Properties

Label 6498.2.a.ca.1.4
Level $6498$
Weight $2$
Character 6498.1
Self dual yes
Analytic conductor $51.887$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6498,2,Mod(1,6498)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6498, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6498.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6498 = 2 \cdot 3^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6498.a (trivial)

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: yes
Analytic conductor: \(51.8867912334\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{20})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 722)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.17557\) of defining polynomial
Character \(\chi\) \(=\) 6498.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +2.45965 q^{5} -2.79360 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +2.45965 q^{5} -2.79360 q^{7} +1.00000 q^{8} +2.45965 q^{10} +1.67853 q^{11} -6.34458 q^{13} -2.79360 q^{14} +1.00000 q^{16} -4.96917 q^{17} +2.45965 q^{20} +1.67853 q^{22} +2.49890 q^{23} +1.04988 q^{25} -6.34458 q^{26} -2.79360 q^{28} +5.93179 q^{29} -7.28408 q^{31} +1.00000 q^{32} -4.96917 q^{34} -6.87129 q^{35} -0.550972 q^{37} +2.45965 q^{40} -2.60741 q^{41} +2.87129 q^{43} +1.67853 q^{44} +2.49890 q^{46} -0.745593 q^{47} +0.804226 q^{49} +1.04988 q^{50} -6.34458 q^{52} -1.47027 q^{53} +4.12860 q^{55} -2.79360 q^{56} +5.93179 q^{58} -4.96261 q^{59} +9.33686 q^{61} -7.28408 q^{62} +1.00000 q^{64} -15.6054 q^{65} -11.5604 q^{67} -4.96917 q^{68} -6.87129 q^{70} -6.99698 q^{71} -6.18619 q^{73} -0.550972 q^{74} -4.68915 q^{77} +5.91158 q^{79} +2.45965 q^{80} -2.60741 q^{82} -15.1773 q^{83} -12.2224 q^{85} +2.87129 q^{86} +1.67853 q^{88} +6.90502 q^{89} +17.7242 q^{91} +2.49890 q^{92} -0.745593 q^{94} -14.3874 q^{97} +0.804226 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 4 q^{4} + 2 q^{5} - 2 q^{7} + 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} + 4 q^{4} + 2 q^{5} - 2 q^{7} + 4 q^{8} + 2 q^{10} - 2 q^{11} - 18 q^{13} - 2 q^{14} + 4 q^{16} - 6 q^{17} + 2 q^{20} - 2 q^{22} + 10 q^{23} + 6 q^{25} - 18 q^{26} - 2 q^{28} - 2 q^{29} - 26 q^{31} + 4 q^{32} - 6 q^{34} - 6 q^{35} - 4 q^{37} + 2 q^{40} - 12 q^{41} - 10 q^{43} - 2 q^{44} + 10 q^{46} + 12 q^{47} - 12 q^{49} + 6 q^{50} - 18 q^{52} + 8 q^{53} - 26 q^{55} - 2 q^{56} - 2 q^{58} - 8 q^{59} - 26 q^{62} + 4 q^{64} - 4 q^{65} - 10 q^{67} - 6 q^{68} - 6 q^{70} - 14 q^{73} - 4 q^{74} - 4 q^{77} - 22 q^{79} + 2 q^{80} - 12 q^{82} + 12 q^{83} - 18 q^{85} - 10 q^{86} - 2 q^{88} - 16 q^{89} + 4 q^{91} + 10 q^{92} + 12 q^{94} - 28 q^{97} - 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 2.45965 1.09999 0.549994 0.835168i \(-0.314630\pi\)
0.549994 + 0.835168i \(0.314630\pi\)
\(6\) 0 0
\(7\) −2.79360 −1.05588 −0.527942 0.849281i \(-0.677036\pi\)
−0.527942 + 0.849281i \(0.677036\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 2.45965 0.777809
\(11\) 1.67853 0.506096 0.253048 0.967454i \(-0.418567\pi\)
0.253048 + 0.967454i \(0.418567\pi\)
\(12\) 0 0
\(13\) −6.34458 −1.75967 −0.879834 0.475280i \(-0.842347\pi\)
−0.879834 + 0.475280i \(0.842347\pi\)
\(14\) −2.79360 −0.746622
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −4.96917 −1.20520 −0.602601 0.798043i \(-0.705869\pi\)
−0.602601 + 0.798043i \(0.705869\pi\)
\(18\) 0 0
\(19\) 0 0
\(20\) 2.45965 0.549994
\(21\) 0 0
\(22\) 1.67853 0.357864
\(23\) 2.49890 0.521057 0.260529 0.965466i \(-0.416103\pi\)
0.260529 + 0.965466i \(0.416103\pi\)
\(24\) 0 0
\(25\) 1.04988 0.209975
\(26\) −6.34458 −1.24427
\(27\) 0 0
\(28\) −2.79360 −0.527942
\(29\) 5.93179 1.10150 0.550752 0.834669i \(-0.314341\pi\)
0.550752 + 0.834669i \(0.314341\pi\)
\(30\) 0 0
\(31\) −7.28408 −1.30826 −0.654130 0.756382i \(-0.726965\pi\)
−0.654130 + 0.756382i \(0.726965\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −4.96917 −0.852206
\(35\) −6.87129 −1.16146
\(36\) 0 0
\(37\) −0.550972 −0.0905792 −0.0452896 0.998974i \(-0.514421\pi\)
−0.0452896 + 0.998974i \(0.514421\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 2.45965 0.388905
\(41\) −2.60741 −0.407209 −0.203605 0.979053i \(-0.565266\pi\)
−0.203605 + 0.979053i \(0.565266\pi\)
\(42\) 0 0
\(43\) 2.87129 0.437867 0.218934 0.975740i \(-0.429742\pi\)
0.218934 + 0.975740i \(0.429742\pi\)
\(44\) 1.67853 0.253048
\(45\) 0 0
\(46\) 2.49890 0.368443
\(47\) −0.745593 −0.108756 −0.0543780 0.998520i \(-0.517318\pi\)
−0.0543780 + 0.998520i \(0.517318\pi\)
\(48\) 0 0
\(49\) 0.804226 0.114889
\(50\) 1.04988 0.148475
\(51\) 0 0
\(52\) −6.34458 −0.879834
\(53\) −1.47027 −0.201957 −0.100979 0.994889i \(-0.532197\pi\)
−0.100979 + 0.994889i \(0.532197\pi\)
