Properties

Label 828.2.q.c.289.2
Level $828$
Weight $2$
Character 828.289
Analytic conductor $6.612$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [828,2,Mod(73,828)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(828, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("828.73");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 828 = 2^{2} \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 828.q (of order \(11\), degree \(10\), 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: \(6.61161328736\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(2\) over \(\Q(\zeta_{11})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 8 x^{19} + 43 x^{18} - 165 x^{17} + 538 x^{16} - 1433 x^{15} + 3444 x^{14} - 7370 x^{13} + 15500 x^{12} - 28190 x^{11} + 41920 x^{10} - 33520 x^{9} - 13837 x^{8} + \cdots + 529 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 276)
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label 289.2
Root \(1.84381 - 0.541390i\) of defining polynomial
Character \(\chi\) \(=\) 828.289
Dual form 828.2.q.c.361.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.21471 - 2.65985i) q^{5} +(0.960219 + 0.281946i) q^{7} +O(q^{10})\) \(q+(1.21471 - 2.65985i) q^{5} +(0.960219 + 0.281946i) q^{7} +(1.65056 - 1.90485i) q^{11} +(0.463267 - 0.136028i) q^{13} +(-0.446491 - 3.10542i) q^{17} +(-0.0786963 + 0.547345i) q^{19} +(-3.87251 + 2.82908i) q^{23} +(-2.32496 - 2.68314i) q^{25} +(0.0173123 + 0.120410i) q^{29} +(6.92752 - 4.45205i) q^{31} +(1.91632 - 2.21155i) q^{35} +(-3.95959 - 8.67029i) q^{37} +(0.578290 - 1.26628i) q^{41} +(5.21923 + 3.35419i) q^{43} -8.69831 q^{47} +(-5.04625 - 3.24303i) q^{49} +(7.41370 + 2.17686i) q^{53} +(-3.06165 - 6.70407i) q^{55} +(-0.227334 + 0.0667513i) q^{59} +(-1.54102 + 0.990353i) q^{61} +(0.200924 - 1.39745i) q^{65} +(1.83195 + 2.11418i) q^{67} +(7.51699 + 8.67507i) q^{71} +(1.94856 - 13.5525i) q^{73} +(2.12196 - 1.36370i) q^{77} +(16.3977 - 4.81479i) q^{79} +(3.34983 + 7.33511i) q^{83} +(-8.80229 - 2.58458i) q^{85} +(-4.15453 - 2.66995i) q^{89} +0.483191 q^{91} +(1.36026 + 0.874186i) q^{95} +(-6.49214 + 14.2158i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 4 q^{5} - 22 q^{13} - 7 q^{17} + 19 q^{19} - 20 q^{23} + 20 q^{25} - 32 q^{29} - 3 q^{31} + 26 q^{35} - 10 q^{37} + 40 q^{41} + 8 q^{43} + 18 q^{47} - 34 q^{49} + 34 q^{53} - 17 q^{55} + 32 q^{59} + 32 q^{61} - 49 q^{65} + 35 q^{67} - 33 q^{71} - q^{73} + 50 q^{77} + 22 q^{79} + 14 q^{83} - 9 q^{85} - 10 q^{89} - 72 q^{91} + 51 q^{95} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/828\mathbb{Z}\right)^\times\).

\(n\) \(415\) \(461\) \(649\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{7}{11}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 1.21471 2.65985i 0.543235 1.18952i −0.416635 0.909074i \(-0.636791\pi\)
0.959870 0.280445i \(-0.0904822\pi\)
\(6\) 0 0
\(7\) 0.960219 + 0.281946i 0.362929 + 0.106566i 0.458113 0.888894i \(-0.348526\pi\)
−0.0951837 + 0.995460i \(0.530344\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.65056 1.90485i 0.497662 0.574333i −0.450235 0.892910i \(-0.648660\pi\)
0.947897 + 0.318578i \(0.103205\pi\)
\(12\) 0 0
\(13\) 0.463267 0.136028i 0.128487 0.0377273i −0.216857 0.976203i \(-0.569580\pi\)
0.345344 + 0.938476i \(0.387762\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.446491 3.10542i −0.108290 0.753174i −0.969529 0.244975i \(-0.921220\pi\)
0.861239 0.508199i \(-0.169689\pi\)
\(18\) 0 0
\(19\) −0.0786963 + 0.547345i −0.0180542 + 0.125569i −0.996855 0.0792435i \(-0.974750\pi\)
0.978801 + 0.204813i \(0.0656586\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.87251 + 2.82908i −0.807473 + 0.589904i
\(24\) 0 0
\(25\) −2.32496 2.68314i −0.464991 0.536628i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.0173123 + 0.120410i 0.00321482 + 0.0223596i 0.991366 0.131126i \(-0.0418592\pi\)
−0.988151 + 0.153486i \(0.950950\pi\)
\(30\) 0 0
\(31\) 6.92752 4.45205i 1.24422 0.799611i 0.258175 0.966098i \(-0.416879\pi\)
0.986044 + 0.166487i \(0.0532424\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.91632 2.21155i 0.323917 0.373821i
\(36\) 0 0
\(37\) −3.95959 8.67029i −0.650952 1.42539i −0.890718 0.454556i \(-0.849798\pi\)
0.239766 0.970831i \(-0.422929\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.578290 1.26628i 0.0903137 0.197760i −0.859085 0.511833i \(-0.828967\pi\)
0.949399 + 0.314074i \(0.101694\pi\)
\(42\) 0 0
\(43\) 5.21923 + 3.35419i 0.795925 + 0.511510i 0.874284 0.485415i \(-0.161332\pi\)
−0.0783589 + 0.996925i \(0.524968\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −8.69831 −1.26878 −0.634389 0.773014i \(-0.718748\pi\)
−0.634389 + 0.773014i \(0.718748\pi\)
\(48\) 0 0
\(49\) −5.04625 3.24303i −0.720892 0.463289i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 7.41370 + 2.17686i 1.01835 + 0.299015i 0.747965 0.663738i \(-0.231031\pi\)
0.270385 + 0.962752i \(0.412849\pi\)
\(54\) 0 0
