Properties

Label 1764.2.l.e.949.2
Level $1764$
Weight $2$
Character 1764.949
Analytic conductor $14.086$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1764,2,Mod(949,1764)]
 
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, 2, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1764.949");
 
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.l (of order \(3\), 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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.309123.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 252)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 949.2
Root \(0.500000 + 2.05195i\) of defining polynomial
Character \(\chi\) \(=\) 1764.949
Dual form 1764.2.l.e.961.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.796790 - 1.53790i) q^{3} +2.46050 q^{5} +(-1.73025 + 2.45076i) q^{9} +O(q^{10})\) \(q+(-0.796790 - 1.53790i) q^{3} +2.46050 q^{5} +(-1.73025 + 2.45076i) q^{9} +4.64766 q^{11} +(-3.55408 + 6.15585i) q^{13} +(-1.96050 - 3.78400i) q^{15} +(2.25729 - 3.90975i) q^{17} +(-2.16372 - 3.74766i) q^{19} +5.86693 q^{23} +1.05408 q^{25} +(5.14766 + 0.708209i) q^{27} +(3.48755 + 6.04061i) q^{29} +(3.69076 + 6.39258i) q^{31} +(-3.70321 - 7.14763i) q^{33} +(0.363327 + 0.629301i) q^{37} +(12.2989 + 0.560893i) q^{39} +(-0.136673 + 0.236725i) q^{41} +(2.41741 + 4.18708i) q^{43} +(-4.25729 + 6.03011i) q^{45} +(-1.83628 + 3.18054i) q^{47} +(-7.81138 - 0.356238i) q^{51} +(-2.52704 + 4.37697i) q^{53} +11.4356 q^{55} +(-4.03950 + 6.31367i) q^{57} +(-4.56654 - 7.90947i) q^{59} +(6.90856 - 11.9660i) q^{61} +(-8.74484 + 15.1465i) q^{65} +(0.663715 + 1.14959i) q^{67} +(-4.67471 - 9.02273i) q^{69} +13.5218 q^{71} +(2.16372 - 3.74766i) q^{73} +(-0.839883 - 1.62107i) q^{75} +(-3.21780 + 5.57339i) q^{79} +(-3.01245 - 8.48087i) q^{81} +(-0.742705 - 1.28640i) q^{83} +(5.55408 - 9.61996i) q^{85} +(6.51099 - 10.1766i) q^{87} +(-4.91741 - 8.51721i) q^{89} +(6.89037 - 10.7695i) q^{93} +(-5.32383 - 9.22115i) q^{95} +(0.246304 + 0.426611i) q^{97} +(-8.04163 + 11.3903i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{3} + 2 q^{5} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{3} + 2 q^{5} - 4 q^{9} + 4 q^{11} - 3 q^{13} + q^{15} - 2 q^{17} - 3 q^{19} + 28 q^{23} - 12 q^{25} + 7 q^{27} - q^{29} + 3 q^{31} - 25 q^{33} + 3 q^{37} + 18 q^{39} - 3 q^{43} - 10 q^{45} - 21 q^{47} - 13 q^{51} - 6 q^{53} + 12 q^{55} - 37 q^{57} - 31 q^{59} - 6 q^{61} - 15 q^{65} - 6 q^{67} + 5 q^{69} + 34 q^{71} + 3 q^{73} - 7 q^{75} + 9 q^{79} - 40 q^{81} - 20 q^{83} + 15 q^{85} + 16 q^{87} - 12 q^{89} + 33 q^{93} - 20 q^{95} + 9 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(e\left(\frac{2}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.796790 1.53790i −0.460027 0.887905i
\(4\) 0 0
\(5\) 2.46050 1.10037 0.550186 0.835042i \(-0.314557\pi\)
0.550186 + 0.835042i \(0.314557\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −1.73025 + 2.45076i −0.576751 + 0.816920i
\(10\) 0 0
\(11\) 4.64766 1.40132 0.700662 0.713494i \(-0.252888\pi\)
0.700662 + 0.713494i \(0.252888\pi\)
\(12\) 0 0
\(13\) −3.55408 + 6.15585i −0.985726 + 1.70733i −0.347059 + 0.937843i \(0.612820\pi\)
−0.638667 + 0.769484i \(0.720514\pi\)
\(14\) 0 0
\(15\) −1.96050 3.78400i −0.506200 0.977025i
\(16\) 0 0
\(17\) 2.25729 3.90975i 0.547474 0.948253i −0.450972 0.892538i \(-0.648923\pi\)
0.998447 0.0557155i \(-0.0177440\pi\)
\(18\) 0 0
\(19\) −2.16372 3.74766i −0.496390 0.859773i 0.503601 0.863936i \(-0.332008\pi\)
−0.999991 + 0.00416311i \(0.998675\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.86693 1.22334 0.611669 0.791114i \(-0.290498\pi\)
0.611669 + 0.791114i \(0.290498\pi\)
\(24\) 0 0
\(25\) 1.05408 0.210817
\(26\) 0 0
\(27\) 5.14766 + 0.708209i 0.990668 + 0.136295i
\(28\) 0 0
\(29\) 3.48755 + 6.04061i 0.647621 + 1.12171i 0.983689 + 0.179875i \(0.0575694\pi\)
−0.336068 + 0.941838i \(0.609097\pi\)
\(30\) 0 0
\(31\) 3.69076 + 6.39258i 0.662880 + 1.14814i 0.979856 + 0.199708i \(0.0639992\pi\)
−0.316976 + 0.948434i \(0.602667\pi\)
\(32\) 0 0
\(33\) −3.70321 7.14763i −0.644646 1.24424i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0.363327 + 0.629301i 0.0597306 + 0.103456i 0.894344 0.447379i \(-0.147643\pi\)
−0.834614 + 0.550835i \(0.814309\pi\)
\(38\) 0 0
\(39\) 12.2989 + 0.560893i 1.96940 + 0.0898148i
\(40\) 0 0
\(41\) −0.136673 + 0.236725i −0.0213448 + 0.0369702i −0.876500 0.481401i \(-0.840128\pi\)
0.855156 + 0.518371i \(0.173461\pi\)
\(42\) 0 0
\(43\) 2.41741 + 4.18708i 0.368652 + 0.638524i 0.989355 0.145522i \(-0.0464862\pi\)
−0.620703 + 0.784046i \(0.713153\pi\)
\(44\) 0 0
\(45\) −4.25729 + 6.03011i −0.634640 + 0.898915i
\(46\) 0 0
\(47\) −1.83628 + 3.18054i −0.267850 + 0.463929i −0.968306 0.249766i \(-0.919646\pi\)
0.700457 + 0.713695i \(0.252980\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −7.81138 0.356238i −1.09381 0.0498833i
\(52\) 0 0
\(53\) −2.52704 + 4.37697i −0.347116 + 0.601222i −0.985736 0.168300i \(-0.946172\pi\)
0.638620 + 0.769522i \(0.279506\pi\)
\(54\) 0 0
\(55\) 11.4356 1.54198
\(56\) 0 0
\(57\) −4.03950 + 6.31367i −0.535044 + 0.836266i
\(58\) 0 0
\(59\) −4.56654 7.90947i −0.594513 1.02973i −0.993615 0.112820i \(-0.964012\pi\)
0.399103 0.916906i \(-0.369322\pi\)
\(60\) 0 0
\(61\) 6.90856 11.9660i 0.884550 1.53209i 0.0383215 0.999265i \(-0.487799\pi\)
