Properties

Label 756.2.t.e.269.6
Level $756$
Weight $2$
Character 756.269
Analytic conductor $6.037$
Analytic rank $0$
Dimension $12$
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] = [756,2,Mod(269,756)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(756, 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("756.269");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 756.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: \(6.03669039281\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.17213603549184.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 5x^{10} + 19x^{8} - 28x^{6} + 31x^{4} - 6x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{9} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 269.6
Root \(-1.56052 - 0.900969i\) of defining polynomial
Character \(\chi\) \(=\) 756.269
Dual form 756.2.t.e.593.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.56052 + 2.70291i) q^{5} +(2.37047 - 1.17511i) q^{7} +O(q^{10})\) \(q+(1.56052 + 2.70291i) q^{5} +(2.37047 - 1.17511i) q^{7} +(3.23975 + 1.87047i) q^{11} -2.93067i q^{13} +(2.71678 - 4.70560i) q^{17} +(-6.07338 + 3.50647i) q^{19} +(-1.44182 + 0.832437i) q^{23} +(-2.37047 + 4.10577i) q^{25} +7.07069i q^{29} +(6.60872 + 3.81555i) q^{31} +(6.87538 + 4.57338i) q^{35} +(-4.03534 - 6.98942i) q^{37} +9.60054 q^{41} -5.74094 q^{43} +(1.56052 + 2.70291i) q^{47} +(4.23825 - 5.57111i) q^{49} +(-3.11266 - 1.79709i) q^{53} +11.6756i q^{55} +(-2.20220 + 3.81431i) q^{59} +(-2.57069 + 1.48419i) q^{61} +(7.92132 - 4.57338i) q^{65} +(4.90850 - 8.50177i) q^{67} +7.33513i q^{71} +(2.46466 + 1.42297i) q^{73} +(9.87772 + 0.626842i) q^{77} +(-2.57338 - 4.45722i) q^{79} -15.6052 q^{83} +16.9584 q^{85} +(2.59808 + 4.50000i) q^{89} +(-3.44385 - 6.94706i) q^{91} +(-18.9553 - 10.9438i) q^{95} +2.39723i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 18 q^{19} - 24 q^{37} - 12 q^{43} + 18 q^{61} + 54 q^{73} + 24 q^{79} - 12 q^{85} + 42 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/756\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(325\) \(379\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{6}\right)\) \(1\)

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
\(5\) 1.56052 + 2.70291i 0.697887 + 1.20878i 0.969198 + 0.246285i \(0.0792098\pi\)
−0.271310 + 0.962492i \(0.587457\pi\)
\(6\) 0 0
\(7\) 2.37047 1.17511i 0.895953 0.444148i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.23975 + 1.87047i 0.976821 + 0.563968i 0.901309 0.433177i \(-0.142608\pi\)
0.0755120 + 0.997145i \(0.475941\pi\)
\(12\) 0 0
\(13\) 2.93067i 0.812821i −0.913691 0.406410i \(-0.866780\pi\)
0.913691 0.406410i \(-0.133220\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.71678 4.70560i 0.658915 1.14127i −0.321982 0.946746i \(-0.604349\pi\)
0.980897 0.194529i \(-0.0623177\pi\)
\(18\) 0 0
\(19\) −6.07338 + 3.50647i −1.39333 + 0.804438i −0.993682 0.112231i \(-0.964200\pi\)
−0.399646 + 0.916670i \(0.630867\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.44182 + 0.832437i −0.300641 + 0.173575i −0.642731 0.766092i \(-0.722199\pi\)
0.342090 + 0.939667i \(0.388865\pi\)
\(24\) 0 0
\(25\) −2.37047 + 4.10577i −0.474094 + 0.821155i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 7.07069i 1.31299i 0.754329 + 0.656497i \(0.227962\pi\)
−0.754329 + 0.656497i \(0.772038\pi\)
\(30\) 0 0
\(31\) 6.60872 + 3.81555i 1.18696 + 0.685292i 0.957615 0.288053i \(-0.0930078\pi\)
0.229347 + 0.973345i \(0.426341\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 6.87538 + 4.57338i 1.16215 + 0.773042i
\(36\) 0 0
\(37\) −4.03534 6.98942i −0.663406 1.14905i −0.979715 0.200397i \(-0.935777\pi\)
0.316308 0.948656i \(-0.397557\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 9.60054 1.49935 0.749677 0.661804i \(-0.230209\pi\)
0.749677 + 0.661804i \(0.230209\pi\)
\(42\) 0 0
\(43\) −5.74094 −0.875485 −0.437742 0.899100i \(-0.644222\pi\)
−0.437742 + 0.899100i \(0.644222\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.56052 + 2.70291i 0.227626 + 0.394259i 0.957104 0.289745i \(-0.0935704\pi\)
−0.729478 + 0.684004i \(0.760237\pi\)
\(48\) 0 0
\(49\) 4.23825 5.57111i 0.605464 0.795872i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −3.11266 1.79709i −0.427556 0.246850i 0.270749 0.962650i \(-0.412729\pi\)
−0.698305 + 0.715800i \(0.746062\pi\)
\(54\) 0 0
\(55\) 11.6756i 1.57434i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2.20220 + 3.81431i −0.286701 + 0.496581i −0.973020 0.230719i \(-0.925892\pi\)
0.686319 + 0.727301i \(0.259225\pi\)
\(60\) 0 0
\(61\) −2.57069 + 1.48419i −0.329143 + 0.190031i −0.655460 0.755229i \(-0.727525\pi\)
0.326318 + 0.945260i \(0.394192\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 7.92132 4.57338i 0.982519 0.567257i
\(66\) 0 0
\(67\) 4.90850 8.50177i 0.599669 1.03866i −0.393201 0.919453i \(-0.628632\pi\)
