Properties

Label 7728.2.a.bg.1.1
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $1$
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] = [7728,2,Mod(1,7728)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7728, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 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("7728.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7728.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: \(61.7083906820\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 483)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -2.61803 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -2.61803 q^{5} -1.00000 q^{7} +1.00000 q^{9} +2.23607 q^{11} -6.85410 q^{13} -2.61803 q^{15} +3.47214 q^{17} +3.76393 q^{19} -1.00000 q^{21} -1.00000 q^{23} +1.85410 q^{25} +1.00000 q^{27} +10.7082 q^{29} +5.00000 q^{31} +2.23607 q^{33} +2.61803 q^{35} -9.47214 q^{37} -6.85410 q^{39} -4.70820 q^{41} -7.32624 q^{43} -2.61803 q^{45} -0.763932 q^{47} +1.00000 q^{49} +3.47214 q^{51} -1.38197 q^{53} -5.85410 q^{55} +3.76393 q^{57} -1.85410 q^{59} +2.09017 q^{61} -1.00000 q^{63} +17.9443 q^{65} +0.145898 q^{67} -1.00000 q^{69} -11.6180 q^{71} +5.47214 q^{73} +1.85410 q^{75} -2.23607 q^{77} +11.9443 q^{79} +1.00000 q^{81} +9.94427 q^{83} -9.09017 q^{85} +10.7082 q^{87} -0.673762 q^{89} +6.85410 q^{91} +5.00000 q^{93} -9.85410 q^{95} +5.18034 q^{97} +2.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 3 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 3 q^{5} - 2 q^{7} + 2 q^{9} - 7 q^{13} - 3 q^{15} - 2 q^{17} + 12 q^{19} - 2 q^{21} - 2 q^{23} - 3 q^{25} + 2 q^{27} + 8 q^{29} + 10 q^{31} + 3 q^{35} - 10 q^{37} - 7 q^{39} + 4 q^{41} + q^{43} - 3 q^{45} - 6 q^{47} + 2 q^{49} - 2 q^{51} - 5 q^{53} - 5 q^{55} + 12 q^{57} + 3 q^{59} - 7 q^{61} - 2 q^{63} + 18 q^{65} + 7 q^{67} - 2 q^{69} - 21 q^{71} + 2 q^{73} - 3 q^{75} + 6 q^{79} + 2 q^{81} + 2 q^{83} - 7 q^{85} + 8 q^{87} - 17 q^{89} + 7 q^{91} + 10 q^{93} - 13 q^{95} - 12 q^{97}+O(q^{100}) \) 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 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −2.61803 −1.17082 −0.585410 0.810737i \(-0.699067\pi\)
−0.585410 + 0.810737i \(0.699067\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.23607 0.674200 0.337100 0.941469i \(-0.390554\pi\)
0.337100 + 0.941469i \(0.390554\pi\)
\(12\) 0 0
\(13\) −6.85410 −1.90099 −0.950493 0.310746i \(-0.899421\pi\)
−0.950493 + 0.310746i \(0.899421\pi\)
\(14\) 0 0
\(15\) −2.61803 −0.675973
\(16\) 0 0
\(17\) 3.47214 0.842117 0.421058 0.907034i \(-0.361659\pi\)
0.421058 + 0.907034i \(0.361659\pi\)
\(18\) 0 0
\(19\) 3.76393 0.863505 0.431753 0.901992i \(-0.357895\pi\)
0.431753 + 0.901992i \(0.357895\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 1.85410 0.370820
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 10.7082 1.98846 0.994232 0.107253i \(-0.0342054\pi\)
0.994232 + 0.107253i \(0.0342054\pi\)
\(30\) 0 0
\(31\) 5.00000 0.898027 0.449013 0.893525i \(-0.351776\pi\)
0.449013 + 0.893525i \(0.351776\pi\)
\(32\) 0 0
\(33\) 2.23607 0.389249
\(34\) 0 0
\(35\) 2.61803 0.442529
\(36\) 0 0
\(37\) −9.47214 −1.55721 −0.778605 0.627515i \(-0.784072\pi\)
−0.778605 + 0.627515i \(0.784072\pi\)
\(38\) 0 0
\(39\) −6.85410 −1.09753
\(40\) 0 0
\(41\) −4.70820 −0.735298 −0.367649 0.929965i \(-0.619837\pi\)
−0.367649 + 0.929965i \(0.619837\pi\)
\(42\) 0 0
\(43\) −7.32624 −1.11724 −0.558620 0.829423i \(-0.688669\pi\)
−0.558620 + 0.829423i \(0.688669\pi\)
\(44\) 0 0
\(45\) −2.61803 −0.390273
\(46\) 0 0
\(47\) −0.763932 −0.111431 −0.0557155 0.998447i \(-0.517744\pi\)
−0.0557155 + 0.998447i \(0.517744\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 3.47214 0.486196
\(52\) 0 0
\(53\) −1.38197 −0.189828 −0.0949138 0.995485i \(-0.530258\pi\)
−0.0949138 + 0.995485i \(0.530258\pi\)
\(54\) 0 0
\(55\) −5.85410 −0.789367
\(56\) 0 0
\(57\) 3.76393 0.498545
\(58\) 0 0
\(59\) −1.85410 −0.241384 −0.120692 0.992690i \(-0.538511\pi\)
−0.120692 + 0.992690i \(0.538511\pi\)
\(60\) 0 0
\(61\) 2.09017 0.267619 0.133809 0.991007i \(-0.457279\pi\)
0.133809 + 0.991007i \(0.457279\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 17.9443 2.22571
