Properties

Label 9450.2.a.ea.1.2
Level $9450$
Weight $2$
Character 9450.1
Self dual yes
Analytic conductor $75.459$
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] = [9450,2,Mod(1,9450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9450, 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("9450.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9450 = 2 \cdot 3^{3} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9450.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: \(75.4586299101\)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1890)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.44949\) of defining polynomial
Character \(\chi\) \(=\) 9450.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} -1.00000 q^{8} +2.44949 q^{11} +1.44949 q^{13} +1.00000 q^{14} +1.00000 q^{16} -1.00000 q^{17} -4.44949 q^{19} -2.44949 q^{22} -1.44949 q^{23} -1.44949 q^{26} -1.00000 q^{28} +2.55051 q^{29} -1.44949 q^{31} -1.00000 q^{32} +1.00000 q^{34} +3.34847 q^{37} +4.44949 q^{38} +4.89898 q^{41} -3.89898 q^{43} +2.44949 q^{44} +1.44949 q^{46} -11.7980 q^{47} +1.00000 q^{49} +1.44949 q^{52} +5.44949 q^{53} +1.00000 q^{56} -2.55051 q^{58} +0.101021 q^{59} +10.8990 q^{61} +1.44949 q^{62} +1.00000 q^{64} -5.00000 q^{67} -1.00000 q^{68} -5.44949 q^{71} -11.7980 q^{73} -3.34847 q^{74} -4.44949 q^{76} -2.44949 q^{77} -6.44949 q^{79} -4.89898 q^{82} +10.8990 q^{83} +3.89898 q^{86} -2.44949 q^{88} +11.0000 q^{89} -1.44949 q^{91} -1.44949 q^{92} +11.7980 q^{94} -9.55051 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8} - 2 q^{13} + 2 q^{14} + 2 q^{16} - 2 q^{17} - 4 q^{19} + 2 q^{23} + 2 q^{26} - 2 q^{28} + 10 q^{29} + 2 q^{31} - 2 q^{32} + 2 q^{34} - 8 q^{37} + 4 q^{38} + 2 q^{43} - 2 q^{46} - 4 q^{47} + 2 q^{49} - 2 q^{52} + 6 q^{53} + 2 q^{56} - 10 q^{58} + 10 q^{59} + 12 q^{61} - 2 q^{62} + 2 q^{64} - 10 q^{67} - 2 q^{68} - 6 q^{71} - 4 q^{73} + 8 q^{74} - 4 q^{76} - 8 q^{79} + 12 q^{83} - 2 q^{86} + 22 q^{89} + 2 q^{91} + 2 q^{92} + 4 q^{94} - 24 q^{97} - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 2.44949 0.738549 0.369274 0.929320i \(-0.379606\pi\)
0.369274 + 0.929320i \(0.379606\pi\)
\(12\) 0 0
\(13\) 1.44949 0.402016 0.201008 0.979590i \(-0.435578\pi\)
0.201008 + 0.979590i \(0.435578\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −1.00000 −0.242536 −0.121268 0.992620i \(-0.538696\pi\)
−0.121268 + 0.992620i \(0.538696\pi\)
\(18\) 0 0
\(19\) −4.44949 −1.02078 −0.510391 0.859942i \(-0.670499\pi\)
−0.510391 + 0.859942i \(0.670499\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −2.44949 −0.522233
\(23\) −1.44949 −0.302240 −0.151120 0.988515i \(-0.548288\pi\)
−0.151120 + 0.988515i \(0.548288\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −1.44949 −0.284268
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) 2.55051 0.473618 0.236809 0.971556i \(-0.423898\pi\)
0.236809 + 0.971556i \(0.423898\pi\)
\(30\) 0 0
\(31\) −1.44949 −0.260336 −0.130168 0.991492i \(-0.541552\pi\)
−0.130168 + 0.991492i \(0.541552\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 1.00000 0.171499
\(35\) 0 0
\(36\) 0 0
\(37\) 3.34847 0.550485 0.275242 0.961375i \(-0.411242\pi\)
0.275242 + 0.961375i \(0.411242\pi\)
\(38\) 4.44949 0.721803
\(39\) 0 0
\(40\) 0 0
\(41\) 4.89898 0.765092 0.382546 0.923936i \(-0.375047\pi\)
0.382546 + 0.923936i \(0.375047\pi\)
\(42\) 0 0
\(43\) −3.89898 −0.594589 −0.297294 0.954786i \(-0.596084\pi\)
−0.297294 + 0.954786i \(0.596084\pi\)
\(44\) 2.44949 0.369274
\(45\) 0 0
\(46\) 1.44949 0.213716
\(47\) −11.7980 −1.72091 −0.860455 0.509527i \(-0.829820\pi\)
−0.860455 + 0.509527i \(0.829820\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 1.44949 0.201008
\(53\) 5.44949 0.748545 0.374272 0.927319i \(-0.377892\pi\)
0.374272 + 0.927319i \(0.377892\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) −2.55051 −0.334898
\(59\) 0.101021 0.0131518 0.00657588 0.999978i \(-0.497907\pi\)
0.00657588 + 0.999978i \(0.497907\pi\)
\(60\) 0 0
\(61\) 10.8990 1.39547 0.697736 0.716355i \(-0.254191\pi\)
0.697736 + 0.716355i \(0.254191\pi\)
\(62\) 1.44949 0.184085
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −5.00000 −0.610847 −0.305424 0.952217i \(-0.598798\pi\)
−0.305424 + 0.952217i \(0.598798\pi\)
\(68\) −1.00000 −0.121268
\(69\) 0 0
\(70\) 0 0
\(71\) −5.44949 −0.646735 −0.323368 0.946273i \(-0.604815\pi\)
