Properties

Label 192.3.l.a.175.1
Level $192$
Weight $3$
Character 192.175
Analytic conductor $5.232$
Analytic rank $0$
Dimension $16$
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] = [192,3,Mod(79,192)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(192, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("192.79");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 192.l (of order \(4\), 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: \(5.23162107572\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 6 x^{14} - 4 x^{13} + 10 x^{12} + 56 x^{11} + 88 x^{10} - 128 x^{9} - 496 x^{8} - 512 x^{7} + 1408 x^{6} + 3584 x^{5} + 2560 x^{4} - 4096 x^{3} - 24576 x^{2} + \cdots + 65536 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{24} \)
Twist minimal: no (minimal twist has level 48)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 175.1
Root \(-1.87459 - 0.697079i\) of defining polynomial
Character \(\chi\) \(=\) 192.175
Dual form 192.3.l.a.79.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.22474 + 1.22474i) q^{3} +(-5.24354 + 5.24354i) q^{5} +5.32796 q^{7} -3.00000i q^{9} +O(q^{10})\) \(q+(-1.22474 + 1.22474i) q^{3} +(-5.24354 + 5.24354i) q^{5} +5.32796 q^{7} -3.00000i q^{9} +(-12.2863 - 12.2863i) q^{11} +(-5.73657 - 5.73657i) q^{13} -12.8440i q^{15} -23.3997 q^{17} +(-11.7492 + 11.7492i) q^{19} +(-6.52540 + 6.52540i) q^{21} -5.80841 q^{23} -29.9894i q^{25} +(3.67423 + 3.67423i) q^{27} +(18.3914 + 18.3914i) q^{29} +16.9053i q^{31} +30.0951 q^{33} +(-27.9374 + 27.9374i) q^{35} +(15.3391 - 15.3391i) q^{37} +14.0517 q^{39} -29.2351i q^{41} +(-33.4099 - 33.4099i) q^{43} +(15.7306 + 15.7306i) q^{45} +18.2125i q^{47} -20.6128 q^{49} +(28.6586 - 28.6586i) q^{51} +(-66.9856 + 66.9856i) q^{53} +128.847 q^{55} -28.7796i q^{57} +(27.1523 + 27.1523i) q^{59} +(65.2399 + 65.2399i) q^{61} -15.9839i q^{63} +60.1599 q^{65} +(37.6951 - 37.6951i) q^{67} +(7.11382 - 7.11382i) q^{69} -42.6559 q^{71} +106.391i q^{73} +(36.7294 + 36.7294i) q^{75} +(-65.4607 - 65.4607i) q^{77} +21.2821i q^{79} -9.00000 q^{81} +(-24.1638 + 24.1638i) q^{83} +(122.697 - 122.697i) q^{85} -45.0495 q^{87} +52.8029i q^{89} +(-30.5643 - 30.5643i) q^{91} +(-20.7047 - 20.7047i) q^{93} -123.215i q^{95} -21.0222 q^{97} +(-36.8588 + 36.8588i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 32 q^{11} + 32 q^{19} + 128 q^{23} + 32 q^{29} - 96 q^{35} - 96 q^{37} - 160 q^{43} + 112 q^{49} + 96 q^{51} - 160 q^{53} + 256 q^{55} + 128 q^{59} - 32 q^{61} - 32 q^{65} - 320 q^{67} + 96 q^{69} - 512 q^{71} - 192 q^{75} + 224 q^{77} - 144 q^{81} + 160 q^{83} + 160 q^{85} + 480 q^{91} - 96 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/192\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(133\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{3}{4}\right)\)

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
\(3\) −1.22474 + 1.22474i −0.408248 + 0.408248i
\(4\) 0 0
\(5\) −5.24354 + 5.24354i −1.04871 + 1.04871i −0.0499563 + 0.998751i \(0.515908\pi\)
−0.998751 + 0.0499563i \(0.984092\pi\)
\(6\) 0 0
\(7\) 5.32796 0.761138 0.380569 0.924753i \(-0.375728\pi\)
0.380569 + 0.924753i \(0.375728\pi\)
\(8\) 0 0
\(9\) 3.00000i 0.333333i
\(10\) 0 0
\(11\) −12.2863 12.2863i −1.11693 1.11693i −0.992189 0.124743i \(-0.960189\pi\)
−0.124743 0.992189i \(-0.539811\pi\)
\(12\) 0 0
\(13\) −5.73657 5.73657i −0.441275 0.441275i 0.451165 0.892440i \(-0.351008\pi\)
−0.892440 + 0.451165i \(0.851008\pi\)
\(14\) 0 0
\(15\) 12.8440i 0.856266i
\(16\) 0 0
\(17\) −23.3997 −1.37645 −0.688226 0.725496i \(-0.741610\pi\)
−0.688226 + 0.725496i \(0.741610\pi\)
\(18\) 0 0
\(19\) −11.7492 + 11.7492i −0.618380 + 0.618380i −0.945116 0.326736i \(-0.894051\pi\)
0.326736 + 0.945116i \(0.394051\pi\)
\(20\) 0 0
\(21\) −6.52540 + 6.52540i −0.310733 + 0.310733i
\(22\) 0 0
\(23\) −5.80841 −0.252540 −0.126270 0.991996i \(-0.540301\pi\)
−0.126270 + 0.991996i \(0.540301\pi\)
\(24\) 0 0
\(25\) 29.9894i 1.19958i
\(26\) 0 0
\(27\) 3.67423 + 3.67423i 0.136083 + 0.136083i
\(28\) 0 0
\(29\) 18.3914 + 18.3914i 0.634185 + 0.634185i 0.949115 0.314930i \(-0.101981\pi\)
−0.314930 + 0.949115i \(0.601981\pi\)
\(30\) 0 0
\(31\) 16.9053i 0.545332i 0.962109 + 0.272666i \(0.0879053\pi\)
−0.962109 + 0.272666i \(0.912095\pi\)
\(32\) 0 0
\(33\) 30.0951 0.911971
\(34\) 0 0
\(35\) −27.9374 + 27.9374i −0.798211 + 0.798211i
\(36\) 0 0
\(37\) 15.3391 15.3391i 0.414571 0.414571i −0.468756 0.883327i \(-0.655298\pi\)
0.883327 + 0.468756i \(0.155298\pi\)
\(38\) 0 0
\(39\) 14.0517 0.360299
\(40\) 0 0
\(41\) 29.2351i 0.713051i −0.934286 0.356526i \(-0.883961\pi\)
0.934286 0.356526i \(-0.116039\pi\)
\(42\) 0 0
\(43\) −33.4099 33.4099i −0.776975 0.776975i 0.202340 0.979315i \(-0.435145\pi\)
−0.979315 + 0.202340i \(0.935145\pi\)
\(44\) 0 0
\(45\) 15.7306 + 15.7306i 0.349569 + 0.349569i
\(46\) 0 0
\(47\) 18.2125i 0.387500i 0.981051 + 0.193750i \(0.0620650\pi\)
−0.981051 + 0.193750i \(0.937935\pi\)
\(48\) 0 0
\(49\) −20.6128 −0.420670
\(50\) 0 0
\(51\) 28.6586 28.6586i 0.561934 0.561934i
\(52\) 0 0
\(53\) −66.9856 + 66.9856i −1.26388 + 1.26388i −0.314681 + 0.949197i \(0.601898\pi\)
