Properties

Label 1472.2.a.s.1.2
Level $1472$
Weight $2$
Character 1472.1
Self dual yes
Analytic conductor $11.754$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1472,2,Mod(1,1472)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1472, 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("1472.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1472 = 2^{6} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1472.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: \(11.7539791775\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 23)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 1472.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.23607 q^{3} +3.23607 q^{5} +1.23607 q^{7} +2.00000 q^{9} +O(q^{10})\) \(q+2.23607 q^{3} +3.23607 q^{5} +1.23607 q^{7} +2.00000 q^{9} -0.763932 q^{11} -3.00000 q^{13} +7.23607 q^{15} +5.23607 q^{17} -2.00000 q^{19} +2.76393 q^{21} -1.00000 q^{23} +5.47214 q^{25} -2.23607 q^{27} +3.00000 q^{29} +6.70820 q^{31} -1.70820 q^{33} +4.00000 q^{35} -3.23607 q^{37} -6.70820 q^{39} +5.47214 q^{41} +6.47214 q^{45} -2.23607 q^{47} -5.47214 q^{49} +11.7082 q^{51} +8.47214 q^{53} -2.47214 q^{55} -4.47214 q^{57} -2.47214 q^{59} -10.9443 q^{61} +2.47214 q^{63} -9.70820 q^{65} -7.23607 q^{67} -2.23607 q^{69} -7.76393 q^{71} +15.4721 q^{73} +12.2361 q^{75} -0.944272 q^{77} -6.94427 q^{79} -11.0000 q^{81} -13.2361 q^{83} +16.9443 q^{85} +6.70820 q^{87} -1.52786 q^{89} -3.70820 q^{91} +15.0000 q^{93} -6.47214 q^{95} +4.29180 q^{97} -1.52786 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - 2 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} - 2 q^{7} + 4 q^{9} - 6 q^{11} - 6 q^{13} + 10 q^{15} + 6 q^{17} - 4 q^{19} + 10 q^{21} - 2 q^{23} + 2 q^{25} + 6 q^{29} + 10 q^{33} + 8 q^{35} - 2 q^{37} + 2 q^{41} + 4 q^{45} - 2 q^{49} + 10 q^{51} + 8 q^{53} + 4 q^{55} + 4 q^{59} - 4 q^{61} - 4 q^{63} - 6 q^{65} - 10 q^{67} - 20 q^{71} + 22 q^{73} + 20 q^{75} + 16 q^{77} + 4 q^{79} - 22 q^{81} - 22 q^{83} + 16 q^{85} - 12 q^{89} + 6 q^{91} + 30 q^{93} - 4 q^{95} + 22 q^{97} - 12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.23607 1.29099 0.645497 0.763763i \(-0.276650\pi\)
0.645497 + 0.763763i \(0.276650\pi\)
\(4\) 0 0
\(5\) 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 \(-0.424951\pi\)
0.233595 + 0.972334i \(0.424951\pi\)
\(8\) 0 0
\(9\) 2.00000 0.666667
\(10\) 0 0
\(11\) −0.763932 −0.230334 −0.115167 0.993346i \(-0.536740\pi\)
−0.115167 + 0.993346i \(0.536740\pi\)
\(12\) 0 0
\(13\) −3.00000 −0.832050 −0.416025 0.909353i \(-0.636577\pi\)
−0.416025 + 0.909353i \(0.636577\pi\)
\(14\) 0 0
\(15\) 7.23607 1.86834
\(16\) 0 0
\(17\) 5.23607 1.26993 0.634967 0.772540i \(-0.281014\pi\)
0.634967 + 0.772540i \(0.281014\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 2.76393 0.603139
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 5.47214 1.09443
\(26\) 0 0
\(27\) −2.23607 −0.430331
\(28\) 0 0
\(29\) 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 \(-0.294205\pi\)
0.602414 + 0.798183i \(0.294205\pi\)
\(32\) 0 0
\(33\) −1.70820 −0.297360
\(34\) 0 0
\(35\) 4.00000 0.676123
\(36\) 0 0
\(37\) −3.23607 −0.532006 −0.266003 0.963972i \(-0.585703\pi\)
−0.266003 + 0.963972i \(0.585703\pi\)
\(38\) 0 0
\(39\) −6.70820 −1.07417
\(40\) 0 0
\(41\) 5.47214 0.854604 0.427302 0.904109i \(-0.359464\pi\)
0.427302 + 0.904109i \(0.359464\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 6.47214 0.964809
\(46\) 0 0
\(47\) −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
\(50\) 0 0
\(51\) 11.7082 1.63948
\(52\) 0 0
\(53\) 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
\(56\) 0 0
\(57\) −4.47214 −0.592349
\(58\) 0 0
\(59\) −2.47214 −0.321845 −0.160922 0.986967i \(-0.551447\pi\)
−0.160922 + 0.986967i \(0.551447\pi\)
\(60\) 0 0
\(61\) −10.9443 −1.40127 −0.700635 0.713520i \(-0.747100\pi\)
−0.700635 + 0.713520i \(0.747100\pi\)
\(62\) 0 0
\(63\) 2.47214 0.311460
\(64\) 0 0
\(65\) −9.70820 −1.20415
\(66\) 0 0
