Properties

Label 2736.2.dc.e.1889.6
Level $2736$
Weight $2$
Character 2736.1889
Analytic conductor $21.847$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2736,2,Mod(449,2736)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2736, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2736.449");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2736.dc (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: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 8 x^{19} - 18 x^{18} + 232 x^{17} + 208 x^{16} - 3152 x^{15} - 2732 x^{14} + 24552 x^{13} + 30728 x^{12} - 110512 x^{11} - 211368 x^{10} + 231024 x^{9} + \cdots + 414864 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 1368)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1889.6
Root \(-2.54394 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 2736.1889
Dual form 2736.2.dc.e.449.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.501439 + 0.289506i) q^{5} +0.198579 q^{7} +O(q^{10})\) \(q+(0.501439 + 0.289506i) q^{5} +0.198579 q^{7} -2.48583i q^{11} +(-4.34042 + 2.50594i) q^{13} +(-4.01349 - 2.31719i) q^{17} +(3.16251 + 2.99976i) q^{19} +(5.14465 - 2.97027i) q^{23} +(-2.33237 - 4.03979i) q^{25} +(4.51838 + 7.82606i) q^{29} -1.41355i q^{31} +(0.0995752 + 0.0574898i) q^{35} -11.6243i q^{37} +(3.60914 - 6.25122i) q^{41} +(4.66185 - 8.07455i) q^{43} +(-1.86260 + 1.07537i) q^{47} -6.96057 q^{49} +(1.62456 + 2.81382i) q^{53} +(0.719661 - 1.24649i) q^{55} +(4.59275 - 7.95487i) q^{59} +(-2.46529 - 4.27001i) q^{61} -2.90194 q^{65} +(-3.17689 + 1.83418i) q^{67} +(8.07974 - 13.9945i) q^{71} +(-5.02871 + 8.70999i) q^{73} -0.493633i q^{77} +(1.26495 + 0.730318i) q^{79} +0.592477i q^{83} +(-1.34168 - 2.32386i) q^{85} +(2.78657 + 4.82649i) q^{89} +(-0.861917 + 0.497628i) q^{91} +(0.717358 + 2.41976i) q^{95} +(-9.32474 - 5.38364i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 4 q^{7} + 6 q^{13} - 12 q^{17} - 4 q^{19} + 14 q^{25} - 12 q^{35} - 8 q^{41} + 2 q^{43} - 36 q^{47} + 32 q^{49} + 8 q^{53} - 12 q^{55} + 8 q^{59} - 2 q^{61} - 8 q^{65} - 30 q^{67} - 4 q^{71} - 22 q^{73} - 54 q^{79} - 4 q^{85} + 32 q^{89} - 18 q^{91} - 32 q^{95} + 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(1\) \(1\)

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
\(5\) 0.501439 + 0.289506i 0.224250 + 0.129471i 0.607917 0.794001i \(-0.292005\pi\)
−0.383666 + 0.923472i \(0.625339\pi\)
\(6\) 0 0
\(7\) 0.198579 0.0750558 0.0375279 0.999296i \(-0.488052\pi\)
0.0375279 + 0.999296i \(0.488052\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.48583i 0.749505i −0.927125 0.374752i \(-0.877728\pi\)
0.927125 0.374752i \(-0.122272\pi\)
\(12\) 0 0
\(13\) −4.34042 + 2.50594i −1.20382 + 0.695024i −0.961402 0.275149i \(-0.911273\pi\)
−0.242415 + 0.970173i \(0.577939\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.01349 2.31719i −0.973415 0.562001i −0.0731391 0.997322i \(-0.523302\pi\)
−0.900276 + 0.435321i \(0.856635\pi\)
\(18\) 0 0
\(19\) 3.16251 + 2.99976i 0.725529 + 0.688191i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.14465 2.97027i 1.07273 0.619344i 0.143807 0.989606i \(-0.454065\pi\)
0.928927 + 0.370262i \(0.120732\pi\)
\(24\) 0 0
\(25\) −2.33237 4.03979i −0.466475 0.807958i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.51838 + 7.82606i 0.839042 + 1.45326i 0.890696 + 0.454599i \(0.150217\pi\)
−0.0516543 + 0.998665i \(0.516449\pi\)
\(30\) 0 0
\(31\) 1.41355i 0.253882i −0.991910 0.126941i \(-0.959484\pi\)
0.991910 0.126941i \(-0.0405158\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.0995752 + 0.0574898i 0.0168313 + 0.00971755i
\(36\) 0 0
\(37\) 11.6243i 1.91102i −0.294951 0.955512i \(-0.595303\pi\)
0.294951 0.955512i \(-0.404697\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.60914 6.25122i 0.563654 0.976277i −0.433520 0.901144i \(-0.642729\pi\)
0.997174 0.0751327i \(-0.0239380\pi\)
\(42\) 0 0
\(43\) 4.66185 8.07455i 0.710925 1.23136i −0.253586 0.967313i \(-0.581610\pi\)
0.964510 0.264045i \(-0.0850567\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.86260 + 1.07537i −0.271689 + 0.156860i −0.629655 0.776875i \(-0.716804\pi\)
0.357966 + 0.933735i \(0.383470\pi\)
\(48\) 0 0
\(49\) −6.96057 −0.994367
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.62456 + 2.81382i 0.223151 + 0.386508i 0.955763 0.294138i \(-0.0950325\pi\)
−0.732612 + 0.680646i \(0.761699\pi\)
\(54\) 0 0
\(55\) 0.719661 1.24649i 0.0970391 0.168077i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.59275 7.95487i 0.597925 1.03564i −0.395202 0.918594i \(-0.629326\pi\)
0.993127 0.117042i \(-0.0373411\pi\)
\(60\) 0 0
\(61\) −2.46529 4.27001i −0.315648 0.546719i 0.663927 0.747798i \(-0.268889\pi\)
−0.979575 + 0.201079i \(0.935555\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.90194 −0.359941
