Properties

Label 624.4.q.c.529.1
Level $624$
Weight $4$
Character 624.529
Analytic conductor $36.817$
Analytic rank $0$
Dimension $2$
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] = [624,4,Mod(289,624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(624, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("624.289");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 624.q (of order \(3\), 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: \(36.8171918436\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 529.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 624.529
Dual form 624.4.q.c.289.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.50000 - 2.59808i) q^{3} +7.00000 q^{5} +(-5.00000 - 8.66025i) q^{7} +(-4.50000 - 7.79423i) q^{9} +O(q^{10})\) \(q+(1.50000 - 2.59808i) q^{3} +7.00000 q^{5} +(-5.00000 - 8.66025i) q^{7} +(-4.50000 - 7.79423i) q^{9} +(-11.0000 + 19.0526i) q^{11} +(-45.5000 - 11.2583i) q^{13} +(10.5000 - 18.1865i) q^{15} +(-18.5000 - 32.0429i) q^{17} +(15.0000 + 25.9808i) q^{19} -30.0000 q^{21} +(-81.0000 + 140.296i) q^{23} -76.0000 q^{25} -27.0000 q^{27} +(56.5000 - 97.8609i) q^{29} -196.000 q^{31} +(33.0000 + 57.1577i) q^{33} +(-35.0000 - 60.6218i) q^{35} +(-6.50000 + 11.2583i) q^{37} +(-97.5000 + 101.325i) q^{39} +(-142.500 + 246.817i) q^{41} +(-123.000 - 213.042i) q^{43} +(-31.5000 - 54.5596i) q^{45} +462.000 q^{47} +(121.500 - 210.444i) q^{49} -111.000 q^{51} -537.000 q^{53} +(-77.0000 + 133.368i) q^{55} +90.0000 q^{57} +(288.000 + 498.831i) q^{59} +(317.500 + 549.926i) q^{61} +(-45.0000 + 77.9423i) q^{63} +(-318.500 - 78.8083i) q^{65} +(101.000 - 174.937i) q^{67} +(243.000 + 420.888i) q^{69} +(-543.000 - 940.504i) q^{71} -805.000 q^{73} +(-114.000 + 197.454i) q^{75} +220.000 q^{77} -884.000 q^{79} +(-40.5000 + 70.1481i) q^{81} -518.000 q^{83} +(-129.500 - 224.301i) q^{85} +(-169.500 - 293.583i) q^{87} +(-97.0000 + 168.009i) q^{89} +(130.000 + 450.333i) q^{91} +(-294.000 + 509.223i) q^{93} +(105.000 + 181.865i) q^{95} +(601.000 + 1040.96i) q^{97} +198.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{3} + 14 q^{5} - 10 q^{7} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{3} + 14 q^{5} - 10 q^{7} - 9 q^{9} - 22 q^{11} - 91 q^{13} + 21 q^{15} - 37 q^{17} + 30 q^{19} - 60 q^{21} - 162 q^{23} - 152 q^{25} - 54 q^{27} + 113 q^{29} - 392 q^{31} + 66 q^{33} - 70 q^{35} - 13 q^{37} - 195 q^{39} - 285 q^{41} - 246 q^{43} - 63 q^{45} + 924 q^{47} + 243 q^{49} - 222 q^{51} - 1074 q^{53} - 154 q^{55} + 180 q^{57} + 576 q^{59} + 635 q^{61} - 90 q^{63} - 637 q^{65} + 202 q^{67} + 486 q^{69} - 1086 q^{71} - 1610 q^{73} - 228 q^{75} + 440 q^{77} - 1768 q^{79} - 81 q^{81} - 1036 q^{83} - 259 q^{85} - 339 q^{87} - 194 q^{89} + 260 q^{91} - 588 q^{93} + 210 q^{95} + 1202 q^{97} + 396 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/624\mathbb{Z}\right)^\times\).

\(n\) \(79\) \(145\) \(209\) \(469\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.50000 2.59808i 0.288675 0.500000i
\(4\) 0 0
\(5\) 7.00000 0.626099 0.313050 0.949737i \(-0.398649\pi\)
0.313050 + 0.949737i \(0.398649\pi\)
\(6\) 0 0
\(7\) −5.00000 8.66025i −0.269975 0.467610i 0.698880 0.715239i \(-0.253682\pi\)
−0.968855 + 0.247629i \(0.920349\pi\)
\(8\) 0 0
\(9\) −4.50000 7.79423i −0.166667 0.288675i
\(10\) 0 0
\(11\) −11.0000 + 19.0526i −0.301511 + 0.522233i −0.976478 0.215615i \(-0.930824\pi\)
0.674967 + 0.737848i \(0.264158\pi\)
\(12\) 0 0
\(13\) −45.5000 11.2583i −0.970725 0.240192i
\(14\) 0 0
\(15\) 10.5000 18.1865i 0.180739 0.313050i
\(16\) 0 0
\(17\) −18.5000 32.0429i −0.263936 0.457150i 0.703348 0.710845i \(-0.251687\pi\)
−0.967284 + 0.253695i \(0.918354\pi\)
\(18\) 0 0
\(19\) 15.0000 + 25.9808i 0.181118 + 0.313705i 0.942261 0.334878i \(-0.108695\pi\)
−0.761144 + 0.648583i \(0.775362\pi\)
\(20\) 0 0
\(21\) −30.0000 −0.311740
\(22\) 0 0
\(23\) −81.0000 + 140.296i −0.734333 + 1.27190i 0.220682 + 0.975346i \(0.429172\pi\)
−0.955015 + 0.296557i \(0.904162\pi\)
\(24\) 0 0
\(25\) −76.0000 −0.608000
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 56.5000 97.8609i 0.361786 0.626631i −0.626469 0.779446i \(-0.715501\pi\)
0.988255 + 0.152815i \(0.0488339\pi\)
\(30\) 0 0
\(31\) −196.000 −1.13557 −0.567785 0.823177i \(-0.692199\pi\)
−0.567785 + 0.823177i \(0.692199\pi\)
\(32\) 0 0
\(33\) 33.0000 + 57.1577i 0.174078 + 0.301511i
\(34\) 0 0
\(35\) −35.0000 60.6218i −0.169031 0.292770i
\(36\) 0 0
\(37\) −6.50000 + 11.2583i −0.0288809 + 0.0500232i −0.880105 0.474780i \(-0.842528\pi\)
0.851224 + 0.524803i \(0.175861\pi\)
\(38\) 0 0
\(39\) −97.5000 + 101.325i −0.400320 + 0.416025i
\(40\) 0 0
\(41\) −142.500 + 246.817i −0.542799 + 0.940156i 0.455943 + 0.890009i \(0.349302\pi\)
−0.998742 + 0.0501465i \(0.984031\pi\)
\(42\) 0 0
\(43\) −123.000 213.042i −0.436217 0.755550i 0.561177 0.827696i \(-0.310349\pi\)
−0.997394 + 0.0721459i \(0.977015\pi\)
\(44\) 0 0
\(45\) −31.5000 54.5596i −0.104350 0.180739i
\(46\) 0 0
\(47\) 462.000 1.43382 0.716911 0.697165i \(-0.245555\pi\)
0.716911 + 0.697165i \(0.245555\pi\)
\(48\) 0 0
\(49\) 121.500 210.444i 0.354227 0.613540i
\(50\) 0 0
\(51\) −111.000 −0.304767
\(52\) 0 0
\(53\) −537.000 −1.39175 −0.695874 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(54\) 0 0
\(55\) −77.0000 + 133.368i −0.188776 + 0.326970i
\(56\) 0 0
\(57\) 90.0000 0.209137