0.846228 0.532820i \(-0.178868\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −8.74484 + 15.1465i −1.08466 + 1.87869i
\(66\) 0 0
\(67\) 0.663715 + 1.14959i 0.0810857 + 0.140445i 0.903717 0.428131i \(-0.140828\pi\)
−0.822631 + 0.568576i \(0.807495\pi\)
\(68\) 0 0
\(69\) −4.67471 9.02273i −0.562768 1.08621i
\(70\) 0 0
\(71\) 13.5218 1.60474 0.802370 0.596826i \(-0.203572\pi\)
0.802370 + 0.596826i \(0.203572\pi\)
\(72\) 0 0
\(73\) 2.16372 3.74766i 0.253244 0.438631i −0.711173 0.703017i \(-0.751836\pi\)
0.964417 + 0.264386i \(0.0851692\pi\)
\(74\) 0 0
\(75\) −0.839883 1.62107i −0.0969814 0.187185i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −3.21780 + 5.57339i −0.362031 + 0.627056i −0.988295 0.152555i \(-0.951250\pi\)
0.626264 + 0.779611i \(0.284583\pi\)
\(80\) 0 0
\(81\) −3.01245 8.48087i −0.334717 0.942319i
\(82\) 0 0
\(83\) −0.742705 1.28640i −0.0815225 0.141201i 0.822382 0.568936i \(-0.192645\pi\)
−0.903904 + 0.427735i \(0.859312\pi\)
\(84\) 0 0
\(85\) 5.55408 9.61996i 0.602425 1.04343i
\(86\) 0 0
\(87\) 6.51099 10.1766i 0.698051 1.09104i
\(88\) 0 0
\(89\) −4.91741 8.51721i −0.521245 0.902822i −0.999695 0.0247073i \(-0.992135\pi\)
0.478450 0.878115i \(-0.341199\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 6.89037 10.7695i 0.714498 1.11675i
\(94\) 0 0
\(95\) −5.32383 9.22115i −0.546214 0.946070i
\(96\) 0 0
\(97\) 0.246304 + 0.426611i 0.0250084 + 0.0433158i 0.878259 0.478186i \(-0.158705\pi\)
−0.853250 + 0.521502i \(0.825372\pi\)
\(98\) 0 0
\(99\) −8.04163 + 11.3903i −0.808214 + 1.14477i
\(100\) 0 0
\(101\) 3.40642 0.338952 0.169476 0.985534i \(-0.445793\pi\)
0.169476 + 0.985534i \(0.445793\pi\)
\(102\) 0 0
\(103\) 5.16225 0.508652 0.254326 0.967119i \(-0.418146\pi\)
0.254326 + 0.967119i \(0.418146\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.88151 + 4.99093i 0.278567 + 0.482491i 0.971029 0.238963i \(-0.0768074\pi\)
−0.692462 + 0.721454i \(0.743474\pi\)
\(108\) 0 0
\(109\) 4.49115 7.77889i 0.430174 0.745083i −0.566714 0.823914i \(-0.691786\pi\)
0.996888 + 0.0788317i \(0.0251190\pi\)
\(110\) 0 0
\(111\) 0.678304 1.06018i 0.0643818 0.100628i
\(112\) 0 0
\(113\) 0.679767 1.17739i 0.0639471 0.110760i −0.832279 0.554356i \(-0.812964\pi\)
0.896226 + 0.443597i \(0.146298\pi\)
\(114\) 0 0
\(115\) 14.4356 1.34613
\(116\) 0 0
\(117\) −8.93706 19.3614i −0.826232 1.78996i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 10.6008 0.963707
\(122\) 0 0
\(123\) 0.472958 + 0.0215693i 0.0426452 + 0.00194484i
\(124\) 0 0
\(125\) −9.70895 −0.868394
\(126\) 0 0
\(127\) −0.820039 −0.0727667 −0.0363833 0.999338i \(-0.511584\pi\)
−0.0363833 + 0.999338i \(0.511584\pi\)
\(128\) 0 0
\(129\) 4.51313 7.05395i 0.397359 0.621066i
\(130\) 0 0
\(131\) 7.78794 0.680435 0.340218 0.940347i \(-0.389499\pi\)
0.340218 + 0.940347i \(0.389499\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 12.6659 + 1.74255i 1.09010 + 0.149975i
\(136\) 0 0
\(137\) −2.99280 −0.255692 −0.127846 0.991794i \(-0.540806\pi\)
−0.127846 + 0.991794i \(0.540806\pi\)
\(138\) 0 0
\(139\) −3.16372 + 5.47972i −0.268343 + 0.464783i −0.968434 0.249270i \(-0.919809\pi\)
0.700091 + 0.714053i \(0.253143\pi\)
\(140\) 0 0
\(141\) 6.35447 + 0.289796i 0.535143 + 0.0244052i
\(142\) 0 0
\(143\) −16.5182 + 28.6103i −1.38132 + 2.39252i
\(144\) 0 0
\(145\) 8.58113 + 14.8629i 0.712624 + 1.23430i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −4.38151 −0.358948 −0.179474 0.983763i \(-0.557440\pi\)
−0.179474 + 0.983763i \(0.557440\pi\)
\(150\) 0 0
\(151\) 6.60078 0.537164 0.268582 0.963257i \(-0.413445\pi\)
0.268582 + 0.963257i \(0.413445\pi\)
\(152\) 0 0
\(153\) 5.67617 + 12.2969i 0.458891 + 0.994149i
\(154\) 0 0
\(155\) 9.08113 + 15.7290i 0.729414 + 1.26338i
\(156\) 0 0
\(157\) 2.89037 + 5.00627i 0.230677 + 0.399544i 0.958007 0.286743i \(-0.0925727\pi\)
−0.727331 + 0.686287i \(0.759239\pi\)
\(158\) 0 0
\(159\) 8.74484 + 0.398809i 0.693511 + 0.0316276i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −3.66372 6.34574i −0.286964 0.497037i 0.686119 0.727489i \(-0.259313\pi\)
−0.973084 + 0.230452i \(0.925979\pi\)
\(164\) 0 0
\(165\) −9.11177 17.5868i −0.709350 1.36913i
\(166\) 0 0
\(167\) 6.01459 10.4176i 0.465423 0.806136i −0.533798 0.845612i \(-0.679236\pi\)
0.999221 + 0.0394762i \(0.0125689\pi\)
\(168\) 0 0
\(169\) −18.7630 32.4985i −1.44331 2.49989i
\(170\) 0 0
\(171\) 12.9284 + 1.18166i 0.988660 + 0.0903637i
\(172\) 0 0
\(173\) −2.44951 + 4.24268i −0.186233 + 0.322565i −0.943991 0.329970i \(-0.892961\pi\)
0.757758 + 0.652535i \(0.226295\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −8.52538 + 13.3251i −0.640807 + 1.00157i
\(178\) 0 0
\(179\) 0.890369 1.54216i 0.0665493 0.115267i −0.830831 0.556525i \(-0.812134\pi\)
0.897380 + 0.441258i \(0.145468\pi\)
\(180\) 0 0
\(181\) −16.9430 −1.25936 −0.629681 0.776854i \(-0.716815\pi\)
−0.629681 + 0.776854i \(0.716815\pi\)
\(182\) 0 0
\(183\) −23.9071 1.09028i −1.76726 0.0805961i
\(184\) 0 0
\(185\) 0.893968 + 1.54840i 0.0657258 + 0.113840i
\(186\) 0 0
\(187\) 10.4911 18.1712i 0.767189 1.32881i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 2.74484 4.75420i 0.198610 0.344002i −0.749468 0.662040i \(-0.769691\pi\)
0.948078 + 0.318038i \(0.103024\pi\)
\(192\) 0 0
\(193\) 2.75370 + 4.76954i 0.198215 + 0.343319i 0.947950 0.318420i \(-0.103152\pi\)
−0.749734 + 0.661739i \(0.769819\pi\)
\(194\) 0 0
\(195\) 30.2616 + 1.38008i 2.16708 + 0.0988296i
\(196\) 0 0
