Properties

Label 9702.2.a.da.1.2
Level $9702$
Weight $2$
Character 9702.1
Self dual yes
Analytic conductor $77.471$
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] = [9702,2,Mod(1,9702)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9702, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("9702.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9702.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: \(77.4708600410\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1386)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 9702.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +0.414214 q^{5} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +0.414214 q^{5} +1.00000 q^{8} +0.414214 q^{10} +1.00000 q^{11} +1.17157 q^{13} +1.00000 q^{16} +2.17157 q^{17} -0.828427 q^{19} +0.414214 q^{20} +1.00000 q^{22} -3.24264 q^{23} -4.82843 q^{25} +1.17157 q^{26} +2.82843 q^{29} +6.48528 q^{31} +1.00000 q^{32} +2.17157 q^{34} +9.65685 q^{37} -0.828427 q^{38} +0.414214 q^{40} -4.65685 q^{41} -2.82843 q^{43} +1.00000 q^{44} -3.24264 q^{46} +9.24264 q^{47} -4.82843 q^{50} +1.17157 q^{52} -5.17157 q^{53} +0.414214 q^{55} +2.82843 q^{58} +3.65685 q^{59} +1.58579 q^{61} +6.48528 q^{62} +1.00000 q^{64} +0.485281 q^{65} -13.4853 q^{67} +2.17157 q^{68} +13.3137 q^{71} +4.82843 q^{73} +9.65685 q^{74} -0.828427 q^{76} +4.75736 q^{79} +0.414214 q^{80} -4.65685 q^{82} +9.82843 q^{83} +0.899495 q^{85} -2.82843 q^{86} +1.00000 q^{88} -12.4853 q^{89} -3.24264 q^{92} +9.24264 q^{94} -0.343146 q^{95} +10.1716 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{8} - 2 q^{10} + 2 q^{11} + 8 q^{13} + 2 q^{16} + 10 q^{17} + 4 q^{19} - 2 q^{20} + 2 q^{22} + 2 q^{23} - 4 q^{25} + 8 q^{26} - 4 q^{31} + 2 q^{32} + 10 q^{34} + 8 q^{37} + 4 q^{38} - 2 q^{40} + 2 q^{41} + 2 q^{44} + 2 q^{46} + 10 q^{47} - 4 q^{50} + 8 q^{52} - 16 q^{53} - 2 q^{55} - 4 q^{59} + 6 q^{61} - 4 q^{62} + 2 q^{64} - 16 q^{65} - 10 q^{67} + 10 q^{68} + 4 q^{71} + 4 q^{73} + 8 q^{74} + 4 q^{76} + 18 q^{79} - 2 q^{80} + 2 q^{82} + 14 q^{83} - 18 q^{85} + 2 q^{88} - 8 q^{89} + 2 q^{92} + 10 q^{94} - 12 q^{95} + 26 q^{97}+O(q^{100}) \) 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
\(5\) 0.414214 0.185242 0.0926210 0.995701i \(-0.470476\pi\)
0.0926210 + 0.995701i \(0.470476\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 0.414214 0.130986
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 1.17157 0.324936 0.162468 0.986714i \(-0.448055\pi\)
0.162468 + 0.986714i \(0.448055\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.17157 0.526684 0.263342 0.964703i \(-0.415175\pi\)
0.263342 + 0.964703i \(0.415175\pi\)
\(18\) 0 0
\(19\) −0.828427 −0.190054 −0.0950271 0.995475i \(-0.530294\pi\)
−0.0950271 + 0.995475i \(0.530294\pi\)
\(20\) 0.414214 0.0926210
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) −3.24264 −0.676137 −0.338069 0.941121i \(-0.609774\pi\)
−0.338069 + 0.941121i \(0.609774\pi\)
\(24\) 0 0
\(25\) −4.82843 −0.965685
\(26\) 1.17157 0.229764
\(27\) 0 0
\(28\) 0 0
\(29\) 2.82843 0.525226 0.262613 0.964901i \(-0.415416\pi\)
0.262613 + 0.964901i \(0.415416\pi\)
\(30\) 0 0
\(31\) 6.48528 1.16479 0.582395 0.812906i \(-0.302116\pi\)
0.582395 + 0.812906i \(0.302116\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 2.17157 0.372422
\(35\) 0 0
\(36\) 0 0
\(37\) 9.65685 1.58758 0.793789 0.608194i \(-0.208106\pi\)
0.793789 + 0.608194i \(0.208106\pi\)
\(38\) −0.828427 −0.134389
\(39\) 0 0
\(40\) 0.414214 0.0654929
\(41\) −4.65685 −0.727278 −0.363639 0.931540i \(-0.618466\pi\)
−0.363639 + 0.931540i \(0.618466\pi\)
\(42\) 0 0
\(43\) −2.82843 −0.431331 −0.215666 0.976467i \(-0.569192\pi\)
−0.215666 + 0.976467i \(0.569192\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) −3.24264 −0.478101
\(47\) 9.24264 1.34818 0.674089 0.738650i \(-0.264536\pi\)
0.674089 + 0.738650i \(0.264536\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −4.82843 −0.682843
\(51\) 0 0
\(52\) 1.17157 0.162468
\(53\) −5.17157 −0.710370 −0.355185 0.934796i \(-0.615582\pi\)
−0.355185 + 0.934796i \(0.615582\pi\)
\(54\) 0 0
\(55\) 0.414214 0.0558525
\(56\) 0 0
\(57\) 0 0
\(58\) 2.82843 0.371391
\(59\) 3.65685 0.476082 0.238041 0.971255i \(-0.423495\pi\)
0.238041 + 0.971255i \(0.423495\pi\)
\(60\) 0 0
\(61\) 1.58579 0.203039 0.101520 0.994834i \(-0.467630\pi\)
0.101520 + 0.994834i \(0.467630\pi\)
\(62\) 6.48528 0.823632
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0.485281 0.0601917