\(58\) 0 0
\(59\) 288.000 + 498.831i 0.635498 + 1.10072i 0.986409 + 0.164307i \(0.0525387\pi\)
−0.350911 + 0.936409i \(0.614128\pi\)
\(60\) 0 0
\(61\) 317.500 + 549.926i 0.666421 + 1.15428i 0.978898 + 0.204350i \(0.0655082\pi\)
−0.312476 + 0.949926i \(0.601159\pi\)
\(62\) 0 0
\(63\) −45.0000 + 77.9423i −0.0899915 + 0.155870i
\(64\) 0 0
\(65\) −318.500 78.8083i −0.607770 0.150384i
\(66\) 0 0
\(67\) 101.000 174.937i 0.184166 0.318985i −0.759129 0.650940i \(-0.774375\pi\)
0.943295 + 0.331955i \(0.107708\pi\)
\(68\) 0 0
\(69\) 243.000 + 420.888i 0.423968 + 0.734333i
\(70\) 0 0
\(71\) −543.000 940.504i −0.907637 1.57207i −0.817338 0.576159i \(-0.804551\pi\)
−0.0902997 0.995915i \(-0.528783\pi\)
\(72\) 0 0
\(73\) −805.000 −1.29066 −0.645330 0.763904i \(-0.723280\pi\)
−0.645330 + 0.763904i \(0.723280\pi\)
\(74\) 0 0
\(75\) −114.000 + 197.454i −0.175514 + 0.304000i
\(76\) 0 0
\(77\) 220.000 0.325602
\(78\) 0 0
\(79\) −884.000 −1.25896 −0.629480 0.777017i \(-0.716732\pi\)
−0.629480 + 0.777017i \(0.716732\pi\)
\(80\) 0 0
\(81\) −40.5000 + 70.1481i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) −518.000 −0.685035 −0.342517 0.939511i \(-0.611280\pi\)
−0.342517 + 0.939511i \(0.611280\pi\)
\(84\) 0 0
\(85\) −129.500 224.301i −0.165250 0.286221i
\(86\) 0 0
\(87\) −169.500 293.583i −0.208877 0.361786i
\(88\) 0 0
\(89\) −97.0000 + 168.009i −0.115528 + 0.200100i −0.917991 0.396602i \(-0.870189\pi\)
0.802463 + 0.596702i \(0.203523\pi\)
\(90\) 0 0
\(91\) 130.000 + 450.333i 0.149755 + 0.518766i
\(92\) 0 0
\(93\) −294.000 + 509.223i −0.327811 + 0.567785i
\(94\) 0 0
\(95\) 105.000 + 181.865i 0.113398 + 0.196410i
\(96\) 0 0
\(97\) 601.000 + 1040.96i 0.629096 + 1.08963i 0.987733 + 0.156149i \(0.0499081\pi\)
−0.358638 + 0.933477i \(0.616759\pi\)
\(98\) 0 0
\(99\) 198.000 0.201008
\(100\) 0 0
\(101\) 214.500 371.525i 0.211322 0.366021i −0.740806 0.671719i \(-0.765556\pi\)
0.952129 + 0.305698i \(0.0988897\pi\)
\(102\) 0 0
\(103\) 1302.00 1.24553 0.622766 0.782408i \(-0.286009\pi\)
0.622766 + 0.782408i \(0.286009\pi\)
\(104\) 0 0
\(105\) −210.000 −0.195180
\(106\) 0 0
\(107\) −669.000 + 1158.74i −0.604436 + 1.04691i 0.387704 + 0.921784i \(0.373268\pi\)
−0.992140 + 0.125130i \(0.960065\pi\)
\(108\) 0 0
\(109\) −1034.00 −0.908617 −0.454308 0.890844i \(-0.650114\pi\)
−0.454308 + 0.890844i \(0.650114\pi\)
\(110\) 0 0
\(111\) 19.5000 + 33.7750i 0.0166744 + 0.0288809i
\(112\) 0 0
\(113\) −538.500 932.709i −0.448299 0.776477i 0.549976 0.835180i \(-0.314637\pi\)
−0.998275 + 0.0587032i \(0.981303\pi\)
\(114\) 0 0
\(115\) −567.000 + 982.073i −0.459765 + 0.796337i
\(116\) 0 0
\(117\) 117.000 + 405.300i 0.0924500 + 0.320256i
\(118\) 0 0
\(119\) −185.000 + 320.429i −0.142512 + 0.246838i
\(120\) 0 0
\(121\) 423.500 + 733.524i 0.318182 + 0.551107i
\(122\) 0 0
\(123\) 427.500 + 740.452i 0.313385 + 0.542799i
\(124\) 0 0
\(125\) −1407.00 −1.00677
\(126\) 0 0
\(127\) −494.000 + 855.633i −0.345161 + 0.597836i −0.985383 0.170354i \(-0.945509\pi\)
0.640222 + 0.768190i \(0.278842\pi\)
\(128\) 0 0
\(129\) −738.000 −0.503700
\(130\) 0 0
\(131\) −560.000 −0.373492 −0.186746 0.982408i \(-0.559794\pi\)
−0.186746 + 0.982408i \(0.559794\pi\)
\(132\) 0 0
\(133\) 150.000 259.808i 0.0977944 0.169385i
\(134\) 0 0
\(135\) −189.000 −0.120493
\(136\) 0 0
\(137\) 259.500 + 449.467i 0.161829 + 0.280296i 0.935525 0.353261i \(-0.114927\pi\)
−0.773696 + 0.633557i \(0.781594\pi\)
\(138\) 0 0
\(139\) −174.000 301.377i −0.106176 0.183903i 0.808042 0.589125i \(-0.200527\pi\)
−0.914218 + 0.405222i \(0.867194\pi\)
\(140\) 0 0
\(141\) 693.000 1200.31i 0.413909 0.716911i
\(142\) 0 0
\(143\) 715.000 743.050i 0.418121 0.434524i
\(144\) 0 0
\(145\) 395.500 685.026i 0.226514 0.392333i
\(146\) 0 0
\(147\) −364.500 631.333i −0.204513 0.354227i
\(148\) 0 0
\(149\) 322.500 + 558.586i 0.177317 + 0.307122i 0.940961 0.338516i \(-0.109925\pi\)
−0.763644 + 0.645638i \(0.776592\pi\)
\(150\) 0 0
\(151\) −2914.00 −1.57045 −0.785225 0.619211i \(-0.787453\pi\)
−0.785225 + 0.619211i \(0.787453\pi\)
\(152\) 0 0
\(153\) −166.500 + 288.386i −0.0879786 + 0.152383i
\(154\) 0 0
\(155\) −1372.00 −0.710979
\(156\) 0 0
\(157\) −2079.00 −1.05683 −0.528415 0.848986i \(-0.677213\pi\)
−0.528415 + 0.848986i \(0.677213\pi\)
\(158\) 0 0
\(159\) −805.500 + 1395.17i −0.401763 + 0.695874i
\(160\) 0 0
\(161\) 1620.00 0.793006
\(162\) 0 0
\(163\) 850.000 + 1472.24i 0.408449 + 0.707454i 0.994716 0.102664i \(-0.0327365\pi\)
−0.586267 + 0.810118i \(0.699403\pi\)
\(164\) 0 0
\(165\) 231.000 + 400.104i 0.108990 + 0.188776i
\(166\) 0 0
\(167\) 1840.00 3186.97i 0.852596 1.47674i −0.0262621 0.999655i \(-0.508360\pi\)
0.878858 0.477084i \(-0.158306\pi\)
\(168\) 0 0
\(169\) 1943.50 + 1024.51i 0.884615 + 0.466321i
\(170\) 0 0
\(171\) 135.000 233.827i 0.0603726 0.104568i
\(172\) 0 0
\(173\) −2073.00 3590.54i −0.911025 1.57794i −0.812619 0.582795i \(-0.801959\pi\)
−0.0984052 0.995146i \(-0.531374\pi\)
\(174\) 0 0
\(175\) 380.000 + 658.179i 0.164145 + 0.284307i
\(176\) 0 0
\(177\) 1728.00 0.733810
\(178\) 0 0
\(179\) 1837.00 3181.78i 0.767060 1.32859i −0.172090 0.985081i \(-0.555052\pi\)
0.939150 0.343506i \(-0.111615\pi\)
\(180\) 0 0
\(181\) −3283.00 −1.34820 −0.674098 0.738642i \(-0.735467\pi\)
−0.674098 + 0.738642i \(0.735467\pi\)
\(182\) 0 0
\(183\) 1905.00 0.769517
\(184\) 0 0
