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Properties

Label 810.2.a.e.1.1
Level $810$
Weight $2$
Character 810.1
Self dual yes
Analytic conductor $6.468$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [810,2,Mod(1,810)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(810, 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("810.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 810.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: \(6.46788256372\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 90)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 810.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -4.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -4.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} -3.00000 q^{11} -4.00000 q^{13} -4.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} +5.00000 q^{19} -1.00000 q^{20} -3.00000 q^{22} -6.00000 q^{23} +1.00000 q^{25} -4.00000 q^{26} -4.00000 q^{28} -6.00000 q^{29} +2.00000 q^{31} +1.00000 q^{32} -3.00000 q^{34} +4.00000 q^{35} -4.00000 q^{37} +5.00000 q^{38} -1.00000 q^{40} +3.00000 q^{41} +11.0000 q^{43} -3.00000 q^{44} -6.00000 q^{46} +9.00000 q^{49} +1.00000 q^{50} -4.00000 q^{52} -6.00000 q^{53} +3.00000 q^{55} -4.00000 q^{56} -6.00000 q^{58} +3.00000 q^{59} -10.0000 q^{61} +2.00000 q^{62} +1.00000 q^{64} +4.00000 q^{65} +5.00000 q^{67} -3.00000 q^{68} +4.00000 q^{70} -6.00000 q^{71} -7.00000 q^{73} -4.00000 q^{74} +5.00000 q^{76} +12.0000 q^{77} +14.0000 q^{79} -1.00000 q^{80} +3.00000 q^{82} -12.0000 q^{83} +3.00000 q^{85} +11.0000 q^{86} -3.00000 q^{88} -6.00000 q^{89} +16.0000 q^{91} -6.00000 q^{92} -5.00000 q^{95} +11.0000 q^{97} +9.00000 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −4.00000 −1.51186 −0.755929 − 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) −3.00000 −0.904534 −0.452267 − 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 − 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) −4.00000 −1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.00000 −0.727607 −0.363803 − 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) 5.00000 1.14708 0.573539 − 0.819178i \(-0.305570\pi\)
0.573539 + 0.819178i \(0.305570\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) −3.00000 −0.639602
\(23\) −6.00000 −1.25109 −0.625543 − 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −4.00000 −0.784465
\(27\) 0 0
\(28\) −4.00000 −0.755929
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 − 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −3.00000 −0.514496
\(35\) 4.00000 0.676123
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 − 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 5.00000 0.811107
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 3.00000 0.468521 0.234261 − 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 0 0
\(43\) 11.0000 1.67748 0.838742 − 0.544529i \(-0.183292\pi\)
0.838742 + 0.544529i \(0.183292\pi\)
\(44\) −3.00000 −0.452267
\(45\) 0 0
\(46\) −6.00000 −0.884652
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −4.00000 −0.554700
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 3.00000 0.404520
\(56\) −4.00000 −0.534522
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) 3.00000 0.390567 0.195283 − 0.980747i \(-0.437437\pi\)
0.195283 + 0.980747i \(0.437437\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 2.00000 0.254000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) 5.00000 0.610847 0.305424 − 0.952217i \(-0.401202\pi\)
0.305424 + 0.952217i \(0.401202\pi\)
\(68\) −3.00000 −0.363803
\(69\) 0 0
\(70\) 4.00000 0.478091
\(71\) −6.00000 −0.712069 −0.356034 − 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) −7.00000 −0.819288 −0.409644 − 0.912245i \(-0.634347\pi\)
−0.409644 + 0.912245i \(0.634347\pi\)
\(74\) −4.00000 −0.464991
\(75\) 0 0
\(76\) 5.00000 0.573539
\(77\) 12.0000 1.36753
\(78\) 0 0
\(79\) 14.0000 1.57512 0.787562 − 0.616236i \(-0.211343\pi\)
0.787562 + 0.616236i \(0.211343\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) 3.00000 0.331295
\(83\) −12.0000 −1.31717 −0.658586 − 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) 3.00000 0.325396
\(86\) 11.0000 1.18616
\(87\) 0 0
\(88\) −3.00000 −0.319801
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 16.0000 1.67726
\(92\) −6.00000 −0.625543
\(93\) 0 0
\(94\) 0 0
\(95\) −5.00000 −0.512989
\(96\) 0 0