\(55\) −3.06165 6.70407i −0.412832 0.903976i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −0.227334 + 0.0667513i −0.0295964 + 0.00869028i −0.296497 0.955034i \(-0.595819\pi\)
0.266901 + 0.963724i \(0.414000\pi\)
\(60\) 0 0
\(61\) −1.54102 + 0.990353i −0.197307 + 0.126802i −0.635564 0.772048i \(-0.719232\pi\)
0.438257 + 0.898850i \(0.355596\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.200924 1.39745i 0.0249215 0.173333i
\(66\) 0 0
\(67\) 1.83195 + 2.11418i 0.223808 + 0.258288i 0.856538 0.516084i \(-0.172611\pi\)
−0.632730 + 0.774373i \(0.718065\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 7.51699 + 8.67507i 0.892103 + 1.02954i 0.999376 + 0.0353118i \(0.0112424\pi\)
−0.107274 + 0.994230i \(0.534212\pi\)
\(72\) 0 0
\(73\) 1.94856 13.5525i 0.228061 1.58620i −0.478200 0.878251i \(-0.658711\pi\)
0.706261 0.707951i \(-0.250380\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.12196 1.36370i 0.241820 0.155408i
\(78\) 0 0
\(79\) 16.3977 4.81479i 1.84488 0.541706i 0.844903 0.534919i \(-0.179658\pi\)
0.999977 0.00678681i \(-0.00216032\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.34983 + 7.33511i 0.367692 + 0.805133i 0.999548 + 0.0300477i \(0.00956591\pi\)
−0.631857 + 0.775085i \(0.717707\pi\)
\(84\) 0 0
\(85\) −8.80229 2.58458i −0.954742 0.280338i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −4.15453 2.66995i −0.440379 0.283015i 0.301604 0.953433i \(-0.402478\pi\)
−0.741983 + 0.670419i \(0.766114\pi\)
\(90\) 0 0
\(91\) 0.483191 0.0506522
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.36026 + 0.874186i 0.139560 + 0.0896896i
\(96\) 0 0
\(97\) −6.49214 + 14.2158i −0.659177 + 1.44340i 0.224109 + 0.974564i \(0.428053\pi\)
−0.883287 + 0.468833i \(0.844675\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1.35100 + 2.95829i 0.134430 + 0.294361i 0.964861 0.262761i \(-0.0846330\pi\)
−0.830431 + 0.557121i \(0.811906\pi\)
\(102\) 0 0
\(103\) 0.917997 1.05943i 0.0904529 0.104388i −0.708719 0.705491i \(-0.750726\pi\)
0.799172 + 0.601103i \(0.205272\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 8.44978 5.43034i 0.816871 0.524971i −0.0642102 0.997936i \(-0.520453\pi\)
0.881081 + 0.472965i \(0.156816\pi\)
\(108\) 0 0
\(109\) 0.961777 + 6.68930i 0.0921215 + 0.640719i 0.982606 + 0.185705i \(0.0594568\pi\)
−0.890484 + 0.455014i \(0.849634\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −6.56447 7.57581i −0.617534 0.712672i 0.357703 0.933835i \(-0.383560\pi\)
−0.975237 + 0.221164i \(0.929015\pi\)
\(114\) 0 0
\(115\) 2.82094 + 13.7368i 0.263054 + 1.28096i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0.446829 3.10777i 0.0409608 0.284889i
\(120\) 0 0
\(121\) 0.661368 + 4.59992i 0.0601244 + 0.418174i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 4.06733 1.19428i 0.363793 0.106819i
\(126\) 0 0
\(127\) −10.9350 + 12.6197i −0.970327 + 1.11982i 0.0224388 + 0.999748i \(0.492857\pi\)
−0.992766 + 0.120069i \(0.961689\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −13.8070 4.05411i −1.20633 0.354209i −0.384057 0.923309i \(-0.625474\pi\)
−0.822268 + 0.569100i \(0.807292\pi\)
\(132\) 0 0
\(133\) −0.229887 + 0.503383i −0.0199338 + 0.0436488i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1.88176 −0.160769 −0.0803847 0.996764i \(-0.525615\pi\)
−0.0803847 + 0.996764i \(0.525615\pi\)
\(138\) 0 0
\(139\) 3.46292 0.293721 0.146860 0.989157i \(-0.453083\pi\)
0.146860 + 0.989157i \(0.453083\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0.505538 1.10697i 0.0422752 0.0925699i
\(144\) 0 0
\(145\) 0.341302 + 0.100215i 0.0283436 + 0.00832242i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −14.1256 + 16.3019i −1.15722 + 1.33550i −0.224678 + 0.974433i \(0.572133\pi\)
−0.932540 + 0.361067i \(0.882412\pi\)
\(150\) 0 0
\(151\) −3.13983 + 0.921937i −0.255516 + 0.0750262i −0.406982 0.913436i \(-0.633419\pi\)
0.151466 + 0.988462i \(0.451601\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −3.42682 23.8341i −0.275249 1.91440i
\(156\) 0 0
\(157\) −3.24424 + 22.5642i −0.258919 + 1.80082i 0.281649 + 0.959518i \(0.409119\pi\)
−0.540567 + 0.841301i \(0.681790\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −4.51610 + 1.62470i −0.355919 + 0.128044i
\(162\) 0 0
\(163\) 7.91542 + 9.13488i 0.619984 + 0.715499i 0.975704 0.219093i \(-0.0703099\pi\)
−0.355720 + 0.934592i \(0.615764\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.18059 + 8.21115i 0.0913565 + 0.635398i 0.983129 + 0.182914i \(0.0585530\pi\)
−0.891772 + 0.452484i \(0.850538\pi\)
\(168\) 0 0
\(169\) −10.7402 + 6.90230i −0.826168 + 0.530946i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −13.5301 + 15.6146i −1.02868 + 1.18716i −0.0465543 + 0.998916i \(0.514824\pi\)
−0.982123 + 0.188241i \(0.939721\pi\)
\(174\) 0 0
\(175\) −1.47597 3.23192i −0.111573 0.244310i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −1.61211 + 3.53002i −0.120494 + 0.263846i −0.960262 0.279100i \(-0.909964\pi\)
0.839768 + 0.542946i \(0.182691\pi\)
\(180\) 0 0
\(181\) 6.70576 + 4.30953i 0.498435 + 0.320325i 0.765590 0.643329i \(-0.222447\pi\)
