Properties

Label 1764.2.w.a.509.2
Level $1764$
Weight $2$
Character 1764.509
Analytic conductor $14.086$
Analytic rank $0$
Dimension $16$
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] = [1764,2,Mod(509,1764)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1764, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1764.509");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1764.w (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: \(14.0856109166\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 3x^{14} - 9x^{12} - 9x^{10} + 225x^{8} - 81x^{6} - 729x^{4} - 2187x^{2} + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 252)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 509.2
Root \(-1.71965 + 0.206851i\) of defining polynomial
Character \(\chi\) \(=\) 1764.509
Dual form 1764.2.w.a.1109.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.03897 + 1.38584i) q^{3} +(2.09336 + 3.62580i) q^{5} +(-0.841101 - 2.87968i) q^{9} +O(q^{10})\) \(q+(-1.03897 + 1.38584i) q^{3} +(2.09336 + 3.62580i) q^{5} +(-0.841101 - 2.87968i) q^{9} +(1.23222 + 0.711425i) q^{11} +(0.850739 + 0.491174i) q^{13} +(-7.19970 - 0.866025i) q^{15} +(0.185474 + 0.321250i) q^{17} +(4.30823 + 2.48736i) q^{19} +(-4.98775 + 2.87968i) q^{23} +(-6.26427 + 10.8500i) q^{25} +(4.86465 + 1.82626i) q^{27} +(7.31732 - 4.22466i) q^{29} +7.25160i q^{31} +(-2.26616 + 0.968518i) q^{33} +(-1.73222 + 3.00030i) q^{37} +(-1.56458 + 0.668674i) q^{39} +(1.06981 - 1.85297i) q^{41} +(3.00875 + 5.21130i) q^{43} +(8.68041 - 9.07785i) q^{45} -8.27085 q^{47} +(-0.637901 - 0.0767308i) q^{51} +(4.30627 - 2.48623i) q^{53} +5.95706i q^{55} +(-7.92317 + 3.38623i) q^{57} -4.55766 q^{59} -7.51035i q^{61} +4.11281i q^{65} -10.0641 q^{67} +(1.19133 - 9.90411i) q^{69} -10.9555i q^{71} +(8.25191 - 4.76424i) q^{73} +(-8.52805 - 19.9541i) q^{75} -8.51105 q^{79} +(-7.58510 + 4.84420i) q^{81} +(-0.972254 - 1.68399i) q^{83} +(-0.776524 + 1.34498i) q^{85} +(-1.74775 + 14.5299i) q^{87} +(-3.90368 + 6.76137i) q^{89} +(-10.0495 - 7.53416i) q^{93} +20.8277i q^{95} +(3.34099 - 1.92892i) q^{97} +(1.01225 - 4.14679i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 6 q^{9} - 6 q^{11} - 12 q^{15} - 6 q^{23} - 8 q^{25} - 12 q^{29} - 2 q^{37} - 36 q^{39} + 4 q^{43} + 12 q^{51} + 36 q^{53} - 42 q^{57} - 28 q^{67} - 40 q^{79} - 18 q^{81} + 6 q^{85} - 6 q^{93} + 90 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(e\left(\frac{5}{6}\right)\)

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
\(3\) −1.03897 + 1.38584i −0.599847 + 0.800115i
\(4\) 0 0
\(5\) 2.09336 + 3.62580i 0.936177 + 1.62151i 0.772521 + 0.634989i \(0.218995\pi\)
0.163656 + 0.986518i \(0.447671\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −0.841101 2.87968i −0.280367 0.959893i
\(10\) 0 0
\(11\) 1.23222 + 0.711425i 0.371529 + 0.214503i 0.674126 0.738616i \(-0.264520\pi\)
−0.302597 + 0.953119i \(0.597854\pi\)
\(12\) 0 0
\(13\) 0.850739 + 0.491174i 0.235952 + 0.136227i 0.613315 0.789838i \(-0.289836\pi\)
−0.377363 + 0.926066i \(0.623169\pi\)
\(14\) 0 0
\(15\) −7.19970 0.866025i −1.85895 0.223607i
\(16\) 0 0
\(17\) 0.185474 + 0.321250i 0.0449840 + 0.0779145i 0.887641 0.460537i \(-0.152343\pi\)
−0.842657 + 0.538451i \(0.819010\pi\)
\(18\) 0 0
\(19\) 4.30823 + 2.48736i 0.988375 + 0.570638i 0.904788 0.425862i \(-0.140029\pi\)
0.0835867 + 0.996501i \(0.473362\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.98775 + 2.87968i −1.04002 + 0.600455i −0.919838 0.392298i \(-0.871680\pi\)
−0.120179 + 0.992752i \(0.538347\pi\)
\(24\) 0 0
\(25\) −6.26427 + 10.8500i −1.25285 + 2.17001i
\(26\) 0 0
\(27\) 4.86465 + 1.82626i 0.936202 + 0.351463i
\(28\) 0 0
\(29\) 7.31732 4.22466i 1.35879 0.784499i 0.369332 0.929298i \(-0.379587\pi\)
0.989461 + 0.144798i \(0.0462534\pi\)
\(30\) 0 0
\(31\) 7.25160i 1.30243i 0.758895 + 0.651213i \(0.225739\pi\)
−0.758895 + 0.651213i \(0.774261\pi\)
\(32\) 0 0
\(33\) −2.26616 + 0.968518i −0.394488 + 0.168597i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1.73222 + 3.00030i −0.284776 + 0.493246i −0.972555 0.232674i \(-0.925252\pi\)
0.687779 + 0.725920i \(0.258586\pi\)
\(38\) 0 0
\(39\) −1.56458 + 0.668674i −0.250533 + 0.107074i
\(40\) 0 0
\(41\) 1.06981 1.85297i 0.167077 0.289386i −0.770314 0.637665i \(-0.779901\pi\)
0.937391 + 0.348279i \(0.113234\pi\)
\(42\) 0 0
\(43\) 3.00875 + 5.21130i 0.458830 + 0.794716i 0.998899 0.0469039i \(-0.0149354\pi\)
−0.540070 + 0.841620i \(0.681602\pi\)
\(44\) 0 0
\(45\) 8.68041 9.07785i 1.29400 1.35325i
\(46\) 0 0
\(47\) −8.27085 −1.20643 −0.603213 0.797580i \(-0.706113\pi\)
−0.603213 + 0.797580i \(0.706113\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −0.637901 0.0767308i −0.0893240 0.0107445i
\(52\) 0 0
\(53\) 4.30627 2.48623i 0.591512 0.341509i −0.174183 0.984713i \(-0.555729\pi\)
0.765695 + 0.643204i \(0.222395\pi\)
\(54\) 0 0
\(55\) 5.95706i 0.803250i
\(56\) 0 0
\(57\) −7.92317 + 3.38623i −1.04945 + 0.448517i
\(58\) 0 0
\(59\) −4.55766 −0.593357 −0.296678 0.954977i \(-0.595879\pi\)
−0.296678 + 0.954977i \(0.595879\pi\)
\(60\) 0 0
\(61\) 7.51035i 0.961602i −0.876830 0.480801i \(-0.840346\pi\)
0.876830 0.480801i \(-0.159654\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.11281i 0.510131i
\(66\) 0 0
\(67\) −10.0641 −1.22953 −0.614763 0.788712i \(-0.710748\pi\)
−0.614763 + 0.788712i \(0.710748\pi\)
\(68\) 0 0
\(69\) 1.19133 9.90411i 0.143419 1.19231i
