Properties

Label 2100.2.bo.g.1349.6
Level $2100$
Weight $2$
Character 2100.1349
Analytic conductor $16.769$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2100,2,Mod(1349,2100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2100, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2100.1349");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2100.bo (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: \(16.7685844245\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 3 x^{18} - 16 x^{16} + 87 x^{14} + 91 x^{12} - 1104 x^{10} + 819 x^{8} + 7047 x^{6} - 11664 x^{4} - 19683 x^{2} + 59049 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 420)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1349.6
Root \(-1.64368 - 0.546177i\) of defining polynomial
Character \(\chi\) \(=\) 2100.1349
Dual form 2100.2.bo.g.1949.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.348838 - 1.69656i) q^{3} +(2.41433 - 1.08214i) q^{7} +(-2.75662 - 1.18365i) q^{9} +O(q^{10})\) \(q+(0.348838 - 1.69656i) q^{3} +(2.41433 - 1.08214i) q^{7} +(-2.75662 - 1.18365i) q^{9} +(1.17086 - 0.675999i) q^{11} -4.94296 q^{13} +(-4.97280 + 2.87105i) q^{17} +(-2.84694 - 1.64368i) q^{19} +(-0.993697 - 4.47354i) q^{21} +(-2.50270 + 4.33480i) q^{23} +(-2.96974 + 4.26388i) q^{27} -5.68630i q^{29} +(-2.45160 + 1.41543i) q^{31} +(-0.738430 - 2.22225i) q^{33} +(-3.33498 - 1.92545i) q^{37} +(-1.72429 + 8.38603i) q^{39} -3.73802 q^{41} -4.06339i q^{43} +(-4.92419 - 2.84298i) q^{47} +(4.65797 - 5.22526i) q^{49} +(3.13620 + 9.43818i) q^{51} +(-0.730773 - 1.26574i) q^{53} +(-3.78172 + 4.25662i) q^{57} +(4.34239 + 7.52123i) q^{59} +(1.65306 + 0.954394i) q^{61} +(-7.93626 + 0.125326i) q^{63} +(4.36371 - 2.51939i) q^{67} +(6.48121 + 5.75812i) q^{69} -3.38259i q^{71} +(-8.16364 - 14.1398i) q^{73} +(2.09533 - 2.89912i) q^{77} +(2.41693 - 4.18625i) q^{79} +(6.19796 + 6.52574i) q^{81} +16.6525i q^{83} +(-9.64714 - 1.98359i) q^{87} +(8.08150 - 13.9976i) q^{89} +(-11.9339 + 5.34895i) q^{91} +(1.54615 + 4.65303i) q^{93} -12.0577 q^{97} +(-4.02778 + 0.477584i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 6 q^{9} + 12 q^{11} - 6 q^{19} + 24 q^{21} + 30 q^{31} + 42 q^{39} + 16 q^{41} + 26 q^{49} + 80 q^{51} + 84 q^{61} - 28 q^{69} - 2 q^{79} - 26 q^{81} + 56 q^{89} - 22 q^{91} - 72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times\).

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(e\left(\frac{5}{6}\right)\)

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
\(3\) 0.348838 1.69656i 0.201402 0.979509i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.41433 1.08214i 0.912531 0.409009i
\(8\) 0 0
\(9\) −2.75662 1.18365i −0.918875 0.394549i
\(10\) 0 0
\(11\) 1.17086 0.675999i 0.353029 0.203821i −0.312990 0.949757i \(-0.601331\pi\)
0.666018 + 0.745935i \(0.267997\pi\)
\(12\) 0 0
\(13\) −4.94296 −1.37093 −0.685466 0.728105i \(-0.740401\pi\)
−0.685466 + 0.728105i \(0.740401\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.97280 + 2.87105i −1.20608 + 0.696332i −0.961901 0.273398i \(-0.911852\pi\)
−0.244181 + 0.969730i \(0.578519\pi\)
\(18\) 0 0
\(19\) −2.84694 1.64368i −0.653133 0.377087i 0.136523 0.990637i \(-0.456407\pi\)
−0.789656 + 0.613550i \(0.789741\pi\)
\(20\) 0 0
\(21\) −0.993697 4.47354i −0.216842 0.976207i
\(22\) 0 0
\(23\) −2.50270 + 4.33480i −0.521849 + 0.903868i 0.477828 + 0.878453i \(0.341424\pi\)
−0.999677 + 0.0254150i \(0.991909\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −2.96974 + 4.26388i −0.571527 + 0.820583i
\(28\) 0 0
\(29\) 5.68630i 1.05592i −0.849270 0.527959i \(-0.822957\pi\)
0.849270 0.527959i \(-0.177043\pi\)
\(30\) 0 0
\(31\) −2.45160 + 1.41543i −0.440320 + 0.254219i −0.703733 0.710464i \(-0.748485\pi\)
0.263414 + 0.964683i \(0.415152\pi\)
\(32\) 0 0
\(33\) −0.738430 2.22225i −0.128544 0.386845i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −3.33498 1.92545i −0.548267 0.316542i 0.200156 0.979764i \(-0.435855\pi\)
−0.748423 + 0.663222i \(0.769189\pi\)
\(38\) 0 0
\(39\) −1.72429 + 8.38603i −0.276108 + 1.34284i
\(40\) 0 0
\(41\) −3.73802 −0.583781 −0.291890 0.956452i \(-0.594284\pi\)
−0.291890 + 0.956452i \(0.594284\pi\)
\(42\) 0 0
\(43\) 4.06339i 0.619661i −0.950792 0.309830i \(-0.899728\pi\)
0.950792 0.309830i \(-0.100272\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.92419 2.84298i −0.718266 0.414691i 0.0958478 0.995396i \(-0.469444\pi\)
−0.814114 + 0.580705i \(0.802777\pi\)
\(48\) 0 0
\(49\) 4.65797 5.22526i 0.665424 0.746466i
\(50\) 0 0
\(51\) 3.13620 + 9.43818i 0.439156 + 1.32161i
\(52\) 0 0
\(53\) −0.730773 1.26574i −0.100379 0.173862i 0.811462 0.584406i \(-0.198672\pi\)
−0.911841 + 0.410544i \(0.865339\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −3.78172 + 4.25662i −0.500902 + 0.563804i
\(58\) 0 0
\(59\) 4.34239 + 7.52123i 0.565330 + 0.979181i 0.997019 + 0.0771582i \(0.0245846\pi\)
−0.431688 + 0.902023i \(0.642082\pi\)
\(60\) 0 0
\(61\) 1.65306 + 0.954394i 0.211653 + 0.122198i 0.602079 0.798436i \(-0.294339\pi\)
−0.390427 + 0.920634i \(0.627672\pi\)
\(62\) 0 0
\(63\) −7.93626 + 0.125326i −0.999875 + 0.0157896i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.36371 2.51939i 0.533113 0.307793i −0.209170 0.977879i \(-0.567076\pi\)
