Properties

Label 7938.2.a.bm.1.1
Level $7938$
Weight $2$
Character 7938.1
Self dual yes
Analytic conductor $63.385$
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] = [7938,2,Mod(1,7938)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7938, 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("7938.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7938 = 2 \cdot 3^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7938.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: \(63.3852491245\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 126)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.44949\) of defining polynomial
Character \(\chi\) \(=\) 7938.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.44949 q^{5} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.44949 q^{5} -1.00000 q^{8} +1.44949 q^{10} -2.00000 q^{11} -4.89898 q^{13} +1.00000 q^{16} +2.00000 q^{17} -2.55051 q^{19} -1.44949 q^{20} +2.00000 q^{22} +1.00000 q^{23} -2.89898 q^{25} +4.89898 q^{26} +6.89898 q^{29} -6.00000 q^{31} -1.00000 q^{32} -2.00000 q^{34} +11.7980 q^{37} +2.55051 q^{38} +1.44949 q^{40} +9.79796 q^{41} +6.89898 q^{43} -2.00000 q^{44} -1.00000 q^{46} +9.79796 q^{47} +2.89898 q^{50} -4.89898 q^{52} +10.8990 q^{53} +2.89898 q^{55} -6.89898 q^{58} -2.00000 q^{59} -6.55051 q^{61} +6.00000 q^{62} +1.00000 q^{64} +7.10102 q^{65} -12.8990 q^{67} +2.00000 q^{68} -0.101021 q^{71} +6.89898 q^{73} -11.7980 q^{74} -2.55051 q^{76} -1.89898 q^{79} -1.44949 q^{80} -9.79796 q^{82} +2.00000 q^{83} -2.89898 q^{85} -6.89898 q^{86} +2.00000 q^{88} -16.8990 q^{89} +1.00000 q^{92} -9.79796 q^{94} +3.69694 q^{95} -2.89898 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{8} - 2 q^{10} - 4 q^{11} + 2 q^{16} + 4 q^{17} - 10 q^{19} + 2 q^{20} + 4 q^{22} + 2 q^{23} + 4 q^{25} + 4 q^{29} - 12 q^{31} - 2 q^{32} - 4 q^{34} + 4 q^{37} + 10 q^{38} - 2 q^{40} + 4 q^{43} - 4 q^{44} - 2 q^{46} - 4 q^{50} + 12 q^{53} - 4 q^{55} - 4 q^{58} - 4 q^{59} - 18 q^{61} + 12 q^{62} + 2 q^{64} + 24 q^{65} - 16 q^{67} + 4 q^{68} - 10 q^{71} + 4 q^{73} - 4 q^{74} - 10 q^{76} + 6 q^{79} + 2 q^{80} + 4 q^{83} + 4 q^{85} - 4 q^{86} + 4 q^{88} - 24 q^{89} + 2 q^{92} - 22 q^{95} + 4 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.44949 −0.648232 −0.324116 0.946017i \(-0.605067\pi\)
−0.324116 + 0.946017i \(0.605067\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 1.44949 0.458369
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) −4.89898 −1.35873 −0.679366 0.733799i \(-0.737745\pi\)
−0.679366 + 0.733799i \(0.737745\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) −2.55051 −0.585127 −0.292564 0.956246i \(-0.594508\pi\)
−0.292564 + 0.956246i \(0.594508\pi\)
\(20\) −1.44949 −0.324116
\(21\) 0 0
\(22\) 2.00000 0.426401
\(23\) 1.00000 0.208514 0.104257 0.994550i \(-0.466753\pi\)
0.104257 + 0.994550i \(0.466753\pi\)
\(24\) 0 0
\(25\) −2.89898 −0.579796
\(26\) 4.89898 0.960769
\(27\) 0 0
\(28\) 0 0
\(29\) 6.89898 1.28111 0.640554 0.767913i \(-0.278705\pi\)
0.640554 + 0.767913i \(0.278705\pi\)
\(30\) 0 0
\(31\) −6.00000 −1.07763 −0.538816 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 0 0
\(37\) 11.7980 1.93957 0.969786 0.243956i \(-0.0784453\pi\)
0.969786 + 0.243956i \(0.0784453\pi\)
\(38\) 2.55051 0.413747
\(39\) 0 0
\(40\) 1.44949 0.229184
\(41\) 9.79796 1.53018 0.765092 0.643921i \(-0.222693\pi\)
0.765092 + 0.643921i \(0.222693\pi\)
\(42\) 0 0
\(43\) 6.89898 1.05208 0.526042 0.850458i \(-0.323675\pi\)
0.526042 + 0.850458i \(0.323675\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 9.79796 1.42918 0.714590 0.699544i \(-0.246613\pi\)
0.714590 + 0.699544i \(0.246613\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 2.89898 0.409978
\(51\) 0 0
\(52\) −4.89898 −0.679366
\(53\) 10.8990 1.49709 0.748545 0.663084i \(-0.230753\pi\)
0.748545 + 0.663084i \(0.230753\pi\)
\(54\) 0 0
\(55\) 2.89898 0.390898
\(56\) 0 0
\(57\) 0 0
\(58\) −6.89898 −0.905880
\(59\) −2.00000 −0.260378 −0.130189 0.991489i \(-0.541558\pi\)
−0.130189 + 0.991489i \(0.541558\pi\)
\(60\) 0 0
\(61\) −6.55051 −0.838707 −0.419353 0.907823i \(-0.637743\pi\)
−0.419353 + 0.907823i \(0.637743\pi\)
\(62\) 6.00000 0.762001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 7.10102 0.880773
\(66\) 0 0
\(67\) −12.8990 −1.57586 −0.787931 0.615764i \(-0.788847\pi\)
−0.787931 + 0.615764i \(0.788847\pi\)
\(68\) 2.00000 0.242536
\(69\) 0 0
\(70\) 0 0
\(71\) −0.101021 −0.0119889 −0.00599446 0.999982i \(-0.501908\pi\)
−0.00599446 + 0.999982i \(0.501908\pi\)
\(72\) 0 0
\(73\) 6.89898 0.807464 0.403732 0.914877i \(-0.367713\pi\)
