Properties

Label 784.2.p.e.31.1
Level $784$
Weight $2$
Character 784.31
Analytic conductor $6.260$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [784,2,Mod(31,784)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(784, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 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("784.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 784.p (of order \(6\), degree \(2\), 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: \(6.26027151847\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 112)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 31.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 784.31
Dual form 784.2.p.e.607.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.00000 - 1.73205i) q^{3} +(-3.00000 + 1.73205i) q^{5} +(-0.500000 - 0.866025i) q^{9} +(3.00000 + 1.73205i) q^{11} +3.46410i q^{13} +6.92820i q^{15} +(1.00000 + 1.73205i) q^{19} +(3.00000 - 1.73205i) q^{23} +(3.50000 - 6.06218i) q^{25} +4.00000 q^{27} +6.00000 q^{29} +(-4.00000 + 6.92820i) q^{31} +(6.00000 - 3.46410i) q^{33} +(1.00000 + 1.73205i) q^{37} +(6.00000 + 3.46410i) q^{39} -6.92820i q^{41} +10.3923i q^{43} +(3.00000 + 1.73205i) q^{45} +(-3.00000 + 5.19615i) q^{53} -12.0000 q^{55} +4.00000 q^{57} +(3.00000 - 5.19615i) q^{59} +(3.00000 - 1.73205i) q^{61} +(-6.00000 - 10.3923i) q^{65} +(3.00000 + 1.73205i) q^{67} -6.92820i q^{69} -3.46410i q^{71} +(-6.00000 - 3.46410i) q^{73} +(-7.00000 - 12.1244i) q^{75} +(-3.00000 + 1.73205i) q^{79} +(5.50000 - 9.52628i) q^{81} +6.00000 q^{83} +(6.00000 - 10.3923i) q^{87} +(-6.00000 + 3.46410i) q^{89} +(8.00000 + 13.8564i) q^{93} +(-6.00000 - 3.46410i) q^{95} +13.8564i q^{97} -3.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 6 q^{5} - q^{9} + 6 q^{11} + 2 q^{19} + 6 q^{23} + 7 q^{25} + 8 q^{27} + 12 q^{29} - 8 q^{31} + 12 q^{33} + 2 q^{37} + 12 q^{39} + 6 q^{45} - 6 q^{53} - 24 q^{55} + 8 q^{57} + 6 q^{59}+ \cdots - 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/784\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{1}{6}\right)\)

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
\(3\) 1.00000 1.73205i 0.577350 1.00000i −0.418432 0.908248i \(-0.637420\pi\)
0.995782 0.0917517i \(-0.0292466\pi\)
\(4\) 0 0
\(5\) −3.00000 + 1.73205i −1.34164 + 0.774597i −0.987048 0.160424i \(-0.948714\pi\)
−0.354593 + 0.935021i \(0.615380\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −0.500000 0.866025i −0.166667 0.288675i
\(10\) 0 0
\(11\) 3.00000 + 1.73205i 0.904534 + 0.522233i 0.878668 0.477432i \(-0.158432\pi\)
0.0258656 + 0.999665i \(0.491766\pi\)
\(12\) 0 0
\(13\) 3.46410i 0.960769i 0.877058 + 0.480384i \(0.159503\pi\)
−0.877058 + 0.480384i \(0.840497\pi\)
\(14\) 0 0
\(15\) 6.92820i 1.78885i
\(16\) 0 0
\(17\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(18\) 0 0
\(19\) 1.00000 + 1.73205i 0.229416 + 0.397360i 0.957635 0.287984i \(-0.0929851\pi\)
−0.728219 + 0.685344i \(0.759652\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.00000 1.73205i 0.625543 0.361158i −0.153481 0.988152i \(-0.549048\pi\)
0.779024 + 0.626994i \(0.215715\pi\)
\(24\) 0 0
\(25\) 3.50000 6.06218i 0.700000 1.21244i
\(26\) 0 0
\(27\) 4.00000 0.769800
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −4.00000 + 6.92820i −0.718421 + 1.24434i 0.243204 + 0.969975i \(0.421802\pi\)
−0.961625 + 0.274367i \(0.911532\pi\)
\(32\) 0 0
\(33\) 6.00000 3.46410i 1.04447 0.603023i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.00000 + 1.73205i 0.164399 + 0.284747i 0.936442 0.350823i \(-0.114098\pi\)
−0.772043 + 0.635571i \(0.780765\pi\)
\(38\) 0 0
\(39\) 6.00000 + 3.46410i 0.960769 + 0.554700i
\(40\) 0 0
\(41\) 6.92820i 1.08200i −0.841021 0.541002i \(-0.818045\pi\)
0.841021 0.541002i \(-0.181955\pi\)
\(42\) 0 0
\(43\) 10.3923i 1.58481i 0.609994 + 0.792406i \(0.291172\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) 3.00000 + 1.73205i 0.447214 + 0.258199i
\(46\) 0 0
\(47\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −3.00000 + 5.19615i −0.412082 + 0.713746i −0.995117 0.0987002i \(-0.968532\pi\)
0.583036 + 0.812447i \(0.301865\pi\)
\(54\) 0 0
\(55\) −12.0000 −1.61808
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) 0 0
\(59\) 3.00000 5.19615i 0.390567 0.676481i −0.601958 0.798528i \(-0.705612\pi\)
0.992524 + 0.122047i \(0.0389457\pi\)
\(60\) 0 0
\(61\) 3.00000 1.73205i 0.384111 0.221766i −0.295495 0.955344i \(-0.595484\pi\)
0.679605 + 0.733578i \(0.262151\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −6.00000 10.3923i −0.744208 1.28901i
\(66\) 0 0
\(67\) 3.00000 + 1.73205i 0.366508 + 0.211604i 0.671932 0.740613i \(-0.265465\pi\)
−0.305424 + 0.952217i \(0.598798\pi\)
\(68\) 0 0
\(69\) 6.92820i 0.834058i
\(70\) 0 0
\(71\) 3.46410i 0.411113i −0.978645 0.205557i \(-0.934100\pi\)
0.978645 0.205557i \(-0.0659005\pi\)
\(72\) 0 0
\(73\) −6.00000 3.46410i −0.702247 0.405442i 0.105937 0.994373i \(-0.466216\pi\)
−0.808184 + 0.588930i \(0.799549\pi\)
\(74\) 0 0