\(66\) 0 0
\(67\) 0.145898 0.0178243 0.00891214 0.999960i \(-0.497163\pi\)
0.00891214 + 0.999960i \(0.497163\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −11.6180 −1.37881 −0.689403 0.724378i \(-0.742127\pi\)
−0.689403 + 0.724378i \(0.742127\pi\)
\(72\) 0 0
\(73\) 5.47214 0.640465 0.320233 0.947339i \(-0.396239\pi\)
0.320233 + 0.947339i \(0.396239\pi\)
\(74\) 0 0
\(75\) 1.85410 0.214093
\(76\) 0 0
\(77\) −2.23607 −0.254824
\(78\) 0 0
\(79\) 11.9443 1.34384 0.671918 0.740626i \(-0.265471\pi\)
0.671918 + 0.740626i \(0.265471\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 9.94427 1.09153 0.545763 0.837940i \(-0.316240\pi\)
0.545763 + 0.837940i \(0.316240\pi\)
\(84\) 0 0
\(85\) −9.09017 −0.985967
\(86\) 0 0
\(87\) 10.7082 1.14804
\(88\) 0 0
\(89\) −0.673762 −0.0714186 −0.0357093 0.999362i \(-0.511369\pi\)
−0.0357093 + 0.999362i \(0.511369\pi\)
\(90\) 0 0
\(91\) 6.85410 0.718505
\(92\) 0 0
\(93\) 5.00000 0.518476
\(94\) 0 0
\(95\) −9.85410 −1.01101
\(96\) 0 0
\(97\) 5.18034 0.525984 0.262992 0.964798i \(-0.415291\pi\)
0.262992 + 0.964798i \(0.415291\pi\)
\(98\) 0 0
\(99\) 2.23607 0.224733
\(100\) 0 0
\(101\) −4.90983 −0.488546 −0.244273 0.969706i \(-0.578549\pi\)
−0.244273 + 0.969706i \(0.578549\pi\)
\(102\) 0 0
\(103\) 6.47214 0.637719 0.318859 0.947802i \(-0.396700\pi\)
0.318859 + 0.947802i \(0.396700\pi\)
\(104\) 0 0
\(105\) 2.61803 0.255494
\(106\) 0 0
\(107\) 0.854102 0.0825692 0.0412846 0.999147i \(-0.486855\pi\)
0.0412846 + 0.999147i \(0.486855\pi\)
\(108\) 0 0
\(109\) −14.0344 −1.34426 −0.672128 0.740435i \(-0.734619\pi\)
−0.672128 + 0.740435i \(0.734619\pi\)
\(110\) 0 0
\(111\) −9.47214 −0.899055
\(112\) 0 0
\(113\) −7.09017 −0.666987 −0.333494 0.942752i \(-0.608228\pi\)
−0.333494 + 0.942752i \(0.608228\pi\)
\(114\) 0 0
\(115\) 2.61803 0.244133
\(116\) 0 0
\(117\) −6.85410 −0.633662
\(118\) 0 0
\(119\) −3.47214 −0.318290
\(120\) 0 0
\(121\) −6.00000 −0.545455
\(122\) 0 0
\(123\) −4.70820 −0.424524
\(124\) 0 0
\(125\) 8.23607 0.736656
\(126\) 0 0
\(127\) 2.67376 0.237258 0.118629 0.992939i \(-0.462150\pi\)
0.118629 + 0.992939i \(0.462150\pi\)
\(128\) 0 0
\(129\) −7.32624 −0.645039
\(130\) 0 0
\(131\) 12.4164 1.08483 0.542413 0.840112i \(-0.317511\pi\)
0.542413 + 0.840112i \(0.317511\pi\)
\(132\) 0 0
\(133\) −3.76393 −0.326374
\(134\) 0 0
\(135\) −2.61803 −0.225324
\(136\) 0 0
\(137\) −19.9443 −1.70395 −0.851977 0.523579i \(-0.824596\pi\)
−0.851977 + 0.523579i \(0.824596\pi\)
\(138\) 0 0
\(139\) −19.8541 −1.68400 −0.842001 0.539475i \(-0.818623\pi\)
−0.842001 + 0.539475i \(0.818623\pi\)
\(140\) 0 0
\(141\) −0.763932 −0.0643347
\(142\) 0 0
\(143\) −15.3262 −1.28164
\(144\) 0 0
\(145\) −28.0344 −2.32813
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) −14.6525 −1.20038 −0.600189 0.799858i \(-0.704908\pi\)
−0.600189 + 0.799858i \(0.704908\pi\)
\(150\) 0 0
\(151\) 1.70820 0.139012 0.0695058 0.997582i \(-0.477858\pi\)
0.0695058 + 0.997582i \(0.477858\pi\)
\(152\) 0 0
\(153\) 3.47214 0.280706
\(154\) 0 0
\(155\) −13.0902 −1.05143
\(156\) 0 0
\(157\) −14.1803 −1.13171 −0.565857 0.824503i \(-0.691455\pi\)
−0.565857 + 0.824503i \(0.691455\pi\)
\(158\) 0 0
\(159\) −1.38197 −0.109597
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) 22.5066 1.76285 0.881426 0.472323i \(-0.156584\pi\)
0.881426 + 0.472323i \(0.156584\pi\)
\(164\) 0 0
\(165\) −5.85410 −0.455741
\(166\) 0 0
\(167\) −11.4721 −0.887741 −0.443870 0.896091i \(-0.646395\pi\)
−0.443870 + 0.896091i \(0.646395\pi\)
\(168\) 0 0
\(169\) 33.9787 2.61375
\(170\) 0 0
\(171\) 3.76393 0.287835
\(172\) 0 0
\(173\) −11.2918 −0.858499 −0.429250 0.903186i \(-0.641222\pi\)
−0.429250 + 0.903186i \(0.641222\pi\)
\(174\) 0 0
\(175\) −1.85410 −0.140157
\(176\) 0 0
\(177\) −1.85410 −0.139363
\(178\) 0 0
\(179\) −20.0344 −1.49744 −0.748722 0.662884i \(-0.769332\pi\)
−0.748722 + 0.662884i \(0.769332\pi\)
\(180\) 0 0
\(181\) −15.1803 −1.12835 −0.564173 0.825657i \(-0.690805\pi\)
−0.564173 + 0.825657i \(0.690805\pi\)
\(182\) 0 0
\(183\) 2.09017 0.154510
\(184\) 0 0
\(185\) 24.7984 1.82321
\(186\) 0 0
\(187\) 7.76393 0.567755
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −21.7082 −1.57075 −0.785375 0.619020i \(-0.787530\pi\)
