Properties

Label 68.2.e.a.21.2
Level $68$
Weight $2$
Character 68.21
Analytic conductor $0.543$
Analytic rank $0$
Dimension $4$
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] = [68,2,Mod(13,68)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(68, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([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("68.13");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 68 = 2^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 68.e (of order \(4\), degree \(2\), 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: \(0.542982733745\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{13})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 7x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 21.2
Root \(-1.30278i\) of defining polynomial
Character \(\chi\) \(=\) 68.21
Dual form 68.2.e.a.13.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.30278 - 1.30278i) q^{3} +(-1.00000 + 1.00000i) q^{5} +(-1.30278 - 1.30278i) q^{7} -0.394449i q^{9} +O(q^{10})\) \(q+(1.30278 - 1.30278i) q^{3} +(-1.00000 + 1.00000i) q^{5} +(-1.30278 - 1.30278i) q^{7} -0.394449i q^{9} +(3.30278 + 3.30278i) q^{11} -4.60555 q^{13} +2.60555i q^{15} +(-3.60555 + 2.00000i) q^{17} -6.60555i q^{19} -3.39445 q^{21} +(-0.697224 - 0.697224i) q^{23} +3.00000i q^{25} +(3.39445 + 3.39445i) q^{27} +(5.60555 - 5.60555i) q^{29} +(0.697224 - 0.697224i) q^{31} +8.60555 q^{33} +2.60555 q^{35} +(3.00000 - 3.00000i) q^{37} +(-6.00000 + 6.00000i) q^{39} +(1.00000 + 1.00000i) q^{41} -10.6056i q^{43} +(0.394449 + 0.394449i) q^{45} -4.00000 q^{47} -3.60555i q^{49} +(-2.09167 + 7.30278i) q^{51} +9.21110i q^{53} -6.60555 q^{55} +(-8.60555 - 8.60555i) q^{57} -1.39445i q^{59} +(8.21110 + 8.21110i) q^{61} +(-0.513878 + 0.513878i) q^{63} +(4.60555 - 4.60555i) q^{65} -5.21110 q^{67} -1.81665 q^{69} +(-7.90833 + 7.90833i) q^{71} +(-7.00000 + 7.00000i) q^{73} +(3.90833 + 3.90833i) q^{75} -8.60555i q^{77} +(3.30278 + 3.30278i) q^{79} +10.0278 q^{81} -3.81665i q^{83} +(1.60555 - 5.60555i) q^{85} -14.6056i q^{87} -13.8167 q^{89} +(6.00000 + 6.00000i) q^{91} -1.81665i q^{93} +(6.60555 + 6.60555i) q^{95} +(-0.394449 + 0.394449i) q^{97} +(1.30278 - 1.30278i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} - 4 q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} - 4 q^{5} + 2 q^{7} + 6 q^{11} - 4 q^{13} - 28 q^{21} - 10 q^{23} + 28 q^{27} + 8 q^{29} + 10 q^{31} + 20 q^{33} - 4 q^{35} + 12 q^{37} - 24 q^{39} + 4 q^{41} + 16 q^{45} - 16 q^{47} - 30 q^{51} - 12 q^{55} - 20 q^{57} + 4 q^{61} + 34 q^{63} + 4 q^{65} + 8 q^{67} + 36 q^{69} - 10 q^{71} - 28 q^{73} - 6 q^{75} + 6 q^{79} - 32 q^{81} - 8 q^{85} - 12 q^{89} + 24 q^{91} + 12 q^{95} - 16 q^{97} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/68\mathbb{Z}\right)^\times\).

\(n\) \(35\) \(37\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{4}\right)\)

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
\(3\) 1.30278 1.30278i 0.752158 0.752158i −0.222724 0.974882i \(-0.571495\pi\)
0.974882 + 0.222724i \(0.0714948\pi\)
\(4\) 0 0
\(5\) −1.00000 + 1.00000i −0.447214 + 0.447214i −0.894427 0.447214i \(-0.852416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) −1.30278 1.30278i −0.492403 0.492403i 0.416660 0.909063i \(-0.363201\pi\)
−0.909063 + 0.416660i \(0.863201\pi\)
\(8\) 0 0
\(9\) 0.394449i 0.131483i
\(10\) 0 0
\(11\) 3.30278 + 3.30278i 0.995824 + 0.995824i 0.999991 0.00416699i \(-0.00132640\pi\)
−0.00416699 + 0.999991i \(0.501326\pi\)
\(12\) 0 0
\(13\) −4.60555 −1.27735 −0.638675 0.769477i \(-0.720517\pi\)
−0.638675 + 0.769477i \(0.720517\pi\)
\(14\) 0 0
\(15\) 2.60555i 0.672750i
\(16\) 0 0
\(17\) −3.60555 + 2.00000i −0.874475 + 0.485071i
\(18\) 0 0
\(19\) 6.60555i 1.51542i −0.652593 0.757709i \(-0.726319\pi\)
0.652593 0.757709i \(-0.273681\pi\)
\(20\) 0 0
\(21\) −3.39445 −0.740729
\(22\) 0 0
\(23\) −0.697224 0.697224i −0.145381 0.145381i 0.630670 0.776051i \(-0.282780\pi\)
−0.776051 + 0.630670i \(0.782780\pi\)
\(24\) 0 0
\(25\) 3.00000i 0.600000i
\(26\) 0 0
\(27\) 3.39445 + 3.39445i 0.653262 + 0.653262i
\(28\) 0 0
\(29\) 5.60555 5.60555i 1.04092 1.04092i 0.0417987 0.999126i \(-0.486691\pi\)
0.999126 0.0417987i \(-0.0133088\pi\)
\(30\) 0 0
\(31\) 0.697224 0.697224i 0.125225 0.125225i −0.641717 0.766942i \(-0.721777\pi\)
0.766942 + 0.641717i \(0.221777\pi\)
\(32\) 0 0
\(33\) 8.60555 1.49803
\(34\) 0 0
\(35\) 2.60555 0.440419
\(36\) 0 0
\(37\) 3.00000 3.00000i 0.493197 0.493197i −0.416115 0.909312i \(-0.636609\pi\)
0.909312 + 0.416115i \(0.136609\pi\)
\(38\) 0 0
\(39\) −6.00000 + 6.00000i −0.960769 + 0.960769i
\(40\) 0 0
\(41\) 1.00000 + 1.00000i 0.156174 + 0.156174i 0.780869 0.624695i \(-0.214777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) 10.6056i 1.61733i −0.588268 0.808666i \(-0.700190\pi\)
0.588268 0.808666i \(-0.299810\pi\)
\(44\) 0 0
\(45\) 0.394449 + 0.394449i 0.0588009 + 0.0588009i
\(46\) 0 0
\(47\) −4.00000 −0.583460 −0.291730 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) 0 0
\(49\) 3.60555i 0.515079i
\(50\) 0 0
\(51\) −2.09167 + 7.30278i −0.292893 + 1.02259i
\(52\) 0 0
\(53\) 9.21110i 1.26524i 0.774461 + 0.632621i \(0.218021\pi\)
−0.774461 + 0.632621i \(0.781979\pi\)
\(54\) 0 0
\(55\) −6.60555 −0.890692
\(56\) 0 0
\(57\) −8.60555 8.60555i −1.13983 1.13983i
