Properties

Label 3380.2.f.a.3041.1
Level $3380$
Weight $2$
Character 3380.3041
Analytic conductor $26.989$
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] = [3380,2,Mod(3041,3380)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3380, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3380.3041");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3380.f (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: \(26.9894358832\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 260)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3041.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3380.3041
Dual form 3380.2.f.a.3041.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} -1.00000i q^{5} +3.00000i q^{7} +6.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -1.00000i q^{5} +3.00000i q^{7} +6.00000 q^{9} +3.00000i q^{11} +3.00000i q^{15} +7.00000 q^{17} -1.00000i q^{19} -9.00000i q^{21} +7.00000 q^{23} -1.00000 q^{25} -9.00000 q^{27} -5.00000 q^{29} +4.00000i q^{31} -9.00000i q^{33} +3.00000 q^{35} -3.00000i q^{37} -7.00000i q^{41} +9.00000 q^{43} -6.00000i q^{45} +8.00000i q^{47} -2.00000 q^{49} -21.0000 q^{51} -6.00000 q^{53} +3.00000 q^{55} +3.00000i q^{57} +5.00000i q^{59} -5.00000 q^{61} +18.0000i q^{63} -13.0000i q^{67} -21.0000 q^{69} +3.00000i q^{71} -14.0000i q^{73} +3.00000 q^{75} -9.00000 q^{77} -8.00000 q^{79} +9.00000 q^{81} -12.0000i q^{83} -7.00000i q^{85} +15.0000 q^{87} +7.00000i q^{89} -12.0000i q^{93} -1.00000 q^{95} +11.0000i q^{97} +18.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} + 12 q^{9} + 14 q^{17} + 14 q^{23} - 2 q^{25} - 18 q^{27} - 10 q^{29} + 6 q^{35} + 18 q^{43} - 4 q^{49} - 42 q^{51} - 12 q^{53} + 6 q^{55} - 10 q^{61} - 42 q^{69} + 6 q^{75} - 18 q^{77} - 16 q^{79} + 18 q^{81} + 30 q^{87} - 2 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3380\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(1691\) \(1861\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) −3.00000 −1.73205 −0.866025 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(4\) 0 0
\(5\) − 1.00000i − 0.447214i
\(6\) 0 0
\(7\) 3.00000i 1.13389i 0.823754 + 0.566947i \(0.191875\pi\)
−0.823754 + 0.566947i \(0.808125\pi\)
\(8\) 0 0
\(9\) 6.00000 2.00000
\(10\) 0 0
\(11\) 3.00000i 0.904534i 0.891883 + 0.452267i \(0.149385\pi\)
−0.891883 + 0.452267i \(0.850615\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) 3.00000i 0.774597i
\(16\) 0 0
\(17\) 7.00000 1.69775 0.848875 0.528594i \(-0.177281\pi\)
0.848875 + 0.528594i \(0.177281\pi\)
\(18\) 0 0
\(19\) − 1.00000i − 0.229416i −0.993399 0.114708i \(-0.963407\pi\)
0.993399 0.114708i \(-0.0365932\pi\)
\(20\) 0 0
\(21\) − 9.00000i − 1.96396i
\(22\) 0 0
\(23\) 7.00000 1.45960 0.729800 0.683660i \(-0.239613\pi\)
0.729800 + 0.683660i \(0.239613\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −9.00000 −1.73205
\(28\) 0 0
\(29\) −5.00000 −0.928477 −0.464238 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 0 0
\(31\) 4.00000i 0.718421i 0.933257 + 0.359211i \(0.116954\pi\)
−0.933257 + 0.359211i \(0.883046\pi\)
\(32\) 0 0
\(33\) − 9.00000i − 1.56670i
\(34\) 0 0
\(35\) 3.00000 0.507093
\(36\) 0 0
\(37\) − 3.00000i − 0.493197i −0.969118 0.246598i \(-0.920687\pi\)
0.969118 0.246598i \(-0.0793129\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 7.00000i − 1.09322i −0.837389 0.546608i \(-0.815919\pi\)
0.837389 0.546608i \(-0.184081\pi\)
\(42\) 0 0
\(43\) 9.00000 1.37249 0.686244 0.727372i \(-0.259258\pi\)
0.686244 + 0.727372i \(0.259258\pi\)
\(44\) 0 0
\(45\) − 6.00000i − 0.894427i
\(46\) 0 0
\(47\) 8.00000i 1.16692i 0.812142 + 0.583460i \(0.198301\pi\)
−0.812142 + 0.583460i \(0.801699\pi\)
\(48\) 0 0
\(49\) −2.00000 −0.285714
\(50\) 0 0
\(51\) −21.0000 −2.94059
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 3.00000 0.404520
\(56\) 0 0
\(57\) 3.00000i 0.397360i
\(58\) 0 0
\(59\) 5.00000i 0.650945i 0.945552 + 0.325472i \(0.105523\pi\)
−0.945552 + 0.325472i \(0.894477\pi\)
\(60\) 0 0
\(61\) −5.00000 −0.640184 −0.320092 0.947386i \(-0.603714\pi\)
−0.320092 + 0.947386i \(0.603714\pi\)
\(62\) 0 0
\(63\) 18.0000i 2.26779i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 13.0000i − 1.58820i −0.607785 0.794101i \(-0.707942\pi\)
0.607785 0.794101i \(-0.292058\pi\)