−0.785375 + 0.619020i \(0.787530\pi\)
\(192\) 0 0
\(193\) −18.1803 −1.30865 −0.654325 0.756214i \(-0.727047\pi\)
−0.654325 + 0.756214i \(0.727047\pi\)
\(194\) 0 0
\(195\) 17.9443 1.28502
\(196\) 0 0
\(197\) −25.3262 −1.80442 −0.902210 0.431297i \(-0.858056\pi\)
−0.902210 + 0.431297i \(0.858056\pi\)
\(198\) 0 0
\(199\) 9.56231 0.677854 0.338927 0.940813i \(-0.389936\pi\)
0.338927 + 0.940813i \(0.389936\pi\)
\(200\) 0 0
\(201\) 0.145898 0.0102909
\(202\) 0 0
\(203\) −10.7082 −0.751569
\(204\) 0 0
\(205\) 12.3262 0.860902
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) 8.41641 0.582175
\(210\) 0 0
\(211\) −15.0000 −1.03264 −0.516321 0.856395i \(-0.672699\pi\)
−0.516321 + 0.856395i \(0.672699\pi\)
\(212\) 0 0
\(213\) −11.6180 −0.796055
\(214\) 0 0
\(215\) 19.1803 1.30809
\(216\) 0 0
\(217\) −5.00000 −0.339422
\(218\) 0 0
\(219\) 5.47214 0.369773
\(220\) 0 0
\(221\) −23.7984 −1.60085
\(222\) 0 0
\(223\) −18.3262 −1.22722 −0.613608 0.789611i \(-0.710282\pi\)
−0.613608 + 0.789611i \(0.710282\pi\)
\(224\) 0 0
\(225\) 1.85410 0.123607
\(226\) 0 0
\(227\) −15.2705 −1.01354 −0.506770 0.862081i \(-0.669161\pi\)
−0.506770 + 0.862081i \(0.669161\pi\)
\(228\) 0 0
\(229\) −5.67376 −0.374933 −0.187466 0.982271i \(-0.560028\pi\)
−0.187466 + 0.982271i \(0.560028\pi\)
\(230\) 0 0
\(231\) −2.23607 −0.147122
\(232\) 0 0
\(233\) 7.09017 0.464492 0.232246 0.972657i \(-0.425392\pi\)
0.232246 + 0.972657i \(0.425392\pi\)
\(234\) 0 0
\(235\) 2.00000 0.130466
\(236\) 0 0
\(237\) 11.9443 0.775864
\(238\) 0 0
\(239\) 6.61803 0.428085 0.214043 0.976824i \(-0.431337\pi\)
0.214043 + 0.976824i \(0.431337\pi\)
\(240\) 0 0
\(241\) 28.7082 1.84926 0.924629 0.380869i \(-0.124375\pi\)
0.924629 + 0.380869i \(0.124375\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −2.61803 −0.167260
\(246\) 0 0
\(247\) −25.7984 −1.64151
\(248\) 0 0
\(249\) 9.94427 0.630193
\(250\) 0 0
\(251\) 12.2918 0.775851 0.387926 0.921691i \(-0.373192\pi\)
0.387926 + 0.921691i \(0.373192\pi\)
\(252\) 0 0
\(253\) −2.23607 −0.140580
\(254\) 0 0
\(255\) −9.09017 −0.569249
\(256\) 0 0
\(257\) 29.7082 1.85315 0.926573 0.376114i \(-0.122740\pi\)
0.926573 + 0.376114i \(0.122740\pi\)
\(258\) 0 0
\(259\) 9.47214 0.588570
\(260\) 0 0
\(261\) 10.7082 0.662821
\(262\) 0 0
\(263\) −5.00000 −0.308313 −0.154157 0.988046i \(-0.549266\pi\)
−0.154157 + 0.988046i \(0.549266\pi\)
\(264\) 0 0
\(265\) 3.61803 0.222254
\(266\) 0 0
\(267\) −0.673762 −0.0412336
\(268\) 0 0
\(269\) −2.85410 −0.174018 −0.0870088 0.996208i \(-0.527731\pi\)
−0.0870088 + 0.996208i \(0.527731\pi\)
\(270\) 0 0
\(271\) 17.1803 1.04363 0.521816 0.853058i \(-0.325255\pi\)
0.521816 + 0.853058i \(0.325255\pi\)
\(272\) 0 0
\(273\) 6.85410 0.414829
\(274\) 0 0
\(275\) 4.14590 0.250007
\(276\) 0 0
\(277\) 21.2148 1.27467 0.637336 0.770586i \(-0.280036\pi\)
0.637336 + 0.770586i \(0.280036\pi\)
\(278\) 0 0
\(279\) 5.00000 0.299342
\(280\) 0 0
\(281\) −21.5967 −1.28835 −0.644177 0.764876i \(-0.722800\pi\)
−0.644177 + 0.764876i \(0.722800\pi\)
\(282\) 0 0
\(283\) −14.3262 −0.851606 −0.425803 0.904816i \(-0.640008\pi\)
−0.425803 + 0.904816i \(0.640008\pi\)
\(284\) 0 0
\(285\) −9.85410 −0.583707
\(286\) 0 0
\(287\) 4.70820 0.277916
\(288\) 0 0
\(289\) −4.94427 −0.290840
\(290\) 0 0
\(291\) 5.18034 0.303677
\(292\) 0 0
\(293\) −9.52786 −0.556624 −0.278312 0.960491i \(-0.589775\pi\)
−0.278312 + 0.960491i \(0.589775\pi\)
\(294\) 0 0
\(295\) 4.85410 0.282617
\(296\) 0 0
\(297\) 2.23607 0.129750
\(298\) 0 0
\(299\) 6.85410 0.396383
\(300\) 0 0
\(301\) 7.32624 0.422277
\(302\) 0 0
\(303\) −4.90983 −0.282062
\(304\) 0 0
\(305\) −5.47214 −0.313334
\(306\) 0 0
\(307\) −25.9443 −1.48072 −0.740359 0.672212i \(-0.765344\pi\)
−0.740359 + 0.672212i \(0.765344\pi\)
\(308\) 0 0
\(309\) 6.47214 0.368187
\(310\) 0 0
\(311\) −7.61803 −0.431979 −0.215990 0.976396i \(-0.569298\pi\)
−0.215990 + 0.976396i \(0.569298\pi\)
\(312\) 0 0
\(313\) −1.52786 −0.0863600 −0.0431800 0.999067i \(-0.513749\pi\)
−0.0431800 + 0.999067i \(0.513749\pi\)
\(314\) 0 0
\(315\) 2.61803 0.147510
\(316\) 0 0
\(317\) −22.1459 −1.24384 −0.621919 0.783082i \(-0.713647\pi\)
−0.621919 + 0.783082i \(0.713647\pi\)
\(318\) 0 0
