Properties

Label 3328.2.b.j.1665.1
Level $3328$
Weight $2$
Character 3328.1665
Analytic conductor $26.574$
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] = [3328,2,Mod(1665,3328)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3328, 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("3328.1665");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3328 = 2^{8} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3328.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: \(26.5742137927\)
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 26)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1665.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3328.1665
Dual form 3328.2.b.j.1665.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} +3.00000i q^{5} -1.00000 q^{7} +2.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} +3.00000i q^{5} -1.00000 q^{7} +2.00000 q^{9} +6.00000i q^{11} +1.00000i q^{13} +3.00000 q^{15} -3.00000 q^{17} -2.00000i q^{19} +1.00000i q^{21} -4.00000 q^{25} -5.00000i q^{27} +6.00000i q^{29} +4.00000 q^{31} +6.00000 q^{33} -3.00000i q^{35} +7.00000i q^{37} +1.00000 q^{39} -1.00000i q^{43} +6.00000i q^{45} -3.00000 q^{47} -6.00000 q^{49} +3.00000i q^{51} -18.0000 q^{55} -2.00000 q^{57} -6.00000i q^{59} +8.00000i q^{61} -2.00000 q^{63} -3.00000 q^{65} -14.0000i q^{67} -3.00000 q^{71} -2.00000 q^{73} +4.00000i q^{75} -6.00000i q^{77} -8.00000 q^{79} +1.00000 q^{81} -12.0000i q^{83} -9.00000i q^{85} +6.00000 q^{87} +6.00000 q^{89} -1.00000i q^{91} -4.00000i q^{93} +6.00000 q^{95} -10.0000 q^{97} +12.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{7} + 4 q^{9} + 6 q^{15} - 6 q^{17} - 8 q^{25} + 8 q^{31} + 12 q^{33} + 2 q^{39} - 6 q^{47} - 12 q^{49} - 36 q^{55} - 4 q^{57} - 4 q^{63} - 6 q^{65} - 6 q^{71} - 4 q^{73} - 16 q^{79} + 2 q^{81} + 12 q^{87} + 12 q^{89} + 12 q^{95} - 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3328\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(769\) \(1535\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(0.234058\pi\)
−0.741620 + 0.670820i \(0.765942\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964 −0.188982 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) 0 0
\(9\) 2.00000 0.666667
\(10\) 0 0
\(11\) 6.00000i 1.80907i 0.426401 + 0.904534i \(0.359781\pi\)
−0.426401 + 0.904534i \(0.640219\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i
\(14\) 0 0
\(15\) 3.00000 0.774597
\(16\) 0 0
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) − 2.00000i − 0.458831i −0.973329 0.229416i \(-0.926318\pi\)
0.973329 0.229416i \(-0.0736815\pi\)
\(20\) 0 0
\(21\) 1.00000i 0.218218i
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) − 5.00000i − 0.962250i
\(28\) 0 0
\(29\) 6.00000i 1.11417i 0.830455 + 0.557086i \(0.188081\pi\)
−0.830455 + 0.557086i \(0.811919\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) 6.00000 1.04447
\(34\) 0 0
\(35\) − 3.00000i − 0.507093i
\(36\) 0 0
\(37\) 7.00000i 1.15079i 0.817875 + 0.575396i \(0.195152\pi\)
−0.817875 + 0.575396i \(0.804848\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 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
\(47\) −3.00000 −0.437595 −0.218797 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) 3.00000i 0.420084i
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) −18.0000 −2.42712
\(56\) 0 0
\(57\) −2.00000 −0.264906
\(58\) 0 0
\(59\) − 6.00000i − 0.781133i −0.920575 0.390567i \(-0.872279\pi\)
0.920575 0.390567i \(-0.127721\pi\)
\(60\) 0 0
\(61\) 8.00000i 1.02430i 0.858898 + 0.512148i \(0.171150\pi\)
−0.858898 + 0.512148i \(0.828850\pi\)
\(62\) 0 0
\(63\) −2.00000 −0.251976
\(64\) 0 0
\(65\) −3.00000 −0.372104
\(66\) 0 0
\(67\) − 14.0000i − 1.71037i −0.518321 0.855186i \(-0.673443\pi\)
0.518321 0.855186i \(-0.326557\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −3.00000 −0.356034 −0.178017 0.984027i \(-0.556968\pi\)
−0.178017 + 0.984027i \(0.556968\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 0 0
\(75\) 4.00000i 0.461880i
\(76\) 0 0
\(77\) − 6.00000i − 0.683763i
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 12.0000i − 1.31717i −0.752506 0.658586i \(-0.771155\pi\)
0.752506 0.658586i \(-0.228845\pi\)
\(84\) 0 0
\(85\) − 9.00000i − 0.976187i