\(97\) 11.0000 1.11688 0.558440 − 0.829545i \(-0.311400\pi\)
0.558440 + 0.829545i \(0.311400\pi\)
\(98\) 9.00000 0.909137
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 12.0000 1.19404 0.597022 − 0.802225i \(-0.296350\pi\)
0.597022 + 0.802225i \(0.296350\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 − 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) −4.00000 −0.392232
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 9.00000 0.870063 0.435031 − 0.900415i \(-0.356737\pi\)
0.435031 + 0.900415i \(0.356737\pi\)
\(108\) 0 0
\(109\) −4.00000 −0.383131 −0.191565 − 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 3.00000 0.286039
\(111\) 0 0
\(112\) −4.00000 −0.377964
\(113\) −18.0000 −1.69330 −0.846649 − 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) 0 0
\(115\) 6.00000 0.559503
\(116\) −6.00000 −0.557086
\(117\) 0 0
\(118\) 3.00000 0.276172
\(119\) 12.0000 1.10004
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) −10.0000 −0.905357
\(123\) 0 0
\(124\) 2.00000 0.179605
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 2.00000 0.177471 0.0887357 − 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 4.00000 0.350823
\(131\) −12.0000 −1.04844 −0.524222 − 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) −20.0000 −1.73422
\(134\) 5.00000 0.431934
\(135\) 0 0
\(136\) −3.00000 −0.257248
\(137\) 9.00000 0.768922 0.384461 − 0.923141i \(-0.374387\pi\)
0.384461 + 0.923141i \(0.374387\pi\)
\(138\) 0 0
\(139\) −1.00000 −0.0848189 −0.0424094 − 0.999100i \(-0.513503\pi\)
−0.0424094 + 0.999100i \(0.513503\pi\)
\(140\) 4.00000 0.338062
\(141\) 0 0
\(142\) −6.00000 −0.503509
\(143\) 12.0000 1.00349
\(144\) 0 0
\(145\) 6.00000 0.498273
\(146\) −7.00000 −0.579324
\(147\) 0 0
\(148\) −4.00000 −0.328798
\(149\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) −10.0000 −0.813788 −0.406894 − 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) 5.00000 0.405554
\(153\) 0 0
\(154\) 12.0000 0.966988
\(155\) −2.00000 −0.160644
\(156\) 0 0
\(157\) 8.00000 0.638470 0.319235 − 0.947676i \(-0.396574\pi\)
0.319235 + 0.947676i \(0.396574\pi\)
\(158\) 14.0000 1.11378
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) 24.0000 1.89146
\(162\) 0 0
\(163\) −16.0000 −1.25322 −0.626608 − 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) 3.00000 0.234261
\(165\) 0 0
\(166\) −12.0000 −0.931381
\(167\) 18.0000 1.39288 0.696441 − 0.717614i \(-0.254766\pi\)
0.696441 + 0.717614i \(0.254766\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 3.00000 0.230089
\(171\) 0 0
\(172\) 11.0000 0.838742
\(173\) 18.0000 1.36851 0.684257 − 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 0 0
\(175\) −4.00000 −0.302372
\(176\) −3.00000 −0.226134
\(177\) 0 0
\(178\) −6.00000 −0.449719
\(179\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −16.0000 −1.18927 −0.594635 − 0.803996i \(-0.702704\pi\)
−0.594635 + 0.803996i \(0.702704\pi\)
\(182\) 16.0000 1.18600
\(183\) 0 0
\(184\) −6.00000 −0.442326
\(185\) 4.00000 0.294086
\(186\) 0 0
\(187\) 9.00000 0.658145
\(188\) 0 0
\(189\) 0 0
\(190\) −5.00000 −0.362738
\(191\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −13.0000 −0.935760 −0.467880 − 0.883792i \(-0.654982\pi\)
−0.467880 + 0.883792i \(0.654982\pi\)
\(194\) 11.0000 0.789754
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 20.0000 1.41776 0.708881 − 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) 12.0000 0.844317
\(203\) 24.0000 1.68447
\(204\) 0 0
\(205\) −3.00000 −0.209529
\(206\) −4.00000 −0.278693
\(207\) 0 0
\(208\) −4.00000 −0.277350
\(209\) −15.0000 −1.03757
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 − 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) 9.00000 0.615227
\(215\) −11.0000 −0.750194
\(216\) 0 0
\(217\) −8.00000 −0.543075
\(218\) −4.00000 −0.270914
\(219\) 0 0
\(220\) 3.00000 0.202260
\(221\) 12.0000 0.807207
\(222\) 0 0
\(223\) −22.0000 −1.47323 −0.736614 − 0.676313i \(-0.763577\pi\)
−0.736614 + 0.676313i \(0.763577\pi\)
\(224\) −4.00000 −0.267261
\(225\) 0 0
\(226\) −18.0000 −1.19734
\(227\) 3.00000 0.199117 0.0995585 − 0.995032i \(-0.468257\pi\)
0.0995585 + 0.995032i \(0.468257\pi\)
\(228\) 0 0
\(229\) 20.0000 1.32164 0.660819 − 0.750546i \(-0.270209\pi\)
0.660819 + 0.750546i \(0.270209\pi\)
\(230\) 6.00000 0.395628
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) −21.0000 −1.37576 −0.687878 − 0.725826i \(-0.741458\pi\)