\(58\) 0 0
\(59\) 1.39445i 0.181542i −0.995872 0.0907709i \(-0.971067\pi\)
0.995872 0.0907709i \(-0.0289331\pi\)
\(60\) 0 0
\(61\) 8.21110 + 8.21110i 1.05132 + 1.05132i 0.998610 + 0.0527143i \(0.0167873\pi\)
0.0527143 + 0.998610i \(0.483213\pi\)
\(62\) 0 0
\(63\) −0.513878 + 0.513878i −0.0647426 + 0.0647426i
\(64\) 0 0
\(65\) 4.60555 4.60555i 0.571248 0.571248i
\(66\) 0 0
\(67\) −5.21110 −0.636638 −0.318319 0.947984i \(-0.603118\pi\)
−0.318319 + 0.947984i \(0.603118\pi\)
\(68\) 0 0
\(69\) −1.81665 −0.218699
\(70\) 0 0
\(71\) −7.90833 + 7.90833i −0.938546 + 0.938546i −0.998218 0.0596723i \(-0.980994\pi\)
0.0596723 + 0.998218i \(0.480994\pi\)
\(72\) 0 0
\(73\) −7.00000 + 7.00000i −0.819288 + 0.819288i −0.986005 0.166717i \(-0.946683\pi\)
0.166717 + 0.986005i \(0.446683\pi\)
\(74\) 0 0
\(75\) 3.90833 + 3.90833i 0.451295 + 0.451295i
\(76\) 0 0
\(77\) 8.60555i 0.980694i
\(78\) 0 0
\(79\) 3.30278 + 3.30278i 0.371591 + 0.371591i 0.868057 0.496465i \(-0.165369\pi\)
−0.496465 + 0.868057i \(0.665369\pi\)
\(80\) 0 0
\(81\) 10.0278 1.11420
\(82\) 0 0
\(83\) 3.81665i 0.418932i −0.977816 0.209466i \(-0.932827\pi\)
0.977816 0.209466i \(-0.0671726\pi\)
\(84\) 0 0
\(85\) 1.60555 5.60555i 0.174146 0.608007i
\(86\) 0 0
\(87\) 14.6056i 1.56588i
\(88\) 0 0
\(89\) −13.8167 −1.46456 −0.732281 0.681002i \(-0.761544\pi\)
−0.732281 + 0.681002i \(0.761544\pi\)
\(90\) 0 0
\(91\) 6.00000 + 6.00000i 0.628971 + 0.628971i
\(92\) 0 0
\(93\) 1.81665i 0.188378i
\(94\) 0 0
\(95\) 6.60555 + 6.60555i 0.677715 + 0.677715i
\(96\) 0 0
\(97\) −0.394449 + 0.394449i −0.0400502 + 0.0400502i −0.726848 0.686798i \(-0.759016\pi\)
0.686798 + 0.726848i \(0.259016\pi\)
\(98\) 0 0
\(99\) 1.30278 1.30278i 0.130934 0.130934i
\(100\) 0 0
\(101\) 3.39445 0.337760 0.168880 0.985637i \(-0.445985\pi\)
0.168880 + 0.985637i \(0.445985\pi\)
\(102\) 0 0
\(103\) 1.21110 0.119333 0.0596667 0.998218i \(-0.480996\pi\)
0.0596667 + 0.998218i \(0.480996\pi\)
\(104\) 0 0
\(105\) 3.39445 3.39445i 0.331264 0.331264i
\(106\) 0 0
\(107\) 9.30278 9.30278i 0.899333 0.899333i −0.0960438 0.995377i \(-0.530619\pi\)
0.995377 + 0.0960438i \(0.0306189\pi\)
\(108\) 0 0
\(109\) 8.21110 + 8.21110i 0.786481 + 0.786481i 0.980916 0.194435i \(-0.0622872\pi\)
−0.194435 + 0.980916i \(0.562287\pi\)
\(110\) 0 0
\(111\) 7.81665i 0.741924i
\(112\) 0 0
\(113\) 2.39445 + 2.39445i 0.225251 + 0.225251i 0.810705 0.585454i \(-0.199084\pi\)
−0.585454 + 0.810705i \(0.699084\pi\)
\(114\) 0 0
\(115\) 1.39445 0.130033
\(116\) 0 0
\(117\) 1.81665i 0.167950i
\(118\) 0 0
\(119\) 7.30278 + 2.09167i 0.669444 + 0.191743i
\(120\) 0 0
\(121\) 10.8167i 0.983332i
\(122\) 0 0
\(123\) 2.60555 0.234935
\(124\) 0 0
\(125\) −8.00000 8.00000i −0.715542 0.715542i
\(126\) 0 0
\(127\) 6.60555i 0.586148i 0.956090 + 0.293074i \(0.0946782\pi\)
−0.956090 + 0.293074i \(0.905322\pi\)
\(128\) 0 0
\(129\) −13.8167 13.8167i −1.21649 1.21649i
\(130\) 0 0
\(131\) −2.69722 + 2.69722i −0.235658 + 0.235658i −0.815049 0.579392i \(-0.803290\pi\)
0.579392 + 0.815049i \(0.303290\pi\)
\(132\) 0 0
\(133\) −8.60555 + 8.60555i −0.746196 + 0.746196i
\(134\) 0 0
\(135\) −6.78890 −0.584295
\(136\) 0 0
\(137\) 4.60555 0.393479 0.196739 0.980456i \(-0.436965\pi\)
0.196739 + 0.980456i \(0.436965\pi\)
\(138\) 0 0
\(139\) −3.30278 + 3.30278i −0.280138 + 0.280138i −0.833164 0.553026i \(-0.813473\pi\)
0.553026 + 0.833164i \(0.313473\pi\)
\(140\) 0 0
\(141\) −5.21110 + 5.21110i −0.438854 + 0.438854i
\(142\) 0 0
\(143\) −15.2111 15.2111i −1.27202 1.27202i
\(144\) 0 0
\(145\) 11.2111i 0.931031i
\(146\) 0 0
\(147\) −4.69722 4.69722i −0.387421 0.387421i
\(148\) 0 0
\(149\) −20.4222 −1.67305 −0.836526 0.547927i \(-0.815417\pi\)
−0.836526 + 0.547927i \(0.815417\pi\)
\(150\) 0 0
\(151\) 2.60555i 0.212037i −0.994364 0.106018i \(-0.966190\pi\)
0.994364 0.106018i \(-0.0338102\pi\)
\(152\) 0 0
\(153\) 0.788897 + 1.42221i 0.0637786 + 0.114978i
\(154\) 0 0
\(155\) 1.39445i 0.112005i
\(156\) 0 0
\(157\) 20.4222 1.62987 0.814935 0.579553i \(-0.196773\pi\)
0.814935 + 0.579553i \(0.196773\pi\)
\(158\) 0 0
\(159\) 12.0000 + 12.0000i 0.951662 + 0.951662i
\(160\) 0 0
\(161\) 1.81665i 0.143172i
\(162\) 0 0
\(163\) −0.697224 0.697224i −0.0546108 0.0546108i 0.679274 0.733885i \(-0.262295\pi\)
−0.733885 + 0.679274i \(0.762295\pi\)
\(164\) 0 0
\(165\) −8.60555 + 8.60555i −0.669941 + 0.669941i
\(166\) 0 0
\(167\) 4.09167 4.09167i 0.316623 0.316623i −0.530845 0.847469i \(-0.678126\pi\)
0.847469 + 0.530845i \(0.178126\pi\)
\(168\) 0 0
\(169\) 8.21110 0.631623
\(170\) 0 0
\(171\) −2.60555 −0.199251
\(172\) 0 0
\(173\) 6.81665 6.81665i 0.518261 0.518261i −0.398784 0.917045i \(-0.630568\pi\)
0.917045 + 0.398784i \(0.130568\pi\)
\(174\) 0 0
\(175\) 3.90833 3.90833i 0.295442 0.295442i
\(176\) 0 0
\(177\) −1.81665 1.81665i −0.136548 0.136548i
\(178\) 0 0
\(179\) 21.0278i 1.57169i 0.618425 + 0.785844i \(0.287771\pi\)
−0.618425 + 0.785844i \(0.712229\pi\)
\(180\) 0 0
\(181\) −13.0000 13.0000i −0.966282 0.966282i 0.0331674 0.999450i \(-0.489441\pi\)
−0.999450 + 0.0331674i \(0.989441\pi\)
\(182\) 0 0
\(183\) 21.3944 1.58152
\(184\) 0 0
\(185\) 6.00000i 0.441129i
\(186\) 0 0
\(187\) −18.5139 5.30278i −1.35387 0.387777i
\(188\) 0 0
\(189\) 8.84441i 0.643336i
\(190\) 0 0
\(191\) 21.2111 1.53478 0.767391 0.641180i \(-0.221555\pi\)
