Properties

Label 7569.2.a.bg.1.1
Level $7569$
Weight $2$
Character 7569.1
Self dual yes
Analytic conductor $60.439$
Analytic rank $1$
Dimension $8$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7569,2,Mod(1,7569)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7569, 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("7569.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7569 = 3^{2} \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7569.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: \(60.4387692899\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: 8.8.14884000000.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 11x^{6} + 36x^{4} - 31x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.30927\) of defining polynomial
Character \(\chi\) \(=\) 7569.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.30927 q^{2} +3.33275 q^{4} -3.03582 q^{5} -4.39250 q^{7} -3.07768 q^{8} +7.01054 q^{10} +5.68046 q^{11} +3.95078 q^{13} +10.1435 q^{14} +0.441718 q^{16} +3.21721 q^{17} -3.61803 q^{19} -10.1176 q^{20} -13.1178 q^{22} -2.69640 q^{23} +4.21619 q^{25} -9.12344 q^{26} -14.6391 q^{28} -0.823684 q^{31} +5.13532 q^{32} -7.42943 q^{34} +13.3348 q^{35} -5.60575 q^{37} +8.35503 q^{38} +9.34328 q^{40} -0.558636 q^{41} -12.7130 q^{43} +18.9316 q^{44} +6.22672 q^{46} -0.129647 q^{47} +12.2941 q^{49} -9.73633 q^{50} +13.1670 q^{52} +6.88596 q^{53} -17.2449 q^{55} +13.5187 q^{56} -0.745036 q^{59} -7.17632 q^{61} +1.90211 q^{62} -12.7423 q^{64} -11.9939 q^{65} +7.06735 q^{67} +10.7222 q^{68} -30.7938 q^{70} -12.6641 q^{71} +8.81434 q^{73} +12.9452 q^{74} -12.0580 q^{76} -24.9514 q^{77} -6.99349 q^{79} -1.34098 q^{80} +1.29004 q^{82} +8.90268 q^{83} -9.76687 q^{85} +29.3577 q^{86} -17.4827 q^{88} -1.51344 q^{89} -17.3538 q^{91} -8.98641 q^{92} +0.299391 q^{94} +10.9837 q^{95} +14.7785 q^{97} -28.3904 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 6 q^{4} + 12 q^{10} + 2 q^{13} - 2 q^{16} - 20 q^{19} - 14 q^{22} + 2 q^{25} - 20 q^{28} - 10 q^{31} - 36 q^{34} - 18 q^{37} + 10 q^{40} - 28 q^{43} - 26 q^{46} + 4 q^{49} + 44 q^{52} - 14 q^{55}+ \cdots + 6 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.30927 −1.63290 −0.816452 0.577414i \(-0.804062\pi\)
−0.816452 + 0.577414i \(0.804062\pi\)
\(3\) 0 0
\(4\) 3.33275 1.66637
\(5\) −3.03582 −1.35766 −0.678829 0.734296i \(-0.737512\pi\)
−0.678829 + 0.734296i \(0.737512\pi\)
\(6\) 0 0
\(7\) −4.39250 −1.66021 −0.830105 0.557608i \(-0.811719\pi\)
−0.830105 + 0.557608i \(0.811719\pi\)
\(8\) −3.07768 −1.08813
\(9\) 0 0
\(10\) 7.01054 2.21693
\(11\) 5.68046 1.71272 0.856362 0.516376i \(-0.172719\pi\)
0.856362 + 0.516376i \(0.172719\pi\)
\(12\) 0 0
\(13\) 3.95078 1.09575 0.547875 0.836560i \(-0.315437\pi\)
0.547875 + 0.836560i \(0.315437\pi\)
\(14\) 10.1435 2.71096
\(15\) 0 0
\(16\) 0.441718 0.110430
\(17\) 3.21721 0.780289 0.390144 0.920754i \(-0.372425\pi\)
0.390144 + 0.920754i \(0.372425\pi\)
\(18\) 0 0
\(19\) −3.61803 −0.830034 −0.415017 0.909814i \(-0.636224\pi\)
−0.415017 + 0.909814i \(0.636224\pi\)
\(20\) −10.1176 −2.26237
\(21\) 0 0
\(22\) −13.1178 −2.79671
\(23\) −2.69640 −0.562238 −0.281119 0.959673i \(-0.590705\pi\)
−0.281119 + 0.959673i \(0.590705\pi\)
\(24\) 0 0
\(25\) 4.21619 0.843237
\(26\) −9.12344 −1.78925
\(27\) 0 0
\(28\) −14.6391 −2.76653
\(29\) 0 0
\(30\) 0 0
\(31\) −0.823684 −0.147938 −0.0739690 0.997261i \(-0.523567\pi\)
−0.0739690 + 0.997261i \(0.523567\pi\)
\(32\) 5.13532 0.907805
\(33\) 0 0
\(34\) −7.42943 −1.27414
\(35\) 13.3348 2.25400
\(36\) 0 0
\(37\) −5.60575 −0.921579 −0.460789 0.887509i \(-0.652434\pi\)
−0.460789 + 0.887509i \(0.652434\pi\)
\(38\) 8.35503 1.35537
\(39\) 0 0
\(40\) 9.34328 1.47730
\(41\) −0.558636 −0.0872442 −0.0436221 0.999048i \(-0.513890\pi\)
−0.0436221 + 0.999048i \(0.513890\pi\)
\(42\) 0 0
\(43\) −12.7130 −1.93871 −0.969354 0.245667i \(-0.920993\pi\)
−0.969354 + 0.245667i \(0.920993\pi\)
\(44\) 18.9316 2.85404
\(45\) 0 0
\(46\) 6.22672 0.918080
\(47\) −0.129647 −0.0189110 −0.00945548 0.999955i \(-0.503010\pi\)
−0.00945548 + 0.999955i \(0.503010\pi\)
\(48\) 0 0
\(49\) 12.2941 1.75630
\(50\) −9.73633 −1.37692
\(51\) 0 0
\(52\) 13.1670 1.82593
\(53\) 6.88596 0.945859 0.472929 0.881100i \(-0.343197\pi\)
0.472929 + 0.881100i \(0.343197\pi\)
\(54\) 0 0
\(55\) −17.2449 −2.32530
\(56\) 13.5187 1.80652
\(57\) 0 0
\(58\) 0 0
\(59\) −0.745036 −0.0969954 −0.0484977 0.998823i \(-0.515443\pi\)
−0.0484977 + 0.998823i \(0.515443\pi\)
\(60\) 0 0