\(70\) 0 0
\(71\) 10.9555i 1.30018i −0.759857 0.650090i \(-0.774731\pi\)
0.759857 0.650090i \(-0.225269\pi\)
\(72\) 0 0
\(73\) 8.25191 4.76424i 0.965813 0.557612i 0.0678555 0.997695i \(-0.478384\pi\)
0.897957 + 0.440083i \(0.145051\pi\)
\(74\) 0 0
\(75\) −8.52805 19.9541i −0.984734 2.30410i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −8.51105 −0.957568 −0.478784 0.877933i \(-0.658922\pi\)
−0.478784 + 0.877933i \(0.658922\pi\)
\(80\) 0 0
\(81\) −7.58510 + 4.84420i −0.842789 + 0.538244i
\(82\) 0 0
\(83\) −0.972254 1.68399i −0.106719 0.184842i 0.807720 0.589566i \(-0.200701\pi\)
−0.914439 + 0.404724i \(0.867368\pi\)
\(84\) 0 0
\(85\) −0.776524 + 1.34498i −0.0842259 + 0.145884i
\(86\) 0 0
\(87\) −1.74775 + 14.5299i −0.187378 + 1.55777i
\(88\) 0 0
\(89\) −3.90368 + 6.76137i −0.413789 + 0.716703i −0.995300 0.0968347i \(-0.969128\pi\)
0.581512 + 0.813538i \(0.302462\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −10.0495 7.53416i −1.04209 0.781256i
\(94\) 0 0
\(95\) 20.8277i 2.13687i
\(96\) 0 0
\(97\) 3.34099 1.92892i 0.339226 0.195852i −0.320704 0.947180i \(-0.603919\pi\)
0.659930 + 0.751327i \(0.270586\pi\)
\(98\) 0 0
\(99\) 1.01225 4.14679i 0.101735 0.416768i
\(100\) 0 0
\(101\) 4.50637 7.80526i 0.448401 0.776653i −0.549882 0.835243i \(-0.685327\pi\)
0.998282 + 0.0585901i \(0.0186605\pi\)
\(102\) 0 0
\(103\) 5.96343 3.44299i 0.587594 0.339248i −0.176551 0.984291i \(-0.556494\pi\)
0.764146 + 0.645044i \(0.223161\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −9.84062 5.68149i −0.951329 0.549250i −0.0578356 0.998326i \(-0.518420\pi\)
−0.893494 + 0.449076i \(0.851753\pi\)
\(108\) 0 0
\(109\) 7.99650 + 13.8503i 0.765926 + 1.32662i 0.939756 + 0.341846i \(0.111052\pi\)
−0.173830 + 0.984776i \(0.555614\pi\)
\(110\) 0 0
\(111\) −2.35821 5.51779i −0.223832 0.523726i
\(112\) 0 0
\(113\) 2.17043 + 1.25310i 0.204177 + 0.117881i 0.598602 0.801046i \(-0.295723\pi\)
−0.394426 + 0.918928i \(0.629056\pi\)
\(114\) 0 0
\(115\) −20.8823 12.0564i −1.94728 1.12426i
\(116\) 0 0
\(117\) 0.698867 2.86298i 0.0646103 0.264683i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −4.48775 7.77301i −0.407977 0.706637i
\(122\) 0 0
\(123\) 1.45642 + 3.40776i 0.131321 + 0.307268i
\(124\) 0 0
\(125\) −31.5199 −2.81922
\(126\) 0 0
\(127\) −1.91140 −0.169609 −0.0848046 0.996398i \(-0.527027\pi\)
−0.0848046 + 0.996398i \(0.527027\pi\)
\(128\) 0 0
\(129\) −10.3480 1.24473i −0.911092 0.109592i
\(130\) 0 0
\(131\) 4.32591 + 7.49270i 0.377957 + 0.654640i 0.990765 0.135592i \(-0.0432935\pi\)
−0.612808 + 0.790232i \(0.709960\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 3.56180 + 21.4612i 0.306551 + 1.84709i
\(136\) 0 0
\(137\) −10.0973 5.82971i −0.862675 0.498065i 0.00223233 0.999998i \(-0.499289\pi\)
−0.864907 + 0.501932i \(0.832623\pi\)
\(138\) 0 0
\(139\) 4.82663 + 2.78666i 0.409390 + 0.236361i 0.690528 0.723306i \(-0.257378\pi\)
−0.281138 + 0.959667i \(0.590712\pi\)
\(140\) 0 0
\(141\) 8.59312 11.4621i 0.723672 0.965280i
\(142\) 0 0
\(143\) 0.698867 + 1.21047i 0.0584422 + 0.101225i
\(144\) 0 0
\(145\) 30.6355 + 17.6874i 2.54414 + 1.46886i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 13.4024 7.73789i 1.09797 0.633913i 0.162282 0.986744i \(-0.448114\pi\)
0.935687 + 0.352832i \(0.114781\pi\)
\(150\) 0 0
\(151\) 8.31732 14.4060i 0.676854 1.17235i −0.299069 0.954231i \(-0.596676\pi\)
0.975923 0.218114i \(-0.0699905\pi\)
\(152\) 0 0
\(153\) 0.769094 0.804308i 0.0621776 0.0650244i
\(154\) 0 0
\(155\) −26.2928 + 15.1802i −2.11189 + 1.21930i
\(156\) 0 0
\(157\) 16.8077i 1.34140i 0.741727 + 0.670702i \(0.234007\pi\)
−0.741727 + 0.670702i \(0.765993\pi\)
\(158\) 0 0
\(159\) −1.02856 + 8.55090i −0.0815699 + 0.678131i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −1.75553 + 3.04066i −0.137503 + 0.238163i −0.926551 0.376169i \(-0.877241\pi\)
0.789048 + 0.614332i \(0.210574\pi\)
\(164\) 0 0
\(165\) −8.25553 6.18918i −0.642692 0.481827i
\(166\) 0 0
\(167\) −4.05253 + 7.01918i −0.313594 + 0.543160i −0.979138 0.203198i \(-0.934866\pi\)
0.665544 + 0.746359i \(0.268200\pi\)
\(168\) 0 0
\(169\) −6.01750 10.4226i −0.462884 0.801739i
\(170\) 0 0
\(171\) 3.53913 14.4984i 0.270644 1.10872i
\(172\) 0 0
\(173\) −13.0969 −0.995737 −0.497868 0.867253i \(-0.665884\pi\)
−0.497868 + 0.867253i \(0.665884\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 4.73525 6.31618i 0.355923 0.474753i
\(178\) 0 0
\(179\) −11.0140 + 6.35893i −0.823225 + 0.475289i −0.851527 0.524310i \(-0.824323\pi\)
0.0283026 + 0.999599i \(0.490990\pi\)
\(180\) 0 0
\(181\) 26.5518i 1.97358i 0.161998 + 0.986791i \(0.448206\pi\)
−0.161998 + 0.986791i \(0.551794\pi\)
\(182\) 0 0
\(183\) 10.4081 + 7.80300i 0.769392 + 0.576814i
\(184\) 0 0
\(185\) −14.5046 −1.06640
\(186\) 0 0
\(187\) 0.527802i 0.0385967i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1.26711i 0.0916848i −0.998949 0.0458424i \(-0.985403\pi\)
0.998949 0.0458424i \(-0.0145972\pi\)
\(192\) 0 0
\(193\) 18.6346 1.34135 0.670676 0.741751i \(-0.266004\pi\)
0.670676 + 0.741751i \(0.266004\pi\)
\(194\) 0 0
\(195\) −5.69969 4.27307i −0.408163 0.306001i
\(196\) 0 0
\(197\) 5.94312i 0.423430i 0.977331 + 0.211715i \(0.0679049\pi\)
−0.977331 + 0.211715i \(0.932095\pi\)
\(198\) 0 0
\(199\) 4.87291 2.81337i 0.345431 0.199435i −0.317240 0.948345i \(-0.602756\pi\)
0.662671 + 0.748910i \(0.269423\pi\)
\(200\) 0 0