\(185\) −45.5000 + 78.8083i −0.0180823 + 0.0313195i
\(186\) 0 0
\(187\) 814.000 0.318319
\(188\) 0 0
\(189\) 135.000 + 233.827i 0.0519566 + 0.0899915i
\(190\) 0 0
\(191\) −298.000 516.151i −0.112893 0.195536i 0.804043 0.594572i \(-0.202678\pi\)
−0.916935 + 0.399036i \(0.869345\pi\)
\(192\) 0 0
\(193\) 196.500 340.348i 0.0732869 0.126937i −0.827053 0.562124i \(-0.809984\pi\)
0.900340 + 0.435187i \(0.143318\pi\)
\(194\) 0 0
\(195\) −682.500 + 709.275i −0.250640 + 0.260473i
\(196\) 0 0
\(197\) 1761.00 3050.14i 0.636884 1.10311i −0.349229 0.937037i \(-0.613557\pi\)
0.986113 0.166077i \(-0.0531101\pi\)
\(198\) 0 0
\(199\) 1009.00 + 1747.64i 0.359428 + 0.622547i 0.987865 0.155313i \(-0.0496386\pi\)
−0.628438 + 0.777860i \(0.716305\pi\)
\(200\) 0 0
\(201\) −303.000 524.811i −0.106328 0.184166i
\(202\) 0 0
\(203\) −1130.00 −0.390692
\(204\) 0 0
\(205\) −997.500 + 1727.72i −0.339846 + 0.588630i
\(206\) 0 0
\(207\) 1458.00 0.489556
\(208\) 0 0
\(209\) −660.000 −0.218436
\(210\) 0 0
\(211\) 80.0000 138.564i 0.0261016 0.0452092i −0.852680 0.522434i \(-0.825024\pi\)
0.878781 + 0.477225i \(0.158357\pi\)
\(212\) 0 0
\(213\) −3258.00 −1.04805
\(214\) 0 0
\(215\) −861.000 1491.30i −0.273115 0.473049i
\(216\) 0 0
\(217\) 980.000 + 1697.41i 0.306575 + 0.531003i
\(218\) 0 0
\(219\) −1207.50 + 2091.45i −0.372581 + 0.645330i
\(220\) 0 0
\(221\) 481.000 + 1666.23i 0.146405 + 0.507163i
\(222\) 0 0
\(223\) 2036.00 3526.46i 0.611393 1.05896i −0.379613 0.925145i \(-0.623943\pi\)
0.991006 0.133818i \(-0.0427239\pi\)
\(224\) 0 0
\(225\) 342.000 + 592.361i 0.101333 + 0.175514i
\(226\) 0 0
\(227\) −2897.00 5017.75i −0.847051 1.46714i −0.883828 0.467812i \(-0.845042\pi\)
0.0367765 0.999324i \(-0.488291\pi\)
\(228\) 0 0
\(229\) 6482.00 1.87049 0.935246 0.353999i \(-0.115178\pi\)
0.935246 + 0.353999i \(0.115178\pi\)
\(230\) 0 0
\(231\) 330.000 571.577i 0.0939931 0.162801i
\(232\) 0 0
\(233\) 6890.00 1.93725 0.968624 0.248530i \(-0.0799474\pi\)
0.968624 + 0.248530i \(0.0799474\pi\)
\(234\) 0 0
\(235\) 3234.00 0.897714
\(236\) 0 0
\(237\) −1326.00 + 2296.70i −0.363430 + 0.629480i
\(238\) 0 0
\(239\) −2466.00 −0.667415 −0.333708 0.942677i \(-0.608300\pi\)
−0.333708 + 0.942677i \(0.608300\pi\)
\(240\) 0 0
\(241\) 1808.50 + 3132.41i 0.483385 + 0.837247i 0.999818 0.0190805i \(-0.00607389\pi\)
−0.516433 + 0.856327i \(0.672741\pi\)
\(242\) 0 0
\(243\) 121.500 + 210.444i 0.0320750 + 0.0555556i
\(244\) 0 0
\(245\) 850.500 1473.11i 0.221781 0.384137i
\(246\) 0 0
\(247\) −390.000 1351.00i −0.100466 0.348024i
\(248\) 0 0
\(249\) −777.000 + 1345.80i −0.197753 + 0.342517i
\(250\) 0 0
\(251\) 2430.00 + 4208.88i 0.611077 + 1.05842i 0.991059 + 0.133422i \(0.0425966\pi\)
−0.379983 + 0.924994i \(0.624070\pi\)
\(252\) 0 0
\(253\) −1782.00 3086.51i −0.442820 0.766986i
\(254\) 0 0
\(255\) −777.000 −0.190814
\(256\) 0 0
\(257\) −282.500 + 489.304i −0.0685676 + 0.118763i −0.898271 0.439442i \(-0.855176\pi\)
0.829703 + 0.558204i \(0.188510\pi\)
\(258\) 0 0
\(259\) 130.000 0.0311884
\(260\) 0 0
\(261\) −1017.00 −0.241190
\(262\) 0 0
\(263\) −249.000 + 431.281i −0.0583802 + 0.101118i −0.893738 0.448588i \(-0.851927\pi\)
0.835358 + 0.549706i \(0.185260\pi\)
\(264\) 0 0
\(265\) −3759.00 −0.871372
\(266\) 0 0
\(267\) 291.000 + 504.027i 0.0667000 + 0.115528i
\(268\) 0 0
\(269\) −2773.00 4802.98i −0.628523 1.08863i −0.987848 0.155422i \(-0.950326\pi\)
0.359325 0.933213i \(-0.383007\pi\)
\(270\) 0 0
\(271\) −1128.00 + 1953.75i −0.252845 + 0.437941i −0.964308 0.264783i \(-0.914700\pi\)
0.711463 + 0.702724i \(0.248033\pi\)
\(272\) 0 0
\(273\) 1365.00 + 337.750i 0.302614 + 0.0748775i
\(274\) 0 0
\(275\) 836.000 1447.99i 0.183319 0.317518i
\(276\) 0 0
\(277\) −1154.50 1999.65i −0.250423 0.433745i 0.713219 0.700941i \(-0.247236\pi\)
−0.963642 + 0.267196i \(0.913903\pi\)
\(278\) 0 0
\(279\) 882.000 + 1527.67i 0.189262 + 0.327811i
\(280\) 0 0
\(281\) 5833.00 1.23832 0.619159 0.785265i \(-0.287473\pi\)
0.619159 + 0.785265i \(0.287473\pi\)
\(282\) 0 0
\(283\) 825.000 1428.94i 0.173290 0.300148i −0.766278 0.642509i \(-0.777893\pi\)
0.939568 + 0.342362i \(0.111227\pi\)
\(284\) 0 0
\(285\) 630.000 0.130940
\(286\) 0 0
\(287\) 2850.00 0.586168
\(288\) 0 0
\(289\) 1772.00 3069.19i 0.360676 0.624709i
\(290\) 0 0
\(291\) 3606.00 0.726417
\(292\) 0 0
\(293\) −1495.50 2590.28i −0.298184 0.516471i 0.677536 0.735489i \(-0.263048\pi\)
−0.975721 + 0.219019i \(0.929714\pi\)
\(294\) 0 0
\(295\) 2016.00 + 3491.81i 0.397885 + 0.689157i
\(296\) 0 0
\(297\) 297.000 514.419i 0.0580259 0.100504i
\(298\) 0 0
\(299\) 5265.00 5471.55i 1.01834 1.05829i
\(300\) 0 0
\(301\) −1230.00 + 2130.42i −0.235535 + 0.407959i
\(302\) 0 0
\(303\) −643.500 1114.57i −0.122007 0.211322i
\(304\) 0 0
\(305\) 2222.50 + 3849.48i 0.417246 + 0.722691i
\(306\) 0 0
\(307\) 2422.00 0.450263 0.225132 0.974328i \(-0.427719\pi\)
0.225132 + 0.974328i \(0.427719\pi\)
\(308\) 0 0
\(309\) 1953.00 3382.70i 0.359554 0.622766i
\(310\) 0 0
\(311\) 3402.00 0.620288 0.310144 0.950690i \(-0.399623\pi\)
0.310144 + 0.950690i \(0.399623\pi\)
\(312\) 0 0
\(313\) 2310.00 0.417153 0.208577 0.978006i \(-0.433117\pi\)
0.208577 + 0.978006i \(0.433117\pi\)
\(314\) 0 0
\(315\) −315.000 + 545.596i −0.0563436 + 0.0975900i
\(316\) 0 0
\(317\) −257.000 −0.0455349 −0.0227674 0.999741i \(-0.507248\pi\)
−0.0227674 + 0.999741i \(0.507248\pi\)