\(75\) −7.00000 12.1244i −0.808290 1.40000i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −3.00000 + 1.73205i −0.337526 + 0.194871i −0.659178 0.751987i \(-0.729095\pi\)
0.321651 + 0.946858i \(0.395762\pi\)
\(80\) 0 0
\(81\) 5.50000 9.52628i 0.611111 1.05848i
\(82\) 0 0
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 6.00000 10.3923i 0.643268 1.11417i
\(88\) 0 0
\(89\) −6.00000 + 3.46410i −0.635999 + 0.367194i −0.783072 0.621932i \(-0.786348\pi\)
0.147073 + 0.989126i \(0.453015\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 8.00000 + 13.8564i 0.829561 + 1.43684i
\(94\) 0 0
\(95\) −6.00000 3.46410i −0.615587 0.355409i
\(96\) 0 0
\(97\) 13.8564i 1.40690i 0.710742 + 0.703452i \(0.248359\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 0 0
\(99\) 3.46410i 0.348155i
\(100\) 0 0
\(101\) −9.00000 5.19615i −0.895533 0.517036i −0.0197851 0.999804i \(-0.506298\pi\)
−0.875748 + 0.482768i \(0.839632\pi\)
\(102\) 0 0
\(103\) 2.00000 + 3.46410i 0.197066 + 0.341328i 0.947576 0.319531i \(-0.103525\pi\)
−0.750510 + 0.660859i \(0.770192\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 9.00000 5.19615i 0.870063 0.502331i 0.00269372 0.999996i \(-0.499143\pi\)
0.867369 + 0.497665i \(0.165809\pi\)
\(108\) 0 0
\(109\) −7.00000 + 12.1244i −0.670478 + 1.16130i 0.307290 + 0.951616i \(0.400578\pi\)
−0.977769 + 0.209687i \(0.932756\pi\)
\(110\) 0 0
\(111\) 4.00000 0.379663
\(112\) 0 0
\(113\) −18.0000 −1.69330 −0.846649 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) 0 0
\(115\) −6.00000 + 10.3923i −0.559503 + 0.969087i
\(116\) 0 0
\(117\) 3.00000 1.73205i 0.277350 0.160128i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0.500000 + 0.866025i 0.0454545 + 0.0787296i
\(122\) 0 0
\(123\) −12.0000 6.92820i −1.08200 0.624695i
\(124\) 0 0
\(125\) 6.92820i 0.619677i
\(126\) 0 0
\(127\) 10.3923i 0.922168i −0.887357 0.461084i \(-0.847461\pi\)
0.887357 0.461084i \(-0.152539\pi\)
\(128\) 0 0
\(129\) 18.0000 + 10.3923i 1.58481 + 0.914991i
\(130\) 0 0
\(131\) 9.00000 + 15.5885i 0.786334 + 1.36197i 0.928199 + 0.372084i \(0.121357\pi\)
−0.141865 + 0.989886i \(0.545310\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −12.0000 + 6.92820i −1.03280 + 0.596285i
\(136\) 0 0
\(137\) 3.00000 5.19615i 0.256307 0.443937i −0.708942 0.705266i \(-0.750827\pi\)
0.965250 + 0.261329i \(0.0841608\pi\)
\(138\) 0 0
\(139\) 2.00000 0.169638 0.0848189 0.996396i \(-0.472969\pi\)
0.0848189 + 0.996396i \(0.472969\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −6.00000 + 10.3923i −0.501745 + 0.869048i
\(144\) 0 0
\(145\) −18.0000 + 10.3923i −1.49482 + 0.863034i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −3.00000 5.19615i −0.245770 0.425685i 0.716578 0.697507i \(-0.245707\pi\)
−0.962348 + 0.271821i \(0.912374\pi\)
\(150\) 0 0
\(151\) −3.00000 1.73205i −0.244137 0.140952i 0.372940 0.927855i \(-0.378350\pi\)
−0.617076 + 0.786903i \(0.711683\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 27.7128i 2.22595i
\(156\) 0 0
\(157\) 9.00000 + 5.19615i 0.718278 + 0.414698i 0.814119 0.580699i \(-0.197221\pi\)
−0.0958404 + 0.995397i \(0.530554\pi\)
\(158\) 0 0
\(159\) 6.00000 + 10.3923i 0.475831 + 0.824163i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −15.0000 + 8.66025i −1.17489 + 0.678323i −0.954827 0.297162i \(-0.903960\pi\)
−0.220064 + 0.975486i \(0.570626\pi\)
\(164\) 0 0
\(165\) −12.0000 + 20.7846i −0.934199 + 1.61808i
\(166\) 0 0
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 1.00000 1.73205i 0.0764719 0.132453i
\(172\) 0 0
\(173\) 3.00000 1.73205i 0.228086 0.131685i −0.381603 0.924326i \(-0.624628\pi\)
0.609689 + 0.792641i \(0.291294\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −6.00000 10.3923i −0.450988 0.781133i
\(178\) 0 0
\(179\) −9.00000 5.19615i −0.672692 0.388379i 0.124404 0.992232i \(-0.460298\pi\)
−0.797096 + 0.603853i \(0.793631\pi\)
\(180\) 0 0
\(181\) 17.3205i 1.28742i −0.765268 0.643712i \(-0.777394\pi\)
0.765268 0.643712i \(-0.222606\pi\)
\(182\) 0 0
\(183\) 6.92820i 0.512148i
\(184\) 0 0
\(185\) −6.00000 3.46410i −0.441129 0.254686i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 21.0000 12.1244i 1.51951 0.877288i 0.519771 0.854306i \(-0.326017\pi\)
0.999736 0.0229818i \(-0.00731599\pi\)
\(192\) 0 0
\(193\) 1.00000 1.73205i 0.0719816 0.124676i −0.827788 0.561041i \(-0.810401\pi\)
0.899770 + 0.436365i \(0.143734\pi\)
\(194\) 0 0
\(195\) −24.0000 −1.71868
\(196\) 0 0
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) 10.0000 17.3205i 0.708881 1.22782i −0.256391 0.966573i \(-0.582534\pi\)
0.965272 0.261245i \(-0.0841331\pi\)
\(200\) 0 0
\(201\) 6.00000 3.46410i 0.423207 0.244339i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 12.0000 + 20.7846i 0.838116 + 1.45166i
\(206\) 0 0
\(207\) −3.00000 1.73205i −0.208514 0.120386i
\(208\) 0 0
\(209\) 6.92820i 0.479234i
\(210\) 0 0
\(211\) 3.46410i 0.238479i −0.992866 0.119239i \(-0.961954\pi\)
0.992866 0.119239i \(-0.0380456\pi\)
\(212\) 0 0
\(213\) −6.00000 3.46410i −0.411113 0.237356i
\(214\) 0 0
