Properties

Label 336.2.bj.f.191.2
Level $336$
Weight $2$
Character 336.191
Analytic conductor $2.683$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [336,2,Mod(95,336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(336, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.95");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 336.bj (of order \(6\), 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: \(2.68297350792\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.303595776.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 191.2
Root \(-0.396143 - 1.68614i\) of defining polynomial
Character \(\chi\) \(=\) 336.191
Dual form 336.2.bj.f.95.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.396143 - 1.68614i) q^{3} +(-2.87228 - 1.65831i) q^{5} +(1.73205 - 2.00000i) q^{7} +(-2.68614 + 1.33591i) q^{9} +O(q^{10})\) \(q+(-0.396143 - 1.68614i) q^{3} +(-2.87228 - 1.65831i) q^{5} +(1.73205 - 2.00000i) q^{7} +(-2.68614 + 1.33591i) q^{9} +(1.65831 + 2.87228i) q^{11} -4.00000 q^{13} +(-1.65831 + 5.50000i) q^{15} +(-2.87228 + 1.65831i) q^{17} +(-6.06218 - 3.50000i) q^{19} +(-4.05842 - 2.12819i) q^{21} +(1.65831 - 2.87228i) q^{23} +(3.00000 + 5.19615i) q^{25} +(3.31662 + 4.00000i) q^{27} -6.63325i q^{29} +(2.59808 - 1.50000i) q^{31} +(4.18614 - 3.93398i) q^{33} +(-8.29156 + 2.87228i) q^{35} +(0.500000 - 0.866025i) q^{37} +(1.58457 + 6.74456i) q^{39} +6.63325i q^{41} -2.00000i q^{43} +(9.93070 + 0.617359i) q^{45} +(4.97494 - 8.61684i) q^{47} +(-1.00000 - 6.92820i) q^{49} +(3.93398 + 4.18614i) q^{51} +(2.87228 - 1.65831i) q^{53} -11.0000i q^{55} +(-3.50000 + 11.6082i) q^{57} +(-1.65831 - 2.87228i) q^{59} +(-1.50000 + 2.59808i) q^{61} +(-1.98072 + 7.68614i) q^{63} +(11.4891 + 6.63325i) q^{65} +(7.79423 - 4.50000i) q^{67} +(-5.50000 - 1.65831i) q^{69} -13.2665 q^{71} +(-3.50000 - 6.06218i) q^{73} +(7.57301 - 7.11684i) q^{75} +(8.61684 + 1.65831i) q^{77} +(-7.79423 - 4.50000i) q^{79} +(5.43070 - 7.17687i) q^{81} +13.2665 q^{83} +11.0000 q^{85} +(-11.1846 + 2.62772i) q^{87} +(2.87228 + 1.65831i) q^{89} +(-6.92820 + 8.00000i) q^{91} +(-3.55842 - 3.78651i) q^{93} +(11.6082 + 20.1060i) q^{95} +8.00000 q^{97} +(-8.29156 - 5.50000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 10 q^{9} - 32 q^{13} + 2 q^{21} + 24 q^{25} + 22 q^{33} + 4 q^{37} + 22 q^{45} - 8 q^{49} - 28 q^{57} - 12 q^{61} - 44 q^{69} - 28 q^{73} - 14 q^{81} + 88 q^{85} + 6 q^{93} + 64 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(e\left(\frac{1}{3}\right)\)

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
\(3\) −0.396143 1.68614i −0.228714 0.973494i
\(4\) 0 0
\(5\) −2.87228 1.65831i −1.28452 0.741620i −0.306851 0.951757i \(-0.599275\pi\)
−0.977672 + 0.210138i \(0.932609\pi\)
\(6\) 0 0
\(7\) 1.73205 2.00000i 0.654654 0.755929i
\(8\) 0 0
\(9\) −2.68614 + 1.33591i −0.895380 + 0.445302i
\(10\) 0 0
\(11\) 1.65831 + 2.87228i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) −1.65831 + 5.50000i −0.428174 + 1.42009i
\(16\) 0 0
\(17\) −2.87228 + 1.65831i −0.696631 + 0.402200i −0.806091 0.591791i \(-0.798421\pi\)
0.109461 + 0.993991i \(0.465088\pi\)
\(18\) 0 0
\(19\) −6.06218 3.50000i −1.39076 0.802955i −0.397360 0.917663i \(-0.630073\pi\)
−0.993399 + 0.114708i \(0.963407\pi\)
\(20\) 0 0
\(21\) −4.05842 2.12819i −0.885620 0.464410i
\(22\) 0 0
\(23\) 1.65831 2.87228i 0.345782 0.598912i −0.639713 0.768613i \(-0.720947\pi\)
0.985496 + 0.169701i \(0.0542803\pi\)
\(24\) 0 0
\(25\) 3.00000 + 5.19615i 0.600000 + 1.03923i
\(26\) 0 0
\(27\) 3.31662 + 4.00000i 0.638285 + 0.769800i
\(28\) 0 0
\(29\) 6.63325i 1.23176i −0.787839 0.615882i \(-0.788800\pi\)
0.787839 0.615882i \(-0.211200\pi\)
\(30\) 0 0
\(31\) 2.59808 1.50000i 0.466628 0.269408i −0.248199 0.968709i \(-0.579839\pi\)
0.714827 + 0.699301i \(0.246505\pi\)
\(32\) 0 0
\(33\) 4.18614 3.93398i 0.728714 0.684819i
\(34\) 0 0
\(35\) −8.29156 + 2.87228i −1.40153 + 0.485504i
\(36\) 0 0
\(37\) 0.500000 0.866025i 0.0821995 0.142374i −0.821995 0.569495i \(-0.807139\pi\)
0.904194 + 0.427121i \(0.140472\pi\)
\(38\) 0 0
\(39\) 1.58457 + 6.74456i 0.253735 + 1.07999i
\(40\) 0 0
\(41\) 6.63325i 1.03594i 0.855399 + 0.517970i \(0.173312\pi\)
−0.855399 + 0.517970i \(0.826688\pi\)
\(42\) 0 0
\(43\) 2.00000i 0.304997i −0.988304 0.152499i \(-0.951268\pi\)
0.988304 0.152499i \(-0.0487319\pi\)
\(44\) 0 0
\(45\) 9.93070 + 0.617359i 1.48038 + 0.0920304i
\(46\) 0 0
\(47\) 4.97494 8.61684i 0.725669 1.25690i −0.233029 0.972470i \(-0.574864\pi\)
0.958698 0.284426i \(-0.0918030\pi\)
\(48\) 0 0
\(49\) −1.00000 6.92820i −0.142857 0.989743i
\(50\) 0 0
\(51\) 3.93398 + 4.18614i 0.550868 + 0.586177i
\(52\) 0 0
\(53\) 2.87228 1.65831i 0.394538 0.227787i −0.289586 0.957152i \(-0.593518\pi\)
0.684125 + 0.729365i \(0.260184\pi\)
\(54\) 0 0
\(55\) 11.0000i 1.48324i
\(56\) 0 0
\(57\) −3.50000 + 11.6082i −0.463586 + 1.53754i
\(58\) 0 0
\(59\) −1.65831 2.87228i −0.215894 0.373939i 0.737655 0.675178i \(-0.235933\pi\)
−0.953549 + 0.301239i \(0.902600\pi\)
\(60\) 0 0
\(61\) −1.50000 + 2.59808i −0.192055 + 0.332650i −0.945931 0.324367i \(-0.894849\pi\)
0.753876 + 0.657017i \(0.228182\pi\)
\(62\) 0 0
\(63\) −1.98072 + 7.68614i −0.249547 + 0.968363i
\(64\) 0 0
\(65\) 11.4891 + 6.63325i 1.42505 + 0.822753i
\(66\) 0 0
\(67\) 7.79423 4.50000i 0.952217 0.549762i 0.0584478 0.998290i \(-0.481385\pi\)
0.893769 + 0.448528i \(0.148052\pi\)
\(68\) 0 0
\(69\) −5.50000 1.65831i −0.662122 0.199637i