\(61\) −7.17632 −0.918833 −0.459417 0.888221i \(-0.651941\pi\)
−0.459417 + 0.888221i \(0.651941\pi\)
\(62\) 1.90211 0.241569
\(63\) 0 0
\(64\) −12.7423 −1.59279
\(65\) −11.9939 −1.48765
\(66\) 0 0
\(67\) 7.06735 0.863414 0.431707 0.902014i \(-0.357911\pi\)
0.431707 + 0.902014i \(0.357911\pi\)
\(68\) 10.7222 1.30025
\(69\) 0 0
\(70\) −30.7938 −3.68056
\(71\) −12.6641 −1.50295 −0.751475 0.659761i \(-0.770657\pi\)
−0.751475 + 0.659761i \(0.770657\pi\)
\(72\) 0 0
\(73\) 8.81434 1.03164 0.515820 0.856697i \(-0.327487\pi\)
0.515820 + 0.856697i \(0.327487\pi\)
\(74\) 12.9452 1.50485
\(75\) 0 0
\(76\) −12.0580 −1.38315
\(77\) −24.9514 −2.84348
\(78\) 0 0
\(79\) −6.99349 −0.786829 −0.393414 0.919361i \(-0.628706\pi\)
−0.393414 + 0.919361i \(0.628706\pi\)
\(80\) −1.34098 −0.149926
\(81\) 0 0
\(82\) 1.29004 0.142461
\(83\) 8.90268 0.977196 0.488598 0.872509i \(-0.337508\pi\)
0.488598 + 0.872509i \(0.337508\pi\)
\(84\) 0 0
\(85\) −9.76687 −1.05937
\(86\) 29.3577 3.16572
\(87\) 0 0
\(88\) −17.4827 −1.86366
\(89\) −1.51344 −0.160425 −0.0802124 0.996778i \(-0.525560\pi\)
−0.0802124 + 0.996778i \(0.525560\pi\)
\(90\) 0 0
\(91\) −17.3538 −1.81917
\(92\) −8.98641 −0.936898
\(93\) 0 0
\(94\) 0.299391 0.0308798
\(95\) 10.9837 1.12690
\(96\) 0 0
\(97\) 14.7785 1.50053 0.750264 0.661138i \(-0.229926\pi\)
0.750264 + 0.661138i \(0.229926\pi\)
\(98\) −28.3904 −2.86786
\(99\) 0 0
\(100\) 14.0515 1.40515
\(101\) 7.59901 0.756130 0.378065 0.925779i \(-0.376590\pi\)
0.378065 + 0.925779i \(0.376590\pi\)
\(102\) 0 0
\(103\) 4.37262 0.430847 0.215423 0.976521i \(-0.430887\pi\)
0.215423 + 0.976521i \(0.430887\pi\)
\(104\) −12.1593 −1.19231
\(105\) 0 0
\(106\) −15.9016 −1.54450
\(107\) 18.2010 1.75955 0.879777 0.475386i \(-0.157692\pi\)
0.879777 + 0.475386i \(0.157692\pi\)
\(108\) 0 0
\(109\) −6.72525 −0.644162 −0.322081 0.946712i \(-0.604382\pi\)
−0.322081 + 0.946712i \(0.604382\pi\)
\(110\) 39.8231 3.79698
\(111\) 0 0
\(112\) −1.94025 −0.183336
\(113\) −4.84586 −0.455860 −0.227930 0.973678i \(-0.573196\pi\)
−0.227930 + 0.973678i \(0.573196\pi\)
\(114\) 0 0
\(115\) 8.18577 0.763327
\(116\) 0 0
\(117\) 0 0
\(118\) 1.72049 0.158384
\(119\) −14.1316 −1.29544
\(120\) 0 0
\(121\) 21.2677 1.93342
\(122\) 16.5721 1.50037
\(123\) 0 0
\(124\) −2.74513 −0.246520
\(125\) 2.37952 0.212831
\(126\) 0 0
\(127\) −5.90048 −0.523583 −0.261792 0.965124i \(-0.584313\pi\)
−0.261792 + 0.965124i \(0.584313\pi\)
\(128\) 19.1548 1.69306
\(129\) 0 0
\(130\) 27.6971 2.42920
\(131\) 0.678840 0.0593106 0.0296553 0.999560i \(-0.490559\pi\)
0.0296553 + 0.999560i \(0.490559\pi\)
\(132\) 0 0
\(133\) 15.8922 1.37803
\(134\) −16.3204 −1.40987
\(135\) 0 0
\(136\) −9.90157 −0.849052
\(137\) 16.9527 1.44836 0.724182 0.689609i \(-0.242218\pi\)
0.724182 + 0.689609i \(0.242218\pi\)
\(138\) 0 0
\(139\) −5.46461 −0.463502 −0.231751 0.972775i \(-0.574445\pi\)
−0.231751 + 0.972775i \(0.574445\pi\)
\(140\) 44.4416 3.75600
\(141\) 0 0
\(142\) 29.2449 2.45417
\(143\) 22.4423 1.87672
\(144\) 0 0
\(145\) 0 0
\(146\) −20.3547 −1.68457
\(147\) 0 0
\(148\) −18.6825 −1.53570
\(149\) 7.55310 0.618774 0.309387 0.950936i \(-0.399876\pi\)
0.309387 + 0.950936i \(0.399876\pi\)
\(150\) 0 0
\(151\) −1.69659 −0.138066 −0.0690331 0.997614i \(-0.521991\pi\)
−0.0690331 + 0.997614i \(0.521991\pi\)
\(152\) 11.1352 0.903181
\(153\) 0 0
\(154\) 57.6197 4.64313
\(155\) 2.50055 0.200849
\(156\) 0 0
\(157\) 18.1863 1.45142 0.725712 0.687999i \(-0.241511\pi\)
0.725712 + 0.687999i \(0.241511\pi\)
\(158\) 16.1499 1.28482
\(159\) 0 0
\(160\) −15.5899 −1.23249
\(161\) 11.8439 0.933432
\(162\) 0 0
\(163\) 11.1453 0.872970 0.436485 0.899712i \(-0.356223\pi\)
0.436485 + 0.899712i \(0.356223\pi\)
\(164\) −1.86179 −0.145382
\(165\) 0 0
\(166\) −20.5587 −1.59567
\(167\) 10.7469 0.831623 0.415811 0.909451i \(-0.363498\pi\)
0.415811 + 0.909451i \(0.363498\pi\)
\(168\) 0 0
\(169\) 2.60869 0.200668
\(170\) 22.5544 1.72984
\(171\) 0 0
\(172\) −42.3691 −3.23061
\(173\) −21.4326 −1.62949 −0.814746 0.579818i \(-0.803124\pi\)
−0.814746 + 0.579818i \(0.803124\pi\)
\(174\) 0 0
\(175\) −18.5196 −1.39995
\(176\) 2.50916 0.189135
\(177\) 0 0
\(178\) 3.49496 0.261958
\(179\) −16.5426 −1.23645 −0.618226 0.786000i \(-0.712148\pi\)
−0.618226 + 0.786000i \(0.712148\pi\)
\(180\) 0 0
\(181\) 14.5336 1.08028 0.540139 0.841576i \(-0.318372\pi\)
0.540139 + 0.841576i \(0.318372\pi\)
\(182\) 40.0747 2.97054
\(183\) 0 0
\(184\) 8.29866 0.611785
\(185\) 17.0180 1.25119
\(186\) 0 0
\(187\) 18.2753 1.33642
\(188\) −0.432081 −0.0315127
