Properties

Label 1521.2.a.s.1.1
Level $1521$
Weight $2$
Character 1521.1
Self dual yes
Analytic conductor $12.145$
Analytic rank $0$
Dimension $3$
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] = [1521,2,Mod(1,1521)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1521, 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("1521.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1521.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: \(12.1452461474\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 507)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.80194\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.04892 q^{2} +2.19806 q^{4} +3.35690 q^{5} -2.24698 q^{7} -0.405813 q^{8} +O(q^{10})\) \(q-2.04892 q^{2} +2.19806 q^{4} +3.35690 q^{5} -2.24698 q^{7} -0.405813 q^{8} -6.87800 q^{10} +4.93900 q^{11} +4.60388 q^{14} -3.56465 q^{16} -0.911854 q^{17} +3.80194 q^{19} +7.37867 q^{20} -10.1196 q^{22} -2.02715 q^{23} +6.26875 q^{25} -4.93900 q^{28} +3.93900 q^{29} +8.82908 q^{31} +8.11529 q^{32} +1.86831 q^{34} -7.54288 q^{35} -8.80194 q^{37} -7.78986 q^{38} -1.36227 q^{40} +6.93900 q^{41} -2.28621 q^{43} +10.8562 q^{44} +4.15346 q^{46} +3.80194 q^{47} -1.95108 q^{49} -12.8442 q^{50} -0.542877 q^{53} +16.5797 q^{55} +0.911854 q^{56} -8.07069 q^{58} -4.71379 q^{59} +3.67994 q^{61} -18.0901 q^{62} -9.49827 q^{64} +1.52111 q^{67} -2.00431 q^{68} +15.4547 q^{70} +2.37867 q^{71} -7.41119 q^{73} +18.0344 q^{74} +8.35690 q^{76} -11.0978 q^{77} -3.74094 q^{79} -11.9661 q^{80} -14.2174 q^{82} +2.30798 q^{83} -3.06100 q^{85} +4.68425 q^{86} -2.00431 q^{88} -10.0586 q^{89} -4.45580 q^{92} -7.78986 q^{94} +12.7627 q^{95} +16.1293 q^{97} +3.99761 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 11 q^{4} + 6 q^{5} - 2 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} + 11 q^{4} + 6 q^{5} - 2 q^{7} + 12 q^{8} - q^{10} + 5 q^{11} + 5 q^{14} + 11 q^{16} + q^{17} + 7 q^{19} + 15 q^{20} - 9 q^{22} + 11 q^{25} - 5 q^{28} + 2 q^{29} + 16 q^{31} + 22 q^{32} + 8 q^{34} - 4 q^{35} - 22 q^{37} + 3 q^{40} + 11 q^{41} - 15 q^{43} + 16 q^{44} + 7 q^{46} + 7 q^{47} - 15 q^{49} - 3 q^{50} + 17 q^{53} + 3 q^{55} - q^{56} - 12 q^{58} - 6 q^{59} - 13 q^{61} + 2 q^{62} - 11 q^{67} + 13 q^{68} + 24 q^{70} - 6 q^{73} - 15 q^{74} + 21 q^{76} - 15 q^{77} + 3 q^{79} - 20 q^{80} - 3 q^{82} + 12 q^{83} - 19 q^{85} - 29 q^{86} + 13 q^{88} + q^{89} - 7 q^{92} + 21 q^{95} + 5 q^{97} - 29 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.04892 −1.44880 −0.724402 0.689378i \(-0.757884\pi\)
−0.724402 + 0.689378i \(0.757884\pi\)
\(3\) 0 0
\(4\) 2.19806 1.09903
\(5\) 3.35690 1.50125 0.750625 0.660729i \(-0.229753\pi\)
0.750625 + 0.660729i \(0.229753\pi\)
\(6\) 0 0
\(7\) −2.24698 −0.849278 −0.424639 0.905363i \(-0.639599\pi\)
−0.424639 + 0.905363i \(0.639599\pi\)
\(8\) −0.405813 −0.143477
\(9\) 0 0
\(10\) −6.87800 −2.17502
\(11\) 4.93900 1.48916 0.744582 0.667531i \(-0.232649\pi\)
0.744582 + 0.667531i \(0.232649\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 4.60388 1.23044
\(15\) 0 0
\(16\) −3.56465 −0.891162
\(17\) −0.911854 −0.221157 −0.110579 0.993867i \(-0.535270\pi\)
−0.110579 + 0.993867i \(0.535270\pi\)
\(18\) 0 0
\(19\) 3.80194 0.872224 0.436112 0.899892i \(-0.356355\pi\)
0.436112 + 0.899892i \(0.356355\pi\)
\(20\) 7.37867 1.64992
\(21\) 0 0
\(22\) −10.1196 −2.15751
\(23\) −2.02715 −0.422689 −0.211345 0.977412i \(-0.567784\pi\)
−0.211345 + 0.977412i \(0.567784\pi\)
\(24\) 0 0
\(25\) 6.26875 1.25375
\(26\) 0 0
\(27\) 0 0
\(28\) −4.93900 −0.933383
\(29\) 3.93900 0.731454 0.365727 0.930722i \(-0.380820\pi\)
0.365727 + 0.930722i \(0.380820\pi\)
\(30\) 0 0
\(31\) 8.82908 1.58575 0.792875 0.609384i \(-0.208583\pi\)
0.792875 + 0.609384i \(0.208583\pi\)
\(32\) 8.11529 1.43459
\(33\) 0 0
\(34\) 1.86831 0.320413
\(35\) −7.54288 −1.27498
\(36\) 0 0
\(37\) −8.80194 −1.44703 −0.723515 0.690309i \(-0.757475\pi\)
−0.723515 + 0.690309i \(0.757475\pi\)
\(38\) −7.78986 −1.26368
\(39\) 0 0
\(40\) −1.36227 −0.215394
\(41\) 6.93900 1.08369 0.541845 0.840478i \(-0.317726\pi\)
0.541845 + 0.840478i \(0.317726\pi\)
\(42\) 0 0
\(43\) −2.28621 −0.348643 −0.174322 0.984689i \(-0.555773\pi\)
−0.174322 + 0.984689i \(0.555773\pi\)
\(44\) 10.8562 1.63664
\(45\) 0 0
\(46\) 4.15346 0.612394
\(47\) 3.80194 0.554570 0.277285 0.960788i \(-0.410565\pi\)
0.277285 + 0.960788i \(0.410565\pi\)
\(48\) 0 0
\(49\) −1.95108 −0.278726
\(50\) −12.8442 −1.81644
\(51\) 0 0
\(52\) 0 0
\(53\) −0.542877 −0.0745698 −0.0372849 0.999305i \(-0.511871\pi\)
−0.0372849 + 0.999305i \(0.511871\pi\)
\(54\) 0 0
\(55\) 16.5797 2.23561