−0.949197 + 0.314681i \(0.898102\pi\)
\(54\) 0 0
\(55\) 128.847 2.34267
\(56\) 0 0
\(57\) 28.7796i 0.504905i
\(58\) 0 0
\(59\) 27.1523 + 27.1523i 0.460209 + 0.460209i 0.898724 0.438515i \(-0.144495\pi\)
−0.438515 + 0.898724i \(0.644495\pi\)
\(60\) 0 0
\(61\) 65.2399 + 65.2399i 1.06951 + 1.06951i 0.997397 + 0.0721103i \(0.0229733\pi\)
0.0721103 + 0.997397i \(0.477027\pi\)
\(62\) 0 0
\(63\) 15.9839i 0.253713i
\(64\) 0 0
\(65\) 60.1599 0.925537
\(66\) 0 0
\(67\) 37.6951 37.6951i 0.562614 0.562614i −0.367435 0.930049i \(-0.619764\pi\)
0.930049 + 0.367435i \(0.119764\pi\)
\(68\) 0 0
\(69\) 7.11382 7.11382i 0.103099 0.103099i
\(70\) 0 0
\(71\) −42.6559 −0.600788 −0.300394 0.953815i \(-0.597118\pi\)
−0.300394 + 0.953815i \(0.597118\pi\)
\(72\) 0 0
\(73\) 106.391i 1.45742i 0.684825 + 0.728708i \(0.259879\pi\)
−0.684825 + 0.728708i \(0.740121\pi\)
\(74\) 0 0
\(75\) 36.7294 + 36.7294i 0.489725 + 0.489725i
\(76\) 0 0
\(77\) −65.4607 65.4607i −0.850139 0.850139i
\(78\) 0 0
\(79\) 21.2821i 0.269394i 0.990887 + 0.134697i \(0.0430061\pi\)
−0.990887 + 0.134697i \(0.956994\pi\)
\(80\) 0 0
\(81\) −9.00000 −0.111111
\(82\) 0 0
\(83\) −24.1638 + 24.1638i −0.291130 + 0.291130i −0.837527 0.546396i \(-0.815999\pi\)
0.546396 + 0.837527i \(0.315999\pi\)
\(84\) 0 0
\(85\) 122.697 122.697i 1.44350 1.44350i
\(86\) 0 0
\(87\) −45.0495 −0.517810
\(88\) 0 0
\(89\) 52.8029i 0.593291i 0.954988 + 0.296645i \(0.0958679\pi\)
−0.954988 + 0.296645i \(0.904132\pi\)
\(90\) 0 0
\(91\) −30.5643 30.5643i −0.335871 0.335871i
\(92\) 0 0
\(93\) −20.7047 20.7047i −0.222631 0.222631i
\(94\) 0 0
\(95\) 123.215i 1.29700i
\(96\) 0 0
\(97\) −21.0222 −0.216724 −0.108362 0.994112i \(-0.534560\pi\)
−0.108362 + 0.994112i \(0.534560\pi\)
\(98\) 0 0
\(99\) −36.8588 + 36.8588i −0.372311 + 0.372311i
\(100\) 0 0
\(101\) −3.24960 + 3.24960i −0.0321743 + 0.0321743i −0.723011 0.690837i \(-0.757242\pi\)
0.690837 + 0.723011i \(0.257242\pi\)
\(102\) 0 0
\(103\) −105.112 −1.02050 −0.510252 0.860025i \(-0.670448\pi\)
−0.510252 + 0.860025i \(0.670448\pi\)
\(104\) 0 0
\(105\) 68.4323i 0.651736i
\(106\) 0 0
\(107\) 99.6160 + 99.6160i 0.930991 + 0.930991i 0.997768 0.0667770i \(-0.0212716\pi\)
−0.0667770 + 0.997768i \(0.521272\pi\)
\(108\) 0 0
\(109\) −108.050 108.050i −0.991282 0.991282i 0.00868078 0.999962i \(-0.497237\pi\)
−0.999962 + 0.00868078i \(0.997237\pi\)
\(110\) 0 0
\(111\) 37.5730i 0.338496i
\(112\) 0 0
\(113\) −23.2835 −0.206048 −0.103024 0.994679i \(-0.532852\pi\)
−0.103024 + 0.994679i \(0.532852\pi\)
\(114\) 0 0
\(115\) 30.4566 30.4566i 0.264840 0.264840i
\(116\) 0 0
\(117\) −17.2097 + 17.2097i −0.147092 + 0.147092i
\(118\) 0 0
\(119\) −124.673 −1.04767
\(120\) 0 0
\(121\) 180.904i 1.49508i
\(122\) 0 0
\(123\) 35.8055 + 35.8055i 0.291102 + 0.291102i
\(124\) 0 0
\(125\) 26.1621 + 26.1621i 0.209297 + 0.209297i
\(126\) 0 0
\(127\) 118.180i 0.930550i 0.885166 + 0.465275i \(0.154045\pi\)
−0.885166 + 0.465275i \(0.845955\pi\)
\(128\) 0 0
\(129\) 81.8373 0.634398
\(130\) 0 0
\(131\) 69.2067 69.2067i 0.528296 0.528296i −0.391768 0.920064i \(-0.628137\pi\)
0.920064 + 0.391768i \(0.128137\pi\)
\(132\) 0 0
\(133\) −62.5994 + 62.5994i −0.470672 + 0.470672i
\(134\) 0 0
\(135\) −38.5320 −0.285422
\(136\) 0 0
\(137\) 124.474i 0.908572i −0.890856 0.454286i \(-0.849894\pi\)
0.890856 0.454286i \(-0.150106\pi\)
\(138\) 0 0
\(139\) −169.014 169.014i −1.21593 1.21593i −0.969046 0.246881i \(-0.920594\pi\)
−0.246881 0.969046i \(-0.579406\pi\)
\(140\) 0 0
\(141\) −22.3057 22.3057i −0.158196 0.158196i
\(142\) 0 0
\(143\) 140.962i 0.985749i
\(144\) 0 0
\(145\) −192.872 −1.33015
\(146\) 0 0
\(147\) 25.2454 25.2454i 0.171738 0.171738i
\(148\) 0 0
\(149\) 146.988 146.988i 0.986495 0.986495i −0.0134145 0.999910i \(-0.504270\pi\)
0.999910 + 0.0134145i \(0.00427011\pi\)
\(150\) 0 0
\(151\) −75.5456 −0.500302 −0.250151 0.968207i \(-0.580480\pi\)
−0.250151 + 0.968207i \(0.580480\pi\)
\(152\) 0 0
\(153\) 70.1991i 0.458817i
\(154\) 0 0
\(155\) −88.6435 88.6435i −0.571893 0.571893i
\(156\) 0 0
\(157\) −81.5356 81.5356i −0.519335 0.519335i 0.398035 0.917370i \(-0.369692\pi\)
−0.917370 + 0.398035i \(0.869692\pi\)
\(158\) 0 0
\(159\) 164.080i 1.03195i
\(160\) 0 0
\(161\) −30.9470 −0.192217
\(162\) 0 0
\(163\) −55.8065 + 55.8065i −0.342371 + 0.342371i −0.857258 0.514887i \(-0.827834\pi\)
0.514887 + 0.857258i \(0.327834\pi\)
\(164\) 0 0
\(165\) −157.805 + 157.805i −0.956391 + 0.956391i
\(166\) 0 0
\(167\) 24.6339 0.147508 0.0737540 0.997276i \(-0.476502\pi\)
0.0737540 + 0.997276i \(0.476502\pi\)
\(168\) 0 0
\(169\) 103.183i 0.610553i
\(170\) 0 0
\(171\) 35.2476 + 35.2476i 0.206127 + 0.206127i
\(172\) 0 0
\(173\) 4.88551 + 4.88551i 0.0282399 + 0.0282399i 0.721086 0.692846i \(-0.243643\pi\)
−0.692846 + 0.721086i \(0.743643\pi\)
\(174\) 0 0
\(175\) 159.782i 0.913042i
\(176\) 0 0
\(177\) −66.5094 −0.375759
\(178\) 0 0
\(179\) 229.504 229.504i 1.28215 1.28215i 0.342702 0.939444i \(-0.388658\pi\)
0.939444 0.342702i \(-0.111342\pi\)
\(180\) 0 0
\(181\) 116.607 116.607i 0.644238 0.644238i −0.307356 0.951595i \(-0.599444\pi\)
0.951595 + 0.307356i \(0.0994443\pi\)