\(86\) 0 0
\(87\) 6.00000 0.643268
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) − 1.00000i − 0.104828i
\(92\) 0 0
\(93\) − 4.00000i − 0.414781i
\(94\) 0 0
\(95\) 6.00000 0.615587
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 0 0
\(99\) 12.0000i 1.20605i
\(100\) 0 0
\(101\) 12.0000i 1.19404i 0.802225 + 0.597022i \(0.203650\pi\)
−0.802225 + 0.597022i \(0.796350\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 0 0
\(105\) −3.00000 −0.292770
\(106\) 0 0
\(107\) 12.0000i 1.16008i 0.814587 + 0.580042i \(0.196964\pi\)
−0.814587 + 0.580042i \(0.803036\pi\)
\(108\) 0 0
\(109\) − 7.00000i − 0.670478i −0.942133 0.335239i \(-0.891183\pi\)
0.942133 0.335239i \(-0.108817\pi\)
\(110\) 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
\(115\) 0 0
\(116\) 0 0
\(117\) 2.00000i 0.184900i
\(118\) 0 0
\(119\) 3.00000 0.275010
\(120\) 0 0
\(121\) −25.0000 −2.27273
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 3.00000i 0.268328i
\(126\) 0 0
\(127\) −20.0000 −1.77471 −0.887357 0.461084i \(-0.847461\pi\)
−0.887357 + 0.461084i \(0.847461\pi\)
\(128\) 0 0
\(129\) −1.00000 −0.0880451
\(130\) 0 0
\(131\) 21.0000i 1.83478i 0.397991 + 0.917389i \(0.369707\pi\)
−0.397991 + 0.917389i \(0.630293\pi\)
\(132\) 0 0
\(133\) 2.00000i 0.173422i
\(134\) 0 0
\(135\) 15.0000 1.29099
\(136\) 0 0
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) − 13.0000i − 1.10265i −0.834292 0.551323i \(-0.814123\pi\)
0.834292 0.551323i \(-0.185877\pi\)
\(140\) 0 0
\(141\) 3.00000i 0.252646i
\(142\) 0 0
\(143\) −6.00000 −0.501745
\(144\) 0 0
\(145\) −18.0000 −1.49482
\(146\) 0 0
\(147\) 6.00000i 0.494872i
\(148\) 0 0
\(149\) 6.00000i 0.491539i 0.969328 + 0.245770i \(0.0790407\pi\)
−0.969328 + 0.245770i \(0.920959\pi\)
\(150\) 0 0
\(151\) 17.0000 1.38344 0.691720 0.722166i \(-0.256853\pi\)
0.691720 + 0.722166i \(0.256853\pi\)
\(152\) 0 0
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) 12.0000i 0.963863i
\(156\) 0 0
\(157\) 14.0000i 1.11732i 0.829396 + 0.558661i \(0.188685\pi\)
−0.829396 + 0.558661i \(0.811315\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 16.0000i 1.25322i 0.779334 + 0.626608i \(0.215557\pi\)
−0.779334 + 0.626608i \(0.784443\pi\)
\(164\) 0 0
\(165\) 18.0000i 1.40130i
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) − 4.00000i − 0.305888i
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 4.00000 0.302372
\(176\) 0 0
\(177\) −6.00000 −0.450988
\(178\) 0 0
\(179\) − 3.00000i − 0.224231i −0.993695 0.112115i \(-0.964237\pi\)
0.993695 0.112115i \(-0.0357626\pi\)
\(180\) 0 0
\(181\) − 20.0000i − 1.48659i −0.668965 0.743294i \(-0.733262\pi\)
0.668965 0.743294i \(-0.266738\pi\)
\(182\) 0 0
\(183\) 8.00000 0.591377
\(184\) 0 0
\(185\) −21.0000 −1.54395
\(186\) 0 0
\(187\) − 18.0000i − 1.31629i
\(188\) 0 0
\(189\) 5.00000i 0.363696i
\(190\) 0 0
\(191\) 18.0000 1.30243 0.651217 0.758891i \(-0.274259\pi\)
0.651217 + 0.758891i \(0.274259\pi\)
\(192\) 0 0
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 0 0
\(195\) 3.00000i 0.214834i
\(196\) 0 0
\(197\) − 3.00000i − 0.213741i −0.994273 0.106871i \(-0.965917\pi\)
0.994273 0.106871i \(-0.0340831\pi\)
\(198\) 0 0
\(199\) 2.00000 0.141776 0.0708881 0.997484i \(-0.477417\pi\)
0.0708881 + 0.997484i \(0.477417\pi\)
\(200\) 0 0
\(201\) −14.0000 −0.987484
\(202\) 0 0
\(203\) − 6.00000i − 0.421117i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 12.0000 0.830057
\(210\) 0 0
\(211\) 13.0000i 0.894957i 0.894295 + 0.447478i \(0.147678\pi\)
−0.894295 + 0.447478i \(0.852322\pi\)
\(212\) 0 0
\(213\) 3.00000i 0.205557i
\(214\) 0 0
\(215\) 3.00000 0.204598
\(216\) 0 0
\(217\) −4.00000 −0.271538
\(218\) 0 0
\(219\) 2.00000i 0.135147i
\(220\) 0 0
\(221\) − 3.00000i − 0.201802i
\(222\) 0 0
\(223\) 19.0000 1.27233 0.636167 0.771551i \(-0.280519\pi\)
0.636167 + 0.771551i \(0.280519\pi\)
\(224\) 0 0
\(225\) −8.00000 −0.533333
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 13.0000i 0.859064i 0.903052 + 0.429532i \(0.141321\pi\)
−0.903052 + 0.429532i \(0.858679\pi\)
\(230\) 0 0
\(231\) −6.00000 −0.394771
\(232\) 0 0
\(233\) 27.0000 1.76883 0.884414 0.466702i \(-0.154558\pi\)
0.884414 + 0.466702i \(0.154558\pi\)
\(234\) 0 0
\(235\) − 9.00000i − 0.587095i
\(236\) 0 0
\(237\) 8.00000i 0.519656i