\(197\) 11.6300 0.828600 0.414300 0.910140i \(-0.364026\pi\)
0.414300 + 0.910140i \(0.364026\pi\)
\(198\) 0 0
\(199\) 2.07373 3.59181i 0.147003 0.254617i −0.783115 0.621876i \(-0.786371\pi\)
0.930118 + 0.367260i \(0.119704\pi\)
\(200\) 0 0
\(201\) 1.23911 1.93671i 0.0873999 0.136605i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −0.336285 + 0.582462i −0.0234871 + 0.0406809i
\(206\) 0 0
\(207\) −10.1513 + 14.3784i −0.705561 + 0.999370i
\(208\) 0 0
\(209\) −10.0562 17.4179i −0.695603 1.20482i
\(210\) 0 0
\(211\) 13.6082 23.5700i 0.936825 1.62263i 0.165478 0.986213i \(-0.447083\pi\)
0.771347 0.636415i \(-0.219583\pi\)
\(212\) 0 0
\(213\) −10.7740 20.7951i −0.738224 1.42486i
\(214\) 0 0
\(215\) 5.94805 + 10.3023i 0.405654 + 0.702613i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −7.48755 0.341470i −0.505962 0.0230744i
\(220\) 0 0
\(221\) 16.0452 + 27.7912i 1.07932 + 1.86944i
\(222\) 0 0
\(223\) 1.60817 + 2.78543i 0.107691 + 0.186526i 0.914834 0.403829i \(-0.132321\pi\)
−0.807144 + 0.590355i \(0.798988\pi\)
\(224\) 0 0
\(225\) −1.82383 + 2.58331i −0.121589 + 0.172221i
\(226\) 0 0
\(227\) 15.9459 1.05837 0.529184 0.848507i \(-0.322498\pi\)
0.529184 + 0.848507i \(0.322498\pi\)
\(228\) 0 0
\(229\) −1.21634 −0.0803778 −0.0401889 0.999192i \(-0.512796\pi\)
−0.0401889 + 0.999192i \(0.512796\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −9.98608 17.2964i −0.654210 1.13313i −0.982091 0.188406i \(-0.939668\pi\)
0.327881 0.944719i \(-0.393665\pi\)
\(234\) 0 0
\(235\) −4.51819 + 7.82573i −0.294734 + 0.510494i
\(236\) 0 0
\(237\) 11.1352 + 0.507822i 0.723310 + 0.0329866i
\(238\) 0 0
\(239\) −3.00739 + 5.20896i −0.194532 + 0.336939i −0.946747 0.321978i \(-0.895652\pi\)
0.752215 + 0.658918i \(0.228985\pi\)
\(240\) 0 0
\(241\) −18.6156 −1.19913 −0.599567 0.800325i \(-0.704660\pi\)
−0.599567 + 0.800325i \(0.704660\pi\)
\(242\) 0 0
\(243\) −10.6424 + 11.3903i −0.682711 + 0.730689i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 30.7601 1.95722
\(248\) 0 0
\(249\) −1.38658 + 2.16720i −0.0878707 + 0.137341i
\(250\) 0 0
\(251\) 6.99707 0.441651 0.220826 0.975313i \(-0.429125\pi\)
0.220826 + 0.975313i \(0.429125\pi\)
\(252\) 0 0
\(253\) 27.2675 1.71429
\(254\) 0 0
\(255\) −19.2199 0.876526i −1.20360 0.0548902i
\(256\) 0 0
\(257\) −17.7778 −1.10895 −0.554475 0.832201i \(-0.687081\pi\)
−0.554475 + 0.832201i \(0.687081\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −20.8384 1.90464i −1.28987 0.117894i
\(262\) 0 0
\(263\) −27.1986 −1.67714 −0.838570 0.544794i \(-0.816608\pi\)
−0.838570 + 0.544794i \(0.816608\pi\)
\(264\) 0 0
\(265\) −6.21780 + 10.7695i −0.381956 + 0.661568i
\(266\) 0 0
\(267\) −9.18044 + 14.3489i −0.561834 + 0.878138i
\(268\) 0 0
\(269\) −11.9481 + 20.6946i −0.728486 + 1.26177i 0.229038 + 0.973418i \(0.426442\pi\)
−0.957523 + 0.288356i \(0.906891\pi\)
\(270\) 0 0
\(271\) −6.13667 10.6290i −0.372776 0.645668i 0.617215 0.786794i \(-0.288261\pi\)
−0.989992 + 0.141127i \(0.954927\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 4.89903 0.295423
\(276\) 0 0
\(277\) 12.7807 0.767920 0.383960 0.923350i \(-0.374560\pi\)
0.383960 + 0.923350i \(0.374560\pi\)
\(278\) 0 0
\(279\) −22.0526 2.01561i −1.32026 0.120672i
\(280\) 0 0
\(281\) 14.2573 + 24.6944i 0.850519 + 1.47314i 0.880741 + 0.473599i \(0.157045\pi\)
−0.0302219 + 0.999543i \(0.509621\pi\)
\(282\) 0 0
\(283\) −0.363327 0.629301i −0.0215975 0.0374080i 0.855025 0.518587i \(-0.173542\pi\)
−0.876622 + 0.481179i \(0.840209\pi\)
\(284\) 0 0
\(285\) −9.93920 + 15.5348i −0.588747 + 0.920203i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −1.69076 2.92848i −0.0994563 0.172263i
\(290\) 0 0
\(291\) 0.459831 0.718710i 0.0269558 0.0421315i
\(292\) 0 0
\(293\) −12.7901 + 22.1531i −0.747204 + 1.29420i 0.201954 + 0.979395i \(0.435271\pi\)
−0.949158 + 0.314800i \(0.898062\pi\)
\(294\) 0 0
\(295\) −11.2360 19.4613i −0.654184 1.13308i
\(296\) 0 0
\(297\) 23.9246 + 3.29152i 1.38825 + 0.190993i
\(298\) 0 0
\(299\) −20.8515 + 36.1159i −1.20588 + 2.08864i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −2.71420 5.23872i −0.155927 0.300957i
\(304\) 0 0
\(305\) 16.9985 29.4423i 0.973333 1.68586i
\(306\) 0 0
\(307\) −6.23405 −0.355796 −0.177898 0.984049i \(-0.556930\pi\)
−0.177898 + 0.984049i \(0.556930\pi\)
\(308\) 0 0
\(309\) −4.11323 7.93901i −0.233993 0.451635i
\(310\) 0 0
\(311\) −14.6192 25.3211i −0.828976 1.43583i −0.898842 0.438273i \(-0.855590\pi\)
0.0698655 0.997556i \(-0.477743\pi\)
\(312\) 0 0
\(313\) −14.2434 + 24.6703i −0.805083 + 1.39445i 0.111151 + 0.993803i \(0.464546\pi\)
−0.916235 + 0.400642i \(0.868787\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −0.809243 + 1.40165i −0.0454516 + 0.0787245i −0.887856 0.460121i \(-0.847806\pi\)
0.842405 + 0.538846i \(0.181139\pi\)
\(318\) 0 0
\(319\) 16.2089 + 28.0747i 0.907527 + 1.57188i
\(320\) 0 0
\(321\) 5.37957 8.40819i 0.300258 0.469300i
\(322\) 0 0
\(323\) −19.5366 −1.08704
\(324\) 0 0
\(325\) −3.74630 + 6.48879i −0.207808 + 0.359933i
\(326\) 0 0
\(327\) −15.5416 0.708777i −0.859454 0.0391954i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 6.99115 12.1090i 0.384268 0.665572i −0.607399 0.794397i \(-0.707787\pi\)
0.991667 + 0.128825i \(0.0411205\pi\)
\(332\) 0 0
\(333\) −2.17091 0.198422i −0.118965 0.0108734i
\(334\) 0 0
\(335\) 1.63307 + 2.82857i 0.0892244 + 0.154541i
\(336\) 0 0