\(201\) 10.4563 13.9472i 0.737527 0.983761i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 8.95800 0.625654
\(206\) 0 0
\(207\) 12.4877 + 11.9410i 0.867959 + 0.829958i
\(208\) 0 0
\(209\) 3.53913 + 6.12996i 0.244807 + 0.424018i
\(210\) 0 0
\(211\) 1.05305 1.82393i 0.0724948 0.125565i −0.827499 0.561467i \(-0.810237\pi\)
0.899994 + 0.435902i \(0.143571\pi\)
\(212\) 0 0
\(213\) 15.1826 + 11.3824i 1.04029 + 0.779909i
\(214\) 0 0
\(215\) −12.5968 + 21.8182i −0.859092 + 1.48799i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −1.97098 + 16.3857i −0.133186 + 1.10724i
\(220\) 0 0
\(221\) 0.364399i 0.0245121i
\(222\) 0 0
\(223\) −5.52351 + 3.18900i −0.369882 + 0.213551i −0.673407 0.739272i \(-0.735170\pi\)
0.303525 + 0.952823i \(0.401836\pi\)
\(224\) 0 0
\(225\) 36.5135 + 8.91312i 2.43423 + 0.594208i
\(226\) 0 0
\(227\) 5.72365 9.91365i 0.379892 0.657992i −0.611154 0.791511i \(-0.709295\pi\)
0.991046 + 0.133520i \(0.0426279\pi\)
\(228\) 0 0
\(229\) −4.88696 + 2.82149i −0.322940 + 0.186449i −0.652702 0.757615i \(-0.726365\pi\)
0.329762 + 0.944064i \(0.393031\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 10.7788 + 6.22316i 0.706145 + 0.407693i 0.809632 0.586938i \(-0.199667\pi\)
−0.103487 + 0.994631i \(0.533000\pi\)
\(234\) 0 0
\(235\) −17.3138 29.9884i −1.12943 1.95623i
\(236\) 0 0
\(237\) 8.84269 11.7950i 0.574394 0.766164i
\(238\) 0 0
\(239\) 3.52450 + 2.03487i 0.227981 + 0.131625i 0.609640 0.792678i \(-0.291314\pi\)
−0.381659 + 0.924303i \(0.624647\pi\)
\(240\) 0 0
\(241\) 4.26195 + 2.46064i 0.274537 + 0.158504i 0.630947 0.775826i \(-0.282666\pi\)
−0.356411 + 0.934329i \(0.616000\pi\)
\(242\) 0 0
\(243\) 1.16737 15.5447i 0.0748871 0.997192i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 2.44345 + 4.23218i 0.155473 + 0.269287i
\(248\) 0 0
\(249\) 3.34388 + 0.402223i 0.211910 + 0.0254899i
\(250\) 0 0
\(251\) 4.65020 0.293518 0.146759 0.989172i \(-0.453116\pi\)
0.146759 + 0.989172i \(0.453116\pi\)
\(252\) 0 0
\(253\) −8.19470 −0.515196
\(254\) 0 0
\(255\) −1.05714 2.47353i −0.0662009 0.154898i
\(256\) 0 0
\(257\) 5.44701 + 9.43450i 0.339775 + 0.588508i 0.984390 0.175999i \(-0.0563156\pi\)
−0.644615 + 0.764507i \(0.722982\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −18.3203 17.5182i −1.13400 1.08435i
\(262\) 0 0
\(263\) −15.6489 9.03488i −0.964951 0.557114i −0.0672574 0.997736i \(-0.521425\pi\)
−0.897693 + 0.440621i \(0.854758\pi\)
\(264\) 0 0
\(265\) 18.0291 + 10.4091i 1.10752 + 0.639427i
\(266\) 0 0
\(267\) −5.31438 12.4347i −0.325235 0.760991i
\(268\) 0 0
\(269\) 1.49205 + 2.58430i 0.0909718 + 0.157568i 0.907920 0.419143i \(-0.137669\pi\)
−0.816948 + 0.576711i \(0.804336\pi\)
\(270\) 0 0
\(271\) 8.97274 + 5.18041i 0.545055 + 0.314688i 0.747125 0.664683i \(-0.231434\pi\)
−0.202070 + 0.979371i \(0.564767\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −15.4380 + 8.91312i −0.930945 + 0.537481i
\(276\) 0 0
\(277\) 5.94345 10.2944i 0.357107 0.618528i −0.630369 0.776296i \(-0.717096\pi\)
0.987476 + 0.157768i \(0.0504297\pi\)
\(278\) 0 0
\(279\) 20.8823 6.09932i 1.25019 0.365157i
\(280\) 0 0
\(281\) −12.0740 + 6.97095i −0.720277 + 0.415852i −0.814855 0.579665i \(-0.803183\pi\)
0.0945775 + 0.995518i \(0.469850\pi\)
\(282\) 0 0
\(283\) 2.53564i 0.150728i 0.997156 + 0.0753642i \(0.0240119\pi\)
−0.997156 + 0.0753642i \(0.975988\pi\)
\(284\) 0 0
\(285\) −28.8638 21.6392i −1.70974 1.28180i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 8.43120 14.6033i 0.495953 0.859016i
\(290\) 0 0
\(291\) −0.797998 + 6.63415i −0.0467795 + 0.388901i
\(292\) 0 0
\(293\) −3.95496 + 6.85020i −0.231052 + 0.400193i −0.958118 0.286374i \(-0.907550\pi\)
0.727066 + 0.686567i \(0.240883\pi\)
\(294\) 0 0
\(295\) −9.54080 16.5251i −0.555487 0.962131i
\(296\) 0 0
\(297\) 4.69509 + 5.71119i 0.272437 + 0.331397i
\(298\) 0 0
\(299\) −5.65769 −0.327193
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 6.13487 + 14.3545i 0.352439 + 0.824645i
\(304\) 0 0
\(305\) 27.2310 15.7218i 1.55924 0.900230i
\(306\) 0 0
\(307\) 13.9676i 0.797170i −0.917131 0.398585i \(-0.869501\pi\)
0.917131 0.398585i \(-0.130499\pi\)
\(308\) 0 0
\(309\) −1.42437 + 11.8415i −0.0810296 + 0.673640i
\(310\) 0 0
\(311\) 32.6327 1.85043 0.925215 0.379443i \(-0.123884\pi\)
0.925215 + 0.379443i \(0.123884\pi\)
\(312\) 0 0
\(313\) 15.7166i 0.888357i −0.895938 0.444178i \(-0.853496\pi\)
0.895938 0.444178i \(-0.146504\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 29.0253i 1.63022i −0.579305 0.815111i \(-0.696676\pi\)
0.579305 0.815111i \(-0.303324\pi\)
\(318\) 0 0
\(319\) 12.0221 0.673109
\(320\) 0 0
\(321\) 18.0977 7.73465i 1.01012 0.431706i
\(322\) 0 0
\(323\) 1.84535i 0.102678i
\(324\) 0 0
\(325\) −10.6585 + 6.15370i −0.591228 + 0.341346i
\(326\) 0 0
\(327\) −27.5024 3.30817i −1.52089 0.182942i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 13.1702 0.723900 0.361950 0.932198i \(-0.382111\pi\)
0.361950 + 0.932198i \(0.382111\pi\)
\(332\) 0 0
\(333\) 10.0969 + 2.46469i 0.553305 + 0.135064i
\(334\) 0 0
\(335\) −21.0677 36.4904i −1.15105 1.99368i
\(336\) 0 0
\(337\) 8.31732 14.4060i 0.453073 0.784746i −0.545502 0.838110i \(-0.683661\pi\)
0.998575 + 0.0533635i \(0.0169942\pi\)
\(338\) 0 0
\(339\) −3.99159 + 1.70594i −0.216793 + 0.0926539i
\(340\) 0 0
\(341\) −5.15896 + 8.93559i −0.279374 + 0.483889i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 38.4042 16.4133i 2.06761 0.883662i
\(346\) 0 0