\(238\) 0 0
\(239\) −15.0000 −0.970269 −0.485135 0.874439i \(-0.661229\pi\)
−0.485135 + 0.874439i \(0.661229\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 0 0
\(243\) − 16.0000i − 1.02640i
\(244\) 0 0
\(245\) − 18.0000i − 1.14998i
\(246\) 0 0
\(247\) 2.00000 0.127257
\(248\) 0 0
\(249\) −12.0000 −0.760469
\(250\) 0 0
\(251\) 24.0000i 1.51487i 0.652913 + 0.757433i \(0.273547\pi\)
−0.652913 + 0.757433i \(0.726453\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −9.00000 −0.563602
\(256\) 0 0
\(257\) 9.00000 0.561405 0.280702 0.959795i \(-0.409433\pi\)
0.280702 + 0.959795i \(0.409433\pi\)
\(258\) 0 0
\(259\) − 7.00000i − 0.434959i
\(260\) 0 0
\(261\) 12.0000i 0.742781i
\(262\) 0 0
\(263\) −12.0000 −0.739952 −0.369976 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 6.00000i − 0.367194i
\(268\) 0 0
\(269\) 24.0000i 1.46331i 0.681677 + 0.731653i \(0.261251\pi\)
−0.681677 + 0.731653i \(0.738749\pi\)
\(270\) 0 0
\(271\) −11.0000 −0.668202 −0.334101 0.942537i \(-0.608433\pi\)
−0.334101 + 0.942537i \(0.608433\pi\)
\(272\) 0 0
\(273\) −1.00000 −0.0605228
\(274\) 0 0
\(275\) − 24.0000i − 1.44725i
\(276\) 0 0
\(277\) 28.0000i 1.68236i 0.540758 + 0.841178i \(0.318138\pi\)
−0.540758 + 0.841178i \(0.681862\pi\)
\(278\) 0 0
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) − 4.00000i − 0.237775i −0.992908 0.118888i \(-0.962067\pi\)
0.992908 0.118888i \(-0.0379328\pi\)
\(284\) 0 0
\(285\) − 6.00000i − 0.355409i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 10.0000i 0.586210i
\(292\) 0 0
\(293\) − 21.0000i − 1.22683i −0.789760 0.613417i \(-0.789795\pi\)
0.789760 0.613417i \(-0.210205\pi\)
\(294\) 0 0
\(295\) 18.0000 1.04800
\(296\) 0 0
\(297\) 30.0000 1.74078
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 1.00000i 0.0576390i
\(302\) 0 0
\(303\) 12.0000 0.689382
\(304\) 0 0
\(305\) −24.0000 −1.37424
\(306\) 0 0
\(307\) − 2.00000i − 0.114146i −0.998370 0.0570730i \(-0.981823\pi\)
0.998370 0.0570730i \(-0.0181768\pi\)
\(308\) 0 0
\(309\) 4.00000i 0.227552i
\(310\) 0 0
\(311\) −30.0000 −1.70114 −0.850572 0.525859i \(-0.823744\pi\)
−0.850572 + 0.525859i \(0.823744\pi\)
\(312\) 0 0
\(313\) 1.00000 0.0565233 0.0282617 0.999601i \(-0.491003\pi\)
0.0282617 + 0.999601i \(0.491003\pi\)
\(314\) 0 0
\(315\) − 6.00000i − 0.338062i
\(316\) 0 0
\(317\) − 6.00000i − 0.336994i −0.985702 0.168497i \(-0.946109\pi\)
0.985702 0.168497i \(-0.0538913\pi\)
\(318\) 0 0
\(319\) −36.0000 −2.01561
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) 6.00000i 0.333849i
\(324\) 0 0
\(325\) − 4.00000i − 0.221880i
\(326\) 0 0
\(327\) −7.00000 −0.387101
\(328\) 0 0
\(329\) 3.00000 0.165395
\(330\) 0 0
\(331\) 8.00000i 0.439720i 0.975531 + 0.219860i \(0.0705600\pi\)
−0.975531 + 0.219860i \(0.929440\pi\)
\(332\) 0 0
\(333\) 14.0000i 0.767195i
\(334\) 0 0
\(335\) 42.0000 2.29471
\(336\) 0 0
\(337\) 23.0000 1.25289 0.626445 0.779466i \(-0.284509\pi\)
0.626445 + 0.779466i \(0.284509\pi\)
\(338\) 0 0
\(339\) 6.00000i 0.325875i
\(340\) 0 0
\(341\) 24.0000i 1.29967i
\(342\) 0 0
\(343\) 13.0000 0.701934
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 3.00000i 0.161048i 0.996753 + 0.0805242i \(0.0256594\pi\)
−0.996753 + 0.0805242i \(0.974341\pi\)
\(348\) 0 0
\(349\) − 19.0000i − 1.01705i −0.861048 0.508523i \(-0.830192\pi\)
0.861048 0.508523i \(-0.169808\pi\)
\(350\) 0 0
\(351\) 5.00000 0.266880
\(352\) 0 0
\(353\) 24.0000 1.27739 0.638696 0.769460i \(-0.279474\pi\)
0.638696 + 0.769460i \(0.279474\pi\)
\(354\) 0 0
\(355\) − 9.00000i − 0.477670i
\(356\) 0 0
\(357\) − 3.00000i − 0.158777i
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 15.0000 0.789474
\(362\) 0 0
\(363\) 25.0000i 1.31216i
\(364\) 0 0
\(365\) − 6.00000i − 0.314054i
\(366\) 0 0
\(367\) −26.0000 −1.35719 −0.678594 0.734513i \(-0.737411\pi\)
−0.678594 + 0.734513i \(0.737411\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 4.00000i 0.207112i 0.994624 + 0.103556i \(0.0330221\pi\)
−0.994624 + 0.103556i \(0.966978\pi\)
\(374\) 0 0
\(375\) 3.00000 0.154919
\(376\) 0 0
\(377\) −6.00000 −0.309016
\(378\) 0 0
\(379\) 20.0000i 1.02733i 0.857991 + 0.513665i \(0.171713\pi\)
−0.857991 + 0.513665i \(0.828287\pi\)
\(380\) 0 0
\(381\) 20.0000i 1.02463i
\(382\) 0 0
\(383\) −21.0000 −1.07305 −0.536525 0.843884i \(-0.680263\pi\)
