Properties

Label 6561.2.a.d.1.11
Level $6561$
Weight $2$
Character 6561.1
Self dual yes
Analytic conductor $52.390$
Analytic rank $0$
Dimension $72$
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] = [6561,2,Mod(1,6561)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6561, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6561.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6561 = 3^{8} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6561.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: \(52.3898487662\)
Analytic rank: \(0\)
Dimension: \(72\)
Twist minimal: no (minimal twist has level 81)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.11
Character \(\chi\) \(=\) 6561.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.11413 q^{2} +2.46953 q^{4} -2.98999 q^{5} -3.51273 q^{7} -0.992637 q^{8} +O(q^{10})\) \(q-2.11413 q^{2} +2.46953 q^{4} -2.98999 q^{5} -3.51273 q^{7} -0.992637 q^{8} +6.32121 q^{10} +1.96066 q^{11} +1.66702 q^{13} +7.42634 q^{14} -2.84049 q^{16} +3.89091 q^{17} +0.0165720 q^{19} -7.38386 q^{20} -4.14509 q^{22} +7.72480 q^{23} +3.94004 q^{25} -3.52430 q^{26} -8.67477 q^{28} -4.62155 q^{29} -10.2650 q^{31} +7.99043 q^{32} -8.22586 q^{34} +10.5030 q^{35} +7.15788 q^{37} -0.0350352 q^{38} +2.96798 q^{40} +5.65673 q^{41} -4.56357 q^{43} +4.84191 q^{44} -16.3312 q^{46} +2.87659 q^{47} +5.33925 q^{49} -8.32975 q^{50} +4.11676 q^{52} -2.80825 q^{53} -5.86236 q^{55} +3.48686 q^{56} +9.77053 q^{58} +4.95688 q^{59} +9.38279 q^{61} +21.7015 q^{62} -11.2118 q^{64} -4.98438 q^{65} -4.48321 q^{67} +9.60870 q^{68} -22.2047 q^{70} +7.70195 q^{71} -4.00919 q^{73} -15.1327 q^{74} +0.0409249 q^{76} -6.88727 q^{77} +8.15157 q^{79} +8.49305 q^{80} -11.9590 q^{82} -5.15607 q^{83} -11.6338 q^{85} +9.64796 q^{86} -1.94623 q^{88} -4.77831 q^{89} -5.85580 q^{91} +19.0766 q^{92} -6.08148 q^{94} -0.0495500 q^{95} -11.6308 q^{97} -11.2878 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q + 9 q^{2} + 63 q^{4} + 18 q^{5} + 27 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 72 q + 9 q^{2} + 63 q^{4} + 18 q^{5} + 27 q^{8} + 36 q^{11} + 36 q^{14} + 45 q^{16} + 36 q^{17} + 54 q^{20} + 54 q^{23} + 36 q^{25} + 45 q^{26} + 9 q^{28} + 54 q^{29} + 63 q^{32} + 72 q^{35} + 54 q^{38} + 72 q^{41} + 90 q^{44} + 90 q^{47} + 18 q^{49} + 45 q^{50} + 45 q^{53} + 9 q^{55} + 108 q^{56} + 18 q^{58} + 108 q^{59} + 72 q^{62} + 9 q^{64} + 72 q^{65} + 108 q^{68} + 126 q^{71} + 90 q^{74} + 72 q^{77} + 144 q^{80} - 18 q^{82} + 108 q^{83} + 90 q^{86} + 108 q^{89} + 72 q^{92} + 144 q^{95} + 81 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.11413 −1.49491 −0.747456 0.664311i \(-0.768725\pi\)
−0.747456 + 0.664311i \(0.768725\pi\)
\(3\) 0 0
\(4\) 2.46953 1.23476
\(5\) −2.98999 −1.33716 −0.668582 0.743638i \(-0.733099\pi\)
−0.668582 + 0.743638i \(0.733099\pi\)
\(6\) 0 0
\(7\) −3.51273 −1.32769 −0.663843 0.747872i \(-0.731076\pi\)
−0.663843 + 0.747872i \(0.731076\pi\)
\(8\) −0.992637 −0.350950
\(9\) 0 0
\(10\) 6.32121 1.99894
\(11\) 1.96066 0.591162 0.295581 0.955318i \(-0.404487\pi\)
0.295581 + 0.955318i \(0.404487\pi\)
\(12\) 0 0
\(13\) 1.66702 0.462349 0.231175 0.972912i \(-0.425743\pi\)
0.231175 + 0.972912i \(0.425743\pi\)
\(14\) 7.42634 1.98477
\(15\) 0 0
\(16\) −2.84049 −0.710123
\(17\) 3.89091 0.943683 0.471842 0.881683i \(-0.343589\pi\)
0.471842 + 0.881683i \(0.343589\pi\)
\(18\) 0 0
\(19\) 0.0165720 0.00380187 0.00190094 0.999998i \(-0.499395\pi\)
0.00190094 + 0.999998i \(0.499395\pi\)
\(20\) −7.38386 −1.65108
\(21\) 0 0
\(22\) −4.14509 −0.883735
\(23\) 7.72480 1.61073 0.805366 0.592778i \(-0.201969\pi\)
0.805366 + 0.592778i \(0.201969\pi\)
\(24\) 0 0
\(25\) 3.94004 0.788009
\(26\) −3.52430 −0.691171
\(27\) 0 0
\(28\) −8.67477 −1.63938
\(29\) −4.62155 −0.858200 −0.429100 0.903257i \(-0.641169\pi\)
−0.429100 + 0.903257i \(0.641169\pi\)
\(30\) 0 0
\(31\) −10.2650 −1.84365 −0.921826 0.387604i \(-0.873303\pi\)
−0.921826 + 0.387604i \(0.873303\pi\)
\(32\) 7.99043 1.41252
\(33\) 0 0
\(34\) −8.22586 −1.41072
\(35\) 10.5030 1.77533
\(36\) 0 0
\(37\) 7.15788 1.17675 0.588374 0.808589i \(-0.299768\pi\)
0.588374 + 0.808589i \(0.299768\pi\)
\(38\) −0.0350352 −0.00568346
\(39\) 0 0
\(40\) 2.96798 0.469278
\(41\) 5.65673 0.883433 0.441716 0.897155i \(-0.354370\pi\)
0.441716 + 0.897155i \(0.354370\pi\)
\(42\) 0 0
\(43\) −4.56357 −0.695938 −0.347969 0.937506i \(-0.613129\pi\)
−0.347969 + 0.937506i \(0.613129\pi\)
\(44\) 4.84191 0.729945
\(45\) 0 0
\(46\) −16.3312 −2.40790
\(47\) 2.87659 0.419594 0.209797 0.977745i \(-0.432720\pi\)
0.209797 + 0.977745i \(0.432720\pi\)
\(48\) 0 0
\(49\) 5.33925 0.762750
\(50\) −8.32975 −1.17800
\(51\) 0 0
\(52\) 4.11676 0.570892