\(318\) 0 0
\(319\) 1243.00 + 2152.94i 0.218165 + 0.377873i
\(320\) 0 0
\(321\) 2007.00 + 3476.23i 0.348971 + 0.604436i
\(322\) 0 0
\(323\) 555.000 961.288i 0.0956069 0.165596i
\(324\) 0 0
\(325\) 3458.00 + 855.633i 0.590201 + 0.146037i
\(326\) 0 0
\(327\) −1551.00 + 2686.41i −0.262295 + 0.454308i
\(328\) 0 0
\(329\) −2310.00 4001.04i −0.387096 0.670469i
\(330\) 0 0
\(331\) 514.000 + 890.274i 0.0853535 + 0.147837i 0.905542 0.424257i \(-0.139465\pi\)
−0.820188 + 0.572094i \(0.806131\pi\)
\(332\) 0 0
\(333\) 117.000 0.0192539
\(334\) 0 0
\(335\) 707.000 1224.56i 0.115306 0.199716i
\(336\) 0 0
\(337\) 2487.00 0.402005 0.201002 0.979591i \(-0.435580\pi\)
0.201002 + 0.979591i \(0.435580\pi\)
\(338\) 0 0
\(339\) −3231.00 −0.517651
\(340\) 0 0
\(341\) 2156.00 3734.30i 0.342387 0.593032i
\(342\) 0 0
\(343\) −5860.00 −0.922479
\(344\) 0 0
\(345\) 1701.00 + 2946.22i 0.265446 + 0.459765i
\(346\) 0 0
\(347\) −1425.00 2468.17i −0.220455 0.381840i 0.734491 0.678618i \(-0.237421\pi\)
−0.954946 + 0.296779i \(0.904088\pi\)
\(348\) 0 0
\(349\) 1009.00 1747.64i 0.154758 0.268049i −0.778213 0.628001i \(-0.783874\pi\)
0.932971 + 0.359952i \(0.117207\pi\)
\(350\) 0 0
\(351\) 1228.50 + 303.975i 0.186816 + 0.0462250i
\(352\) 0 0
\(353\) 2643.50 4578.68i 0.398582 0.690364i −0.594970 0.803748i \(-0.702836\pi\)
0.993551 + 0.113385i \(0.0361692\pi\)
\(354\) 0 0
\(355\) −3801.00 6583.53i −0.568271 0.984274i
\(356\) 0 0
\(357\) 555.000 + 961.288i 0.0822793 + 0.142512i
\(358\) 0 0
\(359\) 7278.00 1.06997 0.534983 0.844863i \(-0.320318\pi\)
0.534983 + 0.844863i \(0.320318\pi\)
\(360\) 0 0
\(361\) 2979.50 5160.65i 0.434393 0.752390i
\(362\) 0 0
\(363\) 2541.00 0.367405
\(364\) 0 0
\(365\) −5635.00 −0.808080
\(366\) 0 0
\(367\) −2101.00 + 3639.04i −0.298832 + 0.517592i −0.975869 0.218357i \(-0.929930\pi\)
0.677037 + 0.735949i \(0.263264\pi\)
\(368\) 0 0
\(369\) 2565.00 0.361866
\(370\) 0 0
\(371\) 2685.00 + 4650.56i 0.375737 + 0.650795i
\(372\) 0 0
\(373\) 791.500 + 1370.92i 0.109872 + 0.190304i 0.915718 0.401821i \(-0.131623\pi\)
−0.805846 + 0.592125i \(0.798289\pi\)
\(374\) 0 0
\(375\) −2110.50 + 3655.49i −0.290629 + 0.503384i
\(376\) 0 0
\(377\) −3672.50 + 3816.57i −0.501707 + 0.521389i
\(378\) 0 0
\(379\) −1026.00 + 1777.08i −0.139056 + 0.240851i −0.927139 0.374717i \(-0.877740\pi\)
0.788084 + 0.615568i \(0.211073\pi\)
\(380\) 0 0
\(381\) 1482.00 + 2566.90i 0.199279 + 0.345161i
\(382\) 0 0
\(383\) −3436.00 5951.33i −0.458411 0.793991i 0.540466 0.841366i \(-0.318248\pi\)
−0.998877 + 0.0473746i \(0.984915\pi\)
\(384\) 0 0
\(385\) 1540.00 0.203859
\(386\) 0 0
\(387\) −1107.00 + 1917.38i −0.145406 + 0.251850i
\(388\) 0 0
\(389\) −11653.0 −1.51884 −0.759422 0.650598i \(-0.774518\pi\)
−0.759422 + 0.650598i \(0.774518\pi\)
\(390\) 0 0
\(391\) 5994.00 0.775268
\(392\) 0 0
\(393\) −840.000 + 1454.92i −0.107818 + 0.186746i
\(394\) 0 0
\(395\) −6188.00 −0.788233
\(396\) 0 0
\(397\) −3067.00 5312.20i −0.387729 0.671566i 0.604415 0.796670i \(-0.293407\pi\)
−0.992144 + 0.125104i \(0.960074\pi\)
\(398\) 0 0
\(399\) −450.000 779.423i −0.0564616 0.0977944i
\(400\) 0 0
\(401\) 5397.50 9348.74i 0.672165 1.16422i −0.305124 0.952313i \(-0.598698\pi\)
0.977289 0.211912i \(-0.0679689\pi\)
\(402\) 0 0
\(403\) 8918.00 + 2206.63i 1.10233 + 0.272755i
\(404\) 0 0
\(405\) −283.500 + 491.036i −0.0347833 + 0.0602464i
\(406\) 0 0
\(407\) −143.000 247.683i −0.0174158 0.0301651i
\(408\) 0 0
\(409\) 4244.50 + 7351.69i 0.513147 + 0.888796i 0.999884 + 0.0152477i \(0.00485367\pi\)
−0.486737 + 0.873549i \(0.661813\pi\)
\(410\) 0 0
\(411\) 1557.00 0.186864
\(412\) 0 0
\(413\) 2880.00 4988.31i 0.343137 0.594331i
\(414\) 0 0
\(415\) −3626.00 −0.428900
\(416\) 0 0
\(417\) −1044.00 −0.122602
\(418\) 0 0
\(419\) 748.000 1295.57i 0.0872129 0.151057i −0.819119 0.573623i \(-0.805537\pi\)
0.906332 + 0.422566i \(0.138871\pi\)
\(420\) 0 0
\(421\) −11695.0 −1.35387 −0.676935 0.736043i \(-0.736692\pi\)
−0.676935 + 0.736043i \(0.736692\pi\)
\(422\) 0 0
\(423\) −2079.00 3600.93i −0.238970 0.413909i
\(424\) 0 0
\(425\) 1406.00 + 2435.26i 0.160473 + 0.277947i
\(426\) 0 0
\(427\) 3175.00 5499.26i 0.359834 0.623250i
\(428\) 0 0
\(429\) −858.000 2972.20i −0.0965609 0.334497i
\(430\) 0 0
\(431\) 5295.00 9171.21i 0.591766 1.02497i −0.402228 0.915539i \(-0.631764\pi\)
0.993995 0.109430i \(-0.0349024\pi\)
\(432\) 0 0
\(433\) 6974.50 + 12080.2i 0.774072 + 1.34073i 0.935315 + 0.353817i \(0.115116\pi\)
−0.161243 + 0.986915i \(0.551550\pi\)
\(434\) 0 0
\(435\) −1186.50 2055.08i −0.130778 0.226514i
\(436\) 0 0
\(437\) −4860.00 −0.532003
\(438\) 0 0
\(439\) −5363.00 + 9288.99i −0.583057 + 1.00988i 0.412058 + 0.911158i \(0.364810\pi\)
−0.995115 + 0.0987266i \(0.968523\pi\)
\(440\) 0 0
\(441\) −2187.00 −0.236152
\(442\) 0 0
\(443\) −16228.0 −1.74044 −0.870221 0.492662i \(-0.836024\pi\)
−0.870221 + 0.492662i \(0.836024\pi\)
\(444\) 0 0
\(445\) −679.000 + 1176.06i −0.0723319 + 0.125282i
\(446\) 0 0
\(447\) 1935.00 0.204748
\(448\) 0 0
\(449\) −3769.00 6528.10i −0.396147 0.686147i 0.597100 0.802167i \(-0.296320\pi\)
−0.993247 + 0.116020i \(0.962986\pi\)
\(450\) 0 0
\(451\) −3135.00 5429.98i −0.327320 0.566935i
\(452\) 0 0
\(453\) −4371.00 + 7570.79i −0.453350 + 0.785225i
\(454\) 0 0
\(455\) 910.000 + 3152.33i 0.0937614 + 0.324799i
\(456\) 0 0
\(457\) −7769.50 + 13457.2i −0.795278 + 1.37746i 0.127385 + 0.991853i \(0.459342\pi\)