\(56\) 0.911854 0.121852
\(57\) 0 0
\(58\) −8.07069 −1.05973
\(59\) −4.71379 −0.613683 −0.306842 0.951761i \(-0.599272\pi\)
−0.306842 + 0.951761i \(0.599272\pi\)
\(60\) 0 0
\(61\) 3.67994 0.471168 0.235584 0.971854i \(-0.424300\pi\)
0.235584 + 0.971854i \(0.424300\pi\)
\(62\) −18.0901 −2.29744
\(63\) 0 0
\(64\) −9.49827 −1.18728
\(65\) 0 0
\(66\) 0 0
\(67\) 1.52111 0.185833 0.0929164 0.995674i \(-0.470381\pi\)
0.0929164 + 0.995674i \(0.470381\pi\)
\(68\) −2.00431 −0.243059
\(69\) 0 0
\(70\) 15.4547 1.84719
\(71\) 2.37867 0.282296 0.141148 0.989989i \(-0.454921\pi\)
0.141148 + 0.989989i \(0.454921\pi\)
\(72\) 0 0
\(73\) −7.41119 −0.867414 −0.433707 0.901054i \(-0.642795\pi\)
−0.433707 + 0.901054i \(0.642795\pi\)
\(74\) 18.0344 2.09646
\(75\) 0 0
\(76\) 8.35690 0.958602
\(77\) −11.0978 −1.26472
\(78\) 0 0
\(79\) −3.74094 −0.420888 −0.210444 0.977606i \(-0.567491\pi\)
−0.210444 + 0.977606i \(0.567491\pi\)
\(80\) −11.9661 −1.33786
\(81\) 0 0
\(82\) −14.2174 −1.57005
\(83\) 2.30798 0.253334 0.126667 0.991945i \(-0.459572\pi\)
0.126667 + 0.991945i \(0.459572\pi\)
\(84\) 0 0
\(85\) −3.06100 −0.332012
\(86\) 4.68425 0.505116
\(87\) 0 0
\(88\) −2.00431 −0.213660
\(89\) −10.0586 −1.06621 −0.533105 0.846049i \(-0.678975\pi\)
−0.533105 + 0.846049i \(0.678975\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −4.45580 −0.464549
\(93\) 0 0
\(94\) −7.78986 −0.803462
\(95\) 12.7627 1.30943
\(96\) 0 0
\(97\) 16.1293 1.63768 0.818841 0.574021i \(-0.194617\pi\)
0.818841 + 0.574021i \(0.194617\pi\)
\(98\) 3.99761 0.403819
\(99\) 0 0
\(100\) 13.7791 1.37791
\(101\) 9.94869 0.989932 0.494966 0.868912i \(-0.335181\pi\)
0.494966 + 0.868912i \(0.335181\pi\)
\(102\) 0 0
\(103\) −10.9879 −1.08267 −0.541336 0.840806i \(-0.682081\pi\)
−0.541336 + 0.840806i \(0.682081\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 1.11231 0.108037
\(107\) 9.87263 0.954423 0.477211 0.878789i \(-0.341648\pi\)
0.477211 + 0.878789i \(0.341648\pi\)
\(108\) 0 0
\(109\) −20.2446 −1.93908 −0.969540 0.244934i \(-0.921234\pi\)
−0.969540 + 0.244934i \(0.921234\pi\)
\(110\) −33.9705 −3.23896
\(111\) 0 0
\(112\) 8.00969 0.756844
\(113\) −9.69202 −0.911749 −0.455874 0.890044i \(-0.650673\pi\)
−0.455874 + 0.890044i \(0.650673\pi\)
\(114\) 0 0
\(115\) −6.80492 −0.634562
\(116\) 8.65817 0.803891
\(117\) 0 0
\(118\) 9.65817 0.889107
\(119\) 2.04892 0.187824
\(120\) 0 0
\(121\) 13.3937 1.21761
\(122\) −7.53989 −0.682630
\(123\) 0 0
\(124\) 19.4069 1.74279
\(125\) 4.25906 0.380942
\(126\) 0 0
\(127\) 13.8116 1.22558 0.612792 0.790244i \(-0.290046\pi\)
0.612792 + 0.790244i \(0.290046\pi\)
\(128\) 3.23059 0.285546
\(129\) 0 0
\(130\) 0 0
\(131\) −2.99462 −0.261641 −0.130821 0.991406i \(-0.541761\pi\)
−0.130821 + 0.991406i \(0.541761\pi\)
\(132\) 0 0
\(133\) −8.54288 −0.740761
\(134\) −3.11662 −0.269235
\(135\) 0 0
\(136\) 0.370042 0.0317309
\(137\) 23.0194 1.96668 0.983339 0.181781i \(-0.0581862\pi\)
0.983339 + 0.181781i \(0.0581862\pi\)
\(138\) 0 0
\(139\) 0.982542 0.0833381 0.0416690 0.999131i \(-0.486732\pi\)
0.0416690 + 0.999131i \(0.486732\pi\)
\(140\) −16.5797 −1.40124
\(141\) 0 0
\(142\) −4.87369 −0.408991
\(143\) 0 0
\(144\) 0 0
\(145\) 13.2228 1.09810
\(146\) 15.1849 1.25671
\(147\) 0 0
\(148\) −19.3472 −1.59033
\(149\) 10.2591 0.840455 0.420228 0.907419i \(-0.361950\pi\)
0.420228 + 0.907419i \(0.361950\pi\)
\(150\) 0 0
\(151\) 20.1685 1.64129 0.820646 0.571438i \(-0.193614\pi\)
0.820646 + 0.571438i \(0.193614\pi\)
\(152\) −1.54288 −0.125144
\(153\) 0 0
\(154\) 22.7385 1.83232
\(155\) 29.6383 2.38061
\(156\) 0 0
\(157\) 10.4383 0.833070 0.416535 0.909120i \(-0.363244\pi\)
0.416535 + 0.909120i \(0.363244\pi\)
\(158\) 7.66487 0.609785
\(159\) 0 0
\(160\) 27.2422 2.15368
\(161\) 4.55496 0.358981
\(162\) 0 0
\(163\) 11.0465 0.865231 0.432615 0.901579i \(-0.357591\pi\)
0.432615 + 0.901579i \(0.357591\pi\)
\(164\) 15.2524 1.19101
\(165\) 0 0
\(166\) −4.72886 −0.367031
\(167\) 8.10992 0.627564 0.313782 0.949495i \(-0.398404\pi\)
0.313782 + 0.949495i \(0.398404\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 6.27173 0.481020
\(171\) 0 0
\(172\) −5.02523 −0.383170
\(173\) 18.0562 1.37279 0.686394 0.727230i \(-0.259192\pi\)
0.686394 + 0.727230i \(0.259192\pi\)
\(174\) 0 0
\(175\) −14.0858 −1.06478
\(176\) −17.6058 −1.32709
\(177\) 0 0
\(178\) 20.6093 1.54473
\(179\) 19.8702 1.48517 0.742585 0.669751i \(-0.233599\pi\)
0.742585 + 0.669751i \(0.233599\pi\)
\(180\) 0 0
\(181\) −10.0828 −0.749446 −0.374723 0.927137i \(-0.622262\pi\)
−0.374723 + 0.927137i \(0.622262\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0.822643 0.0606461
