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Properties

Label 3465.2.a.c.1.1
Level $3465$
Weight $2$
Character 3465.1
Self dual yes
Analytic conductor $27.668$
Analytic rank $0$
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] = [3465,2,Mod(1,3465)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3465, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("3465.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3465 = 3^{2} \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3465.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: \(27.6681643004\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1155)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3465.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +3.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +3.00000 q^{8} +1.00000 q^{10} +1.00000 q^{11} +6.00000 q^{13} -1.00000 q^{14} -1.00000 q^{16} -2.00000 q^{17} +4.00000 q^{19} +1.00000 q^{20} -1.00000 q^{22} +4.00000 q^{23} +1.00000 q^{25} -6.00000 q^{26} -1.00000 q^{28} +2.00000 q^{29} +4.00000 q^{31} -5.00000 q^{32} +2.00000 q^{34} -1.00000 q^{35} -2.00000 q^{37} -4.00000 q^{38} -3.00000 q^{40} -6.00000 q^{41} +4.00000 q^{43} -1.00000 q^{44} -4.00000 q^{46} -8.00000 q^{47} +1.00000 q^{49} -1.00000 q^{50} -6.00000 q^{52} +2.00000 q^{53} -1.00000 q^{55} +3.00000 q^{56} -2.00000 q^{58} +8.00000 q^{59} +10.0000 q^{61} -4.00000 q^{62} +7.00000 q^{64} -6.00000 q^{65} -8.00000 q^{67} +2.00000 q^{68} +1.00000 q^{70} -14.0000 q^{73} +2.00000 q^{74} -4.00000 q^{76} +1.00000 q^{77} -16.0000 q^{79} +1.00000 q^{80} +6.00000 q^{82} +4.00000 q^{83} +2.00000 q^{85} -4.00000 q^{86} +3.00000 q^{88} +6.00000 q^{89} +6.00000 q^{91} -4.00000 q^{92} +8.00000 q^{94} -4.00000 q^{95} +18.0000 q^{97} -1.00000 q^{98} +O(q^{100})\)

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 −0.353553 − 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 3.00000 1.06066
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 6.00000 1.66410 0.832050 − 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) 4.00000 0.834058 0.417029 − 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −6.00000 −1.17670
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) 2.00000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) −5.00000 −0.883883
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) −4.00000 −0.648886
\(39\) 0 0
\(40\) −3.00000 −0.474342
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) −8.00000 −1.16692 −0.583460 − 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −6.00000 −0.832050
\(53\) 2.00000 0.274721 0.137361 − 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 3.00000 0.400892
\(57\) 0 0
\(58\) −2.00000 −0.262613
\(59\) 8.00000 1.04151 0.520756 − 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 − 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) −4.00000 −0.508001
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) −6.00000 −0.744208
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 − 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 2.00000 0.242536
\(69\) 0 0
\(70\) 1.00000 0.119523
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −14.0000 −1.63858 −0.819288 − 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) −16.0000 −1.80014 −0.900070 − 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) 6.00000 0.662589
\(83\) 4.00000 0.439057 0.219529 − 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) 3.00000 0.319801
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 6.00000 0.628971
\(92\) −4.00000 −0.417029
\(93\) 0 0
\(94\) 8.00000 0.825137
\(95\) −4.00000 −0.410391
\(96\) 0 0
\(97\) 18.0000 1.82762 0.913812 − 0.406138i \(-0.133125\pi\)
0.913812 + 0.406138i \(0.133125\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) −2.00000 −0.199007 −0.0995037 − 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) 0 0
\(103\) −16.0000 −1.57653 −0.788263 − 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) 18.0000 1.76505