\(70\) 0 0
\(71\) −13.2665 −1.57444 −0.787222 0.616670i \(-0.788481\pi\)
−0.787222 + 0.616670i \(0.788481\pi\)
\(72\) 0 0
\(73\) −3.50000 6.06218i −0.409644 0.709524i 0.585206 0.810885i \(-0.301014\pi\)
−0.994850 + 0.101361i \(0.967680\pi\)
\(74\) 0 0
\(75\) 7.57301 7.11684i 0.874456 0.821782i
\(76\) 0 0
\(77\) 8.61684 + 1.65831i 0.981981 + 0.188982i
\(78\) 0 0
\(79\) −7.79423 4.50000i −0.876919 0.506290i −0.00727784 0.999974i \(-0.502317\pi\)
−0.869641 + 0.493684i \(0.835650\pi\)
\(80\) 0 0
\(81\) 5.43070 7.17687i 0.603411 0.797430i
\(82\) 0 0
\(83\) 13.2665 1.45619 0.728094 0.685478i \(-0.240407\pi\)
0.728094 + 0.685478i \(0.240407\pi\)
\(84\) 0 0
\(85\) 11.0000 1.19312
\(86\) 0 0
\(87\) −11.1846 + 2.62772i −1.19911 + 0.281721i
\(88\) 0 0
\(89\) 2.87228 + 1.65831i 0.304461 + 0.175781i 0.644445 0.764650i \(-0.277088\pi\)
−0.339984 + 0.940431i \(0.610422\pi\)
\(90\) 0 0
\(91\) −6.92820 + 8.00000i −0.726273 + 0.838628i
\(92\) 0 0
\(93\) −3.55842 3.78651i −0.368991 0.392642i
\(94\) 0 0
\(95\) 11.6082 + 20.1060i 1.19097 + 2.06283i
\(96\) 0 0
\(97\) 8.00000 0.812277 0.406138 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) 0 0
\(99\) −8.29156 5.50000i −0.833333 0.552771i
\(100\) 0 0
\(101\) −2.87228 + 1.65831i −0.285803 + 0.165008i −0.636047 0.771650i \(-0.719432\pi\)
0.350245 + 0.936658i \(0.386098\pi\)
\(102\) 0 0
\(103\) 9.52628 + 5.50000i 0.938652 + 0.541931i 0.889538 0.456862i \(-0.151027\pi\)
0.0491146 + 0.998793i \(0.484360\pi\)
\(104\) 0 0
\(105\) 8.12772 + 12.8429i 0.793184 + 1.25334i
\(106\) 0 0
\(107\) −1.65831 + 2.87228i −0.160315 + 0.277674i −0.934982 0.354696i \(-0.884584\pi\)
0.774667 + 0.632370i \(0.217918\pi\)
\(108\) 0 0
\(109\) −2.50000 4.33013i −0.239457 0.414751i 0.721102 0.692829i \(-0.243636\pi\)
−0.960558 + 0.278078i \(0.910303\pi\)
\(110\) 0 0
\(111\) −1.65831 0.500000i −0.157400 0.0474579i
\(112\) 0 0
\(113\) 6.63325i 0.624004i −0.950082 0.312002i \(-0.899000\pi\)
0.950082 0.312002i \(-0.101000\pi\)
\(114\) 0 0
\(115\) −9.52628 + 5.50000i −0.888330 + 0.512878i
\(116\) 0 0
\(117\) 10.7446 5.34363i 0.993335 0.494019i
\(118\) 0 0
\(119\) −1.65831 + 8.61684i −0.152017 + 0.789905i
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 11.1846 2.62772i 1.00848 0.236933i
\(124\) 0 0
\(125\) 3.31662i 0.296648i
\(126\) 0 0
\(127\) 2.00000i 0.177471i −0.996055 0.0887357i \(-0.971717\pi\)
0.996055 0.0887357i \(-0.0282826\pi\)
\(128\) 0 0
\(129\) −3.37228 + 0.792287i −0.296913 + 0.0697570i
\(130\) 0 0
\(131\) 8.29156 14.3614i 0.724437 1.25476i −0.234768 0.972051i \(-0.575433\pi\)
0.959205 0.282711i \(-0.0912336\pi\)
\(132\) 0 0
\(133\) −17.5000 + 6.06218i −1.51744 + 0.525657i
\(134\) 0 0
\(135\) −2.89303 16.9891i −0.248992 1.46219i
\(136\) 0 0
\(137\) −14.3614 + 8.29156i −1.22698 + 0.708396i −0.966397 0.257056i \(-0.917248\pi\)
−0.260581 + 0.965452i \(0.583914\pi\)
\(138\) 0 0
\(139\) 10.0000i 0.848189i 0.905618 + 0.424094i \(0.139408\pi\)
−0.905618 + 0.424094i \(0.860592\pi\)
\(140\) 0 0
\(141\) −16.5000 4.97494i −1.38955 0.418965i
\(142\) 0 0
\(143\) −6.63325 11.4891i −0.554700 0.960769i
\(144\) 0 0
\(145\) −11.0000 + 19.0526i −0.913500 + 1.58223i
\(146\) 0 0
\(147\) −11.2858 + 4.43070i −0.930836 + 0.365438i
\(148\) 0 0
\(149\) −14.3614 8.29156i −1.17653 0.679271i −0.221322 0.975201i \(-0.571037\pi\)
−0.955210 + 0.295929i \(0.904371\pi\)
\(150\) 0 0
\(151\) 6.06218 3.50000i 0.493333 0.284826i −0.232623 0.972567i \(-0.574731\pi\)
0.725956 + 0.687741i \(0.241398\pi\)
\(152\) 0 0
\(153\) 5.50000 8.29156i 0.444649 0.670333i
\(154\) 0 0
\(155\) −9.94987 −0.799193
\(156\) 0 0
\(157\) −1.50000 2.59808i −0.119713 0.207349i 0.799941 0.600079i \(-0.204864\pi\)
−0.919654 + 0.392730i \(0.871531\pi\)
\(158\) 0 0
\(159\) −3.93398 4.18614i −0.311985 0.331983i
\(160\) 0 0
\(161\) −2.87228 8.29156i −0.226367 0.653467i
\(162\) 0 0
\(163\) 6.06218 + 3.50000i 0.474826 + 0.274141i 0.718258 0.695777i \(-0.244940\pi\)
−0.243432 + 0.969918i \(0.578273\pi\)
\(164\) 0 0
\(165\) −18.5475 + 4.35758i −1.44392 + 0.339237i
\(166\) 0 0
\(167\) −6.63325 −0.513296 −0.256648 0.966505i \(-0.582618\pi\)
−0.256648 + 0.966505i \(0.582618\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 20.9595 + 1.30298i 1.60282 + 0.0996417i
\(172\) 0 0
\(173\) −8.61684 4.97494i −0.655127 0.378237i 0.135291 0.990806i \(-0.456803\pi\)
−0.790418 + 0.612568i \(0.790136\pi\)
\(174\) 0 0
\(175\) 15.5885 + 3.00000i 1.17838 + 0.226779i
\(176\) 0 0
\(177\) −4.18614 + 3.93398i −0.314650 + 0.295696i
\(178\) 0 0
\(179\) 1.65831 + 2.87228i 0.123948 + 0.214684i 0.921321 0.388802i \(-0.127111\pi\)
−0.797373 + 0.603487i \(0.793778\pi\)
\(180\) 0 0
\(181\) 18.0000 1.33793 0.668965 0.743294i \(-0.266738\pi\)
0.668965 + 0.743294i \(0.266738\pi\)
\(182\) 0 0
\(183\) 4.97494 + 1.50000i 0.367758 + 0.110883i
\(184\) 0 0
\(185\) −2.87228 + 1.65831i −0.211174 + 0.121922i
\(186\) 0 0
\(187\) −9.52628 5.50000i −0.696631 0.402200i
\(188\) 0 0
\(189\) 13.7446 + 0.294954i 0.999770 + 0.0214547i
\(190\) 0 0
\(191\) 4.97494 8.61684i 0.359974 0.623493i −0.627982 0.778228i \(-0.716119\pi\)
0.987956 + 0.154735i \(0.0494523\pi\)
\(192\) 0 0
\(193\) 1.50000 + 2.59808i 0.107972 + 0.187014i 0.914949 0.403570i \(-0.132231\pi\)
−0.806976 + 0.590584i \(0.798898\pi\)
\(194\) 0 0
\(195\) 6.63325 22.0000i 0.475017 1.57545i
\(196\) 0 0
\(197\) 19.8997i 1.41780i 0.705310 + 0.708899i \(0.250808\pi\)
−0.705310 + 0.708899i \(0.749192\pi\)