\(347\) 24.7120i 1.32661i −0.748349 0.663305i \(-0.769153\pi\)
0.748349 0.663305i \(-0.230847\pi\)
\(348\) 0 0
\(349\) 26.7994 15.4727i 1.43454 0.828232i 0.437078 0.899424i \(-0.356013\pi\)
0.997463 + 0.0711915i \(0.0226802\pi\)
\(350\) 0 0
\(351\) 3.24153 + 3.94306i 0.173020 + 0.210465i
\(352\) 0 0
\(353\) −11.6758 + 20.2231i −0.621440 + 1.07637i 0.367778 + 0.929914i \(0.380119\pi\)
−0.989218 + 0.146452i \(0.953215\pi\)
\(354\) 0 0
\(355\) 39.7225 22.9338i 2.10825 1.21720i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 18.3902 + 10.6176i 0.970596 + 0.560374i 0.899418 0.437090i \(-0.143991\pi\)
0.0711782 + 0.997464i \(0.477324\pi\)
\(360\) 0 0
\(361\) 2.87387 + 4.97769i 0.151256 + 0.261984i
\(362\) 0 0
\(363\) 15.4348 + 1.85659i 0.810115 + 0.0974458i
\(364\) 0 0
\(365\) 34.5483 + 19.9465i 1.80834 + 1.04405i
\(366\) 0 0
\(367\) 12.0178 + 6.93846i 0.627322 + 0.362185i 0.779714 0.626135i \(-0.215364\pi\)
−0.152392 + 0.988320i \(0.548698\pi\)
\(368\) 0 0
\(369\) −6.23579 1.52218i −0.324622 0.0792417i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 16.0728 + 27.8390i 0.832221 + 1.44145i 0.896273 + 0.443502i \(0.146264\pi\)
−0.0640529 + 0.997947i \(0.520403\pi\)
\(374\) 0 0
\(375\) 32.7480 43.6815i 1.69110 2.25570i
\(376\) 0 0
\(377\) 8.30017 0.427481
\(378\) 0 0
\(379\) 1.95340 0.100339 0.0501696 0.998741i \(-0.484024\pi\)
0.0501696 + 0.998741i \(0.484024\pi\)
\(380\) 0 0
\(381\) 1.98588 2.64889i 0.101740 0.135707i
\(382\) 0 0
\(383\) −0.788167 1.36514i −0.0402734 0.0697557i 0.845186 0.534472i \(-0.179490\pi\)
−0.885460 + 0.464717i \(0.846156\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 12.4762 13.0475i 0.634202 0.663240i
\(388\) 0 0
\(389\) −9.74447 5.62597i −0.494064 0.285248i 0.232195 0.972669i \(-0.425409\pi\)
−0.726259 + 0.687421i \(0.758743\pi\)
\(390\) 0 0
\(391\) −1.85019 1.06821i −0.0935682 0.0540216i
\(392\) 0 0
\(393\) −14.8782 1.78964i −0.750503 0.0902753i
\(394\) 0 0
\(395\) −17.8167 30.8594i −0.896453 1.55270i
\(396\) 0 0
\(397\) −0.548160 0.316480i −0.0275114 0.0158837i 0.486181 0.873858i \(-0.338389\pi\)
−0.513693 + 0.857974i \(0.671723\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −7.38032 + 4.26103i −0.368555 + 0.212786i −0.672827 0.739800i \(-0.734920\pi\)
0.304272 + 0.952585i \(0.401587\pi\)
\(402\) 0 0
\(403\) −3.56180 + 6.16921i −0.177426 + 0.307310i
\(404\) 0 0
\(405\) −33.4424 17.3614i −1.66177 0.862695i
\(406\) 0 0
\(407\) −4.26897 + 2.46469i −0.211605 + 0.122170i
\(408\) 0 0
\(409\) 14.0668i 0.695557i 0.937577 + 0.347778i \(0.113064\pi\)
−0.937577 + 0.347778i \(0.886936\pi\)
\(410\) 0 0
\(411\) 18.5698 7.93644i 0.915982 0.391476i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 4.07054 7.05039i 0.199815 0.346090i
\(416\) 0 0
\(417\) −8.87657 + 3.79370i −0.434687 + 0.185778i
\(418\) 0 0
\(419\) 10.9339 18.9381i 0.534156 0.925185i −0.465048 0.885286i \(-0.653963\pi\)
0.999204 0.0398995i \(-0.0127038\pi\)
\(420\) 0 0
\(421\) 13.3616 + 23.1430i 0.651206 + 1.12792i 0.982831 + 0.184510i \(0.0590698\pi\)
−0.331625 + 0.943411i \(0.607597\pi\)
\(422\) 0 0
\(423\) 6.95662 + 23.8174i 0.338242 + 1.15804i
\(424\) 0 0
\(425\) −4.64743 −0.225433
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −2.40362 0.289123i −0.116048 0.0139590i
\(430\) 0 0
\(431\) 17.2686 9.97000i 0.831797 0.480238i −0.0226706 0.999743i \(-0.507217\pi\)
0.854468 + 0.519505i \(0.173884\pi\)
\(432\) 0 0
\(433\) 20.2826i 0.974719i 0.873201 + 0.487359i \(0.162040\pi\)
−0.873201 + 0.487359i \(0.837960\pi\)
\(434\) 0 0
\(435\) −56.3412 + 24.0793i −2.70135 + 1.15451i
\(436\) 0 0
\(437\) −28.6511 −1.37057
\(438\) 0 0
\(439\) 28.3982i 1.35537i 0.735350 + 0.677687i \(0.237018\pi\)
−0.735350 + 0.677687i \(0.762982\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 12.3056i 0.584658i 0.956318 + 0.292329i \(0.0944302\pi\)
−0.956318 + 0.292329i \(0.905570\pi\)
\(444\) 0 0
\(445\) −32.6871 −1.54952
\(446\) 0 0
\(447\) −3.20118 + 26.6130i −0.151411 + 1.25875i
\(448\) 0 0
\(449\) 36.1924i 1.70803i −0.520251 0.854013i \(-0.674162\pi\)
0.520251 0.854013i \(-0.325838\pi\)
\(450\) 0 0
\(451\) 2.63650 1.52218i 0.124148 0.0716769i
\(452\) 0 0
\(453\) 11.3230 + 26.4938i 0.532002 + 1.24479i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −8.41784 −0.393770 −0.196885 0.980427i \(-0.563083\pi\)
−0.196885 + 0.980427i \(0.563083\pi\)
\(458\) 0 0
\(459\) 0.315579 + 1.90149i 0.0147300 + 0.0887539i
\(460\) 0 0
\(461\) 9.07730 + 15.7224i 0.422772 + 0.732263i 0.996209 0.0869865i \(-0.0277237\pi\)
−0.573437 + 0.819249i \(0.694390\pi\)
\(462\) 0 0
\(463\) −7.64690 + 13.2448i −0.355381 + 0.615539i −0.987183 0.159591i \(-0.948982\pi\)
0.631802 + 0.775130i \(0.282316\pi\)
\(464\) 0 0
\(465\) 6.28007 52.2093i 0.291231 2.42115i
\(466\) 0 0
\(467\) −13.3932 + 23.1977i −0.619763 + 1.07346i 0.369766 + 0.929125i \(0.379438\pi\)
−0.989529 + 0.144336i \(0.953896\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −23.2928 17.4627i −1.07328 0.804637i
\(472\) 0 0
\(473\) 8.56199i 0.393681i
\(474\) 0 0
\(475\) −53.9758 + 31.1630i −2.47658 + 1.42985i
\(476\) 0 0
\(477\) −10.7815 10.3095i −0.493653 0.472040i
\(478\) 0 0
\(479\) −14.2775 + 24.7294i −0.652357 + 1.12992i 0.330193 + 0.943914i \(0.392886\pi\)
−0.982549 + 0.186002i \(0.940447\pi\)
\(480\) 0 0
\(481\) −2.94734 + 1.70165i −0.134387 + 0.0775884i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 13.9877 + 8.07583i 0.635151 + 0.366705i