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) 12.0000 1.16008 0.580042 − 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) 6.00000 0.574696 0.287348 − 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 1.00000 0.0953463
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) 6.00000 0.564433 0.282216 − 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) −4.00000 −0.373002
\(116\) −2.00000 −0.185695
\(117\) 0 0
\(118\) −8.00000 −0.736460
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −10.0000 −0.905357
\(123\) 0 0
\(124\) −4.00000 −0.359211
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 − 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 3.00000 0.265165
\(129\) 0 0
\(130\) 6.00000 0.526235
\(131\) −20.0000 −1.74741 −0.873704 − 0.486458i \(-0.838289\pi\)
−0.873704 + 0.486458i \(0.838289\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) −2.00000 −0.170872 −0.0854358 − 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 0 0
\(139\) −12.0000 −1.01783 −0.508913 − 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 1.00000 0.0845154
\(141\) 0 0
\(142\) 0 0
\(143\) 6.00000 0.501745
\(144\) 0 0
\(145\) −2.00000 −0.166091
\(146\) 14.0000 1.15865
\(147\) 0 0
\(148\) 2.00000 0.164399
\(149\) 10.0000 0.819232 0.409616 − 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 − 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 12.0000 0.973329
\(153\) 0 0
\(154\) −1.00000 −0.0805823
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 − 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 16.0000 1.27289
\(159\) 0 0
\(160\) 5.00000 0.395285
\(161\) 4.00000 0.315244
\(162\) 0 0
\(163\) 24.0000 1.87983 0.939913 − 0.341415i \(-0.110906\pi\)
0.939913 + 0.341415i \(0.110906\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) −4.00000 −0.310460
\(167\) 16.0000 1.23812 0.619059 − 0.785345i \(-0.287514\pi\)
0.619059 + 0.785345i \(0.287514\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) −2.00000 −0.153393
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) 18.0000 1.36851 0.684257 − 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) −6.00000 −0.449719
\(179\) 12.0000 0.896922 0.448461 − 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 − 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) −6.00000 −0.444750
\(183\) 0 0
\(184\) 12.0000 0.884652
\(185\) 2.00000 0.147043
\(186\) 0 0
\(187\) −2.00000 −0.146254
\(188\) 8.00000 0.583460
\(189\) 0 0
\(190\) 4.00000 0.290191
\(191\) 8.00000 0.578860 0.289430 − 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 0 0
\(193\) −18.0000 −1.29567 −0.647834 − 0.761781i \(-0.724325\pi\)
−0.647834 + 0.761781i \(0.724325\pi\)
\(194\) −18.0000 −1.29232
\(195\) 0 0
\(196\) −1.00000 −0.0714286
\(197\) 6.00000 0.427482 0.213741 − 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) 20.0000 1.41776 0.708881 − 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 3.00000 0.212132
\(201\) 0 0
\(202\) 2.00000 0.140720
\(203\) 2.00000 0.140372
\(204\) 0 0
\(205\) 6.00000 0.419058
\(206\) 16.0000 1.11477
\(207\) 0 0
\(208\) −6.00000 −0.416025
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 − 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) −2.00000 −0.137361
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) −4.00000 −0.272798
\(216\) 0 0
\(217\) 4.00000 0.271538
\(218\) −6.00000 −0.406371
\(219\) 0 0
\(220\) 1.00000 0.0674200
\(221\) −12.0000 −0.807207
\(222\) 0 0
\(223\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) −5.00000 −0.334077
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) 28.0000 1.85843 0.929213 − 0.369546i \(-0.120487\pi\)
0.929213 + 0.369546i \(0.120487\pi\)
\(228\) 0 0
\(229\) −2.00000 −0.132164 −0.0660819 − 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 4.00000 0.263752
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) −6.00000 −0.393073 −0.196537 − 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) 8.00000 0.521862
\(236\) −8.00000 −0.520756
\(237\) 0 0
\(238\) 2.00000 0.129641
\(239\) −8.00000 −0.517477 −0.258738 − 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 0 0