\(54\) 0 0
\(55\) 4.12860 0.556700
\(56\) −2.79360 −0.373311
\(57\) 0 0
\(58\) 5.93179 0.778882
\(59\) −4.96261 −0.646077 −0.323038 0.946386i \(-0.604704\pi\)
−0.323038 + 0.946386i \(0.604704\pi\)
\(60\) 0 0
\(61\) 9.33686 1.19546 0.597731 0.801697i \(-0.296069\pi\)
0.597731 + 0.801697i \(0.296069\pi\)
\(62\) −7.28408 −0.925079
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −15.6054 −1.93562
\(66\) 0 0
\(67\) −11.5604 −1.41233 −0.706166 0.708046i \(-0.749577\pi\)
−0.706166 + 0.708046i \(0.749577\pi\)
\(68\) −4.96917 −0.602601
\(69\) 0 0
\(70\) −6.87129 −0.821276
\(71\) −6.99698 −0.830389 −0.415195 0.909733i \(-0.636286\pi\)
−0.415195 + 0.909733i \(0.636286\pi\)
\(72\) 0 0
\(73\) −6.18619 −0.724039 −0.362020 0.932171i \(-0.617913\pi\)
−0.362020 + 0.932171i \(0.617913\pi\)
\(74\) −0.550972 −0.0640492
\(75\) 0 0
\(76\) 0 0
\(77\) −4.68915 −0.534379
\(78\) 0 0
\(79\) 5.91158 0.665105 0.332552 0.943085i \(-0.392090\pi\)
0.332552 + 0.943085i \(0.392090\pi\)
\(80\) 2.45965 0.274997
\(81\) 0 0
\(82\) −2.60741 −0.287941
\(83\) −15.1773 −1.66593 −0.832964 0.553328i \(-0.813358\pi\)
−0.832964 + 0.553328i \(0.813358\pi\)
\(84\) 0 0
\(85\) −12.2224 −1.32571
\(86\) 2.87129 0.309619
\(87\) 0 0
\(88\) 1.67853 0.178932
\(89\) 6.90502 0.731930 0.365965 0.930629i \(-0.380739\pi\)
0.365965 + 0.930629i \(0.380739\pi\)
\(90\) 0 0
\(91\) 17.7242 1.85800
\(92\) 2.49890 0.260529
\(93\) 0 0
\(94\) −0.745593 −0.0769021
\(95\) 0 0
\(96\) 0 0
\(97\) −14.3874 −1.46082 −0.730408 0.683011i \(-0.760670\pi\)
−0.730408 + 0.683011i \(0.760670\pi\)
\(98\) 0.804226 0.0812391
\(99\) 0 0
\(100\) 1.04988 0.104988
\(101\) −11.2829 −1.12269 −0.561347 0.827581i \(-0.689717\pi\)
−0.561347 + 0.827581i \(0.689717\pi\)
\(102\) 0 0
\(103\) −14.8280 −1.46104 −0.730522 0.682889i \(-0.760723\pi\)
−0.730522 + 0.682889i \(0.760723\pi\)
\(104\) −6.34458 −0.622137
\(105\) 0 0
\(106\) −1.47027 −0.142805
\(107\) −2.82849 −0.273440 −0.136720 0.990610i \(-0.543656\pi\)
−0.136720 + 0.990610i \(0.543656\pi\)
\(108\) 0 0
\(109\) −1.72539 −0.165262 −0.0826312 0.996580i \(-0.526332\pi\)
−0.0826312 + 0.996580i \(0.526332\pi\)
\(110\) 4.12860 0.393646
\(111\) 0 0
\(112\) −2.79360 −0.263971
\(113\) −0.427785 −0.0402426 −0.0201213 0.999798i \(-0.506405\pi\)
−0.0201213 + 0.999798i \(0.506405\pi\)
\(114\) 0 0
\(115\) 6.14643 0.573157
\(116\) 5.93179 0.550752
\(117\) 0 0
\(118\) −4.96261 −0.456845
\(119\) 13.8819 1.27255
\(120\) 0 0
\(121\) −8.18253 −0.743867
\(122\) 9.33686 0.845320
\(123\) 0 0
\(124\) −7.28408 −0.654130
\(125\) −9.71592 −0.869018
\(126\) 0 0
\(127\) 1.13632 0.100832 0.0504159 0.998728i \(-0.483945\pi\)
0.0504159 + 0.998728i \(0.483945\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −15.6054 −1.36869
\(131\) 17.8415 1.55882 0.779410 0.626515i \(-0.215519\pi\)
0.779410 + 0.626515i \(0.215519\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −11.5604 −0.998670
\(135\) 0 0
\(136\) −4.96917 −0.426103
\(137\) 15.9156 1.35976 0.679882 0.733321i \(-0.262031\pi\)
0.679882 + 0.733321i \(0.262031\pi\)
\(138\) 0 0
\(139\) 10.1202 0.858382 0.429191 0.903214i \(-0.358799\pi\)
0.429191 + 0.903214i \(0.358799\pi\)
\(140\) −6.87129 −0.580730
\(141\) 0 0
\(142\) −6.99698 −0.587174
\(143\) −10.6496 −0.890562
\(144\) 0 0
\(145\) 14.5901 1.21164
\(146\) −6.18619 −0.511973
\(147\) 0 0
\(148\) −0.550972 −0.0452896
\(149\) −9.39851 −0.769956 −0.384978 0.922926i \(-0.625791\pi\)
−0.384978 + 0.922926i \(0.625791\pi\)
\(150\) 0 0
\(151\) 10.2632 0.835210 0.417605 0.908629i \(-0.362870\pi\)
0.417605 + 0.908629i \(0.362870\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −4.68915 −0.377863
\(155\) −17.9163 −1.43907
\(156\) 0 0
\(157\) 2.97387 0.237341 0.118671 0.992934i \(-0.462137\pi\)
0.118671 + 0.992934i \(0.462137\pi\)
\(158\) 5.91158 0.470300
\(159\) 0 0
\(160\) 2.45965 0.194452
\(161\) −6.98095 −0.550176
\(162\) 0 0
\(163\) 5.59191 0.437992 0.218996 0.975726i \(-0.429722\pi\)
0.218996 + 0.975726i \(0.429722\pi\)
\(164\) −2.60741 −0.203605
\(165\) 0 0
\(166\) −15.1773 −1.17799
\(167\) −7.08361 −0.548146 −0.274073 0.961709i \(-0.588371\pi\)
−0.274073 + 0.961709i \(0.588371\pi\)
\(168\) 0 0
\(169\) 27.2537 2.09643
\(170\) −12.2224 −0.937418
\(171\) 0 0
\(172\) 2.87129 0.218934
\(173\) −23.0208 −1.75024 −0.875120 0.483907i \(-0.839217\pi\)
−0.875120 + 0.483907i \(0.839217\pi\)
\(174\) 0 0
\(175\) −2.93294 −0.221709
\(176\) 1.67853 0.126524
\(177\) 0 0
\(178\) 6.90502 0.517553
\(179\) 8.21471 0.613996 0.306998 0.951710i \(-0.400675\pi\)
0.306998 + 0.951710i \(0.400675\pi\)
\(180\) 0 0
\(181\) 12.9999 0.966274 0.483137 0.875545i \(-0.339497\pi\)
0.483137 + 0.875545i \(0.339497\pi\)
\(182\) 17.7242 1.31381
\(183\) 0 0