\(67\) −7.23607 −0.884026 −0.442013 0.897009i \(-0.645736\pi\)
−0.442013 + 0.897009i \(0.645736\pi\)
\(68\) 0 0
\(69\) −2.23607 −0.269191
\(70\) 0 0
\(71\) −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 \(0.139542\pi\)
\(74\) 0 0
\(75\) 12.2361 1.41290
\(76\) 0 0
\(77\) −0.944272 −0.107610
\(78\) 0 0
\(79\) −6.94427 −0.781292 −0.390646 0.920541i \(-0.627748\pi\)
−0.390646 + 0.920541i \(0.627748\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) −13.2361 −1.45285 −0.726424 0.687247i \(-0.758819\pi\)
−0.726424 + 0.687247i \(0.758819\pi\)
\(84\) 0 0
\(85\) 16.9443 1.83786
\(86\) 0 0
\(87\) 6.70820 0.719195
\(88\) 0 0
\(89\) −1.52786 −0.161953 −0.0809766 0.996716i \(-0.525804\pi\)
−0.0809766 + 0.996716i \(0.525804\pi\)
\(90\) 0 0
\(91\) −3.70820 −0.388725
\(92\) 0 0
\(93\) 15.0000 1.55543
\(94\) 0 0
\(95\) −6.47214 −0.664027
\(96\) 0 0
\(97\) 4.29180 0.435766 0.217883 0.975975i \(-0.430085\pi\)
0.217883 + 0.975975i \(0.430085\pi\)
\(98\) 0 0
\(99\) −1.52786 −0.153556
\(100\) 0 0
\(101\) 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 \(-0.853311\pi\)
−0.895681 + 0.444697i \(0.853311\pi\)
\(104\) 0 0
\(105\) 8.94427 0.872872
\(106\) 0 0
\(107\) −13.4164 −1.29701 −0.648507 0.761209i \(-0.724606\pi\)
−0.648507 + 0.761209i \(0.724606\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) −7.23607 −0.686817
\(112\) 0 0
\(113\) 13.2361 1.24514 0.622572 0.782562i \(-0.286088\pi\)
0.622572 + 0.782562i \(0.286088\pi\)
\(114\) 0 0
\(115\) −3.23607 −0.301765
\(116\) 0 0
\(117\) −6.00000 −0.554700
\(118\) 0 0
\(119\) 6.47214 0.593300
\(120\) 0 0
\(121\) −10.4164 −0.946946
\(122\) 0 0
\(123\) 12.2361 1.10329
\(124\) 0 0
\(125\) 1.52786 0.136656
\(126\) 0 0
\(127\) 20.7082 1.83756 0.918778 0.394775i \(-0.129177\pi\)
0.918778 + 0.394775i \(0.129177\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 5.29180 0.462346 0.231173 0.972913i \(-0.425744\pi\)
0.231173 + 0.972913i \(0.425744\pi\)
\(132\) 0 0
\(133\) −2.47214 −0.214361
\(134\) 0 0
\(135\) −7.23607 −0.622782
\(136\) 0 0
\(137\) 13.8885 1.18658 0.593289 0.804989i \(-0.297829\pi\)
0.593289 + 0.804989i \(0.297829\pi\)
\(138\) 0 0
\(139\) 2.70820 0.229707 0.114853 0.993382i \(-0.463360\pi\)
0.114853 + 0.993382i \(0.463360\pi\)
\(140\) 0 0
\(141\) −5.00000 −0.421076
\(142\) 0 0
\(143\) 2.29180 0.191650
\(144\) 0 0
\(145\) 9.70820 0.806222
\(146\) 0 0
\(147\) −12.2361 −1.00921
\(148\) 0 0
\(149\) 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 \(-0.496942\pi\)
0.00960547 + 0.999954i \(0.496942\pi\)
\(152\) 0 0
\(153\) 10.4721 0.846622
\(154\) 0 0
\(155\) 21.7082 1.74364
\(156\) 0 0
\(157\) −15.4164 −1.23036 −0.615182 0.788385i \(-0.710917\pi\)
−0.615182 + 0.788385i \(0.710917\pi\)
\(158\) 0 0
\(159\) 18.9443 1.50238
\(160\) 0 0
\(161\) −1.23607 −0.0974158
\(162\) 0 0
\(163\) −10.2361 −0.801751 −0.400875 0.916133i \(-0.631294\pi\)
−0.400875 + 0.916133i \(0.631294\pi\)
\(164\) 0 0
\(165\) −5.52786 −0.430344
\(166\) 0 0
\(167\) −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
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 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
\(176\) 0 0
\(177\) −5.52786 −0.415500
\(178\) 0 0
\(179\) −12.7082 −0.949856 −0.474928 0.880025i \(-0.657526\pi\)
−0.474928 + 0.880025i \(0.657526\pi\)
\(180\) 0 0
\(181\) 14.6525 1.08911 0.544555 0.838725i \(-0.316699\pi\)
0.544555 + 0.838725i \(0.316699\pi\)
\(182\) 0 0
\(183\) −24.4721 −1.80903
\(184\) 0 0
\(185\) −10.4721 −0.769927
\(186\) 0 0
\(187\) −4.00000 −0.292509
\(188\) 0 0
\(189\) −2.76393 −0.201046
\(190\) 0 0
\(191\) 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 \(0.592299\pi\)
\(194\) 0 0
\(195\) −21.7082 −1.55456
\(196\) 0 0
\(197\) −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 \(-0.135155\pi\)
0.911203 + 0.411957i \(0.135155\pi\)
\(200\) 0 0
\(201\) −16.1803 −1.14127
\(202\) 0 0
\(203\) 3.70820 0.260265
\(204\) 0 0
\(205\) 17.7082 1.23679
\(206\) 0 0
\(207\) −2.00000 −0.139010
\(208\) 0 0
