Properties

Label 192.3.i.b.113.6
Level $192$
Weight $3$
Character 192.113
Analytic conductor $5.232$
Analytic rank $0$
Dimension $20$
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] = [192,3,Mod(17,192)]
 
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([0, 3, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("192.17");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 192.i (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: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 2 x^{18} + 6 x^{16} - 24 x^{14} - 24 x^{12} + 1216 x^{10} - 384 x^{8} - 6144 x^{6} + 24576 x^{4} - 131072 x^{2} + 1048576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{23} \)
Twist minimal: no (minimal twist has level 48)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 113.6
Root \(0.312316 - 1.97546i\) of defining polynomial
Character \(\chi\) \(=\) 192.113
Dual form 192.3.i.b.17.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.18505 + 2.75602i) q^{3} +(-0.00985921 + 0.00985921i) q^{5} +6.42277i q^{7} +(-6.19134 + 6.53203i) q^{9} +O(q^{10})\) \(q+(1.18505 + 2.75602i) q^{3} +(-0.00985921 + 0.00985921i) q^{5} +6.42277i q^{7} +(-6.19134 + 6.53203i) q^{9} +(-9.07186 + 9.07186i) q^{11} +(12.6098 - 12.6098i) q^{13} +(-0.0388558 - 0.0154886i) q^{15} +19.0155i q^{17} +(2.07165 - 2.07165i) q^{19} +(-17.7013 + 7.61127i) q^{21} -19.5712 q^{23} +24.9998i q^{25} +(-25.3394 - 9.32272i) q^{27} +(-11.1742 - 11.1742i) q^{29} +59.9385 q^{31} +(-35.7528 - 14.2517i) q^{33} +(-0.0633234 - 0.0633234i) q^{35} +(9.32707 + 9.32707i) q^{37} +(49.6962 + 19.8098i) q^{39} +47.2639 q^{41} +(-24.1220 - 24.1220i) q^{43} +(-0.00335893 - 0.125442i) q^{45} -6.29702i q^{47} +7.74808 q^{49} +(-52.4073 + 22.5343i) q^{51} +(20.6409 - 20.6409i) q^{53} -0.178883i q^{55} +(8.16452 + 3.25452i) q^{57} +(60.3533 - 60.3533i) q^{59} +(48.0230 - 48.0230i) q^{61} +(-41.9537 - 39.7655i) q^{63} +0.248646i q^{65} +(23.7768 - 23.7768i) q^{67} +(-23.1928 - 53.9388i) q^{69} -13.5743 q^{71} -31.4516i q^{73} +(-68.9001 + 29.6259i) q^{75} +(-58.2665 - 58.2665i) q^{77} -47.4718 q^{79} +(-4.33472 - 80.8839i) q^{81} +(70.3318 + 70.3318i) q^{83} +(-0.187478 - 0.187478i) q^{85} +(17.5545 - 44.0385i) q^{87} -95.1729 q^{89} +(80.9900 + 80.9900i) q^{91} +(71.0298 + 165.192i) q^{93} +0.0408497i q^{95} +61.6218 q^{97} +(-3.09069 - 115.425i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 6 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 6 q^{3} + 92 q^{13} + 116 q^{15} + 52 q^{19} + 48 q^{21} - 18 q^{27} + 80 q^{31} + 60 q^{33} - 116 q^{37} - 172 q^{43} + 60 q^{45} - 364 q^{49} - 128 q^{51} - 244 q^{61} - 296 q^{63} - 356 q^{67} - 20 q^{69} + 146 q^{75} - 384 q^{79} - 188 q^{81} + 48 q^{85} - 136 q^{91} - 132 q^{93} + 472 q^{97} + 452 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) 1.18505 + 2.75602i 0.395015 + 0.918675i
\(4\) 0 0
\(5\) −0.00985921 + 0.00985921i −0.00197184 + 0.00197184i −0.708092 0.706120i \(-0.750444\pi\)
0.706120 + 0.708092i \(0.250444\pi\)
\(6\) 0 0
\(7\) 6.42277i 0.917538i 0.888556 + 0.458769i \(0.151709\pi\)
−0.888556 + 0.458769i \(0.848291\pi\)
\(8\) 0 0
\(9\) −6.19134 + 6.53203i −0.687926 + 0.725781i
\(10\) 0 0
\(11\) −9.07186 + 9.07186i −0.824715 + 0.824715i −0.986780 0.162065i \(-0.948185\pi\)
0.162065 + 0.986780i \(0.448185\pi\)
\(12\) 0 0
\(13\) 12.6098 12.6098i 0.969987 0.969987i −0.0295753 0.999563i \(-0.509415\pi\)
0.999563 + 0.0295753i \(0.00941548\pi\)
\(14\) 0 0
\(15\) −0.0388558 0.0154886i −0.00259039 0.00103257i
\(16\) 0 0
\(17\) 19.0155i 1.11856i 0.828978 + 0.559281i \(0.188923\pi\)
−0.828978 + 0.559281i \(0.811077\pi\)
\(18\) 0 0
\(19\) 2.07165 2.07165i 0.109034 0.109034i −0.650485 0.759519i \(-0.725434\pi\)
0.759519 + 0.650485i \(0.225434\pi\)
\(20\) 0 0
\(21\) −17.7013 + 7.61127i −0.842919 + 0.362441i
\(22\) 0 0
\(23\) −19.5712 −0.850923 −0.425461 0.904977i \(-0.639888\pi\)
−0.425461 + 0.904977i \(0.639888\pi\)
\(24\) 0 0
\(25\) 24.9998i 0.999992i
\(26\) 0 0
\(27\) −25.3394 9.32272i −0.938497 0.345286i
\(28\) 0 0
\(29\) −11.1742 11.1742i −0.385319 0.385319i 0.487695 0.873014i \(-0.337838\pi\)
−0.873014 + 0.487695i \(0.837838\pi\)
\(30\) 0 0
\(31\) 59.9385 1.93350 0.966750 0.255725i \(-0.0823142\pi\)
0.966750 + 0.255725i \(0.0823142\pi\)
\(32\) 0 0
\(33\) −35.7528 14.2517i −1.08342 0.431870i
\(34\) 0 0
\(35\) −0.0633234 0.0633234i −0.00180924 0.00180924i
\(36\) 0 0
\(37\) 9.32707 + 9.32707i 0.252083 + 0.252083i 0.821824 0.569741i \(-0.192957\pi\)
−0.569741 + 0.821824i \(0.692957\pi\)
\(38\) 0 0
\(39\) 49.6962 + 19.8098i 1.27426 + 0.507943i
\(40\) 0 0
\(41\) 47.2639 1.15278 0.576389 0.817176i \(-0.304461\pi\)
0.576389 + 0.817176i \(0.304461\pi\)
\(42\) 0 0
\(43\) −24.1220 24.1220i −0.560978 0.560978i 0.368607 0.929585i \(-0.379835\pi\)
−0.929585 + 0.368607i \(0.879835\pi\)
\(44\) 0 0
\(45\) −0.00335893 0.125442i −7.46430e−5 0.00278761i
\(46\) 0 0
\(47\) 6.29702i 0.133979i −0.997754 0.0669896i \(-0.978661\pi\)
0.997754 0.0669896i \(-0.0213394\pi\)
\(48\) 0 0
\(49\) 7.74808 0.158124
\(50\) 0 0
\(51\) −52.4073 + 22.5343i −1.02759 + 0.441849i
\(52\) 0 0
\(53\) 20.6409 20.6409i 0.389450 0.389450i −0.485041 0.874491i \(-0.661195\pi\)
0.874491 + 0.485041i \(0.161195\pi\)
\(54\) 0 0
\(55\) 0.178883i 0.00325242i
\(56\) 0 0
\(57\) 8.16452 + 3.25452i 0.143237 + 0.0570968i
\(58\) 0 0
\(59\) 60.3533 60.3533i 1.02294 1.02294i 0.0232062 0.999731i \(-0.492613\pi\)