\(68\) 0 0
\(69\) −21.0000 −2.52810
\(70\) 0 0
\(71\) 3.00000i 0.356034i 0.984027 + 0.178017i \(0.0569683\pi\)
−0.984027 + 0.178017i \(0.943032\pi\)
\(72\) 0 0
\(73\) − 14.0000i − 1.63858i −0.573382 0.819288i \(-0.694369\pi\)
0.573382 0.819288i \(-0.305631\pi\)
\(74\) 0 0
\(75\) 3.00000 0.346410
\(76\) 0 0
\(77\) −9.00000 −1.02565
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) − 12.0000i − 1.31717i −0.752506 0.658586i \(-0.771155\pi\)
0.752506 0.658586i \(-0.228845\pi\)
\(84\) 0 0
\(85\) − 7.00000i − 0.759257i
\(86\) 0 0
\(87\) 15.0000 1.60817
\(88\) 0 0
\(89\) 7.00000i 0.741999i 0.928633 + 0.370999i \(0.120985\pi\)
−0.928633 + 0.370999i \(0.879015\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) − 12.0000i − 1.24434i
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) 11.0000i 1.11688i 0.829545 + 0.558440i \(0.188600\pi\)
−0.829545 + 0.558440i \(0.811400\pi\)
\(98\) 0 0
\(99\) 18.0000i 1.80907i
\(100\) 0 0
\(101\) 9.00000 0.895533 0.447767 0.894150i \(-0.352219\pi\)
0.447767 + 0.894150i \(0.352219\pi\)
\(102\) 0 0
\(103\) 16.0000 1.57653 0.788263 0.615338i \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) 0 0
\(105\) −9.00000 −0.878310
\(106\) 0 0
\(107\) −3.00000 −0.290021 −0.145010 0.989430i \(-0.546322\pi\)
−0.145010 + 0.989430i \(0.546322\pi\)
\(108\) 0 0
\(109\) 14.0000i 1.34096i 0.741929 + 0.670478i \(0.233911\pi\)
−0.741929 + 0.670478i \(0.766089\pi\)
\(110\) 0 0
\(111\) 9.00000i 0.854242i
\(112\) 0 0
\(113\) 13.0000 1.22294 0.611469 0.791269i \(-0.290579\pi\)
0.611469 + 0.791269i \(0.290579\pi\)
\(114\) 0 0
\(115\) − 7.00000i − 0.652753i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 21.0000i 1.92507i
\(120\) 0 0
\(121\) 2.00000 0.181818
\(122\) 0 0
\(123\) 21.0000i 1.89351i
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) −1.00000 −0.0887357 −0.0443678 0.999015i \(-0.514127\pi\)
−0.0443678 + 0.999015i \(0.514127\pi\)
\(128\) 0 0
\(129\) −27.0000 −2.37722
\(130\) 0 0
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 0 0
\(133\) 3.00000 0.260133
\(134\) 0 0
\(135\) 9.00000i 0.774597i
\(136\) 0 0
\(137\) − 3.00000i − 0.256307i −0.991754 0.128154i \(-0.959095\pi\)
0.991754 0.128154i \(-0.0409051\pi\)
\(138\) 0 0
\(139\) 13.0000 1.10265 0.551323 0.834292i \(-0.314123\pi\)
0.551323 + 0.834292i \(0.314123\pi\)
\(140\) 0 0
\(141\) − 24.0000i − 2.02116i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 5.00000i 0.415227i
\(146\) 0 0
\(147\) 6.00000 0.494872
\(148\) 0 0
\(149\) − 3.00000i − 0.245770i −0.992421 0.122885i \(-0.960785\pi\)
0.992421 0.122885i \(-0.0392146\pi\)
\(150\) 0 0
\(151\) − 8.00000i − 0.651031i −0.945537 0.325515i \(-0.894462\pi\)
0.945537 0.325515i \(-0.105538\pi\)
\(152\) 0 0
\(153\) 42.0000 3.39550
\(154\) 0 0
\(155\) 4.00000 0.321288
\(156\) 0 0
\(157\) 6.00000 0.478852 0.239426 0.970915i \(-0.423041\pi\)
0.239426 + 0.970915i \(0.423041\pi\)
\(158\) 0 0
\(159\) 18.0000 1.42749
\(160\) 0 0
\(161\) 21.0000i 1.65503i
\(162\) 0 0
\(163\) 11.0000i 0.861586i 0.902451 + 0.430793i \(0.141766\pi\)
−0.902451 + 0.430793i \(0.858234\pi\)
\(164\) 0 0
\(165\) −9.00000 −0.700649
\(166\) 0 0
\(167\) 1.00000i 0.0773823i 0.999251 + 0.0386912i \(0.0123189\pi\)
−0.999251 + 0.0386912i \(0.987681\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) − 6.00000i − 0.458831i
\(172\) 0 0
\(173\) 15.0000 1.14043 0.570214 0.821496i \(-0.306860\pi\)
0.570214 + 0.821496i \(0.306860\pi\)
\(174\) 0 0
\(175\) − 3.00000i − 0.226779i
\(176\) 0 0
\(177\) − 15.0000i − 1.12747i
\(178\) 0 0
\(179\) −19.0000 −1.42013 −0.710063 0.704138i \(-0.751334\pi\)
−0.710063 + 0.704138i \(0.751334\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 0 0
\(183\) 15.0000 1.10883
\(184\) 0 0
\(185\) −3.00000 −0.220564
\(186\) 0 0
\(187\) 21.0000i 1.53567i
\(188\) 0 0
\(189\) − 27.0000i − 1.96396i
\(190\) 0 0
\(191\) −3.00000 −0.217072 −0.108536 0.994092i \(-0.534616\pi\)
−0.108536 + 0.994092i \(0.534616\pi\)
\(192\) 0 0
\(193\) − 15.0000i − 1.07972i −0.841754 0.539862i \(-0.818476\pi\)
0.841754 0.539862i \(-0.181524\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 23.0000i 1.63868i 0.573306 + 0.819341i \(0.305660\pi\)
−0.573306 + 0.819341i \(0.694340\pi\)
\(198\) 0 0
\(199\) −9.00000 −0.637993 −0.318997 0.947756i \(-0.603346\pi\)