−0.536525 + 0.843884i \(0.680263\pi\)
\(384\) 0 0
\(385\) 18.0000 0.917365
\(386\) 0 0
\(387\) − 2.00000i − 0.101666i
\(388\) 0 0
\(389\) 6.00000i 0.304212i 0.988364 + 0.152106i \(0.0486055\pi\)
−0.988364 + 0.152106i \(0.951394\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 21.0000 1.05931
\(394\) 0 0
\(395\) − 24.0000i − 1.20757i
\(396\) 0 0
\(397\) − 34.0000i − 1.70641i −0.521575 0.853206i \(-0.674655\pi\)
0.521575 0.853206i \(-0.325345\pi\)
\(398\) 0 0
\(399\) 2.00000 0.100125
\(400\) 0 0
\(401\) 36.0000 1.79775 0.898877 0.438201i \(-0.144384\pi\)
0.898877 + 0.438201i \(0.144384\pi\)
\(402\) 0 0
\(403\) 4.00000i 0.199254i
\(404\) 0 0
\(405\) 3.00000i 0.149071i
\(406\) 0 0
\(407\) −42.0000 −2.08186
\(408\) 0 0
\(409\) −32.0000 −1.58230 −0.791149 0.611623i \(-0.790517\pi\)
−0.791149 + 0.611623i \(0.790517\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 6.00000i 0.295241i
\(414\) 0 0
\(415\) 36.0000 1.76717
\(416\) 0 0
\(417\) −13.0000 −0.636613
\(418\) 0 0
\(419\) − 9.00000i − 0.439679i −0.975536 0.219839i \(-0.929447\pi\)
0.975536 0.219839i \(-0.0705533\pi\)
\(420\) 0 0
\(421\) − 17.0000i − 0.828529i −0.910156 0.414265i \(-0.864039\pi\)
0.910156 0.414265i \(-0.135961\pi\)
\(422\) 0 0
\(423\) −6.00000 −0.291730
\(424\) 0 0
\(425\) 12.0000 0.582086
\(426\) 0 0
\(427\) − 8.00000i − 0.387147i
\(428\) 0 0
\(429\) 6.00000i 0.289683i
\(430\) 0 0
\(431\) 33.0000 1.58955 0.794777 0.606902i \(-0.207588\pi\)
0.794777 + 0.606902i \(0.207588\pi\)
\(432\) 0 0
\(433\) −25.0000 −1.20142 −0.600712 0.799466i \(-0.705116\pi\)
−0.600712 + 0.799466i \(0.705116\pi\)
\(434\) 0 0
\(435\) 18.0000i 0.863034i
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 26.0000 1.24091 0.620456 0.784241i \(-0.286947\pi\)
0.620456 + 0.784241i \(0.286947\pi\)
\(440\) 0 0
\(441\) −12.0000 −0.571429
\(442\) 0 0
\(443\) 21.0000i 0.997740i 0.866677 + 0.498870i \(0.166252\pi\)
−0.866677 + 0.498870i \(0.833748\pi\)
\(444\) 0 0
\(445\) 18.0000i 0.853282i
\(446\) 0 0
\(447\) 6.00000 0.283790
\(448\) 0 0
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) − 17.0000i − 0.798730i
\(454\) 0 0
\(455\) 3.00000 0.140642
\(456\) 0 0
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) 0 0
\(459\) 15.0000i 0.700140i
\(460\) 0 0
\(461\) 9.00000i 0.419172i 0.977790 + 0.209586i \(0.0672116\pi\)
−0.977790 + 0.209586i \(0.932788\pi\)
\(462\) 0 0
\(463\) 40.0000 1.85896 0.929479 0.368875i \(-0.120257\pi\)
0.929479 + 0.368875i \(0.120257\pi\)
\(464\) 0 0
\(465\) 12.0000 0.556487
\(466\) 0 0
\(467\) − 36.0000i − 1.66588i −0.553362 0.832941i \(-0.686655\pi\)
0.553362 0.832941i \(-0.313345\pi\)
\(468\) 0 0
\(469\) 14.0000i 0.646460i
\(470\) 0 0
\(471\) 14.0000 0.645086
\(472\) 0 0
\(473\) 6.00000 0.275880
\(474\) 0 0
\(475\) 8.00000i 0.367065i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 21.0000 0.959514 0.479757 0.877401i \(-0.340725\pi\)
0.479757 + 0.877401i \(0.340725\pi\)
\(480\) 0 0
\(481\) −7.00000 −0.319173
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 30.0000i − 1.36223i
\(486\) 0 0
\(487\) −16.0000 −0.725029 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) 0 0
\(489\) 16.0000 0.723545
\(490\) 0 0
\(491\) − 9.00000i − 0.406164i −0.979162 0.203082i \(-0.934904\pi\)
0.979162 0.203082i \(-0.0650959\pi\)
\(492\) 0 0
\(493\) − 18.0000i − 0.810679i
\(494\) 0 0
\(495\) −36.0000 −1.61808
\(496\) 0 0
\(497\) 3.00000 0.134568
\(498\) 0 0
\(499\) 40.0000i 1.79065i 0.445418 + 0.895323i \(0.353055\pi\)
−0.445418 + 0.895323i \(0.646945\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −30.0000 −1.33763 −0.668817 0.743427i \(-0.733199\pi\)
−0.668817 + 0.743427i \(0.733199\pi\)
\(504\) 0 0
\(505\) −36.0000 −1.60198
\(506\) 0 0
\(507\) 1.00000i 0.0444116i
\(508\) 0 0
\(509\) − 18.0000i − 0.797836i −0.916987 0.398918i \(-0.869386\pi\)
0.916987 0.398918i \(-0.130614\pi\)
\(510\) 0 0
\(511\) 2.00000 0.0884748
\(512\) 0 0
\(513\) −10.0000 −0.441511
\(514\) 0 0
\(515\) − 12.0000i − 0.528783i
\(516\) 0 0
\(517\) − 18.0000i − 0.791639i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 9.00000 0.394297 0.197149 0.980374i \(-0.436832\pi\)
0.197149 + 0.980374i \(0.436832\pi\)
\(522\) 0 0
\(523\) 20.0000i 0.874539i 0.899331 + 0.437269i \(0.144054\pi\)