−0.922663 + 0.385608i \(0.873992\pi\)
\(458\) 0 0
\(459\) 499.500 + 865.159i 0.0507945 + 0.0879786i
\(460\) 0 0
\(461\) −2405.50 4166.45i −0.243027 0.420935i 0.718548 0.695477i \(-0.244807\pi\)
−0.961575 + 0.274543i \(0.911474\pi\)
\(462\) 0 0
\(463\) −562.000 −0.0564111 −0.0282056 0.999602i \(-0.508979\pi\)
−0.0282056 + 0.999602i \(0.508979\pi\)
\(464\) 0 0
\(465\) −2058.00 + 3564.56i −0.205242 + 0.355489i
\(466\) 0 0
\(467\) −4914.00 −0.486922 −0.243461 0.969911i \(-0.578283\pi\)
−0.243461 + 0.969911i \(0.578283\pi\)
\(468\) 0 0
\(469\) −2020.00 −0.198880
\(470\) 0 0
\(471\) −3118.50 + 5401.40i −0.305080 + 0.528415i
\(472\) 0 0
\(473\) 5412.00 0.526097
\(474\) 0 0
\(475\) −1140.00 1974.54i −0.110120 0.190733i
\(476\) 0 0
\(477\) 2416.50 + 4185.50i 0.231958 + 0.401763i
\(478\) 0 0
\(479\) −1800.00 + 3117.69i −0.171700 + 0.297392i −0.939014 0.343878i \(-0.888259\pi\)
0.767315 + 0.641271i \(0.221593\pi\)
\(480\) 0 0
\(481\) 422.500 439.075i 0.0400506 0.0416218i
\(482\) 0 0
\(483\) 2430.00 4208.88i 0.228921 0.396503i
\(484\) 0 0
\(485\) 4207.00 + 7286.74i 0.393876 + 0.682214i
\(486\) 0 0
\(487\) −8565.00 14835.0i −0.796955 1.38037i −0.921590 0.388164i \(-0.873109\pi\)
0.124635 0.992203i \(-0.460224\pi\)
\(488\) 0 0
\(489\) 5100.00 0.471636
\(490\) 0 0
\(491\) −5919.00 + 10252.0i −0.544034 + 0.942295i 0.454633 + 0.890679i \(0.349770\pi\)
−0.998667 + 0.0516158i \(0.983563\pi\)
\(492\) 0 0
\(493\) −4181.00 −0.381953
\(494\) 0 0
\(495\) 1386.00 0.125851
\(496\) 0 0
\(497\) −5430.00 + 9405.04i −0.490078 + 0.848840i
\(498\) 0 0
\(499\) −8976.00 −0.805252 −0.402626 0.915364i \(-0.631903\pi\)
−0.402626 + 0.915364i \(0.631903\pi\)
\(500\) 0 0
\(501\) −5520.00 9560.92i −0.492246 0.852596i
\(502\) 0 0
\(503\) 841.000 + 1456.65i 0.0745494 + 0.129123i 0.900890 0.434047i \(-0.142915\pi\)
−0.826341 + 0.563170i \(0.809581\pi\)
\(504\) 0 0
\(505\) 1501.50 2600.67i 0.132309 0.229165i
\(506\) 0 0
\(507\) 5577.00 3512.60i 0.488527 0.307692i
\(508\) 0 0
\(509\) −7583.50 + 13135.0i −0.660379 + 1.14381i 0.320138 + 0.947371i \(0.396271\pi\)
−0.980516 + 0.196438i \(0.937062\pi\)
\(510\) 0 0
\(511\) 4025.00 + 6971.50i 0.348445 + 0.603525i
\(512\) 0 0
\(513\) −405.000 701.481i −0.0348561 0.0603726i
\(514\) 0 0
\(515\) 9114.00 0.779827
\(516\) 0 0
\(517\) −5082.00 + 8802.28i −0.432314 + 0.748789i
\(518\) 0 0
\(519\) −12438.0 −1.05196
\(520\) 0 0
\(521\) −6783.00 −0.570381 −0.285191 0.958471i \(-0.592057\pi\)
−0.285191 + 0.958471i \(0.592057\pi\)
\(522\) 0 0
\(523\) −6959.00 + 12053.3i −0.581828 + 1.00775i 0.413435 + 0.910534i \(0.364329\pi\)
−0.995263 + 0.0972214i \(0.969005\pi\)
\(524\) 0 0
\(525\) 2280.00 0.189538
\(526\) 0 0
\(527\) 3626.00 + 6280.42i 0.299717 + 0.519126i
\(528\) 0 0
\(529\) −7038.50 12191.0i −0.578491 1.00198i
\(530\) 0 0
\(531\) 2592.00 4489.48i 0.211833 0.366905i
\(532\) 0 0
\(533\) 9262.50 9625.87i 0.752727 0.782257i
\(534\) 0 0
\(535\) −4683.00 + 8111.19i −0.378437 + 0.655472i
\(536\) 0 0
\(537\) −5511.00 9545.33i −0.442863 0.767060i
\(538\) 0 0
\(539\) 2673.00 + 4629.77i 0.213607 + 0.369978i
\(540\) 0 0
\(541\) −1335.00 −0.106093 −0.0530463 0.998592i \(-0.516893\pi\)
−0.0530463 + 0.998592i \(0.516893\pi\)
\(542\) 0 0
\(543\) −4924.50 + 8529.48i −0.389191 + 0.674098i
\(544\) 0 0
\(545\) −7238.00 −0.568884
\(546\) 0 0
\(547\) 3806.00 0.297501 0.148750 0.988875i \(-0.452475\pi\)
0.148750 + 0.988875i \(0.452475\pi\)
\(548\) 0 0
\(549\) 2857.50 4949.34i 0.222140 0.384759i
\(550\) 0 0
\(551\) 3390.00 0.262103
\(552\) 0 0
\(553\) 4420.00 + 7655.66i 0.339887 + 0.588702i
\(554\) 0 0
\(555\) 136.500 + 236.425i 0.0104398 + 0.0180823i
\(556\) 0 0
\(557\) 952.500 1649.78i 0.0724573 0.125500i −0.827520 0.561436i \(-0.810249\pi\)
0.899978 + 0.435936i \(0.143583\pi\)
\(558\) 0 0
\(559\) 3198.00 + 11078.2i 0.241970 + 0.838207i
\(560\) 0 0
\(561\) 1221.00 2114.83i 0.0918907 0.159159i
\(562\) 0 0
\(563\) −2400.00 4156.92i −0.179659 0.311178i 0.762105 0.647454i \(-0.224166\pi\)
−0.941764 + 0.336275i \(0.890833\pi\)
\(564\) 0 0
\(565\) −3769.50 6528.97i −0.280680 0.486152i
\(566\) 0 0
\(567\) 810.000 0.0599944
\(568\) 0 0
\(569\) −7339.00 + 12711.5i −0.540715 + 0.936546i 0.458148 + 0.888876i \(0.348513\pi\)
−0.998863 + 0.0476701i \(0.984820\pi\)
\(570\) 0 0
\(571\) 586.000 0.0429481 0.0214740 0.999769i \(-0.493164\pi\)
0.0214740 + 0.999769i \(0.493164\pi\)
\(572\) 0 0
\(573\) −1788.00 −0.130357
\(574\) 0 0
\(575\) 6156.00 10662.5i 0.446475 0.773317i
\(576\) 0 0
\(577\) 8939.00 0.644949 0.322474 0.946578i \(-0.395485\pi\)
0.322474 + 0.946578i \(0.395485\pi\)
\(578\) 0 0
\(579\) −589.500 1021.04i −0.0423122 0.0732869i
\(580\) 0 0
\(581\) 2590.00 + 4486.01i 0.184942 + 0.320329i
\(582\) 0 0
\(583\) 5907.00 10231.2i 0.419628 0.726816i
\(584\) 0 0
\(585\) 819.000 + 2837.10i 0.0578829 + 0.200512i
\(586\) 0 0
\(587\) −6896.00 + 11944.2i −0.484887 + 0.839848i −0.999849 0.0173645i \(-0.994472\pi\)
0.514963 + 0.857213i \(0.327806\pi\)
\(588\) 0 0
\(589\) −2940.00 5092.23i −0.205672 0.356234i
\(590\) 0 0
\(591\) −5283.00 9150.42i −0.367705 0.636884i
\(592\) 0 0
\(593\) 9569.00 0.662650 0.331325 0.943517i \(-0.392504\pi\)
0.331325 + 0.943517i \(0.392504\pi\)
\(594\) 0 0
\(595\) −1295.00 + 2243.01i −0.0892266 + 0.154545i
\(596\) 0 0
\(597\) 6054.00 0.415031
\(598\) 0 0
