Properties

Label 2268.2.t.c.1781.10
Level $2268$
Weight $2$
Character 2268.1781
Analytic conductor $18.110$
Analytic rank $0$
Dimension $32$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2268,2,Mod(1781,2268)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2268, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2268.1781");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2268.t (of order \(6\), degree \(2\), 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: \(18.1100711784\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1781.10
Character \(\chi\) \(=\) 2268.1781
Dual form 2268.2.t.c.2105.10

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.440135 + 0.762336i) q^{5} +(0.391046 + 2.61669i) q^{7} +(4.83660 + 2.79241i) q^{11} -2.59584i q^{13} +(2.35671 - 4.08193i) q^{17} +(0.537797 - 0.310497i) q^{19} +(-3.15847 + 1.82354i) q^{23} +(2.11256 - 3.65906i) q^{25} -5.33890i q^{29} +(9.56034 + 5.51967i) q^{31} +(-1.82269 + 1.44981i) q^{35} +(4.28064 + 7.41428i) q^{37} -6.82696 q^{41} +3.65862 q^{43} +(-3.88689 - 6.73229i) q^{47} +(-6.69417 + 2.04650i) q^{49} +(5.87725 + 3.39323i) q^{53} +4.91616i q^{55} +(-5.35626 + 9.27732i) q^{59} +(-1.77710 + 1.02601i) q^{61} +(1.97890 - 1.14252i) q^{65} +(-4.52099 + 7.83059i) q^{67} -12.3011i q^{71} +(-3.43707 - 1.98439i) q^{73} +(-5.41555 + 13.7479i) q^{77} +(6.01045 + 10.4104i) q^{79} +3.43944 q^{83} +4.14908 q^{85} +(3.40619 + 5.89969i) q^{89} +(6.79251 - 1.01509i) q^{91} +(0.473407 + 0.273322i) q^{95} +16.2182i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q - 8 q^{7} - 16 q^{25} + 24 q^{31} - 4 q^{37} + 8 q^{43} - 4 q^{49} + 12 q^{61} + 4 q^{67} - 36 q^{73} + 28 q^{79} - 24 q^{85} - 36 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1135\) \(1541\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 0.440135 + 0.762336i 0.196834 + 0.340927i 0.947500 0.319755i \(-0.103601\pi\)
−0.750666 + 0.660682i \(0.770267\pi\)
\(6\) 0 0
\(7\) 0.391046 + 2.61669i 0.147802 + 0.989017i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.83660 + 2.79241i 1.45829 + 0.841944i 0.998927 0.0463050i \(-0.0147446\pi\)
0.459362 + 0.888249i \(0.348078\pi\)
\(12\) 0 0
\(13\) 2.59584i 0.719956i −0.932961 0.359978i \(-0.882784\pi\)
0.932961 0.359978i \(-0.117216\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.35671 4.08193i 0.571585 0.990015i −0.424818 0.905279i \(-0.639662\pi\)
0.996403 0.0847359i \(-0.0270047\pi\)
\(18\) 0 0
\(19\) 0.537797 0.310497i 0.123379 0.0712330i −0.437040 0.899442i \(-0.643973\pi\)
0.560419 + 0.828209i \(0.310640\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.15847 + 1.82354i −0.658586 + 0.380235i −0.791738 0.610861i \(-0.790823\pi\)
0.133152 + 0.991096i \(0.457490\pi\)
\(24\) 0 0
\(25\) 2.11256 3.65906i 0.422512 0.731813i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 5.33890i 0.991409i −0.868491 0.495705i \(-0.834910\pi\)
0.868491 0.495705i \(-0.165090\pi\)
\(30\) 0 0
\(31\) 9.56034 + 5.51967i 1.71709 + 0.991361i 0.924123 + 0.382094i \(0.124797\pi\)
0.792965 + 0.609267i \(0.208536\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.82269 + 1.44981i −0.308090 + 0.245062i
\(36\) 0 0
\(37\) 4.28064 + 7.41428i 0.703732 + 1.21890i 0.967147 + 0.254218i \(0.0818179\pi\)
−0.263415 + 0.964683i \(0.584849\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.82696 −1.06619 −0.533096 0.846055i \(-0.678972\pi\)
−0.533096 + 0.846055i \(0.678972\pi\)
\(42\) 0 0
\(43\) 3.65862 0.557935 0.278967 0.960301i \(-0.410008\pi\)
0.278967 + 0.960301i \(0.410008\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.88689 6.73229i −0.566961 0.982005i −0.996864 0.0791303i \(-0.974786\pi\)
0.429903 0.902875i \(-0.358548\pi\)
\(48\) 0 0
\(49\) −6.69417 + 2.04650i −0.956309 + 0.292357i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 5.87725 + 3.39323i 0.807303 + 0.466096i 0.846018 0.533154i \(-0.178993\pi\)
−0.0387156 + 0.999250i \(0.512327\pi\)
\(54\) 0 0
\(55\) 4.91616i 0.662894i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −5.35626 + 9.27732i −0.697326 + 1.20780i 0.272064 + 0.962279i \(0.412294\pi\)
−0.969390 + 0.245525i \(0.921040\pi\)
\(60\) 0 0
\(61\) −1.77710 + 1.02601i −0.227534 + 0.131367i −0.609434 0.792837i \(-0.708603\pi\)
0.381900 + 0.924204i \(0.375270\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.97890 1.14252i 0.245453 0.141712i
\(66\) 0 0
\(67\) −4.52099 + 7.83059i −0.552327 + 0.956659i 0.445779 + 0.895143i \(0.352927\pi\)
−0.998106 + 0.0615158i \(0.980407\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 12.3011i 1.45988i −0.683514 0.729938i \(-0.739549\pi\)
0.683514 0.729938i \(-0.260451\pi\)
\(72\) 0 0
\(73\) −3.43707 1.98439i −0.402278 0.232255i 0.285188 0.958471i \(-0.407944\pi\)