0.742283 + 0.670086i \(0.233743\pi\)
\(68\) 0 0
\(69\) 6.48121 + 5.75812i 0.780246 + 0.693196i
\(70\) 0 0
\(71\) 3.38259i 0.401440i −0.979649 0.200720i \(-0.935672\pi\)
0.979649 0.200720i \(-0.0643281\pi\)
\(72\) 0 0
\(73\) −8.16364 14.1398i −0.955481 1.65494i −0.733263 0.679945i \(-0.762004\pi\)
−0.222218 0.974997i \(-0.571330\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.09533 2.89912i 0.238785 0.330385i
\(78\) 0 0
\(79\) 2.41693 4.18625i 0.271926 0.470990i −0.697429 0.716654i \(-0.745673\pi\)
0.969355 + 0.245664i \(0.0790059\pi\)
\(80\) 0 0
\(81\) 6.19796 + 6.52574i 0.688662 + 0.725083i
\(82\) 0 0
\(83\) 16.6525i 1.82785i 0.405879 + 0.913927i \(0.366965\pi\)
−0.405879 + 0.913927i \(0.633035\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −9.64714 1.98359i −1.03428 0.212664i
\(88\) 0 0
\(89\) 8.08150 13.9976i 0.856637 1.48374i −0.0184813 0.999829i \(-0.505883\pi\)
0.875118 0.483909i \(-0.160784\pi\)
\(90\) 0 0
\(91\) −11.9339 + 5.34895i −1.25102 + 0.560723i
\(92\) 0 0
\(93\) 1.54615 + 4.65303i 0.160328 + 0.482497i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −12.0577 −1.22427 −0.612137 0.790751i \(-0.709690\pi\)
−0.612137 + 0.790751i \(0.709690\pi\)
\(98\) 0 0
\(99\) −4.02778 + 0.477584i −0.404807 + 0.0479990i
\(100\) 0 0
\(101\) 9.01683 + 15.6176i 0.897208 + 1.55401i 0.831048 + 0.556201i \(0.187742\pi\)
0.0661605 + 0.997809i \(0.478925\pi\)
\(102\) 0 0
\(103\) 0.308083 0.533615i 0.0303563 0.0525787i −0.850448 0.526059i \(-0.823669\pi\)
0.880804 + 0.473480i \(0.157002\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 8.85503 15.3374i 0.856048 1.48272i −0.0196209 0.999807i \(-0.506246\pi\)
0.875669 0.482912i \(-0.160421\pi\)
\(108\) 0 0
\(109\) −0.855282 1.48139i −0.0819211 0.141892i 0.822154 0.569265i \(-0.192772\pi\)
−0.904075 + 0.427374i \(0.859439\pi\)
\(110\) 0 0
\(111\) −4.43001 + 4.98632i −0.420478 + 0.473280i
\(112\) 0 0
\(113\) −13.1214 −1.23436 −0.617180 0.786822i \(-0.711725\pi\)
−0.617180 + 0.786822i \(0.711725\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 13.6259 + 5.85073i 1.25971 + 0.540900i
\(118\) 0 0
\(119\) −8.89912 + 12.3129i −0.815781 + 1.12872i
\(120\) 0 0
\(121\) −4.58605 + 7.94327i −0.416914 + 0.722116i
\(122\) 0 0
\(123\) −1.30396 + 6.34177i −0.117574 + 0.571818i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 15.5324i 1.37828i −0.724629 0.689139i \(-0.757989\pi\)
0.724629 0.689139i \(-0.242011\pi\)
\(128\) 0 0
\(129\) −6.89377 1.41746i −0.606963 0.124801i
\(130\) 0 0
\(131\) −4.47822 + 7.75651i −0.391264 + 0.677689i −0.992617 0.121295i \(-0.961295\pi\)
0.601353 + 0.798984i \(0.294629\pi\)
\(132\) 0 0
\(133\) −8.65214 0.887614i −0.750235 0.0769659i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 7.73923 + 13.4047i 0.661207 + 1.14524i 0.980299 + 0.197520i \(0.0632888\pi\)
−0.319092 + 0.947724i \(0.603378\pi\)
\(138\) 0 0
\(139\) 20.1547i 1.70950i −0.519044 0.854748i \(-0.673712\pi\)
0.519044 0.854748i \(-0.326288\pi\)
\(140\) 0 0
\(141\) −6.54103 + 7.36243i −0.550854 + 0.620029i
\(142\) 0 0
\(143\) −5.78754 + 3.34144i −0.483978 + 0.279425i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −7.24009 9.72528i −0.597152 0.802128i
\(148\) 0 0
\(149\) 1.24947 + 0.721384i 0.102361 + 0.0590981i 0.550307 0.834963i \(-0.314511\pi\)
−0.447946 + 0.894061i \(0.647844\pi\)
\(150\) 0 0
\(151\) −8.12108 14.0661i −0.660884 1.14468i −0.980384 0.197098i \(-0.936848\pi\)
0.319500 0.947586i \(-0.396485\pi\)
\(152\) 0 0
\(153\) 17.1065 2.02836i 1.38298 0.163983i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 9.11876 + 15.7942i 0.727756 + 1.26051i 0.957829 + 0.287338i \(0.0927701\pi\)
−0.230073 + 0.973173i \(0.573897\pi\)
\(158\) 0 0
\(159\) −2.40232 + 0.798263i −0.190516 + 0.0633064i
\(160\) 0 0
\(161\) −1.35150 + 13.1739i −0.106513 + 1.03825i
\(162\) 0 0
\(163\) −21.0669 12.1630i −1.65009 0.952679i −0.977033 0.213087i \(-0.931648\pi\)
−0.673055 0.739592i \(-0.735018\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 12.5007i 0.967336i 0.875252 + 0.483668i \(0.160696\pi\)
−0.875252 + 0.483668i \(0.839304\pi\)
\(168\) 0 0
\(169\) 11.4329 0.879453
\(170\) 0 0
\(171\) 5.90241 + 7.90079i 0.451368 + 0.604188i
\(172\) 0 0
\(173\) 12.8534 + 7.42089i 0.977223 + 0.564200i 0.901431 0.432923i \(-0.142518\pi\)
0.0757927 + 0.997124i \(0.475851\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 14.2750 4.74342i 1.07297 0.356538i
\(178\) 0 0
\(179\) −9.93533 + 5.73617i −0.742602 + 0.428741i −0.823015 0.568020i \(-0.807709\pi\)
0.0804127 + 0.996762i \(0.474376\pi\)
\(180\) 0 0
\(181\) 5.91099i 0.439361i −0.975572 0.219680i \(-0.929499\pi\)
0.975572 0.219680i \(-0.0705014\pi\)
\(182\) 0 0
\(183\) 2.19584 2.47158i 0.162321 0.182705i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −3.88165 + 6.72322i −0.283854 + 0.491650i
\(188\) 0 0
\(189\) −2.55584 + 13.5081i −0.185910 + 0.982567i
\(190\) 0 0
\(191\) 9.15764 + 5.28716i 0.662623 + 0.382566i 0.793276 0.608862i \(-0.208374\pi\)
−0.130652 + 0.991428i \(0.541707\pi\)
\(192\) 0 0
\(193\) −10.4483 + 6.03231i −0.752082 + 0.434215i −0.826446 0.563016i \(-0.809641\pi\)
0.0743635 + 0.997231i \(0.476307\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −17.2423 −1.22846 −0.614231 0.789126i \(-0.710534\pi\)
