Properties

Label 725.2.a.c.1.1
Level $725$
Weight $2$
Character 725.1
Self dual yes
Analytic conductor $5.789$
Analytic rank $0$
Dimension $2$
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] = [725,2,Mod(1,725)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(725, 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("725.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 725 = 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 725.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: \(5.78915414654\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 145)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 725.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.414214 q^{2} +2.00000 q^{3} -1.82843 q^{4} -0.828427 q^{6} +4.82843 q^{7} +1.58579 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.414214 q^{2} +2.00000 q^{3} -1.82843 q^{4} -0.828427 q^{6} +4.82843 q^{7} +1.58579 q^{8} +1.00000 q^{9} +0.828427 q^{11} -3.65685 q^{12} +2.00000 q^{13} -2.00000 q^{14} +3.00000 q^{16} -2.82843 q^{17} -0.414214 q^{18} -4.82843 q^{19} +9.65685 q^{21} -0.343146 q^{22} +3.17157 q^{23} +3.17157 q^{24} -0.828427 q^{26} -4.00000 q^{27} -8.82843 q^{28} +1.00000 q^{29} +6.48528 q^{31} -4.41421 q^{32} +1.65685 q^{33} +1.17157 q^{34} -1.82843 q^{36} +8.48528 q^{37} +2.00000 q^{38} +4.00000 q^{39} -6.00000 q^{41} -4.00000 q^{42} +6.00000 q^{43} -1.51472 q^{44} -1.31371 q^{46} +11.6569 q^{47} +6.00000 q^{48} +16.3137 q^{49} -5.65685 q^{51} -3.65685 q^{52} +3.65685 q^{53} +1.65685 q^{54} +7.65685 q^{56} -9.65685 q^{57} -0.414214 q^{58} -3.65685 q^{61} -2.68629 q^{62} +4.82843 q^{63} -4.17157 q^{64} -0.686292 q^{66} -6.48528 q^{67} +5.17157 q^{68} +6.34315 q^{69} -15.3137 q^{71} +1.58579 q^{72} -8.48528 q^{73} -3.51472 q^{74} +8.82843 q^{76} +4.00000 q^{77} -1.65685 q^{78} -2.48528 q^{79} -11.0000 q^{81} +2.48528 q^{82} -7.17157 q^{83} -17.6569 q^{84} -2.48528 q^{86} +2.00000 q^{87} +1.31371 q^{88} -7.65685 q^{89} +9.65685 q^{91} -5.79899 q^{92} +12.9706 q^{93} -4.82843 q^{94} -8.82843 q^{96} +12.4853 q^{97} -6.75736 q^{98} +0.828427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 4 q^{3} + 2 q^{4} + 4 q^{6} + 4 q^{7} + 6 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 4 q^{3} + 2 q^{4} + 4 q^{6} + 4 q^{7} + 6 q^{8} + 2 q^{9} - 4 q^{11} + 4 q^{12} + 4 q^{13} - 4 q^{14} + 6 q^{16} + 2 q^{18} - 4 q^{19} + 8 q^{21} - 12 q^{22} + 12 q^{23} + 12 q^{24} + 4 q^{26} - 8 q^{27} - 12 q^{28} + 2 q^{29} - 4 q^{31} - 6 q^{32} - 8 q^{33} + 8 q^{34} + 2 q^{36} + 4 q^{38} + 8 q^{39} - 12 q^{41} - 8 q^{42} + 12 q^{43} - 20 q^{44} + 20 q^{46} + 12 q^{47} + 12 q^{48} + 10 q^{49} + 4 q^{52} - 4 q^{53} - 8 q^{54} + 4 q^{56} - 8 q^{57} + 2 q^{58} + 4 q^{61} - 28 q^{62} + 4 q^{63} - 14 q^{64} - 24 q^{66} + 4 q^{67} + 16 q^{68} + 24 q^{69} - 8 q^{71} + 6 q^{72} - 24 q^{74} + 12 q^{76} + 8 q^{77} + 8 q^{78} + 12 q^{79} - 22 q^{81} - 12 q^{82} - 20 q^{83} - 24 q^{84} + 12 q^{86} + 4 q^{87} - 20 q^{88} - 4 q^{89} + 8 q^{91} + 28 q^{92} - 8 q^{93} - 4 q^{94} - 12 q^{96} + 8 q^{97} - 22 q^{98} - 4 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.414214 −0.292893 −0.146447 0.989219i \(-0.546784\pi\)
−0.146447 + 0.989219i \(0.546784\pi\)
\(3\) 2.00000 1.15470 0.577350 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) −1.82843 −0.914214
\(5\) 0 0
\(6\) −0.828427 −0.338204
\(7\) 4.82843 1.82497 0.912487 0.409106i \(-0.134159\pi\)
0.912487 + 0.409106i \(0.134159\pi\)
\(8\) 1.58579 0.560660
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0.828427 0.249780 0.124890 0.992171i \(-0.460142\pi\)
0.124890 + 0.992171i \(0.460142\pi\)
\(12\) −3.65685 −1.05564
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) −2.82843 −0.685994 −0.342997 0.939336i \(-0.611442\pi\)
−0.342997 + 0.939336i \(0.611442\pi\)
\(18\) −0.414214 −0.0976311
\(19\) −4.82843 −1.10772 −0.553859 0.832611i \(-0.686845\pi\)
−0.553859 + 0.832611i \(0.686845\pi\)
\(20\) 0 0
\(21\) 9.65685 2.10730
\(22\) −0.343146 −0.0731589
\(23\) 3.17157 0.661319 0.330659 0.943750i \(-0.392729\pi\)
0.330659 + 0.943750i \(0.392729\pi\)
\(24\) 3.17157 0.647395
\(25\) 0 0
\(26\) −0.828427 −0.162468
\(27\) −4.00000 −0.769800
\(28\) −8.82843 −1.66842
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) 6.48528 1.16479 0.582395 0.812906i \(-0.302116\pi\)
0.582395 + 0.812906i \(0.302116\pi\)
\(32\) −4.41421 −0.780330
\(33\) 1.65685 0.288421
\(34\) 1.17157 0.200923
\(35\) 0 0
\(36\) −1.82843 −0.304738
\(37\) 8.48528 1.39497 0.697486 0.716599i \(-0.254302\pi\)
0.697486 + 0.716599i \(0.254302\pi\)
\(38\) 2.00000 0.324443
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) −4.00000 −0.617213
\(43\) 6.00000 0.914991 0.457496 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) −1.51472 −0.228352
\(45\) 0 0
\(46\) −1.31371 −0.193696
\(47\) 11.6569 1.70033 0.850163 0.526519i \(-0.176503\pi\)
