Properties

Label 243.2.a.e.1.2
Level $243$
Weight $2$
Character 243.1
Self dual yes
Analytic conductor $1.940$
Analytic rank $1$
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] = [243,2,Mod(1,243)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(243, 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("243.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 243 = 3^{5} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 243.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(1.94036476912\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.347296\) of defining polynomial
Character \(\chi\) \(=\) 243.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.34730 q^{2} -0.184793 q^{4} -1.65270 q^{5} +2.41147 q^{7} +2.94356 q^{8} +O(q^{10})\) \(q-1.34730 q^{2} -0.184793 q^{4} -1.65270 q^{5} +2.41147 q^{7} +2.94356 q^{8} +2.22668 q^{10} -5.94356 q^{11} -3.22668 q^{13} -3.24897 q^{14} -3.59627 q^{16} -3.00000 q^{17} -6.63816 q^{19} +0.305407 q^{20} +8.00774 q^{22} +2.94356 q^{23} -2.26857 q^{25} +4.34730 q^{26} -0.445622 q^{28} +1.29086 q^{29} -0.588526 q^{31} -1.04189 q^{32} +4.04189 q^{34} -3.98545 q^{35} +0.0418891 q^{37} +8.94356 q^{38} -4.86484 q^{40} +4.90167 q^{41} -5.18479 q^{43} +1.09833 q^{44} -3.96585 q^{46} +3.73648 q^{47} -1.18479 q^{49} +3.05644 q^{50} +0.596267 q^{52} -11.6382 q^{53} +9.82295 q^{55} +7.09833 q^{56} -1.73917 q^{58} +7.34730 q^{59} +11.0496 q^{61} +0.792919 q^{62} +8.59627 q^{64} +5.33275 q^{65} +1.85710 q^{67} +0.554378 q^{68} +5.36959 q^{70} +5.51249 q^{71} +5.55438 q^{73} -0.0564370 q^{74} +1.22668 q^{76} -14.3327 q^{77} -3.78106 q^{79} +5.94356 q^{80} -6.60401 q^{82} -3.98545 q^{83} +4.95811 q^{85} +6.98545 q^{86} -17.4953 q^{88} -8.15064 q^{89} -7.78106 q^{91} -0.543948 q^{92} -5.03415 q^{94} +10.9709 q^{95} +0.260830 q^{97} +1.59627 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{4} - 6 q^{5} - 3 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} + 3 q^{4} - 6 q^{5} - 3 q^{7} - 6 q^{8} - 3 q^{11} - 3 q^{13} + 3 q^{14} + 3 q^{16} - 9 q^{17} - 3 q^{19} + 3 q^{20} - 6 q^{23} + 3 q^{25} + 12 q^{26} - 12 q^{28} - 12 q^{29} - 12 q^{31} + 9 q^{34} + 6 q^{35} - 3 q^{37} + 12 q^{38} + 9 q^{40} + 3 q^{41} - 12 q^{43} + 15 q^{44} + 9 q^{46} + 6 q^{47} + 24 q^{50} - 12 q^{52} - 18 q^{53} + 9 q^{55} + 33 q^{56} + 9 q^{58} + 21 q^{59} + 6 q^{61} + 12 q^{62} + 12 q^{64} - 3 q^{65} + 6 q^{67} - 9 q^{68} + 9 q^{70} + 9 q^{71} + 6 q^{73} - 15 q^{74} - 3 q^{76} - 24 q^{77} + 6 q^{79} + 3 q^{80} + 18 q^{82} + 6 q^{83} + 18 q^{85} + 3 q^{86} - 36 q^{88} - 6 q^{91} - 24 q^{92} - 36 q^{94} - 3 q^{95} + 15 q^{97} - 9 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.34730 −0.952682 −0.476341 0.879261i \(-0.658037\pi\)
−0.476341 + 0.879261i \(0.658037\pi\)
\(3\) 0 0
\(4\) −0.184793 −0.0923963
\(5\) −1.65270 −0.739112 −0.369556 0.929209i \(-0.620490\pi\)
−0.369556 + 0.929209i \(0.620490\pi\)
\(6\) 0 0
\(7\) 2.41147 0.911452 0.455726 0.890120i \(-0.349380\pi\)
0.455726 + 0.890120i \(0.349380\pi\)
\(8\) 2.94356 1.04071
\(9\) 0 0
\(10\) 2.22668 0.704139
\(11\) −5.94356 −1.79205 −0.896026 0.444002i \(-0.853558\pi\)
−0.896026 + 0.444002i \(0.853558\pi\)
\(12\) 0 0
\(13\) −3.22668 −0.894920 −0.447460 0.894304i \(-0.647671\pi\)
−0.447460 + 0.894304i \(0.647671\pi\)
\(14\) −3.24897 −0.868324
\(15\) 0 0
\(16\) −3.59627 −0.899067
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) −6.63816 −1.52290 −0.761449 0.648225i \(-0.775512\pi\)
−0.761449 + 0.648225i \(0.775512\pi\)
\(20\) 0.305407 0.0682911
\(21\) 0 0
\(22\) 8.00774 1.70726
\(23\) 2.94356 0.613775 0.306888 0.951746i \(-0.400712\pi\)
0.306888 + 0.951746i \(0.400712\pi\)
\(24\) 0 0
\(25\) −2.26857 −0.453714
\(26\) 4.34730 0.852575
\(27\) 0 0
\(28\) −0.445622 −0.0842147
\(29\) 1.29086 0.239707 0.119853 0.992792i \(-0.461758\pi\)
0.119853 + 0.992792i \(0.461758\pi\)
\(30\) 0 0
\(31\) −0.588526 −0.105702 −0.0528512 0.998602i \(-0.516831\pi\)
−0.0528512 + 0.998602i \(0.516831\pi\)
\(32\) −1.04189 −0.184182
\(33\) 0 0
\(34\) 4.04189 0.693178
\(35\) −3.98545 −0.673664
\(36\) 0 0
\(37\) 0.0418891 0.00688652 0.00344326 0.999994i \(-0.498904\pi\)
0.00344326 + 0.999994i \(0.498904\pi\)
\(38\) 8.94356 1.45084
\(39\) 0 0
\(40\) −4.86484 −0.769198
\(41\) 4.90167 0.765513 0.382756 0.923849i \(-0.374975\pi\)
0.382756 + 0.923849i \(0.374975\pi\)
\(42\) 0 0
\(43\) −5.18479 −0.790673 −0.395337 0.918536i \(-0.629372\pi\)
−0.395337 + 0.918536i \(0.629372\pi\)
\(44\) 1.09833 0.165579
\(45\) 0 0
\(46\) −3.96585 −0.584733
\(47\) 3.73648 0.545022 0.272511 0.962153i \(-0.412146\pi\)
0.272511 + 0.962153i \(0.412146\pi\)
\(48\) 0 0
\(49\) −1.18479 −0.169256
\(50\) 3.05644 0.432245
\(51\) 0 0
\(52\) 0.596267 0.0826873
\(53\) −11.6382 −1.59862 −0.799312 0.600916i \(-0.794802\pi\)