\(66\) 0 0
\(67\) −3.17689 + 1.83418i −0.388119 + 0.224081i −0.681345 0.731963i \(-0.738605\pi\)
0.293226 + 0.956043i \(0.405271\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8.07974 13.9945i 0.958889 1.66084i 0.233683 0.972313i \(-0.424922\pi\)
0.725206 0.688532i \(-0.241745\pi\)
\(72\) 0 0
\(73\) −5.02871 + 8.70999i −0.588566 + 1.01943i 0.405854 + 0.913938i \(0.366974\pi\)
−0.994420 + 0.105489i \(0.966359\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.493633i 0.0562547i
\(78\) 0 0
\(79\) 1.26495 + 0.730318i 0.142318 + 0.0821672i 0.569468 0.822013i \(-0.307149\pi\)
−0.427150 + 0.904181i \(0.640482\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0.592477i 0.0650328i 0.999471 + 0.0325164i \(0.0103521\pi\)
−0.999471 + 0.0325164i \(0.989648\pi\)
\(84\) 0 0
\(85\) −1.34168 2.32386i −0.145526 0.252058i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.78657 + 4.82649i 0.295376 + 0.511607i 0.975072 0.221887i \(-0.0712216\pi\)
−0.679696 + 0.733494i \(0.737888\pi\)
\(90\) 0 0
\(91\) −0.861917 + 0.497628i −0.0903534 + 0.0521656i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0.717358 + 2.41976i 0.0735994 + 0.248262i
\(96\) 0 0
\(97\) −9.32474 5.38364i −0.946784 0.546626i −0.0547037 0.998503i \(-0.517421\pi\)
−0.892081 + 0.451876i \(0.850755\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 12.3063 7.10502i 1.22452 0.706976i 0.258640 0.965974i \(-0.416726\pi\)
0.965878 + 0.258998i \(0.0833924\pi\)
\(102\) 0 0
\(103\) 18.3772i 1.81076i 0.424607 + 0.905378i \(0.360412\pi\)
−0.424607 + 0.905378i \(0.639588\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 14.0687 1.36007 0.680035 0.733180i \(-0.261965\pi\)
0.680035 + 0.733180i \(0.261965\pi\)
\(108\) 0 0
\(109\) −5.11575 2.95358i −0.490000 0.282902i 0.234574 0.972098i \(-0.424630\pi\)
−0.724574 + 0.689197i \(0.757964\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −3.69467 −0.347566 −0.173783 0.984784i \(-0.555599\pi\)
−0.173783 + 0.984784i \(0.555599\pi\)
\(114\) 0 0
\(115\) 3.43964 0.320748
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −0.796995 0.460145i −0.0730604 0.0421815i
\(120\) 0 0
\(121\) 4.82067 0.438243
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 5.59600i 0.500521i
\(126\) 0 0
\(127\) 7.90854 4.56600i 0.701770 0.405167i −0.106237 0.994341i \(-0.533880\pi\)
0.808006 + 0.589174i \(0.200547\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −13.1576 7.59655i −1.14959 0.663713i −0.200799 0.979632i \(-0.564354\pi\)
−0.948786 + 0.315919i \(0.897687\pi\)
\(132\) 0 0
\(133\) 0.628008 + 0.595689i 0.0544552 + 0.0516527i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −5.50990 + 3.18114i −0.470742 + 0.271783i −0.716550 0.697535i \(-0.754280\pi\)
0.245808 + 0.969319i \(0.420947\pi\)
\(138\) 0 0
\(139\) −9.45810 16.3819i −0.802226 1.38950i −0.918148 0.396238i \(-0.870316\pi\)
0.115922 0.993258i \(-0.463018\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 6.22934 + 10.7895i 0.520924 + 0.902266i
\(144\) 0 0
\(145\) 5.23239i 0.434526i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 3.73882 + 2.15861i 0.306296 + 0.176840i 0.645268 0.763956i \(-0.276746\pi\)
−0.338972 + 0.940797i \(0.610079\pi\)
\(150\) 0 0
\(151\) 13.5536i 1.10298i −0.834182 0.551489i \(-0.814060\pi\)
0.834182 0.551489i \(-0.185940\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0.409232 0.708810i 0.0328703 0.0569330i
\(156\) 0 0
\(157\) −0.860138 + 1.48980i −0.0686465 + 0.118899i −0.898306 0.439371i \(-0.855201\pi\)
0.829659 + 0.558270i \(0.188535\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 1.02162 0.589833i 0.0805150 0.0464853i
\(162\) 0 0
\(163\) −7.69773 −0.602933 −0.301467 0.953477i \(-0.597476\pi\)
−0.301467 + 0.953477i \(0.597476\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −5.95402 10.3127i −0.460736 0.798017i 0.538262 0.842777i \(-0.319081\pi\)
−0.998998 + 0.0447601i \(0.985748\pi\)
\(168\) 0 0
\(169\) 6.05951 10.4954i 0.466116 0.807336i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 7.88358 13.6548i 0.599377 1.03815i −0.393536 0.919309i \(-0.628748\pi\)
0.992913 0.118842i \(-0.0379183\pi\)
\(174\) 0 0
\(175\) −0.463160 0.802217i −0.0350116 0.0606419i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 21.4722 1.60491 0.802454 0.596714i \(-0.203527\pi\)
0.802454 + 0.596714i \(0.203527\pi\)
\(180\) 0 0
\(181\) 6.46463 3.73236i 0.480512 0.277424i −0.240118 0.970744i \(-0.577186\pi\)
0.720630 + 0.693320i \(0.243853\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 3.36530 5.82888i 0.247422 0.428548i
\(186\) 0 0
\(187\) −5.76013 + 9.97684i −0.421223 + 0.729579i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 7.49453i 0.542285i −0.962539 0.271143i \(-0.912598\pi\)
0.962539 0.271143i \(-0.0874015\pi\)
