Properties

Label 460.2.s.a.9.12
Level $460$
Weight $2$
Character 460.9
Analytic conductor $3.673$
Analytic rank $0$
Dimension $120$
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] = [460,2,Mod(9,460)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(460, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 11, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("460.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 460 = 2^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 460.s (of order \(22\), 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: \(3.67311849298\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(12\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label 9.12
Character \(\chi\) \(=\) 460.9
Dual form 460.2.s.a.409.12

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.47974 - 2.14871i) q^{3} +(-2.11264 - 0.732624i) q^{5} +(0.253328 + 0.115691i) q^{7} +(1.10522 - 7.68699i) q^{9} +O(q^{10})\) \(q+(2.47974 - 2.14871i) q^{3} +(-2.11264 - 0.732624i) q^{5} +(0.253328 + 0.115691i) q^{7} +(1.10522 - 7.68699i) q^{9} +(-5.59496 - 1.64283i) q^{11} +(1.41158 - 0.644649i) q^{13} +(-6.81300 + 2.72273i) q^{15} +(-1.71260 + 2.66486i) q^{17} +(5.72284 - 3.67784i) q^{19} +(0.876774 - 0.257444i) q^{21} +(4.03867 + 2.58633i) q^{23} +(3.92652 + 3.09555i) q^{25} +(-8.45463 - 13.1557i) q^{27} +(4.04474 + 2.59939i) q^{29} +(3.92298 - 4.52737i) q^{31} +(-17.4040 + 7.94813i) q^{33} +(-0.450434 - 0.430008i) q^{35} +(5.00986 + 0.720309i) q^{37} +(2.11520 - 4.63164i) q^{39} +(0.607234 + 4.22340i) q^{41} +(-4.03428 + 3.49572i) q^{43} +(-7.96662 + 15.4302i) q^{45} +5.97319i q^{47} +(-4.53323 - 5.23163i) q^{49} +(1.47920 + 10.2880i) q^{51} +(1.47049 + 0.671549i) q^{53} +(10.6166 + 7.56971i) q^{55} +(6.28853 - 21.4168i) q^{57} +(-1.88730 - 4.13261i) q^{59} +(-3.33333 + 3.84687i) q^{61} +(1.16930 - 1.81947i) q^{63} +(-3.45446 + 0.327753i) q^{65} +(-2.08629 - 7.10524i) q^{67} +(15.5721 - 2.26450i) q^{69} +(4.68696 - 1.37622i) q^{71} +(-0.687576 - 1.06989i) q^{73} +(16.3882 - 0.760800i) q^{75} +(-1.22730 - 1.06346i) q^{77} +(2.95425 + 6.46890i) q^{79} +(-26.8785 - 7.89225i) q^{81} +(17.2709 + 2.48318i) q^{83} +(5.57047 - 4.37521i) q^{85} +(15.6152 - 2.24513i) q^{87} +(-8.86092 - 10.2260i) q^{89} +0.432174 q^{91} -19.6560i q^{93} +(-14.7848 + 3.57729i) q^{95} +(3.30296 - 0.474893i) q^{97} +(-18.8121 + 41.1927i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 120 q + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 120 q + 20 q^{9} - 4 q^{11} + 9 q^{15} + 8 q^{19} + 14 q^{25} + 10 q^{29} - 18 q^{31} + 10 q^{35} - 60 q^{39} + 2 q^{41} - 2 q^{45} - 28 q^{49} + 24 q^{51} + 6 q^{55} - 36 q^{61} + 39 q^{65} + 118 q^{69} - 76 q^{71} + 83 q^{75} + 64 q^{79} - 160 q^{81} + 38 q^{85} - 48 q^{89} - 80 q^{91} + 21 q^{95} - 142 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/460\mathbb{Z}\right)^\times\).

\(n\) \(231\) \(277\) \(281\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{5}{11}\right)\)

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
\(3\) 2.47974 2.14871i 1.43168 1.24056i 0.505844 0.862625i \(-0.331181\pi\)
0.925834 0.377931i \(-0.123364\pi\)
\(4\) 0 0
\(5\) −2.11264 0.732624i −0.944803 0.327639i
\(6\) 0 0
\(7\) 0.253328 + 0.115691i 0.0957490 + 0.0437271i 0.462714 0.886508i \(-0.346876\pi\)
−0.366965 + 0.930235i \(0.619603\pi\)
\(8\) 0 0
\(9\) 1.10522 7.68699i 0.368408 2.56233i
\(10\) 0 0
\(11\) −5.59496 1.64283i −1.68694 0.495331i −0.709177 0.705030i \(-0.750933\pi\)
−0.977766 + 0.209699i \(0.932751\pi\)
\(12\) 0 0
\(13\) 1.41158 0.644649i 0.391503 0.178793i −0.209930 0.977716i \(-0.567324\pi\)
0.601434 + 0.798923i \(0.294596\pi\)
\(14\) 0 0
\(15\) −6.81300 + 2.72273i −1.75911 + 0.703007i
\(16\) 0 0
\(17\) −1.71260 + 2.66486i −0.415368 + 0.646324i −0.984391 0.175997i \(-0.943685\pi\)
0.569023 + 0.822321i \(0.307321\pi\)
\(18\) 0 0
\(19\) 5.72284 3.67784i 1.31291 0.843755i 0.318354 0.947972i \(-0.396870\pi\)
0.994555 + 0.104217i \(0.0332336\pi\)
\(20\) 0 0
\(21\) 0.876774 0.257444i 0.191328 0.0561789i
\(22\) 0 0
\(23\) 4.03867 + 2.58633i 0.842122 + 0.539287i
\(24\) 0 0
\(25\) 3.92652 + 3.09555i 0.785305 + 0.619109i
\(26\) 0 0
\(27\) −8.45463 13.1557i −1.62709 2.53181i
\(28\) 0 0
\(29\) 4.04474 + 2.59939i 0.751089 + 0.482695i 0.859325 0.511430i \(-0.170884\pi\)
−0.108236 + 0.994125i \(0.534520\pi\)
\(30\) 0 0
\(31\) 3.92298 4.52737i 0.704589 0.813139i −0.284776 0.958594i \(-0.591919\pi\)
0.989365 + 0.145455i \(0.0464647\pi\)
\(32\) 0 0
\(33\) −17.4040 + 7.94813i −3.02964 + 1.38359i
\(34\) 0 0
\(35\) −0.450434 0.430008i −0.0761372 0.0726847i
\(36\) 0 0
\(37\) 5.00986 + 0.720309i 0.823616 + 0.118418i 0.541221 0.840880i \(-0.317962\pi\)
0.282395 + 0.959298i \(0.408871\pi\)
\(38\) 0 0
\(39\) 2.11520 4.63164i 0.338703 0.741656i
\(40\) 0 0
\(41\) 0.607234 + 4.22340i 0.0948339 + 0.659585i 0.980682 + 0.195609i \(0.0626684\pi\)
−0.885848 + 0.463976i \(0.846423\pi\)
\(42\) 0 0
\(43\) −4.03428 + 3.49572i −0.615222 + 0.533093i −0.905771 0.423767i \(-0.860707\pi\)
0.290549 + 0.956860i \(0.406162\pi\)
\(44\) 0 0
\(45\) −7.96662 + 15.4302i −1.18759 + 2.30019i
\(46\) 0 0
\(47\) 5.97319i 0.871280i 0.900121 + 0.435640i \(0.143478\pi\)
−0.900121 + 0.435640i \(0.856522\pi\)
\(48\) 0 0
\(49\) −4.53323 5.23163i −0.647605 0.747376i
\(50\) 0 0
\(51\) 1.47920 + 10.2880i 0.207129 + 1.44062i
\(52\) 0 0
\(53\) 1.47049 + 0.671549i 0.201987 + 0.0922443i 0.513840 0.857886i \(-0.328222\pi\)
−0.311853 + 0.950130i \(0.600950\pi\)
\(54\) 0 0
\(55\) 10.6166 + 7.56971i 1.43154 + 1.02070i
\(56\) 0 0
\(57\) 6.28853 21.4168i 0.832937 2.83672i