−0.899331 + 0.437269i \(0.855946\pi\)
\(524\) 0 0
\(525\) − 4.00000i − 0.174574i
\(526\) 0 0
\(527\) −12.0000 −0.522728
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) − 12.0000i − 0.520756i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −36.0000 −1.55642
\(536\) 0 0
\(537\) −3.00000 −0.129460
\(538\) 0 0
\(539\) − 36.0000i − 1.55063i
\(540\) 0 0
\(541\) 11.0000i 0.472927i 0.971640 + 0.236463i \(0.0759884\pi\)
−0.971640 + 0.236463i \(0.924012\pi\)
\(542\) 0 0
\(543\) −20.0000 −0.858282
\(544\) 0 0
\(545\) 21.0000 0.899541
\(546\) 0 0
\(547\) − 17.0000i − 0.726868i −0.931620 0.363434i \(-0.881604\pi\)
0.931620 0.363434i \(-0.118396\pi\)
\(548\) 0 0
\(549\) 16.0000i 0.682863i
\(550\) 0 0
\(551\) 12.0000 0.511217
\(552\) 0 0
\(553\) 8.00000 0.340195
\(554\) 0 0
\(555\) 21.0000i 0.891400i
\(556\) 0 0
\(557\) 3.00000i 0.127114i 0.997978 + 0.0635570i \(0.0202445\pi\)
−0.997978 + 0.0635570i \(0.979756\pi\)
\(558\) 0 0
\(559\) 1.00000 0.0422955
\(560\) 0 0
\(561\) −18.0000 −0.759961
\(562\) 0 0
\(563\) − 39.0000i − 1.64365i −0.569737 0.821827i \(-0.692955\pi\)
0.569737 0.821827i \(-0.307045\pi\)
\(564\) 0 0
\(565\) − 18.0000i − 0.757266i
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) −15.0000 −0.628833 −0.314416 0.949285i \(-0.601809\pi\)
−0.314416 + 0.949285i \(0.601809\pi\)
\(570\) 0 0
\(571\) 5.00000i 0.209243i 0.994512 + 0.104622i \(0.0333632\pi\)
−0.994512 + 0.104622i \(0.966637\pi\)
\(572\) 0 0
\(573\) − 18.0000i − 0.751961i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 38.0000 1.58196 0.790980 0.611842i \(-0.209571\pi\)
0.790980 + 0.611842i \(0.209571\pi\)
\(578\) 0 0
\(579\) 4.00000i 0.166234i
\(580\) 0 0
\(581\) 12.0000i 0.497844i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −6.00000 −0.248069
\(586\) 0 0
\(587\) 24.0000i 0.990586i 0.868726 + 0.495293i \(0.164939\pi\)
−0.868726 + 0.495293i \(0.835061\pi\)
\(588\) 0 0
\(589\) − 8.00000i − 0.329634i
\(590\) 0 0
\(591\) −3.00000 −0.123404
\(592\) 0 0
\(593\) 18.0000 0.739171 0.369586 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) 0 0
\(595\) 9.00000i 0.368964i
\(596\) 0 0
\(597\) − 2.00000i − 0.0818546i
\(598\) 0 0
\(599\) 6.00000 0.245153 0.122577 0.992459i \(-0.460884\pi\)
0.122577 + 0.992459i \(0.460884\pi\)
\(600\) 0 0
\(601\) 19.0000 0.775026 0.387513 0.921864i \(-0.373334\pi\)
0.387513 + 0.921864i \(0.373334\pi\)
\(602\) 0 0
\(603\) − 28.0000i − 1.14025i
\(604\) 0 0
\(605\) − 75.0000i − 3.04918i
\(606\) 0 0
\(607\) −14.0000 −0.568242 −0.284121 0.958788i \(-0.591702\pi\)
−0.284121 + 0.958788i \(0.591702\pi\)
\(608\) 0 0
\(609\) −6.00000 −0.243132
\(610\) 0 0
\(611\) − 3.00000i − 0.121367i
\(612\) 0 0
\(613\) − 38.0000i − 1.53481i −0.641165 0.767403i \(-0.721549\pi\)
0.641165 0.767403i \(-0.278451\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 24.0000 0.966204 0.483102 0.875564i \(-0.339510\pi\)
0.483102 + 0.875564i \(0.339510\pi\)
\(618\) 0 0
\(619\) − 28.0000i − 1.12542i −0.826656 0.562708i \(-0.809760\pi\)
0.826656 0.562708i \(-0.190240\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −6.00000 −0.240385
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) − 12.0000i − 0.479234i
\(628\) 0 0
\(629\) − 21.0000i − 0.837325i
\(630\) 0 0
\(631\) 29.0000 1.15447 0.577236 0.816577i \(-0.304131\pi\)
0.577236 + 0.816577i \(0.304131\pi\)
\(632\) 0 0
\(633\) 13.0000 0.516704
\(634\) 0 0
\(635\) − 60.0000i − 2.38103i
\(636\) 0 0
\(637\) − 6.00000i − 0.237729i
\(638\) 0 0
\(639\) −6.00000 −0.237356
\(640\) 0 0
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 0 0
\(643\) − 14.0000i − 0.552106i −0.961142 0.276053i \(-0.910973\pi\)
0.961142 0.276053i \(-0.0890266\pi\)
\(644\) 0 0
\(645\) − 3.00000i − 0.118125i
\(646\) 0 0
\(647\) −6.00000 −0.235884 −0.117942 0.993020i \(-0.537630\pi\)
−0.117942 + 0.993020i \(0.537630\pi\)
\(648\) 0 0
\(649\) 36.0000 1.41312
\(650\) 0 0
\(651\) 4.00000i 0.156772i
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) −63.0000 −2.46161
\(656\) 0 0
\(657\) −4.00000 −0.156055
\(658\) 0 0
\(659\) 36.0000i 1.40236i 0.712984 + 0.701180i \(0.247343\pi\)
−0.712984 + 0.701180i \(0.752657\pi\)
\(660\) 0 0
\(661\) 22.0000i 0.855701i 0.903850 + 0.427850i \(0.140729\pi\)
−0.903850 + 0.427850i \(0.859271\pi\)
\(662\) 0 0