−0.687466 + 0.726216i \(0.741277\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −5.41555 + 13.7479i −0.617159 + 1.56671i
\(78\) 0 0
\(79\) 6.01045 + 10.4104i 0.676228 + 1.17126i 0.976108 + 0.217285i \(0.0697200\pi\)
−0.299880 + 0.953977i \(0.596947\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.43944 0.377527 0.188764 0.982023i \(-0.439552\pi\)
0.188764 + 0.982023i \(0.439552\pi\)
\(84\) 0 0
\(85\) 4.14908 0.450031
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.40619 + 5.89969i 0.361055 + 0.625366i 0.988135 0.153589i \(-0.0490833\pi\)
−0.627080 + 0.778955i \(0.715750\pi\)
\(90\) 0 0
\(91\) 6.79251 1.01509i 0.712049 0.106411i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0.473407 + 0.273322i 0.0485705 + 0.0280422i
\(96\) 0 0
\(97\) 16.2182i 1.64670i 0.567530 + 0.823352i \(0.307899\pi\)
−0.567530 + 0.823352i \(0.692101\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 6.75727 11.7039i 0.672373 1.16458i −0.304856 0.952398i \(-0.598608\pi\)
0.977229 0.212186i \(-0.0680583\pi\)
\(102\) 0 0
\(103\) 10.9579 6.32656i 1.07972 0.623374i 0.148896 0.988853i \(-0.452428\pi\)
0.930820 + 0.365479i \(0.119095\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −2.09662 + 1.21048i −0.202688 + 0.117022i −0.597909 0.801564i \(-0.704001\pi\)
0.395221 + 0.918586i \(0.370668\pi\)
\(108\) 0 0
\(109\) 0.286415 0.496086i 0.0274336 0.0475164i −0.851983 0.523570i \(-0.824600\pi\)
0.879416 + 0.476054i \(0.157933\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 10.6032i 0.997461i 0.866757 + 0.498731i \(0.166200\pi\)
−0.866757 + 0.498731i \(0.833800\pi\)
\(114\) 0 0
\(115\) −2.78030 1.60521i −0.259265 0.149687i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 11.6028 + 4.57055i 1.06362 + 0.418982i
\(120\) 0 0
\(121\) 10.0951 + 17.4853i 0.917739 + 1.58957i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 8.12060 0.726329
\(126\) 0 0
\(127\) 2.85582 0.253413 0.126707 0.991940i \(-0.459559\pi\)
0.126707 + 0.991940i \(0.459559\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −5.80788 10.0595i −0.507437 0.878906i −0.999963 0.00860838i \(-0.997260\pi\)
0.492526 0.870298i \(-0.336074\pi\)
\(132\) 0 0
\(133\) 1.02278 + 1.28583i 0.0886863 + 0.111496i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −6.41744 3.70511i −0.548279 0.316549i 0.200149 0.979766i \(-0.435857\pi\)
−0.748427 + 0.663217i \(0.769191\pi\)
\(138\) 0 0
\(139\) 3.41627i 0.289764i −0.989449 0.144882i \(-0.953720\pi\)
0.989449 0.144882i \(-0.0462802\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 7.24865 12.5550i 0.606162 1.04990i
\(144\) 0 0
\(145\) 4.07004 2.34984i 0.337998 0.195143i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −8.81086 + 5.08695i −0.721814 + 0.416739i −0.815420 0.578870i \(-0.803494\pi\)
0.0936062 + 0.995609i \(0.470161\pi\)
\(150\) 0 0
\(151\) 2.15352 3.73001i 0.175251 0.303544i −0.764997 0.644034i \(-0.777260\pi\)
0.940248 + 0.340490i \(0.110593\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 9.71760i 0.780536i
\(156\) 0 0
\(157\) −10.2160 5.89822i −0.815327 0.470729i 0.0334752 0.999440i \(-0.489343\pi\)
−0.848802 + 0.528710i \(0.822676\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −6.00675 7.55165i −0.473399 0.595153i
\(162\) 0 0
\(163\) 5.81021 + 10.0636i 0.455090 + 0.788240i 0.998693 0.0511028i \(-0.0162736\pi\)
−0.543603 + 0.839342i \(0.682940\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −15.7708 −1.22038 −0.610189 0.792256i \(-0.708907\pi\)
−0.610189 + 0.792256i \(0.708907\pi\)
\(168\) 0 0
\(169\) 6.26163 0.481664
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −1.38670 2.40184i −0.105429 0.182609i 0.808484 0.588518i \(-0.200288\pi\)
−0.913913 + 0.405909i \(0.866955\pi\)
\(174\) 0 0
\(175\) 10.4008 + 4.09706i 0.786223 + 0.309709i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −5.94403 3.43179i −0.444278 0.256504i 0.261133 0.965303i \(-0.415904\pi\)
−0.705410 + 0.708799i \(0.749237\pi\)
\(180\) 0 0
\(181\) 9.85094i 0.732215i −0.930573 0.366107i \(-0.880690\pi\)
0.930573 0.366107i \(-0.119310\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −3.76812 + 6.52657i −0.277038 + 0.479843i
\(186\) 0 0
\(187\) 22.7969 13.1618i 1.66707 0.962485i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 15.5684 8.98842i 1.12649 0.650380i 0.183440 0.983031i \(-0.441277\pi\)
0.943050 + 0.332651i \(0.107943\pi\)
\(192\) 0 0
\(193\) 1.88040 3.25694i 0.135354 0.234440i −0.790379 0.612619i \(-0.790116\pi\)
0.925733 + 0.378179i \(0.123450\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 6.88342i 0.490424i 0.969470 + 0.245212i \(0.0788575\pi\)
−0.969470 + 0.245212i \(0.921142\pi\)
\(198\) 0 0
\(199\) 12.0894 + 6.97981i 0.856993 + 0.494785i 0.863004 0.505197i \(-0.168580\pi\)