0.767391 + 0.641180i \(0.221555\pi\)
\(192\) 0 0
\(193\) 6.21110 + 6.21110i 0.447085 + 0.447085i 0.894384 0.447299i \(-0.147614\pi\)
−0.447299 + 0.894384i \(0.647614\pi\)
\(194\) 0 0
\(195\) 12.0000i 0.859338i
\(196\) 0 0
\(197\) −10.3944 10.3944i −0.740574 0.740574i 0.232115 0.972688i \(-0.425436\pi\)
−0.972688 + 0.232115i \(0.925436\pi\)
\(198\) 0 0
\(199\) 15.1194 15.1194i 1.07179 1.07179i 0.0745723 0.997216i \(-0.476241\pi\)
0.997216 0.0745723i \(-0.0237592\pi\)
\(200\) 0 0
\(201\) −6.78890 + 6.78890i −0.478852 + 0.478852i
\(202\) 0 0
\(203\) −14.6056 −1.02511
\(204\) 0 0
\(205\) −2.00000 −0.139686
\(206\) 0 0
\(207\) −0.275019 + 0.275019i −0.0191152 + 0.0191152i
\(208\) 0 0
\(209\) 21.8167 21.8167i 1.50909 1.50909i
\(210\) 0 0
\(211\) −2.51388 2.51388i −0.173063 0.173063i 0.615261 0.788323i \(-0.289051\pi\)
−0.788323 + 0.615261i \(0.789051\pi\)
\(212\) 0 0
\(213\) 20.6056i 1.41187i
\(214\) 0 0
\(215\) 10.6056 + 10.6056i 0.723293 + 0.723293i
\(216\) 0 0
\(217\) −1.81665 −0.123322
\(218\) 0 0
\(219\) 18.2389i 1.23247i
\(220\) 0 0
\(221\) 16.6056 9.21110i 1.11701 0.619606i
\(222\) 0 0
\(223\) 3.81665i 0.255582i 0.991801 + 0.127791i \(0.0407887\pi\)
−0.991801 + 0.127791i \(0.959211\pi\)
\(224\) 0 0
\(225\) 1.18335 0.0788897
\(226\) 0 0
\(227\) 3.30278 + 3.30278i 0.219213 + 0.219213i 0.808167 0.588954i \(-0.200460\pi\)
−0.588954 + 0.808167i \(0.700460\pi\)
\(228\) 0 0
\(229\) 2.60555i 0.172180i −0.996287 0.0860898i \(-0.972563\pi\)
0.996287 0.0860898i \(-0.0274372\pi\)
\(230\) 0 0
\(231\) −11.2111 11.2111i −0.737636 0.737636i
\(232\) 0 0
\(233\) −1.60555 + 1.60555i −0.105183 + 0.105183i −0.757740 0.652557i \(-0.773696\pi\)
0.652557 + 0.757740i \(0.273696\pi\)
\(234\) 0 0
\(235\) 4.00000 4.00000i 0.260931 0.260931i
\(236\) 0 0
\(237\) 8.60555 0.558991
\(238\) 0 0
\(239\) −29.2111 −1.88951 −0.944755 0.327779i \(-0.893700\pi\)
−0.944755 + 0.327779i \(0.893700\pi\)
\(240\) 0 0
\(241\) 9.00000 9.00000i 0.579741 0.579741i −0.355091 0.934832i \(-0.615550\pi\)
0.934832 + 0.355091i \(0.115550\pi\)
\(242\) 0 0
\(243\) 2.88057 2.88057i 0.184789 0.184789i
\(244\) 0 0
\(245\) 3.60555 + 3.60555i 0.230350 + 0.230350i
\(246\) 0 0
\(247\) 30.4222i 1.93572i
\(248\) 0 0
\(249\) −4.97224 4.97224i −0.315103 0.315103i
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) 4.60555i 0.289549i
\(254\) 0 0
\(255\) −5.21110 9.39445i −0.326332 0.588303i
\(256\) 0 0
\(257\) 26.6056i 1.65961i 0.558054 + 0.829804i \(0.311548\pi\)
−0.558054 + 0.829804i \(0.688452\pi\)
\(258\) 0 0
\(259\) −7.81665 −0.485703
\(260\) 0 0
\(261\) −2.21110 2.21110i −0.136864 0.136864i
\(262\) 0 0
\(263\) 13.0278i 0.803326i −0.915788 0.401663i \(-0.868432\pi\)
0.915788 0.401663i \(-0.131568\pi\)
\(264\) 0 0
\(265\) −9.21110 9.21110i −0.565834 0.565834i
\(266\) 0 0
\(267\) −18.0000 + 18.0000i −1.10158 + 1.10158i
\(268\) 0 0
\(269\) −11.4222 + 11.4222i −0.696424 + 0.696424i −0.963637 0.267213i \(-0.913897\pi\)
0.267213 + 0.963637i \(0.413897\pi\)
\(270\) 0 0
\(271\) −13.2111 −0.802517 −0.401259 0.915965i \(-0.631427\pi\)
−0.401259 + 0.915965i \(0.631427\pi\)
\(272\) 0 0
\(273\) 15.6333 0.946171
\(274\) 0 0
\(275\) −9.90833 + 9.90833i −0.597495 + 0.597495i
\(276\) 0 0
\(277\) −3.60555 + 3.60555i −0.216637 + 0.216637i −0.807079 0.590443i \(-0.798953\pi\)
0.590443 + 0.807079i \(0.298953\pi\)
\(278\) 0 0
\(279\) −0.275019 0.275019i −0.0164650 0.0164650i
\(280\) 0 0
\(281\) 10.4222i 0.621737i −0.950453 0.310868i \(-0.899380\pi\)
0.950453 0.310868i \(-0.100620\pi\)
\(282\) 0 0
\(283\) −23.1194 23.1194i −1.37431 1.37431i −0.853946 0.520361i \(-0.825797\pi\)
−0.520361 0.853946i \(-0.674203\pi\)
\(284\) 0 0
\(285\) 17.2111 1.01950
\(286\) 0 0
\(287\) 2.60555i 0.153801i
\(288\) 0 0
\(289\) 9.00000 14.4222i 0.529412 0.848365i
\(290\) 0 0
\(291\) 1.02776i 0.0602481i
\(292\) 0 0
\(293\) 21.6333 1.26383 0.631916 0.775037i \(-0.282269\pi\)
0.631916 + 0.775037i \(0.282269\pi\)
\(294\) 0 0
\(295\) 1.39445 + 1.39445i 0.0811879 + 0.0811879i
\(296\) 0 0
\(297\) 22.4222i 1.30107i
\(298\) 0 0
\(299\) 3.21110 + 3.21110i 0.185703 + 0.185703i
\(300\) 0 0
\(301\) −13.8167 + 13.8167i −0.796379 + 0.796379i
\(302\) 0 0
\(303\) 4.42221 4.42221i 0.254049 0.254049i
\(304\) 0 0
\(305\) −16.4222 −0.940333
\(306\) 0 0
\(307\) 6.78890 0.387463 0.193731 0.981055i \(-0.437941\pi\)
0.193731 + 0.981055i \(0.437941\pi\)
\(308\) 0 0
\(309\) 1.57779 1.57779i 0.0897576 0.0897576i
\(310\) 0 0
\(311\) 8.09167 8.09167i 0.458837 0.458837i −0.439437 0.898274i \(-0.644822\pi\)
0.898274 + 0.439437i \(0.144822\pi\)
\(312\) 0 0
\(313\) −8.21110 8.21110i −0.464119 0.464119i 0.435884 0.900003i \(-0.356436\pi\)
−0.900003 + 0.435884i \(0.856436\pi\)
\(314\) 0 0
\(315\) 1.02776i 0.0579075i
\(316\) 0 0
\(317\) 12.3944 + 12.3944i 0.696142 + 0.696142i 0.963576 0.267434i \(-0.0861759\pi\)
−0.267434 + 0.963576i \(0.586176\pi\)
\(318\) 0 0
\(319\) 37.0278 2.07316
\(320\) 0 0
\(321\) 24.2389i 1.35288i
\(322\) 0 0
\(323\) 13.2111 + 23.8167i 0.735085 + 1.32519i
\(324\) 0 0
\(325\) 13.8167i 0.766410i
\(326\) 0 0
\(327\) 21.3944 1.18312
\(328\) 0 0
\(329\) 5.21110 + 5.21110i 0.287297 + 0.287297i
\(330\) 0 0
\(331\) 10.6056i 0.582934i −0.956581 0.291467i \(-0.905857\pi\)
0.956581 0.291467i \(-0.0941433\pi\)
\(332\) 0 0
\(333\) −1.18335 1.18335i −0.0648470 0.0648470i