\(209\) 1.52786 0.105685
\(210\) 0 0
\(211\) 3.41641 0.235195 0.117598 0.993061i \(-0.462481\pi\)
0.117598 + 0.993061i \(0.462481\pi\)
\(212\) 0 0
\(213\) −17.3607 −1.18953
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 8.29180 0.562884
\(218\) 0 0
\(219\) 34.5967 2.33783
\(220\) 0 0
\(221\) −15.7082 −1.05665
\(222\) 0 0
\(223\) −4.00000 −0.267860 −0.133930 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) 0 0
\(225\) 10.9443 0.729618
\(226\) 0 0
\(227\) 10.1803 0.675693 0.337846 0.941201i \(-0.390302\pi\)
0.337846 + 0.941201i \(0.390302\pi\)
\(228\) 0 0
\(229\) 12.0000 0.792982 0.396491 0.918039i \(-0.370228\pi\)
0.396491 + 0.918039i \(0.370228\pi\)
\(230\) 0 0
\(231\) −2.11146 −0.138924
\(232\) 0 0
\(233\) −15.4721 −1.01361 −0.506807 0.862060i \(-0.669174\pi\)
−0.506807 + 0.862060i \(0.669174\pi\)
\(234\) 0 0
\(235\) −7.23607 −0.472029
\(236\) 0 0
\(237\) −15.5279 −1.00864
\(238\) 0 0
\(239\) −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 \(0.314038\pi\)
\(242\) 0 0
\(243\) −17.8885 −1.14755
\(244\) 0 0
\(245\) −17.7082 −1.13134
\(246\) 0 0
\(247\) 6.00000 0.381771
\(248\) 0 0
\(249\) −29.5967 −1.87562
\(250\) 0 0
\(251\) 15.7082 0.991493 0.495747 0.868467i \(-0.334895\pi\)
0.495747 + 0.868467i \(0.334895\pi\)
\(252\) 0 0
\(253\) 0.763932 0.0480280
\(254\) 0 0
\(255\) 37.8885 2.37267
\(256\) 0 0
\(257\) 1.47214 0.0918293 0.0459147 0.998945i \(-0.485380\pi\)
0.0459147 + 0.998945i \(0.485380\pi\)
\(258\) 0 0
\(259\) −4.00000 −0.248548
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 0 0
\(263\) 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
\(266\) 0 0
\(267\) −3.41641 −0.209081
\(268\) 0 0
\(269\) −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 \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 0 0
\(273\) −8.29180 −0.501842
\(274\) 0 0
\(275\) −4.18034 −0.252084
\(276\) 0 0
\(277\) −6.52786 −0.392221 −0.196111 0.980582i \(-0.562831\pi\)
−0.196111 + 0.980582i \(0.562831\pi\)
\(278\) 0 0
\(279\) 13.4164 0.803219
\(280\) 0 0
\(281\) −13.2361 −0.789598 −0.394799 0.918768i \(-0.629186\pi\)
−0.394799 + 0.918768i \(0.629186\pi\)
\(282\) 0 0
\(283\) 14.2918 0.849559 0.424780 0.905297i \(-0.360352\pi\)
0.424780 + 0.905297i \(0.360352\pi\)
\(284\) 0 0
\(285\) −14.4721 −0.857255
\(286\) 0 0
\(287\) 6.76393 0.399262
\(288\) 0 0
\(289\) 10.4164 0.612730
\(290\) 0 0
\(291\) 9.59675 0.562571
\(292\) 0 0
\(293\) 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
\(296\) 0 0
\(297\) 1.70820 0.0991200
\(298\) 0 0
\(299\) 3.00000 0.173494
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 10.0000 0.574485
\(304\) 0 0
\(305\) −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 0
\(309\) −40.6525 −2.31264
\(310\) 0 0
\(311\) 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 \(0.695164\pi\)
\(314\) 0 0
\(315\) 8.00000 0.450749
\(316\) 0 0
\(317\) 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
\(320\) 0 0
\(321\) −30.0000 −1.67444
\(322\) 0 0
\(323\) −10.4721 −0.582685
\(324\) 0 0
\(325\) −16.4164 −0.910618
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −2.76393 −0.152381
\(330\) 0 0
\(331\) 11.6525 0.640478 0.320239 0.947337i \(-0.396237\pi\)
0.320239 + 0.947337i \(0.396237\pi\)
\(332\) 0 0
\(333\) −6.47214 −0.354671
\(334\) 0 0
\(335\) −23.4164 −1.27938
\(336\) 0 0
\(337\) −3.41641 −0.186104 −0.0930518 0.995661i \(-0.529662\pi\)
−0.0930518 + 0.995661i \(0.529662\pi\)
\(338\) 0 0
\(339\) 29.5967 1.60747
\(340\) 0 0
\(341\) −5.12461 −0.277513
\(342\) 0 0
\(343\) −15.4164 −0.832408
\(344\) 0 0
\(345\) −7.23607 −0.389577
\(346\) 0 0
\(347\) 25.8885 1.38977 0.694885 0.719121i \(-0.255455\pi\)
0.694885 + 0.719121i \(0.255455\pi\)
\(348\) 0 0
\(349\) 2.41641 0.129347 0.0646737 0.997906i \(-0.479399\pi\)
0.0646737 + 0.997906i \(0.479399\pi\)
\(350\) 0 0
\(351\) 6.70820 0.358057
\(352\) 0 0
\(353\) −35.3607 −1.88206 −0.941030 0.338324i \(-0.890140\pi\)
−0.941030 + 0.338324i \(0.890140\pi\)
\(354\) 0 0
\(355\) −25.1246 −1.33348
\(356\) 0 0