\(337\) −13.8619 + 24.0095i −0.755104 + 1.30788i 0.190219 + 0.981742i \(0.439080\pi\)
−0.945323 + 0.326137i \(0.894253\pi\)
\(338\) 0 0
\(339\) −2.35234 0.107278i −0.127761 0.00582656i
\(340\) 0 0
\(341\) 17.1534 + 29.7106i 0.928909 + 1.60892i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −11.5021 22.2005i −0.619254 1.19523i
\(346\) 0 0
\(347\) −3.76449 6.52029i −0.202089 0.350028i 0.747113 0.664697i \(-0.231440\pi\)
−0.949201 + 0.314670i \(0.898106\pi\)
\(348\) 0 0
\(349\) 15.0541 + 26.0744i 0.805827 + 1.39573i 0.915732 + 0.401791i \(0.131612\pi\)
−0.109905 + 0.993942i \(0.535055\pi\)
\(350\) 0 0
\(351\) −22.6549 + 29.1712i −1.20923 + 1.55705i
\(352\) 0 0
\(353\) −20.3638 −1.08386 −0.541928 0.840425i \(-0.682305\pi\)
−0.541928 + 0.840425i \(0.682305\pi\)
\(354\) 0 0
\(355\) 33.2704 1.76581
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 8.01313 + 13.8791i 0.422917 + 0.732513i 0.996223 0.0868277i \(-0.0276730\pi\)
−0.573307 + 0.819341i \(0.694340\pi\)
\(360\) 0 0
\(361\) 0.136673 0.236725i 0.00719332 0.0124592i
\(362\) 0 0
\(363\) −8.44659 16.3029i −0.443331 0.855680i
\(364\) 0 0
\(365\) 5.32383 9.22115i 0.278662 0.482657i
\(366\) 0 0
\(367\) −13.5979 −0.709802 −0.354901 0.934904i \(-0.615485\pi\)
−0.354901 + 0.934904i \(0.615485\pi\)
\(368\) 0 0
\(369\) −0.343677 0.744547i −0.0178911 0.0387595i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −21.9282 −1.13540 −0.567700 0.823236i \(-0.692167\pi\)
−0.567700 + 0.823236i \(0.692167\pi\)
\(374\) 0 0
\(375\) 7.73599 + 14.9314i 0.399485 + 0.771052i
\(376\) 0 0
\(377\) −49.5801 −2.55351
\(378\) 0 0
\(379\) −29.7965 −1.53054 −0.765271 0.643708i \(-0.777395\pi\)
−0.765271 + 0.643708i \(0.777395\pi\)
\(380\) 0 0
\(381\) 0.653398 + 1.26113i 0.0334746 + 0.0646099i
\(382\) 0 0
\(383\) −0.0219809 −0.00112317 −0.000561587 1.00000i \(-0.500179\pi\)
−0.000561587 1.00000i \(0.500179\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −14.4443 1.32021i −0.734243 0.0671099i
\(388\) 0 0
\(389\) −35.3566 −1.79265 −0.896326 0.443396i \(-0.853773\pi\)
−0.896326 + 0.443396i \(0.853773\pi\)
\(390\) 0 0
\(391\) 13.2434 22.9382i 0.669746 1.16003i
\(392\) 0 0
\(393\) −6.20535 11.9770i −0.313018 0.604162i
\(394\) 0 0
\(395\) −7.91741 + 13.7134i −0.398368 + 0.689994i
\(396\) 0 0
\(397\) 8.47150 + 14.6731i 0.425172 + 0.736420i 0.996436 0.0843464i \(-0.0268802\pi\)
−0.571264 + 0.820766i \(0.693547\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −2.96362 −0.147996 −0.0739982 0.997258i \(-0.523576\pi\)
−0.0739982 + 0.997258i \(0.523576\pi\)
\(402\) 0 0
\(403\) −52.4690 −2.61367
\(404\) 0 0
\(405\) −7.41216 20.8672i −0.368313 1.03690i
\(406\) 0 0
\(407\) 1.68862 + 2.92478i 0.0837018 + 0.144976i
\(408\) 0 0
\(409\) 7.32743 + 12.6915i 0.362318 + 0.627553i 0.988342 0.152251i \(-0.0486521\pi\)
−0.626024 + 0.779804i \(0.715319\pi\)
\(410\) 0 0
\(411\) 2.38463 + 4.60262i 0.117625 + 0.227031i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −1.82743 3.16520i −0.0897050 0.155374i
\(416\) 0 0
\(417\) 10.9481 + 0.499286i 0.536128 + 0.0244502i
\(418\) 0 0
\(419\) 12.6352 21.8848i 0.617270 1.06914i −0.372711 0.927947i \(-0.621572\pi\)
0.989982 0.141196i \(-0.0450949\pi\)
\(420\) 0 0
\(421\) 7.99854 + 13.8539i 0.389825 + 0.675196i 0.992426 0.122846i \(-0.0392022\pi\)
−0.602601 + 0.798043i \(0.705869\pi\)
\(422\) 0 0
\(423\) −4.61750 10.0034i −0.224511 0.486383i
\(424\) 0 0
\(425\) 2.37938 4.12120i 0.115417 0.199908i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 57.1613 + 2.60684i 2.75977 + 0.125860i
\(430\) 0 0
\(431\) −6.51673 + 11.2873i −0.313900 + 0.543690i −0.979203 0.202883i \(-0.934969\pi\)
0.665303 + 0.746573i \(0.268302\pi\)
\(432\) 0 0
\(433\) 23.5467 1.13158 0.565791 0.824549i \(-0.308571\pi\)
0.565791 + 0.824549i \(0.308571\pi\)
\(434\) 0 0
\(435\) 16.0203 25.0395i 0.768116 1.20055i
\(436\) 0 0
\(437\) −12.6944 21.9873i −0.607253 1.05179i
\(438\) 0 0
\(439\) −3.35447 + 5.81012i −0.160100 + 0.277302i −0.934904 0.354900i \(-0.884515\pi\)
0.774804 + 0.632201i \(0.217848\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −17.6228 + 30.5235i −0.837282 + 1.45022i 0.0548760 + 0.998493i \(0.482524\pi\)
−0.892158 + 0.451723i \(0.850810\pi\)
\(444\) 0 0
\(445\) −12.0993 20.9566i −0.573562 0.993439i
\(446\) 0 0
\(447\) 3.49115 + 6.73832i 0.165126 + 0.318711i
\(448\) 0 0
\(449\) −12.9387 −0.610616 −0.305308 0.952254i \(-0.598759\pi\)
−0.305308 + 0.952254i \(0.598759\pi\)
\(450\) 0 0
\(451\) −0.635211 + 1.10022i −0.0299109 + 0.0518072i
\(452\) 0 0
\(453\) −5.25943 10.1513i −0.247110 0.476950i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 14.5993 25.2868i 0.682927 1.18286i −0.291156 0.956675i \(-0.594040\pi\)
0.974083 0.226189i \(-0.0726267\pi\)
\(458\) 0 0
\(459\) 14.3887 18.5274i 0.671608 0.864787i
\(460\) 0 0
\(461\) 9.34348 + 16.1834i 0.435169 + 0.753735i 0.997309 0.0733066i \(-0.0233552\pi\)
−0.562140 + 0.827042i \(0.690022\pi\)
\(462\) 0 0
\(463\) 19.1249 33.1253i 0.888809 1.53946i 0.0475247 0.998870i \(-0.484867\pi\)
0.841285 0.540593i \(-0.181800\pi\)
\(464\) 0 0
\(465\) 16.9538 26.4985i 0.786213 1.22884i
\(466\) 0 0
\(467\) 7.64387 + 13.2396i 0.353716 + 0.612654i 0.986897 0.161349i \(-0.0515846\pi\)
−0.633181 + 0.774004i \(0.718251\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 5.39610 8.43403i 0.248639 0.388620i
\(472\) 0 0
\(473\) 11.2353 + 19.4601i 0.516600 + 0.894778i
