Properties

Label 6075.2.a.bv.1.2
Level $6075$
Weight $2$
Character 6075.1
Self dual yes
Analytic conductor $48.509$
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] = [6075,2,Mod(1,6075)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6075, 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("6075.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6075 = 3^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6075.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: \(48.5091192279\)
Analytic rank: \(0\)
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: no (minimal twist has level 243)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.347296\) of defining polynomial
Character \(\chi\) \(=\) 6075.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.34730 q^{2} -0.184793 q^{4} -2.41147 q^{7} -2.94356 q^{8} +O(q^{10})\) \(q+1.34730 q^{2} -0.184793 q^{4} -2.41147 q^{7} -2.94356 q^{8} -5.94356 q^{11} +3.22668 q^{13} -3.24897 q^{14} -3.59627 q^{16} +3.00000 q^{17} -6.63816 q^{19} -8.00774 q^{22} -2.94356 q^{23} +4.34730 q^{26} +0.445622 q^{28} +1.29086 q^{29} -0.588526 q^{31} +1.04189 q^{32} +4.04189 q^{34} -0.0418891 q^{37} -8.94356 q^{38} +4.90167 q^{41} +5.18479 q^{43} +1.09833 q^{44} -3.96585 q^{46} -3.73648 q^{47} -1.18479 q^{49} -0.596267 q^{52} +11.6382 q^{53} +7.09833 q^{56} +1.73917 q^{58} +7.34730 q^{59} +11.0496 q^{61} -0.792919 q^{62} +8.59627 q^{64} -1.85710 q^{67} -0.554378 q^{68} +5.51249 q^{71} -5.55438 q^{73} -0.0564370 q^{74} +1.22668 q^{76} +14.3327 q^{77} -3.78106 q^{79} +6.60401 q^{82} +3.98545 q^{83} +6.98545 q^{86} +17.4953 q^{88} -8.15064 q^{89} -7.78106 q^{91} +0.543948 q^{92} -5.03415 q^{94} -0.260830 q^{97} -1.59627 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{7} + 6 q^{8} - 3 q^{11} + 3 q^{13} + 3 q^{14} + 3 q^{16} + 9 q^{17} - 3 q^{19} + 6 q^{23} + 12 q^{26} + 12 q^{28} - 12 q^{29} - 12 q^{31} + 9 q^{34} + 3 q^{37} - 12 q^{38} + 3 q^{41} + 12 q^{43} + 15 q^{44} + 9 q^{46} - 6 q^{47} + 12 q^{52} + 18 q^{53} + 33 q^{56} - 9 q^{58} + 21 q^{59} + 6 q^{61} - 12 q^{62} + 12 q^{64} - 6 q^{67} + 9 q^{68} + 9 q^{71} - 6 q^{73} - 15 q^{74} - 3 q^{76} + 24 q^{77} + 6 q^{79} - 18 q^{82} - 6 q^{83} + 3 q^{86} + 36 q^{88} - 6 q^{91} + 24 q^{92} - 36 q^{94} - 15 q^{97} + 9 q^{98}+O(q^{100}) \) 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 \(-0.341963\pi\)
0.476341 + 0.879261i \(0.341963\pi\)
\(3\) 0 0
\(4\) −0.184793 −0.0923963
\(5\) 0 0
\(6\) 0 0
\(7\) −2.41147 −0.911452 −0.455726 0.890120i \(-0.650620\pi\)
−0.455726 + 0.890120i \(0.650620\pi\)
\(8\) −2.94356 −1.04071
\(9\) 0 0
\(10\) 0 0
\(11\) −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 \(-0.352329\pi\)
0.447460 + 0.894304i \(0.352329\pi\)
\(14\) −3.24897 −0.868324
\(15\) 0 0
\(16\) −3.59627 −0.899067
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) −6.63816 −1.52290 −0.761449 0.648225i \(-0.775512\pi\)
−0.761449 + 0.648225i \(0.775512\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −8.00774 −1.70726
\(23\) −2.94356 −0.613775 −0.306888 0.951746i \(-0.599288\pi\)