−0.323368 + 0.946273i \(0.604815\pi\)
\(72\) 0 0
\(73\) −11.7980 −1.38085 −0.690423 0.723406i \(-0.742576\pi\)
−0.690423 + 0.723406i \(0.742576\pi\)
\(74\) −3.34847 −0.389252
\(75\) 0 0
\(76\) −4.44949 −0.510391
\(77\) −2.44949 −0.279145
\(78\) 0 0
\(79\) −6.44949 −0.725624 −0.362812 0.931862i \(-0.618183\pi\)
−0.362812 + 0.931862i \(0.618183\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −4.89898 −0.541002
\(83\) 10.8990 1.19632 0.598159 0.801377i \(-0.295899\pi\)
0.598159 + 0.801377i \(0.295899\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 3.89898 0.420438
\(87\) 0 0
\(88\) −2.44949 −0.261116
\(89\) 11.0000 1.16600 0.582999 0.812473i \(-0.301879\pi\)
0.582999 + 0.812473i \(0.301879\pi\)
\(90\) 0 0
\(91\) −1.44949 −0.151948
\(92\) −1.44949 −0.151120
\(93\) 0 0
\(94\) 11.7980 1.21687
\(95\) 0 0
\(96\) 0 0
\(97\) −9.55051 −0.969707 −0.484854 0.874595i \(-0.661127\pi\)
−0.484854 + 0.874595i \(0.661127\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 0 0
\(101\) 12.2474 1.21867 0.609333 0.792914i \(-0.291437\pi\)
0.609333 + 0.792914i \(0.291437\pi\)
\(102\) 0 0
\(103\) −15.4495 −1.52228 −0.761142 0.648586i \(-0.775361\pi\)
−0.761142 + 0.648586i \(0.775361\pi\)
\(104\) −1.44949 −0.142134
\(105\) 0 0
\(106\) −5.44949 −0.529301
\(107\) −3.34847 −0.323709 −0.161854 0.986815i \(-0.551747\pi\)
−0.161854 + 0.986815i \(0.551747\pi\)
\(108\) 0 0
\(109\) −15.1464 −1.45076 −0.725382 0.688346i \(-0.758337\pi\)
−0.725382 + 0.688346i \(0.758337\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) 6.44949 0.606717 0.303358 0.952877i \(-0.401892\pi\)
0.303358 + 0.952877i \(0.401892\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 2.55051 0.236809
\(117\) 0 0
\(118\) −0.101021 −0.00929969
\(119\) 1.00000 0.0916698
\(120\) 0 0
\(121\) −5.00000 −0.454545
\(122\) −10.8990 −0.986747
\(123\) 0 0
\(124\) −1.44949 −0.130168
\(125\) 0 0
\(126\) 0 0
\(127\) −8.89898 −0.789657 −0.394828 0.918755i \(-0.629196\pi\)
−0.394828 + 0.918755i \(0.629196\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −16.7980 −1.46764 −0.733822 0.679342i \(-0.762266\pi\)
−0.733822 + 0.679342i \(0.762266\pi\)
\(132\) 0 0
\(133\) 4.44949 0.385820
\(134\) 5.00000 0.431934
\(135\) 0 0
\(136\) 1.00000 0.0857493
\(137\) −5.10102 −0.435810 −0.217905 0.975970i \(-0.569922\pi\)
−0.217905 + 0.975970i \(0.569922\pi\)
\(138\) 0 0
\(139\) 22.2474 1.88700 0.943502 0.331367i \(-0.107510\pi\)
0.943502 + 0.331367i \(0.107510\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 5.44949 0.457311
\(143\) 3.55051 0.296909
\(144\) 0 0
\(145\) 0 0
\(146\) 11.7980 0.976406
\(147\) 0 0
\(148\) 3.34847 0.275242
\(149\) 0.550510 0.0450996 0.0225498 0.999746i \(-0.492822\pi\)
0.0225498 + 0.999746i \(0.492822\pi\)
\(150\) 0 0
\(151\) 12.2474 0.996683 0.498342 0.866981i \(-0.333943\pi\)
0.498342 + 0.866981i \(0.333943\pi\)
\(152\) 4.44949 0.360901
\(153\) 0 0
\(154\) 2.44949 0.197386
\(155\) 0 0
\(156\) 0 0
\(157\) −5.44949 −0.434917 −0.217458 0.976070i \(-0.569777\pi\)
−0.217458 + 0.976070i \(0.569777\pi\)
\(158\) 6.44949 0.513094
\(159\) 0 0
\(160\) 0 0
\(161\) 1.44949 0.114236
\(162\) 0 0
\(163\) −4.10102 −0.321217 −0.160608 0.987018i \(-0.551346\pi\)
−0.160608 + 0.987018i \(0.551346\pi\)
\(164\) 4.89898 0.382546
\(165\) 0 0
\(166\) −10.8990 −0.845925
\(167\) −3.55051 −0.274747 −0.137373 0.990519i \(-0.543866\pi\)
−0.137373 + 0.990519i \(0.543866\pi\)
\(168\) 0 0
\(169\) −10.8990 −0.838383
\(170\) 0 0
\(171\) 0 0
\(172\) −3.89898 −0.297294
\(173\) 11.1464 0.847447 0.423724 0.905792i \(-0.360723\pi\)
0.423724 + 0.905792i \(0.360723\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 2.44949 0.184637
\(177\) 0 0
\(178\) −11.0000 −0.824485
\(179\) −0.898979 −0.0671929 −0.0335964 0.999435i \(-0.510696\pi\)
−0.0335964 + 0.999435i \(0.510696\pi\)
\(180\) 0 0
\(181\) 7.44949 0.553716 0.276858 0.960911i \(-0.410707\pi\)
0.276858 + 0.960911i \(0.410707\pi\)
\(182\) 1.44949 0.107443
\(183\) 0 0
\(184\) 1.44949 0.106858
\(185\) 0 0
\(186\) 0 0
\(187\) −2.44949 −0.179124
\(188\) −11.7980 −0.860455
\(189\) 0 0
\(190\) 0 0
\(191\) −22.6969 −1.64229 −0.821146 0.570718i \(-0.806665\pi\)
−0.821146 + 0.570718i \(0.806665\pi\)
\(192\) 0 0
\(193\) −7.20204 −0.518414 −0.259207 0.965822i \(-0.583461\pi\)
−0.259207 + 0.965822i \(0.583461\pi\)
\(194\) 9.55051 0.685687
