Properties

Label 207.2.a.d.1.2
Level $207$
Weight $2$
Character 207.1
Self dual yes
Analytic conductor $1.653$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [207,2,Mod(1,207)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(207, 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("207.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 207 = 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 207.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: \(1.65290332184\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 23)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 207.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.61803 q^{2} +0.618034 q^{4} +3.23607 q^{5} -1.23607 q^{7} -2.23607 q^{8} +5.23607 q^{10} +0.763932 q^{11} +3.00000 q^{13} -2.00000 q^{14} -4.85410 q^{16} -5.23607 q^{17} -2.00000 q^{19} +2.00000 q^{20} +1.23607 q^{22} -1.00000 q^{23} +5.47214 q^{25} +4.85410 q^{26} -0.763932 q^{28} +3.00000 q^{29} -6.70820 q^{31} -3.38197 q^{32} -8.47214 q^{34} -4.00000 q^{35} +3.23607 q^{37} -3.23607 q^{38} -7.23607 q^{40} -5.47214 q^{41} +0.472136 q^{44} -1.61803 q^{46} -2.23607 q^{47} -5.47214 q^{49} +8.85410 q^{50} +1.85410 q^{52} +8.47214 q^{53} +2.47214 q^{55} +2.76393 q^{56} +4.85410 q^{58} +2.47214 q^{59} +10.9443 q^{61} -10.8541 q^{62} +4.23607 q^{64} +9.70820 q^{65} -7.23607 q^{67} -3.23607 q^{68} -6.47214 q^{70} -7.76393 q^{71} +15.4721 q^{73} +5.23607 q^{74} -1.23607 q^{76} -0.944272 q^{77} +6.94427 q^{79} -15.7082 q^{80} -8.85410 q^{82} +13.2361 q^{83} -16.9443 q^{85} -1.70820 q^{88} +1.52786 q^{89} -3.70820 q^{91} -0.618034 q^{92} -3.61803 q^{94} -6.47214 q^{95} +4.29180 q^{97} -8.85410 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{4} + 2 q^{5} + 2 q^{7} + 6 q^{10} + 6 q^{11} + 6 q^{13} - 4 q^{14} - 3 q^{16} - 6 q^{17} - 4 q^{19} + 4 q^{20} - 2 q^{22} - 2 q^{23} + 2 q^{25} + 3 q^{26} - 6 q^{28} + 6 q^{29} - 9 q^{32}+ \cdots - 11 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.61803 1.14412 0.572061 0.820211i \(-0.306144\pi\)
0.572061 + 0.820211i \(0.306144\pi\)
\(3\) 0 0
\(4\) 0.618034 0.309017
\(5\) 3.23607 1.44721 0.723607 0.690212i \(-0.242483\pi\)
0.723607 + 0.690212i \(0.242483\pi\)
\(6\) 0 0
\(7\) −1.23607 −0.467190 −0.233595 0.972334i \(-0.575049\pi\)
−0.233595 + 0.972334i \(0.575049\pi\)
\(8\) −2.23607 −0.790569
\(9\) 0 0
\(10\) 5.23607 1.65579
\(11\) 0.763932 0.230334 0.115167 0.993346i \(-0.463260\pi\)
0.115167 + 0.993346i \(0.463260\pi\)
\(12\) 0 0
\(13\) 3.00000 0.832050 0.416025 0.909353i \(-0.363423\pi\)
0.416025 + 0.909353i \(0.363423\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) −4.85410 −1.21353
\(17\) −5.23607 −1.26993 −0.634967 0.772540i \(-0.718986\pi\)
−0.634967 + 0.772540i \(0.718986\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 2.00000 0.447214
\(21\) 0 0
\(22\) 1.23607 0.263531
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 5.47214 1.09443
\(26\) 4.85410 0.951968
\(27\) 0 0
\(28\) −0.763932 −0.144370
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) −6.70820 −1.20483 −0.602414 0.798183i \(-0.705795\pi\)
−0.602414 + 0.798183i \(0.705795\pi\)
\(32\) −3.38197 −0.597853
\(33\) 0 0
\(34\) −8.47214 −1.45296
\(35\) −4.00000 −0.676123
\(36\) 0 0
\(37\) 3.23607 0.532006 0.266003 0.963972i \(-0.414297\pi\)
0.266003 + 0.963972i \(0.414297\pi\)
\(38\) −3.23607 −0.524960
\(39\) 0 0
\(40\) −7.23607 −1.14412
\(41\) −5.47214 −0.854604 −0.427302 0.904109i \(-0.640536\pi\)
−0.427302 + 0.904109i \(0.640536\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0.472136 0.0711772
\(45\) 0 0
\(46\) −1.61803 −0.238566
\(47\) −2.23607 −0.326164 −0.163082 0.986613i \(-0.552144\pi\)
−0.163082 + 0.986613i \(0.552144\pi\)
\(48\) 0 0
\(49\) −5.47214 −0.781734
\(50\) 8.85410 1.25216
\(51\) 0 0
\(52\) 1.85410 0.257118
\(53\) 8.47214 1.16374 0.581869 0.813283i \(-0.302322\pi\)
0.581869 + 0.813283i \(0.302322\pi\)
\(54\) 0 0
\(55\) 2.47214 0.333343
\(56\) 2.76393 0.369346
\(57\) 0 0
\(58\) 4.85410 0.637375
\(59\) 2.47214 0.321845 0.160922 0.986967i \(-0.448553\pi\)
0.160922 + 0.986967i \(0.448553\pi\)
\(60\) 0 0
\(61\) 10.9443 1.40127 0.700635 0.713520i \(-0.252900\pi\)
0.700635 + 0.713520i \(0.252900\pi\)
\(62\) −10.8541 −1.37847
\(63\) 0 0
\(64\) 4.23607 0.529508
\(65\) 9.70820 1.20415
\(66\) 0 0
\(67\) −7.23607 −0.884026 −0.442013 0.897009i \(-0.645736\pi\)
−0.442013 + 0.897009i \(0.645736\pi\)
\(68\) −3.23607 −0.392431
\(69\) 0 0
\(70\) −6.47214 −0.773568
\(71\) −7.76393 −0.921409 −0.460705 0.887554i \(-0.652403\pi\)
−0.460705 + 0.887554i \(0.652403\pi\)
\(72\) 0 0
\(73\) 15.4721 1.81088 0.905438 0.424478i \(-0.139542\pi\)
0.905438 + 0.424478i \(0.139542\pi\)
\(74\) 5.23607 0.608681
\(75\) 0 0