−0.799312 + 0.600916i \(0.794802\pi\)
\(54\) 0 0
\(55\) 9.82295 1.32453
\(56\) 7.09833 0.948554
\(57\) 0 0
\(58\) −1.73917 −0.228364
\(59\) 7.34730 0.956537 0.478268 0.878214i \(-0.341265\pi\)
0.478268 + 0.878214i \(0.341265\pi\)
\(60\) 0 0
\(61\) 11.0496 1.41476 0.707380 0.706833i \(-0.249877\pi\)
0.707380 + 0.706833i \(0.249877\pi\)
\(62\) 0.792919 0.100701
\(63\) 0 0
\(64\) 8.59627 1.07453
\(65\) 5.33275 0.661446
\(66\) 0 0
\(67\) 1.85710 0.226880 0.113440 0.993545i \(-0.463813\pi\)
0.113440 + 0.993545i \(0.463813\pi\)
\(68\) 0.554378 0.0672282
\(69\) 0 0
\(70\) 5.36959 0.641788
\(71\) 5.51249 0.654212 0.327106 0.944988i \(-0.393927\pi\)
0.327106 + 0.944988i \(0.393927\pi\)
\(72\) 0 0
\(73\) 5.55438 0.650091 0.325045 0.945698i \(-0.394620\pi\)
0.325045 + 0.945698i \(0.394620\pi\)
\(74\) −0.0564370 −0.00656067
\(75\) 0 0
\(76\) 1.22668 0.140710
\(77\) −14.3327 −1.63337
\(78\) 0 0
\(79\) −3.78106 −0.425402 −0.212701 0.977117i \(-0.568226\pi\)
−0.212701 + 0.977117i \(0.568226\pi\)
\(80\) 5.94356 0.664511
\(81\) 0 0
\(82\) −6.60401 −0.729291
\(83\) −3.98545 −0.437460 −0.218730 0.975785i \(-0.570191\pi\)
−0.218730 + 0.975785i \(0.570191\pi\)
\(84\) 0 0
\(85\) 4.95811 0.537783
\(86\) 6.98545 0.753261
\(87\) 0 0
\(88\) −17.4953 −1.86500
\(89\) −8.15064 −0.863967 −0.431983 0.901882i \(-0.642186\pi\)
−0.431983 + 0.901882i \(0.642186\pi\)
\(90\) 0 0
\(91\) −7.78106 −0.815677
\(92\) −0.543948 −0.0567105
\(93\) 0 0
\(94\) −5.03415 −0.519233
\(95\) 10.9709 1.12559
\(96\) 0 0
\(97\) 0.260830 0.0264833 0.0132416 0.999912i \(-0.495785\pi\)
0.0132416 + 0.999912i \(0.495785\pi\)
\(98\) 1.59627 0.161247
\(99\) 0 0
\(100\) 0.419215 0.0419215
\(101\) −11.0273 −1.09726 −0.548631 0.836065i \(-0.684851\pi\)
−0.548631 + 0.836065i \(0.684851\pi\)
\(102\) 0 0
\(103\) −3.90673 −0.384941 −0.192471 0.981303i \(-0.561650\pi\)
−0.192471 + 0.981303i \(0.561650\pi\)
\(104\) −9.49794 −0.931350
\(105\) 0 0
\(106\) 15.6800 1.52298
\(107\) 2.63816 0.255040 0.127520 0.991836i \(-0.459298\pi\)
0.127520 + 0.991836i \(0.459298\pi\)
\(108\) 0 0
\(109\) −8.95811 −0.858031 −0.429016 0.903297i \(-0.641140\pi\)
−0.429016 + 0.903297i \(0.641140\pi\)
\(110\) −13.2344 −1.26185
\(111\) 0 0
\(112\) −8.67230 −0.819456
\(113\) −15.9290 −1.49848 −0.749238 0.662301i \(-0.769580\pi\)
−0.749238 + 0.662301i \(0.769580\pi\)
\(114\) 0 0
\(115\) −4.86484 −0.453648
\(116\) −0.238541 −0.0221480
\(117\) 0 0
\(118\) −9.89899 −0.911275
\(119\) −7.23442 −0.663178
\(120\) 0 0
\(121\) 24.3259 2.21145
\(122\) −14.8871 −1.34782
\(123\) 0 0
\(124\) 0.108755 0.00976650
\(125\) 12.0128 1.07446
\(126\) 0 0
\(127\) 3.59627 0.319117 0.159559 0.987188i \(-0.448993\pi\)
0.159559 + 0.987188i \(0.448993\pi\)
\(128\) −9.49794 −0.839507
\(129\) 0 0
\(130\) −7.18479 −0.630148
\(131\) 17.7074 1.54710 0.773551 0.633734i \(-0.218479\pi\)
0.773551 + 0.633734i \(0.218479\pi\)
\(132\) 0 0
\(133\) −16.0077 −1.38805
\(134\) −2.50206 −0.216145
\(135\) 0 0
\(136\) −8.83069 −0.757225
\(137\) 3.92902 0.335678 0.167839 0.985814i \(-0.446321\pi\)
0.167839 + 0.985814i \(0.446321\pi\)
\(138\) 0 0
\(139\) 11.9659 1.01493 0.507465 0.861672i \(-0.330583\pi\)
0.507465 + 0.861672i \(0.330583\pi\)
\(140\) 0.736482 0.0622441
\(141\) 0 0
\(142\) −7.42696 −0.623256
\(143\) 19.1780 1.60374
\(144\) 0 0
\(145\) −2.13341 −0.177170
\(146\) −7.48339 −0.619330
\(147\) 0 0
\(148\) −0.00774079 −0.000636289 0
\(149\) −20.5253 −1.68150 −0.840748 0.541426i \(-0.817885\pi\)
−0.840748 + 0.541426i \(0.817885\pi\)
\(150\) 0 0
\(151\) 16.0077 1.30269 0.651346 0.758781i \(-0.274205\pi\)
0.651346 + 0.758781i \(0.274205\pi\)
\(152\) −19.5398 −1.58489
\(153\) 0 0
\(154\) 19.3105 1.55608
\(155\) 0.972659 0.0781258
\(156\) 0 0
\(157\) −21.9736 −1.75368 −0.876842 0.480779i \(-0.840354\pi\)
−0.876842 + 0.480779i \(0.840354\pi\)
\(158\) 5.09421 0.405273
\(159\) 0 0
\(160\) 1.72193 0.136131
\(161\) 7.09833 0.559426
\(162\) 0 0
\(163\) 20.5107 1.60652 0.803262 0.595625i \(-0.203096\pi\)
0.803262 + 0.595625i \(0.203096\pi\)
\(164\) −0.905793 −0.0707305
\(165\) 0 0
\(166\) 5.36959 0.416761
\(167\) −4.29086 −0.332037 −0.166018 0.986123i \(-0.553091\pi\)
−0.166018 + 0.986123i \(0.553091\pi\)
\(168\) 0 0
\(169\) −2.58853 −0.199117
\(170\) −6.68004 −0.512336
\(171\) 0 0
\(172\) 0.958111 0.0730553
\(173\) −3.79292 −0.288370 −0.144185 0.989551i \(-0.546056\pi\)
−0.144185 + 0.989551i \(0.546056\pi\)
\(174\) 0 0
\(175\) −5.47060 −0.413538
\(176\) 21.3746 1.61117
\(177\) 0 0
\(178\) 10.9813 0.823086
\(179\) −8.27631 −0.618601 −0.309300 0.950964i \(-0.600095\pi\)
−0.309300 + 0.950964i \(0.600095\pi\)
\(180\) 0 0
\(181\) 6.72193 0.499637 0.249819 0.968293i \(-0.419629\pi\)
0.249819 + 0.968293i \(0.419629\pi\)
\(182\) 10.4834 0.777081