−0.00601088 + 0.999982i \(0.501913\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 13.9703 2.08776i 0.980521 0.146532i
\(204\) 0 0
\(205\) −3.00479 5.20444i −0.209863 0.363494i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 3.46815 0.239897
\(210\) 0 0
\(211\) −2.52094 −0.173549 −0.0867744 0.996228i \(-0.527656\pi\)
−0.0867744 + 0.996228i \(0.527656\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 1.61029 + 2.78910i 0.109821 + 0.190215i
\(216\) 0 0
\(217\) −10.7047 + 27.1749i −0.726685 + 1.84475i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −10.5960 6.11763i −0.712767 0.411516i
\(222\) 0 0
\(223\) 12.3148i 0.824663i −0.911034 0.412332i \(-0.864715\pi\)
0.911034 0.412332i \(-0.135285\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −12.8806 + 22.3098i −0.854913 + 1.48075i 0.0218124 + 0.999762i \(0.493056\pi\)
−0.876726 + 0.480991i \(0.840277\pi\)
\(228\) 0 0
\(229\) −5.01900 + 2.89772i −0.331665 + 0.191487i −0.656580 0.754256i \(-0.727998\pi\)
0.324915 + 0.945743i \(0.394664\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −26.2280 + 15.1428i −1.71826 + 0.992036i −0.796149 + 0.605101i \(0.793133\pi\)
−0.922107 + 0.386935i \(0.873534\pi\)
\(234\) 0 0
\(235\) 3.42151 5.92623i 0.223195 0.386585i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 3.41885i 0.221147i 0.993868 + 0.110573i \(0.0352688\pi\)
−0.993868 + 0.110573i \(0.964731\pi\)
\(240\) 0 0
\(241\) −4.24030 2.44814i −0.273142 0.157699i 0.357173 0.934038i \(-0.383741\pi\)
−0.630315 + 0.776340i \(0.717074\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −4.50646 4.20247i −0.287907 0.268486i
\(246\) 0 0
\(247\) −0.806001 1.39603i −0.0512846 0.0888275i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 27.2710 1.72133 0.860667 0.509169i \(-0.170047\pi\)
0.860667 + 0.509169i \(0.170047\pi\)
\(252\) 0 0
\(253\) −20.3683 −1.28054
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2.56317 + 4.43954i 0.159886 + 0.276931i 0.934827 0.355102i \(-0.115554\pi\)
−0.774941 + 0.632033i \(0.782221\pi\)
\(258\) 0 0
\(259\) −17.7270 + 14.1004i −1.10150 + 0.876159i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −4.13580 2.38780i −0.255024 0.147238i 0.367038 0.930206i \(-0.380372\pi\)
−0.622063 + 0.782967i \(0.713705\pi\)
\(264\) 0 0
\(265\) 5.97392i 0.366975i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 7.34405 12.7203i 0.447775 0.775569i −0.550466 0.834858i \(-0.685550\pi\)
0.998241 + 0.0592889i \(0.0188833\pi\)
\(270\) 0 0
\(271\) 9.13087 5.27171i 0.554661 0.320233i −0.196339 0.980536i \(-0.562905\pi\)
0.751000 + 0.660303i \(0.229572\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 20.4352 11.7983i 1.23229 0.711464i
\(276\) 0 0
\(277\) −13.7907 + 23.8862i −0.828602 + 1.43518i 0.0705327 + 0.997509i \(0.477530\pi\)
−0.899135 + 0.437672i \(0.855803\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 28.5861i 1.70530i −0.522479 0.852652i \(-0.674993\pi\)
0.522479 0.852652i \(-0.325007\pi\)
\(282\) 0 0
\(283\) −16.3146 9.41927i −0.969805 0.559917i −0.0706283 0.997503i \(-0.522500\pi\)
−0.899177 + 0.437585i \(0.855834\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2.66966 17.8641i −0.157585 1.05448i
\(288\) 0 0
\(289\) −2.60813 4.51741i −0.153419 0.265730i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 33.6167 1.96391 0.981954 0.189121i \(-0.0605638\pi\)
0.981954 + 0.189121i \(0.0605638\pi\)
\(294\) 0 0
\(295\) −9.42992 −0.549031
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 4.73362 + 8.19886i 0.273752 + 0.474153i
\(300\) 0 0
\(301\) 1.43069 + 9.57350i 0.0824637 + 0.551807i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −1.56433 0.903164i −0.0895731 0.0517150i
\(306\) 0 0
\(307\) 14.6814i 0.837913i −0.908006 0.418957i \(-0.862396\pi\)
0.908006 0.418957i \(-0.137604\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −5.95735 + 10.3184i −0.337810 + 0.585104i −0.984021 0.178055i \(-0.943020\pi\)
0.646210 + 0.763159i \(0.276353\pi\)
\(312\) 0 0
\(313\) −5.33683 + 3.08122i −0.301655 + 0.174161i −0.643186 0.765710i \(-0.722388\pi\)
0.341531 + 0.939871i \(0.389055\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 3.33379 1.92476i 0.187244 0.108105i −0.403448 0.915003i \(-0.632188\pi\)
0.590692 + 0.806897i \(0.298855\pi\)
\(318\) 0 0
\(319\) 14.9084 25.8221i 0.834711 1.44576i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 2.92700i 0.162863i
\(324\) 0 0
\(325\) −9.49834 5.48387i −0.526873 0.304190i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 16.0964 12.8034i 0.887422 0.705876i
\(330\) 0 0
\(331\) 8.81578 + 15.2694i 0.484559 + 0.839281i 0.999843 0.0177388i \(-0.00564674\pi\)
−0.515284 + 0.857020i \(0.672313\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −7.95939 −0.434868
\(336\) 0 0