−0.687878 + 0.725826i \(0.741458\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 3.00000 0.195283
\(237\) 0 0
\(238\) 12.0000 0.777844
\(239\) −6.00000 −0.388108 −0.194054 − 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) 17.0000 1.09507 0.547533 − 0.836784i \(-0.315567\pi\)
0.547533 + 0.836784i \(0.315567\pi\)
\(242\) −2.00000 −0.128565
\(243\) 0 0
\(244\) −10.0000 −0.640184
\(245\) −9.00000 −0.574989
\(246\) 0 0
\(247\) −20.0000 −1.27257
\(248\) 2.00000 0.127000
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) 21.0000 1.32551 0.662754 − 0.748837i \(-0.269387\pi\)
0.662754 + 0.748837i \(0.269387\pi\)
\(252\) 0 0
\(253\) 18.0000 1.13165
\(254\) 2.00000 0.125491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −9.00000 −0.561405 −0.280702 − 0.959795i \(-0.590567\pi\)
−0.280702 + 0.959795i \(0.590567\pi\)
\(258\) 0 0
\(259\) 16.0000 0.994192
\(260\) 4.00000 0.248069
\(261\) 0 0
\(262\) −12.0000 −0.741362
\(263\) −24.0000 −1.47990 −0.739952 − 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) 6.00000 0.368577
\(266\) −20.0000 −1.22628
\(267\) 0 0
\(268\) 5.00000 0.305424
\(269\) −6.00000 −0.365826 −0.182913 − 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 − 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) −3.00000 −0.181902
\(273\) 0 0
\(274\) 9.00000 0.543710
\(275\) −3.00000 −0.180907
\(276\) 0 0
\(277\) −22.0000 −1.32185 −0.660926 − 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) −1.00000 −0.0599760
\(279\) 0 0
\(280\) 4.00000 0.239046
\(281\) 6.00000 0.357930 0.178965 − 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) 20.0000 1.18888 0.594438 − 0.804141i \(-0.297374\pi\)
0.594438 + 0.804141i \(0.297374\pi\)
\(284\) −6.00000 −0.356034
\(285\) 0 0
\(286\) 12.0000 0.709575
\(287\) −12.0000 −0.708338
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 6.00000 0.352332
\(291\) 0 0
\(292\) −7.00000 −0.409644
\(293\) 18.0000 1.05157 0.525786 − 0.850617i \(-0.323771\pi\)
0.525786 + 0.850617i \(0.323771\pi\)
\(294\) 0 0
\(295\) −3.00000 −0.174667
\(296\) −4.00000 −0.232495
\(297\) 0 0
\(298\) 0 0
\(299\) 24.0000 1.38796
\(300\) 0 0
\(301\) −44.0000 −2.53612
\(302\) −10.0000 −0.575435
\(303\) 0 0
\(304\) 5.00000 0.286770
\(305\) 10.0000 0.572598
\(306\) 0 0
\(307\) −7.00000 −0.399511 −0.199756 − 0.979846i \(-0.564015\pi\)
−0.199756 + 0.979846i \(0.564015\pi\)
\(308\) 12.0000 0.683763
\(309\) 0 0
\(310\) −2.00000 −0.113592
\(311\) 6.00000 0.340229 0.170114 − 0.985424i \(-0.445586\pi\)
0.170114 + 0.985424i \(0.445586\pi\)
\(312\) 0 0
\(313\) −1.00000 −0.0565233 −0.0282617 − 0.999601i \(-0.508997\pi\)
−0.0282617 + 0.999601i \(0.508997\pi\)
\(314\) 8.00000 0.451466
\(315\) 0 0
\(316\) 14.0000 0.787562
\(317\) −24.0000 −1.34797 −0.673987 − 0.738743i \(-0.735420\pi\)
−0.673987 + 0.738743i \(0.735420\pi\)
\(318\) 0 0
\(319\) 18.0000 1.00781
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) 24.0000 1.33747
\(323\) −15.0000 −0.834622
\(324\) 0 0
\(325\) −4.00000 −0.221880
\(326\) −16.0000 −0.886158
\(327\) 0 0
\(328\) 3.00000 0.165647
\(329\) 0 0
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 − 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) −12.0000 −0.658586
\(333\) 0 0
\(334\) 18.0000 0.984916
\(335\) −5.00000 −0.273179
\(336\) 0 0
\(337\) −31.0000 −1.68868 −0.844339 − 0.535810i \(-0.820006\pi\)
−0.844339 + 0.535810i \(0.820006\pi\)
\(338\) 3.00000 0.163178
\(339\) 0 0
\(340\) 3.00000 0.162698
\(341\) −6.00000 −0.324918
\(342\) 0 0
\(343\) −8.00000 −0.431959
\(344\) 11.0000 0.593080
\(345\) 0 0
\(346\) 18.0000 0.967686
\(347\) −21.0000 −1.12734 −0.563670 − 0.826000i \(-0.690611\pi\)
−0.563670 + 0.826000i \(0.690611\pi\)
\(348\) 0 0
\(349\) −16.0000 −0.856460 −0.428230 − 0.903670i \(-0.640863\pi\)
−0.428230 + 0.903670i \(0.640863\pi\)
\(350\) −4.00000 −0.213809
\(351\) 0 0
\(352\) −3.00000 −0.159901
\(353\) −9.00000 −0.479022 −0.239511 − 0.970894i \(-0.576987\pi\)
−0.239511 + 0.970894i \(0.576987\pi\)
\(354\) 0 0
\(355\) 6.00000 0.318447
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) 0 0
\(359\) 24.0000 1.26667 0.633336 − 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) −16.0000 −0.840941
\(363\) 0 0
\(364\) 16.0000 0.838628
\(365\) 7.00000 0.366397
\(366\) 0 0
\(367\) 8.00000 0.417597 0.208798 − 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) −6.00000 −0.312772