\(189\) 0 0
\(190\) −25.3644 −1.84012
\(191\) 5.79567 0.419360 0.209680 0.977770i \(-0.432758\pi\)
0.209680 + 0.977770i \(0.432758\pi\)
\(192\) 0 0
\(193\) −1.75917 −0.126628 −0.0633140 0.997994i \(-0.520167\pi\)
−0.0633140 + 0.997994i \(0.520167\pi\)
\(194\) −34.1276 −2.45022
\(195\) 0 0
\(196\) 40.9730 2.92665
\(197\) 23.9966 1.70969 0.854845 0.518884i \(-0.173652\pi\)
0.854845 + 0.518884i \(0.173652\pi\)
\(198\) 0 0
\(199\) −2.99349 −0.212203 −0.106101 0.994355i \(-0.533837\pi\)
−0.106101 + 0.994355i \(0.533837\pi\)
\(200\) −12.9761 −0.917548
\(201\) 0 0
\(202\) −17.5482 −1.23469
\(203\) 0 0
\(204\) 0 0
\(205\) 1.69592 0.118448
\(206\) −10.0976 −0.703532
\(207\) 0 0
\(208\) 1.74513 0.121003
\(209\) −20.5521 −1.42162
\(210\) 0 0
\(211\) −13.1300 −0.903909 −0.451955 0.892041i \(-0.649273\pi\)
−0.451955 + 0.892041i \(0.649273\pi\)
\(212\) 22.9492 1.57616
\(213\) 0 0
\(214\) −42.0310 −2.87318
\(215\) 38.5942 2.63210
\(216\) 0 0
\(217\) 3.61803 0.245608
\(218\) 15.5304 1.05185
\(219\) 0 0
\(220\) −57.4728 −3.87481
\(221\) 12.7105 0.855002
\(222\) 0 0
\(223\) −3.06343 −0.205142 −0.102571 0.994726i \(-0.532707\pi\)
−0.102571 + 0.994726i \(0.532707\pi\)
\(224\) −22.5569 −1.50715
\(225\) 0 0
\(226\) 11.1904 0.744376
\(227\) −3.82760 −0.254047 −0.127023 0.991900i \(-0.540542\pi\)
−0.127023 + 0.991900i \(0.540542\pi\)
\(228\) 0 0
\(229\) 2.27191 0.150132 0.0750661 0.997179i \(-0.476083\pi\)
0.0750661 + 0.997179i \(0.476083\pi\)
\(230\) −18.9032 −1.24644
\(231\) 0 0
\(232\) 0 0
\(233\) −5.59312 −0.366418 −0.183209 0.983074i \(-0.558648\pi\)
−0.183209 + 0.983074i \(0.558648\pi\)
\(234\) 0 0
\(235\) 0.393585 0.0256746
\(236\) −2.48302 −0.161631
\(237\) 0 0
\(238\) 32.6338 2.11533
\(239\) 2.70501 0.174972 0.0874862 0.996166i \(-0.472117\pi\)
0.0874862 + 0.996166i \(0.472117\pi\)
\(240\) 0 0
\(241\) −24.2707 −1.56341 −0.781706 0.623647i \(-0.785650\pi\)
−0.781706 + 0.623647i \(0.785650\pi\)
\(242\) −49.1129 −3.15710
\(243\) 0 0
\(244\) −23.9169 −1.53112
\(245\) −37.3225 −2.38445
\(246\) 0 0
\(247\) −14.2941 −0.909510
\(248\) 2.53504 0.160975
\(249\) 0 0
\(250\) −5.49496 −0.347532
\(251\) −23.3913 −1.47644 −0.738221 0.674558i \(-0.764334\pi\)
−0.738221 + 0.674558i \(0.764334\pi\)
\(252\) 0 0
\(253\) −15.3168 −0.962958
\(254\) 13.6258 0.854961
\(255\) 0 0
\(256\) −18.7492 −1.17182
\(257\) 23.5267 1.46756 0.733779 0.679388i \(-0.237755\pi\)
0.733779 + 0.679388i \(0.237755\pi\)
\(258\) 0 0
\(259\) 24.6232 1.53001
\(260\) −39.9725 −2.47899
\(261\) 0 0
\(262\) −1.56763 −0.0968484
\(263\) −12.2963 −0.758222 −0.379111 0.925351i \(-0.623770\pi\)
−0.379111 + 0.925351i \(0.623770\pi\)
\(264\) 0 0
\(265\) −20.9045 −1.28415
\(266\) −36.6995 −2.25019
\(267\) 0 0
\(268\) 23.5537 1.43877
\(269\) −23.1823 −1.41345 −0.706725 0.707488i \(-0.749828\pi\)
−0.706725 + 0.707488i \(0.749828\pi\)
\(270\) 0 0
\(271\) −14.0675 −0.854537 −0.427268 0.904125i \(-0.640524\pi\)
−0.427268 + 0.904125i \(0.640524\pi\)
\(272\) 1.42110 0.0861670
\(273\) 0 0
\(274\) −39.1483 −2.36504
\(275\) 23.9499 1.44423
\(276\) 0 0
\(277\) 2.23029 0.134005 0.0670026 0.997753i \(-0.478656\pi\)
0.0670026 + 0.997753i \(0.478656\pi\)
\(278\) 12.6193 0.756854
\(279\) 0 0
\(280\) −41.0404 −2.45263
\(281\) −9.75566 −0.581974 −0.290987 0.956727i \(-0.593984\pi\)
−0.290987 + 0.956727i \(0.593984\pi\)
\(282\) 0 0
\(283\) 27.4962 1.63448 0.817240 0.576297i \(-0.195503\pi\)
0.817240 + 0.576297i \(0.195503\pi\)
\(284\) −42.2062 −2.50448
\(285\) 0 0
\(286\) −51.8254 −3.06450
\(287\) 2.45381 0.144844
\(288\) 0 0
\(289\) −6.64954 −0.391149
\(290\) 0 0
\(291\) 0 0
\(292\) 29.3760 1.71910
\(293\) −13.8751 −0.810592 −0.405296 0.914186i \(-0.632832\pi\)
−0.405296 + 0.914186i \(0.632832\pi\)
\(294\) 0 0
\(295\) 2.26179 0.131687
\(296\) 17.2527 1.00279
\(297\) 0 0
\(298\) −17.4422 −1.01040
\(299\) −10.6529 −0.616072
\(300\) 0 0
\(301\) 55.8417 3.21866
\(302\) 3.91788 0.225449
\(303\) 0 0
\(304\) −1.59815 −0.0916603
\(305\) 21.7860 1.24746
\(306\) 0 0
\(307\) −9.49020 −0.541634 −0.270817 0.962631i \(-0.587294\pi\)
−0.270817 + 0.962631i \(0.587294\pi\)
\(308\) −83.1569 −4.73830
\(309\) 0 0
\(310\) −5.77447 −0.327968
\(311\) −18.7276 −1.06195 −0.530973 0.847389i \(-0.678173\pi\)
−0.530973 + 0.847389i \(0.678173\pi\)
\(312\) 0 0
\(313\) 22.9794 1.29888 0.649438 0.760415i \(-0.275004\pi\)
0.649438 + 0.760415i \(0.275004\pi\)
\(314\) −41.9971 −2.37004
\(315\) 0 0
\(316\) −23.3075 −1.31115
\(317\) −28.7899 −1.61700 −0.808501 0.588495i \(-0.799720\pi\)
