Properties

Label 576.3.o.g.511.3
Level $576$
Weight $3$
Character 576.511
Analytic conductor $15.695$
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] = [576,3,Mod(319,576)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(576, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("576.319");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 576.o (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: \(15.6948632272\)
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: \( x^{16} - 3 x^{15} + 7 x^{14} - 30 x^{13} + 76 x^{12} - 144 x^{11} + 424 x^{10} - 912 x^{9} + 1552 x^{8} - 3648 x^{7} + 6784 x^{6} - 9216 x^{5} + 19456 x^{4} - 30720 x^{3} + \cdots + 65536 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{3} \)
Twist minimal: no (minimal twist has level 36)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 511.3
Root \(-0.710719 - 1.86946i\) of defining polynomial
Character \(\chi\) \(=\) 576.511
Dual form 576.3.o.g.319.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.32245 - 1.89900i) q^{3} +(-1.35609 - 2.34881i) q^{5} +(-10.0431 - 5.79837i) q^{7} +(1.78756 + 8.82069i) q^{9} +O(q^{10})\) \(q+(-2.32245 - 1.89900i) q^{3} +(-1.35609 - 2.34881i) q^{5} +(-10.0431 - 5.79837i) q^{7} +(1.78756 + 8.82069i) q^{9} +(-8.54822 - 4.93532i) q^{11} +(-0.296185 - 0.513008i) q^{13} +(-1.31096 + 8.03023i) q^{15} -8.87968 q^{17} +14.0989i q^{19} +(12.3134 + 32.5383i) q^{21} +(18.2754 - 10.5513i) q^{23} +(8.82205 - 15.2802i) q^{25} +(12.5990 - 23.8802i) q^{27} +(-10.1764 + 17.6260i) q^{29} +(-14.3357 + 8.27670i) q^{31} +(10.4806 + 27.6952i) q^{33} +31.4524i q^{35} +40.6557 q^{37} +(-0.286328 + 1.75389i) q^{39} +(21.2177 + 36.7502i) q^{41} +(-32.2385 - 18.6129i) q^{43} +(18.2941 - 16.1603i) q^{45} +(-1.57134 - 0.907211i) q^{47} +(42.7423 + 74.0318i) q^{49} +(20.6226 + 16.8625i) q^{51} +21.1005 q^{53} +26.7709i q^{55} +(26.7738 - 32.7440i) q^{57} +(76.6879 - 44.2758i) q^{59} +(-36.4925 + 63.2069i) q^{61} +(33.1930 - 98.9519i) q^{63} +(-0.803307 + 1.39137i) q^{65} +(-38.3110 + 22.1189i) q^{67} +(-62.4808 - 10.2002i) q^{69} +111.798i q^{71} -76.2003 q^{73} +(-49.5060 + 18.7345i) q^{75} +(57.2337 + 99.1316i) q^{77} +(8.30434 + 4.79451i) q^{79} +(-74.6092 + 31.5351i) q^{81} +(73.6244 + 42.5070i) q^{83} +(12.0416 + 20.8567i) q^{85} +(57.1060 - 21.6106i) q^{87} +64.7845 q^{89} +6.86958i q^{91} +(49.0114 + 8.00125i) q^{93} +(33.1157 - 19.1193i) q^{95} +(-3.59139 + 6.22047i) q^{97} +(28.2524 - 84.2235i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 6 q^{5} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 6 q^{5} + 18 q^{9} + 46 q^{13} + 12 q^{17} + 66 q^{21} - 30 q^{25} - 42 q^{29} - 168 q^{33} - 56 q^{37} + 84 q^{41} - 174 q^{45} + 58 q^{49} + 72 q^{53} + 366 q^{57} + 34 q^{61} - 30 q^{65} + 54 q^{69} + 116 q^{73} + 330 q^{77} - 102 q^{81} + 140 q^{85} - 384 q^{89} + 486 q^{93} - 148 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(325\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(1\)

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
\(3\) −2.32245 1.89900i −0.774151 0.633001i
\(4\) 0 0
\(5\) −1.35609 2.34881i −0.271218 0.469763i 0.697956 0.716140i \(-0.254093\pi\)
−0.969174 + 0.246378i \(0.920760\pi\)
\(6\) 0 0
\(7\) −10.0431 5.79837i −1.43473 0.828339i −0.437249 0.899340i \(-0.644047\pi\)
−0.997476 + 0.0710013i \(0.977381\pi\)
\(8\) 0 0
\(9\) 1.78756 + 8.82069i 0.198618 + 0.980077i
\(10\) 0 0
\(11\) −8.54822 4.93532i −0.777111 0.448665i 0.0582943 0.998299i \(-0.481434\pi\)
−0.835406 + 0.549634i \(0.814767\pi\)
\(12\) 0 0
\(13\) −0.296185 0.513008i −0.0227835 0.0394622i 0.854409 0.519601i \(-0.173920\pi\)
−0.877192 + 0.480139i \(0.840586\pi\)
\(14\) 0 0
\(15\) −1.31096 + 8.03023i −0.0873972 + 0.535348i
\(16\) 0 0
\(17\) −8.87968 −0.522334 −0.261167 0.965294i \(-0.584107\pi\)
−0.261167 + 0.965294i \(0.584107\pi\)
\(18\) 0 0
\(19\) 14.0989i 0.742046i 0.928624 + 0.371023i \(0.120993\pi\)
−0.928624 + 0.371023i \(0.879007\pi\)
\(20\) 0 0
\(21\) 12.3134 + 32.5383i 0.586354 + 1.54944i
\(22\) 0 0
\(23\) 18.2754 10.5513i 0.794583 0.458753i −0.0469902 0.998895i \(-0.514963\pi\)
0.841574 + 0.540142i \(0.181630\pi\)
\(24\) 0 0
\(25\) 8.82205 15.2802i 0.352882 0.611209i
\(26\) 0 0
\(27\) 12.5990 23.8802i 0.466630 0.884453i
\(28\) 0 0
\(29\) −10.1764 + 17.6260i −0.350910 + 0.607793i −0.986409 0.164308i \(-0.947461\pi\)
0.635499 + 0.772101i \(0.280794\pi\)
\(30\) 0 0
\(31\) −14.3357 + 8.27670i −0.462441 + 0.266990i −0.713070 0.701093i \(-0.752696\pi\)
0.250629 + 0.968083i \(0.419362\pi\)
\(32\) 0 0
\(33\) 10.4806 + 27.6952i 0.317595 + 0.839247i
\(34\) 0 0
\(35\) 31.4524i 0.898641i
\(36\) 0 0
\(37\) 40.6557 1.09880 0.549401 0.835559i \(-0.314856\pi\)
0.549401 + 0.835559i \(0.314856\pi\)
\(38\) 0 0
\(39\) −0.286328 + 1.75389i −0.00734176 + 0.0449716i
\(40\) 0 0
\(41\) 21.2177 + 36.7502i 0.517506 + 0.896346i 0.999793 + 0.0203330i \(0.00647263\pi\)
−0.482288 + 0.876013i \(0.660194\pi\)
\(42\) 0 0
\(43\) −32.2385 18.6129i −0.749732 0.432858i 0.0758649 0.997118i \(-0.475828\pi\)
−0.825597 + 0.564260i \(0.809162\pi\)
\(44\) 0 0
\(45\) 18.2941 16.1603i 0.406535 0.359118i
\(46\) 0 0
\(47\) −1.57134 0.907211i −0.0334327 0.0193024i 0.483191 0.875515i \(-0.339478\pi\)
−0.516623 + 0.856213i \(0.672811\pi\)
\(48\) 0 0
\(49\) 42.7423 + 74.0318i 0.872291 + 1.51085i
\(50\) 0 0
\(51\) 20.6226 + 16.8625i 0.404365 + 0.330638i
\(52\) 0 0
\(53\) 21.1005 0.398122 0.199061 0.979987i \(-0.436211\pi\)
0.199061 + 0.979987i \(0.436211\pi\)
\(54\) 0 0
\(55\) 26.7709i 0.486744i
\(56\) 0 0
