Properties

Label 1815.2.a.h.1.2
Level $1815$
Weight $2$
Character 1815.1
Self dual yes
Analytic conductor $14.493$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1815,2,Mod(1,1815)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1815, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1815.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1815.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: \(14.4928479669\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 1815.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.73205 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.73205 q^{6} -1.73205 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.73205 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.73205 q^{6} -1.73205 q^{8} +1.00000 q^{9} -1.73205 q^{10} -1.00000 q^{12} +3.46410 q^{13} +1.00000 q^{15} -5.00000 q^{16} +1.73205 q^{18} -3.46410 q^{19} -1.00000 q^{20} +1.73205 q^{24} +1.00000 q^{25} +6.00000 q^{26} -1.00000 q^{27} -3.46410 q^{29} +1.73205 q^{30} -8.00000 q^{31} -5.19615 q^{32} +1.00000 q^{36} -2.00000 q^{37} -6.00000 q^{38} -3.46410 q^{39} +1.73205 q^{40} -10.3923 q^{41} -6.92820 q^{43} -1.00000 q^{45} +5.00000 q^{48} -7.00000 q^{49} +1.73205 q^{50} +3.46410 q^{52} -6.00000 q^{53} -1.73205 q^{54} +3.46410 q^{57} -6.00000 q^{58} +12.0000 q^{59} +1.00000 q^{60} -13.8564 q^{61} -13.8564 q^{62} +1.00000 q^{64} -3.46410 q^{65} -8.00000 q^{67} +12.0000 q^{71} -1.73205 q^{72} +3.46410 q^{73} -3.46410 q^{74} -1.00000 q^{75} -3.46410 q^{76} -6.00000 q^{78} +10.3923 q^{79} +5.00000 q^{80} +1.00000 q^{81} -18.0000 q^{82} +17.3205 q^{83} -12.0000 q^{86} +3.46410 q^{87} -6.00000 q^{89} -1.73205 q^{90} +8.00000 q^{93} +3.46410 q^{95} +5.19615 q^{96} -10.0000 q^{97} -12.1244 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{9} - 2 q^{12} + 2 q^{15} - 10 q^{16} - 2 q^{20} + 2 q^{25} + 12 q^{26} - 2 q^{27} - 16 q^{31} + 2 q^{36} - 4 q^{37} - 12 q^{38} - 2 q^{45} + 10 q^{48} - 14 q^{49} - 12 q^{53} - 12 q^{58} + 24 q^{59} + 2 q^{60} + 2 q^{64} - 16 q^{67} + 24 q^{71} - 2 q^{75} - 12 q^{78} + 10 q^{80} + 2 q^{81} - 36 q^{82} - 24 q^{86} - 12 q^{89} + 16 q^{93} - 20 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.73205 1.22474 0.612372 0.790569i \(-0.290215\pi\)
0.612372 + 0.790569i \(0.290215\pi\)
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −1.73205 −0.707107
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) −1.73205 −0.612372
\(9\) 1.00000 0.333333
\(10\) −1.73205 −0.547723
\(11\) 0 0
\(12\) −1.00000 −0.288675
\(13\) 3.46410 0.960769 0.480384 0.877058i \(-0.340497\pi\)
0.480384 + 0.877058i \(0.340497\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) −5.00000 −1.25000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 1.73205 0.408248
\(19\) −3.46410 −0.794719 −0.397360 0.917663i \(-0.630073\pi\)
−0.397360 + 0.917663i \(0.630073\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 1.73205 0.353553
\(25\) 1.00000 0.200000
\(26\) 6.00000 1.17670
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −3.46410 −0.643268 −0.321634 0.946864i \(-0.604232\pi\)
−0.321634 + 0.946864i \(0.604232\pi\)
\(30\) 1.73205 0.316228
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) −5.19615 −0.918559
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) −6.00000 −0.973329
\(39\) −3.46410 −0.554700
\(40\) 1.73205 0.273861
\(41\) −10.3923 −1.62301 −0.811503 0.584349i \(-0.801350\pi\)
−0.811503 + 0.584349i \(0.801350\pi\)
\(42\) 0 0