−0.808501 + 0.588495i \(0.799720\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 38.6833 2.16246
\(321\) 0 0
\(322\) −27.3509 −1.52420
\(323\) −11.6400 −0.647666
\(324\) 0 0
\(325\) 16.6572 0.923977
\(326\) −25.7376 −1.42548
\(327\) 0 0
\(328\) 1.71930 0.0949327
\(329\) 0.569475 0.0313961
\(330\) 0 0
\(331\) 2.33575 0.128385 0.0641923 0.997938i \(-0.479553\pi\)
0.0641923 + 0.997938i \(0.479553\pi\)
\(332\) 29.6704 1.62837
\(333\) 0 0
\(334\) −24.8176 −1.35796
\(335\) −21.4552 −1.17222
\(336\) 0 0
\(337\) −2.36492 −0.128825 −0.0644127 0.997923i \(-0.520517\pi\)
−0.0644127 + 0.997923i \(0.520517\pi\)
\(338\) −6.02417 −0.327672
\(339\) 0 0
\(340\) −32.5505 −1.76530
\(341\) −4.67891 −0.253377
\(342\) 0 0
\(343\) −23.2542 −1.25561
\(344\) 39.1265 2.10956
\(345\) 0 0
\(346\) 49.4938 2.66080
\(347\) 34.2637 1.83937 0.919685 0.392657i \(-0.128444\pi\)
0.919685 + 0.392657i \(0.128444\pi\)
\(348\) 0 0
\(349\) 25.0903 1.34306 0.671528 0.740980i \(-0.265638\pi\)
0.671528 + 0.740980i \(0.265638\pi\)
\(350\) 42.7668 2.28598
\(351\) 0 0
\(352\) 29.1710 1.55482
\(353\) −9.88447 −0.526097 −0.263049 0.964783i \(-0.584728\pi\)
−0.263049 + 0.964783i \(0.584728\pi\)
\(354\) 0 0
\(355\) 38.4458 2.04049
\(356\) −5.04393 −0.267328
\(357\) 0 0
\(358\) 38.2014 2.01901
\(359\) −0.912501 −0.0481600 −0.0240800 0.999710i \(-0.507666\pi\)
−0.0240800 + 0.999710i \(0.507666\pi\)
\(360\) 0 0
\(361\) −5.90983 −0.311044
\(362\) −33.5622 −1.76399
\(363\) 0 0
\(364\) −57.8359 −3.03143
\(365\) −26.7587 −1.40062
\(366\) 0 0
\(367\) −21.9402 −1.14527 −0.572636 0.819810i \(-0.694079\pi\)
−0.572636 + 0.819810i \(0.694079\pi\)
\(368\) −1.19105 −0.0620876
\(369\) 0 0
\(370\) −39.2993 −2.04307
\(371\) −30.2466 −1.57032
\(372\) 0 0
\(373\) −19.1235 −0.990179 −0.495089 0.868842i \(-0.664865\pi\)
−0.495089 + 0.868842i \(0.664865\pi\)
\(374\) −42.2026 −2.18225
\(375\) 0 0
\(376\) 0.399012 0.0205775
\(377\) 0 0
\(378\) 0 0
\(379\) −10.6333 −0.546197 −0.273099 0.961986i \(-0.588049\pi\)
−0.273099 + 0.961986i \(0.588049\pi\)
\(380\) 36.6059 1.87784
\(381\) 0 0
\(382\) −13.3838 −0.684774
\(383\) −24.6435 −1.25922 −0.629612 0.776910i \(-0.716786\pi\)
−0.629612 + 0.776910i \(0.716786\pi\)
\(384\) 0 0
\(385\) 75.7480 3.86048
\(386\) 4.06242 0.206771
\(387\) 0 0
\(388\) 49.2530 2.50044
\(389\) −13.5454 −0.686781 −0.343390 0.939193i \(-0.611575\pi\)
−0.343390 + 0.939193i \(0.611575\pi\)
\(390\) 0 0
\(391\) −8.67489 −0.438708
\(392\) −37.8372 −1.91107
\(393\) 0 0
\(394\) −55.4148 −2.79176
\(395\) 21.2310 1.06825
\(396\) 0 0
\(397\) −31.1851 −1.56514 −0.782568 0.622566i \(-0.786090\pi\)
−0.782568 + 0.622566i \(0.786090\pi\)
\(398\) 6.91279 0.346507
\(399\) 0 0
\(400\) 1.86237 0.0931183
\(401\) 20.3905 1.01825 0.509126 0.860692i \(-0.329969\pi\)
0.509126 + 0.860692i \(0.329969\pi\)
\(402\) 0 0
\(403\) −3.25420 −0.162103
\(404\) 25.3256 1.26000
\(405\) 0 0
\(406\) 0 0
\(407\) −31.8432 −1.57841
\(408\) 0 0
\(409\) 3.54900 0.175487 0.0877434 0.996143i \(-0.472034\pi\)
0.0877434 + 0.996143i \(0.472034\pi\)
\(410\) −3.91634 −0.193414
\(411\) 0 0
\(412\) 14.5728 0.717952
\(413\) 3.27257 0.161033
\(414\) 0 0
\(415\) −27.0269 −1.32670
\(416\) 20.2885 0.994727
\(417\) 0 0
\(418\) 47.4605 2.32137
\(419\) 10.5213 0.514000 0.257000 0.966411i \(-0.417266\pi\)
0.257000 + 0.966411i \(0.417266\pi\)
\(420\) 0 0
\(421\) 8.27884 0.403486 0.201743 0.979439i \(-0.435339\pi\)
0.201743 + 0.979439i \(0.435339\pi\)
\(422\) 30.3209 1.47600
\(423\) 0 0
\(424\) −21.1928 −1.02921
\(425\) 13.5644 0.657969
\(426\) 0 0
\(427\) 31.5220 1.52546
\(428\) 60.6593 2.93208
\(429\) 0 0
\(430\) −89.1247 −4.29797
\(431\) 0.546934 0.0263449 0.0131724 0.999913i \(-0.495807\pi\)
0.0131724 + 0.999913i \(0.495807\pi\)
\(432\) 0 0
\(433\) 2.61555 0.125695 0.0628476 0.998023i \(-0.479982\pi\)
0.0628476 + 0.998023i \(0.479982\pi\)
\(434\) −8.35503 −0.401054
\(435\) 0 0
\(436\) −22.4136 −1.07342
\(437\) 9.75566 0.466676
\(438\) 0 0
\(439\) 33.4898 1.59838 0.799191 0.601078i \(-0.205262\pi\)
0.799191 + 0.601078i \(0.205262\pi\)
\(440\) 53.0742 2.53021
\(441\) 0 0
\(442\) −29.3521 −1.39614
\(443\) −19.1594 −0.910290 −0.455145 0.890417i \(-0.650413\pi\)
−0.455145 + 0.890417i \(0.650413\pi\)
\(444\) 0 0
\(445\) 4.59454 0.217802
\(446\) 7.07429 0.334978
\(447\) 0 0
\(448\) 55.9706 2.64436
\(449\) 26.1010 1.23178 0.615892 0.787831i \(-0.288796\pi\)
0.615892 + 0.787831i \(0.288796\pi\)
\(450\) 0 0
\(451\) −3.17331 −0.149425
\(452\) −16.1500 −0.759634
\(453\) 0 0