\(57\) 26.7738 32.7440i 0.469716 0.574456i
\(58\) 0 0
\(59\) 76.6879 44.2758i 1.29980 0.750437i 0.319427 0.947611i \(-0.396510\pi\)
0.980369 + 0.197174i \(0.0631764\pi\)
\(60\) 0 0
\(61\) −36.4925 + 63.2069i −0.598238 + 1.03618i 0.394843 + 0.918749i \(0.370799\pi\)
−0.993081 + 0.117431i \(0.962534\pi\)
\(62\) 0 0
\(63\) 33.1930 98.9519i 0.526873 1.57066i
\(64\) 0 0
\(65\) −0.803307 + 1.39137i −0.0123586 + 0.0214057i
\(66\) 0 0
\(67\) −38.3110 + 22.1189i −0.571807 + 0.330133i −0.757871 0.652405i \(-0.773760\pi\)
0.186064 + 0.982538i \(0.440427\pi\)
\(68\) 0 0
\(69\) −62.4808 10.2002i −0.905519 0.147829i
\(70\) 0 0
\(71\) 111.798i 1.57462i 0.616557 + 0.787310i \(0.288527\pi\)
−0.616557 + 0.787310i \(0.711473\pi\)
\(72\) 0 0
\(73\) −76.2003 −1.04384 −0.521920 0.852995i \(-0.674784\pi\)
−0.521920 + 0.852995i \(0.674784\pi\)
\(74\) 0 0
\(75\) −49.5060 + 18.7345i −0.660080 + 0.249793i
\(76\) 0 0
\(77\) 57.2337 + 99.1316i 0.743294 + 1.28742i
\(78\) 0 0
\(79\) 8.30434 + 4.79451i 0.105118 + 0.0606901i 0.551637 0.834084i \(-0.314003\pi\)
−0.446519 + 0.894774i \(0.647337\pi\)
\(80\) 0 0
\(81\) −74.6092 + 31.5351i −0.921102 + 0.389322i
\(82\) 0 0
\(83\) 73.6244 + 42.5070i 0.887041 + 0.512133i 0.872973 0.487768i \(-0.162189\pi\)
0.0140672 + 0.999901i \(0.495522\pi\)
\(84\) 0 0
\(85\) 12.0416 + 20.8567i 0.141666 + 0.245373i
\(86\) 0 0
\(87\) 57.1060 21.6106i 0.656391 0.248397i
\(88\) 0 0
\(89\) 64.7845 0.727916 0.363958 0.931415i \(-0.381425\pi\)
0.363958 + 0.931415i \(0.381425\pi\)
\(90\) 0 0
\(91\) 6.86958i 0.0754898i
\(92\) 0 0
\(93\) 49.0114 + 8.00125i 0.527004 + 0.0860350i
\(94\) 0 0
\(95\) 33.1157 19.1193i 0.348586 0.201256i
\(96\) 0 0
\(97\) −3.59139 + 6.22047i −0.0370246 + 0.0641285i −0.883944 0.467593i \(-0.845121\pi\)
0.846919 + 0.531721i \(0.178455\pi\)
\(98\) 0 0
\(99\) 28.2524 84.2235i 0.285378 0.850742i
\(100\) 0 0
\(101\) 55.5037 96.1353i 0.549542 0.951834i −0.448764 0.893650i \(-0.648136\pi\)
0.998306 0.0581840i \(-0.0185310\pi\)
\(102\) 0 0
\(103\) −79.6133 + 45.9648i −0.772945 + 0.446260i −0.833924 0.551879i \(-0.813911\pi\)
0.0609793 + 0.998139i \(0.480578\pi\)
\(104\) 0 0
\(105\) 59.7283 73.0468i 0.568841 0.695683i
\(106\) 0 0
\(107\) 107.741i 1.00693i 0.864016 + 0.503465i \(0.167942\pi\)
−0.864016 + 0.503465i \(0.832058\pi\)
\(108\) 0 0
\(109\) −86.5562 −0.794093 −0.397047 0.917798i \(-0.629965\pi\)
−0.397047 + 0.917798i \(0.629965\pi\)
\(110\) 0 0
\(111\) −94.4209 77.2054i −0.850639 0.695544i
\(112\) 0 0
\(113\) −2.35198 4.07376i −0.0208140 0.0360509i 0.855431 0.517917i \(-0.173292\pi\)
−0.876245 + 0.481866i \(0.839959\pi\)
\(114\) 0 0
\(115\) −49.5662 28.6170i −0.431010 0.248844i
\(116\) 0 0
\(117\) 3.99564 3.52960i 0.0341507 0.0301675i
\(118\) 0 0
\(119\) 89.1793 + 51.4877i 0.749406 + 0.432670i
\(120\) 0 0
\(121\) −11.7852 20.4126i −0.0973987 0.168700i
\(122\) 0 0
\(123\) 20.5116 125.643i 0.166761 1.02149i
\(124\) 0 0
\(125\) −115.658 −0.925267
\(126\) 0 0
\(127\) 8.37118i 0.0659148i −0.999457 0.0329574i \(-0.989507\pi\)
0.999457 0.0329574i \(-0.0104926\pi\)
\(128\) 0 0
\(129\) 39.5264 + 104.449i 0.306406 + 0.809679i
\(130\) 0 0
\(131\) −115.067 + 66.4338i −0.878372 + 0.507129i −0.870121 0.492837i \(-0.835960\pi\)
−0.00825098 + 0.999966i \(0.502626\pi\)
\(132\) 0 0
\(133\) 81.7506 141.596i 0.614666 1.06463i
\(134\) 0 0
\(135\) −73.1756 + 2.79099i −0.542041 + 0.0206740i
\(136\) 0 0
\(137\) 22.5579 39.0715i 0.164656 0.285193i −0.771877 0.635772i \(-0.780682\pi\)
0.936533 + 0.350579i \(0.114015\pi\)
\(138\) 0 0
\(139\) −130.744 + 75.4848i −0.940601 + 0.543056i −0.890149 0.455670i \(-0.849400\pi\)
−0.0504522 + 0.998726i \(0.516066\pi\)
\(140\) 0 0
\(141\) 1.92655 + 5.09093i 0.0136635 + 0.0361059i
\(142\) 0 0
\(143\) 5.84708i 0.0408887i
\(144\) 0 0
\(145\) 55.2003 0.380692
\(146\) 0 0
\(147\) 41.3198 253.103i 0.281087 1.72179i
\(148\) 0 0
\(149\) −71.3914 123.653i −0.479137 0.829889i 0.520577 0.853815i \(-0.325717\pi\)
−0.999714 + 0.0239255i \(0.992384\pi\)
\(150\) 0 0
\(151\) −220.027 127.033i −1.45713 0.841276i −0.458263 0.888817i \(-0.651528\pi\)
−0.998869 + 0.0475407i \(0.984862\pi\)
\(152\) 0 0
\(153\) −15.8730 78.3249i −0.103745 0.511927i
\(154\) 0 0
\(155\) 38.8808 + 22.4479i 0.250844 + 0.144825i
\(156\) 0 0
\(157\) −2.65361 4.59618i −0.0169020 0.0292751i 0.857451 0.514566i \(-0.172047\pi\)
−0.874353 + 0.485291i \(0.838714\pi\)
\(158\) 0 0
\(159\) −49.0048 40.0698i −0.308206 0.252012i
\(160\) 0 0
\(161\) −244.722 −1.52001
\(162\) 0 0
\(163\) 59.5534i 0.365359i 0.983173 + 0.182679i \(0.0584770\pi\)
−0.983173 + 0.182679i \(0.941523\pi\)
\(164\) 0 0
\(165\) 50.8381 62.1742i 0.308110 0.376813i
\(166\) 0 0
\(167\) 85.7434 49.5040i 0.513434 0.296431i −0.220810 0.975317i \(-0.570870\pi\)
0.734244 + 0.678886i \(0.237537\pi\)
\(168\) 0 0
\(169\) 84.3245 146.054i 0.498962 0.864227i
\(170\) 0 0
\(171\) −124.362 + 25.2027i −0.727263 + 0.147384i
\(172\) 0 0
\(173\) −19.2965 + 33.4225i −0.111540 + 0.193193i −0.916391 0.400283i \(-0.868912\pi\)
0.804851 + 0.593477i \(0.202245\pi\)
\(174\) 0 0
\(175\) −177.201 + 102.307i −1.01258 + 0.584612i
\(176\) 0 0
\(177\) −262.184 42.8023i −1.48126 0.241821i
\(178\) 0 0
\(179\) 36.4264i 0.203499i 0.994810 + 0.101750i \(0.0324441\pi\)
−0.994810 + 0.101750i \(0.967556\pi\)
\(180\) 0 0
\(181\) 18.5921 0.102719 0.0513594 0.998680i \(-0.483645\pi\)
0.0513594 + 0.998680i \(0.483645\pi\)
\(182\) 0 0
\(183\) 204.782 77.4956i 1.11903 0.423473i
\(184\) 0 0
\(185\) −55.1327 95.4927i −0.298015 0.516177i
\(186\) 0 0
\(187\) 75.9055 + 43.8240i 0.405912 + 0.234353i
\(188\) 0 0