−0.318997 + 0.947756i \(0.603346\pi\)
\(200\) 0 0
\(201\) 39.0000i 2.75085i
\(202\) 0 0
\(203\) − 15.0000i − 1.05279i
\(204\) 0 0
\(205\) −7.00000 −0.488901
\(206\) 0 0
\(207\) 42.0000 2.91920
\(208\) 0 0
\(209\) 3.00000 0.207514
\(210\) 0 0
\(211\) −5.00000 −0.344214 −0.172107 0.985078i \(-0.555058\pi\)
−0.172107 + 0.985078i \(0.555058\pi\)
\(212\) 0 0
\(213\) − 9.00000i − 0.616670i
\(214\) 0 0
\(215\) − 9.00000i − 0.613795i
\(216\) 0 0
\(217\) −12.0000 −0.814613
\(218\) 0 0
\(219\) 42.0000i 2.83810i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 23.0000i 1.54019i 0.637927 + 0.770097i \(0.279792\pi\)
−0.637927 + 0.770097i \(0.720208\pi\)
\(224\) 0 0
\(225\) −6.00000 −0.400000
\(226\) 0 0
\(227\) 1.00000i 0.0663723i 0.999449 + 0.0331862i \(0.0105654\pi\)
−0.999449 + 0.0331862i \(0.989435\pi\)
\(228\) 0 0
\(229\) 26.0000i 1.71813i 0.511868 + 0.859064i \(0.328954\pi\)
−0.511868 + 0.859064i \(0.671046\pi\)
\(230\) 0 0
\(231\) 27.0000 1.77647
\(232\) 0 0
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 0 0
\(235\) 8.00000 0.521862
\(236\) 0 0
\(237\) 24.0000 1.55897
\(238\) 0 0
\(239\) 16.0000i 1.03495i 0.855697 + 0.517477i \(0.173129\pi\)
−0.855697 + 0.517477i \(0.826871\pi\)
\(240\) 0 0
\(241\) − 1.00000i − 0.0644157i −0.999481 0.0322078i \(-0.989746\pi\)
0.999481 0.0322078i \(-0.0102538\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.00000i 0.127775i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 36.0000i 2.28141i
\(250\) 0 0
\(251\) −5.00000 −0.315597 −0.157799 0.987471i \(-0.550440\pi\)
−0.157799 + 0.987471i \(0.550440\pi\)
\(252\) 0 0
\(253\) 21.0000i 1.32026i
\(254\) 0 0
\(255\) 21.0000i 1.31507i
\(256\) 0 0
\(257\) 19.0000 1.18519 0.592594 0.805502i \(-0.298104\pi\)
0.592594 + 0.805502i \(0.298104\pi\)
\(258\) 0 0
\(259\) 9.00000 0.559233
\(260\) 0 0
\(261\) −30.0000 −1.85695
\(262\) 0 0
\(263\) −7.00000 −0.431638 −0.215819 0.976433i \(-0.569242\pi\)
−0.215819 + 0.976433i \(0.569242\pi\)
\(264\) 0 0
\(265\) 6.00000i 0.368577i
\(266\) 0 0
\(267\) − 21.0000i − 1.28518i
\(268\) 0 0
\(269\) 3.00000 0.182913 0.0914566 0.995809i \(-0.470848\pi\)
0.0914566 + 0.995809i \(0.470848\pi\)
\(270\) 0 0
\(271\) 23.0000i 1.39715i 0.715537 + 0.698575i \(0.246182\pi\)
−0.715537 + 0.698575i \(0.753818\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 3.00000i − 0.180907i
\(276\) 0 0
\(277\) 23.0000 1.38194 0.690968 0.722885i \(-0.257185\pi\)
0.690968 + 0.722885i \(0.257185\pi\)
\(278\) 0 0
\(279\) 24.0000i 1.43684i
\(280\) 0 0
\(281\) − 10.0000i − 0.596550i −0.954480 0.298275i \(-0.903589\pi\)
0.954480 0.298275i \(-0.0964112\pi\)
\(282\) 0 0
\(283\) −1.00000 −0.0594438 −0.0297219 0.999558i \(-0.509462\pi\)
−0.0297219 + 0.999558i \(0.509462\pi\)
\(284\) 0 0
\(285\) 3.00000 0.177705
\(286\) 0 0
\(287\) 21.0000 1.23959
\(288\) 0 0
\(289\) 32.0000 1.88235
\(290\) 0 0
\(291\) − 33.0000i − 1.93449i
\(292\) 0 0
\(293\) 9.00000i 0.525786i 0.964825 + 0.262893i \(0.0846766\pi\)
−0.964825 + 0.262893i \(0.915323\pi\)
\(294\) 0 0
\(295\) 5.00000 0.291111
\(296\) 0 0
\(297\) − 27.0000i − 1.56670i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 27.0000i 1.55625i
\(302\) 0 0
\(303\) −27.0000 −1.55111
\(304\) 0 0
\(305\) 5.00000i 0.286299i
\(306\) 0 0
\(307\) 28.0000i 1.59804i 0.601302 + 0.799022i \(0.294649\pi\)
−0.601302 + 0.799022i \(0.705351\pi\)
\(308\) 0 0
\(309\) −48.0000 −2.73062
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 0 0
\(315\) 18.0000 1.01419
\(316\) 0 0
\(317\) 2.00000i 0.112331i 0.998421 + 0.0561656i \(0.0178875\pi\)
−0.998421 + 0.0561656i \(0.982113\pi\)
\(318\) 0 0
\(319\) − 15.0000i − 0.839839i
\(320\) 0 0
\(321\) 9.00000 0.502331
\(322\) 0 0
\(323\) − 7.00000i − 0.389490i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 42.0000i − 2.32261i
\(328\) 0 0
\(329\) −24.0000 −1.32316
\(330\) 0 0
\(331\) − 13.0000i − 0.714545i −0.934000 0.357272i \(-0.883707\pi\)
0.934000 0.357272i \(-0.116293\pi\)
\(332\) 0 0
\(333\) − 18.0000i − 0.986394i
\(334\) 0 0
\(335\) −13.0000 −0.710266
\(336\) 0 0
\(337\) 18.0000 0.980522 0.490261 0.871576i \(-0.336901\pi\)
0.490261 + 0.871576i \(0.336901\pi\)
\(338\) 0 0
\(339\) −39.0000 −2.11819
\(340\) 0 0
\(341\) −12.0000 −0.649836
\(342\) 0 0