\(53\) −2.80825 −0.385743 −0.192872 0.981224i \(-0.561780\pi\)
−0.192872 + 0.981224i \(0.561780\pi\)
\(54\) 0 0
\(55\) −5.86236 −0.790481
\(56\) 3.48686 0.465952
\(57\) 0 0
\(58\) 9.77053 1.28293
\(59\) 4.95688 0.645331 0.322666 0.946513i \(-0.395421\pi\)
0.322666 + 0.946513i \(0.395421\pi\)
\(60\) 0 0
\(61\) 9.38279 1.20134 0.600672 0.799496i \(-0.294900\pi\)
0.600672 + 0.799496i \(0.294900\pi\)
\(62\) 21.7015 2.75610
\(63\) 0 0
\(64\) −11.2118 −1.40147
\(65\) −4.98438 −0.618237
\(66\) 0 0
\(67\) −4.48321 −0.547711 −0.273856 0.961771i \(-0.588299\pi\)
−0.273856 + 0.961771i \(0.588299\pi\)
\(68\) 9.60870 1.16523
\(69\) 0 0
\(70\) −22.2047 −2.65397
\(71\) 7.70195 0.914053 0.457026 0.889453i \(-0.348914\pi\)
0.457026 + 0.889453i \(0.348914\pi\)
\(72\) 0 0
\(73\) −4.00919 −0.469240 −0.234620 0.972087i \(-0.575385\pi\)
−0.234620 + 0.972087i \(0.575385\pi\)
\(74\) −15.1327 −1.75914
\(75\) 0 0
\(76\) 0.0409249 0.00469441
\(77\) −6.88727 −0.784877
\(78\) 0 0
\(79\) 8.15157 0.917123 0.458562 0.888663i \(-0.348365\pi\)
0.458562 + 0.888663i \(0.348365\pi\)
\(80\) 8.49305 0.949552
\(81\) 0 0
\(82\) −11.9590 −1.32065
\(83\) −5.15607 −0.565953 −0.282976 0.959127i \(-0.591322\pi\)
−0.282976 + 0.959127i \(0.591322\pi\)
\(84\) 0 0
\(85\) −11.6338 −1.26186
\(86\) 9.64796 1.04037
\(87\) 0 0
\(88\) −1.94623 −0.207468
\(89\) −4.77831 −0.506500 −0.253250 0.967401i \(-0.581500\pi\)
−0.253250 + 0.967401i \(0.581500\pi\)
\(90\) 0 0
\(91\) −5.85580 −0.613854
\(92\) 19.0766 1.98887
\(93\) 0 0
\(94\) −6.08148 −0.627257
\(95\) −0.0495500 −0.00508373
\(96\) 0 0
\(97\) −11.6308 −1.18093 −0.590463 0.807065i \(-0.701055\pi\)
−0.590463 + 0.807065i \(0.701055\pi\)
\(98\) −11.2878 −1.14024
\(99\) 0 0
\(100\) 9.73004 0.973004
\(101\) −2.85928 −0.284509 −0.142254 0.989830i \(-0.545435\pi\)
−0.142254 + 0.989830i \(0.545435\pi\)
\(102\) 0 0
\(103\) −2.86090 −0.281893 −0.140947 0.990017i \(-0.545015\pi\)
−0.140947 + 0.990017i \(0.545015\pi\)
\(104\) −1.65475 −0.162261
\(105\) 0 0
\(106\) 5.93700 0.576652
\(107\) −15.7860 −1.52609 −0.763045 0.646345i \(-0.776297\pi\)
−0.763045 + 0.646345i \(0.776297\pi\)
\(108\) 0 0
\(109\) −0.290786 −0.0278522 −0.0139261 0.999903i \(-0.504433\pi\)
−0.0139261 + 0.999903i \(0.504433\pi\)
\(110\) 12.3938 1.18170
\(111\) 0 0
\(112\) 9.97788 0.942821
\(113\) 3.23362 0.304193 0.152097 0.988366i \(-0.451397\pi\)
0.152097 + 0.988366i \(0.451397\pi\)
\(114\) 0 0
\(115\) −23.0971 −2.15381
\(116\) −11.4130 −1.05967
\(117\) 0 0
\(118\) −10.4795 −0.964714
\(119\) −13.6677 −1.25292
\(120\) 0 0
\(121\) −7.15580 −0.650528
\(122\) −19.8364 −1.79590
\(123\) 0 0
\(124\) −25.3497 −2.27647
\(125\) 3.16926 0.283467
\(126\) 0 0
\(127\) 13.1293 1.16503 0.582517 0.812818i \(-0.302068\pi\)
0.582517 + 0.812818i \(0.302068\pi\)
\(128\) 7.72227 0.682558
\(129\) 0 0
\(130\) 10.5376 0.924210
\(131\) 0.000322743 0 2.81982e−5 0 1.40991e−5 1.00000i \(-0.499996\pi\)
1.40991e−5 1.00000i \(0.499996\pi\)
\(132\) 0 0
\(133\) −0.0582128 −0.00504769
\(134\) 9.47807 0.818781
\(135\) 0 0
\(136\) −3.86226 −0.331186
\(137\) −12.8945 −1.10165 −0.550825 0.834620i \(-0.685687\pi\)
−0.550825 + 0.834620i \(0.685687\pi\)
\(138\) 0 0
\(139\) 0.641356 0.0543991 0.0271996 0.999630i \(-0.491341\pi\)
0.0271996 + 0.999630i \(0.491341\pi\)
\(140\) 25.9375 2.19212
\(141\) 0 0
\(142\) −16.2829 −1.36643
\(143\) 3.26847 0.273323
\(144\) 0 0
\(145\) 13.8184 1.14755
\(146\) 8.47593 0.701473
\(147\) 0 0
\(148\) 17.6766 1.45301
\(149\) −12.9227 −1.05867 −0.529336 0.848412i \(-0.677559\pi\)
−0.529336 + 0.848412i \(0.677559\pi\)
\(150\) 0 0
\(151\) −16.7832 −1.36579 −0.682897 0.730514i \(-0.739280\pi\)
−0.682897 + 0.730514i \(0.739280\pi\)
\(152\) −0.0164500 −0.00133427
\(153\) 0 0
\(154\) 14.5606 1.17332
\(155\) 30.6923 2.46527
\(156\) 0 0
\(157\) −8.60099 −0.686434 −0.343217 0.939256i \(-0.611517\pi\)
−0.343217 + 0.939256i \(0.611517\pi\)
\(158\) −17.2334 −1.37102
\(159\) 0 0
\(160\) −23.8913 −1.88877
\(161\) −27.1351 −2.13855
\(162\) 0 0
\(163\) 0.674482 0.0528295 0.0264148 0.999651i \(-0.491591\pi\)
0.0264148 + 0.999651i \(0.491591\pi\)
\(164\) 13.9694 1.09083
\(165\) 0 0
\(166\) 10.9006 0.846050
\(167\) 16.1197 1.24738 0.623690 0.781672i \(-0.285633\pi\)
0.623690 + 0.781672i \(0.285633\pi\)
\(168\) 0 0
\(169\) −10.2210 −0.786233
\(170\) 24.5953 1.88637
\(171\) 0 0
\(172\) −11.2699 −0.859318
\(173\) −16.1302 −1.22635 −0.613177 0.789946i \(-0.710109\pi\)
−0.613177 + 0.789946i \(0.710109\pi\)
\(174\) 0 0
\(175\) −13.8403 −1.04623
\(176\) −5.56925 −0.419798
\(177\) 0 0
\(178\) 10.1020 0.757173
\(179\) 7.93866 0.593363 0.296682 0.954976i \(-0.404120\pi\)
0.296682 + 0.954976i \(0.404120\pi\)
\(180\) 0 0