\(185\) −29.5472 −2.17235
\(186\) 0 0
\(187\) −4.50365 −0.329339
\(188\) 8.35690 0.609489
\(189\) 0 0
\(190\) −26.1497 −1.89710
\(191\) 6.58748 0.476653 0.238327 0.971185i \(-0.423401\pi\)
0.238327 + 0.971185i \(0.423401\pi\)
\(192\) 0 0
\(193\) −10.8672 −0.782242 −0.391121 0.920339i \(-0.627913\pi\)
−0.391121 + 0.920339i \(0.627913\pi\)
\(194\) −33.0476 −2.37268
\(195\) 0 0
\(196\) −4.28860 −0.306329
\(197\) 9.24160 0.658437 0.329218 0.944254i \(-0.393215\pi\)
0.329218 + 0.944254i \(0.393215\pi\)
\(198\) 0 0
\(199\) −1.56465 −0.110915 −0.0554574 0.998461i \(-0.517662\pi\)
−0.0554574 + 0.998461i \(0.517662\pi\)
\(200\) −2.54394 −0.179884
\(201\) 0 0
\(202\) −20.3840 −1.43422
\(203\) −8.85086 −0.621208
\(204\) 0 0
\(205\) 23.2935 1.62689
\(206\) 22.5133 1.56858
\(207\) 0 0
\(208\) 0 0
\(209\) 18.7778 1.29889
\(210\) 0 0
\(211\) 23.2446 1.60022 0.800112 0.599851i \(-0.204774\pi\)
0.800112 + 0.599851i \(0.204774\pi\)
\(212\) −1.19328 −0.0819546
\(213\) 0 0
\(214\) −20.2282 −1.38277
\(215\) −7.67456 −0.523401
\(216\) 0 0
\(217\) −19.8388 −1.34674
\(218\) 41.4795 2.80935
\(219\) 0 0
\(220\) 36.4432 2.45700
\(221\) 0 0
\(222\) 0 0
\(223\) −10.7385 −0.719106 −0.359553 0.933125i \(-0.617071\pi\)
−0.359553 + 0.933125i \(0.617071\pi\)
\(224\) −18.2349 −1.21837
\(225\) 0 0
\(226\) 19.8582 1.32094
\(227\) −6.97584 −0.463003 −0.231501 0.972835i \(-0.574364\pi\)
−0.231501 + 0.972835i \(0.574364\pi\)
\(228\) 0 0
\(229\) 16.8049 1.11050 0.555250 0.831683i \(-0.312622\pi\)
0.555250 + 0.831683i \(0.312622\pi\)
\(230\) 13.9427 0.919356
\(231\) 0 0
\(232\) −1.59850 −0.104947
\(233\) −8.69202 −0.569433 −0.284717 0.958612i \(-0.591900\pi\)
−0.284717 + 0.958612i \(0.591900\pi\)
\(234\) 0 0
\(235\) 12.7627 0.832547
\(236\) −10.3612 −0.674457
\(237\) 0 0
\(238\) −4.19806 −0.272120
\(239\) −22.9191 −1.48252 −0.741258 0.671220i \(-0.765771\pi\)
−0.741258 + 0.671220i \(0.765771\pi\)
\(240\) 0 0
\(241\) −21.9801 −1.41587 −0.707933 0.706280i \(-0.750372\pi\)
−0.707933 + 0.706280i \(0.750372\pi\)
\(242\) −27.4426 −1.76408
\(243\) 0 0
\(244\) 8.08874 0.517828
\(245\) −6.54958 −0.418437
\(246\) 0 0
\(247\) 0 0
\(248\) −3.58296 −0.227518
\(249\) 0 0
\(250\) −8.72646 −0.551910
\(251\) −26.6437 −1.68174 −0.840868 0.541241i \(-0.817955\pi\)
−0.840868 + 0.541241i \(0.817955\pi\)
\(252\) 0 0
\(253\) −10.0121 −0.629454
\(254\) −28.2989 −1.77563
\(255\) 0 0
\(256\) 12.3773 0.773584
\(257\) −23.1444 −1.44371 −0.721853 0.692047i \(-0.756709\pi\)
−0.721853 + 0.692047i \(0.756709\pi\)
\(258\) 0 0
\(259\) 19.7778 1.22893
\(260\) 0 0
\(261\) 0 0
\(262\) 6.13574 0.379067
\(263\) 18.5284 1.14251 0.571255 0.820773i \(-0.306457\pi\)
0.571255 + 0.820773i \(0.306457\pi\)
\(264\) 0 0
\(265\) −1.82238 −0.111948
\(266\) 17.5036 1.07322
\(267\) 0 0
\(268\) 3.34349 0.204236
\(269\) −9.75840 −0.594980 −0.297490 0.954725i \(-0.596149\pi\)
−0.297490 + 0.954725i \(0.596149\pi\)
\(270\) 0 0
\(271\) 22.8019 1.38512 0.692560 0.721361i \(-0.256483\pi\)
0.692560 + 0.721361i \(0.256483\pi\)
\(272\) 3.25044 0.197087
\(273\) 0 0
\(274\) −47.1648 −2.84933
\(275\) 30.9614 1.86704
\(276\) 0 0
\(277\) −23.9705 −1.44025 −0.720123 0.693847i \(-0.755914\pi\)
−0.720123 + 0.693847i \(0.755914\pi\)
\(278\) −2.01315 −0.120741
\(279\) 0 0
\(280\) 3.06100 0.182930
\(281\) −4.12498 −0.246076 −0.123038 0.992402i \(-0.539264\pi\)
−0.123038 + 0.992402i \(0.539264\pi\)
\(282\) 0 0
\(283\) −15.6558 −0.930639 −0.465320 0.885143i \(-0.654061\pi\)
−0.465320 + 0.885143i \(0.654061\pi\)
\(284\) 5.22846 0.310252
\(285\) 0 0
\(286\) 0 0
\(287\) −15.5918 −0.920354
\(288\) 0 0
\(289\) −16.1685 −0.951090
\(290\) −27.0925 −1.59092
\(291\) 0 0
\(292\) −16.2903 −0.953315
\(293\) 22.5948 1.32000 0.660001 0.751265i \(-0.270556\pi\)
0.660001 + 0.751265i \(0.270556\pi\)
\(294\) 0 0
\(295\) −15.8237 −0.921292
\(296\) 3.57194 0.207615
\(297\) 0 0
\(298\) −21.0200 −1.21765
\(299\) 0 0
\(300\) 0 0
\(301\) 5.13706 0.296095
\(302\) −41.3236 −2.37791
\(303\) 0 0
\(304\) −13.5526 −0.777293
\(305\) 12.3532 0.707341
\(306\) 0 0
\(307\) 6.55496 0.374111 0.187056 0.982349i \(-0.440106\pi\)
0.187056 + 0.982349i \(0.440106\pi\)
\(308\) −24.3937 −1.38996
\(309\) 0 0
\(310\) −60.7265 −3.44903
\(311\) 12.0392 0.682682 0.341341 0.939940i \(-0.389119\pi\)
0.341341 + 0.939940i \(0.389119\pi\)
\(312\) 0 0
\(313\) −33.8950 −1.91586 −0.957929 0.287005i \(-0.907340\pi\)
−0.957929 + 0.287005i \(0.907340\pi\)
\(314\) −21.3873 −1.20695
\(315\) 0 0
\(316\) −8.22282 −0.462570
\(317\) −4.49827 −0.252648 −0.126324 0.991989i \(-0.540318\pi\)
−0.126324 + 0.991989i \(0.540318\pi\)
\(318\) 0 0
\(319\) 19.4547 1.08926