−0.614231 + 0.789126i \(0.710534\pi\)
\(198\) 0 0
\(199\) −2.46400 + 1.42259i −0.174668 + 0.100845i −0.584785 0.811188i \(-0.698821\pi\)
0.410117 + 0.912033i \(0.365488\pi\)
\(200\) 0 0
\(201\) −2.75207 8.28216i −0.194116 0.584178i
\(202\) 0 0
\(203\) −6.15334 13.7286i −0.431880 0.963558i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 12.0299 8.98710i 0.836134 0.624647i
\(208\) 0 0
\(209\) −4.44451 −0.307433
\(210\) 0 0
\(211\) 22.4677 1.54674 0.773372 0.633953i \(-0.218569\pi\)
0.773372 + 0.633953i \(0.218569\pi\)
\(212\) 0 0
\(213\) −5.73877 1.17998i −0.393214 0.0808506i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −4.38727 + 6.07027i −0.297828 + 0.412077i
\(218\) 0 0
\(219\) −26.8368 + 8.91758i −1.81347 + 0.602594i
\(220\) 0 0
\(221\) 24.5804 14.1915i 1.65346 0.954623i
\(222\) 0 0
\(223\) −5.27620 −0.353321 −0.176660 0.984272i \(-0.556529\pi\)
−0.176660 + 0.984272i \(0.556529\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −10.8090 + 6.24055i −0.717416 + 0.414200i −0.813801 0.581144i \(-0.802605\pi\)
0.0963851 + 0.995344i \(0.469272\pi\)
\(228\) 0 0
\(229\) 14.0882 + 8.13380i 0.930971 + 0.537496i 0.887119 0.461542i \(-0.152703\pi\)
0.0438526 + 0.999038i \(0.486037\pi\)
\(230\) 0 0
\(231\) −4.18759 4.56617i −0.275523 0.300432i
\(232\) 0 0
\(233\) 4.41551 7.64788i 0.289269 0.501029i −0.684366 0.729139i \(-0.739921\pi\)
0.973636 + 0.228109i \(0.0732543\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −6.25911 5.56079i −0.406573 0.361212i
\(238\) 0 0
\(239\) 12.3333i 0.797773i 0.917000 + 0.398886i \(0.130603\pi\)
−0.917000 + 0.398886i \(0.869397\pi\)
\(240\) 0 0
\(241\) 6.88815 3.97688i 0.443705 0.256173i −0.261463 0.965214i \(-0.584205\pi\)
0.705168 + 0.709040i \(0.250872\pi\)
\(242\) 0 0
\(243\) 13.2334 8.23878i 0.848922 0.528518i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 14.0723 + 8.12466i 0.895401 + 0.516960i
\(248\) 0 0
\(249\) 28.2520 + 5.80903i 1.79040 + 0.368133i
\(250\) 0 0
\(251\) −7.64756 −0.482710 −0.241355 0.970437i \(-0.577592\pi\)
−0.241355 + 0.970437i \(0.577592\pi\)
\(252\) 0 0
\(253\) 6.76728i 0.425455i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 23.8118 + 13.7478i 1.48534 + 0.857563i 0.999861 0.0166841i \(-0.00531095\pi\)
0.485482 + 0.874247i \(0.338644\pi\)
\(258\) 0 0
\(259\) −10.1353 1.03977i −0.629779 0.0646084i
\(260\) 0 0
\(261\) −6.73057 + 15.6750i −0.416612 + 0.970257i
\(262\) 0 0
\(263\) −9.51115 16.4738i −0.586483 1.01582i −0.994689 0.102928i \(-0.967179\pi\)
0.408206 0.912890i \(-0.366155\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −20.9286 18.5936i −1.28081 1.13791i
\(268\) 0 0
\(269\) −4.46010 7.72512i −0.271937 0.471009i 0.697421 0.716662i \(-0.254331\pi\)
−0.969358 + 0.245653i \(0.920998\pi\)
\(270\) 0 0
\(271\) 6.53123 + 3.77081i 0.396744 + 0.229060i 0.685078 0.728470i \(-0.259768\pi\)
−0.288334 + 0.957530i \(0.593101\pi\)
\(272\) 0 0
\(273\) 4.91181 + 22.1125i 0.297276 + 1.33831i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −1.53903 + 0.888562i −0.0924716 + 0.0533885i −0.545523 0.838096i \(-0.683669\pi\)
0.453051 + 0.891485i \(0.350336\pi\)
\(278\) 0 0
\(279\) 8.43350 0.999983i 0.504900 0.0598674i
\(280\) 0 0
\(281\) 25.3819i 1.51416i −0.653324 0.757079i \(-0.726626\pi\)
0.653324 0.757079i \(-0.273374\pi\)
\(282\) 0 0
\(283\) 8.06987 + 13.9774i 0.479704 + 0.830872i 0.999729 0.0232795i \(-0.00741075\pi\)
−0.520025 + 0.854151i \(0.674077\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −9.02481 + 4.04504i −0.532718 + 0.238771i
\(288\) 0 0
\(289\) 7.98584 13.8319i 0.469755 0.813640i
\(290\) 0 0
\(291\) −4.20618 + 20.4566i −0.246571 + 1.19919i
\(292\) 0 0
\(293\) 19.5542i 1.14237i −0.820821 0.571186i \(-0.806484\pi\)
0.820821 0.571186i \(-0.193516\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −0.594790 + 6.99996i −0.0345132 + 0.406179i
\(298\) 0 0
\(299\) 12.3707 21.4268i 0.715419 1.23914i
\(300\) 0 0
\(301\) −4.39713 9.81035i −0.253447 0.565459i
\(302\) 0 0
\(303\) 29.6416 9.84957i 1.70287 0.565843i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −11.8231 −0.674777 −0.337389 0.941365i \(-0.609544\pi\)
−0.337389 + 0.941365i \(0.609544\pi\)
\(308\) 0 0
\(309\) −0.797839 0.708826i −0.0453875 0.0403237i
\(310\) 0 0
\(311\) −11.1689 19.3451i −0.633331 1.09696i −0.986866 0.161540i \(-0.948354\pi\)
0.353535 0.935421i \(-0.384980\pi\)
\(312\) 0 0
\(313\) −6.65398 + 11.5250i −0.376105 + 0.651433i −0.990492 0.137572i \(-0.956070\pi\)
0.614387 + 0.789005i \(0.289404\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 16.9392 29.3395i 0.951400 1.64787i 0.209001 0.977915i \(-0.432979\pi\)
0.742399 0.669958i \(-0.233688\pi\)
\(318\) 0 0
\(319\) −3.84393 6.65788i −0.215219 0.372770i
\(320\) 0 0
\(321\) −22.9318 20.3733i −1.27993 1.13713i
\(322\) 0 0
\(323\) 18.8764 1.05031
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −2.81162 + 0.934271i −0.155483 + 0.0516653i
\(328\) 0 0
\(329\) −14.9651 1.53525i −0.825052 0.0846413i
\(330\) 0 0
\(331\) 0.989824 1.71443i 0.0544057 0.0942334i −0.837540 0.546376i \(-0.816007\pi\)
0.891946 + 0.452143i \(0.149340\pi\)
\(332\) 0 0