\(319\) 23.9443 1.34062
\(320\) 0 0
\(321\) 0.854102 0.0476713
\(322\) 0 0
\(323\) 13.0689 0.727172
\(324\) 0 0
\(325\) −12.7082 −0.704924
\(326\) 0 0
\(327\) −14.0344 −0.776106
\(328\) 0 0
\(329\) 0.763932 0.0421169
\(330\) 0 0
\(331\) −10.9443 −0.601552 −0.300776 0.953695i \(-0.597246\pi\)
−0.300776 + 0.953695i \(0.597246\pi\)
\(332\) 0 0
\(333\) −9.47214 −0.519070
\(334\) 0 0
\(335\) −0.381966 −0.0208690
\(336\) 0 0
\(337\) −18.5623 −1.01115 −0.505577 0.862782i \(-0.668720\pi\)
−0.505577 + 0.862782i \(0.668720\pi\)
\(338\) 0 0
\(339\) −7.09017 −0.385085
\(340\) 0 0
\(341\) 11.1803 0.605449
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 2.61803 0.140950
\(346\) 0 0
\(347\) 3.65248 0.196075 0.0980376 0.995183i \(-0.468743\pi\)
0.0980376 + 0.995183i \(0.468743\pi\)
\(348\) 0 0
\(349\) −6.50658 −0.348289 −0.174145 0.984720i \(-0.555716\pi\)
−0.174145 + 0.984720i \(0.555716\pi\)
\(350\) 0 0
\(351\) −6.85410 −0.365845
\(352\) 0 0
\(353\) −26.5967 −1.41560 −0.707801 0.706412i \(-0.750313\pi\)
−0.707801 + 0.706412i \(0.750313\pi\)
\(354\) 0 0
\(355\) 30.4164 1.61434
\(356\) 0 0
\(357\) −3.47214 −0.183765
\(358\) 0 0
\(359\) −22.7984 −1.20325 −0.601626 0.798778i \(-0.705480\pi\)
−0.601626 + 0.798778i \(0.705480\pi\)
\(360\) 0 0
\(361\) −4.83282 −0.254359
\(362\) 0 0
\(363\) −6.00000 −0.314918
\(364\) 0 0
\(365\) −14.3262 −0.749870
\(366\) 0 0
\(367\) 32.4508 1.69392 0.846960 0.531656i \(-0.178430\pi\)
0.846960 + 0.531656i \(0.178430\pi\)
\(368\) 0 0
\(369\) −4.70820 −0.245099
\(370\) 0 0
\(371\) 1.38197 0.0717481
\(372\) 0 0
\(373\) 2.88854 0.149563 0.0747816 0.997200i \(-0.476174\pi\)
0.0747816 + 0.997200i \(0.476174\pi\)
\(374\) 0 0
\(375\) 8.23607 0.425309
\(376\) 0 0
\(377\) −73.3951 −3.78004
\(378\) 0 0
\(379\) 20.9443 1.07583 0.537917 0.842997i \(-0.319211\pi\)
0.537917 + 0.842997i \(0.319211\pi\)
\(380\) 0 0
\(381\) 2.67376 0.136981
\(382\) 0 0
\(383\) 4.52786 0.231363 0.115682 0.993286i \(-0.463095\pi\)
0.115682 + 0.993286i \(0.463095\pi\)
\(384\) 0 0
\(385\) 5.85410 0.298353
\(386\) 0 0
\(387\) −7.32624 −0.372414
\(388\) 0 0
\(389\) 22.1246 1.12176 0.560881 0.827896i \(-0.310462\pi\)
0.560881 + 0.827896i \(0.310462\pi\)
\(390\) 0 0
\(391\) −3.47214 −0.175593
\(392\) 0 0
\(393\) 12.4164 0.626325
\(394\) 0 0
\(395\) −31.2705 −1.57339
\(396\) 0 0
\(397\) 15.2361 0.764676 0.382338 0.924022i \(-0.375119\pi\)
0.382338 + 0.924022i \(0.375119\pi\)
\(398\) 0 0
\(399\) −3.76393 −0.188432
\(400\) 0 0
\(401\) 6.81966 0.340558 0.170279 0.985396i \(-0.445533\pi\)
0.170279 + 0.985396i \(0.445533\pi\)
\(402\) 0 0
\(403\) −34.2705 −1.70714
\(404\) 0 0
\(405\) −2.61803 −0.130091
\(406\) 0 0
\(407\) −21.1803 −1.04987
\(408\) 0 0
\(409\) 23.6525 1.16954 0.584770 0.811199i \(-0.301185\pi\)
0.584770 + 0.811199i \(0.301185\pi\)
\(410\) 0 0
\(411\) −19.9443 −0.983778
\(412\) 0 0
\(413\) 1.85410 0.0912344
\(414\) 0 0
\(415\) −26.0344 −1.27798
\(416\) 0 0
\(417\) −19.8541 −0.972260
\(418\) 0 0
\(419\) −19.5623 −0.955681 −0.477841 0.878447i \(-0.658580\pi\)
−0.477841 + 0.878447i \(0.658580\pi\)
\(420\) 0 0
\(421\) 10.6738 0.520207 0.260104 0.965581i \(-0.416243\pi\)
0.260104 + 0.965581i \(0.416243\pi\)
\(422\) 0 0
\(423\) −0.763932 −0.0371436
\(424\) 0 0
\(425\) 6.43769 0.312274
\(426\) 0 0
\(427\) −2.09017 −0.101150
\(428\) 0 0
\(429\) −15.3262 −0.739958
\(430\) 0 0
\(431\) −19.9098 −0.959023 −0.479511 0.877536i \(-0.659186\pi\)
−0.479511 + 0.877536i \(0.659186\pi\)
\(432\) 0 0
\(433\) 23.1246 1.11130 0.555649 0.831417i \(-0.312470\pi\)
0.555649 + 0.831417i \(0.312470\pi\)
\(434\) 0 0
\(435\) −28.0344 −1.34415
\(436\) 0 0
\(437\) −3.76393 −0.180053
\(438\) 0 0
\(439\) 17.9443 0.856433 0.428217 0.903676i \(-0.359142\pi\)
0.428217 + 0.903676i \(0.359142\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 4.00000 0.190046 0.0950229 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) 0 0
\(445\) 1.76393 0.0836184
\(446\) 0 0
\(447\) −14.6525 −0.693038
\(448\) 0 0
\(449\) 12.9098 0.609253 0.304626 0.952472i \(-0.401468\pi\)
0.304626 + 0.952472i \(0.401468\pi\)
\(450\) 0 0
\(451\) −10.5279 −0.495738
\(452\) 0 0
\(453\) 1.70820 0.0802584
\(454\) 0 0
\(455\) −17.9443 −0.841240
\(456\) 0 0