\(189\) −264.999 + 166.777i −1.40211 + 0.882419i
\(190\) 0 0
\(191\) 244.973 + 141.435i 1.28258 + 0.740497i 0.977319 0.211772i \(-0.0679233\pi\)
0.305260 + 0.952269i \(0.401257\pi\)
\(192\) 0 0
\(193\) −151.542 262.479i −0.785193 1.35999i −0.928884 0.370372i \(-0.879230\pi\)
0.143691 0.989623i \(-0.454103\pi\)
\(194\) 0 0
\(195\) 4.50786 1.70590i 0.0231172 0.00874822i
\(196\) 0 0
\(197\) −139.184 −0.706520 −0.353260 0.935525i \(-0.614927\pi\)
−0.353260 + 0.935525i \(0.614927\pi\)
\(198\) 0 0
\(199\) 11.2337i 0.0564505i −0.999602 0.0282253i \(-0.991014\pi\)
0.999602 0.0282253i \(-0.00898558\pi\)
\(200\) 0 0
\(201\) 130.979 + 21.3828i 0.651639 + 0.106382i
\(202\) 0 0
\(203\) 204.404 118.013i 1.00692 0.581344i
\(204\) 0 0
\(205\) 57.5462 99.6730i 0.280713 0.486210i
\(206\) 0 0
\(207\) 125.738 + 142.341i 0.607432 + 0.687636i
\(208\) 0 0
\(209\) 69.5825 120.520i 0.332931 0.576653i
\(210\) 0 0
\(211\) −112.017 + 64.6728i −0.530884 + 0.306506i −0.741376 0.671090i \(-0.765827\pi\)
0.210492 + 0.977595i \(0.432493\pi\)
\(212\) 0 0
\(213\) 212.305 259.646i 0.996737 1.21899i
\(214\) 0 0
\(215\) 100.963i 0.469595i
\(216\) 0 0
\(217\) 191.966 0.884634
\(218\) 0 0
\(219\) 176.971 + 144.705i 0.808089 + 0.660752i
\(220\) 0 0
\(221\) 2.63003 + 4.55535i 0.0119006 + 0.0206124i
\(222\) 0 0
\(223\) 209.210 + 120.787i 0.938159 + 0.541647i 0.889383 0.457163i \(-0.151134\pi\)
0.0487765 + 0.998810i \(0.484468\pi\)
\(224\) 0 0
\(225\) 150.552 + 50.5022i 0.669121 + 0.224454i
\(226\) 0 0
\(227\) 330.710 + 190.936i 1.45687 + 0.841126i 0.998856 0.0478181i \(-0.0152268\pi\)
0.458016 + 0.888944i \(0.348560\pi\)
\(228\) 0 0
\(229\) −74.6642 129.322i −0.326044 0.564725i 0.655679 0.755040i \(-0.272383\pi\)
−0.981723 + 0.190315i \(0.939049\pi\)
\(230\) 0 0
\(231\) 55.3289 338.915i 0.239519 1.46717i
\(232\) 0 0
\(233\) −218.934 −0.939631 −0.469816 0.882765i \(-0.655680\pi\)
−0.469816 + 0.882765i \(0.655680\pi\)
\(234\) 0 0
\(235\) 4.92103i 0.0209406i
\(236\) 0 0
\(237\) −10.1816 26.9050i −0.0429605 0.113523i
\(238\) 0 0
\(239\) −218.254 + 126.009i −0.913197 + 0.527235i −0.881458 0.472262i \(-0.843438\pi\)
−0.0317388 + 0.999496i \(0.510104\pi\)
\(240\) 0 0
\(241\) −226.014 + 391.467i −0.937816 + 1.62435i −0.168282 + 0.985739i \(0.553822\pi\)
−0.769534 + 0.638606i \(0.779511\pi\)
\(242\) 0 0
\(243\) 233.162 + 68.4445i 0.959513 + 0.281664i
\(244\) 0 0
\(245\) 115.925 200.787i 0.473162 0.819540i
\(246\) 0 0
\(247\) 7.23284 4.17588i 0.0292828 0.0169064i
\(248\) 0 0
\(249\) −90.2680 238.534i −0.362522 0.957966i
\(250\) 0 0
\(251\) 139.429i 0.555492i −0.960655 0.277746i \(-0.910413\pi\)
0.960655 0.277746i \(-0.0895874\pi\)
\(252\) 0 0
\(253\) −208.297 −0.823306
\(254\) 0 0
\(255\) 11.6409 71.3058i 0.0456505 0.279631i
\(256\) 0 0
\(257\) 235.308 + 407.565i 0.915594 + 1.58586i 0.806029 + 0.591875i \(0.201612\pi\)
0.109564 + 0.993980i \(0.465054\pi\)
\(258\) 0 0
\(259\) −408.308 235.737i −1.57648 0.910181i
\(260\) 0 0
\(261\) −173.665 58.2551i −0.665381 0.223200i
\(262\) 0 0
\(263\) −22.2028 12.8188i −0.0844214 0.0487407i 0.457195 0.889366i \(-0.348854\pi\)
−0.541616 + 0.840626i \(0.682187\pi\)
\(264\) 0 0
\(265\) −28.6141 49.5610i −0.107978 0.187023i
\(266\) 0 0
\(267\) −150.459 123.026i −0.563517 0.460772i
\(268\) 0 0
\(269\) −8.15075 −0.0303002 −0.0151501 0.999885i \(-0.504823\pi\)
−0.0151501 + 0.999885i \(0.504823\pi\)
\(270\) 0 0
\(271\) 401.979i 1.48332i 0.670777 + 0.741659i \(0.265961\pi\)
−0.670777 + 0.741659i \(0.734039\pi\)
\(272\) 0 0
\(273\) 13.0454 15.9543i 0.0477852 0.0584405i
\(274\) 0 0
\(275\) −150.826 + 87.0792i −0.548457 + 0.316652i
\(276\) 0 0
\(277\) −56.2021 + 97.3449i −0.202896 + 0.351426i −0.949460 0.313887i \(-0.898369\pi\)
0.746565 + 0.665313i \(0.231702\pi\)
\(278\) 0 0
\(279\) −98.6321 111.655i −0.353520 0.400198i
\(280\) 0 0
\(281\) 268.867 465.692i 0.956823 1.65727i 0.226681 0.973969i \(-0.427212\pi\)
0.730141 0.683296i \(-0.239454\pi\)
\(282\) 0 0
\(283\) −122.303 + 70.6114i −0.432164 + 0.249510i −0.700268 0.713880i \(-0.746936\pi\)
0.268104 + 0.963390i \(0.413603\pi\)
\(284\) 0 0
\(285\) −113.217 18.4830i −0.397253 0.0648528i
\(286\) 0 0
\(287\) 492.113i 1.71468i
\(288\) 0 0
\(289\) −210.151 −0.727167
\(290\) 0 0
\(291\) 20.1535 7.62667i 0.0692561 0.0262085i
\(292\) 0 0
\(293\) 230.291 + 398.875i 0.785975 + 1.36135i 0.928415 + 0.371545i \(0.121172\pi\)
−0.142440 + 0.989803i \(0.545495\pi\)
\(294\) 0 0
\(295\) −207.991 120.084i −0.705055 0.407064i
\(296\) 0 0
\(297\) −225.556 + 141.953i −0.759447 + 0.477958i
\(298\) 0 0
\(299\) −10.8258 6.25029i −0.0362068 0.0209040i
\(300\) 0 0
\(301\) 215.849 + 373.862i 0.717107 + 1.24206i
\(302\) 0 0
\(303\) −311.466 + 117.868i −1.02794 + 0.389002i
\(304\) 0 0
\(305\) 197.948 0.649011
\(306\) 0 0
\(307\) 210.322i 0.685089i −0.939502 0.342545i \(-0.888711\pi\)
0.939502 0.342545i \(-0.111289\pi\)
\(308\) 0 0
\(309\) 272.185 + 44.4351i 0.880859 + 0.143803i
\(310\) 0 0
\(311\) −110.993 + 64.0821i −0.356892 + 0.206052i −0.667717 0.744416i \(-0.732728\pi\)
0.310824 + 0.950467i \(0.399395\pi\)
\(312\) 0 0
\(313\) −3.62140 + 6.27245i −0.0115700 + 0.0200398i −0.871752 0.489947i \(-0.837016\pi\)
0.860182 + 0.509986i \(0.170350\pi\)
\(314\) 0 0
\(315\) −277.432 + 56.2233i −0.880737 + 0.178487i
\(316\) 0 0
\(317\) 120.145 208.098i 0.379007 0.656460i −0.611911 0.790927i \(-0.709599\pi\)
0.990918 + 0.134467i \(0.0429322\pi\)
\(318\) 0 0
\(319\) 173.980 100.447i 0.545392 0.314882i
\(320\) 0 0
\(321\) 204.602 250.224i 0.637388 0.779515i
\(322\) 0 0
\(323\) 125.194i 0.387596i
\(324\) 0 0