−0.306888 + 0.951746i \(0.599288\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 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
\(35\) 0 0
\(36\) 0 0
\(37\) −0.0418891 −0.00688652 −0.00344326 0.999994i \(-0.501096\pi\)
−0.00344326 + 0.999994i \(0.501096\pi\)
\(38\) −8.94356 −1.45084
\(39\) 0 0
\(40\) 0 0
\(41\) 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 \(-0.370628\pi\)
0.395337 + 0.918536i \(0.370628\pi\)
\(44\) 1.09833 0.165579
\(45\) 0 0
\(46\) −3.96585 −0.584733
\(47\) −3.73648 −0.545022 −0.272511 0.962153i \(-0.587854\pi\)
−0.272511 + 0.962153i \(0.587854\pi\)
\(48\) 0 0
\(49\) −1.18479 −0.169256
\(50\) 0 0
\(51\) 0 0
\(52\) −0.596267 −0.0826873
\(53\) 11.6382 1.59862 0.799312 0.600916i \(-0.205198\pi\)
0.799312 + 0.600916i \(0.205198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 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
\(65\) 0 0
\(66\) 0 0
\(67\) −1.85710 −0.226880 −0.113440 0.993545i \(-0.536187\pi\)
−0.113440 + 0.993545i \(0.536187\pi\)
\(68\) −0.554378 −0.0672282
\(69\) 0 0
\(70\) 0 0
\(71\) 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 \(-0.605380\pi\)
−0.325045 + 0.945698i \(0.605380\pi\)
\(74\) −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 \(0.568226\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 6.60401 0.729291
\(83\) 3.98545 0.437460 0.218730 0.975785i \(-0.429809\pi\)
0.218730 + 0.975785i \(0.429809\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 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
\(95\) 0 0
\(96\) 0 0
\(97\) −0.260830 −0.0264833 −0.0132416 0.999912i \(-0.504215\pi\)
−0.0132416 + 0.999912i \(0.504215\pi\)
\(98\) −1.59627 −0.161247
\(99\) 0 0
\(100\) 0 0
\(101\) −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 \(-0.438350\pi\)
0.192471 + 0.981303i \(0.438350\pi\)
\(104\) −9.49794 −0.931350
\(105\) 0 0
\(106\) 15.6800 1.52298
\(107\) −2.63816 −0.255040 −0.127520 0.991836i \(-0.540702\pi\)
−0.127520 + 0.991836i \(0.540702\pi\)
\(108\) 0 0
\(109\) −8.95811 −0.858031 −0.429016 0.903297i \(-0.641140\pi\)
−0.429016 + 0.903297i \(0.641140\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 8.67230 0.819456
\(113\) 15.9290 1.49848 0.749238 0.662301i \(-0.230420\pi\)
0.749238 + 0.662301i \(0.230420\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −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
\(125\) 0 0
\(126\) 0 0
\(127\) −3.59627 −0.319117 −0.159559 0.987188i \(-0.551007\pi\)
−0.159559 + 0.987188i \(0.551007\pi\)
\(128\) 9.49794 0.839507
\(129\) 0 0
\(130\) 0 0
\(131\) 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 \(-0.553679\pi\)
−0.167839 + 0.985814i \(0.553679\pi\)
\(138\) 0 0
\(139\) 11.9659 1.01493 0.507465 0.861672i \(-0.330583\pi\)
0.507465 + 0.861672i \(0.330583\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 7.42696 0.623256
\(143\) −19.1780 −1.60374
\(144\) 0 0
\(145\) 0 0
\(146\) −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 0
\(156\) 0 0
\(157\) 21.9736 1.75368 0.876842 0.480779i \(-0.159646\pi\)
0.876842 + 0.480779i \(0.159646\pi\)
\(158\) −5.09421 −0.405273
\(159\) 0 0