\(663\) −3.00000 −0.116510
\(664\) 0 0
\(665\) −6.00000 −0.232670
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) − 19.0000i − 0.734582i
\(670\) 0 0
\(671\) −48.0000 −1.85302
\(672\) 0 0
\(673\) −19.0000 −0.732396 −0.366198 0.930537i \(-0.619341\pi\)
−0.366198 + 0.930537i \(0.619341\pi\)
\(674\) 0 0
\(675\) 20.0000i 0.769800i
\(676\) 0 0
\(677\) − 48.0000i − 1.84479i −0.386248 0.922395i \(-0.626229\pi\)
0.386248 0.922395i \(-0.373771\pi\)
\(678\) 0 0
\(679\) 10.0000 0.383765
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 24.0000i 0.918334i 0.888350 + 0.459167i \(0.151852\pi\)
−0.888350 + 0.459167i \(0.848148\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 13.0000 0.495981
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) − 8.00000i − 0.304334i −0.988355 0.152167i \(-0.951375\pi\)
0.988355 0.152167i \(-0.0486252\pi\)
\(692\) 0 0
\(693\) − 12.0000i − 0.455842i
\(694\) 0 0
\(695\) 39.0000 1.47935
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) − 27.0000i − 1.02123i
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 14.0000 0.528020
\(704\) 0 0
\(705\) −9.00000 −0.338960
\(706\) 0 0
\(707\) − 12.0000i − 0.451306i
\(708\) 0 0
\(709\) − 26.0000i − 0.976450i −0.872718 0.488225i \(-0.837644\pi\)
0.872718 0.488225i \(-0.162356\pi\)
\(710\) 0 0
\(711\) −16.0000 −0.600047
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) − 18.0000i − 0.673162i
\(716\) 0 0
\(717\) 15.0000i 0.560185i
\(718\) 0 0
\(719\) −6.00000 −0.223762 −0.111881 0.993722i \(-0.535688\pi\)
−0.111881 + 0.993722i \(0.535688\pi\)
\(720\) 0 0
\(721\) 4.00000 0.148968
\(722\) 0 0
\(723\) 10.0000i 0.371904i
\(724\) 0 0
\(725\) − 24.0000i − 0.891338i
\(726\) 0 0
\(727\) −10.0000 −0.370879 −0.185440 0.982656i \(-0.559371\pi\)
−0.185440 + 0.982656i \(0.559371\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) 3.00000i 0.110959i
\(732\) 0 0
\(733\) 23.0000i 0.849524i 0.905305 + 0.424762i \(0.139642\pi\)
−0.905305 + 0.424762i \(0.860358\pi\)
\(734\) 0 0
\(735\) −18.0000 −0.663940
\(736\) 0 0
\(737\) 84.0000 3.09418
\(738\) 0 0
\(739\) − 20.0000i − 0.735712i −0.929883 0.367856i \(-0.880092\pi\)
0.929883 0.367856i \(-0.119908\pi\)
\(740\) 0 0
\(741\) − 2.00000i − 0.0734718i
\(742\) 0 0
\(743\) −9.00000 −0.330178 −0.165089 0.986279i \(-0.552791\pi\)
−0.165089 + 0.986279i \(0.552791\pi\)
\(744\) 0 0
\(745\) −18.0000 −0.659469
\(746\) 0 0
\(747\) − 24.0000i − 0.878114i
\(748\) 0 0
\(749\) − 12.0000i − 0.438470i
\(750\) 0 0
\(751\) 40.0000 1.45962 0.729810 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) 0 0
\(753\) 24.0000 0.874609
\(754\) 0 0
\(755\) 51.0000i 1.85608i
\(756\) 0 0
\(757\) 16.0000i 0.581530i 0.956795 + 0.290765i \(0.0939098\pi\)
−0.956795 + 0.290765i \(0.906090\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 0 0
\(763\) 7.00000i 0.253417i
\(764\) 0 0
\(765\) − 18.0000i − 0.650791i
\(766\) 0 0
\(767\) 6.00000 0.216647
\(768\) 0 0
\(769\) 32.0000 1.15395 0.576975 0.816762i \(-0.304233\pi\)
0.576975 + 0.816762i \(0.304233\pi\)
\(770\) 0 0
\(771\) − 9.00000i − 0.324127i
\(772\) 0 0
\(773\) 39.0000i 1.40273i 0.712801 + 0.701366i \(0.247426\pi\)
−0.712801 + 0.701366i \(0.752574\pi\)
\(774\) 0 0
\(775\) −16.0000 −0.574737
\(776\) 0 0
\(777\) −7.00000 −0.251124
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) − 18.0000i − 0.644091i
\(782\) 0 0
\(783\) 30.0000 1.07211
\(784\) 0 0
\(785\) −42.0000 −1.49904
\(786\) 0 0
\(787\) 40.0000i 1.42585i 0.701242 + 0.712923i \(0.252629\pi\)
−0.701242 + 0.712923i \(0.747371\pi\)
\(788\) 0 0
\(789\) 12.0000i 0.427211i
\(790\) 0 0
\(791\) 6.00000 0.213335
\(792\) 0 0
\(793\) −8.00000 −0.284088
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 42.0000i − 1.48772i −0.668338 0.743858i \(-0.732994\pi\)
0.668338 0.743858i \(-0.267006\pi\)
\(798\) 0 0
\(799\) 9.00000 0.318397
\(800\) 0 0
\(801\) 12.0000 0.423999
\(802\) 0 0
\(803\) − 12.0000i − 0.423471i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 24.0000 0.844840
\(808\) 0 0
\(809\) 33.0000 1.16022 0.580109 0.814539i \(-0.303010\pi\)
0.580109 + 0.814539i \(0.303010\pi\)
\(810\) 0 0
\(811\) 20.0000i 0.702295i 0.936320 + 0.351147i \(0.114208\pi\)
−0.936320 + 0.351147i \(0.885792\pi\)
\(812\) 0 0
\(813\) 11.0000i 0.385787i
\(814\) 0 0