\(337\) −19.6690 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 30.8264 + 53.3928i 1.66934 + 2.89138i
\(342\) 0 0
\(343\) −7.97278 16.7163i −0.430490 0.902595i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −12.5819 7.26418i −0.675433 0.389962i 0.122699 0.992444i \(-0.460845\pi\)
−0.798132 + 0.602482i \(0.794178\pi\)
\(348\) 0 0
\(349\) 31.1316i 1.66644i −0.552944 0.833218i \(-0.686496\pi\)
0.552944 0.833218i \(-0.313504\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −4.11432 + 7.12621i −0.218983 + 0.379290i −0.954497 0.298219i \(-0.903607\pi\)
0.735514 + 0.677509i \(0.236941\pi\)
\(354\) 0 0
\(355\) 9.37760 5.41416i 0.497711 0.287354i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −25.3084 + 14.6118i −1.33573 + 0.771183i −0.986171 0.165732i \(-0.947001\pi\)
−0.349557 + 0.936915i \(0.613668\pi\)
\(360\) 0 0
\(361\) −9.30718 + 16.1205i −0.489852 + 0.848448i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 3.49360i 0.182863i
\(366\) 0 0
\(367\) 0.491228 + 0.283610i 0.0256419 + 0.0148043i 0.512766 0.858528i \(-0.328621\pi\)
−0.487124 + 0.873333i \(0.661954\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −6.58077 + 16.7059i −0.341657 + 0.867326i
\(372\) 0 0
\(373\) −0.901927 1.56218i −0.0467000 0.0808868i 0.841731 0.539898i \(-0.181537\pi\)
−0.888431 + 0.459011i \(0.848204\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −13.8589 −0.713771
\(378\) 0 0
\(379\) −2.38270 −0.122391 −0.0611956 0.998126i \(-0.519491\pi\)
−0.0611956 + 0.998126i \(0.519491\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −11.6264 20.1375i −0.594080 1.02898i −0.993676 0.112286i \(-0.964183\pi\)
0.399596 0.916691i \(-0.369150\pi\)
\(384\) 0 0
\(385\) −12.8641 + 1.92244i −0.655614 + 0.0979768i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −18.2236 10.5214i −0.923976 0.533458i −0.0390745 0.999236i \(-0.512441\pi\)
−0.884901 + 0.465779i \(0.845774\pi\)
\(390\) 0 0
\(391\) 17.1902i 0.869346i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −5.29082 + 9.16397i −0.266210 + 0.461089i
\(396\) 0 0
\(397\) 23.7334 13.7025i 1.19114 0.687708i 0.232578 0.972578i \(-0.425284\pi\)
0.958566 + 0.284870i \(0.0919505\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 31.8620 18.3955i 1.59111 0.918629i 0.597995 0.801500i \(-0.295964\pi\)
0.993117 0.117129i \(-0.0373690\pi\)
\(402\) 0 0
\(403\) 14.3282 24.8171i 0.713736 1.23623i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 47.8132i 2.37001i
\(408\) 0 0
\(409\) −29.1585 16.8347i −1.44180 0.832422i −0.443828 0.896112i \(-0.646380\pi\)
−0.997970 + 0.0636901i \(0.979713\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −26.3705 10.3878i −1.29761 0.511152i
\(414\) 0 0
\(415\) 1.51382 + 2.62201i 0.0743104 + 0.128709i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −9.19382 −0.449148 −0.224574 0.974457i \(-0.572099\pi\)
−0.224574 + 0.974457i \(0.572099\pi\)
\(420\) 0 0
\(421\) −8.60745 −0.419502 −0.209751 0.977755i \(-0.567265\pi\)
−0.209751 + 0.977755i \(0.567265\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −9.95738 17.2467i −0.483004 0.836587i
\(426\) 0 0
\(427\) −3.37968 4.24890i −0.163554 0.205619i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 6.90554 + 3.98691i 0.332628 + 0.192043i 0.657007 0.753884i \(-0.271822\pi\)
−0.324379 + 0.945927i \(0.605155\pi\)
\(432\) 0 0
\(433\) 1.42915i 0.0686806i −0.999410 0.0343403i \(-0.989067\pi\)
0.999410 0.0343403i \(-0.0109330\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −1.13241 + 1.96139i −0.0541705 + 0.0938260i
\(438\) 0 0
\(439\) 22.0686 12.7413i 1.05328 0.608110i 0.129713 0.991552i \(-0.458594\pi\)
0.923565 + 0.383441i \(0.125261\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −4.81934 + 2.78244i −0.228974 + 0.132198i −0.610098 0.792326i \(-0.708870\pi\)
0.381125 + 0.924524i \(0.375537\pi\)
\(444\) 0 0
\(445\) −2.99837 + 5.19332i −0.142136 + 0.246187i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 26.4132i 1.24652i −0.782016 0.623259i \(-0.785808\pi\)
0.782016 0.623259i \(-0.214192\pi\)
\(450\) 0 0
\(451\) −33.0193 19.0637i −1.55482 0.897674i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 3.76346 + 4.73140i 0.176434 + 0.221811i
\(456\) 0 0
\(457\) −11.1809 19.3659i −0.523021 0.905899i −0.999641 0.0267896i \(-0.991472\pi\)
0.476620 0.879109i \(-0.341862\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −26.5005 −1.23425 −0.617126 0.786864i \(-0.711703\pi\)
−0.617126 + 0.786864i \(0.711703\pi\)
\(462\) 0 0
\(463\) −14.4875 −0.673293 −0.336646 0.941631i \(-0.609293\pi\)
−0.336646 + 0.941631i \(0.609293\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −7.21034 12.4887i −0.333655 0.577907i 0.649571 0.760301i \(-0.274949\pi\)
−0.983226 + 0.182394i \(0.941615\pi\)
\(468\) 0 0