\(215\) −18.0000 31.1769i −1.22759 2.12625i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −12.0000 + 6.92820i −0.810885 + 0.468165i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 16.0000 1.07144 0.535720 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 0 0
\(225\) −7.00000 −0.466667
\(226\) 0 0
\(227\) −3.00000 + 5.19615i −0.199117 + 0.344881i −0.948242 0.317547i \(-0.897141\pi\)
0.749125 + 0.662428i \(0.230474\pi\)
\(228\) 0 0
\(229\) 9.00000 5.19615i 0.594737 0.343371i −0.172231 0.985057i \(-0.555098\pi\)
0.766968 + 0.641685i \(0.221764\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 3.00000 + 5.19615i 0.196537 + 0.340411i 0.947403 0.320043i \(-0.103697\pi\)
−0.750867 + 0.660454i \(0.770364\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 6.92820i 0.450035i
\(238\) 0 0
\(239\) 10.3923i 0.672222i −0.941822 0.336111i \(-0.890888\pi\)
0.941822 0.336111i \(-0.109112\pi\)
\(240\) 0 0
\(241\) −24.0000 13.8564i −1.54598 0.892570i −0.998443 0.0557856i \(-0.982234\pi\)
−0.547533 0.836784i \(-0.684433\pi\)
\(242\) 0 0
\(243\) −5.00000 8.66025i −0.320750 0.555556i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −6.00000 + 3.46410i −0.381771 + 0.220416i
\(248\) 0 0
\(249\) 6.00000 10.3923i 0.380235 0.658586i
\(250\) 0 0
\(251\) −30.0000 −1.89358 −0.946792 0.321847i \(-0.895696\pi\)
−0.946792 + 0.321847i \(0.895696\pi\)
\(252\) 0 0
\(253\) 12.0000 0.754434
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 24.0000 13.8564i 1.49708 0.864339i 0.497085 0.867702i \(-0.334404\pi\)
0.999994 + 0.00336324i \(0.00107055\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −3.00000 5.19615i −0.185695 0.321634i
\(262\) 0 0
\(263\) −21.0000 12.1244i −1.29492 0.747620i −0.315394 0.948961i \(-0.602137\pi\)
−0.979521 + 0.201341i \(0.935470\pi\)
\(264\) 0 0
\(265\) 20.7846i 1.27679i
\(266\) 0 0
\(267\) 13.8564i 0.847998i
\(268\) 0 0
\(269\) −3.00000 1.73205i −0.182913 0.105605i 0.405747 0.913985i \(-0.367011\pi\)
−0.588661 + 0.808380i \(0.700345\pi\)
\(270\) 0 0
\(271\) −4.00000 6.92820i −0.242983 0.420858i 0.718580 0.695444i \(-0.244792\pi\)
−0.961563 + 0.274586i \(0.911459\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 21.0000 12.1244i 1.26635 0.731126i
\(276\) 0 0
\(277\) 1.00000 1.73205i 0.0600842 0.104069i −0.834419 0.551131i \(-0.814196\pi\)
0.894503 + 0.447062i \(0.147530\pi\)
\(278\) 0 0
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) 7.00000 12.1244i 0.416107 0.720718i −0.579437 0.815017i \(-0.696728\pi\)
0.995544 + 0.0942988i \(0.0300609\pi\)
\(284\) 0 0
\(285\) −12.0000 + 6.92820i −0.710819 + 0.410391i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −8.50000 14.7224i −0.500000 0.866025i
\(290\) 0 0
\(291\) 24.0000 + 13.8564i 1.40690 + 0.812277i
\(292\) 0 0
\(293\) 10.3923i 0.607125i 0.952812 + 0.303562i \(0.0981761\pi\)
−0.952812 + 0.303562i \(0.901824\pi\)
\(294\) 0 0
\(295\) 20.7846i 1.21013i
\(296\) 0 0
\(297\) 12.0000 + 6.92820i 0.696311 + 0.402015i
\(298\) 0 0
\(299\) 6.00000 + 10.3923i 0.346989 + 0.601003i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −18.0000 + 10.3923i −1.03407 + 0.597022i
\(304\) 0 0
\(305\) −6.00000 + 10.3923i −0.343559 + 0.595062i
\(306\) 0 0
\(307\) 22.0000 1.25561 0.627803 0.778372i \(-0.283954\pi\)
0.627803 + 0.778372i \(0.283954\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) −6.00000 + 10.3923i −0.340229 + 0.589294i −0.984475 0.175525i \(-0.943838\pi\)
0.644246 + 0.764818i \(0.277171\pi\)
\(312\) 0 0
\(313\) 6.00000 3.46410i 0.339140 0.195803i −0.320752 0.947163i \(-0.603935\pi\)
0.659892 + 0.751361i \(0.270602\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −3.00000 5.19615i −0.168497 0.291845i 0.769395 0.638774i \(-0.220558\pi\)
−0.937892 + 0.346929i \(0.887225\pi\)
\(318\) 0 0
\(319\) 18.0000 + 10.3923i 1.00781 + 0.581857i
\(320\) 0 0
\(321\) 20.7846i 1.16008i
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 21.0000 + 12.1244i 1.16487 + 0.672538i
\(326\) 0 0
\(327\) 14.0000 + 24.2487i 0.774202 + 1.34096i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −3.00000 + 1.73205i −0.164895 + 0.0952021i −0.580176 0.814491i \(-0.697016\pi\)
0.415282 + 0.909693i \(0.363683\pi\)
\(332\) 0 0
\(333\) 1.00000 1.73205i 0.0547997 0.0949158i
\(334\) 0 0
\(335\) −12.0000 −0.655630
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 0 0
\(339\) −18.0000 + 31.1769i −0.977626 + 1.69330i
\(340\) 0 0
\(341\) −24.0000 + 13.8564i −1.29967 + 0.750366i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 12.0000 + 20.7846i 0.646058 + 1.11901i
\(346\) 0 0
\(347\) 15.0000 + 8.66025i 0.805242 + 0.464907i 0.845301 0.534291i \(-0.179421\pi\)
−0.0400587 + 0.999197i \(0.512754\pi\)
\(348\) 0 0
\(349\) 3.46410i 0.185429i 0.995693 + 0.0927146i \(0.0295544\pi\)
−0.995693 + 0.0927146i \(0.970446\pi\)
\(350\) 0 0
\(351\) 13.8564i 0.739600i
\(352\) 0 0
\(353\) 24.0000 + 13.8564i 1.27739 + 0.737502i 0.976368 0.216115i \(-0.0693385\pi\)
0.301023 + 0.953617i \(0.402672\pi\)
\(354\) 0 0