\(182\) 0 0
\(183\) −159.805 −0.873249
\(184\) 0 0
\(185\) 160.863i 0.869528i
\(186\) 0 0
\(187\) 287.495 + 287.495i 1.53740 + 1.53740i
\(188\) 0 0
\(189\) 19.5762 + 19.5762i 0.103578 + 0.103578i
\(190\) 0 0
\(191\) 94.2316i 0.493359i −0.969097 0.246680i \(-0.920660\pi\)
0.969097 0.246680i \(-0.0793395\pi\)
\(192\) 0 0
\(193\) 84.2667 0.436615 0.218308 0.975880i \(-0.429946\pi\)
0.218308 + 0.975880i \(0.429946\pi\)
\(194\) 0 0
\(195\) −73.6805 + 73.6805i −0.377849 + 0.377849i
\(196\) 0 0
\(197\) −56.9578 + 56.9578i −0.289126 + 0.289126i −0.836734 0.547609i \(-0.815538\pi\)
0.547609 + 0.836734i \(0.315538\pi\)
\(198\) 0 0
\(199\) 196.179 0.985827 0.492913 0.870078i \(-0.335932\pi\)
0.492913 + 0.870078i \(0.335932\pi\)
\(200\) 0 0
\(201\) 92.3338i 0.459372i
\(202\) 0 0
\(203\) 97.9886 + 97.9886i 0.482702 + 0.482702i
\(204\) 0 0
\(205\) 153.295 + 153.295i 0.747782 + 0.747782i
\(206\) 0 0
\(207\) 17.4252i 0.0841799i
\(208\) 0 0
\(209\) 288.708 1.38138
\(210\) 0 0
\(211\) −177.340 + 177.340i −0.840475 + 0.840475i −0.988921 0.148445i \(-0.952573\pi\)
0.148445 + 0.988921i \(0.452573\pi\)
\(212\) 0 0
\(213\) 52.2426 52.2426i 0.245271 0.245271i
\(214\) 0 0
\(215\) 350.373 1.62964
\(216\) 0 0
\(217\) 90.0707i 0.415072i
\(218\) 0 0
\(219\) −130.302 130.302i −0.594987 0.594987i
\(220\) 0 0
\(221\) 134.234 + 134.234i 0.607394 + 0.607394i
\(222\) 0 0
\(223\) 377.924i 1.69473i 0.531012 + 0.847364i \(0.321812\pi\)
−0.531012 + 0.847364i \(0.678188\pi\)
\(224\) 0 0
\(225\) −89.9682 −0.399859
\(226\) 0 0
\(227\) −103.909 + 103.909i −0.457750 + 0.457750i −0.897916 0.440166i \(-0.854920\pi\)
0.440166 + 0.897916i \(0.354920\pi\)
\(228\) 0 0
\(229\) −101.055 + 101.055i −0.441290 + 0.441290i −0.892445 0.451156i \(-0.851012\pi\)
0.451156 + 0.892445i \(0.351012\pi\)
\(230\) 0 0
\(231\) 160.345 0.694136
\(232\) 0 0
\(233\) 287.259i 1.23287i −0.787405 0.616436i \(-0.788576\pi\)
0.787405 0.616436i \(-0.211424\pi\)
\(234\) 0 0
\(235\) −95.4979 95.4979i −0.406374 0.406374i
\(236\) 0 0
\(237\) −26.0651 26.0651i −0.109980 0.109980i
\(238\) 0 0
\(239\) 150.941i 0.631554i −0.948833 0.315777i \(-0.897735\pi\)
0.948833 0.315777i \(-0.102265\pi\)
\(240\) 0 0
\(241\) 37.7817 0.156771 0.0783853 0.996923i \(-0.475024\pi\)
0.0783853 + 0.996923i \(0.475024\pi\)
\(242\) 0 0
\(243\) 11.0227 11.0227i 0.0453609 0.0453609i
\(244\) 0 0
\(245\) 108.084 108.084i 0.441159 0.441159i
\(246\) 0 0
\(247\) 134.800 0.545751
\(248\) 0 0
\(249\) 59.1890i 0.237707i
\(250\) 0 0
\(251\) −100.915 100.915i −0.402050 0.402050i 0.476905 0.878955i \(-0.341759\pi\)
−0.878955 + 0.476905i \(0.841759\pi\)
\(252\) 0 0
\(253\) 71.3637 + 71.3637i 0.282070 + 0.282070i
\(254\) 0 0
\(255\) 300.545i 1.17861i
\(256\) 0 0
\(257\) 241.295 0.938891 0.469446 0.882961i \(-0.344454\pi\)
0.469446 + 0.882961i \(0.344454\pi\)
\(258\) 0 0
\(259\) 81.7263 81.7263i 0.315546 0.315546i
\(260\) 0 0
\(261\) 55.1741 55.1741i 0.211395 0.211395i
\(262\) 0 0
\(263\) 118.747 0.451509 0.225754 0.974184i \(-0.427515\pi\)
0.225754 + 0.974184i \(0.427515\pi\)
\(264\) 0 0
\(265\) 702.483i 2.65088i
\(266\) 0 0
\(267\) −64.6700 64.6700i −0.242210 0.242210i
\(268\) 0 0
\(269\) 7.74853 + 7.74853i 0.0288050 + 0.0288050i 0.721363 0.692558i \(-0.243516\pi\)
−0.692558 + 0.721363i \(0.743516\pi\)
\(270\) 0 0
\(271\) 131.899i 0.486712i −0.969937 0.243356i \(-0.921752\pi\)
0.969937 0.243356i \(-0.0782484\pi\)
\(272\) 0 0
\(273\) 74.8668 0.274237
\(274\) 0 0
\(275\) −368.457 + 368.457i −1.33984 + 1.33984i
\(276\) 0 0
\(277\) −202.352 + 202.352i −0.730513 + 0.730513i −0.970721 0.240208i \(-0.922784\pi\)
0.240208 + 0.970721i \(0.422784\pi\)
\(278\) 0 0
\(279\) 50.7158 0.181777
\(280\) 0 0
\(281\) 68.8493i 0.245015i 0.992468 + 0.122508i \(0.0390936\pi\)
−0.992468 + 0.122508i \(0.960906\pi\)
\(282\) 0 0
\(283\) −206.773 206.773i −0.730646 0.730646i 0.240102 0.970748i \(-0.422819\pi\)
−0.970748 + 0.240102i \(0.922819\pi\)
\(284\) 0 0
\(285\) 150.907 + 150.907i 0.529498 + 0.529498i
\(286\) 0 0
\(287\) 155.764i 0.542730i
\(288\) 0 0
\(289\) 258.545 0.894620
\(290\) 0 0
\(291\) 25.7468 25.7468i 0.0884770 0.0884770i
\(292\) 0 0
\(293\) −361.237 + 361.237i −1.23289 + 1.23289i −0.270043 + 0.962848i \(0.587038\pi\)
−0.962848 + 0.270043i \(0.912962\pi\)
\(294\) 0 0
\(295\) −284.749 −0.965250
\(296\) 0 0
\(297\) 90.2852i 0.303990i
\(298\) 0 0
\(299\) 33.3204 + 33.3204i 0.111439 + 0.111439i
\(300\) 0 0
\(301\) −178.007 178.007i −0.591385 0.591385i
\(302\) 0 0
\(303\) 7.95987i 0.0262702i
\(304\) 0 0
\(305\) −684.176 −2.24320
\(306\) 0 0
\(307\) 10.9073 10.9073i 0.0355286 0.0355286i −0.689119 0.724648i \(-0.742002\pi\)
0.724648 + 0.689119i \(0.242002\pi\)
\(308\) 0 0
\(309\) 128.735 128.735i 0.416619 0.416619i
\(310\) 0 0
\(311\) 160.251 0.515278 0.257639 0.966241i \(-0.417055\pi\)
0.257639 + 0.966241i \(0.417055\pi\)
\(312\) 0 0
\(313\) 355.500i 1.13578i 0.823103 + 0.567892i \(0.192241\pi\)
−0.823103 + 0.567892i \(0.807759\pi\)
\(314\) 0 0
\(315\) 83.8121 + 83.8121i 0.266070 + 0.266070i
\(316\) 0 0
\(317\) 72.5192 + 72.5192i 0.228767 + 0.228767i 0.812178 0.583410i \(-0.198282\pi\)