\(333\) 6.91423 + 9.25518i 0.378897 + 0.507181i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 12.0217i 0.654862i −0.944875 0.327431i \(-0.893817\pi\)
0.944875 0.327431i \(-0.106183\pi\)
\(338\) 0 0
\(339\) −4.57724 + 22.2613i −0.248602 + 1.20907i
\(340\) 0 0
\(341\) −1.91366 + 3.31455i −0.103630 + 0.179493i
\(342\) 0 0
\(343\) 5.59143 17.6560i 0.301909 0.953337i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −7.33186 12.6992i −0.393595 0.681726i 0.599326 0.800505i \(-0.295435\pi\)
−0.992921 + 0.118779i \(0.962102\pi\)
\(348\) 0 0
\(349\) 25.0573i 1.34129i −0.741780 0.670644i \(-0.766018\pi\)
0.741780 0.670644i \(-0.233982\pi\)
\(350\) 0 0
\(351\) 14.6793 21.0762i 0.783525 1.12496i
\(352\) 0 0
\(353\) 6.72669 3.88365i 0.358025 0.206706i −0.310189 0.950675i \(-0.600392\pi\)
0.668214 + 0.743969i \(0.267059\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 17.7852 + 19.3931i 0.941293 + 1.02639i
\(358\) 0 0
\(359\) 3.86952 + 2.23407i 0.204225 + 0.117910i 0.598625 0.801030i \(-0.295714\pi\)
−0.394399 + 0.918939i \(0.629047\pi\)
\(360\) 0 0
\(361\) −4.09662 7.09555i −0.215612 0.373450i
\(362\) 0 0
\(363\) 11.8764 + 10.5514i 0.623352 + 0.553806i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0.842240 + 1.45880i 0.0439646 + 0.0761489i 0.887170 0.461442i \(-0.152668\pi\)
−0.843206 + 0.537591i \(0.819334\pi\)
\(368\) 0 0
\(369\) 10.3043 + 4.42450i 0.536421 + 0.230330i
\(370\) 0 0
\(371\) −3.13402 2.26511i −0.162710 0.117599i
\(372\) 0 0
\(373\) 11.9914 + 6.92322i 0.620890 + 0.358471i 0.777215 0.629235i \(-0.216632\pi\)
−0.156325 + 0.987706i \(0.549965\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 28.1072i 1.44759i
\(378\) 0 0
\(379\) 2.15603 0.110748 0.0553739 0.998466i \(-0.482365\pi\)
0.0553739 + 0.998466i \(0.482365\pi\)
\(380\) 0 0
\(381\) −26.3516 5.41829i −1.35004 0.277587i
\(382\) 0 0
\(383\) 3.23197 + 1.86598i 0.165146 + 0.0953471i 0.580295 0.814406i \(-0.302937\pi\)
−0.415149 + 0.909753i \(0.636271\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −4.80962 + 11.2012i −0.244487 + 0.569391i
\(388\) 0 0
\(389\) 11.9512 6.90001i 0.605948 0.349844i −0.165430 0.986222i \(-0.552901\pi\)
0.771378 + 0.636377i \(0.219568\pi\)
\(390\) 0 0
\(391\) 28.7415i 1.45352i
\(392\) 0 0
\(393\) 11.5972 + 10.3033i 0.585001 + 0.519734i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 8.17565 14.1606i 0.410324 0.710702i −0.584601 0.811321i \(-0.698749\pi\)
0.994925 + 0.100619i \(0.0320822\pi\)
\(398\) 0 0
\(399\) −4.52408 + 14.3692i −0.226487 + 0.719361i
\(400\) 0 0
\(401\) −11.9669 6.90910i −0.597599 0.345024i 0.170497 0.985358i \(-0.445463\pi\)
−0.768096 + 0.640334i \(0.778796\pi\)
\(402\) 0 0
\(403\) 12.1182 6.99642i 0.603648 0.348516i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −5.20641 −0.258072
\(408\) 0 0
\(409\) 8.26987 4.77461i 0.408919 0.236089i −0.281406 0.959589i \(-0.590801\pi\)
0.690325 + 0.723499i \(0.257467\pi\)
\(410\) 0 0
\(411\) 25.4417 8.45398i 1.25494 0.417004i
\(412\) 0 0
\(413\) 18.6229 + 13.4597i 0.916375 + 0.662307i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −34.1936 7.03070i −1.67447 0.344295i
\(418\) 0 0
\(419\) −9.71886 −0.474797 −0.237399 0.971412i \(-0.576295\pi\)
−0.237399 + 0.971412i \(0.576295\pi\)
\(420\) 0 0
\(421\) 34.8713 1.69953 0.849763 0.527165i \(-0.176745\pi\)
0.849763 + 0.527165i \(0.176745\pi\)
\(422\) 0 0
\(423\) 10.2090 + 13.6655i 0.496381 + 0.664441i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 5.02381 + 0.515388i 0.243119 + 0.0249414i
\(428\) 0 0
\(429\) 3.65003 + 10.9845i 0.176225 + 0.530338i
\(430\) 0 0
\(431\) −1.57279 + 0.908052i −0.0757587 + 0.0437393i −0.537401 0.843327i \(-0.680594\pi\)
0.461642 + 0.887066i \(0.347260\pi\)
\(432\) 0 0
\(433\) −10.6773 −0.513117 −0.256558 0.966529i \(-0.582589\pi\)
−0.256558 + 0.966529i \(0.582589\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 14.2501 8.22728i 0.681673 0.393564i
\(438\) 0 0
\(439\) −26.6759 15.4013i −1.27317 0.735066i −0.297588 0.954694i \(-0.596182\pi\)
−0.975584 + 0.219628i \(0.929516\pi\)
\(440\) 0 0
\(441\) −19.0251 + 8.89069i −0.905959 + 0.423366i
\(442\) 0 0
\(443\) −8.26640 + 14.3178i −0.392749 + 0.680260i −0.992811 0.119693i \(-0.961809\pi\)
0.600062 + 0.799953i \(0.295142\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 1.65974 1.86816i 0.0785028 0.0883610i
\(448\) 0 0
\(449\) 2.59098i 0.122276i −0.998129 0.0611380i \(-0.980527\pi\)
0.998129 0.0611380i \(-0.0194730\pi\)
\(450\) 0 0
\(451\) −4.37671 + 2.52690i −0.206091 + 0.118987i
\(452\) 0 0
\(453\) −26.6969 + 8.87110i −1.25433 + 0.416800i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 19.2040 + 11.0874i 0.898323 + 0.518647i 0.876656 0.481118i \(-0.159769\pi\)
0.0216672 + 0.999765i \(0.493103\pi\)
\(458\) 0 0
\(459\) 2.52615 29.7297i 0.117910 1.38766i
\(460\) 0 0
\(461\) −30.5304 −1.42194 −0.710971 0.703221i \(-0.751744\pi\)
−0.710971 + 0.703221i \(0.751744\pi\)
\(462\) 0 0
\(463\) 3.54824i 0.164901i −0.996595 0.0824504i \(-0.973725\pi\)
0.996595 0.0824504i \(-0.0262746\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −29.7157 17.1564i −1.37508 0.793901i −0.383515 0.923534i \(-0.625287\pi\)
−0.991562 + 0.129633i \(0.958620\pi\)
\(468\) 0 0
\(469\) 7.80912 10.8048i 0.360592 0.498918i
\(470\) 0 0