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 12.8990 0.919014 0.459507 0.888174i \(-0.348026\pi\)
0.459507 + 0.888174i \(0.348026\pi\)
\(198\) 0 0
\(199\) 9.24745 0.655534 0.327767 0.944759i \(-0.393704\pi\)
0.327767 + 0.944759i \(0.393704\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −12.2474 −0.861727
\(203\) −2.55051 −0.179011
\(204\) 0 0
\(205\) 0 0
\(206\) 15.4495 1.07642
\(207\) 0 0
\(208\) 1.44949 0.100504
\(209\) −10.8990 −0.753898
\(210\) 0 0
\(211\) −15.0000 −1.03264 −0.516321 0.856395i \(-0.672699\pi\)
−0.516321 + 0.856395i \(0.672699\pi\)
\(212\) 5.44949 0.374272
\(213\) 0 0
\(214\) 3.34847 0.228897
\(215\) 0 0
\(216\) 0 0
\(217\) 1.44949 0.0983978
\(218\) 15.1464 1.02585
\(219\) 0 0
\(220\) 0 0
\(221\) −1.44949 −0.0975032
\(222\) 0 0
\(223\) −4.00000 −0.267860 −0.133930 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −6.44949 −0.429014
\(227\) 11.6969 0.776353 0.388177 0.921585i \(-0.373105\pi\)
0.388177 + 0.921585i \(0.373105\pi\)
\(228\) 0 0
\(229\) −3.10102 −0.204921 −0.102461 0.994737i \(-0.532672\pi\)
−0.102461 + 0.994737i \(0.532672\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −2.55051 −0.167449
\(233\) 13.1464 0.861251 0.430626 0.902531i \(-0.358293\pi\)
0.430626 + 0.902531i \(0.358293\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0.101021 0.00657588
\(237\) 0 0
\(238\) −1.00000 −0.0648204
\(239\) 20.4949 1.32570 0.662852 0.748750i \(-0.269346\pi\)
0.662852 + 0.748750i \(0.269346\pi\)
\(240\) 0 0
\(241\) 10.6515 0.686125 0.343063 0.939313i \(-0.388536\pi\)
0.343063 + 0.939313i \(0.388536\pi\)
\(242\) 5.00000 0.321412
\(243\) 0 0
\(244\) 10.8990 0.697736
\(245\) 0 0
\(246\) 0 0
\(247\) −6.44949 −0.410371
\(248\) 1.44949 0.0920427
\(249\) 0 0
\(250\) 0 0
\(251\) 27.5959 1.74184 0.870919 0.491426i \(-0.163524\pi\)
0.870919 + 0.491426i \(0.163524\pi\)
\(252\) 0 0
\(253\) −3.55051 −0.223219
\(254\) 8.89898 0.558372
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 12.0000 0.748539 0.374270 0.927320i \(-0.377893\pi\)
0.374270 + 0.927320i \(0.377893\pi\)
\(258\) 0 0
\(259\) −3.34847 −0.208064
\(260\) 0 0
\(261\) 0 0
\(262\) 16.7980 1.03778
\(263\) −23.9444 −1.47647 −0.738237 0.674541i \(-0.764341\pi\)
−0.738237 + 0.674541i \(0.764341\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −4.44949 −0.272816
\(267\) 0 0
\(268\) −5.00000 −0.305424
\(269\) 3.10102 0.189073 0.0945363 0.995521i \(-0.469863\pi\)
0.0945363 + 0.995521i \(0.469863\pi\)
\(270\) 0 0
\(271\) −4.34847 −0.264151 −0.132075 0.991240i \(-0.542164\pi\)
−0.132075 + 0.991240i \(0.542164\pi\)
\(272\) −1.00000 −0.0606339
\(273\) 0 0
\(274\) 5.10102 0.308164
\(275\) 0 0
\(276\) 0 0
\(277\) −6.20204 −0.372645 −0.186322 0.982489i \(-0.559657\pi\)
−0.186322 + 0.982489i \(0.559657\pi\)
\(278\) −22.2474 −1.33431
\(279\) 0 0
\(280\) 0 0
\(281\) −13.3485 −0.796303 −0.398151 0.917320i \(-0.630348\pi\)
−0.398151 + 0.917320i \(0.630348\pi\)
\(282\) 0 0
\(283\) −17.5505 −1.04327 −0.521635 0.853169i \(-0.674678\pi\)
−0.521635 + 0.853169i \(0.674678\pi\)
\(284\) −5.44949 −0.323368
\(285\) 0 0
\(286\) −3.55051 −0.209946
\(287\) −4.89898 −0.289178
\(288\) 0 0
\(289\) −16.0000 −0.941176
\(290\) 0 0
\(291\) 0 0
\(292\) −11.7980 −0.690423
\(293\) −30.4949 −1.78153 −0.890765 0.454463i \(-0.849831\pi\)
−0.890765 + 0.454463i \(0.849831\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −3.34847 −0.194626
\(297\) 0 0
\(298\) −0.550510 −0.0318902
\(299\) −2.10102 −0.121505
\(300\) 0 0
\(301\) 3.89898 0.224733
\(302\) −12.2474 −0.704761
\(303\) 0 0
\(304\) −4.44949 −0.255196
\(305\) 0 0
\(306\) 0 0
\(307\) 24.4495 1.39541 0.697703 0.716387i \(-0.254206\pi\)
0.697703 + 0.716387i \(0.254206\pi\)
\(308\) −2.44949 −0.139573
\(309\) 0 0
\(310\) 0 0
\(311\) −13.3485 −0.756922 −0.378461 0.925617i \(-0.623547\pi\)
−0.378461 + 0.925617i \(0.623547\pi\)
\(312\) 0 0
\(313\) −28.9444 −1.63603 −0.818017 0.575194i \(-0.804926\pi\)
−0.818017 + 0.575194i \(0.804926\pi\)
\(314\) 5.44949 0.307532
\(315\) 0 0
\(316\) −6.44949 −0.362812
\(317\) 4.20204 0.236010 0.118005 0.993013i \(-0.462350\pi\)
0.118005 + 0.993013i \(0.462350\pi\)
\(318\) 0 0
\(319\) 6.24745 0.349790
\(320\) 0 0
\(321\) 0 0
\(322\) −1.44949 −0.0807769
\(323\) 4.44949 0.247576
\(324\) 0 0
\(325\) 0 0
\(326\) 4.10102 0.227135
\(327\) 0 0
\(328\) −4.89898 −0.270501
\(329\) 11.7980 0.650443
\(330\) 0 0