\(469\) −22.2582 8.76793i −1.02779 0.404865i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 17.6953 + 10.2164i 0.813631 + 0.469750i
\(474\) 0 0
\(475\) 2.62378i 0.120387i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −14.6311 + 25.3418i −0.668513 + 1.15790i 0.309807 + 0.950799i \(0.399736\pi\)
−0.978320 + 0.207099i \(0.933598\pi\)
\(480\) 0 0
\(481\) 19.2463 11.1118i 0.877554 0.506656i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −12.3637 + 7.13818i −0.561407 + 0.324128i
\(486\) 0 0
\(487\) −3.23843 + 5.60913i −0.146747 + 0.254174i −0.930024 0.367500i \(-0.880214\pi\)
0.783276 + 0.621674i \(0.213547\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 35.5282i 1.60337i −0.597750 0.801683i \(-0.703938\pi\)
0.597750 0.801683i \(-0.296062\pi\)
\(492\) 0 0
\(493\) −21.7930 12.5822i −0.981510 0.566675i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 32.1883 4.81031i 1.44384 0.215772i
\(498\) 0 0
\(499\) 16.0108 + 27.7315i 0.716742 + 1.24143i 0.962284 + 0.272047i \(0.0877006\pi\)
−0.245542 + 0.969386i \(0.578966\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −11.2012 −0.499437 −0.249718 0.968319i \(-0.580338\pi\)
−0.249718 + 0.968319i \(0.580338\pi\)
\(504\) 0 0
\(505\) 11.8964 0.529385
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −18.9860 32.8848i −0.841541 1.45759i −0.888591 0.458699i \(-0.848315\pi\)
0.0470504 0.998893i \(-0.485018\pi\)
\(510\) 0 0
\(511\) 3.84849 9.76974i 0.170247 0.432188i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 9.64593 + 5.56908i 0.425050 + 0.245403i
\(516\) 0 0
\(517\) 43.4152i 1.90940i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0.318682 0.551974i 0.0139617 0.0241824i −0.858960 0.512042i \(-0.828889\pi\)
0.872922 + 0.487860i \(0.162222\pi\)
\(522\) 0 0
\(523\) 15.4557 8.92337i 0.675832 0.390192i −0.122451 0.992475i \(-0.539075\pi\)
0.798283 + 0.602283i \(0.205742\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 45.0618 26.0165i 1.96292 1.13330i
\(528\) 0 0
\(529\) −4.84940 + 8.39940i −0.210843 + 0.365191i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 17.7217i 0.767612i
\(534\) 0 0
\(535\) −1.84559 1.06555i −0.0797920 0.0460679i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −38.0917 8.79479i −1.64072 0.378818i
\(540\) 0 0
\(541\) 6.65241 + 11.5223i 0.286009 + 0.495383i 0.972853 0.231422i \(-0.0743378\pi\)
−0.686844 + 0.726805i \(0.741004\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0.504246 0.0215995
\(546\) 0 0
\(547\) −15.3929 −0.658154 −0.329077 0.944303i \(-0.606738\pi\)
−0.329077 + 0.944303i \(0.606738\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −1.65771 2.87125i −0.0706210 0.122319i
\(552\) 0 0
\(553\) −24.8905 + 19.7984i −1.05845 + 0.841915i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 18.1119 + 10.4569i 0.767426 + 0.443074i 0.831956 0.554842i \(-0.187221\pi\)
−0.0645293 + 0.997916i \(0.520555\pi\)
\(558\) 0 0
\(559\) 9.49719i 0.401688i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 9.73637 16.8639i 0.410339 0.710728i −0.584588 0.811331i \(-0.698744\pi\)
0.994927 + 0.100603i \(0.0320771\pi\)
\(564\) 0 0
\(565\) −8.08317 + 4.66682i −0.340062 + 0.196335i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 2.73683 1.58011i 0.114734 0.0662417i −0.441535 0.897244i \(-0.645566\pi\)
0.556269 + 0.831002i \(0.312233\pi\)
\(570\) 0 0
\(571\) 11.5293 19.9693i 0.482486 0.835690i −0.517312 0.855797i \(-0.673067\pi\)
0.999798 + 0.0201068i \(0.00640061\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 15.4094i 0.642615i
\(576\) 0 0
\(577\) −37.6033 21.7103i −1.56545 0.903810i −0.996689 0.0813040i \(-0.974092\pi\)
−0.568756 0.822506i \(-0.692575\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 1.34498 + 8.99995i 0.0557992 + 0.373381i
\(582\) 0 0
\(583\) 18.9506 + 32.8234i 0.784854 + 1.35941i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 10.0485 0.414746 0.207373 0.978262i \(-0.433509\pi\)
0.207373 + 0.978262i \(0.433509\pi\)
\(588\) 0 0
\(589\) 6.85537 0.282470
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 18.9110 + 32.7547i 0.776580 + 1.34508i 0.933902 + 0.357528i \(0.116380\pi\)
−0.157322 + 0.987547i \(0.550286\pi\)
\(594\) 0 0
\(595\) 1.62248 + 10.8569i 0.0665152 + 0.445088i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 31.4823 + 18.1763i 1.28633 + 0.742665i 0.977998 0.208613i \(-0.0668948\pi\)
0.308335 + 0.951278i \(0.400228\pi\)
\(600\) 0 0
\(601\) 3.14659i 0.128352i −0.997939 0.0641760i \(-0.979558\pi\)
0.997939 0.0641760i \(-0.0204419\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −8.88645 + 15.3918i −0.361285 + 0.625765i
\(606\) 0 0
\(607\) −5.26589 + 3.04026i −0.213736 + 0.123400i −0.603046 0.797706i \(-0.706047\pi\)