0.999731 0.0232062i \(-0.00738742\pi\)
\(60\) 0 0
\(61\) 48.0230 48.0230i 0.787262 0.787262i −0.193782 0.981045i \(-0.562076\pi\)
0.981045 + 0.193782i \(0.0620755\pi\)
\(62\) 0 0
\(63\) −41.9537 39.7655i −0.665931 0.631198i
\(64\) 0 0
\(65\) 0.248646i 0.00382532i
\(66\) 0 0
\(67\) 23.7768 23.7768i 0.354878 0.354878i −0.507043 0.861921i \(-0.669262\pi\)
0.861921 + 0.507043i \(0.169262\pi\)
\(68\) 0 0
\(69\) −23.1928 53.9388i −0.336127 0.781721i
\(70\) 0 0
\(71\) −13.5743 −0.191188 −0.0955938 0.995420i \(-0.530475\pi\)
−0.0955938 + 0.995420i \(0.530475\pi\)
\(72\) 0 0
\(73\) 31.4516i 0.430844i −0.976521 0.215422i \(-0.930887\pi\)
0.976521 0.215422i \(-0.0691127\pi\)
\(74\) 0 0
\(75\) −68.9001 + 29.6259i −0.918668 + 0.395012i
\(76\) 0 0
\(77\) −58.2665 58.2665i −0.756707 0.756707i
\(78\) 0 0
\(79\) −47.4718 −0.600909 −0.300455 0.953796i \(-0.597138\pi\)
−0.300455 + 0.953796i \(0.597138\pi\)
\(80\) 0 0
\(81\) −4.33472 80.8839i −0.0535151 0.998567i
\(82\) 0 0
\(83\) 70.3318 + 70.3318i 0.847372 + 0.847372i 0.989804 0.142433i \(-0.0454925\pi\)
−0.142433 + 0.989804i \(0.545493\pi\)
\(84\) 0 0
\(85\) −0.187478 0.187478i −0.00220563 0.00220563i
\(86\) 0 0
\(87\) 17.5545 44.0385i 0.201776 0.506189i
\(88\) 0 0
\(89\) −95.1729 −1.06936 −0.534679 0.845055i \(-0.679568\pi\)
−0.534679 + 0.845055i \(0.679568\pi\)
\(90\) 0 0
\(91\) 80.9900 + 80.9900i 0.890000 + 0.890000i
\(92\) 0 0
\(93\) 71.0298 + 165.192i 0.763761 + 1.77626i
\(94\) 0 0
\(95\) 0.0408497i 0.000429997i
\(96\) 0 0
\(97\) 61.6218 0.635276 0.317638 0.948212i \(-0.397110\pi\)
0.317638 + 0.948212i \(0.397110\pi\)
\(98\) 0 0
\(99\) −3.09069 115.425i −0.0312191 1.16591i
\(100\) 0 0
\(101\) 48.1867 48.1867i 0.477096 0.477096i −0.427106 0.904202i \(-0.640467\pi\)
0.904202 + 0.427106i \(0.140467\pi\)
\(102\) 0 0
\(103\) 4.73669i 0.0459873i −0.999736 0.0229936i \(-0.992680\pi\)
0.999736 0.0229936i \(-0.00731975\pi\)
\(104\) 0 0
\(105\) 0.0994797 0.249562i 0.000947426 0.00237678i
\(106\) 0 0
\(107\) 40.9462 40.9462i 0.382674 0.382674i −0.489390 0.872065i \(-0.662781\pi\)
0.872065 + 0.489390i \(0.162781\pi\)
\(108\) 0 0
\(109\) −120.437 + 120.437i −1.10493 + 1.10493i −0.111123 + 0.993807i \(0.535445\pi\)
−0.993807 + 0.111123i \(0.964555\pi\)
\(110\) 0 0
\(111\) −14.6526 + 36.7586i −0.132006 + 0.331159i
\(112\) 0 0
\(113\) 205.193i 1.81587i 0.419110 + 0.907936i \(0.362342\pi\)
−0.419110 + 0.907936i \(0.637658\pi\)
\(114\) 0 0
\(115\) 0.192957 0.192957i 0.00167789 0.00167789i
\(116\) 0 0
\(117\) 4.29604 + 160.439i 0.0367183 + 1.37128i
\(118\) 0 0
\(119\) −122.132 −1.02632
\(120\) 0 0
\(121\) 43.5974i 0.360309i
\(122\) 0 0
\(123\) 56.0098 + 130.260i 0.455365 + 1.05903i
\(124\) 0 0
\(125\) −0.492959 0.492959i −0.00394367 0.00394367i
\(126\) 0 0
\(127\) −54.1458 −0.426345 −0.213173 0.977015i \(-0.568380\pi\)
−0.213173 + 0.977015i \(0.568380\pi\)
\(128\) 0 0
\(129\) 37.8952 95.0666i 0.293761 0.736951i
\(130\) 0 0
\(131\) −31.2584 31.2584i −0.238614 0.238614i 0.577662 0.816276i \(-0.303965\pi\)
−0.816276 + 0.577662i \(0.803965\pi\)
\(132\) 0 0
\(133\) 13.3057 + 13.3057i 0.100043 + 0.100043i
\(134\) 0 0
\(135\) 0.341742 0.157912i 0.00253142 0.00116972i
\(136\) 0 0
\(137\) 42.9176 0.313267 0.156633 0.987657i \(-0.449936\pi\)
0.156633 + 0.987657i \(0.449936\pi\)
\(138\) 0 0
\(139\) 47.0945 + 47.0945i 0.338809 + 0.338809i 0.855919 0.517110i \(-0.172992\pi\)
−0.517110 + 0.855919i \(0.672992\pi\)
\(140\) 0 0
\(141\) 17.3547 7.46225i 0.123083 0.0529238i
\(142\) 0 0
\(143\) 228.789i 1.59993i
\(144\) 0 0
\(145\) 0.220339 0.00151958
\(146\) 0 0
\(147\) 9.18183 + 21.3539i 0.0624614 + 0.145265i
\(148\) 0 0
\(149\) −131.532 + 131.532i −0.882766 + 0.882766i −0.993815 0.111049i \(-0.964579\pi\)
0.111049 + 0.993815i \(0.464579\pi\)
\(150\) 0 0
\(151\) 145.908i 0.966281i −0.875543 0.483140i \(-0.839496\pi\)
0.875543 0.483140i \(-0.160504\pi\)
\(152\) 0 0
\(153\) −124.210 117.732i −0.811830 0.769488i
\(154\) 0 0
\(155\) −0.590946 + 0.590946i −0.00381256 + 0.00381256i
\(156\) 0 0
\(157\) −55.2586 + 55.2586i −0.351966 + 0.351966i −0.860840 0.508875i \(-0.830062\pi\)
0.508875 + 0.860840i \(0.330062\pi\)
\(158\) 0 0
\(159\) 81.3470 + 32.4263i 0.511617 + 0.203939i
\(160\) 0 0
\(161\) 125.701i 0.780754i
\(162\) 0 0
\(163\) −70.6156 + 70.6156i −0.433225 + 0.433225i −0.889724 0.456499i \(-0.849103\pi\)
0.456499 + 0.889724i \(0.349103\pi\)
\(164\) 0 0
\(165\) 0.493006 0.211984i 0.00298791 0.00128475i
\(166\) 0 0
\(167\) −86.2013 −0.516176 −0.258088 0.966121i \(-0.583092\pi\)
−0.258088 + 0.966121i \(0.583092\pi\)
\(168\) 0 0
\(169\) 149.016i 0.881751i
\(170\) 0 0
\(171\) 0.705790 + 26.3584i 0.00412743 + 0.154142i
\(172\) 0 0
\(173\) −58.2425 58.2425i −0.336662 0.336662i 0.518448 0.855109i \(-0.326510\pi\)
−0.855109 + 0.518448i \(0.826510\pi\)
\(174\) 0 0
\(175\) −160.568 −0.917531
\(176\) 0 0
\(177\) 237.856 + 94.8137i 1.34382 + 0.535671i
\(178\) 0 0
\(179\) −18.9272 18.9272i −0.105738 0.105738i 0.652258 0.757997i \(-0.273822\pi\)
−0.757997 + 0.652258i \(0.773822\pi\)
\(180\) 0 0
\(181\) 24.5109 + 24.5109i 0.135420 + 0.135420i 0.771567 0.636148i \(-0.219473\pi\)
−0.636148 + 0.771567i \(0.719473\pi\)
\(182\) 0 0
\(183\) 189.262 + 75.4431i 1.03422 + 0.412257i
\(184\) 0 0
\(185\) −0.183915 −0.000994136
\(186\) 0 0
\(187\) −172.506 172.506i −0.922494 0.922494i
\(188\) 0 0