\(66\) 0 0
\(67\) −13.4853 −1.64749 −0.823745 0.566961i \(-0.808119\pi\)
−0.823745 + 0.566961i \(0.808119\pi\)
\(68\) 2.17157 0.263342
\(69\) 0 0
\(70\) 0 0
\(71\) 13.3137 1.58005 0.790023 0.613077i \(-0.210068\pi\)
0.790023 + 0.613077i \(0.210068\pi\)
\(72\) 0 0
\(73\) 4.82843 0.565125 0.282562 0.959249i \(-0.408816\pi\)
0.282562 + 0.959249i \(0.408816\pi\)
\(74\) 9.65685 1.12259
\(75\) 0 0
\(76\) −0.828427 −0.0950271
\(77\) 0 0
\(78\) 0 0
\(79\) 4.75736 0.535245 0.267622 0.963524i \(-0.413762\pi\)
0.267622 + 0.963524i \(0.413762\pi\)
\(80\) 0.414214 0.0463105
\(81\) 0 0
\(82\) −4.65685 −0.514264
\(83\) 9.82843 1.07881 0.539405 0.842046i \(-0.318649\pi\)
0.539405 + 0.842046i \(0.318649\pi\)
\(84\) 0 0
\(85\) 0.899495 0.0975639
\(86\) −2.82843 −0.304997
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) −12.4853 −1.32344 −0.661719 0.749752i \(-0.730173\pi\)
−0.661719 + 0.749752i \(0.730173\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −3.24264 −0.338069
\(93\) 0 0
\(94\) 9.24264 0.953306
\(95\) −0.343146 −0.0352060
\(96\) 0 0
\(97\) 10.1716 1.03277 0.516383 0.856358i \(-0.327278\pi\)
0.516383 + 0.856358i \(0.327278\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −4.82843 −0.482843
\(101\) 14.1421 1.40720 0.703598 0.710599i \(-0.251576\pi\)
0.703598 + 0.710599i \(0.251576\pi\)
\(102\) 0 0
\(103\) 9.17157 0.903702 0.451851 0.892093i \(-0.350764\pi\)
0.451851 + 0.892093i \(0.350764\pi\)
\(104\) 1.17157 0.114882
\(105\) 0 0
\(106\) −5.17157 −0.502308
\(107\) −16.6569 −1.61028 −0.805139 0.593086i \(-0.797910\pi\)
−0.805139 + 0.593086i \(0.797910\pi\)
\(108\) 0 0
\(109\) 3.24264 0.310589 0.155294 0.987868i \(-0.450367\pi\)
0.155294 + 0.987868i \(0.450367\pi\)
\(110\) 0.414214 0.0394937
\(111\) 0 0
\(112\) 0 0
\(113\) 3.65685 0.344008 0.172004 0.985096i \(-0.444976\pi\)
0.172004 + 0.985096i \(0.444976\pi\)
\(114\) 0 0
\(115\) −1.34315 −0.125249
\(116\) 2.82843 0.262613
\(117\) 0 0
\(118\) 3.65685 0.336641
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 1.58579 0.143570
\(123\) 0 0
\(124\) 6.48528 0.582395
\(125\) −4.07107 −0.364127
\(126\) 0 0
\(127\) 1.24264 0.110267 0.0551333 0.998479i \(-0.482442\pi\)
0.0551333 + 0.998479i \(0.482442\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 0.485281 0.0425620
\(131\) −15.3137 −1.33796 −0.668982 0.743278i \(-0.733270\pi\)
−0.668982 + 0.743278i \(0.733270\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −13.4853 −1.16495
\(135\) 0 0
\(136\) 2.17157 0.186211
\(137\) −4.82843 −0.412520 −0.206260 0.978497i \(-0.566129\pi\)
−0.206260 + 0.978497i \(0.566129\pi\)
\(138\) 0 0
\(139\) 6.00000 0.508913 0.254457 0.967084i \(-0.418103\pi\)
0.254457 + 0.967084i \(0.418103\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 13.3137 1.11726
\(143\) 1.17157 0.0979718
\(144\) 0 0
\(145\) 1.17157 0.0972938
\(146\) 4.82843 0.399603
\(147\) 0 0
\(148\) 9.65685 0.793789
\(149\) 11.3137 0.926855 0.463428 0.886135i \(-0.346619\pi\)
0.463428 + 0.886135i \(0.346619\pi\)
\(150\) 0 0
\(151\) 1.24264 0.101125 0.0505623 0.998721i \(-0.483899\pi\)
0.0505623 + 0.998721i \(0.483899\pi\)
\(152\) −0.828427 −0.0671943
\(153\) 0 0
\(154\) 0 0
\(155\) 2.68629 0.215768
\(156\) 0 0
\(157\) 9.17157 0.731971 0.365986 0.930621i \(-0.380732\pi\)
0.365986 + 0.930621i \(0.380732\pi\)
\(158\) 4.75736 0.378475
\(159\) 0 0
\(160\) 0.414214 0.0327465
\(161\) 0 0
\(162\) 0 0
\(163\) 3.00000 0.234978 0.117489 0.993074i \(-0.462515\pi\)
0.117489 + 0.993074i \(0.462515\pi\)
\(164\) −4.65685 −0.363639
\(165\) 0 0
\(166\) 9.82843 0.762834
\(167\) 16.4853 1.27567 0.637835 0.770173i \(-0.279830\pi\)
0.637835 + 0.770173i \(0.279830\pi\)
\(168\) 0 0
\(169\) −11.6274 −0.894417
\(170\) 0.899495 0.0689881
\(171\) 0 0
\(172\) −2.82843 −0.215666
\(173\) 12.1421 0.923149 0.461575 0.887101i \(-0.347285\pi\)
0.461575 + 0.887101i \(0.347285\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) −12.4853 −0.935811
\(179\) −3.17157 −0.237054 −0.118527 0.992951i \(-0.537817\pi\)
−0.118527 + 0.992951i \(0.537817\pi\)
\(180\) 0 0
\(181\) 2.34315 0.174165 0.0870823 0.996201i \(-0.472246\pi\)
0.0870823 + 0.996201i \(0.472246\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −3.24264 −0.239051
\(185\) 4.00000 0.294086
\(186\) 0 0
\(187\) 2.17157 0.158801
\(188\) 9.24264 0.674089
\(189\) 0 0
\(190\) −0.343146 −0.0248944
\(191\) −9.31371 −0.673916 −0.336958 0.941520i \(-0.609398\pi\)