\(599\) 5192.00 0.354156 0.177078 0.984197i \(-0.443336\pi\)
0.177078 + 0.984197i \(0.443336\pi\)
\(600\) 0 0
\(601\) 1838.50 3184.38i 0.124782 0.216129i −0.796866 0.604156i \(-0.793510\pi\)
0.921648 + 0.388028i \(0.126844\pi\)
\(602\) 0 0
\(603\) −1818.00 −0.122777
\(604\) 0 0
\(605\) 2964.50 + 5134.66i 0.199213 + 0.345048i
\(606\) 0 0
\(607\) −5480.00 9491.64i −0.366435 0.634685i 0.622570 0.782564i \(-0.286089\pi\)
−0.989005 + 0.147879i \(0.952755\pi\)
\(608\) 0 0
\(609\) −1695.00 + 2935.83i −0.112783 + 0.195346i
\(610\) 0 0
\(611\) −21021.0 5201.35i −1.39185 0.344393i
\(612\) 0 0
\(613\) 13013.5 22540.0i 0.857439 1.48513i −0.0169241 0.999857i \(-0.505387\pi\)
0.874363 0.485272i \(-0.161279\pi\)
\(614\) 0 0
\(615\) 2992.50 + 5183.16i 0.196210 + 0.339846i
\(616\) 0 0
\(617\) −8840.50 15312.2i −0.576832 0.999102i −0.995840 0.0911193i \(-0.970956\pi\)
0.419008 0.907982i \(-0.362378\pi\)
\(618\) 0 0
\(619\) −3192.00 −0.207265 −0.103633 0.994616i \(-0.533047\pi\)
−0.103633 + 0.994616i \(0.533047\pi\)
\(620\) 0 0
\(621\) 2187.00 3788.00i 0.141323 0.244778i
\(622\) 0 0
\(623\) 1940.00 0.124758
\(624\) 0 0
\(625\) −349.000 −0.0223360
\(626\) 0 0
\(627\) −990.000 + 1714.73i −0.0630571 + 0.109218i
\(628\) 0 0
\(629\) 481.000 0.0304908
\(630\) 0 0
\(631\) 3790.00 + 6564.47i 0.239109 + 0.414148i 0.960459 0.278422i \(-0.0898115\pi\)
−0.721350 + 0.692571i \(0.756478\pi\)
\(632\) 0 0
\(633\) −240.000 415.692i −0.0150697 0.0261016i
\(634\) 0 0
\(635\) −3458.00 + 5989.43i −0.216105 + 0.374304i
\(636\) 0 0
\(637\) −7897.50 + 8207.32i −0.491225 + 0.510496i
\(638\) 0 0
\(639\) −4887.00 + 8464.53i −0.302546 + 0.524025i
\(640\) 0 0
\(641\) 13853.5 + 23995.0i 0.853635 + 1.47854i 0.877905 + 0.478835i \(0.158941\pi\)
−0.0242696 + 0.999705i \(0.507726\pi\)
\(642\) 0 0
\(643\) 5608.00 + 9713.34i 0.343947 + 0.595734i 0.985162 0.171628i \(-0.0549026\pi\)
−0.641215 + 0.767361i \(0.721569\pi\)
\(644\) 0 0
\(645\) −5166.00 −0.315366
\(646\) 0 0
\(647\) −1268.00 + 2196.24i −0.0770483 + 0.133452i −0.901975 0.431788i \(-0.857883\pi\)
0.824927 + 0.565239i \(0.191216\pi\)
\(648\) 0 0
\(649\) −12672.0 −0.766440
\(650\) 0 0
\(651\) 5880.00 0.354002
\(652\) 0 0
\(653\) −8865.00 + 15354.6i −0.531262 + 0.920173i 0.468072 + 0.883690i \(0.344949\pi\)
−0.999334 + 0.0364829i \(0.988385\pi\)
\(654\) 0 0
\(655\) −3920.00 −0.233843
\(656\) 0 0
\(657\) 3622.50 + 6274.35i 0.215110 + 0.372581i
\(658\) 0 0
\(659\) 9460.00 + 16385.2i 0.559195 + 0.968554i 0.997564 + 0.0697586i \(0.0222229\pi\)
−0.438369 + 0.898795i \(0.644444\pi\)
\(660\) 0 0
\(661\) −2620.50 + 4538.84i −0.154199 + 0.267081i −0.932767 0.360480i \(-0.882613\pi\)
0.778568 + 0.627560i \(0.215946\pi\)
\(662\) 0 0
\(663\) 5050.50 + 1249.67i 0.295845 + 0.0732026i
\(664\) 0 0
\(665\) 1050.00 1818.65i 0.0612290 0.106052i
\(666\) 0 0
\(667\) 9153.00 + 15853.5i 0.531343 + 0.920313i
\(668\) 0 0
\(669\) −6108.00 10579.4i −0.352988 0.611393i
\(670\) 0 0
\(671\) −13970.0 −0.803735
\(672\) 0 0
\(673\) −10233.5 + 17724.9i −0.586140 + 1.01522i 0.408592 + 0.912717i \(0.366020\pi\)
−0.994732 + 0.102508i \(0.967313\pi\)
\(674\) 0 0
\(675\) 2052.00 0.117010
\(676\) 0 0
\(677\) −70.0000 −0.00397388 −0.00198694 0.999998i \(-0.500632\pi\)
−0.00198694 + 0.999998i \(0.500632\pi\)
\(678\) 0 0
\(679\) 6010.00 10409.6i 0.339680 0.588343i
\(680\) 0 0
\(681\) −17382.0 −0.978091
\(682\) 0 0
\(683\) 3216.00 + 5570.28i 0.180171 + 0.312065i 0.941939 0.335785i \(-0.109002\pi\)
−0.761768 + 0.647850i \(0.775668\pi\)
\(684\) 0 0
\(685\) 1816.50 + 3146.27i 0.101321 + 0.175493i
\(686\) 0 0
\(687\) 9723.00 16840.7i 0.539964 0.935246i
\(688\) 0 0
\(689\) 24433.5 + 6045.72i 1.35100 + 0.334287i
\(690\) 0 0
\(691\) −3333.00 + 5772.93i −0.183492 + 0.317818i −0.943067 0.332601i \(-0.892074\pi\)
0.759575 + 0.650420i \(0.225407\pi\)
\(692\) 0 0
\(693\) −990.000 1714.73i −0.0542669 0.0939931i
\(694\) 0 0
\(695\) −1218.00 2109.64i −0.0664768 0.115141i
\(696\) 0 0
\(697\) 10545.0 0.573056
\(698\) 0 0
\(699\) 10335.0 17900.7i 0.559235 0.968624i
\(700\) 0 0
\(701\) −14054.0 −0.757221 −0.378611 0.925556i \(-0.623598\pi\)
−0.378611 + 0.925556i \(0.623598\pi\)
\(702\) 0 0
\(703\) −390.000 −0.0209234
\(704\) 0 0
\(705\) 4851.00 8402.18i 0.259148 0.448857i
\(706\) 0 0
\(707\) −4290.00 −0.228207
\(708\) 0 0
\(709\) 35.5000 + 61.4878i 0.00188044 + 0.00325701i 0.866964 0.498371i \(-0.166068\pi\)
−0.865084 + 0.501628i \(0.832735\pi\)
\(710\) 0 0
\(711\) 3978.00 + 6890.10i 0.209827 + 0.363430i
\(712\) 0 0
\(713\) 15876.0 27498.0i 0.833886 1.44433i
\(714\) 0 0
\(715\) 5005.00 5201.35i 0.261785 0.272055i
\(716\) 0 0
\(717\) −3699.00 + 6406.86i −0.192666 + 0.333708i
\(718\) 0 0
\(719\) 1968.00 + 3408.68i 0.102078 + 0.176804i 0.912541 0.408986i \(-0.134118\pi\)
−0.810463 + 0.585790i \(0.800784\pi\)
\(720\) 0 0
\(721\) −6510.00 11275.7i −0.336262 0.582423i
\(722\) 0 0
\(723\) 10851.0 0.558165
\(724\) 0 0
\(725\) −4294.00 + 7437.43i −0.219966 + 0.380992i
\(726\) 0 0
\(727\) −34202.0 −1.74482 −0.872409 0.488777i \(-0.837443\pi\)
−0.872409 + 0.488777i \(0.837443\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −4551.00 + 7882.56i −0.230267 + 0.398833i
\(732\) 0 0
\(733\) −27363.0 −1.37882 −0.689410 0.724371i \(-0.742130\pi\)
−0.689410 + 0.724371i \(0.742130\pi\)
\(734\) 0 0
\(735\) −2551.50 4419.33i −0.128046 0.221781i
\(736\) 0 0