\(334\) 0 0
\(335\) 5.21110 5.21110i 0.284713 0.284713i
\(336\) 0 0
\(337\) −1.78890 + 1.78890i −0.0974475 + 0.0974475i −0.754150 0.656702i \(-0.771951\pi\)
0.656702 + 0.754150i \(0.271951\pi\)
\(338\) 0 0
\(339\) 6.23886 0.338848
\(340\) 0 0
\(341\) 4.60555 0.249405
\(342\) 0 0
\(343\) −13.8167 + 13.8167i −0.746029 + 0.746029i
\(344\) 0 0
\(345\) 1.81665 1.81665i 0.0978054 0.0978054i
\(346\) 0 0
\(347\) 12.5139 + 12.5139i 0.671780 + 0.671780i 0.958126 0.286346i \(-0.0924407\pi\)
−0.286346 + 0.958126i \(0.592441\pi\)
\(348\) 0 0
\(349\) 19.6333i 1.05095i −0.850810 0.525473i \(-0.823888\pi\)
0.850810 0.525473i \(-0.176112\pi\)
\(350\) 0 0
\(351\) −15.6333 15.6333i −0.834444 0.834444i
\(352\) 0 0
\(353\) −16.7889 −0.893583 −0.446791 0.894638i \(-0.647433\pi\)
−0.446791 + 0.894638i \(0.647433\pi\)
\(354\) 0 0
\(355\) 15.8167i 0.839461i
\(356\) 0 0
\(357\) 12.2389 6.78890i 0.647749 0.359307i
\(358\) 0 0
\(359\) 9.39445i 0.495820i 0.968783 + 0.247910i \(0.0797437\pi\)
−0.968783 + 0.247910i \(0.920256\pi\)
\(360\) 0 0
\(361\) −24.6333 −1.29649
\(362\) 0 0
\(363\) 14.0917 + 14.0917i 0.739621 + 0.739621i
\(364\) 0 0
\(365\) 14.0000i 0.732793i
\(366\) 0 0
\(367\) 3.90833 + 3.90833i 0.204013 + 0.204013i 0.801717 0.597704i \(-0.203920\pi\)
−0.597704 + 0.801717i \(0.703920\pi\)
\(368\) 0 0
\(369\) 0.394449 0.394449i 0.0205342 0.0205342i
\(370\) 0 0
\(371\) 12.0000 12.0000i 0.623009 0.623009i
\(372\) 0 0
\(373\) −15.0278 −0.778108 −0.389054 0.921215i \(-0.627198\pi\)
−0.389054 + 0.921215i \(0.627198\pi\)
\(374\) 0 0
\(375\) −20.8444 −1.07640
\(376\) 0 0
\(377\) −25.8167 + 25.8167i −1.32963 + 1.32963i
\(378\) 0 0
\(379\) 11.7250 11.7250i 0.602272 0.602272i −0.338643 0.940915i \(-0.609968\pi\)
0.940915 + 0.338643i \(0.109968\pi\)
\(380\) 0 0
\(381\) 8.60555 + 8.60555i 0.440876 + 0.440876i
\(382\) 0 0
\(383\) 26.6056i 1.35948i −0.733453 0.679740i \(-0.762093\pi\)
0.733453 0.679740i \(-0.237907\pi\)
\(384\) 0 0
\(385\) 8.60555 + 8.60555i 0.438580 + 0.438580i
\(386\) 0 0
\(387\) −4.18335 −0.212651
\(388\) 0 0
\(389\) 31.8167i 1.61317i 0.591119 + 0.806584i \(0.298686\pi\)
−0.591119 + 0.806584i \(0.701314\pi\)
\(390\) 0 0
\(391\) 3.90833 + 1.11943i 0.197653 + 0.0566120i
\(392\) 0 0
\(393\) 7.02776i 0.354503i
\(394\) 0 0
\(395\) −6.60555 −0.332361
\(396\) 0 0
\(397\) −9.00000 9.00000i −0.451697 0.451697i 0.444220 0.895918i \(-0.353481\pi\)
−0.895918 + 0.444220i \(0.853481\pi\)
\(398\) 0 0
\(399\) 22.4222i 1.12251i
\(400\) 0 0
\(401\) −7.00000 7.00000i −0.349563 0.349563i 0.510384 0.859947i \(-0.329503\pi\)
−0.859947 + 0.510384i \(0.829503\pi\)
\(402\) 0 0
\(403\) −3.21110 + 3.21110i −0.159956 + 0.159956i
\(404\) 0 0
\(405\) −10.0278 + 10.0278i −0.498283 + 0.498283i
\(406\) 0 0
\(407\) 19.8167 0.982275
\(408\) 0 0
\(409\) 8.78890 0.434583 0.217292 0.976107i \(-0.430278\pi\)
0.217292 + 0.976107i \(0.430278\pi\)
\(410\) 0 0
\(411\) 6.00000 6.00000i 0.295958 0.295958i
\(412\) 0 0
\(413\) −1.81665 + 1.81665i −0.0893917 + 0.0893917i
\(414\) 0 0
\(415\) 3.81665 + 3.81665i 0.187352 + 0.187352i
\(416\) 0 0
\(417\) 8.60555i 0.421416i
\(418\) 0 0
\(419\) −1.30278 1.30278i −0.0636448 0.0636448i 0.674568 0.738213i \(-0.264330\pi\)
−0.738213 + 0.674568i \(0.764330\pi\)
\(420\) 0 0
\(421\) 11.3944 0.555331 0.277666 0.960678i \(-0.410439\pi\)
0.277666 + 0.960678i \(0.410439\pi\)
\(422\) 0 0
\(423\) 1.57779i 0.0767150i
\(424\) 0 0
\(425\) −6.00000 10.8167i −0.291043 0.524685i
\(426\) 0 0
\(427\) 21.3944i 1.03535i
\(428\) 0 0
\(429\) −39.6333 −1.91351
\(430\) 0 0
\(431\) −27.1194 27.1194i −1.30630 1.30630i −0.924067 0.382230i \(-0.875156\pi\)
−0.382230 0.924067i \(-0.624844\pi\)
\(432\) 0 0
\(433\) 21.0278i 1.01053i 0.862964 + 0.505265i \(0.168605\pi\)
−0.862964 + 0.505265i \(0.831395\pi\)
\(434\) 0 0
\(435\) 14.6056 + 14.6056i 0.700283 + 0.700283i
\(436\) 0 0
\(437\) −4.60555 + 4.60555i −0.220313 + 0.220313i
\(438\) 0 0
\(439\) −22.6972 + 22.6972i −1.08328 + 1.08328i −0.0870779 + 0.996202i \(0.527753\pi\)
−0.996202 + 0.0870779i \(0.972247\pi\)
\(440\) 0 0
\(441\) −1.42221 −0.0677241
\(442\) 0 0
\(443\) −2.78890 −0.132505 −0.0662523 0.997803i \(-0.521104\pi\)
−0.0662523 + 0.997803i \(0.521104\pi\)
\(444\) 0 0
\(445\) 13.8167 13.8167i 0.654972 0.654972i
\(446\) 0 0
\(447\) −26.6056 + 26.6056i −1.25840 + 1.25840i
\(448\) 0 0
\(449\) 4.81665 + 4.81665i 0.227312 + 0.227312i 0.811569 0.584257i \(-0.198614\pi\)
−0.584257 + 0.811569i \(0.698614\pi\)
\(450\) 0 0
\(451\) 6.60555i 0.311043i
\(452\) 0 0
\(453\) −3.39445 3.39445i −0.159485 0.159485i
\(454\) 0 0
\(455\) −12.0000 −0.562569
\(456\) 0 0
\(457\) 29.3944i 1.37501i −0.726178 0.687507i \(-0.758705\pi\)
0.726178 0.687507i \(-0.241295\pi\)
\(458\) 0 0
\(459\) −19.0278 5.44996i −0.888140 0.254382i
\(460\) 0 0
\(461\) 14.7889i 0.688788i 0.938825 + 0.344394i \(0.111916\pi\)
−0.938825 + 0.344394i \(0.888084\pi\)
\(462\) 0 0
\(463\) 3.63331 0.168854 0.0844271 0.996430i \(-0.473094\pi\)
0.0844271 + 0.996430i \(0.473094\pi\)
\(464\) 0 0
\(465\) 1.81665 + 1.81665i 0.0842453 + 0.0842453i
\(466\) 0 0
\(467\) 34.2389i 1.58439i −0.610271 0.792193i \(-0.708939\pi\)
0.610271 0.792193i \(-0.291061\pi\)
\(468\) 0 0
\(469\) 6.78890 + 6.78890i 0.313482 + 0.313482i
\(470\) 0 0
\(471\) 26.6056 26.6056i 1.22592 1.22592i
\(472\) 0 0