\(58\) 0 0
\(59\) −1.88730 4.13261i −0.245705 0.538020i 0.746092 0.665843i \(-0.231928\pi\)
−0.991797 + 0.127824i \(0.959201\pi\)
\(60\) 0 0
\(61\) −3.33333 + 3.84687i −0.426789 + 0.492541i −0.927893 0.372846i \(-0.878382\pi\)
0.501104 + 0.865387i \(0.332927\pi\)
\(62\) 0 0
\(63\) 1.16930 1.81947i 0.147318 0.229231i
\(64\) 0 0
\(65\) −3.45446 + 0.327753i −0.428473 + 0.0406528i
\(66\) 0 0
\(67\) −2.08629 7.10524i −0.254881 0.868044i −0.983157 0.182761i \(-0.941497\pi\)
0.728277 0.685283i \(-0.240321\pi\)
\(68\) 0 0
\(69\) 15.5721 2.26450i 1.87466 0.272614i
\(70\) 0 0
\(71\) 4.68696 1.37622i 0.556240 0.163327i 0.00848251 0.999964i \(-0.497300\pi\)
0.547757 + 0.836637i \(0.315482\pi\)
\(72\) 0 0
\(73\) −0.687576 1.06989i −0.0804747 0.125221i 0.798683 0.601752i \(-0.205530\pi\)
−0.879158 + 0.476531i \(0.841894\pi\)
\(74\) 0 0
\(75\) 16.3882 0.760800i 1.89234 0.0878496i
\(76\) 0 0
\(77\) −1.22730 1.06346i −0.139864 0.121193i
\(78\) 0 0
\(79\) 2.95425 + 6.46890i 0.332379 + 0.727808i 0.999859 0.0168199i \(-0.00535421\pi\)
−0.667480 + 0.744628i \(0.732627\pi\)
\(80\) 0 0
\(81\) −26.8785 7.89225i −2.98650 0.876916i
\(82\) 0 0
\(83\) 17.2709 + 2.48318i 1.89573 + 0.272564i 0.988860 0.148851i \(-0.0475574\pi\)
0.906868 + 0.421415i \(0.138466\pi\)
\(84\) 0 0
\(85\) 5.57047 4.37521i 0.604202 0.474558i
\(86\) 0 0
\(87\) 15.6152 2.24513i 1.67413 0.240703i
\(88\) 0 0
\(89\) −8.86092 10.2260i −0.939256 1.08396i −0.996331 0.0855889i \(-0.972723\pi\)
0.0570749 0.998370i \(-0.481823\pi\)
\(90\) 0 0
\(91\) 0.432174 0.0453042
\(92\) 0 0
\(93\) 19.6560i 2.03823i
\(94\) 0 0
\(95\) −14.7848 + 3.57729i −1.51689 + 0.367022i
\(96\) 0 0
\(97\) 3.30296 0.474893i 0.335364 0.0482181i 0.0274243 0.999624i \(-0.491269\pi\)
0.307940 + 0.951406i \(0.400360\pi\)
\(98\) 0 0
\(99\) −18.8121 + 41.1927i −1.89068 + 4.14002i
\(100\) 0 0
\(101\) −1.00808 + 7.01132i −0.100307 + 0.697652i 0.876165 + 0.482011i \(0.160093\pi\)
−0.976473 + 0.215642i \(0.930816\pi\)
\(102\) 0 0
\(103\) −3.79139 + 12.9123i −0.373577 + 1.27229i 0.531507 + 0.847054i \(0.321626\pi\)
−0.905084 + 0.425232i \(0.860192\pi\)
\(104\) 0 0
\(105\) −2.04092 0.0984580i −0.199173 0.00960852i
\(106\) 0 0
\(107\) 0.302058 + 0.261735i 0.0292011 + 0.0253029i 0.669339 0.742957i \(-0.266577\pi\)
−0.640138 + 0.768260i \(0.721123\pi\)
\(108\) 0 0
\(109\) −4.79798 3.08348i −0.459563 0.295344i 0.290301 0.956936i \(-0.406245\pi\)
−0.749864 + 0.661592i \(0.769881\pi\)
\(110\) 0 0
\(111\) 13.9709 8.97854i 1.32606 0.852205i
\(112\) 0 0
\(113\) 2.38700 + 8.12938i 0.224550 + 0.764747i 0.992285 + 0.123979i \(0.0395657\pi\)
−0.767735 + 0.640768i \(0.778616\pi\)
\(114\) 0 0
\(115\) −6.63747 8.42282i −0.618947 0.785432i
\(116\) 0 0
\(117\) −3.39530 11.5633i −0.313895 1.06903i
\(118\) 0 0
\(119\) −0.742152 + 0.476952i −0.0680329 + 0.0437221i
\(120\) 0 0
\(121\) 19.3509 + 12.4360i 1.75917 + 1.13055i
\(122\) 0 0
\(123\) 10.5806 + 9.16817i 0.954023 + 0.826666i
\(124\) 0 0
\(125\) −6.02747 9.41645i −0.539114 0.842233i
\(126\) 0 0
\(127\) −4.46500 + 15.2064i −0.396205 + 1.34935i 0.484130 + 0.874996i \(0.339136\pi\)
−0.880334 + 0.474353i \(0.842682\pi\)
\(128\) 0 0
\(129\) −2.49268 + 17.3370i −0.219468 + 1.52643i
\(130\) 0 0
\(131\) 0.653039 1.42996i 0.0570563 0.124936i −0.878956 0.476902i \(-0.841760\pi\)
0.936013 + 0.351966i \(0.114487\pi\)
\(132\) 0 0
\(133\) 1.87525 0.269620i 0.162605 0.0233790i
\(134\) 0 0
\(135\) 8.22346 + 33.9873i 0.707763 + 2.92516i
\(136\) 0 0
\(137\) 6.77720i 0.579016i 0.957176 + 0.289508i \(0.0934916\pi\)
−0.957176 + 0.289508i \(0.906508\pi\)
\(138\) 0 0
\(139\) 13.2170 1.12105 0.560526 0.828137i \(-0.310599\pi\)
0.560526 + 0.828137i \(0.310599\pi\)
\(140\) 0 0
\(141\) 12.8346 + 14.8120i 1.08087 + 1.24739i
\(142\) 0 0
\(143\) −8.95680 + 1.28779i −0.749005 + 0.107691i
\(144\) 0 0
\(145\) −6.64071 8.45486i −0.551481 0.702138i
\(146\) 0 0
\(147\) −22.4825 3.23249i −1.85432 0.266611i
\(148\) 0 0
\(149\) −15.6849 4.60550i −1.28496 0.377297i −0.433230 0.901283i \(-0.642626\pi\)
−0.851727 + 0.523986i \(0.824444\pi\)
\(150\) 0 0
\(151\) 2.98469 + 6.53556i 0.242891 + 0.531857i 0.991338 0.131338i \(-0.0419274\pi\)
−0.748447 + 0.663195i \(0.769200\pi\)
\(152\) 0 0
\(153\) 18.5920 + 16.1100i 1.50307 + 1.30242i
\(154\) 0 0
\(155\) −11.6047 + 6.69064i −0.932114 + 0.537405i
\(156\) 0 0
\(157\) −4.60896 7.17168i −0.367835 0.572362i 0.607163 0.794578i \(-0.292308\pi\)
−0.974998 + 0.222215i \(0.928671\pi\)
\(158\) 0 0
\(159\) 5.08938 1.49438i 0.403614 0.118512i
\(160\) 0 0
\(161\) 0.723894 + 1.12243i 0.0570509 + 0.0884598i
\(162\) 0 0
\(163\) −4.83815 16.4772i −0.378953 1.29060i −0.899559 0.436798i \(-0.856112\pi\)
0.520607 0.853797i \(-0.325706\pi\)
\(164\) 0 0
\(165\) 42.5914 4.04100i 3.31574 0.314591i
\(166\) 0 0
\(167\) 6.11331 9.51250i 0.473062 0.736100i −0.519939 0.854204i \(-0.674045\pi\)
0.993001 + 0.118104i \(0.0376816\pi\)
\(168\) 0 0
\(169\) −6.93619 + 8.00479i −0.533553 + 0.615753i
\(170\) 0 0
\(171\) −21.9465 48.0562i −1.67829 3.67495i
\(172\) 0 0
\(173\) 3.43430 11.6961i 0.261105 0.889241i −0.719706 0.694279i \(-0.755723\pi\)
0.980811 0.194962i \(-0.0624583\pi\)
\(174\) 0 0
\(175\) 0.636572 + 1.23845i 0.0481203 + 0.0936182i
\(176\) 0 0
\(177\) −13.5598 6.19254i −1.01921 0.465460i
\(178\) 0 0
\(179\) 2.02396 + 14.0769i 0.151278 + 1.05216i 0.914082 + 0.405529i \(0.132913\pi\)
−0.762805 + 0.646629i \(0.776178\pi\)
\(180\) 0 0
\(181\) 1.95257 + 2.25338i 0.145133 + 0.167493i 0.823662 0.567081i \(-0.191928\pi\)
−0.678529 + 0.734574i \(0.737382\pi\)
\(182\) 0 0
\(183\) 16.7016i 1.23462i
\(184\) 0 0
\(185\) −10.0563 5.19210i −0.739356 0.381731i