\(160\) 0 0
\(161\) 7.09833 0.559426
\(162\) 0 0
\(163\) −20.5107 −1.60652 −0.803262 0.595625i \(-0.796904\pi\)
−0.803262 + 0.595625i \(0.796904\pi\)
\(164\) −0.905793 −0.0707305
\(165\) 0 0
\(166\) 5.36959 0.416761
\(167\) 4.29086 0.332037 0.166018 0.986123i \(-0.446909\pi\)
0.166018 + 0.986123i \(0.446909\pi\)
\(168\) 0 0
\(169\) −2.58853 −0.199117
\(170\) 0 0
\(171\) 0 0
\(172\) −0.958111 −0.0730553
\(173\) 3.79292 0.288370 0.144185 0.989551i \(-0.453944\pi\)
0.144185 + 0.989551i \(0.453944\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 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 0
\(186\) 0 0
\(187\) −17.8307 −1.30391
\(188\) 0.690474 0.0503580
\(189\) 0 0
\(190\) 0 0
\(191\) 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 \(-0.277702\pi\)
0.642970 + 0.765892i \(0.277702\pi\)
\(194\) −0.351415 −0.0252301
\(195\) 0 0
\(196\) 0.218941 0.0156386
\(197\) 0.723689 0.0515607 0.0257803 0.999668i \(-0.491793\pi\)
0.0257803 + 0.999668i \(0.491793\pi\)
\(198\) 0 0
\(199\) −10.1925 −0.722530 −0.361265 0.932463i \(-0.617655\pi\)
−0.361265 + 0.932463i \(0.617655\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −14.8571 −1.04534
\(203\) −3.11287 −0.218481
\(204\) 0 0
\(205\) 0 0
\(206\) 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
\(215\) 0 0
\(216\) 0 0
\(217\) 1.41921 0.0963426
\(218\) −12.0692 −0.817431
\(219\) 0 0
\(220\) 0 0
\(221\) 9.68004 0.651150
\(222\) 0 0
\(223\) 10.9486 0.733174 0.366587 0.930384i \(-0.380526\pi\)
0.366587 + 0.930384i \(0.380526\pi\)
\(224\) −2.51249 −0.167873
\(225\) 0 0
\(226\) 21.4611 1.42757
\(227\) −17.3327 −1.15041 −0.575207 0.818008i \(-0.695079\pi\)
−0.575207 + 0.818008i \(0.695079\pi\)
\(228\) 0 0
\(229\) 1.56212 0.103228 0.0516138 0.998667i \(-0.483563\pi\)
0.0516138 + 0.998667i \(0.483563\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −3.79973 −0.249464
\(233\) 16.7888 1.09987 0.549935 0.835207i \(-0.314652\pi\)
0.549935 + 0.835207i \(0.314652\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −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
\(245\) 0 0
\(246\) 0 0
\(247\) −21.4192 −1.36287
\(248\) 1.73236 0.110005
\(249\) 0 0
\(250\) 0 0
\(251\) 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 \(-0.377756\pi\)
0.374669 + 0.927159i \(0.377756\pi\)
\(258\) 0 0
\(259\) 0.101014 0.00627673
\(260\) 0 0
\(261\) 0 0
\(262\) 23.8571 1.47390
\(263\) 16.9017 1.04220 0.521101 0.853495i \(-0.325522\pi\)
0.521101 + 0.853495i \(0.325522\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 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
\(275\) 0 0
\(276\) 0 0
\(277\) −26.4347 −1.58831 −0.794153 0.607717i \(-0.792085\pi\)
−0.794153 + 0.607717i \(0.792085\pi\)
\(278\) 16.1215 0.966906
\(279\) 0 0
\(280\) 0 0
\(281\) 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 \(-0.335904\pi\)
0.492991 + 0.870035i \(0.335904\pi\)
\(284\) −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
\(290\) 0 0
\(291\) 0 0
\(292\) 1.02641 0.0600660
\(293\) 19.3577 1.13089 0.565445 0.824786i \(-0.308704\pi\)