−0.267155 + 0.963654i \(0.586083\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −27.8714 −2.04915
\(186\) 0 0
\(187\) −6.65230 4.27517i −0.486464 0.312632i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 16.2019 + 4.75730i 1.17233 + 0.344226i 0.809211 0.587518i \(-0.199895\pi\)
0.363116 + 0.931744i \(0.381713\pi\)
\(192\) 0 0
\(193\) −8.84842 19.3753i −0.636923 1.39467i −0.902547 0.430591i \(-0.858305\pi\)
0.265624 0.964077i \(-0.414422\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 25.8424 7.58802i 1.84120 0.540624i 0.841196 0.540731i \(-0.181852\pi\)
1.00000 0.000106800i \(3.39955e-5\pi\)
\(198\) 0 0
\(199\) −6.12918 + 3.93898i −0.434486 + 0.279227i −0.739546 0.673106i \(-0.764960\pi\)
0.305060 + 0.952333i \(0.401323\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −0.0173255 + 0.120501i −0.00121601 + 0.00845752i
\(204\) 0 0
\(205\) −2.66565 3.07633i −0.186177 0.214860i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0.912714 + 1.05333i 0.0631338 + 0.0728603i
\(210\) 0 0
\(211\) −2.22131 + 15.4495i −0.152921 + 1.06359i 0.758369 + 0.651825i \(0.225997\pi\)
−0.911290 + 0.411765i \(0.864913\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 15.2615 9.80797i 1.04083 0.668898i
\(216\) 0 0
\(217\) 7.90717 2.32176i 0.536774 0.157611i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −0.629267 1.37790i −0.0423291 0.0926878i
\(222\) 0 0
\(223\) 1.34888 + 0.396066i 0.0903274 + 0.0265225i 0.326584 0.945168i \(-0.394102\pi\)
−0.236257 + 0.971691i \(0.575921\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 3.73511 + 2.40041i 0.247908 + 0.159321i 0.658694 0.752411i \(-0.271109\pi\)
−0.410786 + 0.911732i \(0.634746\pi\)
\(228\) 0 0
\(229\) −1.67523 −0.110702 −0.0553512 0.998467i \(-0.517628\pi\)
−0.0553512 + 0.998467i \(0.517628\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 18.3038 + 11.7632i 1.19912 + 0.770630i 0.978804 0.204798i \(-0.0656538\pi\)
0.220319 + 0.975428i \(0.429290\pi\)
\(234\) 0 0
\(235\) −10.5659 + 23.1362i −0.689245 + 1.50924i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 8.72594 + 19.1071i 0.564434 + 1.23594i 0.949708 + 0.313136i \(0.101380\pi\)
−0.385274 + 0.922802i \(0.625893\pi\)
\(240\) 0 0
\(241\) 17.2462 19.9032i 1.11093 1.28208i 0.155182 0.987886i \(-0.450404\pi\)
0.955746 0.294194i \(-0.0950510\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −14.7557 + 9.48290i −0.942706 + 0.605840i
\(246\) 0 0
\(247\) 0.0379966 + 0.264272i 0.00241766 + 0.0168152i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 17.0613 + 19.6898i 1.07690 + 1.24281i 0.968582 + 0.248693i \(0.0800011\pi\)
0.108318 + 0.994116i \(0.465453\pi\)
\(252\) 0 0
\(253\) −1.00283 + 12.0461i −0.0630476 + 0.757331i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1.12414 7.81860i 0.0701222 0.487711i −0.924252 0.381784i \(-0.875310\pi\)
0.994374 0.105927i \(-0.0337810\pi\)
\(258\) 0 0
\(259\) −1.35752 9.44177i −0.0843523 0.586683i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −11.8421 + 3.47716i −0.730216 + 0.214411i −0.625646 0.780107i \(-0.715165\pi\)
−0.104570 + 0.994518i \(0.533347\pi\)
\(264\) 0 0
\(265\) 14.7956 17.0751i 0.908888 1.04891i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −10.7364 3.15249i −0.654610 0.192211i −0.0624710 0.998047i \(-0.519898\pi\)
−0.592139 + 0.805836i \(0.701716\pi\)
\(270\) 0 0
\(271\) 9.90135 21.6809i 0.601464 1.31702i −0.326797 0.945095i \(-0.605969\pi\)
0.928261 0.371929i \(-0.121303\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −8.94844 −0.539612
\(276\) 0 0
\(277\) −17.8542 −1.07275 −0.536376 0.843979i \(-0.680207\pi\)
−0.536376 + 0.843979i \(0.680207\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0.687581 1.50559i 0.0410176 0.0898161i −0.888014 0.459817i \(-0.847915\pi\)
0.929032 + 0.370000i \(0.120642\pi\)
\(282\) 0 0
\(283\) 1.30907 + 0.384378i 0.0778161 + 0.0228489i 0.320409 0.947279i \(-0.396180\pi\)
−0.242593 + 0.970128i \(0.577998\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0.912307 1.05286i 0.0538518 0.0621483i
\(288\) 0 0
\(289\) 6.86713 2.01637i 0.403949 0.118610i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −1.50932 10.4975i −0.0881752 0.613272i −0.985215 0.171322i \(-0.945196\pi\)
0.897040 0.441950i \(-0.145713\pi\)
\(294\) 0 0
\(295\) −0.0985970 + 0.685757i −0.00574054 + 0.0399263i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −1.40917 + 1.83739i −0.0814946 + 0.106259i
\(300\) 0 0
\(301\) 4.06590 + 4.69230i 0.234355 + 0.270460i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0.762294 + 5.30187i 0.0436488 + 0.303584i
\(306\) 0 0
\(307\) −8.68375 + 5.58071i −0.495608 + 0.318508i −0.764457 0.644674i \(-0.776993\pi\)
0.268849 + 0.963182i \(0.413357\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −4.61609 + 5.32725i −0.261755 + 0.302081i −0.871380 0.490609i \(-0.836774\pi\)
0.609625 + 0.792690i \(0.291320\pi\)
\(312\) 0 0
\(313\) 0.939900 + 2.05809i 0.0531263 + 0.116330i 0.934334 0.356400i \(-0.115996\pi\)