−0.336958 + 0.941520i \(0.609398\pi\)
\(192\) 0 0
\(193\) −9.17157 −0.660184 −0.330092 0.943949i \(-0.607080\pi\)
−0.330092 + 0.943949i \(0.607080\pi\)
\(194\) 10.1716 0.730276
\(195\) 0 0
\(196\) 0 0
\(197\) −3.51472 −0.250413 −0.125207 0.992131i \(-0.539959\pi\)
−0.125207 + 0.992131i \(0.539959\pi\)
\(198\) 0 0
\(199\) 8.34315 0.591430 0.295715 0.955276i \(-0.404442\pi\)
0.295715 + 0.955276i \(0.404442\pi\)
\(200\) −4.82843 −0.341421
\(201\) 0 0
\(202\) 14.1421 0.995037
\(203\) 0 0
\(204\) 0 0
\(205\) −1.92893 −0.134722
\(206\) 9.17157 0.639014
\(207\) 0 0
\(208\) 1.17157 0.0812340
\(209\) −0.828427 −0.0573035
\(210\) 0 0
\(211\) 14.8284 1.02083 0.510416 0.859928i \(-0.329492\pi\)
0.510416 + 0.859928i \(0.329492\pi\)
\(212\) −5.17157 −0.355185
\(213\) 0 0
\(214\) −16.6569 −1.13864
\(215\) −1.17157 −0.0799006
\(216\) 0 0
\(217\) 0 0
\(218\) 3.24264 0.219619
\(219\) 0 0
\(220\) 0.414214 0.0279263
\(221\) 2.54416 0.171138
\(222\) 0 0
\(223\) 13.3137 0.891552 0.445776 0.895145i \(-0.352928\pi\)
0.445776 + 0.895145i \(0.352928\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 3.65685 0.243250
\(227\) −12.3137 −0.817290 −0.408645 0.912694i \(-0.633998\pi\)
−0.408645 + 0.912694i \(0.633998\pi\)
\(228\) 0 0
\(229\) 11.3137 0.747631 0.373815 0.927503i \(-0.378049\pi\)
0.373815 + 0.927503i \(0.378049\pi\)
\(230\) −1.34315 −0.0885644
\(231\) 0 0
\(232\) 2.82843 0.185695
\(233\) 8.31371 0.544649 0.272325 0.962205i \(-0.412208\pi\)
0.272325 + 0.962205i \(0.412208\pi\)
\(234\) 0 0
\(235\) 3.82843 0.249739
\(236\) 3.65685 0.238041
\(237\) 0 0
\(238\) 0 0
\(239\) 20.0000 1.29369 0.646846 0.762620i \(-0.276088\pi\)
0.646846 + 0.762620i \(0.276088\pi\)
\(240\) 0 0
\(241\) −24.9706 −1.60850 −0.804248 0.594294i \(-0.797431\pi\)
−0.804248 + 0.594294i \(0.797431\pi\)
\(242\) 1.00000 0.0642824
\(243\) 0 0
\(244\) 1.58579 0.101520
\(245\) 0 0
\(246\) 0 0
\(247\) −0.970563 −0.0617554
\(248\) 6.48528 0.411816
\(249\) 0 0
\(250\) −4.07107 −0.257477
\(251\) −26.1421 −1.65008 −0.825038 0.565077i \(-0.808847\pi\)
−0.825038 + 0.565077i \(0.808847\pi\)
\(252\) 0 0
\(253\) −3.24264 −0.203863
\(254\) 1.24264 0.0779702
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −25.1716 −1.57016 −0.785080 0.619395i \(-0.787378\pi\)
−0.785080 + 0.619395i \(0.787378\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0.485281 0.0300959
\(261\) 0 0
\(262\) −15.3137 −0.946084
\(263\) 9.31371 0.574308 0.287154 0.957884i \(-0.407291\pi\)
0.287154 + 0.957884i \(0.407291\pi\)
\(264\) 0 0
\(265\) −2.14214 −0.131590
\(266\) 0 0
\(267\) 0 0
\(268\) −13.4853 −0.823745
\(269\) 8.07107 0.492102 0.246051 0.969257i \(-0.420867\pi\)
0.246051 + 0.969257i \(0.420867\pi\)
\(270\) 0 0
\(271\) 13.3137 0.808750 0.404375 0.914593i \(-0.367489\pi\)
0.404375 + 0.914593i \(0.367489\pi\)
\(272\) 2.17157 0.131671
\(273\) 0 0
\(274\) −4.82843 −0.291696
\(275\) −4.82843 −0.291165
\(276\) 0 0
\(277\) −6.82843 −0.410280 −0.205140 0.978733i \(-0.565765\pi\)
−0.205140 + 0.978733i \(0.565765\pi\)
\(278\) 6.00000 0.359856
\(279\) 0 0
\(280\) 0 0
\(281\) 2.51472 0.150016 0.0750078 0.997183i \(-0.476102\pi\)
0.0750078 + 0.997183i \(0.476102\pi\)
\(282\) 0 0
\(283\) 1.85786 0.110439 0.0552193 0.998474i \(-0.482414\pi\)
0.0552193 + 0.998474i \(0.482414\pi\)
\(284\) 13.3137 0.790023
\(285\) 0 0
\(286\) 1.17157 0.0692766
\(287\) 0 0
\(288\) 0 0
\(289\) −12.2843 −0.722604
\(290\) 1.17157 0.0687971
\(291\) 0 0
\(292\) 4.82843 0.282562
\(293\) −12.8284 −0.749445 −0.374722 0.927137i \(-0.622262\pi\)
−0.374722 + 0.927137i \(0.622262\pi\)
\(294\) 0 0
\(295\) 1.51472 0.0881903
\(296\) 9.65685 0.561293
\(297\) 0 0
\(298\) 11.3137 0.655386
\(299\) −3.79899 −0.219701
\(300\) 0 0
\(301\) 0 0
\(302\) 1.24264 0.0715059
\(303\) 0 0
\(304\) −0.828427 −0.0475136
\(305\) 0.656854 0.0376114
\(306\) 0 0
\(307\) 34.6274 1.97629 0.988146 0.153520i \(-0.0490610\pi\)
0.988146 + 0.153520i \(0.0490610\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 2.68629 0.152571
\(311\) −10.2132 −0.579138 −0.289569 0.957157i \(-0.593512\pi\)
−0.289569 + 0.957157i \(0.593512\pi\)
\(312\) 0 0
\(313\) 29.3137 1.65691 0.828454 0.560057i \(-0.189221\pi\)
0.828454 + 0.560057i \(0.189221\pi\)
\(314\) 9.17157 0.517582
\(315\) 0 0
\(316\) 4.75736 0.267622
\(317\) −33.8701 −1.90233 −0.951166 0.308680i \(-0.900113\pi\)
−0.951166 + 0.308680i \(0.900113\pi\)
\(318\) 0 0
\(319\) 2.82843 0.158362
\(320\) 0.414214 0.0231552
\(321\) 0 0