0.565445 + 0.824786i \(0.308704\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 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
\(305\) 0 0
\(306\) 0 0
\(307\) −20.8057 −1.18744 −0.593722 0.804670i \(-0.702342\pi\)
−0.593722 + 0.804670i \(0.702342\pi\)
\(308\) −2.64858 −0.150917
\(309\) 0 0
\(310\) 0 0
\(311\) 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 \(-0.534388\pi\)
−0.107824 + 0.994170i \(0.534388\pi\)
\(314\) 29.6049 1.67070
\(315\) 0 0
\(316\) 0.698711 0.0393056
\(317\) 26.3892 1.48216 0.741082 0.671414i \(-0.234313\pi\)
0.741082 + 0.671414i \(0.234313\pi\)
\(318\) 0 0
\(319\) −7.67230 −0.429567
\(320\) 0 0
\(321\) 0 0
\(322\) 9.56355 0.532956
\(323\) −19.9145 −1.10807
\(324\) 0 0
\(325\) 0 0
\(326\) −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
\(335\) 0 0
\(336\) 0 0
\(337\) 8.01548 0.436631 0.218316 0.975878i \(-0.429944\pi\)
0.218316 + 0.975878i \(0.429944\pi\)
\(338\) −3.48751 −0.189696
\(339\) 0 0
\(340\) 0 0
\(341\) 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 \(-0.321187\pi\)
0.532674 + 0.846320i \(0.321187\pi\)
\(348\) 0 0
\(349\) 11.0933 0.593809 0.296905 0.954907i \(-0.404046\pi\)
0.296905 + 0.954907i \(0.404046\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −6.19253 −0.330063
\(353\) −2.86390 −0.152430 −0.0762151 0.997091i \(-0.524284\pi\)
−0.0762151 + 0.997091i \(0.524284\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 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
\(365\) 0 0
\(366\) 0 0
\(367\) 10.9923 0.573791 0.286896 0.957962i \(-0.407377\pi\)
0.286896 + 0.957962i \(0.407377\pi\)
\(368\) 10.5858 0.551825
\(369\) 0 0
\(370\) 0 0
\(371\) −28.0651 −1.45707
\(372\) 0 0
\(373\) −33.4097 −1.72989 −0.864945 0.501867i \(-0.832647\pi\)
−0.864945 + 0.501867i \(0.832647\pi\)
\(374\) −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 \(0.319258\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 6.75608 0.345671
\(383\) −4.11112 −0.210068 −0.105034 0.994469i \(-0.533495\pi\)
−0.105034 + 0.994469i \(0.533495\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 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
\(395\) 0 0
\(396\) 0 0
\(397\) −22.4020 −1.12432 −0.562162 0.827027i \(-0.690030\pi\)
−0.562162 + 0.827027i \(0.690030\pi\)
\(398\) −13.7324 −0.688341
\(399\) 0 0
\(400\) 0 0
\(401\) 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 \(0.642451\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −0.721934 −0.0355671
\(413\) −17.7178 −0.871837
\(414\) 0 0
\(415\) 0 0
\(416\) 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
\(425\) 0 0
\(426\) 0 0
\(427\) −26.6459 −1.28949
\(428\) 0.487511 0.0235648
\(429\) 0 0
\(430\) 0 0
\(431\) −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 \(-0.294244\pi\)
0.602318 + 0.798256i \(0.294244\pi\)
\(434\) 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 \(0.686834\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 13.0419 0.620339
\(443\) −4.12155 −0.195821 −0.0979103 0.995195i \(-0.531216\pi\)
−0.0979103 + 0.995195i \(0.531216\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 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