\(471\) 29.9767 9.96092i 1.38125 0.458975i
\(472\) 0 0
\(473\) −2.74684 4.75767i −0.126300 0.218758i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0.516282 + 4.35414i 0.0236389 + 0.199362i
\(478\) 0 0
\(479\) 9.78624 + 16.9503i 0.447145 + 0.774477i 0.998199 0.0599923i \(-0.0191076\pi\)
−0.551054 + 0.834469i \(0.685774\pi\)
\(480\) 0 0
\(481\) 16.4847 + 9.51743i 0.751637 + 0.433958i
\(482\) 0 0
\(483\) 21.8788 + 6.88844i 0.995521 + 0.313435i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 21.8468 12.6133i 0.989973 0.571561i 0.0847071 0.996406i \(-0.473005\pi\)
0.905266 + 0.424844i \(0.139671\pi\)
\(488\) 0 0
\(489\) −27.9842 + 31.4984i −1.26549 + 1.42441i
\(490\) 0 0
\(491\) 38.6556i 1.74450i −0.489057 0.872252i \(-0.662659\pi\)
0.489057 0.872252i \(-0.337341\pi\)
\(492\) 0 0
\(493\) 16.3256 + 28.2768i 0.735269 + 1.27352i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −3.66042 8.16669i −0.164192 0.366326i
\(498\) 0 0
\(499\) 2.80910 4.86551i 0.125753 0.217810i −0.796274 0.604936i \(-0.793199\pi\)
0.922027 + 0.387126i \(0.126532\pi\)
\(500\) 0 0
\(501\) 21.2082 + 4.36073i 0.947514 + 0.194823i
\(502\) 0 0
\(503\) 25.8655i 1.15329i −0.816996 0.576644i \(-0.804362\pi\)
0.816996 0.576644i \(-0.195638\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 3.98822 19.3966i 0.177123 0.861432i
\(508\) 0 0
\(509\) −15.2306 + 26.3801i −0.675082 + 1.16928i 0.301362 + 0.953510i \(0.402559\pi\)
−0.976445 + 0.215767i \(0.930775\pi\)
\(510\) 0 0
\(511\) −35.0109 25.3040i −1.54879 1.11939i
\(512\) 0 0
\(513\) 15.4631 7.25769i 0.682714 0.320435i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −7.68740 −0.338092
\(518\) 0 0
\(519\) 17.0737 19.2178i 0.749453 0.843568i
\(520\) 0 0
\(521\) −10.6182 18.3913i −0.465192 0.805737i 0.534018 0.845473i \(-0.320681\pi\)
−0.999210 + 0.0397366i \(0.987348\pi\)
\(522\) 0 0
\(523\) −6.19077 + 10.7227i −0.270704 + 0.468872i −0.969042 0.246895i \(-0.920590\pi\)
0.698339 + 0.715768i \(0.253923\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 8.12754 14.0773i 0.354041 0.613217i
\(528\) 0 0
\(529\) −1.02699 1.77880i −0.0446518 0.0773392i
\(530\) 0 0
\(531\) −3.06784 25.8731i −0.133133 1.12280i
\(532\) 0 0
\(533\) 18.4769 0.800323
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 6.26593 + 18.8569i 0.270395 + 0.813734i
\(538\) 0 0
\(539\) 1.92158 9.26685i 0.0827682 0.399151i
\(540\) 0 0
\(541\) −6.93001 + 12.0031i −0.297944 + 0.516055i −0.975665 0.219264i \(-0.929634\pi\)
0.677721 + 0.735319i \(0.262968\pi\)
\(542\) 0 0
\(543\) −10.0283 2.06198i −0.430358 0.0884879i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 17.7100i 0.757226i 0.925555 + 0.378613i \(0.123599\pi\)
−0.925555 + 0.378613i \(0.876401\pi\)
\(548\) 0 0
\(549\) −3.42720 4.58755i −0.146269 0.195792i
\(550\) 0 0
\(551\) −9.34646 + 16.1885i −0.398173 + 0.689655i
\(552\) 0 0
\(553\) 1.30518 12.7224i 0.0555020 0.541013i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.73857 + 3.01129i 0.0736656 + 0.127593i 0.900505 0.434845i \(-0.143197\pi\)
−0.826840 + 0.562438i \(0.809864\pi\)
\(558\) 0 0
\(559\) 20.0852i 0.849512i
\(560\) 0 0
\(561\) 10.0523 + 8.93076i 0.424407 + 0.377057i
\(562\) 0 0
\(563\) −10.4425 + 6.02897i −0.440098 + 0.254091i −0.703639 0.710557i \(-0.748443\pi\)
0.263541 + 0.964648i \(0.415110\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 22.0256 + 9.04826i 0.924990 + 0.379991i
\(568\) 0 0
\(569\) 11.8065 + 6.81650i 0.494955 + 0.285762i 0.726628 0.687031i \(-0.241087\pi\)
−0.231673 + 0.972794i \(0.574420\pi\)
\(570\) 0 0
\(571\) 22.0474 + 38.1873i 0.922657 + 1.59809i 0.795287 + 0.606233i \(0.207320\pi\)
0.127369 + 0.991855i \(0.459347\pi\)
\(572\) 0 0
\(573\) 12.1645 13.6921i 0.508180 0.571996i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −3.49934 6.06103i −0.145679 0.252324i 0.783947 0.620828i \(-0.213203\pi\)
−0.929626 + 0.368504i \(0.879870\pi\)
\(578\) 0 0
\(579\) 6.58942 + 19.8304i 0.273847 + 0.824123i
\(580\) 0 0
\(581\) 18.0203 + 40.2047i 0.747608 + 1.66797i
\(582\) 0 0
\(583\) −1.71127 0.988003i −0.0708736 0.0409189i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 29.9161i 1.23477i 0.786661 + 0.617385i \(0.211808\pi\)
−0.786661 + 0.617385i \(0.788192\pi\)
\(588\) 0 0
\(589\) 9.30607 0.383450
\(590\) 0 0
\(591\) −6.01476 + 29.2525i −0.247414 + 1.20329i
\(592\) 0 0
\(593\) 20.8583 + 12.0426i 0.856549 + 0.494529i 0.862855 0.505451i \(-0.168674\pi\)
−0.00630583 + 0.999980i \(0.502007\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 1.55397 + 4.67657i 0.0635999 + 0.191399i
\(598\) 0 0
\(599\) 5.14773 2.97205i 0.210331 0.121434i −0.391134 0.920334i \(-0.627917\pi\)
0.601465 + 0.798899i \(0.294584\pi\)
\(600\) 0 0
\(601\) 46.9992i 1.91714i −0.284863 0.958568i \(-0.591948\pi\)
0.284863 0.958568i \(-0.408052\pi\)
\(602\) 0 0
\(603\) −15.0112 + 1.77992i −0.611303 + 0.0724839i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 4.97322 8.61387i 0.201857 0.349626i −0.747270 0.664521i \(-0.768636\pi\)
0.949127 + 0.314894i \(0.101969\pi\)
\(608\) 0 0
\(609\) −25.4379 + 5.65046i −1.03079 + 0.228968i
\(610\) 0 0
\(611\) 24.3401 + 14.0528i 0.984694 + 0.568513i
\(612\) 0 0
\(613\) 28.1873 16.2739i 1.13847 0.657298i 0.192421 0.981312i \(-0.438366\pi\)