\(322\) 0 0
\(323\) −1.79899 −0.100098
\(324\) 0 0
\(325\) −5.65685 −0.313786
\(326\) 3.00000 0.166155
\(327\) 0 0
\(328\) −4.65685 −0.257132
\(329\) 0 0
\(330\) 0 0
\(331\) 6.31371 0.347033 0.173516 0.984831i \(-0.444487\pi\)
0.173516 + 0.984831i \(0.444487\pi\)
\(332\) 9.82843 0.539405
\(333\) 0 0
\(334\) 16.4853 0.902034
\(335\) −5.58579 −0.305184
\(336\) 0 0
\(337\) 9.51472 0.518300 0.259150 0.965837i \(-0.416558\pi\)
0.259150 + 0.965837i \(0.416558\pi\)
\(338\) −11.6274 −0.632448
\(339\) 0 0
\(340\) 0.899495 0.0487820
\(341\) 6.48528 0.351198
\(342\) 0 0
\(343\) 0 0
\(344\) −2.82843 −0.152499
\(345\) 0 0
\(346\) 12.1421 0.652765
\(347\) 4.85786 0.260784 0.130392 0.991463i \(-0.458376\pi\)
0.130392 + 0.991463i \(0.458376\pi\)
\(348\) 0 0
\(349\) 15.7279 0.841896 0.420948 0.907085i \(-0.361697\pi\)
0.420948 + 0.907085i \(0.361697\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) 16.8284 0.895687 0.447843 0.894112i \(-0.352192\pi\)
0.447843 + 0.894112i \(0.352192\pi\)
\(354\) 0 0
\(355\) 5.51472 0.292691
\(356\) −12.4853 −0.661719
\(357\) 0 0
\(358\) −3.17157 −0.167623
\(359\) −8.48528 −0.447836 −0.223918 0.974608i \(-0.571885\pi\)
−0.223918 + 0.974608i \(0.571885\pi\)
\(360\) 0 0
\(361\) −18.3137 −0.963879
\(362\) 2.34315 0.123153
\(363\) 0 0
\(364\) 0 0
\(365\) 2.00000 0.104685
\(366\) 0 0
\(367\) −5.51472 −0.287866 −0.143933 0.989587i \(-0.545975\pi\)
−0.143933 + 0.989587i \(0.545975\pi\)
\(368\) −3.24264 −0.169034
\(369\) 0 0
\(370\) 4.00000 0.207950
\(371\) 0 0
\(372\) 0 0
\(373\) −15.7279 −0.814361 −0.407180 0.913348i \(-0.633488\pi\)
−0.407180 + 0.913348i \(0.633488\pi\)
\(374\) 2.17157 0.112289
\(375\) 0 0
\(376\) 9.24264 0.476653
\(377\) 3.31371 0.170665
\(378\) 0 0
\(379\) 19.3431 0.993591 0.496795 0.867868i \(-0.334510\pi\)
0.496795 + 0.867868i \(0.334510\pi\)
\(380\) −0.343146 −0.0176030
\(381\) 0 0
\(382\) −9.31371 −0.476531
\(383\) 34.2843 1.75184 0.875922 0.482452i \(-0.160254\pi\)
0.875922 + 0.482452i \(0.160254\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −9.17157 −0.466821
\(387\) 0 0
\(388\) 10.1716 0.516383
\(389\) −1.72792 −0.0876091 −0.0438046 0.999040i \(-0.513948\pi\)
−0.0438046 + 0.999040i \(0.513948\pi\)
\(390\) 0 0
\(391\) −7.04163 −0.356111
\(392\) 0 0
\(393\) 0 0
\(394\) −3.51472 −0.177069
\(395\) 1.97056 0.0991498
\(396\) 0 0
\(397\) −12.4853 −0.626618 −0.313309 0.949651i \(-0.601438\pi\)
−0.313309 + 0.949651i \(0.601438\pi\)
\(398\) 8.34315 0.418204
\(399\) 0 0
\(400\) −4.82843 −0.241421
\(401\) −15.7990 −0.788964 −0.394482 0.918904i \(-0.629076\pi\)
−0.394482 + 0.918904i \(0.629076\pi\)
\(402\) 0 0
\(403\) 7.59798 0.378482
\(404\) 14.1421 0.703598
\(405\) 0 0
\(406\) 0 0
\(407\) 9.65685 0.478672
\(408\) 0 0
\(409\) 11.4558 0.566455 0.283228 0.959053i \(-0.408595\pi\)
0.283228 + 0.959053i \(0.408595\pi\)
\(410\) −1.92893 −0.0952632
\(411\) 0 0
\(412\) 9.17157 0.451851
\(413\) 0 0
\(414\) 0 0
\(415\) 4.07107 0.199841
\(416\) 1.17157 0.0574411
\(417\) 0 0
\(418\) −0.828427 −0.0405197
\(419\) 25.7990 1.26036 0.630182 0.776448i \(-0.282980\pi\)
0.630182 + 0.776448i \(0.282980\pi\)
\(420\) 0 0
\(421\) 14.8284 0.722693 0.361347 0.932432i \(-0.382317\pi\)
0.361347 + 0.932432i \(0.382317\pi\)
\(422\) 14.8284 0.721837
\(423\) 0 0
\(424\) −5.17157 −0.251154
\(425\) −10.4853 −0.508611
\(426\) 0 0
\(427\) 0 0
\(428\) −16.6569 −0.805139
\(429\) 0 0
\(430\) −1.17157 −0.0564983
\(431\) −19.1716 −0.923462 −0.461731 0.887020i \(-0.652771\pi\)
−0.461731 + 0.887020i \(0.652771\pi\)
\(432\) 0 0
\(433\) 26.6569 1.28105 0.640523 0.767939i \(-0.278717\pi\)
0.640523 + 0.767939i \(0.278717\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 3.24264 0.155294
\(437\) 2.68629 0.128503
\(438\) 0 0
\(439\) −3.38478 −0.161547 −0.0807733 0.996733i \(-0.525739\pi\)
−0.0807733 + 0.996733i \(0.525739\pi\)
\(440\) 0.414214 0.0197469
\(441\) 0 0
\(442\) 2.54416 0.121013
\(443\) 14.3431 0.681463 0.340732 0.940161i \(-0.389325\pi\)
0.340732 + 0.940161i \(0.389325\pi\)
\(444\) 0 0
\(445\) −5.17157 −0.245156
\(446\) 13.3137 0.630422
\(447\) 0 0
\(448\) 0 0
\(449\) 23.1127 1.09076 0.545378 0.838190i \(-0.316386\pi\)
0.545378 + 0.838190i \(0.316386\pi\)
\(450\) 0 0
\(451\) −4.65685 −0.219283
\(452\) 3.65685 0.172004
\(453\) 0 0
\(454\) −12.3137 −0.577911
\(455\) 0 0
\(456\) 0 0
\(457\) 34.6274 1.61980 0.809901 0.586566i \(-0.199521\pi\)
0.809901 + 0.586566i \(0.199521\pi\)
\(458\) 11.3137 0.528655
\(459\) 0 0