\(241\) −18.0000 −1.15948 −0.579741 − 0.814801i \(-0.696846\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 0 0
\(244\) −10.0000 −0.640184
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) 24.0000 1.52708
\(248\) 12.0000 0.762001
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) 24.0000 1.51487 0.757433 − 0.652913i \(-0.226453\pi\)
0.757433 + 0.652913i \(0.226453\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) −18.0000 −1.12281 −0.561405 − 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) −2.00000 −0.124274
\(260\) 6.00000 0.372104
\(261\) 0 0
\(262\) 20.0000 1.23560
\(263\) −16.0000 −0.986602 −0.493301 − 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) 0 0
\(265\) −2.00000 −0.122859
\(266\) −4.00000 −0.245256
\(267\) 0 0
\(268\) 8.00000 0.488678
\(269\) 10.0000 0.609711 0.304855 − 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 0 0
\(271\) 24.0000 1.45790 0.728948 − 0.684569i \(-0.240010\pi\)
0.728948 + 0.684569i \(0.240010\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) 2.00000 0.120824
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 − 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 12.0000 0.719712
\(279\) 0 0
\(280\) −3.00000 −0.179284
\(281\) 6.00000 0.357930 0.178965 − 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 − 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −6.00000 −0.354787
\(287\) −6.00000 −0.354169
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 2.00000 0.117444
\(291\) 0 0
\(292\) 14.0000 0.819288
\(293\) 10.0000 0.584206 0.292103 − 0.956387i \(-0.405645\pi\)
0.292103 + 0.956387i \(0.405645\pi\)
\(294\) 0 0
\(295\) −8.00000 −0.465778
\(296\) −6.00000 −0.348743
\(297\) 0 0
\(298\) −10.0000 −0.579284
\(299\) 24.0000 1.38796
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) 16.0000 0.920697
\(303\) 0 0
\(304\) −4.00000 −0.229416
\(305\) −10.0000 −0.572598
\(306\) 0 0
\(307\) 12.0000 0.684876 0.342438 − 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) −1.00000 −0.0569803
\(309\) 0 0
\(310\) 4.00000 0.227185
\(311\) −12.0000 −0.680458 −0.340229 − 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) 26.0000 1.46961 0.734803 − 0.678280i \(-0.237274\pi\)
0.734803 + 0.678280i \(0.237274\pi\)
\(314\) 2.00000 0.112867
\(315\) 0 0
\(316\) 16.0000 0.900070
\(317\) −14.0000 −0.786318 −0.393159 − 0.919470i \(-0.628618\pi\)
−0.393159 + 0.919470i \(0.628618\pi\)
\(318\) 0 0
\(319\) 2.00000 0.111979
\(320\) −7.00000 −0.391312
\(321\) 0 0
\(322\) −4.00000 −0.222911
\(323\) −8.00000 −0.445132
\(324\) 0 0
\(325\) 6.00000 0.332820
\(326\) −24.0000 −1.32924
\(327\) 0 0
\(328\) −18.0000 −0.993884
\(329\) −8.00000 −0.441054
\(330\) 0 0
\(331\) −12.0000 −0.659580 −0.329790 − 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) −4.00000 −0.219529
\(333\) 0 0
\(334\) −16.0000 −0.875481
\(335\) 8.00000 0.437087
\(336\) 0 0
\(337\) 22.0000 1.19842 0.599208 − 0.800593i \(-0.295482\pi\)
0.599208 + 0.800593i \(0.295482\pi\)
\(338\) −23.0000 −1.25104
\(339\) 0 0
\(340\) −2.00000 −0.108465
\(341\) 4.00000 0.216612
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 12.0000 0.646997
\(345\) 0 0
\(346\) −18.0000 −0.967686
\(347\) −28.0000 −1.50312 −0.751559 − 0.659665i \(-0.770698\pi\)
−0.751559 + 0.659665i \(0.770698\pi\)
\(348\) 0 0
\(349\) −22.0000 −1.17763 −0.588817 − 0.808267i \(-0.700406\pi\)
−0.588817 + 0.808267i \(0.700406\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 0 0
\(352\) −5.00000 −0.266501
\(353\) 30.0000 1.59674 0.798369 − 0.602168i \(-0.205696\pi\)
0.798369 + 0.602168i \(0.205696\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) 24.0000 1.26667 0.633336 − 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 2.00000 0.105118
\(363\) 0 0
\(364\) −6.00000 −0.314485
\(365\) 14.0000 0.732793
\(366\) 0 0
\(367\) −8.00000 −0.417597 −0.208798 − 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) −4.00000 −0.208514
\(369\) 0 0
\(370\) −2.00000 −0.103975
\(371\) 2.00000 0.103835
\(372\) 0 0