\(43\) −6.92820 −1.05654 −0.528271 0.849076i \(-0.677159\pi\)
−0.528271 + 0.849076i \(0.677159\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 5.00000 0.721688
\(49\) −7.00000 −1.00000
\(50\) 1.73205 0.244949
\(51\) 0 0
\(52\) 3.46410 0.480384
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) −1.73205 −0.235702
\(55\) 0 0
\(56\) 0 0
\(57\) 3.46410 0.458831
\(58\) −6.00000 −0.787839
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 1.00000 0.129099
\(61\) −13.8564 −1.77413 −0.887066 0.461644i \(-0.847260\pi\)
−0.887066 + 0.461644i \(0.847260\pi\)
\(62\) −13.8564 −1.75977
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −3.46410 −0.429669
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) −1.73205 −0.204124
\(73\) 3.46410 0.405442 0.202721 0.979236i \(-0.435021\pi\)
0.202721 + 0.979236i \(0.435021\pi\)
\(74\) −3.46410 −0.402694
\(75\) −1.00000 −0.115470
\(76\) −3.46410 −0.397360
\(77\) 0 0
\(78\) −6.00000 −0.679366
\(79\) 10.3923 1.16923 0.584613 0.811312i \(-0.301246\pi\)
0.584613 + 0.811312i \(0.301246\pi\)
\(80\) 5.00000 0.559017
\(81\) 1.00000 0.111111
\(82\) −18.0000 −1.98777
\(83\) 17.3205 1.90117 0.950586 0.310460i \(-0.100483\pi\)
0.950586 + 0.310460i \(0.100483\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −12.0000 −1.29399
\(87\) 3.46410 0.371391
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) −1.73205 −0.182574
\(91\) 0 0
\(92\) 0 0
\(93\) 8.00000 0.829561
\(94\) 0 0
\(95\) 3.46410 0.355409
\(96\) 5.19615 0.530330
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) −12.1244 −1.22474
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −10.3923 −1.03407 −0.517036 0.855963i \(-0.672965\pi\)
−0.517036 + 0.855963i \(0.672965\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) −10.3923 −1.00939
\(107\) −10.3923 −1.00466 −0.502331 0.864675i \(-0.667524\pi\)
−0.502331 + 0.864675i \(0.667524\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 13.8564 1.32720 0.663602 0.748086i \(-0.269027\pi\)
0.663602 + 0.748086i \(0.269027\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 6.00000 0.561951
\(115\) 0 0
\(116\) −3.46410 −0.321634
\(117\) 3.46410 0.320256
\(118\) 20.7846 1.91338
\(119\) 0 0
\(120\) −1.73205 −0.158114
\(121\) 0 0
\(122\) −24.0000 −2.17286
\(123\) 10.3923 0.937043
\(124\) −8.00000 −0.718421
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −6.92820 −0.614779 −0.307389 0.951584i \(-0.599455\pi\)
−0.307389 + 0.951584i \(0.599455\pi\)
\(128\) 12.1244 1.07165
\(129\) 6.92820 0.609994
\(130\) −6.00000 −0.526235
\(131\) 20.7846 1.81596 0.907980 0.419014i \(-0.137624\pi\)
0.907980 + 0.419014i \(0.137624\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −13.8564 −1.19701
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) 17.3205 1.46911 0.734553 0.678551i \(-0.237392\pi\)
0.734553 + 0.678551i \(0.237392\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 20.7846 1.74421
\(143\) 0 0
\(144\) −5.00000 −0.416667
\(145\) 3.46410 0.287678
\(146\) 6.00000 0.496564
\(147\) 7.00000 0.577350
\(148\) −2.00000 −0.164399
\(149\) 10.3923 0.851371 0.425685 0.904871i \(-0.360033\pi\)
0.425685 + 0.904871i \(0.360033\pi\)
\(150\) −1.73205 −0.141421
\(151\) −3.46410 −0.281905 −0.140952 0.990016i \(-0.545016\pi\)
−0.140952 + 0.990016i \(0.545016\pi\)
\(152\) 6.00000 0.486664
\(153\) 0 0
\(154\) 0 0
\(155\) 8.00000 0.642575
\(156\) −3.46410 −0.277350
\(157\) −14.0000 −1.11732 −0.558661 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) 18.0000 1.43200