\(355\) 6.00000 + 10.3923i 0.318447 + 0.551566i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 27.0000 15.5885i 1.42501 0.822727i 0.428285 0.903644i \(-0.359118\pi\)
0.996721 + 0.0809166i \(0.0257847\pi\)
\(360\) 0 0
\(361\) 7.50000 12.9904i 0.394737 0.683704i
\(362\) 0 0
\(363\) 2.00000 0.104973
\(364\) 0 0
\(365\) 24.0000 1.25622
\(366\) 0 0
\(367\) 4.00000 6.92820i 0.208798 0.361649i −0.742538 0.669804i \(-0.766378\pi\)
0.951336 + 0.308155i \(0.0997115\pi\)
\(368\) 0 0
\(369\) −6.00000 + 3.46410i −0.312348 + 0.180334i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −7.00000 12.1244i −0.362446 0.627775i 0.625917 0.779890i \(-0.284725\pi\)
−0.988363 + 0.152115i \(0.951392\pi\)
\(374\) 0 0
\(375\) 12.0000 + 6.92820i 0.619677 + 0.357771i
\(376\) 0 0
\(377\) 20.7846i 1.07046i
\(378\) 0 0
\(379\) 24.2487i 1.24557i 0.782392 + 0.622786i \(0.213999\pi\)
−0.782392 + 0.622786i \(0.786001\pi\)
\(380\) 0 0
\(381\) −18.0000 10.3923i −0.922168 0.532414i
\(382\) 0 0
\(383\) 12.0000 + 20.7846i 0.613171 + 1.06204i 0.990702 + 0.136047i \(0.0434398\pi\)
−0.377531 + 0.925997i \(0.623227\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 9.00000 5.19615i 0.457496 0.264135i
\(388\) 0 0
\(389\) 9.00000 15.5885i 0.456318 0.790366i −0.542445 0.840091i \(-0.682501\pi\)
0.998763 + 0.0497253i \(0.0158346\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 36.0000 1.81596
\(394\) 0 0
\(395\) 6.00000 10.3923i 0.301893 0.522894i
\(396\) 0 0
\(397\) −9.00000 + 5.19615i −0.451697 + 0.260787i −0.708547 0.705664i \(-0.750649\pi\)
0.256850 + 0.966451i \(0.417315\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −15.0000 25.9808i −0.749064 1.29742i −0.948272 0.317460i \(-0.897170\pi\)
0.199207 0.979957i \(-0.436163\pi\)
\(402\) 0 0
\(403\) −24.0000 13.8564i −1.19553 0.690237i
\(404\) 0 0
\(405\) 38.1051i 1.89346i
\(406\) 0 0
\(407\) 6.92820i 0.343418i
\(408\) 0 0
\(409\) −6.00000 3.46410i −0.296681 0.171289i 0.344270 0.938871i \(-0.388126\pi\)
−0.640951 + 0.767582i \(0.721460\pi\)
\(410\) 0 0
\(411\) −6.00000 10.3923i −0.295958 0.512615i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −18.0000 + 10.3923i −0.883585 + 0.510138i
\(416\) 0 0
\(417\) 2.00000 3.46410i 0.0979404 0.169638i
\(418\) 0 0
\(419\) 30.0000 1.46560 0.732798 0.680446i \(-0.238214\pi\)
0.732798 + 0.680446i \(0.238214\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 12.0000 + 20.7846i 0.579365 + 1.00349i
\(430\) 0 0
\(431\) 33.0000 + 19.0526i 1.58955 + 0.917729i 0.993380 + 0.114874i \(0.0366465\pi\)
0.596174 + 0.802855i \(0.296687\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 41.5692i 1.99309i
\(436\) 0 0
\(437\) 6.00000 + 3.46410i 0.287019 + 0.165710i
\(438\) 0 0
\(439\) −14.0000 24.2487i −0.668184 1.15733i −0.978412 0.206666i \(-0.933739\pi\)
0.310228 0.950662i \(-0.399595\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −3.00000 + 1.73205i −0.142534 + 0.0822922i −0.569571 0.821942i \(-0.692891\pi\)
0.427037 + 0.904234i \(0.359557\pi\)
\(444\) 0 0
\(445\) 12.0000 20.7846i 0.568855 0.985285i
\(446\) 0 0
\(447\) −12.0000 −0.567581
\(448\) 0 0
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 0 0
\(451\) 12.0000 20.7846i 0.565058 0.978709i
\(452\) 0 0
\(453\) −6.00000 + 3.46410i −0.281905 + 0.162758i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 17.0000 + 29.4449i 0.795226 + 1.37737i 0.922695 + 0.385530i \(0.125981\pi\)
−0.127469 + 0.991843i \(0.540685\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 38.1051i 1.77473i −0.461065 0.887366i \(-0.652533\pi\)
0.461065 0.887366i \(-0.347467\pi\)
\(462\) 0 0
\(463\) 24.2487i 1.12693i 0.826139 + 0.563467i \(0.190533\pi\)
−0.826139 + 0.563467i \(0.809467\pi\)
\(464\) 0 0
\(465\) −48.0000 27.7128i −2.22595 1.28515i
\(466\) 0 0
\(467\) −15.0000 25.9808i −0.694117 1.20225i −0.970477 0.241192i \(-0.922462\pi\)
0.276360 0.961054i \(-0.410872\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 18.0000 10.3923i 0.829396 0.478852i
\(472\) 0 0
\(473\) −18.0000 + 31.1769i −0.827641 + 1.43352i
\(474\) 0 0
\(475\) 14.0000 0.642364
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 0 0
\(479\) 12.0000 20.7846i 0.548294 0.949673i −0.450098 0.892979i \(-0.648611\pi\)
0.998392 0.0566937i \(-0.0180558\pi\)
\(480\) 0 0
\(481\) −6.00000 + 3.46410i −0.273576 + 0.157949i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −24.0000 41.5692i −1.08978 1.88756i
\(486\) 0 0
\(487\) 21.0000 + 12.1244i 0.951601 + 0.549407i 0.893578 0.448908i \(-0.148187\pi\)
0.0580230 + 0.998315i \(0.481520\pi\)
\(488\) 0 0
\(489\) 34.6410i 1.56652i
\(490\) 0 0
\(491\) 17.3205i 0.781664i −0.920462 0.390832i \(-0.872187\pi\)
0.920462 0.390832i \(-0.127813\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 6.00000 + 10.3923i 0.269680 + 0.467099i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 9.00000 5.19615i 0.402895 0.232612i −0.284837 0.958576i \(-0.591940\pi\)
0.687733 + 0.725964i \(0.258606\pi\)
\(500\) 0 0
\(501\) 12.0000 20.7846i 0.536120 0.928588i
\(502\) 0 0