\(373\) 26.0000 1.34623 0.673114 − 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(374\) 2.00000 0.103418
\(375\) 0 0
\(376\) −24.0000 −1.23771
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) −4.00000 −0.205466 −0.102733 − 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 4.00000 0.205196
\(381\) 0 0
\(382\) −8.00000 −0.409316
\(383\) −16.0000 −0.817562 −0.408781 − 0.912633i \(-0.634046\pi\)
−0.408781 + 0.912633i \(0.634046\pi\)
\(384\) 0 0
\(385\) −1.00000 −0.0509647
\(386\) 18.0000 0.916176
\(387\) 0 0
\(388\) −18.0000 −0.913812
\(389\) −6.00000 −0.304212 −0.152106 − 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) −8.00000 −0.404577
\(392\) 3.00000 0.151523
\(393\) 0 0
\(394\) −6.00000 −0.302276
\(395\) 16.0000 0.805047
\(396\) 0 0
\(397\) 14.0000 0.702640 0.351320 − 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) −20.0000 −1.00251
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) −18.0000 −0.898877 −0.449439 − 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) 24.0000 1.19553
\(404\) 2.00000 0.0995037
\(405\) 0 0
\(406\) −2.00000 −0.0992583
\(407\) −2.00000 −0.0991363
\(408\) 0 0
\(409\) −10.0000 −0.494468 −0.247234 − 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) −6.00000 −0.296319
\(411\) 0 0
\(412\) 16.0000 0.788263
\(413\) 8.00000 0.393654
\(414\) 0 0
\(415\) −4.00000 −0.196352
\(416\) −30.0000 −1.47087
\(417\) 0 0
\(418\) −4.00000 −0.195646
\(419\) 24.0000 1.17248 0.586238 − 0.810139i \(-0.300608\pi\)
0.586238 + 0.810139i \(0.300608\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 − 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) −4.00000 −0.194717
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) 10.0000 0.483934
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) 4.00000 0.192897
\(431\) 32.0000 1.54139 0.770693 − 0.637207i \(-0.219910\pi\)
0.770693 + 0.637207i \(0.219910\pi\)
\(432\) 0 0
\(433\) 34.0000 1.63394 0.816968 − 0.576683i \(-0.195653\pi\)
0.816968 + 0.576683i \(0.195653\pi\)
\(434\) −4.00000 −0.192006
\(435\) 0 0
\(436\) −6.00000 −0.287348
\(437\) 16.0000 0.765384
\(438\) 0 0
\(439\) −32.0000 −1.52728 −0.763638 − 0.645644i \(-0.776589\pi\)
−0.763638 + 0.645644i \(0.776589\pi\)
\(440\) −3.00000 −0.143019
\(441\) 0 0
\(442\) 12.0000 0.570782
\(443\) 24.0000 1.14027 0.570137 − 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) 0 0
\(445\) −6.00000 −0.284427
\(446\) 0 0
\(447\) 0 0
\(448\) 7.00000 0.330719
\(449\) −18.0000 −0.849473 −0.424736 − 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) −6.00000 −0.282529
\(452\) −6.00000 −0.282216
\(453\) 0 0
\(454\) −28.0000 −1.31411
\(455\) −6.00000 −0.281284
\(456\) 0 0
\(457\) 30.0000 1.40334 0.701670 − 0.712502i \(-0.252438\pi\)
0.701670 + 0.712502i \(0.252438\pi\)
\(458\) 2.00000 0.0934539
\(459\) 0 0
\(460\) 4.00000 0.186501
\(461\) −34.0000 −1.58354 −0.791769 − 0.610821i \(-0.790840\pi\)
−0.791769 + 0.610821i \(0.790840\pi\)
\(462\) 0 0
\(463\) 36.0000 1.67306 0.836531 − 0.547920i \(-0.184580\pi\)
0.836531 + 0.547920i \(0.184580\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) −28.0000 −1.29569 −0.647843 − 0.761774i \(-0.724329\pi\)
−0.647843 + 0.761774i \(0.724329\pi\)
\(468\) 0 0
\(469\) −8.00000 −0.369406
\(470\) −8.00000 −0.369012
\(471\) 0 0
\(472\) 24.0000 1.10469
\(473\) 4.00000 0.183920
\(474\) 0 0
\(475\) 4.00000 0.183533
\(476\) 2.00000 0.0916698
\(477\) 0 0
\(478\) 8.00000 0.365911
\(479\) 16.0000 0.731059 0.365529 − 0.930800i \(-0.380888\pi\)
0.365529 + 0.930800i \(0.380888\pi\)
\(480\) 0 0
\(481\) −12.0000 −0.547153
\(482\) 18.0000 0.819878
\(483\) 0 0
\(484\) −1.00000 −0.0454545
\(485\) −18.0000 −0.817338
\(486\) 0 0
\(487\) −28.0000 −1.26880 −0.634401 − 0.773004i \(-0.718753\pi\)
−0.634401 + 0.773004i \(0.718753\pi\)
\(488\) 30.0000 1.35804
\(489\) 0 0
\(490\) 1.00000 0.0451754
\(491\) −12.0000 −0.541552 −0.270776 − 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 0 0
\(493\) −4.00000 −0.180151
\(494\) −24.0000 −1.07981
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 0 0