\(357\) 14.4721 0.765947
\(358\) 0 0
\(359\) −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
\(362\) 0 0
\(363\) −23.2918 −1.22250
\(364\) 0 0
\(365\) 50.0689 2.62073
\(366\) 0 0
\(367\) −18.1803 −0.949006 −0.474503 0.880254i \(-0.657372\pi\)
−0.474503 + 0.880254i \(0.657372\pi\)
\(368\) 0 0
\(369\) 10.9443 0.569736
\(370\) 0 0
\(371\) 10.4721 0.543686
\(372\) 0 0
\(373\) 5.70820 0.295560 0.147780 0.989020i \(-0.452787\pi\)
0.147780 + 0.989020i \(0.452787\pi\)
\(374\) 0 0
\(375\) 3.41641 0.176423
\(376\) 0 0
\(377\) −9.00000 −0.463524
\(378\) 0 0
\(379\) −20.3607 −1.04586 −0.522929 0.852376i \(-0.675161\pi\)
−0.522929 + 0.852376i \(0.675161\pi\)
\(380\) 0 0
\(381\) 46.3050 2.37227
\(382\) 0 0
\(383\) −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
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −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
\(392\) 0 0
\(393\) 11.8328 0.596887
\(394\) 0 0
\(395\) −22.4721 −1.13070
\(396\) 0 0
\(397\) −2.41641 −0.121276 −0.0606380 0.998160i \(-0.519314\pi\)
−0.0606380 + 0.998160i \(0.519314\pi\)
\(398\) 0 0
\(399\) −5.52786 −0.276739
\(400\) 0 0
\(401\) 8.18034 0.408507 0.204253 0.978918i \(-0.434523\pi\)
0.204253 + 0.978918i \(0.434523\pi\)
\(402\) 0 0
\(403\) −20.1246 −1.00248
\(404\) 0 0
\(405\) −35.5967 −1.76882
\(406\) 0 0
\(407\) 2.47214 0.122539
\(408\) 0 0
\(409\) −23.3607 −1.15511 −0.577556 0.816351i \(-0.695993\pi\)
−0.577556 + 0.816351i \(0.695993\pi\)
\(410\) 0 0
\(411\) 31.0557 1.53187
\(412\) 0 0
\(413\) −3.05573 −0.150363
\(414\) 0 0
\(415\) −42.8328 −2.10258
\(416\) 0 0
\(417\) 6.05573 0.296550
\(418\) 0 0
\(419\) −31.4164 −1.53479 −0.767396 0.641173i \(-0.778448\pi\)
−0.767396 + 0.641173i \(0.778448\pi\)
\(420\) 0 0
\(421\) 23.7082 1.15547 0.577734 0.816225i \(-0.303937\pi\)
0.577734 + 0.816225i \(0.303937\pi\)
\(422\) 0 0
\(423\) −4.47214 −0.217443
\(424\) 0 0
\(425\) 28.6525 1.38985
\(426\) 0 0
\(427\) −13.5279 −0.654659
\(428\) 0 0
\(429\) 5.12461 0.247419
\(430\) 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 \(0.0838897\pi\)
\(434\) 0 0
\(435\) 21.7082 1.04083
\(436\) 0 0
\(437\) 2.00000 0.0956730
\(438\) 0 0
\(439\) 5.29180 0.252564 0.126282 0.991994i \(-0.459696\pi\)
0.126282 + 0.991994i \(0.459696\pi\)
\(440\) 0 0
\(441\) −10.9443 −0.521156
\(442\) 0 0
\(443\) −2.12461 −0.100943 −0.0504717 0.998725i \(-0.516072\pi\)
−0.0504717 + 0.998725i \(0.516072\pi\)
\(444\) 0 0
\(445\) −4.94427 −0.234381
\(446\) 0 0
\(447\) 26.5836 1.25736
\(448\) 0 0
\(449\) 2.94427 0.138949 0.0694744 0.997584i \(-0.477868\pi\)
0.0694744 + 0.997584i \(0.477868\pi\)
\(450\) 0 0
\(451\) −4.18034 −0.196845
\(452\) 0 0
\(453\) 0.527864 0.0248012
\(454\) 0 0
\(455\) −12.0000 −0.562569
\(456\) 0 0
\(457\) 35.1246 1.64306 0.821530 0.570165i \(-0.193121\pi\)
0.821530 + 0.570165i \(0.193121\pi\)
\(458\) 0 0
\(459\) −11.7082 −0.546492
\(460\) 0 0
\(461\) −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 \(-0.346148\pi\)
0.464739 + 0.885448i \(0.346148\pi\)
\(464\) 0 0
\(465\) 48.5410 2.25104
\(466\) 0 0
\(467\) −30.9443 −1.43193 −0.715965 0.698136i \(-0.754013\pi\)
−0.715965 + 0.698136i \(0.754013\pi\)
\(468\) 0 0
\(469\) −8.94427 −0.413008
\(470\) 0 0
\(471\) −34.4721 −1.58839
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −10.9443 −0.502158
\(476\) 0 0
\(477\) 16.9443 0.775825
\(478\) 0 0
\(479\) 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
\(482\) 0 0
\(483\) −2.76393 −0.125763
\(484\) 0 0
\(485\) 13.8885 0.630646
\(486\) 0 0
\(487\) 1.29180 0.0585369 0.0292684 0.999572i \(-0.490682\pi\)
0.0292684 + 0.999572i \(0.490682\pi\)
\(488\) 0 0
\(489\) −22.8885 −1.03506
\(490\) 0 0
\(491\) 39.6525 1.78949 0.894746 0.446576i \(-0.147357\pi\)
0.894746 + 0.446576i \(0.147357\pi\)
\(492\) 0 0
\(493\) 15.7082 0.707462
\(494\) 0 0
\(495\) −4.94427 −0.222228
\(496\) 0 0
\(497\) −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 0
\(501\) −23.4164 −1.04617
\(502\) 0 0