\(486\) 0 0
\(487\) 6.57635 + 11.3906i 0.298003 + 0.516156i 0.975679 0.219204i \(-0.0703462\pi\)
−0.677676 + 0.735361i \(0.737013\pi\)
\(488\) 0 0
\(489\) −2.38994 5.59202i −0.108077 0.252880i
\(490\) 0 0
\(491\) 28.6854 + 16.5615i 1.29455 + 0.747411i 0.979458 0.201649i \(-0.0646301\pi\)
0.315096 + 0.949060i \(0.397963\pi\)
\(492\) 0 0
\(493\) 2.71434 + 1.56712i 0.122248 + 0.0705798i
\(494\) 0 0
\(495\) 17.1544 5.01049i 0.771034 0.225205i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 3.27652 + 5.67511i 0.146677 + 0.254053i 0.929997 0.367566i \(-0.119809\pi\)
−0.783320 + 0.621619i \(0.786475\pi\)
\(500\) 0 0
\(501\) −5.51702 12.9088i −0.246482 0.576724i
\(502\) 0 0
\(503\) −12.4969 −0.557210 −0.278605 0.960406i \(-0.589872\pi\)
−0.278605 + 0.960406i \(0.589872\pi\)
\(504\) 0 0
\(505\) 37.7337 1.67913
\(506\) 0 0
\(507\) 20.6960 + 2.48945i 0.919143 + 0.110560i
\(508\) 0 0
\(509\) −3.30237 5.71987i −0.146375 0.253529i 0.783510 0.621379i \(-0.213427\pi\)
−0.929885 + 0.367850i \(0.880094\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 16.4154 + 19.9680i 0.724760 + 0.881610i
\(514\) 0 0
\(515\) 24.9672 + 14.4148i 1.10018 + 0.635192i
\(516\) 0 0
\(517\) −10.1915 5.88408i −0.448223 0.258782i
\(518\) 0 0
\(519\) 13.6072 18.1502i 0.597290 0.796704i
\(520\) 0 0
\(521\) −7.98920 13.8377i −0.350013 0.606241i 0.636238 0.771493i \(-0.280490\pi\)
−0.986251 + 0.165252i \(0.947156\pi\)
\(522\) 0 0
\(523\) −10.1857 5.88074i −0.445391 0.257147i 0.260491 0.965476i \(-0.416116\pi\)
−0.705882 + 0.708330i \(0.749449\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −2.32957 + 1.34498i −0.101478 + 0.0585882i
\(528\) 0 0
\(529\) 5.08510 8.80765i 0.221091 0.382941i
\(530\) 0 0
\(531\) 3.83345 + 13.1246i 0.166358 + 0.569559i
\(532\) 0 0
\(533\) 1.82026 1.05093i 0.0788444 0.0455208i
\(534\) 0 0
\(535\) 47.5735i 2.05678i
\(536\) 0 0
\(537\) 2.63070 21.8703i 0.113523 0.943775i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 0.205980 0.356768i 0.00885576 0.0153386i −0.861564 0.507650i \(-0.830514\pi\)
0.870419 + 0.492311i \(0.163848\pi\)
\(542\) 0 0
\(543\) −36.7966 27.5864i −1.57909 1.18385i
\(544\) 0 0
\(545\) −33.4790 + 57.9874i −1.43408 + 2.48391i
\(546\) 0 0
\(547\) 11.9166 + 20.6402i 0.509519 + 0.882513i 0.999939 + 0.0110266i \(0.00350995\pi\)
−0.490420 + 0.871486i \(0.663157\pi\)
\(548\) 0 0
\(549\) −21.6274 + 6.31696i −0.923035 + 0.269601i
\(550\) 0 0
\(551\) 42.0329 1.79066
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 15.0698 20.1011i 0.639678 0.853244i
\(556\) 0 0
\(557\) 37.1935 21.4737i 1.57594 0.909869i 0.580522 0.814245i \(-0.302849\pi\)
0.995418 0.0956241i \(-0.0304847\pi\)
\(558\) 0 0
\(559\) 5.91128i 0.250020i
\(560\) 0 0
\(561\) −0.731449 0.548368i −0.0308818 0.0231521i
\(562\) 0 0
\(563\) 31.8908 1.34404 0.672019 0.740534i \(-0.265428\pi\)
0.672019 + 0.740534i \(0.265428\pi\)
\(564\) 0 0
\(565\) 10.4927i 0.441432i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0.495245i 0.0207618i 0.999946 + 0.0103809i \(0.00330440\pi\)
−0.999946 + 0.0103809i \(0.996696\pi\)
\(570\) 0 0
\(571\) 2.68365 0.112307 0.0561535 0.998422i \(-0.482116\pi\)
0.0561535 + 0.998422i \(0.482116\pi\)
\(572\) 0 0
\(573\) 1.75601 + 1.31648i 0.0733584 + 0.0549969i
\(574\) 0 0
\(575\) 72.1564i 3.00913i
\(576\) 0 0
\(577\) −36.3955 + 21.0130i −1.51517 + 0.874781i −0.515324 + 0.856995i \(0.672328\pi\)
−0.999842 + 0.0177861i \(0.994338\pi\)
\(578\) 0 0
\(579\) −19.3608 + 25.8246i −0.804606 + 1.07323i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 7.07505 0.293019
\(584\) 0 0
\(585\) 11.8436 3.45929i 0.489671 0.143024i
\(586\) 0 0
\(587\) −1.71916 2.97768i −0.0709575 0.122902i 0.828364 0.560191i \(-0.189272\pi\)
−0.899321 + 0.437289i \(0.855939\pi\)
\(588\) 0 0
\(589\) −18.0373 + 31.2415i −0.743214 + 1.28728i
\(590\) 0 0
\(591\) −8.23621 6.17470i −0.338793 0.253993i
\(592\) 0 0
\(593\) 4.55126 7.88301i 0.186898 0.323716i −0.757317 0.653048i \(-0.773490\pi\)
0.944214 + 0.329332i \(0.106823\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −1.16390 + 9.67606i −0.0476352 + 0.396015i
\(598\) 0 0
\(599\) 7.75019i 0.316664i −0.987386 0.158332i \(-0.949388\pi\)
0.987386 0.158332i \(-0.0506117\pi\)
\(600\) 0 0
\(601\) 22.0034 12.7037i 0.897536 0.518193i 0.0211361 0.999777i \(-0.493272\pi\)
0.876400 + 0.481584i \(0.159938\pi\)
\(602\) 0 0
\(603\) 8.46492 + 28.9814i 0.344718 + 1.18021i
\(604\) 0 0
\(605\) 18.7889 32.5433i 0.763878 1.32308i
\(606\) 0 0
\(607\) 11.7094 6.76042i 0.475270 0.274397i −0.243173 0.969983i \(-0.578188\pi\)
0.718443 + 0.695586i \(0.244855\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −7.03633 4.06243i −0.284659 0.164348i
\(612\) 0 0
\(613\) −2.41817 4.18840i −0.0976691 0.169168i 0.813050 0.582193i \(-0.197805\pi\)
−0.910719 + 0.413026i \(0.864472\pi\)
\(614\) 0 0
\(615\) −9.30706 + 12.4144i −0.375297 + 0.500595i
\(616\) 0 0
\(617\) 5.30333 + 3.06188i 0.213504 + 0.123267i 0.602939 0.797787i \(-0.293996\pi\)
−0.389435 + 0.921054i \(0.627330\pi\)
\(618\) 0 0
\(619\) 4.94313 + 2.85392i 0.198681 + 0.114709i 0.596040 0.802955i \(-0.296740\pi\)
−0.397359 + 0.917663i \(0.630073\pi\)
\(620\) 0 0
\(621\) −29.5227 + 4.89971i −1.18470 + 0.196618i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −34.6609 60.0344i −1.38644 2.40138i
\(626\) 0 0
\(627\) −12.1722 1.46415i −0.486110 0.0584724i
\(628\) 0 0
\(629\) −1.28513 −0.0512414