\(815\) −48.0000 −1.68137
\(816\) 0 0
\(817\) −2.00000 −0.0699711
\(818\) 0 0
\(819\) − 2.00000i − 0.0698857i
\(820\) 0 0
\(821\) 3.00000i 0.104701i 0.998629 + 0.0523504i \(0.0166713\pi\)
−0.998629 + 0.0523504i \(0.983329\pi\)
\(822\) 0 0
\(823\) 14.0000 0.488009 0.244005 0.969774i \(-0.421539\pi\)
0.244005 + 0.969774i \(0.421539\pi\)
\(824\) 0 0
\(825\) −24.0000 −0.835573
\(826\) 0 0
\(827\) 18.0000i 0.625921i 0.949766 + 0.312961i \(0.101321\pi\)
−0.949766 + 0.312961i \(0.898679\pi\)
\(828\) 0 0
\(829\) 38.0000i 1.31979i 0.751356 + 0.659897i \(0.229400\pi\)
−0.751356 + 0.659897i \(0.770600\pi\)
\(830\) 0 0
\(831\) 28.0000 0.971309
\(832\) 0 0
\(833\) 18.0000 0.623663
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 20.0000i − 0.691301i
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −7.00000 −0.241379
\(842\) 0 0
\(843\) − 6.00000i − 0.206651i
\(844\) 0 0
\(845\) − 3.00000i − 0.103203i
\(846\) 0 0
\(847\) 25.0000 0.859010
\(848\) 0 0
\(849\) −4.00000 −0.137280
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 37.0000i 1.26686i 0.773802 + 0.633428i \(0.218353\pi\)
−0.773802 + 0.633428i \(0.781647\pi\)
\(854\) 0 0
\(855\) 12.0000 0.410391
\(856\) 0 0
\(857\) 42.0000 1.43469 0.717346 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) 0 0
\(859\) − 4.00000i − 0.136478i −0.997669 0.0682391i \(-0.978262\pi\)
0.997669 0.0682391i \(-0.0217381\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 45.0000 1.53182 0.765909 0.642949i \(-0.222289\pi\)
0.765909 + 0.642949i \(0.222289\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 8.00000i 0.271694i
\(868\) 0 0
\(869\) − 48.0000i − 1.62829i
\(870\) 0 0
\(871\) 14.0000 0.474372
\(872\) 0 0
\(873\) −20.0000 −0.676897
\(874\) 0 0
\(875\) − 3.00000i − 0.101419i
\(876\) 0 0
\(877\) − 13.0000i − 0.438979i −0.975615 0.219489i \(-0.929561\pi\)
0.975615 0.219489i \(-0.0704391\pi\)
\(878\) 0 0
\(879\) −21.0000 −0.708312
\(880\) 0 0
\(881\) 21.0000 0.707508 0.353754 0.935339i \(-0.384905\pi\)
0.353754 + 0.935339i \(0.384905\pi\)
\(882\) 0 0
\(883\) − 29.0000i − 0.975928i −0.872864 0.487964i \(-0.837740\pi\)
0.872864 0.487964i \(-0.162260\pi\)
\(884\) 0 0
\(885\) − 18.0000i − 0.605063i
\(886\) 0 0
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 20.0000 0.670778
\(890\) 0 0
\(891\) 6.00000i 0.201008i
\(892\) 0 0
\(893\) 6.00000i 0.200782i
\(894\) 0 0
\(895\) 9.00000 0.300837
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 24.0000i 0.800445i
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 1.00000 0.0332779
\(904\) 0 0
\(905\) 60.0000 1.99447
\(906\) 0 0
\(907\) − 37.0000i − 1.22856i −0.789086 0.614282i \(-0.789446\pi\)
0.789086 0.614282i \(-0.210554\pi\)
\(908\) 0 0
\(909\) 24.0000i 0.796030i
\(910\) 0 0
\(911\) −30.0000 −0.993944 −0.496972 0.867766i \(-0.665555\pi\)
−0.496972 + 0.867766i \(0.665555\pi\)
\(912\) 0 0
\(913\) 72.0000 2.38285
\(914\) 0 0
\(915\) 24.0000i 0.793416i
\(916\) 0 0
\(917\) − 21.0000i − 0.693481i
\(918\) 0 0
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 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
\(927\) −8.00000 −0.262754
\(928\) 0 0
\(929\) 36.0000 1.18112 0.590561 0.806993i \(-0.298907\pi\)
0.590561 + 0.806993i \(0.298907\pi\)
\(930\) 0 0
\(931\) 12.0000i 0.393284i
\(932\) 0 0
\(933\) 30.0000i 0.982156i
\(934\) 0 0
\(935\) 54.0000 1.76599
\(936\) 0 0
\(937\) 34.0000 1.11073 0.555366 0.831606i \(-0.312578\pi\)
0.555366 + 0.831606i \(0.312578\pi\)
\(938\) 0 0
\(939\) − 1.00000i − 0.0326338i
\(940\) 0 0
\(941\) − 21.0000i − 0.684580i −0.939594 0.342290i \(-0.888797\pi\)
0.939594 0.342290i \(-0.111203\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −15.0000 −0.487950
\(946\) 0 0
\(947\) − 6.00000i − 0.194974i −0.995237 0.0974869i \(-0.968920\pi\)
0.995237 0.0974869i \(-0.0310804\pi\)
\(948\) 0 0
\(949\) − 2.00000i − 0.0649227i
\(950\) 0 0
\(951\) −6.00000 −0.194563
\(952\) 0 0
\(953\) −15.0000 −0.485898 −0.242949 0.970039i \(-0.578115\pi\)
−0.242949 + 0.970039i \(0.578115\pi\)
\(954\) 0 0
\(955\) 54.0000i 1.74740i
\(956\) 0 0
\(957\) 36.0000i 1.16371i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 24.0000i 0.773389i
\(964\) 0 0
\(965\) − 12.0000i − 0.386294i
\(966\) 0 0
\(967\) −31.0000 −0.996893 −0.498446 0.866921i \(-0.666096\pi\)