0.992870 0.119204i \(-0.0380344\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 7.33513i 0.870519i 0.900305 + 0.435260i \(0.143343\pi\)
−0.900305 + 0.435260i \(0.856657\pi\)
\(72\) 0 0
\(73\) 2.46466 + 1.42297i 0.288466 + 0.166546i 0.637250 0.770657i \(-0.280072\pi\)
−0.348784 + 0.937203i \(0.613405\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 9.87772 + 0.626842i 1.12567 + 0.0714353i
\(78\) 0 0
\(79\) −2.57338 4.45722i −0.289527 0.501476i 0.684170 0.729323i \(-0.260165\pi\)
−0.973697 + 0.227847i \(0.926831\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −15.6052 −1.71290 −0.856449 0.516232i \(-0.827334\pi\)
−0.856449 + 0.516232i \(0.827334\pi\)
\(84\) 0 0
\(85\) 16.9584 1.83939
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.59808 + 4.50000i 0.275396 + 0.476999i 0.970235 0.242166i \(-0.0778579\pi\)
−0.694839 + 0.719165i \(0.744525\pi\)
\(90\) 0 0
\(91\) −3.44385 6.94706i −0.361013 0.728249i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −18.9553 10.9438i −1.94477 1.12281i
\(96\) 0 0
\(97\) 2.39723i 0.243402i 0.992567 + 0.121701i \(0.0388349\pi\)
−0.992567 + 0.121701i \(0.961165\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 9.87772 17.1087i 0.982870 1.70238i 0.331828 0.943340i \(-0.392335\pi\)
0.651043 0.759041i \(-0.274332\pi\)
\(102\) 0 0
\(103\) 10.6468 6.14691i 1.04906 0.605673i 0.126671 0.991945i \(-0.459571\pi\)
0.922385 + 0.386272i \(0.126237\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −15.7155 + 9.07338i −1.51928 + 0.877156i −0.519537 + 0.854448i \(0.673896\pi\)
−0.999742 + 0.0227086i \(0.992771\pi\)
\(108\) 0 0
\(109\) −5.74094 + 9.94360i −0.549882 + 0.952424i 0.448400 + 0.893833i \(0.351994\pi\)
−0.998282 + 0.0585909i \(0.981339\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.81163i 0.170423i −0.996363 0.0852117i \(-0.972843\pi\)
0.996363 0.0852117i \(-0.0271567\pi\)
\(114\) 0 0
\(115\) −4.50000 2.59808i −0.419627 0.242272i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0.910461 14.3470i 0.0834619 1.31518i
\(120\) 0 0
\(121\) 1.49731 + 2.59342i 0.136119 + 0.235765i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0.808542 0.0723182
\(126\) 0 0
\(127\) −7.47650 −0.663432 −0.331716 0.943379i \(-0.607628\pi\)
−0.331716 + 0.943379i \(0.607628\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −3.52532 6.10603i −0.308009 0.533486i 0.669918 0.742435i \(-0.266329\pi\)
−0.977927 + 0.208949i \(0.932996\pi\)
\(132\) 0 0
\(133\) −10.2763 + 15.4488i −0.891067 + 1.33958i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −12.7048 7.33513i −1.08545 0.626682i −0.153085 0.988213i \(-0.548921\pi\)
−0.932360 + 0.361531i \(0.882254\pi\)
\(138\) 0 0
\(139\) 1.69434i 0.143712i −0.997415 0.0718562i \(-0.977108\pi\)
0.997415 0.0718562i \(-0.0228923\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 5.48172 9.49462i 0.458405 0.793980i
\(144\) 0 0
\(145\) −19.1114 + 11.0340i −1.58712 + 0.916322i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −12.8319 + 7.40850i −1.05123 + 0.606928i −0.922994 0.384815i \(-0.874265\pi\)
−0.128237 + 0.991744i \(0.540932\pi\)
\(150\) 0 0
\(151\) 3.03534 5.25737i 0.247013 0.427839i −0.715683 0.698425i \(-0.753884\pi\)
0.962696 + 0.270587i \(0.0872177\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 23.8170i 1.91303i
\(156\) 0 0
\(157\) −12.1141 6.99408i −0.966810 0.558188i −0.0685479 0.997648i \(-0.521837\pi\)
−0.898262 + 0.439460i \(0.855170\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −2.43960 + 3.66756i −0.192267 + 0.289044i
\(162\) 0 0
\(163\) −0.926624 1.60496i −0.0725788 0.125710i 0.827452 0.561536i \(-0.189790\pi\)
−0.900031 + 0.435826i \(0.856456\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −2.54991 −0.197318 −0.0986588 0.995121i \(-0.531455\pi\)
−0.0986588 + 0.995121i \(0.531455\pi\)
\(168\) 0 0
\(169\) 4.41119 0.339322
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1.95640 + 3.38859i 0.148743 + 0.257630i 0.930763 0.365623i \(-0.119144\pi\)
−0.782020 + 0.623253i \(0.785811\pi\)
\(174\) 0 0
\(175\) −0.794405 + 12.5182i −0.0600514 + 0.946284i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −16.9284 9.77359i −1.26529 0.730513i −0.291193 0.956664i \(-0.594052\pi\)
−0.974092 + 0.226152i \(0.927386\pi\)
\(180\) 0 0
\(181\) 18.1929i 1.35226i −0.736780 0.676132i \(-0.763655\pi\)
0.736780 0.676132i \(-0.236345\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 12.5945 21.8143i 0.925966 1.60382i
\(186\) 0 0
\(187\) 17.6033 10.1633i 1.28728 0.743214i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −13.0609 + 7.54072i −0.945054 + 0.545627i −0.891541 0.452940i \(-0.850375\pi\)
−0.0535132 + 0.998567i \(0.517042\pi\)
\(192\) 0 0
\(193\) 5.57338 9.65337i 0.401180 0.694865i −0.592688 0.805432i \(-0.701933\pi\)