0.403732 + 0.914877i \(0.367713\pi\)
\(74\) −11.7980 −1.37148
\(75\) 0 0
\(76\) −2.55051 −0.292564
\(77\) 0 0
\(78\) 0 0
\(79\) −1.89898 −0.213652 −0.106826 0.994278i \(-0.534069\pi\)
−0.106826 + 0.994278i \(0.534069\pi\)
\(80\) −1.44949 −0.162058
\(81\) 0 0
\(82\) −9.79796 −1.08200
\(83\) 2.00000 0.219529 0.109764 0.993958i \(-0.464990\pi\)
0.109764 + 0.993958i \(0.464990\pi\)
\(84\) 0 0
\(85\) −2.89898 −0.314438
\(86\) −6.89898 −0.743936
\(87\) 0 0
\(88\) 2.00000 0.213201
\(89\) −16.8990 −1.79129 −0.895644 0.444771i \(-0.853285\pi\)
−0.895644 + 0.444771i \(0.853285\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1.00000 0.104257
\(93\) 0 0
\(94\) −9.79796 −1.01058
\(95\) 3.69694 0.379298
\(96\) 0 0
\(97\) −2.89898 −0.294347 −0.147173 0.989111i \(-0.547018\pi\)
−0.147173 + 0.989111i \(0.547018\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −2.89898 −0.289898
\(101\) −17.2474 −1.71619 −0.858093 0.513495i \(-0.828351\pi\)
−0.858093 + 0.513495i \(0.828351\pi\)
\(102\) 0 0
\(103\) −14.0000 −1.37946 −0.689730 0.724066i \(-0.742271\pi\)
−0.689730 + 0.724066i \(0.742271\pi\)
\(104\) 4.89898 0.480384
\(105\) 0 0
\(106\) −10.8990 −1.05860
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) 12.6969 1.21615 0.608073 0.793881i \(-0.291943\pi\)
0.608073 + 0.793881i \(0.291943\pi\)
\(110\) −2.89898 −0.276407
\(111\) 0 0
\(112\) 0 0
\(113\) 6.10102 0.573936 0.286968 0.957940i \(-0.407353\pi\)
0.286968 + 0.957940i \(0.407353\pi\)
\(114\) 0 0
\(115\) −1.44949 −0.135166
\(116\) 6.89898 0.640554
\(117\) 0 0
\(118\) 2.00000 0.184115
\(119\) 0 0
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 6.55051 0.593055
\(123\) 0 0
\(124\) −6.00000 −0.538816
\(125\) 11.4495 1.02407
\(126\) 0 0
\(127\) −3.00000 −0.266207 −0.133103 0.991102i \(-0.542494\pi\)
−0.133103 + 0.991102i \(0.542494\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −7.10102 −0.622801
\(131\) −8.55051 −0.747062 −0.373531 0.927618i \(-0.621853\pi\)
−0.373531 + 0.927618i \(0.621853\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 12.8990 1.11430
\(135\) 0 0
\(136\) −2.00000 −0.171499
\(137\) −7.79796 −0.666225 −0.333112 0.942887i \(-0.608099\pi\)
−0.333112 + 0.942887i \(0.608099\pi\)
\(138\) 0 0
\(139\) −4.55051 −0.385969 −0.192985 0.981202i \(-0.561817\pi\)
−0.192985 + 0.981202i \(0.561817\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0.101021 0.00847745
\(143\) 9.79796 0.819346
\(144\) 0 0
\(145\) −10.0000 −0.830455
\(146\) −6.89898 −0.570964
\(147\) 0 0
\(148\) 11.7980 0.969786
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) −5.00000 −0.406894 −0.203447 0.979086i \(-0.565214\pi\)
−0.203447 + 0.979086i \(0.565214\pi\)
\(152\) 2.55051 0.206874
\(153\) 0 0
\(154\) 0 0
\(155\) 8.69694 0.698555
\(156\) 0 0
\(157\) 8.34847 0.666280 0.333140 0.942877i \(-0.391892\pi\)
0.333140 + 0.942877i \(0.391892\pi\)
\(158\) 1.89898 0.151075
\(159\) 0 0
\(160\) 1.44949 0.114592
\(161\) 0 0
\(162\) 0 0
\(163\) −19.7980 −1.55070 −0.775348 0.631534i \(-0.782425\pi\)
−0.775348 + 0.631534i \(0.782425\pi\)
\(164\) 9.79796 0.765092
\(165\) 0 0
\(166\) −2.00000 −0.155230
\(167\) −10.6969 −0.827754 −0.413877 0.910333i \(-0.635826\pi\)
−0.413877 + 0.910333i \(0.635826\pi\)
\(168\) 0 0
\(169\) 11.0000 0.846154
\(170\) 2.89898 0.222342
\(171\) 0 0
\(172\) 6.89898 0.526042
\(173\) −3.10102 −0.235766 −0.117883 0.993027i \(-0.537611\pi\)
−0.117883 + 0.993027i \(0.537611\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −2.00000 −0.150756
\(177\) 0 0
\(178\) 16.8990 1.26663
\(179\) −20.6969 −1.54696 −0.773481 0.633820i \(-0.781486\pi\)
−0.773481 + 0.633820i \(0.781486\pi\)
\(180\) 0 0
\(181\) 10.3485 0.769196 0.384598 0.923084i \(-0.374340\pi\)
0.384598 + 0.923084i \(0.374340\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −1.00000 −0.0737210
\(185\) −17.1010 −1.25729
\(186\) 0 0
\(187\) −4.00000 −0.292509
\(188\) 9.79796 0.714590
\(189\) 0 0
\(190\) −3.69694 −0.268204
\(191\) −4.10102 −0.296739 −0.148370 0.988932i \(-0.547403\pi\)
−0.148370 + 0.988932i \(0.547403\pi\)
\(192\) 0 0
\(193\) −17.8990 −1.28840 −0.644198 0.764858i \(-0.722809\pi\)
−0.644198 + 0.764858i \(0.722809\pi\)
\(194\) 2.89898 0.208135
\(195\) 0 0
\(196\) 0 0
\(197\) −16.6969 −1.18961 −0.594804 0.803871i \(-0.702770\pi\)
−0.594804 + 0.803871i \(0.702770\pi\)
\(198\) 0 0
\(199\) 2.89898 0.205503 0.102752 0.994707i \(-0.467235\pi\)
0.102752 + 0.994707i \(0.467235\pi\)
\(200\) 2.89898 0.204989
\(201\) 0 0
\(202\) 17.2474 1.21353
\(203\) 0 0
\(204\) 0 0
\(205\) −14.2020 −0.991914
\(206\) 14.0000 0.975426
\(207\) 0 0
\(208\) −4.89898 −0.339683