\(473\) 35.0278 35.0278i 1.61058 1.61058i
\(474\) 0 0
\(475\) 19.8167 0.909250
\(476\) 0 0
\(477\) 3.63331 0.166358
\(478\) 0 0
\(479\) 8.69722 8.69722i 0.397386 0.397386i −0.479924 0.877310i \(-0.659336\pi\)
0.877310 + 0.479924i \(0.159336\pi\)
\(480\) 0 0
\(481\) −13.8167 + 13.8167i −0.629985 + 0.629985i
\(482\) 0 0
\(483\) 2.36669 + 2.36669i 0.107688 + 0.107688i
\(484\) 0 0
\(485\) 0.788897i 0.0358220i
\(486\) 0 0
\(487\) 22.3305 + 22.3305i 1.01189 + 1.01189i 0.999928 + 0.0119646i \(0.00380853\pi\)
0.0119646 + 0.999928i \(0.496191\pi\)
\(488\) 0 0
\(489\) −1.81665 −0.0821519
\(490\) 0 0
\(491\) 11.8167i 0.533278i 0.963796 + 0.266639i \(0.0859132\pi\)
−0.963796 + 0.266639i \(0.914087\pi\)
\(492\) 0 0
\(493\) −9.00000 + 31.4222i −0.405340 + 1.41518i
\(494\) 0 0
\(495\) 2.60555i 0.117111i
\(496\) 0 0
\(497\) 20.6056 0.924285
\(498\) 0 0
\(499\) 6.69722 + 6.69722i 0.299809 + 0.299809i 0.840939 0.541130i \(-0.182003\pi\)
−0.541130 + 0.840939i \(0.682003\pi\)
\(500\) 0 0
\(501\) 10.6611i 0.476301i
\(502\) 0 0
\(503\) 9.48612 + 9.48612i 0.422965 + 0.422965i 0.886223 0.463258i \(-0.153320\pi\)
−0.463258 + 0.886223i \(0.653320\pi\)
\(504\) 0 0
\(505\) −3.39445 + 3.39445i −0.151051 + 0.151051i
\(506\) 0 0
\(507\) 10.6972 10.6972i 0.475080 0.475080i
\(508\) 0 0
\(509\) 11.2111 0.496923 0.248462 0.968642i \(-0.420075\pi\)
0.248462 + 0.968642i \(0.420075\pi\)
\(510\) 0 0
\(511\) 18.2389 0.806840
\(512\) 0 0
\(513\) 22.4222 22.4222i 0.989965 0.989965i
\(514\) 0 0
\(515\) −1.21110 + 1.21110i −0.0533676 + 0.0533676i
\(516\) 0 0
\(517\) −13.2111 13.2111i −0.581024 0.581024i
\(518\) 0 0
\(519\) 17.7611i 0.779628i
\(520\) 0 0
\(521\) 13.0000 + 13.0000i 0.569540 + 0.569540i 0.932000 0.362459i \(-0.118063\pi\)
−0.362459 + 0.932000i \(0.618063\pi\)
\(522\) 0 0
\(523\) −12.0000 −0.524723 −0.262362 0.964970i \(-0.584501\pi\)
−0.262362 + 0.964970i \(0.584501\pi\)
\(524\) 0 0
\(525\) 10.1833i 0.444438i
\(526\) 0 0
\(527\) −1.11943 + 3.90833i −0.0487631 + 0.170249i
\(528\) 0 0
\(529\) 22.0278i 0.957729i
\(530\) 0 0
\(531\) −0.550039 −0.0238696
\(532\) 0 0
\(533\) −4.60555 4.60555i −0.199489 0.199489i
\(534\) 0 0
\(535\) 18.6056i 0.804388i
\(536\) 0 0
\(537\) 27.3944 + 27.3944i 1.18216 + 1.18216i
\(538\) 0 0
\(539\) 11.9083 11.9083i 0.512928 0.512928i
\(540\) 0 0
\(541\) 3.00000 3.00000i 0.128980 0.128980i −0.639670 0.768650i \(-0.720929\pi\)
0.768650 + 0.639670i \(0.220929\pi\)
\(542\) 0 0
\(543\) −33.8722 −1.45359
\(544\) 0 0
\(545\) −16.4222 −0.703450
\(546\) 0 0
\(547\) 6.51388 6.51388i 0.278513 0.278513i −0.554002 0.832515i \(-0.686900\pi\)
0.832515 + 0.554002i \(0.186900\pi\)
\(548\) 0 0
\(549\) 3.23886 3.23886i 0.138231 0.138231i
\(550\) 0 0
\(551\) −37.0278 37.0278i −1.57744 1.57744i
\(552\) 0 0
\(553\) 8.60555i 0.365945i
\(554\) 0 0
\(555\) 7.81665 + 7.81665i 0.331798 + 0.331798i
\(556\) 0 0
\(557\) −28.6056 −1.21206 −0.606028 0.795443i \(-0.707238\pi\)
−0.606028 + 0.795443i \(0.707238\pi\)
\(558\) 0 0
\(559\) 48.8444i 2.06590i
\(560\) 0 0
\(561\) −31.0278 + 17.2111i −1.30999 + 0.726653i
\(562\) 0 0
\(563\) 13.3944i 0.564509i 0.959340 + 0.282254i \(0.0910822\pi\)
−0.959340 + 0.282254i \(0.908918\pi\)
\(564\) 0 0
\(565\) −4.78890 −0.201470
\(566\) 0 0
\(567\) −13.0639 13.0639i −0.548633 0.548633i
\(568\) 0 0
\(569\) 1.57779i 0.0661446i −0.999453 0.0330723i \(-0.989471\pi\)
0.999453 0.0330723i \(-0.0105292\pi\)
\(570\) 0 0
\(571\) −1.30278 1.30278i −0.0545195 0.0545195i 0.679321 0.733841i \(-0.262274\pi\)
−0.733841 + 0.679321i \(0.762274\pi\)
\(572\) 0 0
\(573\) 27.6333 27.6333i 1.15440 1.15440i
\(574\) 0 0
\(575\) 2.09167 2.09167i 0.0872288 0.0872288i
\(576\) 0 0
\(577\) 33.4500 1.39254 0.696270 0.717780i \(-0.254842\pi\)
0.696270 + 0.717780i \(0.254842\pi\)
\(578\) 0 0
\(579\) 16.1833 0.672557
\(580\) 0 0
\(581\) −4.97224 + 4.97224i −0.206283 + 0.206283i
\(582\) 0 0
\(583\) −30.4222 + 30.4222i −1.25996 + 1.25996i
\(584\) 0 0
\(585\) −1.81665 1.81665i −0.0751094 0.0751094i
\(586\) 0 0
\(587\) 2.60555i 0.107543i 0.998553 + 0.0537713i \(0.0171242\pi\)
−0.998553 + 0.0537713i \(0.982876\pi\)
\(588\) 0 0
\(589\) −4.60555 4.60555i −0.189768 0.189768i
\(590\) 0 0
\(591\) −27.0833 −1.11406
\(592\) 0 0
\(593\) 30.4222i 1.24929i 0.780909 + 0.624645i \(0.214756\pi\)
−0.780909 + 0.624645i \(0.785244\pi\)
\(594\) 0 0
\(595\) −9.39445 + 5.21110i −0.385135 + 0.213634i
\(596\) 0 0
\(597\) 39.3944i 1.61231i
\(598\) 0 0
\(599\) −46.0555 −1.88178 −0.940889 0.338716i \(-0.890007\pi\)
−0.940889 + 0.338716i \(0.890007\pi\)
\(600\) 0 0
\(601\) 10.2111 + 10.2111i 0.416520 + 0.416520i 0.884002 0.467483i \(-0.154839\pi\)
−0.467483 + 0.884002i \(0.654839\pi\)
\(602\) 0 0
\(603\) 2.05551i 0.0837070i
\(604\) 0 0
\(605\) −10.8167 10.8167i −0.439760 0.439760i
\(606\) 0 0
\(607\) −8.51388 + 8.51388i −0.345568 + 0.345568i −0.858456 0.512888i \(-0.828576\pi\)
0.512888 + 0.858456i \(0.328576\pi\)
\(608\) 0 0
\(609\) −19.0278 + 19.0278i −0.771044 + 0.771044i
\(610\) 0 0
\(611\) 18.4222 0.745283
\(612\) 0 0
\(613\) 11.5778 0.467623 0.233811 0.972282i \(-0.424880\pi\)
0.233811 + 0.972282i \(0.424880\pi\)
\(614\) 0 0
\(615\) −2.60555 + 2.60555i −0.105066 + 0.105066i
\(616\) 0 0
\(617\) −5.60555 + 5.60555i −0.225671 + 0.225671i −0.810881 0.585210i \(-0.801012\pi\)
0.585210 + 0.810881i \(0.301012\pi\)
\(618\) 0 0