\(76\) −1.23607 −0.141787
\(77\) −0.944272 −0.107610
\(78\) 0 0
\(79\) 6.94427 0.781292 0.390646 0.920541i \(-0.372252\pi\)
0.390646 + 0.920541i \(0.372252\pi\)
\(80\) −15.7082 −1.75623
\(81\) 0 0
\(82\) −8.85410 −0.977772
\(83\) 13.2361 1.45285 0.726424 0.687247i \(-0.241181\pi\)
0.726424 + 0.687247i \(0.241181\pi\)
\(84\) 0 0
\(85\) −16.9443 −1.83786
\(86\) 0 0
\(87\) 0 0
\(88\) −1.70820 −0.182095
\(89\) 1.52786 0.161953 0.0809766 0.996716i \(-0.474196\pi\)
0.0809766 + 0.996716i \(0.474196\pi\)
\(90\) 0 0
\(91\) −3.70820 −0.388725
\(92\) −0.618034 −0.0644345
\(93\) 0 0
\(94\) −3.61803 −0.373172
\(95\) −6.47214 −0.664027
\(96\) 0 0
\(97\) 4.29180 0.435766 0.217883 0.975975i \(-0.430085\pi\)
0.217883 + 0.975975i \(0.430085\pi\)
\(98\) −8.85410 −0.894399
\(99\) 0 0
\(100\) 3.38197 0.338197
\(101\) 4.47214 0.444994 0.222497 0.974933i \(-0.428579\pi\)
0.222497 + 0.974933i \(0.428579\pi\)
\(102\) 0 0
\(103\) 18.1803 1.79136 0.895681 0.444697i \(-0.146689\pi\)
0.895681 + 0.444697i \(0.146689\pi\)
\(104\) −6.70820 −0.657794
\(105\) 0 0
\(106\) 13.7082 1.33146
\(107\) 13.4164 1.29701 0.648507 0.761209i \(-0.275394\pi\)
0.648507 + 0.761209i \(0.275394\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 4.00000 0.381385
\(111\) 0 0
\(112\) 6.00000 0.566947
\(113\) −13.2361 −1.24514 −0.622572 0.782562i \(-0.713912\pi\)
−0.622572 + 0.782562i \(0.713912\pi\)
\(114\) 0 0
\(115\) −3.23607 −0.301765
\(116\) 1.85410 0.172149
\(117\) 0 0
\(118\) 4.00000 0.368230
\(119\) 6.47214 0.593300
\(120\) 0 0
\(121\) −10.4164 −0.946946
\(122\) 17.7082 1.60323
\(123\) 0 0
\(124\) −4.14590 −0.372313
\(125\) 1.52786 0.136656
\(126\) 0 0
\(127\) −20.7082 −1.83756 −0.918778 0.394775i \(-0.870823\pi\)
−0.918778 + 0.394775i \(0.870823\pi\)
\(128\) 13.6180 1.20368
\(129\) 0 0
\(130\) 15.7082 1.37770
\(131\) −5.29180 −0.462346 −0.231173 0.972913i \(-0.574256\pi\)
−0.231173 + 0.972913i \(0.574256\pi\)
\(132\) 0 0
\(133\) 2.47214 0.214361
\(134\) −11.7082 −1.01143
\(135\) 0 0
\(136\) 11.7082 1.00397
\(137\) −13.8885 −1.18658 −0.593289 0.804989i \(-0.702171\pi\)
−0.593289 + 0.804989i \(0.702171\pi\)
\(138\) 0 0
\(139\) 2.70820 0.229707 0.114853 0.993382i \(-0.463360\pi\)
0.114853 + 0.993382i \(0.463360\pi\)
\(140\) −2.47214 −0.208934
\(141\) 0 0
\(142\) −12.5623 −1.05421
\(143\) 2.29180 0.191650
\(144\) 0 0
\(145\) 9.70820 0.806222
\(146\) 25.0344 2.07187
\(147\) 0 0
\(148\) 2.00000 0.164399
\(149\) 11.8885 0.973947 0.486974 0.873417i \(-0.338101\pi\)
0.486974 + 0.873417i \(0.338101\pi\)
\(150\) 0 0
\(151\) −0.236068 −0.0192109 −0.00960547 0.999954i \(-0.503058\pi\)
−0.00960547 + 0.999954i \(0.503058\pi\)
\(152\) 4.47214 0.362738
\(153\) 0 0
\(154\) −1.52786 −0.123119
\(155\) −21.7082 −1.74364
\(156\) 0 0
\(157\) 15.4164 1.23036 0.615182 0.788385i \(-0.289083\pi\)
0.615182 + 0.788385i \(0.289083\pi\)
\(158\) 11.2361 0.893894
\(159\) 0 0
\(160\) −10.9443 −0.865221
\(161\) 1.23607 0.0974158
\(162\) 0 0
\(163\) −10.2361 −0.801751 −0.400875 0.916133i \(-0.631294\pi\)
−0.400875 + 0.916133i \(0.631294\pi\)
\(164\) −3.38197 −0.264087
\(165\) 0 0
\(166\) 21.4164 1.66224
\(167\) −10.4721 −0.810358 −0.405179 0.914237i \(-0.632791\pi\)
−0.405179 + 0.914237i \(0.632791\pi\)
\(168\) 0 0
\(169\) −4.00000 −0.307692
\(170\) −27.4164 −2.10274
\(171\) 0 0
\(172\) 0 0
\(173\) −5.05573 −0.384380 −0.192190 0.981358i \(-0.561559\pi\)
−0.192190 + 0.981358i \(0.561559\pi\)
\(174\) 0 0
\(175\) −6.76393 −0.511305
\(176\) −3.70820 −0.279516
\(177\) 0 0
\(178\) 2.47214 0.185294
\(179\) 12.7082 0.949856 0.474928 0.880025i \(-0.342474\pi\)
0.474928 + 0.880025i \(0.342474\pi\)
\(180\) 0 0
\(181\) −14.6525 −1.08911 −0.544555 0.838725i \(-0.683301\pi\)
−0.544555 + 0.838725i \(0.683301\pi\)
\(182\) −6.00000 −0.444750
\(183\) 0 0
\(184\) 2.23607 0.164845
\(185\) 10.4721 0.769927
\(186\) 0 0
\(187\) −4.00000 −0.292509
\(188\) −1.38197 −0.100790
\(189\) 0 0
\(190\) −10.4721 −0.759729
\(191\) 3.81966 0.276381 0.138190 0.990406i \(-0.455871\pi\)
0.138190 + 0.990406i \(0.455871\pi\)
\(192\) 0 0
\(193\) −7.94427 −0.571841 −0.285921 0.958253i \(-0.592299\pi\)
−0.285921 + 0.958253i \(0.592299\pi\)
\(194\) 6.94427 0.498570
\(195\) 0 0
\(196\) −3.38197 −0.241569
\(197\) −7.47214 −0.532368 −0.266184 0.963922i \(-0.585763\pi\)
−0.266184 + 0.963922i \(0.585763\pi\)
\(198\) 0 0
\(199\) −25.7082 −1.82241 −0.911203 0.411957i \(-0.864845\pi\)
−0.911203 + 0.411957i \(0.864845\pi\)
\(200\) −12.2361 −0.865221
\(201\) 0 0
\(202\) 7.23607 0.509128
\(203\) −3.70820 −0.260265
\(204\) 0 0
\(205\) −17.7082 −1.23679
\(206\) 29.4164 2.04954
\(207\) 0 0
\(208\) −14.5623 −1.00971
\(209\) −1.52786 −0.105685
\(210\) 0 0
\(211\) 3.41641 0.235195 0.117598 0.993061i \(-0.462481\pi\)