\(325\) −10.4518 −0.0321595
\(326\) 0 0
\(327\) 201.023 + 164.371i 0.614748 + 0.502662i
\(328\) 0 0
\(329\) 10.5207 + 18.2224i 0.0319778 + 0.0553872i
\(330\) 0 0
\(331\) 370.385 + 213.842i 1.11899 + 0.646048i 0.941142 0.338011i \(-0.109754\pi\)
0.177845 + 0.984058i \(0.443087\pi\)
\(332\) 0 0
\(333\) 72.6747 + 358.612i 0.218242 + 1.07691i
\(334\) 0 0
\(335\) 103.906 + 59.9904i 0.310168 + 0.179076i
\(336\) 0 0
\(337\) 152.442 + 264.037i 0.452349 + 0.783492i 0.998531 0.0541746i \(-0.0172528\pi\)
−0.546182 + 0.837666i \(0.683919\pi\)
\(338\) 0 0
\(339\) −2.27371 + 13.9275i −0.00670711 + 0.0410842i
\(340\) 0 0
\(341\) 163.393 0.479157
\(342\) 0 0
\(343\) 423.102i 1.23353i
\(344\) 0 0
\(345\) 60.7712 + 160.588i 0.176148 + 0.465473i
\(346\) 0 0
\(347\) −146.406 + 84.5276i −0.421919 + 0.243595i −0.695898 0.718140i \(-0.744994\pi\)
0.273979 + 0.961736i \(0.411660\pi\)
\(348\) 0 0
\(349\) −107.298 + 185.846i −0.307444 + 0.532509i −0.977802 0.209529i \(-0.932807\pi\)
0.670358 + 0.742037i \(0.266140\pi\)
\(350\) 0 0
\(351\) −15.9824 + 0.609584i −0.0455339 + 0.00173671i
\(352\) 0 0
\(353\) −275.895 + 477.865i −0.781574 + 1.35373i 0.149451 + 0.988769i \(0.452249\pi\)
−0.931025 + 0.364956i \(0.881084\pi\)
\(354\) 0 0
\(355\) 262.593 151.608i 0.739698 0.427065i
\(356\) 0 0
\(357\) −109.339 288.930i −0.306272 0.809326i
\(358\) 0 0
\(359\) 554.828i 1.54548i 0.634721 + 0.772741i \(0.281115\pi\)
−0.634721 + 0.772741i \(0.718885\pi\)
\(360\) 0 0
\(361\) 162.222 0.449367
\(362\) 0 0
\(363\) −11.3930 + 69.7876i −0.0313858 + 0.192252i
\(364\) 0 0
\(365\) 103.334 + 178.980i 0.283108 + 0.490357i
\(366\) 0 0
\(367\) −145.642 84.0864i −0.396845 0.229118i 0.288277 0.957547i \(-0.406918\pi\)
−0.685122 + 0.728429i \(0.740251\pi\)
\(368\) 0 0
\(369\) −286.234 + 252.848i −0.775702 + 0.685226i
\(370\) 0 0
\(371\) −211.913 122.348i −0.571195 0.329780i
\(372\) 0 0
\(373\) −171.699 297.391i −0.460318 0.797295i 0.538658 0.842524i \(-0.318931\pi\)
−0.998977 + 0.0452296i \(0.985598\pi\)
\(374\) 0 0
\(375\) 268.611 + 219.636i 0.716296 + 0.585695i
\(376\) 0 0
\(377\) 12.0564 0.0319798
\(378\) 0 0
\(379\) 602.392i 1.58943i 0.606986 + 0.794713i \(0.292379\pi\)
−0.606986 + 0.794713i \(0.707621\pi\)
\(380\) 0 0
\(381\) −15.8969 + 19.4417i −0.0417242 + 0.0510280i
\(382\) 0 0
\(383\) 315.762 182.305i 0.824443 0.475992i −0.0275035 0.999622i \(-0.508756\pi\)
0.851946 + 0.523630i \(0.175422\pi\)
\(384\) 0 0
\(385\) 155.228 268.862i 0.403189 0.698344i
\(386\) 0 0
\(387\) 106.550 317.638i 0.275324 0.820769i
\(388\) 0 0
\(389\) −107.326 + 185.893i −0.275901 + 0.477875i −0.970362 0.241656i \(-0.922310\pi\)
0.694461 + 0.719530i \(0.255643\pi\)
\(390\) 0 0
\(391\) −162.280 + 93.6923i −0.415038 + 0.239622i
\(392\) 0 0
\(393\) 393.395 + 64.2229i 1.00101 + 0.163417i
\(394\) 0 0
\(395\) 26.0071i 0.0658409i
\(396\) 0 0
\(397\) 684.628 1.72450 0.862251 0.506480i \(-0.169054\pi\)
0.862251 + 0.506480i \(0.169054\pi\)
\(398\) 0 0
\(399\) −458.754 + 173.606i −1.14976 + 0.435102i
\(400\) 0 0
\(401\) 95.1918 + 164.877i 0.237386 + 0.411164i 0.959963 0.280125i \(-0.0903761\pi\)
−0.722577 + 0.691290i \(0.757043\pi\)
\(402\) 0 0
\(403\) 8.49203 + 4.90287i 0.0210720 + 0.0121659i
\(404\) 0 0
\(405\) 175.247 + 132.479i 0.432708 + 0.327108i
\(406\) 0 0
\(407\) −347.534 200.649i −0.853892 0.492995i
\(408\) 0 0
\(409\) 188.978 + 327.320i 0.462049 + 0.800293i 0.999063 0.0432806i \(-0.0137809\pi\)
−0.537014 + 0.843574i \(0.680448\pi\)
\(410\) 0 0
\(411\) −126.587 + 47.9040i −0.307997 + 0.116555i
\(412\) 0 0
\(413\) −1026.91 −2.48647
\(414\) 0 0
\(415\) 230.573i 0.555598i
\(416\) 0 0
\(417\) 446.991 + 72.9727i 1.07192 + 0.174994i
\(418\) 0 0
\(419\) −267.326 + 154.341i −0.638009 + 0.368355i −0.783847 0.620954i \(-0.786745\pi\)
0.145838 + 0.989308i \(0.453412\pi\)
\(420\) 0 0
\(421\) −176.834 + 306.286i −0.420034 + 0.727521i −0.995942 0.0899938i \(-0.971315\pi\)
0.575908 + 0.817514i \(0.304649\pi\)
\(422\) 0 0
\(423\) 5.19337 15.4820i 0.0122775 0.0366004i
\(424\) 0 0
\(425\) −78.3369 + 135.684i −0.184322 + 0.319255i
\(426\) 0 0
\(427\) 732.995 423.195i 1.71662 0.991088i
\(428\) 0 0
\(429\) 11.1036 13.5796i 0.0258826 0.0316540i
\(430\) 0 0
\(431\) 472.777i 1.09693i −0.836174 0.548465i \(-0.815213\pi\)
0.836174 0.548465i \(-0.184787\pi\)
\(432\) 0 0
\(433\) 61.4188 0.141845 0.0709224 0.997482i \(-0.477406\pi\)
0.0709224 + 0.997482i \(0.477406\pi\)
\(434\) 0 0
\(435\) −128.200 104.826i −0.294713 0.240978i
\(436\) 0 0
\(437\) 148.762 + 257.663i 0.340416 + 0.589618i
\(438\) 0 0
\(439\) 354.347 + 204.582i 0.807169 + 0.466019i 0.845972 0.533228i \(-0.179021\pi\)
−0.0388030 + 0.999247i \(0.512354\pi\)
\(440\) 0 0
\(441\) −576.607 + 509.353i −1.30750 + 1.15500i
\(442\) 0 0
\(443\) −668.806 386.136i −1.50972 0.871638i −0.999936 0.0113360i \(-0.996392\pi\)
−0.509785 0.860302i \(-0.670275\pi\)
\(444\) 0 0
\(445\) −87.8536 152.167i −0.197424 0.341948i
\(446\) 0 0
\(447\) −69.0155 + 422.752i −0.154397 + 0.945753i
\(448\) 0 0
\(449\) −789.037 −1.75732 −0.878660 0.477448i \(-0.841562\pi\)
−0.878660 + 0.477448i \(0.841562\pi\)
\(450\) 0 0
\(451\) 418.865i 0.928747i
\(452\) 0 0
\(453\) 269.767 + 712.859i 0.595511 + 1.57364i
\(454\) 0 0
\(455\) 16.1354 9.31575i 0.0354623 0.0204742i
\(456\) 0 0
\(457\) 138.165 239.309i 0.302331 0.523653i −0.674332 0.738428i \(-0.735568\pi\)
0.976664 + 0.214775i \(0.0689018\pi\)
\(458\) 0 0
\(459\) −111.875 + 212.049i −0.243737 + 0.461980i
\(460\) 0 0
\(461\) 294.041 509.295i 0.637834 1.10476i −0.348073 0.937467i \(-0.613164\pi\)
0.985907 0.167293i \(-0.0535027\pi\)