\(183\) 0 0
\(184\) 8.66456 0.638760
\(185\) −0.0692302 −0.00508991
\(186\) 0 0
\(187\) 17.8307 1.30391
\(188\) −0.690474 −0.0503580
\(189\) 0 0
\(190\) −14.7811 −1.07233
\(191\) 5.01455 0.362840 0.181420 0.983406i \(-0.441931\pi\)
0.181420 + 0.983406i \(0.441931\pi\)
\(192\) 0 0
\(193\) −17.8648 −1.28594 −0.642970 0.765892i \(-0.722298\pi\)
−0.642970 + 0.765892i \(0.722298\pi\)
\(194\) −0.351415 −0.0252301
\(195\) 0 0
\(196\) 0.218941 0.0156386
\(197\) −0.723689 −0.0515607 −0.0257803 0.999668i \(-0.508207\pi\)
−0.0257803 + 0.999668i \(0.508207\pi\)
\(198\) 0 0
\(199\) −10.1925 −0.722530 −0.361265 0.932463i \(-0.617655\pi\)
−0.361265 + 0.932463i \(0.617655\pi\)
\(200\) −6.67768 −0.472183
\(201\) 0 0
\(202\) 14.8571 1.04534
\(203\) 3.11287 0.218481
\(204\) 0 0
\(205\) −8.10101 −0.565799
\(206\) 5.26352 0.366727
\(207\) 0 0
\(208\) 11.6040 0.804593
\(209\) 39.4543 2.72911
\(210\) 0 0
\(211\) −14.8648 −1.02334 −0.511669 0.859183i \(-0.670973\pi\)
−0.511669 + 0.859183i \(0.670973\pi\)
\(212\) 2.15064 0.147707
\(213\) 0 0
\(214\) −3.55438 −0.242972
\(215\) 8.56893 0.584396
\(216\) 0 0
\(217\) −1.41921 −0.0963426
\(218\) 12.0692 0.817431
\(219\) 0 0
\(220\) −1.81521 −0.122381
\(221\) 9.68004 0.651150
\(222\) 0 0
\(223\) −10.9486 −0.733174 −0.366587 0.930384i \(-0.619474\pi\)
−0.366587 + 0.930384i \(0.619474\pi\)
\(224\) −2.51249 −0.167873
\(225\) 0 0
\(226\) 21.4611 1.42757
\(227\) 17.3327 1.15041 0.575207 0.818008i \(-0.304921\pi\)
0.575207 + 0.818008i \(0.304921\pi\)
\(228\) 0 0
\(229\) 1.56212 0.103228 0.0516138 0.998667i \(-0.483563\pi\)
0.0516138 + 0.998667i \(0.483563\pi\)
\(230\) 6.55438 0.432183
\(231\) 0 0
\(232\) 3.79973 0.249464
\(233\) −16.7888 −1.09987 −0.549935 0.835207i \(-0.685348\pi\)
−0.549935 + 0.835207i \(0.685348\pi\)
\(234\) 0 0
\(235\) −6.17530 −0.402832
\(236\) −1.35773 −0.0883804
\(237\) 0 0
\(238\) 9.74691 0.631798
\(239\) 4.02910 0.260621 0.130310 0.991473i \(-0.458403\pi\)
0.130310 + 0.991473i \(0.458403\pi\)
\(240\) 0 0
\(241\) −3.35235 −0.215944 −0.107972 0.994154i \(-0.534436\pi\)
−0.107972 + 0.994154i \(0.534436\pi\)
\(242\) −32.7743 −2.10681
\(243\) 0 0
\(244\) −2.04189 −0.130719
\(245\) 1.95811 0.125099
\(246\) 0 0
\(247\) 21.4192 1.36287
\(248\) −1.73236 −0.110005
\(249\) 0 0
\(250\) −16.1848 −1.02362
\(251\) 23.1506 1.46126 0.730628 0.682776i \(-0.239227\pi\)
0.730628 + 0.682776i \(0.239227\pi\)
\(252\) 0 0
\(253\) −17.4953 −1.09992
\(254\) −4.84524 −0.304017
\(255\) 0 0
\(256\) −4.39599 −0.274750
\(257\) −12.0128 −0.749337 −0.374669 0.927159i \(-0.622244\pi\)
−0.374669 + 0.927159i \(0.622244\pi\)
\(258\) 0 0
\(259\) 0.101014 0.00627673
\(260\) −0.985452 −0.0611151
\(261\) 0 0
\(262\) −23.8571 −1.47390
\(263\) −16.9017 −1.04220 −0.521101 0.853495i \(-0.674478\pi\)
−0.521101 + 0.853495i \(0.674478\pi\)
\(264\) 0 0
\(265\) 19.2344 1.18156
\(266\) 21.5672 1.32237
\(267\) 0 0
\(268\) −0.343178 −0.0209629
\(269\) 7.91447 0.482554 0.241277 0.970456i \(-0.422434\pi\)
0.241277 + 0.970456i \(0.422434\pi\)
\(270\) 0 0
\(271\) −17.2344 −1.04692 −0.523458 0.852051i \(-0.675358\pi\)
−0.523458 + 0.852051i \(0.675358\pi\)
\(272\) 10.7888 0.654167
\(273\) 0 0
\(274\) −5.29355 −0.319795
\(275\) 13.4834 0.813079
\(276\) 0 0
\(277\) 26.4347 1.58831 0.794153 0.607717i \(-0.207915\pi\)
0.794153 + 0.607717i \(0.207915\pi\)
\(278\) −16.1215 −0.966906
\(279\) 0 0
\(280\) −11.7314 −0.701087
\(281\) 18.9959 1.13320 0.566600 0.823993i \(-0.308259\pi\)
0.566600 + 0.823993i \(0.308259\pi\)
\(282\) 0 0
\(283\) −16.5868 −0.985981 −0.492991 0.870035i \(-0.664096\pi\)
−0.492991 + 0.870035i \(0.664096\pi\)
\(284\) −1.01867 −0.0604467
\(285\) 0 0
\(286\) −25.8384 −1.52786
\(287\) 11.8203 0.697728
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 2.87433 0.168787
\(291\) 0 0
\(292\) −1.02641 −0.0600660
\(293\) −19.3577 −1.13089 −0.565445 0.824786i \(-0.691296\pi\)
−0.565445 + 0.824786i \(0.691296\pi\)
\(294\) 0 0
\(295\) −12.1429 −0.706987
\(296\) 0.123303 0.00716685
\(297\) 0 0
\(298\) 27.6536 1.60193
\(299\) −9.49794 −0.549280
\(300\) 0 0
\(301\) −12.5030 −0.720661
\(302\) −21.5672 −1.24105
\(303\) 0 0
\(304\) 23.8726 1.36919
\(305\) −18.2618 −1.04567
\(306\) 0 0
\(307\) 20.8057 1.18744 0.593722 0.804670i \(-0.297658\pi\)
0.593722 + 0.804670i \(0.297658\pi\)
\(308\) 2.64858 0.150917
\(309\) 0 0
\(310\) −1.31046 −0.0744291
\(311\) 10.6655 0.604785 0.302392 0.953184i \(-0.402215\pi\)
0.302392 + 0.953184i \(0.402215\pi\)
\(312\) 0 0
\(313\) 3.81521 0.215648 0.107824 0.994170i \(-0.465612\pi\)
0.107824 + 0.994170i \(0.465612\pi\)
\(314\) 29.6049 1.67070
\(315\) 0 0
\(316\) 0.698711 0.0393056
\(317\) −26.3892 −1.48216 −0.741082 0.671414i \(-0.765687\pi\)
−0.741082 + 0.671414i \(0.765687\pi\)
\(318\) 0 0