\(630\) 0 0
\(631\) −19.0525 −0.758468 −0.379234 0.925301i \(-0.623812\pi\)
−0.379234 + 0.925301i \(0.623812\pi\)
\(632\) 0 0
\(633\) 1.43360 + 3.35436i 0.0569804 + 0.133324i
\(634\) 0 0
\(635\) −4.00124 6.93035i −0.158784 0.275022i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −31.5483 + 9.21469i −1.24803 + 0.364527i
\(640\) 0 0
\(641\) −13.2820 7.66837i −0.524607 0.302882i 0.214210 0.976788i \(-0.431282\pi\)
−0.738818 + 0.673905i \(0.764616\pi\)
\(642\) 0 0
\(643\) 11.3209 + 6.53612i 0.446453 + 0.257759i 0.706331 0.707882i \(-0.250349\pi\)
−0.259878 + 0.965641i \(0.583682\pi\)
\(644\) 0 0
\(645\) −17.1490 40.1255i −0.675239 1.57994i
\(646\) 0 0
\(647\) 17.8533 + 30.9228i 0.701885 + 1.21570i 0.967804 + 0.251705i \(0.0809913\pi\)
−0.265919 + 0.963995i \(0.585675\pi\)
\(648\) 0 0
\(649\) −5.61605 3.24243i −0.220449 0.127277i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 23.6131 13.6330i 0.924051 0.533501i 0.0391261 0.999234i \(-0.487543\pi\)
0.884925 + 0.465733i \(0.154209\pi\)
\(654\) 0 0
\(655\) −18.1113 + 31.3698i −0.707669 + 1.22572i
\(656\) 0 0
\(657\) −20.6602 19.7556i −0.806030 0.770741i
\(658\) 0 0
\(659\) −14.7911 + 8.53963i −0.576179 + 0.332657i −0.759613 0.650375i \(-0.774612\pi\)
0.183435 + 0.983032i \(0.441278\pi\)
\(660\) 0 0
\(661\) 46.9568i 1.82641i −0.407505 0.913203i \(-0.633601\pi\)
0.407505 0.913203i \(-0.366399\pi\)
\(662\) 0 0
\(663\) −0.504999 0.378598i −0.0196125 0.0147035i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −24.3313 + 42.1431i −0.942112 + 1.63179i
\(668\) 0 0
\(669\) 1.31930 10.9680i 0.0510069 0.424046i
\(670\) 0 0
\(671\) 5.34305 9.25444i 0.206266 0.357264i
\(672\) 0 0
\(673\) 7.76077 + 13.4421i 0.299156 + 0.518153i 0.975943 0.218026i \(-0.0699616\pi\)
−0.676787 + 0.736179i \(0.736628\pi\)
\(674\) 0 0
\(675\) −50.2884 + 41.3414i −1.93560 + 1.59123i
\(676\) 0 0
\(677\) −18.1574 −0.697845 −0.348922 0.937152i \(-0.613452\pi\)
−0.348922 + 0.937152i \(0.613452\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 7.79205 + 18.2320i 0.298592 + 0.698651i
\(682\) 0 0
\(683\) −17.4866 + 10.0959i −0.669104 + 0.386308i −0.795737 0.605642i \(-0.792916\pi\)
0.126633 + 0.991950i \(0.459583\pi\)
\(684\) 0 0
\(685\) 48.8146i 1.86511i
\(686\) 0 0
\(687\) 1.16726 9.70398i 0.0445336 0.370230i
\(688\) 0 0
\(689\) 4.88468 0.186092
\(690\) 0 0
\(691\) 0.0803432i 0.00305640i 0.999999 + 0.00152820i \(0.000486441\pi\)
−0.999999 + 0.00152820i \(0.999514\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 23.3339i 0.885104i
\(696\) 0 0
\(697\) 0.793689 0.0300631
\(698\) 0 0
\(699\) −19.8231 + 8.47207i −0.749780 + 0.320443i
\(700\) 0 0
\(701\) 35.1490i 1.32756i −0.747928 0.663780i \(-0.768951\pi\)
0.747928 0.663780i \(-0.231049\pi\)
\(702\) 0 0
\(703\) −14.9256 + 8.61731i −0.562930 + 0.325008i
\(704\) 0 0
\(705\) 59.5476 + 7.16276i 2.24269 + 0.269765i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 1.56594 0.0588100 0.0294050 0.999568i \(-0.490639\pi\)
0.0294050 + 0.999568i \(0.490639\pi\)
\(710\) 0 0
\(711\) 7.15865 + 24.5091i 0.268470 + 0.919163i
\(712\) 0 0
\(713\) −20.8823 36.1691i −0.782047 1.35455i
\(714\) 0 0
\(715\) −2.92595 + 5.06790i −0.109424 + 0.189529i
\(716\) 0 0
\(717\) −6.48184 + 2.77023i −0.242069 + 0.103456i
\(718\) 0 0
\(719\) 12.9393 22.4116i 0.482556 0.835811i −0.517243 0.855838i \(-0.673042\pi\)
0.999799 + 0.0200268i \(0.00637517\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −7.83807 + 3.34986i −0.291501 + 0.124583i
\(724\) 0 0
\(725\) 105.858i 3.93146i
\(726\) 0 0
\(727\) −0.990545 + 0.571891i −0.0367373 + 0.0212103i −0.518256 0.855225i \(-0.673419\pi\)
0.481519 + 0.876436i \(0.340085\pi\)
\(728\) 0 0
\(729\) 20.3296 + 17.7682i 0.752947 + 0.658081i
\(730\) 0 0
\(731\) −1.11609 + 1.93312i −0.0412800 + 0.0714990i
\(732\) 0 0
\(733\) −20.1408 + 11.6283i −0.743916 + 0.429500i −0.823491 0.567329i \(-0.807977\pi\)
0.0795755 + 0.996829i \(0.474644\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −12.4012 7.15985i −0.456805 0.263736i
\(738\) 0 0
\(739\) 18.0758 + 31.3082i 0.664929 + 1.15169i 0.979305 + 0.202392i \(0.0648714\pi\)
−0.314376 + 0.949299i \(0.601795\pi\)
\(740\) 0 0
\(741\) −8.40378 1.01086i −0.308721 0.0371349i
\(742\) 0 0
\(743\) 6.43940 + 3.71779i 0.236239 + 0.136392i 0.613447 0.789736i \(-0.289783\pi\)
−0.377208 + 0.926129i \(0.623116\pi\)
\(744\) 0 0
\(745\) 56.1121 + 32.3963i 2.05579 + 1.18691i
\(746\) 0 0
\(747\) −4.03159 + 4.21619i −0.147508 + 0.154262i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −3.67798 6.37044i −0.134211 0.232461i 0.791085 0.611707i \(-0.209517\pi\)
−0.925296 + 0.379246i \(0.876183\pi\)
\(752\) 0 0
\(753\) −4.83140 + 6.44443i −0.176066 + 0.234848i
\(754\) 0 0
\(755\) 69.6445 2.53462
\(756\) 0 0
\(757\) 7.65326 0.278163 0.139081 0.990281i \(-0.455585\pi\)
0.139081 + 0.990281i \(0.455585\pi\)
\(758\) 0 0
\(759\) 8.51401 11.3565i 0.309039 0.412216i
\(760\) 0 0
\(761\) −21.7203 37.6207i −0.787362 1.36375i −0.927578 0.373629i \(-0.878113\pi\)
0.140217 0.990121i \(-0.455220\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 4.52624 + 1.10488i 0.163647 + 0.0399469i
\(766\) 0 0
\(767\) −3.87738 2.23860i −0.140004 0.0808313i
\(768\) 0 0
\(769\) 18.8491 + 10.8825i 0.679716 + 0.392434i 0.799748 0.600336i \(-0.204966\pi\)
−0.120032 + 0.992770i \(0.538300\pi\)
\(770\) 0 0
\(771\) −18.7340 2.25344i −0.674687 0.0811557i
\(772\) 0 0