\(192\) 0 0
\(193\) 6.51556 + 3.76176i 0.469001 + 0.270778i 0.715821 0.698284i \(-0.246053\pi\)
−0.246821 + 0.969061i \(0.579386\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 3.30108i 0.235192i 0.993062 + 0.117596i \(0.0375188\pi\)
−0.993062 + 0.117596i \(0.962481\pi\)
\(198\) 0 0
\(199\) −8.08225 13.9989i −0.572936 0.992354i −0.996263 0.0863766i \(-0.972471\pi\)
0.423327 0.905977i \(-0.360862\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0.897255 + 1.55409i 0.0629750 + 0.109076i
\(204\) 0 0
\(205\) 3.61953 2.08974i 0.252799 0.145954i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 7.45687 7.86145i 0.515803 0.543788i
\(210\) 0 0
\(211\) 11.7050 + 6.75788i 0.805805 + 0.465232i 0.845497 0.533980i \(-0.179304\pi\)
−0.0396918 + 0.999212i \(0.512638\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 4.67526 2.69926i 0.318850 0.184088i
\(216\) 0 0
\(217\) 0.280702i 0.0190553i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 23.2270 1.56242
\(222\) 0 0
\(223\) −15.1337 8.73745i −1.01343 0.585103i −0.101234 0.994863i \(-0.532279\pi\)
−0.912193 + 0.409760i \(0.865613\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 16.5976 1.10162 0.550810 0.834630i \(-0.314319\pi\)
0.550810 + 0.834630i \(0.314319\pi\)
\(228\) 0 0
\(229\) −10.5614 −0.697916 −0.348958 0.937138i \(-0.613464\pi\)
−0.348958 + 0.937138i \(0.613464\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −2.58362 1.49165i −0.169258 0.0977214i 0.412978 0.910741i \(-0.364489\pi\)
−0.582236 + 0.813020i \(0.697822\pi\)
\(234\) 0 0
\(235\) −1.24531 −0.0812350
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 21.8616i 1.41411i 0.707158 + 0.707056i \(0.249977\pi\)
−0.707158 + 0.707056i \(0.750023\pi\)
\(240\) 0 0
\(241\) −6.70550 + 3.87142i −0.431939 + 0.249380i −0.700172 0.713974i \(-0.746893\pi\)
0.268233 + 0.963354i \(0.413560\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −3.49030 2.01512i −0.222987 0.128742i
\(246\) 0 0
\(247\) −21.2438 5.09513i −1.35171 0.324196i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −18.5605 + 10.7159i −1.17153 + 0.676383i −0.954040 0.299680i \(-0.903120\pi\)
−0.217490 + 0.976063i \(0.569787\pi\)
\(252\) 0 0
\(253\) −7.38357 12.7887i −0.464201 0.804020i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −7.45876 12.9189i −0.465265 0.805862i 0.533949 0.845517i \(-0.320707\pi\)
−0.999213 + 0.0396549i \(0.987374\pi\)
\(258\) 0 0
\(259\) 2.30834i 0.143433i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −19.7069 11.3778i −1.21518 0.701586i −0.251299 0.967910i \(-0.580858\pi\)
−0.963884 + 0.266324i \(0.914191\pi\)
\(264\) 0 0
\(265\) 1.88128i 0.115566i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −7.57925 + 13.1276i −0.462115 + 0.800407i −0.999066 0.0432067i \(-0.986243\pi\)
0.536951 + 0.843613i \(0.319576\pi\)
\(270\) 0 0
\(271\) −2.62861 + 4.55289i −0.159677 + 0.276568i −0.934752 0.355301i \(-0.884379\pi\)
0.775075 + 0.631869i \(0.217712\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −10.0422 + 5.79787i −0.605568 + 0.349625i
\(276\) 0 0
\(277\) 22.8807 1.37477 0.687384 0.726294i \(-0.258759\pi\)
0.687384 + 0.726294i \(0.258759\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −10.4022 18.0171i −0.620542 1.07481i −0.989385 0.145318i \(-0.953579\pi\)
0.368843 0.929492i \(-0.379754\pi\)
\(282\) 0 0
\(283\) −10.9479 + 18.9622i −0.650783 + 1.12719i 0.332151 + 0.943226i \(0.392226\pi\)
−0.982933 + 0.183962i \(0.941108\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0.716700 1.24136i 0.0423055 0.0732752i
\(288\) 0 0
\(289\) 2.23874 + 3.87761i 0.131691 + 0.228095i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 5.84168 0.341274 0.170637 0.985334i \(-0.445417\pi\)
0.170637 + 0.985334i \(0.445417\pi\)
\(294\) 0 0
\(295\) 4.60596 2.65925i 0.268169 0.154828i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −14.8866 + 25.7844i −0.860917 + 1.49115i
\(300\) 0 0
\(301\) 0.925745 1.60344i 0.0533590 0.0924206i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 2.85487i 0.163469i
\(306\) 0 0
\(307\) −14.3364 8.27711i −0.818220 0.472399i 0.0315824 0.999501i \(-0.489945\pi\)
−0.849802 + 0.527102i \(0.823279\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 16.7502i 0.949817i 0.880035 + 0.474908i \(0.157519\pi\)
−0.880035 + 0.474908i \(0.842481\pi\)
\(312\) 0 0
\(313\) 14.6097 + 25.3047i 0.825789 + 1.43031i 0.901315 + 0.433165i \(0.142603\pi\)
−0.0755253 + 0.997144i \(0.524063\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 7.77821 + 13.4723i 0.436868 + 0.756677i 0.997446 0.0714243i \(-0.0227545\pi\)
−0.560578 + 0.828101i \(0.689421\pi\)
\(318\) 0 0
\(319\) 19.4542 11.2319i 1.08923 0.628866i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −5.74170 19.3676i −0.319477 1.07764i