0.850163 + 0.526519i \(0.176503\pi\)
\(48\) 6.00000 0.866025
\(49\) 16.3137 2.33053
\(50\) 0 0
\(51\) −5.65685 −0.792118
\(52\) −3.65685 −0.507114
\(53\) 3.65685 0.502308 0.251154 0.967947i \(-0.419190\pi\)
0.251154 + 0.967947i \(0.419190\pi\)
\(54\) 1.65685 0.225469
\(55\) 0 0
\(56\) 7.65685 1.02319
\(57\) −9.65685 −1.27908
\(58\) −0.414214 −0.0543889
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −3.65685 −0.468212 −0.234106 0.972211i \(-0.575216\pi\)
−0.234106 + 0.972211i \(0.575216\pi\)
\(62\) −2.68629 −0.341159
\(63\) 4.82843 0.608325
\(64\) −4.17157 −0.521447
\(65\) 0 0
\(66\) −0.686292 −0.0844766
\(67\) −6.48528 −0.792303 −0.396152 0.918185i \(-0.629655\pi\)
−0.396152 + 0.918185i \(0.629655\pi\)
\(68\) 5.17157 0.627145
\(69\) 6.34315 0.763625
\(70\) 0 0
\(71\) −15.3137 −1.81740 −0.908701 0.417447i \(-0.862925\pi\)
−0.908701 + 0.417447i \(0.862925\pi\)
\(72\) 1.58579 0.186887
\(73\) −8.48528 −0.993127 −0.496564 0.868000i \(-0.665405\pi\)
−0.496564 + 0.868000i \(0.665405\pi\)
\(74\) −3.51472 −0.408578
\(75\) 0 0
\(76\) 8.82843 1.01269
\(77\) 4.00000 0.455842
\(78\) −1.65685 −0.187602
\(79\) −2.48528 −0.279616 −0.139808 0.990179i \(-0.544649\pi\)
−0.139808 + 0.990179i \(0.544649\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 2.48528 0.274453
\(83\) −7.17157 −0.787182 −0.393591 0.919286i \(-0.628767\pi\)
−0.393591 + 0.919286i \(0.628767\pi\)
\(84\) −17.6569 −1.92652
\(85\) 0 0
\(86\) −2.48528 −0.267995
\(87\) 2.00000 0.214423
\(88\) 1.31371 0.140042
\(89\) −7.65685 −0.811625 −0.405812 0.913956i \(-0.633011\pi\)
−0.405812 + 0.913956i \(0.633011\pi\)
\(90\) 0 0
\(91\) 9.65685 1.01231
\(92\) −5.79899 −0.604586
\(93\) 12.9706 1.34498
\(94\) −4.82843 −0.498014
\(95\) 0 0
\(96\) −8.82843 −0.901048
\(97\) 12.4853 1.26769 0.633844 0.773461i \(-0.281476\pi\)
0.633844 + 0.773461i \(0.281476\pi\)
\(98\) −6.75736 −0.682596
\(99\) 0.828427 0.0832601
\(100\) 0 0
\(101\) 15.6569 1.55792 0.778958 0.627077i \(-0.215749\pi\)
0.778958 + 0.627077i \(0.215749\pi\)
\(102\) 2.34315 0.232006
\(103\) −16.1421 −1.59053 −0.795266 0.606261i \(-0.792669\pi\)
−0.795266 + 0.606261i \(0.792669\pi\)
\(104\) 3.17157 0.310998
\(105\) 0 0
\(106\) −1.51472 −0.147122
\(107\) −20.1421 −1.94721 −0.973607 0.228232i \(-0.926706\pi\)
−0.973607 + 0.228232i \(0.926706\pi\)
\(108\) 7.31371 0.703762
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 16.9706 1.61077
\(112\) 14.4853 1.36873
\(113\) 2.82843 0.266076 0.133038 0.991111i \(-0.457527\pi\)
0.133038 + 0.991111i \(0.457527\pi\)
\(114\) 4.00000 0.374634
\(115\) 0 0
\(116\) −1.82843 −0.169765
\(117\) 2.00000 0.184900
\(118\) 0 0
\(119\) −13.6569 −1.25192
\(120\) 0 0
\(121\) −10.3137 −0.937610
\(122\) 1.51472 0.137136
\(123\) −12.0000 −1.08200
\(124\) −11.8579 −1.06487
\(125\) 0 0
\(126\) −2.00000 −0.178174
\(127\) −6.00000 −0.532414 −0.266207 0.963916i \(-0.585770\pi\)
−0.266207 + 0.963916i \(0.585770\pi\)
\(128\) 10.5563 0.933058
\(129\) 12.0000 1.05654
\(130\) 0 0
\(131\) −12.1421 −1.06086 −0.530432 0.847728i \(-0.677970\pi\)
−0.530432 + 0.847728i \(0.677970\pi\)
\(132\) −3.02944 −0.263679
\(133\) −23.3137 −2.02155
\(134\) 2.68629 0.232060
\(135\) 0 0
\(136\) −4.48528 −0.384610
\(137\) 5.17157 0.441837 0.220919 0.975292i \(-0.429094\pi\)
0.220919 + 0.975292i \(0.429094\pi\)
\(138\) −2.62742 −0.223661
\(139\) 21.6569 1.83691 0.918455 0.395525i \(-0.129437\pi\)
0.918455 + 0.395525i \(0.129437\pi\)
\(140\) 0 0
\(141\) 23.3137 1.96337
\(142\) 6.34315 0.532305
\(143\) 1.65685 0.138553
\(144\) 3.00000 0.250000
\(145\) 0 0
\(146\) 3.51472 0.290880
\(147\) 32.6274 2.69106
\(148\) −15.5147 −1.27530
\(149\) 9.31371 0.763009 0.381504 0.924367i \(-0.375406\pi\)
0.381504 + 0.924367i \(0.375406\pi\)
\(150\) 0 0
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) −7.65685 −0.621053
\(153\) −2.82843 −0.228665
\(154\) −1.65685 −0.133513
\(155\) 0 0
\(156\) −7.31371 −0.585565
\(157\) −0.485281 −0.0387297 −0.0193648 0.999812i \(-0.506164\pi\)
−0.0193648 + 0.999812i \(0.506164\pi\)
\(158\) 1.02944 0.0818976
\(159\) 7.31371 0.580015
\(160\) 0 0
\(161\) 15.3137 1.20689
\(162\) 4.55635 0.357981
\(163\) 8.34315 0.653486 0.326743 0.945113i \(-0.394049\pi\)
0.326743 + 0.945113i \(0.394049\pi\)
\(164\) 10.9706 0.856657
\(165\) 0 0
\(166\) 2.97056 0.230560
\(167\) 2.48528 0.192317 0.0961584 0.995366i \(-0.469344\pi\)
0.0961584 + 0.995366i \(0.469344\pi\)
\(168\) 15.3137 1.18148
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −4.82843 −0.369239
\(172\) −10.9706 −0.836498
\(173\) −17.3137 −1.31634 −0.658168 0.752871i \(-0.728669\pi\)
−0.658168 + 0.752871i \(0.728669\pi\)
\(174\) −0.828427 −0.0628029
\(175\) 0 0
\(176\) 2.48528 0.187335
\(177\) 0 0
\(178\) 3.17157 0.237719
\(179\) −23.3137 −1.74255 −0.871274 0.490797i \(-0.836706\pi\)
−0.871274 + 0.490797i \(0.836706\pi\)