\(189\) 59.8777 162.749i 0.316813 0.861107i
\(190\) 0 0
\(191\) 156.422i 0.818962i −0.912319 0.409481i \(-0.865710\pi\)
0.912319 0.409481i \(-0.134290\pi\)
\(192\) 0 0
\(193\) −217.972 −1.12939 −0.564695 0.825299i \(-0.691006\pi\)
−0.564695 + 0.825299i \(0.691006\pi\)
\(194\) 0 0
\(195\) −0.685275 + 0.294657i −0.00351423 + 0.00151106i
\(196\) 0 0
\(197\) 245.945 245.945i 1.24845 1.24845i 0.292050 0.956403i \(-0.405663\pi\)
0.956403 0.292050i \(-0.0943374\pi\)
\(198\) 0 0
\(199\) 233.190i 1.17181i 0.810379 + 0.585905i \(0.199261\pi\)
−0.810379 + 0.585905i \(0.800739\pi\)
\(200\) 0 0
\(201\) 93.7060 + 37.3528i 0.466199 + 0.185835i
\(202\) 0 0
\(203\) 71.7695 71.7695i 0.353545 0.353545i
\(204\) 0 0
\(205\) −0.465985 + 0.465985i −0.00227310 + 0.00227310i
\(206\) 0 0
\(207\) 121.172 127.840i 0.585372 0.617583i
\(208\) 0 0
\(209\) 37.5875i 0.179844i
\(210\) 0 0
\(211\) 8.49504 8.49504i 0.0402609 0.0402609i −0.686690 0.726951i \(-0.740937\pi\)
0.726951 + 0.686690i \(0.240937\pi\)
\(212\) 0 0
\(213\) −16.0862 37.4111i −0.0755220 0.175639i
\(214\) 0 0
\(215\) 0.475649 0.00221232
\(216\) 0 0
\(217\) 384.971i 1.77406i
\(218\) 0 0
\(219\) 86.6814 37.2716i 0.395806 0.170190i
\(220\) 0 0
\(221\) 239.783 + 239.783i 1.08499 + 1.08499i
\(222\) 0 0
\(223\) 10.9290 0.0490090 0.0245045 0.999700i \(-0.492199\pi\)
0.0245045 + 0.999700i \(0.492199\pi\)
\(224\) 0 0
\(225\) −163.299 154.782i −0.725775 0.687921i
\(226\) 0 0
\(227\) 99.9027 + 99.9027i 0.440100 + 0.440100i 0.892045 0.451946i \(-0.149270\pi\)
−0.451946 + 0.892045i \(0.649270\pi\)
\(228\) 0 0
\(229\) −231.857 231.857i −1.01248 1.01248i −0.999921 0.0125555i \(-0.996003\pi\)
−0.0125555 0.999921i \(-0.503997\pi\)
\(230\) 0 0
\(231\) 91.5354 229.632i 0.396257 0.994079i
\(232\) 0 0
\(233\) 316.641 1.35897 0.679486 0.733688i \(-0.262203\pi\)
0.679486 + 0.733688i \(0.262203\pi\)
\(234\) 0 0
\(235\) 0.0620836 + 0.0620836i 0.000264186 + 0.000264186i
\(236\) 0 0
\(237\) −56.2562 130.833i −0.237368 0.552040i
\(238\) 0 0
\(239\) 382.691i 1.60122i −0.599187 0.800609i \(-0.704509\pi\)
0.599187 0.800609i \(-0.295491\pi\)
\(240\) 0 0
\(241\) −91.3157 −0.378903 −0.189452 0.981890i \(-0.560671\pi\)
−0.189452 + 0.981890i \(0.560671\pi\)
\(242\) 0 0
\(243\) 217.781 107.798i 0.896219 0.443612i
\(244\) 0 0
\(245\) −0.0763900 + 0.0763900i −0.000311796 + 0.000311796i
\(246\) 0 0
\(247\) 52.2463i 0.211524i
\(248\) 0 0
\(249\) −110.490 + 277.183i −0.443734 + 1.11318i
\(250\) 0 0
\(251\) −128.768 + 128.768i −0.513021 + 0.513021i −0.915451 0.402430i \(-0.868166\pi\)
0.402430 + 0.915451i \(0.368166\pi\)
\(252\) 0 0
\(253\) 177.547 177.547i 0.701769 0.701769i
\(254\) 0 0
\(255\) 0.294524 0.738865i 0.00115500 0.00289751i
\(256\) 0 0
\(257\) 123.915i 0.482159i −0.970505 0.241079i \(-0.922499\pi\)
0.970505 0.241079i \(-0.0775014\pi\)
\(258\) 0 0
\(259\) −59.9056 + 59.9056i −0.231296 + 0.231296i
\(260\) 0 0
\(261\) 142.174 3.80695i 0.544728 0.0145860i
\(262\) 0 0
\(263\) 194.379 0.739085 0.369542 0.929214i \(-0.379514\pi\)
0.369542 + 0.929214i \(0.379514\pi\)
\(264\) 0 0
\(265\) 0.407005i 0.00153587i
\(266\) 0 0
\(267\) −112.784 262.299i −0.422413 0.982393i
\(268\) 0 0
\(269\) −296.636 296.636i −1.10274 1.10274i −0.994079 0.108658i \(-0.965345\pi\)
−0.108658 0.994079i \(-0.534655\pi\)
\(270\) 0 0
\(271\) 278.227 1.02667 0.513334 0.858189i \(-0.328410\pi\)
0.513334 + 0.858189i \(0.328410\pi\)
\(272\) 0 0
\(273\) −127.234 + 319.187i −0.466057 + 1.16918i
\(274\) 0 0
\(275\) −226.795 226.795i −0.824709 0.824709i
\(276\) 0 0
\(277\) −60.1513 60.1513i −0.217153 0.217153i 0.590145 0.807297i \(-0.299071\pi\)
−0.807297 + 0.590145i \(0.799071\pi\)
\(278\) 0 0
\(279\) −371.099 + 391.520i −1.33010 + 1.40330i
\(280\) 0 0
\(281\) −313.645 −1.11617 −0.558087 0.829782i \(-0.688465\pi\)
−0.558087 + 0.829782i \(0.688465\pi\)
\(282\) 0 0
\(283\) −286.980 286.980i −1.01406 1.01406i −0.999900 0.0141627i \(-0.995492\pi\)
−0.0141627 0.999900i \(-0.504508\pi\)
\(284\) 0 0
\(285\) −0.112583 + 0.0484087i −0.000395027 + 0.000169855i
\(286\) 0 0
\(287\) 303.565i 1.05772i
\(288\) 0 0
\(289\) −72.5910 −0.251180
\(290\) 0 0
\(291\) 73.0246 + 169.831i 0.250944 + 0.583612i
\(292\) 0 0
\(293\) −176.501 + 176.501i −0.602394 + 0.602394i −0.940947 0.338553i \(-0.890062\pi\)
0.338553 + 0.940947i \(0.390062\pi\)
\(294\) 0 0
\(295\) 1.19007i 0.00403414i
\(296\) 0 0
\(297\) 314.450 145.301i 1.05876 0.489230i
\(298\) 0 0
\(299\) −246.790 + 246.790i −0.825384 + 0.825384i
\(300\) 0 0
\(301\) 154.930 154.930i 0.514718 0.514718i
\(302\) 0 0
\(303\) 189.907 + 75.7002i 0.626756 + 0.249836i
\(304\) 0 0
\(305\) 0.946938i 0.00310471i
\(306\) 0 0
\(307\) −63.9904 + 63.9904i −0.208438 + 0.208438i −0.803603 0.595165i \(-0.797087\pi\)
0.595165 + 0.803603i \(0.297087\pi\)
\(308\) 0 0
\(309\) 13.0544 5.61319i 0.0422473 0.0181657i
\(310\) 0 0
\(311\) 532.288 1.71154 0.855769 0.517359i \(-0.173085\pi\)
0.855769 + 0.517359i \(0.173085\pi\)
\(312\) 0 0
\(313\) 185.676i 0.593215i −0.954999 0.296607i \(-0.904145\pi\)
0.954999 0.296607i \(-0.0958553\pi\)
\(314\) 0 0
\(315\) 0.805687 0.0215736i 0.00255774 6.84878e-5i
\(316\) 0 0
\(317\) 168.127 + 168.127i 0.530370 + 0.530370i 0.920683 0.390312i \(-0.127633\pi\)
−0.390312 + 0.920683i \(0.627633\pi\)
\(318\) 0 0
\(319\) 202.742 0.635556
\(320\) 0 0
\(321\) 161.372 + 64.3256i 0.502715 + 0.200391i
\(322\) 0 0
\(323\) 39.3936 + 39.3936i 0.121962 + 0.121962i