\(462\) 0 0
\(463\) 677.285 391.031i 1.46282 0.844558i 0.463677 0.886004i \(-0.346530\pi\)
0.999141 + 0.0414459i \(0.0131964\pi\)
\(464\) 0 0
\(465\) −47.6703 125.969i −0.102517 0.270901i
\(466\) 0 0
\(467\) 663.203i 1.42014i 0.704133 + 0.710068i \(0.251336\pi\)
−0.704133 + 0.710068i \(0.748664\pi\)
\(468\) 0 0
\(469\) 513.014 1.09385
\(470\) 0 0
\(471\) −2.56530 + 15.7136i −0.00544649 + 0.0333623i
\(472\) 0 0
\(473\) 183.721 + 318.214i 0.388417 + 0.672758i
\(474\) 0 0
\(475\) 215.434 + 124.381i 0.453546 + 0.261855i
\(476\) 0 0
\(477\) 37.7184 + 186.121i 0.0790743 + 0.390190i
\(478\) 0 0
\(479\) −562.018 324.481i −1.17331 0.677414i −0.218856 0.975757i \(-0.570233\pi\)
−0.954459 + 0.298344i \(0.903566\pi\)
\(480\) 0 0
\(481\) −12.0416 20.8567i −0.0250346 0.0433611i
\(482\) 0 0
\(483\) 568.355 + 464.728i 1.17672 + 0.962170i
\(484\) 0 0
\(485\) 19.4810 0.0401669
\(486\) 0 0
\(487\) 282.104i 0.579269i 0.957137 + 0.289635i \(0.0935338\pi\)
−0.957137 + 0.289635i \(0.906466\pi\)
\(488\) 0 0
\(489\) 113.092 138.310i 0.231272 0.282843i
\(490\) 0 0
\(491\) 652.933 376.971i 1.32980 0.767762i 0.344534 0.938774i \(-0.388037\pi\)
0.985269 + 0.171012i \(0.0547036\pi\)
\(492\) 0 0
\(493\) 90.3630 156.513i 0.183292 0.317471i
\(494\) 0 0
\(495\) −236.138 + 47.8548i −0.477047 + 0.0966763i
\(496\) 0 0
\(497\) 648.247 1122.80i 1.30432 2.25915i
\(498\) 0 0
\(499\) −446.169 + 257.596i −0.894126 + 0.516224i −0.875290 0.483599i \(-0.839329\pi\)
−0.0188362 + 0.999823i \(0.505996\pi\)
\(500\) 0 0
\(501\) −293.143 47.8565i −0.585116 0.0955220i
\(502\) 0 0
\(503\) 523.660i 1.04107i 0.853839 + 0.520537i \(0.174268\pi\)
−0.853839 + 0.520537i \(0.825732\pi\)
\(504\) 0 0
\(505\) −301.072 −0.596182
\(506\) 0 0
\(507\) −473.198 + 179.072i −0.933329 + 0.353198i
\(508\) 0 0
\(509\) 267.685 + 463.645i 0.525905 + 0.910893i 0.999545 + 0.0301749i \(0.00960643\pi\)
−0.473640 + 0.880719i \(0.657060\pi\)
\(510\) 0 0
\(511\) 765.285 + 441.838i 1.49762 + 0.864653i
\(512\) 0 0
\(513\) 336.684 + 177.632i 0.656305 + 0.346261i
\(514\) 0 0
\(515\) 215.925 + 124.665i 0.419273 + 0.242067i
\(516\) 0 0
\(517\) 8.95475 + 15.5101i 0.0173206 + 0.0300002i
\(518\) 0 0
\(519\) 108.285 40.9780i 0.208641 0.0789557i
\(520\) 0 0
\(521\) 177.268 0.340246 0.170123 0.985423i \(-0.445584\pi\)
0.170123 + 0.985423i \(0.445584\pi\)
\(522\) 0 0
\(523\) 444.206i 0.849343i −0.905347 0.424672i \(-0.860390\pi\)
0.905347 0.424672i \(-0.139610\pi\)
\(524\) 0 0
\(525\) 605.822 + 98.9023i 1.15395 + 0.188385i
\(526\) 0 0
\(527\) 127.296 73.4944i 0.241548 0.139458i
\(528\) 0 0
\(529\) −41.8394 + 72.4679i −0.0790914 + 0.136990i
\(530\) 0 0
\(531\) 527.628 + 597.295i 0.993649 + 1.12485i
\(532\) 0 0
\(533\) 12.5688 21.7697i 0.0235812 0.0408438i
\(534\) 0 0
\(535\) 253.065 146.107i 0.473018 0.273097i
\(536\) 0 0
\(537\) 69.1739 84.5986i 0.128815 0.157539i
\(538\) 0 0
\(539\) 843.787i 1.56547i
\(540\) 0 0
\(541\) −571.163 −1.05575 −0.527877 0.849321i \(-0.677012\pi\)
−0.527877 + 0.849321i \(0.677012\pi\)
\(542\) 0 0
\(543\) −43.1793 35.3065i −0.0795198 0.0650211i
\(544\) 0 0
\(545\) 117.378 + 203.304i 0.215372 + 0.373036i
\(546\) 0 0
\(547\) 139.875 + 80.7569i 0.255713 + 0.147636i 0.622377 0.782717i \(-0.286167\pi\)
−0.366664 + 0.930353i \(0.619500\pi\)
\(548\) 0 0
\(549\) −622.762 208.903i −1.13436 0.380515i
\(550\) 0 0
\(551\) −248.507 143.476i −0.451011 0.260391i
\(552\) 0 0
\(553\) −55.6008 96.3034i −0.100544 0.174147i
\(554\) 0 0
\(555\) −53.2979 + 326.475i −0.0960323 + 0.588242i
\(556\) 0 0
\(557\) 568.917 1.02139 0.510697 0.859761i \(-0.329387\pi\)
0.510697 + 0.859761i \(0.329387\pi\)
\(558\) 0 0
\(559\) 22.0515i 0.0394481i
\(560\) 0 0
\(561\) −93.0647 245.924i −0.165891 0.438367i
\(562\) 0 0
\(563\) 250.527 144.642i 0.444985 0.256912i −0.260725 0.965413i \(-0.583962\pi\)
0.705710 + 0.708501i \(0.250628\pi\)
\(564\) 0 0
\(565\) −6.37900 + 11.0487i −0.0112903 + 0.0195553i
\(566\) 0 0
\(567\) 932.159 + 115.903i 1.64402 + 0.204414i
\(568\) 0 0
\(569\) −223.117 + 386.450i −0.392121 + 0.679174i −0.992729 0.120370i \(-0.961592\pi\)
0.600608 + 0.799544i \(0.294925\pi\)
\(570\) 0 0
\(571\) 372.386 214.997i 0.652164 0.376527i −0.137121 0.990554i \(-0.543785\pi\)
0.789285 + 0.614027i \(0.210451\pi\)
\(572\) 0 0
\(573\) −300.351 793.680i −0.524173 1.38513i
\(574\) 0 0
\(575\) 372.337i 0.647542i
\(576\) 0 0
\(577\) 50.9694 0.0883353 0.0441676 0.999024i \(-0.485936\pi\)
0.0441676 + 0.999024i \(0.485936\pi\)
\(578\) 0 0
\(579\) −146.499 + 897.374i −0.253021 + 1.54987i
\(580\) 0 0
\(581\) −492.943 853.803i −0.848440 1.46954i
\(582\) 0 0
\(583\) −180.371 104.137i −0.309385 0.178623i
\(584\) 0 0
\(585\) −13.7088 4.59856i −0.0234339 0.00786079i
\(586\) 0 0
\(587\) −643.771 371.681i −1.09671 0.633188i −0.161358 0.986896i \(-0.551587\pi\)
−0.935356 + 0.353708i \(0.884921\pi\)
\(588\) 0 0
\(589\) −116.692 202.117i −0.198119 0.343152i
\(590\) 0 0
\(591\) 323.249 + 264.312i 0.546953 + 0.447228i
\(592\) 0 0
\(593\) −382.547 −0.645104 −0.322552 0.946552i \(-0.604541\pi\)
−0.322552 + 0.946552i \(0.604541\pi\)
\(594\) 0 0
\(595\) 279.287i 0.469391i
\(596\) 0 0
\(597\) −21.3328 + 26.0896i −0.0357333 + 0.0437012i
\(598\) 0 0
\(599\) −856.248 + 494.355i −1.42946 + 0.825301i −0.997078 0.0763888i \(-0.975661\pi\)
−0.432384 + 0.901689i \(0.642328\pi\)
\(600\) 0 0
\(601\) −263.280 + 456.015i −0.438070 + 0.758760i −0.997541 0.0700905i \(-0.977671\pi\)
0.559470 + 0.828850i \(0.311005\pi\)
\(602\) 0 0
\(603\) −263.587 298.391i −0.437127 0.494844i
\(604\) 0 0
\(605\) −31.9637 + 55.3627i −0.0528325 + 0.0915086i