\(503\) 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
\(506\) 0 0
\(507\) −8.94427 −0.397229
\(508\) 0 0
\(509\) −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
\(512\) 0 0
\(513\) 4.47214 0.197450
\(514\) 0 0
\(515\) −58.8328 −2.59248
\(516\) 0 0
\(517\) 1.70820 0.0751267
\(518\) 0 0
\(519\) −11.3050 −0.496232
\(520\) 0 0
\(521\) 4.58359 0.200811 0.100405 0.994947i \(-0.467986\pi\)
0.100405 + 0.994947i \(0.467986\pi\)
\(522\) 0 0
\(523\) 0.875388 0.0382781 0.0191390 0.999817i \(-0.493907\pi\)
0.0191390 + 0.999817i \(0.493907\pi\)
\(524\) 0 0
\(525\) 15.1246 0.660092
\(526\) 0 0
\(527\) 35.1246 1.53005
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −4.94427 −0.214563
\(532\) 0 0
\(533\) −16.4164 −0.711074
\(534\) 0 0
\(535\) −43.4164 −1.87705
\(536\) 0 0
\(537\) −28.4164 −1.22626
\(538\) 0 0
\(539\) 4.18034 0.180060
\(540\) 0 0
\(541\) 7.58359 0.326044 0.163022 0.986622i \(-0.447876\pi\)
0.163022 + 0.986622i \(0.447876\pi\)
\(542\) 0 0
\(543\) 32.7639 1.40603
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 37.5410 1.60514 0.802569 0.596559i \(-0.203466\pi\)
0.802569 + 0.596559i \(0.203466\pi\)
\(548\) 0 0
\(549\) −21.8885 −0.934180
\(550\) 0 0
\(551\) −6.00000 −0.255609
\(552\) 0 0
\(553\) −8.58359 −0.365011
\(554\) 0 0
\(555\) −23.4164 −0.993971
\(556\) 0 0
\(557\) −19.4164 −0.822700 −0.411350 0.911478i \(-0.634943\pi\)
−0.411350 + 0.911478i \(0.634943\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −8.94427 −0.377627
\(562\) 0 0
\(563\) −15.0557 −0.634523 −0.317262 0.948338i \(-0.602763\pi\)
−0.317262 + 0.948338i \(0.602763\pi\)
\(564\) 0 0
\(565\) 42.8328 1.80199
\(566\) 0 0
\(567\) −13.5967 −0.571010
\(568\) 0 0
\(569\) 0.180340 0.00756024 0.00378012 0.999993i \(-0.498797\pi\)
0.00378012 + 0.999993i \(0.498797\pi\)
\(570\) 0 0
\(571\) −27.7082 −1.15955 −0.579776 0.814776i \(-0.696860\pi\)
−0.579776 + 0.814776i \(0.696860\pi\)
\(572\) 0 0
\(573\) 8.54102 0.356806
\(574\) 0 0
\(575\) −5.47214 −0.228204
\(576\) 0 0
\(577\) −12.8885 −0.536557 −0.268279 0.963341i \(-0.586455\pi\)
−0.268279 + 0.963341i \(0.586455\pi\)
\(578\) 0 0
\(579\) −17.7639 −0.738244
\(580\) 0 0
\(581\) −16.3607 −0.678755
\(582\) 0 0
\(583\) −6.47214 −0.268048
\(584\) 0 0
\(585\) −19.4164 −0.802770
\(586\) 0 0
\(587\) −11.2918 −0.466062 −0.233031 0.972469i \(-0.574864\pi\)
−0.233031 + 0.972469i \(0.574864\pi\)
\(588\) 0 0
\(589\) −13.4164 −0.552813
\(590\) 0 0
\(591\) −16.7082 −0.687284
\(592\) 0 0
\(593\) 14.9443 0.613688 0.306844 0.951760i \(-0.400727\pi\)
0.306844 + 0.951760i \(0.400727\pi\)
\(594\) 0 0
\(595\) 20.9443 0.858631
\(596\) 0 0
\(597\) 57.4853 2.35272
\(598\) 0 0
\(599\) 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
\(603\) −14.4721 −0.589351
\(604\) 0 0
\(605\) −33.7082 −1.37043
\(606\) 0 0
\(607\) −17.5279 −0.711434 −0.355717 0.934594i \(-0.615763\pi\)
−0.355717 + 0.934594i \(0.615763\pi\)
\(608\) 0 0
\(609\) 8.29180 0.336001
\(610\) 0 0
\(611\) 6.70820 0.271385
\(612\) 0 0
\(613\) 7.70820 0.311331 0.155666 0.987810i \(-0.450248\pi\)
0.155666 + 0.987810i \(0.450248\pi\)
\(614\) 0 0
\(615\) 39.5967 1.59669
\(616\) 0 0
\(617\) −16.4721 −0.663143 −0.331572 0.943430i \(-0.607579\pi\)
−0.331572 + 0.943430i \(0.607579\pi\)
\(618\) 0 0
\(619\) −7.41641 −0.298091 −0.149045 0.988830i \(-0.547620\pi\)
−0.149045 + 0.988830i \(0.547620\pi\)
\(620\) 0 0
\(621\) 2.23607 0.0897303
\(622\) 0 0
\(623\) −1.88854 −0.0756629
\(624\) 0 0
\(625\) −22.4164 −0.896656
\(626\) 0 0
\(627\) 3.41641 0.136438
\(628\) 0 0
\(629\) −16.9443 −0.675612
\(630\) 0 0
\(631\) 32.3607 1.28826 0.644129 0.764917i \(-0.277220\pi\)
0.644129 + 0.764917i \(0.277220\pi\)
\(632\) 0 0
\(633\) 7.63932 0.303636
\(634\) 0 0
\(635\) 67.0132 2.65934
\(636\) 0 0
\(637\) 16.4164 0.650442
\(638\) 0 0
\(639\) −15.5279 −0.614273
\(640\) 0 0
\(641\) 45.3050 1.78944 0.894719 0.446629i \(-0.147376\pi\)
0.894719 + 0.446629i \(0.147376\pi\)
\(642\) 0 0
\(643\) 19.5967 0.772820 0.386410 0.922327i \(-0.373715\pi\)