\(209\) 5.10102 0.352845
\(210\) 0 0
\(211\) 12.8990 0.888002 0.444001 0.896026i \(-0.353559\pi\)
0.444001 + 0.896026i \(0.353559\pi\)
\(212\) 10.8990 0.748545
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) −10.0000 −0.681994
\(216\) 0 0
\(217\) 0 0
\(218\) −12.6969 −0.859945
\(219\) 0 0
\(220\) 2.89898 0.195449
\(221\) −9.79796 −0.659082
\(222\) 0 0
\(223\) 11.1010 0.743379 0.371690 0.928357i \(-0.378779\pi\)
0.371690 + 0.928357i \(0.378779\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −6.10102 −0.405834
\(227\) −5.44949 −0.361695 −0.180848 0.983511i \(-0.557884\pi\)
−0.180848 + 0.983511i \(0.557884\pi\)
\(228\) 0 0
\(229\) −1.24745 −0.0824337 −0.0412169 0.999150i \(-0.513123\pi\)
−0.0412169 + 0.999150i \(0.513123\pi\)
\(230\) 1.44949 0.0955765
\(231\) 0 0
\(232\) −6.89898 −0.452940
\(233\) 7.00000 0.458585 0.229293 0.973358i \(-0.426359\pi\)
0.229293 + 0.973358i \(0.426359\pi\)
\(234\) 0 0
\(235\) −14.2020 −0.926439
\(236\) −2.00000 −0.130189
\(237\) 0 0
\(238\) 0 0
\(239\) −6.79796 −0.439723 −0.219862 0.975531i \(-0.570561\pi\)
−0.219862 + 0.975531i \(0.570561\pi\)
\(240\) 0 0
\(241\) −0.898979 −0.0579084 −0.0289542 0.999581i \(-0.509218\pi\)
−0.0289542 + 0.999581i \(0.509218\pi\)
\(242\) 7.00000 0.449977
\(243\) 0 0
\(244\) −6.55051 −0.419353
\(245\) 0 0
\(246\) 0 0
\(247\) 12.4949 0.795031
\(248\) 6.00000 0.381000
\(249\) 0 0
\(250\) −11.4495 −0.724129
\(251\) 17.4495 1.10140 0.550701 0.834703i \(-0.314360\pi\)
0.550701 + 0.834703i \(0.314360\pi\)
\(252\) 0 0
\(253\) −2.00000 −0.125739
\(254\) 3.00000 0.188237
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 8.20204 0.511629 0.255815 0.966726i \(-0.417656\pi\)
0.255815 + 0.966726i \(0.417656\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 7.10102 0.440387
\(261\) 0 0
\(262\) 8.55051 0.528252
\(263\) −25.8990 −1.59700 −0.798500 0.601995i \(-0.794373\pi\)
−0.798500 + 0.601995i \(0.794373\pi\)
\(264\) 0 0
\(265\) −15.7980 −0.970461
\(266\) 0 0
\(267\) 0 0
\(268\) −12.8990 −0.787931
\(269\) −18.3485 −1.11873 −0.559363 0.828923i \(-0.688954\pi\)
−0.559363 + 0.828923i \(0.688954\pi\)
\(270\) 0 0
\(271\) −7.10102 −0.431356 −0.215678 0.976465i \(-0.569196\pi\)
−0.215678 + 0.976465i \(0.569196\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) 7.79796 0.471092
\(275\) 5.79796 0.349630
\(276\) 0 0
\(277\) −18.6969 −1.12339 −0.561695 0.827344i \(-0.689851\pi\)
−0.561695 + 0.827344i \(0.689851\pi\)
\(278\) 4.55051 0.272921
\(279\) 0 0
\(280\) 0 0
\(281\) 19.0000 1.13344 0.566722 0.823909i \(-0.308211\pi\)
0.566722 + 0.823909i \(0.308211\pi\)
\(282\) 0 0
\(283\) 25.4495 1.51282 0.756408 0.654101i \(-0.226953\pi\)
0.756408 + 0.654101i \(0.226953\pi\)
\(284\) −0.101021 −0.00599446
\(285\) 0 0
\(286\) −9.79796 −0.579365
\(287\) 0 0
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 10.0000 0.587220
\(291\) 0 0
\(292\) 6.89898 0.403732
\(293\) 2.75255 0.160806 0.0804029 0.996762i \(-0.474379\pi\)
0.0804029 + 0.996762i \(0.474379\pi\)
\(294\) 0 0
\(295\) 2.89898 0.168785
\(296\) −11.7980 −0.685742
\(297\) 0 0
\(298\) −6.00000 −0.347571
\(299\) −4.89898 −0.283315
\(300\) 0 0
\(301\) 0 0
\(302\) 5.00000 0.287718
\(303\) 0 0
\(304\) −2.55051 −0.146282
\(305\) 9.49490 0.543676
\(306\) 0 0
\(307\) −25.2474 −1.44095 −0.720474 0.693482i \(-0.756076\pi\)
−0.720474 + 0.693482i \(0.756076\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −8.69694 −0.493953
\(311\) 30.6969 1.74066 0.870332 0.492466i \(-0.163904\pi\)
0.870332 + 0.492466i \(0.163904\pi\)
\(312\) 0 0
\(313\) 4.69694 0.265487 0.132743 0.991150i \(-0.457621\pi\)
0.132743 + 0.991150i \(0.457621\pi\)
\(314\) −8.34847 −0.471131
\(315\) 0 0
\(316\) −1.89898 −0.106826
\(317\) −20.6969 −1.16246 −0.581228 0.813741i \(-0.697428\pi\)
−0.581228 + 0.813741i \(0.697428\pi\)
\(318\) 0 0
\(319\) −13.7980 −0.772537
\(320\) −1.44949 −0.0810289
\(321\) 0 0
\(322\) 0 0
\(323\) −5.10102 −0.283828
\(324\) 0 0
\(325\) 14.2020 0.787787
\(326\) 19.7980 1.09651
\(327\) 0 0
\(328\) −9.79796 −0.541002
\(329\) 0 0
\(330\) 0 0
\(331\) 4.69694 0.258167 0.129084 0.991634i \(-0.458796\pi\)
0.129084 + 0.991634i \(0.458796\pi\)
\(332\) 2.00000 0.109764
\(333\) 0 0
\(334\) 10.6969 0.585310
\(335\) 18.6969 1.02152
\(336\) 0 0
\(337\) −23.3939 −1.27435 −0.637173 0.770721i \(-0.719896\pi\)
−0.637173 + 0.770721i \(0.719896\pi\)
\(338\) −11.0000 −0.598321
\(339\) 0 0
\(340\) −2.89898 −0.157219
\(341\) 12.0000 0.649836
\(342\) 0 0
\(343\) 0 0
\(344\) −6.89898 −0.371968
\(345\) 0 0
\(346\) 3.10102 0.166712
\(347\) −19.5959 −1.05196 −0.525982 0.850496i \(-0.676302\pi\)