\(319\) −7.67230 −0.429567
\(320\) −14.2071 −0.794200
\(321\) 0 0
\(322\) −9.56355 −0.532956
\(323\) 19.9145 1.10807
\(324\) 0 0
\(325\) 7.31996 0.406038
\(326\) −27.6340 −1.53051
\(327\) 0 0
\(328\) 14.4284 0.796674
\(329\) 9.01043 0.496761
\(330\) 0 0
\(331\) −1.57161 −0.0863837 −0.0431919 0.999067i \(-0.513753\pi\)
−0.0431919 + 0.999067i \(0.513753\pi\)
\(332\) 0.736482 0.0404197
\(333\) 0 0
\(334\) 5.78106 0.316325
\(335\) −3.06923 −0.167690
\(336\) 0 0
\(337\) −8.01548 −0.436631 −0.218316 0.975878i \(-0.570056\pi\)
−0.218316 + 0.975878i \(0.570056\pi\)
\(338\) 3.48751 0.189696
\(339\) 0 0
\(340\) −0.916222 −0.0496891
\(341\) 3.49794 0.189424
\(342\) 0 0
\(343\) −19.7374 −1.06572
\(344\) −15.2618 −0.822859
\(345\) 0 0
\(346\) 5.11019 0.274725
\(347\) −19.8452 −1.06535 −0.532674 0.846320i \(-0.678813\pi\)
−0.532674 + 0.846320i \(0.678813\pi\)
\(348\) 0 0
\(349\) 11.0933 0.593809 0.296905 0.954907i \(-0.404046\pi\)
0.296905 + 0.954907i \(0.404046\pi\)
\(350\) 7.37052 0.393971
\(351\) 0 0
\(352\) 6.19253 0.330063
\(353\) 2.86390 0.152430 0.0762151 0.997091i \(-0.475716\pi\)
0.0762151 + 0.997091i \(0.475716\pi\)
\(354\) 0 0
\(355\) −9.11051 −0.483536
\(356\) 1.50618 0.0798273
\(357\) 0 0
\(358\) 11.1506 0.589330
\(359\) −28.7888 −1.51941 −0.759707 0.650265i \(-0.774658\pi\)
−0.759707 + 0.650265i \(0.774658\pi\)
\(360\) 0 0
\(361\) 25.0651 1.31922
\(362\) −9.05644 −0.475996
\(363\) 0 0
\(364\) 1.43788 0.0753655
\(365\) −9.17974 −0.480490
\(366\) 0 0
\(367\) −10.9923 −0.573791 −0.286896 0.957962i \(-0.592623\pi\)
−0.286896 + 0.957962i \(0.592623\pi\)
\(368\) −10.5858 −0.551825
\(369\) 0 0
\(370\) 0.0932736 0.00484906
\(371\) −28.0651 −1.45707
\(372\) 0 0
\(373\) 33.4097 1.72989 0.864945 0.501867i \(-0.167353\pi\)
0.864945 + 0.501867i \(0.167353\pi\)
\(374\) −24.0232 −1.24221
\(375\) 0 0
\(376\) 10.9986 0.567208
\(377\) −4.16519 −0.214518
\(378\) 0 0
\(379\) 20.9394 1.07559 0.537794 0.843077i \(-0.319258\pi\)
0.537794 + 0.843077i \(0.319258\pi\)
\(380\) −2.02734 −0.104000
\(381\) 0 0
\(382\) −6.75608 −0.345671
\(383\) 4.11112 0.210068 0.105034 0.994469i \(-0.466505\pi\)
0.105034 + 0.994469i \(0.466505\pi\)
\(384\) 0 0
\(385\) 23.6878 1.20724
\(386\) 24.0692 1.22509
\(387\) 0 0
\(388\) −0.0481994 −0.00244695
\(389\) −17.0942 −0.866711 −0.433355 0.901223i \(-0.642670\pi\)
−0.433355 + 0.901223i \(0.642670\pi\)
\(390\) 0 0
\(391\) −8.83069 −0.446587
\(392\) −3.48751 −0.176146
\(393\) 0 0
\(394\) 0.975023 0.0491209
\(395\) 6.24897 0.314420
\(396\) 0 0
\(397\) 22.4020 1.12432 0.562162 0.827027i \(-0.309970\pi\)
0.562162 + 0.827027i \(0.309970\pi\)
\(398\) 13.7324 0.688341
\(399\) 0 0
\(400\) 8.15839 0.407919
\(401\) 14.5817 0.728176 0.364088 0.931364i \(-0.381381\pi\)
0.364088 + 0.931364i \(0.381381\pi\)
\(402\) 0 0
\(403\) 1.89899 0.0945952
\(404\) 2.03777 0.101383
\(405\) 0 0
\(406\) −4.19396 −0.208143
\(407\) −0.248970 −0.0123410
\(408\) 0 0
\(409\) −17.5030 −0.865467 −0.432734 0.901522i \(-0.642451\pi\)
−0.432734 + 0.901522i \(0.642451\pi\)
\(410\) 10.9145 0.539027
\(411\) 0 0
\(412\) 0.721934 0.0355671
\(413\) 17.7178 0.871837
\(414\) 0 0
\(415\) 6.58677 0.323332
\(416\) 3.36184 0.164828
\(417\) 0 0
\(418\) −53.1566 −2.59998
\(419\) −18.8621 −0.921476 −0.460738 0.887536i \(-0.652415\pi\)
−0.460738 + 0.887536i \(0.652415\pi\)
\(420\) 0 0
\(421\) −32.3337 −1.57585 −0.787924 0.615773i \(-0.788844\pi\)
−0.787924 + 0.615773i \(0.788844\pi\)
\(422\) 20.0273 0.974916
\(423\) 0 0
\(424\) −34.2576 −1.66370
\(425\) 6.80571 0.330126
\(426\) 0 0
\(427\) 26.6459 1.28949
\(428\) −0.487511 −0.0235648
\(429\) 0 0
\(430\) −11.5449 −0.556744
\(431\) −34.3164 −1.65297 −0.826483 0.562962i \(-0.809662\pi\)
−0.826483 + 0.562962i \(0.809662\pi\)
\(432\) 0 0
\(433\) −25.0669 −1.20464 −0.602318 0.798256i \(-0.705756\pi\)
−0.602318 + 0.798256i \(0.705756\pi\)
\(434\) 1.91210 0.0917839
\(435\) 0 0
\(436\) 1.65539 0.0792789
\(437\) −19.5398 −0.934717
\(438\) 0 0
\(439\) −23.2080 −1.10766 −0.553829 0.832630i \(-0.686834\pi\)
−0.553829 + 0.832630i \(0.686834\pi\)
\(440\) 28.9145 1.37844
\(441\) 0 0
\(442\) −13.0419 −0.620339
\(443\) 4.12155 0.195821 0.0979103 0.995195i \(-0.468784\pi\)
0.0979103 + 0.995195i \(0.468784\pi\)
\(444\) 0 0
\(445\) 13.4706 0.638568
\(446\) 14.7510 0.698482
\(447\) 0 0
\(448\) 20.7297 0.979385
\(449\) −18.3414 −0.865585 −0.432793 0.901494i \(-0.642472\pi\)
−0.432793 + 0.901494i \(0.642472\pi\)
\(450\) 0 0
\(451\) −29.1334 −1.37184
\(452\) 2.94356 0.138454
\(453\) 0 0
\(454\) −23.3523 −1.09598
\(455\) 12.8598 0.602876
\(456\) 0 0
\(457\) −19.4611 −0.910352 −0.455176 0.890401i \(-0.650424\pi\)
−0.455176 + 0.890401i \(0.650424\pi\)
\(458\) −2.10464 −0.0983432
\(459\) 0 0
\(460\) 0.898986 0.0419154
\(461\) −27.7493 −1.29241 −0.646206 0.763163i \(-0.723645\pi\)