−0.583410 + 0.812178i \(0.698282\pi\)
\(318\) 0 0
\(319\) 451.922i 1.41668i
\(320\) 0 0
\(321\) −244.008 −0.760151
\(322\) 0 0
\(323\) 274.928 274.928i 0.851170 0.851170i
\(324\) 0 0
\(325\) −172.036 + 172.036i −0.529343 + 0.529343i
\(326\) 0 0
\(327\) 264.667 0.809378
\(328\) 0 0
\(329\) 97.0355i 0.294941i
\(330\) 0 0
\(331\) 248.096 + 248.096i 0.749536 + 0.749536i 0.974392 0.224856i \(-0.0721912\pi\)
−0.224856 + 0.974392i \(0.572191\pi\)
\(332\) 0 0
\(333\) −46.0174 46.0174i −0.138190 0.138190i
\(334\) 0 0
\(335\) 395.312i 1.18003i
\(336\) 0 0
\(337\) −467.271 −1.38656 −0.693280 0.720668i \(-0.743835\pi\)
−0.693280 + 0.720668i \(0.743835\pi\)
\(338\) 0 0
\(339\) 28.5163 28.5163i 0.0841189 0.0841189i
\(340\) 0 0
\(341\) 207.703 207.703i 0.609098 0.609098i
\(342\) 0 0
\(343\) −370.894 −1.08133
\(344\) 0 0
\(345\) 74.6032i 0.216241i
\(346\) 0 0
\(347\) −292.821 292.821i −0.843863 0.843863i 0.145496 0.989359i \(-0.453522\pi\)
−0.989359 + 0.145496i \(0.953522\pi\)
\(348\) 0 0
\(349\) 346.260 + 346.260i 0.992150 + 0.992150i 0.999969 0.00781941i \(-0.00248902\pi\)
−0.00781941 + 0.999969i \(0.502489\pi\)
\(350\) 0 0
\(351\) 42.1550i 0.120100i
\(352\) 0 0
\(353\) 8.01816 0.0227143 0.0113572 0.999936i \(-0.496385\pi\)
0.0113572 + 0.999936i \(0.496385\pi\)
\(354\) 0 0
\(355\) 223.668 223.668i 0.630051 0.630051i
\(356\) 0 0
\(357\) 152.692 152.692i 0.427709 0.427709i
\(358\) 0 0
\(359\) −590.403 −1.64458 −0.822289 0.569071i \(-0.807303\pi\)
−0.822289 + 0.569071i \(0.807303\pi\)
\(360\) 0 0
\(361\) 84.9121i 0.235213i
\(362\) 0 0
\(363\) −221.561 221.561i −0.610362 0.610362i
\(364\) 0 0
\(365\) −557.867 557.867i −1.52840 1.52840i
\(366\) 0 0
\(367\) 397.100i 1.08202i −0.841017 0.541008i \(-0.818043\pi\)
0.841017 0.541008i \(-0.181957\pi\)
\(368\) 0 0
\(369\) −87.7053 −0.237684
\(370\) 0 0
\(371\) −356.897 + 356.897i −0.961986 + 0.961986i
\(372\) 0 0
\(373\) −165.010 + 165.010i −0.442387 + 0.442387i −0.892814 0.450427i \(-0.851272\pi\)
0.450427 + 0.892814i \(0.351272\pi\)
\(374\) 0 0
\(375\) −64.0837 −0.170890
\(376\) 0 0
\(377\) 211.007i 0.559700i
\(378\) 0 0
\(379\) 206.669 + 206.669i 0.545300 + 0.545300i 0.925078 0.379778i \(-0.124000\pi\)
−0.379778 + 0.925078i \(0.624000\pi\)
\(380\) 0 0
\(381\) −144.740 144.740i −0.379895 0.379895i
\(382\) 0 0
\(383\) 598.414i 1.56244i 0.624257 + 0.781219i \(0.285402\pi\)
−0.624257 + 0.781219i \(0.714598\pi\)
\(384\) 0 0
\(385\) 686.492 1.78310
\(386\) 0 0
\(387\) −100.230 + 100.230i −0.258992 + 0.258992i
\(388\) 0 0
\(389\) 186.696 186.696i 0.479939 0.479939i −0.425173 0.905112i \(-0.639787\pi\)
0.905112 + 0.425173i \(0.139787\pi\)
\(390\) 0 0
\(391\) 135.915 0.347609
\(392\) 0 0
\(393\) 169.521i 0.431352i
\(394\) 0 0
\(395\) −111.594 111.594i −0.282515 0.282515i
\(396\) 0 0
\(397\) −57.3727 57.3727i −0.144516 0.144516i 0.631147 0.775663i \(-0.282584\pi\)
−0.775663 + 0.631147i \(0.782584\pi\)
\(398\) 0 0
\(399\) 153.337i 0.384302i
\(400\) 0 0
\(401\) −466.082 −1.16230 −0.581149 0.813797i \(-0.697397\pi\)
−0.581149 + 0.813797i \(0.697397\pi\)
\(402\) 0 0
\(403\) 96.9784 96.9784i 0.240641 0.240641i
\(404\) 0 0
\(405\) 47.1918 47.1918i 0.116523 0.116523i
\(406\) 0 0
\(407\) −376.921 −0.926096
\(408\) 0 0
\(409\) 597.952i 1.46198i −0.682386 0.730992i \(-0.739058\pi\)
0.682386 0.730992i \(-0.260942\pi\)
\(410\) 0 0
\(411\) 152.449 + 152.449i 0.370923 + 0.370923i
\(412\) 0 0
\(413\) 144.667 + 144.667i 0.350282 + 0.350282i
\(414\) 0 0
\(415\) 253.408i 0.610621i
\(416\) 0 0
\(417\) 413.998 0.992800
\(418\) 0 0
\(419\) 4.65301 4.65301i 0.0111050 0.0111050i −0.701532 0.712638i \(-0.747500\pi\)
0.712638 + 0.701532i \(0.247500\pi\)
\(420\) 0 0
\(421\) 34.3754 34.3754i 0.0816519 0.0816519i −0.665101 0.746753i \(-0.731612\pi\)
0.746753 + 0.665101i \(0.231612\pi\)
\(422\) 0 0
\(423\) 54.6375 0.129167
\(424\) 0 0
\(425\) 701.742i 1.65116i
\(426\) 0 0
\(427\) 347.596 + 347.596i 0.814042 + 0.814042i
\(428\) 0 0
\(429\) −172.643 172.643i −0.402430 0.402430i
\(430\) 0 0
\(431\) 423.823i 0.983347i 0.870780 + 0.491674i \(0.163615\pi\)
−0.870780 + 0.491674i \(0.836385\pi\)
\(432\) 0 0
\(433\) 833.377 1.92466 0.962330 0.271885i \(-0.0876472\pi\)
0.962330 + 0.271885i \(0.0876472\pi\)
\(434\) 0 0
\(435\) 236.219 236.219i 0.543031 0.543031i
\(436\) 0 0
\(437\) 68.2443 68.2443i 0.156165 0.156165i
\(438\) 0 0
\(439\) −32.3193 −0.0736203 −0.0368102 0.999322i \(-0.511720\pi\)
−0.0368102 + 0.999322i \(0.511720\pi\)
\(440\) 0 0
\(441\) 61.8384i 0.140223i
\(442\) 0 0
\(443\) −119.527 119.527i −0.269813 0.269813i 0.559212 0.829025i \(-0.311104\pi\)
−0.829025 + 0.559212i \(0.811104\pi\)
\(444\) 0 0
\(445\) −276.874 276.874i −0.622189 0.622189i
\(446\) 0 0
\(447\) 360.045i 0.805470i
\(448\) 0 0
\(449\) −182.359 −0.406146 −0.203073 0.979164i \(-0.565093\pi\)
−0.203073 + 0.979164i \(0.565093\pi\)
\(450\) 0 0
\(451\) −359.190 + 359.190i −0.796430 + 0.796430i
\(452\) 0 0
\(453\) 92.5241 92.5241i 0.204248 0.204248i
\(454\) 0 0
\(455\) 320.530 0.704461
\(456\) 0 0
\(457\) 272.942i 0.597246i 0.954371 + 0.298623i \(0.0965274\pi\)
−0.954371 + 0.298623i \(0.903473\pi\)
\(458\) 0 0