\(320\) −31.8847 −1.78241
\(321\) 0 0
\(322\) −9.33273 −0.520093
\(323\) −3.46681 −0.192899
\(324\) 0 0
\(325\) 0 0
\(326\) −22.6334 −1.25355
\(327\) 0 0
\(328\) −2.81594 −0.155484
\(329\) −8.54288 −0.470984
\(330\) 0 0
\(331\) 11.2131 0.616329 0.308165 0.951333i \(-0.400285\pi\)
0.308165 + 0.951333i \(0.400285\pi\)
\(332\) 5.07308 0.278421
\(333\) 0 0
\(334\) −16.6165 −0.909217
\(335\) 5.10620 0.278981
\(336\) 0 0
\(337\) 7.04892 0.383979 0.191989 0.981397i \(-0.438506\pi\)
0.191989 + 0.981397i \(0.438506\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −6.72827 −0.364891
\(341\) 43.6069 2.36144
\(342\) 0 0
\(343\) 20.1129 1.08599
\(344\) 0.927774 0.0500222
\(345\) 0 0
\(346\) −36.9957 −1.98890
\(347\) 2.70410 0.145164 0.0725819 0.997362i \(-0.476876\pi\)
0.0725819 + 0.997362i \(0.476876\pi\)
\(348\) 0 0
\(349\) 0.415502 0.0222413 0.0111207 0.999938i \(-0.496460\pi\)
0.0111207 + 0.999938i \(0.496460\pi\)
\(350\) 28.8605 1.54266
\(351\) 0 0
\(352\) 40.0814 2.13635
\(353\) −32.1672 −1.71209 −0.856044 0.516904i \(-0.827084\pi\)
−0.856044 + 0.516904i \(0.827084\pi\)
\(354\) 0 0
\(355\) 7.98493 0.423796
\(356\) −22.1094 −1.17180
\(357\) 0 0
\(358\) −40.7125 −2.15172
\(359\) 22.3521 1.17970 0.589850 0.807513i \(-0.299187\pi\)
0.589850 + 0.807513i \(0.299187\pi\)
\(360\) 0 0
\(361\) −4.54527 −0.239225
\(362\) 20.6588 1.08580
\(363\) 0 0
\(364\) 0 0
\(365\) −24.8786 −1.30221
\(366\) 0 0
\(367\) −2.30260 −0.120195 −0.0600974 0.998193i \(-0.519141\pi\)
−0.0600974 + 0.998193i \(0.519141\pi\)
\(368\) 7.22606 0.376685
\(369\) 0 0
\(370\) 60.5397 3.14731
\(371\) 1.21983 0.0633305
\(372\) 0 0
\(373\) −19.2760 −0.998076 −0.499038 0.866580i \(-0.666313\pi\)
−0.499038 + 0.866580i \(0.666313\pi\)
\(374\) 9.22760 0.477148
\(375\) 0 0
\(376\) −1.54288 −0.0795678
\(377\) 0 0
\(378\) 0 0
\(379\) 7.33944 0.377002 0.188501 0.982073i \(-0.439637\pi\)
0.188501 + 0.982073i \(0.439637\pi\)
\(380\) 28.0532 1.43910
\(381\) 0 0
\(382\) −13.4972 −0.690577
\(383\) 19.0901 0.975457 0.487728 0.872995i \(-0.337826\pi\)
0.487728 + 0.872995i \(0.337826\pi\)
\(384\) 0 0
\(385\) −37.2543 −1.89865
\(386\) 22.2661 1.13331
\(387\) 0 0
\(388\) 35.4532 1.79986
\(389\) 23.9879 1.21624 0.608118 0.793847i \(-0.291925\pi\)
0.608118 + 0.793847i \(0.291925\pi\)
\(390\) 0 0
\(391\) 1.84846 0.0934807
\(392\) 0.791775 0.0399907
\(393\) 0 0
\(394\) −18.9353 −0.953946
\(395\) −12.5579 −0.631859
\(396\) 0 0
\(397\) 4.41789 0.221728 0.110864 0.993836i \(-0.464638\pi\)
0.110864 + 0.993836i \(0.464638\pi\)
\(398\) 3.20583 0.160694
\(399\) 0 0
\(400\) −22.3459 −1.11729
\(401\) −20.4088 −1.01917 −0.509583 0.860421i \(-0.670200\pi\)
−0.509583 + 0.860421i \(0.670200\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 21.8678 1.08797
\(405\) 0 0
\(406\) 18.1347 0.900009
\(407\) −43.4728 −2.15487
\(408\) 0 0
\(409\) 22.1491 1.09520 0.547602 0.836739i \(-0.315541\pi\)
0.547602 + 0.836739i \(0.315541\pi\)
\(410\) −47.7265 −2.35704
\(411\) 0 0
\(412\) −24.1521 −1.18989
\(413\) 10.5918 0.521188
\(414\) 0 0
\(415\) 7.74764 0.380317
\(416\) 0 0
\(417\) 0 0
\(418\) −38.4741 −1.88183
\(419\) −14.7560 −0.720878 −0.360439 0.932783i \(-0.617373\pi\)
−0.360439 + 0.932783i \(0.617373\pi\)
\(420\) 0 0
\(421\) −8.47219 −0.412909 −0.206455 0.978456i \(-0.566193\pi\)
−0.206455 + 0.978456i \(0.566193\pi\)
\(422\) −47.6262 −2.31841
\(423\) 0 0
\(424\) 0.220306 0.0106990
\(425\) −5.71618 −0.277276
\(426\) 0 0
\(427\) −8.26875 −0.400153
\(428\) 21.7006 1.04894
\(429\) 0 0
\(430\) 15.7245 0.758305
\(431\) 2.39181 0.115210 0.0576048 0.998339i \(-0.481654\pi\)
0.0576048 + 0.998339i \(0.481654\pi\)
\(432\) 0 0
\(433\) −10.4286 −0.501169 −0.250584 0.968095i \(-0.580623\pi\)
−0.250584 + 0.968095i \(0.580623\pi\)
\(434\) 40.6480 1.95117
\(435\) 0 0
\(436\) −44.4989 −2.13111
\(437\) −7.70709 −0.368680
\(438\) 0 0
\(439\) −32.5502 −1.55353 −0.776767 0.629787i \(-0.783142\pi\)
−0.776767 + 0.629787i \(0.783142\pi\)
\(440\) −6.72827 −0.320758
\(441\) 0 0
\(442\) 0 0
\(443\) −9.58211 −0.455260 −0.227630 0.973748i \(-0.573098\pi\)
−0.227630 + 0.973748i \(0.573098\pi\)
\(444\) 0 0
\(445\) −33.7657 −1.60065
\(446\) 22.0024 1.04184
\(447\) 0 0
\(448\) 21.3424 1.00833
\(449\) −28.3937 −1.33998 −0.669992 0.742369i \(-0.733702\pi\)
−0.669992 + 0.742369i \(0.733702\pi\)
\(450\) 0 0
\(451\) 34.2717 1.61379
\(452\) −21.3037 −1.00204
\(453\) 0 0
\(454\) 14.2929 0.670800
\(455\) 0 0
\(456\) 0 0
\(457\) −4.99569 −0.233688 −0.116844 0.993150i \(-0.537278\pi\)
−0.116844 + 0.993150i \(0.537278\pi\)
\(458\) −34.4319 −1.60890
\(459\) 0 0
\(460\) −14.9576 −0.697404