0.386410 + 0.922327i \(0.373715\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 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
\(650\) 0 0
\(651\) 18.5410 0.726680
\(652\) 0 0
\(653\) −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
\(656\) 0 0
\(657\) 30.9443 1.20725
\(658\) 0 0
\(659\) 20.6525 0.804506 0.402253 0.915528i \(-0.368227\pi\)
0.402253 + 0.915528i \(0.368227\pi\)
\(660\) 0 0
\(661\) 5.05573 0.196645 0.0983225 0.995155i \(-0.468652\pi\)
0.0983225 + 0.995155i \(0.468652\pi\)
\(662\) 0 0
\(663\) −35.1246 −1.36413
\(664\) 0 0
\(665\) −8.00000 −0.310227
\(666\) 0 0
\(667\) −3.00000 −0.116160
\(668\) 0 0
\(669\) −8.94427 −0.345806
\(670\) 0 0
\(671\) 8.36068 0.322760
\(672\) 0 0
\(673\) 3.00000 0.115642 0.0578208 0.998327i \(-0.481585\pi\)
0.0578208 + 0.998327i \(0.481585\pi\)
\(674\) 0 0
\(675\) −12.2361 −0.470966
\(676\) 0 0
\(677\) −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
\(680\) 0 0
\(681\) 22.7639 0.872316
\(682\) 0 0
\(683\) −22.5967 −0.864641 −0.432320 0.901720i \(-0.642305\pi\)
−0.432320 + 0.901720i \(0.642305\pi\)
\(684\) 0 0
\(685\) 44.9443 1.71723
\(686\) 0 0
\(687\) 26.8328 1.02374
\(688\) 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 \(0.342642\pi\)
\(692\) 0 0
\(693\) −1.88854 −0.0717398
\(694\) 0 0
\(695\) 8.76393 0.332435
\(696\) 0 0
\(697\) 28.6525 1.08529
\(698\) 0 0
\(699\) −34.5967 −1.30857
\(700\) 0 0
\(701\) 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
\(704\) 0 0
\(705\) −16.1803 −0.609387
\(706\) 0 0
\(707\) 5.52786 0.207897
\(708\) 0 0
\(709\) −16.0689 −0.603480 −0.301740 0.953390i \(-0.597567\pi\)
−0.301740 + 0.953390i \(0.597567\pi\)
\(710\) 0 0
\(711\) −13.8885 −0.520861
\(712\) 0 0
\(713\) −6.70820 −0.251224
\(714\) 0 0
\(715\) 7.41641 0.277358
\(716\) 0 0
\(717\) −40.7771 −1.52285
\(718\) 0 0
\(719\) 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
\(722\) 0 0
\(723\) 38.2918 1.42409
\(724\) 0 0
\(725\) 16.4164 0.609690
\(726\) 0 0
\(727\) 14.2918 0.530053 0.265027 0.964241i \(-0.414619\pi\)
0.265027 + 0.964241i \(0.414619\pi\)
\(728\) 0 0
\(729\) −7.00000 −0.259259
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 26.7639 0.988548 0.494274 0.869306i \(-0.335434\pi\)
0.494274 + 0.869306i \(0.335434\pi\)
\(734\) 0 0
\(735\) −39.5967 −1.46055
\(736\) 0 0
\(737\) 5.52786 0.203621
\(738\) 0 0
\(739\) 49.1803 1.80913 0.904564 0.426338i \(-0.140197\pi\)
0.904564 + 0.426338i \(0.140197\pi\)
\(740\) 0 0
\(741\) 13.4164 0.492864
\(742\) 0 0
\(743\) −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
\(746\) 0 0
\(747\) −26.4721 −0.968565
\(748\) 0 0
\(749\) −16.5836 −0.605951
\(750\) 0 0
\(751\) 44.3607 1.61874 0.809372 0.587296i \(-0.199808\pi\)
0.809372 + 0.587296i \(0.199808\pi\)
\(752\) 0 0
\(753\) 35.1246 1.28001
\(754\) 0 0
\(755\) 0.763932 0.0278023
\(756\) 0 0
\(757\) 47.5967 1.72993 0.864967 0.501829i \(-0.167339\pi\)
0.864967 + 0.501829i \(0.167339\pi\)
\(758\) 0 0
\(759\) 1.70820 0.0620039
\(760\) 0 0
\(761\) −16.3050 −0.591054 −0.295527 0.955334i \(-0.595495\pi\)
−0.295527 + 0.955334i \(0.595495\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 33.8885 1.22524
\(766\) 0 0
\(767\) 7.41641 0.267791
\(768\) 0 0
\(769\) 17.1246 0.617529 0.308765 0.951138i \(-0.400084\pi\)
0.308765 + 0.951138i \(0.400084\pi\)
\(770\) 0 0
\(771\) 3.29180 0.118551
\(772\) 0 0
\(773\) 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
\(776\) 0 0
\(777\) −8.94427 −0.320874
\(778\) 0 0
\(779\) −10.9443 −0.392119
\(780\) 0 0
\(781\) 5.93112 0.212232
\(782\) 0 0
\(783\) −6.70820 −0.239732
\(784\) 0 0
\(785\) −49.8885 −1.78060
\(786\) 0 0
\(787\) 51.4164 1.83280 0.916399 0.400267i \(-0.131083\pi\)
0.916399 + 0.400267i \(0.131083\pi\)
\(788\) 0 0
\(789\) 33.4164 1.18966
\(790\) 0 0
\(791\) 16.3607 0.581719
\(792\) 0 0
\(793\) 32.8328 1.16593
\(794\) 0 0
\(795\) 61.3050 2.17426
\(796\) 0 0
\(797\) −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
\(800\) 0 0
\(801\) −3.05573 −0.107969
\(802\) 0 0