\(324\) 0 0
\(325\) 315.243 + 315.243i 0.969980 + 0.969980i
\(326\) 0 0
\(327\) −474.652 189.204i −1.45153 0.578607i
\(328\) 0 0
\(329\) 40.4443 0.122931
\(330\) 0 0
\(331\) 241.678 + 241.678i 0.730144 + 0.730144i 0.970648 0.240504i \(-0.0773127\pi\)
−0.240504 + 0.970648i \(0.577313\pi\)
\(332\) 0 0
\(333\) −118.672 + 3.17764i −0.356371 + 0.00954245i
\(334\) 0 0
\(335\) 0.468841i 0.00139953i
\(336\) 0 0
\(337\) 396.856 1.17762 0.588808 0.808273i \(-0.299598\pi\)
0.588808 + 0.808273i \(0.299598\pi\)
\(338\) 0 0
\(339\) −565.518 + 243.163i −1.66819 + 0.717296i
\(340\) 0 0
\(341\) −543.754 + 543.754i −1.59459 + 1.59459i
\(342\) 0 0
\(343\) 364.480i 1.06262i
\(344\) 0 0
\(345\) 0.760456 + 0.303131i 0.00220422 + 0.000878641i
\(346\) 0 0
\(347\) −38.5699 + 38.5699i −0.111153 + 0.111153i −0.760496 0.649343i \(-0.775044\pi\)
0.649343 + 0.760496i \(0.275044\pi\)
\(348\) 0 0
\(349\) 10.4065 10.4065i 0.0298180 0.0298180i −0.692041 0.721859i \(-0.743288\pi\)
0.721859 + 0.692041i \(0.243288\pi\)
\(350\) 0 0
\(351\) −437.084 + 201.968i −1.24525 + 0.575408i
\(352\) 0 0
\(353\) 209.294i 0.592900i −0.955048 0.296450i \(-0.904197\pi\)
0.955048 0.296450i \(-0.0958028\pi\)
\(354\) 0 0
\(355\) 0.133832 0.133832i 0.000376992 0.000376992i
\(356\) 0 0
\(357\) −144.732 336.600i −0.405413 0.942857i
\(358\) 0 0
\(359\) −42.6682 −0.118853 −0.0594264 0.998233i \(-0.518927\pi\)
−0.0594264 + 0.998233i \(0.518927\pi\)
\(360\) 0 0
\(361\) 352.417i 0.976223i
\(362\) 0 0
\(363\) 120.156 51.6649i 0.331007 0.142328i
\(364\) 0 0
\(365\) 0.310088 + 0.310088i 0.000849557 + 0.000849557i
\(366\) 0 0
\(367\) 16.2444 0.0442627 0.0221313 0.999755i \(-0.492955\pi\)
0.0221313 + 0.999755i \(0.492955\pi\)
\(368\) 0 0
\(369\) −292.627 + 308.729i −0.793026 + 0.836664i
\(370\) 0 0
\(371\) 132.571 + 132.571i 0.357335 + 0.357335i
\(372\) 0 0
\(373\) 351.379 + 351.379i 0.942035 + 0.942035i 0.998410 0.0563743i \(-0.0179540\pi\)
−0.0563743 + 0.998410i \(0.517954\pi\)
\(374\) 0 0
\(375\) 0.774428 1.94278i 0.00206514 0.00518076i
\(376\) 0 0
\(377\) −281.811 −0.747509
\(378\) 0 0
\(379\) 170.505 + 170.505i 0.449880 + 0.449880i 0.895315 0.445435i \(-0.146951\pi\)
−0.445435 + 0.895315i \(0.646951\pi\)
\(380\) 0 0
\(381\) −64.1653 149.227i −0.168413 0.391673i
\(382\) 0 0
\(383\) 256.234i 0.669017i 0.942393 + 0.334509i \(0.108570\pi\)
−0.942393 + 0.334509i \(0.891430\pi\)
\(384\) 0 0
\(385\) 1.14892 0.00298422
\(386\) 0 0
\(387\) 306.913 8.21813i 0.793058 0.0212355i
\(388\) 0 0
\(389\) 376.214 376.214i 0.967130 0.967130i −0.0323468 0.999477i \(-0.510298\pi\)
0.999477 + 0.0323468i \(0.0102981\pi\)
\(390\) 0 0
\(391\) 372.158i 0.951810i
\(392\) 0 0
\(393\) 49.1063 123.192i 0.124953 0.313465i
\(394\) 0 0
\(395\) 0.468035 0.468035i 0.00118490 0.00118490i
\(396\) 0 0
\(397\) −312.905 + 312.905i −0.788174 + 0.788174i −0.981195 0.193021i \(-0.938172\pi\)
0.193021 + 0.981195i \(0.438172\pi\)
\(398\) 0 0
\(399\) −20.9030 + 52.4388i −0.0523885 + 0.131426i
\(400\) 0 0
\(401\) 9.22373i 0.0230018i −0.999934 0.0115009i \(-0.996339\pi\)
0.999934 0.0115009i \(-0.00366093\pi\)
\(402\) 0 0
\(403\) 755.814 755.814i 1.87547 1.87547i
\(404\) 0 0
\(405\) 0.840189 + 0.754715i 0.00207454 + 0.00186349i
\(406\) 0 0
\(407\) −169.228 −0.415793
\(408\) 0 0
\(409\) 322.436i 0.788352i −0.919035 0.394176i \(-0.871030\pi\)
0.919035 0.394176i \(-0.128970\pi\)
\(410\) 0 0
\(411\) 50.8592 + 118.282i 0.123745 + 0.287790i
\(412\) 0 0
\(413\) 387.635 + 387.635i 0.938583 + 0.938583i
\(414\) 0 0
\(415\) −1.38683 −0.00334177
\(416\) 0 0
\(417\) −73.9845 + 185.603i −0.177421 + 0.445090i
\(418\) 0 0
\(419\) −226.569 226.569i −0.540738 0.540738i 0.383007 0.923745i \(-0.374888\pi\)
−0.923745 + 0.383007i \(0.874888\pi\)
\(420\) 0 0
\(421\) −498.861 498.861i −1.18494 1.18494i −0.978448 0.206495i \(-0.933794\pi\)
−0.206495 0.978448i \(-0.566206\pi\)
\(422\) 0 0
\(423\) 41.1323 + 38.9870i 0.0972394 + 0.0921677i
\(424\) 0 0
\(425\) −475.385 −1.11855
\(426\) 0 0
\(427\) 308.440 + 308.440i 0.722343 + 0.722343i
\(428\) 0 0
\(429\) −630.549 + 271.126i −1.46981 + 0.631995i
\(430\) 0 0
\(431\) 452.283i 1.04938i −0.851293 0.524690i \(-0.824181\pi\)
0.851293 0.524690i \(-0.175819\pi\)
\(432\) 0 0
\(433\) 379.557 0.876574 0.438287 0.898835i \(-0.355585\pi\)
0.438287 + 0.898835i \(0.355585\pi\)
\(434\) 0 0
\(435\) 0.261111 + 0.607258i 0.000600255 + 0.00139600i
\(436\) 0 0
\(437\) −40.5447 + 40.5447i −0.0927797 + 0.0927797i
\(438\) 0 0
\(439\) 689.509i 1.57063i −0.619094 0.785317i \(-0.712500\pi\)
0.619094 0.785317i \(-0.287500\pi\)
\(440\) 0 0
\(441\) −47.9710 + 50.6107i −0.108778 + 0.114763i
\(442\) 0 0
\(443\) −97.5600 + 97.5600i −0.220226 + 0.220226i −0.808593 0.588368i \(-0.799771\pi\)
0.588368 + 0.808593i \(0.299771\pi\)
\(444\) 0 0
\(445\) 0.938330 0.938330i 0.00210861 0.00210861i
\(446\) 0 0
\(447\) −518.377 206.634i −1.15968 0.462269i
\(448\) 0 0
\(449\) 718.711i 1.60069i −0.599538 0.800347i \(-0.704649\pi\)
0.599538 0.800347i \(-0.295351\pi\)
\(450\) 0 0
\(451\) −428.772 + 428.772i −0.950713 + 0.950713i
\(452\) 0 0
\(453\) 402.127 172.908i 0.887698 0.381695i
\(454\) 0 0
\(455\) −1.59700 −0.00350988
\(456\) 0 0
\(457\) 489.021i 1.07007i −0.844830 0.535034i \(-0.820299\pi\)
0.844830 0.535034i \(-0.179701\pi\)
\(458\) 0 0
\(459\) 177.277 481.843i 0.386224 1.04977i
\(460\) 0 0