\(503\) −36.0000 −1.60516 −0.802580 0.596544i \(-0.796540\pi\)
−0.802580 + 0.596544i \(0.796540\pi\)
\(504\) 0 0
\(505\) 36.0000 1.60198
\(506\) 0 0
\(507\) 1.00000 1.73205i 0.0444116 0.0769231i
\(508\) 0 0
\(509\) 27.0000 15.5885i 1.19675 0.690946i 0.236924 0.971528i \(-0.423861\pi\)
0.959830 + 0.280582i \(0.0905275\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 4.00000 + 6.92820i 0.176604 + 0.305888i
\(514\) 0 0
\(515\) −12.0000 6.92820i −0.528783 0.305293i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 6.92820i 0.304114i
\(520\) 0 0
\(521\) 30.0000 + 17.3205i 1.31432 + 0.758825i 0.982809 0.184625i \(-0.0591070\pi\)
0.331515 + 0.943450i \(0.392440\pi\)
\(522\) 0 0
\(523\) −1.00000 1.73205i −0.0437269 0.0757373i 0.843334 0.537390i \(-0.180590\pi\)
−0.887061 + 0.461653i \(0.847256\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −5.50000 + 9.52628i −0.239130 + 0.414186i
\(530\) 0 0
\(531\) −6.00000 −0.260378
\(532\) 0 0
\(533\) 24.0000 1.03956
\(534\) 0 0
\(535\) −18.0000 + 31.1769i −0.778208 + 1.34790i
\(536\) 0 0
\(537\) −18.0000 + 10.3923i −0.776757 + 0.448461i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 1.00000 + 1.73205i 0.0429934 + 0.0744667i 0.886721 0.462304i \(-0.152977\pi\)
−0.843728 + 0.536771i \(0.819644\pi\)
\(542\) 0 0
\(543\) −30.0000 17.3205i −1.28742 0.743294i
\(544\) 0 0
\(545\) 48.4974i 2.07740i
\(546\) 0 0
\(547\) 17.3205i 0.740571i −0.928918 0.370286i \(-0.879260\pi\)
0.928918 0.370286i \(-0.120740\pi\)
\(548\) 0 0
\(549\) −3.00000 1.73205i −0.128037 0.0739221i
\(550\) 0 0
\(551\) 6.00000 + 10.3923i 0.255609 + 0.442727i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −12.0000 + 6.92820i −0.509372 + 0.294086i
\(556\) 0 0
\(557\) 9.00000 15.5885i 0.381342 0.660504i −0.609912 0.792469i \(-0.708795\pi\)
0.991254 + 0.131965i \(0.0421286\pi\)
\(558\) 0 0
\(559\) −36.0000 −1.52264
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 9.00000 15.5885i 0.379305 0.656975i −0.611656 0.791123i \(-0.709497\pi\)
0.990961 + 0.134148i \(0.0428299\pi\)
\(564\) 0 0
\(565\) 54.0000 31.1769i 2.27180 1.31162i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 9.00000 + 15.5885i 0.377300 + 0.653502i 0.990668 0.136295i \(-0.0435194\pi\)
−0.613369 + 0.789797i \(0.710186\pi\)
\(570\) 0 0
\(571\) 39.0000 + 22.5167i 1.63210 + 0.942293i 0.983444 + 0.181210i \(0.0580014\pi\)
0.648655 + 0.761083i \(0.275332\pi\)
\(572\) 0 0
\(573\) 48.4974i 2.02601i
\(574\) 0 0
\(575\) 24.2487i 1.01124i
\(576\) 0 0
\(577\) 24.0000 + 13.8564i 0.999133 + 0.576850i 0.907992 0.418988i \(-0.137615\pi\)
0.0911414 + 0.995838i \(0.470948\pi\)
\(578\) 0 0
\(579\) −2.00000 3.46410i −0.0831172 0.143963i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −18.0000 + 10.3923i −0.745484 + 0.430405i
\(584\) 0 0
\(585\) −6.00000 + 10.3923i −0.248069 + 0.429669i
\(586\) 0 0
\(587\) 18.0000 0.742940 0.371470 0.928445i \(-0.378854\pi\)
0.371470 + 0.928445i \(0.378854\pi\)
\(588\) 0 0
\(589\) −16.0000 −0.659269
\(590\) 0 0
\(591\) −18.0000 + 31.1769i −0.740421 + 1.28245i
\(592\) 0 0
\(593\) −12.0000 + 6.92820i −0.492781 + 0.284507i −0.725727 0.687982i \(-0.758497\pi\)
0.232946 + 0.972490i \(0.425163\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −20.0000 34.6410i −0.818546 1.41776i
\(598\) 0 0
\(599\) −21.0000 12.1244i −0.858037 0.495388i 0.00531761 0.999986i \(-0.498307\pi\)
−0.863354 + 0.504598i \(0.831641\pi\)
\(600\) 0 0
\(601\) 20.7846i 0.847822i 0.905704 + 0.423911i \(0.139343\pi\)
−0.905704 + 0.423911i \(0.860657\pi\)
\(602\) 0 0
\(603\) 3.46410i 0.141069i
\(604\) 0 0
\(605\) −3.00000 1.73205i −0.121967 0.0704179i
\(606\) 0 0
\(607\) −8.00000 13.8564i −0.324710 0.562414i 0.656744 0.754114i \(-0.271933\pi\)
−0.981454 + 0.191700i \(0.938600\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −19.0000 + 32.9090i −0.767403 + 1.32918i 0.171564 + 0.985173i \(0.445118\pi\)
−0.938967 + 0.344008i \(0.888215\pi\)
\(614\) 0 0
\(615\) 48.0000 1.93555
\(616\) 0 0
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 0 0
\(619\) −13.0000 + 22.5167i −0.522514 + 0.905021i 0.477143 + 0.878826i \(0.341672\pi\)
−0.999657 + 0.0261952i \(0.991661\pi\)
\(620\) 0 0
\(621\) 12.0000 6.92820i 0.481543 0.278019i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 5.50000 + 9.52628i 0.220000 + 0.381051i
\(626\) 0 0
\(627\) 12.0000 + 6.92820i 0.479234 + 0.276686i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 3.46410i 0.137904i −0.997620 0.0689519i \(-0.978035\pi\)
0.997620 0.0689519i \(-0.0219655\pi\)
\(632\) 0 0
\(633\) −6.00000 3.46410i −0.238479 0.137686i
\(634\) 0 0
\(635\) 18.0000 + 31.1769i 0.714308 + 1.23722i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −3.00000 + 1.73205i −0.118678 + 0.0685189i
\(640\) 0 0
\(641\) −15.0000 + 25.9808i −0.592464 + 1.02618i 0.401435 + 0.915888i \(0.368512\pi\)
−0.993899 + 0.110291i \(0.964822\pi\)
\(642\) 0 0
\(643\) −26.0000 −1.02534 −0.512670 0.858586i \(-0.671344\pi\)
−0.512670 + 0.858586i \(0.671344\pi\)
\(644\) 0 0
\(645\) −72.0000 −2.83500
\(646\) 0 0