\(460\) −1.34315 −0.0626245
\(461\) 23.6569 1.10181 0.550905 0.834568i \(-0.314283\pi\)
0.550905 + 0.834568i \(0.314283\pi\)
\(462\) 0 0
\(463\) 23.1127 1.07414 0.537069 0.843538i \(-0.319531\pi\)
0.537069 + 0.843538i \(0.319531\pi\)
\(464\) 2.82843 0.131306
\(465\) 0 0
\(466\) 8.31371 0.385125
\(467\) 2.97056 0.137461 0.0687306 0.997635i \(-0.478105\pi\)
0.0687306 + 0.997635i \(0.478105\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 3.82843 0.176592
\(471\) 0 0
\(472\) 3.65685 0.168320
\(473\) −2.82843 −0.130051
\(474\) 0 0
\(475\) 4.00000 0.183533
\(476\) 0 0
\(477\) 0 0
\(478\) 20.0000 0.914779
\(479\) −5.17157 −0.236295 −0.118148 0.992996i \(-0.537696\pi\)
−0.118148 + 0.992996i \(0.537696\pi\)
\(480\) 0 0
\(481\) 11.3137 0.515861
\(482\) −24.9706 −1.13738
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 4.21320 0.191312
\(486\) 0 0
\(487\) −21.3137 −0.965816 −0.482908 0.875671i \(-0.660420\pi\)
−0.482908 + 0.875671i \(0.660420\pi\)
\(488\) 1.58579 0.0717852
\(489\) 0 0
\(490\) 0 0
\(491\) −15.6274 −0.705255 −0.352628 0.935764i \(-0.614712\pi\)
−0.352628 + 0.935764i \(0.614712\pi\)
\(492\) 0 0
\(493\) 6.14214 0.276628
\(494\) −0.970563 −0.0436677
\(495\) 0 0
\(496\) 6.48528 0.291198
\(497\) 0 0
\(498\) 0 0
\(499\) 14.3431 0.642087 0.321044 0.947064i \(-0.395966\pi\)
0.321044 + 0.947064i \(0.395966\pi\)
\(500\) −4.07107 −0.182064
\(501\) 0 0
\(502\) −26.1421 −1.16678
\(503\) −8.97056 −0.399978 −0.199989 0.979798i \(-0.564091\pi\)
−0.199989 + 0.979798i \(0.564091\pi\)
\(504\) 0 0
\(505\) 5.85786 0.260672
\(506\) −3.24264 −0.144153
\(507\) 0 0
\(508\) 1.24264 0.0551333
\(509\) −25.1716 −1.11571 −0.557855 0.829938i \(-0.688376\pi\)
−0.557855 + 0.829938i \(0.688376\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −25.1716 −1.11027
\(515\) 3.79899 0.167403
\(516\) 0 0
\(517\) 9.24264 0.406491
\(518\) 0 0
\(519\) 0 0
\(520\) 0.485281 0.0212810
\(521\) −29.3137 −1.28426 −0.642128 0.766597i \(-0.721948\pi\)
−0.642128 + 0.766597i \(0.721948\pi\)
\(522\) 0 0
\(523\) 31.4558 1.37547 0.687734 0.725963i \(-0.258606\pi\)
0.687734 + 0.725963i \(0.258606\pi\)
\(524\) −15.3137 −0.668982
\(525\) 0 0
\(526\) 9.31371 0.406097
\(527\) 14.0833 0.613476
\(528\) 0 0
\(529\) −12.4853 −0.542838
\(530\) −2.14214 −0.0930484
\(531\) 0 0
\(532\) 0 0
\(533\) −5.45584 −0.236319
\(534\) 0 0
\(535\) −6.89949 −0.298291
\(536\) −13.4853 −0.582475
\(537\) 0 0
\(538\) 8.07107 0.347968
\(539\) 0 0
\(540\) 0 0
\(541\) −5.44365 −0.234041 −0.117020 0.993130i \(-0.537334\pi\)
−0.117020 + 0.993130i \(0.537334\pi\)
\(542\) 13.3137 0.571873
\(543\) 0 0
\(544\) 2.17157 0.0931054
\(545\) 1.34315 0.0575340
\(546\) 0 0
\(547\) −36.1421 −1.54533 −0.772663 0.634816i \(-0.781076\pi\)
−0.772663 + 0.634816i \(0.781076\pi\)
\(548\) −4.82843 −0.206260
\(549\) 0 0
\(550\) −4.82843 −0.205885
\(551\) −2.34315 −0.0998214
\(552\) 0 0
\(553\) 0 0
\(554\) −6.82843 −0.290112
\(555\) 0 0
\(556\) 6.00000 0.254457
\(557\) −4.48528 −0.190047 −0.0950237 0.995475i \(-0.530293\pi\)
−0.0950237 + 0.995475i \(0.530293\pi\)
\(558\) 0 0
\(559\) −3.31371 −0.140155
\(560\) 0 0
\(561\) 0 0
\(562\) 2.51472 0.106077
\(563\) −25.6569 −1.08131 −0.540654 0.841245i \(-0.681823\pi\)
−0.540654 + 0.841245i \(0.681823\pi\)
\(564\) 0 0
\(565\) 1.51472 0.0637247
\(566\) 1.85786 0.0780919
\(567\) 0 0
\(568\) 13.3137 0.558631
\(569\) −22.0000 −0.922288 −0.461144 0.887325i \(-0.652561\pi\)
−0.461144 + 0.887325i \(0.652561\pi\)
\(570\) 0 0
\(571\) −21.6569 −0.906311 −0.453156 0.891431i \(-0.649702\pi\)
−0.453156 + 0.891431i \(0.649702\pi\)
\(572\) 1.17157 0.0489859
\(573\) 0 0
\(574\) 0 0
\(575\) 15.6569 0.652936
\(576\) 0 0
\(577\) −5.14214 −0.214070 −0.107035 0.994255i \(-0.534136\pi\)
−0.107035 + 0.994255i \(0.534136\pi\)
\(578\) −12.2843 −0.510958
\(579\) 0 0
\(580\) 1.17157 0.0486469
\(581\) 0 0
\(582\) 0 0
\(583\) −5.17157 −0.214185
\(584\) 4.82843 0.199802
\(585\) 0 0
\(586\) −12.8284 −0.529937
\(587\) 28.1421 1.16155 0.580775 0.814064i \(-0.302750\pi\)
0.580775 + 0.814064i \(0.302750\pi\)
\(588\) 0 0
\(589\) −5.37258 −0.221373
\(590\) 1.51472 0.0623600
\(591\) 0 0
\(592\) 9.65685 0.396894
\(593\) −39.9411 −1.64019 −0.820093 0.572231i \(-0.806078\pi\)
−0.820093 + 0.572231i \(0.806078\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 11.3137 0.463428
\(597\) 0 0
\(598\) −3.79899 −0.155352
\(599\) 29.0416 1.18661 0.593304 0.804978i \(-0.297823\pi\)
0.593304 + 0.804978i \(0.297823\pi\)