\(186\) 0 0
\(187\) 13.9599 12.0963i 1.02085 0.884568i
\(188\) 0 0
\(189\) −0.619803 4.31082i −0.0450840 0.313566i
\(190\) 0 0
\(191\) 2.83920 6.21698i 0.205437 0.449845i −0.778667 0.627438i \(-0.784104\pi\)
0.984104 + 0.177593i \(0.0568310\pi\)
\(192\) 0 0
\(193\) −14.1771 2.03836i −1.02049 0.146725i −0.388297 0.921534i \(-0.626937\pi\)
−0.632195 + 0.774810i \(0.717846\pi\)
\(194\) 0 0
\(195\) −7.86191 + 8.23536i −0.563003 + 0.589746i
\(196\) 0 0
\(197\) −22.1233 + 10.1034i −1.57622 + 0.719836i −0.995544 0.0942969i \(-0.969940\pi\)
−0.580678 + 0.814133i \(0.697212\pi\)
\(198\) 0 0
\(199\) −7.89945 + 9.11645i −0.559977 + 0.646248i −0.963178 0.268863i \(-0.913352\pi\)
0.403201 + 0.915111i \(0.367897\pi\)
\(200\) 0 0
\(201\) −20.4405 13.1363i −1.44176 0.926565i
\(202\) 0 0
\(203\) 0.723919 + 1.12644i 0.0508091 + 0.0790606i
\(204\) 0 0
\(205\) 1.81130 9.36742i 0.126507 0.654249i
\(206\) 0 0
\(207\) 24.3447 28.1868i 1.69208 1.95912i
\(208\) 0 0
\(209\) −38.0611 + 11.1757i −2.63274 + 0.773042i
\(210\) 0 0
\(211\) 6.61991 4.25436i 0.455733 0.292882i −0.292563 0.956246i \(-0.594508\pi\)
0.748297 + 0.663364i \(0.230872\pi\)
\(212\) 0 0
\(213\) 8.66535 13.4836i 0.593740 0.923878i
\(214\) 0 0
\(215\) 11.0840 4.42961i 0.755926 0.302097i
\(216\) 0 0
\(217\) 1.51758 0.693055i 0.103020 0.0470476i
\(218\) 0 0
\(219\) −4.00389 1.17565i −0.270558 0.0794429i
\(220\) 0 0
\(221\) −0.699583 + 4.86571i −0.0470591 + 0.327303i
\(222\) 0 0
\(223\) −4.85955 2.21928i −0.325420 0.148614i 0.246005 0.969268i \(-0.420882\pi\)
−0.571425 + 0.820654i \(0.693609\pi\)
\(224\) 0 0
\(225\) 28.1351 26.7619i 1.87568 1.78413i
\(226\) 0 0
\(227\) 10.7149 9.28448i 0.711170 0.616233i −0.222262 0.974987i \(-0.571344\pi\)
0.933432 + 0.358754i \(0.116798\pi\)
\(228\) 0 0
\(229\) 3.61657 0.238990 0.119495 0.992835i \(-0.461873\pi\)
0.119495 + 0.992835i \(0.461873\pi\)
\(230\) 0 0
\(231\) −5.32845 −0.350586
\(232\) 0 0
\(233\) −12.5590 + 10.8824i −0.822765 + 0.712930i −0.960723 0.277508i \(-0.910491\pi\)
0.137959 + 0.990438i \(0.455946\pi\)
\(234\) 0 0
\(235\) 4.37611 12.6192i 0.285466 0.823188i
\(236\) 0 0
\(237\) 21.2255 + 9.69337i 1.37875 + 0.629652i
\(238\) 0 0
\(239\) 2.81795 19.5993i 0.182278 1.26777i −0.669081 0.743189i \(-0.733312\pi\)
0.851359 0.524583i \(-0.175779\pi\)
\(240\) 0 0
\(241\) 2.13804 + 0.627784i 0.137723 + 0.0404391i 0.349868 0.936799i \(-0.386227\pi\)
−0.212145 + 0.977238i \(0.568045\pi\)
\(242\) 0 0
\(243\) −40.9349 + 18.6944i −2.62598 + 1.19924i
\(244\) 0 0
\(245\) 5.74429 + 14.3737i 0.366989 + 0.918304i
\(246\) 0 0
\(247\) 5.70735 8.88081i 0.363150 0.565072i
\(248\) 0 0
\(249\) 48.1629 30.9524i 3.05220 1.96153i
\(250\) 0 0
\(251\) 12.9766 3.81026i 0.819073 0.240501i 0.154756 0.987953i \(-0.450541\pi\)
0.664317 + 0.747451i \(0.268723\pi\)
\(252\) 0 0
\(253\) −18.3473 21.1053i −1.15349 1.32688i
\(254\) 0 0
\(255\) 4.41225 22.8187i 0.276306 1.42896i
\(256\) 0 0
\(257\) 0.440492 + 0.685418i 0.0274771 + 0.0427552i 0.854720 0.519089i \(-0.173729\pi\)
−0.827243 + 0.561844i \(0.810092\pi\)
\(258\) 0 0
\(259\) 1.18581 + 0.762071i 0.0736823 + 0.0473528i
\(260\) 0 0
\(261\) 24.4519 28.2190i 1.51353 1.74671i
\(262\) 0 0
\(263\) 19.2184 8.77675i 1.18506 0.541198i 0.277337 0.960773i \(-0.410548\pi\)
0.907721 + 0.419575i \(0.137821\pi\)
\(264\) 0 0
\(265\) −2.61462 2.49606i −0.160615 0.153332i
\(266\) 0 0
\(267\) −43.9455 6.31841i −2.68942 0.386681i
\(268\) 0 0
\(269\) 6.17597 13.5235i 0.376556 0.824542i −0.622563 0.782570i \(-0.713909\pi\)
0.999119 0.0419722i \(-0.0133641\pi\)
\(270\) 0 0
\(271\) 4.50613 + 31.3408i 0.273728 + 1.90382i 0.408186 + 0.912899i \(0.366162\pi\)
−0.134458 + 0.990919i \(0.542929\pi\)
\(272\) 0 0
\(273\) 1.07168 0.928615i 0.0648610 0.0562023i
\(274\) 0 0
\(275\) −16.8833 23.7700i −1.01810 1.43339i
\(276\) 0 0
\(277\) 1.00562i 0.0604219i −0.999544 0.0302110i \(-0.990382\pi\)
0.999544 0.0302110i \(-0.00961791\pi\)
\(278\) 0 0
\(279\) −30.4661 35.1597i −1.82396 2.10496i
\(280\) 0 0
\(281\) 1.90861 + 13.2747i 0.113858 + 0.791900i 0.964106 + 0.265518i \(0.0855429\pi\)
−0.850248 + 0.526382i \(0.823548\pi\)
\(282\) 0 0
\(283\) −8.13848 3.71672i −0.483782 0.220936i 0.158567 0.987348i \(-0.449313\pi\)
−0.642349 + 0.766412i \(0.722040\pi\)
\(284\) 0 0
\(285\) −28.9759 + 40.6389i −1.71638 + 2.40724i
\(286\) 0 0
\(287\) −0.334781 + 1.14016i −0.0197615 + 0.0673014i
\(288\) 0 0
\(289\) 2.89357 + 6.33603i 0.170210 + 0.372708i
\(290\) 0 0
\(291\) 7.17006 8.27469i 0.420316 0.485071i
\(292\) 0 0
\(293\) −7.35577 + 11.4458i −0.429729 + 0.668671i −0.986828 0.161775i \(-0.948278\pi\)
0.557099 + 0.830446i \(0.311914\pi\)
\(294\) 0 0
\(295\) 0.959542 + 10.1134i 0.0558667 + 0.588825i
\(296\) 0 0
\(297\) 25.6908 + 87.4948i 1.49073 + 5.07697i
\(298\) 0 0
\(299\) 7.36821 + 1.04730i 0.426114 + 0.0605667i
\(300\) 0 0
\(301\) −1.42642 + 0.418835i −0.0822175 + 0.0241412i
\(302\) 0 0
\(303\) 12.5655 + 19.5523i 0.721869 + 1.12325i
\(304\) 0 0
\(305\) 9.86045 5.68499i 0.564608 0.325521i
\(306\) 0 0
\(307\) 23.0744 + 19.9940i 1.31692 + 1.14112i 0.979870 + 0.199635i \(0.0639756\pi\)
0.337053 + 0.941486i \(0.390570\pi\)
\(308\) 0 0
\(309\) 18.3431 + 40.1657i 1.04350 + 2.28495i
\(310\) 0 0
\(311\) −27.2678 8.00654i −1.54621 0.454009i −0.606247 0.795276i \(-0.707326\pi\)
−0.939966 + 0.341267i \(0.889144\pi\)
\(312\) 0 0
\(313\) −6.90156 0.992295i −0.390099 0.0560878i −0.0555254 0.998457i \(-0.517683\pi\)
−0.334574 + 0.942369i \(0.608592\pi\)
\(314\) 0 0
\(315\) −3.80330 + 2.98723i −0.214292 + 0.168311i
\(316\) 0 0
\(317\) −26.1434 + 3.75886i −1.46836 + 0.211119i −0.829631 0.558312i \(-0.811449\pi\)