\(324\) 0 0
\(325\) 20.2470 + 11.6896i 1.12310 + 0.648422i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −0.369874 + 0.213547i −0.0203918 + 0.0117732i
\(330\) 0 0
\(331\) 30.0226i 1.65019i −0.564995 0.825094i \(-0.691122\pi\)
0.564995 0.825094i \(-0.308878\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −2.12402 −0.116048
\(336\) 0 0
\(337\) 4.16548 + 2.40494i 0.226908 + 0.131005i 0.609145 0.793059i \(-0.291513\pi\)
−0.382237 + 0.924064i \(0.624846\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −3.51385 −0.190285
\(342\) 0 0
\(343\) −2.77228 −0.149689
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −3.64059 2.10190i −0.195437 0.112836i 0.399088 0.916913i \(-0.369327\pi\)
−0.594525 + 0.804077i \(0.702660\pi\)
\(348\) 0 0
\(349\) 30.7302 1.64495 0.822476 0.568800i \(-0.192592\pi\)
0.822476 + 0.568800i \(0.192592\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 2.32845i 0.123931i −0.998078 0.0619653i \(-0.980263\pi\)
0.998078 0.0619653i \(-0.0197368\pi\)
\(354\) 0 0
\(355\) 8.10299 4.67827i 0.430062 0.248297i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 25.0394 + 14.4565i 1.32153 + 0.762985i 0.983972 0.178321i \(-0.0570666\pi\)
0.337556 + 0.941306i \(0.390400\pi\)
\(360\) 0 0
\(361\) 1.00293 + 18.9735i 0.0527858 + 0.998606i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −5.04318 + 2.91168i −0.263972 + 0.152404i
\(366\) 0 0
\(367\) 14.3220 + 24.8064i 0.747601 + 1.29488i 0.948970 + 0.315368i \(0.102128\pi\)
−0.201368 + 0.979516i \(0.564539\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0.322604 + 0.558766i 0.0167488 + 0.0290097i
\(372\) 0 0
\(373\) 17.0680i 0.883745i −0.897078 0.441873i \(-0.854314\pi\)
0.897078 0.441873i \(-0.145686\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −39.2233 22.6456i −2.02011 1.16631i
\(378\) 0 0
\(379\) 31.8562i 1.63634i 0.574975 + 0.818171i \(0.305012\pi\)
−0.574975 + 0.818171i \(0.694988\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 2.84259 4.92351i 0.145249 0.251579i −0.784217 0.620487i \(-0.786935\pi\)
0.929466 + 0.368908i \(0.120268\pi\)
\(384\) 0 0
\(385\) 0.142910 0.247527i 0.00728335 0.0126151i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 29.0407 16.7667i 1.47242 0.850104i 0.472904 0.881114i \(-0.343206\pi\)
0.999519 + 0.0310100i \(0.00987237\pi\)
\(390\) 0 0
\(391\) −27.5307 −1.39229
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0.422863 + 0.732420i 0.0212765 + 0.0368520i
\(396\) 0 0
\(397\) 1.89375 3.28008i 0.0950448 0.164622i −0.814582 0.580048i \(-0.803034\pi\)
0.909627 + 0.415425i \(0.136367\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −12.0794 + 20.9221i −0.603216 + 1.04480i 0.389115 + 0.921189i \(0.372781\pi\)
−0.992331 + 0.123611i \(0.960552\pi\)
\(402\) 0 0
\(403\) 3.54228 + 6.13541i 0.176454 + 0.305627i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −28.8960 −1.43232
\(408\) 0 0
\(409\) 7.04034 4.06474i 0.348122 0.200988i −0.315736 0.948847i \(-0.602251\pi\)
0.663858 + 0.747859i \(0.268918\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0.912023 1.57967i 0.0448777 0.0777305i
\(414\) 0 0
\(415\) −0.171526 + 0.297091i −0.00841986 + 0.0145836i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 9.73708i 0.475687i −0.971303 0.237844i \(-0.923559\pi\)
0.971303 0.237844i \(-0.0764406\pi\)
\(420\) 0 0
\(421\) −15.1997 8.77556i −0.740789 0.427695i 0.0815671 0.996668i \(-0.474007\pi\)
−0.822356 + 0.568973i \(0.807341\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 21.6182i 1.04864i
\(426\) 0 0
\(427\) −0.489555 0.847935i −0.0236913 0.0410344i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 1.74207 + 3.01736i 0.0839127 + 0.145341i 0.904927 0.425566i \(-0.139925\pi\)
−0.821015 + 0.570907i \(0.806592\pi\)
\(432\) 0 0
\(433\) −22.5893 + 13.0419i −1.08557 + 0.626756i −0.932394 0.361443i \(-0.882284\pi\)
−0.153178 + 0.988199i \(0.548951\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 25.1801 + 6.03921i 1.20453 + 0.288894i
\(438\) 0 0
\(439\) 0.261898 + 0.151207i 0.0124997 + 0.00721671i 0.506237 0.862394i \(-0.331036\pi\)
−0.493737 + 0.869611i \(0.664369\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −4.28403 + 2.47339i −0.203541 + 0.117514i −0.598306 0.801268i \(-0.704159\pi\)
0.394765 + 0.918782i \(0.370826\pi\)
\(444\) 0 0
\(445\) 3.22692i 0.152971i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −18.5597 −0.875889 −0.437944 0.899002i \(-0.644293\pi\)
−0.437944 + 0.899002i \(0.644293\pi\)
\(450\) 0 0
\(451\) −15.5394 8.97170i −0.731724 0.422461i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −0.576265 −0.0270157
\(456\) 0 0
\(457\) −39.3948 −1.84281 −0.921405 0.388603i \(-0.872958\pi\)
−0.921405 + 0.388603i \(0.872958\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 21.3706 + 12.3383i 0.995327 + 0.574652i 0.906862 0.421427i \(-0.138471\pi\)