0.117598 + 0.993061i \(0.462481\pi\)
\(212\) 5.23607 0.359615
\(213\) 0 0
\(214\) 21.7082 1.48394
\(215\) 0 0
\(216\) 0 0
\(217\) 8.29180 0.562884
\(218\) 0 0
\(219\) 0 0
\(220\) 1.52786 0.103009
\(221\) −15.7082 −1.05665
\(222\) 0 0
\(223\) 4.00000 0.267860 0.133930 0.990991i \(-0.457240\pi\)
0.133930 + 0.990991i \(0.457240\pi\)
\(224\) 4.18034 0.279311
\(225\) 0 0
\(226\) −21.4164 −1.42460
\(227\) −10.1803 −0.675693 −0.337846 0.941201i \(-0.609698\pi\)
−0.337846 + 0.941201i \(0.609698\pi\)
\(228\) 0 0
\(229\) −12.0000 −0.792982 −0.396491 0.918039i \(-0.629772\pi\)
−0.396491 + 0.918039i \(0.629772\pi\)
\(230\) −5.23607 −0.345256
\(231\) 0 0
\(232\) −6.70820 −0.440415
\(233\) 15.4721 1.01361 0.506807 0.862060i \(-0.330826\pi\)
0.506807 + 0.862060i \(0.330826\pi\)
\(234\) 0 0
\(235\) −7.23607 −0.472029
\(236\) 1.52786 0.0994555
\(237\) 0 0
\(238\) 10.4721 0.678808
\(239\) −18.2361 −1.17959 −0.589797 0.807552i \(-0.700792\pi\)
−0.589797 + 0.807552i \(0.700792\pi\)
\(240\) 0 0
\(241\) 17.1246 1.10309 0.551547 0.834144i \(-0.314038\pi\)
0.551547 + 0.834144i \(0.314038\pi\)
\(242\) −16.8541 −1.08342
\(243\) 0 0
\(244\) 6.76393 0.433016
\(245\) −17.7082 −1.13134
\(246\) 0 0
\(247\) −6.00000 −0.381771
\(248\) 15.0000 0.952501
\(249\) 0 0
\(250\) 2.47214 0.156352
\(251\) −15.7082 −0.991493 −0.495747 0.868467i \(-0.665105\pi\)
−0.495747 + 0.868467i \(0.665105\pi\)
\(252\) 0 0
\(253\) −0.763932 −0.0480280
\(254\) −33.5066 −2.10239
\(255\) 0 0
\(256\) 13.5623 0.847644
\(257\) −1.47214 −0.0918293 −0.0459147 0.998945i \(-0.514620\pi\)
−0.0459147 + 0.998945i \(0.514620\pi\)
\(258\) 0 0
\(259\) −4.00000 −0.248548
\(260\) 6.00000 0.372104
\(261\) 0 0
\(262\) −8.56231 −0.528981
\(263\) 14.9443 0.921503 0.460752 0.887529i \(-0.347580\pi\)
0.460752 + 0.887529i \(0.347580\pi\)
\(264\) 0 0
\(265\) 27.4164 1.68418
\(266\) 4.00000 0.245256
\(267\) 0 0
\(268\) −4.47214 −0.273179
\(269\) −9.94427 −0.606313 −0.303156 0.952941i \(-0.598040\pi\)
−0.303156 + 0.952941i \(0.598040\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 25.4164 1.54110
\(273\) 0 0
\(274\) −22.4721 −1.35759
\(275\) 4.18034 0.252084
\(276\) 0 0
\(277\) 6.52786 0.392221 0.196111 0.980582i \(-0.437169\pi\)
0.196111 + 0.980582i \(0.437169\pi\)
\(278\) 4.38197 0.262813
\(279\) 0 0
\(280\) 8.94427 0.534522
\(281\) 13.2361 0.789598 0.394799 0.918768i \(-0.370814\pi\)
0.394799 + 0.918768i \(0.370814\pi\)
\(282\) 0 0
\(283\) 14.2918 0.849559 0.424780 0.905297i \(-0.360352\pi\)
0.424780 + 0.905297i \(0.360352\pi\)
\(284\) −4.79837 −0.284731
\(285\) 0 0
\(286\) 3.70820 0.219271
\(287\) 6.76393 0.399262
\(288\) 0 0
\(289\) 10.4164 0.612730
\(290\) 15.7082 0.922417
\(291\) 0 0
\(292\) 9.56231 0.559592
\(293\) 10.4721 0.611789 0.305894 0.952065i \(-0.401045\pi\)
0.305894 + 0.952065i \(0.401045\pi\)
\(294\) 0 0
\(295\) 8.00000 0.465778
\(296\) −7.23607 −0.420588
\(297\) 0 0
\(298\) 19.2361 1.11432
\(299\) −3.00000 −0.173494
\(300\) 0 0
\(301\) 0 0
\(302\) −0.381966 −0.0219797
\(303\) 0 0
\(304\) 9.70820 0.556804
\(305\) 35.4164 2.02794
\(306\) 0 0
\(307\) 18.4721 1.05426 0.527130 0.849785i \(-0.323268\pi\)
0.527130 + 0.849785i \(0.323268\pi\)
\(308\) −0.583592 −0.0332532
\(309\) 0 0
\(310\) −35.1246 −1.99494
\(311\) 9.18034 0.520569 0.260285 0.965532i \(-0.416184\pi\)
0.260285 + 0.965532i \(0.416184\pi\)
\(312\) 0 0
\(313\) −20.3607 −1.15085 −0.575427 0.817853i \(-0.695164\pi\)
−0.575427 + 0.817853i \(0.695164\pi\)
\(314\) 24.9443 1.40769
\(315\) 0 0
\(316\) 4.29180 0.241432
\(317\) 1.41641 0.0795534 0.0397767 0.999209i \(-0.487335\pi\)
0.0397767 + 0.999209i \(0.487335\pi\)
\(318\) 0 0
\(319\) 2.29180 0.128316
\(320\) 13.7082 0.766312
\(321\) 0 0
\(322\) 2.00000 0.111456
\(323\) 10.4721 0.582685
\(324\) 0 0
\(325\) 16.4164 0.910618
\(326\) −16.5623 −0.917301
\(327\) 0 0
\(328\) 12.2361 0.675624
\(329\) 2.76393 0.152381
\(330\) 0 0
\(331\) 11.6525 0.640478 0.320239 0.947337i \(-0.396237\pi\)
0.320239 + 0.947337i \(0.396237\pi\)
\(332\) 8.18034 0.448954
\(333\) 0 0
\(334\) −16.9443 −0.927149
\(335\) −23.4164 −1.27938
\(336\) 0 0
\(337\) −3.41641 −0.186104 −0.0930518 0.995661i \(-0.529662\pi\)
−0.0930518 + 0.995661i \(0.529662\pi\)
\(338\) −6.47214 −0.352038
\(339\) 0 0
\(340\) −10.4721 −0.567931
\(341\) −5.12461 −0.277513
\(342\) 0 0
\(343\) 15.4164 0.832408
\(344\) 0 0
\(345\) 0 0
\(346\) −8.18034 −0.439778
\(347\) −25.8885 −1.38977 −0.694885 0.719121i \(-0.744545\pi\)
−0.694885 + 0.719121i \(0.744545\pi\)
\(348\) 0 0
\(349\) −2.41641 −0.129347 −0.0646737 0.997906i \(-0.520601\pi\)
−0.0646737 + 0.997906i \(0.520601\pi\)
\(350\) −10.9443 −0.584996
\(351\) 0 0
\(352\) −2.58359 −0.137706
\(353\) 35.3607 1.88206 0.941030 0.338324i \(-0.109860\pi\)