−0.881207 + 0.472730i \(0.843269\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.31835 2.88679i 0.0740460 0.162138i −0.868989 0.494831i \(-0.835230\pi\)
0.943035 + 0.332693i \(0.107957\pi\)
\(318\) 0 0
\(319\) 0.257938 + 0.165766i 0.0144417 + 0.00928114i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 1.73487 0.0965308
\(324\) 0 0
\(325\) −1.44206 0.926754i −0.0799909 0.0514071i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −8.35228 2.45245i −0.460476 0.135208i
\(330\) 0 0
\(331\) −8.63436 18.9066i −0.474587 1.03920i −0.983916 0.178629i \(-0.942834\pi\)
0.509329 0.860572i \(-0.329894\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 7.84869 2.30458i 0.428820 0.125913i
\(336\) 0 0
\(337\) −21.9636 + 14.1152i −1.19644 + 0.768902i −0.978336 0.207024i \(-0.933622\pi\)
−0.218099 + 0.975927i \(0.569986\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 2.95381 20.5442i 0.159958 1.11253i
\(342\) 0 0
\(343\) −8.51864 9.83104i −0.459963 0.530826i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −7.52379 8.68291i −0.403898 0.466123i 0.516967 0.856006i \(-0.327061\pi\)
−0.920865 + 0.389882i \(0.872516\pi\)
\(348\) 0 0
\(349\) 4.15887 28.9256i 0.222619 1.54835i −0.505456 0.862852i \(-0.668676\pi\)
0.728075 0.685497i \(-0.240415\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 27.4206 17.6221i 1.45945 0.937932i 0.460721 0.887545i \(-0.347591\pi\)
0.998730 0.0503875i \(-0.0160456\pi\)
\(354\) 0 0
\(355\) 32.2053 9.45634i 1.70928 0.501890i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 12.1092 + 26.5155i 0.639101 + 1.39944i 0.900778 + 0.434280i \(0.142997\pi\)
−0.261678 + 0.965155i \(0.584276\pi\)
\(360\) 0 0
\(361\) 17.9370 + 5.26677i 0.944051 + 0.277198i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −33.6807 21.6453i −1.76293 1.13296i
\(366\) 0 0
\(367\) −22.4367 −1.17119 −0.585593 0.810605i \(-0.699138\pi\)
−0.585593 + 0.810605i \(0.699138\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 6.50502 + 4.18053i 0.337724 + 0.217042i
\(372\) 0 0
\(373\) −5.65832 + 12.3900i −0.292977 + 0.641530i −0.997688 0.0679649i \(-0.978349\pi\)
0.704711 + 0.709495i \(0.251077\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0.0243993 + 0.0534271i 0.00125663 + 0.00275163i
\(378\) 0 0
\(379\) −3.03505 + 3.50264i −0.155900 + 0.179918i −0.828327 0.560246i \(-0.810707\pi\)
0.672426 + 0.740164i \(0.265252\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −25.6052 + 16.4555i −1.30837 + 0.840836i −0.994097 0.108498i \(-0.965396\pi\)
−0.314269 + 0.949334i \(0.601759\pi\)
\(384\) 0 0
\(385\) −1.04967 7.30060i −0.0534960 0.372073i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 1.96796 + 2.27115i 0.0997797 + 0.115152i 0.803443 0.595382i \(-0.202999\pi\)
−0.703663 + 0.710534i \(0.748454\pi\)
\(390\) 0 0
\(391\) 10.5145 + 10.7626i 0.531742 + 0.544287i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 7.11182 49.4638i 0.357835 2.48879i
\(396\) 0 0
\(397\) 1.38627 + 9.64170i 0.0695747 + 0.483903i 0.994582 + 0.103954i \(0.0331494\pi\)
−0.925007 + 0.379949i \(0.875942\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 19.1560 5.62470i 0.956603 0.280884i 0.234069 0.972220i \(-0.424796\pi\)
0.722533 + 0.691336i \(0.242978\pi\)
\(402\) 0 0
\(403\) 2.60369 3.00482i 0.129699 0.149681i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −23.0511 6.76841i −1.14260 0.335498i
\(408\) 0 0
\(409\) 10.1075 22.1323i 0.499784 1.09437i −0.476756 0.879036i \(-0.658187\pi\)
0.976540 0.215338i \(-0.0690853\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −0.237111 −0.0116675
\(414\) 0 0
\(415\) 23.5793 1.15746
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −5.43702 + 11.9054i −0.265616 + 0.581618i −0.994702 0.102805i \(-0.967218\pi\)
0.729086 + 0.684422i \(0.239946\pi\)
\(420\) 0 0
\(421\) 5.86675 + 1.72263i 0.285928 + 0.0839560i 0.421552 0.906804i \(-0.361486\pi\)
−0.135624 + 0.990760i \(0.543304\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −7.29420 + 8.41795i −0.353821 + 0.408331i
\(426\) 0 0
\(427\) −1.75894 + 0.516472i −0.0851212 + 0.0249938i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −3.47655 24.1799i −0.167459 1.16471i −0.884112 0.467275i \(-0.845236\pi\)
0.716653 0.697430i \(-0.245673\pi\)
\(432\) 0 0
\(433\) 2.34204 16.2892i 0.112551 0.782810i −0.852872 0.522121i \(-0.825141\pi\)
0.965423 0.260690i \(-0.0839499\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −1.24373 2.34223i −0.0594957 0.112044i
\(438\) 0 0
\(439\) 0.691787 + 0.798365i 0.0330172 + 0.0381039i 0.772018 0.635601i \(-0.219247\pi\)
−0.739001 + 0.673705i \(0.764702\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −2.83450 19.7144i −0.134671 0.936659i −0.939353 0.342953i \(-0.888573\pi\)
0.804681 0.593707i \(-0.202336\pi\)
\(444\) 0 0
\(445\) −12.1482 + 7.80719i −0.575881 + 0.370096i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 16.0130 18.4800i 0.755702 0.872127i −0.239406 0.970920i \(-0.576953\pi\)
0.995108 + 0.0987927i \(0.0314981\pi\)
\(450\) 0 0