\(184\) 2.49890 0.184222
\(185\) −1.35520 −0.0996361
\(186\) 0 0
\(187\) −8.34092 −0.609948
\(188\) −0.745593 −0.0543780
\(189\) 0 0
\(190\) 0 0
\(191\) −6.57479 −0.475735 −0.237868 0.971298i \(-0.576448\pi\)
−0.237868 + 0.971298i \(0.576448\pi\)
\(192\) 0 0
\(193\) −26.8826 −1.93505 −0.967527 0.252769i \(-0.918659\pi\)
−0.967527 + 0.252769i \(0.918659\pi\)
\(194\) −14.3874 −1.03295
\(195\) 0 0
\(196\) 0.804226 0.0574447
\(197\) 9.84940 0.701741 0.350870 0.936424i \(-0.385886\pi\)
0.350870 + 0.936424i \(0.385886\pi\)
\(198\) 0 0
\(199\) −20.4661 −1.45080 −0.725402 0.688326i \(-0.758346\pi\)
−0.725402 + 0.688326i \(0.758346\pi\)
\(200\) 1.04988 0.0742374
\(201\) 0 0
\(202\) −11.2829 −0.793864
\(203\) −16.5711 −1.16306
\(204\) 0 0
\(205\) −6.41332 −0.447926
\(206\) −14.8280 −1.03311
\(207\) 0 0
\(208\) −6.34458 −0.439917
\(209\) 0 0
\(210\) 0 0
\(211\) −8.11630 −0.558749 −0.279374 0.960182i \(-0.590127\pi\)
−0.279374 + 0.960182i \(0.590127\pi\)
\(212\) −1.47027 −0.100979
\(213\) 0 0
\(214\) −2.82849 −0.193351
\(215\) 7.06236 0.481649
\(216\) 0 0
\(217\) 20.3488 1.38137
\(218\) −1.72539 −0.116858
\(219\) 0 0
\(220\) 4.12860 0.278350
\(221\) 31.5273 2.12076
\(222\) 0 0
\(223\) −10.7540 −0.720143 −0.360071 0.932925i \(-0.617248\pi\)
−0.360071 + 0.932925i \(0.617248\pi\)
\(224\) −2.79360 −0.186656
\(225\) 0 0
\(226\) −0.427785 −0.0284558
\(227\) 27.0936 1.79827 0.899133 0.437676i \(-0.144198\pi\)
0.899133 + 0.437676i \(0.144198\pi\)
\(228\) 0 0
\(229\) 9.46557 0.625503 0.312751 0.949835i \(-0.398749\pi\)
0.312751 + 0.949835i \(0.398749\pi\)
\(230\) 6.14643 0.405283
\(231\) 0 0
\(232\) 5.93179 0.389441
\(233\) −22.8621 −1.49775 −0.748873 0.662713i \(-0.769405\pi\)
−0.748873 + 0.662713i \(0.769405\pi\)
\(234\) 0 0
\(235\) −1.83390 −0.119630
\(236\) −4.96261 −0.323038
\(237\) 0 0
\(238\) 13.8819 0.899831
\(239\) −8.66611 −0.560564 −0.280282 0.959918i \(-0.590428\pi\)
−0.280282 + 0.959918i \(0.590428\pi\)
\(240\) 0 0
\(241\) −5.65826 −0.364480 −0.182240 0.983254i \(-0.558335\pi\)
−0.182240 + 0.983254i \(0.558335\pi\)
\(242\) −8.18253 −0.525993
\(243\) 0 0
\(244\) 9.33686 0.597731
\(245\) 1.97811 0.126377
\(246\) 0 0
\(247\) 0 0
\(248\) −7.28408 −0.462539
\(249\) 0 0
\(250\) −9.71592 −0.614489
\(251\) 13.5552 0.855599 0.427799 0.903874i \(-0.359289\pi\)
0.427799 + 0.903874i \(0.359289\pi\)
\(252\) 0 0
\(253\) 4.19449 0.263705
\(254\) 1.13632 0.0712988
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 10.0393 0.626231 0.313116 0.949715i \(-0.398627\pi\)
0.313116 + 0.949715i \(0.398627\pi\)
\(258\) 0 0
\(259\) 1.53920 0.0956411
\(260\) −15.6054 −0.967808
\(261\) 0 0
\(262\) 17.8415 1.10225
\(263\) 18.8262 1.16087 0.580436 0.814306i \(-0.302882\pi\)
0.580436 + 0.814306i \(0.302882\pi\)
\(264\) 0 0
\(265\) −3.61635 −0.222151
\(266\) 0 0
\(267\) 0 0
\(268\) −11.5604 −0.706166
\(269\) −16.3827 −0.998870 −0.499435 0.866351i \(-0.666459\pi\)
−0.499435 + 0.866351i \(0.666459\pi\)
\(270\) 0 0
\(271\) −2.76684 −0.168073 −0.0840367 0.996463i \(-0.526781\pi\)
−0.0840367 + 0.996463i \(0.526781\pi\)
\(272\) −4.96917 −0.301300
\(273\) 0 0
\(274\) 15.9156 0.961499
\(275\) 1.76225 0.106268
\(276\) 0 0
\(277\) 24.5321 1.47399 0.736996 0.675897i \(-0.236244\pi\)
0.736996 + 0.675897i \(0.236244\pi\)
\(278\) 10.1202 0.606967
\(279\) 0 0
\(280\) −6.87129 −0.410638
\(281\) 10.1704 0.606713 0.303356 0.952877i \(-0.401893\pi\)
0.303356 + 0.952877i \(0.401893\pi\)
\(282\) 0 0
\(283\) 27.6678 1.64468 0.822340 0.568997i \(-0.192668\pi\)
0.822340 + 0.568997i \(0.192668\pi\)
\(284\) −6.99698 −0.415195
\(285\) 0 0
\(286\) −10.6496 −0.629722
\(287\) 7.28408 0.429966
\(288\) 0 0
\(289\) 7.69270 0.452512
\(290\) 14.5901 0.856761
\(291\) 0 0
\(292\) −6.18619 −0.362020
\(293\) 23.8386 1.39267 0.696333 0.717719i \(-0.254814\pi\)
0.696333 + 0.717719i \(0.254814\pi\)
\(294\) 0 0
\(295\) −12.2063 −0.710677
\(296\) −0.550972 −0.0320246
\(297\) 0 0
\(298\) −9.39851 −0.544441
\(299\) −15.8545 −0.916889
\(300\) 0 0
\(301\) −8.02124 −0.462337
\(302\) 10.2632 0.590583
\(303\) 0 0
\(304\) 0 0
\(305\) 22.9654 1.31500
\(306\) 0 0
\(307\) −25.9884 −1.48324 −0.741619 0.670821i \(-0.765942\pi\)
−0.741619 + 0.670821i \(0.765942\pi\)
\(308\) −4.68915 −0.267189
\(309\) 0 0
\(310\) −17.9163 −1.01758
\(311\) 23.1043 1.31013 0.655063 0.755574i \(-0.272642\pi\)
0.655063 + 0.755574i \(0.272642\pi\)
\(312\) 0 0
\(313\) 12.2647 0.693242 0.346621 0.938005i \(-0.387329\pi\)
0.346621 + 0.938005i \(0.387329\pi\)
\(314\) 2.97387 0.167825
\(315\) 0 0
\(316\) 5.91158 0.332552
\(317\) −0.272305 −0.0152942 −0.00764708 0.999971i \(-0.502434\pi\)
−0.00764708 + 0.999971i \(0.502434\pi\)
\(318\) 0 0
\(319\) 9.95669 0.557468
\(320\) 2.45965 0.137499