0.993869 + 0.110567i \(0.0352667\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 20.3696i 1.45127i −0.688079 0.725636i \(-0.741546\pi\)
0.688079 0.725636i \(-0.258454\pi\)
\(198\) 0 0
\(199\) 6.42931 + 3.71197i 0.455762 + 0.263134i 0.710261 0.703939i \(-0.248577\pi\)
−0.254499 + 0.967073i \(0.581910\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 8.30881 + 16.7608i 0.583164 + 1.17638i
\(204\) 0 0
\(205\) 14.9819 + 25.9494i 1.04638 + 1.81238i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −26.2349 −1.81471
\(210\) 0 0
\(211\) −13.5526 −0.932997 −0.466499 0.884522i \(-0.654485\pi\)
−0.466499 + 0.884522i \(0.654485\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −8.95887 15.5172i −0.610990 1.05827i
\(216\) 0 0
\(217\) 20.1494 + 1.27869i 1.36783 + 0.0868029i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −13.7905 7.96197i −0.927652 0.535580i
\(222\) 0 0
\(223\) 8.79200i 0.588756i −0.955689 0.294378i \(-0.904888\pi\)
0.955689 0.294378i \(-0.0951125\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 6.35241 11.0027i 0.421624 0.730274i −0.574475 0.818522i \(-0.694794\pi\)
0.996099 + 0.0882484i \(0.0281269\pi\)
\(228\) 0 0
\(229\) 24.2201 13.9835i 1.60051 0.924056i 0.609126 0.793073i \(-0.291520\pi\)
0.991385 0.130982i \(-0.0418131\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 14.2737 8.24094i 0.935103 0.539882i 0.0466810 0.998910i \(-0.485136\pi\)
0.888422 + 0.459028i \(0.151802\pi\)
\(234\) 0 0
\(235\) −4.87047 + 8.43590i −0.317714 + 0.550297i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 5.52350i 0.357285i 0.983914 + 0.178643i \(0.0571706\pi\)
−0.983914 + 0.178643i \(0.942829\pi\)
\(240\) 0 0
\(241\) −18.6414 10.7626i −1.20080 0.693280i −0.240064 0.970757i \(-0.577168\pi\)
−0.960732 + 0.277477i \(0.910502\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 21.6721 + 2.76175i 1.38458 + 0.176442i
\(246\) 0 0
\(247\) 10.2763 + 17.7990i 0.653864 + 1.13253i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −0.808542 −0.0510347 −0.0255174 0.999674i \(-0.508123\pi\)
−0.0255174 + 0.999674i \(0.508123\pi\)
\(252\) 0 0
\(253\) −6.22819 −0.391563
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 5.48172 + 9.49462i 0.341941 + 0.592258i 0.984793 0.173732i \(-0.0555825\pi\)
−0.642853 + 0.765990i \(0.722249\pi\)
\(258\) 0 0
\(259\) −17.7790 11.8262i −1.10473 0.734847i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −6.35241 3.66756i −0.391706 0.226152i 0.291193 0.956664i \(-0.405948\pi\)
−0.682899 + 0.730513i \(0.739281\pi\)
\(264\) 0 0
\(265\) 11.2176i 0.689093i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −4.28569 + 7.42303i −0.261303 + 0.452590i −0.966588 0.256333i \(-0.917486\pi\)
0.705285 + 0.708924i \(0.250819\pi\)
\(270\) 0 0
\(271\) 6.79619 3.92378i 0.412839 0.238353i −0.279170 0.960242i \(-0.590059\pi\)
0.692009 + 0.721889i \(0.256726\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −15.3594 + 8.86778i −0.926210 + 0.534747i
\(276\) 0 0
\(277\) 1.13491 1.96572i 0.0681900 0.118109i −0.829915 0.557890i \(-0.811611\pi\)
0.898105 + 0.439782i \(0.144944\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 28.9584i 1.72751i 0.503910 + 0.863756i \(0.331894\pi\)
−0.503910 + 0.863756i \(0.668106\pi\)
\(282\) 0 0
\(283\) 2.11679 + 1.22213i 0.125830 + 0.0726479i 0.561594 0.827413i \(-0.310188\pi\)
−0.435764 + 0.900061i \(0.643522\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 22.7578 11.2817i 1.34335 0.665935i
\(288\) 0 0
\(289\) −6.26175 10.8457i −0.368338 0.637980i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −8.55460 −0.499765 −0.249883 0.968276i \(-0.580392\pi\)
−0.249883 + 0.968276i \(0.580392\pi\)
\(294\) 0 0
\(295\) −13.7463 −0.800341
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 2.43960 + 4.22550i 0.141086 + 0.244367i
\(300\) 0 0
\(301\) −13.6087 + 6.74621i −0.784394 + 0.388845i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −8.02324 4.63222i −0.459409 0.265240i
\(306\) 0 0
\(307\) 23.8000i 1.35834i 0.733982 + 0.679169i \(0.237660\pi\)
−0.733982 + 0.679169i \(0.762340\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 8.55460 14.8170i 0.485087 0.840195i −0.514766 0.857331i \(-0.672121\pi\)
0.999853 + 0.0171354i \(0.00545463\pi\)
\(312\) 0 0
\(313\) 1.35325 0.781298i 0.0764901 0.0441616i −0.461267 0.887261i \(-0.652605\pi\)
0.537757 + 0.843100i \(0.319272\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 22.5511 13.0199i 1.26660 0.731271i 0.292256 0.956340i \(-0.405594\pi\)
0.974343 + 0.225069i \(0.0722608\pi\)
\(318\) 0 0
\(319\) −13.2255 + 22.9072i −0.740486 + 1.28256i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 38.1051i 2.12023i
\(324\) 0 0
\(325\) 12.0327 + 6.94706i 0.667452 + 0.385353i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 6.87538 + 4.57338i 0.379052 + 0.252138i
\(330\) 0 0
\(331\) 15.1087 + 26.1691i 0.830450 + 1.43838i 0.897682 + 0.440645i \(0.145250\pi\)