\(474\) 0 0
\(475\) −2.28074 3.95035i −0.104647 0.181255i
\(476\) 0 0
\(477\) −6.35447 13.7664i −0.290951 0.630322i
\(478\) 0 0
\(479\) 11.0321 0.504070 0.252035 0.967718i \(-0.418900\pi\)
0.252035 + 0.967718i \(0.418900\pi\)
\(480\) 0 0
\(481\) −5.16518 −0.235512
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0.606032 + 1.04968i 0.0275185 + 0.0476635i
\(486\) 0 0
\(487\) −8.30039 + 14.3767i −0.376126 + 0.651470i −0.990495 0.137549i \(-0.956078\pi\)
0.614368 + 0.789019i \(0.289411\pi\)
\(488\) 0 0
\(489\) −6.83988 + 10.6906i −0.309310 + 0.483447i
\(490\) 0 0
\(491\) 13.3633 23.1460i 0.603079 1.04456i −0.389273 0.921122i \(-0.627274\pi\)
0.992352 0.123440i \(-0.0393928\pi\)
\(492\) 0 0
\(493\) 31.4897 1.41822
\(494\) 0 0
\(495\) −19.7865 + 28.0259i −0.889336 + 1.25967i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 1.23697 0.0553744 0.0276872 0.999617i \(-0.491186\pi\)
0.0276872 + 0.999617i \(0.491186\pi\)
\(500\) 0 0
\(501\) −20.8135 0.949201i −0.929879 0.0424072i
\(502\) 0 0
\(503\) −1.07179 −0.0477889 −0.0238944 0.999714i \(-0.507607\pi\)
−0.0238944 + 0.999714i \(0.507607\pi\)
\(504\) 0 0
\(505\) 8.38151 0.372973
\(506\) 0 0
\(507\) −35.0292 + 54.7501i −1.55570 + 2.43154i
\(508\) 0 0
\(509\) 20.0689 0.889537 0.444768 0.895646i \(-0.353286\pi\)
0.444768 + 0.895646i \(0.353286\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −8.48395 20.8241i −0.374575 0.919406i
\(514\) 0 0
\(515\) 12.7017 0.559706
\(516\) 0 0
\(517\) −8.53443 + 14.7821i −0.375344 + 0.650115i
\(518\) 0 0
\(519\) 8.47656 + 0.386574i 0.372080 + 0.0169687i
\(520\) 0 0
\(521\) 15.4430 26.7480i 0.676570 1.17185i −0.299438 0.954116i \(-0.596799\pi\)
0.976007 0.217737i \(-0.0698676\pi\)
\(522\) 0 0
\(523\) 3.69961 + 6.40792i 0.161773 + 0.280199i 0.935505 0.353315i \(-0.114946\pi\)
−0.773732 + 0.633513i \(0.781612\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 33.3245 1.45164
\(528\) 0 0
\(529\) 11.4208 0.496557
\(530\) 0 0
\(531\) 27.2855 + 2.49390i 1.18409 + 0.108226i
\(532\) 0 0
\(533\) −0.971495 1.68268i −0.0420801 0.0728849i
\(534\) 0 0
\(535\) 7.08998 + 12.2802i 0.306527 + 0.530920i
\(536\) 0 0
\(537\) −3.08113 0.140515i −0.132960 0.00606367i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −11.3348 19.6325i −0.487322 0.844067i 0.512572 0.858644i \(-0.328693\pi\)
−0.999894 + 0.0145779i \(0.995360\pi\)
\(542\) 0 0
\(543\) 13.5000 + 26.0566i 0.579340 + 1.11819i
\(544\) 0 0
\(545\) 11.0505 19.1400i 0.473351 0.819868i
\(546\) 0 0
\(547\) 3.07373 + 5.32386i 0.131423 + 0.227632i 0.924225 0.381847i \(-0.124712\pi\)
−0.792802 + 0.609479i \(0.791379\pi\)
\(548\) 0 0
\(549\) 17.3722 + 37.6354i 0.741427 + 1.60624i
\(550\) 0 0
\(551\) 15.0921 26.1403i 0.642946 1.11361i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 1.66897 2.60858i 0.0708439 0.110728i
\(556\) 0 0
\(557\) −14.8370 + 25.6984i −0.628662 + 1.08887i 0.359158 + 0.933277i \(0.383064\pi\)
−0.987820 + 0.155598i \(0.950270\pi\)
\(558\) 0 0
\(559\) −34.3667 −1.45356
\(560\) 0 0
\(561\) −36.3047 1.65568i −1.53278 0.0699027i
\(562\) 0 0
\(563\) 14.6555 + 25.3841i 0.617657 + 1.06981i 0.989912 + 0.141683i \(0.0452514\pi\)
−0.372255 + 0.928131i \(0.621415\pi\)
\(564\) 0 0
\(565\) 1.67257 2.89698i 0.0703655 0.121877i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 18.4430 31.9442i 0.773170 1.33917i −0.162647 0.986684i \(-0.552003\pi\)
0.935817 0.352486i \(-0.114664\pi\)
\(570\) 0 0
\(571\) −16.1893 28.0407i −0.677501 1.17347i −0.975731 0.218972i \(-0.929730\pi\)
0.298230 0.954494i \(-0.403604\pi\)
\(572\) 0 0
\(573\) −9.49854 0.433181i −0.396807 0.0180964i
\(574\) 0 0
\(575\) 6.18423 0.257900
\(576\) 0 0
\(577\) −11.5093 + 19.9348i −0.479140 + 0.829895i −0.999714 0.0239220i \(-0.992385\pi\)
0.520574 + 0.853817i \(0.325718\pi\)
\(578\) 0 0
\(579\) 5.14095 8.03522i 0.213650 0.333932i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −11.7448 + 20.3427i −0.486422 + 0.842507i
\(584\) 0 0
\(585\) −21.9897 47.6388i −0.909161 1.96962i
\(586\) 0 0
\(587\) −2.87052 4.97189i −0.118479 0.205212i 0.800686 0.599084i \(-0.204469\pi\)
−0.919165 + 0.393872i \(0.871135\pi\)
\(588\) 0 0
\(589\) 15.9715 27.6634i 0.658094 1.13985i
\(590\) 0 0
\(591\) −9.26663 17.8857i −0.381178 0.735718i
\(592\) 0 0
\(593\) −13.8727 24.0282i −0.569682 0.986718i −0.996597 0.0824263i \(-0.973733\pi\)
0.426915 0.904292i \(-0.359600\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −7.17617 0.327269i −0.293701 0.0133942i
\(598\) 0 0
\(599\) −2.05408 3.55778i −0.0839276 0.145367i 0.821006 0.570919i \(-0.193413\pi\)
−0.904934 + 0.425552i \(0.860080\pi\)
\(600\) 0 0
\(601\) −7.80924 13.5260i −0.318546 0.551737i 0.661639 0.749822i \(-0.269861\pi\)
−0.980185 + 0.198085i \(0.936528\pi\)
\(602\) 0 0
\(603\) −3.96576 0.362471i −0.161498 0.0147610i
\(604\) 0 0
\(605\) 26.0833 1.06044
\(606\) 0 0
\(607\) −0.561476 −0.0227896 −0.0113948 0.999935i \(-0.503627\pi\)
−0.0113948 + 0.999935i \(0.503627\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −13.0526 22.6078i −0.528053 0.914614i
\(612\) 0 0
\(613\) 10.1008 17.4951i 0.407967 0.706619i −0.586695 0.809808i \(-0.699571\pi\)
0.994662 + 0.103189i \(0.0329047\pi\)
\(614\) 0 0
\(615\) 1.16372 + 0.0530713i 0.0469255 + 0.00214004i
\(616\) 0 0
\(617\) 11.4569 19.8439i 0.461238 0.798887i −0.537785 0.843082i \(-0.680739\pi\)
0.999023 + 0.0441948i \(0.0140722\pi\)
\(618\) 0 0