0.946052 + 0.324014i \(0.105033\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 7.69843 0.309927 0.154964 0.987920i \(-0.450474\pi\)
0.154964 + 0.987920i \(0.450474\pi\)
\(618\) 0 0
\(619\) −2.76170 + 1.59447i −0.111002 + 0.0640872i −0.554473 0.832202i \(-0.687080\pi\)
0.443471 + 0.896289i \(0.353747\pi\)
\(620\) 0 0
\(621\) −11.0507 23.5444i −0.443448 0.944805i
\(622\) 0 0
\(623\) 4.36414 42.5400i 0.174845 1.70433i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −1.55041 + 7.54037i −0.0619175 + 0.301133i
\(628\) 0 0
\(629\) 22.1122 0.881673
\(630\) 0 0
\(631\) 7.97461 0.317464 0.158732 0.987322i \(-0.449259\pi\)
0.158732 + 0.987322i \(0.449259\pi\)
\(632\) 0 0
\(633\) 7.83760 38.1179i 0.311516 1.51505i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −23.0242 + 25.8283i −0.912251 + 1.02335i
\(638\) 0 0
\(639\) −4.00380 + 9.32454i −0.158388 + 0.368873i
\(640\) 0 0
\(641\) −23.7774 + 13.7279i −0.939152 + 0.542219i −0.889694 0.456557i \(-0.849083\pi\)
−0.0494573 + 0.998776i \(0.515749\pi\)
\(642\) 0 0
\(643\) −29.0919 −1.14727 −0.573637 0.819110i \(-0.694468\pi\)
−0.573637 + 0.819110i \(0.694468\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 37.4122 21.5999i 1.47082 0.849181i 0.471361 0.881940i \(-0.343763\pi\)
0.999463 + 0.0327597i \(0.0104296\pi\)
\(648\) 0 0
\(649\) 10.1687 + 5.87089i 0.399156 + 0.230453i
\(650\) 0 0
\(651\) 8.76813 + 9.56081i 0.343650 + 0.374718i
\(652\) 0 0
\(653\) −3.62080 + 6.27142i −0.141693 + 0.245420i −0.928134 0.372246i \(-0.878588\pi\)
0.786441 + 0.617665i \(0.211921\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 5.76750 + 48.6411i 0.225012 + 1.89767i
\(658\) 0 0
\(659\) 5.29324i 0.206196i 0.994671 + 0.103098i \(0.0328755\pi\)
−0.994671 + 0.103098i \(0.967125\pi\)
\(660\) 0 0
\(661\) 43.2638 24.9783i 1.68276 0.971545i 0.722954 0.690896i \(-0.242784\pi\)
0.959810 0.280649i \(-0.0905497\pi\)
\(662\) 0 0
\(663\) −15.5021 46.6526i −0.602053 1.81184i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 24.6490 + 14.2311i 0.954411 + 0.551030i
\(668\) 0 0
\(669\) −1.84054 + 8.95139i −0.0711593 + 0.346081i
\(670\) 0 0
\(671\) 2.58068 0.0996259
\(672\) 0 0
\(673\) 21.8855i 0.843625i 0.906683 + 0.421812i \(0.138606\pi\)
−0.906683 + 0.421812i \(0.861394\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −11.9033 6.87236i −0.457480 0.264126i 0.253504 0.967334i \(-0.418417\pi\)
−0.710984 + 0.703208i \(0.751750\pi\)
\(678\) 0 0
\(679\) −29.1113 + 13.0481i −1.11719 + 0.500739i
\(680\) 0 0
\(681\) 6.81690 + 20.5150i 0.261224 + 0.786135i
\(682\) 0 0
\(683\) 21.4513 + 37.1547i 0.820811 + 1.42169i 0.905079 + 0.425243i \(0.139811\pi\)
−0.0842682 + 0.996443i \(0.526855\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 18.7139 21.0640i 0.713982 0.803642i
\(688\) 0 0
\(689\) 3.61218 + 6.25649i 0.137613 + 0.238353i
\(690\) 0 0
\(691\) −20.5494 11.8642i −0.781735 0.451335i 0.0553098 0.998469i \(-0.482385\pi\)
−0.837045 + 0.547134i \(0.815719\pi\)
\(692\) 0 0
\(693\) −9.20757 + 5.51164i −0.349767 + 0.209370i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 18.5884 10.7320i 0.704087 0.406505i
\(698\) 0 0
\(699\) −11.4348 10.1590i −0.432503 0.384250i
\(700\) 0 0
\(701\) 21.2721i 0.803436i 0.915763 + 0.401718i \(0.131587\pi\)
−0.915763 + 0.401718i \(0.868413\pi\)
\(702\) 0 0
\(703\) 6.32966 + 10.9633i 0.238728 + 0.413488i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 38.6700 + 27.9486i 1.45433 + 1.05112i
\(708\) 0 0
\(709\) −22.8069 + 39.5026i −0.856529 + 1.48355i 0.0186897 + 0.999825i \(0.494051\pi\)
−0.875219 + 0.483727i \(0.839283\pi\)
\(710\) 0 0
\(711\) −11.6176 + 8.67913i −0.435695 + 0.325493i
\(712\) 0 0
\(713\) 14.1696i 0.530655i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 20.9241 + 4.30231i 0.781426 + 0.160673i
\(718\) 0 0
\(719\) 9.50850 16.4692i 0.354607 0.614198i −0.632443 0.774607i \(-0.717948\pi\)
0.987051 + 0.160409i \(0.0512812\pi\)
\(720\) 0 0
\(721\) 0.166370 1.62171i 0.00619593 0.0603956i
\(722\) 0 0
\(723\) −4.34416 13.0734i −0.161561 0.486207i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −34.0540 −1.26299 −0.631496 0.775379i \(-0.717559\pi\)
−0.631496 + 0.775379i \(0.717559\pi\)
\(728\) 0 0
\(729\) −9.36126 25.3252i −0.346714 0.937971i
\(730\) 0 0
\(731\) 11.6662 + 20.2064i 0.431489 + 0.747361i
\(732\) 0 0
\(733\) −3.87885 + 6.71837i −0.143269 + 0.248148i −0.928726 0.370768i \(-0.879095\pi\)
0.785457 + 0.618916i \(0.212428\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 3.40621 5.89973i 0.125469 0.217319i
\(738\) 0 0
\(739\) −4.19659 7.26871i −0.154374 0.267384i 0.778457 0.627698i \(-0.216003\pi\)
−0.932831 + 0.360314i \(0.882669\pi\)
\(740\) 0 0
\(741\) 18.6929 21.0403i 0.686702 0.772936i
\(742\) 0 0
\(743\) −14.7540 −0.541273 −0.270636 0.962682i \(-0.587234\pi\)
−0.270636 + 0.962682i \(0.587234\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 19.7107 45.9048i 0.721178 1.67957i
\(748\) 0 0
\(749\) 4.78186 46.6118i 0.174725 1.70316i
\(750\) 0 0
\(751\) −20.6067 + 35.6918i −0.751948 + 1.30241i 0.194929 + 0.980817i \(0.437552\pi\)
−0.946877 + 0.321595i \(0.895781\pi\)
\(752\) 0 0