\(181\) −13.8345 −1.02831 −0.514153 0.857698i \(-0.671894\pi\)
−0.514153 + 0.857698i \(0.671894\pi\)
\(182\) 12.3799 0.917658
\(183\) 0 0
\(184\) −7.66792 −0.565287
\(185\) −21.4020 −1.57351
\(186\) 0 0
\(187\) 7.62875 0.557870
\(188\) 7.10382 0.518100
\(189\) 0 0
\(190\) 0.104755 0.00759972
\(191\) 14.3742 1.04008 0.520042 0.854141i \(-0.325916\pi\)
0.520042 + 0.854141i \(0.325916\pi\)
\(192\) 0 0
\(193\) −4.18390 −0.301163 −0.150582 0.988598i \(-0.548115\pi\)
−0.150582 + 0.988598i \(0.548115\pi\)
\(194\) 24.5889 1.76538
\(195\) 0 0
\(196\) 13.1854 0.941815
\(197\) 0.236490 0.0168492 0.00842461 0.999965i \(-0.497318\pi\)
0.00842461 + 0.999965i \(0.497318\pi\)
\(198\) 0 0
\(199\) 26.0899 1.84946 0.924731 0.380622i \(-0.124290\pi\)
0.924731 + 0.380622i \(0.124290\pi\)
\(200\) −3.91103 −0.276552
\(201\) 0 0
\(202\) 6.04487 0.425316
\(203\) 16.2342 1.13942
\(204\) 0 0
\(205\) −16.9136 −1.18130
\(206\) 6.04831 0.421406
\(207\) 0 0
\(208\) −4.73517 −0.328325
\(209\) 0.0324920 0.00224752
\(210\) 0 0
\(211\) −11.7991 −0.812285 −0.406143 0.913810i \(-0.633126\pi\)
−0.406143 + 0.913810i \(0.633126\pi\)
\(212\) −6.93505 −0.476301
\(213\) 0 0
\(214\) 33.3736 2.28137
\(215\) 13.6450 0.930583
\(216\) 0 0
\(217\) 36.0582 2.44779
\(218\) 0.614757 0.0416366
\(219\) 0 0
\(220\) −14.4773 −0.976056
\(221\) 6.48623 0.436311
\(222\) 0 0
\(223\) −26.3361 −1.76359 −0.881797 0.471630i \(-0.843666\pi\)
−0.881797 + 0.471630i \(0.843666\pi\)
\(224\) −28.0682 −1.87539
\(225\) 0 0
\(226\) −6.83628 −0.454742
\(227\) 1.01951 0.0676673 0.0338337 0.999427i \(-0.489228\pi\)
0.0338337 + 0.999427i \(0.489228\pi\)
\(228\) 0 0
\(229\) 26.0815 1.72352 0.861758 0.507320i \(-0.169364\pi\)
0.861758 + 0.507320i \(0.169364\pi\)
\(230\) 48.8301 3.21976
\(231\) 0 0
\(232\) 4.58752 0.301185
\(233\) −4.48925 −0.294101 −0.147050 0.989129i \(-0.546978\pi\)
−0.147050 + 0.989129i \(0.546978\pi\)
\(234\) 0 0
\(235\) −8.60099 −0.561067
\(236\) 12.2412 0.796831
\(237\) 0 0
\(238\) 28.8952 1.87300
\(239\) 1.52499 0.0986431 0.0493216 0.998783i \(-0.484294\pi\)
0.0493216 + 0.998783i \(0.484294\pi\)
\(240\) 0 0
\(241\) 0.842169 0.0542488 0.0271244 0.999632i \(-0.491365\pi\)
0.0271244 + 0.999632i \(0.491365\pi\)
\(242\) 15.1283 0.972482
\(243\) 0 0
\(244\) 23.1711 1.48337
\(245\) −15.9643 −1.01992
\(246\) 0 0
\(247\) 0.0276259 0.00175779
\(248\) 10.1894 0.647030
\(249\) 0 0
\(250\) −6.70022 −0.423759
\(251\) −14.9225 −0.941899 −0.470950 0.882160i \(-0.656089\pi\)
−0.470950 + 0.882160i \(0.656089\pi\)
\(252\) 0 0
\(253\) 15.1457 0.952203
\(254\) −27.7569 −1.74162
\(255\) 0 0
\(256\) 6.09774 0.381109
\(257\) 9.22403 0.575379 0.287690 0.957724i \(-0.407113\pi\)
0.287690 + 0.957724i \(0.407113\pi\)
\(258\) 0 0
\(259\) −25.1437 −1.56235
\(260\) −12.3091 −0.763376
\(261\) 0 0
\(262\) −0.000682320 0 −4.21539e−5 0
\(263\) −0.238033 −0.0146777 −0.00733886 0.999973i \(-0.502336\pi\)
−0.00733886 + 0.999973i \(0.502336\pi\)
\(264\) 0 0
\(265\) 8.39665 0.515802
\(266\) 0.123069 0.00754585
\(267\) 0 0
\(268\) −11.0714 −0.676294
\(269\) 14.4205 0.879234 0.439617 0.898185i \(-0.355114\pi\)
0.439617 + 0.898185i \(0.355114\pi\)
\(270\) 0 0
\(271\) 16.6640 1.01227 0.506133 0.862456i \(-0.331075\pi\)
0.506133 + 0.862456i \(0.331075\pi\)
\(272\) −11.0521 −0.670132
\(273\) 0 0
\(274\) 27.2606 1.64687
\(275\) 7.72509 0.465841
\(276\) 0 0
\(277\) 24.8172 1.49112 0.745561 0.666438i \(-0.232182\pi\)
0.745561 + 0.666438i \(0.232182\pi\)
\(278\) −1.35591 −0.0813219
\(279\) 0 0
\(280\) −10.4257 −0.623054
\(281\) 19.3251 1.15284 0.576418 0.817155i \(-0.304450\pi\)
0.576418 + 0.817155i \(0.304450\pi\)
\(282\) 0 0
\(283\) 1.11626 0.0663547 0.0331773 0.999449i \(-0.489437\pi\)
0.0331773 + 0.999449i \(0.489437\pi\)
\(284\) 19.0202 1.12864
\(285\) 0 0
\(286\) −6.90995 −0.408594
\(287\) −19.8705 −1.17292
\(288\) 0 0
\(289\) −1.86084 −0.109461
\(290\) −29.2138 −1.71549
\(291\) 0 0
\(292\) −9.90080 −0.579400
\(293\) 8.03767 0.469566 0.234783 0.972048i \(-0.424562\pi\)
0.234783 + 0.972048i \(0.424562\pi\)
\(294\) 0 0
\(295\) −14.8210 −0.862914
\(296\) −7.10518 −0.412980
\(297\) 0 0
\(298\) 27.3203 1.58262
\(299\) 12.8774 0.744721
\(300\) 0 0
\(301\) 16.0306 0.923987
\(302\) 35.4817 2.04174
\(303\) 0 0
\(304\) −0.0470726 −0.00269980
\(305\) −28.0545 −1.60639
\(306\) 0 0
\(307\) 16.7923 0.958388 0.479194 0.877709i \(-0.340929\pi\)
0.479194 + 0.877709i \(0.340929\pi\)
\(308\) −17.0083 −0.969137
\(309\) 0 0
\(310\) −64.8874 −3.68536
\(311\) 5.15497 0.292312 0.146156 0.989262i \(-0.453310\pi\)
0.146156 + 0.989262i \(0.453310\pi\)
\(312\) 0 0
\(313\) 27.9400 1.57926 0.789632 0.613580i \(-0.210271\pi\)
0.789632 + 0.613580i \(0.210271\pi\)
\(314\) 18.1836 1.02616