−0.638731 + 0.769430i \(0.720540\pi\)
\(318\) 0 0
\(319\) −18.3598 21.1883i −1.02795 1.18632i
\(320\) 0 0
\(321\) 1.31142 0.0731962
\(322\) 0 0
\(323\) 21.5493i 1.19903i
\(324\) 0 0
\(325\) 7.53816 + 1.83839i 0.418142 + 0.101976i
\(326\) 0 0
\(327\) −18.5232 + 2.66324i −1.02434 + 0.147277i
\(328\) 0 0
\(329\) −0.691045 + 1.51318i −0.0380986 + 0.0834242i
\(330\) 0 0
\(331\) −2.15416 + 14.9825i −0.118403 + 0.823513i 0.840911 + 0.541173i \(0.182020\pi\)
−0.959314 + 0.282340i \(0.908889\pi\)
\(332\) 0 0
\(333\) 11.0740 37.7147i 0.606853 2.06675i
\(334\) 0 0
\(335\) −0.797889 + 16.5393i −0.0435933 + 0.903639i
\(336\) 0 0
\(337\) −16.4883 14.2872i −0.898175 0.778273i 0.0776150 0.996983i \(-0.475270\pi\)
−0.975790 + 0.218711i \(0.929815\pi\)
\(338\) 0 0
\(339\) 23.3868 + 15.0298i 1.27020 + 0.816305i
\(340\) 0 0
\(341\) −29.3866 + 18.8856i −1.59137 + 1.02271i
\(342\) 0 0
\(343\) −1.09237 3.72027i −0.0589825 0.200876i
\(344\) 0 0
\(345\) −34.5574 6.62443i −1.86051 0.356647i
\(346\) 0 0
\(347\) −3.75520 12.7890i −0.201590 0.686552i −0.996780 0.0801910i \(-0.974447\pi\)
0.795190 0.606361i \(-0.207371\pi\)
\(348\) 0 0
\(349\) 3.59758 2.31202i 0.192574 0.123760i −0.440801 0.897605i \(-0.645306\pi\)
0.633375 + 0.773845i \(0.281669\pi\)
\(350\) 0 0
\(351\) −20.4152 13.1201i −1.08968 0.700297i
\(352\) 0 0
\(353\) −13.0815 11.3352i −0.696260 0.603313i 0.233115 0.972449i \(-0.425108\pi\)
−0.929375 + 0.369136i \(0.879654\pi\)
\(354\) 0 0
\(355\) −10.9101 0.526326i −0.579049 0.0279345i
\(356\) 0 0
\(357\) −0.815513 + 2.77738i −0.0431615 + 0.146995i
\(358\) 0 0
\(359\) 1.01985 7.09320i 0.0538255 0.374365i −0.945049 0.326929i \(-0.893986\pi\)
0.998874 0.0474355i \(-0.0151048\pi\)
\(360\) 0 0
\(361\) 11.3314 24.8124i 0.596391 1.30591i
\(362\) 0 0
\(363\) 74.7065 10.7412i 3.92107 0.563765i
\(364\) 0 0
\(365\) 0.668777 + 2.76403i 0.0350054 + 0.144676i
\(366\) 0 0
\(367\) 21.8856i 1.14242i 0.820805 + 0.571208i \(0.193525\pi\)
−0.820805 + 0.571208i \(0.806475\pi\)
\(368\) 0 0
\(369\) 33.1364 1.72501
\(370\) 0 0
\(371\) 0.294823 + 0.340244i 0.0153065 + 0.0176646i
\(372\) 0 0
\(373\) 10.7075 1.53951i 0.554414 0.0797127i 0.140587 0.990068i \(-0.455101\pi\)
0.413827 + 0.910356i \(0.364192\pi\)
\(374\) 0 0
\(375\) −35.1797 10.3991i −1.81667 0.537006i
\(376\) 0 0
\(377\) 7.38518 + 1.06183i 0.380356 + 0.0546870i
\(378\) 0 0
\(379\) 2.47246 + 0.725980i 0.127002 + 0.0372911i 0.344616 0.938744i \(-0.388009\pi\)
−0.217614 + 0.976035i \(0.569827\pi\)
\(380\) 0 0
\(381\) 21.6020 + 47.3019i 1.10671 + 2.42335i
\(382\) 0 0
\(383\) −21.8827 18.9615i −1.11816 0.968887i −0.118445 0.992961i \(-0.537791\pi\)
−0.999710 + 0.0240737i \(0.992336\pi\)
\(384\) 0 0
\(385\) 1.81373 + 3.14586i 0.0924362 + 0.160328i
\(386\) 0 0
\(387\) 22.4128 + 34.8750i 1.13931 + 1.77280i
\(388\) 0 0
\(389\) 7.25482 2.13021i 0.367834 0.108006i −0.0925914 0.995704i \(-0.529515\pi\)
0.460425 + 0.887698i \(0.347697\pi\)
\(390\) 0 0
\(391\) −13.8089 + 6.33316i −0.698345 + 0.320282i
\(392\) 0 0
\(393\) −1.45319 4.94911i −0.0733037 0.249649i
\(394\) 0 0
\(395\) −1.50200 15.8308i −0.0755739 0.796535i
\(396\) 0 0
\(397\) −8.85092 + 13.7723i −0.444215 + 0.691212i −0.989091 0.147304i \(-0.952940\pi\)
0.544876 + 0.838517i \(0.316577\pi\)
\(398\) 0 0
\(399\) 4.07079 4.69795i 0.203795 0.235191i
\(400\) 0 0
\(401\) 2.16105 + 4.73205i 0.107918 + 0.236307i 0.955885 0.293742i \(-0.0949007\pi\)
−0.847967 + 0.530049i \(0.822173\pi\)
\(402\) 0 0
\(403\) 2.61906 8.91971i 0.130465 0.444322i
\(404\) 0 0
\(405\) 51.0027 + 36.3654i 2.53434 + 1.80701i
\(406\) 0 0
\(407\) −26.8466 12.2604i −1.33074 0.607727i
\(408\) 0 0
\(409\) −3.34288 23.2503i −0.165295 1.14965i −0.888453 0.458968i \(-0.848219\pi\)
0.723158 0.690683i \(-0.242690\pi\)
\(410\) 0 0
\(411\) 14.5622 + 16.8057i 0.718301 + 0.828964i
\(412\) 0 0
\(413\) 1.26525i 0.0622588i
\(414\) 0 0
\(415\) −34.6680 17.8991i −1.70179 0.878635i
\(416\) 0 0
\(417\) 32.7747 28.3995i 1.60498 1.39073i
\(418\) 0 0
\(419\) −3.71731 25.8545i −0.181603 1.26307i −0.852974 0.521953i \(-0.825204\pi\)
0.671372 0.741121i \(-0.265706\pi\)
\(420\) 0 0
\(421\) −4.33407 + 9.49029i −0.211230 + 0.462528i −0.985358 0.170501i \(-0.945461\pi\)
0.774128 + 0.633029i \(0.218189\pi\)
\(422\) 0 0
\(423\) 45.9159 + 6.60171i 2.23251 + 0.320986i
\(424\) 0 0
\(425\) −14.9738 + 5.16221i −0.726336 + 0.250404i
\(426\) 0 0
\(427\) −1.28948 + 0.588884i −0.0624021 + 0.0284981i
\(428\) 0 0
\(429\) −19.4434 + 22.4389i −0.938738 + 1.08336i
\(430\) 0 0
\(431\) 32.3034 + 20.7601i 1.55600 + 0.999980i 0.983676 + 0.179951i \(0.0575940\pi\)
0.572323 + 0.820028i \(0.306042\pi\)
\(432\) 0 0
\(433\) 6.01435 + 9.35851i 0.289031 + 0.449741i 0.955158 0.296097i \(-0.0956853\pi\)
−0.666127 + 0.745839i \(0.732049\pi\)
\(434\) 0 0
\(435\) −34.6342 6.69693i −1.66058 0.321093i
\(436\) 0 0
\(437\) 32.6248 0.0524672i 1.56066 0.00250985i
\(438\) 0 0
\(439\) 11.2714 3.30959i 0.537956 0.157958i −0.00145800 0.999999i \(-0.500464\pi\)
0.539414 + 0.842041i \(0.318646\pi\)
\(440\) 0 0
\(441\) −45.2257 + 29.0648i −2.15361 + 1.38404i
\(442\) 0 0
\(443\) −1.24589 + 1.93864i −0.0591938 + 0.0921073i −0.869589 0.493776i \(-0.835616\pi\)
0.810395 + 0.585884i \(0.199252\pi\)
\(444\) 0 0
\(445\) 11.2281 + 28.0957i 0.532264 + 1.33186i
\(446\) 0 0
\(447\) −48.7903 + 22.2818i −2.30770 + 1.05389i
\(448\) 0 0
\(449\) −11.8645 3.48372i −0.559918 0.164407i −0.0104860 0.999945i \(-0.503338\pi\)
−0.549432 + 0.835538i \(0.685156\pi\)
\(450\) 0 0
\(451\) 3.54088 24.6273i 0.166733 1.15966i
\(452\) 0 0
\(453\) 21.4443 + 9.79326i 1.00754 + 0.460128i