−0.0672312 + 0.997737i \(0.521417\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 30.6393 1.67401
\(336\) 0 0
\(337\) −22.2228 −1.21055 −0.605277 0.796015i \(-0.706938\pi\)
−0.605277 + 0.796015i \(0.706938\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 14.2737 + 24.7228i 0.772966 + 1.33882i
\(342\) 0 0
\(343\) 3.50000 18.1865i 0.188982 0.981981i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 31.6853 + 18.2935i 1.70095 + 0.982047i 0.944796 + 0.327658i \(0.106259\pi\)
0.756159 + 0.654388i \(0.227074\pi\)
\(348\) 0 0
\(349\) 9.19369i 0.492127i −0.969254 0.246063i \(-0.920863\pi\)
0.969254 0.246063i \(-0.0791372\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0.277180 0.480090i 0.0147528 0.0255526i −0.858555 0.512722i \(-0.828637\pi\)
0.873308 + 0.487169i \(0.161971\pi\)
\(354\) 0 0
\(355\) −19.8262 + 11.4466i −1.05226 + 0.607524i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −12.3487 + 7.12953i −0.651740 + 0.376282i −0.789123 0.614236i \(-0.789464\pi\)
0.137383 + 0.990518i \(0.456131\pi\)
\(360\) 0 0
\(361\) 15.0906 26.1377i 0.794242 1.37567i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 8.88231i 0.464922i
\(366\) 0 0
\(367\) −12.0814 6.97522i −0.630646 0.364104i 0.150356 0.988632i \(-0.451958\pi\)
−0.781002 + 0.624528i \(0.785291\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −9.49023 0.602252i −0.492708 0.0312674i
\(372\) 0 0
\(373\) −8.21744 14.2330i −0.425483 0.736958i 0.570983 0.820962i \(-0.306562\pi\)
−0.996465 + 0.0840042i \(0.973229\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 20.7218 1.06723
\(378\) 0 0
\(379\) 34.5217 1.77326 0.886630 0.462479i \(-0.153040\pi\)
0.886630 + 0.462479i \(0.153040\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 1.96480 + 3.40312i 0.100396 + 0.173892i 0.911848 0.410528i \(-0.134656\pi\)
−0.811452 + 0.584420i \(0.801322\pi\)
\(384\) 0 0
\(385\) 13.7201 + 27.6768i 0.699242 + 1.41054i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −18.5992 10.7383i −0.943017 0.544451i −0.0521120 0.998641i \(-0.516595\pi\)
−0.890905 + 0.454190i \(0.849929\pi\)
\(390\) 0 0
\(391\) 9.04618i 0.457485i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 8.03163 13.9112i 0.404115 0.699948i
\(396\) 0 0
\(397\) −15.8913 + 9.17483i −0.797560 + 0.460472i −0.842617 0.538513i \(-0.818986\pi\)
0.0450569 + 0.998984i \(0.485653\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 4.78349 2.76175i 0.238876 0.137915i −0.375784 0.926707i \(-0.622627\pi\)
0.614660 + 0.788792i \(0.289293\pi\)
\(402\) 0 0
\(403\) 11.1821 19.3680i 0.557020 0.964787i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 30.1919i 1.49656i
\(408\) 0 0
\(409\) −8.64406 4.99065i −0.427421 0.246772i 0.270826 0.962628i \(-0.412703\pi\)
−0.698248 + 0.715856i \(0.746037\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −0.738012 + 11.6295i −0.0363152 + 0.572252i
\(414\) 0 0
\(415\) −24.3523 42.1795i −1.19541 2.07051i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 10.2960 0.502991 0.251495 0.967858i \(-0.419078\pi\)
0.251495 + 0.967858i \(0.419078\pi\)
\(420\) 0 0
\(421\) 10.4873 0.511118 0.255559 0.966794i \(-0.417741\pi\)
0.255559 + 0.966794i \(0.417741\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 12.8801 + 22.3089i 0.624775 + 1.08214i
\(426\) 0 0
\(427\) −4.34966 + 6.53905i −0.210495 + 0.316447i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −5.26668 3.04072i −0.253687 0.146466i 0.367764 0.929919i \(-0.380123\pi\)
−0.621451 + 0.783453i \(0.713457\pi\)
\(432\) 0 0
\(433\) 25.6452i 1.23243i 0.787579 + 0.616214i \(0.211334\pi\)
−0.787579 + 0.616214i \(0.788666\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 5.83782 10.1114i 0.279261 0.483694i
\(438\) 0 0
\(439\) 24.7228 14.2737i 1.17996 0.681248i 0.223951 0.974600i \(-0.428104\pi\)
0.956004 + 0.293353i \(0.0947710\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −18.3702 + 10.6060i −0.872794 + 0.503908i −0.868276 0.496082i \(-0.834772\pi\)
−0.00451818 + 0.999990i \(0.501438\pi\)
\(444\) 0 0
\(445\) −8.10872 + 14.0447i −0.384390 + 0.665783i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 22.5633i 1.06483i 0.846484 + 0.532414i \(0.178715\pi\)
−0.846484 + 0.532414i \(0.821285\pi\)
\(450\) 0 0
\(451\) 31.1033 + 17.9575i 1.46460 + 0.845587i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 13.4030 20.1494i 0.628344 0.944620i
\(456\) 0 0
\(457\) −6.72013 11.6396i −0.314354 0.544478i 0.664946 0.746892i \(-0.268455\pi\)
−0.979300 + 0.202414i \(0.935121\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 19.9928 0.931160 0.465580 0.885006i \(-0.345846\pi\)
0.465580 + 0.885006i \(0.345846\pi\)
\(462\) 0 0
\(463\) 10.0108 0.465239 0.232620 0.972568i \(-0.425270\pi\)
0.232620 + 0.972568i \(0.425270\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 16.7699 + 29.0463i 0.776018 + 1.34410i 0.934221 + 0.356695i \(0.116097\pi\)