\(159\) 6.00000 0.475831
\(160\) 5.19615 0.410792
\(161\) 0 0
\(162\) 1.73205 0.136083
\(163\) −16.0000 −1.25322 −0.626608 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) −10.3923 −0.811503
\(165\) 0 0
\(166\) 30.0000 2.32845
\(167\) −17.3205 −1.34030 −0.670151 0.742225i \(-0.733770\pi\)
−0.670151 + 0.742225i \(0.733770\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) −3.46410 −0.264906
\(172\) −6.92820 −0.528271
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 6.00000 0.454859
\(175\) 0 0
\(176\) 0 0
\(177\) −12.0000 −0.901975
\(178\) −10.3923 −0.778936
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) 13.8564 1.02430
\(184\) 0 0
\(185\) 2.00000 0.147043
\(186\) 13.8564 1.01600
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 6.00000 0.435286
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 24.2487 1.74546 0.872730 0.488203i \(-0.162347\pi\)
0.872730 + 0.488203i \(0.162347\pi\)
\(194\) −17.3205 −1.24354
\(195\) 3.46410 0.248069
\(196\) −7.00000 −0.500000
\(197\) 13.8564 0.987228 0.493614 0.869681i \(-0.335676\pi\)
0.493614 + 0.869681i \(0.335676\pi\)
\(198\) 0 0
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) −1.73205 −0.122474
\(201\) 8.00000 0.564276
\(202\) −18.0000 −1.26648
\(203\) 0 0
\(204\) 0 0
\(205\) 10.3923 0.725830
\(206\) −6.92820 −0.482711
\(207\) 0 0
\(208\) −17.3205 −1.20096
\(209\) 0 0
\(210\) 0 0
\(211\) −3.46410 −0.238479 −0.119239 0.992866i \(-0.538046\pi\)
−0.119239 + 0.992866i \(0.538046\pi\)
\(212\) −6.00000 −0.412082
\(213\) −12.0000 −0.822226
\(214\) −18.0000 −1.23045
\(215\) 6.92820 0.472500
\(216\) 1.73205 0.117851
\(217\) 0 0
\(218\) 24.0000 1.62549
\(219\) −3.46410 −0.234082
\(220\) 0 0
\(221\) 0 0
\(222\) 3.46410 0.232495
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) −10.3923 −0.691286
\(227\) 10.3923 0.689761 0.344881 0.938647i \(-0.387919\pi\)
0.344881 + 0.938647i \(0.387919\pi\)
\(228\) 3.46410 0.229416
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 6.00000 0.392232
\(235\) 0 0
\(236\) 12.0000 0.781133
\(237\) −10.3923 −0.675053
\(238\) 0 0
\(239\) 13.8564 0.896296 0.448148 0.893959i \(-0.352084\pi\)
0.448148 + 0.893959i \(0.352084\pi\)
\(240\) −5.00000 −0.322749
\(241\) 13.8564 0.892570 0.446285 0.894891i \(-0.352747\pi\)
0.446285 + 0.894891i \(0.352747\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) −13.8564 −0.887066
\(245\) 7.00000 0.447214
\(246\) 18.0000 1.14764
\(247\) −12.0000 −0.763542
\(248\) 13.8564 0.879883
\(249\) −17.3205 −1.09764
\(250\) −1.73205 −0.109545
\(251\) −24.0000 −1.51487 −0.757433 0.652913i \(-0.773547\pi\)
−0.757433 + 0.652913i \(0.773547\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −12.0000 −0.752947
\(255\) 0 0
\(256\) 19.0000 1.18750
\(257\) −30.0000 −1.87135 −0.935674 0.352865i \(-0.885208\pi\)
−0.935674 + 0.352865i \(0.885208\pi\)
\(258\) 12.0000 0.747087
\(259\) 0 0
\(260\) −3.46410 −0.214834
\(261\) −3.46410 −0.214423
\(262\) 36.0000 2.22409
\(263\) 3.46410 0.213606 0.106803 0.994280i \(-0.465939\pi\)
0.106803 + 0.994280i \(0.465939\pi\)
\(264\) 0 0
\(265\) 6.00000 0.368577
\(266\) 0 0
\(267\) 6.00000 0.367194
\(268\) −8.00000 −0.488678
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 1.73205 0.105409
\(271\) −24.2487 −1.47300 −0.736502 0.676435i \(-0.763524\pi\)
−0.736502 + 0.676435i \(0.763524\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 10.3923 0.627822
\(275\) 0 0
\(276\) 0 0
\(277\) −10.3923 −0.624413 −0.312207 0.950014i \(-0.601068\pi\)