\(369\) 0 0
\(370\) 4.00000 0.207950
\(371\) 24.0000 1.24602
\(372\) 0 0
\(373\) −10.0000 −0.517780 −0.258890 − 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 9.00000 0.465379
\(375\) 0 0
\(376\) 0 0
\(377\) 24.0000 1.23606
\(378\) 0 0
\(379\) 29.0000 1.48963 0.744815 − 0.667271i \(-0.232538\pi\)
0.744815 + 0.667271i \(0.232538\pi\)
\(380\) −5.00000 −0.256495
\(381\) 0 0
\(382\) 0 0
\(383\) −12.0000 −0.613171 −0.306586 − 0.951843i \(-0.599187\pi\)
−0.306586 + 0.951843i \(0.599187\pi\)
\(384\) 0 0
\(385\) −12.0000 −0.611577
\(386\) −13.0000 −0.661683
\(387\) 0 0
\(388\) 11.0000 0.558440
\(389\) −36.0000 −1.82527 −0.912636 − 0.408773i \(-0.865957\pi\)
−0.912636 + 0.408773i \(0.865957\pi\)
\(390\) 0 0
\(391\) 18.0000 0.910299
\(392\) 9.00000 0.454569
\(393\) 0 0
\(394\) 0 0
\(395\) −14.0000 −0.704416
\(396\) 0 0
\(397\) 8.00000 0.401508 0.200754 − 0.979642i \(-0.435661\pi\)
0.200754 + 0.979642i \(0.435661\pi\)
\(398\) 20.0000 1.00251
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −33.0000 −1.64794 −0.823971 − 0.566632i \(-0.808246\pi\)
−0.823971 + 0.566632i \(0.808246\pi\)
\(402\) 0 0
\(403\) −8.00000 −0.398508
\(404\) 12.0000 0.597022
\(405\) 0 0
\(406\) 24.0000 1.19110
\(407\) 12.0000 0.594818
\(408\) 0 0
\(409\) −31.0000 −1.53285 −0.766426 − 0.642333i \(-0.777967\pi\)
−0.766426 + 0.642333i \(0.777967\pi\)
\(410\) −3.00000 −0.148159
\(411\) 0 0
\(412\) −4.00000 −0.197066
\(413\) −12.0000 −0.590481
\(414\) 0 0
\(415\) 12.0000 0.589057
\(416\) −4.00000 −0.196116
\(417\) 0 0
\(418\) −15.0000 −0.733674
\(419\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 2.00000 0.0974740 0.0487370 − 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) −4.00000 −0.194717
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) −3.00000 −0.145521
\(426\) 0 0
\(427\) 40.0000 1.93574
\(428\) 9.00000 0.435031
\(429\) 0 0
\(430\) −11.0000 −0.530467
\(431\) −24.0000 −1.15604 −0.578020 − 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 0 0
\(433\) −13.0000 −0.624740 −0.312370 − 0.949960i \(-0.601123\pi\)
−0.312370 + 0.949960i \(0.601123\pi\)
\(434\) −8.00000 −0.384012
\(435\) 0 0
\(436\) −4.00000 −0.191565
\(437\) −30.0000 −1.43509
\(438\) 0 0
\(439\) −10.0000 −0.477274 −0.238637 − 0.971109i \(-0.576701\pi\)
−0.238637 + 0.971109i \(0.576701\pi\)
\(440\) 3.00000 0.143019
\(441\) 0 0
\(442\) 12.0000 0.570782
\(443\) −3.00000 −0.142534 −0.0712672 − 0.997457i \(-0.522704\pi\)
−0.0712672 + 0.997457i \(0.522704\pi\)
\(444\) 0 0
\(445\) 6.00000 0.284427
\(446\) −22.0000 −1.04173
\(447\) 0 0
\(448\) −4.00000 −0.188982
\(449\) 15.0000 0.707894 0.353947 − 0.935266i \(-0.384839\pi\)
0.353947 + 0.935266i \(0.384839\pi\)
\(450\) 0 0
\(451\) −9.00000 −0.423793
\(452\) −18.0000 −0.846649
\(453\) 0 0
\(454\) 3.00000 0.140797
\(455\) −16.0000 −0.750092
\(456\) 0 0
\(457\) −1.00000 −0.0467780 −0.0233890 − 0.999726i \(-0.507446\pi\)
−0.0233890 + 0.999726i \(0.507446\pi\)
\(458\) 20.0000 0.934539
\(459\) 0 0
\(460\) 6.00000 0.279751
\(461\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 20.0000 0.929479 0.464739 − 0.885448i \(-0.346148\pi\)
0.464739 + 0.885448i \(0.346148\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) −21.0000 −0.972806
\(467\) −21.0000 −0.971764 −0.485882 − 0.874024i \(-0.661502\pi\)
−0.485882 + 0.874024i \(0.661502\pi\)
\(468\) 0 0
\(469\) −20.0000 −0.923514
\(470\) 0 0
\(471\) 0 0
\(472\) 3.00000 0.138086
\(473\) −33.0000 −1.51734
\(474\) 0 0
\(475\) 5.00000 0.229416
\(476\) 12.0000 0.550019
\(477\) 0 0
\(478\) −6.00000 −0.274434
\(479\) 6.00000 0.274147 0.137073 − 0.990561i \(-0.456230\pi\)
0.137073 + 0.990561i \(0.456230\pi\)
\(480\) 0 0
\(481\) 16.0000 0.729537
\(482\) 17.0000 0.774329
\(483\) 0 0
\(484\) −2.00000 −0.0909091
\(485\) −11.0000 −0.499484
\(486\) 0 0
\(487\) 2.00000 0.0906287 0.0453143 − 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) −10.0000 −0.452679
\(489\) 0 0
\(490\) −9.00000 −0.406579
\(491\) 33.0000 1.48927 0.744635 − 0.667472i \(-0.232624\pi\)
0.744635 + 0.667472i \(0.232624\pi\)
\(492\) 0 0
\(493\) 18.0000 0.810679
\(494\) −20.0000 −0.899843
\(495\) 0 0
\(496\) 2.00000 0.0898027
\(497\) 24.0000 1.07655
\(498\) 0 0
\(499\) −31.0000 −1.38775 −0.693875 − 0.720095i \(-0.744098\pi\)
−0.693875 + 0.720095i \(0.744098\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) 21.0000 0.937276
\(503\) −6.00000 −0.267527 −0.133763 − 0.991013i \(-0.542706\pi\)