\(455\) 0 0
\(456\) 0 0
\(457\) 19.4611 0.910352 0.455176 0.890401i \(-0.349576\pi\)
0.455176 + 0.890401i \(0.349576\pi\)
\(458\) 2.10464 0.0983432
\(459\) 0 0
\(460\) 0 0
\(461\) −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 \(-0.855666\pi\)
−0.898946 + 0.438059i \(0.855666\pi\)
\(464\) −4.64227 −0.215512
\(465\) 0 0
\(466\) 22.6195 1.04783
\(467\) 29.7638 1.37731 0.688653 0.725091i \(-0.258202\pi\)
0.688653 + 0.725091i \(0.258202\pi\)
\(468\) 0 0
\(469\) 4.47834 0.206791
\(470\) 0 0
\(471\) 0 0
\(472\) −21.6272 −0.995474
\(473\) −30.8161 −1.41693
\(474\) 0 0
\(475\) 0 0
\(476\) 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 0
\(486\) 0 0
\(487\) 0.763823 0.0346121 0.0173061 0.999850i \(-0.494491\pi\)
0.0173061 + 0.999850i \(0.494491\pi\)
\(488\) −32.5253 −1.47235
\(489\) 0 0
\(490\) 0 0
\(491\) 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 \(0.435684\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 31.1908 1.39211
\(503\) 18.3618 0.818714 0.409357 0.912374i \(-0.365753\pi\)
0.409357 + 0.912374i \(0.365753\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 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
\(515\) 0 0
\(516\) 0 0
\(517\) 22.2080 0.976707
\(518\) 0.136096 0.00597973
\(519\) 0 0
\(520\) 0 0
\(521\) 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 \(-0.340091\pi\)
0.481504 + 0.876444i \(0.340091\pi\)
\(524\) −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
\(530\) 0 0
\(531\) 0 0
\(532\) −2.95811 −0.128250
\(533\) 15.8161 0.685073
\(534\) 0 0
\(535\) 0 0
\(536\) 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
\(545\) 0 0
\(546\) 0 0
\(547\) 27.4192 1.17236 0.586180 0.810180i \(-0.300631\pi\)
0.586180 + 0.810180i \(0.300631\pi\)
\(548\) 0.726053 0.0310154
\(549\) 0 0
\(550\) 0 0
\(551\) −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 \(-0.285954\pi\)
0.622901 + 0.782301i \(0.285954\pi\)
\(558\) 0 0
\(559\) 16.7297 0.707590
\(560\) 0 0
\(561\) 0 0
\(562\) 25.5931 1.07958
\(563\) −10.3705 −0.437065 −0.218533 0.975830i \(-0.570127\pi\)
−0.218533 + 0.975830i \(0.570127\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 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
\(575\) 0 0
\(576\) 0 0
\(577\) −19.3432 −0.805267 −0.402634 0.915361i \(-0.631905\pi\)
−0.402634 + 0.915361i \(0.631905\pi\)
\(578\) −10.7784 −0.448321
\(579\) 0 0
\(580\) 0 0
\(581\) −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 \(-0.728841\pi\)
−0.658577 + 0.752514i \(0.728841\pi\)
\(588\) 0 0
\(589\) 3.90673 0.160974
\(590\) 0 0
\(591\) 0 0
\(592\) 0.150644 0.00619144
\(593\) 31.6783 1.30087 0.650436 0.759561i \(-0.274586\pi\)
0.650436 + 0.759561i \(0.274586\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 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
\(605\) 0 0
\(606\) 0 0
\(607\) 33.1242 1.34447 0.672236 0.740337i \(-0.265334\pi\)
0.672236 + 0.740337i \(0.265334\pi\)
\(608\) −6.91622 −0.280490
\(609\) 0 0
\(610\) 0 0
\(611\) −12.0564 −0.487751
\(612\) 0 0
\(613\) −17.6800 −0.714090 −0.357045 0.934087i \(-0.616216\pi\)