−0.646206 + 0.763163i \(0.723645\pi\)
\(462\) 0 0
\(463\) 38.6860 1.79789 0.898946 0.438059i \(-0.144334\pi\)
0.898946 + 0.438059i \(0.144334\pi\)
\(464\) −4.64227 −0.215512
\(465\) 0 0
\(466\) 22.6195 1.04783
\(467\) −29.7638 −1.37731 −0.688653 0.725091i \(-0.741798\pi\)
−0.688653 + 0.725091i \(0.741798\pi\)
\(468\) 0 0
\(469\) 4.47834 0.206791
\(470\) 8.31996 0.383771
\(471\) 0 0
\(472\) 21.6272 0.995474
\(473\) 30.8161 1.41693
\(474\) 0 0
\(475\) 15.0591 0.690960
\(476\) 1.33687 0.0612752
\(477\) 0 0
\(478\) −5.42839 −0.248289
\(479\) 37.6759 1.72146 0.860728 0.509064i \(-0.170008\pi\)
0.860728 + 0.509064i \(0.170008\pi\)
\(480\) 0 0
\(481\) −0.135163 −0.00616289
\(482\) 4.51661 0.205726
\(483\) 0 0
\(484\) −4.49525 −0.204330
\(485\) −0.431074 −0.0195741
\(486\) 0 0
\(487\) −0.763823 −0.0346121 −0.0173061 0.999850i \(-0.505509\pi\)
−0.0173061 + 0.999850i \(0.505509\pi\)
\(488\) 32.5253 1.47235
\(489\) 0 0
\(490\) −2.63816 −0.119180
\(491\) 0.497941 0.0224717 0.0112359 0.999937i \(-0.496423\pi\)
0.0112359 + 0.999937i \(0.496423\pi\)
\(492\) 0 0
\(493\) −3.87258 −0.174412
\(494\) −28.8580 −1.29838
\(495\) 0 0
\(496\) 2.11650 0.0950335
\(497\) 13.2932 0.596283
\(498\) 0 0
\(499\) 8.96585 0.401367 0.200683 0.979656i \(-0.435684\pi\)
0.200683 + 0.979656i \(0.435684\pi\)
\(500\) −2.21987 −0.0992758
\(501\) 0 0
\(502\) −31.1908 −1.39211
\(503\) −18.3618 −0.818714 −0.409357 0.912374i \(-0.634247\pi\)
−0.409357 + 0.912374i \(0.634247\pi\)
\(504\) 0 0
\(505\) 18.2249 0.810999
\(506\) 23.5713 1.04787
\(507\) 0 0
\(508\) −0.664563 −0.0294852
\(509\) 28.3705 1.25750 0.628751 0.777607i \(-0.283567\pi\)
0.628751 + 0.777607i \(0.283567\pi\)
\(510\) 0 0
\(511\) 13.3942 0.592526
\(512\) 24.9186 1.10126
\(513\) 0 0
\(514\) 16.1848 0.713881
\(515\) 6.45666 0.284514
\(516\) 0 0
\(517\) −22.2080 −0.976707
\(518\) −0.136096 −0.00597973
\(519\) 0 0
\(520\) 15.6973 0.688371
\(521\) 32.6382 1.42990 0.714952 0.699174i \(-0.246449\pi\)
0.714952 + 0.699174i \(0.246449\pi\)
\(522\) 0 0
\(523\) −22.0232 −0.963008 −0.481504 0.876444i \(-0.659909\pi\)
−0.481504 + 0.876444i \(0.659909\pi\)
\(524\) −3.27219 −0.142946
\(525\) 0 0
\(526\) 22.7716 0.992887
\(527\) 1.76558 0.0769098
\(528\) 0 0
\(529\) −14.3354 −0.623280
\(530\) −25.9145 −1.12565
\(531\) 0 0
\(532\) 2.95811 0.128250
\(533\) −15.8161 −0.685073
\(534\) 0 0
\(535\) −4.36009 −0.188503
\(536\) 5.46648 0.236116
\(537\) 0 0
\(538\) −10.6631 −0.459720
\(539\) 7.04189 0.303316
\(540\) 0 0
\(541\) −15.7870 −0.678738 −0.339369 0.940653i \(-0.610214\pi\)
−0.339369 + 0.940653i \(0.610214\pi\)
\(542\) 23.2199 0.997379
\(543\) 0 0
\(544\) 3.12567 0.134012
\(545\) 14.8051 0.634181
\(546\) 0 0
\(547\) −27.4192 −1.17236 −0.586180 0.810180i \(-0.699369\pi\)
−0.586180 + 0.810180i \(0.699369\pi\)
\(548\) −0.726053 −0.0310154
\(549\) 0 0
\(550\) −18.1661 −0.774606
\(551\) −8.56893 −0.365048
\(552\) 0 0
\(553\) −9.11793 −0.387734
\(554\) −35.6154 −1.51315
\(555\) 0 0
\(556\) −2.21120 −0.0937758
\(557\) −29.4020 −1.24580 −0.622901 0.782301i \(-0.714046\pi\)
−0.622901 + 0.782301i \(0.714046\pi\)
\(558\) 0 0
\(559\) 16.7297 0.707590
\(560\) 14.3327 0.605669
\(561\) 0 0
\(562\) −25.5931 −1.07958
\(563\) 10.3705 0.437065 0.218533 0.975830i \(-0.429873\pi\)
0.218533 + 0.975830i \(0.429873\pi\)
\(564\) 0 0
\(565\) 26.3259 1.10754
\(566\) 22.3473 0.939327
\(567\) 0 0
\(568\) 16.2264 0.680843
\(569\) 32.9691 1.38214 0.691069 0.722788i \(-0.257140\pi\)
0.691069 + 0.722788i \(0.257140\pi\)
\(570\) 0 0
\(571\) −0.737415 −0.0308599 −0.0154299 0.999881i \(-0.504912\pi\)
−0.0154299 + 0.999881i \(0.504912\pi\)
\(572\) −3.54395 −0.148180
\(573\) 0 0
\(574\) −15.9254 −0.664713
\(575\) −6.67768 −0.278479
\(576\) 0 0
\(577\) 19.3432 0.805267 0.402634 0.915361i \(-0.368095\pi\)
0.402634 + 0.915361i \(0.368095\pi\)
\(578\) 10.7784 0.448321
\(579\) 0 0
\(580\) 0.394238 0.0163698
\(581\) −9.61081 −0.398724
\(582\) 0 0
\(583\) 69.1721 2.86482
\(584\) 16.3497 0.676554
\(585\) 0 0
\(586\) 26.0806 1.07738
\(587\) 31.9121 1.31715 0.658577 0.752514i \(-0.271159\pi\)
0.658577 + 0.752514i \(0.271159\pi\)
\(588\) 0 0
\(589\) 3.90673 0.160974
\(590\) 16.3601 0.673534
\(591\) 0 0
\(592\) −0.150644 −0.00619144
\(593\) −31.6783 −1.30087 −0.650436 0.759561i \(-0.725414\pi\)
−0.650436 + 0.759561i \(0.725414\pi\)
\(594\) 0 0
\(595\) 11.9564 0.490163
\(596\) 3.79292 0.155364
\(597\) 0 0
\(598\) 12.7965 0.523289
\(599\) 12.6236 0.515787 0.257893 0.966173i \(-0.416972\pi\)
0.257893 + 0.966173i \(0.416972\pi\)
\(600\) 0 0
\(601\) −8.90848 −0.363385 −0.181692 0.983355i \(-0.558157\pi\)
−0.181692 + 0.983355i \(0.558157\pi\)
\(602\) 16.8452 0.686561
\(603\) 0 0
\(604\) −2.95811 −0.120364
\(605\) −40.2036 −1.63451
\(606\) 0 0