\(315\) 0 0
\(316\) 20.1305 1.13243
\(317\) 9.19431 0.516404 0.258202 0.966091i \(-0.416870\pi\)
0.258202 + 0.966091i \(0.416870\pi\)
\(318\) 0 0
\(319\) −9.06130 −0.507335
\(320\) 33.5231 1.87400
\(321\) 0 0
\(322\) 57.3670 3.19694
\(323\) 0.0644800 0.00358776
\(324\) 0 0
\(325\) 6.56814 0.364335
\(326\) −1.42594 −0.0789755
\(327\) 0 0
\(328\) −5.61508 −0.310041
\(329\) −10.1047 −0.557090
\(330\) 0 0
\(331\) 16.9510 0.931711 0.465855 0.884861i \(-0.345747\pi\)
0.465855 + 0.884861i \(0.345747\pi\)
\(332\) −12.7331 −0.698817
\(333\) 0 0
\(334\) −34.0791 −1.86472
\(335\) 13.4048 0.732380
\(336\) 0 0
\(337\) 13.0457 0.710644 0.355322 0.934744i \(-0.384371\pi\)
0.355322 + 0.934744i \(0.384371\pi\)
\(338\) 21.6085 1.17535
\(339\) 0 0
\(340\) −28.7299 −1.55810
\(341\) −20.1262 −1.08990
\(342\) 0 0
\(343\) 5.83377 0.314994
\(344\) 4.52997 0.244240
\(345\) 0 0
\(346\) 34.1012 1.83329
\(347\) 14.3935 0.772682 0.386341 0.922356i \(-0.373739\pi\)
0.386341 + 0.922356i \(0.373739\pi\)
\(348\) 0 0
\(349\) 8.13327 0.435364 0.217682 0.976020i \(-0.430150\pi\)
0.217682 + 0.976020i \(0.430150\pi\)
\(350\) 29.2601 1.56402
\(351\) 0 0
\(352\) 15.6665 0.835029
\(353\) 27.1058 1.44270 0.721348 0.692572i \(-0.243523\pi\)
0.721348 + 0.692572i \(0.243523\pi\)
\(354\) 0 0
\(355\) −23.0287 −1.22224
\(356\) −11.8002 −0.625408
\(357\) 0 0
\(358\) −16.7833 −0.887026
\(359\) −24.9323 −1.31587 −0.657937 0.753073i \(-0.728571\pi\)
−0.657937 + 0.753073i \(0.728571\pi\)
\(360\) 0 0
\(361\) −18.9997 −0.999986
\(362\) 29.2478 1.53723
\(363\) 0 0
\(364\) −14.4610 −0.757965
\(365\) 11.9874 0.627451
\(366\) 0 0
\(367\) 13.0921 0.683403 0.341702 0.939809i \(-0.388997\pi\)
0.341702 + 0.939809i \(0.388997\pi\)
\(368\) −21.9422 −1.14382
\(369\) 0 0
\(370\) 45.2465 2.35225
\(371\) 9.86462 0.512146
\(372\) 0 0
\(373\) −28.7755 −1.48994 −0.744969 0.667099i \(-0.767536\pi\)
−0.744969 + 0.667099i \(0.767536\pi\)
\(374\) −16.1281 −0.833966
\(375\) 0 0
\(376\) −2.85541 −0.147257
\(377\) −7.70423 −0.396788
\(378\) 0 0
\(379\) −20.9830 −1.07782 −0.538912 0.842362i \(-0.681164\pi\)
−0.538912 + 0.842362i \(0.681164\pi\)
\(380\) −0.122365 −0.00627720
\(381\) 0 0
\(382\) −30.3890 −1.55483
\(383\) −24.4951 −1.25164 −0.625820 0.779968i \(-0.715236\pi\)
−0.625820 + 0.779968i \(0.715236\pi\)
\(384\) 0 0
\(385\) 20.5929 1.04951
\(386\) 8.84528 0.450213
\(387\) 0 0
\(388\) −28.7225 −1.45816
\(389\) 15.7668 0.799411 0.399705 0.916644i \(-0.369112\pi\)
0.399705 + 0.916644i \(0.369112\pi\)
\(390\) 0 0
\(391\) 30.0565 1.52002
\(392\) −5.29994 −0.267687
\(393\) 0 0
\(394\) −0.499970 −0.0251881
\(395\) −24.3731 −1.22634
\(396\) 0 0
\(397\) −25.8264 −1.29619 −0.648095 0.761560i \(-0.724434\pi\)
−0.648095 + 0.761560i \(0.724434\pi\)
\(398\) −55.1572 −2.76478
\(399\) 0 0
\(400\) −11.1917 −0.559583
\(401\) −16.3786 −0.817909 −0.408954 0.912555i \(-0.634106\pi\)
−0.408954 + 0.912555i \(0.634106\pi\)
\(402\) 0 0
\(403\) −17.1120 −0.852411
\(404\) −7.06106 −0.351301
\(405\) 0 0
\(406\) −34.3212 −1.70333
\(407\) 14.0342 0.695649
\(408\) 0 0
\(409\) −25.3561 −1.25378 −0.626888 0.779109i \(-0.715672\pi\)
−0.626888 + 0.779109i \(0.715672\pi\)
\(410\) 35.7574 1.76593
\(411\) 0 0
\(412\) −7.06508 −0.348071
\(413\) −17.4122 −0.856797
\(414\) 0 0
\(415\) 15.4166 0.756772
\(416\) 13.3202 0.653078
\(417\) 0 0
\(418\) −0.0686922 −0.00335985
\(419\) 6.35976 0.310695 0.155347 0.987860i \(-0.450350\pi\)
0.155347 + 0.987860i \(0.450350\pi\)
\(420\) 0 0
\(421\) 26.2648 1.28007 0.640033 0.768347i \(-0.278921\pi\)
0.640033 + 0.768347i \(0.278921\pi\)
\(422\) 24.9448 1.21430
\(423\) 0 0
\(424\) 2.78757 0.135377
\(425\) 15.3303 0.743631
\(426\) 0 0
\(427\) −32.9592 −1.59501
\(428\) −38.9839 −1.88436
\(429\) 0 0
\(430\) −28.8473 −1.39114
\(431\) 12.3288 0.593858 0.296929 0.954900i \(-0.404038\pi\)
0.296929 + 0.954900i \(0.404038\pi\)
\(432\) 0 0
\(433\) −6.75112 −0.324438 −0.162219 0.986755i \(-0.551865\pi\)
−0.162219 + 0.986755i \(0.551865\pi\)
\(434\) −76.2316 −3.65923
\(435\) 0 0
\(436\) −0.718103 −0.0343909
\(437\) 0.128015 0.00612380
\(438\) 0 0
\(439\) −4.46643 −0.213171 −0.106586 0.994304i \(-0.533992\pi\)
−0.106586 + 0.994304i \(0.533992\pi\)
\(440\) 5.81920 0.277419
\(441\) 0 0
\(442\) −13.7127 −0.652247
\(443\) −28.2113 −1.34036 −0.670179 0.742199i \(-0.733783\pi\)
−0.670179 + 0.742199i \(0.733783\pi\)
\(444\) 0 0
\(445\) 14.2871 0.677274
\(446\) 55.6777 2.63642
\(447\) 0 0
\(448\) 39.3840 1.86072
\(449\) 26.2299 1.23787 0.618933 0.785444i \(-0.287565\pi\)
0.618933 + 0.785444i \(0.287565\pi\)
\(450\) 0 0
\(451\) 11.0909 0.522252
\(452\) 7.98551 0.375607
\(453\) 0 0
\(454\) −2.15537 −0.101157
\(455\) 17.5088 0.820824
\(456\) 0 0