\(461\) 459.082 + 459.082i 0.995840 + 0.995840i 0.999991 0.00415179i \(-0.00132156\pi\)
−0.00415179 + 0.999991i \(0.501322\pi\)
\(462\) 0 0
\(463\) −587.611 −1.26914 −0.634569 0.772866i \(-0.718822\pi\)
−0.634569 + 0.772866i \(0.718822\pi\)
\(464\) 0 0
\(465\) −2.32896 0.928364i −0.00500852 0.00199648i
\(466\) 0 0
\(467\) −89.5077 89.5077i −0.191665 0.191665i 0.604750 0.796415i \(-0.293273\pi\)
−0.796415 + 0.604750i \(0.793273\pi\)
\(468\) 0 0
\(469\) 152.713 + 152.713i 0.325614 + 0.325614i
\(470\) 0 0
\(471\) −217.778 86.8101i −0.462374 0.184310i
\(472\) 0 0
\(473\) 437.664 0.925293
\(474\) 0 0
\(475\) 51.7909 + 51.7909i 0.109033 + 0.109033i
\(476\) 0 0
\(477\) 7.03213 + 262.621i 0.0147424 + 0.550568i
\(478\) 0 0
\(479\) 439.291i 0.917101i −0.888668 0.458550i \(-0.848369\pi\)
0.888668 0.458550i \(-0.151631\pi\)
\(480\) 0 0
\(481\) 235.226 0.489034
\(482\) 0 0
\(483\) 346.436 148.962i 0.717259 0.308410i
\(484\) 0 0
\(485\) −0.607542 + 0.607542i −0.00125266 + 0.00125266i
\(486\) 0 0
\(487\) 499.716i 1.02611i 0.858355 + 0.513056i \(0.171487\pi\)
−0.858355 + 0.513056i \(0.828513\pi\)
\(488\) 0 0
\(489\) −278.301 110.936i −0.569123 0.226862i
\(490\) 0 0
\(491\) 359.246 359.246i 0.731663 0.731663i −0.239286 0.970949i \(-0.576913\pi\)
0.970949 + 0.239286i \(0.0769134\pi\)
\(492\) 0 0
\(493\) 212.484 212.484i 0.431003 0.431003i
\(494\) 0 0
\(495\) 1.16847 + 1.10752i 0.00236054 + 0.00223742i
\(496\) 0 0
\(497\) 87.1846i 0.175422i
\(498\) 0 0
\(499\) 64.4682 64.4682i 0.129195 0.129195i −0.639553 0.768747i \(-0.720880\pi\)
0.768747 + 0.639553i \(0.220880\pi\)
\(500\) 0 0
\(501\) −102.152 237.573i −0.203897 0.474197i
\(502\) 0 0
\(503\) 597.277 1.18743 0.593714 0.804676i \(-0.297661\pi\)
0.593714 + 0.804676i \(0.297661\pi\)
\(504\) 0 0
\(505\) 0.950165i 0.00188151i
\(506\) 0 0
\(507\) 410.691 176.591i 0.810042 0.348305i
\(508\) 0 0
\(509\) −359.574 359.574i −0.706433 0.706433i 0.259350 0.965783i \(-0.416492\pi\)
−0.965783 + 0.259350i \(0.916492\pi\)
\(510\) 0 0
\(511\) 202.006 0.395316
\(512\) 0 0
\(513\) −71.8079 + 33.1810i −0.139976 + 0.0646804i
\(514\) 0 0
\(515\) 0.0467000 + 0.0467000i 9.06796e−5 + 9.06796e-5i
\(516\) 0 0
\(517\) 57.1257 + 57.1257i 0.110495 + 0.110495i
\(518\) 0 0
\(519\) 91.4977 229.538i 0.176296 0.442269i
\(520\) 0 0
\(521\) 862.399 1.65528 0.827639 0.561261i \(-0.189684\pi\)
0.827639 + 0.561261i \(0.189684\pi\)
\(522\) 0 0
\(523\) 256.574 + 256.574i 0.490581 + 0.490581i 0.908489 0.417908i \(-0.137237\pi\)
−0.417908 + 0.908489i \(0.637237\pi\)
\(524\) 0 0
\(525\) −190.280 442.529i −0.362438 0.842912i
\(526\) 0 0
\(527\) 1139.76i 2.16274i
\(528\) 0 0
\(529\) −145.967 −0.275930
\(530\) 0 0
\(531\) 20.5617 + 767.897i 0.0387227 + 1.44613i
\(532\) 0 0
\(533\) 595.990 595.990i 1.11818 1.11818i
\(534\) 0 0
\(535\) 0.807394i 0.00150915i
\(536\) 0 0
\(537\) 29.7342 74.5933i 0.0553709 0.138907i
\(538\) 0 0
\(539\) −70.2895 + 70.2895i −0.130407 + 0.130407i
\(540\) 0 0
\(541\) −431.469 + 431.469i −0.797540 + 0.797540i −0.982707 0.185167i \(-0.940717\pi\)
0.185167 + 0.982707i \(0.440717\pi\)
\(542\) 0 0
\(543\) −38.5062 + 96.5993i −0.0709137 + 0.177899i
\(544\) 0 0
\(545\) 2.37483i 0.00435749i
\(546\) 0 0
\(547\) −335.381 + 335.381i −0.613127 + 0.613127i −0.943760 0.330632i \(-0.892738\pi\)
0.330632 + 0.943760i \(0.392738\pi\)
\(548\) 0 0
\(549\) 16.3609 + 611.014i 0.0298014 + 1.11296i
\(550\) 0 0
\(551\) −46.2983 −0.0840259
\(552\) 0 0
\(553\) 304.900i 0.551357i
\(554\) 0 0
\(555\) −0.217948 0.506874i −0.000392699 0.000913287i
\(556\) 0 0
\(557\) −118.642 118.642i −0.213001 0.213001i 0.592540 0.805541i \(-0.298125\pi\)
−0.805541 + 0.592540i \(0.798125\pi\)
\(558\) 0 0
\(559\) −608.350 −1.08828
\(560\) 0 0
\(561\) 271.004 679.860i 0.483073 1.21187i
\(562\) 0 0
\(563\) −290.766 290.766i −0.516459 0.516459i 0.400039 0.916498i \(-0.368997\pi\)
−0.916498 + 0.400039i \(0.868997\pi\)
\(564\) 0 0
\(565\) −2.02305 2.02305i −0.00358061 0.00358061i
\(566\) 0 0
\(567\) 519.499 27.8409i 0.916223 0.0491021i
\(568\) 0 0
\(569\) −669.398 −1.17645 −0.588223 0.808699i \(-0.700172\pi\)
−0.588223 + 0.808699i \(0.700172\pi\)
\(570\) 0 0
\(571\) −454.971 454.971i −0.796798 0.796798i 0.185792 0.982589i \(-0.440515\pi\)
−0.982589 + 0.185792i \(0.940515\pi\)
\(572\) 0 0
\(573\) 431.102 185.367i 0.752359 0.323502i
\(574\) 0 0
\(575\) 489.277i 0.850916i
\(576\) 0 0
\(577\) −288.393 −0.499814 −0.249907 0.968270i \(-0.580400\pi\)
−0.249907 + 0.968270i \(0.580400\pi\)
\(578\) 0 0
\(579\) −258.307 600.737i −0.446126 1.03754i
\(580\) 0 0
\(581\) −451.725 + 451.725i −0.777496 + 0.777496i
\(582\) 0 0
\(583\) 374.502i 0.642371i
\(584\) 0 0
\(585\) −1.62416 1.53945i −0.00277635 0.00263154i
\(586\) 0 0
\(587\) −393.610 + 393.610i −0.670545 + 0.670545i −0.957842 0.287297i \(-0.907243\pi\)
0.287297 + 0.957842i \(0.407243\pi\)
\(588\) 0 0
\(589\) 124.172 124.172i 0.210818 0.210818i
\(590\) 0 0
\(591\) 969.287 + 386.375i 1.64008 + 0.653764i
\(592\) 0 0
\(593\) 707.638i 1.19332i 0.802495 + 0.596659i \(0.203506\pi\)
−0.802495 + 0.596659i \(0.796494\pi\)
\(594\) 0 0
\(595\) 1.20413 1.20413i 0.00202375 0.00202375i
\(596\) 0 0
\(597\) −642.678 + 276.341i −1.07651 + 0.462883i
\(598\) 0 0
\(599\) −996.581 −1.66374 −0.831870 0.554970i \(-0.812730\pi\)
−0.831870 + 0.554970i \(0.812730\pi\)
\(600\) 0 0
\(601\) 214.386i 0.356716i 0.983966 + 0.178358i \(0.0570785\pi\)
−0.983966 + 0.178358i \(0.942921\pi\)