0.941030 + 0.338324i \(0.109860\pi\)
\(354\) 0 0
\(355\) −25.1246 −1.33348
\(356\) 0.944272 0.0500463
\(357\) 0 0
\(358\) 20.5623 1.08675
\(359\) −15.8885 −0.838565 −0.419283 0.907856i \(-0.637718\pi\)
−0.419283 + 0.907856i \(0.637718\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) −23.7082 −1.24608
\(363\) 0 0
\(364\) −2.29180 −0.120123
\(365\) 50.0689 2.62073
\(366\) 0 0
\(367\) 18.1803 0.949006 0.474503 0.880254i \(-0.342628\pi\)
0.474503 + 0.880254i \(0.342628\pi\)
\(368\) 4.85410 0.253038
\(369\) 0 0
\(370\) 16.9443 0.880891
\(371\) −10.4721 −0.543686
\(372\) 0 0
\(373\) −5.70820 −0.295560 −0.147780 0.989020i \(-0.547213\pi\)
−0.147780 + 0.989020i \(0.547213\pi\)
\(374\) −6.47214 −0.334666
\(375\) 0 0
\(376\) 5.00000 0.257855
\(377\) 9.00000 0.463524
\(378\) 0 0
\(379\) −20.3607 −1.04586 −0.522929 0.852376i \(-0.675161\pi\)
−0.522929 + 0.852376i \(0.675161\pi\)
\(380\) −4.00000 −0.205196
\(381\) 0 0
\(382\) 6.18034 0.316214
\(383\) −24.9443 −1.27459 −0.637296 0.770619i \(-0.719947\pi\)
−0.637296 + 0.770619i \(0.719947\pi\)
\(384\) 0 0
\(385\) −3.05573 −0.155734
\(386\) −12.8541 −0.654257
\(387\) 0 0
\(388\) 2.65248 0.134659
\(389\) −34.4721 −1.74781 −0.873903 0.486100i \(-0.838419\pi\)
−0.873903 + 0.486100i \(0.838419\pi\)
\(390\) 0 0
\(391\) 5.23607 0.264799
\(392\) 12.2361 0.618015
\(393\) 0 0
\(394\) −12.0902 −0.609094
\(395\) 22.4721 1.13070
\(396\) 0 0
\(397\) 2.41641 0.121276 0.0606380 0.998160i \(-0.480686\pi\)
0.0606380 + 0.998160i \(0.480686\pi\)
\(398\) −41.5967 −2.08506
\(399\) 0 0
\(400\) −26.5623 −1.32812
\(401\) −8.18034 −0.408507 −0.204253 0.978918i \(-0.565477\pi\)
−0.204253 + 0.978918i \(0.565477\pi\)
\(402\) 0 0
\(403\) −20.1246 −1.00248
\(404\) 2.76393 0.137511
\(405\) 0 0
\(406\) −6.00000 −0.297775
\(407\) 2.47214 0.122539
\(408\) 0 0
\(409\) −23.3607 −1.15511 −0.577556 0.816351i \(-0.695993\pi\)
−0.577556 + 0.816351i \(0.695993\pi\)
\(410\) −28.6525 −1.41504
\(411\) 0 0
\(412\) 11.2361 0.553561
\(413\) −3.05573 −0.150363
\(414\) 0 0
\(415\) 42.8328 2.10258
\(416\) −10.1459 −0.497444
\(417\) 0 0
\(418\) −2.47214 −0.120916
\(419\) 31.4164 1.53479 0.767396 0.641173i \(-0.221552\pi\)
0.767396 + 0.641173i \(0.221552\pi\)
\(420\) 0 0
\(421\) −23.7082 −1.15547 −0.577734 0.816225i \(-0.696063\pi\)
−0.577734 + 0.816225i \(0.696063\pi\)
\(422\) 5.52786 0.269092
\(423\) 0 0
\(424\) −18.9443 −0.920015
\(425\) −28.6525 −1.38985
\(426\) 0 0
\(427\) −13.5279 −0.654659
\(428\) 8.29180 0.400799
\(429\) 0 0
\(430\) 0 0
\(431\) 26.4721 1.27512 0.637559 0.770402i \(-0.279944\pi\)
0.637559 + 0.770402i \(0.279944\pi\)
\(432\) 0 0
\(433\) 40.1803 1.93094 0.965472 0.260507i \(-0.0838897\pi\)
0.965472 + 0.260507i \(0.0838897\pi\)
\(434\) 13.4164 0.644008
\(435\) 0 0
\(436\) 0 0
\(437\) 2.00000 0.0956730
\(438\) 0 0
\(439\) −5.29180 −0.252564 −0.126282 0.991994i \(-0.540304\pi\)
−0.126282 + 0.991994i \(0.540304\pi\)
\(440\) −5.52786 −0.263531
\(441\) 0 0
\(442\) −25.4164 −1.20894
\(443\) 2.12461 0.100943 0.0504717 0.998725i \(-0.483928\pi\)
0.0504717 + 0.998725i \(0.483928\pi\)
\(444\) 0 0
\(445\) 4.94427 0.234381
\(446\) 6.47214 0.306465
\(447\) 0 0
\(448\) −5.23607 −0.247381
\(449\) −2.94427 −0.138949 −0.0694744 0.997584i \(-0.522132\pi\)
−0.0694744 + 0.997584i \(0.522132\pi\)
\(450\) 0 0
\(451\) −4.18034 −0.196845
\(452\) −8.18034 −0.384771
\(453\) 0 0
\(454\) −16.4721 −0.773076
\(455\) −12.0000 −0.562569
\(456\) 0 0
\(457\) 35.1246 1.64306 0.821530 0.570165i \(-0.193121\pi\)
0.821530 + 0.570165i \(0.193121\pi\)
\(458\) −19.4164 −0.907269
\(459\) 0 0
\(460\) −2.00000 −0.0932505
\(461\) −7.47214 −0.348012 −0.174006 0.984745i \(-0.555671\pi\)
−0.174006 + 0.984745i \(0.555671\pi\)
\(462\) 0 0
\(463\) −20.0000 −0.929479 −0.464739 0.885448i \(-0.653852\pi\)
−0.464739 + 0.885448i \(0.653852\pi\)
\(464\) −14.5623 −0.676038
\(465\) 0 0
\(466\) 25.0344 1.15970
\(467\) 30.9443 1.43193 0.715965 0.698136i \(-0.245987\pi\)
0.715965 + 0.698136i \(0.245987\pi\)
\(468\) 0 0
\(469\) 8.94427 0.413008
\(470\) −11.7082 −0.540059
\(471\) 0 0
\(472\) −5.52786 −0.254441
\(473\) 0 0
\(474\) 0 0
\(475\) −10.9443 −0.502158
\(476\) 4.00000 0.183340
\(477\) 0 0
\(478\) −29.5066 −1.34960
\(479\) 17.5967 0.804016 0.402008 0.915636i \(-0.368312\pi\)
0.402008 + 0.915636i \(0.368312\pi\)
\(480\) 0 0
\(481\) 9.70820 0.442656
\(482\) 27.7082 1.26207
\(483\) 0 0
\(484\) −6.43769 −0.292622
\(485\) 13.8885 0.630646
\(486\) 0 0
\(487\) −1.29180 −0.0585369 −0.0292684 0.999572i \(-0.509318\pi\)
−0.0292684 + 0.999572i \(0.509318\pi\)
\(488\) −24.4721 −1.10780
\(489\) 0 0
\(490\) −28.6525 −1.29439
\(491\) −39.6525 −1.78949 −0.894746 0.446576i \(-0.852643\pi\)
−0.894746 + 0.446576i \(0.852643\pi\)
\(492\) 0 0