0.389311 + 0.921107i \(0.372713\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −17.4759 + 10.0897i −0.707000 + 0.408187i
\(612\) 0 0
\(613\) 23.6217 40.9140i 0.954072 1.65250i 0.217593 0.976040i \(-0.430179\pi\)
0.736478 0.676461i \(-0.236487\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 28.1759i 1.13432i 0.823608 + 0.567160i \(0.191958\pi\)
−0.823608 + 0.567160i \(0.808042\pi\)
\(618\) 0 0
\(619\) −30.8831 17.8304i −1.24130 0.716663i −0.271939 0.962314i \(-0.587665\pi\)
−0.969358 + 0.245651i \(0.920998\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −14.1057 + 11.2200i −0.565133 + 0.449520i
\(624\) 0 0
\(625\) −6.98865 12.1047i −0.279546 0.484188i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 40.3528 1.60897
\(630\) 0 0
\(631\) 12.7686 0.508311 0.254155 0.967163i \(-0.418203\pi\)
0.254155 + 0.967163i \(0.418203\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 1.25695 + 2.17710i 0.0498805 + 0.0863956i
\(636\) 0 0
\(637\) 5.31237 + 17.3770i 0.210484 + 0.688500i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 10.6617 + 6.15551i 0.421110 + 0.243128i 0.695552 0.718476i \(-0.255160\pi\)
−0.274442 + 0.961604i \(0.588493\pi\)
\(642\) 0 0
\(643\) 45.3728i 1.78933i −0.446739 0.894664i \(-0.647415\pi\)
0.446739 0.894664i \(-0.352585\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 23.9646 41.5079i 0.942145 1.63184i 0.180776 0.983524i \(-0.442139\pi\)
0.761369 0.648319i \(-0.224527\pi\)
\(648\) 0 0
\(649\) −51.8122 + 29.9138i −2.03381 + 1.17422i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 8.07258 4.66071i 0.315905 0.182388i −0.333661 0.942693i \(-0.608284\pi\)
0.649566 + 0.760305i \(0.274951\pi\)
\(654\) 0 0
\(655\) 5.11250 8.85511i 0.199762 0.345998i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 11.8913i 0.463218i −0.972809 0.231609i \(-0.925601\pi\)
0.972809 0.231609i \(-0.0743990\pi\)
\(660\) 0 0
\(661\) −8.32726 4.80775i −0.323893 0.187000i 0.329234 0.944249i \(-0.393210\pi\)
−0.653126 + 0.757249i \(0.726543\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −0.530075 + 1.34564i −0.0205554 + 0.0521818i
\(666\) 0 0
\(667\) 9.73571 + 16.8627i 0.376968 + 0.652928i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −11.4601 −0.442414
\(672\) 0 0
\(673\) −22.5795 −0.870377 −0.435188 0.900339i \(-0.643318\pi\)
−0.435188 + 0.900339i \(0.643318\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −21.4679 37.1834i −0.825077 1.42907i −0.901861 0.432027i \(-0.857799\pi\)
0.0767843 0.997048i \(-0.475535\pi\)
\(678\) 0 0
\(679\) −42.4380 + 6.34205i −1.62862 + 0.243386i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −1.66677 0.962311i −0.0637772 0.0368218i 0.467772 0.883849i \(-0.345057\pi\)
−0.531550 + 0.847027i \(0.678390\pi\)
\(684\) 0 0
\(685\) 6.52300i 0.249231i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 8.80828 15.2564i 0.335569 0.581222i
\(690\) 0 0
\(691\) 5.60142 3.23398i 0.213088 0.123026i −0.389658 0.920960i \(-0.627407\pi\)
0.602746 + 0.797933i \(0.294073\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 2.60434 1.50362i 0.0987884 0.0570355i
\(696\) 0 0
\(697\) −16.0891 + 27.8672i −0.609420 + 1.05555i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 26.1067i 0.986036i 0.870019 + 0.493018i \(0.164106\pi\)
−0.870019 + 0.493018i \(0.835894\pi\)
\(702\) 0 0
\(703\) 4.60423 + 2.65825i 0.173652 + 0.100258i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 33.2680 + 13.1049i 1.25117 + 0.492861i
\(708\) 0 0
\(709\) 17.4171 + 30.1673i 0.654113 + 1.13296i 0.982115 + 0.188280i \(0.0602913\pi\)
−0.328002 + 0.944677i \(0.606375\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −40.2614 −1.50780
\(714\) 0 0
\(715\) 12.7615 0.477255
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 18.5506 + 32.1306i 0.691822 + 1.19827i 0.971240 + 0.238101i \(0.0765250\pi\)
−0.279418 + 0.960169i \(0.590142\pi\)
\(720\) 0 0
\(721\) 20.8397 + 26.1995i 0.776111 + 0.975721i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −19.5354 11.2788i −0.725526 0.418883i
\(726\) 0 0
\(727\) 5.97941i 0.221764i −0.993834 0.110882i \(-0.964632\pi\)
0.993834 0.110882i \(-0.0353676\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 8.62230 14.9343i 0.318907 0.552364i
\(732\) 0 0
\(733\) 13.7934 7.96362i 0.509471 0.294143i −0.223145 0.974785i \(-0.571632\pi\)
0.732616 + 0.680642i \(0.238299\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −43.7325 + 25.2490i −1.61091 + 0.930057i
\(738\) 0 0
\(739\) 3.41719 5.91875i 0.125704 0.217725i −0.796304 0.604896i \(-0.793215\pi\)
0.922008 + 0.387171i \(0.126548\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 23.9809i 0.879775i −0.898053 0.439887i \(-0.855018\pi\)
0.898053 0.439887i \(-0.144982\pi\)
\(744\) 0 0
