Properties

Label 1764.2.w.a.509.7
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 509.7
Root \(1.71965 - 0.206851i\) of defining polynomial
Character \(\chi\) \(=\) 1764.509
Dual form 1764.2.w.a.1109.7

$q$-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 \(-0.781005\pi\)
−0.163656 0.986518i \(-0.552329\pi\)
\(6\) 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 0.377363 0.926066i \(-0.376831\pi\)
−0.613315 + 0.789838i \(0.710164\pi\)
\(14\) 0 0
\(15\) −7.19970 0.866025i −1.85895 0.223607i
\(16\) 0 0
\(17\) −0.185474 0.321250i −0.0449840 0.0779145i 0.842657 0.538451i \(-0.180990\pi\)
−0.887641 + 0.460537i \(0.847657\pi\)
\(18\) 0 0
\(19\) −4.30823 2.48736i −0.988375 0.570638i −0.0835867 0.996501i \(-0.526638\pi\)
−0.904788 + 0.425862i \(0.859971\pi\)
\(20\) 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 \(-0.774261\pi\)
0.758895 0.651213i \(-0.225739\pi\)
\(32\) 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 −0.937391 0.348279i \(-0.886766\pi\)
0.770314 + 0.637665i \(0.220099\pi\)
\(42\) 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 \(-0.293887\pi\)
0.603213 + 0.797580i \(0.293887\pi\)
\(48\) 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 \(-0.404121\pi\)
0.296678 + 0.954977i \(0.404121\pi\)
\(60\) 0 0
\(61\) 7.51035i 0.961602i 0.876830 + 0.480801i \(0.159654\pi\)
−0.876830 + 0.480801i \(0.840346\pi\)
\(62\) 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 −0.897957 0.440083i \(-0.854949\pi\)
−0.0678555 + 0.997695i \(0.521616\pi\)
\(74\) 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 0.914439 0.404724i \(-0.132632\pi\)
−0.807720 + 0.589566i \(0.799299\pi\)
\(84\) 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 −0.581512 0.813538i \(-0.697538\pi\)
0.995300 + 0.0968347i \(0.0308718\pi\)
\(90\) 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 −0.659930 0.751327i \(-0.729414\pi\)
0.320704 + 0.947180i \(0.396081\pi\)
\(98\) 0 0
\(99\) 1.01225 4.14679i 0.101735 0.416768i
\(100\) 0 0
\(101\) −4.50637 + 7.80526i −0.448401 + 0.776653i −0.998282 0.0585901i \(-0.981340\pi\)
0.549882 + 0.835243i \(0.314673\pi\)
\(102\) 0 0
\(103\) −5.96343 + 3.44299i −0.587594 + 0.339248i −0.764146 0.645044i \(-0.776839\pi\)
0.176551 + 0.984291i \(0.443506\pi\)
\(104\) 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 0.612808 0.790232i \(-0.290040\pi\)
−0.990765 + 0.135592i \(0.956707\pi\)
\(132\) 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 0.281138 0.959667i \(-0.409288\pi\)
−0.690528 + 0.723306i \(0.742622\pi\)
\(140\) 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 \(-0.765993\pi\)
0.741727 0.670702i \(-0.234007\pi\)
\(158\) 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 −0.665544 0.746359i \(-0.731800\pi\)
0.979138 + 0.203198i \(0.0651336\pi\)
\(168\) 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 \(-0.334116\pi\)
0.497868 + 0.867253i \(0.334116\pi\)
\(174\) 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 \(-0.551794\pi\)
0.161998 0.986791i \(-0.448206\pi\)
\(182\) 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 −0.662671 0.748910i \(-0.730577\pi\)
0.317240 + 0.948345i \(0.397244\pi\)
\(200\) 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 −0.303525 0.952823i \(-0.598164\pi\)
0.673407 + 0.739272i \(0.264830\pi\)
\(224\) 0 0
\(225\) 36.5135 + 8.91312i 2.43423 + 0.594208i
\(226\) 0 0
\(227\) −5.72365 + 9.91365i −0.379892 + 0.657992i −0.991046 0.133520i \(-0.957372\pi\)
0.611154 + 0.791511i \(0.290705\pi\)
\(228\) 0 0