\(619\) 39.7031 1.59580 0.797901 0.602788i \(-0.205944\pi\)
0.797901 + 0.602788i \(0.205944\pi\)
\(620\) 0 0
\(621\) 30.2010 + 4.15501i 1.21192 + 0.166735i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −29.1593 −1.16637
\(626\) 0 0
\(627\) −18.7742 + 29.3438i −0.749770 + 1.17188i
\(628\) 0 0
\(629\) 3.28054 0.130804
\(630\) 0 0
\(631\) −31.0364 −1.23554 −0.617769 0.786359i \(-0.711963\pi\)
−0.617769 + 0.786359i \(0.711963\pi\)
\(632\) 0 0
\(633\) −47.0911 2.14759i −1.87170 0.0853592i
\(634\) 0 0
\(635\) −2.01771 −0.0800703
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −23.3961 + 33.1387i −0.925536 + 1.31095i
\(640\) 0 0
\(641\) −29.5864 −1.16859 −0.584296 0.811541i \(-0.698629\pi\)
−0.584296 + 0.811541i \(0.698629\pi\)
\(642\) 0 0
\(643\) −12.8442 + 22.2467i −0.506524 + 0.877325i 0.493447 + 0.869776i \(0.335737\pi\)
−0.999972 + 0.00754978i \(0.997597\pi\)
\(644\) 0 0
\(645\) 11.1046 17.3563i 0.437242 0.683403i
\(646\) 0 0
\(647\) −8.50885 + 14.7378i −0.334518 + 0.579401i −0.983392 0.181494i \(-0.941907\pi\)
0.648874 + 0.760895i \(0.275240\pi\)
\(648\) 0 0
\(649\) −21.2237 36.7606i −0.833104 1.44298i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1.47102 0.0575653 0.0287827 0.999586i \(-0.490837\pi\)
0.0287827 + 0.999586i \(0.490837\pi\)
\(654\) 0 0
\(655\) 19.1623 0.748731
\(656\) 0 0
\(657\) 5.44085 + 11.7872i 0.212268 + 0.459861i
\(658\) 0 0
\(659\) 20.7003 + 35.8539i 0.806369 + 1.39667i 0.915363 + 0.402629i \(0.131904\pi\)
−0.108995 + 0.994042i \(0.534763\pi\)
\(660\) 0 0
\(661\) −19.1352 33.1432i −0.744273 1.28912i −0.950533 0.310622i \(-0.899463\pi\)
0.206260 0.978497i \(-0.433871\pi\)
\(662\) 0 0
\(663\) 29.9552 46.8196i 1.16337 1.81832i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 20.4612 + 35.4398i 0.792260 + 1.37223i
\(668\) 0 0
\(669\) 3.00233 4.69260i 0.116077 0.181426i
\(670\) 0 0
\(671\) 32.1086 55.6138i 1.23954 2.14695i
\(672\) 0 0
\(673\) 15.2448 + 26.4048i 0.587645 + 1.01783i 0.994540 + 0.104357i \(0.0332783\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(674\) 0 0
\(675\) 5.42607 + 0.746512i 0.208850 + 0.0287333i
\(676\) 0 0
\(677\) −22.4626 + 38.9064i −0.863309 + 1.49530i 0.00540665 + 0.999985i \(0.498279\pi\)
−0.868716 + 0.495310i \(0.835054\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −12.7055 24.5232i −0.486877 0.939730i
\(682\) 0 0
\(683\) 24.1986 41.9133i 0.925935 1.60377i 0.135884 0.990725i \(-0.456613\pi\)
0.790051 0.613041i \(-0.210054\pi\)
\(684\) 0 0
\(685\) −7.36381 −0.281357
\(686\) 0 0
\(687\) 0.969165 + 1.87060i 0.0369759 + 0.0713679i
\(688\) 0 0
\(689\) −17.9626 31.1122i −0.684322 1.18528i
\(690\) 0 0
\(691\) −9.19076 + 15.9189i −0.349633 + 0.605582i −0.986184 0.165652i \(-0.947027\pi\)
0.636551 + 0.771234i \(0.280360\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −7.78434 + 13.4829i −0.295277 + 0.511434i
\(696\) 0 0
\(697\) 0.617023 + 1.06871i 0.0233714 + 0.0404805i
\(698\) 0 0
\(699\) −18.6433 + 29.1392i −0.705153 + 1.10214i
\(700\) 0 0
\(701\) −27.0292 −1.02088 −0.510439 0.859914i \(-0.670517\pi\)
−0.510439 + 0.859914i \(0.670517\pi\)
\(702\) 0 0
\(703\) 1.57227 2.72325i 0.0592994 0.102710i
\(704\) 0 0
\(705\) 15.6352 + 0.713045i 0.588856 + 0.0268548i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −2.49261 + 4.31732i −0.0936119 + 0.162141i −0.909028 0.416734i \(-0.863175\pi\)
0.815417 + 0.578875i \(0.196508\pi\)
\(710\) 0 0
\(711\) −8.09144 17.5294i −0.303453 0.657405i
\(712\) 0 0
\(713\) 21.6534 + 37.5048i 0.810926 + 1.40457i
\(714\) 0 0
\(715\) −40.6431 + 70.3959i −1.51997 + 2.63266i
\(716\) 0 0
\(717\) 10.4071 + 0.474616i 0.388660 + 0.0177249i
\(718\) 0 0
\(719\) −7.84708 13.5915i −0.292647 0.506879i 0.681788 0.731550i \(-0.261203\pi\)
−0.974435 + 0.224671i \(0.927869\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 14.8327 + 28.6288i 0.551634 + 1.06472i
\(724\) 0 0
\(725\) 3.67617 + 6.36731i 0.136529 + 0.236476i
\(726\) 0 0
\(727\) −10.9071 18.8916i −0.404522 0.700652i 0.589744 0.807590i \(-0.299229\pi\)
−0.994266 + 0.106938i \(0.965895\pi\)
\(728\) 0 0
\(729\) 25.9969 + 7.29124i 0.962847 + 0.270046i
\(730\) 0 0
\(731\) 21.8272 0.807309
\(732\) 0 0
\(733\) −24.0148 −0.887006 −0.443503 0.896273i \(-0.646264\pi\)
−0.443503 + 0.896273i \(0.646264\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 3.08472 + 5.34290i 0.113627 + 0.196808i
\(738\) 0 0
\(739\) −9.35447 + 16.2024i −0.344110 + 0.596016i −0.985192 0.171457i \(-0.945153\pi\)
0.641082 + 0.767473i \(0.278486\pi\)
\(740\) 0 0
\(741\) −24.5093 47.3059i −0.900373 1.73782i
\(742\) 0 0
\(743\) 20.1534 34.9067i 0.739356 1.28060i −0.213429 0.976959i \(-0.568463\pi\)
0.952785 0.303644i \(-0.0982035\pi\)
\(744\) 0 0
\(745\) −10.7807 −0.394976
\(746\) 0 0
\(747\) 4.43773 + 0.405610i 0.162368 + 0.0148405i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −21.1259 −0.770894 −0.385447 0.922730i \(-0.625953\pi\)
−0.385447 + 0.922730i \(0.625953\pi\)
\(752\) 0 0
\(753\) −5.57520 10.7608i −0.203171 0.392145i
\(754\) 0 0
\(755\) 16.2412 0.591079
\(756\) 0 0
\(757\) 8.85934 0.321998 0.160999 0.986955i \(-0.448528\pi\)
0.160999 + 0.986955i \(0.448528\pi\)
\(758\) 0 0
\(759\) −21.7265 41.9346i −0.788621 1.52213i
\(760\) 0 0
\(761\) 1.38910 0.0503549 0.0251774 0.999683i \(-0.491985\pi\)
0.0251774 + 0.999683i \(0.491985\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 13.9662 + 30.2567i 0.504950 + 1.09393i