\(493\) −15.7082 −0.707462
\(494\) −9.70820 −0.436793
\(495\) 0 0
\(496\) 32.5623 1.46209
\(497\) 9.59675 0.430473
\(498\) 0 0
\(499\) 32.7082 1.46422 0.732110 0.681186i \(-0.238536\pi\)
0.732110 + 0.681186i \(0.238536\pi\)
\(500\) 0.944272 0.0422291
\(501\) 0 0
\(502\) −25.4164 −1.13439
\(503\) 9.05573 0.403775 0.201887 0.979409i \(-0.435292\pi\)
0.201887 + 0.979409i \(0.435292\pi\)
\(504\) 0 0
\(505\) 14.4721 0.644002
\(506\) −1.23607 −0.0549499
\(507\) 0 0
\(508\) −12.7984 −0.567836
\(509\) −34.3050 −1.52054 −0.760270 0.649607i \(-0.774933\pi\)
−0.760270 + 0.649607i \(0.774933\pi\)
\(510\) 0 0
\(511\) −19.1246 −0.846023
\(512\) −5.29180 −0.233867
\(513\) 0 0
\(514\) −2.38197 −0.105064
\(515\) 58.8328 2.59248
\(516\) 0 0
\(517\) −1.70820 −0.0751267
\(518\) −6.47214 −0.284369
\(519\) 0 0
\(520\) −21.7082 −0.951968
\(521\) −4.58359 −0.200811 −0.100405 0.994947i \(-0.532014\pi\)
−0.100405 + 0.994947i \(0.532014\pi\)
\(522\) 0 0
\(523\) 0.875388 0.0382781 0.0191390 0.999817i \(-0.493907\pi\)
0.0191390 + 0.999817i \(0.493907\pi\)
\(524\) −3.27051 −0.142873
\(525\) 0 0
\(526\) 24.1803 1.05431
\(527\) 35.1246 1.53005
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 44.3607 1.92690
\(531\) 0 0
\(532\) 1.52786 0.0662413
\(533\) −16.4164 −0.711074
\(534\) 0 0
\(535\) 43.4164 1.87705
\(536\) 16.1803 0.698884
\(537\) 0 0
\(538\) −16.0902 −0.693696
\(539\) −4.18034 −0.180060
\(540\) 0 0
\(541\) −7.58359 −0.326044 −0.163022 0.986622i \(-0.552124\pi\)
−0.163022 + 0.986622i \(0.552124\pi\)
\(542\) 12.9443 0.556004
\(543\) 0 0
\(544\) 17.7082 0.759233
\(545\) 0 0
\(546\) 0 0
\(547\) 37.5410 1.60514 0.802569 0.596559i \(-0.203466\pi\)
0.802569 + 0.596559i \(0.203466\pi\)
\(548\) −8.58359 −0.366673
\(549\) 0 0
\(550\) 6.76393 0.288415
\(551\) −6.00000 −0.255609
\(552\) 0 0
\(553\) −8.58359 −0.365011
\(554\) 10.5623 0.448749
\(555\) 0 0
\(556\) 1.67376 0.0709833
\(557\) −19.4164 −0.822700 −0.411350 0.911478i \(-0.634943\pi\)
−0.411350 + 0.911478i \(0.634943\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 19.4164 0.820493
\(561\) 0 0
\(562\) 21.4164 0.903397
\(563\) 15.0557 0.634523 0.317262 0.948338i \(-0.397237\pi\)
0.317262 + 0.948338i \(0.397237\pi\)
\(564\) 0 0
\(565\) −42.8328 −1.80199
\(566\) 23.1246 0.972000
\(567\) 0 0
\(568\) 17.3607 0.728438
\(569\) −0.180340 −0.00756024 −0.00378012 0.999993i \(-0.501203\pi\)
−0.00378012 + 0.999993i \(0.501203\pi\)
\(570\) 0 0
\(571\) −27.7082 −1.15955 −0.579776 0.814776i \(-0.696860\pi\)
−0.579776 + 0.814776i \(0.696860\pi\)
\(572\) 1.41641 0.0592230
\(573\) 0 0
\(574\) 10.9443 0.456805
\(575\) −5.47214 −0.228204
\(576\) 0 0
\(577\) −12.8885 −0.536557 −0.268279 0.963341i \(-0.586455\pi\)
−0.268279 + 0.963341i \(0.586455\pi\)
\(578\) 16.8541 0.701038
\(579\) 0 0
\(580\) 6.00000 0.249136
\(581\) −16.3607 −0.678755
\(582\) 0 0
\(583\) 6.47214 0.268048
\(584\) −34.5967 −1.43162
\(585\) 0 0
\(586\) 16.9443 0.699961
\(587\) 11.2918 0.466062 0.233031 0.972469i \(-0.425136\pi\)
0.233031 + 0.972469i \(0.425136\pi\)
\(588\) 0 0
\(589\) 13.4164 0.552813
\(590\) 12.9443 0.532907
\(591\) 0 0
\(592\) −15.7082 −0.645603
\(593\) −14.9443 −0.613688 −0.306844 0.951760i \(-0.599273\pi\)
−0.306844 + 0.951760i \(0.599273\pi\)
\(594\) 0 0
\(595\) 20.9443 0.858631
\(596\) 7.34752 0.300966
\(597\) 0 0
\(598\) −4.85410 −0.198499
\(599\) 1.88854 0.0771638 0.0385819 0.999255i \(-0.487716\pi\)
0.0385819 + 0.999255i \(0.487716\pi\)
\(600\) 0 0
\(601\) 11.1115 0.453246 0.226623 0.973983i \(-0.427232\pi\)
0.226623 + 0.973983i \(0.427232\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −0.145898 −0.00593651
\(605\) −33.7082 −1.37043
\(606\) 0 0
\(607\) 17.5279 0.711434 0.355717 0.934594i \(-0.384237\pi\)
0.355717 + 0.934594i \(0.384237\pi\)
\(608\) 6.76393 0.274314
\(609\) 0 0
\(610\) 57.3050 2.32021
\(611\) −6.70820 −0.271385
\(612\) 0 0
\(613\) −7.70820 −0.311331 −0.155666 0.987810i \(-0.549752\pi\)
−0.155666 + 0.987810i \(0.549752\pi\)
\(614\) 29.8885 1.20620
\(615\) 0 0
\(616\) 2.11146 0.0850730
\(617\) 16.4721 0.663143 0.331572 0.943430i \(-0.392421\pi\)
0.331572 + 0.943430i \(0.392421\pi\)
\(618\) 0 0
\(619\) −7.41641 −0.298091 −0.149045 0.988830i \(-0.547620\pi\)
−0.149045 + 0.988830i \(0.547620\pi\)
\(620\) −13.4164 −0.538816
\(621\) 0 0
\(622\) 14.8541 0.595595
\(623\) −1.88854 −0.0756629
\(624\) 0 0
\(625\) −22.4164 −0.896656
\(626\) −32.9443 −1.31672
\(627\) 0 0
\(628\) 9.52786 0.380203
\(629\) −16.9443 −0.675612
\(630\) 0 0
\(631\) −32.3607 −1.28826 −0.644129 0.764917i \(-0.722780\pi\)
−0.644129 + 0.764917i \(0.722780\pi\)
\(632\) −15.5279 −0.617665
\(633\) 0 0
\(634\) 2.29180 0.0910188
\(635\) −67.0132 −2.65934
\(636\) 0 0
\(637\) −16.4164 −0.650442
\(638\) 3.70820 0.146809