\(600\) 0 0
\(601\) 20.0000 0.815817 0.407909 0.913023i \(-0.366258\pi\)
0.407909 + 0.913023i \(0.366258\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 1.24264 0.0505623
\(605\) 0.414214 0.0168402
\(606\) 0 0
\(607\) −46.2132 −1.87574 −0.937868 0.346992i \(-0.887203\pi\)
−0.937868 + 0.346992i \(0.887203\pi\)
\(608\) −0.828427 −0.0335972
\(609\) 0 0
\(610\) 0.656854 0.0265953
\(611\) 10.8284 0.438071
\(612\) 0 0
\(613\) −30.8995 −1.24802 −0.624009 0.781417i \(-0.714497\pi\)
−0.624009 + 0.781417i \(0.714497\pi\)
\(614\) 34.6274 1.39745
\(615\) 0 0
\(616\) 0 0
\(617\) −24.4853 −0.985740 −0.492870 0.870103i \(-0.664052\pi\)
−0.492870 + 0.870103i \(0.664052\pi\)
\(618\) 0 0
\(619\) −0.514719 −0.0206883 −0.0103441 0.999946i \(-0.503293\pi\)
−0.0103441 + 0.999946i \(0.503293\pi\)
\(620\) 2.68629 0.107884
\(621\) 0 0
\(622\) −10.2132 −0.409512
\(623\) 0 0
\(624\) 0 0
\(625\) 22.4558 0.898234
\(626\) 29.3137 1.17161
\(627\) 0 0
\(628\) 9.17157 0.365986
\(629\) 20.9706 0.836151
\(630\) 0 0
\(631\) −1.02944 −0.0409812 −0.0204906 0.999790i \(-0.506523\pi\)
−0.0204906 + 0.999790i \(0.506523\pi\)
\(632\) 4.75736 0.189238
\(633\) 0 0
\(634\) −33.8701 −1.34515
\(635\) 0.514719 0.0204260
\(636\) 0 0
\(637\) 0 0
\(638\) 2.82843 0.111979
\(639\) 0 0
\(640\) 0.414214 0.0163732
\(641\) −27.3137 −1.07883 −0.539413 0.842041i \(-0.681354\pi\)
−0.539413 + 0.842041i \(0.681354\pi\)
\(642\) 0 0
\(643\) −17.6569 −0.696318 −0.348159 0.937435i \(-0.613193\pi\)
−0.348159 + 0.937435i \(0.613193\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −1.79899 −0.0707803
\(647\) −42.6985 −1.67865 −0.839325 0.543629i \(-0.817050\pi\)
−0.839325 + 0.543629i \(0.817050\pi\)
\(648\) 0 0
\(649\) 3.65685 0.143544
\(650\) −5.65685 −0.221880
\(651\) 0 0
\(652\) 3.00000 0.117489
\(653\) 44.3553 1.73576 0.867879 0.496775i \(-0.165483\pi\)
0.867879 + 0.496775i \(0.165483\pi\)
\(654\) 0 0
\(655\) −6.34315 −0.247847
\(656\) −4.65685 −0.181820
\(657\) 0 0
\(658\) 0 0
\(659\) −36.9411 −1.43902 −0.719511 0.694481i \(-0.755634\pi\)
−0.719511 + 0.694481i \(0.755634\pi\)
\(660\) 0 0
\(661\) 38.9706 1.51578 0.757890 0.652383i \(-0.226231\pi\)
0.757890 + 0.652383i \(0.226231\pi\)
\(662\) 6.31371 0.245389
\(663\) 0 0
\(664\) 9.82843 0.381417
\(665\) 0 0
\(666\) 0 0
\(667\) −9.17157 −0.355125
\(668\) 16.4853 0.637835
\(669\) 0 0
\(670\) −5.58579 −0.215798
\(671\) 1.58579 0.0612186
\(672\) 0 0
\(673\) 24.6274 0.949317 0.474659 0.880170i \(-0.342572\pi\)
0.474659 + 0.880170i \(0.342572\pi\)
\(674\) 9.51472 0.366493
\(675\) 0 0
\(676\) −11.6274 −0.447208
\(677\) −42.7696 −1.64377 −0.821884 0.569655i \(-0.807077\pi\)
−0.821884 + 0.569655i \(0.807077\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0.899495 0.0344941
\(681\) 0 0
\(682\) 6.48528 0.248334
\(683\) 17.6569 0.675621 0.337810 0.941214i \(-0.390314\pi\)
0.337810 + 0.941214i \(0.390314\pi\)
\(684\) 0 0
\(685\) −2.00000 −0.0764161
\(686\) 0 0
\(687\) 0 0
\(688\) −2.82843 −0.107833
\(689\) −6.05887 −0.230825
\(690\) 0 0
\(691\) −29.1421 −1.10862 −0.554310 0.832311i \(-0.687018\pi\)
−0.554310 + 0.832311i \(0.687018\pi\)
\(692\) 12.1421 0.461575
\(693\) 0 0
\(694\) 4.85786 0.184402
\(695\) 2.48528 0.0942721
\(696\) 0 0
\(697\) −10.1127 −0.383046
\(698\) 15.7279 0.595311
\(699\) 0 0
\(700\) 0 0
\(701\) 28.1421 1.06291 0.531457 0.847085i \(-0.321645\pi\)
0.531457 + 0.847085i \(0.321645\pi\)
\(702\) 0 0
\(703\) −8.00000 −0.301726
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) 16.8284 0.633346
\(707\) 0 0
\(708\) 0 0
\(709\) 3.85786 0.144885 0.0724426 0.997373i \(-0.476921\pi\)
0.0724426 + 0.997373i \(0.476921\pi\)
\(710\) 5.51472 0.206964
\(711\) 0 0
\(712\) −12.4853 −0.467906
\(713\) −21.0294 −0.787559
\(714\) 0 0
\(715\) 0.485281 0.0181485
\(716\) −3.17157 −0.118527
\(717\) 0 0
\(718\) −8.48528 −0.316668
\(719\) 16.2132 0.604650 0.302325 0.953205i \(-0.402237\pi\)
0.302325 + 0.953205i \(0.402237\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −18.3137 −0.681566
\(723\) 0 0
\(724\) 2.34315 0.0870823
\(725\) −13.6569 −0.507203
\(726\) 0 0
\(727\) 32.4853 1.20481 0.602406 0.798190i \(-0.294209\pi\)
0.602406 + 0.798190i \(0.294209\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 2.00000 0.0740233
\(731\) −6.14214 −0.227175
\(732\) 0 0
\(733\) 14.7574 0.545076 0.272538 0.962145i \(-0.412137\pi\)
0.272538 + 0.962145i \(0.412137\pi\)
\(734\) −5.51472 −0.203552
\(735\) 0 0
\(736\) −3.24264 −0.119525