\(607\) −33.1242 −1.34447 −0.672236 0.740337i \(-0.734666\pi\)
−0.672236 + 0.740337i \(0.734666\pi\)
\(608\) 6.91622 0.280490
\(609\) 0 0
\(610\) 24.6040 0.996187
\(611\) −12.0564 −0.487751
\(612\) 0 0
\(613\) 17.6800 0.714090 0.357045 0.934087i \(-0.383784\pi\)
0.357045 + 0.934087i \(0.383784\pi\)
\(614\) −28.0315 −1.13126
\(615\) 0 0
\(616\) −42.1893 −1.69986
\(617\) 25.7324 1.03595 0.517973 0.855397i \(-0.326687\pi\)
0.517973 + 0.855397i \(0.326687\pi\)
\(618\) 0 0
\(619\) 27.7948 1.11717 0.558583 0.829448i \(-0.311345\pi\)
0.558583 + 0.829448i \(0.311345\pi\)
\(620\) −0.179740 −0.00721854
\(621\) 0 0
\(622\) −14.3696 −0.576168
\(623\) −19.6551 −0.787464
\(624\) 0 0
\(625\) −8.51073 −0.340429
\(626\) −5.14022 −0.205444
\(627\) 0 0
\(628\) 4.06056 0.162034
\(629\) −0.125667 −0.00501068
\(630\) 0 0
\(631\) 26.8138 1.06744 0.533720 0.845661i \(-0.320794\pi\)
0.533720 + 0.845661i \(0.320794\pi\)
\(632\) −11.1298 −0.442719
\(633\) 0 0
\(634\) 35.5541 1.41203
\(635\) −5.94356 −0.235863
\(636\) 0 0
\(637\) 3.82295 0.151471
\(638\) 10.3369 0.409240
\(639\) 0 0
\(640\) 15.6973 0.620490
\(641\) −12.6905 −0.501244 −0.250622 0.968085i \(-0.580635\pi\)
−0.250622 + 0.968085i \(0.580635\pi\)
\(642\) 0 0
\(643\) 15.4766 0.610337 0.305168 0.952298i \(-0.401287\pi\)
0.305168 + 0.952298i \(0.401287\pi\)
\(644\) −1.31172 −0.0516889
\(645\) 0 0
\(646\) −26.8307 −1.05564
\(647\) −11.1506 −0.438377 −0.219189 0.975683i \(-0.570341\pi\)
−0.219189 + 0.975683i \(0.570341\pi\)
\(648\) 0 0
\(649\) −43.6691 −1.71416
\(650\) −9.86215 −0.386825
\(651\) 0 0
\(652\) −3.79023 −0.148437
\(653\) −44.5921 −1.74503 −0.872513 0.488591i \(-0.837511\pi\)
−0.872513 + 0.488591i \(0.837511\pi\)
\(654\) 0 0
\(655\) −29.2651 −1.14348
\(656\) −17.6277 −0.688247
\(657\) 0 0
\(658\) −12.1397 −0.473255
\(659\) 14.0966 0.549124 0.274562 0.961569i \(-0.411467\pi\)
0.274562 + 0.961569i \(0.411467\pi\)
\(660\) 0 0
\(661\) 36.1147 1.40470 0.702350 0.711831i \(-0.252134\pi\)
0.702350 + 0.711831i \(0.252134\pi\)
\(662\) 2.11743 0.0822962
\(663\) 0 0
\(664\) −11.7314 −0.455268
\(665\) 26.4561 1.02592
\(666\) 0 0
\(667\) 3.79973 0.147126
\(668\) 0.792919 0.0306789
\(669\) 0 0
\(670\) 4.13516 0.159755
\(671\) −65.6742 −2.53532
\(672\) 0 0
\(673\) 2.24216 0.0864290 0.0432145 0.999066i \(-0.486240\pi\)
0.0432145 + 0.999066i \(0.486240\pi\)
\(674\) 10.7992 0.415971
\(675\) 0 0
\(676\) 0.478340 0.0183977
\(677\) −35.1762 −1.35193 −0.675966 0.736933i \(-0.736273\pi\)
−0.675966 + 0.736933i \(0.736273\pi\)
\(678\) 0 0
\(679\) 0.628984 0.0241382
\(680\) 14.5945 0.559674
\(681\) 0 0
\(682\) −4.71276 −0.180461
\(683\) 17.7638 0.679714 0.339857 0.940477i \(-0.389621\pi\)
0.339857 + 0.940477i \(0.389621\pi\)
\(684\) 0 0
\(685\) −6.49350 −0.248104
\(686\) 26.5921 1.01529
\(687\) 0 0
\(688\) 18.6459 0.710868
\(689\) 37.5526 1.43064
\(690\) 0 0
\(691\) −44.0306 −1.67500 −0.837502 0.546434i \(-0.815985\pi\)
−0.837502 + 0.546434i \(0.815985\pi\)
\(692\) 0.700903 0.0266443
\(693\) 0 0
\(694\) 26.7374 1.01494
\(695\) −19.7760 −0.750147
\(696\) 0 0
\(697\) −14.7050 −0.556992
\(698\) −14.9459 −0.565712
\(699\) 0 0
\(700\) 1.01093 0.0382094
\(701\) 30.1052 1.13706 0.568530 0.822663i \(-0.307512\pi\)
0.568530 + 0.822663i \(0.307512\pi\)
\(702\) 0 0
\(703\) −0.278066 −0.0104875
\(704\) −51.0925 −1.92562
\(705\) 0 0
\(706\) −3.85853 −0.145218
\(707\) −26.5921 −1.00010
\(708\) 0 0
\(709\) −24.7374 −0.929033 −0.464517 0.885564i \(-0.653772\pi\)
−0.464517 + 0.885564i \(0.653772\pi\)
\(710\) 12.2746 0.460656
\(711\) 0 0
\(712\) −23.9919 −0.899136
\(713\) −1.73236 −0.0648775
\(714\) 0 0
\(715\) −31.6955 −1.18535
\(716\) 1.52940 0.0571564
\(717\) 0 0
\(718\) 38.7870 1.44752
\(719\) 43.5526 1.62424 0.812119 0.583491i \(-0.198314\pi\)
0.812119 + 0.583491i \(0.198314\pi\)
\(720\) 0 0
\(721\) −9.42097 −0.350855
\(722\) −33.7701 −1.25679
\(723\) 0 0
\(724\) −1.24216 −0.0461646
\(725\) −2.92841 −0.108758
\(726\) 0 0
\(727\) 20.5371 0.761680 0.380840 0.924641i \(-0.375635\pi\)
0.380840 + 0.924641i \(0.375635\pi\)
\(728\) −22.9040 −0.848880
\(729\) 0 0
\(730\) 12.3678 0.457754
\(731\) 15.5544 0.575299
\(732\) 0 0
\(733\) 14.0060 0.517323 0.258661 0.965968i \(-0.416719\pi\)
0.258661 + 0.965968i \(0.416719\pi\)
\(734\) 14.8098 0.546641
\(735\) 0 0
\(736\) −3.06687 −0.113046
\(737\) −11.0378 −0.406581
\(738\) 0 0
\(739\) −41.9813 −1.54431 −0.772154 0.635435i \(-0.780821\pi\)
−0.772154 + 0.635435i \(0.780821\pi\)
\(740\) 0.0127932 0.000470288 0
\(741\) 0 0
\(742\) 37.8120 1.38812
\(743\) −27.8621 −1.02216 −0.511082 0.859532i \(-0.670755\pi\)
−0.511082 + 0.859532i \(0.670755\pi\)
\(744\) 0 0
\(745\) 33.9222 1.24281
\(746\) −45.0128 −1.64804
\(747\) 0 0
\(748\) −3.29498 −0.120476
\(749\) 6.36184 0.232457