\(454\) 8.83898 0.414834
\(455\) 52.6830 2.46982
\(456\) 0 0
\(457\) 10.3432 0.483835 0.241918 0.970297i \(-0.422224\pi\)
0.241918 + 0.970297i \(0.422224\pi\)
\(458\) −5.24647 −0.245152
\(459\) 0 0
\(460\) 27.2811 1.27199
\(461\) 33.4724 1.55897 0.779483 0.626424i \(-0.215482\pi\)
0.779483 + 0.626424i \(0.215482\pi\)
\(462\) 0 0
\(463\) −11.2612 −0.523351 −0.261675 0.965156i \(-0.584275\pi\)
−0.261675 + 0.965156i \(0.584275\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 12.9161 0.598325
\(467\) −30.6211 −1.41697 −0.708487 0.705724i \(-0.750622\pi\)
−0.708487 + 0.705724i \(0.750622\pi\)
\(468\) 0 0
\(469\) −31.0433 −1.43345
\(470\) −0.908895 −0.0419242
\(471\) 0 0
\(472\) 2.29298 0.105543
\(473\) −72.2155 −3.32047
\(474\) 0 0
\(475\) −15.2543 −0.699915
\(476\) −47.0971 −2.15869
\(477\) 0 0
\(478\) −6.24660 −0.285713
\(479\) 1.29283 0.0590710 0.0295355 0.999564i \(-0.490597\pi\)
0.0295355 + 0.999564i \(0.490597\pi\)
\(480\) 0 0
\(481\) −22.1471 −1.00982
\(482\) 56.0477 2.55290
\(483\) 0 0
\(484\) 70.8798 3.22181
\(485\) −44.8648 −2.03721
\(486\) 0 0
\(487\) −0.933841 −0.0423164 −0.0211582 0.999776i \(-0.506735\pi\)
−0.0211582 + 0.999776i \(0.506735\pi\)
\(488\) 22.0864 0.999806
\(489\) 0 0
\(490\) 86.1880 3.89358
\(491\) −7.91557 −0.357225 −0.178612 0.983920i \(-0.557161\pi\)
−0.178612 + 0.983920i \(0.557161\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 33.0089 1.48514
\(495\) 0 0
\(496\) −0.363836 −0.0163367
\(497\) 55.6270 2.49521
\(498\) 0 0
\(499\) 3.89505 0.174367 0.0871833 0.996192i \(-0.472213\pi\)
0.0871833 + 0.996192i \(0.472213\pi\)
\(500\) 7.93033 0.354655
\(501\) 0 0
\(502\) 54.0168 2.41089
\(503\) 18.2986 0.815896 0.407948 0.913005i \(-0.366244\pi\)
0.407948 + 0.913005i \(0.366244\pi\)
\(504\) 0 0
\(505\) −23.0692 −1.02657
\(506\) 35.3707 1.57242
\(507\) 0 0
\(508\) −19.6648 −0.872486
\(509\) 0.297945 0.0132062 0.00660309 0.999978i \(-0.497898\pi\)
0.00660309 + 0.999978i \(0.497898\pi\)
\(510\) 0 0
\(511\) −38.7170 −1.71274
\(512\) 4.98730 0.220410
\(513\) 0 0
\(514\) −54.3297 −2.39638
\(515\) −13.2745 −0.584943
\(516\) 0 0
\(517\) −0.736455 −0.0323893
\(518\) −56.8618 −2.49837
\(519\) 0 0
\(520\) 36.9133 1.61875
\(521\) 21.6023 0.946415 0.473208 0.880951i \(-0.343096\pi\)
0.473208 + 0.880951i \(0.343096\pi\)
\(522\) 0 0
\(523\) −28.4828 −1.24546 −0.622732 0.782435i \(-0.713977\pi\)
−0.622732 + 0.782435i \(0.713977\pi\)
\(524\) 2.26240 0.0988336
\(525\) 0 0
\(526\) 28.3955 1.23810
\(527\) −2.64997 −0.115434
\(528\) 0 0
\(529\) −15.7294 −0.683889
\(530\) 48.2742 2.09690
\(531\) 0 0
\(532\) 52.9648 2.29631
\(533\) −2.20705 −0.0955979
\(534\) 0 0
\(535\) −55.2548 −2.38887
\(536\) −21.7511 −0.939503
\(537\) 0 0
\(538\) 53.5343 2.30803
\(539\) 69.8360 3.00805
\(540\) 0 0
\(541\) 31.9456 1.37345 0.686725 0.726917i \(-0.259048\pi\)
0.686725 + 0.726917i \(0.259048\pi\)
\(542\) 32.4856 1.39538
\(543\) 0 0
\(544\) 16.5214 0.708350
\(545\) 20.4166 0.874552
\(546\) 0 0
\(547\) −29.4737 −1.26020 −0.630102 0.776513i \(-0.716987\pi\)
−0.630102 + 0.776513i \(0.716987\pi\)
\(548\) 56.4990 2.41352
\(549\) 0 0
\(550\) −55.3069 −2.35829
\(551\) 0 0
\(552\) 0 0
\(553\) 30.7189 1.30630
\(554\) −5.15035 −0.218818
\(555\) 0 0
\(556\) −18.2122 −0.772368
\(557\) −7.13081 −0.302142 −0.151071 0.988523i \(-0.548272\pi\)
−0.151071 + 0.988523i \(0.548272\pi\)
\(558\) 0 0
\(559\) −50.2262 −2.12434
\(560\) 5.89024 0.248908
\(561\) 0 0
\(562\) 22.5285 0.950307
\(563\) −17.2395 −0.726558 −0.363279 0.931680i \(-0.618343\pi\)
−0.363279 + 0.931680i \(0.618343\pi\)
\(564\) 0 0
\(565\) 14.7111 0.618903
\(566\) −63.4963 −2.66895
\(567\) 0 0
\(568\) 38.9760 1.63540
\(569\) 25.0950 1.05204 0.526019 0.850473i \(-0.323684\pi\)
0.526019 + 0.850473i \(0.323684\pi\)
\(570\) 0 0
\(571\) 15.8448 0.663082 0.331541 0.943441i \(-0.392431\pi\)
0.331541 + 0.943441i \(0.392431\pi\)
\(572\) 74.7945 3.12731
\(573\) 0 0
\(574\) −5.66652 −0.236516
\(575\) −11.3685 −0.474100
\(576\) 0 0
\(577\) 10.7137 0.446017 0.223009 0.974816i \(-0.428412\pi\)
0.223009 + 0.974816i \(0.428412\pi\)
\(578\) 15.3556 0.638709
\(579\) 0 0
\(580\) 0 0
\(581\) −39.1050 −1.62235
\(582\) 0 0
\(583\) 39.1154 1.62000
\(584\) −27.1277 −1.12255
\(585\) 0 0
\(586\) 32.0414 1.32362
\(587\) 42.1108 1.73810 0.869049 0.494726i \(-0.164732\pi\)
0.869049 + 0.494726i \(0.164732\pi\)
\(588\) 0 0
\(589\) 2.98012 0.122794
\(590\) −5.22310 −0.215032
\(591\) 0 0
\(592\) −2.47616 −0.101770
\(593\) −42.1378 −1.73039 −0.865196 0.501433i \(-0.832806\pi\)