\(229\) 4.88696 2.82149i 0.322940 0.186449i −0.329762 0.944064i \(-0.606969\pi\)
0.652702 + 0.757615i \(0.273635\pi\)
\(230\) 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 0.356411 0.934329i \(-0.384000\pi\)
−0.630947 + 0.775826i \(0.717334\pi\)
\(242\) 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 \(-0.546884\pi\)
−0.146759 + 0.989172i \(0.546884\pi\)
\(252\) 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 0.644615 0.764507i \(-0.277018\pi\)
−0.984390 + 0.175999i \(0.943684\pi\)
\(258\) 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 0.816948 0.576711i \(-0.195664\pi\)
−0.907920 + 0.419143i \(0.862331\pi\)
\(270\) 0 0
\(271\) −8.97274 5.18041i −0.545055 0.314688i 0.202070 0.979371i \(-0.435233\pi\)
−0.747125 + 0.664683i \(0.768566\pi\)
\(272\) 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 \(-0.975988\pi\)
0.997156 0.0753642i \(-0.0240119\pi\)
\(284\) 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 −0.727066 0.686567i \(-0.759117\pi\)
0.958118 + 0.286374i \(0.0924501\pi\)
\(294\) 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 \(0.130499\pi\)
−0.917131 + 0.398585i \(0.869501\pi\)
\(308\) 0 0
\(309\) −1.42437 + 11.8415i −0.0810296 + 0.673640i
\(310\) 0 0
\(311\) −32.6327 −1.85043 −0.925215 0.379443i \(-0.876116\pi\)
−0.925215 + 0.379443i \(0.876116\pi\)
\(312\) 0 0
\(313\) 15.7166i 0.888357i 0.895938 + 0.444178i \(0.146504\pi\)
−0.895938 + 0.444178i \(0.853496\pi\)
\(314\) 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 −0.997463 0.0711915i \(-0.977320\pi\)
−0.437078 + 0.899424i \(0.643987\pi\)
\(350\) 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 \(-0.619881\pi\)
0.989218 0.146452i \(-0.0467853\pi\)
\(354\) 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 0.152392 0.988320i \(-0.451302\pi\)
−0.779714 + 0.626135i \(0.784636\pi\)
\(368\) 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 0.885460 0.464717i \(-0.153844\pi\)
−0.845186 + 0.534472i \(0.820510\pi\)
\(384\) 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 0.513693 0.857974i \(-0.328277\pi\)
−0.486181 + 0.873858i \(0.661611\pi\)
\(398\) 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 \(-0.886936\pi\)
0.937577 0.347778i \(-0.113064\pi\)
\(410\) 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 \(0.346037\pi\)
−0.999204 + 0.0398995i \(0.987296\pi\)
\(420\) 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 \(-0.837960\pi\)
0.873201 0.487359i \(-0.162040\pi\)
\(434\) 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 \(-0.762982\pi\)
0.735350 0.677687i \(-0.237018\pi\)
\(440\) 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 0.573437 0.819249i \(-0.305610\pi\)
−0.996209 + 0.0869865i \(0.972276\pi\)
\(462\) 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 \(-0.620562\pi\)
0.989529 0.144336i \(-0.0461045\pi\)
\(468\) 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 \(-0.607114\pi\)
0.982549 0.186002i \(-0.0595529\pi\)
\(480\) 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 \(-0.410128\pi\)
0.278605 + 0.960406i \(0.410128\pi\)
\(504\) 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 0.929885 0.367850i \(-0.119906\pi\)
−0.783510 + 0.621379i \(0.786573\pi\)
\(510\) 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 0.986251 0.165252i \(-0.0528438\pi\)
−0.636238 + 0.771493i \(0.719510\pi\)
\(522\) 0 0
\(523\) 10.1857 + 5.88074i 0.445391 + 0.257147i 0.705882 0.708330i \(-0.250551\pi\)
−0.260491 + 0.965476i \(0.583884\pi\)