\(180\) 0 0
\(181\) −6.00000 −0.445976 −0.222988 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) −4.00000 −0.296500
\(183\) −7.31371 −0.540645
\(184\) 5.02944 0.370775
\(185\) 0 0
\(186\) −5.37258 −0.393937
\(187\) −2.34315 −0.171348
\(188\) −21.3137 −1.55446
\(189\) −19.3137 −1.40487
\(190\) 0 0
\(191\) −20.8284 −1.50709 −0.753546 0.657395i \(-0.771658\pi\)
−0.753546 + 0.657395i \(0.771658\pi\)
\(192\) −8.34315 −0.602115
\(193\) −4.48528 −0.322858 −0.161429 0.986884i \(-0.551610\pi\)
−0.161429 + 0.986884i \(0.551610\pi\)
\(194\) −5.17157 −0.371297
\(195\) 0 0
\(196\) −29.8284 −2.13060
\(197\) 19.6569 1.40049 0.700246 0.713901i \(-0.253073\pi\)
0.700246 + 0.713901i \(0.253073\pi\)
\(198\) −0.343146 −0.0243863
\(199\) 12.0000 0.850657 0.425329 0.905039i \(-0.360158\pi\)
0.425329 + 0.905039i \(0.360158\pi\)
\(200\) 0 0
\(201\) −12.9706 −0.914873
\(202\) −6.48528 −0.456303
\(203\) 4.82843 0.338889
\(204\) 10.3431 0.724165
\(205\) 0 0
\(206\) 6.68629 0.465856
\(207\) 3.17157 0.220440
\(208\) 6.00000 0.416025
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) −0.828427 −0.0570313 −0.0285156 0.999593i \(-0.509078\pi\)
−0.0285156 + 0.999593i \(0.509078\pi\)
\(212\) −6.68629 −0.459216
\(213\) −30.6274 −2.09856
\(214\) 8.34315 0.570326
\(215\) 0 0
\(216\) −6.34315 −0.431596
\(217\) 31.3137 2.12571
\(218\) −0.828427 −0.0561082
\(219\) −16.9706 −1.14676
\(220\) 0 0
\(221\) −5.65685 −0.380521
\(222\) −7.02944 −0.471785
\(223\) 17.7990 1.19191 0.595954 0.803018i \(-0.296774\pi\)
0.595954 + 0.803018i \(0.296774\pi\)
\(224\) −21.3137 −1.42408
\(225\) 0 0
\(226\) −1.17157 −0.0779319
\(227\) −20.1421 −1.33688 −0.668440 0.743766i \(-0.733038\pi\)
−0.668440 + 0.743766i \(0.733038\pi\)
\(228\) 17.6569 1.16935
\(229\) −2.00000 −0.132164 −0.0660819 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 0 0
\(231\) 8.00000 0.526361
\(232\) 1.58579 0.104112
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) −0.828427 −0.0541560
\(235\) 0 0
\(236\) 0 0
\(237\) −4.97056 −0.322873
\(238\) 5.65685 0.366679
\(239\) −0.686292 −0.0443925 −0.0221963 0.999754i \(-0.507066\pi\)
−0.0221963 + 0.999754i \(0.507066\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 4.27208 0.274620
\(243\) −10.0000 −0.641500
\(244\) 6.68629 0.428046
\(245\) 0 0
\(246\) 4.97056 0.316912
\(247\) −9.65685 −0.614451
\(248\) 10.2843 0.653052
\(249\) −14.3431 −0.908960
\(250\) 0 0
\(251\) 8.82843 0.557245 0.278623 0.960401i \(-0.410122\pi\)
0.278623 + 0.960401i \(0.410122\pi\)
\(252\) −8.82843 −0.556139
\(253\) 2.62742 0.165184
\(254\) 2.48528 0.155940
\(255\) 0 0
\(256\) 3.97056 0.248160
\(257\) −6.68629 −0.417079 −0.208540 0.978014i \(-0.566871\pi\)
−0.208540 + 0.978014i \(0.566871\pi\)
\(258\) −4.97056 −0.309454
\(259\) 40.9706 2.54579
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) 5.02944 0.310720
\(263\) 19.6569 1.21209 0.606047 0.795429i \(-0.292754\pi\)
0.606047 + 0.795429i \(0.292754\pi\)
\(264\) 2.62742 0.161706
\(265\) 0 0
\(266\) 9.65685 0.592100
\(267\) −15.3137 −0.937184
\(268\) 11.8579 0.724334
\(269\) −21.3137 −1.29952 −0.649760 0.760140i \(-0.725131\pi\)
−0.649760 + 0.760140i \(0.725131\pi\)
\(270\) 0 0
\(271\) −9.79899 −0.595246 −0.297623 0.954683i \(-0.596194\pi\)
−0.297623 + 0.954683i \(0.596194\pi\)
\(272\) −8.48528 −0.514496
\(273\) 19.3137 1.16892
\(274\) −2.14214 −0.129411
\(275\) 0 0
\(276\) −11.5980 −0.698116
\(277\) 3.65685 0.219719 0.109860 0.993947i \(-0.464960\pi\)
0.109860 + 0.993947i \(0.464960\pi\)
\(278\) −8.97056 −0.538019
\(279\) 6.48528 0.388264
\(280\) 0 0
\(281\) −29.3137 −1.74871 −0.874355 0.485288i \(-0.838715\pi\)
−0.874355 + 0.485288i \(0.838715\pi\)
\(282\) −9.65685 −0.575057
\(283\) −4.82843 −0.287020 −0.143510 0.989649i \(-0.545839\pi\)
−0.143510 + 0.989649i \(0.545839\pi\)
\(284\) 28.0000 1.66149
\(285\) 0 0
\(286\) −0.686292 −0.0405813
\(287\) −28.9706 −1.71008
\(288\) −4.41421 −0.260110
\(289\) −9.00000 −0.529412
\(290\) 0 0
\(291\) 24.9706 1.46380
\(292\) 15.5147 0.907930
\(293\) −8.48528 −0.495715 −0.247858 0.968796i \(-0.579727\pi\)
−0.247858 + 0.968796i \(0.579727\pi\)
\(294\) −13.5147 −0.788194
\(295\) 0 0
\(296\) 13.4558 0.782105
\(297\) −3.31371 −0.192281
\(298\) −3.85786 −0.223480
\(299\) 6.34315 0.366834
\(300\) 0 0
\(301\) 28.9706 1.66984
\(302\) 4.97056 0.286024
\(303\) 31.3137 1.79893
\(304\) −14.4853 −0.830788
\(305\) 0 0
\(306\) 1.17157 0.0669744
\(307\) 22.9706 1.31100 0.655500 0.755195i \(-0.272458\pi\)
0.655500 + 0.755195i \(0.272458\pi\)
\(308\) −7.31371 −0.416737
\(309\) −32.2843 −1.83659
\(310\) 0 0
\(311\) 14.4853 0.821385 0.410692 0.911774i \(-0.365287\pi\)
0.410692 + 0.911774i \(0.365287\pi\)
\(312\) 6.34315 0.359110
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) 0.201010 0.0113437
\(315\) 0 0
\(316\) 4.54416 0.255629
\(317\) −2.82843 −0.158860 −0.0794301 0.996840i \(-0.525310\pi\)
