\) 0 0
\(897\) 4.00000 0.133556
\(898\) 0 0
\(899\) 32.0000i 1.06726i
\(900\) 0 0
\(901\) 28.0000i 0.932815i
\(902\) 0 0
\(903\) 3.00000 0.0998337
\(904\) 0 0
\(905\) −30.0000 −0.997234
\(906\) 0 0
\(907\) − 13.0000i − 0.431658i −0.976431 0.215829i \(-0.930755\pi\)
0.976431 0.215829i \(-0.0692454\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 8.00000 0.265052 0.132526 0.991180i \(-0.457691\pi\)
0.132526 + 0.991180i \(0.457691\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) − 30.0000i − 0.991769i
\(916\) 0 0
\(917\) 45.0000i 1.48603i
\(918\) 0 0
\(919\) −60.0000 −1.97922 −0.989609 0.143787i \(-0.954072\pi\)
−0.989609 + 0.143787i \(0.954072\pi\)
\(920\) 0 0
\(921\) 2.00000 0.0659022
\(922\) 0 0
\(923\) − 3.00000i − 0.0987462i
\(924\) 0 0
\(925\) 28.0000i 0.920634i
\(926\) 0 0
\(927\) 12.0000 0.394132
\(928\) 0 0
\(929\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) − 8.00000i − 0.262189i
\(932\) 0 0
\(933\) 32.0000i 1.04763i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −22.0000 −0.718709 −0.359354 0.933201i \(-0.617003\pi\)
−0.359354 + 0.933201i \(0.617003\pi\)
\(938\) 0 0
\(939\) − 29.0000i − 0.946379i
\(940\) 0 0
\(941\) 25.0000i 0.814977i 0.913210 + 0.407488i \(0.133595\pi\)
−0.913210 + 0.407488i \(0.866405\pi\)
\(942\) 0 0
\(943\) −8.00000 −0.260516
\(944\) 0 0
\(945\) 45.0000 1.46385
\(946\) 0 0
\(947\) 32.0000i 1.03986i 0.854209 + 0.519930i \(0.174042\pi\)
−0.854209 + 0.519930i \(0.825958\pi\)
\(948\) 0 0
\(949\) − 14.0000i − 0.454459i
\(950\) 0 0
\(951\) 18.0000 0.583690
\(952\) 0 0
\(953\) 9.00000 0.291539 0.145769 0.989319i \(-0.453434\pi\)
0.145769 + 0.989319i \(0.453434\pi\)
\(954\) 0 0
\(955\) 36.0000i 1.16493i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 36.0000 1.16250
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) − 16.0000i − 0.515593i
\(964\) 0 0
\(965\) 18.0000i 0.579441i
\(966\) 0 0
\(967\) 7.00000 0.225105 0.112552 0.993646i \(-0.464097\pi\)
0.112552 + 0.993646i \(0.464097\pi\)
\(968\) 0 0
\(969\) 28.0000 0.899490
\(970\) 0 0
\(971\) − 15.0000i − 0.481373i −0.970603 0.240686i \(-0.922627\pi\)
0.970603 0.240686i \(-0.0773725\pi\)
\(972\) 0 0
\(973\) − 33.0000i − 1.05793i
\(974\) 0 0
\(975\) 4.00000 0.128103
\(976\) 0 0
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) − 2.00000i − 0.0638551i
\(982\) 0 0
\(983\) −59.0000 −1.88181 −0.940904 0.338674i \(-0.890022\pi\)
−0.940904 + 0.338674i \(0.890022\pi\)
\(984\) 0 0
\(985\) −51.0000 −1.62500
\(986\) 0 0
\(987\) 21.0000i 0.668437i
\(988\) 0 0
\(989\) 4.00000i 0.127193i
\(990\) 0 0
\(991\) 8.00000 0.254128 0.127064 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) 0 0
\(993\) 10.0000 0.317340
\(994\) 0 0
\(995\) − 30.0000i − 0.951064i
\(996\) 0 0
\(997\) − 52.0000i − 1.64686i −0.567420 0.823428i \(-0.692059\pi\)
0.567420 0.823428i \(-0.307941\pi\)
\(998\) 0 0
\(999\) −35.0000 −1.10735
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 416.2.b.a.209.1 2
3.2 odd 2 3744.2.g.a.1873.2 2
4.3 odd 2 104.2.b.a.53.1 2
8.3 odd 2 104.2.b.a.53.2 yes 2
8.5 even 2 inner 416.2.b.a.209.2 2
12.11 even 2 936.2.g.a.469.2 2
16.3 odd 4 3328.2.a.c.1.1 1
16.5 even 4 3328.2.a.f.1.1 1
16.11 odd 4 3328.2.a.j.1.1 1
16.13 even 4 3328.2.a.g.1.1 1
24.5 odd 2 3744.2.g.a.1873.1 2
24.11 even 2 936.2.g.a.469.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
104.2.b.a.53.1 2 4.3 odd 2
104.2.b.a.53.2 yes 2 8.3 odd 2
416.2.b.a.209.1 2 1.1 even 1 trivial
416.2.b.a.209.2 2 8.5 even 2 inner
936.2.g.a.469.1 2 24.11 even 2
936.2.g.a.469.2 2 12.11 even 2
3328.2.a.c.1.1 1 16.3 odd 4
3328.2.a.f.1.1 1 16.5 even 4
3328.2.a.g.1.1 1 16.13 even 4
3328.2.a.j.1.1 1 16.11 odd 4
3744.2.g.a.1873.1 2 24.5 odd 2
3744.2.g.a.1873.2 2 3.2 odd 2