\(461\) 1.35258 0.0629961 0.0314981 0.999504i \(-0.489972\pi\)
0.0314981 + 0.999504i \(0.489972\pi\)
\(462\) 0 0
\(463\) 3.36467 0.156369 0.0781846 0.996939i \(-0.475088\pi\)
0.0781846 + 0.996939i \(0.475088\pi\)
\(464\) −14.0411 −0.651844
\(465\) 0 0
\(466\) 17.8092 0.824997
\(467\) −6.91079 −0.319793 −0.159897 0.987134i \(-0.551116\pi\)
−0.159897 + 0.987134i \(0.551116\pi\)
\(468\) 0 0
\(469\) −3.41789 −0.157824
\(470\) −26.1497 −1.20620
\(471\) 0 0
\(472\) 1.91292 0.0880492
\(473\) −11.2916 −0.519188
\(474\) 0 0
\(475\) 23.8334 1.09355
\(476\) 4.50365 0.206424
\(477\) 0 0
\(478\) 46.9594 2.14787
\(479\) 3.51573 0.160638 0.0803189 0.996769i \(-0.474406\pi\)
0.0803189 + 0.996769i \(0.474406\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 45.0355 2.05131
\(483\) 0 0
\(484\) 29.4403 1.33819
\(485\) 54.1444 2.45857
\(486\) 0 0
\(487\) 12.8479 0.582193 0.291096 0.956694i \(-0.405980\pi\)
0.291096 + 0.956694i \(0.405980\pi\)
\(488\) −1.49337 −0.0676016
\(489\) 0 0
\(490\) 13.4196 0.606234
\(491\) −28.6708 −1.29390 −0.646948 0.762534i \(-0.723955\pi\)
−0.646948 + 0.762534i \(0.723955\pi\)
\(492\) 0 0
\(493\) −3.59179 −0.161766
\(494\) 0 0
\(495\) 0 0
\(496\) −31.4726 −1.41316
\(497\) −5.34481 −0.239748
\(498\) 0 0
\(499\) 33.5555 1.50215 0.751076 0.660215i \(-0.229535\pi\)
0.751076 + 0.660215i \(0.229535\pi\)
\(500\) 9.36168 0.418667
\(501\) 0 0
\(502\) 54.5907 2.43650
\(503\) −21.5633 −0.961461 −0.480730 0.876868i \(-0.659628\pi\)
−0.480730 + 0.876868i \(0.659628\pi\)
\(504\) 0 0
\(505\) 33.3967 1.48613
\(506\) 20.5139 0.911955
\(507\) 0 0
\(508\) 30.3588 1.34695
\(509\) −8.89008 −0.394046 −0.197023 0.980399i \(-0.563127\pi\)
−0.197023 + 0.980399i \(0.563127\pi\)
\(510\) 0 0
\(511\) 16.6528 0.736676
\(512\) −31.8213 −1.40632
\(513\) 0 0
\(514\) 47.4209 2.09165
\(515\) −36.8853 −1.62536
\(516\) 0 0
\(517\) 18.7778 0.825846
\(518\) −40.5230 −1.78048
\(519\) 0 0
\(520\) 0 0
\(521\) −19.3478 −0.847642 −0.423821 0.905746i \(-0.639312\pi\)
−0.423821 + 0.905746i \(0.639312\pi\)
\(522\) 0 0
\(523\) −12.5948 −0.550731 −0.275366 0.961340i \(-0.588799\pi\)
−0.275366 + 0.961340i \(0.588799\pi\)
\(524\) −6.58237 −0.287552
\(525\) 0 0
\(526\) −37.9632 −1.65527
\(527\) −8.05084 −0.350700
\(528\) 0 0
\(529\) −18.8907 −0.821334
\(530\) 3.73391 0.162191
\(531\) 0 0
\(532\) −18.7778 −0.814120
\(533\) 0 0
\(534\) 0 0
\(535\) 33.1414 1.43283
\(536\) −0.617285 −0.0266627
\(537\) 0 0
\(538\) 19.9941 0.862009
\(539\) −9.63640 −0.415069
\(540\) 0 0
\(541\) −29.0019 −1.24689 −0.623445 0.781867i \(-0.714267\pi\)
−0.623445 + 0.781867i \(0.714267\pi\)
\(542\) −46.7193 −2.00677
\(543\) 0 0
\(544\) −7.39996 −0.317271
\(545\) −67.9590 −2.91104
\(546\) 0 0
\(547\) 27.7006 1.18439 0.592197 0.805793i \(-0.298261\pi\)
0.592197 + 0.805793i \(0.298261\pi\)
\(548\) 50.5980 2.16144
\(549\) 0 0
\(550\) −63.4373 −2.70497
\(551\) 14.9758 0.637992
\(552\) 0 0
\(553\) 8.40581 0.357452
\(554\) 49.1135 2.08663
\(555\) 0 0
\(556\) 2.15969 0.0915912
\(557\) 6.80971 0.288537 0.144268 0.989539i \(-0.453917\pi\)
0.144268 + 0.989539i \(0.453917\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 26.8877 1.13621
\(561\) 0 0
\(562\) 8.45175 0.356515
\(563\) 19.2524 0.811390 0.405695 0.914008i \(-0.367030\pi\)
0.405695 + 0.914008i \(0.367030\pi\)
\(564\) 0 0
\(565\) −32.5351 −1.36876
\(566\) 32.0774 1.34831
\(567\) 0 0
\(568\) −0.965294 −0.0405028
\(569\) −7.84846 −0.329025 −0.164512 0.986375i \(-0.552605\pi\)
−0.164512 + 0.986375i \(0.552605\pi\)
\(570\) 0 0
\(571\) 29.8568 1.24947 0.624735 0.780837i \(-0.285207\pi\)
0.624735 + 0.780837i \(0.285207\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 31.9463 1.33341
\(575\) −12.7077 −0.529947
\(576\) 0 0
\(577\) −8.97823 −0.373769 −0.186884 0.982382i \(-0.559839\pi\)
−0.186884 + 0.982382i \(0.559839\pi\)
\(578\) 33.1280 1.37794
\(579\) 0 0
\(580\) 29.0646 1.20684
\(581\) −5.18598 −0.215151
\(582\) 0 0
\(583\) −2.68127 −0.111047
\(584\) 3.00756 0.124454
\(585\) 0 0
\(586\) −46.2948 −1.91242
\(587\) 23.8538 0.984553 0.492277 0.870439i \(-0.336165\pi\)
0.492277 + 0.870439i \(0.336165\pi\)
\(588\) 0 0
\(589\) 33.5676 1.38313
\(590\) 32.4215 1.33477
\(591\) 0 0
\(592\) 31.3758 1.28954
\(593\) 11.9866 0.492230 0.246115 0.969241i \(-0.420846\pi\)
0.246115 + 0.969241i \(0.420846\pi\)
\(594\) 0 0
\(595\) 6.87800 0.281971
\(596\) 22.5501 0.923686
\(597\) 0 0
\(598\) 0 0
\(599\) −29.1142 −1.18958 −0.594788 0.803883i \(-0.702764\pi\)
−0.594788 + 0.803883i \(0.702764\pi\)
\(600\) 0 0
\(601\) 37.1366 1.51483 0.757417 0.652932i \(-0.226461\pi\)
0.757417 + 0.652932i \(0.226461\pi\)
\(602\) −10.5254 −0.428984
\(603\) 0 0