−0.133763 + 0.991013i \(0.542706\pi\)
\(504\) 0 0
\(505\) −12.0000 −0.533993
\(506\) 18.0000 0.800198
\(507\) 0 0
\(508\) 2.00000 0.0887357
\(509\) 12.0000 0.531891 0.265945 − 0.963988i \(-0.414316\pi\)
0.265945 + 0.963988i \(0.414316\pi\)
\(510\) 0 0
\(511\) 28.0000 1.23865
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −9.00000 −0.396973
\(515\) 4.00000 0.176261
\(516\) 0 0
\(517\) 0 0
\(518\) 16.0000 0.703000
\(519\) 0 0
\(520\) 4.00000 0.175412
\(521\) 3.00000 0.131432 0.0657162 − 0.997838i \(-0.479067\pi\)
0.0657162 + 0.997838i \(0.479067\pi\)
\(522\) 0 0
\(523\) 20.0000 0.874539 0.437269 − 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) −6.00000 −0.261364
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 6.00000 0.260623
\(531\) 0 0
\(532\) −20.0000 −0.867110
\(533\) −12.0000 −0.519778
\(534\) 0 0
\(535\) −9.00000 −0.389104
\(536\) 5.00000 0.215967
\(537\) 0 0
\(538\) −6.00000 −0.258678
\(539\) −27.0000 −1.16297
\(540\) 0 0
\(541\) 8.00000 0.343947 0.171973 − 0.985102i \(-0.444986\pi\)
0.171973 + 0.985102i \(0.444986\pi\)
\(542\) −16.0000 −0.687259
\(543\) 0 0
\(544\) −3.00000 −0.128624
\(545\) 4.00000 0.171341
\(546\) 0 0
\(547\) −1.00000 −0.0427569 −0.0213785 − 0.999771i \(-0.506805\pi\)
−0.0213785 + 0.999771i \(0.506805\pi\)
\(548\) 9.00000 0.384461
\(549\) 0 0
\(550\) −3.00000 −0.127920
\(551\) −30.0000 −1.27804
\(552\) 0 0
\(553\) −56.0000 −2.38136
\(554\) −22.0000 −0.934690
\(555\) 0 0
\(556\) −1.00000 −0.0424094
\(557\) 12.0000 0.508456 0.254228 − 0.967144i \(-0.418179\pi\)
0.254228 + 0.967144i \(0.418179\pi\)
\(558\) 0 0
\(559\) −44.0000 −1.86100
\(560\) 4.00000 0.169031
\(561\) 0 0
\(562\) 6.00000 0.253095
\(563\) 3.00000 0.126435 0.0632175 − 0.998000i \(-0.479864\pi\)
0.0632175 + 0.998000i \(0.479864\pi\)
\(564\) 0 0
\(565\) 18.0000 0.757266
\(566\) 20.0000 0.840663
\(567\) 0 0
\(568\) −6.00000 −0.251754
\(569\) 39.0000 1.63497 0.817483 − 0.575953i \(-0.195369\pi\)
0.817483 + 0.575953i \(0.195369\pi\)
\(570\) 0 0
\(571\) 29.0000 1.21361 0.606806 − 0.794850i \(-0.292450\pi\)
0.606806 + 0.794850i \(0.292450\pi\)
\(572\) 12.0000 0.501745
\(573\) 0 0
\(574\) −12.0000 −0.500870
\(575\) −6.00000 −0.250217
\(576\) 0 0
\(577\) −7.00000 −0.291414 −0.145707 − 0.989328i \(-0.546546\pi\)
−0.145707 + 0.989328i \(0.546546\pi\)
\(578\) −8.00000 −0.332756
\(579\) 0 0
\(580\) 6.00000 0.249136
\(581\) 48.0000 1.99138
\(582\) 0 0
\(583\) 18.0000 0.745484
\(584\) −7.00000 −0.289662
\(585\) 0 0
\(586\) 18.0000 0.743573
\(587\) −39.0000 −1.60970 −0.804851 − 0.593477i \(-0.797755\pi\)
−0.804851 + 0.593477i \(0.797755\pi\)
\(588\) 0 0
\(589\) 10.0000 0.412043
\(590\) −3.00000 −0.123508
\(591\) 0 0
\(592\) −4.00000 −0.164399
\(593\) 18.0000 0.739171 0.369586 − 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) 0 0
\(595\) −12.0000 −0.491952
\(596\) 0 0
\(597\) 0 0
\(598\) 24.0000 0.981433
\(599\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −25.0000 −1.01977 −0.509886 − 0.860242i \(-0.670312\pi\)
−0.509886 + 0.860242i \(0.670312\pi\)
\(602\) −44.0000 −1.79331
\(603\) 0 0
\(604\) −10.0000 −0.406894
\(605\) 2.00000 0.0813116
\(606\) 0 0
\(607\) 8.00000 0.324710 0.162355 − 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) 5.00000 0.202777
\(609\) 0 0
\(610\) 10.0000 0.404888
\(611\) 0 0
\(612\) 0 0
\(613\) 2.00000 0.0807792 0.0403896 − 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) −7.00000 −0.282497
\(615\) 0 0
\(616\) 12.0000 0.483494
\(617\) 39.0000 1.57008 0.785040 − 0.619445i \(-0.212642\pi\)
0.785040 + 0.619445i \(0.212642\pi\)
\(618\) 0 0
\(619\) −19.0000 −0.763674 −0.381837 − 0.924230i \(-0.624709\pi\)
−0.381837 + 0.924230i \(0.624709\pi\)
\(620\) −2.00000 −0.0803219
\(621\) 0 0
\(622\) 6.00000 0.240578
\(623\) 24.0000 0.961540
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −1.00000 −0.0399680
\(627\) 0 0
\(628\) 8.00000 0.319235
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) −34.0000 −1.35352 −0.676759 − 0.736204i \(-0.736616\pi\)
−0.676759 + 0.736204i \(0.736616\pi\)
\(632\) 14.0000 0.556890
\(633\) 0 0
\(634\) −24.0000 −0.953162
\(635\) −2.00000 −0.0793676
\(636\) 0 0
\(637\) −36.0000 −1.42637
\(638\) 18.0000 0.712627
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) −9.00000 −0.355479 −0.177739 − 0.984078i \(-0.556878\pi\)
−0.177739 + 0.984078i \(0.556878\pi\)
\(642\) 0 0