−0.0794301 + 0.996840i \(0.525310\pi\)
\(318\) −3.02944 −0.169882
\(319\) 0.828427 0.0463830
\(320\) 0 0
\(321\) −40.2843 −2.24845
\(322\) −6.34315 −0.353490
\(323\) 13.6569 0.759888
\(324\) 20.1127 1.11737
\(325\) 0 0
\(326\) −3.45584 −0.191402
\(327\) 4.00000 0.221201
\(328\) −9.51472 −0.525362
\(329\) 56.2843 3.10305
\(330\) 0 0
\(331\) 21.7990 1.19818 0.599090 0.800681i \(-0.295529\pi\)
0.599090 + 0.800681i \(0.295529\pi\)
\(332\) 13.1127 0.719653
\(333\) 8.48528 0.464991
\(334\) −1.02944 −0.0563283
\(335\) 0 0
\(336\) 28.9706 1.58047
\(337\) 1.17157 0.0638196 0.0319098 0.999491i \(-0.489841\pi\)
0.0319098 + 0.999491i \(0.489841\pi\)
\(338\) 3.72792 0.202772
\(339\) 5.65685 0.307238
\(340\) 0 0
\(341\) 5.37258 0.290942
\(342\) 2.00000 0.108148
\(343\) 44.9706 2.42818
\(344\) 9.51472 0.512999
\(345\) 0 0
\(346\) 7.17157 0.385546
\(347\) 8.14214 0.437093 0.218546 0.975827i \(-0.429869\pi\)
0.218546 + 0.975827i \(0.429869\pi\)
\(348\) −3.65685 −0.196028
\(349\) 20.6274 1.10416 0.552080 0.833791i \(-0.313834\pi\)
0.552080 + 0.833791i \(0.313834\pi\)
\(350\) 0 0
\(351\) −8.00000 −0.427008
\(352\) −3.65685 −0.194911
\(353\) 4.34315 0.231162 0.115581 0.993298i \(-0.463127\pi\)
0.115581 + 0.993298i \(0.463127\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 14.0000 0.741999
\(357\) −27.3137 −1.44559
\(358\) 9.65685 0.510381
\(359\) −3.85786 −0.203610 −0.101805 0.994804i \(-0.532462\pi\)
−0.101805 + 0.994804i \(0.532462\pi\)
\(360\) 0 0
\(361\) 4.31371 0.227037
\(362\) 2.48528 0.130623
\(363\) −20.6274 −1.08266
\(364\) −17.6569 −0.925471
\(365\) 0 0
\(366\) 3.02944 0.158351
\(367\) −18.0000 −0.939592 −0.469796 0.882775i \(-0.655673\pi\)
−0.469796 + 0.882775i \(0.655673\pi\)
\(368\) 9.51472 0.495989
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) 17.6569 0.916698
\(372\) −23.7157 −1.22960
\(373\) 6.97056 0.360922 0.180461 0.983582i \(-0.442241\pi\)
0.180461 + 0.983582i \(0.442241\pi\)
\(374\) 0.970563 0.0501866
\(375\) 0 0
\(376\) 18.4853 0.953306
\(377\) 2.00000 0.103005
\(378\) 8.00000 0.411476
\(379\) 22.4853 1.15499 0.577496 0.816394i \(-0.304030\pi\)
0.577496 + 0.816394i \(0.304030\pi\)
\(380\) 0 0
\(381\) −12.0000 −0.614779
\(382\) 8.62742 0.441417
\(383\) −2.48528 −0.126992 −0.0634960 0.997982i \(-0.520225\pi\)
−0.0634960 + 0.997982i \(0.520225\pi\)
\(384\) 21.1127 1.07740
\(385\) 0 0
\(386\) 1.85786 0.0945628
\(387\) 6.00000 0.304997
\(388\) −22.8284 −1.15894
\(389\) −29.3137 −1.48626 −0.743132 0.669145i \(-0.766661\pi\)
−0.743132 + 0.669145i \(0.766661\pi\)
\(390\) 0 0
\(391\) −8.97056 −0.453661
\(392\) 25.8701 1.30664
\(393\) −24.2843 −1.22498
\(394\) −8.14214 −0.410195
\(395\) 0 0
\(396\) −1.51472 −0.0761175
\(397\) 19.6569 0.986549 0.493275 0.869874i \(-0.335800\pi\)
0.493275 + 0.869874i \(0.335800\pi\)
\(398\) −4.97056 −0.249152
\(399\) −46.6274 −2.33429
\(400\) 0 0
\(401\) −6.68629 −0.333897 −0.166949 0.985966i \(-0.553391\pi\)
−0.166949 + 0.985966i \(0.553391\pi\)
\(402\) 5.37258 0.267960
\(403\) 12.9706 0.646110
\(404\) −28.6274 −1.42427
\(405\) 0 0
\(406\) −2.00000 −0.0992583
\(407\) 7.02944 0.348436
\(408\) −8.97056 −0.444109
\(409\) −2.97056 −0.146885 −0.0734424 0.997299i \(-0.523399\pi\)
−0.0734424 + 0.997299i \(0.523399\pi\)
\(410\) 0 0
\(411\) 10.3431 0.510190
\(412\) 29.5147 1.45409
\(413\) 0 0
\(414\) −1.31371 −0.0645653
\(415\) 0 0
\(416\) −8.82843 −0.432849
\(417\) 43.3137 2.12108
\(418\) 1.65685 0.0810394
\(419\) 28.9706 1.41530 0.707652 0.706561i \(-0.249754\pi\)
0.707652 + 0.706561i \(0.249754\pi\)
\(420\) 0 0
\(421\) 18.9706 0.924569 0.462284 0.886732i \(-0.347030\pi\)
0.462284 + 0.886732i \(0.347030\pi\)
\(422\) 0.343146 0.0167041
\(423\) 11.6569 0.566776
\(424\) 5.79899 0.281624
\(425\) 0 0
\(426\) 12.6863 0.614653
\(427\) −17.6569 −0.854475
\(428\) 36.8284 1.78017
\(429\) 3.31371 0.159987
\(430\) 0 0
\(431\) 3.31371 0.159616 0.0798079 0.996810i \(-0.474569\pi\)
0.0798079 + 0.996810i \(0.474569\pi\)
\(432\) −12.0000 −0.577350
\(433\) 29.1716 1.40190 0.700948 0.713212i \(-0.252760\pi\)
0.700948 + 0.713212i \(0.252760\pi\)
\(434\) −12.9706 −0.622607
\(435\) 0 0
\(436\) −3.65685 −0.175132
\(437\) −15.3137 −0.732554
\(438\) 7.02944 0.335880
\(439\) −10.3431 −0.493651 −0.246826 0.969060i \(-0.579388\pi\)
−0.246826 + 0.969060i \(0.579388\pi\)
\(440\) 0 0
\(441\) 16.3137 0.776843
\(442\) 2.34315 0.111452
\(443\) 7.65685 0.363788 0.181894 0.983318i \(-0.441777\pi\)
0.181894 + 0.983318i \(0.441777\pi\)
\(444\) −31.0294 −1.47259
\(445\) 0 0
\(446\) −7.37258 −0.349102
\(447\) 18.6274 0.881047
\(448\) −20.1421 −0.951626
\(449\) 11.6569 0.550121 0.275060 0.961427i \(-0.411302\pi\)
0.275060 + 0.961427i \(0.411302\pi\)
\(450\) 0 0
\(451\) −4.97056 −0.234055
\(452\) −5.17157 −0.243250
\(453\) −24.0000 −1.12762
\(454\) 8.34315 0.391563
\(455\) 0 0
\(456\) −15.3137 −0.717130