\(198\) 0 0
\(199\) −0.866025 + 0.500000i −0.0613909 + 0.0354441i −0.530381 0.847759i \(-0.677951\pi\)
0.468990 + 0.883203i \(0.344618\pi\)
\(200\) 0 0
\(201\) −10.6753 11.3595i −0.752975 0.801239i
\(202\) 0 0
\(203\) −13.2665 11.4891i −0.931126 0.806379i
\(204\) 0 0
\(205\) 11.0000 19.0526i 0.768273 1.33069i
\(206\) 0 0
\(207\) −0.617359 + 9.93070i −0.0429094 + 0.690232i
\(208\) 0 0
\(209\) 23.2164i 1.60591i
\(210\) 0 0
\(211\) 26.0000i 1.78991i 0.446153 + 0.894957i \(0.352794\pi\)
−0.446153 + 0.894957i \(0.647206\pi\)
\(212\) 0 0
\(213\) 5.25544 + 22.3692i 0.360097 + 1.53271i
\(214\) 0 0
\(215\) −3.31662 + 5.74456i −0.226192 + 0.391776i
\(216\) 0 0
\(217\) 1.50000 7.79423i 0.101827 0.529107i
\(218\) 0 0
\(219\) −8.83518 + 8.30298i −0.597026 + 0.561064i
\(220\) 0 0
\(221\) 11.4891 6.63325i 0.772842 0.446201i
\(222\) 0 0
\(223\) 14.0000i 0.937509i 0.883328 + 0.468755i \(0.155297\pi\)
−0.883328 + 0.468755i \(0.844703\pi\)
\(224\) 0 0
\(225\) −15.0000 9.94987i −1.00000 0.663325i
\(226\) 0 0
\(227\) 1.65831 + 2.87228i 0.110066 + 0.190640i 0.915797 0.401642i \(-0.131560\pi\)
−0.805731 + 0.592282i \(0.798227\pi\)
\(228\) 0 0
\(229\) −14.5000 + 25.1147i −0.958187 + 1.65963i −0.231287 + 0.972886i \(0.574293\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 0 0
\(231\) −0.617359 15.1861i −0.0406192 0.999175i
\(232\) 0 0
\(233\) −14.3614 8.29156i −0.940847 0.543198i −0.0506213 0.998718i \(-0.516120\pi\)
−0.890226 + 0.455520i \(0.849453\pi\)
\(234\) 0 0
\(235\) −28.5788 + 16.5000i −1.86428 + 1.07634i
\(236\) 0 0
\(237\) −4.50000 + 14.9248i −0.292306 + 0.969471i
\(238\) 0 0
\(239\) 6.63325 0.429069 0.214535 0.976716i \(-0.431177\pi\)
0.214535 + 0.976716i \(0.431177\pi\)
\(240\) 0 0
\(241\) −9.50000 16.4545i −0.611949 1.05993i −0.990912 0.134515i \(-0.957053\pi\)
0.378963 0.925412i \(-0.376281\pi\)
\(242\) 0 0
\(243\) −14.2525 6.31386i −0.914302 0.405034i
\(244\) 0 0
\(245\) −8.61684 + 21.5581i −0.550510 + 1.37729i
\(246\) 0 0
\(247\) 24.2487 + 14.0000i 1.54291 + 0.890799i
\(248\) 0 0
\(249\) −5.25544 22.3692i −0.333050 1.41759i
\(250\) 0 0
\(251\) 19.8997 1.25606 0.628031 0.778189i \(-0.283861\pi\)
0.628031 + 0.778189i \(0.283861\pi\)
\(252\) 0 0
\(253\) 11.0000 0.691564
\(254\) 0 0
\(255\) −4.35758 18.5475i −0.272882 1.16149i
\(256\) 0 0
\(257\) 20.1060 + 11.6082i 1.25418 + 0.724099i 0.971936 0.235245i \(-0.0755893\pi\)
0.282240 + 0.959344i \(0.408923\pi\)
\(258\) 0 0
\(259\) −0.866025 2.50000i −0.0538122 0.155342i
\(260\) 0 0
\(261\) 8.86141 + 17.8178i 0.548507 + 1.10290i
\(262\) 0 0
\(263\) −1.65831 2.87228i −0.102256 0.177112i 0.810358 0.585935i \(-0.199273\pi\)
−0.912614 + 0.408823i \(0.865939\pi\)
\(264\) 0 0
\(265\) −11.0000 −0.675725
\(266\) 0 0
\(267\) 1.65831 5.50000i 0.101487 0.336595i
\(268\) 0 0
\(269\) 14.3614 8.29156i 0.875630 0.505545i 0.00641522 0.999979i \(-0.497958\pi\)
0.869215 + 0.494434i \(0.164625\pi\)
\(270\) 0 0
\(271\) −14.7224 8.50000i −0.894324 0.516338i −0.0189696 0.999820i \(-0.506039\pi\)
−0.875354 + 0.483482i \(0.839372\pi\)
\(272\) 0 0
\(273\) 16.2337 + 8.51278i 0.982507 + 0.515217i
\(274\) 0 0
\(275\) −9.94987 + 17.2337i −0.600000 + 1.03923i
\(276\) 0 0
\(277\) 6.50000 + 11.2583i 0.390547 + 0.676448i 0.992522 0.122068i \(-0.0389525\pi\)
−0.601975 + 0.798515i \(0.705619\pi\)
\(278\) 0 0
\(279\) −4.97494 + 7.50000i −0.297842 + 0.449013i
\(280\) 0 0
\(281\) 6.63325i 0.395706i 0.980232 + 0.197853i \(0.0633969\pi\)
−0.980232 + 0.197853i \(0.936603\pi\)
\(282\) 0 0
\(283\) 12.9904 7.50000i 0.772198 0.445829i −0.0614601 0.998110i \(-0.519576\pi\)
0.833658 + 0.552281i \(0.186242\pi\)
\(284\) 0 0
\(285\) 29.3030 27.5379i 1.73576 1.63120i
\(286\) 0 0
\(287\) 13.2665 + 11.4891i 0.783097 + 0.678182i
\(288\) 0 0
\(289\) −3.00000 + 5.19615i −0.176471 + 0.305656i
\(290\) 0 0
\(291\) −3.16915 13.4891i −0.185779 0.790747i
\(292\) 0 0
\(293\) 13.2665i 0.775037i 0.921862 + 0.387519i \(0.126668\pi\)
−0.921862 + 0.387519i \(0.873332\pi\)
\(294\) 0 0
\(295\) 11.0000i 0.640445i
\(296\) 0 0
\(297\) −5.98913 + 16.1595i −0.347524 + 0.937671i
\(298\) 0 0
\(299\) −6.63325 + 11.4891i −0.383611 + 0.664433i
\(300\) 0 0
\(301\) −4.00000 3.46410i −0.230556 0.199667i
\(302\) 0 0
\(303\) 3.93398 + 4.18614i 0.226001 + 0.240487i
\(304\) 0 0
\(305\) 8.61684 4.97494i 0.493399 0.284864i
\(306\) 0 0
\(307\) 6.00000i 0.342438i −0.985233 0.171219i \(-0.945229\pi\)
0.985233 0.171219i \(-0.0547706\pi\)
\(308\) 0 0
\(309\) 5.50000 18.2414i 0.312884 1.03772i
\(310\) 0 0
\(311\) 4.97494 + 8.61684i 0.282103 + 0.488616i 0.971902 0.235384i \(-0.0756347\pi\)
−0.689800 + 0.724000i \(0.742301\pi\)
\(312\) 0 0
\(313\) 4.50000 7.79423i 0.254355 0.440556i −0.710365 0.703833i \(-0.751470\pi\)
0.964720 + 0.263278i \(0.0848035\pi\)
\(314\) 0 0
\(315\) 18.4352 18.7921i 1.03871 1.05882i
\(316\) 0 0
\(317\) 25.8505 + 14.9248i 1.45191 + 0.838261i 0.998590 0.0530866i \(-0.0169059\pi\)
0.453321 + 0.891348i \(0.350239\pi\)
\(318\) 0 0
\(319\) 19.0526 11.0000i 1.06674 0.615882i
\(320\) 0 0
\(321\) 5.50000 + 1.65831i 0.306980 + 0.0925580i
\(322\) 0 0
\(323\) 23.2164 1.29179
\(324\) 0 0
\(325\) −12.0000 20.7846i −0.665640 1.15292i
\(326\) 0 0
\(327\) −6.31084 + 5.93070i −0.348990 + 0.327969i
\(328\) 0 0
\(329\) −8.61684 24.8747i −0.475062 1.37139i
\(330\) 0 0
\(331\) −14.7224 8.50000i −0.809218 0.467202i 0.0374662 0.999298i \(-0.488071\pi\)
−0.846684 + 0.532096i \(0.821405\pi\)
\(332\) 0 0