\(602\) 0 0
\(603\) 8.10051 + 302.521i 0.0134337 + 0.501693i
\(604\) 0 0
\(605\) 0.429836 + 0.429836i 0.000710474 + 0.000710474i
\(606\) 0 0
\(607\) 989.981 1.63094 0.815470 0.578799i \(-0.196478\pi\)
0.815470 + 0.578799i \(0.196478\pi\)
\(608\) 0 0
\(609\) 282.849 + 112.748i 0.464448 + 0.185137i
\(610\) 0 0
\(611\) −79.4044 79.4044i −0.129958 0.129958i
\(612\) 0 0
\(613\) 277.427 + 277.427i 0.452572 + 0.452572i 0.896207 0.443636i \(-0.146312\pi\)
−0.443636 + 0.896207i \(0.646312\pi\)
\(614\) 0 0
\(615\) −1.83648 0.732052i −0.00298614 0.00119033i
\(616\) 0 0
\(617\) 294.951 0.478040 0.239020 0.971015i \(-0.423174\pi\)
0.239020 + 0.971015i \(0.423174\pi\)
\(618\) 0 0
\(619\) −717.374 717.374i −1.15892 1.15892i −0.984707 0.174218i \(-0.944260\pi\)
−0.174218 0.984707i \(-0.555740\pi\)
\(620\) 0 0
\(621\) 495.924 + 182.457i 0.798589 + 0.293812i
\(622\) 0 0
\(623\) 611.273i 0.981177i
\(624\) 0 0
\(625\) −624.985 −0.999977
\(626\) 0 0
\(627\) −103.592 + 44.5428i −0.165218 + 0.0710412i
\(628\) 0 0
\(629\) −177.359 + 177.359i −0.281970 + 0.281970i
\(630\) 0 0
\(631\) 526.114i 0.833779i −0.908957 0.416889i \(-0.863120\pi\)
0.908957 0.416889i \(-0.136880\pi\)
\(632\) 0 0
\(633\) 33.4795 + 13.3455i 0.0528903 + 0.0210830i
\(634\) 0 0
\(635\) 0.533835 0.533835i 0.000840686 0.000840686i
\(636\) 0 0
\(637\) 97.7020 97.7020i 0.153378 0.153378i
\(638\) 0 0
\(639\) 84.0431 88.6678i 0.131523 0.138760i
\(640\) 0 0
\(641\) 1025.84i 1.60037i 0.599754 + 0.800184i \(0.295265\pi\)
−0.599754 + 0.800184i \(0.704735\pi\)
\(642\) 0 0
\(643\) 366.197 366.197i 0.569514 0.569514i −0.362479 0.931992i \(-0.618069\pi\)
0.931992 + 0.362479i \(0.118069\pi\)
\(644\) 0 0
\(645\) 0.563665 + 1.31090i 0.000873900 + 0.00203240i
\(646\) 0 0
\(647\) −90.9084 −0.140508 −0.0702538 0.997529i \(-0.522381\pi\)
−0.0702538 + 0.997529i \(0.522381\pi\)
\(648\) 0 0
\(649\) 1095.03i 1.68726i
\(650\) 0 0
\(651\) −1060.99 + 456.208i −1.62978 + 0.700780i
\(652\) 0 0
\(653\) −291.274 291.274i −0.446056 0.446056i 0.447985 0.894041i \(-0.352142\pi\)
−0.894041 + 0.447985i \(0.852142\pi\)
\(654\) 0 0
\(655\) 0.616367 0.000941019
\(656\) 0 0
\(657\) 205.443 + 194.728i 0.312698 + 0.296389i
\(658\) 0 0
\(659\) 817.853 + 817.853i 1.24105 + 1.24105i 0.959565 + 0.281486i \(0.0908273\pi\)
0.281486 + 0.959565i \(0.409173\pi\)
\(660\) 0 0
\(661\) −673.995 673.995i −1.01966 1.01966i −0.999803 0.0198568i \(-0.993679\pi\)
−0.0198568 0.999803i \(-0.506321\pi\)
\(662\) 0 0
\(663\) −376.694 + 945.001i −0.568166 + 1.42534i
\(664\) 0 0
\(665\) −0.262368 −0.000394538
\(666\) 0 0
\(667\) 218.694 + 218.694i 0.327877 + 0.327877i
\(668\) 0 0
\(669\) 12.9514 + 30.1206i 0.0193593 + 0.0450233i
\(670\) 0 0
\(671\) 871.316i 1.29853i
\(672\) 0 0
\(673\) −526.059 −0.781662 −0.390831 0.920462i \(-0.627812\pi\)
−0.390831 + 0.920462i \(0.627812\pi\)
\(674\) 0 0
\(675\) 233.066 633.481i 0.345283 0.938490i
\(676\) 0 0
\(677\) −143.663 + 143.663i −0.212205 + 0.212205i −0.805204 0.592998i \(-0.797944\pi\)
0.592998 + 0.805204i \(0.297944\pi\)
\(678\) 0 0
\(679\) 395.782i 0.582890i
\(680\) 0 0
\(681\) −156.945 + 393.723i −0.230462 + 0.578155i
\(682\) 0 0
\(683\) 50.6262 50.6262i 0.0741232 0.0741232i −0.669073 0.743196i \(-0.733309\pi\)
0.743196 + 0.669073i \(0.233309\pi\)
\(684\) 0 0
\(685\) −0.423133 + 0.423133i −0.000617713 + 0.000617713i
\(686\) 0 0
\(687\) 364.243 913.765i 0.530193 1.33008i
\(688\) 0 0
\(689\) 520.556i 0.755523i
\(690\) 0 0
\(691\) −396.186 + 396.186i −0.573351 + 0.573351i −0.933063 0.359712i \(-0.882875\pi\)
0.359712 + 0.933063i \(0.382875\pi\)
\(692\) 0 0
\(693\) 741.345 19.8508i 1.06976 0.0286447i
\(694\) 0 0
\(695\) −0.928629 −0.00133616
\(696\) 0 0
\(697\) 898.749i 1.28945i
\(698\) 0 0
\(699\) 375.233 + 872.669i 0.536815 + 1.24845i
\(700\) 0 0
\(701\) −525.886 525.886i −0.750195 0.750195i 0.224321 0.974515i \(-0.427984\pi\)
−0.974515 + 0.224321i \(0.927984\pi\)
\(702\) 0 0
\(703\) 38.6449 0.0549713
\(704\) 0 0
\(705\) −0.0975321 + 0.244676i −0.000138343 + 0.000347058i
\(706\) 0 0
\(707\) 309.492 + 309.492i 0.437753 + 0.437753i
\(708\) 0 0
\(709\) −99.4062 99.4062i −0.140206 0.140206i 0.633520 0.773726i \(-0.281609\pi\)
−0.773726 + 0.633520i \(0.781609\pi\)
\(710\) 0 0
\(711\) 293.914 310.087i 0.413381 0.436128i
\(712\) 0 0
\(713\) −1173.07 −1.64526
\(714\) 0 0
\(715\) −2.25568 2.25568i −0.00315480 0.00315480i
\(716\) 0 0
\(717\) 1054.71 453.506i 1.47100 0.632505i
\(718\) 0 0
\(719\) 551.765i 0.767406i 0.923456 + 0.383703i \(0.125351\pi\)
−0.923456 + 0.383703i \(0.874649\pi\)
\(720\) 0 0
\(721\) 30.4226 0.0421951
\(722\) 0 0
\(723\) −108.213 251.668i −0.149673 0.348089i
\(724\) 0 0
\(725\) 279.354 279.354i 0.385316 0.385316i
\(726\) 0 0
\(727\) 75.0947i 0.103294i −0.998665 0.0516470i \(-0.983553\pi\)
0.998665 0.0516470i \(-0.0164471\pi\)
\(728\) 0 0
\(729\) 555.174 + 472.465i 0.761555 + 0.648100i
\(730\) 0 0
\(731\) 458.694 458.694i 0.627488 0.627488i
\(732\) 0 0
\(733\) 442.709 442.709i 0.603968 0.603968i −0.337395 0.941363i \(-0.609546\pi\)
0.941363 + 0.337395i \(0.109546\pi\)
\(734\) 0 0
\(735\) −0.301058 0.120007i −0.000409603 0.000163275i
\(736\) 0 0
\(737\) 431.400i 0.585346i
\(738\) 0 0
\(739\) −283.395 + 283.395i −0.383485 + 0.383485i −0.872356 0.488871i \(-0.837409\pi\)
0.488871 + 0.872356i \(0.337409\pi\)
\(740\) 0 0
\(741\) 143.992 61.9143i 0.194321 0.0835550i
\(742\) 0 0