\(773\) −7.25734 12.5701i −0.261028 0.452114i 0.705487 0.708723i \(-0.250728\pi\)
−0.966515 + 0.256609i \(0.917395\pi\)
\(774\) 0 0
\(775\) −78.6801 45.4260i −2.82627 1.63175i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 9.21800 5.32202i 0.330269 0.190681i
\(780\) 0 0
\(781\) 7.79402 13.4996i 0.278892 0.483055i
\(782\) 0 0
\(783\) 43.3115 7.18816i 1.54783 0.256884i
\(784\) 0 0
\(785\) −60.9415 + 35.1846i −2.17509 + 1.25579i
\(786\) 0 0
\(787\) 13.1972i 0.470430i 0.971943 + 0.235215i \(0.0755795\pi\)
−0.971943 + 0.235215i \(0.924421\pi\)
\(788\) 0 0
\(789\) 28.7795 12.2999i 1.02458 0.437888i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 3.68889 6.38935i 0.130996 0.226892i
\(794\) 0 0
\(795\) −33.1570 + 14.1707i −1.17596 + 0.502584i
\(796\) 0 0
\(797\) 24.8899 43.1106i 0.881646 1.52706i 0.0321352 0.999484i \(-0.489769\pi\)
0.849511 0.527572i \(-0.176897\pi\)
\(798\) 0 0
\(799\) −1.53402 2.65701i −0.0542698 0.0939981i
\(800\) 0 0
\(801\) 22.7539 + 5.55434i 0.803971 + 0.196253i
\(802\) 0 0
\(803\) 13.5576 0.478437
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −5.13162 0.617264i −0.180642 0.0217287i
\(808\) 0 0
\(809\) −24.4123 + 14.0944i −0.858290 + 0.495534i −0.863439 0.504453i \(-0.831694\pi\)
0.00514934 + 0.999987i \(0.498361\pi\)
\(810\) 0 0
\(811\) 33.5981i 1.17979i −0.807480 0.589894i \(-0.799169\pi\)
0.807480 0.589894i \(-0.200831\pi\)
\(812\) 0 0
\(813\) −16.5016 + 7.05250i −0.578736 + 0.247342i
\(814\) 0 0
\(815\) −14.6998 −0.514910
\(816\) 0 0
\(817\) 29.9353i 1.04730i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 54.1696i 1.89053i −0.326299 0.945267i \(-0.605802\pi\)
0.326299 0.945267i \(-0.394198\pi\)
\(822\) 0 0
\(823\) 28.5390 0.994808 0.497404 0.867519i \(-0.334287\pi\)
0.497404 + 0.867519i \(0.334287\pi\)
\(824\) 0 0
\(825\) 3.68738 30.6550i 0.128378 1.06727i
\(826\) 0 0
\(827\) 37.2062i 1.29379i 0.762581 + 0.646893i \(0.223932\pi\)
−0.762581 + 0.646893i \(0.776068\pi\)
\(828\) 0 0
\(829\) −1.92557 + 1.11173i −0.0668777 + 0.0386119i −0.533066 0.846074i \(-0.678960\pi\)
0.466188 + 0.884686i \(0.345627\pi\)
\(830\) 0 0
\(831\) 8.09128 + 18.9321i 0.280684 + 0.656749i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −33.9335 −1.17432
\(836\) 0 0
\(837\) −13.2433 + 35.2765i −0.457755 + 1.21933i
\(838\) 0 0
\(839\) 27.8383 + 48.2173i 0.961084 + 1.66465i 0.719787 + 0.694195i \(0.244240\pi\)
0.241297 + 0.970451i \(0.422427\pi\)
\(840\) 0 0
\(841\) 21.1955 36.7116i 0.730878 1.26592i
\(842\) 0 0
\(843\) 2.88390 23.9753i 0.0993267 0.825752i
\(844\) 0 0
\(845\) 25.1935 43.6364i 0.866683 1.50114i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −3.51399 2.63445i −0.120600 0.0904140i
\(850\) 0 0
\(851\) 19.9530i 0.683980i
\(852\) 0 0
\(853\) 16.2574 9.38622i 0.556643 0.321378i −0.195154 0.980773i \(-0.562521\pi\)
0.751797 + 0.659395i \(0.229187\pi\)
\(854\) 0 0
\(855\) 59.9770 17.5182i 2.05117 0.599109i
\(856\) 0 0
\(857\) −11.8516 + 20.5276i −0.404844 + 0.701210i −0.994303 0.106588i \(-0.966007\pi\)
0.589460 + 0.807798i \(0.299341\pi\)
\(858\) 0 0
\(859\) −14.5237 + 8.38527i −0.495543 + 0.286102i −0.726871 0.686774i \(-0.759026\pi\)
0.231328 + 0.972876i \(0.425693\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 29.2235 + 16.8722i 0.994778 + 0.574335i 0.906699 0.421778i \(-0.138594\pi\)
0.0880791 + 0.996113i \(0.471927\pi\)
\(864\) 0 0
\(865\) −27.4164 47.4866i −0.932186 1.61459i
\(866\) 0 0
\(867\) 11.4781 + 26.8566i 0.389815 + 0.912097i
\(868\) 0 0
\(869\) −10.4875 6.05497i −0.355765 0.205401i
\(870\) 0 0
\(871\) −8.56192 4.94323i −0.290110 0.167495i
\(872\) 0 0
\(873\) −8.36478 7.99855i −0.283105 0.270710i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 19.7087 + 34.1365i 0.665515 + 1.15271i 0.979145 + 0.203161i \(0.0651215\pi\)
−0.313630 + 0.949545i \(0.601545\pi\)
\(878\) 0 0
\(879\) −5.38420 12.5981i −0.181605 0.424922i
\(880\) 0 0
\(881\) −26.2496 −0.884372 −0.442186 0.896923i \(-0.645797\pi\)
−0.442186 + 0.896923i \(0.645797\pi\)
\(882\) 0 0
\(883\) −43.5087 −1.46418 −0.732091 0.681206i \(-0.761456\pi\)
−0.732091 + 0.681206i \(0.761456\pi\)
\(884\) 0 0
\(885\) 32.8138 + 3.94705i 1.10302 + 0.132679i
\(886\) 0 0
\(887\) −2.83888 4.91708i −0.0953202 0.165099i 0.814422 0.580273i \(-0.197054\pi\)
−0.909742 + 0.415174i \(0.863721\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −12.7928 + 0.572912i −0.428576 + 0.0191933i
\(892\) 0 0
\(893\) −35.6327 20.5725i −1.19240 0.688434i
\(894\) 0 0
\(895\) −46.1124 26.6230i −1.54137 0.889909i
\(896\) 0 0
\(897\) 5.87815 7.84066i 0.196266 0.261792i
\(898\) 0 0
\(899\) 30.6355 + 53.0623i 1.02175 + 1.76973i
\(900\) 0 0
\(901\) 1.59740 + 0.922259i 0.0532171 + 0.0307249i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −96.2716 + 55.5824i −3.20018 + 1.84762i
\(906\) 0 0
\(907\) −7.32741 + 12.6914i −0.243303 + 0.421412i −0.961653 0.274269i \(-0.911564\pi\)
0.718350 + 0.695681i \(0.244897\pi\)
\(908\) 0 0
\(909\) −26.2670 6.41189i −0.871220 0.212669i
\(910\) 0 0
\(911\) 7.19133 4.15192i 0.238260 0.137559i −0.376117 0.926572i \(-0.622741\pi\)
0.614377 + 0.789013i \(0.289408\pi\)
\(912\) 0 0
\(913\) 2.76674i 0.0915658i
\(914\) 0 0
\(915\) −6.50416 + 54.0723i −0.215021 + 1.78757i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 18.5554 32.1388i 0.612085 1.06016i −0.378804 0.925477i \(-0.623665\pi\)
0.990889 0.134685i \(-0.0430021\pi\)
\(920\) 0 0