\(457\) −19.6569 −0.919509 −0.459754 0.888046i \(-0.652063\pi\)
−0.459754 + 0.888046i \(0.652063\pi\)
\(458\) 0.828427 0.0387099
\(459\) 11.3137 0.528079
\(460\) 0 0
\(461\) −35.6569 −1.66071 −0.830353 0.557238i \(-0.811861\pi\)
−0.830353 + 0.557238i \(0.811861\pi\)
\(462\) −3.31371 −0.154168
\(463\) −21.7990 −1.01308 −0.506542 0.862215i \(-0.669077\pi\)
−0.506542 + 0.862215i \(0.669077\pi\)
\(464\) 3.00000 0.139272
\(465\) 0 0
\(466\) 7.45584 0.345385
\(467\) −10.9706 −0.507657 −0.253829 0.967249i \(-0.581690\pi\)
−0.253829 + 0.967249i \(0.581690\pi\)
\(468\) −3.65685 −0.169038
\(469\) −31.3137 −1.44593
\(470\) 0 0
\(471\) −0.970563 −0.0447212
\(472\) 0 0
\(473\) 4.97056 0.228547
\(474\) 2.05887 0.0945672
\(475\) 0 0
\(476\) 24.9706 1.14452
\(477\) 3.65685 0.167436
\(478\) 0.284271 0.0130023
\(479\) 7.17157 0.327678 0.163839 0.986487i \(-0.447612\pi\)
0.163839 + 0.986487i \(0.447612\pi\)
\(480\) 0 0
\(481\) 16.9706 0.773791
\(482\) −4.14214 −0.188669
\(483\) 30.6274 1.39360
\(484\) 18.8579 0.857176
\(485\) 0 0
\(486\) 4.14214 0.187891
\(487\) −9.79899 −0.444035 −0.222017 0.975043i \(-0.571264\pi\)
−0.222017 + 0.975043i \(0.571264\pi\)
\(488\) −5.79899 −0.262508
\(489\) 16.6863 0.754580
\(490\) 0 0
\(491\) −7.45584 −0.336478 −0.168239 0.985746i \(-0.553808\pi\)
−0.168239 + 0.985746i \(0.553808\pi\)
\(492\) 21.9411 0.989182
\(493\) −2.82843 −0.127386
\(494\) 4.00000 0.179969
\(495\) 0 0
\(496\) 19.4558 0.873593
\(497\) −73.9411 −3.31671
\(498\) 5.94113 0.266228
\(499\) 36.0000 1.61158 0.805791 0.592200i \(-0.201741\pi\)
0.805791 + 0.592200i \(0.201741\pi\)
\(500\) 0 0
\(501\) 4.97056 0.222068
\(502\) −3.65685 −0.163213
\(503\) 30.0000 1.33763 0.668817 0.743427i \(-0.266801\pi\)
0.668817 + 0.743427i \(0.266801\pi\)
\(504\) 7.65685 0.341063
\(505\) 0 0
\(506\) −1.08831 −0.0483814
\(507\) −18.0000 −0.799408
\(508\) 10.9706 0.486740
\(509\) 0.627417 0.0278098 0.0139049 0.999903i \(-0.495574\pi\)
0.0139049 + 0.999903i \(0.495574\pi\)
\(510\) 0 0
\(511\) −40.9706 −1.81243
\(512\) −22.7574 −1.00574
\(513\) 19.3137 0.852721
\(514\) 2.76955 0.122160
\(515\) 0 0
\(516\) −21.9411 −0.965904
\(517\) 9.65685 0.424708
\(518\) −16.9706 −0.745644
\(519\) −34.6274 −1.51997
\(520\) 0 0
\(521\) −21.3137 −0.933771 −0.466885 0.884318i \(-0.654624\pi\)
−0.466885 + 0.884318i \(0.654624\pi\)
\(522\) −0.414214 −0.0181296
\(523\) −2.48528 −0.108674 −0.0543369 0.998523i \(-0.517304\pi\)
−0.0543369 + 0.998523i \(0.517304\pi\)
\(524\) 22.2010 0.969856
\(525\) 0 0
\(526\) −8.14214 −0.355014
\(527\) −18.3431 −0.799040
\(528\) 4.97056 0.216316
\(529\) −12.9411 −0.562658
\(530\) 0 0
\(531\) 0 0
\(532\) 42.6274 1.84813
\(533\) −12.0000 −0.519778
\(534\) 6.34315 0.274495
\(535\) 0 0
\(536\) −10.2843 −0.444213
\(537\) −46.6274 −2.01212
\(538\) 8.82843 0.380621
\(539\) 13.5147 0.582120
\(540\) 0 0
\(541\) −5.02944 −0.216232 −0.108116 0.994138i \(-0.534482\pi\)
−0.108116 + 0.994138i \(0.534482\pi\)
\(542\) 4.05887 0.174344
\(543\) −12.0000 −0.514969
\(544\) 12.4853 0.535302
\(545\) 0 0
\(546\) −8.00000 −0.342368
\(547\) 2.48528 0.106263 0.0531315 0.998588i \(-0.483080\pi\)
0.0531315 + 0.998588i \(0.483080\pi\)
\(548\) −9.45584 −0.403934
\(549\) −3.65685 −0.156071
\(550\) 0 0
\(551\) −4.82843 −0.205698
\(552\) 10.0589 0.428134
\(553\) −12.0000 −0.510292
\(554\) −1.51472 −0.0643542
\(555\) 0 0
\(556\) −39.5980 −1.67933
\(557\) 27.9411 1.18390 0.591952 0.805973i \(-0.298358\pi\)
0.591952 + 0.805973i \(0.298358\pi\)
\(558\) −2.68629 −0.113720
\(559\) 12.0000 0.507546
\(560\) 0 0
\(561\) −4.68629 −0.197855
\(562\) 12.1421 0.512185
\(563\) −7.65685 −0.322698 −0.161349 0.986897i \(-0.551584\pi\)
−0.161349 + 0.986897i \(0.551584\pi\)
\(564\) −42.6274 −1.79494
\(565\) 0 0
\(566\) 2.00000 0.0840663
\(567\) −53.1127 −2.23052
\(568\) −24.2843 −1.01895
\(569\) 27.6569 1.15944 0.579718 0.814817i \(-0.303163\pi\)
0.579718 + 0.814817i \(0.303163\pi\)
\(570\) 0 0
\(571\) 28.0000 1.17176 0.585882 0.810397i \(-0.300748\pi\)
0.585882 + 0.810397i \(0.300748\pi\)
\(572\) −3.02944 −0.126667
\(573\) −41.6569 −1.74024
\(574\) 12.0000 0.500870
\(575\) 0 0
\(576\) −4.17157 −0.173816
\(577\) −23.7990 −0.990765 −0.495382 0.868675i \(-0.664972\pi\)
−0.495382 + 0.868675i \(0.664972\pi\)
\(578\) 3.72792 0.155061
\(579\) −8.97056 −0.372804
\(580\) 0 0
\(581\) −34.6274 −1.43659
\(582\) −10.3431 −0.428737
\(583\) 3.02944 0.125466
\(584\) −13.4558 −0.556807
\(585\) 0 0
\(586\) 3.51472 0.145192
\(587\) 29.7990 1.22994 0.614968 0.788552i \(-0.289169\pi\)
0.614968 + 0.788552i \(0.289169\pi\)
\(588\) −59.6569 −2.46021
\(589\) −31.3137 −1.29026
\(590\) 0 0
\(591\) 39.3137 1.61715
\(592\) 25.4558 1.04623
\(593\) 7.65685 0.314429 0.157215 0.987564i \(-0.449749\pi\)
0.157215 + 0.987564i \(0.449749\pi\)
\(594\) 1.37258 0.0563178
\(595\) 0 0
\(596\) −17.0294 −0.697553