\(457\) −32.7984 −1.53424 −0.767122 0.641502i \(-0.778312\pi\)
−0.767122 + 0.641502i \(0.778312\pi\)
\(458\) 0 0
\(459\) 3.47214 0.162065
\(460\) 0 0
\(461\) 6.14590 0.286243 0.143122 0.989705i \(-0.454286\pi\)
0.143122 + 0.989705i \(0.454286\pi\)
\(462\) 0 0
\(463\) −14.5967 −0.678368 −0.339184 0.940720i \(-0.610151\pi\)
−0.339184 + 0.940720i \(0.610151\pi\)
\(464\) 0 0
\(465\) −13.0902 −0.607042
\(466\) 0 0
\(467\) −18.8885 −0.874058 −0.437029 0.899448i \(-0.643969\pi\)
−0.437029 + 0.899448i \(0.643969\pi\)
\(468\) 0 0
\(469\) −0.145898 −0.00673695
\(470\) 0 0
\(471\) −14.1803 −0.653396
\(472\) 0 0
\(473\) −16.3820 −0.753244
\(474\) 0 0
\(475\) 6.97871 0.320205
\(476\) 0 0
\(477\) −1.38197 −0.0632759
\(478\) 0 0
\(479\) 5.65248 0.258268 0.129134 0.991627i \(-0.458780\pi\)
0.129134 + 0.991627i \(0.458780\pi\)
\(480\) 0 0
\(481\) 64.9230 2.96023
\(482\) 0 0
\(483\) 1.00000 0.0455016
\(484\) 0 0
\(485\) −13.5623 −0.615833
\(486\) 0 0
\(487\) −17.7639 −0.804961 −0.402480 0.915429i \(-0.631852\pi\)
−0.402480 + 0.915429i \(0.631852\pi\)
\(488\) 0 0
\(489\) 22.5066 1.01778
\(490\) 0 0
\(491\) −28.0902 −1.26769 −0.633846 0.773459i \(-0.718525\pi\)
−0.633846 + 0.773459i \(0.718525\pi\)
\(492\) 0 0
\(493\) 37.1803 1.67452
\(494\) 0 0
\(495\) −5.85410 −0.263122
\(496\) 0 0
\(497\) 11.6180 0.521140
\(498\) 0 0
\(499\) −12.7984 −0.572934 −0.286467 0.958090i \(-0.592481\pi\)
−0.286467 + 0.958090i \(0.592481\pi\)
\(500\) 0 0
\(501\) −11.4721 −0.512537
\(502\) 0 0
\(503\) 21.0902 0.940364 0.470182 0.882569i \(-0.344188\pi\)
0.470182 + 0.882569i \(0.344188\pi\)
\(504\) 0 0
\(505\) 12.8541 0.572000
\(506\) 0 0
\(507\) 33.9787 1.50905
\(508\) 0 0
\(509\) 25.7082 1.13950 0.569748 0.821819i \(-0.307041\pi\)
0.569748 + 0.821819i \(0.307041\pi\)
\(510\) 0 0
\(511\) −5.47214 −0.242073
\(512\) 0 0
\(513\) 3.76393 0.166182
\(514\) 0 0
\(515\) −16.9443 −0.746654
\(516\) 0 0
\(517\) −1.70820 −0.0751267
\(518\) 0 0
\(519\) −11.2918 −0.495655
\(520\) 0 0
\(521\) −3.52786 −0.154559 −0.0772793 0.997009i \(-0.524623\pi\)
−0.0772793 + 0.997009i \(0.524623\pi\)
\(522\) 0 0
\(523\) −32.8328 −1.43568 −0.717839 0.696209i \(-0.754869\pi\)
−0.717839 + 0.696209i \(0.754869\pi\)
\(524\) 0 0
\(525\) −1.85410 −0.0809196
\(526\) 0 0
\(527\) 17.3607 0.756243
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −1.85410 −0.0804612
\(532\) 0 0
\(533\) 32.2705 1.39779
\(534\) 0 0
\(535\) −2.23607 −0.0966736
\(536\) 0 0
\(537\) −20.0344 −0.864550
\(538\) 0 0
\(539\) 2.23607 0.0963143
\(540\) 0 0
\(541\) −23.2361 −0.998997 −0.499498 0.866315i \(-0.666482\pi\)
−0.499498 + 0.866315i \(0.666482\pi\)
\(542\) 0 0
\(543\) −15.1803 −0.651451
\(544\) 0 0
\(545\) 36.7426 1.57388
\(546\) 0 0
\(547\) 26.6180 1.13810 0.569052 0.822301i \(-0.307310\pi\)
0.569052 + 0.822301i \(0.307310\pi\)
\(548\) 0 0
\(549\) 2.09017 0.0892063
\(550\) 0 0
\(551\) 40.3050 1.71705
\(552\) 0 0
\(553\) −11.9443 −0.507922
\(554\) 0 0
\(555\) 24.7984 1.05263
\(556\) 0 0
\(557\) −17.2361 −0.730316 −0.365158 0.930946i \(-0.618985\pi\)
−0.365158 + 0.930946i \(0.618985\pi\)
\(558\) 0 0
\(559\) 50.2148 2.12386
\(560\) 0 0
\(561\) 7.76393 0.327793
\(562\) 0 0
\(563\) −8.56231 −0.360858 −0.180429 0.983588i \(-0.557749\pi\)
−0.180429 + 0.983588i \(0.557749\pi\)
\(564\) 0 0
\(565\) 18.5623 0.780922
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 27.7771 1.16448 0.582238 0.813018i \(-0.302177\pi\)
0.582238 + 0.813018i \(0.302177\pi\)
\(570\) 0 0
\(571\) 44.7771 1.87386 0.936931 0.349513i \(-0.113653\pi\)
0.936931 + 0.349513i \(0.113653\pi\)
\(572\) 0 0
\(573\) −21.7082 −0.906873
\(574\) 0 0
\(575\) −1.85410 −0.0773214
\(576\) 0 0
\(577\) −17.4164 −0.725055 −0.362527 0.931973i \(-0.618086\pi\)
−0.362527 + 0.931973i \(0.618086\pi\)
\(578\) 0 0
\(579\) −18.1803 −0.755549
\(580\) 0 0
\(581\) −9.94427 −0.412558
\(582\) 0 0
\(583\) −3.09017 −0.127982
\(584\) 0 0
\(585\) 17.9443 0.741904
\(586\) 0 0
\(587\) −2.56231 −0.105758 −0.0528788 0.998601i \(-0.516840\pi\)
−0.0528788 + 0.998601i \(0.516840\pi\)
\(588\) 0 0
\(589\) 18.8197 0.775451
\(590\) 0 0
\(591\) −25.3262 −1.04178
\(592\) 0 0
\(593\) −24.2918 −0.997545 −0.498772 0.866733i \(-0.666216\pi\)
−0.498772 + 0.866733i \(0.666216\pi\)