\(331\) −13.0000 −0.714545 −0.357272 0.934000i \(-0.616293\pi\)
−0.357272 + 0.934000i \(0.616293\pi\)
\(332\) 10.8990 0.598159
\(333\) 0 0
\(334\) 3.55051 0.194275
\(335\) 0 0
\(336\) 0 0
\(337\) −22.7980 −1.24188 −0.620942 0.783857i \(-0.713250\pi\)
−0.620942 + 0.783857i \(0.713250\pi\)
\(338\) 10.8990 0.592826
\(339\) 0 0
\(340\) 0 0
\(341\) −3.55051 −0.192271
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 3.89898 0.210219
\(345\) 0 0
\(346\) −11.1464 −0.599236
\(347\) 13.5959 0.729867 0.364934 0.931034i \(-0.381092\pi\)
0.364934 + 0.931034i \(0.381092\pi\)
\(348\) 0 0
\(349\) −20.1464 −1.07841 −0.539207 0.842173i \(-0.681276\pi\)
−0.539207 + 0.842173i \(0.681276\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −2.44949 −0.130558
\(353\) −15.4949 −0.824710 −0.412355 0.911023i \(-0.635294\pi\)
−0.412355 + 0.911023i \(0.635294\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 11.0000 0.582999
\(357\) 0 0
\(358\) 0.898979 0.0475125
\(359\) 21.4495 1.13206 0.566030 0.824384i \(-0.308478\pi\)
0.566030 + 0.824384i \(0.308478\pi\)
\(360\) 0 0
\(361\) 0.797959 0.0419978
\(362\) −7.44949 −0.391536
\(363\) 0 0
\(364\) −1.44949 −0.0759739
\(365\) 0 0
\(366\) 0 0
\(367\) 15.0454 0.785364 0.392682 0.919674i \(-0.371547\pi\)
0.392682 + 0.919674i \(0.371547\pi\)
\(368\) −1.44949 −0.0755599
\(369\) 0 0
\(370\) 0 0
\(371\) −5.44949 −0.282923
\(372\) 0 0
\(373\) −27.3485 −1.41605 −0.708025 0.706187i \(-0.750414\pi\)
−0.708025 + 0.706187i \(0.750414\pi\)
\(374\) 2.44949 0.126660
\(375\) 0 0
\(376\) 11.7980 0.608433
\(377\) 3.69694 0.190402
\(378\) 0 0
\(379\) 10.0000 0.513665 0.256833 0.966456i \(-0.417321\pi\)
0.256833 + 0.966456i \(0.417321\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 22.6969 1.16128
\(383\) −7.10102 −0.362845 −0.181423 0.983405i \(-0.558070\pi\)
−0.181423 + 0.983405i \(0.558070\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 7.20204 0.366574
\(387\) 0 0
\(388\) −9.55051 −0.484854
\(389\) −28.8990 −1.46524 −0.732618 0.680640i \(-0.761702\pi\)
−0.732618 + 0.680640i \(0.761702\pi\)
\(390\) 0 0
\(391\) 1.44949 0.0733038
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) −12.8990 −0.649841
\(395\) 0 0
\(396\) 0 0
\(397\) 22.6969 1.13913 0.569563 0.821947i \(-0.307112\pi\)
0.569563 + 0.821947i \(0.307112\pi\)
\(398\) −9.24745 −0.463533
\(399\) 0 0
\(400\) 0 0
\(401\) −8.69694 −0.434304 −0.217152 0.976138i \(-0.569677\pi\)
−0.217152 + 0.976138i \(0.569677\pi\)
\(402\) 0 0
\(403\) −2.10102 −0.104659
\(404\) 12.2474 0.609333
\(405\) 0 0
\(406\) 2.55051 0.126580
\(407\) 8.20204 0.406560
\(408\) 0 0
\(409\) −9.59592 −0.474488 −0.237244 0.971450i \(-0.576244\pi\)
−0.237244 + 0.971450i \(0.576244\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −15.4495 −0.761142
\(413\) −0.101021 −0.00497089
\(414\) 0 0
\(415\) 0 0
\(416\) −1.44949 −0.0710671
\(417\) 0 0
\(418\) 10.8990 0.533087
\(419\) −38.5959 −1.88553 −0.942767 0.333452i \(-0.891786\pi\)
−0.942767 + 0.333452i \(0.891786\pi\)
\(420\) 0 0
\(421\) 19.7980 0.964893 0.482447 0.875925i \(-0.339748\pi\)
0.482447 + 0.875925i \(0.339748\pi\)
\(422\) 15.0000 0.730189
\(423\) 0 0
\(424\) −5.44949 −0.264651
\(425\) 0 0
\(426\) 0 0
\(427\) −10.8990 −0.527439
\(428\) −3.34847 −0.161854
\(429\) 0 0
\(430\) 0 0
\(431\) −7.79796 −0.375614 −0.187807 0.982206i \(-0.560138\pi\)
−0.187807 + 0.982206i \(0.560138\pi\)
\(432\) 0 0
\(433\) −23.5959 −1.13395 −0.566974 0.823736i \(-0.691886\pi\)
−0.566974 + 0.823736i \(0.691886\pi\)
\(434\) −1.44949 −0.0695777
\(435\) 0 0
\(436\) −15.1464 −0.725382
\(437\) 6.44949 0.308521
\(438\) 0 0
\(439\) 6.34847 0.302996 0.151498 0.988458i \(-0.451590\pi\)
0.151498 + 0.988458i \(0.451590\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 1.44949 0.0689452
\(443\) −27.5505 −1.30896 −0.654482 0.756077i \(-0.727113\pi\)
−0.654482 + 0.756077i \(0.727113\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 4.00000 0.189405
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) 32.0000 1.51017 0.755087 0.655625i \(-0.227595\pi\)
0.755087 + 0.655625i \(0.227595\pi\)
\(450\) 0 0
\(451\) 12.0000 0.565058
\(452\) 6.44949 0.303358
\(453\) 0 0
\(454\) −11.6969 −0.548965
\(455\) 0 0
\(456\) 0 0
\(457\) −0.101021 −0.00472554 −0.00236277 0.999997i \(-0.500752\pi\)
−0.00236277 + 0.999997i \(0.500752\pi\)
\(458\) 3.10102 0.144901
\(459\) 0 0
\(460\) 0 0