\(750\) 0 0
\(751\) −52.9050 −1.93053 −0.965265 0.261273i \(-0.915858\pi\)
−0.965265 + 0.261273i \(0.915858\pi\)
\(752\) −13.4374 −0.490011
\(753\) 0 0
\(754\) 5.61175 0.204368
\(755\) −26.4561 −0.962834
\(756\) 0 0
\(757\) −41.4858 −1.50783 −0.753913 0.656975i \(-0.771836\pi\)
−0.753913 + 0.656975i \(0.771836\pi\)
\(758\) −28.2116 −1.02469
\(759\) 0 0
\(760\) 32.2935 1.17141
\(761\) 45.3874 1.64529 0.822647 0.568553i \(-0.192497\pi\)
0.822647 + 0.568553i \(0.192497\pi\)
\(762\) 0 0
\(763\) −21.6023 −0.782054
\(764\) −0.926651 −0.0335251
\(765\) 0 0
\(766\) −5.53890 −0.200128
\(767\) −23.7074 −0.856024
\(768\) 0 0
\(769\) 5.11650 0.184506 0.0922528 0.995736i \(-0.470593\pi\)
0.0922528 + 0.995736i \(0.470593\pi\)
\(770\) −31.9145 −1.15012
\(771\) 0 0
\(772\) 3.30129 0.118816
\(773\) 52.6427 1.89343 0.946713 0.322077i \(-0.104381\pi\)
0.946713 + 0.322077i \(0.104381\pi\)
\(774\) 0 0
\(775\) 1.33511 0.0479587
\(776\) 0.767769 0.0275613
\(777\) 0 0
\(778\) 23.0310 0.825700
\(779\) −32.5381 −1.16580
\(780\) 0 0
\(781\) −32.7638 −1.17238
\(782\) 11.8976 0.425456
\(783\) 0 0
\(784\) 4.26083 0.152172
\(785\) 36.3158 1.29617
\(786\) 0 0
\(787\) −20.7624 −0.740099 −0.370050 0.929012i \(-0.620659\pi\)
−0.370050 + 0.929012i \(0.620659\pi\)
\(788\) 0.133732 0.00476401
\(789\) 0 0
\(790\) −8.41921 −0.299542
\(791\) −38.4124 −1.36579
\(792\) 0 0
\(793\) −35.6536 −1.26610
\(794\) −30.1821 −1.07112
\(795\) 0 0
\(796\) 1.88350 0.0667590
\(797\) 45.5800 1.61453 0.807263 0.590192i \(-0.200948\pi\)
0.807263 + 0.590192i \(0.200948\pi\)
\(798\) 0 0
\(799\) −11.2094 −0.396562
\(800\) 2.36360 0.0835658
\(801\) 0 0
\(802\) −19.6459 −0.693721
\(803\) −33.0128 −1.16500
\(804\) 0 0
\(805\) −11.7314 −0.413479
\(806\) −2.55850 −0.0901192
\(807\) 0 0
\(808\) −32.4597 −1.14193
\(809\) 4.21120 0.148058 0.0740290 0.997256i \(-0.476414\pi\)
0.0740290 + 0.997256i \(0.476414\pi\)
\(810\) 0 0
\(811\) 11.3618 0.398968 0.199484 0.979901i \(-0.436073\pi\)
0.199484 + 0.979901i \(0.436073\pi\)
\(812\) −0.575236 −0.0201868
\(813\) 0 0
\(814\) 0.335437 0.0117571
\(815\) −33.8982 −1.18740
\(816\) 0 0
\(817\) 34.4175 1.20411
\(818\) 23.5817 0.824515
\(819\) 0 0
\(820\) 1.49701 0.0522778
\(821\) −1.64227 −0.0573158 −0.0286579 0.999589i \(-0.509123\pi\)
−0.0286579 + 0.999589i \(0.509123\pi\)
\(822\) 0 0
\(823\) 11.2189 0.391068 0.195534 0.980697i \(-0.437356\pi\)
0.195534 + 0.980697i \(0.437356\pi\)
\(824\) −11.4997 −0.400611
\(825\) 0 0
\(826\) −23.8711 −0.830583
\(827\) 9.61318 0.334283 0.167141 0.985933i \(-0.446546\pi\)
0.167141 + 0.985933i \(0.446546\pi\)
\(828\) 0 0
\(829\) 33.4938 1.16329 0.581644 0.813443i \(-0.302410\pi\)
0.581644 + 0.813443i \(0.302410\pi\)
\(830\) −8.87433 −0.308033
\(831\) 0 0
\(832\) −27.7374 −0.961622
\(833\) 3.55438 0.123152
\(834\) 0 0
\(835\) 7.09152 0.245412
\(836\) −7.29086 −0.252160
\(837\) 0 0
\(838\) 25.4129 0.877874
\(839\) 31.9941 1.10456 0.552280 0.833659i \(-0.313758\pi\)
0.552280 + 0.833659i \(0.313758\pi\)
\(840\) 0 0
\(841\) −27.3337 −0.942541
\(842\) 43.5631 1.50128
\(843\) 0 0
\(844\) 2.74691 0.0945526
\(845\) 4.27807 0.147170
\(846\) 0 0
\(847\) 58.6614 2.01563
\(848\) 41.8539 1.43727
\(849\) 0 0
\(850\) −9.16931 −0.314505
\(851\) 0.123303 0.00422678
\(852\) 0 0
\(853\) 35.2422 1.20667 0.603334 0.797488i \(-0.293838\pi\)
0.603334 + 0.797488i \(0.293838\pi\)
\(854\) −35.8999 −1.22847
\(855\) 0 0
\(856\) 7.76558 0.265422
\(857\) 21.2490 0.725851 0.362925 0.931818i \(-0.381778\pi\)
0.362925 + 0.931818i \(0.381778\pi\)
\(858\) 0 0
\(859\) 51.8384 1.76870 0.884352 0.466820i \(-0.154601\pi\)
0.884352 + 0.466820i \(0.154601\pi\)
\(860\) −1.58347 −0.0539960
\(861\) 0 0
\(862\) 46.2344 1.57475
\(863\) 22.6783 0.771978 0.385989 0.922503i \(-0.373860\pi\)
0.385989 + 0.922503i \(0.373860\pi\)
\(864\) 0 0
\(865\) 6.26857 0.213138
\(866\) 33.7725 1.14764
\(867\) 0 0
\(868\) 0.262260 0.00890170
\(869\) 22.4730 0.762343
\(870\) 0 0
\(871\) −5.99226 −0.203040
\(872\) −26.3688 −0.892959
\(873\) 0 0
\(874\) 26.3259 0.890488
\(875\) 28.9685 0.979315
\(876\) 0 0
\(877\) 1.13341 0.0382725 0.0191362 0.999817i \(-0.493908\pi\)
0.0191362 + 0.999817i \(0.493908\pi\)
\(878\) 31.2681 1.05525
\(879\) 0 0
\(880\) −35.3259 −1.19084
\(881\) 30.8289 1.03865 0.519327 0.854576i \(-0.326183\pi\)
0.519327 + 0.854576i \(0.326183\pi\)
\(882\) 0 0
\(883\) −9.33511 −0.314152 −0.157076 0.987587i \(-0.550207\pi\)
−0.157076 + 0.987587i \(0.550207\pi\)
\(884\) −1.78880 −0.0601639
\(885\) 0 0
\(886\) −5.55295 −0.186555
\(887\) 14.0942 0.473237 0.236619 0.971603i \(-0.423961\pi\)
0.236619 + 0.971603i \(0.423961\pi\)
\(888\) 0 0
\(889\) 8.67230 0.290860
\(890\) −18.1489 −0.608352
\(891\) 0 0
\(892\) 2.02322 0.0677425
\(893\) −24.8033 −0.830012
\(894\) 0 0