\(745\) −7.75594 4.47789i −0.284156 0.164057i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −3.98734 5.01286i −0.145694 0.183166i
\(750\) 0 0
\(751\) −3.13906 5.43702i −0.114546 0.198400i 0.803052 0.595909i \(-0.203208\pi\)
−0.917598 + 0.397509i \(0.869875\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 3.79137 0.137982
\(756\) 0 0
\(757\) −43.9573 −1.59766 −0.798828 0.601559i \(-0.794546\pi\)
−0.798828 + 0.601559i \(0.794546\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −7.85711 13.6089i −0.284820 0.493323i 0.687745 0.725952i \(-0.258601\pi\)
−0.972566 + 0.232629i \(0.925267\pi\)
\(762\) 0 0
\(763\) 1.41011 + 0.555469i 0.0510493 + 0.0201093i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 24.0824 + 13.9040i 0.869566 + 0.502044i
\(768\) 0 0
\(769\) 33.9448i 1.22408i 0.790827 + 0.612040i \(0.209651\pi\)
−0.790827 + 0.612040i \(0.790349\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 24.4346 42.3220i 0.878852 1.52222i 0.0262509 0.999655i \(-0.491643\pi\)
0.852602 0.522562i \(-0.175024\pi\)
\(774\) 0 0
\(775\) 40.3936 23.3213i 1.45098 0.837725i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −3.67152 + 2.11975i −0.131546 + 0.0759481i
\(780\) 0 0
\(781\) 34.3498 59.4956i 1.22913 2.12892i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 10.3841i 0.370623i
\(786\) 0 0
\(787\) 31.4579 + 18.1622i 1.12135 + 0.647413i 0.941746 0.336325i \(-0.109184\pi\)
0.179607 + 0.983738i \(0.442517\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −27.7452 + 4.14633i −0.986506 + 0.147426i
\(792\) 0 0
\(793\) 2.66335 + 4.61306i 0.0945783 + 0.163814i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −28.0307 −0.992898 −0.496449 0.868066i \(-0.665363\pi\)
−0.496449 + 0.868066i \(0.665363\pi\)
\(798\) 0 0
\(799\) −36.6410 −1.29627
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −11.0825 19.1954i −0.391092 0.677391i
\(804\) 0 0
\(805\) 3.11311 7.90291i 0.109723 0.278541i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −10.9591 6.32725i −0.385302 0.222454i 0.294820 0.955553i \(-0.404740\pi\)
−0.680123 + 0.733098i \(0.738074\pi\)
\(810\) 0 0
\(811\) 33.7585i 1.18542i 0.805415 + 0.592711i \(0.201943\pi\)
−0.805415 + 0.592711i \(0.798057\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −5.11455 + 8.85866i −0.179155 + 0.310305i
\(816\) 0 0
\(817\) 1.96760 1.13599i 0.0688375 0.0397434i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 5.08428 2.93541i 0.177443 0.102447i −0.408648 0.912692i \(-0.634000\pi\)
0.586091 + 0.810246i \(0.300666\pi\)
\(822\) 0 0
\(823\) −15.4509 + 26.7617i −0.538583 + 0.932853i 0.460398 + 0.887713i \(0.347707\pi\)
−0.998981 + 0.0451404i \(0.985626\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 44.0586i 1.53207i −0.642800 0.766034i \(-0.722227\pi\)
0.642800 0.766034i \(-0.277773\pi\)
\(828\) 0 0
\(829\) 5.47912 + 3.16337i 0.190298 + 0.109868i 0.592122 0.805848i \(-0.298290\pi\)
−0.401824 + 0.915717i \(0.631624\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −7.42252 + 32.1481i −0.257175 + 1.11387i
\(834\) 0 0
\(835\) −6.94127 12.0226i −0.240212 0.416060i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 10.5967 0.365840 0.182920 0.983128i \(-0.441445\pi\)
0.182920 + 0.983128i \(0.441445\pi\)
\(840\) 0 0
\(841\) 0.496127 0.0171078
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 2.75596 + 4.77347i 0.0948080 + 0.164212i
\(846\) 0 0
\(847\) −41.8060 + 33.2534i −1.43647 + 1.14260i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −27.0405 15.6118i −0.926936 0.535167i
\(852\) 0 0
\(853\) 22.0216i 0.754007i −0.926212 0.377003i \(-0.876955\pi\)
0.926212 0.377003i \(-0.123045\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1.76205 3.05197i 0.0601906 0.104253i −0.834360 0.551220i \(-0.814162\pi\)
0.894551 + 0.446967i \(0.147496\pi\)
\(858\) 0 0
\(859\) 32.0730 18.5174i 1.09432 0.631805i 0.159595 0.987183i \(-0.448981\pi\)
0.934723 + 0.355378i \(0.115648\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −13.5411 + 7.81797i −0.460945 + 0.266127i −0.712442 0.701731i \(-0.752411\pi\)
0.251496 + 0.967858i \(0.419077\pi\)
\(864\) 0 0
\(865\) 1.22067 2.11427i 0.0415042 0.0718873i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 67.1346i 2.27738i
\(870\) 0 0
\(871\) 20.3269 + 11.7358i 0.688752 + 0.397651i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 3.17553 + 21.2491i 0.107353 + 0.718352i
\(876\) 0 0
\(877\) 1.92641 + 3.33664i 0.0650502 + 0.112670i 0.896716 0.442606i \(-0.145946\pi\)
−0.831666 + 0.555276i \(0.812613\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −44.3493 −1.49417 −0.747084 0.664730i \(-0.768547\pi\)
−0.747084 + 0.664730i \(0.768547\pi\)
\(882\) 0 0
\(883\) −20.5569 −0.691795 −0.345897 0.938272i \(-0.612425\pi\)