\(498\) 0 0
\(499\) 20.0000 0.895323 0.447661 − 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) −24.0000 −1.07117
\(503\) −16.0000 −0.713405 −0.356702 − 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) 0 0
\(505\) 2.00000 0.0889988
\(506\) −4.00000 −0.177822
\(507\) 0 0
\(508\) −8.00000 −0.354943
\(509\) −22.0000 −0.975133 −0.487566 − 0.873086i \(-0.662115\pi\)
−0.487566 + 0.873086i \(0.662115\pi\)
\(510\) 0 0
\(511\) −14.0000 −0.619324
\(512\) 11.0000 0.486136
\(513\) 0 0
\(514\) 18.0000 0.793946
\(515\) 16.0000 0.705044
\(516\) 0 0
\(517\) −8.00000 −0.351840
\(518\) 2.00000 0.0878750
\(519\) 0 0
\(520\) −18.0000 −0.789352
\(521\) 14.0000 0.613351 0.306676 − 0.951814i \(-0.400783\pi\)
0.306676 + 0.951814i \(0.400783\pi\)
\(522\) 0 0
\(523\) 20.0000 0.874539 0.437269 − 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 20.0000 0.873704
\(525\) 0 0
\(526\) 16.0000 0.697633
\(527\) −8.00000 −0.348485
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 2.00000 0.0868744
\(531\) 0 0
\(532\) −4.00000 −0.173422
\(533\) −36.0000 −1.55933
\(534\) 0 0
\(535\) −12.0000 −0.518805
\(536\) −24.0000 −1.03664
\(537\) 0 0
\(538\) −10.0000 −0.431131
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) 30.0000 1.28980 0.644900 − 0.764267i \(-0.276899\pi\)
0.644900 + 0.764267i \(0.276899\pi\)
\(542\) −24.0000 −1.03089
\(543\) 0 0
\(544\) 10.0000 0.428746
\(545\) −6.00000 −0.257012
\(546\) 0 0
\(547\) −12.0000 −0.513083 −0.256541 − 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) 2.00000 0.0854358
\(549\) 0 0
\(550\) −1.00000 −0.0426401
\(551\) 8.00000 0.340811
\(552\) 0 0
\(553\) −16.0000 −0.680389
\(554\) −2.00000 −0.0849719
\(555\) 0 0
\(556\) 12.0000 0.508913
\(557\) −2.00000 −0.0847427 −0.0423714 − 0.999102i \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) 0 0
\(559\) 24.0000 1.01509
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) −6.00000 −0.253095
\(563\) 44.0000 1.85438 0.927189 − 0.374593i \(-0.122217\pi\)
0.927189 + 0.374593i \(0.122217\pi\)
\(564\) 0 0
\(565\) −6.00000 −0.252422
\(566\) 4.00000 0.168133
\(567\) 0 0
\(568\) 0 0
\(569\) 30.0000 1.25767 0.628833 − 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 0 0
\(571\) −12.0000 −0.502184 −0.251092 − 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) −6.00000 −0.250873
\(573\) 0 0
\(574\) 6.00000 0.250435
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) 34.0000 1.41544 0.707719 − 0.706494i \(-0.249724\pi\)
0.707719 + 0.706494i \(0.249724\pi\)
\(578\) 13.0000 0.540729
\(579\) 0 0
\(580\) 2.00000 0.0830455
\(581\) 4.00000 0.165948
\(582\) 0 0
\(583\) 2.00000 0.0828315
\(584\) −42.0000 −1.73797
\(585\) 0 0
\(586\) −10.0000 −0.413096
\(587\) 12.0000 0.495293 0.247647 − 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 0 0
\(589\) 16.0000 0.659269
\(590\) 8.00000 0.329355
\(591\) 0 0
\(592\) 2.00000 0.0821995
\(593\) −42.0000 −1.72473 −0.862367 − 0.506284i \(-0.831019\pi\)
−0.862367 + 0.506284i \(0.831019\pi\)
\(594\) 0 0
\(595\) 2.00000 0.0819920
\(596\) −10.0000 −0.409616
\(597\) 0 0
\(598\) −24.0000 −0.981433
\(599\) −32.0000 −1.30748 −0.653742 − 0.756717i \(-0.726802\pi\)
−0.653742 + 0.756717i \(0.726802\pi\)
\(600\) 0 0
\(601\) 6.00000 0.244745 0.122373 − 0.992484i \(-0.460950\pi\)
0.122373 + 0.992484i \(0.460950\pi\)
\(602\) −4.00000 −0.163028
\(603\) 0 0
\(604\) 16.0000 0.651031
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) 8.00000 0.324710 0.162355 − 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) −20.0000 −0.811107
\(609\) 0 0
\(610\) 10.0000 0.404888
\(611\) −48.0000 −1.94187
\(612\) 0 0
\(613\) 18.0000 0.727013 0.363507 − 0.931592i \(-0.381579\pi\)
0.363507 + 0.931592i \(0.381579\pi\)
\(614\) −12.0000 −0.484281
\(615\) 0 0
\(616\) 3.00000 0.120873
\(617\) −18.0000 −0.724653 −0.362326 − 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 0 0
\(619\) 24.0000 0.964641 0.482321 − 0.875995i \(-0.339794\pi\)
0.482321 + 0.875995i \(0.339794\pi\)
\(620\) 4.00000 0.160644
\(621\) 0 0
\(622\) 12.0000 0.481156