\(643\) 23.0000 0.907031 0.453516 − 0.891248i \(-0.350170\pi\)
0.453516 + 0.891248i \(0.350170\pi\)
\(644\) 24.0000 0.945732
\(645\) 0 0
\(646\) −15.0000 −0.590167
\(647\) −6.00000 −0.235884 −0.117942 − 0.993020i \(-0.537630\pi\)
−0.117942 + 0.993020i \(0.537630\pi\)
\(648\) 0 0
\(649\) −9.00000 −0.353281
\(650\) −4.00000 −0.156893
\(651\) 0 0
\(652\) −16.0000 −0.626608
\(653\) −42.0000 −1.64359 −0.821794 − 0.569785i \(-0.807026\pi\)
−0.821794 + 0.569785i \(0.807026\pi\)
\(654\) 0 0
\(655\) 12.0000 0.468879
\(656\) 3.00000 0.117130
\(657\) 0 0
\(658\) 0 0
\(659\) 36.0000 1.40236 0.701180 − 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 0 0
\(661\) 32.0000 1.24466 0.622328 − 0.782757i \(-0.286187\pi\)
0.622328 + 0.782757i \(0.286187\pi\)
\(662\) −4.00000 −0.155464
\(663\) 0 0
\(664\) −12.0000 −0.465690
\(665\) 20.0000 0.775567
\(666\) 0 0
\(667\) 36.0000 1.39393
\(668\) 18.0000 0.696441
\(669\) 0 0
\(670\) −5.00000 −0.193167
\(671\) 30.0000 1.15814
\(672\) 0 0
\(673\) 14.0000 0.539660 0.269830 − 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) −31.0000 −1.19408
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) 36.0000 1.38359 0.691796 − 0.722093i \(-0.256820\pi\)
0.691796 + 0.722093i \(0.256820\pi\)
\(678\) 0 0
\(679\) −44.0000 −1.68857
\(680\) 3.00000 0.115045
\(681\) 0 0
\(682\) −6.00000 −0.229752
\(683\) 21.0000 0.803543 0.401771 − 0.915740i \(-0.368395\pi\)
0.401771 + 0.915740i \(0.368395\pi\)
\(684\) 0 0
\(685\) −9.00000 −0.343872
\(686\) −8.00000 −0.305441
\(687\) 0 0
\(688\) 11.0000 0.419371
\(689\) 24.0000 0.914327
\(690\) 0 0
\(691\) 8.00000 0.304334 0.152167 − 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) 18.0000 0.684257
\(693\) 0 0
\(694\) −21.0000 −0.797149
\(695\) 1.00000 0.0379322
\(696\) 0 0
\(697\) −9.00000 −0.340899
\(698\) −16.0000 −0.605609
\(699\) 0 0
\(700\) −4.00000 −0.151186
\(701\) −42.0000 −1.58632 −0.793159 − 0.609015i \(-0.791565\pi\)
−0.793159 + 0.609015i \(0.791565\pi\)
\(702\) 0 0
\(703\) −20.0000 −0.754314
\(704\) −3.00000 −0.113067
\(705\) 0 0
\(706\) −9.00000 −0.338719
\(707\) −48.0000 −1.80523
\(708\) 0 0
\(709\) −34.0000 −1.27690 −0.638448 − 0.769665i \(-0.720423\pi\)
−0.638448 + 0.769665i \(0.720423\pi\)
\(710\) 6.00000 0.225176
\(711\) 0 0
\(712\) −6.00000 −0.224860
\(713\) −12.0000 −0.449404
\(714\) 0 0
\(715\) −12.0000 −0.448775
\(716\) 0 0
\(717\) 0 0
\(718\) 24.0000 0.895672
\(719\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) 6.00000 0.223297
\(723\) 0 0
\(724\) −16.0000 −0.594635
\(725\) −6.00000 −0.222834
\(726\) 0 0
\(727\) −28.0000 −1.03846 −0.519231 − 0.854634i \(-0.673782\pi\)
−0.519231 + 0.854634i \(0.673782\pi\)
\(728\) 16.0000 0.592999
\(729\) 0 0
\(730\) 7.00000 0.259082
\(731\) −33.0000 −1.22055
\(732\) 0 0
\(733\) 32.0000 1.18195 0.590973 − 0.806691i \(-0.298744\pi\)
0.590973 + 0.806691i \(0.298744\pi\)
\(734\) 8.00000 0.295285
\(735\) 0 0
\(736\) −6.00000 −0.221163
\(737\) −15.0000 −0.552532
\(738\) 0 0
\(739\) 29.0000 1.06678 0.533391 − 0.845869i \(-0.320917\pi\)
0.533391 + 0.845869i \(0.320917\pi\)
\(740\) 4.00000 0.147043
\(741\) 0 0
\(742\) 24.0000 0.881068
\(743\) 6.00000 0.220119 0.110059 − 0.993925i \(-0.464896\pi\)
0.110059 + 0.993925i \(0.464896\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −10.0000 −0.366126
\(747\) 0 0
\(748\) 9.00000 0.329073
\(749\) −36.0000 −1.31541
\(750\) 0 0
\(751\) −28.0000 −1.02173 −0.510867 − 0.859660i \(-0.670676\pi\)
−0.510867 + 0.859660i \(0.670676\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 24.0000 0.874028
\(755\) 10.0000 0.363937
\(756\) 0 0
\(757\) 38.0000 1.38113 0.690567 − 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) 29.0000 1.05333
\(759\) 0 0
\(760\) −5.00000 −0.181369
\(761\) −18.0000 −0.652499 −0.326250 − 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) 0 0
\(763\) 16.0000 0.579239
\(764\) 0 0
\(765\) 0 0
\(766\) −12.0000 −0.433578
\(767\) −12.0000 −0.433295
\(768\) 0 0
\(769\) 50.0000 1.80305 0.901523 − 0.432731i \(-0.142450\pi\)
0.901523 + 0.432731i \(0.142450\pi\)
\(770\) −12.0000 −0.432450
\(771\) 0 0
\(772\) −13.0000 −0.467880
\(773\) 30.0000 1.07903 0.539513 − 0.841978i \(-0.318609\pi\)
0.539513 + 0.841978i \(0.318609\pi\)
\(774\) 0 0
\(775\) 2.00000 0.0718421
\(776\) 11.0000 0.394877
\(777\) 0 0
\(778\) −36.0000 −1.29066
\(779\) 15.0000 0.537431
\(780\) 0 0
\(781\) 18.0000 0.644091
\(782\) 18.0000 0.643679
\(783\) 0 0