\(743\) 835.949 1.12510 0.562550 0.826763i \(-0.309820\pi\)
0.562550 + 0.826763i \(0.309820\pi\)
\(744\) 0 0
\(745\) 2.59361i 0.00348135i
\(746\) 0 0
\(747\) −894.857 + 23.9613i −1.19794 + 0.0320768i
\(748\) 0 0
\(749\) 262.988 + 262.988i 0.351118 + 0.351118i
\(750\) 0 0
\(751\) −753.712 −1.00361 −0.501806 0.864980i \(-0.667331\pi\)
−0.501806 + 0.864980i \(0.667331\pi\)
\(752\) 0 0
\(753\) −507.485 202.292i −0.673951 0.268648i
\(754\) 0 0
\(755\) 1.43854 + 1.43854i 0.00190535 + 0.00190535i
\(756\) 0 0
\(757\) −335.789 335.789i −0.443578 0.443578i 0.449634 0.893213i \(-0.351554\pi\)
−0.893213 + 0.449634i \(0.851554\pi\)
\(758\) 0 0
\(759\) 699.727 + 278.923i 0.921906 + 0.367488i
\(760\) 0 0
\(761\) 1094.53 1.43828 0.719138 0.694868i \(-0.244537\pi\)
0.719138 + 0.694868i \(0.244537\pi\)
\(762\) 0 0
\(763\) −773.541 773.541i −1.01381 1.01381i
\(764\) 0 0
\(765\) 2.38535 0.0638720i 0.00311811 8.34928e-5i
\(766\) 0 0
\(767\) 1522.09i 1.98447i
\(768\) 0 0
\(769\) 290.367 0.377590 0.188795 0.982016i \(-0.439542\pi\)
0.188795 + 0.982016i \(0.439542\pi\)
\(770\) 0 0
\(771\) 341.512 146.845i 0.442947 0.190460i
\(772\) 0 0
\(773\) −193.239 + 193.239i −0.249986 + 0.249986i −0.820965 0.570979i \(-0.806564\pi\)
0.570979 + 0.820965i \(0.306564\pi\)
\(774\) 0 0
\(775\) 1498.45i 1.93348i
\(776\) 0 0
\(777\) −236.092 94.1104i −0.303851 0.121120i
\(778\) 0 0
\(779\) 97.9143 97.9143i 0.125692 0.125692i
\(780\) 0 0
\(781\) 123.144 123.144i 0.157675 0.157675i
\(782\) 0 0
\(783\) 178.975 + 387.323i 0.228575 + 0.494666i
\(784\) 0 0
\(785\) 1.08961i 0.00138804i
\(786\) 0 0
\(787\) −483.899 + 483.899i −0.614865 + 0.614865i −0.944210 0.329345i \(-0.893172\pi\)
0.329345 + 0.944210i \(0.393172\pi\)
\(788\) 0 0
\(789\) 230.348 + 535.714i 0.291950 + 0.678979i
\(790\) 0 0
\(791\) −1317.91 −1.66613
\(792\) 0 0
\(793\) 1211.12i 1.52727i
\(794\) 0 0
\(795\) −1.12172 + 0.482320i −0.00141096 + 0.000606691i
\(796\) 0 0
\(797\) −872.325 872.325i −1.09451 1.09451i −0.995041 0.0994694i \(-0.968285\pi\)
−0.0994694 0.995041i \(-0.531715\pi\)
\(798\) 0 0
\(799\) 119.741 0.149864
\(800\) 0 0
\(801\) 589.248 621.672i 0.735640 0.776120i
\(802\) 0 0
\(803\) 285.325 + 285.325i 0.355324 + 0.355324i
\(804\) 0 0
\(805\) 1.23932 + 1.23932i 0.00153952 + 0.00153952i
\(806\) 0 0
\(807\) 466.009 1169.06i 0.577459 1.44865i
\(808\) 0 0
\(809\) −146.162 −0.180670 −0.0903349 0.995911i \(-0.528794\pi\)
−0.0903349 + 0.995911i \(0.528794\pi\)
\(810\) 0 0
\(811\) 375.179 + 375.179i 0.462613 + 0.462613i 0.899511 0.436898i \(-0.143923\pi\)
−0.436898 + 0.899511i \(0.643923\pi\)
\(812\) 0 0
\(813\) 329.711 + 766.800i 0.405549 + 0.943173i
\(814\) 0 0
\(815\) 1.39243i 0.00170850i
\(816\) 0 0
\(817\) −99.9449 −0.122332
\(818\) 0 0
\(819\) −1030.47 + 27.5925i −1.25820 + 0.0336904i
\(820\) 0 0
\(821\) 671.154 671.154i 0.817484 0.817484i −0.168259 0.985743i \(-0.553814\pi\)
0.985743 + 0.168259i \(0.0538144\pi\)
\(822\) 0 0
\(823\) 675.121i 0.820317i −0.912014 0.410159i \(-0.865473\pi\)
0.912014 0.410159i \(-0.134527\pi\)
\(824\) 0 0
\(825\) 356.290 893.814i 0.431867 1.08341i
\(826\) 0 0
\(827\) 1052.16 1052.16i 1.27226 1.27226i 0.327355 0.944901i \(-0.393843\pi\)
0.944901 0.327355i \(-0.106157\pi\)
\(828\) 0 0
\(829\) −95.3529 + 95.3529i −0.115022 + 0.115022i −0.762275 0.647253i \(-0.775918\pi\)
0.647253 + 0.762275i \(0.275918\pi\)
\(830\) 0 0
\(831\) 94.4964 237.060i 0.113714 0.285271i
\(832\) 0 0
\(833\) 147.334i 0.176872i
\(834\) 0 0
\(835\) 0.849877 0.849877i 0.00101782 0.00101782i
\(836\) 0 0
\(837\) −1518.81 558.790i −1.81458 0.667610i
\(838\) 0 0
\(839\) −581.969 −0.693646 −0.346823 0.937931i \(-0.612740\pi\)
−0.346823 + 0.937931i \(0.612740\pi\)
\(840\) 0 0
\(841\) 591.273i 0.703059i
\(842\) 0 0
\(843\) −371.684 864.413i −0.440906 1.02540i
\(844\) 0 0
\(845\) 1.46918 + 1.46918i 0.00173867 + 0.00173867i
\(846\) 0 0
\(847\) 280.016 0.330598
\(848\) 0 0
\(849\) 450.839 1131.01i 0.531024 1.33216i
\(850\) 0 0
\(851\) −182.542 182.542i −0.214503 0.214503i
\(852\) 0 0
\(853\) 595.516 + 595.516i 0.698143 + 0.698143i 0.964010 0.265866i \(-0.0856580\pi\)
−0.265866 + 0.964010i \(0.585658\pi\)
\(854\) 0 0
\(855\) −0.266831 0.252914i −0.000312083 0.000295806i
\(856\) 0 0
\(857\) 731.802 0.853912 0.426956 0.904273i \(-0.359586\pi\)
0.426956 + 0.904273i \(0.359586\pi\)
\(858\) 0 0
\(859\) −303.614 303.614i −0.353451 0.353451i 0.507941 0.861392i \(-0.330407\pi\)
−0.861392 + 0.507941i \(0.830407\pi\)
\(860\) 0 0
\(861\) −836.632 + 359.738i −0.971698 + 0.417814i
\(862\) 0 0
\(863\) 1423.90i 1.64995i 0.565173 + 0.824973i \(0.308810\pi\)
−0.565173 + 0.824973i \(0.691190\pi\)
\(864\) 0 0
\(865\) 1.14845 0.00132769
\(866\) 0 0
\(867\) −86.0236 200.062i −0.0992198 0.230753i
\(868\) 0 0
\(869\) 430.658 430.658i 0.495579 0.495579i
\(870\) 0 0
\(871\) 599.643i 0.688453i
\(872\) 0 0
\(873\) −381.521 + 402.515i −0.437023 + 0.461071i
\(874\) 0 0
\(875\) 3.16616 3.16616i 0.00361847 0.00361847i
\(876\) 0 0
\(877\) −524.721 + 524.721i −0.598314 + 0.598314i −0.939864 0.341550i \(-0.889048\pi\)
0.341550 + 0.939864i \(0.389048\pi\)
\(878\) 0 0
\(879\) −695.604 277.280i −0.791358 0.315449i
\(880\) 0 0
\(881\) 51.0313i 0.0579243i 0.999581 + 0.0289622i \(0.00922023\pi\)
−0.999581 + 0.0289622i \(0.990780\pi\)
\(882\) 0 0
\(883\) 935.183 935.183i 1.05910 1.05910i 0.0609574 0.998140i \(-0.480585\pi\)
0.998140 0.0609574i \(-0.0194154\pi\)