\(647\) −6.00000 + 10.3923i −0.235884 + 0.408564i −0.959529 0.281609i \(-0.909132\pi\)
0.723645 + 0.690172i \(0.242465\pi\)
\(648\) 0 0
\(649\) 18.0000 10.3923i 0.706562 0.407934i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 9.00000 + 15.5885i 0.352197 + 0.610023i 0.986634 0.162951i \(-0.0521013\pi\)
−0.634437 + 0.772975i \(0.718768\pi\)
\(654\) 0 0
\(655\) −54.0000 31.1769i −2.10995 1.21818i
\(656\) 0 0
\(657\) 6.92820i 0.270295i
\(658\) 0 0
\(659\) 3.46410i 0.134942i −0.997721 0.0674711i \(-0.978507\pi\)
0.997721 0.0674711i \(-0.0214931\pi\)
\(660\) 0 0
\(661\) 3.00000 + 1.73205i 0.116686 + 0.0673690i 0.557207 0.830373i \(-0.311873\pi\)
−0.440521 + 0.897742i \(0.645206\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 18.0000 10.3923i 0.696963 0.402392i
\(668\) 0 0
\(669\) 16.0000 27.7128i 0.618596 1.07144i
\(670\) 0 0
\(671\) 12.0000 0.463255
\(672\) 0 0
\(673\) −14.0000 −0.539660 −0.269830 0.962908i \(-0.586968\pi\)
−0.269830 + 0.962908i \(0.586968\pi\)
\(674\) 0 0
\(675\) 14.0000 24.2487i 0.538860 0.933333i
\(676\) 0 0
\(677\) −3.00000 + 1.73205i −0.115299 + 0.0665681i −0.556541 0.830820i \(-0.687872\pi\)
0.441241 + 0.897389i \(0.354538\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 6.00000 + 10.3923i 0.229920 + 0.398234i
\(682\) 0 0
\(683\) −33.0000 19.0526i −1.26271 0.729026i −0.289112 0.957295i \(-0.593360\pi\)
−0.973598 + 0.228269i \(0.926693\pi\)
\(684\) 0 0
\(685\) 20.7846i 0.794139i
\(686\) 0 0
\(687\) 20.7846i 0.792982i
\(688\) 0 0
\(689\) −18.0000 10.3923i −0.685745 0.395915i
\(690\) 0 0
\(691\) 5.00000 + 8.66025i 0.190209 + 0.329452i 0.945319 0.326146i \(-0.105750\pi\)
−0.755110 + 0.655598i \(0.772417\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −6.00000 + 3.46410i −0.227593 + 0.131401i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 12.0000 0.453882
\(700\) 0 0
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 0 0
\(703\) −2.00000 + 3.46410i −0.0754314 + 0.130651i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 5.00000 + 8.66025i 0.187779 + 0.325243i 0.944509 0.328484i \(-0.106538\pi\)
−0.756730 + 0.653727i \(0.773204\pi\)
\(710\) 0 0
\(711\) 3.00000 + 1.73205i 0.112509 + 0.0649570i
\(712\) 0 0
\(713\) 27.7128i 1.03785i
\(714\) 0 0
\(715\) 41.5692i 1.55460i
\(716\) 0 0
\(717\) −18.0000 10.3923i −0.672222 0.388108i
\(718\) 0 0
\(719\) −24.0000 41.5692i −0.895049 1.55027i −0.833744 0.552151i \(-0.813807\pi\)
−0.0613050 0.998119i \(-0.519526\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −48.0000 + 27.7128i −1.78514 + 1.03065i
\(724\) 0 0
\(725\) 21.0000 36.3731i 0.779920 1.35086i
\(726\) 0 0
\(727\) −28.0000 −1.03846 −0.519231 0.854634i \(-0.673782\pi\)
−0.519231 + 0.854634i \(0.673782\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −45.0000 + 25.9808i −1.66211 + 0.959621i −0.690409 + 0.723419i \(0.742569\pi\)
−0.971704 + 0.236202i \(0.924097\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 6.00000 + 10.3923i 0.221013 + 0.382805i
\(738\) 0 0
\(739\) −9.00000 5.19615i −0.331070 0.191144i 0.325246 0.945629i \(-0.394553\pi\)
−0.656316 + 0.754486i \(0.727886\pi\)
\(740\) 0 0
\(741\) 13.8564i 0.509028i
\(742\) 0 0
\(743\) 24.2487i 0.889599i −0.895630 0.444799i \(-0.853275\pi\)
0.895630 0.444799i \(-0.146725\pi\)
\(744\) 0 0
\(745\) 18.0000 + 10.3923i 0.659469 + 0.380745i
\(746\) 0 0
\(747\) −3.00000 5.19615i −0.109764 0.190117i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 39.0000 22.5167i 1.42313 0.821645i 0.426565 0.904457i \(-0.359724\pi\)
0.996565 + 0.0828123i \(0.0263902\pi\)
\(752\) 0 0
\(753\) −30.0000 + 51.9615i −1.09326 + 1.89358i
\(754\) 0 0
\(755\) 12.0000 0.436725
\(756\) 0 0
\(757\) −10.0000 −0.363456 −0.181728 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) 0 0
\(759\) 12.0000 20.7846i 0.435572 0.754434i
\(760\) 0 0
\(761\) −18.0000 + 10.3923i −0.652499 + 0.376721i −0.789413 0.613862i \(-0.789615\pi\)
0.136914 + 0.990583i \(0.456282\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 18.0000 + 10.3923i 0.649942 + 0.375244i
\(768\) 0 0
\(769\) 13.8564i 0.499675i 0.968288 + 0.249837i \(0.0803772\pi\)
−0.968288 + 0.249837i \(0.919623\pi\)
\(770\) 0 0
\(771\) 55.4256i 1.99611i
\(772\) 0 0
\(773\) 3.00000 + 1.73205i 0.107903 + 0.0622975i 0.552980 0.833194i \(-0.313491\pi\)
−0.445078 + 0.895492i \(0.646824\pi\)
\(774\) 0 0
\(775\) 28.0000 + 48.4974i 1.00579 + 1.74208i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 12.0000 6.92820i 0.429945 0.248229i
\(780\) 0 0
\(781\) 6.00000 10.3923i 0.214697 0.371866i
\(782\) 0 0
\(783\) 24.0000 0.857690
\(784\) 0 0
\(785\) −36.0000 −1.28490
\(786\) 0 0
\(787\) 17.0000 29.4449i 0.605985 1.04960i −0.385911 0.922536i \(-0.626113\pi\)
0.991895 0.127060i \(-0.0405540\pi\)
\(788\) 0 0
\(789\) −42.0000 + 24.2487i −1.49524 + 0.863277i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 6.00000 + 10.3923i 0.213066 + 0.369042i
\(794\) 0 0
\(795\) −36.0000 20.7846i −1.27679 0.737154i
\(796\) 0 0
\(797\) 17.3205i 0.613524i 0.951786 + 0.306762i \(0.0992455\pi\)