\(619\) 32.5139 + 32.5139i 1.30684 + 1.30684i 0.923681 + 0.383162i \(0.125165\pi\)
0.383162 + 0.923681i \(0.374835\pi\)
\(620\) 0 0
\(621\) 4.73338i 0.189944i
\(622\) 0 0
\(623\) 18.0000 + 18.0000i 0.721155 + 0.721155i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 56.8444i 2.27015i
\(628\) 0 0
\(629\) −4.81665 + 16.8167i −0.192053 + 0.670524i
\(630\) 0 0
\(631\) 26.2389i 1.04455i −0.852776 0.522276i \(-0.825083\pi\)
0.852776 0.522276i \(-0.174917\pi\)
\(632\) 0 0
\(633\) −6.55004 −0.260341
\(634\) 0 0
\(635\) −6.60555 6.60555i −0.262133 0.262133i
\(636\) 0 0
\(637\) 16.6056i 0.657936i
\(638\) 0 0
\(639\) 3.11943 + 3.11943i 0.123403 + 0.123403i
\(640\) 0 0
\(641\) 9.18335 9.18335i 0.362720 0.362720i −0.502093 0.864814i \(-0.667437\pi\)
0.864814 + 0.502093i \(0.167437\pi\)
\(642\) 0 0
\(643\) −21.1194 + 21.1194i −0.832869 + 0.832869i −0.987908 0.155039i \(-0.950450\pi\)
0.155039 + 0.987908i \(0.450450\pi\)
\(644\) 0 0
\(645\) 27.6333 1.08806
\(646\) 0 0
\(647\) −2.78890 −0.109643 −0.0548214 0.998496i \(-0.517459\pi\)
−0.0548214 + 0.998496i \(0.517459\pi\)
\(648\) 0 0
\(649\) 4.60555 4.60555i 0.180784 0.180784i
\(650\) 0 0
\(651\) −2.36669 + 2.36669i −0.0927580 + 0.0927580i
\(652\) 0 0
\(653\) 3.18335 + 3.18335i 0.124574 + 0.124574i 0.766645 0.642071i \(-0.221925\pi\)
−0.642071 + 0.766645i \(0.721925\pi\)
\(654\) 0 0
\(655\) 5.39445i 0.210779i
\(656\) 0 0
\(657\) 2.76114 + 2.76114i 0.107722 + 0.107722i
\(658\) 0 0
\(659\) −32.0000 −1.24654 −0.623272 0.782006i \(-0.714197\pi\)
−0.623272 + 0.782006i \(0.714197\pi\)
\(660\) 0 0
\(661\) 13.2111i 0.513852i 0.966431 + 0.256926i \(0.0827097\pi\)
−0.966431 + 0.256926i \(0.917290\pi\)
\(662\) 0 0
\(663\) 9.63331 33.6333i 0.374127 1.30621i
\(664\) 0 0
\(665\) 17.2111i 0.667418i
\(666\) 0 0
\(667\) −7.81665 −0.302662
\(668\) 0 0
\(669\) 4.97224 + 4.97224i 0.192238 + 0.192238i
\(670\) 0 0
\(671\) 54.2389i 2.09387i
\(672\) 0 0
\(673\) −1.60555 1.60555i −0.0618895 0.0618895i 0.675485 0.737374i \(-0.263934\pi\)
−0.737374 + 0.675485i \(0.763934\pi\)
\(674\) 0 0
\(675\) −10.1833 + 10.1833i −0.391957 + 0.391957i
\(676\) 0 0
\(677\) 20.0278 20.0278i 0.769729 0.769729i −0.208329 0.978059i \(-0.566803\pi\)
0.978059 + 0.208329i \(0.0668026\pi\)
\(678\) 0 0
\(679\) 1.02776 0.0394417
\(680\) 0 0
\(681\) 8.60555 0.329765
\(682\) 0 0
\(683\) 31.7250 31.7250i 1.21392 1.21392i 0.244197 0.969726i \(-0.421476\pi\)
0.969726 0.244197i \(-0.0785244\pi\)
\(684\) 0 0
\(685\) −4.60555 + 4.60555i −0.175969 + 0.175969i
\(686\) 0 0
\(687\) −3.39445 3.39445i −0.129506 0.129506i
\(688\) 0 0
\(689\) 42.4222i 1.61616i
\(690\) 0 0
\(691\) 12.5139 + 12.5139i 0.476050 + 0.476050i 0.903866 0.427816i \(-0.140717\pi\)
−0.427816 + 0.903866i \(0.640717\pi\)
\(692\) 0 0
\(693\) −3.39445 −0.128944
\(694\) 0 0
\(695\) 6.60555i 0.250563i
\(696\) 0 0
\(697\) −5.60555 1.60555i −0.212325 0.0608146i
\(698\) 0 0
\(699\) 4.18335i 0.158229i
\(700\) 0 0
\(701\) 13.8167 0.521848 0.260924 0.965359i \(-0.415973\pi\)
0.260924 + 0.965359i \(0.415973\pi\)
\(702\) 0 0
\(703\) −19.8167 19.8167i −0.747399 0.747399i
\(704\) 0 0
\(705\) 10.4222i 0.392523i
\(706\) 0 0
\(707\) −4.42221 4.42221i −0.166314 0.166314i
\(708\) 0 0
\(709\) 21.6056 21.6056i 0.811414 0.811414i −0.173432 0.984846i \(-0.555486\pi\)
0.984846 + 0.173432i \(0.0554858\pi\)
\(710\) 0 0
\(711\) 1.30278 1.30278i 0.0488579 0.0488579i
\(712\) 0 0
\(713\) −0.972244 −0.0364108
\(714\) 0 0
\(715\) 30.4222 1.13773
\(716\) 0 0
\(717\) −38.0555 + 38.0555i −1.42121 + 1.42121i
\(718\) 0 0
\(719\) 27.7250 27.7250i 1.03397 1.03397i 0.0345649 0.999402i \(-0.488995\pi\)
0.999402 0.0345649i \(-0.0110045\pi\)
\(720\) 0 0
\(721\) −1.57779 1.57779i −0.0587602 0.0587602i
\(722\) 0 0
\(723\) 23.4500i 0.872113i
\(724\) 0 0
\(725\) 16.8167 + 16.8167i 0.624555 + 0.624555i
\(726\) 0 0
\(727\) 22.4222 0.831594 0.415797 0.909458i \(-0.363503\pi\)
0.415797 + 0.909458i \(0.363503\pi\)
\(728\) 0 0
\(729\) 22.5778i 0.836215i
\(730\) 0 0
\(731\) 21.2111 + 38.2389i 0.784521 + 1.41432i
\(732\) 0 0
\(733\) 26.0555i 0.962382i 0.876616 + 0.481191i \(0.159796\pi\)
−0.876616 + 0.481191i \(0.840204\pi\)
\(734\) 0 0
\(735\) 9.39445 0.346519
\(736\) 0 0
\(737\) −17.2111 17.2111i −0.633979 0.633979i
\(738\) 0 0
\(739\) 44.6611i 1.64288i −0.570292 0.821442i \(-0.693170\pi\)
0.570292 0.821442i \(-0.306830\pi\)
\(740\) 0 0
\(741\) 39.6333 + 39.6333i 1.45597 + 1.45597i
\(742\) 0 0
\(743\) −17.7250 + 17.7250i −0.650266 + 0.650266i −0.953057 0.302791i \(-0.902082\pi\)
0.302791 + 0.953057i \(0.402082\pi\)
\(744\) 0 0
\(745\) 20.4222 20.4222i 0.748212 0.748212i
\(746\) 0 0
\(747\) −1.50547 −0.0550824
\(748\) 0 0
\(749\) −24.2389 −0.885669
\(750\) 0 0
\(751\) 5.30278 5.30278i 0.193501 0.193501i −0.603706 0.797207i \(-0.706310\pi\)
0.797207 + 0.603706i \(0.206310\pi\)
\(752\) 0 0
\(753\) −15.6333 + 15.6333i −0.569709 + 0.569709i
\(754\) 0 0
\(755\) 2.60555 + 2.60555i 0.0948257 + 0.0948257i
\(756\) 0 0
\(757\) 2.97224i 0.108028i 0.998540 + 0.0540140i \(0.0172016\pi\)
−0.998540 + 0.0540140i \(0.982798\pi\)
\(758\) 0 0
\(759\) −6.00000 6.00000i −0.217786 0.217786i
\(760\) 0 0
\(761\) 33.4500 1.21256 0.606280 0.795251i \(-0.292661\pi\)
0.606280 + 0.795251i \(0.292661\pi\)
\(762\) 0 0
\(763\) 21.3944i 0.774531i
\(764\) 0 0
\(765\) −2.21110 0.633308i −0.0799426 0.0228973i