\(753\) −2.66776 + 12.9745i −0.0972185 + 0.472818i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 11.8681i 0.431354i −0.976465 0.215677i \(-0.930804\pi\)
0.976465 0.215677i \(-0.0691958\pi\)
\(758\) 0 0
\(759\) 11.4811 + 2.36068i 0.416737 + 0.0856874i
\(760\) 0 0
\(761\) 17.7705 30.7795i 0.644181 1.11575i −0.340309 0.940314i \(-0.610532\pi\)
0.984490 0.175441i \(-0.0561351\pi\)
\(762\) 0 0
\(763\) −3.66800 2.65104i −0.132790 0.0959739i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −21.4643 37.1772i −0.775029 1.34239i
\(768\) 0 0
\(769\) 6.25608i 0.225600i 0.993618 + 0.112800i \(0.0359820\pi\)
−0.993618 + 0.112800i \(0.964018\pi\)
\(770\) 0 0
\(771\) 31.6304 35.6025i 1.13914 1.28219i
\(772\) 0 0
\(773\) −17.3483 + 10.0161i −0.623977 + 0.360253i −0.778416 0.627749i \(-0.783976\pi\)
0.154439 + 0.988002i \(0.450643\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −5.29962 + 16.8325i −0.190123 + 0.603862i
\(778\) 0 0
\(779\) 10.6419 + 6.14412i 0.381286 + 0.220136i
\(780\) 0 0
\(781\) −2.28663 3.96056i −0.0818220 0.141720i
\(782\) 0 0
\(783\) 24.2457 + 16.8868i 0.866469 + 0.603486i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −0.738053 1.27835i −0.0263088 0.0455681i 0.852571 0.522611i \(-0.175042\pi\)
−0.878880 + 0.477043i \(0.841709\pi\)
\(788\) 0 0
\(789\) −31.2666 + 10.3895i −1.11312 + 0.369878i
\(790\) 0 0
\(791\) −31.6794 + 14.1991i −1.12639 + 0.504863i
\(792\) 0 0
\(793\) −8.17101 4.71754i −0.290161 0.167525i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 22.9470i 0.812823i −0.913690 0.406411i \(-0.866780\pi\)
0.913690 0.406411i \(-0.133220\pi\)
\(798\) 0 0
\(799\) 32.6493 1.15505
\(800\) 0 0
\(801\) −38.8458 + 29.0204i −1.37255 + 1.02538i
\(802\) 0 0
\(803\) −19.1170 11.0372i −0.674625 0.389495i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −14.6620 + 4.87201i −0.516126 + 0.171503i
\(808\) 0 0
\(809\) −19.5613 + 11.2937i −0.687739 + 0.397066i −0.802764 0.596296i \(-0.796638\pi\)
0.115026 + 0.993363i \(0.463305\pi\)
\(810\) 0 0
\(811\) 41.9366i 1.47259i 0.676659 + 0.736296i \(0.263427\pi\)
−0.676659 + 0.736296i \(0.736573\pi\)
\(812\) 0 0
\(813\) 8.67573 9.76522i 0.304271 0.342481i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −6.67892 + 11.5682i −0.233666 + 0.404721i
\(818\) 0 0
\(819\) 39.2287 0.619483i 1.37076 0.0216465i
\(820\) 0 0
\(821\) 0.447025 + 0.258090i 0.0156013 + 0.00900741i 0.507780 0.861487i \(-0.330466\pi\)
−0.492179 + 0.870494i \(0.663799\pi\)
\(822\) 0 0
\(823\) 17.8125 10.2841i 0.620906 0.358480i −0.156316 0.987707i \(-0.549962\pi\)
0.777222 + 0.629227i \(0.216628\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −24.0587 −0.836603 −0.418301 0.908308i \(-0.637374\pi\)
−0.418301 + 0.908308i \(0.637374\pi\)
\(828\) 0 0
\(829\) 39.3074 22.6941i 1.36520 0.788200i 0.374891 0.927069i \(-0.377680\pi\)
0.990311 + 0.138869i \(0.0443468\pi\)
\(830\) 0 0
\(831\) 0.970625 + 2.92103i 0.0336706 + 0.101329i
\(832\) 0 0
\(833\) −8.16117 + 39.3574i −0.282768 + 1.36365i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 1.24539 14.6568i 0.0430471 0.506612i
\(838\) 0 0
\(839\) −39.8758 −1.37666 −0.688332 0.725395i \(-0.741657\pi\)
−0.688332 + 0.725395i \(0.741657\pi\)
\(840\) 0 0
\(841\) −3.33396 −0.114964
\(842\) 0 0
\(843\) −43.0619 8.85416i −1.48313 0.304954i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −2.47654 + 24.1404i −0.0850949 + 0.829474i
\(848\) 0 0
\(849\) 26.5286 8.81516i 0.910459 0.302535i
\(850\) 0 0
\(851\) 16.6929 9.63764i 0.572225 0.330374i
\(852\) 0 0
\(853\) −16.9504 −0.580372 −0.290186 0.956970i \(-0.593717\pi\)
−0.290186 + 0.956970i \(0.593717\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0.546711 0.315644i 0.0186753 0.0107822i −0.490633 0.871366i \(-0.663235\pi\)
0.509309 + 0.860584i \(0.329901\pi\)
\(858\) 0 0
\(859\) 5.59642 + 3.23109i 0.190947 + 0.110243i 0.592426 0.805625i \(-0.298170\pi\)
−0.401479 + 0.915868i \(0.631504\pi\)
\(860\) 0 0
\(861\) 3.71446 + 16.7222i 0.126588 + 0.569891i
\(862\) 0 0
\(863\) −3.64782 + 6.31821i −0.124173 + 0.215074i −0.921409 0.388593i \(-0.872961\pi\)
0.797236 + 0.603667i \(0.206295\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −20.6808 18.3735i −0.702358 0.623998i
\(868\) 0 0
\(869\) 6.53538i 0.221698i
\(870\) 0 0
\(871\) −21.5697 + 12.4533i −0.730861 + 0.421963i
\(872\) 0 0
\(873\) 33.2386 + 14.2721i 1.12496 + 0.483037i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −25.7109 14.8442i −0.868196 0.501253i −0.00144772 0.999999i \(-0.500461\pi\)
−0.866748 + 0.498746i \(0.833794\pi\)
\(878\) 0 0
\(879\) −33.1749 6.82126i −1.11896 0.230075i
\(880\) 0 0
\(881\) −21.6272 −0.728638 −0.364319 0.931274i \(-0.618698\pi\)
−0.364319 + 0.931274i \(0.618698\pi\)
\(882\) 0 0
\(883\) 29.9462i 1.00777i 0.863771 + 0.503884i \(0.168096\pi\)
−0.863771 + 0.503884i \(0.831904\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −10.2160 5.89821i −0.343020 0.198043i 0.318587 0.947894i \(-0.396792\pi\)
−0.661607 + 0.749851i \(0.730125\pi\)
\(888\) 0 0
\(889\) −16.8082 37.5003i −0.563728 1.25772i
\(890\) 0 0
\(891\) 11.6684 + 3.45095i 0.390905 + 0.115611i
\(892\) 0 0
\(893\) 9.34591 + 16.1876i 0.312749 + 0.541697i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −32.0364 28.4622i −1.06966 0.950324i
\(898\) 0 0