\(623\) 6.00000 0.240385
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −26.0000 −1.03917
\(627\) 0 0
\(628\) 2.00000 0.0798087
\(629\) 4.00000 0.159490
\(630\) 0 0
\(631\) −24.0000 −0.955425 −0.477712 − 0.878516i \(-0.658534\pi\)
−0.477712 + 0.878516i \(0.658534\pi\)
\(632\) −48.0000 −1.90934
\(633\) 0 0
\(634\) 14.0000 0.556011
\(635\) −8.00000 −0.317470
\(636\) 0 0
\(637\) 6.00000 0.237729
\(638\) −2.00000 −0.0791808
\(639\) 0 0
\(640\) −3.00000 −0.118585
\(641\) −18.0000 −0.710957 −0.355479 − 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 0 0
\(643\) −20.0000 −0.788723 −0.394362 − 0.918955i \(-0.629034\pi\)
−0.394362 + 0.918955i \(0.629034\pi\)
\(644\) −4.00000 −0.157622
\(645\) 0 0
\(646\) 8.00000 0.314756
\(647\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 8.00000 0.314027
\(650\) −6.00000 −0.235339
\(651\) 0 0
\(652\) −24.0000 −0.939913
\(653\) 10.0000 0.391330 0.195665 − 0.980671i \(-0.437313\pi\)
0.195665 + 0.980671i \(0.437313\pi\)
\(654\) 0 0
\(655\) 20.0000 0.781465
\(656\) 6.00000 0.234261
\(657\) 0 0
\(658\) 8.00000 0.311872
\(659\) −12.0000 −0.467454 −0.233727 − 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 − 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) 12.0000 0.466393
\(663\) 0 0
\(664\) 12.0000 0.465690
\(665\) −4.00000 −0.155113
\(666\) 0 0
\(667\) 8.00000 0.309761
\(668\) −16.0000 −0.619059
\(669\) 0 0
\(670\) −8.00000 −0.309067
\(671\) 10.0000 0.386046
\(672\) 0 0
\(673\) −10.0000 −0.385472 −0.192736 − 0.981251i \(-0.561736\pi\)
−0.192736 + 0.981251i \(0.561736\pi\)
\(674\) −22.0000 −0.847408
\(675\) 0 0
\(676\) −23.0000 −0.884615
\(677\) 2.00000 0.0768662 0.0384331 − 0.999261i \(-0.487763\pi\)
0.0384331 + 0.999261i \(0.487763\pi\)
\(678\) 0 0
\(679\) 18.0000 0.690777
\(680\) 6.00000 0.230089
\(681\) 0 0
\(682\) −4.00000 −0.153168
\(683\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 2.00000 0.0764161
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) −4.00000 −0.152499
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) −24.0000 −0.913003 −0.456502 − 0.889723i \(-0.650898\pi\)
−0.456502 + 0.889723i \(0.650898\pi\)
\(692\) −18.0000 −0.684257
\(693\) 0 0
\(694\) 28.0000 1.06287
\(695\) 12.0000 0.455186
\(696\) 0 0
\(697\) 12.0000 0.454532
\(698\) 22.0000 0.832712
\(699\) 0 0
\(700\) −1.00000 −0.0377964
\(701\) 2.00000 0.0755390 0.0377695 − 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) 0 0
\(703\) −8.00000 −0.301726
\(704\) 7.00000 0.263822
\(705\) 0 0
\(706\) −30.0000 −1.12906
\(707\) −2.00000 −0.0752177
\(708\) 0 0
\(709\) −26.0000 −0.976450 −0.488225 − 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 18.0000 0.674579
\(713\) 16.0000 0.599205
\(714\) 0 0
\(715\) −6.00000 −0.224387
\(716\) −12.0000 −0.448461
\(717\) 0 0
\(718\) −24.0000 −0.895672
\(719\) −44.0000 −1.64092 −0.820462 − 0.571702i \(-0.806283\pi\)
−0.820462 + 0.571702i \(0.806283\pi\)
\(720\) 0 0
\(721\) −16.0000 −0.595871
\(722\) 3.00000 0.111648
\(723\) 0 0
\(724\) 2.00000 0.0743294
\(725\) 2.00000 0.0742781
\(726\) 0 0
\(727\) 16.0000 0.593407 0.296704 − 0.954970i \(-0.404113\pi\)
0.296704 + 0.954970i \(0.404113\pi\)
\(728\) 18.0000 0.667124
\(729\) 0 0
\(730\) −14.0000 −0.518163
\(731\) −8.00000 −0.295891
\(732\) 0 0
\(733\) −10.0000 −0.369358 −0.184679 − 0.982799i \(-0.559125\pi\)
−0.184679 + 0.982799i \(0.559125\pi\)
\(734\) 8.00000 0.295285
\(735\) 0 0
\(736\) −20.0000 −0.737210
\(737\) −8.00000 −0.294684
\(738\) 0 0
\(739\) 20.0000 0.735712 0.367856 − 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) −2.00000 −0.0735215
\(741\) 0 0
\(742\) −2.00000 −0.0734223
\(743\) 48.0000 1.76095 0.880475 − 0.474093i \(-0.157224\pi\)
0.880475 + 0.474093i \(0.157224\pi\)
\(744\) 0 0
\(745\) −10.0000 −0.366372
\(746\) −26.0000 −0.951928
\(747\) 0 0
\(748\) 2.00000 0.0731272
\(749\) 12.0000 0.438470
\(750\) 0 0
\(751\) 8.00000 0.291924 0.145962 − 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) 8.00000 0.291730
\(753\) 0 0
\(754\) −12.0000 −0.437014
\(755\) 16.0000 0.582300
\(756\) 0 0