\(454\) 0 0
\(455\) −0.913030 0.316621i −0.0428035 0.0148434i
\(456\) 0 0
\(457\) 11.3544 9.83861i 0.531135 0.460231i −0.347531 0.937669i \(-0.612980\pi\)
0.878666 + 0.477438i \(0.158434\pi\)
\(458\) 0 0
\(459\) 49.5375 2.31221
\(460\) 0 0
\(461\) −0.718745 −0.0334753 −0.0167377 0.999860i \(-0.505328\pi\)
−0.0167377 + 0.999860i \(0.505328\pi\)
\(462\) 0 0
\(463\) −4.43312 + 3.84132i −0.206024 + 0.178521i −0.751761 0.659436i \(-0.770795\pi\)
0.545737 + 0.837957i \(0.316250\pi\)
\(464\) 0 0
\(465\) −14.4005 + 41.5262i −0.667806 + 1.92573i
\(466\) 0 0
\(467\) 20.0466 + 9.15498i 0.927646 + 0.423642i 0.821177 0.570674i \(-0.193318\pi\)
0.106470 + 0.994316i \(0.466045\pi\)
\(468\) 0 0
\(469\) 0.293498 2.04132i 0.0135525 0.0942595i
\(470\) 0 0
\(471\) −26.8388 7.88060i −1.23667 0.363119i
\(472\) 0 0
\(473\) 28.3145 12.9308i 1.30190 0.594559i
\(474\) 0 0
\(475\) 33.8558 + 3.27416i 1.55341 + 0.150229i
\(476\) 0 0
\(477\) 6.78740 10.5614i 0.310774 0.483574i
\(478\) 0 0
\(479\) −5.63822 + 3.62346i −0.257617 + 0.165560i −0.663074 0.748554i \(-0.730749\pi\)
0.405457 + 0.914114i \(0.367112\pi\)
\(480\) 0 0
\(481\) 7.53619 2.21282i 0.343621 0.100896i
\(482\) 0 0
\(483\) 4.20684 + 1.22789i 0.191418 + 0.0558711i
\(484\) 0 0
\(485\) −7.32588 1.41654i −0.332651 0.0643219i
\(486\) 0 0
\(487\) −9.45626 14.7142i −0.428504 0.666766i 0.558123 0.829758i \(-0.311522\pi\)
−0.986627 + 0.162993i \(0.947885\pi\)
\(488\) 0 0
\(489\) −47.4020 30.4634i −2.14359 1.37760i
\(490\) 0 0
\(491\) −12.2402 + 14.1260i −0.552395 + 0.637497i −0.961439 0.275017i \(-0.911317\pi\)
0.409045 + 0.912514i \(0.365862\pi\)
\(492\) 0 0
\(493\) −13.8541 + 6.32694i −0.623956 + 0.284951i
\(494\) 0 0
\(495\) 69.9220 73.2433i 3.14276 3.29204i
\(496\) 0 0
\(497\) 1.34655 + 0.193605i 0.0604012 + 0.00868439i
\(498\) 0 0
\(499\) 0.767209 1.67995i 0.0343450 0.0752050i −0.891682 0.452662i \(-0.850474\pi\)
0.926027 + 0.377457i \(0.123202\pi\)
\(500\) 0 0
\(501\) −5.28015 36.7242i −0.235900 1.64072i
\(502\) 0 0
\(503\) −21.1649 + 18.3395i −0.943698 + 0.817719i −0.983392 0.181493i \(-0.941907\pi\)
0.0396945 + 0.999212i \(0.487362\pi\)
\(504\) 0 0
\(505\) 7.26636 14.0739i 0.323349 0.626279i
\(506\) 0 0
\(507\) 34.7536i 1.54346i
\(508\) 0 0
\(509\) 26.4140 + 30.4833i 1.17078 + 1.35115i 0.924146 + 0.382041i \(0.124779\pi\)
0.246632 + 0.969109i \(0.420676\pi\)
\(510\) 0 0
\(511\) −0.0504057 0.350580i −0.00222982 0.0155087i
\(512\) 0 0
\(513\) −96.7689 44.1929i −4.27245 1.95116i
\(514\) 0 0
\(515\) 17.4697 24.5014i 0.769807 1.07966i
\(516\) 0 0
\(517\) 9.81293 33.4198i 0.431572 1.46980i
\(518\) 0 0
\(519\) −16.6154 36.3826i −0.729335 1.59702i
\(520\) 0 0
\(521\) 7.35414 8.48713i 0.322191 0.371828i −0.571430 0.820651i \(-0.693611\pi\)
0.893621 + 0.448823i \(0.148157\pi\)
\(522\) 0 0
\(523\) −14.3133 + 22.2720i −0.625879 + 0.973886i 0.373059 + 0.927808i \(0.378309\pi\)
−0.998938 + 0.0460786i \(0.985328\pi\)
\(524\) 0 0
\(525\) 4.23960 + 1.70323i 0.185031 + 0.0743352i
\(526\) 0 0
\(527\) 5.34629 + 18.2078i 0.232888 + 0.793145i
\(528\) 0 0
\(529\) 9.62179 + 20.8907i 0.418339 + 0.908291i
\(530\) 0 0
\(531\) −33.8532 + 9.94020i −1.46910 + 0.431368i
\(532\) 0 0
\(533\) 3.57977 + 5.57024i 0.155057 + 0.241274i
\(534\) 0 0
\(535\) −0.446388 0.774248i −0.0192990 0.0334737i
\(536\) 0 0
\(537\) 35.2660 + 30.5582i 1.52184 + 1.31868i
\(538\) 0 0
\(539\) 16.7686 + 36.7181i 0.722274 + 1.58156i
\(540\) 0 0
\(541\) 26.9513 + 7.91361i 1.15873 + 0.340233i 0.803937 0.594714i \(-0.202735\pi\)
0.354788 + 0.934947i \(0.384553\pi\)
\(542\) 0 0
\(543\) 9.68371 + 1.39231i 0.415568 + 0.0597496i
\(544\) 0 0
\(545\) 7.87740 + 10.0294i 0.337431 + 0.429612i
\(546\) 0 0
\(547\) 10.4782 1.50653i 0.448014 0.0644147i 0.0853845 0.996348i \(-0.472788\pi\)
0.362630 + 0.931933i \(0.381879\pi\)
\(548\) 0 0
\(549\) 25.8868 + 29.8749i 1.10482 + 1.27503i
\(550\) 0 0
\(551\) 32.7075 1.39339
\(552\) 0 0
\(553\) 1.98053i 0.0842209i
\(554\) 0 0
\(555\) −36.0934 + 8.73305i −1.53208 + 0.370697i
\(556\) 0 0
\(557\) −17.6790 + 2.54185i −0.749082 + 0.107702i −0.506274 0.862373i \(-0.668978\pi\)
−0.242808 + 0.970074i \(0.578068\pi\)
\(558\) 0 0
\(559\) −3.44121 + 7.53521i −0.145548 + 0.318705i
\(560\) 0 0
\(561\) 8.62544 59.9913i 0.364166 2.53283i
\(562\) 0 0
\(563\) 3.27239 11.1447i 0.137915 0.469695i −0.861351 0.508011i \(-0.830381\pi\)
0.999266 + 0.0383158i \(0.0121993\pi\)
\(564\) 0 0
\(565\) 0.912895 18.9232i 0.0384058 0.796107i
\(566\) 0 0
\(567\) −5.89602 5.10893i −0.247610 0.214555i
\(568\) 0 0
\(569\) −20.4857 13.1654i −0.858806 0.551921i 0.0355039 0.999370i \(-0.488696\pi\)
−0.894310 + 0.447449i \(0.852333\pi\)
\(570\) 0 0
\(571\) 11.6193 7.46725i 0.486251 0.312495i −0.274445 0.961603i \(-0.588494\pi\)
0.760696 + 0.649108i \(0.224858\pi\)
\(572\) 0 0
\(573\) −6.31799 21.5171i −0.263938 0.898890i
\(574\) 0 0
\(575\) 7.85185 + 22.6572i 0.327445 + 0.944870i
\(576\) 0 0
\(577\) 6.57537 + 22.3937i 0.273736 + 0.932260i 0.975528 + 0.219877i \(0.0705657\pi\)
−0.701791 + 0.712383i \(0.747616\pi\)
\(578\) 0 0
\(579\) −39.5354 + 25.4079i −1.64304 + 1.05591i
\(580\) 0 0
\(581\) 4.08792 + 2.62715i 0.169596 + 0.108992i
\(582\) 0 0
\(583\) −7.12407 6.17304i −0.295049 0.255661i
\(584\) 0 0
\(585\) −1.29851 + 26.9166i −0.0536869 + 1.11287i
\(586\) 0 0
\(587\) −4.53359 + 15.4400i −0.187121 + 0.637277i 0.811479 + 0.584382i \(0.198663\pi\)
−0.998600 + 0.0528950i \(0.983155\pi\)
\(588\) 0 0
\(589\) 5.79966 40.3375i 0.238971 1.66208i
\(590\) 0 0
\(591\) −33.1509 + 72.5903i −1.36364 + 2.98597i
\(592\) 0 0
\(593\) 4.66631 0.670914i 0.191622 0.0275511i −0.0458354 0.998949i \(-0.514595\pi\)