\(921\) 19.3568 + 14.5118i 0.637828 + 0.478180i
\(922\) 0 0
\(923\) 5.38106 9.32027i 0.177120 0.306781i
\(924\) 0 0
\(925\) −21.7022 37.5894i −0.713566 1.23593i
\(926\) 0 0
\(927\) −14.9305 14.2769i −0.490383 0.468914i
\(928\) 0 0
\(929\) 7.88987 0.258858 0.129429 0.991589i \(-0.458686\pi\)
0.129429 + 0.991589i \(0.458686\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −33.9042 + 45.2237i −1.10998 + 1.48056i
\(934\) 0 0
\(935\) −1.91370 + 1.10488i −0.0625848 + 0.0361333i
\(936\) 0 0
\(937\) 13.2688i 0.433472i 0.976230 + 0.216736i \(0.0695411\pi\)
−0.976230 + 0.216736i \(0.930459\pi\)
\(938\) 0 0
\(939\) 21.7807 + 16.3290i 0.710787 + 0.532878i
\(940\) 0 0
\(941\) −7.92922 −0.258485 −0.129243 0.991613i \(-0.541255\pi\)
−0.129243 + 0.991613i \(0.541255\pi\)
\(942\) 0 0
\(943\) 12.3229i 0.401288i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 30.9635i 1.00618i −0.864235 0.503089i \(-0.832197\pi\)
0.864235 0.503089i \(-0.167803\pi\)
\(948\) 0 0
\(949\) 9.36029 0.303848
\(950\) 0 0
\(951\) 40.2244 + 30.1563i 1.30436 + 0.977884i
\(952\) 0 0
\(953\) 34.1087i 1.10489i 0.833549 + 0.552445i \(0.186305\pi\)
−0.833549 + 0.552445i \(0.813695\pi\)
\(954\) 0 0
\(955\) 4.59428 2.65251i 0.148668 0.0858332i
\(956\) 0 0
\(957\) −12.4906 + 16.6607i −0.403762 + 0.538564i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −21.5856 −0.696311
\(962\) 0 0
\(963\) −8.08390 + 33.1165i −0.260500 + 1.06717i
\(964\) 0 0
\(965\) 39.0089 + 67.5655i 1.25574 + 2.17501i
\(966\) 0 0
\(967\) 29.1066 50.4142i 0.936007 1.62121i 0.163177 0.986597i \(-0.447826\pi\)
0.772829 0.634614i \(-0.218841\pi\)
\(968\) 0 0
\(969\) −2.55737 1.91726i −0.0821544 0.0615913i
\(970\) 0 0
\(971\) 4.91896 8.51988i 0.157857 0.273416i −0.776239 0.630439i \(-0.782875\pi\)
0.934096 + 0.357023i \(0.116208\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 2.54580 21.1645i 0.0815308 0.677806i
\(976\) 0 0
\(977\) 33.5274i 1.07264i 0.844016 + 0.536318i \(0.180185\pi\)
−0.844016 + 0.536318i \(0.819815\pi\)
\(978\) 0 0
\(979\) −9.62041 + 5.55434i −0.307470 + 0.177518i
\(980\) 0 0
\(981\) 33.1587 34.6769i 1.05867 1.10715i
\(982\) 0 0
\(983\) 24.2359 41.9779i 0.773006 1.33889i −0.162902 0.986642i \(-0.552086\pi\)
0.935908 0.352243i \(-0.114581\pi\)
\(984\) 0 0
\(985\) −21.5486 + 12.4411i −0.686594 + 0.396405i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −30.0138 17.3285i −0.954382 0.551013i
\(990\) 0 0
\(991\) 26.5005 + 45.9003i 0.841818 + 1.45807i 0.888357 + 0.459154i \(0.151848\pi\)
−0.0465389 + 0.998916i \(0.514819\pi\)
\(992\) 0 0
\(993\) −13.6834 + 18.2518i −0.434229 + 0.579203i
\(994\) 0 0
\(995\) 20.4015 + 11.7788i 0.646770 + 0.373413i
\(996\) 0 0
\(997\) −7.57384 4.37276i −0.239866 0.138487i 0.375249 0.926924i \(-0.377557\pi\)
−0.615115 + 0.788437i \(0.710890\pi\)
\(998\) 0 0
\(999\) −13.9060 + 11.4319i −0.439966 + 0.361690i
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 1764.2.w.a.509.2 16
3.2 odd 2 5292.2.w.a.1097.1 16
7.2 even 3 252.2.x.a.41.4 16
7.3 odd 6 1764.2.bm.b.1697.1 16
7.4 even 3 1764.2.bm.b.1697.8 16
7.5 odd 6 252.2.x.a.41.5 yes 16
7.6 odd 2 inner 1764.2.w.a.509.7 16
9.2 odd 6 1764.2.bm.b.1685.1 16
9.7 even 3 5292.2.bm.b.4625.1 16
21.2 odd 6 756.2.x.a.125.1 16
21.5 even 6 756.2.x.a.125.8 16
21.11 odd 6 5292.2.bm.b.2285.8 16
21.17 even 6 5292.2.bm.b.2285.1 16
21.20 even 2 5292.2.w.a.1097.8 16
28.19 even 6 1008.2.cc.c.545.4 16
28.23 odd 6 1008.2.cc.c.545.5 16
63.2 odd 6 252.2.x.a.209.5 yes 16
63.5 even 6 2268.2.f.b.1133.2 16
63.11 odd 6 inner 1764.2.w.a.1109.7 16
63.16 even 3 756.2.x.a.629.8 16
63.20 even 6 1764.2.bm.b.1685.8 16
63.23 odd 6 2268.2.f.b.1133.15 16
63.25 even 3 5292.2.w.a.521.8 16
63.34 odd 6 5292.2.bm.b.4625.8 16
63.38 even 6 inner 1764.2.w.a.1109.2 16
63.40 odd 6 2268.2.f.b.1133.16 16
63.47 even 6 252.2.x.a.209.4 yes 16
63.52 odd 6 5292.2.w.a.521.1 16
63.58 even 3 2268.2.f.b.1133.1 16
63.61 odd 6 756.2.x.a.629.1 16
84.23 even 6 3024.2.cc.c.881.1 16
84.47 odd 6 3024.2.cc.c.881.8 16
252.47 odd 6 1008.2.cc.c.209.5 16
252.79 odd 6 3024.2.cc.c.2897.8 16
252.187 even 6 3024.2.cc.c.2897.1 16
252.191 even 6 1008.2.cc.c.209.4 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.2.x.a.41.4 16 7.2 even 3
252.2.x.a.41.5 yes 16 7.5 odd 6
252.2.x.a.209.4 yes 16 63.47 even 6
252.2.x.a.209.5 yes 16 63.2 odd 6
756.2.x.a.125.1 16 21.2 odd 6
756.2.x.a.125.8 16 21.5 even 6
756.2.x.a.629.1 16 63.61 odd 6
756.2.x.a.629.8 16 63.16 even 3
1008.2.cc.c.209.4 16 252.191 even 6
1008.2.cc.c.209.5 16 252.47 odd 6
1008.2.cc.c.545.4 16 28.19 even 6
1008.2.cc.c.545.5 16 28.23 odd 6
1764.2.w.a.509.2 16 1.1 even 1 trivial
1764.2.w.a.509.7 16 7.6 odd 2 inner
1764.2.w.a.1109.2 16 63.38 even 6 inner
1764.2.w.a.1109.7 16 63.11 odd 6 inner
1764.2.bm.b.1685.1 16 9.2 odd 6
1764.2.bm.b.1685.8 16 63.20 even 6
1764.2.bm.b.1697.1 16 7.3 odd 6
1764.2.bm.b.1697.8 16 7.4 even 3
2268.2.f.b.1133.1 16 63.58 even 3
2268.2.f.b.1133.2 16 63.5 even 6
2268.2.f.b.1133.15 16 63.23 odd 6
2268.2.f.b.1133.16 16 63.40 odd 6
3024.2.cc.c.881.1 16 84.23 even 6
3024.2.cc.c.881.8 16 84.47 odd 6
3024.2.cc.c.2897.1 16 252.187 even 6
3024.2.cc.c.2897.8 16 252.79 odd 6
5292.2.w.a.521.1 16 63.52 odd 6
5292.2.w.a.521.8 16 63.25 even 3
5292.2.w.a.1097.1 16 3.2 odd 2
5292.2.w.a.1097.8 16 21.20 even 2
5292.2.bm.b.2285.1 16 21.17 even 6
5292.2.bm.b.2285.8 16 21.11 odd 6
5292.2.bm.b.4625.1 16 9.7 even 3
5292.2.bm.b.4625.8 16 63.34 odd 6