\(343\) 15.0000i 0.809924i
\(344\) 0 0
\(345\) 21.0000i 1.13060i
\(346\) 0 0
\(347\) −13.0000 −0.697877 −0.348938 0.937146i \(-0.613458\pi\)
−0.348938 + 0.937146i \(0.613458\pi\)
\(348\) 0 0
\(349\) − 25.0000i − 1.33822i −0.743164 0.669110i \(-0.766676\pi\)
0.743164 0.669110i \(-0.233324\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 21.0000i − 1.11772i −0.829263 0.558859i \(-0.811239\pi\)
0.829263 0.558859i \(-0.188761\pi\)
\(354\) 0 0
\(355\) 3.00000 0.159223
\(356\) 0 0
\(357\) − 63.0000i − 3.33431i
\(358\) 0 0
\(359\) − 8.00000i − 0.422224i −0.977462 0.211112i \(-0.932292\pi\)
0.977462 0.211112i \(-0.0677085\pi\)
\(360\) 0 0
\(361\) 18.0000 0.947368
\(362\) 0 0
\(363\) −6.00000 −0.314918
\(364\) 0 0
\(365\) −14.0000 −0.732793
\(366\) 0 0
\(367\) 9.00000 0.469796 0.234898 0.972020i \(-0.424524\pi\)
0.234898 + 0.972020i \(0.424524\pi\)
\(368\) 0 0
\(369\) − 42.0000i − 2.18643i
\(370\) 0 0
\(371\) − 18.0000i − 0.934513i
\(372\) 0 0
\(373\) −27.0000 −1.39801 −0.699004 0.715118i \(-0.746373\pi\)
−0.699004 + 0.715118i \(0.746373\pi\)
\(374\) 0 0
\(375\) − 3.00000i − 0.154919i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 9.00000i 0.462299i 0.972918 + 0.231149i \(0.0742486\pi\)
−0.972918 + 0.231149i \(0.925751\pi\)
\(380\) 0 0
\(381\) 3.00000 0.153695
\(382\) 0 0
\(383\) 13.0000i 0.664269i 0.943232 + 0.332134i \(0.107769\pi\)
−0.943232 + 0.332134i \(0.892231\pi\)
\(384\) 0 0
\(385\) 9.00000i 0.458682i
\(386\) 0 0
\(387\) 54.0000 2.74497
\(388\) 0 0
\(389\) −18.0000 −0.912636 −0.456318 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) 0 0
\(391\) 49.0000 2.47804
\(392\) 0 0
\(393\) −12.0000 −0.605320
\(394\) 0 0
\(395\) 8.00000i 0.402524i
\(396\) 0 0
\(397\) 33.0000i 1.65622i 0.560564 + 0.828111i \(0.310584\pi\)
−0.560564 + 0.828111i \(0.689416\pi\)
\(398\) 0 0
\(399\) −9.00000 −0.450564
\(400\) 0 0
\(401\) 15.0000i 0.749064i 0.927214 + 0.374532i \(0.122197\pi\)
−0.927214 + 0.374532i \(0.877803\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) − 9.00000i − 0.447214i
\(406\) 0 0
\(407\) 9.00000 0.446113
\(408\) 0 0
\(409\) 1.00000i 0.0494468i 0.999694 + 0.0247234i \(0.00787051\pi\)
−0.999694 + 0.0247234i \(0.992129\pi\)
\(410\) 0 0
\(411\) 9.00000i 0.443937i
\(412\) 0 0
\(413\) −15.0000 −0.738102
\(414\) 0 0
\(415\) −12.0000 −0.589057
\(416\) 0 0
\(417\) −39.0000 −1.90984
\(418\) 0 0
\(419\) −33.0000 −1.61216 −0.806078 0.591810i \(-0.798414\pi\)
−0.806078 + 0.591810i \(0.798414\pi\)
\(420\) 0 0
\(421\) − 34.0000i − 1.65706i −0.559946 0.828529i \(-0.689178\pi\)
0.559946 0.828529i \(-0.310822\pi\)
\(422\) 0 0
\(423\) 48.0000i 2.33384i
\(424\) 0 0
\(425\) −7.00000 −0.339550
\(426\) 0 0
\(427\) − 15.0000i − 0.725901i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 9.00000i 0.433515i 0.976226 + 0.216757i \(0.0695480\pi\)
−0.976226 + 0.216757i \(0.930452\pi\)
\(432\) 0 0
\(433\) −1.00000 −0.0480569 −0.0240285 0.999711i \(-0.507649\pi\)
−0.0240285 + 0.999711i \(0.507649\pi\)
\(434\) 0 0
\(435\) − 15.0000i − 0.719195i
\(436\) 0 0
\(437\) − 7.00000i − 0.334855i
\(438\) 0 0
\(439\) −3.00000 −0.143182 −0.0715911 0.997434i \(-0.522808\pi\)
−0.0715911 + 0.997434i \(0.522808\pi\)
\(440\) 0 0
\(441\) −12.0000 −0.571429
\(442\) 0 0
\(443\) 24.0000 1.14027 0.570137 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) 0 0
\(445\) 7.00000 0.331832
\(446\) 0 0
\(447\) 9.00000i 0.425685i
\(448\) 0 0
\(449\) − 21.0000i − 0.991051i −0.868593 0.495526i \(-0.834975\pi\)
0.868593 0.495526i \(-0.165025\pi\)
\(450\) 0 0
\(451\) 21.0000 0.988851
\(452\) 0 0
\(453\) 24.0000i 1.12762i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 23.0000i 1.07589i 0.842978 + 0.537947i \(0.180800\pi\)
−0.842978 + 0.537947i \(0.819200\pi\)
\(458\) 0 0
\(459\) −63.0000 −2.94059
\(460\) 0 0
\(461\) − 11.0000i − 0.512321i −0.966634 0.256161i \(-0.917542\pi\)
0.966634 0.256161i \(-0.0824576\pi\)
\(462\) 0 0
\(463\) − 28.0000i − 1.30127i −0.759390 0.650635i \(-0.774503\pi\)
0.759390 0.650635i \(-0.225497\pi\)
\(464\) 0 0
\(465\) −12.0000 −0.556487
\(466\) 0 0
\(467\) −20.0000 −0.925490 −0.462745 0.886492i \(-0.653135\pi\)
−0.462745 + 0.886492i \(0.653135\pi\)
\(468\) 0 0
\(469\) 39.0000 1.80085
\(470\) 0 0
\(471\) −18.0000 −0.829396
\(472\) 0 0
\(473\) 27.0000i 1.24146i