0.0884645 + 0.996079i \(0.471804\pi\)
\(462\) 0 0
\(463\) 20.5320 0.954205 0.477102 0.878848i \(-0.341687\pi\)
0.477102 + 0.878848i \(0.341687\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 12.4282i 0.575111i 0.957764 + 0.287555i \(0.0928425\pi\)
−0.957764 + 0.287555i \(0.907157\pi\)
\(468\) 0 0
\(469\) −0.630864 + 0.364229i −0.0291306 + 0.0168185i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −20.0719 11.5885i −0.922909 0.532842i
\(474\) 0 0
\(475\) 4.74223 19.7724i 0.217588 0.907221i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 11.1284 6.42499i 0.508470 0.293565i −0.223735 0.974650i \(-0.571825\pi\)
0.732204 + 0.681085i \(0.238492\pi\)
\(480\) 0 0
\(481\) 29.1299 + 50.4544i 1.32821 + 2.30052i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −3.11719 5.39913i −0.141544 0.245162i
\(486\) 0 0
\(487\) 11.7617i 0.532975i 0.963838 + 0.266487i \(0.0858630\pi\)
−0.963838 + 0.266487i \(0.914137\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 12.7058 + 7.33571i 0.573406 + 0.331056i 0.758509 0.651663i \(-0.225928\pi\)
−0.185103 + 0.982719i \(0.559262\pi\)
\(492\) 0 0
\(493\) 41.8798i 1.88617i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 1.60447 2.77902i 0.0719702 0.124656i
\(498\) 0 0
\(499\) −3.33608 + 5.77826i −0.149343 + 0.258670i −0.930985 0.365057i \(-0.881049\pi\)
0.781642 + 0.623728i \(0.214383\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 5.07351 2.92919i 0.226216 0.130606i −0.382609 0.923910i \(-0.624974\pi\)
0.608825 + 0.793304i \(0.291641\pi\)
\(504\) 0 0
\(505\) 8.22777 0.366131
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 5.56376 + 9.63672i 0.246609 + 0.427140i 0.962583 0.270987i \(-0.0873502\pi\)
−0.715973 + 0.698128i \(0.754017\pi\)
\(510\) 0 0
\(511\) −0.998597 + 1.72962i −0.0441753 + 0.0765139i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −5.32029 + 9.21502i −0.234440 + 0.406062i
\(516\) 0 0
\(517\) 2.67319 + 4.63011i 0.117567 + 0.203632i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −31.9945 −1.40170 −0.700852 0.713307i \(-0.747197\pi\)
−0.700852 + 0.713307i \(0.747197\pi\)
\(522\) 0 0
\(523\) 7.69847 4.44471i 0.336631 0.194354i −0.322150 0.946688i \(-0.604406\pi\)
0.658781 + 0.752335i \(0.271072\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −3.27547 + 5.67328i −0.142682 + 0.247132i
\(528\) 0 0
\(529\) 6.14498 10.6434i 0.267173 0.462757i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 36.1772i 1.56701i
\(534\) 0 0
\(535\) 7.05458 + 4.07296i 0.304996 + 0.176090i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 17.3028i 0.745283i
\(540\) 0 0
\(541\) 17.0404 + 29.5149i 0.732626 + 1.26895i 0.955757 + 0.294157i \(0.0950389\pi\)
−0.223131 + 0.974788i \(0.571628\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −1.71016 2.96208i −0.0732551 0.126882i
\(546\) 0 0
\(547\) 1.75459 1.01302i 0.0750210 0.0433134i −0.462020 0.886869i \(-0.652875\pi\)
0.537041 + 0.843556i \(0.319542\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −9.18686 + 38.3040i −0.391373 + 1.63181i
\(552\) 0 0
\(553\) 0.251192 + 0.145026i 0.0106818 + 0.00616713i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −15.9880 + 9.23067i −0.677433 + 0.391116i −0.798887 0.601481i \(-0.794577\pi\)
0.121454 + 0.992597i \(0.461244\pi\)
\(558\) 0 0
\(559\) 46.7293i 1.97644i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −26.3102 −1.10884 −0.554421 0.832237i \(-0.687060\pi\)
−0.554421 + 0.832237i \(0.687060\pi\)
\(564\) 0 0
\(565\) −1.85265 1.06963i −0.0779417 0.0449996i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 27.3661 1.14725 0.573624 0.819119i \(-0.305537\pi\)
0.573624 + 0.819119i \(0.305537\pi\)
\(570\) 0 0
\(571\) 42.5121 1.77908 0.889539 0.456860i \(-0.151026\pi\)
0.889539 + 0.456860i \(0.151026\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −23.9985 13.8555i −1.00081 0.577816i
\(576\) 0 0
\(577\) −30.8052 −1.28244 −0.641219 0.767358i \(-0.721571\pi\)
−0.641219 + 0.767358i \(0.721571\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0.117654i 0.00488109i
\(582\) 0 0
\(583\) 6.99467 4.03838i 0.289690 0.167253i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −20.0210 11.5592i −0.826357 0.477097i 0.0262466 0.999655i \(-0.491644\pi\)
−0.852604 + 0.522558i \(0.824978\pi\)
\(588\) 0 0
\(589\) 4.24031 4.47037i 0.174719 0.184199i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −27.8849 + 16.0993i −1.14509 + 0.661121i −0.947687 0.319201i \(-0.896586\pi\)
−0.197408 + 0.980321i \(0.563252\pi\)
\(594\) 0 0
\(595\) −0.266429 0.461469i −0.0109225 0.0189184i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −3.17822 5.50484i −0.129859 0.224922i 0.793763 0.608227i \(-0.208119\pi\)