−0.357045 + 0.934087i \(0.616216\pi\)
\(614\) −28.0315 −1.13126
\(615\) 0 0
\(616\) −42.1893 −1.69986
\(617\) −25.7324 −1.03595 −0.517973 0.855397i \(-0.673313\pi\)
−0.517973 + 0.855397i \(0.673313\pi\)
\(618\) 0 0
\(619\) 27.7948 1.11717 0.558583 0.829448i \(-0.311345\pi\)
0.558583 + 0.829448i \(0.311345\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 14.3696 0.576168
\(623\) 19.6551 0.787464
\(624\) 0 0
\(625\) 0 0
\(626\) −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
\(635\) 0 0
\(636\) 0 0
\(637\) −3.82295 −0.151471
\(638\) −10.3369 −0.409240
\(639\) 0 0
\(640\) 0 0
\(641\) −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 \(-0.598713\pi\)
−0.305168 + 0.952298i \(0.598713\pi\)
\(644\) −1.31172 −0.0516889
\(645\) 0 0
\(646\) −26.8307 −1.05564
\(647\) 11.1506 0.438377 0.219189 0.975683i \(-0.429659\pi\)
0.219189 + 0.975683i \(0.429659\pi\)
\(648\) 0 0
\(649\) −43.6691 −1.71416
\(650\) 0 0
\(651\) 0 0
\(652\) 3.79023 0.148437
\(653\) 44.5921 1.74503 0.872513 0.488591i \(-0.162489\pi\)
0.872513 + 0.488591i \(0.162489\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −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
\(665\) 0 0
\(666\) 0 0
\(667\) −3.79973 −0.147126
\(668\) −0.792919 −0.0306789
\(669\) 0 0
\(670\) 0 0
\(671\) −65.6742 −2.53532
\(672\) 0 0
\(673\) −2.24216 −0.0864290 −0.0432145 0.999066i \(-0.513760\pi\)
−0.0432145 + 0.999066i \(0.513760\pi\)
\(674\) 10.7992 0.415971
\(675\) 0 0
\(676\) 0.478340 0.0183977
\(677\) 35.1762 1.35193 0.675966 0.736933i \(-0.263727\pi\)
0.675966 + 0.736933i \(0.263727\pi\)
\(678\) 0 0
\(679\) 0.628984 0.0241382
\(680\) 0 0
\(681\) 0 0
\(682\) 4.71276 0.180461
\(683\) −17.7638 −0.679714 −0.339857 0.940477i \(-0.610379\pi\)
−0.339857 + 0.940477i \(0.610379\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 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
\(695\) 0 0
\(696\) 0 0
\(697\) 14.7050 0.556992
\(698\) 14.9459 0.565712
\(699\) 0 0
\(700\) 0 0
\(701\) 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 \(0.653772\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 23.9919 0.899136
\(713\) 1.73236 0.0648775
\(714\) 0 0
\(715\) 0 0
\(716\) 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
\(725\) 0 0
\(726\) 0 0
\(727\) −20.5371 −0.761680 −0.380840 0.924641i \(-0.624365\pi\)
−0.380840 + 0.924641i \(0.624365\pi\)
\(728\) 22.9040 0.848880
\(729\) 0 0
\(730\) 0 0
\(731\) 15.5544 0.575299
\(732\) 0 0
\(733\) −14.0060 −0.517323 −0.258661 0.965968i \(-0.583281\pi\)
−0.258661 + 0.965968i \(0.583281\pi\)
\(734\) 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 0
\(741\) 0 0
\(742\) −37.8120 −1.38812
\(743\) 27.8621 1.02216 0.511082 0.859532i \(-0.329245\pi\)
0.511082 + 0.859532i \(0.329245\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −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
\(755\) 0 0
\(756\) 0 0
\(757\) 41.4858 1.50783 0.753913 0.656975i \(-0.228164\pi\)
0.753913 + 0.656975i \(0.228164\pi\)
\(758\) 28.2116 1.02469
\(759\) 0 0
\(760\) 0 0