\(606\) 0 0
\(607\) −447.631 + 258.440i −0.737448 + 0.425766i −0.821141 0.570726i \(-0.806662\pi\)
0.0836928 + 0.996492i \(0.473329\pi\)
\(608\) 0 0
\(609\) −698.826 114.085i −1.14750 0.187332i
\(610\) 0 0
\(611\) 1.07481i 0.00175910i
\(612\) 0 0
\(613\) −762.957 −1.24463 −0.622314 0.782768i \(-0.713807\pi\)
−0.622314 + 0.782768i \(0.713807\pi\)
\(614\) 0 0
\(615\) −322.928 + 122.205i −0.525086 + 0.198708i
\(616\) 0 0
\(617\) 60.9168 + 105.511i 0.0987307 + 0.171007i 0.911160 0.412054i \(-0.135188\pi\)
−0.812429 + 0.583060i \(0.801855\pi\)
\(618\) 0 0
\(619\) 265.675 + 153.388i 0.429200 + 0.247799i 0.699006 0.715116i \(-0.253626\pi\)
−0.269806 + 0.962915i \(0.586959\pi\)
\(620\) 0 0
\(621\) −21.7158 569.357i −0.0349692 0.916839i
\(622\) 0 0
\(623\) −650.636 375.645i −1.04436 0.602961i
\(624\) 0 0
\(625\) −63.7082 110.346i −0.101933 0.176553i
\(626\) 0 0
\(627\) −390.471 + 147.765i −0.622760 + 0.235670i
\(628\) 0 0
\(629\) −361.010 −0.573942
\(630\) 0 0
\(631\) 1071.11i 1.69749i −0.528805 0.848744i \(-0.677360\pi\)
0.528805 0.848744i \(-0.322640\pi\)
\(632\) 0 0
\(633\) 382.967 + 62.5205i 0.605003 + 0.0987685i
\(634\) 0 0
\(635\) −19.6623 + 11.3521i −0.0309643 + 0.0178773i
\(636\) 0 0
\(637\) 25.3193 43.8543i 0.0397477 0.0688450i
\(638\) 0 0
\(639\) −986.136 + 199.846i −1.54325 + 0.312748i
\(640\) 0 0
\(641\) 527.259 913.240i 0.822557 1.42471i −0.0812143 0.996697i \(-0.525880\pi\)
0.903772 0.428015i \(-0.140787\pi\)
\(642\) 0 0
\(643\) −42.0680 + 24.2880i −0.0654246 + 0.0377729i −0.532355 0.846521i \(-0.678693\pi\)
0.466931 + 0.884294i \(0.345360\pi\)
\(644\) 0 0
\(645\) 191.729 234.482i 0.297254 0.363537i
\(646\) 0 0
\(647\) 539.373i 0.833653i −0.908986 0.416826i \(-0.863142\pi\)
0.908986 0.416826i \(-0.136858\pi\)
\(648\) 0 0
\(649\) −874.061 −1.34678
\(650\) 0 0
\(651\) −445.831 364.543i −0.684840 0.559974i
\(652\) 0 0
\(653\) 276.457 + 478.838i 0.423365 + 0.733290i 0.996266 0.0863348i \(-0.0275155\pi\)
−0.572901 + 0.819624i \(0.694182\pi\)
\(654\) 0 0
\(655\) 312.081 + 180.180i 0.476460 + 0.275084i
\(656\) 0 0
\(657\) −136.213 672.139i −0.207326 1.02304i
\(658\) 0 0
\(659\) −734.162 423.869i −1.11406 0.643200i −0.174178 0.984714i \(-0.555727\pi\)
−0.939877 + 0.341514i \(0.889060\pi\)
\(660\) 0 0
\(661\) 359.447 + 622.580i 0.543792 + 0.941876i 0.998682 + 0.0513280i \(0.0163454\pi\)
−0.454890 + 0.890548i \(0.650321\pi\)
\(662\) 0 0
\(663\) 2.54250 15.5740i 0.00383485 0.0234902i
\(664\) 0 0
\(665\) −443.444 −0.666833
\(666\) 0 0
\(667\) 429.497i 0.643923i
\(668\) 0 0
\(669\) −256.504 677.812i −0.383414 1.01317i
\(670\) 0 0
\(671\) 623.893 360.205i 0.929796 0.536818i
\(672\) 0 0
\(673\) 288.488 499.675i 0.428659 0.742460i −0.568095 0.822963i \(-0.692319\pi\)
0.996754 + 0.0805033i \(0.0256528\pi\)
\(674\) 0 0
\(675\) −253.746 403.188i −0.375921 0.597316i
\(676\) 0 0
\(677\) 101.021 174.974i 0.149219 0.258454i −0.781720 0.623629i \(-0.785657\pi\)
0.930939 + 0.365175i \(0.118991\pi\)
\(678\) 0 0
\(679\) 72.1372 41.6484i 0.106240 0.0613379i
\(680\) 0 0
\(681\) −405.471 1071.46i −0.595405 1.57336i
\(682\) 0 0
\(683\) 568.249i 0.831990i −0.909367 0.415995i \(-0.863433\pi\)
0.909367 0.415995i \(-0.136567\pi\)
\(684\) 0 0
\(685\) −122.362 −0.178631
\(686\) 0 0
\(687\) −72.1793 + 442.132i −0.105065 + 0.643569i
\(688\) 0 0
\(689\) −6.24965 10.8247i −0.00907060 0.0157107i
\(690\) 0 0
\(691\) −351.376 202.867i −0.508504 0.293585i 0.223714 0.974655i \(-0.428182\pi\)
−0.732218 + 0.681070i \(0.761515\pi\)
\(692\) 0 0
\(693\) −772.100 + 682.045i −1.11414 + 0.984191i
\(694\) 0 0
\(695\) 354.600 + 204.728i 0.510215 + 0.294573i
\(696\) 0 0
\(697\) −188.407 326.330i −0.270311 0.468192i
\(698\) 0 0
\(699\) 508.464 + 415.757i 0.727416 + 0.594788i
\(700\) 0 0
\(701\) −83.5164 −0.119139 −0.0595695 0.998224i \(-0.518973\pi\)
−0.0595695 + 0.998224i \(0.518973\pi\)
\(702\) 0 0
\(703\) 573.200i 0.815363i
\(704\) 0 0
\(705\) 9.34506 11.4289i 0.0132554 0.0162112i
\(706\) 0 0
\(707\) −1114.86 + 643.663i −1.57688 + 0.910414i
\(708\) 0 0
\(709\) 173.908 301.217i 0.245286 0.424848i −0.716926 0.697149i \(-0.754452\pi\)
0.962212 + 0.272302i \(0.0877849\pi\)
\(710\) 0 0
\(711\) −27.4464 + 81.8205i −0.0386025 + 0.115078i
\(712\) 0 0
\(713\) −174.660 + 302.520i −0.244965 + 0.424292i
\(714\) 0 0
\(715\) 13.7337 7.92916i 0.0192080 0.0110897i
\(716\) 0 0
\(717\) 746.177 + 121.816i 1.04069 + 0.169896i
\(718\) 0 0
\(719\) 536.277i 0.745865i −0.927858 0.372933i \(-0.878352\pi\)
0.927858 0.372933i \(-0.121648\pi\)
\(720\) 0 0
\(721\) 1066.08 1.47862
\(722\) 0 0
\(723\) 1268.30 479.963i 1.75422 0.663849i
\(724\) 0 0
\(725\) 179.553 + 310.995i 0.247659 + 0.428959i
\(726\) 0 0
\(727\) −815.055 470.573i −1.12112 0.647280i −0.179435 0.983770i \(-0.557427\pi\)
−0.941687 + 0.336490i \(0.890760\pi\)
\(728\) 0 0
\(729\) −411.530 601.734i −0.564514 0.825424i
\(730\) 0 0
\(731\) 286.267 + 165.277i 0.391611 + 0.226096i
\(732\) 0 0
\(733\) −311.063 538.777i −0.424370 0.735030i 0.571991 0.820260i \(-0.306171\pi\)
−0.996361 + 0.0852294i \(0.972838\pi\)
\(734\) 0 0
\(735\) −650.525 + 246.178i −0.885068 + 0.334935i
\(736\) 0 0
\(737\) 436.655 0.592477
\(738\) 0 0
\(739\) 444.439i 0.601406i −0.953718 0.300703i \(-0.902779\pi\)
0.953718 0.300703i \(-0.0972213\pi\)
\(740\) 0 0
\(741\) −24.7279 4.03691i −0.0333710 0.00544792i
\(742\) 0 0
\(743\) 66.2270 38.2362i 0.0891346 0.0514619i −0.454770 0.890609i \(-0.650279\pi\)
0.543905 + 0.839147i \(0.316945\pi\)
\(744\) 0 0
\(745\) −193.626 + 335.370i −0.259901 + 0.450161i
\(746\) 0 0