\(737\) −13.4853 −0.496737
\(738\) 0 0
\(739\) −34.0000 −1.25071 −0.625355 0.780340i \(-0.715046\pi\)
−0.625355 + 0.780340i \(0.715046\pi\)
\(740\) 4.00000 0.147043
\(741\) 0 0
\(742\) 0 0
\(743\) −13.6569 −0.501021 −0.250511 0.968114i \(-0.580599\pi\)
−0.250511 + 0.968114i \(0.580599\pi\)
\(744\) 0 0
\(745\) 4.68629 0.171692
\(746\) −15.7279 −0.575840
\(747\) 0 0
\(748\) 2.17157 0.0794006
\(749\) 0 0
\(750\) 0 0
\(751\) 29.7990 1.08738 0.543690 0.839286i \(-0.317027\pi\)
0.543690 + 0.839286i \(0.317027\pi\)
\(752\) 9.24264 0.337044
\(753\) 0 0
\(754\) 3.31371 0.120678
\(755\) 0.514719 0.0187325
\(756\) 0 0
\(757\) 19.6569 0.714441 0.357220 0.934020i \(-0.383725\pi\)
0.357220 + 0.934020i \(0.383725\pi\)
\(758\) 19.3431 0.702575
\(759\) 0 0
\(760\) −0.343146 −0.0124472
\(761\) −24.4558 −0.886524 −0.443262 0.896392i \(-0.646179\pi\)
−0.443262 + 0.896392i \(0.646179\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −9.31371 −0.336958
\(765\) 0 0
\(766\) 34.2843 1.23874
\(767\) 4.28427 0.154696
\(768\) 0 0
\(769\) −26.4853 −0.955084 −0.477542 0.878609i \(-0.658472\pi\)
−0.477542 + 0.878609i \(0.658472\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −9.17157 −0.330092
\(773\) −31.2426 −1.12372 −0.561860 0.827232i \(-0.689914\pi\)
−0.561860 + 0.827232i \(0.689914\pi\)
\(774\) 0 0
\(775\) −31.3137 −1.12482
\(776\) 10.1716 0.365138
\(777\) 0 0
\(778\) −1.72792 −0.0619490
\(779\) 3.85786 0.138222
\(780\) 0 0
\(781\) 13.3137 0.476402
\(782\) −7.04163 −0.251808
\(783\) 0 0
\(784\) 0 0
\(785\) 3.79899 0.135592
\(786\) 0 0
\(787\) 38.7696 1.38199 0.690993 0.722862i \(-0.257174\pi\)
0.690993 + 0.722862i \(0.257174\pi\)
\(788\) −3.51472 −0.125207
\(789\) 0 0
\(790\) 1.97056 0.0701095
\(791\) 0 0
\(792\) 0 0
\(793\) 1.85786 0.0659747
\(794\) −12.4853 −0.443086
\(795\) 0 0
\(796\) 8.34315 0.295715
\(797\) −52.8995 −1.87380 −0.936898 0.349602i \(-0.886317\pi\)
−0.936898 + 0.349602i \(0.886317\pi\)
\(798\) 0 0
\(799\) 20.0711 0.710063
\(800\) −4.82843 −0.170711
\(801\) 0 0
\(802\) −15.7990 −0.557882
\(803\) 4.82843 0.170391
\(804\) 0 0
\(805\) 0 0
\(806\) 7.59798 0.267627
\(807\) 0 0
\(808\) 14.1421 0.497519
\(809\) −9.34315 −0.328488 −0.164244 0.986420i \(-0.552518\pi\)
−0.164244 + 0.986420i \(0.552518\pi\)
\(810\) 0 0
\(811\) 7.65685 0.268869 0.134434 0.990923i \(-0.457078\pi\)
0.134434 + 0.990923i \(0.457078\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 9.65685 0.338473
\(815\) 1.24264 0.0435278
\(816\) 0 0
\(817\) 2.34315 0.0819763
\(818\) 11.4558 0.400544
\(819\) 0 0
\(820\) −1.92893 −0.0673612
\(821\) 21.5147 0.750869 0.375434 0.926849i \(-0.377494\pi\)
0.375434 + 0.926849i \(0.377494\pi\)
\(822\) 0 0
\(823\) −21.5147 −0.749956 −0.374978 0.927034i \(-0.622350\pi\)
−0.374978 + 0.927034i \(0.622350\pi\)
\(824\) 9.17157 0.319507
\(825\) 0 0
\(826\) 0 0
\(827\) 2.17157 0.0755130 0.0377565 0.999287i \(-0.487979\pi\)
0.0377565 + 0.999287i \(0.487979\pi\)
\(828\) 0 0
\(829\) 28.0000 0.972480 0.486240 0.873825i \(-0.338368\pi\)
0.486240 + 0.873825i \(0.338368\pi\)
\(830\) 4.07107 0.141309
\(831\) 0 0
\(832\) 1.17157 0.0406170
\(833\) 0 0
\(834\) 0 0
\(835\) 6.82843 0.236307
\(836\) −0.828427 −0.0286518
\(837\) 0 0
\(838\) 25.7990 0.891211
\(839\) −8.21320 −0.283551 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(840\) 0 0
\(841\) −21.0000 −0.724138
\(842\) 14.8284 0.511021
\(843\) 0 0
\(844\) 14.8284 0.510416
\(845\) −4.81623 −0.165683
\(846\) 0 0
\(847\) 0 0
\(848\) −5.17157 −0.177593
\(849\) 0 0
\(850\) −10.4853 −0.359642
\(851\) −31.3137 −1.07342
\(852\) 0 0
\(853\) 23.2426 0.795813 0.397906 0.917426i \(-0.369737\pi\)
0.397906 + 0.917426i \(0.369737\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −16.6569 −0.569320
\(857\) −21.6863 −0.740790 −0.370395 0.928874i \(-0.620778\pi\)
−0.370395 + 0.928874i \(0.620778\pi\)
\(858\) 0 0
\(859\) 50.9411 1.73809 0.869044 0.494734i \(-0.164735\pi\)
0.869044 + 0.494734i \(0.164735\pi\)
\(860\) −1.17157 −0.0399503
\(861\) 0 0
\(862\) −19.1716 −0.652986
\(863\) −39.1838 −1.33383 −0.666915 0.745133i \(-0.732386\pi\)
−0.666915 + 0.745133i \(0.732386\pi\)
\(864\) 0 0
\(865\) 5.02944 0.171006
\(866\) 26.6569 0.905837
\(867\) 0 0
\(868\) 0 0
\(869\) 4.75736 0.161382
\(870\) 0 0
\(871\) −15.7990 −0.535328
\(872\) 3.24264 0.109810
\(873\) 0 0
\(874\) 2.68629 0.0908652
\(875\) 0 0
\(876\) 0 0
\(877\) 24.5563 0.829209 0.414604 0.910002i \(-0.363920\pi\)