\(884\) 0 0
\(885\) −3.27987 + 1.41029i −0.00370606 + 0.00159355i
\(886\) 0 0
\(887\) 1077.88 1.21520 0.607598 0.794245i \(-0.292133\pi\)
0.607598 + 0.794245i \(0.292133\pi\)
\(888\) 0 0
\(889\) 347.766i 0.391188i
\(890\) 0 0
\(891\) 773.092 + 694.444i 0.867668 + 0.779398i
\(892\) 0 0
\(893\) −13.0452 13.0452i −0.0146083 0.0146083i
\(894\) 0 0
\(895\) 0.373214 0.000416999
\(896\) 0 0
\(897\) −972.616 387.702i −1.08430 0.432220i
\(898\) 0 0
\(899\) −669.767 669.767i −0.745013 0.745013i
\(900\) 0 0
\(901\) 392.497 + 392.497i 0.435624 + 0.435624i
\(902\) 0 0
\(903\) 610.591 + 243.392i 0.676180 + 0.269537i
\(904\) 0 0
\(905\) −0.483317 −0.000534052
\(906\) 0 0
\(907\) 1091.36 + 1091.36i 1.20327 + 1.20327i 0.973167 + 0.230101i \(0.0739056\pi\)
0.230101 + 0.973167i \(0.426094\pi\)
\(908\) 0 0
\(909\) 16.4167 + 613.096i 0.0180602 + 0.674473i
\(910\) 0 0
\(911\) 599.270i 0.657816i −0.944362 0.328908i \(-0.893319\pi\)
0.944362 0.328908i \(-0.106681\pi\)
\(912\) 0 0
\(913\) −1276.08 −1.39768
\(914\) 0 0
\(915\) −2.60978 + 1.12216i −0.00285222 + 0.00122641i
\(916\) 0 0
\(917\) 200.766 200.766i 0.218937 0.218937i
\(918\) 0 0
\(919\) 1271.46i 1.38353i −0.722123 0.691765i \(-0.756834\pi\)
0.722123 0.691765i \(-0.243166\pi\)
\(920\) 0 0
\(921\) −252.190 100.528i −0.273822 0.109150i
\(922\) 0 0
\(923\) −171.170 + 171.170i −0.185449 + 0.185449i
\(924\) 0 0
\(925\) −233.175 + 233.175i −0.252081 + 0.252081i
\(926\) 0 0
\(927\) 30.9402 + 29.3264i 0.0333767 + 0.0316358i
\(928\) 0 0
\(929\) 1274.81i 1.37224i 0.727488 + 0.686120i \(0.240688\pi\)
−0.727488 + 0.686120i \(0.759312\pi\)
\(930\) 0 0
\(931\) 16.0513 16.0513i 0.0172409 0.0172409i
\(932\) 0 0
\(933\) 630.785 + 1467.00i 0.676083 + 1.57235i
\(934\) 0 0
\(935\) 3.40156 0.00363803
\(936\) 0 0
\(937\) 416.118i 0.444096i 0.975036 + 0.222048i \(0.0712742\pi\)
−0.975036 + 0.222048i \(0.928726\pi\)
\(938\) 0 0
\(939\) 511.728 220.035i 0.544971 0.234329i
\(940\) 0 0
\(941\) 59.8023 + 59.8023i 0.0635518 + 0.0635518i 0.738168 0.674617i \(-0.235691\pi\)
−0.674617 + 0.738168i \(0.735691\pi\)
\(942\) 0 0
\(943\) −925.012 −0.980925
\(944\) 0 0
\(945\) 1.01423 + 2.19493i 0.00107326 + 0.00232267i
\(946\) 0 0
\(947\) 438.459 + 438.459i 0.462998 + 0.462998i 0.899637 0.436639i \(-0.143831\pi\)
−0.436639 + 0.899637i \(0.643831\pi\)
\(948\) 0 0
\(949\) −396.600 396.600i −0.417913 0.417913i
\(950\) 0 0
\(951\) −264.125 + 662.602i −0.277733 + 0.696742i
\(952\) 0 0
\(953\) 874.202 0.917316 0.458658 0.888613i \(-0.348330\pi\)
0.458658 + 0.888613i \(0.348330\pi\)
\(954\) 0 0
\(955\) 1.54219 + 1.54219i 0.00161486 + 0.00161486i
\(956\) 0 0
\(957\) 240.259 + 558.763i 0.251054 + 0.583869i
\(958\) 0 0
\(959\) 275.649i 0.287434i
\(960\) 0 0
\(961\) 2631.62 2.73842
\(962\) 0 0
\(963\) 13.9499 + 520.973i 0.0144859 + 0.540989i
\(964\) 0 0
\(965\) 2.14904 2.14904i 0.00222698 0.00222698i
\(966\) 0 0
\(967\) 78.0123i 0.0806746i 0.999186 + 0.0403373i \(0.0128432\pi\)
−0.999186 + 0.0403373i \(0.987157\pi\)
\(968\) 0 0
\(969\) −61.8865 + 155.253i −0.0638663 + 0.160220i
\(970\) 0 0
\(971\) 545.451 545.451i 0.561742 0.561742i −0.368060 0.929802i \(-0.619978\pi\)
0.929802 + 0.368060i \(0.119978\pi\)
\(972\) 0 0
\(973\) −302.477 + 302.477i −0.310870 + 0.310870i
\(974\) 0 0
\(975\) −495.241 + 1242.40i −0.507939 + 1.27425i
\(976\) 0 0
\(977\) 1711.35i 1.75164i −0.482640 0.875819i \(-0.660322\pi\)
0.482640 0.875819i \(-0.339678\pi\)
\(978\) 0 0
\(979\) 863.396 863.396i 0.881916 0.881916i
\(980\) 0 0
\(981\) −41.0318 1532.37i −0.0418265 1.56205i
\(982\) 0 0
\(983\) −1349.18 −1.37252 −0.686259 0.727358i \(-0.740748\pi\)
−0.686259 + 0.727358i \(0.740748\pi\)
\(984\) 0 0
\(985\) 4.84965i 0.00492351i
\(986\) 0 0
\(987\) 47.9283 + 111.465i 0.0485596 + 0.112934i
\(988\) 0 0
\(989\) 472.098 + 472.098i 0.477349 + 0.477349i
\(990\) 0 0
\(991\) 923.093 0.931476 0.465738 0.884923i \(-0.345789\pi\)
0.465738 + 0.884923i \(0.345789\pi\)
\(992\) 0 0
\(993\) −379.671 + 952.469i −0.382347 + 0.959183i
\(994\) 0 0
\(995\) −2.29907 2.29907i −0.00231063 0.00231063i
\(996\) 0 0
\(997\) 371.389 + 371.389i 0.372507 + 0.372507i 0.868390 0.495883i \(-0.165155\pi\)
−0.495883 + 0.868390i \(0.665155\pi\)
\(998\) 0 0
\(999\) −149.389 323.296i −0.149538 0.323620i
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.i.b.113.6 20
3.2 odd 2 inner 192.3.i.b.113.2 20
4.3 odd 2 48.3.i.b.5.5 20
8.3 odd 2 384.3.i.d.353.6 20
8.5 even 2 384.3.i.c.353.5 20
12.11 even 2 48.3.i.b.5.6 yes 20
16.3 odd 4 48.3.i.b.29.6 yes 20
16.5 even 4 384.3.i.c.161.9 20
16.11 odd 4 384.3.i.d.161.2 20
16.13 even 4 inner 192.3.i.b.17.2 20
24.5 odd 2 384.3.i.c.353.9 20
24.11 even 2 384.3.i.d.353.2 20
48.5 odd 4 384.3.i.c.161.5 20
48.11 even 4 384.3.i.d.161.6 20
48.29 odd 4 inner 192.3.i.b.17.6 20
48.35 even 4 48.3.i.b.29.5 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
48.3.i.b.5.5 20 4.3 odd 2
48.3.i.b.5.6 yes 20 12.11 even 2
48.3.i.b.29.5 yes 20 48.35 even 4
48.3.i.b.29.6 yes 20 16.3 odd 4
192.3.i.b.17.2 20 16.13 even 4 inner
192.3.i.b.17.6 20 48.29 odd 4 inner
192.3.i.b.113.2 20 3.2 odd 2 inner
192.3.i.b.113.6 20 1.1 even 1 trivial
384.3.i.c.161.5 20 48.5 odd 4
384.3.i.c.161.9 20 16.5 even 4
384.3.i.c.353.5 20 8.5 even 2
384.3.i.c.353.9 20 24.5 odd 2
384.3.i.d.161.2 20 16.11 odd 4
384.3.i.d.161.6 20 48.11 even 4
384.3.i.d.353.2 20 24.11 even 2
384.3.i.d.353.6 20 8.3 odd 2