−0.525982 + 0.850496i \(0.676302\pi\)
\(348\) 0 0
\(349\) −11.1010 −0.594224 −0.297112 0.954843i \(-0.596023\pi\)
−0.297112 + 0.954843i \(0.596023\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 2.00000 0.106600
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) 0.146428 0.00777160
\(356\) −16.8990 −0.895644
\(357\) 0 0
\(358\) 20.6969 1.09387
\(359\) 8.79796 0.464339 0.232169 0.972675i \(-0.425418\pi\)
0.232169 + 0.972675i \(0.425418\pi\)
\(360\) 0 0
\(361\) −12.4949 −0.657626
\(362\) −10.3485 −0.543903
\(363\) 0 0
\(364\) 0 0
\(365\) −10.0000 −0.523424
\(366\) 0 0
\(367\) 13.7980 0.720248 0.360124 0.932905i \(-0.382734\pi\)
0.360124 + 0.932905i \(0.382734\pi\)
\(368\) 1.00000 0.0521286
\(369\) 0 0
\(370\) 17.1010 0.889040
\(371\) 0 0
\(372\) 0 0
\(373\) −6.89898 −0.357216 −0.178608 0.983920i \(-0.557159\pi\)
−0.178608 + 0.983920i \(0.557159\pi\)
\(374\) 4.00000 0.206835
\(375\) 0 0
\(376\) −9.79796 −0.505291
\(377\) −33.7980 −1.74068
\(378\) 0 0
\(379\) 22.4949 1.15549 0.577743 0.816219i \(-0.303934\pi\)
0.577743 + 0.816219i \(0.303934\pi\)
\(380\) 3.69694 0.189649
\(381\) 0 0
\(382\) 4.10102 0.209826
\(383\) −2.89898 −0.148131 −0.0740655 0.997253i \(-0.523597\pi\)
−0.0740655 + 0.997253i \(0.523597\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 17.8990 0.911034
\(387\) 0 0
\(388\) −2.89898 −0.147173
\(389\) 24.8990 1.26243 0.631214 0.775609i \(-0.282557\pi\)
0.631214 + 0.775609i \(0.282557\pi\)
\(390\) 0 0
\(391\) 2.00000 0.101144
\(392\) 0 0
\(393\) 0 0
\(394\) 16.6969 0.841180
\(395\) 2.75255 0.138496
\(396\) 0 0
\(397\) −38.6969 −1.94214 −0.971072 0.238788i \(-0.923250\pi\)
−0.971072 + 0.238788i \(0.923250\pi\)
\(398\) −2.89898 −0.145313
\(399\) 0 0
\(400\) −2.89898 −0.144949
\(401\) 19.8990 0.993708 0.496854 0.867834i \(-0.334489\pi\)
0.496854 + 0.867834i \(0.334489\pi\)
\(402\) 0 0
\(403\) 29.3939 1.46421
\(404\) −17.2474 −0.858093
\(405\) 0 0
\(406\) 0 0
\(407\) −23.5959 −1.16961
\(408\) 0 0
\(409\) 13.7980 0.682265 0.341133 0.940015i \(-0.389189\pi\)
0.341133 + 0.940015i \(0.389189\pi\)
\(410\) 14.2020 0.701389
\(411\) 0 0
\(412\) −14.0000 −0.689730
\(413\) 0 0
\(414\) 0 0
\(415\) −2.89898 −0.142305
\(416\) 4.89898 0.240192
\(417\) 0 0
\(418\) −5.10102 −0.249499
\(419\) 29.4495 1.43870 0.719351 0.694647i \(-0.244439\pi\)
0.719351 + 0.694647i \(0.244439\pi\)
\(420\) 0 0
\(421\) 22.8990 1.11603 0.558014 0.829832i \(-0.311564\pi\)
0.558014 + 0.829832i \(0.311564\pi\)
\(422\) −12.8990 −0.627912
\(423\) 0 0
\(424\) −10.8990 −0.529301
\(425\) −5.79796 −0.281242
\(426\) 0 0
\(427\) 0 0
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) 10.0000 0.482243
\(431\) −31.5959 −1.52192 −0.760961 0.648798i \(-0.775272\pi\)
−0.760961 + 0.648798i \(0.775272\pi\)
\(432\) 0 0
\(433\) 7.79796 0.374746 0.187373 0.982289i \(-0.440003\pi\)
0.187373 + 0.982289i \(0.440003\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 12.6969 0.608073
\(437\) −2.55051 −0.122007
\(438\) 0 0
\(439\) −2.20204 −0.105098 −0.0525488 0.998618i \(-0.516735\pi\)
−0.0525488 + 0.998618i \(0.516735\pi\)
\(440\) −2.89898 −0.138203
\(441\) 0 0
\(442\) 9.79796 0.466041
\(443\) 14.8990 0.707872 0.353936 0.935270i \(-0.384843\pi\)
0.353936 + 0.935270i \(0.384843\pi\)
\(444\) 0 0
\(445\) 24.4949 1.16117
\(446\) −11.1010 −0.525649
\(447\) 0 0
\(448\) 0 0
\(449\) −20.5959 −0.971981 −0.485991 0.873964i \(-0.661541\pi\)
−0.485991 + 0.873964i \(0.661541\pi\)
\(450\) 0 0
\(451\) −19.5959 −0.922736
\(452\) 6.10102 0.286968
\(453\) 0 0
\(454\) 5.44949 0.255757
\(455\) 0 0
\(456\) 0 0
\(457\) −17.4949 −0.818377 −0.409188 0.912450i \(-0.634188\pi\)
−0.409188 + 0.912450i \(0.634188\pi\)
\(458\) 1.24745 0.0582895
\(459\) 0 0
\(460\) −1.44949 −0.0675828
\(461\) 5.65153 0.263218 0.131609 0.991302i \(-0.457986\pi\)
0.131609 + 0.991302i \(0.457986\pi\)
\(462\) 0 0
\(463\) 3.69694 0.171811 0.0859057 0.996303i \(-0.472622\pi\)
0.0859057 + 0.996303i \(0.472622\pi\)
\(464\) 6.89898 0.320277
\(465\) 0 0
\(466\) −7.00000 −0.324269
\(467\) 10.0000 0.462745 0.231372 0.972865i \(-0.425678\pi\)
0.231372 + 0.972865i \(0.425678\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 14.2020 0.655091
\(471\) 0 0
\(472\) 2.00000 0.0920575
\(473\) −13.7980 −0.634431
\(474\) 0 0
\(475\) 7.39388 0.339254
\(476\) 0 0
\(477\) 0 0
\(478\) 6.79796 0.310931
\(479\) 9.59592 0.438449 0.219224 0.975674i \(-0.429647\pi\)
0.219224 + 0.975674i \(0.429647\pi\)
\(480\) 0 0
\(481\) −57.7980 −2.63536
\(482\) 0.898979 0.0409474
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) 4.20204 0.190805
\(486\) 0 0
\(487\) −36.3939 −1.64916 −0.824582 0.565742i \(-0.808590\pi\)