\(766\) 0 0
\(767\) 64.9194 2.34410
\(768\) 0 0
\(769\) 18.9626 32.8443i 0.683810 1.18439i −0.289999 0.957027i \(-0.593655\pi\)
0.973809 0.227367i \(-0.0730118\pi\)
\(770\) 0 0
\(771\) 14.1652 + 27.3404i 0.510146 + 0.984642i
\(772\) 0 0
\(773\) −0.657981 + 1.13966i −0.0236659 + 0.0409906i −0.877616 0.479365i \(-0.840867\pi\)
0.853950 + 0.520355i \(0.174200\pi\)
\(774\) 0 0
\(775\) 3.89037 + 6.73832i 0.139746 + 0.242047i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1.18289 0.0423813
\(780\) 0 0
\(781\) 62.8447 2.24876
\(782\) 0 0
\(783\) 13.6747 + 33.5649i 0.488694 + 1.19951i
\(784\) 0 0
\(785\) 7.11177 + 12.3179i 0.253830 + 0.439646i
\(786\) 0 0
\(787\) −6.12928 10.6162i −0.218485 0.378428i 0.735860 0.677134i \(-0.236778\pi\)
−0.954345 + 0.298706i \(0.903445\pi\)
\(788\) 0 0
\(789\) 21.6716 + 41.8287i 0.771529 + 1.48914i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 49.1072 + 85.0561i 1.74385 + 3.02043i
\(794\) 0 0
\(795\) 21.5167 + 0.981271i 0.763120 + 0.0348021i
\(796\) 0 0
\(797\) −10.7178 + 18.5638i −0.379644 + 0.657563i −0.991010 0.133785i \(-0.957287\pi\)
0.611366 + 0.791348i \(0.290620\pi\)
\(798\) 0 0
\(799\) 8.29007 + 14.3588i 0.293282 + 0.507979i
\(800\) 0 0
\(801\) 29.3820 + 2.68552i 1.03816 + 0.0948882i
\(802\) 0 0
\(803\) 10.0562 17.4179i 0.354876 0.614664i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 41.3463 + 1.88560i 1.45546 + 0.0663762i
\(808\) 0 0
\(809\) −13.3478 + 23.1190i −0.469282 + 0.812820i −0.999383 0.0351140i \(-0.988821\pi\)
0.530101 + 0.847934i \(0.322154\pi\)
\(810\) 0 0
\(811\) 38.2852 1.34438 0.672188 0.740381i \(-0.265355\pi\)
0.672188 + 0.740381i \(0.265355\pi\)
\(812\) 0 0
\(813\) −11.4567 + 17.9067i −0.401804 + 0.628014i
\(814\) 0 0
\(815\) −9.01459 15.6137i −0.315767 0.546925i
\(816\) 0 0
\(817\) 10.4612 18.1193i 0.365990 0.633914i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 5.24990 9.09310i 0.183223 0.317351i −0.759753 0.650211i \(-0.774680\pi\)
0.942976 + 0.332860i \(0.108014\pi\)
\(822\) 0 0
\(823\) −8.00000 13.8564i −0.278862 0.483004i 0.692240 0.721668i \(-0.256624\pi\)
−0.971102 + 0.238664i \(0.923291\pi\)
\(824\) 0 0
\(825\) −3.90350 7.53420i −0.135902 0.262307i
\(826\) 0 0
\(827\) −48.7817 −1.69631 −0.848153 0.529752i \(-0.822285\pi\)
−0.848153 + 0.529752i \(0.822285\pi\)
\(828\) 0 0
\(829\) 3.10963 5.38604i 0.108002 0.187065i −0.806959 0.590608i \(-0.798888\pi\)
0.914961 + 0.403543i \(0.132221\pi\)
\(830\) 0 0
\(831\) −10.1836 19.6555i −0.353264 0.681840i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 14.7989 25.6325i 0.512138 0.887049i
\(836\) 0 0
\(837\) 14.4715 + 35.5207i 0.500208 + 1.22777i
\(838\) 0 0
\(839\) 21.0366 + 36.4364i 0.726263 + 1.25792i 0.958452 + 0.285254i \(0.0920779\pi\)
−0.232189 + 0.972671i \(0.574589\pi\)
\(840\) 0 0
\(841\) −9.82597 + 17.0191i −0.338826 + 0.586865i
\(842\) 0 0
\(843\) 26.6173 41.6025i 0.916749 1.43286i
\(844\) 0 0
\(845\) −46.1665 79.9628i −1.58818 2.75080i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −0.678304 + 1.06018i −0.0232793 + 0.0363853i
\(850\) 0 0
\(851\) 2.13161 + 3.69206i 0.0730707 + 0.126562i
\(852\) 0 0
\(853\) 6.72519 + 11.6484i 0.230266 + 0.398833i 0.957886 0.287147i \(-0.0927070\pi\)
−0.727620 + 0.685980i \(0.759374\pi\)
\(854\) 0 0
\(855\) 31.8104 + 2.90748i 1.08789 + 0.0994336i
\(856\) 0 0
\(857\) 41.3786 1.41347 0.706733 0.707481i \(-0.250168\pi\)
0.706733 + 0.707481i \(0.250168\pi\)
\(858\) 0 0
\(859\) −39.7630 −1.35670 −0.678349 0.734740i \(-0.737304\pi\)
−0.678349 + 0.734740i \(0.737304\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −26.6929 46.2334i −0.908637 1.57380i −0.815960 0.578109i \(-0.803791\pi\)
−0.0926768 0.995696i \(-0.529542\pi\)
\(864\) 0 0
\(865\) −6.02704 + 10.4391i −0.204926 + 0.354942i
\(866\) 0 0
\(867\) −3.15652 + 4.93359i −0.107201 + 0.167554i
\(868\) 0 0
\(869\) −14.9552 + 25.9033i −0.507322 + 0.878708i
\(870\) 0 0
\(871\) −9.43560 −0.319713
\(872\) 0 0
\(873\) −1.47169 0.134513i −0.0498092 0.00455257i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −6.85349 −0.231426 −0.115713 0.993283i \(-0.536915\pi\)
−0.115713 + 0.993283i \(0.536915\pi\)
\(878\) 0 0
\(879\) 44.2601 + 2.01848i 1.49286 + 0.0680818i
\(880\) 0 0
\(881\) −12.5103 −0.421483 −0.210742 0.977542i \(-0.567588\pi\)
−0.210742 + 0.977542i \(0.567588\pi\)
\(882\) 0 0
\(883\) 6.69124 0.225178 0.112589 0.993642i \(-0.464086\pi\)
0.112589 + 0.993642i \(0.464086\pi\)
\(884\) 0 0
\(885\) −20.9768 + 32.7864i −0.705126 + 1.10210i
\(886\) 0 0
\(887\) 32.1416 1.07921 0.539605 0.841918i \(-0.318574\pi\)
0.539605 + 0.841918i \(0.318574\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −14.0009 39.4162i −0.469047 1.32049i
\(892\) 0 0
\(893\) 15.8928 0.531832
\(894\) 0 0
\(895\) 2.19076 3.79450i 0.0732289 0.126836i
\(896\) 0 0
\(897\) 72.1569 + 3.29072i 2.40925 + 0.109874i
\(898\) 0 0
\(899\) −25.7434 + 44.5888i −0.858590 + 1.48712i
\(900\) 0 0
\(901\) 11.4086 + 19.7602i 0.380074 + 0.658308i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −41.6883 −1.38577
\(906\) 0 0
\(907\) −31.4031 −1.04272 −0.521362 0.853336i \(-0.674576\pi\)
−0.521362 + 0.853336i \(0.674576\pi\)
\(908\) 0 0
\(909\) −5.89397 + 8.34832i −0.195491 + 0.276896i
\(910\) 0 0
\(911\) −22.8982 39.6609i −0.758653 1.31402i −0.943538 0.331265i \(-0.892525\pi\)
0.184885 0.982760i \(-0.440809\pi\)
\(912\) 0 0