\(461\) 10.0000 0.465746 0.232873 0.972507i \(-0.425187\pi\)
0.232873 + 0.972507i \(0.425187\pi\)
\(462\) 0 0
\(463\) −3.75255 −0.174396 −0.0871979 0.996191i \(-0.527791\pi\)
−0.0871979 + 0.996191i \(0.527791\pi\)
\(464\) 2.55051 0.118404
\(465\) 0 0
\(466\) −13.1464 −0.608997
\(467\) −12.2020 −0.564643 −0.282322 0.959320i \(-0.591105\pi\)
−0.282322 + 0.959320i \(0.591105\pi\)
\(468\) 0 0
\(469\) 5.00000 0.230879
\(470\) 0 0
\(471\) 0 0
\(472\) −0.101021 −0.00464985
\(473\) −9.55051 −0.439133
\(474\) 0 0
\(475\) 0 0
\(476\) 1.00000 0.0458349
\(477\) 0 0
\(478\) −20.4949 −0.937415
\(479\) −36.0000 −1.64488 −0.822441 0.568850i \(-0.807388\pi\)
−0.822441 + 0.568850i \(0.807388\pi\)
\(480\) 0 0
\(481\) 4.85357 0.221304
\(482\) −10.6515 −0.485164
\(483\) 0 0
\(484\) −5.00000 −0.227273
\(485\) 0 0
\(486\) 0 0
\(487\) −39.3939 −1.78511 −0.892553 0.450942i \(-0.851088\pi\)
−0.892553 + 0.450942i \(0.851088\pi\)
\(488\) −10.8990 −0.493374
\(489\) 0 0
\(490\) 0 0
\(491\) −4.40408 −0.198753 −0.0993767 0.995050i \(-0.531685\pi\)
−0.0993767 + 0.995050i \(0.531685\pi\)
\(492\) 0 0
\(493\) −2.55051 −0.114869
\(494\) 6.44949 0.290176
\(495\) 0 0
\(496\) −1.44949 −0.0650840
\(497\) 5.44949 0.244443
\(498\) 0 0
\(499\) 2.89898 0.129776 0.0648881 0.997893i \(-0.479331\pi\)
0.0648881 + 0.997893i \(0.479331\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −27.5959 −1.23167
\(503\) 6.65153 0.296577 0.148289 0.988944i \(-0.452624\pi\)
0.148289 + 0.988944i \(0.452624\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 3.55051 0.157839
\(507\) 0 0
\(508\) −8.89898 −0.394828
\(509\) 13.1464 0.582705 0.291353 0.956616i \(-0.405895\pi\)
0.291353 + 0.956616i \(0.405895\pi\)
\(510\) 0 0
\(511\) 11.7980 0.521911
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −12.0000 −0.529297
\(515\) 0 0
\(516\) 0 0
\(517\) −28.8990 −1.27098
\(518\) 3.34847 0.147123
\(519\) 0 0
\(520\) 0 0
\(521\) 3.00000 0.131432 0.0657162 0.997838i \(-0.479067\pi\)
0.0657162 + 0.997838i \(0.479067\pi\)
\(522\) 0 0
\(523\) 3.59592 0.157239 0.0786193 0.996905i \(-0.474949\pi\)
0.0786193 + 0.996905i \(0.474949\pi\)
\(524\) −16.7980 −0.733822
\(525\) 0 0
\(526\) 23.9444 1.04402
\(527\) 1.44949 0.0631408
\(528\) 0 0
\(529\) −20.8990 −0.908651
\(530\) 0 0
\(531\) 0 0
\(532\) 4.44949 0.192910
\(533\) 7.10102 0.307579
\(534\) 0 0
\(535\) 0 0
\(536\) 5.00000 0.215967
\(537\) 0 0
\(538\) −3.10102 −0.133694
\(539\) 2.44949 0.105507
\(540\) 0 0
\(541\) −9.75255 −0.419295 −0.209647 0.977777i \(-0.567232\pi\)
−0.209647 + 0.977777i \(0.567232\pi\)
\(542\) 4.34847 0.186783
\(543\) 0 0
\(544\) 1.00000 0.0428746
\(545\) 0 0
\(546\) 0 0
\(547\) −0.696938 −0.0297989 −0.0148995 0.999889i \(-0.504743\pi\)
−0.0148995 + 0.999889i \(0.504743\pi\)
\(548\) −5.10102 −0.217905
\(549\) 0 0
\(550\) 0 0
\(551\) −11.3485 −0.483461
\(552\) 0 0
\(553\) 6.44949 0.274260
\(554\) 6.20204 0.263499
\(555\) 0 0
\(556\) 22.2474 0.943502
\(557\) −33.0454 −1.40018 −0.700089 0.714055i \(-0.746857\pi\)
−0.700089 + 0.714055i \(0.746857\pi\)
\(558\) 0 0
\(559\) −5.65153 −0.239034
\(560\) 0 0
\(561\) 0 0
\(562\) 13.3485 0.563071
\(563\) 9.69694 0.408677 0.204339 0.978900i \(-0.434496\pi\)
0.204339 + 0.978900i \(0.434496\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 17.5505 0.737703
\(567\) 0 0
\(568\) 5.44949 0.228656
\(569\) 31.8434 1.33494 0.667472 0.744635i \(-0.267377\pi\)
0.667472 + 0.744635i \(0.267377\pi\)
\(570\) 0 0
\(571\) −19.4949 −0.815836 −0.407918 0.913019i \(-0.633745\pi\)
−0.407918 + 0.913019i \(0.633745\pi\)
\(572\) 3.55051 0.148454
\(573\) 0 0
\(574\) 4.89898 0.204479
\(575\) 0 0
\(576\) 0 0
\(577\) 19.5505 0.813898 0.406949 0.913451i \(-0.366593\pi\)
0.406949 + 0.913451i \(0.366593\pi\)
\(578\) 16.0000 0.665512
\(579\) 0 0
\(580\) 0 0
\(581\) −10.8990 −0.452166
\(582\) 0 0
\(583\) 13.3485 0.552837
\(584\) 11.7980 0.488203
\(585\) 0 0
\(586\) 30.4949 1.25973
\(587\) 35.4949 1.46503 0.732516 0.680750i \(-0.238346\pi\)
0.732516 + 0.680750i \(0.238346\pi\)
\(588\) 0 0
\(589\) 6.44949 0.265747
\(590\) 0 0
\(591\) 0 0
\(592\) 3.34847 0.137621
\(593\) −16.8990 −0.693958 −0.346979 0.937873i \(-0.612792\pi\)
−0.346979 + 0.937873i \(0.612792\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0.550510 0.0225498
\(597\) 0 0
\(598\) 2.10102 0.0859171
\(599\) −5.44949 −0.222660 −0.111330 0.993783i \(-0.535511\pi\)