−0.345897 + 0.938272i \(0.612425\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 2.05890 + 3.56612i 0.0691311 + 0.119739i 0.898519 0.438934i \(-0.144644\pi\)
−0.829388 + 0.558673i \(0.811311\pi\)
\(888\) 0 0
\(889\) 1.11676 + 7.47282i 0.0374549 + 0.250630i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −4.18072 2.41374i −0.139902 0.0807726i
\(894\) 0 0
\(895\) 6.04180i 0.201955i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 29.4690 51.0417i 0.982845 1.70234i
\(900\) 0 0
\(901\) 27.7019 15.9937i 0.922884 0.532828i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 7.50973 4.33575i 0.249632 0.144125i
\(906\) 0 0
\(907\) −21.1401 + 36.6158i −0.701946 + 1.21581i 0.265836 + 0.964018i \(0.414352\pi\)
−0.967782 + 0.251789i \(0.918981\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 19.2152i 0.636628i 0.947985 + 0.318314i \(0.103117\pi\)
−0.947985 + 0.318314i \(0.896883\pi\)
\(912\) 0 0
\(913\) 16.6352 + 9.60433i 0.550544 + 0.317857i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 24.0516 19.1312i 0.794253 0.631767i
\(918\) 0 0
\(919\) 6.02706 + 10.4392i 0.198814 + 0.344357i 0.948144 0.317840i \(-0.102958\pi\)
−0.749330 + 0.662197i \(0.769624\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −31.9317 −1.05105
\(924\) 0 0
\(925\) 36.1724 1.18934
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 18.2907 + 31.6805i 0.600099 + 1.03940i 0.992806 + 0.119738i \(0.0382055\pi\)
−0.392706 + 0.919664i \(0.628461\pi\)
\(930\) 0 0
\(931\) −2.96467 + 3.17912i −0.0971632 + 0.104191i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 20.0674 + 11.5859i 0.656275 + 0.378901i
\(936\) 0 0
\(937\) 39.4119i 1.28753i −0.765222 0.643766i \(-0.777371\pi\)
0.765222 0.643766i \(-0.222629\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −6.21752 + 10.7691i −0.202685 + 0.351061i −0.949393 0.314091i \(-0.898300\pi\)
0.746707 + 0.665153i \(0.231634\pi\)
\(942\) 0 0
\(943\) 21.5627 12.4492i 0.702179 0.405403i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 25.8637 14.9324i 0.840459 0.485239i −0.0169614 0.999856i \(-0.505399\pi\)
0.857420 + 0.514617i \(0.172066\pi\)
\(948\) 0 0
\(949\) −5.15116 + 8.92206i −0.167214 + 0.289622i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 22.4851i 0.728365i −0.931328 0.364183i \(-0.881348\pi\)
0.931328 0.364183i \(-0.118652\pi\)
\(954\) 0 0
\(955\) 13.7044 + 7.91224i 0.443464 + 0.256034i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 7.18562 18.2413i 0.232036 0.589043i
\(960\) 0 0
\(961\) 45.4334 + 78.6930i 1.46559 + 2.53848i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 3.31051 0.106569
\(966\) 0 0
\(967\) −3.82744 −0.123082 −0.0615411 0.998105i \(-0.519602\pi\)
−0.0615411 + 0.998105i \(0.519602\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 29.4317 + 50.9773i 0.944509 + 1.63594i 0.756731 + 0.653727i \(0.226795\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(972\) 0 0
\(973\) 8.93932 1.33592i 0.286581 0.0428276i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 9.18043 + 5.30032i 0.293708 + 0.169572i 0.639613 0.768697i \(-0.279095\pi\)
−0.345905 + 0.938270i \(0.612428\pi\)
\(978\) 0 0
\(979\) 38.0459i 1.21595i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 16.3230 28.2723i 0.520624 0.901747i −0.479089 0.877766i \(-0.659033\pi\)
0.999712 0.0239802i \(-0.00763386\pi\)
\(984\) 0 0
\(985\) −5.24749 + 3.02964i −0.167199 + 0.0965323i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −11.5556 + 6.67165i −0.367448 + 0.212146i
\(990\) 0 0
\(991\) 16.3697 28.3531i 0.520000 0.900666i −0.479730 0.877416i \(-0.659265\pi\)
0.999730 0.0232498i \(-0.00740131\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 12.2882i 0.389563i
\(996\) 0 0
\(997\) −39.3141 22.6980i −1.24509 0.718854i −0.274965 0.961454i \(-0.588666\pi\)
−0.970126 + 0.242601i \(0.922000\pi\)
\(998\) 0 0
\(999\) 0 0
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 2268.2.t.c.1781.10 yes 32
3.2 odd 2 inner 2268.2.t.c.1781.7 32
7.5 odd 6 inner 2268.2.t.c.2105.7 yes 32
9.2 odd 6 2268.2.bm.j.1025.10 32
9.4 even 3 2268.2.w.j.269.10 32
9.5 odd 6 2268.2.w.j.269.7 32
9.7 even 3 2268.2.bm.j.1025.7 32
21.5 even 6 inner 2268.2.t.c.2105.10 yes 32
63.5 even 6 2268.2.bm.j.593.7 32
63.40 odd 6 2268.2.bm.j.593.10 32
63.47 even 6 2268.2.w.j.1349.10 32
63.61 odd 6 2268.2.w.j.1349.7 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2268.2.t.c.1781.7 32 3.2 odd 2 inner
2268.2.t.c.1781.10 yes 32 1.1 even 1 trivial
2268.2.t.c.2105.7 yes 32 7.5 odd 6 inner
2268.2.t.c.2105.10 yes 32 21.5 even 6 inner
2268.2.w.j.269.7 32 9.5 odd 6
2268.2.w.j.269.10 32 9.4 even 3
2268.2.w.j.1349.7 32 63.61 odd 6
2268.2.w.j.1349.10 32 63.47 even 6
2268.2.bm.j.593.7 32 63.5 even 6
2268.2.bm.j.593.10 32 63.40 odd 6
2268.2.bm.j.1025.7 32 9.7 even 3
2268.2.bm.j.1025.10 32 9.2 odd 6