−0.158203 + 0.987407i \(0.550570\pi\)
\(468\) 0 0
\(469\) 1.64496 25.9212i 0.0759574 1.19693i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −18.5992 10.7383i −0.855192 0.493745i
\(474\) 0 0
\(475\) 33.2479i 1.52552i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −4.15860 + 7.20291i −0.190011 + 0.329109i −0.945254 0.326336i \(-0.894186\pi\)
0.755242 + 0.655446i \(0.227519\pi\)
\(480\) 0 0
\(481\) −20.4837 + 11.8262i −0.933975 + 0.539231i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −6.47950 + 3.74094i −0.294219 + 0.169867i
\(486\) 0 0
\(487\) −1.36509 + 2.36441i −0.0618583 + 0.107142i −0.895296 0.445472i \(-0.853036\pi\)
0.833438 + 0.552613i \(0.186369\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 10.4004i 0.469365i 0.972072 + 0.234683i \(0.0754051\pi\)
−0.972072 + 0.234683i \(0.924595\pi\)
\(492\) 0 0
\(493\) 33.2718 + 19.2095i 1.49849 + 0.865151i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 8.61955 + 17.3877i 0.386640 + 0.779945i
\(498\) 0 0
\(499\) 5.37585 + 9.31124i 0.240656 + 0.416828i 0.960901 0.276891i \(-0.0893041\pi\)
−0.720245 + 0.693719i \(0.755971\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −17.3466 −0.773447 −0.386723 0.922196i \(-0.626393\pi\)
−0.386723 + 0.922196i \(0.626393\pi\)
\(504\) 0 0
\(505\) 61.6577 2.74373
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −0.689842 1.19484i −0.0305767 0.0529604i 0.850332 0.526246i \(-0.176401\pi\)
−0.880909 + 0.473286i \(0.843068\pi\)
\(510\) 0 0
\(511\) 7.51453 + 0.476874i 0.332423 + 0.0210956i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 33.2290 + 19.1848i 1.46425 + 0.845383i
\(516\) 0 0
\(517\) 11.6756i 0.513494i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 2.19381 3.79978i 0.0961123 0.166471i −0.813960 0.580921i \(-0.802693\pi\)
0.910072 + 0.414450i \(0.136026\pi\)
\(522\) 0 0
\(523\) −30.6494 + 17.6955i −1.34021 + 0.773769i −0.986837 0.161715i \(-0.948297\pi\)
−0.353369 + 0.935484i \(0.614964\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 35.9088 20.7320i 1.56421 0.903099i
\(528\) 0 0
\(529\) −10.1141 + 17.5181i −0.439743 + 0.761658i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 28.1360i 1.21871i
\(534\) 0 0
\(535\) −49.0490 28.3184i −2.12057 1.22431i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 24.1514 10.1215i 1.04028 0.435962i
\(540\) 0 0
\(541\) −23.1114 40.0301i −0.993637 1.72103i −0.594359 0.804200i \(-0.702594\pi\)
−0.399278 0.916830i \(-0.630739\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −35.8355 −1.53502
\(546\) 0 0
\(547\) 4.28813 0.183347 0.0916735 0.995789i \(-0.470778\pi\)
0.0916735 + 0.995789i \(0.470778\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −24.7931 42.9429i −1.05622 1.82943i
\(552\) 0 0
\(553\) −11.3378 7.54171i −0.482133 0.320706i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 9.49023 + 5.47919i 0.402114 + 0.232161i 0.687396 0.726283i \(-0.258754\pi\)
−0.285282 + 0.958444i \(0.592087\pi\)
\(558\) 0 0
\(559\) 16.8248i 0.711612i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −16.3572 + 28.3315i −0.689375 + 1.19403i 0.282666 + 0.959218i \(0.408781\pi\)
−0.972040 + 0.234813i \(0.924552\pi\)
\(564\) 0 0
\(565\) 4.89666 2.82709i 0.206004 0.118936i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −31.8124 + 18.3669i −1.33364 + 0.769980i −0.985856 0.167594i \(-0.946400\pi\)
−0.347788 + 0.937573i \(0.613067\pi\)
\(570\) 0 0
\(571\) −7.10603 + 12.3080i −0.297378 + 0.515074i −0.975535 0.219843i \(-0.929446\pi\)
0.678157 + 0.734917i \(0.262779\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 7.89307i 0.329164i
\(576\) 0 0
\(577\) 22.1848 + 12.8084i 0.923565 + 0.533220i 0.884771 0.466027i \(-0.154315\pi\)
0.0387941 + 0.999247i \(0.487648\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −36.9917 + 18.3378i −1.53468 + 0.760781i
\(582\) 0 0
\(583\) −6.72282 11.6443i −0.278431 0.482256i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 14.1013 0.582022 0.291011 0.956720i \(-0.406008\pi\)
0.291011 + 0.956720i \(0.406008\pi\)
\(588\) 0 0
\(589\) −53.5163 −2.20510
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −18.9721 32.8606i −0.779090 1.34942i −0.932467 0.361255i \(-0.882348\pi\)
0.153377 0.988168i \(-0.450985\pi\)
\(594\) 0 0
\(595\) 40.1993 19.9279i 1.64801 0.816964i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 16.9032 + 9.75906i 0.690646 + 0.398744i 0.803854 0.594827i \(-0.202779\pi\)
−0.113208 + 0.993571i \(0.536113\pi\)
\(600\) 0 0
\(601\) 38.1555i 1.55639i 0.628020 + 0.778197i \(0.283865\pi\)
−0.628020 + 0.778197i \(0.716135\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −4.67318 + 8.09419i −0.189992 + 0.329075i
\(606\) 0 0
\(607\) −29.4756 + 17.0177i −1.19638 + 0.690729i −0.959746 0.280870i \(-0.909377\pi\)