−0.312207 + 0.950014i \(0.601068\pi\)
\(278\) 30.0000 1.79928
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) 3.46410 0.206651 0.103325 0.994648i \(-0.467052\pi\)
0.103325 + 0.994648i \(0.467052\pi\)
\(282\) 0 0
\(283\) 20.7846 1.23552 0.617758 0.786368i \(-0.288041\pi\)
0.617758 + 0.786368i \(0.288041\pi\)
\(284\) 12.0000 0.712069
\(285\) −3.46410 −0.205196
\(286\) 0 0
\(287\) 0 0
\(288\) −5.19615 −0.306186
\(289\) −17.0000 −1.00000
\(290\) 6.00000 0.352332
\(291\) 10.0000 0.586210
\(292\) 3.46410 0.202721
\(293\) 13.8564 0.809500 0.404750 0.914427i \(-0.367359\pi\)
0.404750 + 0.914427i \(0.367359\pi\)
\(294\) 12.1244 0.707107
\(295\) −12.0000 −0.698667
\(296\) 3.46410 0.201347
\(297\) 0 0
\(298\) 18.0000 1.04271
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) 0 0
\(302\) −6.00000 −0.345261
\(303\) 10.3923 0.597022
\(304\) 17.3205 0.993399
\(305\) 13.8564 0.793416
\(306\) 0 0
\(307\) −13.8564 −0.790827 −0.395413 0.918503i \(-0.629399\pi\)
−0.395413 + 0.918503i \(0.629399\pi\)
\(308\) 0 0
\(309\) 4.00000 0.227552
\(310\) 13.8564 0.786991
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 6.00000 0.339683
\(313\) −14.0000 −0.791327 −0.395663 0.918396i \(-0.629485\pi\)
−0.395663 + 0.918396i \(0.629485\pi\)
\(314\) −24.2487 −1.36843
\(315\) 0 0
\(316\) 10.3923 0.584613
\(317\) 30.0000 1.68497 0.842484 0.538721i \(-0.181092\pi\)
0.842484 + 0.538721i \(0.181092\pi\)
\(318\) 10.3923 0.582772
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) 10.3923 0.580042
\(322\) 0 0
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 3.46410 0.192154
\(326\) −27.7128 −1.53487
\(327\) −13.8564 −0.766261
\(328\) 18.0000 0.993884
\(329\) 0 0
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 17.3205 0.950586
\(333\) −2.00000 −0.109599
\(334\) −30.0000 −1.64153
\(335\) 8.00000 0.437087
\(336\) 0 0
\(337\) 17.3205 0.943508 0.471754 0.881730i \(-0.343621\pi\)
0.471754 + 0.881730i \(0.343621\pi\)
\(338\) −1.73205 −0.0942111
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) 0 0
\(342\) −6.00000 −0.324443
\(343\) 0 0
\(344\) 12.0000 0.646997
\(345\) 0 0
\(346\) 0 0
\(347\) −10.3923 −0.557888 −0.278944 0.960307i \(-0.589984\pi\)
−0.278944 + 0.960307i \(0.589984\pi\)
\(348\) 3.46410 0.185695
\(349\) −27.7128 −1.48343 −0.741716 0.670714i \(-0.765988\pi\)
−0.741716 + 0.670714i \(0.765988\pi\)
\(350\) 0 0
\(351\) −3.46410 −0.184900
\(352\) 0 0
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) −20.7846 −1.10469
\(355\) −12.0000 −0.636894
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) −20.7846 −1.09850
\(359\) −6.92820 −0.365657 −0.182828 0.983145i \(-0.558525\pi\)
−0.182828 + 0.983145i \(0.558525\pi\)
\(360\) 1.73205 0.0912871
\(361\) −7.00000 −0.368421
\(362\) 3.46410 0.182069
\(363\) 0 0
\(364\) 0 0
\(365\) −3.46410 −0.181319
\(366\) 24.0000 1.25450
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 0 0
\(369\) −10.3923 −0.541002
\(370\) 3.46410 0.180090
\(371\) 0 0
\(372\) 8.00000 0.414781
\(373\) −38.1051 −1.97301 −0.986504 0.163737i \(-0.947645\pi\)
−0.986504 + 0.163737i \(0.947645\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 3.46410 0.177705
\(381\) 6.92820 0.354943
\(382\) 0 0
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) −12.1244 −0.618718
\(385\) 0 0
\(386\) 42.0000 2.13774
\(387\) −6.92820 −0.352180
\(388\) −10.0000 −0.507673
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 6.00000 0.303822
\(391\) 0 0
\(392\) 12.1244 0.612372