−0.951786 + 0.306762i \(0.900754\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 6.00000 + 3.46410i 0.212000 + 0.122398i
\(802\) 0 0
\(803\) −12.0000 20.7846i −0.423471 0.733473i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −6.00000 + 3.46410i −0.211210 + 0.121942i
\(808\) 0 0
\(809\) −15.0000 + 25.9808i −0.527372 + 0.913435i 0.472119 + 0.881535i \(0.343489\pi\)
−0.999491 + 0.0319002i \(0.989844\pi\)
\(810\) 0 0
\(811\) −22.0000 −0.772524 −0.386262 0.922389i \(-0.626234\pi\)
−0.386262 + 0.922389i \(0.626234\pi\)
\(812\) 0 0
\(813\) −16.0000 −0.561144
\(814\) 0 0
\(815\) 30.0000 51.9615i 1.05085 1.82013i
\(816\) 0 0
\(817\) −18.0000 + 10.3923i −0.629740 + 0.363581i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −15.0000 25.9808i −0.523504 0.906735i −0.999626 0.0273557i \(-0.991291\pi\)
0.476122 0.879379i \(-0.342042\pi\)
\(822\) 0 0
\(823\) 3.00000 + 1.73205i 0.104573 + 0.0603755i 0.551375 0.834258i \(-0.314104\pi\)
−0.446801 + 0.894633i \(0.647437\pi\)
\(824\) 0 0
\(825\) 48.4974i 1.68846i
\(826\) 0 0
\(827\) 24.2487i 0.843210i 0.906780 + 0.421605i \(0.138533\pi\)
−0.906780 + 0.421605i \(0.861467\pi\)
\(828\) 0 0
\(829\) −3.00000 1.73205i −0.104194 0.0601566i 0.446997 0.894535i \(-0.352493\pi\)
−0.551192 + 0.834379i \(0.685827\pi\)
\(830\) 0 0
\(831\) −2.00000 3.46410i −0.0693792 0.120168i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −36.0000 + 20.7846i −1.24583 + 0.719281i
\(836\) 0 0
\(837\) −16.0000 + 27.7128i −0.553041 + 0.957895i
\(838\) 0 0
\(839\) 12.0000 0.414286 0.207143 0.978311i \(-0.433583\pi\)
0.207143 + 0.978311i \(0.433583\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) −6.00000 + 10.3923i −0.206651 + 0.357930i
\(844\) 0 0
\(845\) −3.00000 + 1.73205i −0.103203 + 0.0595844i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −14.0000 24.2487i −0.480479 0.832214i
\(850\) 0 0
\(851\) 6.00000 + 3.46410i 0.205677 + 0.118748i
\(852\) 0 0
\(853\) 3.46410i 0.118609i −0.998240 0.0593043i \(-0.981112\pi\)
0.998240 0.0593043i \(-0.0188882\pi\)
\(854\) 0 0
\(855\) 6.92820i 0.236940i
\(856\) 0 0
\(857\) −30.0000 17.3205i −1.02478 0.591657i −0.109295 0.994009i \(-0.534859\pi\)
−0.915485 + 0.402352i \(0.868193\pi\)
\(858\) 0 0
\(859\) 7.00000 + 12.1244i 0.238837 + 0.413678i 0.960381 0.278691i \(-0.0899005\pi\)
−0.721544 + 0.692369i \(0.756567\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −27.0000 + 15.5885i −0.919091 + 0.530637i −0.883345 0.468724i \(-0.844714\pi\)
−0.0357458 + 0.999361i \(0.511381\pi\)
\(864\) 0 0
\(865\) −6.00000 + 10.3923i −0.204006 + 0.353349i
\(866\) 0 0
\(867\) −34.0000 −1.15470
\(868\) 0 0
\(869\) −12.0000 −0.407072
\(870\) 0 0
\(871\) −6.00000 + 10.3923i −0.203302 + 0.352130i
\(872\) 0 0
\(873\) 12.0000 6.92820i 0.406138 0.234484i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −19.0000 32.9090i −0.641584 1.11126i −0.985079 0.172102i \(-0.944944\pi\)
0.343495 0.939155i \(-0.388389\pi\)
\(878\) 0 0
\(879\) 18.0000 + 10.3923i 0.607125 + 0.350524i
\(880\) 0 0
\(881\) 13.8564i 0.466834i 0.972377 + 0.233417i \(0.0749907\pi\)
−0.972377 + 0.233417i \(0.925009\pi\)
\(882\) 0 0
\(883\) 3.46410i 0.116576i −0.998300 0.0582882i \(-0.981436\pi\)
0.998300 0.0582882i \(-0.0185642\pi\)
\(884\) 0 0
\(885\) 36.0000 + 20.7846i 1.21013 + 0.698667i
\(886\) 0 0
\(887\) 6.00000 + 10.3923i 0.201460 + 0.348939i 0.948999 0.315279i \(-0.102098\pi\)
−0.747539 + 0.664218i \(0.768765\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 33.0000 19.0526i 1.10554 0.638285i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 36.0000 1.20335
\(896\) 0 0
\(897\) 24.0000 0.801337
\(898\) 0 0
\(899\) −24.0000 + 41.5692i −0.800445 + 1.38641i
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 30.0000 + 51.9615i 0.997234 + 1.72726i
\(906\) 0 0
\(907\) −21.0000 12.1244i −0.697294 0.402583i 0.109045 0.994037i \(-0.465221\pi\)
−0.806339 + 0.591454i \(0.798554\pi\)
\(908\) 0 0
\(909\) 10.3923i 0.344691i
\(910\) 0 0
\(911\) 10.3923i 0.344312i −0.985070 0.172156i \(-0.944927\pi\)
0.985070 0.172156i \(-0.0550734\pi\)
\(912\) 0 0
\(913\) 18.0000 + 10.3923i 0.595713 + 0.343935i
\(914\) 0 0
\(915\) 12.0000 + 20.7846i 0.396708 + 0.687118i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −27.0000 + 15.5885i −0.890648 + 0.514216i −0.874154 0.485648i \(-0.838584\pi\)
−0.0164935 + 0.999864i \(0.505250\pi\)
\(920\) 0 0
\(921\) 22.0000 38.1051i 0.724925 1.25561i
\(922\) 0 0
\(923\) 12.0000 0.394985
\(924\) 0 0
\(925\) 14.0000 0.460317
\(926\) 0 0
\(927\) 2.00000 3.46410i 0.0656886 0.113776i
\(928\) 0 0
\(929\) 36.0000 20.7846i 1.18112 0.681921i 0.224848 0.974394i \(-0.427811\pi\)
0.956274 + 0.292473i \(0.0944781\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 12.0000 + 20.7846i 0.392862 + 0.680458i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 48.4974i 1.58434i −0.610299 0.792171i \(-0.708951\pi\)
0.610299 0.792171i \(-0.291049\pi\)
\(938\) 0 0
\(939\) 13.8564i 0.452187i
\(940\) 0 0