−0.865196 + 0.501433i \(0.832806\pi\)
\(594\) 0 0
\(595\) 42.9010 1.75877
\(596\) 25.1726 1.03111
\(597\) 0 0
\(598\) 24.6004 1.00599
\(599\) 28.2376 1.15376 0.576879 0.816830i \(-0.304270\pi\)
0.576879 + 0.816830i \(0.304270\pi\)
\(600\) 0 0
\(601\) −13.6703 −0.557621 −0.278811 0.960346i \(-0.589940\pi\)
−0.278811 + 0.960346i \(0.589940\pi\)
\(602\) −128.954 −5.25577
\(603\) 0 0
\(604\) −5.65429 −0.230070
\(605\) −64.5648 −2.62493
\(606\) 0 0
\(607\) 16.0259 0.650471 0.325235 0.945633i \(-0.394556\pi\)
0.325235 + 0.945633i \(0.394556\pi\)
\(608\) −18.5798 −0.753509
\(609\) 0 0
\(610\) −50.3098 −2.03698
\(611\) −0.512207 −0.0207217
\(612\) 0 0
\(613\) −19.2847 −0.778903 −0.389451 0.921047i \(-0.627335\pi\)
−0.389451 + 0.921047i \(0.627335\pi\)
\(614\) 21.9155 0.884437
\(615\) 0 0
\(616\) 76.7927 3.09406
\(617\) −18.8384 −0.758405 −0.379202 0.925314i \(-0.623802\pi\)
−0.379202 + 0.925314i \(0.623802\pi\)
\(618\) 0 0
\(619\) −43.8366 −1.76194 −0.880970 0.473172i \(-0.843109\pi\)
−0.880970 + 0.473172i \(0.843109\pi\)
\(620\) 8.33372 0.334690
\(621\) 0 0
\(622\) 43.2472 1.73406
\(623\) 6.64781 0.266339
\(624\) 0 0
\(625\) −28.3047 −1.13219
\(626\) −53.0659 −2.12094
\(627\) 0 0
\(628\) 60.6103 2.41862
\(629\) −18.0349 −0.719098
\(630\) 0 0
\(631\) 4.64369 0.184863 0.0924313 0.995719i \(-0.470536\pi\)
0.0924313 + 0.995719i \(0.470536\pi\)
\(632\) 21.5237 0.856169
\(633\) 0 0
\(634\) 66.4837 2.64041
\(635\) 17.9128 0.710847
\(636\) 0 0
\(637\) 48.5712 1.92446
\(638\) 0 0
\(639\) 0 0
\(640\) −58.1505 −2.29860
\(641\) 45.8894 1.81252 0.906261 0.422718i \(-0.138924\pi\)
0.906261 + 0.422718i \(0.138924\pi\)
\(642\) 0 0
\(643\) 19.8523 0.782898 0.391449 0.920200i \(-0.371974\pi\)
0.391449 + 0.920200i \(0.371974\pi\)
\(644\) 39.4728 1.55545
\(645\) 0 0
\(646\) 26.8799 1.05758
\(647\) 19.9161 0.782982 0.391491 0.920182i \(-0.371959\pi\)
0.391491 + 0.920182i \(0.371959\pi\)
\(648\) 0 0
\(649\) −4.23215 −0.166126
\(650\) −38.4661 −1.50877
\(651\) 0 0
\(652\) 37.1446 1.45469
\(653\) −22.3060 −0.872900 −0.436450 0.899729i \(-0.643764\pi\)
−0.436450 + 0.899729i \(0.643764\pi\)
\(654\) 0 0
\(655\) −2.06084 −0.0805235
\(656\) −0.246760 −0.00963434
\(657\) 0 0
\(658\) −1.31507 −0.0512669
\(659\) 6.11978 0.238393 0.119196 0.992871i \(-0.461968\pi\)
0.119196 + 0.992871i \(0.461968\pi\)
\(660\) 0 0
\(661\) 30.5599 1.18864 0.594322 0.804227i \(-0.297421\pi\)
0.594322 + 0.804227i \(0.297421\pi\)
\(662\) −5.39390 −0.209640
\(663\) 0 0
\(664\) −27.3996 −1.06331
\(665\) −48.2459 −1.87089
\(666\) 0 0
\(667\) 0 0
\(668\) 35.8168 1.38580
\(669\) 0 0
\(670\) 49.5459 1.91412
\(671\) −40.7648 −1.57371
\(672\) 0 0
\(673\) −38.5751 −1.48696 −0.743481 0.668757i \(-0.766826\pi\)
−0.743481 + 0.668757i \(0.766826\pi\)
\(674\) 5.46125 0.210359
\(675\) 0 0
\(676\) 8.69410 0.334388
\(677\) −17.6382 −0.677890 −0.338945 0.940806i \(-0.610070\pi\)
−0.338945 + 0.940806i \(0.610070\pi\)
\(678\) 0 0
\(679\) −64.9145 −2.49119
\(680\) 30.0593 1.15272
\(681\) 0 0
\(682\) 10.8049 0.413740
\(683\) −4.65446 −0.178098 −0.0890489 0.996027i \(-0.528383\pi\)
−0.0890489 + 0.996027i \(0.528383\pi\)
\(684\) 0 0
\(685\) −51.4652 −1.96638
\(686\) 53.7003 2.05029
\(687\) 0 0
\(688\) −5.61555 −0.214091
\(689\) 27.2049 1.03643
\(690\) 0 0
\(691\) 9.86464 0.375268 0.187634 0.982239i \(-0.439918\pi\)
0.187634 + 0.982239i \(0.439918\pi\)
\(692\) −71.4296 −2.71535
\(693\) 0 0
\(694\) −79.1242 −3.00351
\(695\) 16.5895 0.629277
\(696\) 0 0
\(697\) −1.79725 −0.0680757
\(698\) −57.9405 −2.19308
\(699\) 0 0
\(700\) −61.7212 −2.33284
\(701\) 17.9403 0.677597 0.338798 0.940859i \(-0.389980\pi\)
0.338798 + 0.940859i \(0.389980\pi\)
\(702\) 0 0
\(703\) 20.2818 0.764942
\(704\) −72.3822 −2.72801
\(705\) 0 0
\(706\) 22.8260 0.859066
\(707\) −33.3787 −1.25533
\(708\) 0 0
\(709\) −12.6352 −0.474524 −0.237262 0.971446i \(-0.576250\pi\)
−0.237262 + 0.971446i \(0.576250\pi\)
\(710\) −88.7820 −3.33193
\(711\) 0 0
\(712\) 4.65790 0.174562
\(713\) 2.22098 0.0831763
\(714\) 0 0
\(715\) −68.1307 −2.54794
\(716\) −55.1323 −2.06039
\(717\) 0 0
\(718\) 2.10722 0.0786406
\(719\) −45.4419 −1.69470 −0.847348 0.531039i \(-0.821802\pi\)
−0.847348 + 0.531039i \(0.821802\pi\)
\(720\) 0 0
\(721\) −19.2067 −0.715296
\(722\) 13.6474 0.507904
\(723\) 0 0
\(724\) 48.4370 1.80015
\(725\) 0 0
\(726\) 0 0
\(727\) −39.0300 −1.44754 −0.723771 0.690041i \(-0.757593\pi\)
−0.723771 + 0.690041i \(0.757593\pi\)
\(728\) 53.4096 1.97949
\(729\) 0 0
\(730\) 61.7932 2.28707