\(639\) 0 0
\(640\) 44.0689 1.74198
\(641\) −45.3050 −1.78944 −0.894719 0.446629i \(-0.852624\pi\)
−0.894719 + 0.446629i \(0.852624\pi\)
\(642\) 0 0
\(643\) 19.5967 0.772820 0.386410 0.922327i \(-0.373715\pi\)
0.386410 + 0.922327i \(0.373715\pi\)
\(644\) 0.763932 0.0301031
\(645\) 0 0
\(646\) 16.9443 0.666663
\(647\) 6.70820 0.263727 0.131863 0.991268i \(-0.457904\pi\)
0.131863 + 0.991268i \(0.457904\pi\)
\(648\) 0 0
\(649\) 1.88854 0.0741318
\(650\) 26.5623 1.04186
\(651\) 0 0
\(652\) −6.32624 −0.247755
\(653\) −24.3050 −0.951126 −0.475563 0.879682i \(-0.657756\pi\)
−0.475563 + 0.879682i \(0.657756\pi\)
\(654\) 0 0
\(655\) −17.1246 −0.669114
\(656\) 26.5623 1.03708
\(657\) 0 0
\(658\) 4.47214 0.174342
\(659\) −20.6525 −0.804506 −0.402253 0.915528i \(-0.631773\pi\)
−0.402253 + 0.915528i \(0.631773\pi\)
\(660\) 0 0
\(661\) −5.05573 −0.196645 −0.0983225 0.995155i \(-0.531348\pi\)
−0.0983225 + 0.995155i \(0.531348\pi\)
\(662\) 18.8541 0.732785
\(663\) 0 0
\(664\) −29.5967 −1.14858
\(665\) 8.00000 0.310227
\(666\) 0 0
\(667\) −3.00000 −0.116160
\(668\) −6.47214 −0.250414
\(669\) 0 0
\(670\) −37.8885 −1.46376
\(671\) 8.36068 0.322760
\(672\) 0 0
\(673\) 3.00000 0.115642 0.0578208 0.998327i \(-0.481585\pi\)
0.0578208 + 0.998327i \(0.481585\pi\)
\(674\) −5.52786 −0.212925
\(675\) 0 0
\(676\) −2.47214 −0.0950822
\(677\) −18.0000 −0.691796 −0.345898 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) 0 0
\(679\) −5.30495 −0.203585
\(680\) 37.8885 1.45296
\(681\) 0 0
\(682\) −8.29180 −0.317509
\(683\) 22.5967 0.864641 0.432320 0.901720i \(-0.357695\pi\)
0.432320 + 0.901720i \(0.357695\pi\)
\(684\) 0 0
\(685\) −44.9443 −1.71723
\(686\) 24.9443 0.952377
\(687\) 0 0
\(688\) 0 0
\(689\) 25.4164 0.968288
\(690\) 0 0
\(691\) 24.9443 0.948925 0.474462 0.880276i \(-0.342642\pi\)
0.474462 + 0.880276i \(0.342642\pi\)
\(692\) −3.12461 −0.118780
\(693\) 0 0
\(694\) −41.8885 −1.59007
\(695\) 8.76393 0.332435
\(696\) 0 0
\(697\) 28.6525 1.08529
\(698\) −3.90983 −0.147989
\(699\) 0 0
\(700\) −4.18034 −0.158002
\(701\) 26.1803 0.988818 0.494409 0.869229i \(-0.335385\pi\)
0.494409 + 0.869229i \(0.335385\pi\)
\(702\) 0 0
\(703\) −6.47214 −0.244101
\(704\) 3.23607 0.121964
\(705\) 0 0
\(706\) 57.2148 2.15331
\(707\) −5.52786 −0.207897
\(708\) 0 0
\(709\) 16.0689 0.603480 0.301740 0.953390i \(-0.402433\pi\)
0.301740 + 0.953390i \(0.402433\pi\)
\(710\) −40.6525 −1.52566
\(711\) 0 0
\(712\) −3.41641 −0.128035
\(713\) 6.70820 0.251224
\(714\) 0 0
\(715\) 7.41641 0.277358
\(716\) 7.85410 0.293522
\(717\) 0 0
\(718\) −25.7082 −0.959422
\(719\) 20.9443 0.781090 0.390545 0.920584i \(-0.372287\pi\)
0.390545 + 0.920584i \(0.372287\pi\)
\(720\) 0 0
\(721\) −22.4721 −0.836906
\(722\) −24.2705 −0.903255
\(723\) 0 0
\(724\) −9.05573 −0.336553
\(725\) 16.4164 0.609690
\(726\) 0 0
\(727\) −14.2918 −0.530053 −0.265027 0.964241i \(-0.585381\pi\)
−0.265027 + 0.964241i \(0.585381\pi\)
\(728\) 8.29180 0.307314
\(729\) 0 0
\(730\) 81.0132 2.99843
\(731\) 0 0
\(732\) 0 0
\(733\) −26.7639 −0.988548 −0.494274 0.869306i \(-0.664566\pi\)
−0.494274 + 0.869306i \(0.664566\pi\)
\(734\) 29.4164 1.08578
\(735\) 0 0
\(736\) 3.38197 0.124661
\(737\) −5.52786 −0.203621
\(738\) 0 0
\(739\) 49.1803 1.80913 0.904564 0.426338i \(-0.140197\pi\)
0.904564 + 0.426338i \(0.140197\pi\)
\(740\) 6.47214 0.237920
\(741\) 0 0
\(742\) −16.9443 −0.622044
\(743\) −0.875388 −0.0321149 −0.0160574 0.999871i \(-0.505111\pi\)
−0.0160574 + 0.999871i \(0.505111\pi\)
\(744\) 0 0
\(745\) 38.4721 1.40951
\(746\) −9.23607 −0.338156
\(747\) 0 0
\(748\) −2.47214 −0.0903902
\(749\) −16.5836 −0.605951
\(750\) 0 0
\(751\) −44.3607 −1.61874 −0.809372 0.587296i \(-0.800192\pi\)
−0.809372 + 0.587296i \(0.800192\pi\)
\(752\) 10.8541 0.395808
\(753\) 0 0
\(754\) 14.5623 0.530328
\(755\) −0.763932 −0.0278023
\(756\) 0 0
\(757\) −47.5967 −1.72993 −0.864967 0.501829i \(-0.832661\pi\)
−0.864967 + 0.501829i \(0.832661\pi\)
\(758\) −32.9443 −1.19659
\(759\) 0 0
\(760\) 14.4721 0.524960
\(761\) 16.3050 0.591054 0.295527 0.955334i \(-0.404505\pi\)
0.295527 + 0.955334i \(0.404505\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 2.36068 0.0854064
\(765\) 0 0
\(766\) −40.3607 −1.45829
\(767\) 7.41641 0.267791
\(768\) 0 0
\(769\) 17.1246 0.617529 0.308765 0.951138i \(-0.400084\pi\)
0.308765 + 0.951138i \(0.400084\pi\)
\(770\) −4.94427 −0.178179
\(771\) 0 0
\(772\) −4.90983 −0.176709
\(773\) 14.4721 0.520527 0.260263 0.965538i \(-0.416191\pi\)
0.260263 + 0.965538i \(0.416191\pi\)
\(774\) 0 0
\(775\) −36.7082 −1.31860
\(776\) −9.59675 −0.344503
\(777\) 0 0
\(778\) −55.7771 −1.99971
\(779\) 10.9443 0.392119
\(780\) 0 0
\(781\) −5.93112 −0.212232
\(782\) 8.47214 0.302963
\(783\) 0 0