\(941\) 21.0000 + 12.1244i 0.684580 + 0.395243i 0.801579 0.597889i \(-0.203994\pi\)
−0.116998 + 0.993132i \(0.537327\pi\)
\(942\) 0 0
\(943\) −12.0000 20.7846i −0.390774 0.676840i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 21.0000 12.1244i 0.682408 0.393989i −0.118354 0.992972i \(-0.537762\pi\)
0.800762 + 0.598983i \(0.204428\pi\)
\(948\) 0 0
\(949\) 12.0000 20.7846i 0.389536 0.674697i
\(950\) 0 0
\(951\) −12.0000 −0.389127
\(952\) 0 0
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 0 0
\(955\) −42.0000 + 72.7461i −1.35909 + 2.35401i
\(956\) 0 0
\(957\) 36.0000 20.7846i 1.16371 0.671871i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −16.5000 28.5788i −0.532258 0.921898i
\(962\) 0 0
\(963\) −9.00000 5.19615i −0.290021 0.167444i
\(964\) 0 0
\(965\) 6.92820i 0.223027i
\(966\) 0 0
\(967\) 24.2487i 0.779786i −0.920860 0.389893i \(-0.872512\pi\)
0.920860 0.389893i \(-0.127488\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −9.00000 15.5885i −0.288824 0.500257i 0.684706 0.728820i \(-0.259931\pi\)
−0.973529 + 0.228562i \(0.926597\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 42.0000 24.2487i 1.34508 0.776580i
\(976\) 0 0
\(977\) 15.0000 25.9808i 0.479893 0.831198i −0.519841 0.854263i \(-0.674009\pi\)
0.999734 + 0.0230645i \(0.00734232\pi\)
\(978\) 0 0
\(979\) −24.0000 −0.767043
\(980\) 0 0
\(981\) 14.0000 0.446986
\(982\) 0 0
\(983\) −18.0000 + 31.1769i −0.574111 + 0.994389i 0.422027 + 0.906583i \(0.361319\pi\)
−0.996138 + 0.0878058i \(0.972015\pi\)
\(984\) 0 0
\(985\) 54.0000 31.1769i 1.72058 0.993379i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 18.0000 + 31.1769i 0.572367 + 0.991368i
\(990\) 0 0
\(991\) 3.00000 + 1.73205i 0.0952981 + 0.0550204i 0.546892 0.837203i \(-0.315811\pi\)
−0.451594 + 0.892224i \(0.649144\pi\)
\(992\) 0 0
\(993\) 6.92820i 0.219860i
\(994\) 0 0
\(995\) 69.2820i 2.19639i
\(996\) 0 0
\(997\) −45.0000 25.9808i −1.42516 0.822819i −0.428430 0.903575i \(-0.640933\pi\)
−0.996734 + 0.0807558i \(0.974267\pi\)
\(998\) 0 0
\(999\) 4.00000 + 6.92820i 0.126554 + 0.219199i
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 784.2.p.e.31.1 2
4.3 odd 2 784.2.p.a.31.1 2
7.2 even 3 784.2.p.f.607.1 2
7.3 odd 6 112.2.f.b.111.2 yes 2
7.4 even 3 112.2.f.a.111.1 2
7.5 odd 6 784.2.p.a.607.1 2
7.6 odd 2 784.2.p.b.31.1 2
21.11 odd 6 1008.2.b.g.559.2 2
21.17 even 6 1008.2.b.b.559.1 2
28.3 even 6 112.2.f.a.111.2 yes 2
28.11 odd 6 112.2.f.b.111.1 yes 2
28.19 even 6 inner 784.2.p.e.607.1 2
28.23 odd 6 784.2.p.b.607.1 2
28.27 even 2 784.2.p.f.31.1 2
35.3 even 12 2800.2.e.b.2799.3 4
35.4 even 6 2800.2.k.e.2351.2 2
35.17 even 12 2800.2.e.b.2799.2 4
35.18 odd 12 2800.2.e.c.2799.1 4
35.24 odd 6 2800.2.k.b.2351.2 2
35.32 odd 12 2800.2.e.c.2799.4 4
56.3 even 6 448.2.f.c.447.1 2
56.11 odd 6 448.2.f.a.447.2 2
56.45 odd 6 448.2.f.a.447.1 2
56.53 even 6 448.2.f.c.447.2 2
84.11 even 6 1008.2.b.b.559.2 2
84.59 odd 6 1008.2.b.g.559.1 2
112.3 even 12 1792.2.e.a.895.2 4
112.11 odd 12 1792.2.e.c.895.2 4
112.45 odd 12 1792.2.e.c.895.4 4
112.53 even 12 1792.2.e.a.895.4 4
112.59 even 12 1792.2.e.a.895.3 4
112.67 odd 12 1792.2.e.c.895.3 4
112.101 odd 12 1792.2.e.c.895.1 4
112.109 even 12 1792.2.e.a.895.1 4
140.3 odd 12 2800.2.e.c.2799.2 4
140.39 odd 6 2800.2.k.b.2351.1 2
140.59 even 6 2800.2.k.e.2351.1 2
140.67 even 12 2800.2.e.b.2799.1 4
140.87 odd 12 2800.2.e.c.2799.3 4
140.123 even 12 2800.2.e.b.2799.4 4
168.11 even 6 4032.2.b.b.3583.1 2
168.53 odd 6 4032.2.b.h.3583.1 2
168.59 odd 6 4032.2.b.h.3583.2 2
168.101 even 6 4032.2.b.b.3583.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.2.f.a.111.1 2 7.4 even 3
112.2.f.a.111.2 yes 2 28.3 even 6
112.2.f.b.111.1 yes 2 28.11 odd 6
112.2.f.b.111.2 yes 2 7.3 odd 6
448.2.f.a.447.1 2 56.45 odd 6
448.2.f.a.447.2 2 56.11 odd 6
448.2.f.c.447.1 2 56.3 even 6
448.2.f.c.447.2 2 56.53 even 6
784.2.p.a.31.1 2 4.3 odd 2
784.2.p.a.607.1 2 7.5 odd 6
784.2.p.b.31.1 2 7.6 odd 2
784.2.p.b.607.1 2 28.23 odd 6
784.2.p.e.31.1 2 1.1 even 1 trivial
784.2.p.e.607.1 2 28.19 even 6 inner
784.2.p.f.31.1 2 28.27 even 2
784.2.p.f.607.1 2 7.2 even 3
1008.2.b.b.559.1 2 21.17 even 6
1008.2.b.b.559.2 2 84.11 even 6
1008.2.b.g.559.1 2 84.59 odd 6
1008.2.b.g.559.2 2 21.11 odd 6
1792.2.e.a.895.1 4 112.109 even 12
1792.2.e.a.895.2 4 112.3 even 12
1792.2.e.a.895.3 4 112.59 even 12
1792.2.e.a.895.4 4 112.53 even 12
1792.2.e.c.895.1 4 112.101 odd 12
1792.2.e.c.895.2 4 112.11 odd 12
1792.2.e.c.895.3 4 112.67 odd 12
1792.2.e.c.895.4 4 112.45 odd 12
2800.2.e.b.2799.1 4 140.67 even 12
2800.2.e.b.2799.2 4 35.17 even 12
2800.2.e.b.2799.3 4 35.3 even 12
2800.2.e.b.2799.4 4 140.123 even 12
2800.2.e.c.2799.1 4 35.18 odd 12
2800.2.e.c.2799.2 4 140.3 odd 12
2800.2.e.c.2799.3 4 140.87 odd 12
2800.2.e.c.2799.4 4 35.32 odd 12
2800.2.k.b.2351.1 2 140.39 odd 6
2800.2.k.b.2351.2 2 35.24 odd 6
2800.2.k.e.2351.1 2 140.59 even 6
2800.2.k.e.2351.2 2 35.4 even 6
4032.2.b.b.3583.1 2 168.11 even 6
4032.2.b.b.3583.2 2 168.101 even 6
4032.2.b.h.3583.1 2 168.53 odd 6
4032.2.b.h.3583.2 2 168.59 odd 6