\(393\) −20.7846 −1.04844
\(394\) 24.0000 1.20910
\(395\) −10.3923 −0.522894
\(396\) 0 0
\(397\) −22.0000 −1.10415 −0.552074 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) 27.7128 1.38912
\(399\) 0 0
\(400\) −5.00000 −0.250000
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) 13.8564 0.691095
\(403\) −27.7128 −1.38047
\(404\) −10.3923 −0.517036
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 34.6410 1.71289 0.856444 0.516240i \(-0.172669\pi\)
0.856444 + 0.516240i \(0.172669\pi\)
\(410\) 18.0000 0.888957
\(411\) −6.00000 −0.295958
\(412\) −4.00000 −0.197066
\(413\) 0 0
\(414\) 0 0
\(415\) −17.3205 −0.850230
\(416\) −18.0000 −0.882523
\(417\) −17.3205 −0.848189
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) −6.00000 −0.292075
\(423\) 0 0
\(424\) 10.3923 0.504695
\(425\) 0 0
\(426\) −20.7846 −1.00702
\(427\) 0 0
\(428\) −10.3923 −0.502331
\(429\) 0 0
\(430\) 12.0000 0.578691
\(431\) 20.7846 1.00116 0.500580 0.865690i \(-0.333120\pi\)
0.500580 + 0.865690i \(0.333120\pi\)
\(432\) 5.00000 0.240563
\(433\) −34.0000 −1.63394 −0.816968 0.576683i \(-0.804347\pi\)
−0.816968 + 0.576683i \(0.804347\pi\)
\(434\) 0 0
\(435\) −3.46410 −0.166091
\(436\) 13.8564 0.663602
\(437\) 0 0
\(438\) −6.00000 −0.286691
\(439\) 24.2487 1.15733 0.578664 0.815566i \(-0.303574\pi\)
0.578664 + 0.815566i \(0.303574\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) 0 0
\(443\) 36.0000 1.71041 0.855206 0.518289i \(-0.173431\pi\)
0.855206 + 0.518289i \(0.173431\pi\)
\(444\) 2.00000 0.0949158
\(445\) 6.00000 0.284427
\(446\) −27.7128 −1.31224
\(447\) −10.3923 −0.491539
\(448\) 0 0
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 1.73205 0.0816497
\(451\) 0 0
\(452\) −6.00000 −0.282216
\(453\) 3.46410 0.162758
\(454\) 18.0000 0.844782
\(455\) 0 0
\(456\) −6.00000 −0.280976
\(457\) −17.3205 −0.810219 −0.405110 0.914268i \(-0.632767\pi\)
−0.405110 + 0.914268i \(0.632767\pi\)
\(458\) 17.3205 0.809334
\(459\) 0 0
\(460\) 0 0
\(461\) −3.46410 −0.161339 −0.0806696 0.996741i \(-0.525706\pi\)
−0.0806696 + 0.996741i \(0.525706\pi\)
\(462\) 0 0
\(463\) 8.00000 0.371792 0.185896 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) 17.3205 0.804084
\(465\) −8.00000 −0.370991
\(466\) 0 0
\(467\) 36.0000 1.66588 0.832941 0.553362i \(-0.186655\pi\)
0.832941 + 0.553362i \(0.186655\pi\)
\(468\) 3.46410 0.160128
\(469\) 0 0
\(470\) 0 0
\(471\) 14.0000 0.645086
\(472\) −20.7846 −0.956689
\(473\) 0 0
\(474\) −18.0000 −0.826767
\(475\) −3.46410 −0.158944
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) 24.0000 1.09773
\(479\) −13.8564 −0.633115 −0.316558 0.948573i \(-0.602527\pi\)
−0.316558 + 0.948573i \(0.602527\pi\)
\(480\) −5.19615 −0.237171
\(481\) −6.92820 −0.315899
\(482\) 24.0000 1.09317
\(483\) 0 0
\(484\) 0 0
\(485\) 10.0000 0.454077
\(486\) −1.73205 −0.0785674
\(487\) −20.0000 −0.906287 −0.453143 0.891438i \(-0.649697\pi\)
−0.453143 + 0.891438i \(0.649697\pi\)
\(488\) 24.0000 1.08643
\(489\) 16.0000 0.723545
\(490\) 12.1244 0.547723
\(491\) 6.92820 0.312665 0.156333 0.987704i \(-0.450033\pi\)
0.156333 + 0.987704i \(0.450033\pi\)
\(492\) 10.3923 0.468521
\(493\) 0 0
\(494\) −20.7846 −0.935144
\(495\) 0 0
\(496\) 40.0000 1.79605
\(497\) 0 0
\(498\) −30.0000 −1.34433
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 17.3205 0.773823
\(502\) −41.5692 −1.85533
\(503\) −3.46410 −0.154457 −0.0772283 0.997013i \(-0.524607\pi\)
−0.0772283 + 0.997013i \(0.524607\pi\)
\(504\) 0 0
\(505\) 10.3923 0.462451