\(731\) −40.9003 −1.51275
\(732\) 0 0
\(733\) −20.8126 −0.768730 −0.384365 0.923181i \(-0.625580\pi\)
−0.384365 + 0.923181i \(0.625580\pi\)
\(734\) 50.6661 1.87012
\(735\) 0 0
\(736\) −13.8469 −0.510402
\(737\) 40.1458 1.47879
\(738\) 0 0
\(739\) −39.5723 −1.45569 −0.727845 0.685742i \(-0.759478\pi\)
−0.727845 + 0.685742i \(0.759478\pi\)
\(740\) 56.7168 2.08495
\(741\) 0 0
\(742\) 69.8476 2.56419
\(743\) 36.0498 1.32254 0.661270 0.750148i \(-0.270018\pi\)
0.661270 + 0.750148i \(0.270018\pi\)
\(744\) 0 0
\(745\) −22.9298 −0.840084
\(746\) 44.1615 1.61687
\(747\) 0 0
\(748\) 60.9069 2.22698
\(749\) −79.9478 −2.92123
\(750\) 0 0
\(751\) −10.0995 −0.368537 −0.184268 0.982876i \(-0.558992\pi\)
−0.184268 + 0.982876i \(0.558992\pi\)
\(752\) −0.0572674 −0.00208833
\(753\) 0 0
\(754\) 0 0
\(755\) 5.15052 0.187447
\(756\) 0 0
\(757\) −27.3486 −0.994000 −0.497000 0.867750i \(-0.665565\pi\)
−0.497000 + 0.867750i \(0.665565\pi\)
\(758\) 24.5553 0.891888
\(759\) 0 0
\(760\) −33.8043 −1.22621
\(761\) −20.1609 −0.730834 −0.365417 0.930844i \(-0.619074\pi\)
−0.365417 + 0.930844i \(0.619074\pi\)
\(762\) 0 0
\(763\) 29.5407 1.06944
\(764\) 19.3155 0.698810
\(765\) 0 0
\(766\) 56.9086 2.05619
\(767\) −2.94347 −0.106283
\(768\) 0 0
\(769\) −4.25483 −0.153433 −0.0767165 0.997053i \(-0.524444\pi\)
−0.0767165 + 0.997053i \(0.524444\pi\)
\(770\) −174.923 −6.30379
\(771\) 0 0
\(772\) −5.86288 −0.211010
\(773\) −40.4349 −1.45434 −0.727171 0.686457i \(-0.759165\pi\)
−0.727171 + 0.686457i \(0.759165\pi\)
\(774\) 0 0
\(775\) −3.47281 −0.124747
\(776\) −45.4835 −1.63276
\(777\) 0 0
\(778\) 31.2801 1.12145
\(779\) 2.02116 0.0724157
\(780\) 0 0
\(781\) −71.9379 −2.57414
\(782\) 20.0327 0.716368
\(783\) 0 0
\(784\) 5.43051 0.193947
\(785\) −55.2102 −1.97054
\(786\) 0 0
\(787\) 24.7692 0.882927 0.441463 0.897279i \(-0.354459\pi\)
0.441463 + 0.897279i \(0.354459\pi\)
\(788\) 79.9748 2.84898
\(789\) 0 0
\(790\) −49.0281 −1.74434
\(791\) 21.2854 0.756823
\(792\) 0 0
\(793\) −28.3521 −1.00681
\(794\) 72.0150 2.55572
\(795\) 0 0
\(796\) −9.97655 −0.353609
\(797\) −1.60802 −0.0569589 −0.0284794 0.999594i \(-0.509067\pi\)
−0.0284794 + 0.999594i \(0.509067\pi\)
\(798\) 0 0
\(799\) −0.417102 −0.0147560
\(800\) 21.6515 0.765495
\(801\) 0 0
\(802\) −47.0872 −1.66271
\(803\) 50.0695 1.76692
\(804\) 0 0
\(805\) −35.9560 −1.26728
\(806\) 7.51484 0.264699
\(807\) 0 0
\(808\) −23.3873 −0.822764
\(809\) 2.09411 0.0736249 0.0368124 0.999322i \(-0.488280\pi\)
0.0368124 + 0.999322i \(0.488280\pi\)
\(810\) 0 0
\(811\) 26.6256 0.934952 0.467476 0.884006i \(-0.345163\pi\)
0.467476 + 0.884006i \(0.345163\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 73.5348 2.57739
\(815\) −33.8352 −1.18520
\(816\) 0 0
\(817\) 45.9959 1.60919
\(818\) −8.19561 −0.286553
\(819\) 0 0
\(820\) 5.65206 0.197379
\(821\) −6.39578 −0.223214 −0.111607 0.993752i \(-0.535600\pi\)
−0.111607 + 0.993752i \(0.535600\pi\)
\(822\) 0 0
\(823\) 19.8845 0.693131 0.346565 0.938026i \(-0.387348\pi\)
0.346565 + 0.938026i \(0.387348\pi\)
\(824\) −13.4575 −0.468816
\(825\) 0 0
\(826\) −7.55726 −0.262951
\(827\) 40.8266 1.41968 0.709839 0.704364i \(-0.248768\pi\)
0.709839 + 0.704364i \(0.248768\pi\)
\(828\) 0 0
\(829\) −4.59538 −0.159604 −0.0798021 0.996811i \(-0.525429\pi\)
−0.0798021 + 0.996811i \(0.525429\pi\)
\(830\) 62.4126 2.16637
\(831\) 0 0
\(832\) −50.3421 −1.74530
\(833\) 39.5526 1.37042
\(834\) 0 0
\(835\) −32.6257 −1.12906
\(836\) −68.4950 −2.36895
\(837\) 0 0
\(838\) −24.2966 −0.839313
\(839\) −38.9935 −1.34620 −0.673102 0.739549i \(-0.735039\pi\)
−0.673102 + 0.739549i \(0.735039\pi\)
\(840\) 0 0
\(841\) 0 0
\(842\) −19.1181 −0.658854
\(843\) 0 0
\(844\) −43.7591 −1.50625
\(845\) −7.91950 −0.272439
\(846\) 0 0
\(847\) −93.4183 −3.20989
\(848\) 3.04165 0.104451
\(849\) 0 0
\(850\) −31.3239 −1.07440
\(851\) 15.1153 0.518146
\(852\) 0 0
\(853\) −40.7481 −1.39519 −0.697595 0.716492i \(-0.745746\pi\)
−0.697595 + 0.716492i \(0.745746\pi\)
\(854\) −72.7929 −2.49092
\(855\) 0 0
\(856\) −56.0168 −1.91462
\(857\) −44.6261 −1.52440 −0.762200 0.647342i \(-0.775881\pi\)
−0.762200 + 0.647342i \(0.775881\pi\)
\(858\) 0 0
\(859\) −18.5044 −0.631361 −0.315680 0.948866i \(-0.602233\pi\)
−0.315680 + 0.948866i \(0.602233\pi\)
\(860\) 128.625 4.38607
\(861\) 0 0
\(862\) −1.26302 −0.0430186
\(863\) 9.37702 0.319197 0.159599 0.987182i \(-0.448980\pi\)
0.159599 + 0.987182i \(0.448980\pi\)
\(864\) 0 0
\(865\) 65.0655 2.21229
\(866\) −6.04002 −0.205248
\(867\) 0 0