−0.824582 + 0.565742i \(0.808590\pi\)
\(488\) 6.55051 0.296528
\(489\) 0 0
\(490\) 0 0
\(491\) −15.7980 −0.712952 −0.356476 0.934304i \(-0.616022\pi\)
−0.356476 + 0.934304i \(0.616022\pi\)
\(492\) 0 0
\(493\) 13.7980 0.621429
\(494\) −12.4949 −0.562172
\(495\) 0 0
\(496\) −6.00000 −0.269408
\(497\) 0 0
\(498\) 0 0
\(499\) −25.3939 −1.13679 −0.568393 0.822757i \(-0.692435\pi\)
−0.568393 + 0.822757i \(0.692435\pi\)
\(500\) 11.4495 0.512037
\(501\) 0 0
\(502\) −17.4495 −0.778809
\(503\) −24.4949 −1.09217 −0.546087 0.837729i \(-0.683883\pi\)
−0.546087 + 0.837729i \(0.683883\pi\)
\(504\) 0 0
\(505\) 25.0000 1.11249
\(506\) 2.00000 0.0889108
\(507\) 0 0
\(508\) −3.00000 −0.133103
\(509\) −7.10102 −0.314747 −0.157374 0.987539i \(-0.550303\pi\)
−0.157374 + 0.987539i \(0.550303\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −8.20204 −0.361777
\(515\) 20.2929 0.894210
\(516\) 0 0
\(517\) −19.5959 −0.861827
\(518\) 0 0
\(519\) 0 0
\(520\) −7.10102 −0.311400
\(521\) −9.30306 −0.407575 −0.203787 0.979015i \(-0.565325\pi\)
−0.203787 + 0.979015i \(0.565325\pi\)
\(522\) 0 0
\(523\) 14.3485 0.627415 0.313707 0.949520i \(-0.398429\pi\)
0.313707 + 0.949520i \(0.398429\pi\)
\(524\) −8.55051 −0.373531
\(525\) 0 0
\(526\) 25.8990 1.12925
\(527\) −12.0000 −0.522728
\(528\) 0 0
\(529\) −22.0000 −0.956522
\(530\) 15.7980 0.686219
\(531\) 0 0
\(532\) 0 0
\(533\) −48.0000 −2.07911
\(534\) 0 0
\(535\) −17.3939 −0.752003
\(536\) 12.8990 0.557151
\(537\) 0 0
\(538\) 18.3485 0.791059
\(539\) 0 0
\(540\) 0 0
\(541\) −18.4949 −0.795158 −0.397579 0.917568i \(-0.630149\pi\)
−0.397579 + 0.917568i \(0.630149\pi\)
\(542\) 7.10102 0.305015
\(543\) 0 0
\(544\) −2.00000 −0.0857493
\(545\) −18.4041 −0.788344
\(546\) 0 0
\(547\) −7.59592 −0.324778 −0.162389 0.986727i \(-0.551920\pi\)
−0.162389 + 0.986727i \(0.551920\pi\)
\(548\) −7.79796 −0.333112
\(549\) 0 0
\(550\) −5.79796 −0.247226
\(551\) −17.5959 −0.749611
\(552\) 0 0
\(553\) 0 0
\(554\) 18.6969 0.794357
\(555\) 0 0
\(556\) −4.55051 −0.192985
\(557\) 12.8990 0.546547 0.273274 0.961936i \(-0.411894\pi\)
0.273274 + 0.961936i \(0.411894\pi\)
\(558\) 0 0
\(559\) −33.7980 −1.42950
\(560\) 0 0
\(561\) 0 0
\(562\) −19.0000 −0.801467
\(563\) −39.9444 −1.68346 −0.841728 0.539902i \(-0.818461\pi\)
−0.841728 + 0.539902i \(0.818461\pi\)
\(564\) 0 0
\(565\) −8.84337 −0.372043
\(566\) −25.4495 −1.06972
\(567\) 0 0
\(568\) 0.101021 0.00423873
\(569\) 30.0000 1.25767 0.628833 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 0 0
\(571\) 33.7980 1.41440 0.707200 0.707013i \(-0.249958\pi\)
0.707200 + 0.707013i \(0.249958\pi\)
\(572\) 9.79796 0.409673
\(573\) 0 0
\(574\) 0 0
\(575\) −2.89898 −0.120896
\(576\) 0 0
\(577\) 15.5959 0.649267 0.324633 0.945840i \(-0.394759\pi\)
0.324633 + 0.945840i \(0.394759\pi\)
\(578\) 13.0000 0.540729
\(579\) 0 0
\(580\) −10.0000 −0.415227
\(581\) 0 0
\(582\) 0 0
\(583\) −21.7980 −0.902779
\(584\) −6.89898 −0.285482
\(585\) 0 0
\(586\) −2.75255 −0.113707
\(587\) 16.1464 0.666434 0.333217 0.942850i \(-0.391866\pi\)
0.333217 + 0.942850i \(0.391866\pi\)
\(588\) 0 0
\(589\) 15.3031 0.630552
\(590\) −2.89898 −0.119349
\(591\) 0 0
\(592\) 11.7980 0.484893
\(593\) 14.6969 0.603531 0.301765 0.953382i \(-0.402424\pi\)
0.301765 + 0.953382i \(0.402424\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) 0 0
\(598\) 4.89898 0.200334
\(599\) 33.7980 1.38095 0.690474 0.723358i \(-0.257402\pi\)
0.690474 + 0.723358i \(0.257402\pi\)
\(600\) 0 0
\(601\) −16.6969 −0.681082 −0.340541 0.940230i \(-0.610610\pi\)
−0.340541 + 0.940230i \(0.610610\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −5.00000 −0.203447
\(605\) 10.1464 0.412511
\(606\) 0 0
\(607\) −20.6969 −0.840063 −0.420031 0.907510i \(-0.637981\pi\)
−0.420031 + 0.907510i \(0.637981\pi\)
\(608\) 2.55051 0.103437
\(609\) 0 0
\(610\) −9.49490 −0.384437
\(611\) −48.0000 −1.94187
\(612\) 0 0
\(613\) −14.6969 −0.593604 −0.296802 0.954939i \(-0.595920\pi\)
−0.296802 + 0.954939i \(0.595920\pi\)
\(614\) 25.2474 1.01890
\(615\) 0 0
\(616\) 0 0
\(617\) 15.3939 0.619734 0.309867 0.950780i \(-0.399715\pi\)
0.309867 + 0.950780i \(0.399715\pi\)
\(618\) 0 0
\(619\) −30.1464 −1.21169 −0.605844 0.795584i \(-0.707164\pi\)
−0.605844 + 0.795584i \(0.707164\pi\)
\(620\) 8.69694 0.349277
\(621\) 0 0
\(622\) −30.6969 −1.23084
\(623\) 0 0
\(624\) 0 0
\(625\) −2.10102 −0.0840408
\(626\) −4.69694 −0.187727
\(627\) 0 0
\(628\) 8.34847 0.333140
\(629\) 23.5959 0.940831
\(630\) 0 0
\(631\) 27.8990 1.11064 0.555320 0.831636i \(-0.312596\pi\)
0.555320 + 0.831636i \(0.312596\pi\)
\(632\) 1.89898 0.0755373