\(506\) 0 0
\(507\) 1.00000 0.0444116
\(508\) −6.92820 −0.307389
\(509\) −42.0000 −1.86162 −0.930809 0.365507i \(-0.880896\pi\)
−0.930809 + 0.365507i \(0.880896\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 8.66025 0.382733
\(513\) 3.46410 0.152944
\(514\) −51.9615 −2.29192
\(515\) 4.00000 0.176261
\(516\) 6.92820 0.304997
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 6.00000 0.263117
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) −6.00000 −0.262613
\(523\) −13.8564 −0.605898 −0.302949 0.953007i \(-0.597971\pi\)
−0.302949 + 0.953007i \(0.597971\pi\)
\(524\) 20.7846 0.907980
\(525\) 0 0
\(526\) 6.00000 0.261612
\(527\) 0 0
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 10.3923 0.451413
\(531\) 12.0000 0.520756
\(532\) 0 0
\(533\) −36.0000 −1.55933
\(534\) 10.3923 0.449719
\(535\) 10.3923 0.449299
\(536\) 13.8564 0.598506
\(537\) 12.0000 0.517838
\(538\) 10.3923 0.448044
\(539\) 0 0
\(540\) 1.00000 0.0430331
\(541\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(542\) −42.0000 −1.80405
\(543\) −2.00000 −0.0858282
\(544\) 0 0
\(545\) −13.8564 −0.593543
\(546\) 0 0
\(547\) −13.8564 −0.592457 −0.296229 0.955117i \(-0.595729\pi\)
−0.296229 + 0.955117i \(0.595729\pi\)
\(548\) 6.00000 0.256307
\(549\) −13.8564 −0.591377
\(550\) 0 0
\(551\) 12.0000 0.511217
\(552\) 0 0
\(553\) 0 0
\(554\) −18.0000 −0.764747
\(555\) −2.00000 −0.0848953
\(556\) 17.3205 0.734553
\(557\) 6.92820 0.293557 0.146779 0.989169i \(-0.453109\pi\)
0.146779 + 0.989169i \(0.453109\pi\)
\(558\) −13.8564 −0.586588
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) 0 0
\(562\) 6.00000 0.253095
\(563\) 3.46410 0.145994 0.0729972 0.997332i \(-0.476744\pi\)
0.0729972 + 0.997332i \(0.476744\pi\)
\(564\) 0 0
\(565\) 6.00000 0.252422
\(566\) 36.0000 1.51319
\(567\) 0 0
\(568\) −20.7846 −0.872103
\(569\) 45.0333 1.88790 0.943948 0.330096i \(-0.107081\pi\)
0.943948 + 0.330096i \(0.107081\pi\)
\(570\) −6.00000 −0.251312
\(571\) −31.1769 −1.30471 −0.652357 0.757912i \(-0.726220\pi\)
−0.652357 + 0.757912i \(0.726220\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −26.0000 −1.08239 −0.541197 0.840896i \(-0.682029\pi\)
−0.541197 + 0.840896i \(0.682029\pi\)
\(578\) −29.4449 −1.22474
\(579\) −24.2487 −1.00774
\(580\) 3.46410 0.143839
\(581\) 0 0
\(582\) 17.3205 0.717958
\(583\) 0 0
\(584\) −6.00000 −0.248282
\(585\) −3.46410 −0.143223
\(586\) 24.0000 0.991431
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 7.00000 0.288675
\(589\) 27.7128 1.14189
\(590\) −20.7846 −0.855689
\(591\) −13.8564 −0.569976
\(592\) 10.0000 0.410997
\(593\) 34.6410 1.42254 0.711268 0.702921i \(-0.248121\pi\)
0.711268 + 0.702921i \(0.248121\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 10.3923 0.425685
\(597\) −16.0000 −0.654836
\(598\) 0 0
\(599\) 12.0000 0.490307 0.245153 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) 1.73205 0.0707107
\(601\) −34.6410 −1.41304 −0.706518 0.707695i \(-0.749735\pi\)
−0.706518 + 0.707695i \(0.749735\pi\)
\(602\) 0 0
\(603\) −8.00000 −0.325785
\(604\) −3.46410 −0.140952
\(605\) 0 0
\(606\) 18.0000 0.731200
\(607\) −6.92820 −0.281207 −0.140604 0.990066i \(-0.544904\pi\)
−0.140604 + 0.990066i \(0.544904\pi\)
\(608\) 18.0000 0.729996
\(609\) 0 0
\(610\) 24.0000 0.971732
\(611\) 0 0
\(612\) 0 0
\(613\) 17.3205 0.699569 0.349784 0.936830i \(-0.386255\pi\)
0.349784 + 0.936830i \(0.386255\pi\)
\(614\) −24.0000 −0.968561
\(615\) −10.3923 −0.419058