\(803\) −11.8197 −0.417107
\(804\) 0 0
\(805\) −4.00000 −0.140981
\(806\) 0 0
\(807\) −22.2361 −0.782747
\(808\) 0 0
\(809\) 47.8885 1.68367 0.841836 0.539734i \(-0.181475\pi\)
0.841836 + 0.539734i \(0.181475\pi\)
\(810\) 0 0
\(811\) −55.6525 −1.95422 −0.977111 0.212728i \(-0.931765\pi\)
−0.977111 + 0.212728i \(0.931765\pi\)
\(812\) 0 0
\(813\) −17.8885 −0.627379
\(814\) 0 0
\(815\) −33.1246 −1.16030
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −7.41641 −0.259150
\(820\) 0 0
\(821\) 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 \(-0.659367\pi\)
−0.480010 + 0.877263i \(0.659367\pi\)
\(824\) 0 0
\(825\) −9.34752 −0.325439
\(826\) 0 0
\(827\) 10.4721 0.364152 0.182076 0.983284i \(-0.441718\pi\)
0.182076 + 0.983284i \(0.441718\pi\)
\(828\) 0 0
\(829\) 40.2492 1.39791 0.698957 0.715164i \(-0.253648\pi\)
0.698957 + 0.715164i \(0.253648\pi\)
\(830\) 0 0
\(831\) −14.5967 −0.506356
\(832\) 0 0
\(833\) −28.6525 −0.992749
\(834\) 0 0
\(835\) −33.8885 −1.17276
\(836\) 0 0
\(837\) −15.0000 −0.518476
\(838\) 0 0
\(839\) 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
\(842\) 0 0
\(843\) −29.5967 −1.01937
\(844\) 0 0
\(845\) −12.9443 −0.445296
\(846\) 0 0
\(847\) −12.8754 −0.442404
\(848\) 0 0
\(849\) 31.9574 1.09678
\(850\) 0 0
\(851\) 3.23607 0.110931
\(852\) 0 0
\(853\) 37.4164 1.28111 0.640557 0.767911i \(-0.278704\pi\)
0.640557 + 0.767911i \(0.278704\pi\)
\(854\) 0 0
\(855\) −12.9443 −0.442685
\(856\) 0 0
\(857\) −7.47214 −0.255243 −0.127622 0.991823i \(-0.540734\pi\)
−0.127622 + 0.991823i \(0.540734\pi\)
\(858\) 0 0
\(859\) −3.29180 −0.112315 −0.0561573 0.998422i \(-0.517885\pi\)
−0.0561573 + 0.998422i \(0.517885\pi\)
\(860\) 0 0
\(861\) 15.1246 0.515445
\(862\) 0 0
\(863\) −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
\(866\) 0 0
\(867\) 23.2918 0.791031
\(868\) 0 0
\(869\) 5.30495 0.179958
\(870\) 0 0
\(871\) 21.7082 0.735554
\(872\) 0 0
\(873\) 8.58359 0.290511
\(874\) 0 0
\(875\) 1.88854 0.0638444
\(876\) 0 0
\(877\) 27.5279 0.929550 0.464775 0.885429i \(-0.346135\pi\)
0.464775 + 0.885429i \(0.346135\pi\)
\(878\) 0 0
\(879\) 23.4164 0.789816
\(880\) 0 0
\(881\) 21.8197 0.735123 0.367562 0.929999i \(-0.380193\pi\)
0.367562 + 0.929999i \(0.380193\pi\)
\(882\) 0 0
\(883\) 4.00000 0.134611 0.0673054 0.997732i \(-0.478560\pi\)
0.0673054 + 0.997732i \(0.478560\pi\)
\(884\) 0 0
\(885\) −17.8885 −0.601317
\(886\) 0 0
\(887\) 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
\(890\) 0 0
\(891\) 8.40325 0.281520
\(892\) 0 0
\(893\) 4.47214 0.149654
\(894\) 0 0
\(895\) −41.1246 −1.37464
\(896\) 0 0
\(897\) 6.70820 0.223980
\(898\) 0 0
\(899\) 20.1246 0.671193
\(900\) 0 0
\(901\) 44.3607 1.47787
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 47.4164 1.57617
\(906\) 0 0
\(907\) 40.2492 1.33645 0.668227 0.743958i \(-0.267054\pi\)
0.668227 + 0.743958i \(0.267054\pi\)
\(908\) 0 0
\(909\) 8.94427 0.296663
\(910\) 0 0
\(911\) 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
\(914\) 0 0
\(915\) −79.1935 −2.61806
\(916\) 0 0
\(917\) 6.54102 0.216003
\(918\) 0 0
\(919\) −0.875388 −0.0288764 −0.0144382 0.999896i \(-0.504596\pi\)
−0.0144382 + 0.999896i \(0.504596\pi\)
\(920\) 0 0
\(921\) 41.3050 1.36104
\(922\) 0 0
\(923\) 23.2918 0.766659
\(924\) 0 0
\(925\) −17.7082 −0.582242
\(926\) 0 0
\(927\) −36.3607 −1.19424
\(928\) 0 0
\(929\) −41.9443 −1.37615 −0.688073 0.725641i \(-0.741543\pi\)
−0.688073 + 0.725641i \(0.741543\pi\)
\(930\) 0 0
\(931\) 10.9443 0.358684
\(932\) 0 0
\(933\) 20.5279 0.672052
\(934\) 0 0
\(935\) −12.9443 −0.423323
\(936\) 0 0
\(937\) 11.8197 0.386131 0.193066 0.981186i \(-0.438157\pi\)
0.193066 + 0.981186i \(0.438157\pi\)
\(938\) 0 0
\(939\) −45.5279 −1.48575
\(940\) 0 0
\(941\) 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
\(944\) 0 0
\(945\) −8.94427 −0.290957
\(946\) 0 0
\(947\) −33.1803 −1.07822 −0.539108 0.842237i \(-0.681239\pi\)
−0.539108 + 0.842237i \(0.681239\pi\)
\(948\) 0 0
\(949\) −46.4164 −1.50674