\(594\) 0 0
\(595\) 9.09017 0.372661
\(596\) 0 0
\(597\) 9.56231 0.391359
\(598\) 0 0
\(599\) −7.72949 −0.315818 −0.157909 0.987454i \(-0.550475\pi\)
−0.157909 + 0.987454i \(0.550475\pi\)
\(600\) 0 0
\(601\) 12.5066 0.510154 0.255077 0.966921i \(-0.417899\pi\)
0.255077 + 0.966921i \(0.417899\pi\)
\(602\) 0 0
\(603\) 0.145898 0.00594143
\(604\) 0 0
\(605\) 15.7082 0.638629
\(606\) 0 0
\(607\) 8.38197 0.340214 0.170107 0.985426i \(-0.445589\pi\)
0.170107 + 0.985426i \(0.445589\pi\)
\(608\) 0 0
\(609\) −10.7082 −0.433918
\(610\) 0 0
\(611\) 5.23607 0.211829
\(612\) 0 0
\(613\) −6.34752 −0.256374 −0.128187 0.991750i \(-0.540916\pi\)
−0.128187 + 0.991750i \(0.540916\pi\)
\(614\) 0 0
\(615\) 12.3262 0.497042
\(616\) 0 0
\(617\) −4.74265 −0.190932 −0.0954659 0.995433i \(-0.530434\pi\)
−0.0954659 + 0.995433i \(0.530434\pi\)
\(618\) 0 0
\(619\) −0.270510 −0.0108727 −0.00543635 0.999985i \(-0.501730\pi\)
−0.00543635 + 0.999985i \(0.501730\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 0.673762 0.0269937
\(624\) 0 0
\(625\) −30.8328 −1.23331
\(626\) 0 0
\(627\) 8.41641 0.336119
\(628\) 0 0
\(629\) −32.8885 −1.31135
\(630\) 0 0
\(631\) 33.7639 1.34412 0.672060 0.740496i \(-0.265409\pi\)
0.672060 + 0.740496i \(0.265409\pi\)
\(632\) 0 0
\(633\) −15.0000 −0.596196
\(634\) 0 0
\(635\) −7.00000 −0.277787
\(636\) 0 0
\(637\) −6.85410 −0.271569
\(638\) 0 0
\(639\) −11.6180 −0.459602
\(640\) 0 0
\(641\) −26.3820 −1.04203 −0.521013 0.853549i \(-0.674446\pi\)
−0.521013 + 0.853549i \(0.674446\pi\)
\(642\) 0 0
\(643\) 35.5066 1.40024 0.700121 0.714024i \(-0.253129\pi\)
0.700121 + 0.714024i \(0.253129\pi\)
\(644\) 0 0
\(645\) 19.1803 0.755225
\(646\) 0 0
\(647\) 9.09017 0.357371 0.178686 0.983906i \(-0.442815\pi\)
0.178686 + 0.983906i \(0.442815\pi\)
\(648\) 0 0
\(649\) −4.14590 −0.162741
\(650\) 0 0
\(651\) −5.00000 −0.195965
\(652\) 0 0
\(653\) −20.9098 −0.818265 −0.409132 0.912475i \(-0.634169\pi\)
−0.409132 + 0.912475i \(0.634169\pi\)
\(654\) 0 0
\(655\) −32.5066 −1.27014
\(656\) 0 0
\(657\) 5.47214 0.213488
\(658\) 0 0
\(659\) −20.5967 −0.802335 −0.401168 0.916005i \(-0.631396\pi\)
−0.401168 + 0.916005i \(0.631396\pi\)
\(660\) 0 0
\(661\) 29.1803 1.13498 0.567492 0.823379i \(-0.307914\pi\)
0.567492 + 0.823379i \(0.307914\pi\)
\(662\) 0 0
\(663\) −23.7984 −0.924252
\(664\) 0 0
\(665\) 9.85410 0.382126
\(666\) 0 0
\(667\) −10.7082 −0.414623
\(668\) 0 0
\(669\) −18.3262 −0.708533
\(670\) 0 0
\(671\) 4.67376 0.180429
\(672\) 0 0
\(673\) 4.05573 0.156337 0.0781684 0.996940i \(-0.475093\pi\)
0.0781684 + 0.996940i \(0.475093\pi\)
\(674\) 0 0
\(675\) 1.85410 0.0713644
\(676\) 0 0
\(677\) −23.7984 −0.914646 −0.457323 0.889301i \(-0.651192\pi\)
−0.457323 + 0.889301i \(0.651192\pi\)
\(678\) 0 0
\(679\) −5.18034 −0.198803
\(680\) 0 0
\(681\) −15.2705 −0.585167
\(682\) 0 0
\(683\) −27.2918 −1.04429 −0.522146 0.852856i \(-0.674868\pi\)
−0.522146 + 0.852856i \(0.674868\pi\)
\(684\) 0 0
\(685\) 52.2148 1.99502
\(686\) 0 0
\(687\) −5.67376 −0.216468
\(688\) 0 0
\(689\) 9.47214 0.360860
\(690\) 0 0
\(691\) −3.79837 −0.144497 −0.0722485 0.997387i \(-0.523017\pi\)
−0.0722485 + 0.997387i \(0.523017\pi\)
\(692\) 0 0
\(693\) −2.23607 −0.0849412
\(694\) 0 0
\(695\) 51.9787 1.97166
\(696\) 0 0
\(697\) −16.3475 −0.619207
\(698\) 0 0
\(699\) 7.09017 0.268175
\(700\) 0 0
\(701\) −7.20163 −0.272002 −0.136001 0.990709i \(-0.543425\pi\)
−0.136001 + 0.990709i \(0.543425\pi\)
\(702\) 0 0
\(703\) −35.6525 −1.34466
\(704\) 0 0
\(705\) 2.00000 0.0753244
\(706\) 0 0
\(707\) 4.90983 0.184653
\(708\) 0 0
\(709\) 3.32624 0.124919 0.0624597 0.998047i \(-0.480105\pi\)
0.0624597 + 0.998047i \(0.480105\pi\)
\(710\) 0 0
\(711\) 11.9443 0.447945
\(712\) 0 0
\(713\) −5.00000 −0.187251
\(714\) 0 0
\(715\) 40.1246 1.50058
\(716\) 0 0
\(717\) 6.61803 0.247155
\(718\) 0 0
\(719\) −22.7082 −0.846873 −0.423437 0.905926i \(-0.639176\pi\)
−0.423437 + 0.905926i \(0.639176\pi\)
\(720\) 0 0
\(721\) −6.47214 −0.241035
\(722\) 0 0
\(723\) 28.7082 1.06767
\(724\) 0 0
\(725\) 19.8541 0.737363
\(726\) 0 0
\(727\) 25.7639 0.955531 0.477766 0.878487i \(-0.341447\pi\)
0.477766 + 0.878487i \(0.341447\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −25.4377 −0.940847