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 6.92820 0.278693
\(619\) −28.0000 −1.12542 −0.562708 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) 8.00000 0.321288
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 17.3205 0.693375
\(625\) 1.00000 0.0400000
\(626\) −24.2487 −0.969173
\(627\) 0 0
\(628\) −14.0000 −0.558661
\(629\) 0 0
\(630\) 0 0
\(631\) 16.0000 0.636950 0.318475 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) −18.0000 −0.716002
\(633\) 3.46410 0.137686
\(634\) 51.9615 2.06366
\(635\) 6.92820 0.274937
\(636\) 6.00000 0.237915
\(637\) −24.2487 −0.960769
\(638\) 0 0
\(639\) 12.0000 0.474713
\(640\) −12.1244 −0.479257
\(641\) −6.00000 −0.236986 −0.118493 0.992955i \(-0.537806\pi\)
−0.118493 + 0.992955i \(0.537806\pi\)
\(642\) 18.0000 0.710403
\(643\) 8.00000 0.315489 0.157745 0.987480i \(-0.449578\pi\)
0.157745 + 0.987480i \(0.449578\pi\)
\(644\) 0 0
\(645\) −6.92820 −0.272798
\(646\) 0 0
\(647\) −24.0000 −0.943537 −0.471769 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) −1.73205 −0.0680414
\(649\) 0 0
\(650\) 6.00000 0.235339
\(651\) 0 0
\(652\) −16.0000 −0.626608
\(653\) −18.0000 −0.704394 −0.352197 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) −24.0000 −0.938474
\(655\) −20.7846 −0.812122
\(656\) 51.9615 2.02876
\(657\) 3.46410 0.135147
\(658\) 0 0
\(659\) 13.8564 0.539769 0.269884 0.962893i \(-0.413014\pi\)
0.269884 + 0.962893i \(0.413014\pi\)
\(660\) 0 0
\(661\) −38.0000 −1.47803 −0.739014 0.673690i \(-0.764708\pi\)
−0.739014 + 0.673690i \(0.764708\pi\)
\(662\) 34.6410 1.34636
\(663\) 0 0
\(664\) −30.0000 −1.16423
\(665\) 0 0
\(666\) −3.46410 −0.134231
\(667\) 0 0
\(668\) −17.3205 −0.670151
\(669\) 16.0000 0.618596
\(670\) 13.8564 0.535320
\(671\) 0 0
\(672\) 0 0
\(673\) −31.1769 −1.20178 −0.600891 0.799331i \(-0.705187\pi\)
−0.600891 + 0.799331i \(0.705187\pi\)
\(674\) 30.0000 1.15556
\(675\) −1.00000 −0.0384900
\(676\) −1.00000 −0.0384615
\(677\) −34.6410 −1.33136 −0.665681 0.746236i \(-0.731859\pi\)
−0.665681 + 0.746236i \(0.731859\pi\)
\(678\) 10.3923 0.399114
\(679\) 0 0
\(680\) 0 0
\(681\) −10.3923 −0.398234
\(682\) 0 0
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) −3.46410 −0.132453
\(685\) −6.00000 −0.229248
\(686\) 0 0
\(687\) −10.0000 −0.381524
\(688\) 34.6410 1.32068
\(689\) −20.7846 −0.791831
\(690\) 0 0
\(691\) −28.0000 −1.06517 −0.532585 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −18.0000 −0.683271
\(695\) −17.3205 −0.657004
\(696\) −6.00000 −0.227429
\(697\) 0 0
\(698\) −48.0000 −1.81683
\(699\) 0 0
\(700\) 0 0
\(701\) 45.0333 1.70089 0.850443 0.526068i \(-0.176334\pi\)
0.850443 + 0.526068i \(0.176334\pi\)
\(702\) −6.00000 −0.226455
\(703\) 6.92820 0.261302
\(704\) 0 0
\(705\) 0 0
\(706\) −31.1769 −1.17336
\(707\) 0 0
\(708\) −12.0000 −0.450988
\(709\) −26.0000 −0.976450 −0.488225 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) −20.7846 −0.780033
\(711\) 10.3923 0.389742
\(712\) 10.3923 0.389468
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) −13.8564 −0.517477
\(718\) −12.0000 −0.447836
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 5.00000 0.186339
\(721\) 0 0
\(722\) −12.1244 −0.451222
\(723\) −13.8564 −0.515325
\(724\) 2.00000 0.0743294
\(725\) −3.46410 −0.128654
\(726\) 0 0
\(727\) −8.00000 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −6.00000 −0.222070
\(731\) 0 0
\(732\) 13.8564 0.512148
\(733\) −45.0333 −1.66334 −0.831672 0.555267i \(-0.812616\pi\)
−0.831672 + 0.555267i \(0.812616\pi\)