0.237458 + 0.971398i \(0.423686\pi\)
\(594\) 0 0
\(595\) 1.91733 0.463911i 0.0786028 0.0190185i
\(596\) 0 0
\(597\) 39.5800i 1.61990i
\(598\) 0 0
\(599\) −16.8596 −0.688865 −0.344432 0.938811i \(-0.611929\pi\)
−0.344432 + 0.938811i \(0.611929\pi\)
\(600\) 0 0
\(601\) 18.8072 + 21.7046i 0.767160 + 0.885349i 0.996113 0.0880880i \(-0.0280757\pi\)
−0.228953 + 0.973437i \(0.573530\pi\)
\(602\) 0 0
\(603\) −56.9238 + 8.18440i −2.31811 + 0.333295i
\(604\) 0 0
\(605\) −31.7705 40.4498i −1.29166 1.64452i
\(606\) 0 0
\(607\) −7.76602 1.11658i −0.315213 0.0453208i −0.0171073 0.999854i \(-0.505446\pi\)
−0.298106 + 0.954533i \(0.596355\pi\)
\(608\) 0 0
\(609\) 4.21552 + 1.23779i 0.170821 + 0.0501577i
\(610\) 0 0
\(611\) 3.85061 + 8.43167i 0.155779 + 0.341109i
\(612\) 0 0
\(613\) −22.1144 19.1623i −0.893193 0.773956i 0.0816938 0.996657i \(-0.473967\pi\)
−0.974887 + 0.222701i \(0.928513\pi\)
\(614\) 0 0
\(615\) −15.6363 27.1207i −0.630516 1.09361i
\(616\) 0 0
\(617\) −3.95727 6.15763i −0.159314 0.247897i 0.752416 0.658689i \(-0.228889\pi\)
−0.911729 + 0.410792i \(0.865252\pi\)
\(618\) 0 0
\(619\) −2.39420 + 0.703001i −0.0962311 + 0.0282560i −0.329494 0.944158i \(-0.606878\pi\)
0.233263 + 0.972414i \(0.425060\pi\)
\(620\) 0 0
\(621\) −0.120612 74.9979i −0.00483998 3.00956i
\(622\) 0 0
\(623\) −1.06166 3.61567i −0.0425344 0.144859i
\(624\) 0 0
\(625\) 5.83519 + 24.3095i 0.233407 + 0.972379i
\(626\) 0 0
\(627\) −70.3682 + 109.495i −2.81023 + 4.37281i
\(628\) 0 0
\(629\) −10.4994 + 12.1170i −0.418640 + 0.483136i
\(630\) 0 0
\(631\) 13.8649 + 30.3599i 0.551954 + 1.20861i 0.955863 + 0.293811i \(0.0949238\pi\)
−0.403910 + 0.914799i \(0.632349\pi\)
\(632\) 0 0
\(633\) 7.27428 24.7739i 0.289127 0.984675i
\(634\) 0 0
\(635\) 20.5735 28.8545i 0.816436 1.14506i
\(636\) 0 0
\(637\) −9.77161 4.46254i −0.387165 0.176812i
\(638\) 0 0
\(639\) −5.39883 37.5497i −0.213574 1.48544i
\(640\) 0 0
\(641\) −7.92266 9.14324i −0.312926 0.361136i 0.577398 0.816463i \(-0.304068\pi\)
−0.890325 + 0.455326i \(0.849523\pi\)
\(642\) 0 0
\(643\) 21.0550i 0.830329i 0.909746 + 0.415164i \(0.136276\pi\)
−0.909746 + 0.415164i \(0.863724\pi\)
\(644\) 0 0
\(645\) 17.9676 34.8006i 0.707474 1.37027i
\(646\) 0 0
\(647\) 17.4257 15.0995i 0.685076 0.593622i −0.241196 0.970476i \(-0.577540\pi\)
0.926272 + 0.376854i \(0.122994\pi\)
\(648\) 0 0
\(649\) 3.77019 + 26.2223i 0.147993 + 1.02931i
\(650\) 0 0
\(651\) 2.27403 4.97942i 0.0891261 0.195159i
\(652\) 0 0
\(653\) 0.810443 + 0.116524i 0.0317151 + 0.00455994i 0.158154 0.987414i \(-0.449446\pi\)
−0.126439 + 0.991974i \(0.540355\pi\)
\(654\) 0 0
\(655\) −2.42726 + 2.54255i −0.0948408 + 0.0993458i
\(656\) 0 0
\(657\) −8.98416 + 4.10293i −0.350505 + 0.160070i
\(658\) 0 0
\(659\) 30.4079 35.0926i 1.18452 1.36701i 0.269810 0.962914i \(-0.413039\pi\)
0.914715 0.404101i \(-0.132415\pi\)
\(660\) 0 0
\(661\) −2.98598 1.91897i −0.116141 0.0746393i 0.481281 0.876567i \(-0.340172\pi\)
−0.597422 + 0.801927i \(0.703808\pi\)
\(662\) 0 0
\(663\) 8.72019 + 13.5689i 0.338664 + 0.526972i
\(664\) 0 0
\(665\) −4.15926 0.804241i −0.161289 0.0311871i
\(666\) 0 0
\(667\) 9.61248 + 20.9591i 0.372197 + 0.811541i
\(668\) 0 0
\(669\) −16.8190 + 4.93850i −0.650260 + 0.190934i
\(670\) 0 0
\(671\) 24.9696 16.0470i 0.963940 0.619487i
\(672\) 0 0
\(673\) 1.79496 2.79302i 0.0691908 0.107663i −0.804934 0.593364i \(-0.797799\pi\)
0.874125 + 0.485701i \(0.161436\pi\)
\(674\) 0 0
\(675\) 7.52664 77.8277i 0.289701 2.99559i
\(676\) 0 0
\(677\) 33.5606 15.3266i 1.28984 0.589049i 0.351959 0.936015i \(-0.385515\pi\)
0.937878 + 0.346966i \(0.112788\pi\)
\(678\) 0 0
\(679\) 0.891672 + 0.261819i 0.0342192 + 0.0100477i
\(680\) 0 0
\(681\) 6.62044 46.0462i 0.253696 1.76449i
\(682\) 0 0
\(683\) 31.3900 + 14.3353i 1.20110 + 0.548526i 0.912561 0.408941i \(-0.134102\pi\)
0.288544 + 0.957467i \(0.406829\pi\)
\(684\) 0 0
\(685\) 4.96514 14.3178i 0.189708 0.547056i
\(686\) 0 0
\(687\) 8.96814 7.77094i 0.342156 0.296480i
\(688\) 0 0
\(689\) 2.50863 0.0955711
\(690\) 0 0
\(691\) −50.4793 −1.92032 −0.960161 0.279448i \(-0.909849\pi\)
−0.960161 + 0.279448i \(0.909849\pi\)
\(692\) 0 0
\(693\) −9.53126 + 8.25888i −0.362062 + 0.313729i
\(694\) 0 0
\(695\) −27.9228 9.68309i −1.05917 0.367301i
\(696\) 0 0
\(697\) −12.2947 5.61482i −0.465697 0.212677i
\(698\) 0 0
\(699\) −7.75986 + 53.9710i −0.293505 + 2.04137i
\(700\) 0 0
\(701\) 23.1195 + 6.78850i 0.873212 + 0.256398i 0.687481 0.726202i \(-0.258716\pi\)
0.185731 + 0.982601i \(0.440535\pi\)
\(702\) 0 0
\(703\) 31.3198 14.3033i 1.18125 0.539458i
\(704\) 0 0
\(705\) −16.2634 40.6954i −0.612516 1.53268i
\(706\) 0 0
\(707\) −1.06652 + 1.65954i −0.0401107 + 0.0624134i
\(708\) 0 0
\(709\) −31.6380 + 20.3325i −1.18819 + 0.763602i −0.976874 0.213816i \(-0.931411\pi\)
−0.211315 + 0.977418i \(0.567774\pi\)
\(710\) 0 0
\(711\) 52.9915 15.5597i 1.98734 0.583534i
\(712\) 0 0
\(713\) 27.5529 8.13842i 1.03187 0.304786i
\(714\) 0 0
\(715\) 19.8660 + 3.84132i 0.742946 + 0.143657i
\(716\) 0 0
\(717\) −35.1253 54.6560i −1.31178 2.04117i
\(718\) 0 0
\(719\) 28.7807 + 18.4962i 1.07334 + 0.689792i 0.953009 0.302942i \(-0.0979688\pi\)
0.120329 + 0.992734i \(0.461605\pi\)
\(720\) 0 0
\(721\) −2.45430 + 2.83242i −0.0914030 + 0.105485i
\(722\) 0 0
\(723\) 6.65070 3.03727i 0.247342 0.112957i
\(724\) 0 0
\(725\) 7.83521 + 22.7273i 0.290992 + 0.844069i
\(726\) 0 0
\(727\) −22.1962 3.19134i −0.823213 0.118360i −0.282180 0.959362i \(-0.591058\pi\)
−0.541033 + 0.841001i \(0.681967\pi\)
\(728\) 0 0
\(729\) −26.4279 + 57.8689i −0.978809 + 2.14329i
\(730\) 0 0