\(913\) −3.45185 5.97877i −0.114239 0.197868i
\(914\) 0 0
\(915\) −58.8235 2.68265i −1.94465 0.0886857i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −13.5900 23.5385i −0.448292 0.776465i 0.549983 0.835176i \(-0.314634\pi\)
−0.998275 + 0.0587112i \(0.981301\pi\)
\(920\) 0 0
\(921\) 4.96722 + 9.58732i 0.163676 + 0.315913i
\(922\) 0 0
\(923\) −48.0576 + 83.2381i −1.58183 + 2.73982i
\(924\) 0 0
\(925\) 0.382977 + 0.663336i 0.0125922 + 0.0218104i
\(926\) 0 0
\(927\) −8.93200 + 12.6514i −0.293365 + 0.415528i
\(928\) 0 0
\(929\) 20.3338 35.2192i 0.667132 1.15551i −0.311571 0.950223i \(-0.600855\pi\)
0.978703 0.205283i \(-0.0658115\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −27.2929 + 42.6584i −0.893529 + 1.39657i
\(934\) 0 0
\(935\) 25.8135 44.7103i 0.844192 1.46218i
\(936\) 0 0
\(937\) 16.4150 0.536254 0.268127 0.963384i \(-0.413595\pi\)
0.268127 + 0.963384i \(0.413595\pi\)
\(938\) 0 0
\(939\) 49.2893 + 2.24784i 1.60849 + 0.0733555i
\(940\) 0 0
\(941\) −3.66878 6.35451i −0.119599 0.207151i 0.800010 0.599987i \(-0.204827\pi\)
−0.919609 + 0.392836i \(0.871494\pi\)
\(942\) 0 0
\(943\) −0.801851 + 1.38885i −0.0261119 + 0.0452271i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −29.5562 + 51.1929i −0.960448 + 1.66354i −0.239071 + 0.971002i \(0.576843\pi\)
−0.721377 + 0.692543i \(0.756490\pi\)
\(948\) 0 0
\(949\) 15.3801 + 26.6390i 0.499258 + 0.864740i
\(950\) 0 0
\(951\) 2.80039 + 0.127712i 0.0908088 + 0.00414134i
\(952\) 0 0
\(953\) −16.9354 −0.548592 −0.274296 0.961645i \(-0.588445\pi\)
−0.274296 + 0.961645i \(0.588445\pi\)
\(954\) 0 0
\(955\) 6.75370 11.6977i 0.218544 0.378530i
\(956\) 0 0
\(957\) 30.2609 47.2973i 0.978196 1.52891i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −11.7434 + 20.3401i −0.378819 + 0.656133i
\(962\) 0 0
\(963\) −17.2173 1.57367i −0.554820 0.0507107i
\(964\) 0 0
\(965\) 6.77548 + 11.7355i 0.218110 + 0.377778i
\(966\) 0 0
\(967\) 3.55555 6.15839i 0.114339 0.198040i −0.803177 0.595741i \(-0.796858\pi\)
0.917515 + 0.397701i \(0.130192\pi\)
\(968\) 0 0
\(969\) 15.5665 + 30.0452i 0.500069 + 0.965192i
\(970\) 0 0
\(971\) 0.735508 + 1.27394i 0.0236036 + 0.0408826i 0.877586 0.479419i \(-0.159153\pi\)
−0.853982 + 0.520302i \(0.825819\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 12.9641 + 0.591229i 0.415184 + 0.0189345i
\(976\) 0 0
\(977\) 9.71634 + 16.8292i 0.310853 + 0.538413i 0.978547 0.206022i \(-0.0660519\pi\)
−0.667694 + 0.744436i \(0.732719\pi\)
\(978\) 0 0
\(979\) −22.8545 39.5851i −0.730432 1.26515i
\(980\) 0 0
\(981\) 11.2934 + 24.4662i 0.360570 + 0.781145i
\(982\) 0 0
\(983\) 7.74436 0.247007 0.123503 0.992344i \(-0.460587\pi\)
0.123503 + 0.992344i \(0.460587\pi\)
\(984\) 0 0
\(985\) 28.6156 0.911768
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 14.1828 + 24.5653i 0.450986 + 0.781130i
\(990\) 0 0
\(991\) 7.23551 12.5323i 0.229843 0.398101i −0.727918 0.685664i \(-0.759512\pi\)
0.957762 + 0.287563i \(0.0928452\pi\)
\(992\) 0 0
\(993\) −24.1929 1.10332i −0.767738 0.0350127i
\(994\) 0 0
\(995\) 5.10243 8.83767i 0.161758 0.280173i
\(996\) 0 0
\(997\) −55.3097 −1.75168 −0.875838 0.482605i \(-0.839691\pi\)
−0.875838 + 0.482605i \(0.839691\pi\)
\(998\) 0 0
\(999\) 1.42461 + 3.49674i 0.0450726 + 0.110632i
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.l.e.949.2 6
3.2 odd 2 5292.2.l.e.361.1 6
7.2 even 3 1764.2.i.g.373.3 6
7.3 odd 6 1764.2.j.e.589.2 6
7.4 even 3 252.2.j.a.85.2 6
7.5 odd 6 1764.2.i.d.373.1 6
7.6 odd 2 1764.2.l.f.949.2 6
9.2 odd 6 5292.2.i.f.2125.3 6
9.7 even 3 1764.2.i.g.1537.3 6
21.2 odd 6 5292.2.i.f.1549.3 6
21.5 even 6 5292.2.i.e.1549.1 6
21.11 odd 6 756.2.j.b.253.3 6
21.17 even 6 5292.2.j.d.1765.1 6
21.20 even 2 5292.2.l.f.361.3 6
28.11 odd 6 1008.2.r.j.337.2 6
63.2 odd 6 5292.2.l.e.3313.1 6
63.4 even 3 2268.2.a.i.1.3 3
63.11 odd 6 756.2.j.b.505.3 6
63.16 even 3 inner 1764.2.l.e.961.2 6
63.20 even 6 5292.2.i.e.2125.1 6
63.25 even 3 252.2.j.a.169.2 yes 6
63.32 odd 6 2268.2.a.h.1.1 3
63.34 odd 6 1764.2.i.d.1537.1 6
63.38 even 6 5292.2.j.d.3529.1 6
63.47 even 6 5292.2.l.f.3313.3 6
63.52 odd 6 1764.2.j.e.1177.2 6
63.61 odd 6 1764.2.l.f.961.2 6
84.11 even 6 3024.2.r.j.1009.3 6
252.11 even 6 3024.2.r.j.2017.3 6
252.67 odd 6 9072.2.a.by.1.3 3
252.95 even 6 9072.2.a.bv.1.1 3
252.151 odd 6 1008.2.r.j.673.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.2.j.a.85.2 6 7.4 even 3
252.2.j.a.169.2 yes 6 63.25 even 3
756.2.j.b.253.3 6 21.11 odd 6
756.2.j.b.505.3 6 63.11 odd 6
1008.2.r.j.337.2 6 28.11 odd 6
1008.2.r.j.673.2 6 252.151 odd 6
1764.2.i.d.373.1 6 7.5 odd 6
1764.2.i.d.1537.1 6 63.34 odd 6
1764.2.i.g.373.3 6 7.2 even 3
1764.2.i.g.1537.3 6 9.7 even 3
1764.2.j.e.589.2 6 7.3 odd 6
1764.2.j.e.1177.2 6 63.52 odd 6
1764.2.l.e.949.2 6 1.1 even 1 trivial
1764.2.l.e.961.2 6 63.16 even 3 inner
1764.2.l.f.949.2 6 7.6 odd 2
1764.2.l.f.961.2 6 63.61 odd 6
2268.2.a.h.1.1 3 63.32 odd 6
2268.2.a.i.1.3 3 63.4 even 3
3024.2.r.j.1009.3 6 84.11 even 6
3024.2.r.j.2017.3 6 252.11 even 6
5292.2.i.e.1549.1 6 21.5 even 6
5292.2.i.e.2125.1 6 63.20 even 6
5292.2.i.f.1549.3 6 21.2 odd 6
5292.2.i.f.2125.3 6 9.2 odd 6
5292.2.j.d.1765.1 6 21.17 even 6
5292.2.j.d.3529.1 6 63.38 even 6
5292.2.l.e.361.1 6 3.2 odd 2
5292.2.l.e.3313.1 6 63.2 odd 6
5292.2.l.f.361.3 6 21.20 even 2
5292.2.l.f.3313.3 6 63.47 even 6
9072.2.a.bv.1.1 3 252.95 even 6
9072.2.a.by.1.3 3 252.67 odd 6