\(457\) −24.1349 −1.12898 −0.564492 0.825438i \(-0.690928\pi\)
−0.564492 + 0.825438i \(0.690928\pi\)
\(458\) −55.1396 −2.57650
\(459\) 0 0
\(460\) −57.0388 −2.65945
\(461\) 18.9067 0.880572 0.440286 0.897858i \(-0.354877\pi\)
0.440286 + 0.897858i \(0.354877\pi\)
\(462\) 0 0
\(463\) 1.69898 0.0789581 0.0394790 0.999220i \(-0.487430\pi\)
0.0394790 + 0.999220i \(0.487430\pi\)
\(464\) 13.1275 0.609428
\(465\) 0 0
\(466\) 9.49084 0.439655
\(467\) −1.53809 −0.0711744 −0.0355872 0.999367i \(-0.511330\pi\)
−0.0355872 + 0.999367i \(0.511330\pi\)
\(468\) 0 0
\(469\) 15.7483 0.727189
\(470\) 18.1836 0.838746
\(471\) 0 0
\(472\) −4.92039 −0.226479
\(473\) −8.94762 −0.411412
\(474\) 0 0
\(475\) 0.0652943 0.00299591
\(476\) −33.7527 −1.54705
\(477\) 0 0
\(478\) −3.22401 −0.147463
\(479\) −15.8904 −0.726052 −0.363026 0.931779i \(-0.618256\pi\)
−0.363026 + 0.931779i \(0.618256\pi\)
\(480\) 0 0
\(481\) 11.9324 0.544068
\(482\) −1.78045 −0.0810973
\(483\) 0 0
\(484\) −17.6714 −0.803248
\(485\) 34.7759 1.57909
\(486\) 0 0
\(487\) 39.0750 1.77066 0.885330 0.464964i \(-0.153933\pi\)
0.885330 + 0.464964i \(0.153933\pi\)
\(488\) −9.31371 −0.421612
\(489\) 0 0
\(490\) 33.7505 1.52469
\(491\) 38.5841 1.74128 0.870638 0.491925i \(-0.163706\pi\)
0.870638 + 0.491925i \(0.163706\pi\)
\(492\) 0 0
\(493\) −17.9820 −0.809869
\(494\) −0.0584045 −0.00262774
\(495\) 0 0
\(496\) 29.1577 1.30922
\(497\) −27.0548 −1.21358
\(498\) 0 0
\(499\) 5.32929 0.238572 0.119286 0.992860i \(-0.461940\pi\)
0.119286 + 0.992860i \(0.461940\pi\)
\(500\) 7.82657 0.350015
\(501\) 0 0
\(502\) 31.5480 1.40806
\(503\) 22.6519 1.01000 0.505000 0.863119i \(-0.331492\pi\)
0.505000 + 0.863119i \(0.331492\pi\)
\(504\) 0 0
\(505\) 8.54921 0.380435
\(506\) −32.0200 −1.42346
\(507\) 0 0
\(508\) 32.4231 1.43854
\(509\) −31.2829 −1.38659 −0.693295 0.720654i \(-0.743842\pi\)
−0.693295 + 0.720654i \(0.743842\pi\)
\(510\) 0 0
\(511\) 14.0832 0.623003
\(512\) −28.3359 −1.25228
\(513\) 0 0
\(514\) −19.5008 −0.860141
\(515\) 8.55408 0.376938
\(516\) 0 0
\(517\) 5.64003 0.248048
\(518\) 53.1569 2.33558
\(519\) 0 0
\(520\) 4.94768 0.216970
\(521\) 28.7603 1.26001 0.630006 0.776590i \(-0.283052\pi\)
0.630006 + 0.776590i \(0.283052\pi\)
\(522\) 0 0
\(523\) −7.36746 −0.322156 −0.161078 0.986942i \(-0.551497\pi\)
−0.161078 + 0.986942i \(0.551497\pi\)
\(524\) 0.000797023 0 3.48181e−5 0
\(525\) 0 0
\(526\) 0.503231 0.0219419
\(527\) −39.9402 −1.73982
\(528\) 0 0
\(529\) 36.6725 1.59446
\(530\) −17.7516 −0.771079
\(531\) 0 0
\(532\) −0.143758 −0.00623270
\(533\) 9.42990 0.408454
\(534\) 0 0
\(535\) 47.2000 2.04063
\(536\) 4.45020 0.192219
\(537\) 0 0
\(538\) −30.4868 −1.31438
\(539\) 10.4685 0.450909
\(540\) 0 0
\(541\) −25.2403 −1.08517 −0.542583 0.840002i \(-0.682554\pi\)
−0.542583 + 0.840002i \(0.682554\pi\)
\(542\) −35.2298 −1.51325
\(543\) 0 0
\(544\) 31.0900 1.33297
\(545\) 0.869446 0.0372430
\(546\) 0 0
\(547\) 18.1106 0.774355 0.387178 0.922005i \(-0.373450\pi\)
0.387178 + 0.922005i \(0.373450\pi\)
\(548\) −31.8433 −1.36028
\(549\) 0 0
\(550\) −16.3318 −0.696391
\(551\) −0.0765882 −0.00326277
\(552\) 0 0
\(553\) −28.6342 −1.21765
\(554\) −52.4667 −2.22910
\(555\) 0 0
\(556\) 1.58385 0.0671700
\(557\) −9.10466 −0.385777 −0.192888 0.981221i \(-0.561786\pi\)
−0.192888 + 0.981221i \(0.561786\pi\)
\(558\) 0 0
\(559\) −7.60758 −0.321766
\(560\) −29.8338 −1.26071
\(561\) 0 0
\(562\) −40.8556 −1.72339
\(563\) 17.6895 0.745522 0.372761 0.927927i \(-0.378411\pi\)
0.372761 + 0.927927i \(0.378411\pi\)
\(564\) 0 0
\(565\) −9.66849 −0.406757
\(566\) −2.35991 −0.0991944
\(567\) 0 0
\(568\) −7.64524 −0.320787
\(569\) 41.7760 1.75134 0.875670 0.482909i \(-0.160420\pi\)
0.875670 + 0.482909i \(0.160420\pi\)
\(570\) 0 0
\(571\) 19.7073 0.824726 0.412363 0.911020i \(-0.364704\pi\)
0.412363 + 0.911020i \(0.364704\pi\)
\(572\) 8.07157 0.337489
\(573\) 0 0
\(574\) 42.0088 1.75341
\(575\) 30.4360 1.26927
\(576\) 0 0
\(577\) −7.80701 −0.325010 −0.162505 0.986708i \(-0.551957\pi\)
−0.162505 + 0.986708i \(0.551957\pi\)
\(578\) 3.93406 0.163635
\(579\) 0 0
\(580\) 34.1249 1.41696
\(581\) 18.1119 0.751407
\(582\) 0 0
\(583\) −5.50603 −0.228037
\(584\) 3.97967 0.164680
\(585\) 0 0
\(586\) −16.9926 −0.701960
\(587\) −20.9714 −0.865582 −0.432791 0.901494i \(-0.642471\pi\)
−0.432791 + 0.901494i \(0.642471\pi\)
\(588\) 0 0
\(589\) −0.170112 −0.00700933
\(590\) 31.3335 1.28998
\(591\) 0 0
\(592\) −20.3319 −0.835636
\(593\) −23.6392 −0.970747 −0.485373 0.874307i \(-0.661316\pi\)
−0.485373 + 0.874307i \(0.661316\pi\)
\(594\) 0 0
\(595\) 40.8663 1.67535
\(596\) −31.9131 −1.30721
\(597\) 0 0
\(598\) −27.2245 −1.11329
\(599\) 1.59764 0.0652777 0.0326389 0.999467i \(-0.489609\pi\)