−0.111330 + 0.993783i \(0.535511\pi\)
\(600\) 0 0
\(601\) −13.3031 −0.542643 −0.271322 0.962489i \(-0.587461\pi\)
−0.271322 + 0.962489i \(0.587461\pi\)
\(602\) −3.89898 −0.158911
\(603\) 0 0
\(604\) 12.2474 0.498342
\(605\) 0 0
\(606\) 0 0
\(607\) 23.0454 0.935384 0.467692 0.883891i \(-0.345086\pi\)
0.467692 + 0.883891i \(0.345086\pi\)
\(608\) 4.44949 0.180451
\(609\) 0 0
\(610\) 0 0
\(611\) −17.1010 −0.691833
\(612\) 0 0
\(613\) 48.2474 1.94870 0.974348 0.225046i \(-0.0722534\pi\)
0.974348 + 0.225046i \(0.0722534\pi\)
\(614\) −24.4495 −0.986701
\(615\) 0 0
\(616\) 2.44949 0.0986928
\(617\) 38.8990 1.56601 0.783007 0.622013i \(-0.213685\pi\)
0.783007 + 0.622013i \(0.213685\pi\)
\(618\) 0 0
\(619\) 43.7980 1.76039 0.880194 0.474614i \(-0.157412\pi\)
0.880194 + 0.474614i \(0.157412\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 13.3485 0.535225
\(623\) −11.0000 −0.440706
\(624\) 0 0
\(625\) 0 0
\(626\) 28.9444 1.15685
\(627\) 0 0
\(628\) −5.44949 −0.217458
\(629\) −3.34847 −0.133512
\(630\) 0 0
\(631\) 4.89898 0.195025 0.0975126 0.995234i \(-0.468911\pi\)
0.0975126 + 0.995234i \(0.468911\pi\)
\(632\) 6.44949 0.256547
\(633\) 0 0
\(634\) −4.20204 −0.166884
\(635\) 0 0
\(636\) 0 0
\(637\) 1.44949 0.0574309
\(638\) −6.24745 −0.247339
\(639\) 0 0
\(640\) 0 0
\(641\) −5.55051 −0.219232 −0.109616 0.993974i \(-0.534962\pi\)
−0.109616 + 0.993974i \(0.534962\pi\)
\(642\) 0 0
\(643\) −10.8536 −0.428023 −0.214012 0.976831i \(-0.568653\pi\)
−0.214012 + 0.976831i \(0.568653\pi\)
\(644\) 1.44949 0.0571179
\(645\) 0 0
\(646\) −4.44949 −0.175063
\(647\) −13.5959 −0.534511 −0.267255 0.963626i \(-0.586117\pi\)
−0.267255 + 0.963626i \(0.586117\pi\)
\(648\) 0 0
\(649\) 0.247449 0.00971321
\(650\) 0 0
\(651\) 0 0
\(652\) −4.10102 −0.160608
\(653\) −15.2474 −0.596679 −0.298339 0.954460i \(-0.596433\pi\)
−0.298339 + 0.954460i \(0.596433\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 4.89898 0.191273
\(657\) 0 0
\(658\) −11.7980 −0.459932
\(659\) −23.7526 −0.925268 −0.462634 0.886549i \(-0.653096\pi\)
−0.462634 + 0.886549i \(0.653096\pi\)
\(660\) 0 0
\(661\) 34.4949 1.34170 0.670848 0.741595i \(-0.265930\pi\)
0.670848 + 0.741595i \(0.265930\pi\)
\(662\) 13.0000 0.505259
\(663\) 0 0
\(664\) −10.8990 −0.422962
\(665\) 0 0
\(666\) 0 0
\(667\) −3.69694 −0.143146
\(668\) −3.55051 −0.137373
\(669\) 0 0
\(670\) 0 0
\(671\) 26.6969 1.03062
\(672\) 0 0
\(673\) −47.4949 −1.83079 −0.915397 0.402553i \(-0.868123\pi\)
−0.915397 + 0.402553i \(0.868123\pi\)
\(674\) 22.7980 0.878145
\(675\) 0 0
\(676\) −10.8990 −0.419192
\(677\) −13.7980 −0.530299 −0.265149 0.964207i \(-0.585421\pi\)
−0.265149 + 0.964207i \(0.585421\pi\)
\(678\) 0 0
\(679\) 9.55051 0.366515
\(680\) 0 0
\(681\) 0 0
\(682\) 3.55051 0.135956
\(683\) 5.55051 0.212384 0.106192 0.994346i \(-0.466134\pi\)
0.106192 + 0.994346i \(0.466134\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) −3.89898 −0.148647
\(689\) 7.89898 0.300927
\(690\) 0 0
\(691\) −18.2929 −0.695893 −0.347947 0.937514i \(-0.613121\pi\)
−0.347947 + 0.937514i \(0.613121\pi\)
\(692\) 11.1464 0.423724
\(693\) 0 0
\(694\) −13.5959 −0.516094
\(695\) 0 0
\(696\) 0 0
\(697\) −4.89898 −0.185562
\(698\) 20.1464 0.762554
\(699\) 0 0
\(700\) 0 0
\(701\) −39.3939 −1.48789 −0.743943 0.668243i \(-0.767047\pi\)
−0.743943 + 0.668243i \(0.767047\pi\)
\(702\) 0 0
\(703\) −14.8990 −0.561926
\(704\) 2.44949 0.0923186
\(705\) 0 0
\(706\) 15.4949 0.583158
\(707\) −12.2474 −0.460613
\(708\) 0 0
\(709\) 23.1464 0.869282 0.434641 0.900604i \(-0.356875\pi\)
0.434641 + 0.900604i \(0.356875\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −11.0000 −0.412242
\(713\) 2.10102 0.0786838
\(714\) 0 0
\(715\) 0 0
\(716\) −0.898979 −0.0335964
\(717\) 0 0
\(718\) −21.4495 −0.800488
\(719\) −15.1464 −0.564866 −0.282433 0.959287i \(-0.591142\pi\)
−0.282433 + 0.959287i \(0.591142\pi\)
\(720\) 0 0
\(721\) 15.4495 0.575369
\(722\) −0.797959 −0.0296970
\(723\) 0 0
\(724\) 7.44949 0.276858
\(725\) 0 0
\(726\) 0 0
\(727\) 3.65153 0.135428 0.0677139 0.997705i \(-0.478429\pi\)
0.0677139 + 0.997705i \(0.478429\pi\)
\(728\) 1.44949 0.0537217
\(729\) 0 0
\(730\) 0 0
\(731\) 3.89898 0.144209
\(732\) 0 0
\(733\) −37.4495 −1.38323 −0.691614 0.722267i \(-0.743100\pi\)
−0.691614 + 0.722267i \(0.743100\pi\)
\(734\) −15.0454 −0.555336
\(735\) 0 0
\(736\) 1.44949 0.0534289
\(737\) −12.2474 −0.451141