\(474\) 0 0
\(475\) 1.00000i 0.0458831i
\(476\) 0 0
\(477\) −36.0000 −1.64833
\(478\) 0 0
\(479\) − 1.00000i − 0.0456912i −0.999739 0.0228456i \(-0.992727\pi\)
0.999739 0.0228456i \(-0.00727261\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) − 63.0000i − 2.86660i
\(484\) 0 0
\(485\) 11.0000 0.499484
\(486\) 0 0
\(487\) 17.0000i 0.770344i 0.922845 + 0.385172i \(0.125858\pi\)
−0.922845 + 0.385172i \(0.874142\pi\)
\(488\) 0 0
\(489\) − 33.0000i − 1.49231i
\(490\) 0 0
\(491\) −23.0000 −1.03798 −0.518988 0.854782i \(-0.673691\pi\)
−0.518988 + 0.854782i \(0.673691\pi\)
\(492\) 0 0
\(493\) −35.0000 −1.57632
\(494\) 0 0
\(495\) 18.0000 0.809040
\(496\) 0 0
\(497\) −9.00000 −0.403705
\(498\) 0 0
\(499\) 16.0000i 0.716258i 0.933672 + 0.358129i \(0.116585\pi\)
−0.933672 + 0.358129i \(0.883415\pi\)
\(500\) 0 0
\(501\) − 3.00000i − 0.134030i
\(502\) 0 0
\(503\) 11.0000 0.490466 0.245233 0.969464i \(-0.421136\pi\)
0.245233 + 0.969464i \(0.421136\pi\)
\(504\) 0 0
\(505\) − 9.00000i − 0.400495i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 15.0000i − 0.664863i −0.943127 0.332432i \(-0.892131\pi\)
0.943127 0.332432i \(-0.107869\pi\)
\(510\) 0 0
\(511\) 42.0000 1.85797
\(512\) 0 0
\(513\) 9.00000i 0.397360i
\(514\) 0 0
\(515\) − 16.0000i − 0.705044i
\(516\) 0 0
\(517\) −24.0000 −1.05552
\(518\) 0 0
\(519\) −45.0000 −1.97528
\(520\) 0 0
\(521\) −34.0000 −1.48957 −0.744784 0.667306i \(-0.767447\pi\)
−0.744784 + 0.667306i \(0.767447\pi\)
\(522\) 0 0
\(523\) −23.0000 −1.00572 −0.502860 0.864368i \(-0.667719\pi\)
−0.502860 + 0.864368i \(0.667719\pi\)
\(524\) 0 0
\(525\) 9.00000i 0.392792i
\(526\) 0 0
\(527\) 28.0000i 1.21970i
\(528\) 0 0
\(529\) 26.0000 1.13043
\(530\) 0 0
\(531\) 30.0000i 1.30189i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 3.00000i 0.129701i
\(536\) 0 0
\(537\) 57.0000 2.45973
\(538\) 0 0
\(539\) − 6.00000i − 0.258438i
\(540\) 0 0
\(541\) 10.0000i 0.429934i 0.976621 + 0.214967i \(0.0689643\pi\)
−0.976621 + 0.214967i \(0.931036\pi\)
\(542\) 0 0
\(543\) −42.0000 −1.80239
\(544\) 0 0
\(545\) 14.0000 0.599694
\(546\) 0 0
\(547\) −16.0000 −0.684111 −0.342055 0.939680i \(-0.611123\pi\)
−0.342055 + 0.939680i \(0.611123\pi\)
\(548\) 0 0
\(549\) −30.0000 −1.28037
\(550\) 0 0
\(551\) 5.00000i 0.213007i
\(552\) 0 0
\(553\) − 24.0000i − 1.02058i
\(554\) 0 0
\(555\) 9.00000 0.382029
\(556\) 0 0
\(557\) − 19.0000i − 0.805056i −0.915408 0.402528i \(-0.868132\pi\)
0.915408 0.402528i \(-0.131868\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) − 63.0000i − 2.65986i
\(562\) 0 0
\(563\) 45.0000 1.89652 0.948262 0.317489i \(-0.102840\pi\)
0.948262 + 0.317489i \(0.102840\pi\)
\(564\) 0 0
\(565\) − 13.0000i − 0.546914i
\(566\) 0 0
\(567\) 27.0000i 1.13389i
\(568\) 0 0
\(569\) −19.0000 −0.796521 −0.398261 0.917272i \(-0.630386\pi\)
−0.398261 + 0.917272i \(0.630386\pi\)
\(570\) 0 0
\(571\) 40.0000 1.67395 0.836974 0.547243i \(-0.184323\pi\)
0.836974 + 0.547243i \(0.184323\pi\)
\(572\) 0 0
\(573\) 9.00000 0.375980
\(574\) 0 0
\(575\) −7.00000 −0.291920
\(576\) 0 0
\(577\) − 2.00000i − 0.0832611i −0.999133 0.0416305i \(-0.986745\pi\)
0.999133 0.0416305i \(-0.0132552\pi\)
\(578\) 0 0
\(579\) 45.0000i 1.87014i
\(580\) 0 0
\(581\) 36.0000 1.49353
\(582\) 0 0
\(583\) − 18.0000i − 0.745484i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 37.0000i − 1.52715i −0.645717 0.763577i \(-0.723441\pi\)
0.645717 0.763577i \(-0.276559\pi\)
\(588\) 0 0
\(589\) 4.00000 0.164817
\(590\) 0 0
\(591\) − 69.0000i − 2.83828i
\(592\) 0 0
\(593\) 26.0000i 1.06769i 0.845582 + 0.533846i \(0.179254\pi\)
−0.845582 + 0.533846i \(0.820746\pi\)
\(594\) 0 0
\(595\) 21.0000 0.860916
\(596\) 0 0
\(597\) 27.0000 1.10504
\(598\) 0 0
\(599\) 8.00000 0.326871 0.163436 0.986554i \(-0.447742\pi\)
0.163436 + 0.986554i \(0.447742\pi\)
\(600\) 0 0
\(601\) −13.0000 −0.530281 −0.265141 0.964210i \(-0.585418\pi\)
−0.265141 + 0.964210i \(0.585418\pi\)
\(602\) 0 0
\(603\) − 78.0000i − 3.17641i
\(604\) 0 0
\(605\) − 2.00000i − 0.0813116i
\(606\) 0 0
\(607\) −9.00000 −0.365299 −0.182649 0.983178i \(-0.558467\pi\)
−0.182649 + 0.983178i \(0.558467\pi\)
\(608\) 0 0
\(609\) 45.0000i 1.82349i
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 31.0000i 1.25208i 0.779792 + 0.626039i \(0.215325\pi\)