\(747\) −243.333 + 725.402i −0.325747 + 0.971087i
\(748\) 0 0
\(749\) 624.725 1082.06i 0.834079 1.44467i
\(750\) 0 0
\(751\) 949.025 547.920i 1.26368 0.729587i 0.289897 0.957058i \(-0.406379\pi\)
0.973785 + 0.227471i \(0.0730457\pi\)
\(752\) 0 0
\(753\) −264.775 + 323.816i −0.351627 + 0.430034i
\(754\) 0 0
\(755\) 689.070i 0.912676i
\(756\) 0 0
\(757\) 346.346 0.457525 0.228762 0.973482i \(-0.426532\pi\)
0.228762 + 0.973482i \(0.426532\pi\)
\(758\) 0 0
\(759\) 483.759 + 395.556i 0.637363 + 0.521154i
\(760\) 0 0
\(761\) −106.565 184.576i −0.140033 0.242544i 0.787476 0.616345i \(-0.211387\pi\)
−0.927509 + 0.373802i \(0.878054\pi\)
\(762\) 0 0
\(763\) 869.291 + 501.885i 1.13931 + 0.657779i
\(764\) 0 0
\(765\) −162.445 + 143.498i −0.212347 + 0.187579i
\(766\) 0 0
\(767\) −45.4277 26.2277i −0.0592277 0.0341952i
\(768\) 0 0
\(769\) −270.786 469.015i −0.352127 0.609902i 0.634495 0.772927i \(-0.281208\pi\)
−0.986622 + 0.163025i \(0.947875\pi\)
\(770\) 0 0
\(771\) 227.477 1393.40i 0.295041 1.80726i
\(772\) 0 0
\(773\) −1255.73 −1.62449 −0.812245 0.583317i \(-0.801755\pi\)
−0.812245 + 0.583317i \(0.801755\pi\)
\(774\) 0 0
\(775\) 292.070i 0.376864i
\(776\) 0 0
\(777\) 500.611 + 1322.87i 0.644287 + 1.70253i
\(778\) 0 0
\(779\) −518.136 + 299.146i −0.665130 + 0.384013i
\(780\) 0 0
\(781\) 551.759 955.675i 0.706478 1.22366i
\(782\) 0 0
\(783\) 292.701 + 465.084i 0.373820 + 0.593977i
\(784\) 0 0
\(785\) −7.19706 + 12.4657i −0.00916822 + 0.0158798i
\(786\) 0 0
\(787\) 859.448 496.202i 1.09206 0.630499i 0.157933 0.987450i \(-0.449517\pi\)
0.934123 + 0.356951i \(0.116184\pi\)
\(788\) 0 0
\(789\) 27.2220 + 71.9343i 0.0345019 + 0.0911715i
\(790\) 0 0
\(791\) 54.5507i 0.0689643i
\(792\) 0 0
\(793\) 43.2342 0.0545198
\(794\) 0 0
\(795\) −27.6618 + 169.441i −0.0347947 + 0.213134i
\(796\) 0 0
\(797\) −319.262 552.978i −0.400580 0.693824i 0.593216 0.805043i \(-0.297858\pi\)
−0.993796 + 0.111219i \(0.964525\pi\)
\(798\) 0 0
\(799\) 13.9530 + 8.05574i 0.0174630 + 0.0100823i
\(800\) 0 0
\(801\) 115.807 + 571.445i 0.144577 + 0.713414i
\(802\) 0 0
\(803\) 651.377 + 376.073i 0.811179 + 0.468335i
\(804\) 0 0
\(805\) 331.865 + 574.806i 0.412254 + 0.714045i
\(806\) 0 0
\(807\) 18.9297 + 15.4783i 0.0234569 + 0.0191801i
\(808\) 0 0
\(809\) −174.260 −0.215401 −0.107701 0.994183i \(-0.534349\pi\)
−0.107701 + 0.994183i \(0.534349\pi\)
\(810\) 0 0
\(811\) 1182.19i 1.45770i 0.684675 + 0.728849i \(0.259944\pi\)
−0.684675 + 0.728849i \(0.740056\pi\)
\(812\) 0 0
\(813\) 763.360 933.577i 0.938942 1.14831i
\(814\) 0 0
\(815\) 139.880 80.7597i 0.171632 0.0990917i
\(816\) 0 0
\(817\) 262.421 454.527i 0.321201 0.556336i
\(818\) 0 0
\(819\) −60.5944 + 12.2798i −0.0739858 + 0.0149937i
\(820\) 0 0
\(821\) −293.955 + 509.144i −0.358045 + 0.620151i −0.987634 0.156777i \(-0.949890\pi\)
0.629590 + 0.776928i \(0.283223\pi\)
\(822\) 0 0
\(823\) −98.0750 + 56.6236i −0.119168 + 0.0688015i −0.558399 0.829573i \(-0.688584\pi\)
0.439231 + 0.898374i \(0.355251\pi\)
\(824\) 0 0
\(825\) 515.649 + 84.1813i 0.625029 + 0.102038i
\(826\) 0 0
\(827\) 800.560i 0.968030i −0.875060 0.484015i \(-0.839178\pi\)
0.875060 0.484015i \(-0.160822\pi\)
\(828\) 0 0
\(829\) −1162.87 −1.40274 −0.701369 0.712798i \(-0.747427\pi\)
−0.701369 + 0.712798i \(0.747427\pi\)
\(830\) 0 0
\(831\) 315.385 119.351i 0.379525 0.143623i
\(832\) 0 0
\(833\) −379.538 657.378i −0.455627 0.789170i
\(834\) 0 0
\(835\) −232.551 134.264i −0.278505 0.160795i
\(836\) 0 0
\(837\) 17.0344 + 446.617i 0.0203517 + 0.533592i
\(838\) 0 0
\(839\) 1355.92 + 782.842i 1.61612 + 0.933065i 0.987912 + 0.155017i \(0.0495432\pi\)
0.628205 + 0.778048i \(0.283790\pi\)
\(840\) 0 0
\(841\) 213.383 + 369.590i 0.253725 + 0.439464i
\(842\) 0 0
\(843\) −1508.78 + 570.966i −1.78978 + 0.677303i
\(844\) 0 0
\(845\) −457.406 −0.541309
\(846\) 0 0
\(847\) 273.341i 0.322717i
\(848\) 0 0
\(849\) 418.133 + 68.2615i 0.492501 + 0.0804022i
\(850\) 0 0
\(851\) 743.000 428.971i 0.873091 0.504079i
\(852\) 0 0
\(853\) 68.8088 119.180i 0.0806668 0.139719i −0.822870 0.568230i \(-0.807628\pi\)
0.903537 + 0.428511i \(0.140962\pi\)
\(854\) 0 0
\(855\) 227.842 + 257.926i 0.266482 + 0.301668i
\(856\) 0 0
\(857\) −384.489 + 665.955i −0.448646 + 0.777077i −0.998298 0.0583164i \(-0.981427\pi\)
0.549653 + 0.835393i \(0.314760\pi\)
\(858\) 0 0
\(859\) 178.353 102.972i 0.207629 0.119875i −0.392580 0.919718i \(-0.628417\pi\)
0.600209 + 0.799843i \(0.295084\pi\)
\(860\) 0 0
\(861\) −934.525 + 1142.91i −1.08540 + 1.32742i
\(862\) 0 0
\(863\) 772.757i 0.895431i 0.894176 + 0.447716i \(0.147762\pi\)
−0.894176 + 0.447716i \(0.852238\pi\)
\(864\) 0 0
\(865\) 104.671 0.121007
\(866\) 0 0
\(867\) 488.066 + 399.078i 0.562937 + 0.460298i
\(868\) 0 0
\(869\) −47.3249 81.9692i −0.0544591 0.0943258i
\(870\) 0 0
\(871\) 22.6943 + 13.1026i 0.0260555 + 0.0150432i
\(872\) 0 0
\(873\) −61.2887 20.5590i −0.0702047 0.0235499i
\(874\) 0 0
\(875\) 1161.57 + 670.630i 1.32750 + 0.766435i
\(876\) 0 0
\(877\) 200.096 + 346.577i 0.228160 + 0.395185i 0.957263 0.289219i \(-0.0933957\pi\)
−0.729103 + 0.684404i \(0.760062\pi\)
\(878\) 0 0
\(879\) 222.627 1363.69i 0.253273 1.55141i
\(880\) 0 0
\(881\) −728.323 −0.826700 −0.413350 0.910572i \(-0.635641\pi\)
−0.413350 + 0.910572i \(0.635641\pi\)
\(882\) 0 0
\(883\) 383.413i 0.434216i −0.976148 0.217108i \(-0.930338\pi\)
0.976148 0.217108i \(-0.0696625\pi\)
\(884\) 0 0
\(885\) 255.010 + 673.865i 0.288147 + 0.761429i
\(886\) 0 0
\(887\) −1255.91 + 725.098i −1.41590 + 0.817473i −0.995936 0.0900662i \(-0.971292\pi\)
−0.419968 + 0.907539i \(0.637959\pi\)