\(633\) 0 0
\(634\) 20.6969 0.821980
\(635\) 4.34847 0.172564
\(636\) 0 0
\(637\) 0 0
\(638\) 13.7980 0.546266
\(639\) 0 0
\(640\) 1.44949 0.0572961
\(641\) 7.49490 0.296031 0.148015 0.988985i \(-0.452712\pi\)
0.148015 + 0.988985i \(0.452712\pi\)
\(642\) 0 0
\(643\) 39.3939 1.55354 0.776771 0.629783i \(-0.216856\pi\)
0.776771 + 0.629783i \(0.216856\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 5.10102 0.200697
\(647\) −50.6969 −1.99310 −0.996551 0.0829807i \(-0.973556\pi\)
−0.996551 + 0.0829807i \(0.973556\pi\)
\(648\) 0 0
\(649\) 4.00000 0.157014
\(650\) −14.2020 −0.557050
\(651\) 0 0
\(652\) −19.7980 −0.775348
\(653\) −9.79796 −0.383424 −0.191712 0.981451i \(-0.561404\pi\)
−0.191712 + 0.981451i \(0.561404\pi\)
\(654\) 0 0
\(655\) 12.3939 0.484269
\(656\) 9.79796 0.382546
\(657\) 0 0
\(658\) 0 0
\(659\) 24.6969 0.962056 0.481028 0.876705i \(-0.340264\pi\)
0.481028 + 0.876705i \(0.340264\pi\)
\(660\) 0 0
\(661\) −4.55051 −0.176994 −0.0884972 0.996076i \(-0.528206\pi\)
−0.0884972 + 0.996076i \(0.528206\pi\)
\(662\) −4.69694 −0.182552
\(663\) 0 0
\(664\) −2.00000 −0.0776151
\(665\) 0 0
\(666\) 0 0
\(667\) 6.89898 0.267130
\(668\) −10.6969 −0.413877
\(669\) 0 0
\(670\) −18.6969 −0.722326
\(671\) 13.1010 0.505759
\(672\) 0 0
\(673\) −8.59592 −0.331348 −0.165674 0.986181i \(-0.552980\pi\)
−0.165674 + 0.986181i \(0.552980\pi\)
\(674\) 23.3939 0.901098
\(675\) 0 0
\(676\) 11.0000 0.423077
\(677\) 14.6969 0.564849 0.282425 0.959289i \(-0.408861\pi\)
0.282425 + 0.959289i \(0.408861\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 2.89898 0.111171
\(681\) 0 0
\(682\) −12.0000 −0.459504
\(683\) −51.7980 −1.98199 −0.990997 0.133885i \(-0.957255\pi\)
−0.990997 + 0.133885i \(0.957255\pi\)
\(684\) 0 0
\(685\) 11.3031 0.431868
\(686\) 0 0
\(687\) 0 0
\(688\) 6.89898 0.263021
\(689\) −53.3939 −2.03414
\(690\) 0 0
\(691\) −51.0454 −1.94186 −0.970929 0.239366i \(-0.923060\pi\)
−0.970929 + 0.239366i \(0.923060\pi\)
\(692\) −3.10102 −0.117883
\(693\) 0 0
\(694\) 19.5959 0.743851
\(695\) 6.59592 0.250197
\(696\) 0 0
\(697\) 19.5959 0.742248
\(698\) 11.1010 0.420180
\(699\) 0 0
\(700\) 0 0
\(701\) 7.39388 0.279263 0.139631 0.990204i \(-0.455408\pi\)
0.139631 + 0.990204i \(0.455408\pi\)
\(702\) 0 0
\(703\) −30.0908 −1.13490
\(704\) −2.00000 −0.0753778
\(705\) 0 0
\(706\) 6.00000 0.225813
\(707\) 0 0
\(708\) 0 0
\(709\) 27.5959 1.03639 0.518193 0.855264i \(-0.326605\pi\)
0.518193 + 0.855264i \(0.326605\pi\)
\(710\) −0.146428 −0.00549535
\(711\) 0 0
\(712\) 16.8990 0.633316
\(713\) −6.00000 −0.224702
\(714\) 0 0
\(715\) −14.2020 −0.531126
\(716\) −20.6969 −0.773481
\(717\) 0 0
\(718\) −8.79796 −0.328337
\(719\) 9.79796 0.365402 0.182701 0.983169i \(-0.441516\pi\)
0.182701 + 0.983169i \(0.441516\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 12.4949 0.465012
\(723\) 0 0
\(724\) 10.3485 0.384598
\(725\) −20.0000 −0.742781
\(726\) 0 0
\(727\) 8.49490 0.315058 0.157529 0.987514i \(-0.449647\pi\)
0.157529 + 0.987514i \(0.449647\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 10.0000 0.370117
\(731\) 13.7980 0.510336
\(732\) 0 0
\(733\) 17.4495 0.644512 0.322256 0.946653i \(-0.395559\pi\)
0.322256 + 0.946653i \(0.395559\pi\)
\(734\) −13.7980 −0.509292
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 25.7980 0.950280
\(738\) 0 0
\(739\) 13.5959 0.500134 0.250067 0.968229i \(-0.419547\pi\)
0.250067 + 0.968229i \(0.419547\pi\)
\(740\) −17.1010 −0.628646
\(741\) 0 0
\(742\) 0 0
\(743\) −36.0000 −1.32071 −0.660356 0.750953i \(-0.729595\pi\)
−0.660356 + 0.750953i \(0.729595\pi\)
\(744\) 0 0
\(745\) −8.69694 −0.318631
\(746\) 6.89898 0.252590
\(747\) 0 0
\(748\) −4.00000 −0.146254
\(749\) 0 0
\(750\) 0 0
\(751\) 1.40408 0.0512357 0.0256178 0.999672i \(-0.491845\pi\)
0.0256178 + 0.999672i \(0.491845\pi\)
\(752\) 9.79796 0.357295
\(753\) 0 0
\(754\) 33.7980 1.23085
\(755\) 7.24745 0.263762
\(756\) 0 0
\(757\) −35.3939 −1.28641 −0.643206 0.765693i \(-0.722396\pi\)
−0.643206 + 0.765693i \(0.722396\pi\)
\(758\) −22.4949 −0.817051
\(759\) 0 0
\(760\) −3.69694 −0.134102
\(761\) 2.00000 0.0724999 0.0362500 0.999343i \(-0.488459\pi\)
0.0362500 + 0.999343i \(0.488459\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −4.10102 −0.148370
\(765\) 0 0
\(766\) 2.89898 0.104744
\(767\) 9.79796 0.353784
\(768\) 0 0
\(769\) 34.0908 1.22935 0.614673 0.788782i \(-0.289288\pi\)
0.614673 + 0.788782i \(0.289288\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −17.8990 −0.644198
\(773\) 33.9444 1.22089 0.610447 0.792057i \(-0.290990\pi\)
0.610447 + 0.792057i \(0.290990\pi\)