\(761\) 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 \(0.470593\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −3.30129 −0.118816
\(773\) −52.6427 −1.89343 −0.946713 0.322077i \(-0.895619\pi\)
−0.946713 + 0.322077i \(0.895619\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 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
\(785\) 0 0
\(786\) 0 0
\(787\) 20.7624 0.740099 0.370050 0.929012i \(-0.379341\pi\)
0.370050 + 0.929012i \(0.379341\pi\)
\(788\) −0.133732 −0.00476401
\(789\) 0 0
\(790\) 0 0
\(791\) −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 \(-0.799052\pi\)
−0.807263 + 0.590192i \(0.799052\pi\)
\(798\) 0 0
\(799\) −11.2094 −0.396562
\(800\) 0 0
\(801\) 0 0
\(802\) 19.6459 0.693721
\(803\) 33.0128 1.16500
\(804\) 0 0
\(805\) 0 0
\(806\) −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
\(815\) 0 0
\(816\) 0 0
\(817\) −34.4175 −1.20411
\(818\) −23.5817 −0.824515
\(819\) 0 0
\(820\) 0 0
\(821\) −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 \(-0.562644\pi\)
−0.195534 + 0.980697i \(0.562644\pi\)
\(824\) −11.4997 −0.400611
\(825\) 0 0
\(826\) −23.8711 −0.830583
\(827\) −9.61318 −0.334283 −0.167141 0.985933i \(-0.553454\pi\)
−0.167141 + 0.985933i \(0.553454\pi\)
\(828\) 0 0
\(829\) 33.4938 1.16329 0.581644 0.813443i \(-0.302410\pi\)
0.581644 + 0.813443i \(0.302410\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 27.7374 0.961622
\(833\) −3.55438 −0.123152
\(834\) 0 0
\(835\) 0 0
\(836\) −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
\(845\) 0 0
\(846\) 0 0
\(847\) −58.6614 −2.01563
\(848\) −41.8539 −1.43727
\(849\) 0 0
\(850\) 0 0
\(851\) 0.123303 0.00422678
\(852\) 0 0
\(853\) −35.2422 −1.20667 −0.603334 0.797488i \(-0.706162\pi\)
−0.603334 + 0.797488i \(0.706162\pi\)
\(854\) −35.8999 −1.22847
\(855\) 0 0
\(856\) 7.76558 0.265422
\(857\) −21.2490 −0.725851 −0.362925 0.931818i \(-0.618222\pi\)
−0.362925 + 0.931818i \(0.618222\pi\)
\(858\) 0 0
\(859\) 51.8384 1.76870 0.884352 0.466820i \(-0.154601\pi\)
0.884352 + 0.466820i \(0.154601\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −46.2344 −1.57475
\(863\) −22.6783 −0.771978 −0.385989 0.922503i \(-0.626140\pi\)
−0.385989 + 0.922503i \(0.626140\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 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
\(875\) 0 0
\(876\) 0 0
\(877\) −1.13341 −0.0382725 −0.0191362 0.999817i \(-0.506092\pi\)
−0.0191362 + 0.999817i \(0.506092\pi\)
\(878\) −31.2681 −1.05525
\(879\) 0 0
\(880\) 0 0
\(881\) 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 \(-0.449793\pi\)
0.157076 + 0.987587i \(0.449793\pi\)
\(884\) −1.78880 −0.0601639
\(885\) 0 0
\(886\) −5.55295 −0.186555
\(887\) −14.0942 −0.473237 −0.236619 0.971603i \(-0.576039\pi\)
−0.236619 + 0.971603i \(0.576039\pi\)
\(888\) 0 0
\(889\) 8.67230 0.290860
\(890\) 0 0
\(891\) 0 0
\(892\) −2.02322 −0.0677425
\(893\) 24.8033 0.830012
\(894\) 0 0
\(895\) 0 0
\(896\) −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
\(905\) 0 0
\(906\) 0 0
\(907\) 8.73917 0.290179 0.145090 0.989419i \(-0.453653\pi\)
0.145090 + 0.989419i \(0.453653\pi\)
\(908\) 3.20296 0.106294