\(899\) 8.04855 + 13.9405i 0.268434 + 0.464942i
\(900\) 0 0
\(901\) 7.26798 + 4.19617i 0.242131 + 0.139795i
\(902\) 0 0
\(903\) −18.1777 + 4.03778i −0.604917 + 0.134369i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −24.6739 + 14.2455i −0.819284 + 0.473014i −0.850169 0.526509i \(-0.823501\pi\)
0.0308855 + 0.999523i \(0.490167\pi\)
\(908\) 0 0
\(909\) −6.37028 53.7246i −0.211289 1.78193i
\(910\) 0 0
\(911\) 14.4884i 0.480023i −0.970770 0.240011i \(-0.922849\pi\)
0.970770 0.240011i \(-0.0771511\pi\)
\(912\) 0 0
\(913\) 11.2571 + 19.4979i 0.372555 + 0.645285i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −2.41831 + 23.5728i −0.0798597 + 0.778442i
\(918\) 0 0
\(919\) −2.83403 + 4.90869i −0.0934861 + 0.161923i −0.908976 0.416849i \(-0.863134\pi\)
0.815490 + 0.578772i \(0.196468\pi\)
\(920\) 0 0
\(921\) −4.12433 + 20.0585i −0.135901 + 0.660950i
\(922\) 0 0
\(923\) 16.7200i 0.550346i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −1.48088 + 1.10632i −0.0486385 + 0.0363362i
\(928\) 0 0
\(929\) −8.51159 + 14.7425i −0.279256 + 0.483686i −0.971200 0.238265i \(-0.923421\pi\)
0.691944 + 0.721951i \(0.256755\pi\)
\(930\) 0 0
\(931\) −21.8496 + 7.21979i −0.716092 + 0.236619i
\(932\) 0 0
\(933\) −36.7163 + 12.2004i −1.20204 + 0.399424i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −11.1936 −0.365678 −0.182839 0.983143i \(-0.558529\pi\)
−0.182839 + 0.983143i \(0.558529\pi\)
\(938\) 0 0
\(939\) 17.2317 + 15.3092i 0.562336 + 0.499598i
\(940\) 0 0
\(941\) 11.0101 + 19.0700i 0.358919 + 0.621665i 0.987780 0.155852i \(-0.0498122\pi\)
−0.628862 + 0.777517i \(0.716479\pi\)
\(942\) 0 0
\(943\) 9.35513 16.2036i 0.304645 0.527661i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −14.3035 + 24.7743i −0.464800 + 0.805057i −0.999192 0.0401795i \(-0.987207\pi\)
0.534393 + 0.845236i \(0.320540\pi\)
\(948\) 0 0
\(949\) 40.3526 + 69.8927i 1.30990 + 2.26881i
\(950\) 0 0
\(951\) −43.8672 38.9731i −1.42249 1.26379i
\(952\) 0 0
\(953\) −45.2210 −1.46485 −0.732426 0.680846i \(-0.761612\pi\)
−0.732426 + 0.680846i \(0.761612\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −12.6364 + 4.19893i −0.408477 + 0.135732i
\(958\) 0 0
\(959\) 33.1908 + 23.9886i 1.07179 + 0.774631i
\(960\) 0 0
\(961\) −11.4931 + 19.9067i −0.370746 + 0.642150i
\(962\) 0 0
\(963\) −42.5640 + 31.7981i −1.37161 + 1.02468i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 54.5961i 1.75569i −0.478943 0.877846i \(-0.658980\pi\)
0.478943 0.877846i \(-0.341020\pi\)
\(968\) 0 0
\(969\) 6.58479 32.0249i 0.211534 1.02879i
\(970\) 0 0
\(971\) 12.1591 21.0603i 0.390206 0.675856i −0.602271 0.798292i \(-0.705737\pi\)
0.992476 + 0.122436i \(0.0390706\pi\)
\(972\) 0 0
\(973\) −21.8101 48.6600i −0.699198 1.55997i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 4.58898 + 7.94835i 0.146815 + 0.254290i 0.930048 0.367437i \(-0.119765\pi\)
−0.783234 + 0.621727i \(0.786431\pi\)
\(978\) 0 0
\(979\) 21.8523i 0.698403i
\(980\) 0 0
\(981\) 0.604246 + 5.09599i 0.0192921 + 0.162703i
\(982\) 0 0
\(983\) 14.3720 8.29768i 0.458396 0.264655i −0.252974 0.967473i \(-0.581409\pi\)
0.711369 + 0.702818i \(0.248075\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −7.82504 + 24.8536i −0.249074 + 0.791099i
\(988\) 0 0
\(989\) 17.6140 + 10.1694i 0.560092 + 0.323369i
\(990\) 0 0
\(991\) −9.44914 16.3664i −0.300162 0.519895i 0.676011 0.736892i \(-0.263707\pi\)
−0.976172 + 0.216996i \(0.930374\pi\)
\(992\) 0 0
\(993\) −2.56334 2.27735i −0.0813450 0.0722696i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 7.37103 + 12.7670i 0.233443 + 0.404335i 0.958819 0.284018i \(-0.0916675\pi\)
−0.725376 + 0.688353i \(0.758334\pi\)
\(998\) 0 0
\(999\) 18.1139 8.50184i 0.573099 0.268986i
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 2100.2.bo.g.1349.6 20
3.2 odd 2 2100.2.bo.h.1349.2 20
5.2 odd 4 2100.2.bi.j.1601.1 10
5.3 odd 4 420.2.bh.b.341.5 yes 10
5.4 even 2 inner 2100.2.bo.g.1349.5 20
7.3 odd 6 2100.2.bo.h.1949.9 20
15.2 even 4 2100.2.bi.k.1601.2 10
15.8 even 4 420.2.bh.a.341.4 yes 10
15.14 odd 2 2100.2.bo.h.1349.9 20
21.17 even 6 inner 2100.2.bo.g.1949.5 20
35.3 even 12 420.2.bh.a.101.4 10
35.17 even 12 2100.2.bi.k.101.2 10
35.23 odd 12 2940.2.d.a.881.4 10
35.24 odd 6 2100.2.bo.h.1949.2 20
35.33 even 12 2940.2.d.b.881.7 10
105.17 odd 12 2100.2.bi.j.101.1 10
105.23 even 12 2940.2.d.b.881.8 10
105.38 odd 12 420.2.bh.b.101.5 yes 10
105.59 even 6 inner 2100.2.bo.g.1949.6 20
105.68 odd 12 2940.2.d.a.881.3 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
420.2.bh.a.101.4 10 35.3 even 12
420.2.bh.a.341.4 yes 10 15.8 even 4
420.2.bh.b.101.5 yes 10 105.38 odd 12
420.2.bh.b.341.5 yes 10 5.3 odd 4
2100.2.bi.j.101.1 10 105.17 odd 12
2100.2.bi.j.1601.1 10 5.2 odd 4
2100.2.bi.k.101.2 10 35.17 even 12
2100.2.bi.k.1601.2 10 15.2 even 4
2100.2.bo.g.1349.5 20 5.4 even 2 inner
2100.2.bo.g.1349.6 20 1.1 even 1 trivial
2100.2.bo.g.1949.5 20 21.17 even 6 inner
2100.2.bo.g.1949.6 20 105.59 even 6 inner
2100.2.bo.h.1349.2 20 3.2 odd 2
2100.2.bo.h.1349.9 20 15.14 odd 2
2100.2.bo.h.1949.2 20 35.24 odd 6
2100.2.bo.h.1949.9 20 7.3 odd 6
2940.2.d.a.881.3 10 105.68 odd 12
2940.2.d.a.881.4 10 35.23 odd 12
2940.2.d.b.881.7 10 35.33 even 12
2940.2.d.b.881.8 10 105.23 even 12