\(784\) 9.00000 0.321429
\(785\) −8.00000 −0.285532
\(786\) 0 0
\(787\) −4.00000 −0.142585 −0.0712923 − 0.997455i \(-0.522712\pi\)
−0.0712923 + 0.997455i \(0.522712\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) −14.0000 −0.498098
\(791\) 72.0000 2.56003
\(792\) 0 0
\(793\) 40.0000 1.42044
\(794\) 8.00000 0.283909
\(795\) 0 0
\(796\) 20.0000 0.708881
\(797\) 48.0000 1.70025 0.850124 − 0.526583i \(-0.176527\pi\)
0.850124 + 0.526583i \(0.176527\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) −33.0000 −1.16527
\(803\) 21.0000 0.741074
\(804\) 0 0
\(805\) −24.0000 −0.845889
\(806\) −8.00000 −0.281788
\(807\) 0 0
\(808\) 12.0000 0.422159
\(809\) 39.0000 1.37117 0.685583 − 0.727994i \(-0.259547\pi\)
0.685583 + 0.727994i \(0.259547\pi\)
\(810\) 0 0
\(811\) 35.0000 1.22902 0.614508 − 0.788911i \(-0.289355\pi\)
0.614508 + 0.788911i \(0.289355\pi\)
\(812\) 24.0000 0.842235
\(813\) 0 0
\(814\) 12.0000 0.420600
\(815\) 16.0000 0.560456
\(816\) 0 0
\(817\) 55.0000 1.92421
\(818\) −31.0000 −1.08389
\(819\) 0 0
\(820\) −3.00000 −0.104765
\(821\) 48.0000 1.67521 0.837606 − 0.546275i \(-0.183955\pi\)
0.837606 + 0.546275i \(0.183955\pi\)
\(822\) 0 0
\(823\) −4.00000 −0.139431 −0.0697156 − 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) −4.00000 −0.139347
\(825\) 0 0
\(826\) −12.0000 −0.417533
\(827\) 36.0000 1.25184 0.625921 − 0.779886i \(-0.284723\pi\)
0.625921 + 0.779886i \(0.284723\pi\)
\(828\) 0 0
\(829\) 44.0000 1.52818 0.764092 − 0.645108i \(-0.223188\pi\)
0.764092 + 0.645108i \(0.223188\pi\)
\(830\) 12.0000 0.416526
\(831\) 0 0
\(832\) −4.00000 −0.138675
\(833\) −27.0000 −0.935495
\(834\) 0 0
\(835\) −18.0000 −0.622916
\(836\) −15.0000 −0.518786
\(837\) 0 0
\(838\) 0 0
\(839\) 30.0000 1.03572 0.517858 − 0.855467i \(-0.326730\pi\)
0.517858 + 0.855467i \(0.326730\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 2.00000 0.0689246
\(843\) 0 0
\(844\) −4.00000 −0.137686
\(845\) −3.00000 −0.103203
\(846\) 0 0
\(847\) 8.00000 0.274883
\(848\) −6.00000 −0.206041
\(849\) 0 0
\(850\) −3.00000 −0.102899
\(851\) 24.0000 0.822709
\(852\) 0 0
\(853\) 44.0000 1.50653 0.753266 − 0.657716i \(-0.228477\pi\)
0.753266 + 0.657716i \(0.228477\pi\)
\(854\) 40.0000 1.36877
\(855\) 0 0
\(856\) 9.00000 0.307614
\(857\) 18.0000 0.614868 0.307434 − 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) −19.0000 −0.648272 −0.324136 − 0.946011i \(-0.605073\pi\)
−0.324136 + 0.946011i \(0.605073\pi\)
\(860\) −11.0000 −0.375097
\(861\) 0 0
\(862\) −24.0000 −0.817443
\(863\) 6.00000 0.204242 0.102121 − 0.994772i \(-0.467437\pi\)
0.102121 + 0.994772i \(0.467437\pi\)
\(864\) 0 0
\(865\) −18.0000 −0.612018
\(866\) −13.0000 −0.441758
\(867\) 0 0
\(868\) −8.00000 −0.271538
\(869\) −42.0000 −1.42475
\(870\) 0 0
\(871\) −20.0000 −0.677674
\(872\) −4.00000 −0.135457
\(873\) 0 0
\(874\) −30.0000 −1.01477
\(875\) 4.00000 0.135225
\(876\) 0 0
\(877\) 2.00000 0.0675352 0.0337676 − 0.999430i \(-0.489249\pi\)
0.0337676 + 0.999430i \(0.489249\pi\)
\(878\) −10.0000 −0.337484
\(879\) 0 0
\(880\) 3.00000 0.101130
\(881\) −30.0000 −1.01073 −0.505363 − 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 0 0
\(883\) 41.0000 1.37976 0.689880 − 0.723924i \(-0.257663\pi\)
0.689880 + 0.723924i \(0.257663\pi\)
\(884\) 12.0000 0.403604
\(885\) 0 0
\(886\) −3.00000 −0.100787
\(887\) 42.0000 1.41022 0.705111 − 0.709097i \(-0.250897\pi\)
0.705111 + 0.709097i \(0.250897\pi\)
\(888\) 0 0
\(889\) −8.00000 −0.268311
\(890\) 6.00000 0.201120
\(891\) 0 0
\(892\) −22.0000 −0.736614
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) −4.00000 −0.133631
\(897\) 0 0
\(898\) 15.0000 0.500556
\(899\) −12.0000 −0.400222
\(900\) 0 0
\(901\) 18.0000 0.599667
\(902\) −9.00000 −0.299667
\(903\) 0 0
\(904\) −18.0000 −0.598671
\(905\) 16.0000 0.531858
\(906\) 0 0
\(907\) 5.00000 0.166022 0.0830111 − 0.996549i \(-0.473546\pi\)
0.0830111 + 0.996549i \(0.473546\pi\)
\(908\) 3.00000 0.0995585
\(909\) 0 0
\(910\) −16.0000 −0.530395
\(911\) 30.0000 0.993944 0.496972 − 0.867766i \(-0.334445\pi\)
0.496972 + 0.867766i \(0.334445\pi\)
\(912\) 0 0
\(913\) 36.0000 1.19143
\(914\) −1.00000 −0.0330771
\(915\) 0 0
\(916\) 20.0000 0.660819
\(917\) 48.0000 1.58510
\(918\) 0 0
\(919\) −34.0000 −1.12156 −0.560778 − 0.827966i \(-0.689498\pi\)
−0.560778 + 0.827966i \(0.689498\pi\)
\(920\) 6.00000 0.197814
\(921\) 0 0
\(922\) 0 0
\(923\) 24.0000 0.789970