\(731\) −2.40650 16.7376i −0.0890078 0.619063i
\(732\) 0 0
\(733\) −27.4605 + 23.7947i −1.01428 + 0.878877i −0.992666 0.120890i \(-0.961425\pi\)
−0.0216118 + 0.999766i \(0.506880\pi\)
\(734\) 0 0
\(735\) 45.1292 + 23.3003i 1.66462 + 0.859445i
\(736\) 0 0
\(737\) 43.1809i 1.59059i
\(738\) 0 0
\(739\) −19.2182 22.1790i −0.706952 0.815866i 0.282722 0.959202i \(-0.408763\pi\)
−0.989674 + 0.143336i \(0.954217\pi\)
\(740\) 0 0
\(741\) −4.92951 34.2855i −0.181090 1.25951i
\(742\) 0 0
\(743\) 15.4257 + 7.04470i 0.565915 + 0.258445i 0.677763 0.735280i \(-0.262950\pi\)
−0.111848 + 0.993725i \(0.535677\pi\)
\(744\) 0 0
\(745\) 29.7625 + 21.2209i 1.09041 + 0.777474i
\(746\) 0 0
\(747\) 38.1764 130.017i 1.39680 4.75707i
\(748\) 0 0
\(749\) 0.0462395 + 0.101250i 0.00168955 + 0.00369961i
\(750\) 0 0
\(751\) 24.1142 27.8293i 0.879940 1.01550i −0.119802 0.992798i \(-0.538226\pi\)
0.999742 0.0227070i \(-0.00722848\pi\)
\(752\) 0 0
\(753\) 23.9913 37.3312i 0.874293 1.36043i
\(754\) 0 0
\(755\) −1.51748 15.9940i −0.0552267 0.582080i
\(756\) 0 0
\(757\) 12.0234 + 40.9480i 0.436998 + 1.48828i 0.824208 + 0.566287i \(0.191621\pi\)
−0.387210 + 0.921992i \(0.626561\pi\)
\(758\) 0 0
\(759\) −90.8455 12.9125i −3.29748 0.468695i
\(760\) 0 0
\(761\) 8.47008 2.48704i 0.307040 0.0901551i −0.124583 0.992209i \(-0.539759\pi\)
0.431623 + 0.902054i \(0.357941\pi\)
\(762\) 0 0
\(763\) −0.858733 1.33622i −0.0310882 0.0483742i
\(764\) 0 0
\(765\) −27.4756 47.6557i −0.993383 1.72300i
\(766\) 0 0
\(767\) −5.32816 4.61688i −0.192389 0.166706i
\(768\) 0 0
\(769\) −2.97806 6.52104i −0.107392 0.235155i 0.848305 0.529507i \(-0.177623\pi\)
−0.955697 + 0.294353i \(0.904896\pi\)
\(770\) 0 0
\(771\) 2.56507 + 0.753171i 0.0923786 + 0.0271248i
\(772\) 0 0
\(773\) 16.2283 + 2.33328i 0.583691 + 0.0839221i 0.427834 0.903857i \(-0.359277\pi\)
0.155857 + 0.987780i \(0.450186\pi\)
\(774\) 0 0
\(775\) 29.4184 5.63303i 1.05674 0.202344i
\(776\) 0 0
\(777\) 4.57795 0.658210i 0.164233 0.0236132i
\(778\) 0 0
\(779\) 19.0081 + 21.9365i 0.681036 + 0.785958i
\(780\) 0 0
\(781\) −28.4842 −1.01925
\(782\) 0 0
\(783\) 75.1881i 2.68700i
\(784\) 0 0
\(785\) 4.48294 + 18.5278i 0.160003 + 0.661287i
\(786\) 0 0
\(787\) 37.2713 5.35880i 1.32858 0.191021i 0.558784 0.829314i \(-0.311268\pi\)
0.769794 + 0.638293i \(0.220359\pi\)
\(788\) 0 0
\(789\) 28.7980 63.0587i 1.02523 2.24495i
\(790\) 0 0
\(791\) −0.335802 + 2.33555i −0.0119397 + 0.0830427i
\(792\) 0 0
\(793\) −2.22540 + 7.57901i −0.0790262 + 0.269139i
\(794\) 0 0
\(795\) −11.8469 0.571516i −0.420165 0.0202696i
\(796\) 0 0
\(797\) −3.52833 3.05731i −0.124980 0.108296i 0.590128 0.807309i \(-0.299077\pi\)
−0.715108 + 0.699014i \(0.753623\pi\)
\(798\) 0 0
\(799\) −15.9178 10.2297i −0.563130 0.361901i
\(800\) 0 0
\(801\) −88.4008 + 56.8118i −3.12349 + 2.00735i
\(802\) 0 0
\(803\) 2.08932 + 7.11556i 0.0737303 + 0.251103i
\(804\) 0 0
\(805\) −0.707013 2.90163i −0.0249189 0.102269i
\(806\) 0 0
\(807\) −13.7432 46.8051i −0.483784 1.64762i
\(808\) 0 0
\(809\) −3.82825 + 2.46026i −0.134594 + 0.0864983i −0.606205 0.795308i \(-0.707309\pi\)
0.471611 + 0.881807i \(0.343673\pi\)
\(810\) 0 0
\(811\) −33.4528 21.4988i −1.17469 0.754926i −0.200286 0.979737i \(-0.564187\pi\)
−0.974402 + 0.224811i \(0.927823\pi\)
\(812\) 0 0
\(813\) 78.5162 + 68.0347i 2.75368 + 2.38608i
\(814\) 0 0
\(815\) −1.85032 + 38.3550i −0.0648140 + 1.34352i
\(816\) 0 0
\(817\) −10.2308 + 34.8429i −0.357930 + 1.21900i
\(818\) 0 0
\(819\) 0.477649 3.32212i 0.0166904 0.116084i
\(820\) 0 0
\(821\) 17.2830 37.8445i 0.603181 1.32078i −0.323961 0.946071i \(-0.605015\pi\)
0.927142 0.374711i \(-0.122258\pi\)
\(822\) 0 0
\(823\) −52.0811 + 7.48813i −1.81543 + 0.261020i −0.964468 0.264201i \(-0.914892\pi\)
−0.850966 + 0.525221i \(0.823983\pi\)
\(824\) 0 0
\(825\) −92.9410 22.6663i −3.23579 0.789139i
\(826\) 0 0
\(827\) 51.8808i 1.80407i −0.431659 0.902037i \(-0.642072\pi\)
0.431659 0.902037i \(-0.357928\pi\)
\(828\) 0 0
\(829\) −18.2043 −0.632260 −0.316130 0.948716i \(-0.602384\pi\)
−0.316130 + 0.948716i \(0.602384\pi\)
\(830\) 0 0
\(831\) −2.16078 2.49368i −0.0749568 0.0865047i
\(832\) 0 0
\(833\) 21.7052 3.12074i 0.752041 0.108127i
\(834\) 0 0
\(835\) −19.8843 + 15.6178i −0.688126 + 0.540475i
\(836\) 0 0
\(837\) −92.7279 13.3323i −3.20514 0.460830i
\(838\) 0 0
\(839\) 32.2079 + 9.45708i 1.11194 + 0.326495i 0.785585 0.618753i \(-0.212362\pi\)
0.326353 + 0.945248i \(0.394180\pi\)
\(840\) 0 0
\(841\) −2.44399 5.35160i −0.0842756 0.184538i
\(842\) 0 0
\(843\) 33.2562 + 28.8167i 1.14540 + 0.992499i
\(844\) 0 0
\(845\) 20.5182 11.8297i 0.705848 0.406952i
\(846\) 0 0
\(847\) 3.46338 + 5.38912i 0.119003 + 0.185172i
\(848\) 0 0
\(849\) −28.1674 + 8.27071i −0.966704 + 0.283850i
\(850\) 0 0
\(851\) 18.3702 + 15.8663i 0.629724 + 0.543888i
\(852\) 0 0
\(853\) 9.07323 + 30.9006i 0.310662 + 1.05802i 0.955816 + 0.293964i \(0.0949747\pi\)
−0.645155 + 0.764052i \(0.723207\pi\)
\(854\) 0 0
\(855\) 11.1581 + 117.604i 0.381598 + 4.02198i
\(856\) 0 0
\(857\) −5.09449 + 7.92718i −0.174024 + 0.270787i −0.917299 0.398199i \(-0.869635\pi\)
0.743275 + 0.668987i \(0.233272\pi\)
\(858\) 0 0
\(859\) −11.0252 + 12.7238i −0.376176 + 0.434130i −0.911994 0.410203i \(-0.865458\pi\)
0.535818 + 0.844333i \(0.320003\pi\)
\(860\) 0 0
\(861\) 1.61970 + 3.54664i 0.0551991 + 0.120869i
\(862\) 0 0
\(863\) 14.2874 48.6585i 0.486349 1.65635i −0.241336 0.970442i \(-0.577586\pi\)
0.727685 0.685911i \(-0.240596\pi\)
\(864\) 0 0
\(865\) −15.8243 + 22.1937i −0.538043 + 0.754609i
\(866\) 0 0
\(867\) 20.7896 + 9.49427i 0.706050 + 0.322442i
\(868\) 0 0
\(869\) −5.90160 41.0465i −0.200198 1.39241i