\(604\) 44.3317 1.80383
\(605\) 44.9614 1.82794
\(606\) 0 0
\(607\) −14.2325 −0.577680 −0.288840 0.957377i \(-0.593269\pi\)
−0.288840 + 0.957377i \(0.593269\pi\)
\(608\) 30.8538 1.25129
\(609\) 0 0
\(610\) −25.3106 −1.02480
\(611\) 0 0
\(612\) 0 0
\(613\) −30.3139 −1.22437 −0.612184 0.790715i \(-0.709709\pi\)
−0.612184 + 0.790715i \(0.709709\pi\)
\(614\) −13.4306 −0.542014
\(615\) 0 0
\(616\) 4.50365 0.181457
\(617\) −24.9638 −1.00500 −0.502501 0.864576i \(-0.667587\pi\)
−0.502501 + 0.864576i \(0.667587\pi\)
\(618\) 0 0
\(619\) −35.6122 −1.43138 −0.715688 0.698420i \(-0.753887\pi\)
−0.715688 + 0.698420i \(0.753887\pi\)
\(620\) 65.1469 2.61636
\(621\) 0 0
\(622\) −24.6674 −0.989072
\(623\) 22.6015 0.905509
\(624\) 0 0
\(625\) −17.0465 −0.681861
\(626\) 69.4480 2.77570
\(627\) 0 0
\(628\) 22.9441 0.915570
\(629\) 8.02608 0.320021
\(630\) 0 0
\(631\) −23.8829 −0.950763 −0.475382 0.879780i \(-0.657690\pi\)
−0.475382 + 0.879780i \(0.657690\pi\)
\(632\) 1.51812 0.0603877
\(633\) 0 0
\(634\) 9.21659 0.366037
\(635\) 46.3642 1.83991
\(636\) 0 0
\(637\) 0 0
\(638\) −39.8611 −1.57812
\(639\) 0 0
\(640\) 10.8447 0.428676
\(641\) 24.6577 0.973920 0.486960 0.873424i \(-0.338106\pi\)
0.486960 + 0.873424i \(0.338106\pi\)
\(642\) 0 0
\(643\) −47.8165 −1.88570 −0.942850 0.333218i \(-0.891866\pi\)
−0.942850 + 0.333218i \(0.891866\pi\)
\(644\) 10.0121 0.394531
\(645\) 0 0
\(646\) 7.10321 0.279472
\(647\) 5.38942 0.211880 0.105940 0.994373i \(-0.466215\pi\)
0.105940 + 0.994373i \(0.466215\pi\)
\(648\) 0 0
\(649\) −23.2814 −0.913876
\(650\) 0 0
\(651\) 0 0
\(652\) 24.2809 0.950915
\(653\) 5.03790 0.197148 0.0985741 0.995130i \(-0.468572\pi\)
0.0985741 + 0.995130i \(0.468572\pi\)
\(654\) 0 0
\(655\) −10.0526 −0.392789
\(656\) −24.7351 −0.965743
\(657\) 0 0
\(658\) 17.5036 0.682363
\(659\) 43.9812 1.71326 0.856632 0.515927i \(-0.172553\pi\)
0.856632 + 0.515927i \(0.172553\pi\)
\(660\) 0 0
\(661\) 38.2194 1.48656 0.743280 0.668980i \(-0.233269\pi\)
0.743280 + 0.668980i \(0.233269\pi\)
\(662\) −22.9748 −0.892940
\(663\) 0 0
\(664\) −0.936608 −0.0363474
\(665\) −28.6775 −1.11207
\(666\) 0 0
\(667\) −7.98493 −0.309178
\(668\) 17.8261 0.689713
\(669\) 0 0
\(670\) −10.4622 −0.404189
\(671\) 18.1752 0.701647
\(672\) 0 0
\(673\) −6.66487 −0.256912 −0.128456 0.991715i \(-0.541002\pi\)
−0.128456 + 0.991715i \(0.541002\pi\)
\(674\) −14.4426 −0.556310
\(675\) 0 0
\(676\) 0 0
\(677\) −4.80194 −0.184553 −0.0922767 0.995733i \(-0.529414\pi\)
−0.0922767 + 0.995733i \(0.529414\pi\)
\(678\) 0 0
\(679\) −36.2422 −1.39085
\(680\) 1.24219 0.0476360
\(681\) 0 0
\(682\) −89.3469 −3.42127
\(683\) 11.2591 0.430816 0.215408 0.976524i \(-0.430892\pi\)
0.215408 + 0.976524i \(0.430892\pi\)
\(684\) 0 0
\(685\) 77.2737 2.95247
\(686\) −41.2097 −1.57339
\(687\) 0 0
\(688\) 8.14952 0.310698
\(689\) 0 0
\(690\) 0 0
\(691\) 24.7144 0.940179 0.470090 0.882619i \(-0.344222\pi\)
0.470090 + 0.882619i \(0.344222\pi\)
\(692\) 39.6887 1.50874
\(693\) 0 0
\(694\) −5.54048 −0.210314
\(695\) 3.29829 0.125111
\(696\) 0 0
\(697\) −6.32736 −0.239666
\(698\) −0.851329 −0.0322233
\(699\) 0 0
\(700\) −30.9614 −1.17023
\(701\) −25.8920 −0.977927 −0.488964 0.872304i \(-0.662625\pi\)
−0.488964 + 0.872304i \(0.662625\pi\)
\(702\) 0 0
\(703\) −33.4644 −1.26213
\(704\) −46.9120 −1.76806
\(705\) 0 0
\(706\) 65.9079 2.48048
\(707\) −22.3545 −0.840728
\(708\) 0 0
\(709\) −0.0851621 −0.00319833 −0.00159916 0.999999i \(-0.500509\pi\)
−0.00159916 + 0.999999i \(0.500509\pi\)
\(710\) −16.3605 −0.613998
\(711\) 0 0
\(712\) 4.08192 0.152976
\(713\) −17.8979 −0.670280
\(714\) 0 0
\(715\) 0 0
\(716\) 43.6760 1.63225
\(717\) 0 0
\(718\) −45.7976 −1.70915
\(719\) −27.0508 −1.00883 −0.504413 0.863463i \(-0.668291\pi\)
−0.504413 + 0.863463i \(0.668291\pi\)
\(720\) 0 0
\(721\) 24.6896 0.919490
\(722\) 9.31288 0.346590
\(723\) 0 0
\(724\) −22.1626 −0.823665
\(725\) 24.6926 0.917061
\(726\) 0 0
\(727\) −47.1584 −1.74901 −0.874503 0.485019i \(-0.838813\pi\)
−0.874503 + 0.485019i \(0.838813\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 50.9742 1.88664
\(731\) 2.08469 0.0771050
\(732\) 0 0
\(733\) 9.68186 0.357608 0.178804 0.983885i \(-0.442777\pi\)
0.178804 + 0.983885i \(0.442777\pi\)
\(734\) 4.71784 0.174139
\(735\) 0 0
\(736\) −16.4509 −0.606388
\(737\) 7.51275 0.276736
\(738\) 0 0
\(739\) −36.3139 −1.33583 −0.667915 0.744238i \(-0.732813\pi\)
−0.667915 + 0.744238i \(0.732813\pi\)
\(740\) −64.9466 −2.38748
\(741\) 0 0
\(742\) −2.49934 −0.0917535
\(743\) −41.7536 −1.53179 −0.765896 0.642965i \(-0.777704\pi\)
−0.765896 + 0.642965i \(0.777704\pi\)
\(744\) 0 0