\(321\) 0 0
\(322\) −6.98095 −0.389033
\(323\) 0 0
\(324\) 0 0
\(325\) −6.66102 −0.369487
\(326\) 5.59191 0.309707
\(327\) 0 0
\(328\) −2.60741 −0.143970
\(329\) 2.08289 0.114834
\(330\) 0 0
\(331\) 19.9930 1.09891 0.549457 0.835522i \(-0.314835\pi\)
0.549457 + 0.835522i \(0.314835\pi\)
\(332\) −15.1773 −0.832964
\(333\) 0 0
\(334\) −7.08361 −0.387598
\(335\) −28.4346 −1.55355
\(336\) 0 0
\(337\) 17.2824 0.941433 0.470717 0.882284i \(-0.343995\pi\)
0.470717 + 0.882284i \(0.343995\pi\)
\(338\) 27.2537 1.48240
\(339\) 0 0
\(340\) −12.2224 −0.662854
\(341\) −12.2266 −0.662105
\(342\) 0 0
\(343\) 17.3085 0.934573
\(344\) 2.87129 0.154809
\(345\) 0 0
\(346\) −23.0208 −1.23761
\(347\) 18.5456 0.995579 0.497789 0.867298i \(-0.334145\pi\)
0.497789 + 0.867298i \(0.334145\pi\)
\(348\) 0 0
\(349\) −20.1210 −1.07705 −0.538527 0.842609i \(-0.681019\pi\)
−0.538527 + 0.842609i \(0.681019\pi\)
\(350\) −2.93294 −0.156772
\(351\) 0 0
\(352\) 1.67853 0.0894660
\(353\) −24.0557 −1.28035 −0.640177 0.768227i \(-0.721139\pi\)
−0.640177 + 0.768227i \(0.721139\pi\)
\(354\) 0 0
\(355\) −17.2101 −0.913419
\(356\) 6.90502 0.365965
\(357\) 0 0
\(358\) 8.21471 0.434161
\(359\) 10.8535 0.572824 0.286412 0.958107i \(-0.407537\pi\)
0.286412 + 0.958107i \(0.407537\pi\)
\(360\) 0 0
\(361\) 0 0
\(362\) 12.9999 0.683259
\(363\) 0 0
\(364\) 17.7242 0.929002
\(365\) −15.2159 −0.796435
\(366\) 0 0
\(367\) −32.6121 −1.70234 −0.851168 0.524893i \(-0.824105\pi\)
−0.851168 + 0.524893i \(0.824105\pi\)
\(368\) 2.49890 0.130264
\(369\) 0 0
\(370\) −1.35520 −0.0704534
\(371\) 4.10736 0.213243
\(372\) 0 0
\(373\) 5.30198 0.274526 0.137263 0.990535i \(-0.456169\pi\)
0.137263 + 0.990535i \(0.456169\pi\)
\(374\) −8.34092 −0.431299
\(375\) 0 0
\(376\) −0.745593 −0.0384510
\(377\) −37.6347 −1.93828
\(378\) 0 0
\(379\) −24.6656 −1.26699 −0.633494 0.773748i \(-0.718380\pi\)
−0.633494 + 0.773748i \(0.718380\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −6.57479 −0.336396
\(383\) 3.64416 0.186208 0.0931039 0.995656i \(-0.470321\pi\)
0.0931039 + 0.995656i \(0.470321\pi\)
\(384\) 0 0
\(385\) −11.5337 −0.587810
\(386\) −26.8826 −1.36829
\(387\) 0 0
\(388\) −14.3874 −0.730408
\(389\) −2.79353 −0.141638 −0.0708189 0.997489i \(-0.522561\pi\)
−0.0708189 + 0.997489i \(0.522561\pi\)
\(390\) 0 0
\(391\) −12.4175 −0.627979
\(392\) 0.804226 0.0406196
\(393\) 0 0
\(394\) 9.84940 0.496206
\(395\) 14.5404 0.731608
\(396\) 0 0
\(397\) −39.5101 −1.98296 −0.991478 0.130273i \(-0.958415\pi\)
−0.991478 + 0.130273i \(0.958415\pi\)
\(398\) −20.4661 −1.02587
\(399\) 0 0
\(400\) 1.04988 0.0524938
\(401\) −18.6215 −0.929912 −0.464956 0.885334i \(-0.653930\pi\)
−0.464956 + 0.885334i \(0.653930\pi\)
\(402\) 0 0
\(403\) 46.2144 2.30210
\(404\) −11.2829 −0.561347
\(405\) 0 0
\(406\) −16.5711 −0.822408
\(407\) −0.924824 −0.0458418
\(408\) 0 0
\(409\) 17.1433 0.847681 0.423840 0.905737i \(-0.360682\pi\)
0.423840 + 0.905737i \(0.360682\pi\)
\(410\) −6.41332 −0.316731
\(411\) 0 0
\(412\) −14.8280 −0.730522
\(413\) 13.8636 0.682182
\(414\) 0 0
\(415\) −37.3309 −1.83250
\(416\) −6.34458 −0.311068
\(417\) 0 0
\(418\) 0 0
\(419\) 18.5042 0.903989 0.451995 0.892021i \(-0.350713\pi\)
0.451995 + 0.892021i \(0.350713\pi\)
\(420\) 0 0
\(421\) 16.1996 0.789520 0.394760 0.918784i \(-0.370828\pi\)
0.394760 + 0.918784i \(0.370828\pi\)
\(422\) −8.11630 −0.395095
\(423\) 0 0
\(424\) −1.47027 −0.0714027
\(425\) −5.21702 −0.253062
\(426\) 0 0
\(427\) −26.0835 −1.26227
\(428\) −2.82849 −0.136720
\(429\) 0 0
\(430\) 7.06236 0.340577
\(431\) 12.6997 0.611721 0.305861 0.952076i \(-0.401056\pi\)
0.305861 + 0.952076i \(0.401056\pi\)
\(432\) 0 0
\(433\) −14.2350 −0.684092 −0.342046 0.939683i \(-0.611120\pi\)
−0.342046 + 0.939683i \(0.611120\pi\)
\(434\) 20.3488 0.976775
\(435\) 0 0
\(436\) −1.72539 −0.0826312
\(437\) 0 0
\(438\) 0 0
\(439\) −28.1900 −1.34544 −0.672718 0.739899i \(-0.734873\pi\)
−0.672718 + 0.739899i \(0.734873\pi\)
\(440\) 4.12860 0.196823
\(441\) 0 0
\(442\) 31.5273 1.49960
\(443\) 32.9944 1.56761 0.783805 0.621007i \(-0.213276\pi\)
0.783805 + 0.621007i \(0.213276\pi\)
\(444\) 0 0
\(445\) 16.9839 0.805115
\(446\) −10.7540 −0.509218
\(447\) 0 0
\(448\) −2.79360 −0.131985
\(449\) −17.1975 −0.811601 −0.405801 0.913962i \(-0.633007\pi\)
−0.405801 + 0.913962i \(0.633007\pi\)
\(450\) 0 0
\(451\) −4.37662 −0.206087
\(452\) −0.427785 −0.0201213
\(453\) 0 0
\(454\) 27.0936 1.27157
\(455\) 43.5954 2.04378
\(456\) 0 0
\(457\) −31.1517 −1.45722 −0.728608 0.684931i \(-0.759832\pi\)
−0.728608 + 0.684931i \(0.759832\pi\)
\(458\) 9.46557 0.442297
\(459\) 0 0
\(460\) 6.14643 0.286579
\(461\) −15.5410 −0.723816 −0.361908 0.932214i \(-0.617875\pi\)