\(524\) 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 \(-0.734572\pi\)
−0.672019 + 0.740534i \(0.734572\pi\)
\(564\) 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 \(-0.327672\pi\)
0.999842 0.0177861i \(-0.00566180\pi\)
\(578\) 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 0.899321 0.437289i \(-0.144061\pi\)
−0.828364 + 0.560191i \(0.810728\pi\)
\(588\) 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 −0.944214 0.329332i \(-0.893177\pi\)
0.757317 + 0.653048i \(0.226510\pi\)
\(594\) 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 −0.876400 0.481584i \(-0.840062\pi\)
−0.0211361 + 0.999777i \(0.506728\pi\)
\(602\) 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 −0.718443 0.695586i \(-0.755145\pi\)
0.243173 + 0.969983i \(0.421812\pi\)
\(608\) 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 0.397359 0.917663i \(-0.369927\pi\)
−0.596040 + 0.802955i \(0.703260\pi\)
\(620\) 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 0.259878 0.965641i \(-0.416318\pi\)
−0.706331 + 0.707882i \(0.749651\pi\)
\(644\) 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 \(-0.919009\pi\)
0.265919 0.963995i \(-0.414325\pi\)
\(648\) 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 \(0.366399\pi\)
−0.407505 + 0.913203i \(0.633601\pi\)
\(662\) 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 \(-0.386548\pi\)
0.348922 + 0.937152i \(0.386548\pi\)
\(678\) 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 \(-0.999514\pi\)
0.999999 0.00152820i \(-0.000486441\pi\)
\(692\) 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 −0.999799 0.0200268i \(-0.993625\pi\)
0.517243 + 0.855838i \(0.326958\pi\)
\(720\) 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 −0.481519 0.876436i \(-0.659915\pi\)
0.518256 + 0.855225i \(0.326581\pi\)
\(728\) 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 −0.0795755 0.996829i \(-0.525356\pi\)
0.823491 + 0.567329i \(0.192023\pi\)
\(734\) 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 \(0.121887\pi\)
−0.140217 + 0.990121i \(0.544780\pi\)
\(762\) 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 0.120032 0.992770i \(-0.461700\pi\)
−0.799748 + 0.600336i \(0.795034\pi\)
\(770\) 0 0
\(771\) −18.7340 2.25344i −0.674687 0.0811557i
\(772\) 0 0
\(773\) 7.25734 + 12.5701i 0.261028 + 0.452114i 0.966515 0.256609i \(-0.0826051\pi\)
−0.705487 + 0.708723i \(0.749272\pi\)
\(774\) 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 \(-0.924421\pi\)
0.971943 0.235215i \(-0.0755795\pi\)
\(788\) 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 \(0.510231\pi\)
−0.849511 + 0.527572i \(0.823103\pi\)
\(798\) 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 \(0.200831\pi\)
−0.807480 + 0.589894i \(0.799169\pi\)
\(812\) 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 −0.466188 0.884686i \(-0.654373\pi\)
0.533066 + 0.846074i \(0.321040\pi\)
\(830\) 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 \(-0.755760\pi\)
−0.241297 0.970451i \(-0.577573\pi\)
\(840\) 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 −0.751797 0.659395i \(-0.770813\pi\)
0.195154 + 0.980773i \(0.437479\pi\)
\(854\) 0 0
\(855\) 59.9770 17.5182i 2.05117 0.599109i
\(856\) 0 0
\(857\) 11.8516 20.5276i 0.404844 0.701210i −0.589460 0.807798i \(-0.700659\pi\)
0.994303 + 0.106588i \(0.0339926\pi\)
\(858\) 0 0
\(859\) 14.5237 8.38527i 0.495543 0.286102i −0.231328 0.972876i \(-0.574307\pi\)
0.726871 + 0.686774i \(0.240974\pi\)
\(860\) 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 \(-0.354203\pi\)