\(766\) 0 0
\(767\) 6.42221i 0.231892i
\(768\) 0 0
\(769\) −11.3944 −0.410894 −0.205447 0.978668i \(-0.565865\pi\)
−0.205447 + 0.978668i \(0.565865\pi\)
\(770\) 0 0
\(771\) 34.6611 + 34.6611i 1.24829 + 1.24829i
\(772\) 0 0
\(773\) 21.3944i 0.769505i −0.923020 0.384752i \(-0.874287\pi\)
0.923020 0.384752i \(-0.125713\pi\)
\(774\) 0 0
\(775\) 2.09167 + 2.09167i 0.0751351 + 0.0751351i
\(776\) 0 0
\(777\) −10.1833 + 10.1833i −0.365326 + 0.365326i
\(778\) 0 0
\(779\) 6.60555 6.60555i 0.236668 0.236668i
\(780\) 0 0
\(781\) −52.2389 −1.86925
\(782\) 0 0
\(783\) 38.0555 1.35999
\(784\) 0 0
\(785\) −20.4222 + 20.4222i −0.728900 + 0.728900i
\(786\) 0 0
\(787\) −17.7250 + 17.7250i −0.631827 + 0.631827i −0.948526 0.316699i \(-0.897426\pi\)
0.316699 + 0.948526i \(0.397426\pi\)
\(788\) 0 0
\(789\) −16.9722 16.9722i −0.604228 0.604228i
\(790\) 0 0
\(791\) 6.23886i 0.221828i
\(792\) 0 0
\(793\) −37.8167 37.8167i −1.34291 1.34291i
\(794\) 0 0
\(795\) −24.0000 −0.851192
\(796\) 0 0
\(797\) 39.6333i 1.40388i −0.712234 0.701942i \(-0.752317\pi\)
0.712234 0.701942i \(-0.247683\pi\)
\(798\) 0 0
\(799\) 14.4222 8.00000i 0.510221 0.283020i
\(800\) 0 0
\(801\) 5.44996i 0.192565i
\(802\) 0 0
\(803\) −46.2389 −1.63173
\(804\) 0 0
\(805\) −1.81665 1.81665i −0.0640286 0.0640286i
\(806\) 0 0
\(807\) 29.7611i 1.04764i
\(808\) 0 0
\(809\) 0.816654 + 0.816654i 0.0287120 + 0.0287120i 0.721317 0.692605i \(-0.243537\pi\)
−0.692605 + 0.721317i \(0.743537\pi\)
\(810\) 0 0
\(811\) −21.7250 + 21.7250i −0.762867 + 0.762867i −0.976840 0.213972i \(-0.931360\pi\)
0.213972 + 0.976840i \(0.431360\pi\)
\(812\) 0 0
\(813\) −17.2111 + 17.2111i −0.603620 + 0.603620i
\(814\) 0 0
\(815\) 1.39445 0.0488454
\(816\) 0 0
\(817\) −70.0555 −2.45093
\(818\) 0 0
\(819\) 2.36669 2.36669i 0.0826989 0.0826989i
\(820\) 0 0
\(821\) −8.81665 + 8.81665i −0.307703 + 0.307703i −0.844018 0.536315i \(-0.819816\pi\)
0.536315 + 0.844018i \(0.319816\pi\)
\(822\) 0 0
\(823\) 2.09167 + 2.09167i 0.0729111 + 0.0729111i 0.742622 0.669711i \(-0.233582\pi\)
−0.669711 + 0.742622i \(0.733582\pi\)
\(824\) 0 0
\(825\) 25.8167i 0.898821i
\(826\) 0 0
\(827\) 2.69722 + 2.69722i 0.0937917 + 0.0937917i 0.752446 0.658654i \(-0.228874\pi\)
−0.658654 + 0.752446i \(0.728874\pi\)
\(828\) 0 0
\(829\) 38.8444 1.34912 0.674561 0.738219i \(-0.264333\pi\)
0.674561 + 0.738219i \(0.264333\pi\)
\(830\) 0 0
\(831\) 9.39445i 0.325890i
\(832\) 0 0
\(833\) 7.21110 + 13.0000i 0.249850 + 0.450423i
\(834\) 0 0
\(835\) 8.18335i 0.283196i
\(836\) 0 0
\(837\) 4.73338 0.163610
\(838\) 0 0
\(839\) −4.09167 4.09167i −0.141260 0.141260i 0.632940 0.774201i \(-0.281848\pi\)
−0.774201 + 0.632940i \(0.781848\pi\)
\(840\) 0 0
\(841\) 33.8444i 1.16705i
\(842\) 0 0
\(843\) −13.5778 13.5778i −0.467644 0.467644i
\(844\) 0 0
\(845\) −8.21110 + 8.21110i −0.282471 + 0.282471i
\(846\) 0 0
\(847\) 14.0917 14.0917i 0.484196 0.484196i
\(848\) 0 0
\(849\) −60.2389 −2.06739
\(850\) 0 0
\(851\) −4.18335 −0.143403
\(852\) 0 0
\(853\) 0.577795 0.577795i 0.0197833 0.0197833i −0.697146 0.716929i \(-0.745547\pi\)
0.716929 + 0.697146i \(0.245547\pi\)
\(854\) 0 0
\(855\) 2.60555 2.60555i 0.0891080 0.0891080i
\(856\) 0 0
\(857\) 24.6333 + 24.6333i 0.841458 + 0.841458i 0.989049 0.147591i \(-0.0471519\pi\)
−0.147591 + 0.989049i \(0.547152\pi\)
\(858\) 0 0
\(859\) 19.8167i 0.676136i −0.941122 0.338068i \(-0.890227\pi\)
0.941122 0.338068i \(-0.109773\pi\)
\(860\) 0 0
\(861\) −3.39445 3.39445i −0.115683 0.115683i
\(862\) 0 0
\(863\) 48.4777 1.65020 0.825100 0.564986i \(-0.191119\pi\)
0.825100 + 0.564986i \(0.191119\pi\)
\(864\) 0 0
\(865\) 13.6333i 0.463546i
\(866\) 0 0
\(867\) −7.06392 30.5139i −0.239903 1.03631i
\(868\) 0 0
\(869\) 21.8167i 0.740079i
\(870\) 0 0
\(871\) 24.0000 0.813209
\(872\) 0 0
\(873\) 0.155590 + 0.155590i 0.00526592 + 0.00526592i
\(874\) 0 0
\(875\) 20.8444i 0.704670i
\(876\) 0 0
\(877\) 16.2111 + 16.2111i 0.547410 + 0.547410i 0.925691 0.378281i \(-0.123485\pi\)
−0.378281 + 0.925691i \(0.623485\pi\)
\(878\) 0 0
\(879\) 28.1833 28.1833i 0.950601 0.950601i
\(880\) 0 0
\(881\) −9.78890 + 9.78890i −0.329796 + 0.329796i −0.852509 0.522713i \(-0.824920\pi\)
0.522713 + 0.852509i \(0.324920\pi\)
\(882\) 0 0
\(883\) 54.4222 1.83145 0.915727 0.401802i \(-0.131616\pi\)
0.915727 + 0.401802i \(0.131616\pi\)
\(884\) 0 0
\(885\) 3.63331 0.122132
\(886\) 0 0
\(887\) −15.3028 + 15.3028i −0.513817 + 0.513817i −0.915694 0.401877i \(-0.868358\pi\)
0.401877 + 0.915694i \(0.368358\pi\)
\(888\) 0 0
\(889\) 8.60555 8.60555i 0.288621 0.288621i
\(890\) 0 0
\(891\) 33.1194 + 33.1194i 1.10954 + 1.10954i
\(892\) 0 0
\(893\) 26.4222i 0.884185i
\(894\) 0 0
\(895\) −21.0278 21.0278i −0.702880 0.702880i
\(896\) 0 0
\(897\) 8.36669 0.279356
\(898\) 0 0
\(899\) 7.81665i 0.260700i
\(900\) 0 0
\(901\) −18.4222 33.2111i −0.613733 1.10642i
\(902\) 0 0
\(903\) 36.0000i 1.19800i
\(904\) 0 0
\(905\) 26.0000 0.864269
\(906\) 0 0
\(907\) −29.5416 29.5416i −0.980914 0.980914i 0.0189074 0.999821i \(-0.493981\pi\)
−0.999821 + 0.0189074i \(0.993981\pi\)
\(908\) 0 0
\(909\) 1.33894i 0.0444097i
\(910\) 0 0
\(911\) −27.7250 27.7250i −0.918570 0.918570i 0.0783559 0.996925i \(-0.475033\pi\)
−0.996925 + 0.0783559i \(0.975033\pi\)
\(912\) 0 0
\(913\) 12.6056 12.6056i 0.417183 0.417183i
\(914\) 0 0