\(333\) −0.186141 + 2.99422i −0.0102004 + 0.164082i
\(334\) 0 0
\(335\) −29.8496 −1.63086
\(336\) 0 0
\(337\) 28.0000 1.52526 0.762629 0.646837i \(-0.223908\pi\)
0.762629 + 0.646837i \(0.223908\pi\)
\(338\) 0 0
\(339\) −11.1846 + 2.62772i −0.607464 + 0.142718i
\(340\) 0 0
\(341\) 8.61684 + 4.97494i 0.466628 + 0.269408i
\(342\) 0 0
\(343\) −15.5885 10.0000i −0.841698 0.539949i
\(344\) 0 0
\(345\) 13.0475 + 13.8839i 0.702456 + 0.747482i
\(346\) 0 0
\(347\) −14.9248 25.8505i −0.801206 1.38773i −0.918823 0.394670i \(-0.870859\pi\)
0.117617 0.993059i \(-0.462474\pi\)
\(348\) 0 0
\(349\) −8.00000 −0.428230 −0.214115 0.976808i \(-0.568687\pi\)
−0.214115 + 0.976808i \(0.568687\pi\)
\(350\) 0 0
\(351\) −13.2665 16.0000i −0.708113 0.854017i
\(352\) 0 0
\(353\) 31.5951 18.2414i 1.68164 0.970894i 0.721063 0.692869i \(-0.243654\pi\)
0.960574 0.278024i \(-0.0896796\pi\)
\(354\) 0 0
\(355\) 38.1051 + 22.0000i 2.02241 + 1.16764i
\(356\) 0 0
\(357\) 15.1861 0.617359i 0.803736 0.0326741i
\(358\) 0 0
\(359\) 18.2414 31.5951i 0.962746 1.66753i 0.247194 0.968966i \(-0.420491\pi\)
0.715552 0.698559i \(-0.246175\pi\)
\(360\) 0 0
\(361\) 15.0000 + 25.9808i 0.789474 + 1.36741i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 23.2164i 1.21520i
\(366\) 0 0
\(367\) 0.866025 0.500000i 0.0452062 0.0260998i −0.477227 0.878780i \(-0.658358\pi\)
0.522433 + 0.852680i \(0.325025\pi\)
\(368\) 0 0
\(369\) −8.86141 17.8178i −0.461306 0.927560i
\(370\) 0 0
\(371\) 1.65831 8.61684i 0.0860953 0.447364i
\(372\) 0 0
\(373\) 6.50000 11.2583i 0.336557 0.582934i −0.647225 0.762299i \(-0.724071\pi\)
0.983783 + 0.179364i \(0.0574041\pi\)
\(374\) 0 0
\(375\) −5.59230 + 1.31386i −0.288785 + 0.0678474i
\(376\) 0 0
\(377\) 26.5330i 1.36652i
\(378\) 0 0
\(379\) 24.0000i 1.23280i −0.787434 0.616399i \(-0.788591\pi\)
0.787434 0.616399i \(-0.211409\pi\)
\(380\) 0 0
\(381\) −3.37228 + 0.792287i −0.172767 + 0.0405901i
\(382\) 0 0
\(383\) −8.29156 + 14.3614i −0.423679 + 0.733834i −0.996296 0.0859894i \(-0.972595\pi\)
0.572617 + 0.819823i \(0.305928\pi\)
\(384\) 0 0
\(385\) −22.0000 19.0526i −1.12122 0.971008i
\(386\) 0 0
\(387\) 2.67181 + 5.37228i 0.135816 + 0.273088i
\(388\) 0 0
\(389\) −2.87228 + 1.65831i −0.145630 + 0.0840798i −0.571045 0.820919i \(-0.693462\pi\)
0.425414 + 0.904999i \(0.360128\pi\)
\(390\) 0 0
\(391\) 11.0000i 0.556294i
\(392\) 0 0
\(393\) −27.5000 8.29156i −1.38719 0.418254i
\(394\) 0 0
\(395\) 14.9248 + 25.8505i 0.750949 + 1.30068i
\(396\) 0 0
\(397\) −6.50000 + 11.2583i −0.326226 + 0.565039i −0.981760 0.190126i \(-0.939110\pi\)
0.655534 + 0.755166i \(0.272444\pi\)
\(398\) 0 0
\(399\) 17.1542 + 27.1060i 0.858784 + 1.35700i
\(400\) 0 0
\(401\) −14.3614 8.29156i −0.717174 0.414061i 0.0965374 0.995329i \(-0.469223\pi\)
−0.813712 + 0.581269i \(0.802557\pi\)
\(402\) 0 0
\(403\) −10.3923 + 6.00000i −0.517678 + 0.298881i
\(404\) 0 0
\(405\) −27.5000 + 11.6082i −1.36649 + 0.576815i
\(406\) 0 0
\(407\) 3.31662 0.164399
\(408\) 0 0
\(409\) 7.50000 + 12.9904i 0.370851 + 0.642333i 0.989697 0.143180i \(-0.0457327\pi\)
−0.618846 + 0.785513i \(0.712399\pi\)
\(410\) 0 0
\(411\) 19.6699 + 20.9307i 0.970245 + 1.03244i
\(412\) 0 0
\(413\) −8.61684 1.65831i −0.424007 0.0816002i
\(414\) 0 0
\(415\) −38.1051 22.0000i −1.87051 1.07994i
\(416\) 0 0
\(417\) 16.8614 3.96143i 0.825707 0.193992i
\(418\) 0 0
\(419\) −26.5330 −1.29622 −0.648111 0.761546i \(-0.724441\pi\)
−0.648111 + 0.761546i \(0.724441\pi\)
\(420\) 0 0
\(421\) 8.00000 0.389896 0.194948 0.980814i \(-0.437546\pi\)
0.194948 + 0.980814i \(0.437546\pi\)
\(422\) 0 0
\(423\) −1.85208 + 29.7921i −0.0900510 + 1.44854i
\(424\) 0 0
\(425\) −17.2337 9.94987i −0.835957 0.482640i
\(426\) 0 0
\(427\) 2.59808 + 7.50000i 0.125730 + 0.362950i
\(428\) 0 0
\(429\) −16.7446 + 15.7359i −0.808435 + 0.759738i
\(430\) 0 0
\(431\) −11.6082 20.1060i −0.559147 0.968470i −0.997568 0.0697009i \(-0.977796\pi\)
0.438421 0.898770i \(-0.355538\pi\)
\(432\) 0 0
\(433\) −20.0000 −0.961139 −0.480569 0.876957i \(-0.659570\pi\)
−0.480569 + 0.876957i \(0.659570\pi\)
\(434\) 0 0
\(435\) 36.4829 + 11.0000i 1.74922 + 0.527410i
\(436\) 0 0
\(437\) −20.1060 + 11.6082i −0.961799 + 0.555295i
\(438\) 0 0
\(439\) 7.79423 + 4.50000i 0.371998 + 0.214773i 0.674331 0.738429i \(-0.264432\pi\)
−0.302333 + 0.953202i \(0.597765\pi\)
\(440\) 0 0
\(441\) 11.9416 + 17.2742i 0.568647 + 0.822582i
\(442\) 0 0
\(443\) −14.9248 + 25.8505i −0.709099 + 1.22820i 0.256092 + 0.966652i \(0.417565\pi\)
−0.965192 + 0.261544i \(0.915768\pi\)
\(444\) 0 0
\(445\) −5.50000 9.52628i −0.260725 0.451589i
\(446\) 0 0
\(447\) −8.29156 + 27.5000i −0.392177 + 1.30071i
\(448\) 0 0
\(449\) 19.8997i 0.939127i −0.882899 0.469564i \(-0.844411\pi\)
0.882899 0.469564i \(-0.155589\pi\)
\(450\) 0 0
\(451\) −19.0526 + 11.0000i −0.897150 + 0.517970i
\(452\) 0 0
\(453\) −8.30298 8.83518i −0.390108 0.415113i
\(454\) 0 0
\(455\) 33.1662 11.4891i 1.55486 0.538619i
\(456\) 0 0
\(457\) −1.50000 + 2.59808i −0.0701670 + 0.121533i −0.898974 0.438001i \(-0.855687\pi\)
0.828807 + 0.559534i \(0.189020\pi\)
\(458\) 0 0
\(459\) −16.1595 5.98913i −0.754262 0.279548i
\(460\) 0 0
\(461\) 33.1662i 1.54471i −0.635193 0.772353i \(-0.719080\pi\)
0.635193 0.772353i \(-0.280920\pi\)
\(462\) 0 0
\(463\) 38.0000i 1.76601i −0.469364 0.883005i \(-0.655517\pi\)
0.469364 0.883005i \(-0.344483\pi\)
\(464\) 0 0
\(465\) 3.94158 + 16.7769i 0.182786 + 0.778010i
\(466\) 0 0