−0.498446 + 0.866921i \(0.666096\pi\)
\(968\) 0 0
\(969\) 6.00000 0.192748
\(970\) 0 0
\(971\) − 3.00000i − 0.0962746i −0.998841 0.0481373i \(-0.984672\pi\)
0.998841 0.0481373i \(-0.0153285\pi\)
\(972\) 0 0
\(973\) 13.0000i 0.416761i
\(974\) 0 0
\(975\) −4.00000 −0.128103
\(976\) 0 0
\(977\) −54.0000 −1.72761 −0.863807 0.503824i \(-0.831926\pi\)
−0.863807 + 0.503824i \(0.831926\pi\)
\(978\) 0 0
\(979\) 36.0000i 1.15056i
\(980\) 0 0
\(981\) − 14.0000i − 0.446986i
\(982\) 0 0
\(983\) 39.0000 1.24391 0.621953 0.783054i \(-0.286339\pi\)
0.621953 + 0.783054i \(0.286339\pi\)
\(984\) 0 0
\(985\) 9.00000 0.286764
\(986\) 0 0
\(987\) − 3.00000i − 0.0954911i
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −2.00000 −0.0635321 −0.0317660 0.999495i \(-0.510113\pi\)
−0.0317660 + 0.999495i \(0.510113\pi\)
\(992\) 0 0
\(993\) 8.00000 0.253872
\(994\) 0 0
\(995\) 6.00000i 0.190213i
\(996\) 0 0
\(997\) 46.0000i 1.45683i 0.685134 + 0.728417i \(0.259744\pi\)
−0.685134 + 0.728417i \(0.740256\pi\)
\(998\) 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 3328.2.b.j.1665.1 2
4.3 odd 2 3328.2.b.m.1665.2 2
8.3 odd 2 3328.2.b.m.1665.1 2
8.5 even 2 inner 3328.2.b.j.1665.2 2
16.3 odd 4 832.2.a.d.1.1 1
16.5 even 4 208.2.a.a.1.1 1
16.11 odd 4 26.2.a.a.1.1 1
16.13 even 4 832.2.a.i.1.1 1
48.5 odd 4 1872.2.a.q.1.1 1
48.11 even 4 234.2.a.e.1.1 1
48.29 odd 4 7488.2.a.h.1.1 1
48.35 even 4 7488.2.a.g.1.1 1
80.27 even 4 650.2.b.d.599.1 2
80.43 even 4 650.2.b.d.599.2 2
80.59 odd 4 650.2.a.j.1.1 1
80.69 even 4 5200.2.a.x.1.1 1
112.11 odd 12 1274.2.f.p.79.1 2
112.27 even 4 1274.2.a.d.1.1 1
112.59 even 12 1274.2.f.r.79.1 2
112.75 even 12 1274.2.f.r.1145.1 2
112.107 odd 12 1274.2.f.p.1145.1 2
144.11 even 12 2106.2.e.b.1405.1 2
144.43 odd 12 2106.2.e.ba.1405.1 2
144.59 even 12 2106.2.e.b.703.1 2
144.139 odd 12 2106.2.e.ba.703.1 2
176.43 even 4 3146.2.a.n.1.1 1
208.5 odd 4 2704.2.f.d.337.2 2
208.11 even 12 338.2.e.a.147.2 4
208.21 odd 4 2704.2.f.d.337.1 2
208.43 odd 12 338.2.c.a.315.1 2
208.59 even 12 338.2.e.a.23.1 4
208.75 odd 12 338.2.c.a.191.1 2
208.107 odd 12 338.2.c.d.191.1 2
208.123 even 12 338.2.e.a.23.2 4
208.139 odd 12 338.2.c.d.315.1 2
208.155 odd 4 338.2.a.f.1.1 1
208.171 even 12 338.2.e.a.147.1 4
208.181 even 4 2704.2.a.f.1.1 1
208.187 even 4 338.2.b.c.337.2 2
208.203 even 4 338.2.b.c.337.1 2
240.59 even 4 5850.2.a.p.1.1 1
240.107 odd 4 5850.2.e.a.5149.2 2
240.203 odd 4 5850.2.e.a.5149.1 2
272.203 odd 4 7514.2.a.c.1.1 1
304.75 even 4 9386.2.a.j.1.1 1
624.155 even 4 3042.2.a.a.1.1 1
624.203 odd 4 3042.2.b.a.1351.2 2
624.395 odd 4 3042.2.b.a.1351.1 2
1040.779 odd 4 8450.2.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
26.2.a.a.1.1 1 16.11 odd 4
208.2.a.a.1.1 1 16.5 even 4
234.2.a.e.1.1 1 48.11 even 4
338.2.a.f.1.1 1 208.155 odd 4
338.2.b.c.337.1 2 208.203 even 4
338.2.b.c.337.2 2 208.187 even 4
338.2.c.a.191.1 2 208.75 odd 12
338.2.c.a.315.1 2 208.43 odd 12
338.2.c.d.191.1 2 208.107 odd 12
338.2.c.d.315.1 2 208.139 odd 12
338.2.e.a.23.1 4 208.59 even 12
338.2.e.a.23.2 4 208.123 even 12
338.2.e.a.147.1 4 208.171 even 12
338.2.e.a.147.2 4 208.11 even 12
650.2.a.j.1.1 1 80.59 odd 4
650.2.b.d.599.1 2 80.27 even 4
650.2.b.d.599.2 2 80.43 even 4
832.2.a.d.1.1 1 16.3 odd 4
832.2.a.i.1.1 1 16.13 even 4
1274.2.a.d.1.1 1 112.27 even 4
1274.2.f.p.79.1 2 112.11 odd 12
1274.2.f.p.1145.1 2 112.107 odd 12
1274.2.f.r.79.1 2 112.59 even 12
1274.2.f.r.1145.1 2 112.75 even 12
1872.2.a.q.1.1 1 48.5 odd 4
2106.2.e.b.703.1 2 144.59 even 12
2106.2.e.b.1405.1 2 144.11 even 12
2106.2.e.ba.703.1 2 144.139 odd 12
2106.2.e.ba.1405.1 2 144.43 odd 12
2704.2.a.f.1.1 1 208.181 even 4
2704.2.f.d.337.1 2 208.21 odd 4
2704.2.f.d.337.2 2 208.5 odd 4
3042.2.a.a.1.1 1 624.155 even 4
3042.2.b.a.1351.1 2 624.395 odd 4
3042.2.b.a.1351.2 2 624.203 odd 4
3146.2.a.n.1.1 1 176.43 even 4
3328.2.b.j.1665.1 2 1.1 even 1 trivial
3328.2.b.j.1665.2 2 8.5 even 2 inner
3328.2.b.m.1665.1 2 8.3 odd 2
3328.2.b.m.1665.2 2 4.3 odd 2
5200.2.a.x.1.1 1 80.69 even 4
5850.2.a.p.1.1 1 240.59 even 4
5850.2.e.a.5149.1 2 240.203 odd 4
5850.2.e.a.5149.2 2 240.107 odd 4
7488.2.a.g.1.1 1 48.35 even 4
7488.2.a.h.1.1 1 48.29 odd 4
7514.2.a.c.1.1 1 272.203 odd 4
8450.2.a.c.1.1 1 1040.779 odd 4
9386.2.a.j.1.1 1 304.75 even 4