0.0326389 + 0.999467i \(0.489609\pi\)
\(600\) 0 0
\(601\) 31.7194 1.29386 0.646931 0.762548i \(-0.276052\pi\)
0.646931 + 0.762548i \(0.276052\pi\)
\(602\) −33.8906 −1.38128
\(603\) 0 0
\(604\) −41.4465 −1.68643
\(605\) 21.3958 0.869862
\(606\) 0 0
\(607\) 10.6373 0.431755 0.215877 0.976421i \(-0.430739\pi\)
0.215877 + 0.976421i \(0.430739\pi\)
\(608\) 0.132417 0.00537023
\(609\) 0 0
\(610\) 59.3107 2.40142
\(611\) 4.79535 0.193999
\(612\) 0 0
\(613\) −45.3797 −1.83287 −0.916434 0.400185i \(-0.868946\pi\)
−0.916434 + 0.400185i \(0.868946\pi\)
\(614\) −35.5011 −1.43271
\(615\) 0 0
\(616\) 6.83656 0.275453
\(617\) −23.0932 −0.929699 −0.464850 0.885390i \(-0.653892\pi\)
−0.464850 + 0.885390i \(0.653892\pi\)
\(618\) 0 0
\(619\) 1.96634 0.0790337 0.0395168 0.999219i \(-0.487418\pi\)
0.0395168 + 0.999219i \(0.487418\pi\)
\(620\) 75.7955 3.04402
\(621\) 0 0
\(622\) −10.8983 −0.436980
\(623\) 16.7849 0.672473
\(624\) 0 0
\(625\) −29.1763 −1.16705
\(626\) −59.0688 −2.36086
\(627\) 0 0
\(628\) −21.2404 −0.847583
\(629\) 27.8506 1.11048
\(630\) 0 0
\(631\) 48.7084 1.93905 0.969526 0.244990i \(-0.0787845\pi\)
0.969526 + 0.244990i \(0.0787845\pi\)
\(632\) −8.09155 −0.321865
\(633\) 0 0
\(634\) −19.4379 −0.771979
\(635\) −39.2564 −1.55784
\(636\) 0 0
\(637\) 8.90065 0.352657
\(638\) 19.1567 0.758422
\(639\) 0 0
\(640\) −23.0895 −0.912693
\(641\) 24.6679 0.974325 0.487162 0.873311i \(-0.338032\pi\)
0.487162 + 0.873311i \(0.338032\pi\)
\(642\) 0 0
\(643\) −10.5915 −0.417686 −0.208843 0.977949i \(-0.566970\pi\)
−0.208843 + 0.977949i \(0.566970\pi\)
\(644\) −67.0109 −2.64060
\(645\) 0 0
\(646\) −0.136319 −0.00536339
\(647\) 12.2785 0.482716 0.241358 0.970436i \(-0.422407\pi\)
0.241358 + 0.970436i \(0.422407\pi\)
\(648\) 0 0
\(649\) 9.71877 0.381495
\(650\) −13.8859 −0.544649
\(651\) 0 0
\(652\) 1.66565 0.0652320
\(653\) 2.86324 0.112047 0.0560235 0.998429i \(-0.482158\pi\)
0.0560235 + 0.998429i \(0.482158\pi\)
\(654\) 0 0
\(655\) −0.000965000 0 −3.77057e−5 0
\(656\) −16.0679 −0.627346
\(657\) 0 0
\(658\) 21.3626 0.832800
\(659\) 46.7125 1.81966 0.909831 0.414979i \(-0.136211\pi\)
0.909831 + 0.414979i \(0.136211\pi\)
\(660\) 0 0
\(661\) 23.8425 0.927366 0.463683 0.886001i \(-0.346528\pi\)
0.463683 + 0.886001i \(0.346528\pi\)
\(662\) −35.8365 −1.39283
\(663\) 0 0
\(664\) 5.11811 0.198621
\(665\) 0.174056 0.00674959
\(666\) 0 0
\(667\) −35.7005 −1.38233
\(668\) 39.8080 1.54022
\(669\) 0 0
\(670\) −28.3393 −1.09484
\(671\) 18.3965 0.710188
\(672\) 0 0
\(673\) 4.51325 0.173973 0.0869864 0.996209i \(-0.472276\pi\)
0.0869864 + 0.996209i \(0.472276\pi\)
\(674\) −27.5802 −1.06235
\(675\) 0 0
\(676\) −25.2411 −0.970812
\(677\) −12.6120 −0.484720 −0.242360 0.970186i \(-0.577922\pi\)
−0.242360 + 0.970186i \(0.577922\pi\)
\(678\) 0 0
\(679\) 40.8557 1.56790
\(680\) 11.5481 0.442850
\(681\) 0 0
\(682\) 42.5494 1.62930
\(683\) 18.0420 0.690358 0.345179 0.938537i \(-0.387818\pi\)
0.345179 + 0.938537i \(0.387818\pi\)
\(684\) 0 0
\(685\) 38.5544 1.47309
\(686\) −12.3333 −0.470888
\(687\) 0 0
\(688\) 12.9628 0.494202
\(689\) −4.68142 −0.178348
\(690\) 0 0
\(691\) −33.2733 −1.26578 −0.632888 0.774243i \(-0.718131\pi\)
−0.632888 + 0.774243i \(0.718131\pi\)
\(692\) −39.8339 −1.51426
\(693\) 0 0
\(694\) −30.4296 −1.15509
\(695\) −1.91765 −0.0727406
\(696\) 0 0
\(697\) 22.0098 0.833681
\(698\) −17.1948 −0.650831
\(699\) 0 0
\(700\) −34.1790 −1.29184
\(701\) −18.2677 −0.689961 −0.344981 0.938610i \(-0.612115\pi\)
−0.344981 + 0.938610i \(0.612115\pi\)
\(702\) 0 0
\(703\) 0.118620 0.00447384
\(704\) −21.9825 −0.828498
\(705\) 0 0
\(706\) −57.3051 −2.15671
\(707\) 10.0439 0.377738
\(708\) 0 0
\(709\) 1.97841 0.0743006 0.0371503 0.999310i \(-0.488172\pi\)
0.0371503 + 0.999310i \(0.488172\pi\)
\(710\) 48.6857 1.82714
\(711\) 0 0
\(712\) 4.74313 0.177756
\(713\) −79.2952 −2.96963
\(714\) 0 0
\(715\) −9.77269 −0.365478
\(716\) 19.6047 0.732663
\(717\) 0 0
\(718\) 52.7099 1.96712
\(719\) −13.9854 −0.521566 −0.260783 0.965397i \(-0.583981\pi\)
−0.260783 + 0.965397i \(0.583981\pi\)
\(720\) 0 0
\(721\) 10.0496 0.374266
\(722\) 40.1678 1.49489
\(723\) 0 0
\(724\) −34.1645 −1.26972
\(725\) −18.2091 −0.676269
\(726\) 0 0
\(727\) 37.3803 1.38636 0.693180 0.720764i \(-0.256209\pi\)
0.693180 + 0.720764i \(0.256209\pi\)
\(728\) 5.81268 0.215432
\(729\) 0 0
\(730\) −25.3429 −0.937984
\(731\) −17.7564 −0.656745
\(732\) 0 0
\(733\) 26.6081 0.982793 0.491397 0.870936i \(-0.336487\pi\)
0.491397 + 0.870936i \(0.336487\pi\)
\(734\) −27.6784 −1.02163
\(735\) 0 0
\(736\) 61.7245 2.27520
\(737\) −8.79006 −0.323786
\(738\) 0 0
\(739\) 43.5914 1.60353 0.801767 0.597636i \(-0.203893\pi\)
0.801767 + 0.597636i \(0.203893\pi\)