\(597\) 24.0000 0.982255
\(598\) −2.62742 −0.107443
\(599\) −37.7990 −1.54442 −0.772212 0.635364i \(-0.780850\pi\)
−0.772212 + 0.635364i \(0.780850\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) −12.0000 −0.489083
\(603\) −6.48528 −0.264101
\(604\) 21.9411 0.892772
\(605\) 0 0
\(606\) −12.9706 −0.526893
\(607\) −9.02944 −0.366494 −0.183247 0.983067i \(-0.558661\pi\)
−0.183247 + 0.983067i \(0.558661\pi\)
\(608\) 21.3137 0.864385
\(609\) 9.65685 0.391315
\(610\) 0 0
\(611\) 23.3137 0.943172
\(612\) 5.17157 0.209048
\(613\) 2.00000 0.0807792 0.0403896 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) −9.51472 −0.383983
\(615\) 0 0
\(616\) 6.34315 0.255573
\(617\) 9.17157 0.369234 0.184617 0.982811i \(-0.440896\pi\)
0.184617 + 0.982811i \(0.440896\pi\)
\(618\) 13.3726 0.537924
\(619\) −9.79899 −0.393855 −0.196927 0.980418i \(-0.563096\pi\)
−0.196927 + 0.980418i \(0.563096\pi\)
\(620\) 0 0
\(621\) −12.6863 −0.509083
\(622\) −6.00000 −0.240578
\(623\) −36.9706 −1.48119
\(624\) 12.0000 0.480384
\(625\) 0 0
\(626\) −2.48528 −0.0993318
\(627\) −8.00000 −0.319489
\(628\) 0.887302 0.0354072
\(629\) −24.0000 −0.956943
\(630\) 0 0
\(631\) 36.9706 1.47177 0.735887 0.677104i \(-0.236765\pi\)
0.735887 + 0.677104i \(0.236765\pi\)
\(632\) −3.94113 −0.156770
\(633\) −1.65685 −0.0658540
\(634\) 1.17157 0.0465291
\(635\) 0 0
\(636\) −13.3726 −0.530257
\(637\) 32.6274 1.29275
\(638\) −0.343146 −0.0135853
\(639\) −15.3137 −0.605801
\(640\) 0 0
\(641\) 0.627417 0.0247815 0.0123907 0.999923i \(-0.496056\pi\)
0.0123907 + 0.999923i \(0.496056\pi\)
\(642\) 16.6863 0.658555
\(643\) −19.4558 −0.767264 −0.383632 0.923486i \(-0.625327\pi\)
−0.383632 + 0.923486i \(0.625327\pi\)
\(644\) −28.0000 −1.10335
\(645\) 0 0
\(646\) −5.65685 −0.222566
\(647\) 41.1127 1.61631 0.808153 0.588972i \(-0.200467\pi\)
0.808153 + 0.588972i \(0.200467\pi\)
\(648\) −17.4437 −0.685251
\(649\) 0 0
\(650\) 0 0
\(651\) 62.6274 2.45456
\(652\) −15.2548 −0.597425
\(653\) 17.1716 0.671976 0.335988 0.941866i \(-0.390930\pi\)
0.335988 + 0.941866i \(0.390930\pi\)
\(654\) −1.65685 −0.0647881
\(655\) 0 0
\(656\) −18.0000 −0.702782
\(657\) −8.48528 −0.331042
\(658\) −23.3137 −0.908863
\(659\) 1.79899 0.0700787 0.0350393 0.999386i \(-0.488844\pi\)
0.0350393 + 0.999386i \(0.488844\pi\)
\(660\) 0 0
\(661\) 26.0000 1.01128 0.505641 0.862744i \(-0.331256\pi\)
0.505641 + 0.862744i \(0.331256\pi\)
\(662\) −9.02944 −0.350939
\(663\) −11.3137 −0.439388
\(664\) −11.3726 −0.441342
\(665\) 0 0
\(666\) −3.51472 −0.136193
\(667\) 3.17157 0.122804
\(668\) −4.54416 −0.175819
\(669\) 35.5980 1.37630
\(670\) 0 0
\(671\) −3.02944 −0.116950
\(672\) −42.6274 −1.64439
\(673\) −22.9706 −0.885450 −0.442725 0.896657i \(-0.645988\pi\)
−0.442725 + 0.896657i \(0.645988\pi\)
\(674\) −0.485281 −0.0186923
\(675\) 0 0
\(676\) 16.4558 0.632917
\(677\) 36.7696 1.41317 0.706584 0.707629i \(-0.250235\pi\)
0.706584 + 0.707629i \(0.250235\pi\)
\(678\) −2.34315 −0.0899880
\(679\) 60.2843 2.31350
\(680\) 0 0
\(681\) −40.2843 −1.54370
\(682\) −2.22540 −0.0852148
\(683\) −11.8579 −0.453729 −0.226864 0.973926i \(-0.572847\pi\)
−0.226864 + 0.973926i \(0.572847\pi\)
\(684\) 8.82843 0.337563
\(685\) 0 0
\(686\) −18.6274 −0.711198
\(687\) −4.00000 −0.152610
\(688\) 18.0000 0.686244
\(689\) 7.31371 0.278630
\(690\) 0 0
\(691\) −44.9706 −1.71076 −0.855380 0.518000i \(-0.826677\pi\)
−0.855380 + 0.518000i \(0.826677\pi\)
\(692\) 31.6569 1.20341
\(693\) 4.00000 0.151947
\(694\) −3.37258 −0.128022
\(695\) 0 0
\(696\) 3.17157 0.120218
\(697\) 16.9706 0.642806
\(698\) −8.54416 −0.323401
\(699\) −36.0000 −1.36165
\(700\) 0 0
\(701\) 6.68629 0.252538 0.126269 0.991996i \(-0.459700\pi\)
0.126269 + 0.991996i \(0.459700\pi\)
\(702\) 3.31371 0.125068
\(703\) −40.9706 −1.54523
\(704\) −3.45584 −0.130247
\(705\) 0 0
\(706\) −1.79899 −0.0677059
\(707\) 75.5980 2.84315
\(708\) 0 0
\(709\) −22.0000 −0.826227 −0.413114 0.910679i \(-0.635559\pi\)
−0.413114 + 0.910679i \(0.635559\pi\)
\(710\) 0 0
\(711\) −2.48528 −0.0932053
\(712\) −12.1421 −0.455046
\(713\) 20.5685 0.770298
\(714\) 11.3137 0.423405
\(715\) 0 0
\(716\) 42.6274 1.59306
\(717\) −1.37258 −0.0512601
\(718\) 1.59798 0.0596361
\(719\) −34.6274 −1.29138 −0.645692 0.763598i \(-0.723431\pi\)
−0.645692 + 0.763598i \(0.723431\pi\)
\(720\) 0 0
\(721\) −77.9411 −2.90268
\(722\) −1.78680 −0.0664977
\(723\) 20.0000 0.743808
\(724\) 10.9706 0.407718
\(725\) 0 0
\(726\) 8.54416 0.317103
\(727\) 23.9411 0.887927 0.443964 0.896045i \(-0.353572\pi\)
0.443964 + 0.896045i \(0.353572\pi\)
\(728\) 15.3137 0.567564
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −16.9706 −0.627679
\(732\) 13.3726 0.494265
\(733\) 22.8284 0.843187 0.421594 0.906785i \(-0.361471\pi\)
0.421594 + 0.906785i \(0.361471\pi\)
\(734\) 7.45584 0.275200
\(735\) 0 0
\(736\) −14.0000 −0.516047
\(737\) −5.37258 −0.197902