\(868\) 12.0580 0.409275
\(869\) −39.7263 −1.34762
\(870\) 0 0
\(871\) 27.9216 0.946086
\(872\) 20.6982 0.700929
\(873\) 0 0
\(874\) −22.5285 −0.762037
\(875\) −10.4520 −0.353343
\(876\) 0 0
\(877\) −5.93854 −0.200530 −0.100265 0.994961i \(-0.531969\pi\)
−0.100265 + 0.994961i \(0.531969\pi\)
\(878\) −77.3372 −2.61000
\(879\) 0 0
\(880\) −7.61736 −0.256781
\(881\) −1.02910 −0.0346712 −0.0173356 0.999850i \(-0.505518\pi\)
−0.0173356 + 0.999850i \(0.505518\pi\)
\(882\) 0 0
\(883\) −28.3201 −0.953046 −0.476523 0.879162i \(-0.658103\pi\)
−0.476523 + 0.879162i \(0.658103\pi\)
\(884\) 42.3610 1.42475
\(885\) 0 0
\(886\) 44.2443 1.48642
\(887\) 11.9079 0.399827 0.199913 0.979814i \(-0.435934\pi\)
0.199913 + 0.979814i \(0.435934\pi\)
\(888\) 0 0
\(889\) 25.9179 0.869258
\(890\) −10.6101 −0.355650
\(891\) 0 0
\(892\) −10.2096 −0.341844
\(893\) 0.469067 0.0156967
\(894\) 0 0
\(895\) 50.2203 1.67868
\(896\) −84.1376 −2.81084
\(897\) 0 0
\(898\) −60.2744 −2.01138
\(899\) 0 0
\(900\) 0 0
\(901\) 22.1536 0.738043
\(902\) 7.32804 0.243997
\(903\) 0 0
\(904\) 14.9140 0.496033
\(905\) −44.1215 −1.46665
\(906\) 0 0
\(907\) 17.7879 0.590637 0.295318 0.955399i \(-0.404574\pi\)
0.295318 + 0.955399i \(0.404574\pi\)
\(908\) −12.7564 −0.423337
\(909\) 0 0
\(910\) −121.660 −4.03298
\(911\) −41.3338 −1.36945 −0.684725 0.728802i \(-0.740078\pi\)
−0.684725 + 0.728802i \(0.740078\pi\)
\(912\) 0 0
\(913\) 50.5714 1.67367
\(914\) −23.8853 −0.790057
\(915\) 0 0
\(916\) 7.57172 0.250177
\(917\) −2.98181 −0.0984680
\(918\) 0 0
\(919\) 26.3691 0.869837 0.434918 0.900470i \(-0.356777\pi\)
0.434918 + 0.900470i \(0.356777\pi\)
\(920\) −25.1932 −0.830595
\(921\) 0 0
\(922\) −77.2970 −2.54564
\(923\) −50.0331 −1.64686
\(924\) 0 0
\(925\) −23.6349 −0.777110
\(926\) 26.0051 0.854581
\(927\) 0 0
\(928\) 0 0
\(929\) 40.6550 1.33385 0.666923 0.745127i \(-0.267611\pi\)
0.666923 + 0.745127i \(0.267611\pi\)
\(930\) 0 0
\(931\) −44.4804 −1.45778
\(932\) −18.6405 −0.610589
\(933\) 0 0
\(934\) 70.7125 2.31378
\(935\) −55.4804 −1.81440
\(936\) 0 0
\(937\) −14.8322 −0.484548 −0.242274 0.970208i \(-0.577893\pi\)
−0.242274 + 0.970208i \(0.577893\pi\)
\(938\) 71.6876 2.34068
\(939\) 0 0
\(940\) 1.31172 0.0427835
\(941\) −36.7391 −1.19766 −0.598830 0.800876i \(-0.704368\pi\)
−0.598830 + 0.800876i \(0.704368\pi\)
\(942\) 0 0
\(943\) 1.50630 0.0490520
\(944\) −0.329096 −0.0107112
\(945\) 0 0
\(946\) 166.765 5.42201
\(947\) 24.7731 0.805019 0.402509 0.915416i \(-0.368138\pi\)
0.402509 + 0.915416i \(0.368138\pi\)
\(948\) 0 0
\(949\) 34.8235 1.13042
\(950\) 35.2264 1.14289
\(951\) 0 0
\(952\) 43.4926 1.40960
\(953\) −25.3471 −0.821073 −0.410537 0.911844i \(-0.634659\pi\)
−0.410537 + 0.911844i \(0.634659\pi\)
\(954\) 0 0
\(955\) −17.5946 −0.569347
\(956\) 9.01511 0.291569
\(957\) 0 0
\(958\) −2.98551 −0.0964573
\(959\) −74.4646 −2.40459
\(960\) 0 0
\(961\) −30.3215 −0.978114
\(962\) 51.1437 1.64894
\(963\) 0 0
\(964\) −80.8881 −2.60523
\(965\) 5.34053 0.171918
\(966\) 0 0
\(967\) 16.1414 0.519072 0.259536 0.965733i \(-0.416430\pi\)
0.259536 + 0.965733i \(0.416430\pi\)
\(968\) −65.4552 −2.10381
\(969\) 0 0
\(970\) 103.605 3.32656
\(971\) −39.1025 −1.25486 −0.627430 0.778673i \(-0.715893\pi\)
−0.627430 + 0.778673i \(0.715893\pi\)
\(972\) 0 0
\(973\) 24.0033 0.769510
\(974\) 2.15650 0.0690986
\(975\) 0 0
\(976\) −3.16991 −0.101466
\(977\) 0.922826 0.0295238 0.0147619 0.999891i \(-0.495301\pi\)
0.0147619 + 0.999891i \(0.495301\pi\)
\(978\) 0 0
\(979\) −8.59707 −0.274763
\(980\) −124.387 −3.97339
\(981\) 0 0
\(982\) 18.2792 0.583314
\(983\) 31.5174 1.00525 0.502625 0.864505i \(-0.332368\pi\)
0.502625 + 0.864505i \(0.332368\pi\)
\(984\) 0 0
\(985\) −72.8494 −2.32117
\(986\) 0 0
\(987\) 0 0
\(988\) −47.6385 −1.51558
\(989\) 34.2792 1.09001
\(990\) 0 0
\(991\) −35.3884 −1.12415 −0.562075 0.827086i \(-0.689997\pi\)
−0.562075 + 0.827086i \(0.689997\pi\)
\(992\) −4.22988 −0.134299
\(993\) 0 0
\(994\) −128.458 −4.07444
\(995\) 9.08768 0.288099
\(996\) 0 0
\(997\) 23.5611 0.746188 0.373094 0.927794i \(-0.378297\pi\)
0.373094 + 0.927794i \(0.378297\pi\)
\(998\) −8.99475 −0.284724
\(999\) 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 7569.2.a.bg.1.1 yes 8
3.2 odd 2 inner 7569.2.a.bg.1.8 yes 8
29.28 even 2 7569.2.a.bf.1.8 yes 8
87.86 odd 2 7569.2.a.bf.1.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7569.2.a.bf.1.1 8 87.86 odd 2
7569.2.a.bf.1.8 yes 8 29.28 even 2
7569.2.a.bg.1.1 yes 8 1.1 even 1 trivial
7569.2.a.bg.1.8 yes 8 3.2 odd 2 inner