\(915\) −21.3944 + 21.3944i −0.707279 + 0.707279i
\(916\) 0 0
\(917\) 7.02776 0.232077
\(918\) 0 0
\(919\) −16.8444 −0.555646 −0.277823 0.960632i \(-0.589613\pi\)
−0.277823 + 0.960632i \(0.589613\pi\)
\(920\) 0 0
\(921\) 8.84441 8.84441i 0.291433 0.291433i
\(922\) 0 0
\(923\) 36.4222 36.4222i 1.19885 1.19885i
\(924\) 0 0
\(925\) 9.00000 + 9.00000i 0.295918 + 0.295918i
\(926\) 0 0
\(927\) 0.477718i 0.0156903i
\(928\) 0 0
\(929\) 1.00000 + 1.00000i 0.0328089 + 0.0328089i 0.723321 0.690512i \(-0.242615\pi\)
−0.690512 + 0.723321i \(0.742615\pi\)
\(930\) 0 0
\(931\) −23.8167 −0.780559
\(932\) 0 0
\(933\) 21.0833i 0.690235i
\(934\) 0 0
\(935\) 23.8167 13.2111i 0.778888 0.432049i
\(936\) 0 0
\(937\) 22.4222i 0.732502i −0.930516 0.366251i \(-0.880641\pi\)
0.930516 0.366251i \(-0.119359\pi\)
\(938\) 0 0
\(939\) −21.3944 −0.698181
\(940\) 0 0
\(941\) 20.0278 + 20.0278i 0.652886 + 0.652886i 0.953687 0.300801i \(-0.0972539\pi\)
−0.300801 + 0.953687i \(0.597254\pi\)
\(942\) 0 0
\(943\) 1.39445i 0.0454095i
\(944\) 0 0
\(945\) 8.84441 + 8.84441i 0.287709 + 0.287709i
\(946\) 0 0
\(947\) −41.7250 + 41.7250i −1.35588 + 1.35588i −0.476948 + 0.878931i \(0.658257\pi\)
−0.878931 + 0.476948i \(0.841743\pi\)
\(948\) 0 0
\(949\) 32.2389 32.2389i 1.04652 1.04652i
\(950\) 0 0
\(951\) 32.2944 1.04722
\(952\) 0 0
\(953\) −0.238859 −0.00773740 −0.00386870 0.999993i \(-0.501231\pi\)
−0.00386870 + 0.999993i \(0.501231\pi\)
\(954\) 0 0
\(955\) −21.2111 + 21.2111i −0.686375 + 0.686375i
\(956\) 0 0
\(957\) 48.2389 48.2389i 1.55934 1.55934i
\(958\) 0 0
\(959\) −6.00000 6.00000i −0.193750 0.193750i
\(960\) 0 0
\(961\) 30.0278i 0.968637i
\(962\) 0 0
\(963\) −3.66947 3.66947i −0.118247 0.118247i
\(964\) 0 0
\(965\) −12.4222 −0.399885
\(966\) 0 0
\(967\) 0.183346i 0.00589602i 0.999996 + 0.00294801i \(0.000938381\pi\)
−0.999996 + 0.00294801i \(0.999062\pi\)
\(968\) 0 0
\(969\) 48.2389 + 13.8167i 1.54966 + 0.443855i
\(970\) 0 0
\(971\) 18.9722i 0.608848i 0.952537 + 0.304424i \(0.0984640\pi\)
−0.952537 + 0.304424i \(0.901536\pi\)
\(972\) 0 0
\(973\) 8.60555 0.275881
\(974\) 0 0
\(975\) −18.0000 18.0000i −0.576461 0.576461i
\(976\) 0 0
\(977\) 40.8444i 1.30673i −0.757044 0.653364i \(-0.773357\pi\)
0.757044 0.653364i \(-0.226643\pi\)
\(978\) 0 0
\(979\) −45.6333 45.6333i −1.45845 1.45845i
\(980\) 0 0
\(981\) 3.23886 3.23886i 0.103409 0.103409i
\(982\) 0 0
\(983\) −17.1194 + 17.1194i −0.546025 + 0.546025i −0.925289 0.379264i \(-0.876177\pi\)
0.379264 + 0.925289i \(0.376177\pi\)
\(984\) 0 0
\(985\) 20.7889 0.662389
\(986\) 0 0
\(987\) 13.5778 0.432186
\(988\) 0 0
\(989\) −7.39445 + 7.39445i −0.235130 + 0.235130i
\(990\) 0 0
\(991\) −5.48612 + 5.48612i −0.174272 + 0.174272i −0.788854 0.614581i \(-0.789325\pi\)
0.614581 + 0.788854i \(0.289325\pi\)
\(992\) 0 0
\(993\) −13.8167 13.8167i −0.438458 0.438458i
\(994\) 0 0
\(995\) 30.2389i 0.958636i
\(996\) 0 0
\(997\) −11.2389 11.2389i −0.355938 0.355938i 0.506375 0.862313i \(-0.330985\pi\)
−0.862313 + 0.506375i \(0.830985\pi\)
\(998\) 0 0
\(999\) 20.3667 0.644374
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 68.2.e.a.21.2 yes 4
3.2 odd 2 612.2.k.e.361.1 4
4.3 odd 2 272.2.o.g.225.1 4
5.2 odd 4 1700.2.m.b.1449.1 4
5.3 odd 4 1700.2.m.a.1449.2 4
5.4 even 2 1700.2.o.c.701.1 4
8.3 odd 2 1088.2.o.s.769.2 4
8.5 even 2 1088.2.o.t.769.1 4
12.11 even 2 2448.2.be.u.1585.2 4
17.2 even 8 1156.2.b.a.577.2 4
17.3 odd 16 1156.2.h.e.733.3 16
17.4 even 4 1156.2.e.c.829.1 4
17.5 odd 16 1156.2.h.e.1001.3 16
17.6 odd 16 1156.2.h.e.977.3 16
17.7 odd 16 1156.2.h.e.757.2 16
17.8 even 8 1156.2.a.h.1.2 4
17.9 even 8 1156.2.a.h.1.3 4
17.10 odd 16 1156.2.h.e.757.3 16
17.11 odd 16 1156.2.h.e.977.2 16
17.12 odd 16 1156.2.h.e.1001.2 16
17.13 even 4 inner 68.2.e.a.13.2 4
17.14 odd 16 1156.2.h.e.733.2 16
17.15 even 8 1156.2.b.a.577.3 4
17.16 even 2 1156.2.e.c.905.1 4
51.47 odd 4 612.2.k.e.217.1 4
68.43 odd 8 4624.2.a.bq.1.2 4
68.47 odd 4 272.2.o.g.81.1 4
68.59 odd 8 4624.2.a.bq.1.3 4
85.13 odd 4 1700.2.m.b.149.1 4
85.47 odd 4 1700.2.m.a.149.2 4
85.64 even 4 1700.2.o.c.1101.1 4
136.13 even 4 1088.2.o.t.897.1 4
136.115 odd 4 1088.2.o.s.897.2 4
204.47 even 4 2448.2.be.u.1441.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
68.2.e.a.13.2 4 17.13 even 4 inner
68.2.e.a.21.2 yes 4 1.1 even 1 trivial
272.2.o.g.81.1 4 68.47 odd 4
272.2.o.g.225.1 4 4.3 odd 2
612.2.k.e.217.1 4 51.47 odd 4
612.2.k.e.361.1 4 3.2 odd 2
1088.2.o.s.769.2 4 8.3 odd 2
1088.2.o.s.897.2 4 136.115 odd 4
1088.2.o.t.769.1 4 8.5 even 2
1088.2.o.t.897.1 4 136.13 even 4
1156.2.a.h.1.2 4 17.8 even 8
1156.2.a.h.1.3 4 17.9 even 8
1156.2.b.a.577.2 4 17.2 even 8
1156.2.b.a.577.3 4 17.15 even 8
1156.2.e.c.829.1 4 17.4 even 4
1156.2.e.c.905.1 4 17.16 even 2
1156.2.h.e.733.2 16 17.14 odd 16
1156.2.h.e.733.3 16 17.3 odd 16
1156.2.h.e.757.2 16 17.7 odd 16
1156.2.h.e.757.3 16 17.10 odd 16
1156.2.h.e.977.2 16 17.11 odd 16
1156.2.h.e.977.3 16 17.6 odd 16
1156.2.h.e.1001.2 16 17.12 odd 16
1156.2.h.e.1001.3 16 17.5 odd 16
1700.2.m.a.149.2 4 85.47 odd 4
1700.2.m.a.1449.2 4 5.3 odd 4
1700.2.m.b.149.1 4 85.13 odd 4
1700.2.m.b.1449.1 4 5.2 odd 4
1700.2.o.c.701.1 4 5.4 even 2
1700.2.o.c.1101.1 4 85.64 even 4
2448.2.be.u.1441.2 4 204.47 even 4
2448.2.be.u.1585.2 4 12.11 even 2
4624.2.a.bq.1.2 4 68.43 odd 8
4624.2.a.bq.1.3 4 68.59 odd 8