\(459\) −85.9759 85.9759i −0.187311 0.187311i
\(460\) 0 0
\(461\) 188.323 + 188.323i 0.408510 + 0.408510i 0.881219 0.472709i \(-0.156724\pi\)
−0.472709 + 0.881219i \(0.656724\pi\)
\(462\) 0 0
\(463\) 116.023i 0.250590i 0.992120 + 0.125295i \(0.0399877\pi\)
−0.992120 + 0.125295i \(0.960012\pi\)
\(464\) 0 0
\(465\) 217.131 0.466949
\(466\) 0 0
\(467\) 271.914 271.914i 0.582257 0.582257i −0.353266 0.935523i \(-0.614929\pi\)
0.935523 + 0.353266i \(0.114929\pi\)
\(468\) 0 0
\(469\) 200.838 200.838i 0.428227 0.428227i
\(470\) 0 0
\(471\) 199.721 0.424035
\(472\) 0 0
\(473\) 820.966i 1.73566i
\(474\) 0 0
\(475\) 352.352 + 352.352i 0.741793 + 0.741793i
\(476\) 0 0
\(477\) 200.957 + 200.957i 0.421293 + 0.421293i
\(478\) 0 0
\(479\) 775.808i 1.61964i −0.586678 0.809820i \(-0.699565\pi\)
0.586678 0.809820i \(-0.300435\pi\)
\(480\) 0 0
\(481\) −175.988 −0.365880
\(482\) 0 0
\(483\) 37.9022 37.9022i 0.0784725 0.0784725i
\(484\) 0 0
\(485\) 110.231 110.231i 0.227280 0.227280i
\(486\) 0 0
\(487\) −174.891 −0.359118 −0.179559 0.983747i \(-0.557467\pi\)
−0.179559 + 0.983747i \(0.557467\pi\)
\(488\) 0 0
\(489\) 136.697i 0.279545i
\(490\) 0 0
\(491\) 348.578 + 348.578i 0.709934 + 0.709934i 0.966521 0.256587i \(-0.0825980\pi\)
−0.256587 + 0.966521i \(0.582598\pi\)
\(492\) 0 0
\(493\) −430.352 430.352i −0.872926 0.872926i
\(494\) 0 0
\(495\) 386.541i 0.780890i
\(496\) 0 0
\(497\) −227.269 −0.457282
\(498\) 0 0
\(499\) 607.544 607.544i 1.21752 1.21752i 0.249027 0.968496i \(-0.419889\pi\)
0.968496 0.249027i \(-0.0801109\pi\)
\(500\) 0 0
\(501\) −30.1702 + 30.1702i −0.0602199 + 0.0602199i
\(502\) 0 0
\(503\) 130.935 0.260309 0.130154 0.991494i \(-0.458453\pi\)
0.130154 + 0.991494i \(0.458453\pi\)
\(504\) 0 0
\(505\) 34.0789i 0.0674829i
\(506\) 0 0
\(507\) 126.373 + 126.373i 0.249257 + 0.249257i
\(508\) 0 0
\(509\) −61.5539 61.5539i −0.120931 0.120931i 0.644051 0.764982i \(-0.277252\pi\)
−0.764982 + 0.644051i \(0.777252\pi\)
\(510\) 0 0
\(511\) 566.849i 1.10929i
\(512\) 0 0
\(513\) −86.3387 −0.168302
\(514\) 0 0
\(515\) 551.159 551.159i 1.07021 1.07021i
\(516\) 0 0
\(517\) 223.763 223.763i 0.432811 0.432811i
\(518\) 0 0
\(519\) −11.9670 −0.0230578
\(520\) 0 0
\(521\) 32.5929i 0.0625584i 0.999511 + 0.0312792i \(0.00995810\pi\)
−0.999511 + 0.0312792i \(0.990042\pi\)
\(522\) 0 0
\(523\) 226.407 + 226.407i 0.432900 + 0.432900i 0.889614 0.456713i \(-0.150974\pi\)
−0.456713 + 0.889614i \(0.650974\pi\)
\(524\) 0 0
\(525\) 195.693 + 195.693i 0.372748 + 0.372748i
\(526\) 0 0
\(527\) 395.578i 0.750623i
\(528\) 0 0
\(529\) −495.262 −0.936224
\(530\) 0 0
\(531\) 81.4570 81.4570i 0.153403 0.153403i
\(532\) 0 0
\(533\) −167.709 + 167.709i −0.314652 + 0.314652i
\(534\) 0 0
\(535\) −1044.68 −1.95267
\(536\) 0 0
\(537\) 562.168i 1.04687i
\(538\) 0 0
\(539\) 253.254 + 253.254i 0.469859 + 0.469859i
\(540\) 0 0
\(541\) 510.912 + 510.912i 0.944385 + 0.944385i 0.998533 0.0541480i \(-0.0172443\pi\)
−0.0541480 + 0.998533i \(0.517244\pi\)
\(542\) 0 0
\(543\) 285.628i 0.526018i
\(544\) 0 0
\(545\) 1133.13 2.07913
\(546\) 0 0
\(547\) −512.889 + 512.889i −0.937639 + 0.937639i −0.998167 0.0605271i \(-0.980722\pi\)
0.0605271 + 0.998167i \(0.480722\pi\)
\(548\) 0 0
\(549\) 195.720 195.720i 0.356502 0.356502i
\(550\) 0 0
\(551\) −432.168 −0.784334
\(552\) 0 0
\(553\) 113.390i 0.205046i
\(554\) 0 0
\(555\) −197.016 197.016i −0.354983 0.354983i
\(556\) 0 0
\(557\) 566.691 + 566.691i 1.01740 + 1.01740i 0.999846 + 0.0175529i \(0.00558754\pi\)
0.0175529 + 0.999846i \(0.494412\pi\)
\(558\) 0 0
\(559\) 383.317i 0.685720i
\(560\) 0 0
\(561\) −704.215 −1.25529
\(562\) 0 0
\(563\) −548.653 + 548.653i −0.974517 + 0.974517i −0.999683 0.0251665i \(-0.991988\pi\)
0.0251665 + 0.999683i \(0.491988\pi\)
\(564\) 0 0
\(565\) 122.088 122.088i 0.216085 0.216085i
\(566\) 0 0
\(567\) −47.9517 −0.0845708
\(568\) 0 0
\(569\) 551.224i 0.968760i −0.874858 0.484380i \(-0.839045\pi\)
0.874858 0.484380i \(-0.160955\pi\)
\(570\) 0 0
\(571\) −458.387 458.387i −0.802780 0.802780i 0.180749 0.983529i \(-0.442148\pi\)
−0.983529 + 0.180749i \(0.942148\pi\)
\(572\) 0 0
\(573\) 115.410 + 115.410i 0.201413 + 0.201413i
\(574\) 0 0
\(575\) 174.191i 0.302941i
\(576\) 0 0
\(577\) −718.488 −1.24521 −0.622607 0.782535i \(-0.713926\pi\)
−0.622607 + 0.782535i \(0.713926\pi\)
\(578\) 0 0
\(579\) −103.205 + 103.205i −0.178247 + 0.178247i
\(580\) 0 0
\(581\) −128.744 + 128.744i −0.221590 + 0.221590i
\(582\) 0 0
\(583\) 1646.00 2.82333
\(584\) 0 0
\(585\) 180.480i 0.308512i
\(586\) 0 0
\(587\) −3.02450 3.02450i −0.00515247 0.00515247i 0.704526 0.709678i \(-0.251160\pi\)
−0.709678 + 0.704526i \(0.751160\pi\)
\(588\) 0 0
\(589\) −198.624 198.624i −0.337222 0.337222i
\(590\) 0 0
\(591\) 139.517i 0.236070i
\(592\) 0 0
\(593\) 576.193 0.971657 0.485829 0.874054i \(-0.338518\pi\)
0.485829 + 0.874054i \(0.338518\pi\)
\(594\) 0 0
\(595\) 653.726 653.726i 1.09870 1.09870i
\(596\) 0 0
\(597\) −240.270 + 240.270i −0.402462 + 0.402462i
\(598\) 0 0
\(599\) 1101.40 1.83873 0.919365 0.393406i \(-0.128703\pi\)
0.919365 + 0.393406i \(0.128703\pi\)
\(600\) 0 0
\(601\) 7.11053i 0.0118312i −0.999983 0.00591558i \(-0.998117\pi\)