\(732\) 0 0
\(733\) −38.5410 −1.42355 −0.711773 0.702410i \(-0.752107\pi\)
−0.711773 + 0.702410i \(0.752107\pi\)
\(734\) 0 0
\(735\) −2.61803 −0.0965676
\(736\) 0 0
\(737\) 0.326238 0.0120171
\(738\) 0 0
\(739\) 13.5836 0.499681 0.249840 0.968287i \(-0.419622\pi\)
0.249840 + 0.968287i \(0.419622\pi\)
\(740\) 0 0
\(741\) −25.7984 −0.947727
\(742\) 0 0
\(743\) 48.3951 1.77544 0.887722 0.460379i \(-0.152287\pi\)
0.887722 + 0.460379i \(0.152287\pi\)
\(744\) 0 0
\(745\) 38.3607 1.40543
\(746\) 0 0
\(747\) 9.94427 0.363842
\(748\) 0 0
\(749\) −0.854102 −0.0312082
\(750\) 0 0
\(751\) 20.9098 0.763011 0.381505 0.924367i \(-0.375406\pi\)
0.381505 + 0.924367i \(0.375406\pi\)
\(752\) 0 0
\(753\) 12.2918 0.447938
\(754\) 0 0
\(755\) −4.47214 −0.162758
\(756\) 0 0
\(757\) −5.58359 −0.202939 −0.101470 0.994839i \(-0.532354\pi\)
−0.101470 + 0.994839i \(0.532354\pi\)
\(758\) 0 0
\(759\) −2.23607 −0.0811641
\(760\) 0 0
\(761\) 18.1115 0.656540 0.328270 0.944584i \(-0.393534\pi\)
0.328270 + 0.944584i \(0.393534\pi\)
\(762\) 0 0
\(763\) 14.0344 0.508081
\(764\) 0 0
\(765\) −9.09017 −0.328656
\(766\) 0 0
\(767\) 12.7082 0.458867
\(768\) 0 0
\(769\) −45.9574 −1.65727 −0.828634 0.559791i \(-0.810881\pi\)
−0.828634 + 0.559791i \(0.810881\pi\)
\(770\) 0 0
\(771\) 29.7082 1.06991
\(772\) 0 0
\(773\) 40.0132 1.43917 0.719587 0.694403i \(-0.244331\pi\)
0.719587 + 0.694403i \(0.244331\pi\)
\(774\) 0 0
\(775\) 9.27051 0.333007
\(776\) 0 0
\(777\) 9.47214 0.339811
\(778\) 0 0
\(779\) −17.7214 −0.634934
\(780\) 0 0
\(781\) −25.9787 −0.929591
\(782\) 0 0
\(783\) 10.7082 0.382680
\(784\) 0 0
\(785\) 37.1246 1.32503
\(786\) 0 0
\(787\) −22.7984 −0.812674 −0.406337 0.913723i \(-0.633194\pi\)
−0.406337 + 0.913723i \(0.633194\pi\)
\(788\) 0 0
\(789\) −5.00000 −0.178005
\(790\) 0 0
\(791\) 7.09017 0.252097
\(792\) 0 0
\(793\) −14.3262 −0.508740
\(794\) 0 0
\(795\) 3.61803 0.128318
\(796\) 0 0
\(797\) 9.70820 0.343882 0.171941 0.985107i \(-0.444996\pi\)
0.171941 + 0.985107i \(0.444996\pi\)
\(798\) 0 0
\(799\) −2.65248 −0.0938378
\(800\) 0 0
\(801\) −0.673762 −0.0238062
\(802\) 0 0
\(803\) 12.2361 0.431801
\(804\) 0 0
\(805\) −2.61803 −0.0922736
\(806\) 0 0
\(807\) −2.85410 −0.100469
\(808\) 0 0
\(809\) −22.5623 −0.793248 −0.396624 0.917981i \(-0.629818\pi\)
−0.396624 + 0.917981i \(0.629818\pi\)
\(810\) 0 0
\(811\) −41.5410 −1.45870 −0.729351 0.684139i \(-0.760178\pi\)
−0.729351 + 0.684139i \(0.760178\pi\)
\(812\) 0 0
\(813\) 17.1803 0.602541
\(814\) 0 0
\(815\) −58.9230 −2.06398
\(816\) 0 0
\(817\) −27.5755 −0.964743
\(818\) 0 0
\(819\) 6.85410 0.239502
\(820\) 0 0
\(821\) 25.7082 0.897223 0.448611 0.893727i \(-0.351919\pi\)
0.448611 + 0.893727i \(0.351919\pi\)
\(822\) 0 0
\(823\) −54.9787 −1.91644 −0.958219 0.286036i \(-0.907662\pi\)
−0.958219 + 0.286036i \(0.907662\pi\)
\(824\) 0 0
\(825\) 4.14590 0.144342
\(826\) 0 0
\(827\) 16.1459 0.561448 0.280724 0.959789i \(-0.409425\pi\)
0.280724 + 0.959789i \(0.409425\pi\)
\(828\) 0 0
\(829\) −23.0689 −0.801215 −0.400608 0.916250i \(-0.631201\pi\)
−0.400608 + 0.916250i \(0.631201\pi\)
\(830\) 0 0
\(831\) 21.2148 0.735933
\(832\) 0 0
\(833\) 3.47214 0.120302
\(834\) 0 0
\(835\) 30.0344 1.03938
\(836\) 0 0
\(837\) 5.00000 0.172825
\(838\) 0 0
\(839\) 36.2705 1.25220 0.626099 0.779744i \(-0.284651\pi\)
0.626099 + 0.779744i \(0.284651\pi\)
\(840\) 0 0
\(841\) 85.6656 2.95399
\(842\) 0 0
\(843\) −21.5967 −0.743832
\(844\) 0 0
\(845\) −88.9574 −3.06023
\(846\) 0 0
\(847\) 6.00000 0.206162
\(848\) 0 0
\(849\) −14.3262 −0.491675
\(850\) 0 0
\(851\) 9.47214 0.324701
\(852\) 0 0
\(853\) −11.3475 −0.388532 −0.194266 0.980949i \(-0.562232\pi\)
−0.194266 + 0.980949i \(0.562232\pi\)
\(854\) 0 0
\(855\) −9.85410 −0.337003
\(856\) 0 0
\(857\) −49.9574 −1.70651 −0.853257 0.521491i \(-0.825376\pi\)
−0.853257 + 0.521491i \(0.825376\pi\)
\(858\) 0 0
\(859\) 25.4164 0.867197 0.433598 0.901106i \(-0.357244\pi\)
0.433598 + 0.901106i \(0.357244\pi\)
\(860\) 0 0
\(861\) 4.70820 0.160455
\(862\) 0 0
\(863\) 17.5967 0.599000 0.299500 0.954096i \(-0.403180\pi\)
0.299500 + 0.954096i \(0.403180\pi\)
\(864\) 0 0
\(865\) 29.5623 1.00515
\(866\) 0 0
\(867\) −4.94427 −0.167916