−0.361908 + 0.932214i \(0.617875\pi\)
\(462\) 0 0
\(463\) 20.6648 0.960374 0.480187 0.877166i \(-0.340569\pi\)
0.480187 + 0.877166i \(0.340569\pi\)
\(464\) 5.93179 0.275376
\(465\) 0 0
\(466\) −22.8621 −1.05907
\(467\) 28.4830 1.31803 0.659017 0.752128i \(-0.270972\pi\)
0.659017 + 0.752128i \(0.270972\pi\)
\(468\) 0 0
\(469\) 32.2953 1.49126
\(470\) −1.83390 −0.0845914
\(471\) 0 0
\(472\) −4.96261 −0.228423
\(473\) 4.81955 0.221603
\(474\) 0 0
\(475\) 0 0
\(476\) 13.8819 0.636276
\(477\) 0 0
\(478\) −8.66611 −0.396379
\(479\) 10.4726 0.478507 0.239254 0.970957i \(-0.423097\pi\)
0.239254 + 0.970957i \(0.423097\pi\)
\(480\) 0 0
\(481\) 3.49568 0.159389
\(482\) −5.65826 −0.257727
\(483\) 0 0
\(484\) −8.18253 −0.371933
\(485\) −35.3879 −1.60688
\(486\) 0 0
\(487\) −2.29894 −0.104175 −0.0520875 0.998643i \(-0.516587\pi\)
−0.0520875 + 0.998643i \(0.516587\pi\)
\(488\) 9.33686 0.422660
\(489\) 0 0
\(490\) 1.97811 0.0893621
\(491\) −12.1612 −0.548826 −0.274413 0.961612i \(-0.588484\pi\)
−0.274413 + 0.961612i \(0.588484\pi\)
\(492\) 0 0
\(493\) −29.4761 −1.32754
\(494\) 0 0
\(495\) 0 0
\(496\) −7.28408 −0.327065
\(497\) 19.5468 0.876794
\(498\) 0 0
\(499\) 15.1348 0.677528 0.338764 0.940871i \(-0.389991\pi\)
0.338764 + 0.940871i \(0.389991\pi\)
\(500\) −9.71592 −0.434509
\(501\) 0 0
\(502\) 13.5552 0.605000
\(503\) 0.573683 0.0255793 0.0127896 0.999918i \(-0.495929\pi\)
0.0127896 + 0.999918i \(0.495929\pi\)
\(504\) 0 0
\(505\) −27.7520 −1.23495
\(506\) 4.19449 0.186468
\(507\) 0 0
\(508\) 1.13632 0.0504159
\(509\) −26.2163 −1.16202 −0.581008 0.813898i \(-0.697341\pi\)
−0.581008 + 0.813898i \(0.697341\pi\)
\(510\) 0 0
\(511\) 17.2818 0.764501
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 10.0393 0.442813
\(515\) −36.4716 −1.60713
\(516\) 0 0
\(517\) −1.25150 −0.0550410
\(518\) 1.53920 0.0676285
\(519\) 0 0
\(520\) −15.6054 −0.684344
\(521\) −12.9637 −0.567948 −0.283974 0.958832i \(-0.591653\pi\)
−0.283974 + 0.958832i \(0.591653\pi\)
\(522\) 0 0
\(523\) −17.7793 −0.777432 −0.388716 0.921358i \(-0.627081\pi\)
−0.388716 + 0.921358i \(0.627081\pi\)
\(524\) 17.8415 0.779410
\(525\) 0 0
\(526\) 18.8262 0.820861
\(527\) 36.1959 1.57672
\(528\) 0 0
\(529\) −16.7555 −0.728499
\(530\) −3.61635 −0.157084
\(531\) 0 0
\(532\) 0 0
\(533\) 16.5429 0.716554
\(534\) 0 0
\(535\) −6.95709 −0.300781
\(536\) −11.5604 −0.499335
\(537\) 0 0
\(538\) −16.3827 −0.706307
\(539\) 1.34992 0.0581451
\(540\) 0 0
\(541\) −5.34418 −0.229764 −0.114882 0.993379i \(-0.536649\pi\)
−0.114882 + 0.993379i \(0.536649\pi\)
\(542\) −2.76684 −0.118846
\(543\) 0 0
\(544\) −4.96917 −0.213052
\(545\) −4.24385 −0.181787
\(546\) 0 0
\(547\) −5.10112 −0.218108 −0.109054 0.994036i \(-0.534782\pi\)
−0.109054 + 0.994036i \(0.534782\pi\)
\(548\) 15.9156 0.679882
\(549\) 0 0
\(550\) 1.76225 0.0751426
\(551\) 0 0
\(552\) 0 0
\(553\) −16.5146 −0.702273
\(554\) 24.5321 1.04227
\(555\) 0 0
\(556\) 10.1202 0.429191
\(557\) 10.7235 0.454370 0.227185 0.973852i \(-0.427048\pi\)
0.227185 + 0.973852i \(0.427048\pi\)
\(558\) 0 0
\(559\) −18.2171 −0.770502
\(560\) −6.87129 −0.290365
\(561\) 0 0
\(562\) 10.1704 0.429011
\(563\) 9.76285 0.411455 0.205728 0.978609i \(-0.434044\pi\)
0.205728 + 0.978609i \(0.434044\pi\)
\(564\) 0 0
\(565\) −1.05220 −0.0442664
\(566\) 27.6678 1.16296
\(567\) 0 0
\(568\) −6.99698 −0.293587
\(569\) −8.84194 −0.370674 −0.185337 0.982675i \(-0.559338\pi\)
−0.185337 + 0.982675i \(0.559338\pi\)
\(570\) 0 0
\(571\) 22.8411 0.955871 0.477935 0.878395i \(-0.341385\pi\)
0.477935 + 0.878395i \(0.341385\pi\)
\(572\) −10.6496 −0.445281
\(573\) 0 0
\(574\) 7.28408 0.304032
\(575\) 2.62354 0.109409
\(576\) 0 0
\(577\) −14.0176 −0.583560 −0.291780 0.956486i \(-0.594247\pi\)
−0.291780 + 0.956486i \(0.594247\pi\)
\(578\) 7.69270 0.319974
\(579\) 0 0
\(580\) 14.5901 0.605821
\(581\) 42.3994 1.75903
\(582\) 0 0
\(583\) −2.46790 −0.102210
\(584\) −6.18619 −0.255986
\(585\) 0 0
\(586\) 23.8386 0.984763
\(587\) −21.0356 −0.868232 −0.434116 0.900857i \(-0.642939\pi\)
−0.434116 + 0.900857i \(0.642939\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) −12.2063 −0.502525
\(591\) 0 0
\(592\) −0.550972 −0.0226448
\(593\) 34.9193 1.43396 0.716981 0.697093i \(-0.245523\pi\)
0.716981 + 0.697093i \(0.245523\pi\)
\(594\) 0 0
\(595\) 34.1446 1.39979
\(596\) −9.39851 −0.384978
\(597\) 0 0
\(598\) −15.8545 −0.648338
\(599\) 21.5106 0.878901 0.439450 0.898267i \(-0.355173\pi\)
0.439450 + 0.898267i \(0.355173\pi\)
\(600\) 0 0
\(601\) 14.3417 0.585012 0.292506 0.956264i \(-0.405511\pi\)
0.292506 + 0.956264i \(0.405511\pi\)
\(602\) −8.02124 −0.326921
\(603\) 0 0
\(604\) 10.2632 0.417605
\(605\) −20.1262 −0.818245
\(606\) 0 0