−0.779792 + 0.626039i \(0.784675\pi\)
\(614\) 0 0
\(615\) 21.0000 0.846802
\(616\) 0 0
\(617\) − 29.0000i − 1.16750i −0.811935 0.583748i \(-0.801586\pi\)
0.811935 0.583748i \(-0.198414\pi\)
\(618\) 0 0
\(619\) 12.0000i 0.482321i 0.970485 + 0.241160i \(0.0775280\pi\)
−0.970485 + 0.241160i \(0.922472\pi\)
\(620\) 0 0
\(621\) −63.0000 −2.52810
\(622\) 0 0
\(623\) −21.0000 −0.841347
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −9.00000 −0.359425
\(628\) 0 0
\(629\) − 21.0000i − 0.837325i
\(630\) 0 0
\(631\) − 15.0000i − 0.597141i −0.954388 0.298570i \(-0.903490\pi\)
0.954388 0.298570i \(-0.0965097\pi\)
\(632\) 0 0
\(633\) 15.0000 0.596196
\(634\) 0 0
\(635\) 1.00000i 0.0396838i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 18.0000i 0.712069i
\(640\) 0 0
\(641\) 21.0000 0.829450 0.414725 0.909947i \(-0.363878\pi\)
0.414725 + 0.909947i \(0.363878\pi\)
\(642\) 0 0
\(643\) − 7.00000i − 0.276053i −0.990429 0.138027i \(-0.955924\pi\)
0.990429 0.138027i \(-0.0440759\pi\)
\(644\) 0 0
\(645\) 27.0000i 1.06312i
\(646\) 0 0
\(647\) −17.0000 −0.668339 −0.334169 0.942513i \(-0.608456\pi\)
−0.334169 + 0.942513i \(0.608456\pi\)
\(648\) 0 0
\(649\) −15.0000 −0.588802
\(650\) 0 0
\(651\) 36.0000 1.41095
\(652\) 0 0
\(653\) −11.0000 −0.430463 −0.215232 0.976563i \(-0.569051\pi\)
−0.215232 + 0.976563i \(0.569051\pi\)
\(654\) 0 0
\(655\) − 4.00000i − 0.156293i
\(656\) 0 0
\(657\) − 84.0000i − 3.27715i
\(658\) 0 0
\(659\) −15.0000 −0.584317 −0.292159 0.956370i \(-0.594373\pi\)
−0.292159 + 0.956370i \(0.594373\pi\)
\(660\) 0 0
\(661\) − 17.0000i − 0.661223i −0.943767 0.330612i \(-0.892745\pi\)
0.943767 0.330612i \(-0.107255\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 3.00000i − 0.116335i
\(666\) 0 0
\(667\) −35.0000 −1.35521
\(668\) 0 0
\(669\) − 69.0000i − 2.66769i
\(670\) 0 0
\(671\) − 15.0000i − 0.579069i
\(672\) 0 0
\(673\) −13.0000 −0.501113 −0.250557 0.968102i \(-0.580614\pi\)
−0.250557 + 0.968102i \(0.580614\pi\)
\(674\) 0 0
\(675\) 9.00000 0.346410
\(676\) 0 0
\(677\) 22.0000 0.845529 0.422764 0.906240i \(-0.361060\pi\)
0.422764 + 0.906240i \(0.361060\pi\)
\(678\) 0 0
\(679\) −33.0000 −1.26642
\(680\) 0 0
\(681\) − 3.00000i − 0.114960i
\(682\) 0 0
\(683\) 31.0000i 1.18618i 0.805135 + 0.593091i \(0.202093\pi\)
−0.805135 + 0.593091i \(0.797907\pi\)
\(684\) 0 0
\(685\) −3.00000 −0.114624
\(686\) 0 0
\(687\) − 78.0000i − 2.97589i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 25.0000i 0.951045i 0.879704 + 0.475522i \(0.157741\pi\)
−0.879704 + 0.475522i \(0.842259\pi\)
\(692\) 0 0
\(693\) −54.0000 −2.05129
\(694\) 0 0
\(695\) − 13.0000i − 0.493118i
\(696\) 0 0
\(697\) − 49.0000i − 1.85601i
\(698\) 0 0
\(699\) 54.0000 2.04247
\(700\) 0 0
\(701\) −2.00000 −0.0755390 −0.0377695 0.999286i \(-0.512025\pi\)
−0.0377695 + 0.999286i \(0.512025\pi\)
\(702\) 0 0
\(703\) −3.00000 −0.113147
\(704\) 0 0
\(705\) −24.0000 −0.903892
\(706\) 0 0
\(707\) 27.0000i 1.01544i
\(708\) 0 0
\(709\) 35.0000i 1.31445i 0.753693 + 0.657226i \(0.228270\pi\)
−0.753693 + 0.657226i \(0.771730\pi\)
\(710\) 0 0
\(711\) −48.0000 −1.80014
\(712\) 0 0
\(713\) 28.0000i 1.04861i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 48.0000i − 1.79259i
\(718\) 0 0
\(719\) −1.00000 −0.0372937 −0.0186469 0.999826i \(-0.505936\pi\)
−0.0186469 + 0.999826i \(0.505936\pi\)
\(720\) 0 0
\(721\) 48.0000i 1.78761i
\(722\) 0 0
\(723\) 3.00000i 0.111571i
\(724\) 0 0
\(725\) 5.00000 0.185695
\(726\) 0 0
\(727\) 28.0000 1.03846 0.519231 0.854634i \(-0.326218\pi\)
0.519231 + 0.854634i \(0.326218\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 63.0000 2.33014
\(732\) 0 0
\(733\) − 14.0000i − 0.517102i −0.965998 0.258551i \(-0.916755\pi\)
0.965998 0.258551i \(-0.0832450\pi\)
\(734\) 0 0
\(735\) − 6.00000i − 0.221313i
\(736\) 0 0
\(737\) 39.0000 1.43658
\(738\) 0 0
\(739\) 39.0000i 1.43464i 0.696745 + 0.717319i \(0.254631\pi\)
−0.696745 + 0.717319i \(0.745369\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 1.00000i − 0.0366864i −0.999832 0.0183432i \(-0.994161\pi\)
0.999832 0.0183432i \(-0.00583916\pi\)
\(744\) 0 0
\(745\) −3.00000 −0.109911
\(746\) 0 0
\(747\) − 72.0000i − 2.63434i
\(748\) 0 0
\(749\) − 9.00000i − 0.328853i
\(750\) 0 0