\(870\) 0 0
\(871\) −7.52536 8.68473i −0.254987 0.294271i
\(872\) 0 0
\(873\) 25.9147i 0.877078i
\(874\) 0 0
\(875\) −0.437529 3.08278i −0.0147912 0.104217i
\(876\) 0 0
\(877\) 16.5117 14.3075i 0.557562 0.483130i −0.329897 0.944017i \(-0.607014\pi\)
0.887459 + 0.460887i \(0.152469\pi\)
\(878\) 0 0
\(879\) 6.35327 + 44.1880i 0.214291 + 1.49042i
\(880\) 0 0
\(881\) 1.10516 2.41997i 0.0372338 0.0815307i −0.890101 0.455763i \(-0.849366\pi\)
0.927335 + 0.374232i \(0.122094\pi\)
\(882\) 0 0
\(883\) −48.8626 7.02539i −1.64436 0.236423i −0.742936 0.669362i \(-0.766567\pi\)
−0.901423 + 0.432939i \(0.857476\pi\)
\(884\) 0 0
\(885\) 24.1101 + 23.0168i 0.810454 + 0.773702i
\(886\) 0 0
\(887\) −49.8558 + 22.7684i −1.67399 + 0.764488i −0.674334 + 0.738426i \(0.735569\pi\)
−0.999660 + 0.0260611i \(0.991704\pi\)
\(888\) 0 0
\(889\) −2.89036 + 3.33565i −0.0969394 + 0.111874i
\(890\) 0 0
\(891\) 137.419 + 88.3136i 4.60370 + 2.95862i
\(892\) 0 0
\(893\) 21.9685 + 34.1836i 0.735147 + 1.14391i
\(894\) 0 0
\(895\) 6.03719 31.2223i 0.201801 1.04365i
\(896\) 0 0
\(897\) 20.5216 13.2351i 0.685195 0.441907i
\(898\) 0 0
\(899\) 27.6358 8.11462i 0.921707 0.270638i
\(900\) 0 0
\(901\) −4.30795 + 2.76855i −0.143519 + 0.0922337i
\(902\) 0 0
\(903\) −2.63720 + 4.10356i −0.0877605 + 0.136558i
\(904\) 0 0
\(905\) −2.47420 6.19109i −0.0822450 0.205799i
\(906\) 0 0
\(907\) 49.4265 22.5723i 1.64118 0.749502i 0.641338 0.767258i \(-0.278380\pi\)
0.999842 + 0.0177566i \(0.00565239\pi\)
\(908\) 0 0
\(909\) 52.7818 + 15.4981i 1.75066 + 0.514041i
\(910\) 0 0
\(911\) −1.26965 + 8.83058i −0.0420652 + 0.292570i 0.957919 + 0.287037i \(0.0926703\pi\)
−0.999985 + 0.00553302i \(0.998239\pi\)
\(912\) 0 0
\(913\) −92.5505 42.2664i −3.06297 1.39881i
\(914\) 0 0
\(915\) 12.2360 35.2845i 0.404509 1.16647i
\(916\) 0 0
\(917\) 0.330866 0.286697i 0.0109262 0.00946758i
\(918\) 0 0
\(919\) 27.4603 0.905831 0.452915 0.891554i \(-0.350384\pi\)
0.452915 + 0.891554i \(0.350384\pi\)
\(920\) 0 0
\(921\) 100.180 3.30103
\(922\) 0 0
\(923\) 5.72886 4.96409i 0.188568 0.163395i
\(924\) 0 0
\(925\) 17.4416 + 18.3366i 0.573476 + 0.602903i
\(926\) 0 0
\(927\) 95.0664 + 43.4154i 3.12239 + 1.42595i
\(928\) 0 0
\(929\) −4.13309 + 28.7463i −0.135602 + 0.943135i 0.802469 + 0.596694i \(0.203519\pi\)
−0.938072 + 0.346442i \(0.887390\pi\)
\(930\) 0 0
\(931\) −45.1841 13.2672i −1.48085 0.434816i
\(932\) 0 0
\(933\) −84.8206 + 38.7363i −2.77690 + 1.26817i
\(934\) 0 0
\(935\) −38.3542 + 15.3278i −1.25432 + 0.501273i
\(936\) 0 0
\(937\) −20.7639 + 32.3093i −0.678327 + 1.05550i 0.315961 + 0.948772i \(0.397673\pi\)
−0.994289 + 0.106726i \(0.965963\pi\)
\(938\) 0 0
\(939\) −19.2462 + 12.3688i −0.628077 + 0.403640i
\(940\) 0 0
\(941\) 24.6530 7.23878i 0.803666 0.235978i 0.145997 0.989285i \(-0.453361\pi\)
0.657669 + 0.753307i \(0.271543\pi\)
\(942\) 0 0
\(943\) −8.47070 + 18.6275i −0.275844 + 0.606593i
\(944\) 0 0
\(945\) −1.84879 + 9.56131i −0.0601411 + 0.311030i
\(946\) 0 0
\(947\) −11.6195 18.0803i −0.377583 0.587530i 0.599504 0.800372i \(-0.295365\pi\)
−0.977087 + 0.212842i \(0.931728\pi\)
\(948\) 0 0
\(949\) −1.66028 1.06699i −0.0538948 0.0346361i
\(950\) 0 0
\(951\) −56.7522 + 65.4955i −1.84032 + 2.12384i
\(952\) 0 0
\(953\) 3.41027 1.55742i 0.110470 0.0504498i −0.359414 0.933178i \(-0.617023\pi\)
0.469883 + 0.882729i \(0.344296\pi\)
\(954\) 0 0
\(955\) −10.5529 + 11.0542i −0.341485 + 0.357705i
\(956\) 0 0
\(957\) −91.0549 13.0917i −2.94339 0.423195i
\(958\) 0 0
\(959\) −0.784062 + 1.71686i −0.0253187 + 0.0554402i
\(960\) 0 0
\(961\) −0.695472 4.83711i −0.0224346 0.156036i
\(962\) 0 0
\(963\) 2.34580 2.03265i 0.0755923 0.0655011i
\(964\) 0 0
\(965\) 28.4579 + 14.6928i 0.916091 + 0.472979i
\(966\) 0 0
\(967\) 0.0817874i 0.00263010i 0.999999 + 0.00131505i \(0.000418594\pi\)
−0.999999 + 0.00131505i \(0.999581\pi\)
\(968\) 0 0
\(969\) 46.3030 + 53.4366i 1.48747 + 1.71663i
\(970\) 0 0
\(971\) 1.43638 + 9.99021i 0.0460955 + 0.320601i 0.999803 + 0.0198644i \(0.00632346\pi\)
−0.953707 + 0.300737i \(0.902767\pi\)
\(972\) 0 0
\(973\) 3.34824 + 1.52909i 0.107340 + 0.0490204i
\(974\) 0 0
\(975\) 22.6428 11.6386i 0.725151 0.372732i
\(976\) 0 0
\(977\) 3.07577 10.4751i 0.0984026 0.335129i −0.895547 0.444967i \(-0.853215\pi\)
0.993949 + 0.109839i \(0.0350335\pi\)
\(978\) 0 0
\(979\) 32.7768 + 71.7712i 1.04755 + 2.29382i
\(980\) 0 0
\(981\) −29.0055 + 33.4741i −0.926075 + 1.06875i
\(982\) 0 0
\(983\) 13.1835 20.5140i 0.420490 0.654295i −0.564792 0.825233i \(-0.691044\pi\)
0.985282 + 0.170939i \(0.0546800\pi\)
\(984\) 0 0
\(985\) 54.1407 5.13677i 1.72507 0.163671i
\(986\) 0 0
\(987\) 1.53776 + 5.23714i 0.0489475 + 0.166700i
\(988\) 0 0
\(989\) −25.3342 + 3.68411i −0.805582 + 0.117148i
\(990\) 0 0
\(991\) −35.4252 + 10.4018i −1.12532 + 0.330424i −0.790866 0.611989i \(-0.790370\pi\)
−0.334453 + 0.942412i \(0.608552\pi\)
\(992\) 0 0
\(993\) 26.8513 + 41.7814i 0.852099 + 1.32589i
\(994\) 0 0
\(995\) 23.3676 13.4725i 0.740804 0.427106i
\(996\) 0 0
\(997\) 21.3232 + 18.4766i 0.675312 + 0.585161i 0.923522 0.383545i \(-0.125297\pi\)
−0.248210 + 0.968706i \(0.579842\pi\)
\(998\) 0 0
\(999\) −32.8804 71.9980i −1.04029 2.27791i
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 460.2.s.a.9.12 yes 120
5.4 even 2 inner 460.2.s.a.9.1 120
23.18 even 11 inner 460.2.s.a.409.1 yes 120
115.64 even 22 inner 460.2.s.a.409.12 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
460.2.s.a.9.1 120 5.4 even 2 inner
460.2.s.a.9.12 yes 120 1.1 even 1 trivial
460.2.s.a.409.1 yes 120 23.18 even 11 inner
460.2.s.a.409.12 yes 120 115.64 even 22 inner