\(607\) 10.3488 0.420046 0.210023 0.977696i \(-0.432646\pi\)
0.210023 + 0.977696i \(0.432646\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 22.9654 0.929842
\(611\) 4.73047 0.191375
\(612\) 0 0
\(613\) 19.5674 0.790320 0.395160 0.918612i \(-0.370689\pi\)
0.395160 + 0.918612i \(0.370689\pi\)
\(614\) −25.9884 −1.04881
\(615\) 0 0
\(616\) −4.68915 −0.188931
\(617\) 5.92522 0.238540 0.119270 0.992862i \(-0.461945\pi\)
0.119270 + 0.992862i \(0.461945\pi\)
\(618\) 0 0
\(619\) 8.03958 0.323138 0.161569 0.986861i \(-0.448345\pi\)
0.161569 + 0.986861i \(0.448345\pi\)
\(620\) −17.9163 −0.719535
\(621\) 0 0
\(622\) 23.1043 0.926400
\(623\) −19.2899 −0.772833
\(624\) 0 0
\(625\) −29.1471 −1.16589
\(626\) 12.2647 0.490196
\(627\) 0 0
\(628\) 2.97387 0.118671
\(629\) 2.73788 0.109166
\(630\) 0 0
\(631\) −12.9006 −0.513567 −0.256783 0.966469i \(-0.582663\pi\)
−0.256783 + 0.966469i \(0.582663\pi\)
\(632\) 5.91158 0.235150
\(633\) 0 0
\(634\) −0.272305 −0.0108146
\(635\) 2.79494 0.110914
\(636\) 0 0
\(637\) −5.10247 −0.202167
\(638\) 9.95669 0.394189
\(639\) 0 0
\(640\) 2.45965 0.0972262
\(641\) −28.8639 −1.14006 −0.570028 0.821625i \(-0.693068\pi\)
−0.570028 + 0.821625i \(0.693068\pi\)
\(642\) 0 0
\(643\) 26.3394 1.03873 0.519363 0.854554i \(-0.326169\pi\)
0.519363 + 0.854554i \(0.326169\pi\)
\(644\) −6.98095 −0.275088
\(645\) 0 0
\(646\) 0 0
\(647\) 3.04601 0.119751 0.0598756 0.998206i \(-0.480930\pi\)
0.0598756 + 0.998206i \(0.480930\pi\)
\(648\) 0 0
\(649\) −8.32990 −0.326977
\(650\) −6.66102 −0.261267
\(651\) 0 0
\(652\) 5.59191 0.218996
\(653\) 15.1973 0.594717 0.297359 0.954766i \(-0.403894\pi\)
0.297359 + 0.954766i \(0.403894\pi\)
\(654\) 0 0
\(655\) 43.8838 1.71468
\(656\) −2.60741 −0.101802
\(657\) 0 0
\(658\) 2.08289 0.0811996
\(659\) 17.8741 0.696275 0.348138 0.937443i \(-0.386814\pi\)
0.348138 + 0.937443i \(0.386814\pi\)
\(660\) 0 0
\(661\) −37.8525 −1.47229 −0.736146 0.676823i \(-0.763356\pi\)
−0.736146 + 0.676823i \(0.763356\pi\)
\(662\) 19.9930 0.777050
\(663\) 0 0
\(664\) −15.1773 −0.588994
\(665\) 0 0
\(666\) 0 0
\(667\) 14.8230 0.573947
\(668\) −7.08361 −0.274073
\(669\) 0 0
\(670\) −28.4346 −1.09853
\(671\) 15.6722 0.605019
\(672\) 0 0
\(673\) 8.28461 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(674\) 17.2824 0.665694
\(675\) 0 0
\(676\) 27.2537 1.04822
\(677\) −18.3233 −0.704223 −0.352111 0.935958i \(-0.614536\pi\)
−0.352111 + 0.935958i \(0.614536\pi\)
\(678\) 0 0
\(679\) 40.1926 1.54245
\(680\) −12.2224 −0.468709
\(681\) 0 0
\(682\) −12.2266 −0.468179
\(683\) −49.2217 −1.88342 −0.941709 0.336429i \(-0.890781\pi\)
−0.941709 + 0.336429i \(0.890781\pi\)
\(684\) 0 0
\(685\) 39.1469 1.49573
\(686\) 17.3085 0.660843
\(687\) 0 0
\(688\) 2.87129 0.109467
\(689\) 9.32825 0.355378
\(690\) 0 0
\(691\) 16.6779 0.634458 0.317229 0.948349i \(-0.397248\pi\)
0.317229 + 0.948349i \(0.397248\pi\)
\(692\) −23.0208 −0.875120
\(693\) 0 0
\(694\) 18.5456 0.703981
\(695\) 24.8921 0.944210
\(696\) 0 0
\(697\) 12.9567 0.490770
\(698\) −20.1210 −0.761592
\(699\) 0 0
\(700\) −2.93294 −0.110855
\(701\) 30.9066 1.16733 0.583663 0.811996i \(-0.301619\pi\)
0.583663 + 0.811996i \(0.301619\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 1.67853 0.0632620
\(705\) 0 0
\(706\) −24.0557 −0.905348
\(707\) 31.5200 1.18543
\(708\) 0 0
\(709\) 40.1056 1.50620 0.753098 0.657908i \(-0.228558\pi\)
0.753098 + 0.657908i \(0.228558\pi\)
\(710\) −17.2101 −0.645884
\(711\) 0 0
\(712\) 6.90502 0.258776
\(713\) −18.2022 −0.681678
\(714\) 0 0
\(715\) −26.1942 −0.979608
\(716\) 8.21471 0.306998
\(717\) 0 0
\(718\) 10.8535 0.405048
\(719\) 45.6529 1.70256 0.851282 0.524708i \(-0.175825\pi\)
0.851282 + 0.524708i \(0.175825\pi\)
\(720\) 0 0
\(721\) 41.4235 1.54269
\(722\) 0 0
\(723\) 0 0
\(724\) 12.9999 0.483137
\(725\) 6.22764 0.231289
\(726\) 0 0
\(727\) −15.1811 −0.563034 −0.281517 0.959556i \(-0.590838\pi\)
−0.281517 + 0.959556i \(0.590838\pi\)
\(728\) 17.7242 0.656904
\(729\) 0 0
\(730\) −15.2159 −0.563164
\(731\) −14.2679 −0.527719
\(732\) 0 0
\(733\) −26.9986 −0.997217 −0.498608 0.866827i \(-0.666155\pi\)
−0.498608 + 0.866827i \(0.666155\pi\)
\(734\) −32.6121 −1.20373
\(735\) 0 0
\(736\) 2.49890 0.0921108
\(737\) −19.4046 −0.714776
\(738\) 0 0
\(739\) −5.20782 −0.191573 −0.0957864 0.995402i \(-0.530537\pi\)
−0.0957864 + 0.995402i \(0.530537\pi\)
\(740\) −1.35520 −0.0498181
\(741\) 0 0
\(742\) 4.10736 0.150786
\(743\) 22.1079 0.811059 0.405529 0.914082i \(-0.367087\pi\)
0.405529 + 0.914082i \(0.367087\pi\)
\(744\) 0 0
\(745\) −23.1170 −0.846943
\(746\) 5.30198 0.194119
\(747\) 0 0
\(748\) −8.34092 −0.304974
\(749\) 7.90167 0.288721
\(750\) 0 0
\(751\) 12.4960 0.455984 0.227992 0.973663i \(-0.426784\pi\)