\(738\) 2.48528 0.0914845
\(739\) 14.4853 0.532850 0.266425 0.963856i \(-0.414158\pi\)
0.266425 + 0.963856i \(0.414158\pi\)
\(740\) 0 0
\(741\) −19.3137 −0.709507
\(742\) −7.31371 −0.268495
\(743\) 52.6274 1.93071 0.965356 0.260935i \(-0.0840309\pi\)
0.965356 + 0.260935i \(0.0840309\pi\)
\(744\) 20.5685 0.754079
\(745\) 0 0
\(746\) −2.88730 −0.105712
\(747\) −7.17157 −0.262394
\(748\) 4.28427 0.156648
\(749\) −97.2548 −3.55361
\(750\) 0 0
\(751\) 16.1421 0.589035 0.294517 0.955646i \(-0.404841\pi\)
0.294517 + 0.955646i \(0.404841\pi\)
\(752\) 34.9706 1.27525
\(753\) 17.6569 0.643452
\(754\) −0.828427 −0.0301695
\(755\) 0 0
\(756\) 35.3137 1.28435
\(757\) −19.5147 −0.709275 −0.354637 0.935004i \(-0.615396\pi\)
−0.354637 + 0.935004i \(0.615396\pi\)
\(758\) −9.31371 −0.338289
\(759\) 5.25483 0.190738
\(760\) 0 0
\(761\) 8.62742 0.312744 0.156372 0.987698i \(-0.450020\pi\)
0.156372 + 0.987698i \(0.450020\pi\)
\(762\) 4.97056 0.180064
\(763\) 9.65685 0.349602
\(764\) 38.0833 1.37780
\(765\) 0 0
\(766\) 1.02944 0.0371951
\(767\) 0 0
\(768\) 7.94113 0.286551
\(769\) −15.6569 −0.564601 −0.282300 0.959326i \(-0.591097\pi\)
−0.282300 + 0.959326i \(0.591097\pi\)
\(770\) 0 0
\(771\) −13.3726 −0.481602
\(772\) 8.20101 0.295161
\(773\) 8.48528 0.305194 0.152597 0.988288i \(-0.451236\pi\)
0.152597 + 0.988288i \(0.451236\pi\)
\(774\) −2.48528 −0.0893316
\(775\) 0 0
\(776\) 19.7990 0.710742
\(777\) 81.9411 2.93962
\(778\) 12.1421 0.435317
\(779\) 28.9706 1.03798
\(780\) 0 0
\(781\) −12.6863 −0.453951
\(782\) 3.71573 0.132874
\(783\) −4.00000 −0.142948
\(784\) 48.9411 1.74790
\(785\) 0 0
\(786\) 10.0589 0.358788
\(787\) −17.7990 −0.634465 −0.317233 0.948348i \(-0.602754\pi\)
−0.317233 + 0.948348i \(0.602754\pi\)
\(788\) −35.9411 −1.28035
\(789\) 39.3137 1.39961
\(790\) 0 0
\(791\) 13.6569 0.485582
\(792\) 1.31371 0.0466806
\(793\) −7.31371 −0.259717
\(794\) −8.14214 −0.288954
\(795\) 0 0
\(796\) −21.9411 −0.777683
\(797\) 5.85786 0.207496 0.103748 0.994604i \(-0.466916\pi\)
0.103748 + 0.994604i \(0.466916\pi\)
\(798\) 19.3137 0.683698
\(799\) −32.9706 −1.16641
\(800\) 0 0
\(801\) −7.65685 −0.270542
\(802\) 2.76955 0.0977963
\(803\) −7.02944 −0.248063
\(804\) 23.7157 0.836389
\(805\) 0 0
\(806\) −5.37258 −0.189241
\(807\) −42.6274 −1.50056
\(808\) 24.8284 0.873461
\(809\) 42.2843 1.48664 0.743318 0.668938i \(-0.233251\pi\)
0.743318 + 0.668938i \(0.233251\pi\)
\(810\) 0 0
\(811\) 37.6569 1.32231 0.661155 0.750249i \(-0.270066\pi\)
0.661155 + 0.750249i \(0.270066\pi\)
\(812\) −8.82843 −0.309817
\(813\) −19.5980 −0.687331
\(814\) −2.91169 −0.102055
\(815\) 0 0
\(816\) −16.9706 −0.594089
\(817\) −28.9706 −1.01355
\(818\) 1.23045 0.0430216
\(819\) 9.65685 0.337438
\(820\) 0 0
\(821\) −22.6863 −0.791757 −0.395879 0.918303i \(-0.629560\pi\)
−0.395879 + 0.918303i \(0.629560\pi\)
\(822\) −4.28427 −0.149431
\(823\) 30.9706 1.07957 0.539783 0.841804i \(-0.318506\pi\)
0.539783 + 0.841804i \(0.318506\pi\)
\(824\) −25.5980 −0.891748
\(825\) 0 0
\(826\) 0 0
\(827\) −17.3137 −0.602057 −0.301028 0.953615i \(-0.597330\pi\)
−0.301028 + 0.953615i \(0.597330\pi\)
\(828\) −5.79899 −0.201529
\(829\) 20.6274 0.716420 0.358210 0.933641i \(-0.383387\pi\)
0.358210 + 0.933641i \(0.383387\pi\)
\(830\) 0 0
\(831\) 7.31371 0.253710
\(832\) −8.34315 −0.289247
\(833\) −46.1421 −1.59873
\(834\) −17.9411 −0.621250
\(835\) 0 0
\(836\) 7.31371 0.252950
\(837\) −25.9411 −0.896656
\(838\) −12.0000 −0.414533
\(839\) 2.48528 0.0858014 0.0429007 0.999079i \(-0.486340\pi\)
0.0429007 + 0.999079i \(0.486340\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) −7.85786 −0.270800
\(843\) −58.6274 −2.01924
\(844\) 1.51472 0.0521388
\(845\) 0 0
\(846\) −4.82843 −0.166005
\(847\) −49.7990 −1.71111
\(848\) 10.9706 0.376731
\(849\) −9.65685 −0.331422
\(850\) 0 0
\(851\) 26.9117 0.922521
\(852\) 56.0000 1.91853
\(853\) −51.1127 −1.75007 −0.875033 0.484064i \(-0.839160\pi\)
−0.875033 + 0.484064i \(0.839160\pi\)
\(854\) 7.31371 0.250270
\(855\) 0 0
\(856\) −31.9411 −1.09173
\(857\) −3.37258 −0.115205 −0.0576026 0.998340i \(-0.518346\pi\)
−0.0576026 + 0.998340i \(0.518346\pi\)
\(858\) −1.37258 −0.0468592
\(859\) −56.4264 −1.92524 −0.962622 0.270848i \(-0.912696\pi\)
−0.962622 + 0.270848i \(0.912696\pi\)
\(860\) 0 0
\(861\) −57.9411 −1.97463
\(862\) −1.37258 −0.0467504
\(863\) 36.1421 1.23029 0.615146 0.788413i \(-0.289097\pi\)
0.615146 + 0.788413i \(0.289097\pi\)
\(864\) 17.6569 0.600698
\(865\) 0 0
\(866\) −12.0833 −0.410606
\(867\) −18.0000 −0.611312
\(868\) −57.2548 −1.94336
\(869\) −2.05887 −0.0698425
\(870\) 0 0
\(871\) −12.9706 −0.439491
\(872\) 3.17157 0.107403
\(873\) 12.4853 0.422563
\(874\) 6.34315 0.214560
\(875\) 0 0
\(876\) 31.0294 1.04839
\(877\) −38.2843 −1.29277 −0.646384 0.763012i \(-0.723720\pi\)
−0.646384 + 0.763012i \(0.723720\pi\)