0.414604 + 0.910002i \(0.363920\pi\)
\(878\) −3.38478 −0.114231
\(879\) 0 0
\(880\) 0.414214 0.0139631
\(881\) −26.0000 −0.875962 −0.437981 0.898984i \(-0.644306\pi\)
−0.437981 + 0.898984i \(0.644306\pi\)
\(882\) 0 0
\(883\) 0.514719 0.0173217 0.00866083 0.999962i \(-0.497243\pi\)
0.00866083 + 0.999962i \(0.497243\pi\)
\(884\) 2.54416 0.0855692
\(885\) 0 0
\(886\) 14.3431 0.481867
\(887\) −8.34315 −0.280135 −0.140068 0.990142i \(-0.544732\pi\)
−0.140068 + 0.990142i \(0.544732\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −5.17157 −0.173352
\(891\) 0 0
\(892\) 13.3137 0.445776
\(893\) −7.65685 −0.256227
\(894\) 0 0
\(895\) −1.31371 −0.0439124
\(896\) 0 0
\(897\) 0 0
\(898\) 23.1127 0.771281
\(899\) 18.3431 0.611778
\(900\) 0 0
\(901\) −11.2304 −0.374140
\(902\) −4.65685 −0.155056
\(903\) 0 0
\(904\) 3.65685 0.121625
\(905\) 0.970563 0.0322626
\(906\) 0 0
\(907\) 10.8579 0.360529 0.180265 0.983618i \(-0.442305\pi\)
0.180265 + 0.983618i \(0.442305\pi\)
\(908\) −12.3137 −0.408645
\(909\) 0 0
\(910\) 0 0
\(911\) 50.2132 1.66364 0.831819 0.555047i \(-0.187300\pi\)
0.831819 + 0.555047i \(0.187300\pi\)
\(912\) 0 0
\(913\) 9.82843 0.325273
\(914\) 34.6274 1.14537
\(915\) 0 0
\(916\) 11.3137 0.373815
\(917\) 0 0
\(918\) 0 0
\(919\) −22.8995 −0.755385 −0.377692 0.925931i \(-0.623282\pi\)
−0.377692 + 0.925931i \(0.623282\pi\)
\(920\) −1.34315 −0.0442822
\(921\) 0 0
\(922\) 23.6569 0.779097
\(923\) 15.5980 0.513414
\(924\) 0 0
\(925\) −46.6274 −1.53310
\(926\) 23.1127 0.759530
\(927\) 0 0
\(928\) 2.82843 0.0928477
\(929\) −7.45584 −0.244618 −0.122309 0.992492i \(-0.539030\pi\)
−0.122309 + 0.992492i \(0.539030\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 8.31371 0.272325
\(933\) 0 0
\(934\) 2.97056 0.0971998
\(935\) 0.899495 0.0294166
\(936\) 0 0
\(937\) −11.6569 −0.380813 −0.190406 0.981705i \(-0.560981\pi\)
−0.190406 + 0.981705i \(0.560981\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 3.82843 0.124870
\(941\) −11.3726 −0.370736 −0.185368 0.982669i \(-0.559348\pi\)
−0.185368 + 0.982669i \(0.559348\pi\)
\(942\) 0 0
\(943\) 15.1005 0.491740
\(944\) 3.65685 0.119020
\(945\) 0 0
\(946\) −2.82843 −0.0919601
\(947\) 12.6274 0.410336 0.205168 0.978727i \(-0.434226\pi\)
0.205168 + 0.978727i \(0.434226\pi\)
\(948\) 0 0
\(949\) 5.65685 0.183629
\(950\) 4.00000 0.129777
\(951\) 0 0
\(952\) 0 0
\(953\) −6.31371 −0.204521 −0.102261 0.994758i \(-0.532608\pi\)
−0.102261 + 0.994758i \(0.532608\pi\)
\(954\) 0 0
\(955\) −3.85786 −0.124838
\(956\) 20.0000 0.646846
\(957\) 0 0
\(958\) −5.17157 −0.167086
\(959\) 0 0
\(960\) 0 0
\(961\) 11.0589 0.356738
\(962\) 11.3137 0.364769
\(963\) 0 0
\(964\) −24.9706 −0.804248
\(965\) −3.79899 −0.122294
\(966\) 0 0
\(967\) 3.10051 0.0997055 0.0498528 0.998757i \(-0.484125\pi\)
0.0498528 + 0.998757i \(0.484125\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) 4.21320 0.135278
\(971\) 10.0000 0.320915 0.160458 0.987043i \(-0.448703\pi\)
0.160458 + 0.987043i \(0.448703\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −21.3137 −0.682935
\(975\) 0 0
\(976\) 1.58579 0.0507598
\(977\) 31.9411 1.02189 0.510944 0.859614i \(-0.329296\pi\)
0.510944 + 0.859614i \(0.329296\pi\)
\(978\) 0 0
\(979\) −12.4853 −0.399031
\(980\) 0 0
\(981\) 0 0
\(982\) −15.6274 −0.498691
\(983\) −36.7574 −1.17238 −0.586189 0.810174i \(-0.699372\pi\)
−0.586189 + 0.810174i \(0.699372\pi\)
\(984\) 0 0
\(985\) −1.45584 −0.0463871
\(986\) 6.14214 0.195605
\(987\) 0 0
\(988\) −0.970563 −0.0308777
\(989\) 9.17157 0.291639
\(990\) 0 0
\(991\) 22.3431 0.709753 0.354877 0.934913i \(-0.384523\pi\)
0.354877 + 0.934913i \(0.384523\pi\)
\(992\) 6.48528 0.205908
\(993\) 0 0
\(994\) 0 0
\(995\) 3.45584 0.109558
\(996\) 0 0
\(997\) −47.7990 −1.51381 −0.756905 0.653525i \(-0.773289\pi\)
−0.756905 + 0.653525i \(0.773289\pi\)
\(998\) 14.3431 0.454024
\(999\) 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 9702.2.a.da.1.2 2
3.2 odd 2 9702.2.a.cv.1.1 2
7.3 odd 6 1386.2.k.q.793.2 4
7.5 odd 6 1386.2.k.q.991.2 yes 4
7.6 odd 2 9702.2.a.ds.1.1 2
21.5 even 6 1386.2.k.u.991.1 yes 4
21.17 even 6 1386.2.k.u.793.1 yes 4
21.20 even 2 9702.2.a.cj.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1386.2.k.q.793.2 4 7.3 odd 6
1386.2.k.q.991.2 yes 4 7.5 odd 6
1386.2.k.u.793.1 yes 4 21.17 even 6
1386.2.k.u.991.1 yes 4 21.5 even 6
9702.2.a.cj.1.2 2 21.20 even 2
9702.2.a.cv.1.1 2 3.2 odd 2
9702.2.a.da.1.2 2 1.1 even 1 trivial
9702.2.a.ds.1.1 2 7.6 odd 2