−0.236632 + 0.971599i \(0.576044\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 7.92132 4.57338i 0.320462 0.185019i
\(612\) 0 0
\(613\) −10.6676 + 18.4768i −0.430859 + 0.746269i −0.996947 0.0780750i \(-0.975123\pi\)
0.566089 + 0.824344i \(0.308456\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 21.6233i 0.870519i 0.900305 + 0.435260i \(0.143343\pi\)
−0.900305 + 0.435260i \(0.856657\pi\)
\(618\) 0 0
\(619\) −17.1902 9.92474i −0.690931 0.398909i 0.113030 0.993592i \(-0.463944\pi\)
−0.803961 + 0.594682i \(0.797278\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 11.4466 + 7.61410i 0.458600 + 0.305052i
\(624\) 0 0
\(625\) 13.1141 + 22.7143i 0.524564 + 0.908571i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −43.8525 −1.74851
\(630\) 0 0
\(631\) −26.3803 −1.05018 −0.525092 0.851045i \(-0.675969\pi\)
−0.525092 + 0.851045i \(0.675969\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −11.6673 20.2083i −0.463001 0.801941i
\(636\) 0 0
\(637\) −16.3271 12.4209i −0.646902 0.492134i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −30.7441 17.7501i −1.21432 0.701087i −0.250621 0.968085i \(-0.580635\pi\)
−0.963697 + 0.266999i \(0.913968\pi\)
\(642\) 0 0
\(643\) 6.48881i 0.255894i 0.991781 + 0.127947i \(0.0408387\pi\)
−0.991781 + 0.127947i \(0.959161\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 21.8076 37.7718i 0.857343 1.48496i −0.0171104 0.999854i \(-0.505447\pi\)
0.874454 0.485109i \(-0.161220\pi\)
\(648\) 0 0
\(649\) −14.2691 + 8.23828i −0.560112 + 0.323381i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 21.7119 12.5353i 0.849650 0.490546i −0.0108825 0.999941i \(-0.503464\pi\)
0.860533 + 0.509395i \(0.170131\pi\)
\(654\) 0 0
\(655\) 11.0027 19.0572i 0.429911 0.744627i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 13.4657i 0.524551i −0.964993 0.262276i \(-0.915527\pi\)
0.964993 0.262276i \(-0.0844729\pi\)
\(660\) 0 0
\(661\) 40.4783 + 23.3702i 1.57442 + 0.908994i 0.995617 + 0.0935285i \(0.0298146\pi\)
0.578806 + 0.815465i \(0.303519\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −57.7931 3.66756i −2.24112 0.142222i
\(666\) 0 0
\(667\) −5.88590 10.1947i −0.227903 0.394740i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −11.1045 −0.428685
\(672\) 0 0
\(673\) −37.9275 −1.46200 −0.730999 0.682378i \(-0.760946\pi\)
−0.730999 + 0.682378i \(0.760946\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −1.56891 2.71744i −0.0602983 0.104440i 0.834300 0.551310i \(-0.185872\pi\)
−0.894599 + 0.446870i \(0.852539\pi\)
\(678\) 0 0
\(679\) 2.81700 + 5.68257i 0.108107 + 0.218077i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −12.1449 7.01184i −0.464711 0.268301i 0.249312 0.968423i \(-0.419795\pi\)
−0.714023 + 0.700122i \(0.753129\pi\)
\(684\) 0 0
\(685\) 45.7866i 1.74941i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −5.26668 + 9.12216i −0.200645 + 0.347527i
\(690\) 0 0
\(691\) 28.6141 16.5204i 1.08853 0.628464i 0.155346 0.987860i \(-0.450351\pi\)
0.933185 + 0.359396i \(0.117017\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 4.57965 2.64406i 0.173716 0.100295i
\(696\) 0 0
\(697\) 26.0825 45.1763i 0.987946 1.71117i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 32.4058i 1.22395i −0.790877 0.611975i \(-0.790375\pi\)
0.790877 0.611975i \(-0.209625\pi\)
\(702\) 0 0
\(703\) 49.0163 + 28.2996i 1.84869 + 1.06734i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 3.31028 52.1631i 0.124496 1.96179i
\(708\) 0 0
\(709\) 23.7528 + 41.1410i 0.892055 + 1.54508i 0.837408 + 0.546578i \(0.184070\pi\)
0.0546463 + 0.998506i \(0.482597\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −12.7048 −0.475799
\(714\) 0 0
\(715\) 34.2174 1.27966
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −23.1139 40.0344i −0.862003 1.49303i −0.869993 0.493065i \(-0.835877\pi\)
0.00798976 0.999968i \(-0.497457\pi\)
\(720\) 0 0
\(721\) 18.0145 27.0821i 0.670896 1.00859i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −29.0306 16.7608i −1.07817 0.622482i
\(726\) 0 0
\(727\) 14.2330i 0.527874i −0.964540 0.263937i \(-0.914979\pi\)
0.964540 0.263937i \(-0.0850210\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −15.5968 + 27.0145i −0.576870 + 0.999169i
\(732\) 0 0
\(733\) 34.7201 20.0457i 1.28242 0.740404i 0.305127 0.952312i \(-0.401301\pi\)
0.977290 + 0.211908i \(0.0679676\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 31.8046 18.3624i 1.17154 0.676388i
\(738\) 0 0
\(739\) 2.99731 5.19150i 0.110258 0.190972i −0.805616 0.592438i \(-0.798166\pi\)
0.915874 + 0.401465i \(0.131499\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 8.32437i 0.305392i −0.988273 0.152696i \(-0.951205\pi\)
0.988273 0.152696i \(-0.0487955\pi\)
\(744\) 0 0
\(745\) −40.0490 23.1223i −1.46728 0.847135i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −26.5910 + 39.9756i −0.971616 + 1.46068i