\(467\) 1.65831 2.87228i 0.0767375 0.132913i −0.825103 0.564982i \(-0.808883\pi\)
0.901841 + 0.432069i \(0.142216\pi\)
\(468\) 0 0
\(469\) 4.50000 23.3827i 0.207791 1.07971i
\(470\) 0 0
\(471\) −3.78651 + 3.55842i −0.174473 + 0.163963i
\(472\) 0 0
\(473\) 5.74456 3.31662i 0.264135 0.152499i
\(474\) 0 0
\(475\) 42.0000i 1.92709i
\(476\) 0 0
\(477\) −5.50000 + 8.29156i −0.251828 + 0.379645i
\(478\) 0 0
\(479\) −8.29156 14.3614i −0.378851 0.656189i 0.612044 0.790824i \(-0.290347\pi\)
−0.990895 + 0.134634i \(0.957014\pi\)
\(480\) 0 0
\(481\) −2.00000 + 3.46410i −0.0911922 + 0.157949i
\(482\) 0 0
\(483\) −12.8429 + 8.12772i −0.584372 + 0.369824i
\(484\) 0 0
\(485\) −22.9783 13.2665i −1.04339 0.602401i
\(486\) 0 0
\(487\) 14.7224 8.50000i 0.667137 0.385172i −0.127854 0.991793i \(-0.540809\pi\)
0.794991 + 0.606621i \(0.207476\pi\)
\(488\) 0 0
\(489\) 3.50000 11.6082i 0.158275 0.524940i
\(490\) 0 0
\(491\) 6.63325 0.299354 0.149677 0.988735i \(-0.452177\pi\)
0.149677 + 0.988735i \(0.452177\pi\)
\(492\) 0 0
\(493\) 11.0000 + 19.0526i 0.495415 + 0.858084i
\(494\) 0 0
\(495\) 14.6950 + 29.5475i 0.660490 + 1.32806i
\(496\) 0 0
\(497\) −22.9783 + 26.5330i −1.03072 + 1.19017i
\(498\) 0 0
\(499\) 30.3109 + 17.5000i 1.35690 + 0.783408i 0.989205 0.146538i \(-0.0468131\pi\)
0.367697 + 0.929946i \(0.380146\pi\)
\(500\) 0 0
\(501\) 2.62772 + 11.1846i 0.117398 + 0.499691i
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 11.0000 0.489494
\(506\) 0 0
\(507\) −1.18843 5.05842i −0.0527801 0.224652i
\(508\) 0 0
\(509\) −14.3614 8.29156i −0.636558 0.367517i 0.146729 0.989177i \(-0.453125\pi\)
−0.783287 + 0.621660i \(0.786459\pi\)
\(510\) 0 0
\(511\) −18.1865 3.50000i −0.804525 0.154831i
\(512\) 0 0
\(513\) −6.10597 35.8569i −0.269585 1.58312i
\(514\) 0 0
\(515\) −18.2414 31.5951i −0.803814 1.39225i
\(516\) 0 0
\(517\) 33.0000 1.45134
\(518\) 0 0
\(519\) −4.97494 + 16.5000i −0.218376 + 0.724270i
\(520\) 0 0
\(521\) −20.1060 + 11.6082i −0.880859 + 0.508564i −0.870941 0.491387i \(-0.836490\pi\)
−0.00991713 + 0.999951i \(0.503157\pi\)
\(522\) 0 0
\(523\) −16.4545 9.50000i −0.719504 0.415406i 0.0950659 0.995471i \(-0.469694\pi\)
−0.814570 + 0.580065i \(0.803027\pi\)
\(524\) 0 0
\(525\) −1.11684 27.4728i −0.0487431 1.19901i
\(526\) 0 0
\(527\) −4.97494 + 8.61684i −0.216712 + 0.375356i
\(528\) 0 0
\(529\) 6.00000 + 10.3923i 0.260870 + 0.451839i
\(530\) 0 0
\(531\) 8.29156 + 5.50000i 0.359823 + 0.238680i
\(532\) 0 0
\(533\) 26.5330i 1.14927i
\(534\) 0 0
\(535\) 9.52628 5.50000i 0.411857 0.237786i
\(536\) 0 0
\(537\) 4.18614 3.93398i 0.180645 0.169764i
\(538\) 0 0
\(539\) 18.2414 14.3614i 0.785714 0.618590i
\(540\) 0 0
\(541\) −18.5000 + 32.0429i −0.795377 + 1.37763i 0.127222 + 0.991874i \(0.459394\pi\)
−0.922599 + 0.385759i \(0.873939\pi\)
\(542\) 0 0
\(543\) −7.13058 30.3505i −0.306003 1.30247i
\(544\) 0 0
\(545\) 16.5831i 0.710343i
\(546\) 0 0
\(547\) 40.0000i 1.71028i −0.518400 0.855138i \(-0.673472\pi\)
0.518400 0.855138i \(-0.326528\pi\)
\(548\) 0 0
\(549\) 0.558422 8.98266i 0.0238329 0.383371i
\(550\) 0 0
\(551\) −23.2164 + 40.2119i −0.989051 + 1.71309i
\(552\) 0 0
\(553\) −22.5000 + 7.79423i −0.956797 + 0.331444i
\(554\) 0 0
\(555\) 3.93398 + 4.18614i 0.166988 + 0.177692i
\(556\) 0 0
\(557\) −2.87228 + 1.65831i −0.121702 + 0.0702650i −0.559615 0.828752i \(-0.689051\pi\)
0.437913 + 0.899017i \(0.355718\pi\)
\(558\) 0 0
\(559\) 8.00000i 0.338364i
\(560\) 0 0
\(561\) −5.50000 + 18.2414i −0.232210 + 0.770154i
\(562\) 0 0
\(563\) 4.97494 + 8.61684i 0.209669 + 0.363157i 0.951610 0.307308i \(-0.0994282\pi\)
−0.741942 + 0.670465i \(0.766095\pi\)
\(564\) 0 0
\(565\) −11.0000 + 19.0526i −0.462773 + 0.801547i
\(566\) 0 0
\(567\) −4.94749 23.2921i −0.207775 0.978177i
\(568\) 0 0
\(569\) 25.8505 + 14.9248i 1.08371 + 0.625681i 0.931895 0.362728i \(-0.118155\pi\)
0.151816 + 0.988409i \(0.451488\pi\)
\(570\) 0 0
\(571\) −4.33013 + 2.50000i −0.181210 + 0.104622i −0.587861 0.808962i \(-0.700030\pi\)
0.406651 + 0.913584i \(0.366697\pi\)
\(572\) 0 0
\(573\) −16.5000 4.97494i −0.689297 0.207831i
\(574\) 0 0
\(575\) 19.8997 0.829877
\(576\) 0 0
\(577\) −17.5000 30.3109i −0.728535 1.26186i −0.957503 0.288425i \(-0.906868\pi\)
0.228968 0.973434i \(-0.426465\pi\)
\(578\) 0 0
\(579\) 3.78651 3.55842i 0.157362 0.147883i
\(580\) 0 0
\(581\) 22.9783 26.5330i 0.953298 1.10077i
\(582\) 0 0
\(583\) 9.52628 + 5.50000i 0.394538 + 0.227787i
\(584\) 0 0
\(585\) −39.7228 2.46943i −1.64234 0.102099i
\(586\) 0 0
\(587\) 26.5330 1.09513 0.547567 0.836762i \(-0.315554\pi\)
0.547567 + 0.836762i \(0.315554\pi\)
\(588\) 0 0
\(589\) −21.0000 −0.865290
\(590\) 0 0
\(591\) 33.5538 7.88316i 1.38022 0.324270i
\(592\) 0 0
\(593\) −2.87228 1.65831i −0.117950 0.0680987i 0.439864 0.898064i \(-0.355027\pi\)
−0.557815 + 0.829966i \(0.688360\pi\)
\(594\) 0 0
\(595\) 19.0526 22.0000i 0.781079 0.901912i
\(596\) 0 0
\(597\) 1.18614 + 1.26217i 0.0485455 + 0.0516571i
\(598\) 0 0
\(599\) 4.97494 + 8.61684i 0.203270 + 0.352075i 0.949580 0.313524i \(-0.101510\pi\)
−0.746310 + 0.665599i \(0.768176\pi\)
\(600\) 0 0
\(601\) −16.0000 −0.652654 −0.326327 0.945257i \(-0.605811\pi\)
−0.326327 + 0.945257i \(0.605811\pi\)
\(602\) 0 0
\(603\) −14.9248 + 22.5000i −0.607785 + 0.916271i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −6.06218 3.50000i −0.246056 0.142061i 0.371901 0.928272i \(-0.378706\pi\)
−0.617957 + 0.786212i \(0.712039\pi\)
\(608\) 0 0