\(757\) −10.0000 −0.363456 −0.181728 − 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) 4.00000 0.145287
\(759\) 0 0
\(760\) −12.0000 −0.435286
\(761\) 42.0000 1.52250 0.761249 − 0.648459i \(-0.224586\pi\)
0.761249 + 0.648459i \(0.224586\pi\)
\(762\) 0 0
\(763\) 6.00000 0.217215
\(764\) −8.00000 −0.289430
\(765\) 0 0
\(766\) 16.0000 0.578103
\(767\) 48.0000 1.73318
\(768\) 0 0
\(769\) −10.0000 −0.360609 −0.180305 − 0.983611i \(-0.557708\pi\)
−0.180305 + 0.983611i \(0.557708\pi\)
\(770\) 1.00000 0.0360375
\(771\) 0 0
\(772\) 18.0000 0.647834
\(773\) −54.0000 −1.94225 −0.971123 − 0.238581i \(-0.923318\pi\)
−0.971123 + 0.238581i \(0.923318\pi\)
\(774\) 0 0
\(775\) 4.00000 0.143684
\(776\) 54.0000 1.93849
\(777\) 0 0
\(778\) 6.00000 0.215110
\(779\) −24.0000 −0.859889
\(780\) 0 0
\(781\) 0 0
\(782\) 8.00000 0.286079
\(783\) 0 0
\(784\) −1.00000 −0.0357143
\(785\) 2.00000 0.0713831
\(786\) 0 0
\(787\) −44.0000 −1.56843 −0.784215 − 0.620489i \(-0.786934\pi\)
−0.784215 + 0.620489i \(0.786934\pi\)
\(788\) −6.00000 −0.213741
\(789\) 0 0
\(790\) −16.0000 −0.569254
\(791\) 6.00000 0.213335
\(792\) 0 0
\(793\) 60.0000 2.13066
\(794\) −14.0000 −0.496841
\(795\) 0 0
\(796\) −20.0000 −0.708881
\(797\) −30.0000 −1.06265 −0.531327 − 0.847167i \(-0.678307\pi\)
−0.531327 + 0.847167i \(0.678307\pi\)
\(798\) 0 0
\(799\) 16.0000 0.566039
\(800\) −5.00000 −0.176777
\(801\) 0 0
\(802\) 18.0000 0.635602
\(803\) −14.0000 −0.494049
\(804\) 0 0
\(805\) −4.00000 −0.140981
\(806\) −24.0000 −0.845364
\(807\) 0 0
\(808\) −6.00000 −0.211079
\(809\) 30.0000 1.05474 0.527372 − 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) −12.0000 −0.421377 −0.210688 − 0.977553i \(-0.567571\pi\)
−0.210688 + 0.977553i \(0.567571\pi\)
\(812\) −2.00000 −0.0701862
\(813\) 0 0
\(814\) 2.00000 0.0701000
\(815\) −24.0000 −0.840683
\(816\) 0 0
\(817\) 16.0000 0.559769
\(818\) 10.0000 0.349642
\(819\) 0 0
\(820\) −6.00000 −0.209529
\(821\) 18.0000 0.628204 0.314102 − 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) 0 0
\(823\) −4.00000 −0.139431 −0.0697156 − 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) −48.0000 −1.67216
\(825\) 0 0
\(826\) −8.00000 −0.278356
\(827\) 36.0000 1.25184 0.625921 − 0.779886i \(-0.284723\pi\)
0.625921 + 0.779886i \(0.284723\pi\)
\(828\) 0 0
\(829\) 30.0000 1.04194 0.520972 − 0.853574i \(-0.325570\pi\)
0.520972 + 0.853574i \(0.325570\pi\)
\(830\) 4.00000 0.138842
\(831\) 0 0
\(832\) 42.0000 1.45609
\(833\) −2.00000 −0.0692959
\(834\) 0 0
\(835\) −16.0000 −0.553703
\(836\) −4.00000 −0.138343
\(837\) 0 0
\(838\) −24.0000 −0.829066
\(839\) 36.0000 1.24286 0.621429 − 0.783470i \(-0.286552\pi\)
0.621429 + 0.783470i \(0.286552\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 26.0000 0.896019
\(843\) 0 0
\(844\) −4.00000 −0.137686
\(845\) −23.0000 −0.791224
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) −2.00000 −0.0686803
\(849\) 0 0
\(850\) 2.00000 0.0685994
\(851\) −8.00000 −0.274236
\(852\) 0 0
\(853\) 54.0000 1.84892 0.924462 − 0.381273i \(-0.124514\pi\)
0.924462 + 0.381273i \(0.124514\pi\)
\(854\) −10.0000 −0.342193
\(855\) 0 0
\(856\) 36.0000 1.23045
\(857\) 30.0000 1.02478 0.512390 − 0.858753i \(-0.328760\pi\)
0.512390 + 0.858753i \(0.328760\pi\)
\(858\) 0 0
\(859\) 8.00000 0.272956 0.136478 − 0.990643i \(-0.456422\pi\)
0.136478 + 0.990643i \(0.456422\pi\)
\(860\) 4.00000 0.136399
\(861\) 0 0
\(862\) −32.0000 −1.08992
\(863\) 12.0000 0.408485 0.204242 − 0.978920i \(-0.434527\pi\)
0.204242 + 0.978920i \(0.434527\pi\)
\(864\) 0 0
\(865\) −18.0000 −0.612018
\(866\) −34.0000 −1.15537
\(867\) 0 0
\(868\) −4.00000 −0.135769
\(869\) −16.0000 −0.542763
\(870\) 0 0
\(871\) −48.0000 −1.62642
\(872\) 18.0000 0.609557
\(873\) 0 0
\(874\) −16.0000 −0.541208
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) −22.0000 −0.742887 −0.371444 − 0.928456i \(-0.621137\pi\)
−0.371444 + 0.928456i \(0.621137\pi\)
\(878\) 32.0000 1.07995
\(879\) 0 0
\(880\) 1.00000 0.0337100
\(881\) −42.0000 −1.41502 −0.707508 − 0.706705i \(-0.750181\pi\)