0.442186 + 0.896923i \(0.354203\pi\)
\(882\) 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 0.909742 0.415174i \(-0.136279\pi\)
−0.814422 + 0.580273i \(0.802946\pi\)
\(888\) 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 \(-0.541314\pi\)
−0.129429 + 0.991589i \(0.541314\pi\)
\(930\) 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 \(-0.930459\pi\)
0.976230 0.216736i \(-0.0695411\pi\)
\(938\) 0 0
\(939\) 21.7807 + 16.3290i 0.710787 + 0.532878i
\(940\) 0 0
\(941\) 7.92922 0.258485 0.129243 0.991613i \(-0.458745\pi\)
0.129243 + 0.991613i \(0.458745\pi\)
\(942\) 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 −0.934096 0.357023i \(-0.883792\pi\)
0.776239 + 0.630439i \(0.217125\pi\)
\(972\) 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 \(0.447914\pi\)
−0.935908 + 0.352243i \(0.885419\pi\)
\(984\) 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 0.615115 0.788437i \(-0.289110\pi\)
−0.375249 + 0.926924i \(0.622443\pi\)
\(998\) 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.509.7 16
3.2 odd 2 5292.2.w.a.1097.8 16
7.2 even 3 252.2.x.a.41.5 yes 16
7.3 odd 6 1764.2.bm.b.1697.8 16
7.4 even 3 1764.2.bm.b.1697.1 16
7.5 odd 6 252.2.x.a.41.4 16
7.6 odd 2 inner 1764.2.w.a.509.2 16
9.2 odd 6 1764.2.bm.b.1685.8 16
9.7 even 3 5292.2.bm.b.4625.8 16
21.2 odd 6 756.2.x.a.125.8 16
21.5 even 6 756.2.x.a.125.1 16
21.11 odd 6 5292.2.bm.b.2285.1 16
21.17 even 6 5292.2.bm.b.2285.8 16
21.20 even 2 5292.2.w.a.1097.1 16
28.19 even 6 1008.2.cc.c.545.5 16
28.23 odd 6 1008.2.cc.c.545.4 16
63.2 odd 6 252.2.x.a.209.4 yes 16
63.5 even 6 2268.2.f.b.1133.15 16
63.11 odd 6 inner 1764.2.w.a.1109.2 16
63.16 even 3 756.2.x.a.629.1 16
63.20 even 6 1764.2.bm.b.1685.1 16
63.23 odd 6 2268.2.f.b.1133.2 16
63.25 even 3 5292.2.w.a.521.1 16
63.34 odd 6 5292.2.bm.b.4625.1 16
63.38 even 6 inner 1764.2.w.a.1109.7 16
63.40 odd 6 2268.2.f.b.1133.1 16
63.47 even 6 252.2.x.a.209.5 yes 16
63.52 odd 6 5292.2.w.a.521.8 16
63.58 even 3 2268.2.f.b.1133.16 16
63.61 odd 6 756.2.x.a.629.8 16
84.23 even 6 3024.2.cc.c.881.8 16
84.47 odd 6 3024.2.cc.c.881.1 16
252.47 odd 6 1008.2.cc.c.209.4 16
252.79 odd 6 3024.2.cc.c.2897.1 16
252.187 even 6 3024.2.cc.c.2897.8 16
252.191 even 6 1008.2.cc.c.209.5 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.2.x.a.41.4 16 7.5 odd 6
252.2.x.a.41.5 yes 16 7.2 even 3
252.2.x.a.209.4 yes 16 63.2 odd 6
252.2.x.a.209.5 yes 16 63.47 even 6
756.2.x.a.125.1 16 21.5 even 6
756.2.x.a.125.8 16 21.2 odd 6
756.2.x.a.629.1 16 63.16 even 3
756.2.x.a.629.8 16 63.61 odd 6
1008.2.cc.c.209.4 16 252.47 odd 6
1008.2.cc.c.209.5 16 252.191 even 6
1008.2.cc.c.545.4 16 28.23 odd 6
1008.2.cc.c.545.5 16 28.19 even 6
1764.2.w.a.509.2 16 7.6 odd 2 inner
1764.2.w.a.509.7 16 1.1 even 1 trivial
1764.2.w.a.1109.2 16 63.11 odd 6 inner
1764.2.w.a.1109.7 16 63.38 even 6 inner
1764.2.bm.b.1685.1 16 63.20 even 6
1764.2.bm.b.1685.8 16 9.2 odd 6
1764.2.bm.b.1697.1 16 7.4 even 3
1764.2.bm.b.1697.8 16 7.3 odd 6
2268.2.f.b.1133.1 16 63.40 odd 6
2268.2.f.b.1133.2 16 63.23 odd 6
2268.2.f.b.1133.15 16 63.5 even 6
2268.2.f.b.1133.16 16 63.58 even 3
3024.2.cc.c.881.1 16 84.47 odd 6
3024.2.cc.c.881.8 16 84.23 even 6
3024.2.cc.c.2897.1 16 252.79 odd 6
3024.2.cc.c.2897.8 16 252.187 even 6
5292.2.w.a.521.1 16 63.25 even 3
5292.2.w.a.521.8 16 63.52 odd 6
5292.2.w.a.1097.1 16 21.20 even 2
5292.2.w.a.1097.8 16 3.2 odd 2
5292.2.bm.b.2285.1 16 21.11 odd 6
5292.2.bm.b.2285.8 16 21.17 even 6
5292.2.bm.b.4625.1 16 63.34 odd 6
5292.2.bm.b.4625.8 16 9.7 even 3