−0.923622 + 0.383305i \(0.874786\pi\)
\(600\) 0 0
\(601\) 3.82631i 0.156079i −0.996950 0.0780393i \(-0.975134\pi\)
0.996950 0.0780393i \(-0.0248660\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 2.41727 + 1.39561i 0.0982760 + 0.0567397i
\(606\) 0 0
\(607\) 15.1802i 0.616146i 0.951363 + 0.308073i \(0.0996842\pi\)
−0.951363 + 0.308073i \(0.900316\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 5.38966 9.33516i 0.218042 0.377660i
\(612\) 0 0
\(613\) 3.68177 6.37702i 0.148705 0.257565i −0.782044 0.623223i \(-0.785823\pi\)
0.930749 + 0.365658i \(0.119156\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −8.93545 + 5.15888i −0.359728 + 0.207689i −0.668961 0.743297i \(-0.733261\pi\)
0.309234 + 0.950986i \(0.399927\pi\)
\(618\) 0 0
\(619\) 13.5742 0.545594 0.272797 0.962072i \(-0.412051\pi\)
0.272797 + 0.962072i \(0.412051\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0.553355 + 0.958439i 0.0221697 + 0.0383991i
\(624\) 0 0
\(625\) −10.0418 + 17.3929i −0.401672 + 0.695716i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −26.9357 + 46.6541i −1.07400 + 1.86022i
\(630\) 0 0
\(631\) 11.8122 + 20.4593i 0.470236 + 0.814472i 0.999421 0.0340341i \(-0.0108355\pi\)
−0.529185 + 0.848507i \(0.677502\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 5.28753 0.209829
\(636\) 0 0
\(637\) 30.2118 17.4428i 1.19703 0.691108i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −4.65587 + 8.06420i −0.183896 + 0.318517i −0.943204 0.332215i \(-0.892204\pi\)
0.759308 + 0.650731i \(0.225538\pi\)
\(642\) 0 0
\(643\) −6.52083 + 11.2944i −0.257156 + 0.445408i −0.965479 0.260481i \(-0.916119\pi\)
0.708323 + 0.705889i \(0.249452\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 39.2031i 1.54123i −0.637298 0.770617i \(-0.719948\pi\)
0.637298 0.770617i \(-0.280052\pi\)
\(648\) 0 0
\(649\) −19.7744 11.4168i −0.776214 0.448147i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 3.23276i 0.126508i −0.997997 0.0632539i \(-0.979852\pi\)
0.997997 0.0632539i \(-0.0201478\pi\)
\(654\) 0 0
\(655\) −4.39849 7.61841i −0.171863 0.297676i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −21.0326 36.4295i −0.819312 1.41909i −0.906190 0.422872i \(-0.861022\pi\)
0.0868772 0.996219i \(-0.472311\pi\)
\(660\) 0 0
\(661\) −31.6185 + 18.2549i −1.22982 + 0.710035i −0.966991 0.254809i \(-0.917987\pi\)
−0.262825 + 0.964844i \(0.584654\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0.142452 + 0.480513i 0.00552406 + 0.0186335i
\(666\) 0 0
\(667\) 46.4910 + 26.8416i 1.80014 + 1.03931i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −10.6145 + 6.12829i −0.409769 + 0.236580i
\(672\) 0 0
\(673\) 27.9597i 1.07777i −0.842381 0.538883i \(-0.818847\pi\)
0.842381 0.538883i \(-0.181153\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −40.3400 −1.55039 −0.775196 0.631720i \(-0.782349\pi\)
−0.775196 + 0.631720i \(0.782349\pi\)
\(678\) 0 0
\(679\) −1.85170 1.06908i −0.0710617 0.0410275i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 49.5412 1.89564 0.947820 0.318805i \(-0.103282\pi\)
0.947820 + 0.318805i \(0.103282\pi\)
\(684\) 0 0
\(685\) −3.68384 −0.140752
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −14.1026 8.14212i −0.537265 0.310190i
\(690\) 0 0
\(691\) −17.1533 −0.652544 −0.326272 0.945276i \(-0.605793\pi\)
−0.326272 + 0.945276i \(0.605793\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 10.9527i 0.415460i
\(696\) 0 0
\(697\) −28.9705 + 16.7261i −1.09734 + 0.633548i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −2.98145 1.72134i −0.112608 0.0650143i 0.442638 0.896700i \(-0.354043\pi\)
−0.555246 + 0.831686i \(0.687376\pi\)
\(702\) 0 0
\(703\) 34.8701 36.7620i 1.31515 1.38650i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 2.44376 1.41091i 0.0919072 0.0530626i
\(708\) 0 0
\(709\) 3.80569 + 6.59164i 0.142926 + 0.247554i 0.928597 0.371090i \(-0.121016\pi\)
−0.785671 + 0.618644i \(0.787682\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −4.19863 7.27224i −0.157240 0.272348i
\(714\) 0 0
\(715\) 7.21372i 0.269778i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 8.05303 + 4.64942i 0.300328 + 0.173394i 0.642590 0.766210i \(-0.277860\pi\)
−0.342263 + 0.939604i \(0.611193\pi\)
\(720\) 0 0
\(721\) 3.64932i 0.135908i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 21.0771 36.5066i 0.782784 1.35582i
\(726\) 0 0
\(727\) 15.3374 26.5652i 0.568834 0.985249i −0.427848 0.903851i \(-0.640728\pi\)
0.996682 0.0813982i \(-0.0259385\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −37.4206 + 21.6048i −1.38405 + 0.799081i
\(732\) 0 0
\(733\) 8.63651 0.318997 0.159498 0.987198i \(-0.449012\pi\)
0.159498 + 0.987198i \(0.449012\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 4.55945 + 7.89720i 0.167949 + 0.290897i
\(738\) 0 0