\(737\) 2222.00 + 3848.62i 0.111056 + 0.192355i
\(738\) 0 0
\(739\) 10888.0 18858.6i 0.541978 0.938733i −0.456813 0.889563i \(-0.651009\pi\)
0.998790 0.0491701i \(-0.0156576\pi\)
\(740\) 0 0
\(741\) −4095.00 1013.25i −0.203014 0.0502330i
\(742\) 0 0
\(743\) 1242.00 2151.21i 0.0613251 0.106218i −0.833733 0.552168i \(-0.813801\pi\)
0.895058 + 0.445950i \(0.147134\pi\)
\(744\) 0 0
\(745\) 2257.50 + 3910.10i 0.111018 + 0.192289i
\(746\) 0 0
\(747\) 2331.00 + 4037.41i 0.114172 + 0.197753i
\(748\) 0 0
\(749\) 13380.0 0.652730
\(750\) 0 0
\(751\) 16453.0 28497.4i 0.799439 1.38467i −0.120543 0.992708i \(-0.538464\pi\)
0.919982 0.391960i \(-0.128203\pi\)
\(752\) 0 0
\(753\) 14580.0 0.705611
\(754\) 0 0
\(755\) −20398.0 −0.983257
\(756\) 0 0
\(757\) 1957.00 3389.62i 0.0939609 0.162745i −0.815214 0.579160i \(-0.803380\pi\)
0.909174 + 0.416415i \(0.136714\pi\)
\(758\) 0 0
\(759\) −10692.0 −0.511324
\(760\) 0 0
\(761\) 16519.0 + 28611.7i 0.786877 + 1.36291i 0.927871 + 0.372900i \(0.121637\pi\)
−0.140995 + 0.990010i \(0.545030\pi\)
\(762\) 0 0
\(763\) 5170.00 + 8954.70i 0.245303 + 0.424878i
\(764\) 0 0
\(765\) −1165.50 + 2018.71i −0.0550833 + 0.0954071i
\(766\) 0 0
\(767\) −7488.00 25939.2i −0.352511 1.22113i
\(768\) 0 0
\(769\) 8793.00 15229.9i 0.412332 0.714181i −0.582812 0.812607i \(-0.698048\pi\)
0.995144 + 0.0984263i \(0.0313809\pi\)
\(770\) 0 0
\(771\) 847.500 + 1467.91i 0.0395875 + 0.0685676i
\(772\) 0 0
\(773\) −9157.00 15860.4i −0.426073 0.737980i 0.570447 0.821334i \(-0.306770\pi\)
−0.996520 + 0.0833544i \(0.973437\pi\)
\(774\) 0 0
\(775\) 14896.0 0.690426
\(776\) 0 0
\(777\) 195.000 337.750i 0.00900333 0.0155942i
\(778\) 0 0
\(779\) −8550.00 −0.393242
\(780\) 0 0
\(781\) 23892.0 1.09465
\(782\) 0 0
\(783\) −1525.50 + 2642.24i −0.0696257 + 0.120595i
\(784\) 0 0
\(785\) −14553.0 −0.661680
\(786\) 0 0
\(787\) −21034.0 36432.0i −0.952708 1.65014i −0.739528 0.673125i \(-0.764951\pi\)
−0.213180 0.977013i \(-0.568382\pi\)
\(788\) 0 0
\(789\) 747.000 + 1293.84i 0.0337058 + 0.0583802i
\(790\) 0 0
\(791\) −5385.00 + 9327.09i −0.242059 + 0.419258i
\(792\) 0 0
\(793\) −8255.00 28596.2i −0.369664 1.28055i
\(794\) 0 0
\(795\) −5638.50 + 9766.17i −0.251543 + 0.435686i
\(796\) 0 0
\(797\) 2141.00 + 3708.32i 0.0951545 + 0.164812i 0.909673 0.415325i \(-0.136332\pi\)
−0.814519 + 0.580137i \(0.802999\pi\)
\(798\) 0 0
\(799\) −8547.00 14803.8i −0.378437 0.655472i
\(800\) 0 0
\(801\) 1746.00 0.0770186
\(802\) 0 0
\(803\) 8855.00 15337.3i 0.389148 0.674025i
\(804\) 0 0
\(805\) 11340.0 0.496500
\(806\) 0 0
\(807\) −16638.0 −0.725756
\(808\) 0 0
\(809\) −20110.5 + 34832.4i −0.873977 + 1.51377i −0.0161288 + 0.999870i \(0.505134\pi\)
−0.857848 + 0.513903i \(0.828199\pi\)
\(810\) 0 0
\(811\) 7084.00 0.306724 0.153362 0.988170i \(-0.450990\pi\)
0.153362 + 0.988170i \(0.450990\pi\)
\(812\) 0 0
\(813\) 3384.00 + 5861.26i 0.145980 + 0.252845i
\(814\) 0 0
\(815\) 5950.00 + 10305.7i 0.255729 + 0.442936i
\(816\) 0 0
\(817\) 3690.00 6391.27i 0.158013 0.273687i
\(818\) 0 0
\(819\) 2925.00 3039.75i 0.124796 0.129692i
\(820\) 0 0
\(821\) −8669.00 + 15015.1i −0.368514 + 0.638285i −0.989333 0.145668i \(-0.953467\pi\)
0.620819 + 0.783954i \(0.286800\pi\)
\(822\) 0 0
\(823\) 17748.0 + 30740.4i 0.751709 + 1.30200i 0.946994 + 0.321251i \(0.104103\pi\)
−0.195285 + 0.980747i \(0.562563\pi\)
\(824\) 0 0
\(825\) −2508.00 4343.98i −0.105839 0.183319i
\(826\) 0 0
\(827\) 14992.0 0.630378 0.315189 0.949029i \(-0.397932\pi\)
0.315189 + 0.949029i \(0.397932\pi\)
\(828\) 0 0
\(829\) 10329.5 17891.2i 0.432760 0.749563i −0.564349 0.825536i \(-0.690873\pi\)
0.997110 + 0.0759730i \(0.0242063\pi\)
\(830\) 0 0
\(831\) −6927.00 −0.289164
\(832\) 0 0
\(833\) −8991.00 −0.373973
\(834\) 0 0
\(835\) 12880.0 22308.8i 0.533809 0.924585i
\(836\) 0 0
\(837\) 5292.00 0.218540
\(838\) 0 0
\(839\) 14358.0 + 24868.8i 0.590814 + 1.02332i 0.994123 + 0.108256i \(0.0345268\pi\)
−0.403309 + 0.915064i \(0.632140\pi\)
\(840\) 0 0
\(841\) 5810.00 + 10063.2i 0.238222 + 0.412613i
\(842\) 0 0
\(843\) 8749.50 15154.6i 0.357472 0.619159i
\(844\) 0 0
\(845\) 13604.5 + 7171.56i 0.553857 + 0.291963i
\(846\) 0 0
\(847\) 4235.00 7335.24i 0.171802 0.297570i
\(848\) 0 0
\(849\) −2475.00 4286.83i −0.100049 0.173290i
\(850\) 0 0
\(851\) −1053.00 1823.85i −0.0424164 0.0734674i
\(852\) 0 0
\(853\) 13377.0 0.536952 0.268476 0.963286i \(-0.413480\pi\)
0.268476 + 0.963286i \(0.413480\pi\)
\(854\) 0 0
\(855\) 945.000 1636.79i 0.0377992 0.0654701i
\(856\) 0 0
\(857\) −27419.0 −1.09290 −0.546450 0.837492i \(-0.684021\pi\)
−0.546450 + 0.837492i \(0.684021\pi\)
\(858\) 0 0
\(859\) −2422.00 −0.0962021 −0.0481010 0.998842i \(-0.515317\pi\)
−0.0481010 + 0.998842i \(0.515317\pi\)
\(860\) 0 0
\(861\) 4275.00 7404.52i 0.169212 0.293084i
\(862\) 0 0
\(863\) 34522.0 1.36169 0.680847 0.732425i \(-0.261612\pi\)
0.680847 + 0.732425i \(0.261612\pi\)
\(864\) 0 0
\(865\) −14511.0 25133.8i −0.570392 0.987947i
\(866\) 0 0
\(867\) −5316.00 9207.58i −0.208236 0.360676i
\(868\) 0 0
\(869\) 9724.00 16842.5i 0.379590 0.657470i
\(870\) 0 0
\(871\) −6565.00 + 6822.55i −0.255392 + 0.265411i
\(872\) 0 0
\(873\) 5409.00 9368.66i 0.209699 0.363209i
\(874\) 0 0
\(875\) 7035.00 + 12185.0i 0.271802 + 0.470774i
\(876\) 0 0
\(877\) −6866.50 11893.1i −0.264385 0.457927i 0.703018 0.711172i \(-0.251835\pi\)
−0.967402 + 0.253245i \(0.918502\pi\)