\(784\) 26.5623 0.948654
\(785\) 49.8885 1.78060
\(786\) 0 0
\(787\) 51.4164 1.83280 0.916399 0.400267i \(-0.131083\pi\)
0.916399 + 0.400267i \(0.131083\pi\)
\(788\) −4.61803 −0.164511
\(789\) 0 0
\(790\) 36.3607 1.29365
\(791\) 16.3607 0.581719
\(792\) 0 0
\(793\) 32.8328 1.16593
\(794\) 3.90983 0.138755
\(795\) 0 0
\(796\) −15.8885 −0.563155
\(797\) −10.3607 −0.366994 −0.183497 0.983020i \(-0.558742\pi\)
−0.183497 + 0.983020i \(0.558742\pi\)
\(798\) 0 0
\(799\) 11.7082 0.414206
\(800\) −18.5066 −0.654306
\(801\) 0 0
\(802\) −13.2361 −0.467382
\(803\) 11.8197 0.417107
\(804\) 0 0
\(805\) 4.00000 0.140981
\(806\) −32.5623 −1.14696
\(807\) 0 0
\(808\) −10.0000 −0.351799
\(809\) −47.8885 −1.68367 −0.841836 0.539734i \(-0.818525\pi\)
−0.841836 + 0.539734i \(0.818525\pi\)
\(810\) 0 0
\(811\) −55.6525 −1.95422 −0.977111 0.212728i \(-0.931765\pi\)
−0.977111 + 0.212728i \(0.931765\pi\)
\(812\) −2.29180 −0.0804263
\(813\) 0 0
\(814\) 4.00000 0.140200
\(815\) −33.1246 −1.16030
\(816\) 0 0
\(817\) 0 0
\(818\) −37.7984 −1.32159
\(819\) 0 0
\(820\) −10.9443 −0.382191
\(821\) 21.0557 0.734850 0.367425 0.930053i \(-0.380239\pi\)
0.367425 + 0.930053i \(0.380239\pi\)
\(822\) 0 0
\(823\) 27.5410 0.960020 0.480010 0.877263i \(-0.340633\pi\)
0.480010 + 0.877263i \(0.340633\pi\)
\(824\) −40.6525 −1.41620
\(825\) 0 0
\(826\) −4.94427 −0.172033
\(827\) −10.4721 −0.364152 −0.182076 0.983284i \(-0.558282\pi\)
−0.182076 + 0.983284i \(0.558282\pi\)
\(828\) 0 0
\(829\) −40.2492 −1.39791 −0.698957 0.715164i \(-0.746352\pi\)
−0.698957 + 0.715164i \(0.746352\pi\)
\(830\) 69.3050 2.40561
\(831\) 0 0
\(832\) 12.7082 0.440578
\(833\) 28.6525 0.992749
\(834\) 0 0
\(835\) −33.8885 −1.17276
\(836\) −0.944272 −0.0326583
\(837\) 0 0
\(838\) 50.8328 1.75599
\(839\) 0.875388 0.0302218 0.0151109 0.999886i \(-0.495190\pi\)
0.0151109 + 0.999886i \(0.495190\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) −38.3607 −1.32200
\(843\) 0 0
\(844\) 2.11146 0.0726793
\(845\) −12.9443 −0.445296
\(846\) 0 0
\(847\) 12.8754 0.442404
\(848\) −41.1246 −1.41222
\(849\) 0 0
\(850\) −46.3607 −1.59016
\(851\) −3.23607 −0.110931
\(852\) 0 0
\(853\) −37.4164 −1.28111 −0.640557 0.767911i \(-0.721296\pi\)
−0.640557 + 0.767911i \(0.721296\pi\)
\(854\) −21.8885 −0.749011
\(855\) 0 0
\(856\) −30.0000 −1.02538
\(857\) 7.47214 0.255243 0.127622 0.991823i \(-0.459266\pi\)
0.127622 + 0.991823i \(0.459266\pi\)
\(858\) 0 0
\(859\) −3.29180 −0.112315 −0.0561573 0.998422i \(-0.517885\pi\)
−0.0561573 + 0.998422i \(0.517885\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 42.8328 1.45889
\(863\) −45.5410 −1.55023 −0.775117 0.631818i \(-0.782309\pi\)
−0.775117 + 0.631818i \(0.782309\pi\)
\(864\) 0 0
\(865\) −16.3607 −0.556280
\(866\) 65.0132 2.20924
\(867\) 0 0
\(868\) 5.12461 0.173941
\(869\) 5.30495 0.179958
\(870\) 0 0
\(871\) −21.7082 −0.735554
\(872\) 0 0
\(873\) 0 0
\(874\) 3.23607 0.109462
\(875\) −1.88854 −0.0638444
\(876\) 0 0
\(877\) −27.5279 −0.929550 −0.464775 0.885429i \(-0.653865\pi\)
−0.464775 + 0.885429i \(0.653865\pi\)
\(878\) −8.56231 −0.288964
\(879\) 0 0
\(880\) −12.0000 −0.404520
\(881\) −21.8197 −0.735123 −0.367562 0.929999i \(-0.619807\pi\)
−0.367562 + 0.929999i \(0.619807\pi\)
\(882\) 0 0
\(883\) 4.00000 0.134611 0.0673054 0.997732i \(-0.478560\pi\)
0.0673054 + 0.997732i \(0.478560\pi\)
\(884\) −9.70820 −0.326522
\(885\) 0 0
\(886\) 3.43769 0.115492
\(887\) 35.0689 1.17750 0.588749 0.808316i \(-0.299621\pi\)
0.588749 + 0.808316i \(0.299621\pi\)
\(888\) 0 0
\(889\) 25.5967 0.858487
\(890\) 8.00000 0.268161
\(891\) 0 0
\(892\) 2.47214 0.0827732
\(893\) 4.47214 0.149654
\(894\) 0 0
\(895\) 41.1246 1.37464
\(896\) −16.8328 −0.562345
\(897\) 0 0
\(898\) −4.76393 −0.158974
\(899\) −20.1246 −0.671193
\(900\) 0 0
\(901\) −44.3607 −1.47787
\(902\) −6.76393 −0.225214
\(903\) 0 0
\(904\) 29.5967 0.984373
\(905\) −47.4164 −1.57617
\(906\) 0 0
\(907\) 40.2492 1.33645 0.668227 0.743958i \(-0.267054\pi\)
0.668227 + 0.743958i \(0.267054\pi\)
\(908\) −6.29180 −0.208801
\(909\) 0 0
\(910\) −19.4164 −0.643648
\(911\) 31.3050 1.03718 0.518590 0.855023i \(-0.326457\pi\)
0.518590 + 0.855023i \(0.326457\pi\)
\(912\) 0 0
\(913\) 10.1115 0.334640
\(914\) 56.8328 1.87986
\(915\) 0 0
\(916\) −7.41641 −0.245045
\(917\) 6.54102 0.216003
\(918\) 0 0
\(919\) 0.875388 0.0288764 0.0144382 0.999896i \(-0.495404\pi\)
0.0144382 + 0.999896i \(0.495404\pi\)
\(920\) 7.23607 0.238566
\(921\) 0 0
\(922\) −12.0902 −0.398169
\(923\) −23.2918 −0.766659
\(924\) 0 0
\(925\) 17.7082 0.582242
\(926\) −32.3607 −1.06344
\(927\) 0 0
\(928\) −10.1459 −0.333055
\(929\) 41.9443 1.37615 0.688073 0.725641i \(-0.258457\pi\)
0.688073 + 0.725641i \(0.258457\pi\)
\(930\) 0 0