\(750\) 0 0
\(751\) −5.47650 9.48558i −0.199840 0.346134i 0.748636 0.662981i \(-0.230709\pi\)
−0.948477 + 0.316847i \(0.897376\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 18.9469 0.689548
\(756\) 0 0
\(757\) −3.28275 −0.119314 −0.0596568 0.998219i \(-0.519001\pi\)
−0.0596568 + 0.998219i \(0.519001\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 23.4616 + 40.6367i 0.850483 + 1.47308i 0.880773 + 0.473538i \(0.157023\pi\)
−0.0302909 + 0.999541i \(0.509643\pi\)
\(762\) 0 0
\(763\) −1.92394 + 30.3172i −0.0696511 + 1.09756i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 11.1785 + 6.45390i 0.403632 + 0.233037i
\(768\) 0 0
\(769\) 4.52166i 0.163055i 0.996671 + 0.0815276i \(0.0259799\pi\)
−0.996671 + 0.0815276i \(0.974020\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −20.8635 + 36.1367i −0.750409 + 1.29975i 0.197216 + 0.980360i \(0.436810\pi\)
−0.947625 + 0.319386i \(0.896523\pi\)
\(774\) 0 0
\(775\) −31.3315 + 18.0893i −1.12546 + 0.649786i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −58.3077 + 33.6640i −2.08909 + 1.20614i
\(780\) 0 0
\(781\) −13.7201 + 23.7640i −0.490945 + 0.850341i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 43.6577i 1.55821i
\(786\) 0 0
\(787\) 6.86957 + 3.96615i 0.244874 + 0.141378i 0.617415 0.786638i \(-0.288180\pi\)
−0.372541 + 0.928016i \(0.621513\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −2.12885 4.29440i −0.0756933 0.152691i
\(792\) 0 0
\(793\) 4.34966 + 7.53383i 0.154461 + 0.267534i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 17.5840 0.622857 0.311429 0.950270i \(-0.399193\pi\)
0.311429 + 0.950270i \(0.399193\pi\)
\(798\) 0 0
\(799\) 16.9584 0.599944
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 5.32324 + 9.22013i 0.187853 + 0.325371i
\(804\) 0 0
\(805\) −13.7201 0.870682i −0.483571 0.0306875i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 45.9338 + 26.5199i 1.61495 + 0.932390i 0.988200 + 0.153168i \(0.0489475\pi\)
0.626747 + 0.779223i \(0.284386\pi\)
\(810\) 0 0
\(811\) 0.966384i 0.0339343i −0.999856 0.0169672i \(-0.994599\pi\)
0.999856 0.0169672i \(-0.00540108\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 2.89204 5.00916i 0.101304 0.175463i
\(816\) 0 0
\(817\) 34.8669 20.1304i 1.21984 0.704274i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −38.5209 + 22.2400i −1.34439 + 0.776183i −0.987448 0.157945i \(-0.949513\pi\)
−0.356940 + 0.934127i \(0.616180\pi\)
\(822\) 0 0
\(823\) −13.0327 + 22.5732i −0.454290 + 0.786853i −0.998647 0.0520005i \(-0.983440\pi\)
0.544357 + 0.838853i \(0.316774\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 13.4166i 0.466540i −0.972412 0.233270i \(-0.925057\pi\)
0.972412 0.233270i \(-0.0749426\pi\)
\(828\) 0 0
\(829\) −0.407601 0.235328i −0.0141566 0.00817329i 0.492905 0.870083i \(-0.335935\pi\)
−0.507062 + 0.861910i \(0.669268\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −14.7010 35.0789i −0.509359 1.21541i
\(834\) 0 0
\(835\) −3.97919 6.89216i −0.137706 0.238513i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 23.4309 0.808923 0.404461 0.914555i \(-0.367459\pi\)
0.404461 + 0.914555i \(0.367459\pi\)
\(840\) 0 0
\(841\) −20.9946 −0.723953
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 6.88377 + 11.9230i 0.236809 + 0.410165i
\(846\) 0 0
\(847\) 6.59688 + 4.38812i 0.226671 + 0.150778i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 11.6365 + 6.71834i 0.398894 + 0.230302i
\(852\) 0 0
\(853\) 18.9678i 0.649446i −0.945809 0.324723i \(-0.894729\pi\)
0.945809 0.324723i \(-0.105271\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 6.59820 11.4284i 0.225390 0.390387i −0.731046 0.682328i \(-0.760968\pi\)
0.956436 + 0.291941i \(0.0943010\pi\)
\(858\) 0 0
\(859\) 7.78544 4.49493i 0.265636 0.153365i −0.361267 0.932462i \(-0.617656\pi\)
0.626903 + 0.779098i \(0.284322\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −12.0178 + 6.93847i −0.409090 + 0.236188i −0.690399 0.723429i \(-0.742565\pi\)
0.281309 + 0.959617i \(0.409232\pi\)
\(864\) 0 0
\(865\) −6.10603 + 10.5760i −0.207611 + 0.359593i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 19.2537i 0.653137i
\(870\) 0 0
\(871\) −24.9159 14.3852i −0.844242 0.487423i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 1.91662 0.950123i 0.0647937 0.0321200i
\(876\) 0 0
\(877\) 5.06531 + 8.77338i 0.171043 + 0.296256i 0.938785 0.344504i \(-0.111953\pi\)
−0.767742 + 0.640760i \(0.778620\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −30.7189 −1.03495 −0.517473 0.855700i \(-0.673127\pi\)
−0.517473 + 0.855700i \(0.673127\pi\)
\(882\) 0 0
\(883\) −35.4403 −1.19266 −0.596330 0.802740i \(-0.703375\pi\)
−0.596330 + 0.802740i \(0.703375\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −18.1384 31.4166i −0.609027 1.05487i −0.991401 0.130858i \(-0.958227\pi\)