\(609\) −14.1168 + 26.9205i −0.572043 + 1.09087i
\(610\) 0 0
\(611\) −19.8997 + 34.4674i −0.805058 + 1.39440i
\(612\) 0 0
\(613\) −0.500000 0.866025i −0.0201948 0.0349784i 0.855751 0.517387i \(-0.173095\pi\)
−0.875946 + 0.482409i \(0.839762\pi\)
\(614\) 0 0
\(615\) −36.4829 11.0000i −1.47113 0.443563i
\(616\) 0 0
\(617\) 6.63325i 0.267045i 0.991046 + 0.133522i \(0.0426288\pi\)
−0.991046 + 0.133522i \(0.957371\pi\)
\(618\) 0 0
\(619\) 21.6506 12.5000i 0.870212 0.502417i 0.00279365 0.999996i \(-0.499111\pi\)
0.867419 + 0.497579i \(0.165777\pi\)
\(620\) 0 0
\(621\) 16.9891 2.89303i 0.681750 0.116093i
\(622\) 0 0
\(623\) 8.29156 2.87228i 0.332194 0.115076i
\(624\) 0 0
\(625\) 9.50000 16.4545i 0.380000 0.658179i
\(626\) 0 0
\(627\) −39.1461 + 9.19702i −1.56334 + 0.367293i
\(628\) 0 0
\(629\) 3.31662i 0.132242i
\(630\) 0 0
\(631\) 18.0000i 0.716569i 0.933613 + 0.358284i \(0.116638\pi\)
−0.933613 + 0.358284i \(0.883362\pi\)
\(632\) 0 0
\(633\) 43.8397 10.2997i 1.74247 0.409378i
\(634\) 0 0
\(635\) −3.31662 + 5.74456i −0.131616 + 0.227966i
\(636\) 0 0
\(637\) 4.00000 + 27.7128i 0.158486 + 1.09802i
\(638\) 0 0
\(639\) 35.6357 17.7228i 1.40973 0.701104i
\(640\) 0 0
\(641\) −31.5951 + 18.2414i −1.24793 + 0.720493i −0.970696 0.240309i \(-0.922751\pi\)
−0.277235 + 0.960802i \(0.589418\pi\)
\(642\) 0 0
\(643\) 2.00000i 0.0788723i 0.999222 + 0.0394362i \(0.0125562\pi\)
−0.999222 + 0.0394362i \(0.987444\pi\)
\(644\) 0 0
\(645\) 11.0000 + 3.31662i 0.433125 + 0.130592i
\(646\) 0 0
\(647\) −1.65831 2.87228i −0.0651950 0.112921i 0.831585 0.555397i \(-0.187434\pi\)
−0.896780 + 0.442476i \(0.854100\pi\)
\(648\) 0 0
\(649\) 5.50000 9.52628i 0.215894 0.373939i
\(650\) 0 0
\(651\) −13.7364 + 0.558422i −0.538371 + 0.0218863i
\(652\) 0 0
\(653\) 2.87228 + 1.65831i 0.112401 + 0.0648948i 0.555147 0.831753i \(-0.312662\pi\)
−0.442746 + 0.896647i \(0.645995\pi\)
\(654\) 0 0
\(655\) −47.6314 + 27.5000i −1.86111 + 1.07451i
\(656\) 0 0
\(657\) 17.5000 + 11.6082i 0.682740 + 0.452878i
\(658\) 0 0
\(659\) 26.5330 1.03358 0.516789 0.856113i \(-0.327127\pi\)
0.516789 + 0.856113i \(0.327127\pi\)
\(660\) 0 0
\(661\) −19.5000 33.7750i −0.758462 1.31369i −0.943635 0.330989i \(-0.892618\pi\)
0.185173 0.982706i \(-0.440716\pi\)
\(662\) 0 0
\(663\) −15.7359 16.7446i −0.611133 0.650305i
\(664\) 0 0
\(665\) 60.3179 + 11.6082i 2.33903 + 0.450146i
\(666\) 0 0
\(667\) −19.0526 11.0000i −0.737718 0.425922i
\(668\) 0 0
\(669\) 23.6060 5.54601i 0.912659 0.214421i
\(670\) 0 0
\(671\) −9.94987 −0.384111
\(672\) 0 0
\(673\) 40.0000 1.54189 0.770943 0.636904i \(-0.219785\pi\)
0.770943 + 0.636904i \(0.219785\pi\)
\(674\) 0 0
\(675\) −10.8347 + 29.2337i −0.417029 + 1.12521i
\(676\) 0 0
\(677\) 8.61684 + 4.97494i 0.331172 + 0.191202i 0.656361 0.754447i \(-0.272095\pi\)
−0.325189 + 0.945649i \(0.605428\pi\)
\(678\) 0 0
\(679\) 13.8564 16.0000i 0.531760 0.614024i
\(680\) 0 0
\(681\) 4.18614 3.93398i 0.160413 0.150751i
\(682\) 0 0
\(683\) −11.6082 20.1060i −0.444175 0.769334i 0.553819 0.832637i \(-0.313170\pi\)
−0.997994 + 0.0633033i \(0.979836\pi\)
\(684\) 0 0
\(685\) 55.0000 2.10144
\(686\) 0 0
\(687\) 48.0911 + 14.5000i 1.83479 + 0.553210i
\(688\) 0 0
\(689\) −11.4891 + 6.63325i −0.437701 + 0.252707i
\(690\) 0 0
\(691\) −28.5788 16.5000i −1.08719 0.627690i −0.154363 0.988014i \(-0.549333\pi\)
−0.932827 + 0.360325i \(0.882666\pi\)
\(692\) 0 0
\(693\) −25.3614 + 7.05684i −0.963400 + 0.268067i
\(694\) 0 0
\(695\) 16.5831 28.7228i 0.629034 1.08952i
\(696\) 0 0
\(697\) −11.0000 19.0526i −0.416655 0.721667i
\(698\) 0 0
\(699\) −8.29156 + 27.5000i −0.313616 + 1.04015i
\(700\) 0 0
\(701\) 13.2665i 0.501069i −0.968108 0.250534i \(-0.919394\pi\)
0.968108 0.250534i \(-0.0806063\pi\)
\(702\) 0 0
\(703\) −6.06218 + 3.50000i −0.228639 + 0.132005i
\(704\) 0 0
\(705\) 39.1426 + 41.6516i 1.47420 + 1.56869i
\(706\) 0 0
\(707\) −1.65831 + 8.61684i −0.0623673 + 0.324070i
\(708\) 0 0
\(709\) 16.5000 28.5788i 0.619671 1.07330i −0.369875 0.929081i \(-0.620600\pi\)
0.989546 0.144219i \(-0.0460671\pi\)
\(710\) 0 0
\(711\) 26.9480 + 1.67527i 1.01063 + 0.0628274i
\(712\) 0 0
\(713\) 9.94987i 0.372626i
\(714\) 0 0
\(715\) 44.0000i 1.64551i
\(716\) 0 0
\(717\) −2.62772 11.1846i −0.0981340 0.417696i
\(718\) 0 0
\(719\) −11.6082 + 20.1060i −0.432912 + 0.749826i −0.997123 0.0758050i \(-0.975847\pi\)
0.564210 + 0.825631i \(0.309181\pi\)
\(720\) 0 0
\(721\) 27.5000 9.52628i 1.02415 0.354777i
\(722\) 0 0
\(723\) −23.9812 + 22.5367i −0.891871 + 0.838148i
\(724\) 0 0
\(725\) 34.4674 19.8997i 1.28009 0.739058i
\(726\) 0 0
\(727\) 10.0000i 0.370879i 0.982656 + 0.185440i \(0.0593710\pi\)
−0.982656 + 0.185440i \(0.940629\pi\)
\(728\) 0 0
\(729\) −5.00000 + 26.5330i −0.185185 + 0.982704i
\(730\) 0 0
\(731\) 3.31662 + 5.74456i 0.122670 + 0.212470i
\(732\) 0 0
\(733\) −3.50000 + 6.06218i −0.129275 + 0.223912i −0.923396 0.383849i \(-0.874598\pi\)
0.794121 + 0.607760i \(0.207932\pi\)
\(734\) 0 0
\(735\) 39.7634 + 5.98913i 1.46670 + 0.220912i
\(736\) 0 0
\(737\) 25.8505 + 14.9248i 0.952217 + 0.549762i
\(738\) 0 0
\(739\) 26.8468 15.5000i 0.987575 0.570177i 0.0830265 0.996547i \(-0.473541\pi\)
0.904549 + 0.426371i \(0.140208\pi\)
\(740\) 0 0
\(741\) 14.0000 46.4327i 0.514303 1.70575i
\(742\) 0 0
\(743\) −13.2665 −0.486701 −0.243350 0.969938i \(-0.578246\pi\)
−0.243350 + 0.969938i \(0.578246\pi\)
\(744\) 0 0
\(745\) 27.5000 + 47.6314i 1.00752 + 1.74508i
\(746\) 0 0