\(931\) 10.9443 0.358684
\(932\) 9.56231 0.313224
\(933\) 0 0
\(934\) 50.0689 1.63830
\(935\) −12.9443 −0.423323
\(936\) 0 0
\(937\) 11.8197 0.386131 0.193066 0.981186i \(-0.438157\pi\)
0.193066 + 0.981186i \(0.438157\pi\)
\(938\) 14.4721 0.472532
\(939\) 0 0
\(940\) −4.47214 −0.145865
\(941\) 24.6525 0.803648 0.401824 0.915717i \(-0.368376\pi\)
0.401824 + 0.915717i \(0.368376\pi\)
\(942\) 0 0
\(943\) 5.47214 0.178197
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) 0 0
\(947\) 33.1803 1.07822 0.539108 0.842237i \(-0.318761\pi\)
0.539108 + 0.842237i \(0.318761\pi\)
\(948\) 0 0
\(949\) 46.4164 1.50674
\(950\) −17.7082 −0.574530
\(951\) 0 0
\(952\) −14.4721 −0.469045
\(953\) −11.5279 −0.373424 −0.186712 0.982415i \(-0.559783\pi\)
−0.186712 + 0.982415i \(0.559783\pi\)
\(954\) 0 0
\(955\) 12.3607 0.399982
\(956\) −11.2705 −0.364514
\(957\) 0 0
\(958\) 28.4721 0.919893
\(959\) 17.1672 0.554357
\(960\) 0 0
\(961\) 14.0000 0.451613
\(962\) 15.7082 0.506453
\(963\) 0 0
\(964\) 10.5836 0.340875
\(965\) −25.7082 −0.827576
\(966\) 0 0
\(967\) −39.5410 −1.27155 −0.635777 0.771873i \(-0.719320\pi\)
−0.635777 + 0.771873i \(0.719320\pi\)
\(968\) 23.2918 0.748627
\(969\) 0 0
\(970\) 22.4721 0.721537
\(971\) −7.52786 −0.241581 −0.120790 0.992678i \(-0.538543\pi\)
−0.120790 + 0.992678i \(0.538543\pi\)
\(972\) 0 0
\(973\) −3.34752 −0.107317
\(974\) −2.09017 −0.0669734
\(975\) 0 0
\(976\) −53.1246 −1.70048
\(977\) 54.6525 1.74849 0.874244 0.485487i \(-0.161358\pi\)
0.874244 + 0.485487i \(0.161358\pi\)
\(978\) 0 0
\(979\) 1.16718 0.0373034
\(980\) −10.9443 −0.349602
\(981\) 0 0
\(982\) −64.1591 −2.04740
\(983\) 31.5279 1.00558 0.502791 0.864408i \(-0.332306\pi\)
0.502791 + 0.864408i \(0.332306\pi\)
\(984\) 0 0
\(985\) −24.1803 −0.770450
\(986\) −25.4164 −0.809423
\(987\) 0 0
\(988\) −3.70820 −0.117974
\(989\) 0 0
\(990\) 0 0
\(991\) 24.0000 0.762385 0.381193 0.924496i \(-0.375513\pi\)
0.381193 + 0.924496i \(0.375513\pi\)
\(992\) 22.6869 0.720310
\(993\) 0 0
\(994\) 15.5279 0.492514
\(995\) −83.1935 −2.63741
\(996\) 0 0
\(997\) −36.8328 −1.16651 −0.583253 0.812290i \(-0.698221\pi\)
−0.583253 + 0.812290i \(0.698221\pi\)
\(998\) 52.9230 1.67525
\(999\) 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 207.2.a.d.1.2 2
3.2 odd 2 23.2.a.a.1.1 2
4.3 odd 2 3312.2.a.ba.1.2 2
5.4 even 2 5175.2.a.be.1.1 2
12.11 even 2 368.2.a.h.1.1 2
15.2 even 4 575.2.b.d.24.1 4
15.8 even 4 575.2.b.d.24.4 4
15.14 odd 2 575.2.a.f.1.2 2
21.20 even 2 1127.2.a.c.1.1 2
23.22 odd 2 4761.2.a.w.1.2 2
24.5 odd 2 1472.2.a.t.1.1 2
24.11 even 2 1472.2.a.s.1.2 2
33.32 even 2 2783.2.a.c.1.2 2
39.38 odd 2 3887.2.a.i.1.2 2
51.50 odd 2 6647.2.a.b.1.1 2
57.56 even 2 8303.2.a.e.1.2 2
60.59 even 2 9200.2.a.bt.1.2 2
69.2 odd 22 529.2.c.o.487.1 20
69.5 even 22 529.2.c.n.255.1 20
69.8 odd 22 529.2.c.o.501.2 20
69.11 even 22 529.2.c.n.466.1 20
69.14 even 22 529.2.c.n.334.1 20
69.17 even 22 529.2.c.n.266.2 20
69.20 even 22 529.2.c.n.170.2 20
69.26 odd 22 529.2.c.o.170.2 20
69.29 odd 22 529.2.c.o.266.2 20
69.32 odd 22 529.2.c.o.334.1 20
69.35 odd 22 529.2.c.o.466.1 20
69.38 even 22 529.2.c.n.501.2 20
69.41 odd 22 529.2.c.o.255.1 20
69.44 even 22 529.2.c.n.487.1 20
69.50 odd 22 529.2.c.o.177.2 20
69.53 even 22 529.2.c.n.118.2 20
69.56 even 22 529.2.c.n.399.2 20
69.59 odd 22 529.2.c.o.399.2 20
69.62 odd 22 529.2.c.o.118.2 20
69.65 even 22 529.2.c.n.177.2 20
69.68 even 2 529.2.a.a.1.1 2
276.275 odd 2 8464.2.a.bb.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
23.2.a.a.1.1 2 3.2 odd 2
207.2.a.d.1.2 2 1.1 even 1 trivial
368.2.a.h.1.1 2 12.11 even 2
529.2.a.a.1.1 2 69.68 even 2
529.2.c.n.118.2 20 69.53 even 22
529.2.c.n.170.2 20 69.20 even 22
529.2.c.n.177.2 20 69.65 even 22
529.2.c.n.255.1 20 69.5 even 22
529.2.c.n.266.2 20 69.17 even 22
529.2.c.n.334.1 20 69.14 even 22
529.2.c.n.399.2 20 69.56 even 22
529.2.c.n.466.1 20 69.11 even 22
529.2.c.n.487.1 20 69.44 even 22
529.2.c.n.501.2 20 69.38 even 22
529.2.c.o.118.2 20 69.62 odd 22
529.2.c.o.170.2 20 69.26 odd 22
529.2.c.o.177.2 20 69.50 odd 22
529.2.c.o.255.1 20 69.41 odd 22
529.2.c.o.266.2 20 69.29 odd 22
529.2.c.o.334.1 20 69.32 odd 22
529.2.c.o.399.2 20 69.59 odd 22
529.2.c.o.466.1 20 69.35 odd 22
529.2.c.o.487.1 20 69.2 odd 22
529.2.c.o.501.2 20 69.8 odd 22
575.2.a.f.1.2 2 15.14 odd 2
575.2.b.d.24.1 4 15.2 even 4
575.2.b.d.24.4 4 15.8 even 4
1127.2.a.c.1.1 2 21.20 even 2
1472.2.a.s.1.2 2 24.11 even 2
1472.2.a.t.1.1 2 24.5 odd 2
2783.2.a.c.1.2 2 33.32 even 2
3312.2.a.ba.1.2 2 4.3 odd 2
3887.2.a.i.1.2 2 39.38 odd 2
4761.2.a.w.1.2 2 23.22 odd 2
5175.2.a.be.1.1 2 5.4 even 2
6647.2.a.b.1.1 2 51.50 odd 2
8303.2.a.e.1.2 2 57.56 even 2
8464.2.a.bb.1.1 2 276.275 odd 2
9200.2.a.bt.1.2 2 60.59 even 2