\(739\) −15.0042 + 25.9880i −0.551938 + 0.955985i 0.446197 + 0.894935i \(0.352778\pi\)
−0.998135 + 0.0610498i \(0.980555\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −11.2894 + 19.5538i −0.414167 + 0.717358i −0.995341 0.0964216i \(-0.969260\pi\)
0.581174 + 0.813779i \(0.302594\pi\)
\(744\) 0 0
\(745\) 1.24986 + 2.16482i 0.0457913 + 0.0793128i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 2.79374 0.102081
\(750\) 0 0
\(751\) −15.1212 + 8.73023i −0.551780 + 0.318571i −0.749840 0.661619i \(-0.769869\pi\)
0.198059 + 0.980190i \(0.436536\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 3.92385 6.79632i 0.142804 0.247343i
\(756\) 0 0
\(757\) −9.82310 + 17.0141i −0.357027 + 0.618388i −0.987463 0.157853i \(-0.949543\pi\)
0.630436 + 0.776241i \(0.282876\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 41.6922i 1.51134i 0.654953 + 0.755670i \(0.272688\pi\)
−0.654953 + 0.755670i \(0.727312\pi\)
\(762\) 0 0
\(763\) −1.01588 0.586519i −0.0367773 0.0212334i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 46.0366i 1.66229i
\(768\) 0 0
\(769\) −5.69363 9.86165i −0.205317 0.355620i 0.744916 0.667158i \(-0.232489\pi\)
−0.950234 + 0.311538i \(0.899156\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 9.94814 + 17.2307i 0.357810 + 0.619745i 0.987595 0.157025i \(-0.0501903\pi\)
−0.629785 + 0.776770i \(0.716857\pi\)
\(774\) 0 0
\(775\) −5.71045 + 3.29693i −0.205126 + 0.118429i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 30.1661 8.94299i 1.08081 0.320416i
\(780\) 0 0
\(781\) −34.7880 20.0848i −1.24481 0.718692i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −0.862613 + 0.498030i −0.0307880 + 0.0177755i
\(786\) 0 0
\(787\) 35.6451i 1.27061i 0.772262 + 0.635305i \(0.219125\pi\)
−0.772262 + 0.635305i \(0.780875\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −0.733685 −0.0260868
\(792\) 0 0
\(793\) 21.4008 + 12.3558i 0.759966 + 0.438766i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −0.904874 −0.0320523 −0.0160261 0.999872i \(-0.505101\pi\)
−0.0160261 + 0.999872i \(0.505101\pi\)
\(798\) 0 0
\(799\) 9.96739 0.352621
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 21.6515 + 12.5005i 0.764065 + 0.441133i
\(804\) 0 0
\(805\) 0.683040 0.0240740
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 10.9234i 0.384048i 0.981390 + 0.192024i \(0.0615051\pi\)
−0.981390 + 0.192024i \(0.938495\pi\)
\(810\) 0 0
\(811\) 27.9919 16.1611i 0.982929 0.567494i 0.0797760 0.996813i \(-0.474579\pi\)
0.903153 + 0.429318i \(0.141246\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −3.85994 2.22854i −0.135208 0.0780623i
\(816\) 0 0
\(817\) 38.9648 11.5515i 1.36321 0.404134i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 13.3396 7.70160i 0.465554 0.268788i −0.248823 0.968549i \(-0.580044\pi\)
0.714377 + 0.699761i \(0.246710\pi\)
\(822\) 0 0
\(823\) 10.5572 + 18.2857i 0.368002 + 0.637398i 0.989253 0.146213i \(-0.0467086\pi\)
−0.621251 + 0.783612i \(0.713375\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 6.58148 + 11.3995i 0.228861 + 0.396398i 0.957471 0.288531i \(-0.0931667\pi\)
−0.728610 + 0.684929i \(0.759833\pi\)
\(828\) 0 0
\(829\) 2.92198i 0.101485i 0.998712 + 0.0507423i \(0.0161587\pi\)
−0.998712 + 0.0507423i \(0.983841\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 27.9362 + 16.1290i 0.967931 + 0.558835i
\(834\) 0 0
\(835\) 6.89489i 0.238607i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 15.5472 26.9286i 0.536750 0.929678i −0.462326 0.886710i \(-0.652985\pi\)
0.999076 0.0429685i \(-0.0136815\pi\)
\(840\) 0 0
\(841\) −26.3315 + 45.6075i −0.907983 + 1.57267i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 6.07694 3.50852i 0.209053 0.120697i
\(846\) 0 0
\(847\) 0.957283 0.0328926
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −34.5273 59.8030i −1.18358 2.05002i
\(852\) 0 0
\(853\) 16.4673 28.5222i 0.563829 0.976580i −0.433329 0.901236i \(-0.642661\pi\)
0.997158 0.0753440i \(-0.0240055\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −13.5980 + 23.5524i −0.464499 + 0.804535i −0.999179 0.0405194i \(-0.987099\pi\)
0.534680 + 0.845054i \(0.320432\pi\)
\(858\) 0 0
\(859\) 9.09525 + 15.7534i 0.310326 + 0.537500i 0.978433 0.206565i \(-0.0662285\pi\)
−0.668107 + 0.744065i \(0.732895\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 17.4199 0.592980 0.296490 0.955036i \(-0.404184\pi\)
0.296490 + 0.955036i \(0.404184\pi\)
\(864\) 0 0
\(865\) 7.90626 4.56468i 0.268821 0.155204i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 1.81544 3.14444i 0.0615847 0.106668i
\(870\) 0 0
\(871\) 9.19270 15.9222i 0.311483 0.539504i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 1.11125i 0.0375670i
\(876\) 0 0
\(877\) −1.65733 0.956862i −0.0559642 0.0323109i 0.471757 0.881729i \(-0.343620\pi\)