\(868\) 0 0
\(869\) 26.7082 0.906014
\(870\) 0 0
\(871\) −1.00000 −0.0338837
\(872\) 0 0
\(873\) 5.18034 0.175328
\(874\) 0 0
\(875\) −8.23607 −0.278430
\(876\) 0 0
\(877\) −17.5279 −0.591874 −0.295937 0.955207i \(-0.595632\pi\)
−0.295937 + 0.955207i \(0.595632\pi\)
\(878\) 0 0
\(879\) −9.52786 −0.321367
\(880\) 0 0
\(881\) 26.5410 0.894190 0.447095 0.894487i \(-0.352459\pi\)
0.447095 + 0.894487i \(0.352459\pi\)
\(882\) 0 0
\(883\) 34.7426 1.16918 0.584592 0.811328i \(-0.301255\pi\)
0.584592 + 0.811328i \(0.301255\pi\)
\(884\) 0 0
\(885\) 4.85410 0.163169
\(886\) 0 0
\(887\) −6.67376 −0.224083 −0.112041 0.993704i \(-0.535739\pi\)
−0.112041 + 0.993704i \(0.535739\pi\)
\(888\) 0 0
\(889\) −2.67376 −0.0896751
\(890\) 0 0
\(891\) 2.23607 0.0749111
\(892\) 0 0
\(893\) −2.87539 −0.0962212
\(894\) 0 0
\(895\) 52.4508 1.75324
\(896\) 0 0
\(897\) 6.85410 0.228852
\(898\) 0 0
\(899\) 53.5410 1.78569
\(900\) 0 0
\(901\) −4.79837 −0.159857
\(902\) 0 0
\(903\) 7.32624 0.243802
\(904\) 0 0
\(905\) 39.7426 1.32109
\(906\) 0 0
\(907\) 16.3262 0.542104 0.271052 0.962565i \(-0.412628\pi\)
0.271052 + 0.962565i \(0.412628\pi\)
\(908\) 0 0
\(909\) −4.90983 −0.162849
\(910\) 0 0
\(911\) 18.2918 0.606034 0.303017 0.952985i \(-0.402006\pi\)
0.303017 + 0.952985i \(0.402006\pi\)
\(912\) 0 0
\(913\) 22.2361 0.735906
\(914\) 0 0
\(915\) −5.47214 −0.180903
\(916\) 0 0
\(917\) −12.4164 −0.410026
\(918\) 0 0
\(919\) −36.4853 −1.20354 −0.601769 0.798670i \(-0.705537\pi\)
−0.601769 + 0.798670i \(0.705537\pi\)
\(920\) 0 0
\(921\) −25.9443 −0.854893
\(922\) 0 0
\(923\) 79.6312 2.62109
\(924\) 0 0
\(925\) −17.5623 −0.577445
\(926\) 0 0
\(927\) 6.47214 0.212573
\(928\) 0 0
\(929\) 33.7426 1.10706 0.553530 0.832829i \(-0.313280\pi\)
0.553530 + 0.832829i \(0.313280\pi\)
\(930\) 0 0
\(931\) 3.76393 0.123358
\(932\) 0 0
\(933\) −7.61803 −0.249403
\(934\) 0 0
\(935\) −20.3262 −0.664739
\(936\) 0 0
\(937\) −33.3607 −1.08985 −0.544923 0.838486i \(-0.683441\pi\)
−0.544923 + 0.838486i \(0.683441\pi\)
\(938\) 0 0
\(939\) −1.52786 −0.0498600
\(940\) 0 0
\(941\) −34.8328 −1.13552 −0.567759 0.823195i \(-0.692189\pi\)
−0.567759 + 0.823195i \(0.692189\pi\)
\(942\) 0 0
\(943\) 4.70820 0.153320
\(944\) 0 0
\(945\) 2.61803 0.0851647
\(946\) 0 0
\(947\) 46.7214 1.51824 0.759120 0.650951i \(-0.225630\pi\)
0.759120 + 0.650951i \(0.225630\pi\)
\(948\) 0 0
\(949\) −37.5066 −1.21752
\(950\) 0 0
\(951\) −22.1459 −0.718130
\(952\) 0 0
\(953\) 48.9230 1.58477 0.792385 0.610021i \(-0.208839\pi\)
0.792385 + 0.610021i \(0.208839\pi\)
\(954\) 0 0
\(955\) 56.8328 1.83907
\(956\) 0 0
\(957\) 23.9443 0.774008
\(958\) 0 0
\(959\) 19.9443 0.644034
\(960\) 0 0
\(961\) −6.00000 −0.193548
\(962\) 0 0
\(963\) 0.854102 0.0275231
\(964\) 0 0
\(965\) 47.5967 1.53219
\(966\) 0 0
\(967\) 16.4721 0.529708 0.264854 0.964288i \(-0.414676\pi\)
0.264854 + 0.964288i \(0.414676\pi\)
\(968\) 0 0
\(969\) 13.0689 0.419833
\(970\) 0 0
\(971\) 14.0902 0.452175 0.226088 0.974107i \(-0.427406\pi\)
0.226088 + 0.974107i \(0.427406\pi\)
\(972\) 0 0
\(973\) 19.8541 0.636493
\(974\) 0 0
\(975\) −12.7082 −0.406988
\(976\) 0 0
\(977\) 53.9787 1.72693 0.863466 0.504407i \(-0.168289\pi\)
0.863466 + 0.504407i \(0.168289\pi\)
\(978\) 0 0
\(979\) −1.50658 −0.0481504
\(980\) 0 0
\(981\) −14.0344 −0.448085
\(982\) 0 0
\(983\) 59.3607 1.89331 0.946656 0.322246i \(-0.104438\pi\)
0.946656 + 0.322246i \(0.104438\pi\)
\(984\) 0 0
\(985\) 66.3050 2.11265
\(986\) 0 0
\(987\) 0.763932 0.0243162
\(988\) 0 0
\(989\) 7.32624 0.232961
\(990\) 0 0
\(991\) −2.67376 −0.0849349 −0.0424674 0.999098i \(-0.513522\pi\)
−0.0424674 + 0.999098i \(0.513522\pi\)
\(992\) 0 0
\(993\) −10.9443 −0.347306
\(994\) 0 0
\(995\) −25.0344 −0.793645
\(996\) 0 0
\(997\) −25.5410 −0.808892 −0.404446 0.914562i \(-0.632536\pi\)
−0.404446 + 0.914562i \(0.632536\pi\)
\(998\) 0 0
\(999\) −9.47214 −0.299685
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 7728.2.a.bg.1.1 2
4.3 odd 2 483.2.a.g.1.2 2
12.11 even 2 1449.2.a.f.1.1 2
28.27 even 2 3381.2.a.u.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.a.g.1.2 2 4.3 odd 2
1449.2.a.f.1.1 2 12.11 even 2
3381.2.a.u.1.2 2 28.27 even 2
7728.2.a.bg.1.1 2 1.1 even 1 trivial