\(738\) 0 0
\(739\) 43.7980 1.61113 0.805567 0.592505i \(-0.201861\pi\)
0.805567 + 0.592505i \(0.201861\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 5.44949 0.200057
\(743\) 40.3485 1.48024 0.740121 0.672474i \(-0.234768\pi\)
0.740121 + 0.672474i \(0.234768\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 27.3485 1.00130
\(747\) 0 0
\(748\) −2.44949 −0.0895622
\(749\) 3.34847 0.122350
\(750\) 0 0
\(751\) −46.4495 −1.69497 −0.847483 0.530823i \(-0.821883\pi\)
−0.847483 + 0.530823i \(0.821883\pi\)
\(752\) −11.7980 −0.430227
\(753\) 0 0
\(754\) −3.69694 −0.134635
\(755\) 0 0
\(756\) 0 0
\(757\) 43.5959 1.58452 0.792260 0.610183i \(-0.208904\pi\)
0.792260 + 0.610183i \(0.208904\pi\)
\(758\) −10.0000 −0.363216
\(759\) 0 0
\(760\) 0 0
\(761\) −29.4949 −1.06919 −0.534595 0.845109i \(-0.679536\pi\)
−0.534595 + 0.845109i \(0.679536\pi\)
\(762\) 0 0
\(763\) 15.1464 0.548338
\(764\) −22.6969 −0.821146
\(765\) 0 0
\(766\) 7.10102 0.256570
\(767\) 0.146428 0.00528722
\(768\) 0 0
\(769\) −20.7423 −0.747988 −0.373994 0.927431i \(-0.622012\pi\)
−0.373994 + 0.927431i \(0.622012\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −7.20204 −0.259207
\(773\) 38.6969 1.39183 0.695916 0.718123i \(-0.254999\pi\)
0.695916 + 0.718123i \(0.254999\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 9.55051 0.342843
\(777\) 0 0
\(778\) 28.8990 1.03608
\(779\) −21.7980 −0.780993
\(780\) 0 0
\(781\) −13.3485 −0.477646
\(782\) −1.44949 −0.0518336
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 0 0
\(786\) 0 0
\(787\) −52.2929 −1.86404 −0.932020 0.362408i \(-0.881955\pi\)
−0.932020 + 0.362408i \(0.881955\pi\)
\(788\) 12.8990 0.459507
\(789\) 0 0
\(790\) 0 0
\(791\) −6.44949 −0.229317
\(792\) 0 0
\(793\) 15.7980 0.561002
\(794\) −22.6969 −0.805484
\(795\) 0 0
\(796\) 9.24745 0.327767
\(797\) 2.49490 0.0883738 0.0441869 0.999023i \(-0.485930\pi\)
0.0441869 + 0.999023i \(0.485930\pi\)
\(798\) 0 0
\(799\) 11.7980 0.417382
\(800\) 0 0
\(801\) 0 0
\(802\) 8.69694 0.307100
\(803\) −28.8990 −1.01982
\(804\) 0 0
\(805\) 0 0
\(806\) 2.10102 0.0740053
\(807\) 0 0
\(808\) −12.2474 −0.430864
\(809\) 6.20204 0.218052 0.109026 0.994039i \(-0.465227\pi\)
0.109026 + 0.994039i \(0.465227\pi\)
\(810\) 0 0
\(811\) −45.1918 −1.58690 −0.793450 0.608635i \(-0.791717\pi\)
−0.793450 + 0.608635i \(0.791717\pi\)
\(812\) −2.55051 −0.0895054
\(813\) 0 0
\(814\) −8.20204 −0.287481
\(815\) 0 0
\(816\) 0 0
\(817\) 17.3485 0.606946
\(818\) 9.59592 0.335513
\(819\) 0 0
\(820\) 0 0
\(821\) −53.4495 −1.86540 −0.932700 0.360653i \(-0.882554\pi\)
−0.932700 + 0.360653i \(0.882554\pi\)
\(822\) 0 0
\(823\) −55.8434 −1.94658 −0.973289 0.229585i \(-0.926263\pi\)
−0.973289 + 0.229585i \(0.926263\pi\)
\(824\) 15.4495 0.538208
\(825\) 0 0
\(826\) 0.101021 0.00351495
\(827\) −36.2929 −1.26203 −0.631013 0.775772i \(-0.717361\pi\)
−0.631013 + 0.775772i \(0.717361\pi\)
\(828\) 0 0
\(829\) −6.89898 −0.239611 −0.119806 0.992797i \(-0.538227\pi\)
−0.119806 + 0.992797i \(0.538227\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 1.44949 0.0502520
\(833\) −1.00000 −0.0346479
\(834\) 0 0
\(835\) 0 0
\(836\) −10.8990 −0.376949
\(837\) 0 0
\(838\) 38.5959 1.33327
\(839\) −42.4495 −1.46552 −0.732760 0.680488i \(-0.761768\pi\)
−0.732760 + 0.680488i \(0.761768\pi\)
\(840\) 0 0
\(841\) −22.4949 −0.775686
\(842\) −19.7980 −0.682283
\(843\) 0 0
\(844\) −15.0000 −0.516321
\(845\) 0 0
\(846\) 0 0
\(847\) 5.00000 0.171802
\(848\) 5.44949 0.187136
\(849\) 0 0
\(850\) 0 0
\(851\) −4.85357 −0.166378
\(852\) 0 0
\(853\) 13.2474 0.453584 0.226792 0.973943i \(-0.427176\pi\)
0.226792 + 0.973943i \(0.427176\pi\)
\(854\) 10.8990 0.372955
\(855\) 0 0
\(856\) 3.34847 0.114448
\(857\) −0.101021 −0.00345080 −0.00172540 0.999999i \(-0.500549\pi\)
−0.00172540 + 0.999999i \(0.500549\pi\)
\(858\) 0 0
\(859\) 20.9444 0.714613 0.357307 0.933987i \(-0.383695\pi\)
0.357307 + 0.933987i \(0.383695\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 7.79796 0.265600
\(863\) 0.348469 0.0118620 0.00593102 0.999982i \(-0.498112\pi\)
0.00593102 + 0.999982i \(0.498112\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 23.5959 0.801822
\(867\) 0 0
\(868\) 1.44949 0.0491989
\(869\) −15.7980 −0.535909
\(870\) 0 0
\(871\) −7.24745 −0.245570
\(872\) 15.1464 0.512923
\(873\) 0 0
\(874\) −6.44949 −0.218157
\(875\) 0 0
\(876\) 0 0