\(950\) 0 0
\(951\) 3.16718 0.102703
\(952\) 0 0
\(953\) 11.5279 0.373424 0.186712 0.982415i \(-0.440217\pi\)
0.186712 + 0.982415i \(0.440217\pi\)
\(954\) 0 0
\(955\) 12.3607 0.399982
\(956\) 0 0
\(957\) −5.12461 −0.165655
\(958\) 0 0
\(959\) 17.1672 0.554357
\(960\) 0 0
\(961\) 14.0000 0.451613
\(962\) 0 0
\(963\) −26.8328 −0.864675
\(964\) 0 0
\(965\) −25.7082 −0.827576
\(966\) 0 0
\(967\) 39.5410 1.27155 0.635777 0.771873i \(-0.280680\pi\)
0.635777 + 0.771873i \(0.280680\pi\)
\(968\) 0 0
\(969\) −23.4164 −0.752243
\(970\) 0 0
\(971\) 7.52786 0.241581 0.120790 0.992678i \(-0.461457\pi\)
0.120790 + 0.992678i \(0.461457\pi\)
\(972\) 0 0
\(973\) 3.34752 0.107317
\(974\) 0 0
\(975\) −36.7082 −1.17560
\(976\) 0 0
\(977\) −54.6525 −1.74849 −0.874244 0.485487i \(-0.838642\pi\)
−0.874244 + 0.485487i \(0.838642\pi\)
\(978\) 0 0
\(979\) 1.16718 0.0373034
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 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
\(986\) 0 0
\(987\) −6.18034 −0.196722
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −24.0000 −0.762385 −0.381193 0.924496i \(-0.624487\pi\)
−0.381193 + 0.924496i \(0.624487\pi\)
\(992\) 0 0
\(993\) 26.0557 0.826854
\(994\) 0 0
\(995\) 83.1935 2.63741
\(996\) 0 0
\(997\) 36.8328 1.16651 0.583253 0.812290i \(-0.301779\pi\)
0.583253 + 0.812290i \(0.301779\pi\)
\(998\) 0 0
\(999\) 7.23607 0.228939
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 1472.2.a.s.1.2 2
4.3 odd 2 1472.2.a.t.1.1 2
8.3 odd 2 23.2.a.a.1.1 2
8.5 even 2 368.2.a.h.1.1 2
24.5 odd 2 3312.2.a.ba.1.2 2
24.11 even 2 207.2.a.d.1.2 2
40.3 even 4 575.2.b.d.24.4 4
40.19 odd 2 575.2.a.f.1.2 2
40.27 even 4 575.2.b.d.24.1 4
40.29 even 2 9200.2.a.bt.1.2 2
56.27 even 2 1127.2.a.c.1.1 2
88.43 even 2 2783.2.a.c.1.2 2
104.51 odd 2 3887.2.a.i.1.2 2
120.59 even 2 5175.2.a.be.1.1 2
136.67 odd 2 6647.2.a.b.1.1 2
152.75 even 2 8303.2.a.e.1.2 2
184.3 odd 22 529.2.c.o.170.2 20
184.11 even 22 529.2.c.n.466.1 20
184.19 even 22 529.2.c.n.177.2 20
184.27 odd 22 529.2.c.o.177.2 20
184.35 odd 22 529.2.c.o.466.1 20
184.43 even 22 529.2.c.n.170.2 20
184.45 odd 2 8464.2.a.bb.1.1 2
184.51 even 22 529.2.c.n.255.1 20
184.59 odd 22 529.2.c.o.399.2 20
184.67 even 22 529.2.c.n.487.1 20
184.75 odd 22 529.2.c.o.266.2 20
184.83 even 22 529.2.c.n.334.1 20
184.91 even 2 529.2.a.a.1.1 2
184.99 even 22 529.2.c.n.118.2 20
184.107 even 22 529.2.c.n.501.2 20
184.123 odd 22 529.2.c.o.501.2 20
184.131 odd 22 529.2.c.o.118.2 20
184.147 odd 22 529.2.c.o.334.1 20
184.155 even 22 529.2.c.n.266.2 20
184.163 odd 22 529.2.c.o.487.1 20
184.171 even 22 529.2.c.n.399.2 20
184.179 odd 22 529.2.c.o.255.1 20
552.275 odd 2 4761.2.a.w.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
23.2.a.a.1.1 2 8.3 odd 2
207.2.a.d.1.2 2 24.11 even 2
368.2.a.h.1.1 2 8.5 even 2
529.2.a.a.1.1 2 184.91 even 2
529.2.c.n.118.2 20 184.99 even 22
529.2.c.n.170.2 20 184.43 even 22
529.2.c.n.177.2 20 184.19 even 22
529.2.c.n.255.1 20 184.51 even 22
529.2.c.n.266.2 20 184.155 even 22
529.2.c.n.334.1 20 184.83 even 22
529.2.c.n.399.2 20 184.171 even 22
529.2.c.n.466.1 20 184.11 even 22
529.2.c.n.487.1 20 184.67 even 22
529.2.c.n.501.2 20 184.107 even 22
529.2.c.o.118.2 20 184.131 odd 22
529.2.c.o.170.2 20 184.3 odd 22
529.2.c.o.177.2 20 184.27 odd 22
529.2.c.o.255.1 20 184.179 odd 22
529.2.c.o.266.2 20 184.75 odd 22
529.2.c.o.334.1 20 184.147 odd 22
529.2.c.o.399.2 20 184.59 odd 22
529.2.c.o.466.1 20 184.35 odd 22
529.2.c.o.487.1 20 184.163 odd 22
529.2.c.o.501.2 20 184.123 odd 22
575.2.a.f.1.2 2 40.19 odd 2
575.2.b.d.24.1 4 40.27 even 4
575.2.b.d.24.4 4 40.3 even 4
1127.2.a.c.1.1 2 56.27 even 2
1472.2.a.s.1.2 2 1.1 even 1 trivial
1472.2.a.t.1.1 2 4.3 odd 2
2783.2.a.c.1.2 2 88.43 even 2
3312.2.a.ba.1.2 2 24.5 odd 2
3887.2.a.i.1.2 2 104.51 odd 2
4761.2.a.w.1.2 2 552.275 odd 2
5175.2.a.be.1.1 2 120.59 even 2
6647.2.a.b.1.1 2 136.67 odd 2
8303.2.a.e.1.2 2 152.75 even 2
8464.2.a.bb.1.1 2 184.45 odd 2
9200.2.a.bt.1.2 2 40.29 even 2