\(740\) −52.8528 −1.94291
\(741\) 0 0
\(742\) −20.8550 −0.765613
\(743\) −50.8670 −1.86613 −0.933065 0.359709i \(-0.882876\pi\)
−0.933065 + 0.359709i \(0.882876\pi\)
\(744\) 0 0
\(745\) 38.6389 1.41562
\(746\) 60.8350 2.22733
\(747\) 0 0
\(748\) 18.8394 0.688837
\(749\) 55.4519 2.02617
\(750\) 0 0
\(751\) 12.6722 0.462416 0.231208 0.972904i \(-0.425732\pi\)
0.231208 + 0.972904i \(0.425732\pi\)
\(752\) −8.17095 −0.297964
\(753\) 0 0
\(754\) 16.2877 0.593163
\(755\) 50.1815 1.82629
\(756\) 0 0
\(757\) −14.9877 −0.544737 −0.272368 0.962193i \(-0.587807\pi\)
−0.272368 + 0.962193i \(0.587807\pi\)
\(758\) 44.3607 1.61125
\(759\) 0 0
\(760\) 0.0491852 0.00178413
\(761\) −10.5506 −0.382458 −0.191229 0.981545i \(-0.561247\pi\)
−0.191229 + 0.981545i \(0.561247\pi\)
\(762\) 0 0
\(763\) 1.02145 0.0369790
\(764\) 35.4976 1.28426
\(765\) 0 0
\(766\) 51.7857 1.87109
\(767\) 8.26324 0.298368
\(768\) 0 0
\(769\) −39.3336 −1.41840 −0.709202 0.705005i \(-0.750945\pi\)
−0.709202 + 0.705005i \(0.750945\pi\)
\(770\) −43.5359 −1.56893
\(771\) 0 0
\(772\) −10.3322 −0.371865
\(773\) 48.4610 1.74302 0.871511 0.490376i \(-0.163140\pi\)
0.871511 + 0.490376i \(0.163140\pi\)
\(774\) 0 0
\(775\) −40.4446 −1.45281
\(776\) 11.5451 0.414446
\(777\) 0 0
\(778\) −33.3331 −1.19505
\(779\) 0.0937432 0.00335870
\(780\) 0 0
\(781\) 15.1009 0.540353
\(782\) −63.5432 −2.27230
\(783\) 0 0
\(784\) −15.1661 −0.541646
\(785\) 25.7169 0.917875
\(786\) 0 0
\(787\) 19.1310 0.681946 0.340973 0.940073i \(-0.389243\pi\)
0.340973 + 0.940073i \(0.389243\pi\)
\(788\) 0.584019 0.0208048
\(789\) 0 0
\(790\) 51.5278 1.83328
\(791\) −11.3588 −0.403873
\(792\) 0 0
\(793\) 15.6413 0.555440
\(794\) 54.6002 1.93769
\(795\) 0 0
\(796\) 64.4296 2.28365
\(797\) 53.1941 1.88423 0.942116 0.335288i \(-0.108834\pi\)
0.942116 + 0.335288i \(0.108834\pi\)
\(798\) 0 0
\(799\) 11.1926 0.395964
\(800\) 31.4826 1.11308
\(801\) 0 0
\(802\) 34.6264 1.22270
\(803\) −7.86066 −0.277397
\(804\) 0 0
\(805\) 81.1337 2.85959
\(806\) 36.1770 1.27428
\(807\) 0 0
\(808\) 2.83822 0.0998484
\(809\) −52.4325 −1.84343 −0.921715 0.387868i \(-0.873211\pi\)
−0.921715 + 0.387868i \(0.873211\pi\)
\(810\) 0 0
\(811\) −16.1664 −0.567679 −0.283839 0.958872i \(-0.591608\pi\)
−0.283839 + 0.958872i \(0.591608\pi\)
\(812\) 40.0909 1.40691
\(813\) 0 0
\(814\) −29.6700 −1.03993
\(815\) −2.01670 −0.0706418
\(816\) 0 0
\(817\) −0.0756273 −0.00264587
\(818\) 53.6059 1.87429
\(819\) 0 0
\(820\) −41.7685 −1.45862
\(821\) 56.3605 1.96700 0.983498 0.180918i \(-0.0579069\pi\)
0.983498 + 0.180918i \(0.0579069\pi\)
\(822\) 0 0
\(823\) −39.8461 −1.38895 −0.694473 0.719518i \(-0.744363\pi\)
−0.694473 + 0.719518i \(0.744363\pi\)
\(824\) 2.83984 0.0989305
\(825\) 0 0
\(826\) 36.8115 1.28084
\(827\) −26.8004 −0.931942 −0.465971 0.884800i \(-0.654295\pi\)
−0.465971 + 0.884800i \(0.654295\pi\)
\(828\) 0 0
\(829\) 52.9647 1.83954 0.919769 0.392460i \(-0.128376\pi\)
0.919769 + 0.392460i \(0.128376\pi\)
\(830\) −32.5926 −1.13131
\(831\) 0 0
\(832\) −18.6903 −0.647970
\(833\) 20.7745 0.719794
\(834\) 0 0
\(835\) −48.1978 −1.66795
\(836\) 0.0802399 0.00277516
\(837\) 0 0
\(838\) −13.4453 −0.464461
\(839\) 36.5919 1.26329 0.631647 0.775256i \(-0.282379\pi\)
0.631647 + 0.775256i \(0.282379\pi\)
\(840\) 0 0
\(841\) −7.64129 −0.263493
\(842\) −55.5270 −1.91359
\(843\) 0 0
\(844\) −29.1382 −1.00298
\(845\) 30.5608 1.05132
\(846\) 0 0
\(847\) 25.1364 0.863696
\(848\) 7.97682 0.273925
\(849\) 0 0
\(850\) −32.4103 −1.11166
\(851\) 55.2932 1.89543
\(852\) 0 0
\(853\) 54.8342 1.87749 0.938743 0.344617i \(-0.111991\pi\)
0.938743 + 0.344617i \(0.111991\pi\)
\(854\) 69.6799 2.38440
\(855\) 0 0
\(856\) 15.6698 0.535582
\(857\) 43.3706 1.48151 0.740757 0.671774i \(-0.234467\pi\)
0.740757 + 0.671774i \(0.234467\pi\)
\(858\) 0 0
\(859\) −11.2926 −0.385298 −0.192649 0.981268i \(-0.561708\pi\)
−0.192649 + 0.981268i \(0.561708\pi\)
\(860\) 33.6968 1.14905
\(861\) 0 0
\(862\) −26.0646 −0.887765
\(863\) 11.5449 0.392991 0.196496 0.980505i \(-0.437044\pi\)
0.196496 + 0.980505i \(0.437044\pi\)
\(864\) 0 0
\(865\) 48.2291 1.63984
\(866\) 14.2727 0.485007
\(867\) 0 0
\(868\) 89.0467 3.02244
\(869\) 15.9825 0.542168
\(870\) 0 0
\(871\) −7.47362 −0.253234
\(872\) 0.288645 0.00977474
\(873\) 0 0
\(874\) −0.270640 −0.00915454
\(875\) −11.1327 −0.376356
\(876\) 0 0
\(877\) −33.8656 −1.14356 −0.571780 0.820407i \(-0.693747\pi\)
−0.571780 + 0.820407i \(0.693747\pi\)
\(878\) 9.44259 0.318672
\(879\) 0 0
\(880\) 16.6520 0.561339
\(881\) 25.2356 0.850210 0.425105 0.905144i \(-0.360237\pi\)
0.425105 + 0.905144i \(0.360237\pi\)
\(882\) 0 0
\(883\) −6.56124 −0.220803 −0.110402 0.993887i \(-0.535214\pi\)