\(747\) −35.6357 + 17.7228i −1.30384 + 0.648444i
\(748\) 0 0
\(749\) 2.87228 + 8.29156i 0.104951 + 0.302967i
\(750\) 0 0
\(751\) 4.33013 + 2.50000i 0.158009 + 0.0912263i 0.576919 0.816801i \(-0.304255\pi\)
−0.418911 + 0.908027i \(0.637588\pi\)
\(752\) 0 0
\(753\) −7.88316 33.5538i −0.287278 1.22277i
\(754\) 0 0
\(755\) −23.2164 −0.844930
\(756\) 0 0
\(757\) −48.0000 −1.74459 −0.872295 0.488980i \(-0.837369\pi\)
−0.872295 + 0.488980i \(0.837369\pi\)
\(758\) 0 0
\(759\) −4.35758 18.5475i −0.158170 0.673233i
\(760\) 0 0
\(761\) 20.1060 + 11.6082i 0.728841 + 0.420796i 0.817998 0.575221i \(-0.195084\pi\)
−0.0891571 + 0.996018i \(0.528417\pi\)
\(762\) 0 0
\(763\) −12.9904 2.50000i −0.470283 0.0905061i
\(764\) 0 0
\(765\) −29.5475 + 14.6950i −1.06829 + 0.531298i
\(766\) 0 0
\(767\) 6.63325 + 11.4891i 0.239513 + 0.414848i
\(768\) 0 0
\(769\) −16.0000 −0.576975 −0.288487 0.957484i \(-0.593152\pi\)
−0.288487 + 0.957484i \(0.593152\pi\)
\(770\) 0 0
\(771\) 11.6082 38.5000i 0.418059 1.38654i
\(772\) 0 0
\(773\) −14.3614 + 8.29156i −0.516544 + 0.298227i −0.735519 0.677504i \(-0.763062\pi\)
0.218976 + 0.975730i \(0.429728\pi\)
\(774\) 0 0
\(775\) 15.5885 + 9.00000i 0.559954 + 0.323290i
\(776\) 0 0
\(777\) −3.87228 + 2.45060i −0.138917 + 0.0879148i
\(778\) 0 0
\(779\) 23.2164 40.2119i 0.831813 1.44074i
\(780\) 0 0
\(781\) −22.0000 38.1051i −0.787222 1.36351i
\(782\) 0 0
\(783\) 26.5330 22.0000i 0.948212 0.786216i
\(784\) 0 0
\(785\) 9.94987i 0.355126i
\(786\) 0 0
\(787\) 38.9711 22.5000i 1.38917 0.802038i 0.395949 0.918272i \(-0.370416\pi\)
0.993222 + 0.116234i \(0.0370822\pi\)
\(788\) 0 0
\(789\) −4.18614 + 3.93398i −0.149031 + 0.140054i
\(790\) 0 0
\(791\) −13.2665 11.4891i −0.471702 0.408506i
\(792\) 0 0
\(793\) 6.00000 10.3923i 0.213066 0.369042i
\(794\) 0 0
\(795\) 4.35758 + 18.5475i 0.154547 + 0.657814i
\(796\) 0 0
\(797\) 6.63325i 0.234962i 0.993075 + 0.117481i \(0.0374819\pi\)
−0.993075 + 0.117481i \(0.962518\pi\)
\(798\) 0 0
\(799\) 33.0000i 1.16746i
\(800\) 0 0
\(801\) −9.93070 0.617359i −0.350884 0.0218133i
\(802\) 0 0
\(803\) 11.6082 20.1060i 0.409644 0.709524i
\(804\) 0 0
\(805\) −5.50000 + 28.5788i −0.193850 + 1.00727i
\(806\) 0 0
\(807\) −19.6699 20.9307i −0.692414 0.736796i
\(808\) 0 0
\(809\) 2.87228 1.65831i 0.100984 0.0583032i −0.448657 0.893704i \(-0.648098\pi\)
0.549641 + 0.835401i \(0.314764\pi\)
\(810\) 0 0
\(811\) 30.0000i 1.05344i 0.850038 + 0.526721i \(0.176579\pi\)
−0.850038 + 0.526721i \(0.823421\pi\)
\(812\) 0 0
\(813\) −8.50000 + 28.1913i −0.298108 + 0.988712i
\(814\) 0 0
\(815\) −11.6082 20.1060i −0.406617 0.704281i
\(816\) 0 0
\(817\) −7.00000 + 12.1244i −0.244899 + 0.424178i
\(818\) 0 0
\(819\) 7.92287 30.7446i 0.276847 1.07430i
\(820\) 0 0
\(821\) 31.5951 + 18.2414i 1.10268 + 0.636631i 0.936923 0.349537i \(-0.113661\pi\)
0.165754 + 0.986167i \(0.446994\pi\)
\(822\) 0 0
\(823\) −12.9904 + 7.50000i −0.452816 + 0.261434i −0.709019 0.705190i \(-0.750862\pi\)
0.256203 + 0.966623i \(0.417529\pi\)
\(824\) 0 0
\(825\) 33.0000 + 9.94987i 1.14891 + 0.346410i
\(826\) 0 0
\(827\) −39.7995 −1.38396 −0.691982 0.721915i \(-0.743262\pi\)
−0.691982 + 0.721915i \(0.743262\pi\)
\(828\) 0 0
\(829\) −14.5000 25.1147i −0.503606 0.872271i −0.999991 0.00416865i \(-0.998673\pi\)
0.496385 0.868102i \(-0.334660\pi\)
\(830\) 0 0
\(831\) 16.4082 15.4198i 0.569194 0.534908i
\(832\) 0 0
\(833\) 14.3614 + 18.2414i 0.497593 + 0.632028i
\(834\) 0 0
\(835\) 19.0526 + 11.0000i 0.659341 + 0.380671i
\(836\) 0 0
\(837\) 14.6168 + 5.41737i 0.505232 + 0.187252i
\(838\) 0 0
\(839\) −13.2665 −0.458010 −0.229005 0.973425i \(-0.573547\pi\)
−0.229005 + 0.973425i \(0.573547\pi\)
\(840\) 0 0
\(841\) −15.0000 −0.517241
\(842\) 0 0
\(843\) 11.1846 2.62772i 0.385218 0.0905034i
\(844\) 0 0
\(845\) −8.61684 4.97494i −0.296428 0.171143i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −17.7921 18.9325i −0.610624 0.649763i
\(850\) 0 0
\(851\) −1.65831 2.87228i −0.0568462 0.0984605i
\(852\) 0 0
\(853\) −20.0000 −0.684787 −0.342393 0.939557i \(-0.611238\pi\)
−0.342393 + 0.939557i \(0.611238\pi\)
\(854\) 0 0
\(855\) −58.0409 38.5000i −1.98496 1.31667i
\(856\) 0 0
\(857\) 14.3614 8.29156i 0.490576 0.283234i −0.234237 0.972179i \(-0.575259\pi\)
0.724814 + 0.688945i \(0.241926\pi\)
\(858\) 0 0
\(859\) 14.7224 + 8.50000i 0.502323 + 0.290016i 0.729672 0.683797i \(-0.239673\pi\)
−0.227349 + 0.973813i \(0.573006\pi\)
\(860\) 0 0
\(861\) 14.1168 26.9205i 0.481101 0.917449i
\(862\) 0 0
\(863\) −18.2414 + 31.5951i −0.620946 + 1.07551i 0.368364 + 0.929682i \(0.379918\pi\)
−0.989310 + 0.145828i \(0.953415\pi\)
\(864\) 0 0
\(865\) 16.5000 + 28.5788i 0.561017 + 0.971710i
\(866\) 0 0
\(867\) 9.94987 + 3.00000i 0.337915 + 0.101885i
\(868\) 0 0
\(869\) 29.8496i 1.01258i
\(870\) 0 0
\(871\) −31.1769 + 18.0000i −1.05639 + 0.609907i
\(872\) 0 0
\(873\) −21.4891 + 10.6873i −0.727297 + 0.361709i
\(874\) 0 0
\(875\) −6.63325 5.74456i −0.224245 0.194202i
\(876\) 0 0
\(877\) 11.5000 19.9186i 0.388327 0.672603i −0.603897 0.797062i \(-0.706386\pi\)
0.992225 + 0.124459i \(0.0397196\pi\)
\(878\) 0 0
\(879\) 22.3692 5.25544i 0.754494 0.177262i
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 6.00000i 0.201916i 0.994891 + 0.100958i \(0.0321908\pi\)
−0.994891 + 0.100958i \(0.967809\pi\)
\(884\) 0 0
\(885\) 18.5475 4.35758i 0.623469 0.146478i
\(886\) 0 0
\(887\) 11.6082 20.1060i 0.389765 0.675092i −0.602653 0.798003i \(-0.705890\pi\)