\(451\) −1.45756 3.19162i −0.0686340 0.150288i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0.586937 1.28521i 0.0275160 0.0602517i
\(456\) 0 0
\(457\) 8.21331 + 5.27838i 0.384203 + 0.246912i 0.718458 0.695571i \(-0.244848\pi\)
−0.334255 + 0.942483i \(0.608485\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −28.8585 −1.34408 −0.672038 0.740517i \(-0.734581\pi\)
−0.672038 + 0.740517i \(0.734581\pi\)
\(462\) 0 0
\(463\) 17.0636 + 10.9661i 0.793014 + 0.509639i 0.873329 0.487130i \(-0.161956\pi\)
−0.0803151 + 0.996770i \(0.525593\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 26.5386 + 7.79244i 1.22806 + 0.360591i 0.830518 0.556993i \(-0.188045\pi\)
0.397543 + 0.917584i \(0.369863\pi\)
\(468\) 0 0
\(469\) 1.16299 + 2.54659i 0.0537018 + 0.117591i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 15.0039 4.40553i 0.689878 0.202567i
\(474\) 0 0
\(475\) 1.65157 1.06140i 0.0757792 0.0487003i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 2.91070 20.2444i 0.132993 0.924988i −0.808630 0.588318i \(-0.799790\pi\)
0.941623 0.336670i \(-0.109301\pi\)
\(480\) 0 0
\(481\) −3.01375 3.47805i −0.137415 0.158585i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 29.9258 + 34.5362i 1.35886 + 1.56821i
\(486\) 0 0
\(487\) −0.443196 + 3.08250i −0.0200831 + 0.139681i −0.997396 0.0721195i \(-0.977024\pi\)
0.977313 + 0.211801i \(0.0679328\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 4.62679 2.97346i 0.208804 0.134190i −0.432059 0.901845i \(-0.642213\pi\)
0.640863 + 0.767655i \(0.278576\pi\)
\(492\) 0 0
\(493\) 0.366193 0.107524i 0.0164925 0.00484264i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 4.77206 + 10.4494i 0.214056 + 0.468718i
\(498\) 0 0
\(499\) −8.04956 2.36356i −0.360348 0.105808i 0.0965470 0.995328i \(-0.469220\pi\)
−0.456895 + 0.889521i \(0.651038\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 1.42347 + 0.914807i 0.0634693 + 0.0407892i 0.571991 0.820260i \(-0.306171\pi\)
−0.508521 + 0.861049i \(0.669808\pi\)
\(504\) 0 0
\(505\) 9.50967 0.423175
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 12.2843 + 7.89467i 0.544494 + 0.349925i 0.783794 0.621021i \(-0.213282\pi\)
−0.239300 + 0.970946i \(0.576918\pi\)
\(510\) 0 0
\(511\) 5.69212 12.4640i 0.251805 0.551375i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −1.70281 3.72863i −0.0750346 0.164303i
\(516\) 0 0
\(517\) −14.3571 + 16.5689i −0.631423 + 0.728701i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 21.6860 13.9368i 0.950083 0.610581i 0.0288462 0.999584i \(-0.490817\pi\)
0.921237 + 0.389003i \(0.127180\pi\)
\(522\) 0 0
\(523\) −2.00234 13.9266i −0.0875563 0.608967i −0.985604 0.169070i \(-0.945924\pi\)
0.898048 0.439898i \(-0.144985\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −16.9185 19.5250i −0.736983 0.850524i
\(528\) 0 0
\(529\) 6.99260 21.9113i 0.304026 0.952664i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0.0956541 0.665289i 0.00414324 0.0288169i
\(534\) 0 0
\(535\) −4.17984 29.0714i −0.180710 1.25687i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −14.5066 + 4.25952i −0.624843 + 0.183470i
\(540\) 0 0
\(541\) 11.8199 13.6409i 0.508177 0.586468i −0.442454 0.896791i \(-0.645892\pi\)
0.950631 + 0.310324i \(0.100437\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 18.9608 + 5.56739i 0.812191 + 0.238481i
\(546\) 0 0
\(547\) −6.77596 + 14.8373i −0.289719 + 0.634396i −0.997394 0.0721416i \(-0.977017\pi\)
0.707675 + 0.706538i \(0.249744\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −0.0672682 −0.00286572
\(552\) 0 0
\(553\) 17.1029 0.727287
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −9.28190 + 20.3245i −0.393287 + 0.861178i 0.604620 + 0.796514i \(0.293325\pi\)
−0.997907 + 0.0646639i \(0.979402\pi\)
\(558\) 0 0
\(559\) 2.87416 + 0.843930i 0.121564 + 0.0356944i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −9.19015 + 10.6060i −0.387318 + 0.446989i −0.915606 0.402076i \(-0.868289\pi\)
0.528288 + 0.849065i \(0.322834\pi\)
\(564\) 0 0
\(565\) −28.1244 + 8.25807i −1.18320 + 0.347420i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 5.30762 + 36.9153i 0.222507 + 1.54757i 0.728509 + 0.685037i \(0.240214\pi\)
−0.506002 + 0.862532i \(0.668877\pi\)
\(570\) 0 0
\(571\) 5.24484 36.4787i 0.219490 1.52659i −0.520438 0.853899i \(-0.674231\pi\)
0.739928 0.672686i \(-0.234860\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 16.5942 + 3.81299i 0.692027 + 0.159013i
\(576\) 0 0
\(577\) −13.7946 15.9198i −0.574278 0.662752i 0.392087 0.919928i \(-0.371753\pi\)
−0.966364 + 0.257177i \(0.917208\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 1.14847 + 7.98779i 0.0476466 + 0.331389i
\(582\) 0 0
\(583\) 16.3833 10.5289i 0.678528 0.436064i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0.0269562 0.0311092i 0.00111260 0.00128401i −0.755193 0.655503i \(-0.772457\pi\)
0.756306 + 0.654218i \(0.227002\pi\)
\(588\) 0 0
\(589\) 1.89163 + 4.14210i 0.0779434 + 0.170672i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −6.17509 + 13.5216i −0.253580 + 0.555264i −0.993018 0.117961i \(-0.962364\pi\)