0.227992 + 0.973663i \(0.426784\pi\)
\(752\) −0.745593 −0.0271890
\(753\) 0 0
\(754\) −37.6347 −1.37057
\(755\) 25.2440 0.918722
\(756\) 0 0
\(757\) −30.9988 −1.12667 −0.563336 0.826228i \(-0.690482\pi\)
−0.563336 + 0.826228i \(0.690482\pi\)
\(758\) −24.6656 −0.895895
\(759\) 0 0
\(760\) 0 0
\(761\) −1.96398 −0.0711941 −0.0355971 0.999366i \(-0.511333\pi\)
−0.0355971 + 0.999366i \(0.511333\pi\)
\(762\) 0 0
\(763\) 4.82006 0.174498
\(764\) −6.57479 −0.237868
\(765\) 0 0
\(766\) 3.64416 0.131669
\(767\) 31.4857 1.13688
\(768\) 0 0
\(769\) −22.3322 −0.805318 −0.402659 0.915350i \(-0.631914\pi\)
−0.402659 + 0.915350i \(0.631914\pi\)
\(770\) −11.5337 −0.415645
\(771\) 0 0
\(772\) −26.8826 −0.967527
\(773\) 27.3553 0.983901 0.491951 0.870623i \(-0.336284\pi\)
0.491951 + 0.870623i \(0.336284\pi\)
\(774\) 0 0
\(775\) −7.64738 −0.274702
\(776\) −14.3874 −0.516477
\(777\) 0 0
\(778\) −2.79353 −0.100153
\(779\) 0 0
\(780\) 0 0
\(781\) −11.7447 −0.420257
\(782\) −12.4175 −0.444049
\(783\) 0 0
\(784\) 0.804226 0.0287224
\(785\) 7.31469 0.261072
\(786\) 0 0
\(787\) −7.78962 −0.277670 −0.138835 0.990316i \(-0.544336\pi\)
−0.138835 + 0.990316i \(0.544336\pi\)
\(788\) 9.84940 0.350870
\(789\) 0 0
\(790\) 14.5404 0.517325
\(791\) 1.19506 0.0424915
\(792\) 0 0
\(793\) −59.2384 −2.10362
\(794\) −39.5101 −1.40216
\(795\) 0 0
\(796\) −20.4661 −0.725402
\(797\) −6.09655 −0.215951 −0.107975 0.994154i \(-0.534437\pi\)
−0.107975 + 0.994154i \(0.534437\pi\)
\(798\) 0 0
\(799\) 3.70498 0.131073
\(800\) 1.04988 0.0371187
\(801\) 0 0
\(802\) −18.6215 −0.657547
\(803\) −10.3837 −0.366433
\(804\) 0 0
\(805\) −17.1707 −0.605187
\(806\) 46.2144 1.62783
\(807\) 0 0
\(808\) −11.2829 −0.396932
\(809\) −56.8455 −1.99858 −0.999291 0.0376413i \(-0.988016\pi\)
−0.999291 + 0.0376413i \(0.988016\pi\)
\(810\) 0 0
\(811\) 20.9209 0.734631 0.367316 0.930096i \(-0.380277\pi\)
0.367316 + 0.930096i \(0.380277\pi\)
\(812\) −16.5711 −0.581530
\(813\) 0 0
\(814\) −0.924824 −0.0324151
\(815\) 13.7541 0.481786
\(816\) 0 0
\(817\) 0 0
\(818\) 17.1433 0.599401
\(819\) 0 0
\(820\) −6.41332 −0.223963
\(821\) −16.2582 −0.567416 −0.283708 0.958911i \(-0.591565\pi\)
−0.283708 + 0.958911i \(0.591565\pi\)
\(822\) 0 0
\(823\) −17.5328 −0.611157 −0.305578 0.952167i \(-0.598850\pi\)
−0.305578 + 0.952167i \(0.598850\pi\)
\(824\) −14.8280 −0.516557
\(825\) 0 0
\(826\) 13.8636 0.482375
\(827\) −17.0254 −0.592030 −0.296015 0.955183i \(-0.595658\pi\)
−0.296015 + 0.955183i \(0.595658\pi\)
\(828\) 0 0
\(829\) 5.76207 0.200125 0.100062 0.994981i \(-0.468096\pi\)
0.100062 + 0.994981i \(0.468096\pi\)
\(830\) −37.3309 −1.29577
\(831\) 0 0
\(832\) −6.34458 −0.219959
\(833\) −3.99634 −0.138465
\(834\) 0 0
\(835\) −17.4232 −0.602954
\(836\) 0 0
\(837\) 0 0
\(838\) 18.5042 0.639217
\(839\) 8.89954 0.307246 0.153623 0.988130i \(-0.450906\pi\)
0.153623 + 0.988130i \(0.450906\pi\)
\(840\) 0 0
\(841\) 6.18608 0.213313
\(842\) 16.1996 0.558275
\(843\) 0 0
\(844\) −8.11630 −0.279374
\(845\) 67.0344 2.30605
\(846\) 0 0
\(847\) 22.8588 0.785436
\(848\) −1.47027 −0.0504893
\(849\) 0 0
\(850\) −5.21702 −0.178942
\(851\) −1.37683 −0.0471970
\(852\) 0 0
\(853\) 2.78915 0.0954987 0.0477493 0.998859i \(-0.484795\pi\)
0.0477493 + 0.998859i \(0.484795\pi\)
\(854\) −26.0835 −0.892559
\(855\) 0 0
\(856\) −2.82849 −0.0966757
\(857\) 16.3945 0.560025 0.280012 0.959996i \(-0.409661\pi\)
0.280012 + 0.959996i \(0.409661\pi\)
\(858\) 0 0
\(859\) 9.52505 0.324990 0.162495 0.986709i \(-0.448046\pi\)
0.162495 + 0.986709i \(0.448046\pi\)
\(860\) 7.06236 0.240825
\(861\) 0 0
\(862\) 12.6997 0.432552
\(863\) −50.6764 −1.72505 −0.862523 0.506018i \(-0.831117\pi\)
−0.862523 + 0.506018i \(0.831117\pi\)
\(864\) 0 0
\(865\) −56.6231 −1.92524
\(866\) −14.2350 −0.483726
\(867\) 0 0
\(868\) 20.3488 0.690684
\(869\) 9.92278 0.336607
\(870\) 0 0
\(871\) 73.3461 2.48524
\(872\) −1.72539 −0.0584291
\(873\) 0 0
\(874\) 0 0
\(875\) 27.1424 0.917582
\(876\) 0 0
\(877\) −19.2578 −0.650291 −0.325146 0.945664i \(-0.605413\pi\)
−0.325146 + 0.945664i \(0.605413\pi\)
\(878\) −28.1900 −0.951367
\(879\) 0 0
\(880\) 4.12860 0.139175
\(881\) −19.1076 −0.643750 −0.321875 0.946782i \(-0.604313\pi\)
−0.321875 + 0.946782i \(0.604313\pi\)
\(882\) 0 0
\(883\) 49.9980 1.68257 0.841283 0.540595i \(-0.181801\pi\)
0.841283 + 0.540595i \(0.181801\pi\)
\(884\) 31.5273 1.06038
\(885\) 0 0
\(886\) 32.9944 1.10847
\(887\) 39.6195 1.33029 0.665146 0.746713i \(-0.268369\pi\)
0.665146 + 0.746713i \(0.268369\pi\)
\(888\) 0 0
\(889\) −3.17442 −0.106467
\(890\) 16.9839 0.569302
\(891\) 0 0
\(892\) −10.7540 −0.360071
\(893\) 0 0
\(894\) 0 0
\(895\) 20.2053 0.675389
\(896\) −2.79360 −0.0933278
\(897\) 0 0