\(877\) −13.3031 −0.449212 −0.224606 0.974450i \(-0.572110\pi\)
−0.224606 + 0.974450i \(0.572110\pi\)
\(878\) −6.34847 −0.214250
\(879\) 0 0
\(880\) 0 0
\(881\) 13.2929 0.447848 0.223924 0.974607i \(-0.428113\pi\)
0.223924 + 0.974607i \(0.428113\pi\)
\(882\) 0 0
\(883\) −2.10102 −0.0707050 −0.0353525 0.999375i \(-0.511255\pi\)
−0.0353525 + 0.999375i \(0.511255\pi\)
\(884\) −1.44949 −0.0487516
\(885\) 0 0
\(886\) 27.5505 0.925577
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 8.89898 0.298462
\(890\) 0 0
\(891\) 0 0
\(892\) −4.00000 −0.133930
\(893\) 52.4949 1.75667
\(894\) 0 0
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) −32.0000 −1.06785
\(899\) −3.69694 −0.123300
\(900\) 0 0
\(901\) −5.44949 −0.181549
\(902\) −12.0000 −0.399556
\(903\) 0 0
\(904\) −6.44949 −0.214507
\(905\) 0 0
\(906\) 0 0
\(907\) 18.2020 0.604389 0.302194 0.953246i \(-0.402281\pi\)
0.302194 + 0.953246i \(0.402281\pi\)
\(908\) 11.6969 0.388177
\(909\) 0 0
\(910\) 0 0
\(911\) −6.20204 −0.205483 −0.102741 0.994708i \(-0.532761\pi\)
−0.102741 + 0.994708i \(0.532761\pi\)
\(912\) 0 0
\(913\) 26.6969 0.883540
\(914\) 0.101021 0.00334146
\(915\) 0 0
\(916\) −3.10102 −0.102461
\(917\) 16.7980 0.554717
\(918\) 0 0
\(919\) −17.5959 −0.580436 −0.290218 0.956961i \(-0.593728\pi\)
−0.290218 + 0.956961i \(0.593728\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −10.0000 −0.329332
\(923\) −7.89898 −0.259998
\(924\) 0 0
\(925\) 0 0
\(926\) 3.75255 0.123316
\(927\) 0 0
\(928\) −2.55051 −0.0837246
\(929\) −31.5959 −1.03663 −0.518314 0.855190i \(-0.673440\pi\)
−0.518314 + 0.855190i \(0.673440\pi\)
\(930\) 0 0
\(931\) −4.44949 −0.145826
\(932\) 13.1464 0.430626
\(933\) 0 0
\(934\) 12.2020 0.399263
\(935\) 0 0
\(936\) 0 0
\(937\) 34.9444 1.14158 0.570792 0.821095i \(-0.306636\pi\)
0.570792 + 0.821095i \(0.306636\pi\)
\(938\) −5.00000 −0.163256
\(939\) 0 0
\(940\) 0 0
\(941\) −29.5959 −0.964799 −0.482400 0.875951i \(-0.660235\pi\)
−0.482400 + 0.875951i \(0.660235\pi\)
\(942\) 0 0
\(943\) −7.10102 −0.231241
\(944\) 0.101021 0.00328794
\(945\) 0 0
\(946\) 9.55051 0.310514
\(947\) −28.0454 −0.911353 −0.455677 0.890145i \(-0.650603\pi\)
−0.455677 + 0.890145i \(0.650603\pi\)
\(948\) 0 0
\(949\) −17.1010 −0.555123
\(950\) 0 0
\(951\) 0 0
\(952\) −1.00000 −0.0324102
\(953\) 46.2929 1.49957 0.749786 0.661680i \(-0.230156\pi\)
0.749786 + 0.661680i \(0.230156\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 20.4949 0.662852
\(957\) 0 0
\(958\) 36.0000 1.16311
\(959\) 5.10102 0.164721
\(960\) 0 0
\(961\) −28.8990 −0.932225
\(962\) −4.85357 −0.156485
\(963\) 0 0
\(964\) 10.6515 0.343063
\(965\) 0 0
\(966\) 0 0
\(967\) −24.8990 −0.800697 −0.400349 0.916363i \(-0.631111\pi\)
−0.400349 + 0.916363i \(0.631111\pi\)
\(968\) 5.00000 0.160706
\(969\) 0 0
\(970\) 0 0
\(971\) 9.49490 0.304706 0.152353 0.988326i \(-0.451315\pi\)
0.152353 + 0.988326i \(0.451315\pi\)
\(972\) 0 0
\(973\) −22.2474 −0.713220
\(974\) 39.3939 1.26226
\(975\) 0 0
\(976\) 10.8990 0.348868
\(977\) −35.3939 −1.13235 −0.566175 0.824285i \(-0.691577\pi\)
−0.566175 + 0.824285i \(0.691577\pi\)
\(978\) 0 0
\(979\) 26.9444 0.861146
\(980\) 0 0
\(981\) 0 0
\(982\) 4.40408 0.140540
\(983\) 34.6969 1.10666 0.553330 0.832962i \(-0.313357\pi\)
0.553330 + 0.832962i \(0.313357\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 2.55051 0.0812248
\(987\) 0 0
\(988\) −6.44949 −0.205186
\(989\) 5.65153 0.179708
\(990\) 0 0
\(991\) 27.3939 0.870195 0.435098 0.900383i \(-0.356714\pi\)
0.435098 + 0.900383i \(0.356714\pi\)
\(992\) 1.44949 0.0460213
\(993\) 0 0
\(994\) −5.44949 −0.172847
\(995\) 0 0
\(996\) 0 0
\(997\) −49.7423 −1.57536 −0.787678 0.616087i \(-0.788717\pi\)
−0.787678 + 0.616087i \(0.788717\pi\)
\(998\) −2.89898 −0.0917656
\(999\) 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 9450.2.a.ea.1.2 2
3.2 odd 2 9450.2.a.ep.1.1 2
5.2 odd 4 1890.2.g.r.379.2 yes 4
5.3 odd 4 1890.2.g.r.379.4 yes 4
5.4 even 2 9450.2.a.ev.1.2 2
15.2 even 4 1890.2.g.m.379.3 yes 4
15.8 even 4 1890.2.g.m.379.1 4
15.14 odd 2 9450.2.a.ek.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1890.2.g.m.379.1 4 15.8 even 4
1890.2.g.m.379.3 yes 4 15.2 even 4
1890.2.g.r.379.2 yes 4 5.2 odd 4
1890.2.g.r.379.4 yes 4 5.3 odd 4
9450.2.a.ea.1.2 2 1.1 even 1 trivial
9450.2.a.ek.1.1 2 15.14 odd 2
9450.2.a.ep.1.1 2 3.2 odd 2
9450.2.a.ev.1.2 2 5.4 even 2