\(878\) 0 0
\(879\) −8973.00 −0.344314
\(880\) 0 0
\(881\) 11379.5 19709.9i 0.435170 0.753737i −0.562139 0.827043i \(-0.690021\pi\)
0.997310 + 0.0733055i \(0.0233548\pi\)
\(882\) 0 0
\(883\) 2168.00 0.0826263 0.0413131 0.999146i \(-0.486846\pi\)
0.0413131 + 0.999146i \(0.486846\pi\)
\(884\) 0 0
\(885\) 12096.0 0.459438
\(886\) 0 0
\(887\) 7944.00 13759.4i 0.300714 0.520852i −0.675584 0.737283i \(-0.736108\pi\)
0.976298 + 0.216431i \(0.0694417\pi\)
\(888\) 0 0
\(889\) 9880.00 0.372739
\(890\) 0 0
\(891\) −891.000 1543.26i −0.0335013 0.0580259i
\(892\) 0 0
\(893\) 6930.00 + 12003.1i 0.259690 + 0.449797i
\(894\) 0 0
\(895\) 12859.0 22272.4i 0.480256 0.831827i
\(896\) 0 0
\(897\) −6318.00 21886.2i −0.235175 0.814670i
\(898\) 0 0
\(899\) −11074.0 + 19180.7i −0.410833 + 0.711583i
\(900\) 0 0
\(901\) 9934.50 + 17207.1i 0.367332 + 0.636238i
\(902\) 0 0
\(903\) 3690.00 + 6391.27i 0.135986 + 0.235535i
\(904\) 0 0
\(905\) −22981.0 −0.844104
\(906\) 0 0
\(907\) −5814.00 + 10070.1i −0.212845 + 0.368659i −0.952604 0.304214i \(-0.901606\pi\)
0.739759 + 0.672872i \(0.234940\pi\)
\(908\) 0 0
\(909\) −3861.00 −0.140882
\(910\) 0 0
\(911\) 12584.0 0.457658 0.228829 0.973467i \(-0.426510\pi\)
0.228829 + 0.973467i \(0.426510\pi\)
\(912\) 0 0
\(913\) 5698.00 9869.23i 0.206546 0.357748i
\(914\) 0 0
\(915\) 13335.0 0.481794
\(916\) 0 0
\(917\) 2800.00 + 4849.74i 0.100833 + 0.174648i
\(918\) 0 0
\(919\) 8592.00 + 14881.8i 0.308405 + 0.534173i 0.978014 0.208541i \(-0.0668716\pi\)
−0.669609 + 0.742714i \(0.733538\pi\)
\(920\) 0 0
\(921\) 3633.00 6292.54i 0.129980 0.225132i
\(922\) 0 0
\(923\) 14118.0 + 48906.2i 0.503467 + 1.74406i
\(924\) 0 0
\(925\) 494.000 855.633i 0.0175596 0.0304141i
\(926\) 0 0
\(927\) −5859.00 10148.1i −0.207589 0.359554i
\(928\) 0 0
\(929\) −6388.50 11065.2i −0.225619 0.390783i 0.730886 0.682499i \(-0.239107\pi\)
−0.956505 + 0.291716i \(0.905774\pi\)
\(930\) 0 0
\(931\) 7290.00 0.256627
\(932\) 0 0
\(933\) 5103.00 8838.66i 0.179062 0.310144i
\(934\) 0 0
\(935\) 5698.00 0.199299
\(936\) 0 0
\(937\) 9191.00 0.320445 0.160222 0.987081i \(-0.448779\pi\)
0.160222 + 0.987081i \(0.448779\pi\)
\(938\) 0 0
\(939\) 3465.00 6001.56i 0.120422 0.208577i
\(940\) 0 0
\(941\) 50498.0 1.74940 0.874701 0.484662i \(-0.161058\pi\)
0.874701 + 0.484662i \(0.161058\pi\)
\(942\) 0 0
\(943\) −23085.0 39984.4i −0.797191 1.38078i
\(944\) 0 0
\(945\) 945.000 + 1636.79i 0.0325300 + 0.0563436i
\(946\) 0 0
\(947\) 780.000 1351.00i 0.0267651 0.0463586i −0.852333 0.523000i \(-0.824813\pi\)
0.879098 + 0.476642i \(0.158146\pi\)
\(948\) 0 0
\(949\) 36627.5 + 9062.96i 1.25288 + 0.310006i
\(950\) 0 0
\(951\) −385.500 + 667.706i −0.0131448 + 0.0227674i
\(952\) 0 0
\(953\) 10749.0 + 18617.8i 0.365366 + 0.632833i 0.988835 0.149015i \(-0.0476103\pi\)
−0.623468 + 0.781849i \(0.714277\pi\)
\(954\) 0 0
\(955\) −2086.00 3613.06i −0.0706821 0.122425i
\(956\) 0 0
\(957\) 7458.00 0.251915
\(958\) 0 0
\(959\) 2595.00 4494.67i 0.0873795 0.151346i
\(960\) 0 0
\(961\) 8625.00 0.289517
\(962\) 0 0
\(963\) 12042.0 0.402957
\(964\) 0 0
\(965\) 1375.50 2382.44i 0.0458849 0.0794749i
\(966\) 0 0
\(967\) 418.000 0.0139007 0.00695035 0.999976i \(-0.497788\pi\)
0.00695035 + 0.999976i \(0.497788\pi\)
\(968\) 0 0
\(969\) −1665.00 2883.86i −0.0551987 0.0956069i
\(970\) 0 0
\(971\) 9066.00 + 15702.8i 0.299631 + 0.518976i 0.976052 0.217539i \(-0.0698031\pi\)
−0.676420 + 0.736516i \(0.736470\pi\)
\(972\) 0 0
\(973\) −1740.00 + 3013.77i −0.0573297 + 0.0992980i
\(974\) 0 0
\(975\) 7410.00 7700.70i 0.243395 0.252943i
\(976\) 0 0
\(977\) −6250.50 + 10826.2i −0.204679 + 0.354514i −0.950030 0.312158i \(-0.898948\pi\)
0.745352 + 0.666672i \(0.232282\pi\)
\(978\) 0 0
\(979\) −2134.00 3696.20i −0.0696659 0.120665i
\(980\) 0 0
\(981\) 4653.00 + 8059.23i 0.151436 + 0.262295i
\(982\) 0 0
\(983\) 43708.0 1.41818 0.709089 0.705119i \(-0.249106\pi\)
0.709089 + 0.705119i \(0.249106\pi\)
\(984\) 0 0
\(985\) 12327.0 21351.0i 0.398752 0.690659i
\(986\) 0 0
\(987\) −13860.0 −0.446979
\(988\) 0 0
\(989\) 39852.0 1.28131
\(990\) 0 0
\(991\) −19807.0 + 34306.7i −0.634904 + 1.09969i 0.351631 + 0.936139i \(0.385627\pi\)
−0.986535 + 0.163548i \(0.947706\pi\)
\(992\) 0 0
\(993\) 3084.00 0.0985577
\(994\) 0 0
\(995\) 7063.00 + 12233.5i 0.225037 + 0.389776i
\(996\) 0 0
\(997\) 18251.5 + 31612.5i 0.579770 + 1.00419i 0.995505 + 0.0947056i \(0.0301910\pi\)
−0.415735 + 0.909486i \(0.636476\pi\)
\(998\) 0 0
\(999\) 175.500 303.975i 0.00555813 0.00962697i
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 624.4.q.c.529.1 2
4.3 odd 2 39.4.e.b.22.1 yes 2
12.11 even 2 117.4.g.a.100.1 2
13.3 even 3 inner 624.4.q.c.289.1 2
52.3 odd 6 39.4.e.b.16.1 2
52.7 even 12 507.4.b.d.337.1 2
52.19 even 12 507.4.b.d.337.2 2
52.35 odd 6 507.4.a.b.1.1 1
52.43 odd 6 507.4.a.d.1.1 1
156.35 even 6 1521.4.a.h.1.1 1
156.95 even 6 1521.4.a.e.1.1 1
156.107 even 6 117.4.g.a.55.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.e.b.16.1 2 52.3 odd 6
39.4.e.b.22.1 yes 2 4.3 odd 2
117.4.g.a.55.1 2 156.107 even 6
117.4.g.a.100.1 2 12.11 even 2
507.4.a.b.1.1 1 52.35 odd 6
507.4.a.d.1.1 1 52.43 odd 6
507.4.b.d.337.1 2 52.7 even 12
507.4.b.d.337.2 2 52.19 even 12
624.4.q.c.289.1 2 13.3 even 3 inner
624.4.q.c.529.1 2 1.1 even 1 trivial
1521.4.a.e.1.1 1 156.95 even 6
1521.4.a.h.1.1 1 156.35 even 6