\(909\) 0 0
\(910\) 0 0
\(911\) 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 \(0.512394\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −37.3865 −1.23126
\(923\) 17.7870 0.585468
\(924\) 0 0
\(925\) 0 0
\(926\) −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
\(935\) 0 0
\(936\) 0 0
\(937\) −1.93313 −0.0631527 −0.0315764 0.999501i \(-0.510053\pi\)
−0.0315764 + 0.999501i \(0.510053\pi\)
\(938\) 6.03365 0.197006
\(939\) 0 0
\(940\) 0 0
\(941\) −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 \(-0.501633\pi\)
−0.00513165 + 0.999987i \(0.501633\pi\)
\(948\) 0 0
\(949\) −17.9222 −0.581779
\(950\) 0 0
\(951\) 0 0
\(952\) 21.2950 0.690174
\(953\) 3.25133 0.105321 0.0526605 0.998612i \(-0.483230\pi\)
0.0526605 + 0.998612i \(0.483230\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −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
\(965\) 0 0
\(966\) 0 0
\(967\) −10.9676 −0.352694 −0.176347 0.984328i \(-0.556428\pi\)
−0.176347 + 0.984328i \(0.556428\pi\)
\(968\) −71.6049 −2.30147
\(969\) 0 0
\(970\) 0 0
\(971\) −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 \(-0.796572\pi\)
−0.802640 + 0.596464i \(0.796572\pi\)
\(978\) 0 0
\(979\) 48.4439 1.54827
\(980\) 0 0
\(981\) 0 0
\(982\) 0.670874 0.0214084
\(983\) −14.6946 −0.468685 −0.234342 0.972154i \(-0.575294\pi\)
−0.234342 + 0.972154i \(0.575294\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 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
\(995\) 0 0
\(996\) 0 0
\(997\) 45.8863 1.45323 0.726617 0.687043i \(-0.241092\pi\)
0.726617 + 0.687043i \(0.241092\pi\)
\(998\) 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 6075.2.a.bv.1.2 3
3.2 odd 2 6075.2.a.bq.1.2 3
5.4 even 2 243.2.a.e.1.2 3
15.14 odd 2 243.2.a.f.1.2 yes 3
20.19 odd 2 3888.2.a.bd.1.2 3
45.4 even 6 243.2.c.f.163.2 6
45.14 odd 6 243.2.c.e.163.2 6
45.29 odd 6 243.2.c.e.82.2 6
45.34 even 6 243.2.c.f.82.2 6
60.59 even 2 3888.2.a.bk.1.2 3
135.4 even 18 729.2.e.h.406.1 6
135.14 odd 18 729.2.e.b.163.1 6
135.29 odd 18 729.2.e.b.568.1 6
135.34 even 18 729.2.e.h.325.1 6
135.49 even 18 729.2.e.a.649.1 6
135.59 odd 18 729.2.e.i.649.1 6
135.74 odd 18 729.2.e.c.325.1 6
135.79 even 18 729.2.e.g.568.1 6
135.94 even 18 729.2.e.g.163.1 6
135.104 odd 18 729.2.e.c.406.1 6
135.119 odd 18 729.2.e.i.82.1 6
135.124 even 18 729.2.e.a.82.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
243.2.a.e.1.2 3 5.4 even 2
243.2.a.f.1.2 yes 3 15.14 odd 2
243.2.c.e.82.2 6 45.29 odd 6
243.2.c.e.163.2 6 45.14 odd 6
243.2.c.f.82.2 6 45.34 even 6
243.2.c.f.163.2 6 45.4 even 6
729.2.e.a.82.1 6 135.124 even 18
729.2.e.a.649.1 6 135.49 even 18
729.2.e.b.163.1 6 135.14 odd 18
729.2.e.b.568.1 6 135.29 odd 18
729.2.e.c.325.1 6 135.74 odd 18
729.2.e.c.406.1 6 135.104 odd 18
729.2.e.g.163.1 6 135.94 even 18
729.2.e.g.568.1 6 135.79 even 18
729.2.e.h.325.1 6 135.34 even 18
729.2.e.h.406.1 6 135.4 even 18
729.2.e.i.82.1 6 135.119 odd 18
729.2.e.i.649.1 6 135.59 odd 18
3888.2.a.bd.1.2 3 20.19 odd 2
3888.2.a.bk.1.2 3 60.59 even 2
6075.2.a.bq.1.2 3 3.2 odd 2
6075.2.a.bv.1.2 3 1.1 even 1 trivial