0.999983 0.00591558i \(-0.00188300\pi\)
\(602\) 0 0
\(603\) −113.085 113.085i −0.187538 0.187538i
\(604\) 0 0
\(605\) −948.578 948.578i −1.56790 1.56790i
\(606\) 0 0
\(607\) 528.384i 0.870485i −0.900313 0.435242i \(-0.856663\pi\)
0.900313 0.435242i \(-0.143337\pi\)
\(608\) 0 0
\(609\) −240.022 −0.394125
\(610\) 0 0
\(611\) 104.477 104.477i 0.170994 0.170994i
\(612\) 0 0
\(613\) −642.364 + 642.364i −1.04790 + 1.04790i −0.0491093 + 0.998793i \(0.515638\pi\)
−0.998793 + 0.0491093i \(0.984362\pi\)
\(614\) 0 0
\(615\) −375.496 −0.610562
\(616\) 0 0
\(617\) 1068.16i 1.73122i −0.500717 0.865611i \(-0.666930\pi\)
0.500717 0.865611i \(-0.333070\pi\)
\(618\) 0 0
\(619\) −691.136 691.136i −1.11654 1.11654i −0.992246 0.124290i \(-0.960335\pi\)
−0.124290 0.992246i \(-0.539665\pi\)
\(620\) 0 0
\(621\) −21.3415 21.3415i −0.0343663 0.0343663i
\(622\) 0 0
\(623\) 281.332i 0.451576i
\(624\) 0 0
\(625\) 475.371 0.760594
\(626\) 0 0
\(627\) −353.593 + 353.593i −0.563945 + 0.563945i
\(628\) 0 0
\(629\) −358.931 + 358.931i −0.570637 + 0.570637i
\(630\) 0 0
\(631\) −486.622 −0.771191 −0.385596 0.922668i \(-0.626004\pi\)
−0.385596 + 0.922668i \(0.626004\pi\)
\(632\) 0 0
\(633\) 434.393i 0.686245i
\(634\) 0 0
\(635\) −619.681 619.681i −0.975875 0.975875i
\(636\) 0 0
\(637\) 118.247 + 118.247i 0.185631 + 0.185631i
\(638\) 0 0
\(639\) 127.968i 0.200263i
\(640\) 0 0
\(641\) −691.017 −1.07803 −0.539015 0.842296i \(-0.681203\pi\)
−0.539015 + 0.842296i \(0.681203\pi\)
\(642\) 0 0
\(643\) −652.605 + 652.605i −1.01494 + 1.01494i −0.0150512 + 0.999887i \(0.504791\pi\)
−0.999887 + 0.0150512i \(0.995209\pi\)
\(644\) 0 0
\(645\) −429.117 + 429.117i −0.665298 + 0.665298i
\(646\) 0 0
\(647\) −1156.72 −1.78782 −0.893911 0.448245i \(-0.852049\pi\)
−0.893911 + 0.448245i \(0.852049\pi\)
\(648\) 0 0
\(649\) 667.201i 1.02804i
\(650\) 0 0
\(651\) −110.314 110.314i −0.169453 0.169453i
\(652\) 0 0
\(653\) −209.105 209.105i −0.320222 0.320222i 0.528630 0.848852i \(-0.322706\pi\)
−0.848852 + 0.528630i \(0.822706\pi\)
\(654\) 0 0
\(655\) 725.776i 1.10806i
\(656\) 0 0
\(657\) 319.174 0.485805
\(658\) 0 0
\(659\) −533.902 + 533.902i −0.810170 + 0.810170i −0.984659 0.174489i \(-0.944173\pi\)
0.174489 + 0.984659i \(0.444173\pi\)
\(660\) 0 0
\(661\) 283.120 283.120i 0.428320 0.428320i −0.459736 0.888056i \(-0.652056\pi\)
0.888056 + 0.459736i \(0.152056\pi\)
\(662\) 0 0
\(663\) −328.805 −0.495935
\(664\) 0 0
\(665\) 656.484i 0.987195i
\(666\) 0 0
\(667\) −106.825 106.825i −0.160157 0.160157i
\(668\) 0 0
\(669\) −462.861 462.861i −0.691870 0.691870i
\(670\) 0 0
\(671\) 1603.11i 2.38913i
\(672\) 0 0
\(673\) −397.854 −0.591164 −0.295582 0.955317i \(-0.595514\pi\)
−0.295582 + 0.955317i \(0.595514\pi\)
\(674\) 0 0
\(675\) 110.188 110.188i 0.163242 0.163242i
\(676\) 0 0
\(677\) 289.959 289.959i 0.428299 0.428299i −0.459749 0.888049i \(-0.652061\pi\)
0.888049 + 0.459749i \(0.152061\pi\)
\(678\) 0 0
\(679\) −112.005 −0.164956
\(680\) 0 0
\(681\) 254.525i 0.373751i
\(682\) 0 0
\(683\) 150.197 + 150.197i 0.219908 + 0.219908i 0.808460 0.588551i \(-0.200302\pi\)
−0.588551 + 0.808460i \(0.700302\pi\)
\(684\) 0 0
\(685\) 652.686 + 652.686i 0.952826 + 0.952826i
\(686\) 0 0
\(687\) 247.534i 0.360312i
\(688\) 0 0
\(689\) 768.535 1.11544
\(690\) 0 0
\(691\) −791.212 + 791.212i −1.14502 + 1.14502i −0.157506 + 0.987518i \(0.550345\pi\)
−0.987518 + 0.157506i \(0.949655\pi\)
\(692\) 0 0
\(693\) −196.382 + 196.382i −0.283380 + 0.283380i
\(694\) 0 0
\(695\) 1772.46 2.55030
\(696\) 0 0
\(697\) 684.092i 0.981481i
\(698\) 0 0
\(699\) 351.819 + 351.819i 0.503318 + 0.503318i
\(700\) 0 0
\(701\) 900.201 + 900.201i 1.28417 + 1.28417i 0.938274 + 0.345893i \(0.112424\pi\)
0.345893 + 0.938274i \(0.387576\pi\)
\(702\) 0 0
\(703\) 360.445i 0.512724i
\(704\) 0 0
\(705\) 233.921 0.331803
\(706\) 0 0
\(707\) −17.3138 + 17.3138i −0.0244891 + 0.0244891i
\(708\) 0 0
\(709\) 128.490 128.490i 0.181227 0.181227i −0.610663 0.791891i \(-0.709097\pi\)
0.791891 + 0.610663i \(0.209097\pi\)
\(710\) 0 0
\(711\) 63.8463 0.0897979
\(712\) 0 0
\(713\) 98.1928i 0.137718i
\(714\) 0 0
\(715\) −739.140 739.140i −1.03376 1.03376i
\(716\) 0 0
\(717\) 184.865 + 184.865i 0.257831 + 0.257831i
\(718\) 0 0
\(719\) 1246.14i 1.73315i 0.499045 + 0.866576i \(0.333684\pi\)
−0.499045 + 0.866576i \(0.666316\pi\)
\(720\) 0 0
\(721\) −560.033 −0.776745
\(722\) 0 0
\(723\) −46.2730 + 46.2730i −0.0640014 + 0.0640014i
\(724\) 0 0
\(725\) 551.546 551.546i 0.760753 0.760753i
\(726\) 0 0
\(727\) 1130.07 1.55443 0.777216 0.629234i \(-0.216631\pi\)
0.777216 + 0.629234i \(0.216631\pi\)
\(728\) 0 0
\(729\) 27.0000i 0.0370370i
\(730\) 0 0
\(731\) 781.782 + 781.782i 1.06947 + 1.06947i
\(732\) 0 0
\(733\) −708.087 708.087i −0.966012 0.966012i 0.0334292 0.999441i \(-0.489357\pi\)
−0.999441 + 0.0334292i \(0.989357\pi\)
\(734\) 0 0
\(735\) 264.751i 0.360205i
\(736\) 0 0
\(737\) −926.264 −1.25680
\(738\) 0 0
\(739\) 32.7516 32.7516i 0.0443188 0.0443188i −0.684600 0.728919i \(-0.740023\pi\)
0.728919 + 0.684600i \(0.240023\pi\)
\(740\) 0 0
\(741\) −165.096 + 165.096i −0.222802 + 0.222802i