0.739438 + 0.673225i \(0.235091\pi\)
\(594\) 0 0
\(595\) −7.72341 4.96354i −0.316629 0.203485i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −44.0448 −1.79962 −0.899811 0.436280i \(-0.856296\pi\)
−0.899811 + 0.436280i \(0.856296\pi\)
\(600\) 0 0
\(601\) 37.1322 + 23.8634i 1.51465 + 0.973409i 0.992723 + 0.120422i \(0.0384248\pi\)
0.521932 + 0.852987i \(0.325212\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 13.0384 + 3.82843i 0.530088 + 0.155648i
\(606\) 0 0
\(607\) −8.82155 19.3165i −0.358056 0.784032i −0.999853 0.0171579i \(-0.994538\pi\)
0.641797 0.766874i \(-0.278189\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −4.02964 + 1.18321i −0.163022 + 0.0478676i
\(612\) 0 0
\(613\) −28.7218 + 18.4584i −1.16006 + 0.745527i −0.971618 0.236557i \(-0.923981\pi\)
−0.188445 + 0.982084i \(0.560345\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 5.83677 40.5956i 0.234980 1.63432i −0.441077 0.897469i \(-0.645404\pi\)
0.676057 0.736849i \(-0.263687\pi\)
\(618\) 0 0
\(619\) −4.61337 5.32411i −0.185427 0.213994i 0.655424 0.755262i \(-0.272490\pi\)
−0.840851 + 0.541267i \(0.817945\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −3.23648 3.73510i −0.129667 0.149643i
\(624\) 0 0
\(625\) 4.29035 29.8400i 0.171614 1.19360i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −25.1569 + 16.1674i −1.00307 + 0.644636i
\(630\) 0 0
\(631\) −22.4420 + 6.58958i −0.893403 + 0.262327i −0.696040 0.718003i \(-0.745056\pi\)
−0.197364 + 0.980330i \(0.563238\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 20.2835 + 44.4148i 0.804928 + 1.76255i
\(636\) 0 0
\(637\) −2.77890 0.815959i −0.110104 0.0323295i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −13.9034 8.93518i −0.549152 0.352919i 0.236457 0.971642i \(-0.424014\pi\)
−0.785609 + 0.618723i \(0.787650\pi\)
\(642\) 0 0
\(643\) −34.1400 −1.34635 −0.673176 0.739482i \(-0.735070\pi\)
−0.673176 + 0.739482i \(0.735070\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 8.12978 + 5.22469i 0.319614 + 0.205404i 0.690604 0.723233i \(-0.257345\pi\)
−0.370989 + 0.928637i \(0.620981\pi\)
\(648\) 0 0
\(649\) −0.248077 + 0.543213i −0.00973788 + 0.0213230i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −2.47632 5.42237i −0.0969057 0.212194i 0.854971 0.518677i \(-0.173575\pi\)
−0.951876 + 0.306483i \(0.900848\pi\)
\(654\) 0 0
\(655\) −27.5549 + 31.8000i −1.07666 + 1.24253i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −31.7171 + 20.3833i −1.23552 + 0.794021i −0.984742 0.174023i \(-0.944323\pi\)
−0.250780 + 0.968044i \(0.580687\pi\)
\(660\) 0 0
\(661\) 4.57689 + 31.8330i 0.178020 + 1.23816i 0.861336 + 0.508036i \(0.169628\pi\)
−0.683316 + 0.730123i \(0.739463\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1.05967 + 1.22293i 0.0410924 + 0.0474232i
\(666\) 0 0
\(667\) −0.407692 0.417310i −0.0157859 0.0161583i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −0.657073 + 4.57004i −0.0253660 + 0.176424i
\(672\) 0 0
\(673\) 4.44939 + 30.9462i 0.171511 + 1.19289i 0.875693 + 0.482868i \(0.160405\pi\)
−0.704181 + 0.710020i \(0.748686\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 5.24755 1.54082i 0.201680 0.0592185i −0.179332 0.983789i \(-0.557394\pi\)
0.381012 + 0.924570i \(0.375576\pi\)
\(678\) 0 0
\(679\) −10.2420 + 11.8199i −0.393051 + 0.453605i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −31.9778 9.38953i −1.22360 0.359280i −0.394767 0.918781i \(-0.629175\pi\)
−0.828830 + 0.559501i \(0.810993\pi\)
\(684\) 0 0
\(685\) −2.28579 + 5.00518i −0.0873356 + 0.191238i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 3.73064 0.142126
\(690\) 0 0
\(691\) 23.4533 0.892206 0.446103 0.894982i \(-0.352811\pi\)
0.446103 + 0.894982i \(0.352811\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 4.20644 9.21083i 0.159560 0.349387i
\(696\) 0 0
\(697\) −4.19052 1.23045i −0.158727 0.0466066i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 9.67059 11.1605i 0.365253 0.421525i −0.543140 0.839642i \(-0.682765\pi\)
0.908393 + 0.418118i \(0.137310\pi\)
\(702\) 0 0
\(703\) 5.05724 1.48494i 0.190737 0.0560056i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0.463184 + 3.22152i 0.0174198 + 0.121158i
\(708\) 0 0
\(709\) −1.84594 + 12.8388i −0.0693258 + 0.482172i 0.925350 + 0.379115i \(0.123771\pi\)
−0.994675 + 0.103057i \(0.967138\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −14.2317 + 36.8391i −0.532980 + 1.37963i
\(714\) 0 0
\(715\) −2.33030 2.68931i −0.0871482 0.100574i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −2.59209 18.0284i −0.0966687 0.672345i −0.979320 0.202317i \(-0.935153\pi\)
0.882651 0.470028i \(-0.155756\pi\)
\(720\) 0 0
\(721\) 1.18018 0.758455i 0.0439522 0.0282463i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0.282827 0.326399i 0.0105039 0.0121222i
\(726\) 0 0
\(727\) 3.56341 + 7.80279i 0.132160 + 0.289389i 0.964130 0.265431i \(-0.0855143\pi\)
−0.831970 + 0.554820i \(0.812787\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 8.08583 17.7055i 0.299065 0.654861i