\(898\) −17.1975 −0.573889
\(899\) −43.2076 −1.44105
\(900\) 0 0
\(901\) 7.30603 0.243399
\(902\) −4.37662 −0.145726
\(903\) 0 0
\(904\) −0.427785 −0.0142279
\(905\) 31.9752 1.06289
\(906\) 0 0
\(907\) −55.2545 −1.83470 −0.917348 0.398087i \(-0.869674\pi\)
−0.917348 + 0.398087i \(0.869674\pi\)
\(908\) 27.0936 0.899133
\(909\) 0 0
\(910\) 43.5954 1.44517
\(911\) 27.7087 0.918031 0.459015 0.888428i \(-0.348202\pi\)
0.459015 + 0.888428i \(0.348202\pi\)
\(912\) 0 0
\(913\) −25.4756 −0.843120
\(914\) −31.1517 −1.03041
\(915\) 0 0
\(916\) 9.46557 0.312751
\(917\) −49.8421 −1.64593
\(918\) 0 0
\(919\) 27.2500 0.898894 0.449447 0.893307i \(-0.351621\pi\)
0.449447 + 0.893307i \(0.351621\pi\)
\(920\) 6.14643 0.202642
\(921\) 0 0
\(922\) −15.5410 −0.511815
\(923\) 44.3929 1.46121
\(924\) 0 0
\(925\) −0.578452 −0.0190194
\(926\) 20.6648 0.679087
\(927\) 0 0
\(928\) 5.93179 0.194720
\(929\) −44.4410 −1.45806 −0.729031 0.684481i \(-0.760029\pi\)
−0.729031 + 0.684481i \(0.760029\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −22.8621 −0.748873
\(933\) 0 0
\(934\) 28.4830 0.931991
\(935\) −20.5157 −0.670936
\(936\) 0 0
\(937\) −18.2097 −0.594885 −0.297442 0.954740i \(-0.596134\pi\)
−0.297442 + 0.954740i \(0.596134\pi\)
\(938\) 32.2953 1.05448
\(939\) 0 0
\(940\) −1.83390 −0.0598152
\(941\) −40.2680 −1.31270 −0.656350 0.754457i \(-0.727901\pi\)
−0.656350 + 0.754457i \(0.727901\pi\)
\(942\) 0 0
\(943\) −6.51567 −0.212180
\(944\) −4.96261 −0.161519
\(945\) 0 0
\(946\) 4.81955 0.156697
\(947\) −6.36561 −0.206854 −0.103427 0.994637i \(-0.532981\pi\)
−0.103427 + 0.994637i \(0.532981\pi\)
\(948\) 0 0
\(949\) 39.2488 1.27407
\(950\) 0 0
\(951\) 0 0
\(952\) 13.8819 0.449915
\(953\) −32.6944 −1.05907 −0.529537 0.848287i \(-0.677634\pi\)
−0.529537 + 0.848287i \(0.677634\pi\)
\(954\) 0 0
\(955\) −16.1717 −0.523303
\(956\) −8.66611 −0.280282
\(957\) 0 0
\(958\) 10.4726 0.338356
\(959\) −44.4620 −1.43575
\(960\) 0 0
\(961\) 22.0578 0.711542
\(962\) 3.49568 0.112705
\(963\) 0 0
\(964\) −5.65826 −0.182240
\(965\) −66.1218 −2.12854
\(966\) 0 0
\(967\) 8.55936 0.275250 0.137625 0.990484i \(-0.456053\pi\)
0.137625 + 0.990484i \(0.456053\pi\)
\(968\) −8.18253 −0.262997
\(969\) 0 0
\(970\) −35.3879 −1.13624
\(971\) −41.8789 −1.34396 −0.671978 0.740571i \(-0.734555\pi\)
−0.671978 + 0.740571i \(0.734555\pi\)
\(972\) 0 0
\(973\) −28.2718 −0.906351
\(974\) −2.29894 −0.0736628
\(975\) 0 0
\(976\) 9.33686 0.298866
\(977\) −6.18021 −0.197722 −0.0988612 0.995101i \(-0.531520\pi\)
−0.0988612 + 0.995101i \(0.531520\pi\)
\(978\) 0 0
\(979\) 11.5903 0.370427
\(980\) 1.97811 0.0631885
\(981\) 0 0
\(982\) −12.1612 −0.388079
\(983\) 33.5742 1.07085 0.535425 0.844583i \(-0.320152\pi\)
0.535425 + 0.844583i \(0.320152\pi\)
\(984\) 0 0
\(985\) 24.2261 0.771907
\(986\) −29.4761 −0.938710
\(987\) 0 0
\(988\) 0 0
\(989\) 7.17507 0.228154
\(990\) 0 0
\(991\) 46.1746 1.46679 0.733393 0.679805i \(-0.237936\pi\)
0.733393 + 0.679805i \(0.237936\pi\)
\(992\) −7.28408 −0.231270
\(993\) 0 0
\(994\) 19.5468 0.619987
\(995\) −50.3394 −1.59587
\(996\) 0 0
\(997\) −0.819130 −0.0259421 −0.0129711 0.999916i \(-0.504129\pi\)
−0.0129711 + 0.999916i \(0.504129\pi\)
\(998\) 15.1348 0.479085
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 6498.2.a.ca.1.4 4
3.2 odd 2 722.2.a.m.1.4 4
12.11 even 2 5776.2.a.bv.1.1 4
19.18 odd 2 6498.2.a.bx.1.4 4
57.2 even 18 722.2.e.r.99.1 24
57.5 odd 18 722.2.e.s.595.1 24
57.8 even 6 722.2.c.m.653.4 8
57.11 odd 6 722.2.c.n.653.1 8
57.14 even 18 722.2.e.r.595.4 24
57.17 odd 18 722.2.e.s.99.4 24
57.23 odd 18 722.2.e.s.415.1 24
57.26 odd 6 722.2.c.n.429.1 8
57.29 even 18 722.2.e.r.423.1 24
57.32 even 18 722.2.e.r.245.1 24
57.35 odd 18 722.2.e.s.389.4 24
57.41 even 18 722.2.e.r.389.1 24
57.44 odd 18 722.2.e.s.245.4 24
57.47 odd 18 722.2.e.s.423.4 24
57.50 even 6 722.2.c.m.429.4 8
57.53 even 18 722.2.e.r.415.4 24
57.56 even 2 722.2.a.n.1.1 yes 4
228.227 odd 2 5776.2.a.bt.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
722.2.a.m.1.4 4 3.2 odd 2
722.2.a.n.1.1 yes 4 57.56 even 2
722.2.c.m.429.4 8 57.50 even 6
722.2.c.m.653.4 8 57.8 even 6
722.2.c.n.429.1 8 57.26 odd 6
722.2.c.n.653.1 8 57.11 odd 6
722.2.e.r.99.1 24 57.2 even 18
722.2.e.r.245.1 24 57.32 even 18
722.2.e.r.389.1 24 57.41 even 18
722.2.e.r.415.4 24 57.53 even 18
722.2.e.r.423.1 24 57.29 even 18
722.2.e.r.595.4 24 57.14 even 18
722.2.e.s.99.4 24 57.17 odd 18
722.2.e.s.245.4 24 57.44 odd 18
722.2.e.s.389.4 24 57.35 odd 18
722.2.e.s.415.1 24 57.23 odd 18
722.2.e.s.423.4 24 57.47 odd 18
722.2.e.s.595.1 24 57.5 odd 18
5776.2.a.bt.1.4 4 228.227 odd 2
5776.2.a.bv.1.1 4 12.11 even 2
6498.2.a.bx.1.4 4 19.18 odd 2
6498.2.a.ca.1.4 4 1.1 even 1 trivial