\(751\) 13.0000 0.474377 0.237188 0.971464i \(-0.423774\pi\)
0.237188 + 0.971464i \(0.423774\pi\)
\(752\) 0 0
\(753\) 15.0000 0.546630
\(754\) 0 0
\(755\) −8.00000 −0.291150
\(756\) 0 0
\(757\) 1.00000 0.0363456 0.0181728 0.999835i \(-0.494215\pi\)
0.0181728 + 0.999835i \(0.494215\pi\)
\(758\) 0 0
\(759\) − 63.0000i − 2.28676i
\(760\) 0 0
\(761\) 51.0000i 1.84875i 0.381487 + 0.924374i \(0.375412\pi\)
−0.381487 + 0.924374i \(0.624588\pi\)
\(762\) 0 0
\(763\) −42.0000 −1.52050
\(764\) 0 0
\(765\) − 42.0000i − 1.51851i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 5.00000i 0.180305i 0.995928 + 0.0901523i \(0.0287354\pi\)
−0.995928 + 0.0901523i \(0.971265\pi\)
\(770\) 0 0
\(771\) −57.0000 −2.05280
\(772\) 0 0
\(773\) − 25.0000i − 0.899188i −0.893233 0.449594i \(-0.851569\pi\)
0.893233 0.449594i \(-0.148431\pi\)
\(774\) 0 0
\(775\) − 4.00000i − 0.143684i
\(776\) 0 0
\(777\) −27.0000 −0.968620
\(778\) 0 0
\(779\) −7.00000 −0.250801
\(780\) 0 0
\(781\) −9.00000 −0.322045
\(782\) 0 0
\(783\) 45.0000 1.60817
\(784\) 0 0
\(785\) − 6.00000i − 0.214149i
\(786\) 0 0
\(787\) − 1.00000i − 0.0356462i −0.999841 0.0178231i \(-0.994326\pi\)
0.999841 0.0178231i \(-0.00567356\pi\)
\(788\) 0 0
\(789\) 21.0000 0.747620
\(790\) 0 0
\(791\) 39.0000i 1.38668i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) − 18.0000i − 0.638394i
\(796\) 0 0
\(797\) 7.00000 0.247953 0.123976 0.992285i \(-0.460435\pi\)
0.123976 + 0.992285i \(0.460435\pi\)
\(798\) 0 0
\(799\) 56.0000i 1.98114i
\(800\) 0 0
\(801\) 42.0000i 1.48400i
\(802\) 0 0
\(803\) 42.0000 1.48215
\(804\) 0 0
\(805\) 21.0000 0.740153
\(806\) 0 0
\(807\) −9.00000 −0.316815
\(808\) 0 0
\(809\) −25.0000 −0.878953 −0.439477 0.898254i \(-0.644836\pi\)
−0.439477 + 0.898254i \(0.644836\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) − 69.0000i − 2.41994i
\(814\) 0 0
\(815\) 11.0000 0.385313
\(816\) 0 0
\(817\) − 9.00000i − 0.314870i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 15.0000i − 0.523504i −0.965135 0.261752i \(-0.915700\pi\)
0.965135 0.261752i \(-0.0843002\pi\)
\(822\) 0 0
\(823\) −47.0000 −1.63832 −0.819159 0.573567i \(-0.805559\pi\)
−0.819159 + 0.573567i \(0.805559\pi\)
\(824\) 0 0
\(825\) 9.00000i 0.313340i
\(826\) 0 0
\(827\) 12.0000i 0.417281i 0.977992 + 0.208640i \(0.0669038\pi\)
−0.977992 + 0.208640i \(0.933096\pi\)
\(828\) 0 0
\(829\) 13.0000 0.451509 0.225754 0.974184i \(-0.427515\pi\)
0.225754 + 0.974184i \(0.427515\pi\)
\(830\) 0 0
\(831\) −69.0000 −2.39358
\(832\) 0 0
\(833\) −14.0000 −0.485071
\(834\) 0 0
\(835\) 1.00000 0.0346064
\(836\) 0 0
\(837\) − 36.0000i − 1.24434i
\(838\) 0 0
\(839\) 13.0000i 0.448810i 0.974496 + 0.224405i \(0.0720438\pi\)
−0.974496 + 0.224405i \(0.927956\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 0 0
\(843\) 30.0000i 1.03325i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 6.00000i 0.206162i
\(848\) 0 0
\(849\) 3.00000 0.102960
\(850\) 0 0
\(851\) − 21.0000i − 0.719871i
\(852\) 0 0
\(853\) 10.0000i 0.342393i 0.985237 + 0.171197i \(0.0547634\pi\)
−0.985237 + 0.171197i \(0.945237\pi\)
\(854\) 0 0
\(855\) −6.00000 −0.205196
\(856\) 0 0
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 0 0
\(859\) 44.0000 1.50126 0.750630 0.660722i \(-0.229750\pi\)
0.750630 + 0.660722i \(0.229750\pi\)
\(860\) 0 0
\(861\) −63.0000 −2.14703
\(862\) 0 0
\(863\) − 16.0000i − 0.544646i −0.962206 0.272323i \(-0.912208\pi\)
0.962206 0.272323i \(-0.0877920\pi\)
\(864\) 0 0
\(865\) − 15.0000i − 0.510015i
\(866\) 0 0
\(867\) −96.0000 −3.26033
\(868\) 0 0
\(869\) − 24.0000i − 0.814144i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 66.0000i 2.23376i
\(874\) 0 0
\(875\) −3.00000 −0.101419
\(876\) 0 0
\(877\) − 5.00000i − 0.168838i −0.996430 0.0844190i \(-0.973097\pi\)
0.996430 0.0844190i \(-0.0269034\pi\)
\(878\) 0 0
\(879\) − 27.0000i − 0.910687i
\(880\) 0 0
\(881\) −11.0000 −0.370599 −0.185300 0.982682i \(-0.559326\pi\)
−0.185300 + 0.982682i \(0.559326\pi\)
\(882\) 0 0
\(883\) 44.0000 1.48072 0.740359 0.672212i \(-0.234656\pi\)
0.740359 + 0.672212i \(0.234656\pi\)
\(884\) 0 0
\(885\) −15.0000 −0.504219
\(886\) 0 0
\(887\) −19.0000 −0.637958 −0.318979 0.947762i \(-0.603340\pi\)
−0.318979 + 0.947762i \(0.603340\pi\)
\(888\) 0 0
\(889\) − 3.00000i − 0.100617i
\(890\) 0 0