\(732\) 0 0
\(733\) −2.64399 1.69919i −0.0976579 0.0627609i 0.490900 0.871216i \(-0.336668\pi\)
−0.588558 + 0.808455i \(0.700304\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 7.05093 0.259724
\(738\) 0 0
\(739\) 18.9339 + 12.1681i 0.696494 + 0.447610i 0.840389 0.541983i \(-0.182326\pi\)
−0.143895 + 0.989593i \(0.545963\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 4.72970 + 1.38877i 0.173516 + 0.0509489i 0.367336 0.930088i \(-0.380270\pi\)
−0.193821 + 0.981037i \(0.562088\pi\)
\(744\) 0 0
\(745\) 26.2019 + 57.3741i 0.959962 + 2.10202i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 9.64470 2.83194i 0.352410 0.103477i
\(750\) 0 0
\(751\) 0.881369 0.566421i 0.0321616 0.0206690i −0.524461 0.851434i \(-0.675733\pi\)
0.556623 + 0.830765i \(0.312097\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −1.36178 + 9.47135i −0.0495601 + 0.344698i
\(756\) 0 0
\(757\) 1.85765 + 2.14384i 0.0675173 + 0.0779192i 0.788503 0.615030i \(-0.210856\pi\)
−0.720986 + 0.692950i \(0.756311\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −19.1081 22.0519i −0.692666 0.799380i 0.295076 0.955474i \(-0.404655\pi\)
−0.987742 + 0.156094i \(0.950110\pi\)
\(762\) 0 0
\(763\) −0.962505 + 6.69437i −0.0348450 + 0.242352i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −0.0962364 + 0.0618474i −0.00347490 + 0.00223318i
\(768\) 0 0
\(769\) −19.4993 + 5.72551i −0.703163 + 0.206467i −0.613719 0.789525i \(-0.710327\pi\)
−0.0894441 + 0.995992i \(0.528509\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 9.32220 + 20.4128i 0.335296 + 0.734196i 0.999916 0.0129858i \(-0.00413362\pi\)
−0.664620 + 0.747182i \(0.731406\pi\)
\(774\) 0 0
\(775\) −28.0516 8.23670i −1.00764 0.295871i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0.647582 + 0.416175i 0.0232020 + 0.0149110i
\(780\) 0 0
\(781\) 28.9319 1.03526
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 56.0765 + 36.0382i 2.00145 + 1.28626i
\(786\) 0 0
\(787\) −5.82947 + 12.7648i −0.207798 + 0.455015i −0.984621 0.174705i \(-0.944103\pi\)
0.776823 + 0.629720i \(0.216830\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −4.16737 9.12526i −0.148175 0.324457i
\(792\) 0 0
\(793\) −0.579189 + 0.668420i −0.0205676 + 0.0237363i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −44.3146 + 28.4792i −1.56970 + 1.00879i −0.590224 + 0.807240i \(0.700960\pi\)
−0.979479 + 0.201547i \(0.935403\pi\)
\(798\) 0 0
\(799\) 3.88372 + 27.0119i 0.137396 + 0.955611i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −22.5993 26.0809i −0.797510 0.920376i
\(804\) 0 0
\(805\) −1.16430 + 13.9857i −0.0410363 + 0.492931i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 7.51836 52.2913i 0.264331 1.83847i −0.234927 0.972013i \(-0.575485\pi\)
0.499259 0.866453i \(-0.333606\pi\)
\(810\) 0 0
\(811\) −6.05545 42.1166i −0.212636 1.47891i −0.764307 0.644852i \(-0.776919\pi\)
0.551672 0.834061i \(-0.313990\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 33.9123 9.95756i 1.18790 0.348798i
\(816\) 0 0
\(817\) −2.24663 + 2.59275i −0.0785998 + 0.0907090i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 9.90317 + 2.90783i 0.345623 + 0.101484i 0.449937 0.893060i \(-0.351446\pi\)
−0.104314 + 0.994544i \(0.533265\pi\)
\(822\) 0 0
\(823\) −22.5958 + 49.4779i −0.787640 + 1.72469i −0.104355 + 0.994540i \(0.533278\pi\)
−0.683285 + 0.730151i \(0.739450\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −27.7576 −0.965227 −0.482613 0.875834i \(-0.660312\pi\)
−0.482613 + 0.875834i \(0.660312\pi\)
\(828\) 0 0
\(829\) −31.3925 −1.09031 −0.545153 0.838336i \(-0.683529\pi\)
−0.545153 + 0.838336i \(0.683529\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −7.81784 + 17.1187i −0.270872 + 0.593127i
\(834\) 0 0
\(835\) 23.2745 + 6.83400i 0.805446 + 0.236500i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 2.24487 2.59072i 0.0775017 0.0894417i −0.715674 0.698434i \(-0.753881\pi\)
0.793176 + 0.608992i \(0.208426\pi\)
\(840\) 0 0
\(841\) 27.8111 8.16607i 0.959003 0.281589i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 5.31283 + 36.9515i 0.182767 + 1.27117i
\(846\) 0 0
\(847\) −0.661869 + 4.60340i −0.0227421 + 0.158175i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 39.8625 + 22.3738i 1.36647 + 0.766962i
\(852\) 0 0
\(853\) −21.3027 24.5846i −0.729390 0.841760i 0.263013 0.964792i \(-0.415284\pi\)
−0.992403 + 0.123032i \(0.960738\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −5.10290 35.4914i −0.174312 1.21236i −0.869645 0.493677i \(-0.835652\pi\)
0.695334 0.718687i \(-0.255257\pi\)
\(858\) 0 0
\(859\) 7.63096 4.90412i 0.260365 0.167326i −0.403946 0.914783i \(-0.632362\pi\)
0.664311 + 0.747456i \(0.268725\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 27.7686 32.0467i 0.945255 1.09088i −0.0504891 0.998725i \(-0.516078\pi\)
0.995744 0.0921581i \(-0.0293765\pi\)
\(864\) 0 0
\(865\) 25.0973 + 54.9553i 0.853332 + 1.86854i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 17.8939 39.1821i 0.607008 1.32916i
\(870\) 0 0