\(742\) 0 0
\(743\) −708.128 −0.953066 −0.476533 0.879157i \(-0.658107\pi\)
−0.476533 + 0.879157i \(0.658107\pi\)
\(744\) 0 0
\(745\) 1541.47i 2.06909i
\(746\) 0 0
\(747\) 72.4914 + 72.4914i 0.0970434 + 0.0970434i
\(748\) 0 0
\(749\) 530.751 + 530.751i 0.708612 + 0.708612i
\(750\) 0 0
\(751\) 1242.37i 1.65429i −0.561990 0.827144i \(-0.689964\pi\)
0.561990 0.827144i \(-0.310036\pi\)
\(752\) 0 0
\(753\) 247.189 0.328272
\(754\) 0 0
\(755\) 396.127 396.127i 0.524671 0.524671i
\(756\) 0 0
\(757\) −311.304 + 311.304i −0.411233 + 0.411233i −0.882168 0.470935i \(-0.843917\pi\)
0.470935 + 0.882168i \(0.343917\pi\)
\(758\) 0 0
\(759\) −174.805 −0.230309
\(760\) 0 0
\(761\) 179.137i 0.235397i 0.993049 + 0.117699i \(0.0375517\pi\)
−0.993049 + 0.117699i \(0.962448\pi\)
\(762\) 0 0
\(763\) −575.685 575.685i −0.754502 0.754502i
\(764\) 0 0
\(765\) −368.091 368.091i −0.481165 0.481165i
\(766\) 0 0
\(767\) 311.523i 0.406158i
\(768\) 0 0
\(769\) −967.409 −1.25801 −0.629005 0.777402i \(-0.716537\pi\)
−0.629005 + 0.777402i \(0.716537\pi\)
\(770\) 0 0
\(771\) −295.525 + 295.525i −0.383301 + 0.383301i
\(772\) 0 0
\(773\) −96.7342 + 96.7342i −0.125141 + 0.125141i −0.766904 0.641762i \(-0.778204\pi\)
0.641762 + 0.766904i \(0.278204\pi\)
\(774\) 0 0
\(775\) 506.979 0.654166
\(776\) 0 0
\(777\) 200.188i 0.257642i
\(778\) 0 0
\(779\) 343.489 + 343.489i 0.440936 + 0.440936i
\(780\) 0 0
\(781\) 524.082 + 524.082i 0.671039 + 0.671039i
\(782\) 0 0
\(783\) 135.148i 0.172603i
\(784\) 0 0
\(785\) 855.070 1.08926
\(786\) 0 0
\(787\) 381.038 381.038i 0.484166 0.484166i −0.422293 0.906459i \(-0.638775\pi\)
0.906459 + 0.422293i \(0.138775\pi\)
\(788\) 0 0
\(789\) −145.435 + 145.435i −0.184328 + 0.184328i
\(790\) 0 0
\(791\) −124.054 −0.156831
\(792\) 0 0
\(793\) 748.507i 0.943893i
\(794\) 0 0
\(795\) 860.362 + 860.362i 1.08222 + 1.08222i
\(796\) 0 0
\(797\) −371.148 371.148i −0.465681 0.465681i 0.434831 0.900512i \(-0.356808\pi\)
−0.900512 + 0.434831i \(0.856808\pi\)
\(798\) 0 0
\(799\) 426.167i 0.533375i
\(800\) 0 0
\(801\) 158.409 0.197764
\(802\) 0 0
\(803\) 1307.15 1307.15i 1.62783 1.62783i
\(804\) 0 0
\(805\) 162.272 162.272i 0.201580 0.201580i
\(806\) 0 0
\(807\) −18.9799 −0.0235191
\(808\) 0 0
\(809\) 309.566i 0.382653i −0.981526 0.191326i \(-0.938721\pi\)
0.981526 0.191326i \(-0.0612789\pi\)
\(810\) 0 0
\(811\) 27.2916 + 27.2916i 0.0336517 + 0.0336517i 0.723732 0.690081i \(-0.242425\pi\)
−0.690081 + 0.723732i \(0.742425\pi\)
\(812\) 0 0
\(813\) 161.543 + 161.543i 0.198699 + 0.198699i
\(814\) 0 0
\(815\) 585.247i 0.718095i
\(816\) 0 0
\(817\) 785.081 0.960931
\(818\) 0 0
\(819\) −91.6928 + 91.6928i −0.111957 + 0.111957i
\(820\) 0 0
\(821\) −879.903 + 879.903i −1.07175 + 1.07175i −0.0745264 + 0.997219i \(0.523745\pi\)
−0.997219 + 0.0745264i \(0.976255\pi\)
\(822\) 0 0
\(823\) −68.6842 −0.0834559 −0.0417280 0.999129i \(-0.513286\pi\)
−0.0417280 + 0.999129i \(0.513286\pi\)
\(824\) 0 0
\(825\) 902.533i 1.09398i
\(826\) 0 0
\(827\) −942.097 942.097i −1.13917 1.13917i −0.988599 0.150575i \(-0.951887\pi\)
−0.150575 0.988599i \(-0.548113\pi\)
\(828\) 0 0
\(829\) −568.532 568.532i −0.685805 0.685805i 0.275497 0.961302i \(-0.411158\pi\)
−0.961302 + 0.275497i \(0.911158\pi\)
\(830\) 0 0
\(831\) 495.660i 0.596462i
\(832\) 0 0
\(833\) 482.333 0.579031
\(834\) 0 0
\(835\) −129.169 + 129.169i −0.154693 + 0.154693i
\(836\) 0 0
\(837\) −62.1140 + 62.1140i −0.0742102 + 0.0742102i
\(838\) 0 0
\(839\) 1346.87 1.60533 0.802666 0.596429i \(-0.203414\pi\)
0.802666 + 0.596429i \(0.203414\pi\)
\(840\) 0 0
\(841\) 164.515i 0.195618i
\(842\) 0 0
\(843\) −84.3228 84.3228i −0.100027 0.100027i
\(844\) 0 0
\(845\) 541.046 + 541.046i 0.640291 + 0.640291i
\(846\) 0 0
\(847\) 963.851i 1.13796i
\(848\) 0 0
\(849\) 506.488 0.596570
\(850\) 0 0
\(851\) −89.0960 + 89.0960i −0.104696 + 0.104696i
\(852\) 0 0
\(853\) 74.4816 74.4816i 0.0873172 0.0873172i −0.662099 0.749416i \(-0.730334\pi\)
0.749416 + 0.662099i \(0.230334\pi\)
\(854\) 0 0
\(855\) −369.645 −0.432333
\(856\) 0 0
\(857\) 53.7221i 0.0626862i 0.999509 + 0.0313431i \(0.00997845\pi\)
−0.999509 + 0.0313431i \(0.990022\pi\)
\(858\) 0 0
\(859\) 537.704 + 537.704i 0.625965 + 0.625965i 0.947050 0.321085i \(-0.104048\pi\)
−0.321085 + 0.947050i \(0.604048\pi\)
\(860\) 0 0
\(861\) 190.771 + 190.771i 0.221569 + 0.221569i
\(862\) 0 0
\(863\) 1390.97i 1.61178i 0.592064 + 0.805891i \(0.298313\pi\)
−0.592064 + 0.805891i \(0.701687\pi\)
\(864\) 0 0
\(865\) −51.2347 −0.0592309
\(866\) 0 0
\(867\) −316.652 + 316.652i −0.365227 + 0.365227i
\(868\) 0 0
\(869\) 261.477 261.477i 0.300895 0.300895i
\(870\) 0 0
\(871\) −432.482 −0.496535
\(872\) 0 0
\(873\) 63.0666i 0.0722412i
\(874\) 0 0
\(875\) 139.391 + 139.391i 0.159303 + 0.159303i
\(876\) 0 0
\(877\) 940.115 + 940.115i 1.07197 + 1.07197i 0.997201 + 0.0747652i \(0.0238207\pi\)
0.0747652 + 0.997201i \(0.476179\pi\)
\(878\) 0 0
\(879\) 884.847i 1.00665i
\(880\) 0 0
\(881\) −140.985 −0.160029 −0.0800143 0.996794i \(-0.525497\pi\)
−0.0800143 + 0.996794i \(0.525497\pi\)
\(882\) 0 0
\(883\) 482.231 482.231i 0.546127 0.546127i −0.379191 0.925318i \(-0.623798\pi\)