\(891\) 27.0000i 0.904534i
\(892\) 0 0
\(893\) 8.00000 0.267710
\(894\) 0 0
\(895\) 19.0000i 0.635100i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 20.0000i − 0.667037i
\(900\) 0 0
\(901\) −42.0000 −1.39922
\(902\) 0 0
\(903\) − 81.0000i − 2.69551i
\(904\) 0 0
\(905\) − 14.0000i − 0.465376i
\(906\) 0 0
\(907\) −17.0000 −0.564476 −0.282238 0.959344i \(-0.591077\pi\)
−0.282238 + 0.959344i \(0.591077\pi\)
\(908\) 0 0
\(909\) 54.0000 1.79107
\(910\) 0 0
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 0 0
\(913\) 36.0000 1.19143
\(914\) 0 0
\(915\) − 15.0000i − 0.495885i
\(916\) 0 0
\(917\) 12.0000i 0.396275i
\(918\) 0 0
\(919\) −43.0000 −1.41844 −0.709220 0.704988i \(-0.750953\pi\)
−0.709220 + 0.704988i \(0.750953\pi\)
\(920\) 0 0
\(921\) − 84.0000i − 2.76789i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 3.00000i 0.0986394i
\(926\) 0 0
\(927\) 96.0000 3.15305
\(928\) 0 0
\(929\) 53.0000i 1.73887i 0.494044 + 0.869437i \(0.335518\pi\)
−0.494044 + 0.869437i \(0.664482\pi\)
\(930\) 0 0
\(931\) 2.00000i 0.0655474i
\(932\) 0 0
\(933\) −72.0000 −2.35717
\(934\) 0 0
\(935\) 21.0000 0.686773
\(936\) 0 0
\(937\) 50.0000 1.63343 0.816714 0.577042i \(-0.195793\pi\)
0.816714 + 0.577042i \(0.195793\pi\)
\(938\) 0 0
\(939\) 18.0000 0.587408
\(940\) 0 0
\(941\) − 42.0000i − 1.36916i −0.728937 0.684580i \(-0.759985\pi\)
0.728937 0.684580i \(-0.240015\pi\)
\(942\) 0 0
\(943\) − 49.0000i − 1.59566i
\(944\) 0 0
\(945\) −27.0000 −0.878310
\(946\) 0 0
\(947\) − 55.0000i − 1.78726i −0.448805 0.893630i \(-0.648150\pi\)
0.448805 0.893630i \(-0.351850\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) − 6.00000i − 0.194563i
\(952\) 0 0
\(953\) 31.0000 1.00419 0.502094 0.864813i \(-0.332563\pi\)
0.502094 + 0.864813i \(0.332563\pi\)
\(954\) 0 0
\(955\) 3.00000i 0.0970777i
\(956\) 0 0
\(957\) 45.0000i 1.45464i
\(958\) 0 0
\(959\) 9.00000 0.290625
\(960\) 0 0
\(961\) 15.0000 0.483871
\(962\) 0 0
\(963\) −18.0000 −0.580042
\(964\) 0 0
\(965\) −15.0000 −0.482867
\(966\) 0 0
\(967\) 48.0000i 1.54358i 0.635880 + 0.771788i \(0.280637\pi\)
−0.635880 + 0.771788i \(0.719363\pi\)
\(968\) 0 0
\(969\) 21.0000i 0.674617i
\(970\) 0 0
\(971\) −43.0000 −1.37994 −0.689968 0.723840i \(-0.742375\pi\)
−0.689968 + 0.723840i \(0.742375\pi\)
\(972\) 0 0
\(973\) 39.0000i 1.25028i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 3.00000i 0.0959785i 0.998848 + 0.0479893i \(0.0152813\pi\)
−0.998848 + 0.0479893i \(0.984719\pi\)
\(978\) 0 0
\(979\) −21.0000 −0.671163
\(980\) 0 0
\(981\) 84.0000i 2.68191i
\(982\) 0 0
\(983\) 36.0000i 1.14822i 0.818778 + 0.574111i \(0.194652\pi\)
−0.818778 + 0.574111i \(0.805348\pi\)
\(984\) 0 0
\(985\) 23.0000 0.732841
\(986\) 0 0
\(987\) 72.0000 2.29179
\(988\) 0 0
\(989\) 63.0000 2.00328
\(990\) 0 0
\(991\) −13.0000 −0.412959 −0.206479 0.978451i \(-0.566201\pi\)
−0.206479 + 0.978451i \(0.566201\pi\)
\(992\) 0 0
\(993\) 39.0000i 1.23763i
\(994\) 0 0
\(995\) 9.00000i 0.285319i
\(996\) 0 0
\(997\) 13.0000 0.411714 0.205857 0.978582i \(-0.434002\pi\)
0.205857 + 0.978582i \(0.434002\pi\)
\(998\) 0 0
\(999\) 27.0000i 0.854242i
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 3380.2.f.a.3041.1 2
13.5 odd 4 3380.2.a.a.1.1 1
13.7 odd 12 260.2.i.d.81.1 yes 2
13.8 odd 4 3380.2.a.b.1.1 1
13.11 odd 12 260.2.i.d.61.1 2
13.12 even 2 inner 3380.2.f.a.3041.2 2
39.11 even 12 2340.2.q.a.1621.1 2
39.20 even 12 2340.2.q.a.2161.1 2
52.7 even 12 1040.2.q.b.81.1 2
52.11 even 12 1040.2.q.b.321.1 2
65.7 even 12 1300.2.bb.e.549.2 4
65.24 odd 12 1300.2.i.a.1101.1 2
65.33 even 12 1300.2.bb.e.549.1 4
65.37 even 12 1300.2.bb.e.1049.1 4
65.59 odd 12 1300.2.i.a.601.1 2
65.63 even 12 1300.2.bb.e.1049.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
260.2.i.d.61.1 2 13.11 odd 12
260.2.i.d.81.1 yes 2 13.7 odd 12
1040.2.q.b.81.1 2 52.7 even 12
1040.2.q.b.321.1 2 52.11 even 12
1300.2.i.a.601.1 2 65.59 odd 12
1300.2.i.a.1101.1 2 65.24 odd 12
1300.2.bb.e.549.1 4 65.33 even 12
1300.2.bb.e.549.2 4 65.7 even 12
1300.2.bb.e.1049.1 4 65.37 even 12
1300.2.bb.e.1049.2 4 65.63 even 12
2340.2.q.a.1621.1 2 39.11 even 12
2340.2.q.a.2161.1 2 39.20 even 12
3380.2.a.a.1.1 1 13.5 odd 4
3380.2.a.b.1.1 1 13.8 odd 4
3380.2.f.a.3041.1 2 1.1 even 1 trivial
3380.2.f.a.3041.2 2 13.12 even 2 inner