\(895\) 13.6783 0.457215
\(896\) −22.9040 −0.765170
\(897\) 0 0
\(898\) 24.7113 0.824628
\(899\) −0.759704 −0.0253376
\(900\) 0 0
\(901\) 34.9145 1.16317
\(902\) 39.2513 1.30693
\(903\) 0 0
\(904\) −46.8881 −1.55947
\(905\) −11.1094 −0.369288
\(906\) 0 0
\(907\) −8.73917 −0.290179 −0.145090 0.989419i \(-0.546347\pi\)
−0.145090 + 0.989419i \(0.546347\pi\)
\(908\) −3.20296 −0.106294
\(909\) 0 0
\(910\) −17.3259 −0.574349
\(911\) 20.6509 0.684196 0.342098 0.939664i \(-0.388862\pi\)
0.342098 + 0.939664i \(0.388862\pi\)
\(912\) 0 0
\(913\) 23.6878 0.783951
\(914\) 26.2199 0.867276
\(915\) 0 0
\(916\) −0.288668 −0.00953785
\(917\) 42.7009 1.41011
\(918\) 0 0
\(919\) −2.36009 −0.0778522 −0.0389261 0.999242i \(-0.512394\pi\)
−0.0389261 + 0.999242i \(0.512394\pi\)
\(920\) −14.3200 −0.472115
\(921\) 0 0
\(922\) 37.3865 1.23126
\(923\) −17.7870 −0.585468
\(924\) 0 0
\(925\) −0.0950283 −0.00312451
\(926\) −52.1215 −1.71282
\(927\) 0 0
\(928\) −1.34493 −0.0441496
\(929\) 26.8203 0.879944 0.439972 0.898011i \(-0.354988\pi\)
0.439972 + 0.898011i \(0.354988\pi\)
\(930\) 0 0
\(931\) 7.86484 0.257760
\(932\) 3.10244 0.101624
\(933\) 0 0
\(934\) 40.1007 1.31213
\(935\) −29.4688 −0.963734
\(936\) 0 0
\(937\) 1.93313 0.0631527 0.0315764 0.999501i \(-0.489947\pi\)
0.0315764 + 0.999501i \(0.489947\pi\)
\(938\) −6.03365 −0.197006
\(939\) 0 0
\(940\) 1.14115 0.0372202
\(941\) −11.9103 −0.388266 −0.194133 0.980975i \(-0.562189\pi\)
−0.194133 + 0.980975i \(0.562189\pi\)
\(942\) 0 0
\(943\) 14.4284 0.469853
\(944\) −26.4228 −0.859990
\(945\) 0 0
\(946\) −41.5185 −1.34988
\(947\) 0.315836 0.0102633 0.00513165 0.999987i \(-0.498367\pi\)
0.00513165 + 0.999987i \(0.498367\pi\)
\(948\) 0 0
\(949\) −17.9222 −0.581779
\(950\) −20.2891 −0.658265
\(951\) 0 0
\(952\) −21.2950 −0.690174
\(953\) −3.25133 −0.105321 −0.0526605 0.998612i \(-0.516770\pi\)
−0.0526605 + 0.998612i \(0.516770\pi\)
\(954\) 0 0
\(955\) −8.28756 −0.268179
\(956\) −0.744547 −0.0240804
\(957\) 0 0
\(958\) −50.7606 −1.64000
\(959\) 9.47472 0.305955
\(960\) 0 0
\(961\) −30.6536 −0.988827
\(962\) 0.182104 0.00587127
\(963\) 0 0
\(964\) 0.619489 0.0199524
\(965\) 29.5253 0.950452
\(966\) 0 0
\(967\) 10.9676 0.352694 0.176347 0.984328i \(-0.443572\pi\)
0.176347 + 0.984328i \(0.443572\pi\)
\(968\) 71.6049 2.30147
\(969\) 0 0
\(970\) 0.580785 0.0186479
\(971\) −23.3868 −0.750519 −0.375259 0.926920i \(-0.622446\pi\)
−0.375259 + 0.926920i \(0.622446\pi\)
\(972\) 0 0
\(973\) 28.8553 0.925060
\(974\) 1.02910 0.0329744
\(975\) 0 0
\(976\) −39.7374 −1.27196
\(977\) 50.1762 1.60528 0.802640 0.596464i \(-0.203428\pi\)
0.802640 + 0.596464i \(0.203428\pi\)
\(978\) 0 0
\(979\) 48.4439 1.54827
\(980\) −0.361844 −0.0115587
\(981\) 0 0
\(982\) −0.670874 −0.0214084
\(983\) 14.6946 0.468685 0.234342 0.972154i \(-0.424706\pi\)
0.234342 + 0.972154i \(0.424706\pi\)
\(984\) 0 0
\(985\) 1.19604 0.0381091
\(986\) 5.21751 0.166159
\(987\) 0 0
\(988\) −3.95811 −0.125924
\(989\) −15.2618 −0.485296
\(990\) 0 0
\(991\) 2.00000 0.0635321 0.0317660 0.999495i \(-0.489887\pi\)
0.0317660 + 0.999495i \(0.489887\pi\)
\(992\) 0.613179 0.0194684
\(993\) 0 0
\(994\) −17.9099 −0.568068
\(995\) 16.8452 0.534030
\(996\) 0 0
\(997\) −45.8863 −1.45323 −0.726617 0.687043i \(-0.758908\pi\)
−0.726617 + 0.687043i \(0.758908\pi\)
\(998\) −12.0797 −0.382375
\(999\) 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 243.2.a.e.1.2 3
3.2 odd 2 243.2.a.f.1.2 yes 3
4.3 odd 2 3888.2.a.bd.1.2 3
5.4 even 2 6075.2.a.bv.1.2 3
9.2 odd 6 243.2.c.e.82.2 6
9.4 even 3 243.2.c.f.163.2 6
9.5 odd 6 243.2.c.e.163.2 6
9.7 even 3 243.2.c.f.82.2 6
12.11 even 2 3888.2.a.bk.1.2 3
15.14 odd 2 6075.2.a.bq.1.2 3
27.2 odd 18 729.2.e.b.568.1 6
27.4 even 9 729.2.e.h.406.1 6
27.5 odd 18 729.2.e.i.649.1 6
27.7 even 9 729.2.e.h.325.1 6
27.11 odd 18 729.2.e.i.82.1 6
27.13 even 9 729.2.e.g.163.1 6
27.14 odd 18 729.2.e.b.163.1 6
27.16 even 9 729.2.e.a.82.1 6
27.20 odd 18 729.2.e.c.325.1 6
27.22 even 9 729.2.e.a.649.1 6
27.23 odd 18 729.2.e.c.406.1 6
27.25 even 9 729.2.e.g.568.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
243.2.a.e.1.2 3 1.1 even 1 trivial
243.2.a.f.1.2 yes 3 3.2 odd 2
243.2.c.e.82.2 6 9.2 odd 6
243.2.c.e.163.2 6 9.5 odd 6
243.2.c.f.82.2 6 9.7 even 3
243.2.c.f.163.2 6 9.4 even 3
729.2.e.a.82.1 6 27.16 even 9
729.2.e.a.649.1 6 27.22 even 9
729.2.e.b.163.1 6 27.14 odd 18
729.2.e.b.568.1 6 27.2 odd 18
729.2.e.c.325.1 6 27.20 odd 18
729.2.e.c.406.1 6 27.23 odd 18
729.2.e.g.163.1 6 27.13 even 9
729.2.e.g.568.1 6 27.25 even 9
729.2.e.h.325.1 6 27.7 even 9
729.2.e.h.406.1 6 27.4 even 9
729.2.e.i.82.1 6 27.11 odd 18
729.2.e.i.649.1 6 27.5 odd 18
3888.2.a.bd.1.2 3 4.3 odd 2
3888.2.a.bk.1.2 3 12.11 even 2
6075.2.a.bq.1.2 3 15.14 odd 2
6075.2.a.bv.1.2 3 5.4 even 2