0.925318 + 0.379191i \(0.123798\pi\)
\(884\) 0 0
\(885\) 348.744 348.744i 0.394062 0.394062i
\(886\) 0 0
\(887\) −266.180 −0.300091 −0.150045 0.988679i \(-0.547942\pi\)
−0.150045 + 0.988679i \(0.547942\pi\)
\(888\) 0 0
\(889\) 629.658i 0.708277i
\(890\) 0 0
\(891\) 110.576 + 110.576i 0.124104 + 0.124104i
\(892\) 0 0
\(893\) −213.982 213.982i −0.239622 0.239622i
\(894\) 0 0
\(895\) 2406.83i 2.68919i
\(896\) 0 0
\(897\) −81.6180 −0.0909899
\(898\) 0 0
\(899\) −310.911 + 310.911i −0.345841 + 0.345841i
\(900\) 0 0
\(901\) 1567.44 1567.44i 1.73967 1.73967i
\(902\) 0 0
\(903\) 436.026 0.482864
\(904\) 0 0
\(905\) 1222.87i 1.35124i
\(906\) 0 0
\(907\) 303.117 + 303.117i 0.334197 + 0.334197i 0.854178 0.519981i \(-0.174061\pi\)
−0.519981 + 0.854178i \(0.674061\pi\)
\(908\) 0 0
\(909\) 9.74881 + 9.74881i 0.0107248 + 0.0107248i
\(910\) 0 0
\(911\) 296.228i 0.325168i 0.986695 + 0.162584i \(0.0519829\pi\)
−0.986695 + 0.162584i \(0.948017\pi\)
\(912\) 0 0
\(913\) 593.765 0.650346
\(914\) 0 0
\(915\) 837.941 837.941i 0.915783 0.915783i
\(916\) 0 0
\(917\) 368.731 368.731i 0.402106 0.402106i
\(918\) 0 0
\(919\) 228.052 0.248153 0.124076 0.992273i \(-0.460403\pi\)
0.124076 + 0.992273i \(0.460403\pi\)
\(920\) 0 0
\(921\) 26.7172i 0.0290090i
\(922\) 0 0
\(923\) 244.699 + 244.699i 0.265113 + 0.265113i
\(924\) 0 0
\(925\) −460.011 460.011i −0.497309 0.497309i
\(926\) 0 0
\(927\) 315.336i 0.340168i
\(928\) 0 0
\(929\) −574.026 −0.617897 −0.308948 0.951079i \(-0.599977\pi\)
−0.308948 + 0.951079i \(0.599977\pi\)
\(930\) 0 0
\(931\) 242.184 242.184i 0.260133 0.260133i
\(932\) 0 0
\(933\) −196.267 + 196.267i −0.210361 + 0.210361i
\(934\) 0 0
\(935\) −3014.98 −3.22457
\(936\) 0 0
\(937\) 1098.22i 1.17206i 0.810291 + 0.586028i \(0.199309\pi\)
−0.810291 + 0.586028i \(0.800691\pi\)
\(938\) 0 0
\(939\) −435.397 435.397i −0.463682 0.463682i
\(940\) 0 0
\(941\) −857.669 857.669i −0.911444 0.911444i 0.0849418 0.996386i \(-0.472930\pi\)
−0.996386 + 0.0849418i \(0.972930\pi\)
\(942\) 0 0
\(943\) 169.810i 0.180074i
\(944\) 0 0
\(945\) −205.297 −0.217245
\(946\) 0 0
\(947\) −1041.67 + 1041.67i −1.09997 + 1.09997i −0.105556 + 0.994413i \(0.533662\pi\)
−0.994413 + 0.105556i \(0.966338\pi\)
\(948\) 0 0
\(949\) 610.322 610.322i 0.643121 0.643121i
\(950\) 0 0
\(951\) −177.635 −0.186788
\(952\) 0 0
\(953\) 910.089i 0.954973i −0.878639 0.477486i \(-0.841548\pi\)
0.878639 0.477486i \(-0.158452\pi\)
\(954\) 0 0
\(955\) 494.107 + 494.107i 0.517389 + 0.517389i
\(956\) 0 0
\(957\) 553.489 + 553.489i 0.578359 + 0.578359i
\(958\) 0 0
\(959\) 663.195i 0.691548i
\(960\) 0 0
\(961\) 675.212 0.702614
\(962\) 0 0
\(963\) 298.848 298.848i 0.310330 0.310330i
\(964\) 0 0
\(965\) −441.856 + 441.856i −0.457882 + 0.457882i
\(966\) 0 0
\(967\) 695.071 0.718791 0.359396 0.933185i \(-0.382983\pi\)
0.359396 + 0.933185i \(0.382983\pi\)
\(968\) 0 0
\(969\) 673.433i 0.694977i
\(970\) 0 0
\(971\) 1208.40 + 1208.40i 1.24449 + 1.24449i 0.958120 + 0.286366i \(0.0924472\pi\)
0.286366 + 0.958120i \(0.407553\pi\)
\(972\) 0 0
\(973\) −900.500 900.500i −0.925488 0.925488i
\(974\) 0 0
\(975\) 421.401i 0.432206i
\(976\) 0 0
\(977\) 141.036 0.144356 0.0721780 0.997392i \(-0.477005\pi\)
0.0721780 + 0.997392i \(0.477005\pi\)
\(978\) 0 0
\(979\) 648.750 648.750i 0.662666 0.662666i
\(980\) 0 0
\(981\) −324.149 + 324.149i −0.330427 + 0.330427i
\(982\) 0 0
\(983\) −1692.71 −1.72199 −0.860994 0.508616i \(-0.830157\pi\)
−0.860994 + 0.508616i \(0.830157\pi\)
\(984\) 0 0
\(985\) 597.320i 0.606417i
\(986\) 0 0
\(987\) −118.844 118.844i −0.120409 0.120409i
\(988\) 0 0
\(989\) 194.059 + 194.059i 0.196217 + 0.196217i
\(990\) 0 0
\(991\) 1532.62i 1.54654i −0.634079 0.773268i \(-0.718621\pi\)
0.634079 0.773268i \(-0.281379\pi\)
\(992\) 0 0
\(993\) −607.710 −0.611994
\(994\) 0 0
\(995\) −1028.67 + 1028.67i −1.03384 + 1.03384i
\(996\) 0 0
\(997\) 1131.91 1131.91i 1.13532 1.13532i 0.146039 0.989279i \(-0.453348\pi\)
0.989279 0.146039i \(-0.0466524\pi\)
\(998\) 0 0
\(999\) 112.719 0.112832
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 192.3.l.a.175.1 16
3.2 odd 2 576.3.m.c.559.8 16
4.3 odd 2 48.3.l.a.19.7 16
8.3 odd 2 384.3.l.a.223.4 16
8.5 even 2 384.3.l.b.223.8 16
12.11 even 2 144.3.m.c.19.2 16
16.3 odd 4 384.3.l.b.31.8 16
16.5 even 4 48.3.l.a.43.7 yes 16
16.11 odd 4 inner 192.3.l.a.79.1 16
16.13 even 4 384.3.l.a.31.4 16
24.5 odd 2 1152.3.m.c.991.1 16
24.11 even 2 1152.3.m.f.991.1 16
48.5 odd 4 144.3.m.c.91.2 16
48.11 even 4 576.3.m.c.271.8 16
48.29 odd 4 1152.3.m.f.415.1 16
48.35 even 4 1152.3.m.c.415.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
48.3.l.a.19.7 16 4.3 odd 2
48.3.l.a.43.7 yes 16 16.5 even 4
144.3.m.c.19.2 16 12.11 even 2
144.3.m.c.91.2 16 48.5 odd 4
192.3.l.a.79.1 16 16.11 odd 4 inner
192.3.l.a.175.1 16 1.1 even 1 trivial
384.3.l.a.31.4 16 16.13 even 4
384.3.l.a.223.4 16 8.3 odd 2
384.3.l.b.31.8 16 16.3 odd 4
384.3.l.b.223.8 16 8.5 even 2
576.3.m.c.271.8 16 48.11 even 4
576.3.m.c.559.8 16 3.2 odd 2
1152.3.m.c.415.1 16 48.35 even 4
1152.3.m.c.991.1 16 24.5 odd 2
1152.3.m.f.415.1 16 48.29 odd 4
1152.3.m.f.991.1 16 24.11 even 2