\(745\) 34.4386 1.26173
\(746\) 39.4950 1.44602
\(747\) 0 0
\(748\) −9.89930 −0.361954
\(749\) −22.1836 −0.810571
\(750\) 0 0
\(751\) −22.1927 −0.809823 −0.404911 0.914356i \(-0.632698\pi\)
−0.404911 + 0.914356i \(0.632698\pi\)
\(752\) −13.5526 −0.494211
\(753\) 0 0
\(754\) 0 0
\(755\) 67.7036 2.46399
\(756\) 0 0
\(757\) 9.23729 0.335735 0.167868 0.985810i \(-0.446312\pi\)
0.167868 + 0.985810i \(0.446312\pi\)
\(758\) −15.0379 −0.546201
\(759\) 0 0
\(760\) −5.17928 −0.187872
\(761\) −7.50173 −0.271937 −0.135969 0.990713i \(-0.543415\pi\)
−0.135969 + 0.990713i \(0.543415\pi\)
\(762\) 0 0
\(763\) 45.4892 1.64682
\(764\) 14.4797 0.523857
\(765\) 0 0
\(766\) −39.1140 −1.41325
\(767\) 0 0
\(768\) 0 0
\(769\) −5.14005 −0.185355 −0.0926774 0.995696i \(-0.529543\pi\)
−0.0926774 + 0.995696i \(0.529543\pi\)
\(770\) 76.3309 2.75078
\(771\) 0 0
\(772\) −23.8869 −0.859708
\(773\) −8.50173 −0.305786 −0.152893 0.988243i \(-0.548859\pi\)
−0.152893 + 0.988243i \(0.548859\pi\)
\(774\) 0 0
\(775\) 55.3473 1.98813
\(776\) −6.54548 −0.234969
\(777\) 0 0
\(778\) −49.1493 −1.76209
\(779\) 26.3817 0.945221
\(780\) 0 0
\(781\) 11.7482 0.420385
\(782\) −3.78735 −0.135435
\(783\) 0 0
\(784\) 6.95492 0.248390
\(785\) 35.0404 1.25065
\(786\) 0 0
\(787\) 7.78554 0.277525 0.138762 0.990326i \(-0.455688\pi\)
0.138762 + 0.990326i \(0.455688\pi\)
\(788\) 20.3136 0.723643
\(789\) 0 0
\(790\) 25.7302 0.915439
\(791\) 21.7778 0.774329
\(792\) 0 0
\(793\) 0 0
\(794\) −9.05190 −0.321240
\(795\) 0 0
\(796\) −3.43919 −0.121899
\(797\) −21.3840 −0.757462 −0.378731 0.925507i \(-0.623639\pi\)
−0.378731 + 0.925507i \(0.623639\pi\)
\(798\) 0 0
\(799\) −3.46681 −0.122647
\(800\) 50.8727 1.79862
\(801\) 0 0
\(802\) 41.8159 1.47657
\(803\) −36.6039 −1.29172
\(804\) 0 0
\(805\) 15.2905 0.538920
\(806\) 0 0
\(807\) 0 0
\(808\) −4.03731 −0.142032
\(809\) −2.28621 −0.0803788 −0.0401894 0.999192i \(-0.512796\pi\)
−0.0401894 + 0.999192i \(0.512796\pi\)
\(810\) 0 0
\(811\) −17.8079 −0.625320 −0.312660 0.949865i \(-0.601220\pi\)
−0.312660 + 0.949865i \(0.601220\pi\)
\(812\) −19.4547 −0.682727
\(813\) 0 0
\(814\) 89.0721 3.12198
\(815\) 37.0820 1.29893
\(816\) 0 0
\(817\) −8.69202 −0.304095
\(818\) −45.3818 −1.58674
\(819\) 0 0
\(820\) 51.2006 1.78800
\(821\) 13.9054 0.485302 0.242651 0.970114i \(-0.421983\pi\)
0.242651 + 0.970114i \(0.421983\pi\)
\(822\) 0 0
\(823\) −7.24831 −0.252660 −0.126330 0.991988i \(-0.540320\pi\)
−0.126330 + 0.991988i \(0.540320\pi\)
\(824\) 4.45904 0.155338
\(825\) 0 0
\(826\) −21.7017 −0.755099
\(827\) 54.1191 1.88191 0.940953 0.338536i \(-0.109932\pi\)
0.940953 + 0.338536i \(0.109932\pi\)
\(828\) 0 0
\(829\) −5.79178 −0.201157 −0.100578 0.994929i \(-0.532069\pi\)
−0.100578 + 0.994929i \(0.532069\pi\)
\(830\) −15.8743 −0.551004
\(831\) 0 0
\(832\) 0 0
\(833\) 1.77910 0.0616422
\(834\) 0 0
\(835\) 27.2241 0.942130
\(836\) 41.2747 1.42752
\(837\) 0 0
\(838\) 30.2338 1.04441
\(839\) −5.29350 −0.182752 −0.0913760 0.995816i \(-0.529127\pi\)
−0.0913760 + 0.995816i \(0.529127\pi\)
\(840\) 0 0
\(841\) −13.4843 −0.464975
\(842\) 17.3588 0.598224
\(843\) 0 0
\(844\) 51.0930 1.75870
\(845\) 0 0
\(846\) 0 0
\(847\) −30.0954 −1.03409
\(848\) 1.93516 0.0664538
\(849\) 0 0
\(850\) 11.7120 0.401718
\(851\) 17.8428 0.611644
\(852\) 0 0
\(853\) 13.5961 0.465522 0.232761 0.972534i \(-0.425224\pi\)
0.232761 + 0.972534i \(0.425224\pi\)
\(854\) 16.9420 0.579743
\(855\) 0 0
\(856\) −4.00644 −0.136937
\(857\) 23.8323 0.814097 0.407048 0.913407i \(-0.366558\pi\)
0.407048 + 0.913407i \(0.366558\pi\)
\(858\) 0 0
\(859\) −26.9861 −0.920754 −0.460377 0.887723i \(-0.652286\pi\)
−0.460377 + 0.887723i \(0.652286\pi\)
\(860\) −16.8692 −0.575234
\(861\) 0 0
\(862\) −4.90063 −0.166916
\(863\) −27.0291 −0.920080 −0.460040 0.887898i \(-0.652165\pi\)
−0.460040 + 0.887898i \(0.652165\pi\)
\(864\) 0 0
\(865\) 60.6128 2.06090
\(866\) 21.3674 0.726095
\(867\) 0 0
\(868\) −43.6069 −1.48011
\(869\) −18.4765 −0.626772
\(870\) 0 0
\(871\) 0 0
\(872\) 8.21552 0.278213
\(873\) 0 0
\(874\) 15.7912 0.534145
\(875\) −9.57002 −0.323526
\(876\) 0 0
\(877\) −40.8780 −1.38035 −0.690176 0.723642i \(-0.742467\pi\)
−0.690176 + 0.723642i \(0.742467\pi\)
\(878\) 66.6926 2.25077
\(879\) 0 0
\(880\) −59.1008 −1.99229
\(881\) 22.4964 0.757921 0.378961 0.925413i \(-0.376282\pi\)
0.378961 + 0.925413i \(0.376282\pi\)
\(882\) 0 0
\(883\) −4.16315 −0.140101 −0.0700505 0.997543i \(-0.522316\pi\)
−0.0700505 + 0.997543i \(0.522316\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 19.6329 0.659582
\(887\) 26.3002 0.883075 0.441537 0.897243i \(-0.354433\pi\)
0.441537 + 0.897243i \(0.354433\pi\)