\(774\) 0 0
\(775\) 17.3939 0.624807
\(776\) 2.89898 0.104067
\(777\) 0 0
\(778\) −24.8990 −0.892672
\(779\) −24.9898 −0.895352
\(780\) 0 0
\(781\) 0.202041 0.00722960
\(782\) −2.00000 −0.0715199
\(783\) 0 0
\(784\) 0 0
\(785\) −12.1010 −0.431904
\(786\) 0 0
\(787\) 11.3939 0.406148 0.203074 0.979163i \(-0.434907\pi\)
0.203074 + 0.979163i \(0.434907\pi\)
\(788\) −16.6969 −0.594804
\(789\) 0 0
\(790\) −2.75255 −0.0979314
\(791\) 0 0
\(792\) 0 0
\(793\) 32.0908 1.13958
\(794\) 38.6969 1.37330
\(795\) 0 0
\(796\) 2.89898 0.102752
\(797\) −17.9444 −0.635623 −0.317811 0.948154i \(-0.602948\pi\)
−0.317811 + 0.948154i \(0.602948\pi\)
\(798\) 0 0
\(799\) 19.5959 0.693254
\(800\) 2.89898 0.102494
\(801\) 0 0
\(802\) −19.8990 −0.702657
\(803\) −13.7980 −0.486919
\(804\) 0 0
\(805\) 0 0
\(806\) −29.3939 −1.03536
\(807\) 0 0
\(808\) 17.2474 0.606763
\(809\) 16.2020 0.569633 0.284817 0.958582i \(-0.408067\pi\)
0.284817 + 0.958582i \(0.408067\pi\)
\(810\) 0 0
\(811\) 2.00000 0.0702295 0.0351147 0.999383i \(-0.488820\pi\)
0.0351147 + 0.999383i \(0.488820\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 23.5959 0.827036
\(815\) 28.6969 1.00521
\(816\) 0 0
\(817\) −17.5959 −0.615603
\(818\) −13.7980 −0.482434
\(819\) 0 0
\(820\) −14.2020 −0.495957
\(821\) −0.404082 −0.0141026 −0.00705128 0.999975i \(-0.502245\pi\)
−0.00705128 + 0.999975i \(0.502245\pi\)
\(822\) 0 0
\(823\) −13.3939 −0.466881 −0.233441 0.972371i \(-0.574998\pi\)
−0.233441 + 0.972371i \(0.574998\pi\)
\(824\) 14.0000 0.487713
\(825\) 0 0
\(826\) 0 0
\(827\) 36.4949 1.26905 0.634526 0.772902i \(-0.281195\pi\)
0.634526 + 0.772902i \(0.281195\pi\)
\(828\) 0 0
\(829\) −1.30306 −0.0452572 −0.0226286 0.999744i \(-0.507204\pi\)
−0.0226286 + 0.999744i \(0.507204\pi\)
\(830\) 2.89898 0.100625
\(831\) 0 0
\(832\) −4.89898 −0.169842
\(833\) 0 0
\(834\) 0 0
\(835\) 15.5051 0.536576
\(836\) 5.10102 0.176422
\(837\) 0 0
\(838\) −29.4495 −1.01732
\(839\) −35.1010 −1.21182 −0.605911 0.795533i \(-0.707191\pi\)
−0.605911 + 0.795533i \(0.707191\pi\)
\(840\) 0 0
\(841\) 18.5959 0.641239
\(842\) −22.8990 −0.789151
\(843\) 0 0
\(844\) 12.8990 0.444001
\(845\) −15.9444 −0.548504
\(846\) 0 0
\(847\) 0 0
\(848\) 10.8990 0.374272
\(849\) 0 0
\(850\) 5.79796 0.198868
\(851\) 11.7980 0.404429
\(852\) 0 0
\(853\) −24.8434 −0.850621 −0.425310 0.905048i \(-0.639835\pi\)
−0.425310 + 0.905048i \(0.639835\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) −34.8990 −1.19213 −0.596063 0.802938i \(-0.703269\pi\)
−0.596063 + 0.802938i \(0.703269\pi\)
\(858\) 0 0
\(859\) 10.0000 0.341196 0.170598 0.985341i \(-0.445430\pi\)
0.170598 + 0.985341i \(0.445430\pi\)
\(860\) −10.0000 −0.340997
\(861\) 0 0
\(862\) 31.5959 1.07616
\(863\) −11.8990 −0.405046 −0.202523 0.979278i \(-0.564914\pi\)
−0.202523 + 0.979278i \(0.564914\pi\)
\(864\) 0 0
\(865\) 4.49490 0.152831
\(866\) −7.79796 −0.264985
\(867\) 0 0
\(868\) 0 0
\(869\) 3.79796 0.128837
\(870\) 0 0
\(871\) 63.1918 2.14117
\(872\) −12.6969 −0.429973
\(873\) 0 0
\(874\) 2.55051 0.0862723
\(875\) 0 0
\(876\) 0 0
\(877\) 22.4949 0.759599 0.379799 0.925069i \(-0.375993\pi\)
0.379799 + 0.925069i \(0.375993\pi\)
\(878\) 2.20204 0.0743153
\(879\) 0 0
\(880\) 2.89898 0.0977246
\(881\) 19.5959 0.660203 0.330102 0.943945i \(-0.392917\pi\)
0.330102 + 0.943945i \(0.392917\pi\)
\(882\) 0 0
\(883\) −19.7980 −0.666254 −0.333127 0.942882i \(-0.608104\pi\)
−0.333127 + 0.942882i \(0.608104\pi\)
\(884\) −9.79796 −0.329541
\(885\) 0 0
\(886\) −14.8990 −0.500541
\(887\) −14.2020 −0.476858 −0.238429 0.971160i \(-0.576632\pi\)
−0.238429 + 0.971160i \(0.576632\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −24.4949 −0.821071
\(891\) 0 0
\(892\) 11.1010 0.371690
\(893\) −24.9898 −0.836252
\(894\) 0 0
\(895\) 30.0000 1.00279
\(896\) 0 0
\(897\) 0 0
\(898\) 20.5959 0.687295
\(899\) −41.3939 −1.38056
\(900\) 0 0
\(901\) 21.7980 0.726195
\(902\) 19.5959 0.652473
\(903\) 0 0
\(904\) −6.10102 −0.202917
\(905\) −15.0000 −0.498617
\(906\) 0 0
\(907\) 2.69694 0.0895504 0.0447752 0.998997i \(-0.485743\pi\)
0.0447752 + 0.998997i \(0.485743\pi\)
\(908\) −5.44949 −0.180848
\(909\) 0 0
\(910\) 0 0
\(911\) −51.9898 −1.72250 −0.861249 0.508183i \(-0.830318\pi\)
−0.861249 + 0.508183i \(0.830318\pi\)
\(912\) 0 0
\(913\) −4.00000 −0.132381
\(914\) 17.4949 0.578680
\(915\) 0 0
\(916\) −1.24745 −0.0412169
\(917\) 0 0
\(918\) 0 0
\(919\) −25.6969 −0.847664 −0.423832 0.905741i \(-0.639315\pi\)
−0.423832 + 0.905741i \(0.639315\pi\)
\(920\) 1.44949 0.0477883
\(921\) 0 0
\(922\) −5.65153 −0.186123
\(923\) 0.494897 0.0162897
\(924\) 0 0
\(925\) −34.2020 −1.12456