0.992418 + 0.122911i \(0.0392230\pi\)
\(888\) 0 0
\(889\) −4.00000 3.46410i −0.134156 0.116182i
\(890\) 0 0
\(891\) 29.6198 + 3.69702i 0.992300 + 0.123855i
\(892\) 0 0
\(893\) −60.3179 + 34.8246i −2.01846 + 1.16536i
\(894\) 0 0
\(895\) 11.0000i 0.367689i
\(896\) 0 0
\(897\) 22.0000 + 6.63325i 0.734559 + 0.221478i
\(898\) 0 0
\(899\) −9.94987 17.2337i −0.331847 0.574776i
\(900\) 0 0
\(901\) −5.50000 + 9.52628i −0.183232 + 0.317366i
\(902\) 0 0
\(903\) −4.25639 + 8.11684i −0.141644 + 0.270112i
\(904\) 0 0
\(905\) −51.7011 29.8496i −1.71860 0.992235i
\(906\) 0 0
\(907\) 45.8993 26.5000i 1.52406 0.879918i 0.524469 0.851430i \(-0.324264\pi\)
0.999594 0.0284883i \(-0.00906934\pi\)
\(908\) 0 0
\(909\) 5.50000 8.29156i 0.182423 0.275014i
\(910\) 0 0
\(911\) 26.5330 0.879077 0.439539 0.898224i \(-0.355142\pi\)
0.439539 + 0.898224i \(0.355142\pi\)
\(912\) 0 0
\(913\) 22.0000 + 38.1051i 0.728094 + 1.26110i
\(914\) 0 0
\(915\) −11.8020 12.5584i −0.390160 0.415169i
\(916\) 0 0
\(917\) −14.3614 41.4578i −0.474255 1.36906i
\(918\) 0 0
\(919\) −19.9186 11.5000i −0.657053 0.379350i 0.134100 0.990968i \(-0.457186\pi\)
−0.791153 + 0.611618i \(0.790519\pi\)
\(920\) 0 0
\(921\) −10.1168 + 2.37686i −0.333361 + 0.0783202i
\(922\) 0 0
\(923\) 53.0660 1.74669
\(924\) 0 0
\(925\) 6.00000 0.197279
\(926\) 0 0
\(927\) −32.9364 2.04755i −1.08177 0.0672503i
\(928\) 0 0
\(929\) −20.1060 11.6082i −0.659655 0.380852i 0.132490 0.991184i \(-0.457703\pi\)
−0.792146 + 0.610332i \(0.791036\pi\)
\(930\) 0 0
\(931\) −18.1865 + 45.5000i −0.596040 + 1.49120i
\(932\) 0 0
\(933\) 12.5584 11.8020i 0.411144 0.386379i
\(934\) 0 0
\(935\) 18.2414 + 31.5951i 0.596559 + 1.03327i
\(936\) 0 0
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) 0 0
\(939\) −14.9248 4.50000i −0.487053 0.146852i
\(940\) 0 0
\(941\) −8.61684 + 4.97494i −0.280901 + 0.162178i −0.633831 0.773471i \(-0.718519\pi\)
0.352930 + 0.935650i \(0.385185\pi\)
\(942\) 0 0
\(943\) 19.0526 + 11.0000i 0.620437 + 0.358209i
\(944\) 0 0
\(945\) −38.9891 23.6400i −1.26832 0.769008i
\(946\) 0 0
\(947\) −4.97494 + 8.61684i −0.161664 + 0.280010i −0.935465 0.353418i \(-0.885019\pi\)
0.773802 + 0.633428i \(0.218353\pi\)
\(948\) 0 0
\(949\) 14.0000 + 24.2487i 0.454459 + 0.787146i
\(950\) 0 0
\(951\) 14.9248 49.5000i 0.483970 1.60515i
\(952\) 0 0
\(953\) 19.8997i 0.644616i −0.946635 0.322308i \(-0.895541\pi\)
0.946635 0.322308i \(-0.104459\pi\)
\(954\) 0 0
\(955\) −28.5788 + 16.5000i −0.924789 + 0.533927i
\(956\) 0 0
\(957\) −26.0951 27.7677i −0.843535 0.897603i
\(958\) 0 0
\(959\) −8.29156 + 43.0842i −0.267749 + 1.39126i
\(960\) 0 0
\(961\) −11.0000 + 19.0526i −0.354839 + 0.614599i
\(962\) 0 0
\(963\) 0.617359 9.93070i 0.0198941 0.320012i
\(964\) 0 0
\(965\) 9.94987i 0.320298i
\(966\) 0 0
\(967\) 2.00000i 0.0643157i −0.999483 0.0321578i \(-0.989762\pi\)
0.999483 0.0321578i \(-0.0102379\pi\)
\(968\) 0 0
\(969\) −9.19702 39.1461i −0.295451 1.25755i
\(970\) 0 0
\(971\) 8.29156 14.3614i 0.266089 0.460879i −0.701759 0.712414i \(-0.747602\pi\)
0.967848 + 0.251535i \(0.0809352\pi\)
\(972\) 0 0
\(973\) 20.0000 + 17.3205i 0.641171 + 0.555270i
\(974\) 0 0
\(975\) −30.2921 + 28.4674i −0.970122 + 0.911686i
\(976\) 0 0
\(977\) −2.87228 + 1.65831i −0.0918924 + 0.0530541i −0.545242 0.838279i \(-0.683562\pi\)
0.453350 + 0.891333i \(0.350229\pi\)
\(978\) 0 0
\(979\) 11.0000i 0.351562i
\(980\) 0 0
\(981\) 12.5000 + 8.29156i 0.399094 + 0.264729i
\(982\) 0 0
\(983\) 28.1913 + 48.8288i 0.899163 + 1.55740i 0.828566 + 0.559891i \(0.189157\pi\)
0.0705972 + 0.997505i \(0.477510\pi\)
\(984\) 0 0
\(985\) 33.0000 57.1577i 1.05147 1.82120i
\(986\) 0 0
\(987\) −38.5287 + 24.3832i −1.22638 + 0.776124i
\(988\) 0 0
\(989\) −5.74456 3.31662i −0.182666 0.105463i
\(990\) 0 0
\(991\) −37.2391 + 21.5000i −1.18294 + 0.682970i −0.956693 0.291100i \(-0.905979\pi\)
−0.226246 + 0.974070i \(0.572645\pi\)
\(992\) 0 0
\(993\) −8.50000 + 28.1913i −0.269739 + 0.894624i
\(994\) 0 0
\(995\) 3.31662 0.105144
\(996\) 0 0
\(997\) 2.50000 + 4.33013i 0.0791758 + 0.137136i 0.902895 0.429862i \(-0.141438\pi\)
−0.823719 + 0.566999i \(0.808104\pi\)
\(998\) 0 0
\(999\) 5.12241 0.872281i 0.162066 0.0275978i
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 336.2.bj.f.191.2 yes 8
3.2 odd 2 inner 336.2.bj.f.191.4 yes 8
4.3 odd 2 inner 336.2.bj.f.191.3 yes 8
7.2 even 3 2352.2.h.i.2255.4 4
7.4 even 3 inner 336.2.bj.f.95.1 8
7.5 odd 6 2352.2.h.j.2255.1 4
12.11 even 2 inner 336.2.bj.f.191.1 yes 8
21.2 odd 6 2352.2.h.i.2255.2 4
21.5 even 6 2352.2.h.j.2255.3 4
21.11 odd 6 inner 336.2.bj.f.95.3 yes 8
28.11 odd 6 inner 336.2.bj.f.95.4 yes 8
28.19 even 6 2352.2.h.j.2255.4 4
28.23 odd 6 2352.2.h.i.2255.1 4
84.11 even 6 inner 336.2.bj.f.95.2 yes 8
84.23 even 6 2352.2.h.i.2255.3 4
84.47 odd 6 2352.2.h.j.2255.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
336.2.bj.f.95.1 8 7.4 even 3 inner
336.2.bj.f.95.2 yes 8 84.11 even 6 inner
336.2.bj.f.95.3 yes 8 21.11 odd 6 inner
336.2.bj.f.95.4 yes 8 28.11 odd 6 inner
336.2.bj.f.191.1 yes 8 12.11 even 2 inner
336.2.bj.f.191.2 yes 8 1.1 even 1 trivial
336.2.bj.f.191.3 yes 8 4.3 odd 2 inner
336.2.bj.f.191.4 yes 8 3.2 odd 2 inner
2352.2.h.i.2255.1 4 28.23 odd 6
2352.2.h.i.2255.2 4 21.2 odd 6
2352.2.h.i.2255.3 4 84.23 even 6
2352.2.h.i.2255.4 4 7.2 even 3
2352.2.h.j.2255.1 4 7.5 odd 6
2352.2.h.j.2255.2 4 84.47 odd 6
2352.2.h.j.2255.3 4 21.5 even 6
2352.2.h.j.2255.4 4 28.19 even 6