\(878\) 4.28427 0.144587
\(879\) −16.9706 −0.572403
\(880\) 0 0
\(881\) 29.3137 0.987604 0.493802 0.869574i \(-0.335607\pi\)
0.493802 + 0.869574i \(0.335607\pi\)
\(882\) −6.75736 −0.227532
\(883\) 14.4853 0.487469 0.243734 0.969842i \(-0.421628\pi\)
0.243734 + 0.969842i \(0.421628\pi\)
\(884\) 10.3431 0.347878
\(885\) 0 0
\(886\) −3.17157 −0.106551
\(887\) 6.68629 0.224504 0.112252 0.993680i \(-0.464194\pi\)
0.112252 + 0.993680i \(0.464194\pi\)
\(888\) 26.9117 0.903097
\(889\) −28.9706 −0.971641
\(890\) 0 0
\(891\) −9.11270 −0.305287
\(892\) −32.5442 −1.08966
\(893\) −56.2843 −1.88348
\(894\) −7.71573 −0.258053
\(895\) 0 0
\(896\) 50.9706 1.70281
\(897\) 12.6863 0.423583
\(898\) −4.82843 −0.161127
\(899\) 6.48528 0.216296
\(900\) 0 0
\(901\) −10.3431 −0.344580
\(902\) 2.05887 0.0685530
\(903\) 57.9411 1.92816
\(904\) 4.48528 0.149178
\(905\) 0 0
\(906\) 9.94113 0.330272
\(907\) 10.0000 0.332045 0.166022 0.986122i \(-0.446908\pi\)
0.166022 + 0.986122i \(0.446908\pi\)
\(908\) 36.8284 1.22219
\(909\) 15.6569 0.519305
\(910\) 0 0
\(911\) 32.1421 1.06492 0.532458 0.846456i \(-0.321268\pi\)
0.532458 + 0.846456i \(0.321268\pi\)
\(912\) −28.9706 −0.959311
\(913\) −5.94113 −0.196623
\(914\) 8.14214 0.269318
\(915\) 0 0
\(916\) 3.65685 0.120826
\(917\) −58.6274 −1.93605
\(918\) −4.68629 −0.154671
\(919\) −36.0000 −1.18753 −0.593765 0.804638i \(-0.702359\pi\)
−0.593765 + 0.804638i \(0.702359\pi\)
\(920\) 0 0
\(921\) 45.9411 1.51381
\(922\) 14.7696 0.486409
\(923\) −30.6274 −1.00811
\(924\) −14.6274 −0.481207
\(925\) 0 0
\(926\) 9.02944 0.296726
\(927\) −16.1421 −0.530177
\(928\) −4.41421 −0.144904
\(929\) −4.62742 −0.151821 −0.0759103 0.997115i \(-0.524186\pi\)
−0.0759103 + 0.997115i \(0.524186\pi\)
\(930\) 0 0
\(931\) −78.7696 −2.58157
\(932\) 32.9117 1.07806
\(933\) 28.9706 0.948454
\(934\) 4.54416 0.148689
\(935\) 0 0
\(936\) 3.17157 0.103666
\(937\) −19.6569 −0.642161 −0.321081 0.947052i \(-0.604046\pi\)
−0.321081 + 0.947052i \(0.604046\pi\)
\(938\) 12.9706 0.423504
\(939\) 12.0000 0.391605
\(940\) 0 0
\(941\) −27.9411 −0.910855 −0.455427 0.890273i \(-0.650514\pi\)
−0.455427 + 0.890273i \(0.650514\pi\)
\(942\) 0.402020 0.0130985
\(943\) −19.0294 −0.619684
\(944\) 0 0
\(945\) 0 0
\(946\) −2.05887 −0.0669398
\(947\) −44.9117 −1.45943 −0.729717 0.683749i \(-0.760348\pi\)
−0.729717 + 0.683749i \(0.760348\pi\)
\(948\) 9.08831 0.295175
\(949\) −16.9706 −0.550888
\(950\) 0 0
\(951\) −5.65685 −0.183436
\(952\) −21.6569 −0.701903
\(953\) −29.3137 −0.949564 −0.474782 0.880103i \(-0.657473\pi\)
−0.474782 + 0.880103i \(0.657473\pi\)
\(954\) −1.51472 −0.0490408
\(955\) 0 0
\(956\) 1.25483 0.0405842
\(957\) 1.65685 0.0535585
\(958\) −2.97056 −0.0959745
\(959\) 24.9706 0.806342
\(960\) 0 0
\(961\) 11.0589 0.356738
\(962\) −7.02944 −0.226638
\(963\) −20.1421 −0.649071
\(964\) −18.2843 −0.588897
\(965\) 0 0
\(966\) −12.6863 −0.408175
\(967\) −14.9706 −0.481421 −0.240710 0.970597i \(-0.577380\pi\)
−0.240710 + 0.970597i \(0.577380\pi\)
\(968\) −16.3553 −0.525681
\(969\) 27.3137 0.877443
\(970\) 0 0
\(971\) 28.1421 0.903124 0.451562 0.892240i \(-0.350867\pi\)
0.451562 + 0.892240i \(0.350867\pi\)
\(972\) 18.2843 0.586468
\(973\) 104.569 3.35231
\(974\) 4.05887 0.130055
\(975\) 0 0
\(976\) −10.9706 −0.351159
\(977\) 2.68629 0.0859421 0.0429710 0.999076i \(-0.486318\pi\)
0.0429710 + 0.999076i \(0.486318\pi\)
\(978\) −6.91169 −0.221011
\(979\) −6.34315 −0.202728
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) 3.08831 0.0985520
\(983\) 9.31371 0.297061 0.148531 0.988908i \(-0.452546\pi\)
0.148531 + 0.988908i \(0.452546\pi\)
\(984\) −19.0294 −0.606636
\(985\) 0 0
\(986\) 1.17157 0.0373105
\(987\) 112.569 3.58310
\(988\) 17.6569 0.561739
\(989\) 19.0294 0.605101
\(990\) 0 0
\(991\) 52.0000 1.65183 0.825917 0.563791i \(-0.190658\pi\)
0.825917 + 0.563791i \(0.190658\pi\)
\(992\) −28.6274 −0.908921
\(993\) 43.5980 1.38354
\(994\) 30.6274 0.971443
\(995\) 0 0
\(996\) 26.2254 0.830983
\(997\) −6.82843 −0.216258 −0.108129 0.994137i \(-0.534486\pi\)
−0.108129 + 0.994137i \(0.534486\pi\)
\(998\) −14.9117 −0.472021
\(999\) −33.9411 −1.07385
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 725.2.a.c.1.1 2
3.2 odd 2 6525.2.a.p.1.2 2
5.2 odd 4 725.2.b.c.349.2 4
5.3 odd 4 725.2.b.c.349.3 4
5.4 even 2 145.2.a.b.1.2 2
15.14 odd 2 1305.2.a.n.1.1 2
20.19 odd 2 2320.2.a.k.1.2 2
35.34 odd 2 7105.2.a.e.1.2 2
40.19 odd 2 9280.2.a.w.1.2 2
40.29 even 2 9280.2.a.be.1.1 2
145.144 even 2 4205.2.a.d.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.b.1.2 2 5.4 even 2
725.2.a.c.1.1 2 1.1 even 1 trivial
725.2.b.c.349.2 4 5.2 odd 4
725.2.b.c.349.3 4 5.3 odd 4
1305.2.a.n.1.1 2 15.14 odd 2
2320.2.a.k.1.2 2 20.19 odd 2
4205.2.a.d.1.1 2 145.144 even 2
6525.2.a.p.1.2 2 3.2 odd 2
7105.2.a.e.1.2 2 35.34 odd 2
9280.2.a.w.1.2 2 40.19 odd 2
9280.2.a.be.1.1 2 40.29 even 2