\(926\) −3.69694 −0.121489
\(927\) 0 0
\(928\) −6.89898 −0.226470
\(929\) 34.2929 1.12511 0.562556 0.826759i \(-0.309818\pi\)
0.562556 + 0.826759i \(0.309818\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 7.00000 0.229293
\(933\) 0 0
\(934\) −10.0000 −0.327210
\(935\) 5.79796 0.189614
\(936\) 0 0
\(937\) −45.5959 −1.48955 −0.744777 0.667314i \(-0.767444\pi\)
−0.744777 + 0.667314i \(0.767444\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −14.2020 −0.463220
\(941\) −1.44949 −0.0472520 −0.0236260 0.999721i \(-0.507521\pi\)
−0.0236260 + 0.999721i \(0.507521\pi\)
\(942\) 0 0
\(943\) 9.79796 0.319065
\(944\) −2.00000 −0.0650945
\(945\) 0 0
\(946\) 13.7980 0.448610
\(947\) −52.4949 −1.70585 −0.852927 0.522029i \(-0.825175\pi\)
−0.852927 + 0.522029i \(0.825175\pi\)
\(948\) 0 0
\(949\) −33.7980 −1.09713
\(950\) −7.39388 −0.239889
\(951\) 0 0
\(952\) 0 0
\(953\) 3.39388 0.109938 0.0549692 0.998488i \(-0.482494\pi\)
0.0549692 + 0.998488i \(0.482494\pi\)
\(954\) 0 0
\(955\) 5.94439 0.192356
\(956\) −6.79796 −0.219862
\(957\) 0 0
\(958\) −9.59592 −0.310030
\(959\) 0 0
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 57.7980 1.86348
\(963\) 0 0
\(964\) −0.898979 −0.0289542
\(965\) 25.9444 0.835179
\(966\) 0 0
\(967\) 24.5959 0.790951 0.395476 0.918476i \(-0.370580\pi\)
0.395476 + 0.918476i \(0.370580\pi\)
\(968\) 7.00000 0.224989
\(969\) 0 0
\(970\) −4.20204 −0.134919
\(971\) −0.0556128 −0.00178470 −0.000892350 1.00000i \(-0.500284\pi\)
−0.000892350 1.00000i \(0.500284\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 36.3939 1.16614
\(975\) 0 0
\(976\) −6.55051 −0.209677
\(977\) 37.5959 1.20280 0.601400 0.798948i \(-0.294610\pi\)
0.601400 + 0.798948i \(0.294610\pi\)
\(978\) 0 0
\(979\) 33.7980 1.08019
\(980\) 0 0
\(981\) 0 0
\(982\) 15.7980 0.504133
\(983\) −33.1918 −1.05866 −0.529328 0.848418i \(-0.677556\pi\)
−0.529328 + 0.848418i \(0.677556\pi\)
\(984\) 0 0
\(985\) 24.2020 0.771141
\(986\) −13.7980 −0.439417
\(987\) 0 0
\(988\) 12.4949 0.397516
\(989\) 6.89898 0.219375
\(990\) 0 0
\(991\) −1.79796 −0.0571140 −0.0285570 0.999592i \(-0.509091\pi\)
−0.0285570 + 0.999592i \(0.509091\pi\)
\(992\) 6.00000 0.190500
\(993\) 0 0
\(994\) 0 0
\(995\) −4.20204 −0.133214
\(996\) 0 0
\(997\) −52.1464 −1.65149 −0.825747 0.564041i \(-0.809246\pi\)
−0.825747 + 0.564041i \(0.809246\pi\)
\(998\) 25.3939 0.803829
\(999\) 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 7938.2.a.bm.1.1 2
3.2 odd 2 7938.2.a.bn.1.2 2
7.6 odd 2 1134.2.a.i.1.2 2
9.2 odd 6 882.2.f.j.589.2 4
9.4 even 3 2646.2.f.k.883.2 4
9.5 odd 6 882.2.f.j.295.1 4
9.7 even 3 2646.2.f.k.1765.2 4
21.20 even 2 1134.2.a.p.1.1 2
28.27 even 2 9072.2.a.bd.1.2 2
63.2 odd 6 882.2.h.l.67.1 4
63.4 even 3 2646.2.h.n.667.1 4
63.5 even 6 882.2.e.m.655.1 4
63.11 odd 6 882.2.e.n.373.2 4
63.13 odd 6 378.2.f.d.127.1 4
63.16 even 3 2646.2.h.n.361.1 4
63.20 even 6 126.2.f.c.85.1 yes 4
63.23 odd 6 882.2.e.n.655.2 4
63.25 even 3 2646.2.e.k.1549.2 4
63.31 odd 6 2646.2.h.m.667.2 4
63.32 odd 6 882.2.h.l.79.1 4
63.34 odd 6 378.2.f.d.253.1 4
63.38 even 6 882.2.e.m.373.1 4
63.40 odd 6 2646.2.e.l.2125.1 4
63.41 even 6 126.2.f.c.43.2 4
63.47 even 6 882.2.h.k.67.2 4
63.52 odd 6 2646.2.e.l.1549.1 4
63.58 even 3 2646.2.e.k.2125.2 4
63.59 even 6 882.2.h.k.79.2 4
63.61 odd 6 2646.2.h.m.361.2 4
84.83 odd 2 9072.2.a.bk.1.1 2
252.83 odd 6 1008.2.r.e.337.2 4
252.139 even 6 3024.2.r.e.2017.1 4
252.167 odd 6 1008.2.r.e.673.1 4
252.223 even 6 3024.2.r.e.1009.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.2.f.c.43.2 4 63.41 even 6
126.2.f.c.85.1 yes 4 63.20 even 6
378.2.f.d.127.1 4 63.13 odd 6
378.2.f.d.253.1 4 63.34 odd 6
882.2.e.m.373.1 4 63.38 even 6
882.2.e.m.655.1 4 63.5 even 6
882.2.e.n.373.2 4 63.11 odd 6
882.2.e.n.655.2 4 63.23 odd 6
882.2.f.j.295.1 4 9.5 odd 6
882.2.f.j.589.2 4 9.2 odd 6
882.2.h.k.67.2 4 63.47 even 6
882.2.h.k.79.2 4 63.59 even 6
882.2.h.l.67.1 4 63.2 odd 6
882.2.h.l.79.1 4 63.32 odd 6
1008.2.r.e.337.2 4 252.83 odd 6
1008.2.r.e.673.1 4 252.167 odd 6
1134.2.a.i.1.2 2 7.6 odd 2
1134.2.a.p.1.1 2 21.20 even 2
2646.2.e.k.1549.2 4 63.25 even 3
2646.2.e.k.2125.2 4 63.58 even 3
2646.2.e.l.1549.1 4 63.52 odd 6
2646.2.e.l.2125.1 4 63.40 odd 6
2646.2.f.k.883.2 4 9.4 even 3
2646.2.f.k.1765.2 4 9.7 even 3
2646.2.h.m.361.2 4 63.61 odd 6
2646.2.h.m.667.2 4 63.31 odd 6
2646.2.h.n.361.1 4 63.16 even 3
2646.2.h.n.667.1 4 63.4 even 3
3024.2.r.e.1009.1 4 252.223 even 6
3024.2.r.e.2017.1 4 252.139 even 6
7938.2.a.bm.1.1 2 1.1 even 1 trivial
7938.2.a.bn.1.2 2 3.2 odd 2
9072.2.a.bd.1.2 2 28.27 even 2
9072.2.a.bk.1.1 2 84.83 odd 2