\(888\) 0 0
\(889\) −31.0344 −1.04086
\(890\) 69.1831 2.31902
\(891\) 0 0
\(892\) −23.6040 −0.790320
\(893\) 14.4547 0.483709
\(894\) 0 0
\(895\) 66.7023 2.22961
\(896\) −7.25906 −0.242508
\(897\) 0 0
\(898\) 58.1764 1.94137
\(899\) 34.7778 1.15990
\(900\) 0 0
\(901\) 0.495024 0.0164916
\(902\) −70.2199 −2.33807
\(903\) 0 0
\(904\) 3.93315 0.130815
\(905\) −33.8468 −1.12511
\(906\) 0 0
\(907\) −57.9114 −1.92292 −0.961458 0.274952i \(-0.911338\pi\)
−0.961458 + 0.274952i \(0.911338\pi\)
\(908\) −15.3333 −0.508854
\(909\) 0 0
\(910\) 0 0
\(911\) 0.286799 0.00950208 0.00475104 0.999989i \(-0.498488\pi\)
0.00475104 + 0.999989i \(0.498488\pi\)
\(912\) 0 0
\(913\) 11.3991 0.377255
\(914\) 10.2358 0.338569
\(915\) 0 0
\(916\) 36.9383 1.22047
\(917\) 6.72886 0.222206
\(918\) 0 0
\(919\) −31.1239 −1.02668 −0.513342 0.858184i \(-0.671593\pi\)
−0.513342 + 0.858184i \(0.671593\pi\)
\(920\) 2.76153 0.0910449
\(921\) 0 0
\(922\) −2.77133 −0.0912690
\(923\) 0 0
\(924\) 0 0
\(925\) −55.1771 −1.81421
\(926\) −6.89392 −0.226548
\(927\) 0 0
\(928\) 31.9661 1.04934
\(929\) 7.62671 0.250224 0.125112 0.992143i \(-0.460071\pi\)
0.125112 + 0.992143i \(0.460071\pi\)
\(930\) 0 0
\(931\) −7.41789 −0.243112
\(932\) −19.1056 −0.625825
\(933\) 0 0
\(934\) 14.1596 0.463317
\(935\) −15.1183 −0.494421
\(936\) 0 0
\(937\) −5.67324 −0.185337 −0.0926683 0.995697i \(-0.529540\pi\)
−0.0926683 + 0.995697i \(0.529540\pi\)
\(938\) 7.00298 0.228656
\(939\) 0 0
\(940\) 28.0532 0.914995
\(941\) 41.5394 1.35415 0.677073 0.735916i \(-0.263248\pi\)
0.677073 + 0.735916i \(0.263248\pi\)
\(942\) 0 0
\(943\) −14.0664 −0.458064
\(944\) 16.8030 0.546891
\(945\) 0 0
\(946\) 23.1355 0.752201
\(947\) −47.3110 −1.53740 −0.768700 0.639610i \(-0.779096\pi\)
−0.768700 + 0.639610i \(0.779096\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −48.8327 −1.58434
\(951\) 0 0
\(952\) −0.831478 −0.0269483
\(953\) 34.3435 1.11249 0.556247 0.831017i \(-0.312241\pi\)
0.556247 + 0.831017i \(0.312241\pi\)
\(954\) 0 0
\(955\) 22.1135 0.715576
\(956\) −50.3777 −1.62933
\(957\) 0 0
\(958\) −7.20344 −0.232733
\(959\) −51.7241 −1.67026
\(960\) 0 0
\(961\) 46.9527 1.51460
\(962\) 0 0
\(963\) 0 0
\(964\) −48.3137 −1.55608
\(965\) −36.4802 −1.17434
\(966\) 0 0
\(967\) −48.5096 −1.55996 −0.779982 0.625802i \(-0.784772\pi\)
−0.779982 + 0.625802i \(0.784772\pi\)
\(968\) −5.43535 −0.174699
\(969\) 0 0
\(970\) −110.937 −3.56198
\(971\) −41.4650 −1.33068 −0.665338 0.746542i \(-0.731712\pi\)
−0.665338 + 0.746542i \(0.731712\pi\)
\(972\) 0 0
\(973\) −2.20775 −0.0707772
\(974\) −26.3242 −0.843483
\(975\) 0 0
\(976\) −13.1177 −0.419887
\(977\) −2.09677 −0.0670816 −0.0335408 0.999437i \(-0.510678\pi\)
−0.0335408 + 0.999437i \(0.510678\pi\)
\(978\) 0 0
\(979\) −49.6795 −1.58776
\(980\) −14.3964 −0.459876
\(981\) 0 0
\(982\) 58.7442 1.87460
\(983\) 25.2336 0.804826 0.402413 0.915458i \(-0.368172\pi\)
0.402413 + 0.915458i \(0.368172\pi\)
\(984\) 0 0
\(985\) 31.0231 0.988478
\(986\) 7.35929 0.234367
\(987\) 0 0
\(988\) 0 0
\(989\) 4.63448 0.147368
\(990\) 0 0
\(991\) 12.0489 0.382746 0.191373 0.981517i \(-0.438706\pi\)
0.191373 + 0.981517i \(0.438706\pi\)
\(992\) 71.6506 2.27491
\(993\) 0 0
\(994\) 10.9511 0.347347
\(995\) −5.25236 −0.166511
\(996\) 0 0
\(997\) −1.43403 −0.0454160 −0.0227080 0.999742i \(-0.507229\pi\)
−0.0227080 + 0.999742i \(0.507229\pi\)
\(998\) −68.7525 −2.17632
\(999\) 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 1521.2.a.s.1.1 3
3.2 odd 2 507.2.a.i.1.3 3
12.11 even 2 8112.2.a.cg.1.2 3
13.5 odd 4 1521.2.b.k.1351.4 6
13.8 odd 4 1521.2.b.k.1351.3 6
13.12 even 2 1521.2.a.n.1.3 3
39.2 even 12 507.2.j.i.316.4 12
39.5 even 4 507.2.b.f.337.3 6
39.8 even 4 507.2.b.f.337.4 6
39.11 even 12 507.2.j.i.316.3 12
39.17 odd 6 507.2.e.i.484.3 6
39.20 even 12 507.2.j.i.361.4 12
39.23 odd 6 507.2.e.i.22.3 6
39.29 odd 6 507.2.e.l.22.1 6
39.32 even 12 507.2.j.i.361.3 12
39.35 odd 6 507.2.e.l.484.1 6
39.38 odd 2 507.2.a.l.1.1 yes 3
156.155 even 2 8112.2.a.cp.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.2.a.i.1.3 3 3.2 odd 2
507.2.a.l.1.1 yes 3 39.38 odd 2
507.2.b.f.337.3 6 39.5 even 4
507.2.b.f.337.4 6 39.8 even 4
507.2.e.i.22.3 6 39.23 odd 6
507.2.e.i.484.3 6 39.17 odd 6
507.2.e.l.22.1 6 39.29 odd 6
507.2.e.l.484.1 6 39.35 odd 6
507.2.j.i.316.3 12 39.11 even 12
507.2.j.i.316.4 12 39.2 even 12
507.2.j.i.361.3 12 39.32 even 12
507.2.j.i.361.4 12 39.20 even 12
1521.2.a.n.1.3 3 13.12 even 2
1521.2.a.s.1.1 3 1.1 even 1 trivial
1521.2.b.k.1351.3 6 13.8 odd 4
1521.2.b.k.1351.4 6 13.5 odd 4
8112.2.a.cg.1.2 3 12.11 even 2
8112.2.a.cp.1.2 3 156.155 even 2