\(888\) 0 0
\(889\) −48.5392 + 84.0724i −0.0545998 + 0.0945696i
\(890\) 0 0
\(891\) 793.412 + 98.6511i 0.890474 + 0.110720i
\(892\) 0 0
\(893\) 12.7907 22.1541i 0.0143232 0.0248086i
\(894\) 0 0
\(895\) 85.5589 49.3974i 0.0955965 0.0551927i
\(896\) 0 0
\(897\) 13.2731 + 35.0743i 0.0147972 + 0.0391018i
\(898\) 0 0
\(899\) 336.907i 0.374758i
\(900\) 0 0
\(901\) −187.365 −0.207952
\(902\) 0 0
\(903\) 208.666 1278.17i 0.231080 1.41547i
\(904\) 0 0
\(905\) −25.2125 43.6694i −0.0278592 0.0482535i
\(906\) 0 0
\(907\) −533.290 307.895i −0.587971 0.339465i 0.176324 0.984332i \(-0.443580\pi\)
−0.764295 + 0.644867i \(0.776913\pi\)
\(908\) 0 0
\(909\) 947.196 + 317.733i 1.04202 + 0.349541i
\(910\) 0 0
\(911\) −227.031 131.076i −0.249211 0.143882i 0.370192 0.928955i \(-0.379292\pi\)
−0.619403 + 0.785073i \(0.712625\pi\)
\(912\) 0 0
\(913\) −419.572 726.720i −0.459553 0.795969i
\(914\) 0 0
\(915\) −459.726 375.905i −0.502433 0.410825i
\(916\) 0 0
\(917\) 1540.83 1.68030
\(918\) 0 0
\(919\) 566.458i 0.616385i −0.951324 0.308193i \(-0.900276\pi\)
0.951324 0.308193i \(-0.0997241\pi\)
\(920\) 0 0
\(921\) −399.403 + 488.464i −0.433663 + 0.530362i
\(922\) 0 0
\(923\) 57.3533 33.1130i 0.0621379 0.0358754i
\(924\) 0 0
\(925\) 358.667 621.229i 0.387748 0.671599i
\(926\) 0 0
\(927\) −547.755 620.080i −0.590890 0.668910i
\(928\) 0 0
\(929\) 298.745 517.442i 0.321577 0.556988i −0.659236 0.751936i \(-0.729120\pi\)
0.980814 + 0.194948i \(0.0624537\pi\)
\(930\) 0 0
\(931\) −1043.77 + 602.618i −1.12112 + 0.647281i
\(932\) 0 0
\(933\) 379.469 + 61.9495i 0.406719 + 0.0663982i
\(934\) 0 0
\(935\) 237.717i 0.254243i
\(936\) 0 0
\(937\) 457.785 0.488564 0.244282 0.969704i \(-0.421448\pi\)
0.244282 + 0.969704i \(0.421448\pi\)
\(938\) 0 0
\(939\) 20.3220 7.69041i 0.0216421 0.00819000i
\(940\) 0 0
\(941\) 677.858 + 1174.09i 0.720360 + 1.24770i 0.960856 + 0.277049i \(0.0893566\pi\)
−0.240496 + 0.970650i \(0.577310\pi\)
\(942\) 0 0
\(943\) 775.526 + 447.750i 0.822403 + 0.474814i
\(944\) 0 0
\(945\) 751.091 + 396.269i 0.794805 + 0.419332i
\(946\) 0 0
\(947\) −567.925 327.891i −0.599709 0.346242i 0.169218 0.985579i \(-0.445876\pi\)
−0.768927 + 0.639336i \(0.779209\pi\)
\(948\) 0 0
\(949\) 22.5694 + 39.0914i 0.0237823 + 0.0411922i
\(950\) 0 0
\(951\) −674.210 + 255.141i −0.708949 + 0.268287i
\(952\) 0 0
\(953\) 554.778 0.582139 0.291069 0.956702i \(-0.405989\pi\)
0.291069 + 0.956702i \(0.405989\pi\)
\(954\) 0 0
\(955\) 767.193i 0.803344i
\(956\) 0 0
\(957\) −594.810 97.1045i −0.621536 0.101468i
\(958\) 0 0
\(959\) −453.102 + 261.599i −0.472474 + 0.272783i
\(960\) 0 0
\(961\) −343.493 + 594.947i −0.357432 + 0.619091i
\(962\) 0 0
\(963\) −950.354 + 192.595i −0.986869 + 0.199995i
\(964\) 0 0
\(965\) −411.009 + 711.889i −0.425917 + 0.737709i
\(966\) 0 0
\(967\) −502.000 + 289.830i −0.519131 + 0.299720i −0.736579 0.676351i \(-0.763560\pi\)
0.217448 + 0.976072i \(0.430227\pi\)
\(968\) 0 0
\(969\) −237.743 + 290.756i −0.245349 + 0.300058i
\(970\) 0 0
\(971\) 260.660i 0.268445i 0.990951 + 0.134223i \(0.0428537\pi\)
−0.990951 + 0.134223i \(0.957146\pi\)
\(972\) 0 0
\(973\) 1750.76 1.79934
\(974\) 0 0
\(975\) 24.2739 + 19.8481i 0.0248963 + 0.0203570i
\(976\) 0 0
\(977\) 638.953 + 1106.70i 0.653995 + 1.13275i 0.982145 + 0.188127i \(0.0602417\pi\)
−0.328149 + 0.944626i \(0.606425\pi\)
\(978\) 0 0
\(979\) −553.793 319.732i −0.565672 0.326591i
\(980\) 0 0
\(981\) −154.725 763.486i −0.157721 0.778273i
\(982\) 0 0
\(983\) −378.325 218.426i −0.384868 0.222204i 0.295066 0.955477i \(-0.404658\pi\)
−0.679934 + 0.733273i \(0.737992\pi\)
\(984\) 0 0
\(985\) 188.746 + 326.918i 0.191621 + 0.331897i
\(986\) 0 0
\(987\) 10.1706 62.2994i 0.0103045 0.0631200i
\(988\) 0 0
\(989\) −785.562 −0.794300
\(990\) 0 0
\(991\) 1344.50i 1.35671i 0.734734 + 0.678356i \(0.237307\pi\)
−0.734734 + 0.678356i \(0.762693\pi\)
\(992\) 0 0
\(993\) −454.114 1200.00i −0.457316 1.20846i
\(994\) 0 0
\(995\) −26.3858 + 15.2338i −0.0265184 + 0.0153104i
\(996\) 0 0
\(997\) −942.397 + 1632.28i −0.945233 + 1.63719i −0.189949 + 0.981794i \(0.560832\pi\)
−0.755284 + 0.655398i \(0.772501\pi\)
\(998\) 0 0
\(999\) 512.221 970.868i 0.512734 0.971839i
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 576.3.o.g.511.3 16
3.2 odd 2 1728.3.o.g.127.5 16
4.3 odd 2 inner 576.3.o.g.511.6 16
8.3 odd 2 36.3.f.c.7.7 yes 16
8.5 even 2 36.3.f.c.7.6 16
9.4 even 3 inner 576.3.o.g.319.6 16
9.5 odd 6 1728.3.o.g.1279.6 16
12.11 even 2 1728.3.o.g.127.6 16
24.5 odd 2 108.3.f.c.19.3 16
24.11 even 2 108.3.f.c.19.2 16
36.23 even 6 1728.3.o.g.1279.5 16
36.31 odd 6 inner 576.3.o.g.319.3 16
72.5 odd 6 108.3.f.c.91.2 16
72.11 even 6 324.3.d.g.163.8 8
72.13 even 6 36.3.f.c.31.7 yes 16
72.29 odd 6 324.3.d.g.163.7 8
72.43 odd 6 324.3.d.i.163.1 8
72.59 even 6 108.3.f.c.91.3 16
72.61 even 6 324.3.d.i.163.2 8
72.67 odd 6 36.3.f.c.31.6 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
36.3.f.c.7.6 16 8.5 even 2
36.3.f.c.7.7 yes 16 8.3 odd 2
36.3.f.c.31.6 yes 16 72.67 odd 6
36.3.f.c.31.7 yes 16 72.13 even 6
108.3.f.c.19.2 16 24.11 even 2
108.3.f.c.19.3 16 24.5 odd 2
108.3.f.c.91.2 16 72.5 odd 6
108.3.f.c.91.3 16 72.59 even 6
324.3.d.g.163.7 8 72.29 odd 6
324.3.d.g.163.8 8 72.11 even 6
324.3.d.i.163.1 8 72.43 odd 6
324.3.d.i.163.2 8 72.61 even 6
576.3.o.g.319.3 16 36.31 odd 6 inner
576.3.o.g.319.6 16 9.4 even 3 inner
576.3.o.g.511.3 16 1.1 even 1 trivial
576.3.o.g.511.6 16 4.3 odd 2 inner
1728.3.o.g.127.5 16 3.2 odd 2
1728.3.o.g.127.6 16 12.11 even 2
1728.3.o.g.1279.5 16 36.23 even 6
1728.3.o.g.1279.6 16 9.5 odd 6