−0.707508 + 0.706705i \(0.750181\pi\)
\(882\) 0 0
\(883\) 16.0000 0.538443 0.269221 − 0.963078i \(-0.413234\pi\)
0.269221 + 0.963078i \(0.413234\pi\)
\(884\) 12.0000 0.403604
\(885\) 0 0
\(886\) −24.0000 −0.806296
\(887\) 8.00000 0.268614 0.134307 − 0.990940i \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(888\) 0 0
\(889\) 8.00000 0.268311
\(890\) 6.00000 0.201120
\(891\) 0 0
\(892\) 0 0
\(893\) −32.0000 −1.07084
\(894\) 0 0
\(895\) −12.0000 −0.401116
\(896\) 3.00000 0.100223
\(897\) 0 0
\(898\) 18.0000 0.600668
\(899\) 8.00000 0.266815
\(900\) 0 0
\(901\) −4.00000 −0.133259
\(902\) 6.00000 0.199778
\(903\) 0 0
\(904\) 18.0000 0.598671
\(905\) 2.00000 0.0664822
\(906\) 0 0
\(907\) −32.0000 −1.06254 −0.531271 − 0.847202i \(-0.678286\pi\)
−0.531271 + 0.847202i \(0.678286\pi\)
\(908\) −28.0000 −0.929213
\(909\) 0 0
\(910\) 6.00000 0.198898
\(911\) 48.0000 1.59031 0.795155 − 0.606406i \(-0.207389\pi\)
0.795155 + 0.606406i \(0.207389\pi\)
\(912\) 0 0
\(913\) 4.00000 0.132381
\(914\) −30.0000 −0.992312
\(915\) 0 0
\(916\) 2.00000 0.0660819
\(917\) −20.0000 −0.660458
\(918\) 0 0
\(919\) −24.0000 −0.791687 −0.395843 − 0.918318i \(-0.629548\pi\)
−0.395843 + 0.918318i \(0.629548\pi\)
\(920\) −12.0000 −0.395628
\(921\) 0 0
\(922\) 34.0000 1.11973
\(923\) 0 0
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) −36.0000 −1.18303
\(927\) 0 0
\(928\) −10.0000 −0.328266
\(929\) −10.0000 −0.328089 −0.164045 − 0.986453i \(-0.552454\pi\)
−0.164045 + 0.986453i \(0.552454\pi\)
\(930\) 0 0
\(931\) 4.00000 0.131095
\(932\) 6.00000 0.196537
\(933\) 0 0
\(934\) 28.0000 0.916188
\(935\) 2.00000 0.0654070
\(936\) 0 0
\(937\) 10.0000 0.326686 0.163343 − 0.986569i \(-0.447772\pi\)
0.163343 + 0.986569i \(0.447772\pi\)
\(938\) 8.00000 0.261209
\(939\) 0 0
\(940\) −8.00000 −0.260931
\(941\) −18.0000 −0.586783 −0.293392 − 0.955992i \(-0.594784\pi\)
−0.293392 + 0.955992i \(0.594784\pi\)
\(942\) 0 0
\(943\) −24.0000 −0.781548
\(944\) −8.00000 −0.260378
\(945\) 0 0
\(946\) −4.00000 −0.130051
\(947\) −32.0000 −1.03986 −0.519930 − 0.854209i \(-0.674042\pi\)
−0.519930 + 0.854209i \(0.674042\pi\)
\(948\) 0 0
\(949\) −84.0000 −2.72676
\(950\) −4.00000 −0.129777
\(951\) 0 0
\(952\) −6.00000 −0.194461
\(953\) −46.0000 −1.49009 −0.745043 − 0.667016i \(-0.767571\pi\)
−0.745043 + 0.667016i \(0.767571\pi\)
\(954\) 0 0
\(955\) −8.00000 −0.258874
\(956\) 8.00000 0.258738
\(957\) 0 0
\(958\) −16.0000 −0.516937
\(959\) −2.00000 −0.0645834
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 12.0000 0.386896
\(963\) 0 0
\(964\) 18.0000 0.579741
\(965\) 18.0000 0.579441
\(966\) 0 0
\(967\) −56.0000 −1.80084 −0.900419 − 0.435023i \(-0.856740\pi\)
−0.900419 + 0.435023i \(0.856740\pi\)
\(968\) 3.00000 0.0964237
\(969\) 0 0
\(970\) 18.0000 0.577945
\(971\) −32.0000 −1.02693 −0.513464 − 0.858111i \(-0.671638\pi\)
−0.513464 + 0.858111i \(0.671638\pi\)
\(972\) 0 0
\(973\) −12.0000 −0.384702
\(974\) 28.0000 0.897178
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) −26.0000 −0.831814 −0.415907 − 0.909407i \(-0.636536\pi\)
−0.415907 + 0.909407i \(0.636536\pi\)
\(978\) 0 0
\(979\) 6.00000 0.191761
\(980\) 1.00000 0.0319438
\(981\) 0 0
\(982\) 12.0000 0.382935
\(983\) 16.0000 0.510321 0.255160 − 0.966899i \(-0.417872\pi\)
0.255160 + 0.966899i \(0.417872\pi\)
\(984\) 0 0
\(985\) −6.00000 −0.191176
\(986\) 4.00000 0.127386
\(987\) 0 0
\(988\) −24.0000 −0.763542
\(989\) 16.0000 0.508770
\(990\) 0 0
\(991\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) −20.0000 −0.635001
\(993\) 0 0
\(994\) 0 0
\(995\) −20.0000 −0.634043
\(996\) 0 0
\(997\) −10.0000 −0.316703 −0.158352 − 0.987383i \(-0.550618\pi\)
−0.158352 + 0.987383i \(0.550618\pi\)
\(998\) −20.0000 −0.633089
\(999\) 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 3465.2.a.c.1.1 1
3.2 odd 2 1155.2.a.m.1.1 ✓ 1
15.14 odd 2 5775.2.a.d.1.1 1
21.20 even 2 8085.2.a.r.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1155.2.a.m.1.1 ✓ 1 3.2 odd 2
3465.2.a.c.1.1 1 1.1 even 1 trivial
5775.2.a.d.1.1 1 15.14 odd 2
8085.2.a.r.1.1 1 21.20 even 2