\(871\) 1.13627 + 0.730236i 0.0385010 + 0.0247431i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 4.24225 0.143414
\(876\) 0 0
\(877\) 34.8182 + 22.3763i 1.17573 + 0.755594i 0.974596 0.223970i \(-0.0719017\pi\)
0.201132 + 0.979564i \(0.435538\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 17.0646 + 5.01061i 0.574920 + 0.168812i 0.556252 0.831014i \(-0.312239\pi\)
0.0186687 + 0.999826i \(0.494057\pi\)
\(882\) 0 0
\(883\) 3.41762 + 7.48354i 0.115012 + 0.251841i 0.958380 0.285495i \(-0.0921580\pi\)
−0.843368 + 0.537336i \(0.819431\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −42.8259 + 12.5748i −1.43795 + 0.422221i −0.905539 0.424263i \(-0.860533\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(888\) 0 0
\(889\) −14.0581 + 9.03459i −0.471493 + 0.303010i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0.684525 4.76097i 0.0229067 0.159320i
\(894\) 0 0
\(895\) 7.43107 + 8.57591i 0.248393 + 0.286661i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0.656002 + 0.757067i 0.0218789 + 0.0252496i
\(900\) 0 0
\(901\) 3.44990 23.9946i 0.114933 0.799375i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 19.6083 12.6015i 0.651800 0.418887i
\(906\) 0 0
\(907\) −39.2401 + 11.5219i −1.30295 + 0.382580i −0.858310 0.513131i \(-0.828486\pi\)
−0.444637 + 0.895711i \(0.646667\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −0.360612 0.789631i −0.0119476 0.0261616i 0.903564 0.428453i \(-0.140941\pi\)
−0.915512 + 0.402291i \(0.868214\pi\)
\(912\) 0 0
\(913\) 19.5014 + 5.72611i 0.645401 + 0.189507i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −12.1147 7.78567i −0.400064 0.257105i
\(918\) 0 0
\(919\) −53.6469 −1.76965 −0.884824 0.465926i \(-0.845721\pi\)
−0.884824 + 0.465926i \(0.845721\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 4.66243 + 2.99636i 0.153466 + 0.0986264i
\(924\) 0 0
\(925\) −14.0577 + 30.7822i −0.462216 + 1.01211i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1.81781 + 3.98044i 0.0596403 + 0.130594i 0.937105 0.349046i \(-0.113494\pi\)
−0.877465 + 0.479640i \(0.840767\pi\)
\(930\) 0 0
\(931\) 2.17217 2.50682i 0.0711901 0.0821578i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −19.4519 + 12.5010i −0.636146 + 0.408826i
\(936\) 0 0
\(937\) −0.811889 5.64681i −0.0265232 0.184473i 0.972253 0.233932i \(-0.0751594\pi\)
−0.998776 + 0.0494590i \(0.984250\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −7.27395 8.39458i −0.237124 0.273656i 0.624698 0.780866i \(-0.285222\pi\)
−0.861822 + 0.507211i \(0.830677\pi\)
\(942\) 0 0
\(943\) 1.34297 + 6.53970i 0.0437332 + 0.212962i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −2.12760 + 14.7978i −0.0691376 + 0.480863i 0.925608 + 0.378484i \(0.123554\pi\)
−0.994745 + 0.102379i \(0.967355\pi\)
\(948\) 0 0
\(949\) −0.940813 6.54350i −0.0305401 0.212411i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −28.3258 + 8.31722i −0.917564 + 0.269421i −0.706221 0.707991i \(-0.749602\pi\)
−0.211343 + 0.977412i \(0.567784\pi\)
\(954\) 0 0
\(955\) 32.3343 37.3158i 1.04631 1.20751i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −1.80690 0.530554i −0.0583478 0.0171325i
\(960\) 0 0
\(961\) 15.2919 33.4847i 0.493288 1.08015i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −62.2837 −2.00498
\(966\) 0 0
\(967\) −44.5676 −1.43320 −0.716598 0.697486i \(-0.754302\pi\)
−0.716598 + 0.697486i \(0.754302\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −22.1020 + 48.3967i −0.709288 + 1.55312i 0.119046 + 0.992889i \(0.462017\pi\)
−0.828334 + 0.560235i \(0.810711\pi\)
\(972\) 0 0
\(973\) 3.32516 + 0.976355i 0.106600 + 0.0313005i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −24.1039 + 27.8174i −0.771151 + 0.889956i −0.996437 0.0843367i \(-0.973123\pi\)
0.225286 + 0.974293i \(0.427668\pi\)
\(978\) 0 0
\(979\) −11.9431 + 3.50682i −0.381705 + 0.112079i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −5.86774 40.8111i −0.187152 1.30167i −0.839337 0.543611i \(-0.817057\pi\)
0.652185 0.758059i \(-0.273852\pi\)
\(984\) 0 0
\(985\) 11.2081 77.9541i 0.357120 2.48382i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −29.7008 + 1.77648i −0.944430 + 0.0564888i
\(990\) 0 0
\(991\) 26.4437 + 30.5176i 0.840011 + 0.969425i 0.999843 0.0177197i \(-0.00564066\pi\)
−0.159832 + 0.987144i \(0.551095\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 3.03191 + 21.0874i 0.0961180 + 0.668515i
\(996\) 0 0
\(997\) 22.2576 14.3041i 0.704905 0.453015i −0.138451 0.990369i \(-0.544212\pi\)
0.843357 + 0.537354i \(0.180576\pi\)
\(998\) 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 828.2.q.c.289.2 20
3.2 odd 2 276.2.i.a.13.1 20
23.16 even 11 inner 828.2.q.c.361.2 20
69.50 odd 22 6348.2.a.s.1.2 10
69.62 odd 22 276.2.i.a.85.1 yes 20
69.65 even 22 6348.2.a.t.1.9 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
276.2.i.a.13.1 20 3.2 odd 2
276.2.i.a.85.1 yes 20 69.62 odd 22
828.2.q.c.289.2 20 1.1 even 1 trivial
828.2.q.c.361.2 20 23.16 even 11 inner
6348.2.a.s.1.2 10 69.50 odd 22
6348.2.a.t.1.9 10 69.65 even 22