\(924\) 0 0
\(925\) −4.00000 −0.131519
\(926\) 20.0000 0.657241
\(927\) 0 0
\(928\) −6.00000 −0.196960
\(929\) −30.0000 −0.984268 −0.492134 − 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 0 0
\(931\) 45.0000 1.47482
\(932\) −21.0000 −0.687878
\(933\) 0 0
\(934\) −21.0000 −0.687141
\(935\) −9.00000 −0.294331
\(936\) 0 0
\(937\) −10.0000 −0.326686 −0.163343 − 0.986569i \(-0.552228\pi\)
−0.163343 + 0.986569i \(0.552228\pi\)
\(938\) −20.0000 −0.653023
\(939\) 0 0
\(940\) 0 0
\(941\) −24.0000 −0.782378 −0.391189 − 0.920310i \(-0.627936\pi\)
−0.391189 + 0.920310i \(0.627936\pi\)
\(942\) 0 0
\(943\) −18.0000 −0.586161
\(944\) 3.00000 0.0976417
\(945\) 0 0
\(946\) −33.0000 −1.07292
\(947\) −27.0000 −0.877382 −0.438691 − 0.898638i \(-0.644558\pi\)
−0.438691 + 0.898638i \(0.644558\pi\)
\(948\) 0 0
\(949\) 28.0000 0.908918
\(950\) 5.00000 0.162221
\(951\) 0 0
\(952\) 12.0000 0.388922
\(953\) −51.0000 −1.65205 −0.826026 − 0.563632i \(-0.809404\pi\)
−0.826026 + 0.563632i \(0.809404\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −6.00000 −0.194054
\(957\) 0 0
\(958\) 6.00000 0.193851
\(959\) −36.0000 −1.16250
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 16.0000 0.515861
\(963\) 0 0
\(964\) 17.0000 0.547533
\(965\) 13.0000 0.418485
\(966\) 0 0
\(967\) −22.0000 −0.707472 −0.353736 − 0.935345i \(-0.615089\pi\)
−0.353736 + 0.935345i \(0.615089\pi\)
\(968\) −2.00000 −0.0642824
\(969\) 0 0
\(970\) −11.0000 −0.353189
\(971\) −60.0000 −1.92549 −0.962746 − 0.270408i \(-0.912841\pi\)
−0.962746 + 0.270408i \(0.912841\pi\)
\(972\) 0 0
\(973\) 4.00000 0.128234
\(974\) 2.00000 0.0640841
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) 9.00000 0.287936 0.143968 − 0.989582i \(-0.454014\pi\)
0.143968 + 0.989582i \(0.454014\pi\)
\(978\) 0 0
\(979\) 18.0000 0.575282
\(980\) −9.00000 −0.287494
\(981\) 0 0
\(982\) 33.0000 1.05307
\(983\) −36.0000 −1.14822 −0.574111 − 0.818778i \(-0.694652\pi\)
−0.574111 + 0.818778i \(0.694652\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 18.0000 0.573237
\(987\) 0 0
\(988\) −20.0000 −0.636285
\(989\) −66.0000 −2.09868
\(990\) 0 0
\(991\) 20.0000 0.635321 0.317660 − 0.948205i \(-0.397103\pi\)
0.317660 + 0.948205i \(0.397103\pi\)
\(992\) 2.00000 0.0635001
\(993\) 0 0
\(994\) 24.0000 0.761234
\(995\) −20.0000 −0.634043
\(996\) 0 0
\(997\) 26.0000 0.823428 0.411714 − 0.911313i \(-0.364930\pi\)
0.411714 + 0.911313i \(0.364930\pi\)
\(998\) −31.0000 −0.981288
\(999\) 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 810.2.a.e.1.1 1
3.2 odd 2 810.2.a.a.1.1 1
4.3 odd 2 6480.2.a.l.1.1 1
5.2 odd 4 4050.2.c.d.649.2 2
5.3 odd 4 4050.2.c.d.649.1 2
5.4 even 2 4050.2.a.q.1.1 1
9.2 odd 6 90.2.e.b.31.1 ✓ 2
9.4 even 3 270.2.e.a.181.1 2
9.5 odd 6 90.2.e.b.61.1 yes 2
9.7 even 3 270.2.e.a.91.1 2
12.11 even 2 6480.2.a.z.1.1 1
15.2 even 4 4050.2.c.p.649.1 2
15.8 even 4 4050.2.c.p.649.2 2
15.14 odd 2 4050.2.a.bi.1.1 1
36.7 odd 6 2160.2.q.d.1441.1 2
36.11 even 6 720.2.q.c.481.1 2
36.23 even 6 720.2.q.c.241.1 2
36.31 odd 6 2160.2.q.d.721.1 2
45.2 even 12 450.2.j.a.49.1 4
45.4 even 6 1350.2.e.g.451.1 2
45.7 odd 12 1350.2.j.c.199.2 4
45.13 odd 12 1350.2.j.c.1099.2 4
45.14 odd 6 450.2.e.d.151.1 2
45.22 odd 12 1350.2.j.c.1099.1 4
45.23 even 12 450.2.j.a.349.1 4
45.29 odd 6 450.2.e.d.301.1 2
45.32 even 12 450.2.j.a.349.2 4
45.34 even 6 1350.2.e.g.901.1 2
45.38 even 12 450.2.j.a.49.2 4
45.43 odd 12 1350.2.j.c.199.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.2.e.b.31.1 ✓ 2 9.2 odd 6
90.2.e.b.61.1 yes 2 9.5 odd 6
270.2.e.a.91.1 2 9.7 even 3
270.2.e.a.181.1 2 9.4 even 3
450.2.e.d.151.1 2 45.14 odd 6
450.2.e.d.301.1 2 45.29 odd 6
450.2.j.a.49.1 4 45.2 even 12
450.2.j.a.49.2 4 45.38 even 12
450.2.j.a.349.1 4 45.23 even 12
450.2.j.a.349.2 4 45.32 even 12
720.2.q.c.241.1 2 36.23 even 6
720.2.q.c.481.1 2 36.11 even 6
810.2.a.a.1.1 1 3.2 odd 2
810.2.a.e.1.1 1 1.1 even 1 trivial
1350.2.e.g.451.1 2 45.4 even 6
1350.2.e.g.901.1 2 45.34 even 6
1350.2.j.c.199.1 4 45.43 odd 12
1350.2.j.c.199.2 4 45.7 odd 12
1350.2.j.c.1099.1 4 45.22 odd 12
1350.2.j.c.1099.2 4 45.13 odd 12
2160.2.q.d.721.1 2 36.31 odd 6
2160.2.q.d.1441.1 2 36.7 odd 6
4050.2.a.q.1.1 1 5.4 even 2
4050.2.a.bi.1.1 1 15.14 odd 2
4050.2.c.d.649.1 2 5.3 odd 4
4050.2.c.d.649.2 2 5.2 odd 4
4050.2.c.p.649.1 2 15.2 even 4
4050.2.c.p.649.2 2 15.8 even 4
6480.2.a.l.1.1 1 4.3 odd 2
6480.2.a.z.1.1 1 12.11 even 2