0.382375 0.924007i \(-0.375106\pi\)
\(888\) 0 0
\(889\) −17.7228 + 8.78568i −0.594404 + 0.294662i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −18.9553 10.9438i −0.634315 0.366222i
\(894\) 0 0
\(895\) 61.0077i 2.03926i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −26.9785 + 46.7282i −0.899785 + 1.55847i
\(900\) 0 0
\(901\) −16.9128 + 9.76460i −0.563447 + 0.325306i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 49.1736 28.3904i 1.63459 0.943728i
\(906\) 0 0
\(907\) −3.92931 + 6.80577i −0.130471 + 0.225982i −0.923858 0.382735i \(-0.874982\pi\)
0.793387 + 0.608717i \(0.208316\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 28.9584i 0.959434i −0.877423 0.479717i \(-0.840739\pi\)
0.877423 0.479717i \(-0.159261\pi\)
\(912\) 0 0
\(913\) −50.5570 29.1891i −1.67319 0.966019i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −15.5319 10.3315i −0.512908 0.341177i
\(918\) 0 0
\(919\) 17.1993 + 29.7901i 0.567353 + 0.982684i 0.996826 + 0.0796049i \(0.0253659\pi\)
−0.429473 + 0.903080i \(0.641301\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 21.4968 0.707576
\(924\) 0 0
\(925\) 38.2626 1.25807
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −3.05052 5.28365i −0.100084 0.173351i 0.811635 0.584165i \(-0.198578\pi\)
−0.911719 + 0.410814i \(0.865245\pi\)
\(930\) 0 0
\(931\) −6.20560 + 48.6967i −0.203380 + 1.59597i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 54.9409 + 31.7201i 1.79676 + 1.03736i
\(936\) 0 0
\(937\) 50.7128i 1.65671i 0.560201 + 0.828357i \(0.310724\pi\)
−0.560201 + 0.828357i \(0.689276\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −17.5616 + 30.4177i −0.572493 + 0.991587i 0.423816 + 0.905748i \(0.360690\pi\)
−0.996309 + 0.0858389i \(0.972643\pi\)
\(942\) 0 0
\(943\) −13.8423 + 7.99185i −0.450767 + 0.260250i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 34.2973 19.8016i 1.11451 0.643465i 0.174519 0.984654i \(-0.444163\pi\)
0.939995 + 0.341189i \(0.110830\pi\)
\(948\) 0 0
\(949\) 4.17025 7.22309i 0.135372 0.234471i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 12.5942i 0.407966i −0.978974 0.203983i \(-0.934611\pi\)
0.978974 0.203983i \(-0.0653887\pi\)
\(954\) 0 0
\(955\) −40.7637 23.5349i −1.31908 0.761573i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −38.7359 2.45819i −1.25085 0.0793790i
\(960\) 0 0
\(961\) 13.6168 + 23.5850i 0.439251 + 0.760805i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 34.7895 1.11992
\(966\) 0 0
\(967\) 51.8407 1.66708 0.833542 0.552456i \(-0.186309\pi\)
0.833542 + 0.552456i \(0.186309\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −0.918852 1.59150i −0.0294874 0.0510736i 0.850905 0.525319i \(-0.176054\pi\)
−0.880392 + 0.474246i \(0.842721\pi\)
\(972\) 0 0
\(973\) −1.99103 4.01639i −0.0638296 0.128760i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 23.2808 + 13.4412i 0.744818 + 0.430021i 0.823818 0.566854i \(-0.191840\pi\)
−0.0790007 + 0.996875i \(0.525173\pi\)
\(978\) 0 0
\(979\) 19.4385i 0.621257i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −8.19850 + 14.2002i −0.261492 + 0.452917i −0.966638 0.256145i \(-0.917548\pi\)
0.705147 + 0.709061i \(0.250881\pi\)
\(984\) 0 0
\(985\) 55.0570 31.7872i 1.75426 1.01282i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 8.27742 4.77897i 0.263207 0.151962i
\(990\) 0 0
\(991\) 1.17941 2.04279i 0.0374651 0.0648915i −0.846685 0.532095i \(-0.821405\pi\)
0.884150 + 0.467203i \(0.154738\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 23.1704i 0.734552i
\(996\) 0 0
\(997\) −13.1114 7.56988i −0.415242 0.239740i 0.277797 0.960640i \(-0.410396\pi\)
−0.693040 + 0.720899i \(0.743729\pi\)
\(998\) 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 756.2.t.e.269.6 yes 12
3.2 odd 2 inner 756.2.t.e.269.1 12
7.3 odd 6 5292.2.f.e.2645.11 12
7.4 even 3 5292.2.f.e.2645.1 12
7.5 odd 6 inner 756.2.t.e.593.1 yes 12
9.2 odd 6 2268.2.bm.i.1025.6 12
9.4 even 3 2268.2.w.i.269.6 12
9.5 odd 6 2268.2.w.i.269.1 12
9.7 even 3 2268.2.bm.i.1025.1 12
21.5 even 6 inner 756.2.t.e.593.6 yes 12
21.11 odd 6 5292.2.f.e.2645.12 12
21.17 even 6 5292.2.f.e.2645.2 12
63.5 even 6 2268.2.bm.i.593.1 12
63.40 odd 6 2268.2.bm.i.593.6 12
63.47 even 6 2268.2.w.i.1349.6 12
63.61 odd 6 2268.2.w.i.1349.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
756.2.t.e.269.1 12 3.2 odd 2 inner
756.2.t.e.269.6 yes 12 1.1 even 1 trivial
756.2.t.e.593.1 yes 12 7.5 odd 6 inner
756.2.t.e.593.6 yes 12 21.5 even 6 inner
2268.2.w.i.269.1 12 9.5 odd 6
2268.2.w.i.269.6 12 9.4 even 3
2268.2.w.i.1349.1 12 63.61 odd 6
2268.2.w.i.1349.6 12 63.47 even 6
2268.2.bm.i.593.1 12 63.5 even 6
2268.2.bm.i.593.6 12 63.40 odd 6
2268.2.bm.i.1025.1 12 9.7 even 3
2268.2.bm.i.1025.6 12 9.2 odd 6
5292.2.f.e.2645.1 12 7.4 even 3
5292.2.f.e.2645.2 12 21.17 even 6
5292.2.f.e.2645.11 12 7.3 odd 6
5292.2.f.e.2645.12 12 21.11 odd 6