−0.110402 + 0.993887i \(0.535214\pi\)
\(884\) 16.0179 0.538741
\(885\) 0 0
\(886\) 59.6422 2.00372
\(887\) 47.7246 1.60243 0.801217 0.598374i \(-0.204186\pi\)
0.801217 + 0.598374i \(0.204186\pi\)
\(888\) 0 0
\(889\) −46.1195 −1.54680
\(890\) −30.2047 −1.01247
\(891\) 0 0
\(892\) −65.0376 −2.17762
\(893\) 0.0476708 0.00159524
\(894\) 0 0
\(895\) −23.7365 −0.793424
\(896\) −27.1262 −0.906223
\(897\) 0 0
\(898\) −55.4533 −1.85050
\(899\) 47.4403 1.58222
\(900\) 0 0
\(901\) −10.9266 −0.364019
\(902\) −23.4476 −0.780721
\(903\) 0 0
\(904\) −3.20981 −0.106757
\(905\) 41.3649 1.37502
\(906\) 0 0
\(907\) 20.1207 0.668098 0.334049 0.942556i \(-0.391585\pi\)
0.334049 + 0.942556i \(0.391585\pi\)
\(908\) 2.51771 0.0835531
\(909\) 0 0
\(910\) −37.0157 −1.22706
\(911\) 43.6896 1.44750 0.723751 0.690061i \(-0.242416\pi\)
0.723751 + 0.690061i \(0.242416\pi\)
\(912\) 0 0
\(913\) −10.1093 −0.334570
\(914\) 51.0243 1.68773
\(915\) 0 0
\(916\) 64.4090 2.12813
\(917\) −0.00113371 −3.74384e−5 0
\(918\) 0 0
\(919\) −12.5424 −0.413734 −0.206867 0.978369i \(-0.566327\pi\)
−0.206867 + 0.978369i \(0.566327\pi\)
\(920\) 22.9270 0.755881
\(921\) 0 0
\(922\) −39.9711 −1.31638
\(923\) 12.8393 0.422612
\(924\) 0 0
\(925\) 28.2024 0.927288
\(926\) −3.59185 −0.118035
\(927\) 0 0
\(928\) −36.9282 −1.21223
\(929\) −31.6414 −1.03812 −0.519060 0.854738i \(-0.673718\pi\)
−0.519060 + 0.854738i \(0.673718\pi\)
\(930\) 0 0
\(931\) 0.0884818 0.00289988
\(932\) −11.0863 −0.363145
\(933\) 0 0
\(934\) 3.25172 0.106399
\(935\) −22.8099 −0.745963
\(936\) 0 0
\(937\) 35.3437 1.15463 0.577315 0.816522i \(-0.304101\pi\)
0.577315 + 0.816522i \(0.304101\pi\)
\(938\) −33.2939 −1.08708
\(939\) 0 0
\(940\) −21.2404 −0.692784
\(941\) 27.8837 0.908982 0.454491 0.890751i \(-0.349821\pi\)
0.454491 + 0.890751i \(0.349821\pi\)
\(942\) 0 0
\(943\) 43.6971 1.42297
\(944\) −14.0800 −0.458265
\(945\) 0 0
\(946\) 18.9164 0.615025
\(947\) −17.6519 −0.573609 −0.286804 0.957989i \(-0.592593\pi\)
−0.286804 + 0.957989i \(0.592593\pi\)
\(948\) 0 0
\(949\) −6.68341 −0.216953
\(950\) −0.138040 −0.00447862
\(951\) 0 0
\(952\) 13.5671 0.439711
\(953\) 1.16016 0.0375813 0.0187907 0.999823i \(-0.494018\pi\)
0.0187907 + 0.999823i \(0.494018\pi\)
\(954\) 0 0
\(955\) −42.9789 −1.39076
\(956\) 3.76599 0.121801
\(957\) 0 0
\(958\) 33.5943 1.08538
\(959\) 45.2948 1.46265
\(960\) 0 0
\(961\) 74.3707 2.39905
\(962\) −25.2265 −0.813335
\(963\) 0 0
\(964\) 2.07976 0.0669845
\(965\) 12.5098 0.402705
\(966\) 0 0
\(967\) −37.5213 −1.20660 −0.603302 0.797513i \(-0.706149\pi\)
−0.603302 + 0.797513i \(0.706149\pi\)
\(968\) 7.10312 0.228303
\(969\) 0 0
\(970\) −73.5205 −2.36060
\(971\) 13.1664 0.422531 0.211265 0.977429i \(-0.432242\pi\)
0.211265 + 0.977429i \(0.432242\pi\)
\(972\) 0 0
\(973\) −2.25291 −0.0722250
\(974\) −82.6096 −2.64698
\(975\) 0 0
\(976\) −26.6518 −0.853102
\(977\) −26.8056 −0.857587 −0.428793 0.903403i \(-0.641061\pi\)
−0.428793 + 0.903403i \(0.641061\pi\)
\(978\) 0 0
\(979\) −9.36866 −0.299424
\(980\) −39.4243 −1.25936
\(981\) 0 0
\(982\) −81.5716 −2.60305
\(983\) −12.0965 −0.385819 −0.192910 0.981217i \(-0.561792\pi\)
−0.192910 + 0.981217i \(0.561792\pi\)
\(984\) 0 0
\(985\) −0.707103 −0.0225302
\(986\) 38.0162 1.21068
\(987\) 0 0
\(988\) 0.0682228 0.00217046
\(989\) −35.2527 −1.12097
\(990\) 0 0
\(991\) −24.9307 −0.791950 −0.395975 0.918261i \(-0.629593\pi\)
−0.395975 + 0.918261i \(0.629593\pi\)
\(992\) −82.0220 −2.60420
\(993\) 0 0
\(994\) 57.1973 1.81419
\(995\) −78.0084 −2.47303
\(996\) 0 0
\(997\) −33.4776 −1.06025 −0.530124 0.847920i \(-0.677855\pi\)
−0.530124 + 0.847920i \(0.677855\pi\)
\(998\) −11.2668 −0.356644
\(999\) 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 6561.2.a.d.1.11 72
3.2 odd 2 6561.2.a.c.1.62 72
81.5 odd 54 729.2.g.c.460.8 144
81.11 odd 54 81.2.g.a.40.1 144
81.16 even 27 729.2.g.b.271.1 144
81.22 even 27 243.2.g.a.235.8 144
81.32 odd 54 729.2.g.d.703.1 144
81.38 odd 54 729.2.g.d.28.1 144
81.43 even 27 729.2.g.a.28.8 144
81.49 even 27 729.2.g.a.703.8 144
81.59 odd 54 81.2.g.a.79.1 yes 144
81.65 odd 54 729.2.g.c.271.8 144
81.70 even 27 243.2.g.a.91.8 144
81.76 even 27 729.2.g.b.460.1 144
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
81.2.g.a.40.1 144 81.11 odd 54
81.2.g.a.79.1 yes 144 81.59 odd 54
243.2.g.a.91.8 144 81.70 even 27
243.2.g.a.235.8 144 81.22 even 27
729.2.g.a.28.8 144 81.43 even 27
729.2.g.a.703.8 144 81.49 even 27
729.2.g.b.271.1 144 81.16 even 27
729.2.g.b.460.1 144 81.76 even 27
729.2.g.c.271.8 144 81.65 odd 54
729.2.g.c.460.8 144 81.5 odd 54
729.2.g.d.28.1 144 81.38 odd 54
729.2.g.d.703.1 144 81.32 odd 54
6561.2.a.c.1.62 72 3.2 odd 2
6561.2.a.d.1.11 72 1.1 even 1 trivial