−0.527721 + 0.849418i \(0.676953\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 13.2316i 0.445785i −0.974843 0.222893i \(-0.928450\pi\)
0.974843 0.222893i \(-0.0715499\pi\)
\(882\) 0 0
\(883\) −24.2127 41.9377i −0.814823 1.41131i −0.909455 0.415802i \(-0.863501\pi\)
0.0946321 0.995512i \(-0.469833\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −24.8340 43.0138i −0.833846 1.44426i −0.894967 0.446133i \(-0.852801\pi\)
0.0611211 0.998130i \(-0.480532\pi\)
\(888\) 0 0
\(889\) 1.57047 0.906712i 0.0526719 0.0304101i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −9.11636 2.18647i −0.305067 0.0731675i
\(894\) 0 0
\(895\) 10.7670 + 6.21633i 0.359901 + 0.207789i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 11.0626 6.38697i 0.368957 0.213017i
\(900\) 0 0
\(901\) 15.0577i 0.501644i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 4.32216 0.143673
\(906\) 0 0
\(907\) 49.6301 + 28.6540i 1.64794 + 0.951439i 0.977889 + 0.209125i \(0.0670616\pi\)
0.670052 + 0.742314i \(0.266272\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 33.1726 1.09906 0.549528 0.835475i \(-0.314808\pi\)
0.549528 + 0.835475i \(0.314808\pi\)
\(912\) 0 0
\(913\) 1.47280 0.0487424
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −2.61282 1.50851i −0.0862831 0.0498156i
\(918\) 0 0
\(919\) 2.32773 0.0767847 0.0383924 0.999263i \(-0.487776\pi\)
0.0383924 + 0.999263i \(0.487776\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 80.9895i 2.66580i
\(924\) 0 0
\(925\) −46.9597 + 27.1122i −1.54403 + 0.891444i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −45.5417 26.2935i −1.49418 0.862663i −0.494199 0.869349i \(-0.664538\pi\)
−0.999978 + 0.00668592i \(0.997872\pi\)
\(930\) 0 0
\(931\) −22.0129 20.8800i −0.721442 0.684314i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −5.77671 + 3.33518i −0.188919 + 0.109072i
\(936\) 0 0
\(937\) −12.0863 20.9342i −0.394844 0.683889i 0.598238 0.801319i \(-0.295868\pi\)
−0.993081 + 0.117429i \(0.962535\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 14.9700 + 25.9289i 0.488009 + 0.845257i 0.999905 0.0137909i \(-0.00438992\pi\)
−0.511896 + 0.859048i \(0.671057\pi\)
\(942\) 0 0
\(943\) 42.8805i 1.39638i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −23.4249 13.5244i −0.761206 0.439482i 0.0685227 0.997650i \(-0.478171\pi\)
−0.829729 + 0.558167i \(0.811505\pi\)
\(948\) 0 0
\(949\) 50.4067i 1.63627i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −3.04170 + 5.26838i −0.0985304 + 0.170660i −0.911077 0.412237i \(-0.864748\pi\)
0.812546 + 0.582897i \(0.198081\pi\)
\(954\) 0 0
\(955\) 2.16971 3.75805i 0.0702102 0.121608i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −1.09415 + 0.631708i −0.0353320 + 0.0203989i
\(960\) 0 0
\(961\) 29.0019 0.935544
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 2.17810 + 3.77259i 0.0701157 + 0.121444i
\(966\) 0 0
\(967\) −25.9763 + 44.9923i −0.835343 + 1.44686i 0.0584083 + 0.998293i \(0.481397\pi\)
−0.893751 + 0.448563i \(0.851936\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −14.3108 + 24.7870i −0.459254 + 0.795451i −0.998922 0.0464267i \(-0.985217\pi\)
0.539668 + 0.841878i \(0.318550\pi\)
\(972\) 0 0
\(973\) −1.87818 3.25310i −0.0602117 0.104290i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −50.5032 −1.61574 −0.807870 0.589361i \(-0.799380\pi\)
−0.807870 + 0.589361i \(0.799380\pi\)
\(978\) 0 0
\(979\) 11.9978 6.92694i 0.383452 0.221386i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 18.1280 31.3986i 0.578193 1.00146i −0.417494 0.908680i \(-0.637091\pi\)
0.995687 0.0927799i \(-0.0295753\pi\)
\(984\) 0 0
\(985\) −0.955681 + 1.65529i −0.0304505 + 0.0527419i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 55.3877i 1.76123i
\(990\) 0 0
\(991\) −44.4701 25.6748i −1.41264 0.815588i −0.417004 0.908905i \(-0.636920\pi\)
−0.995637 + 0.0933161i \(0.970253\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 9.35944i 0.296714i
\(996\) 0 0
\(997\) −4.24043 7.34464i −0.134296 0.232607i 0.791032 0.611774i \(-0.209544\pi\)
−0.925328 + 0.379167i \(0.876211\pi\)
\(998\) 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 2736.2.dc.e.1889.6 20
3.2 odd 2 2736.2.dc.f.1889.5 20
4.3 odd 2 1368.2.cu.a.521.6 yes 20
12.11 even 2 1368.2.cu.b.521.5 yes 20
19.12 odd 6 2736.2.dc.f.449.5 20
57.50 even 6 inner 2736.2.dc.e.449.6 20
76.31 even 6 1368.2.cu.b.449.5 yes 20
228.107 odd 6 1368.2.cu.a.449.6 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1368.2.cu.a.449.6 20 228.107 odd 6
1368.2.cu.a.521.6 yes 20 4.3 odd 2
1368.2.cu.b.449.5 yes 20 76.31 even 6
1368.2.cu.b.521.5 yes 20 12.11 even 2
2736.2.dc.e.449.6 20 57.50 even 6 inner
2736.2.dc.e.1889.6 20 1.1 even 1 trivial
2736.2.dc.f.449.5 20 19.12 odd 6
2736.2.dc.f.1889.5 20 3.2 odd 2