\(734\) 13.8564 0.511449
\(735\) −7.00000 −0.258199
\(736\) 0 0
\(737\) 0 0
\(738\) −18.0000 −0.662589
\(739\) 24.2487 0.892003 0.446002 0.895032i \(-0.352848\pi\)
0.446002 + 0.895032i \(0.352848\pi\)
\(740\) 2.00000 0.0735215
\(741\) 12.0000 0.440831
\(742\) 0 0
\(743\) −31.1769 −1.14377 −0.571885 0.820334i \(-0.693788\pi\)
−0.571885 + 0.820334i \(0.693788\pi\)
\(744\) −13.8564 −0.508001
\(745\) −10.3923 −0.380745
\(746\) −66.0000 −2.41643
\(747\) 17.3205 0.633724
\(748\) 0 0
\(749\) 0 0
\(750\) 1.73205 0.0632456
\(751\) 16.0000 0.583848 0.291924 0.956441i \(-0.405705\pi\)
0.291924 + 0.956441i \(0.405705\pi\)
\(752\) 0 0
\(753\) 24.0000 0.874609
\(754\) −20.7846 −0.756931
\(755\) 3.46410 0.126072
\(756\) 0 0
\(757\) 34.0000 1.23575 0.617876 0.786276i \(-0.287994\pi\)
0.617876 + 0.786276i \(0.287994\pi\)
\(758\) 34.6410 1.25822
\(759\) 0 0
\(760\) −6.00000 −0.217643
\(761\) −10.3923 −0.376721 −0.188360 0.982100i \(-0.560317\pi\)
−0.188360 + 0.982100i \(0.560317\pi\)
\(762\) 12.0000 0.434714
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 41.5692 1.50196
\(767\) 41.5692 1.50098
\(768\) −19.0000 −0.685603
\(769\) −6.92820 −0.249837 −0.124919 0.992167i \(-0.539867\pi\)
−0.124919 + 0.992167i \(0.539867\pi\)
\(770\) 0 0
\(771\) 30.0000 1.08042
\(772\) 24.2487 0.872730
\(773\) 6.00000 0.215805 0.107903 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) −12.0000 −0.431331
\(775\) −8.00000 −0.287368
\(776\) 17.3205 0.621770
\(777\) 0 0
\(778\) −10.3923 −0.372582
\(779\) 36.0000 1.28983
\(780\) 3.46410 0.124035
\(781\) 0 0
\(782\) 0 0
\(783\) 3.46410 0.123797
\(784\) 35.0000 1.25000
\(785\) 14.0000 0.499681
\(786\) −36.0000 −1.28408
\(787\) 48.4974 1.72875 0.864373 0.502851i \(-0.167715\pi\)
0.864373 + 0.502851i \(0.167715\pi\)
\(788\) 13.8564 0.493614
\(789\) −3.46410 −0.123325
\(790\) −18.0000 −0.640411
\(791\) 0 0
\(792\) 0 0
\(793\) −48.0000 −1.70453
\(794\) −38.1051 −1.35230
\(795\) −6.00000 −0.212798
\(796\) 16.0000 0.567105
\(797\) 6.00000 0.212531 0.106265 0.994338i \(-0.466111\pi\)
0.106265 + 0.994338i \(0.466111\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −5.19615 −0.183712
\(801\) −6.00000 −0.212000
\(802\) 51.9615 1.83483
\(803\) 0 0
\(804\) 8.00000 0.282138
\(805\) 0 0
\(806\) −48.0000 −1.69073
\(807\) −6.00000 −0.211210
\(808\) 18.0000 0.633238
\(809\) 38.1051 1.33970 0.669852 0.742494i \(-0.266357\pi\)
0.669852 + 0.742494i \(0.266357\pi\)
\(810\) −1.73205 −0.0608581
\(811\) 31.1769 1.09477 0.547385 0.836881i \(-0.315623\pi\)
0.547385 + 0.836881i \(0.315623\pi\)
\(812\) 0 0
\(813\) 24.2487 0.850439
\(814\) 0 0
\(815\) 16.0000 0.560456
\(816\) 0 0
\(817\) 24.0000 0.839654
\(818\) 60.0000 2.09785
\(819\) 0 0
\(820\) 10.3923 0.362915
\(821\) 24.2487 0.846286 0.423143 0.906063i \(-0.360927\pi\)
0.423143 + 0.906063i \(0.360927\pi\)
\(822\) −10.3923 −0.362473
\(823\) −52.0000 −1.81261 −0.906303 0.422628i \(-0.861108\pi\)
−0.906303 + 0.422628i \(0.861108\pi\)
\(824\) 6.92820 0.241355
\(825\) 0 0
\(826\) 0 0
\(827\) −38.1051 −1.32504 −0.662522 0.749042i \(-0.730514\pi\)
−0.662522 + 0.749042i \(0.730514\pi\)
\(828\) 0 0
\(829\) 50.0000 1.73657 0.868286 0.496064i \(-0.165222\pi\)
0.868286 + 0.496064i \(0.165222\pi\)
\(830\) −30.0000 −1.04132
\(831\) 10.3923 0.360505
\(832\) 3.46410 0.120096
\(833\) 0 0
\(834\) −30.0000 −1.03882
\(835\) 17.3205 0.599401
\(836\) 0 0
\(837\) 8.00000 0.276520
\(838\) −20.7846 −0.717992
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −17.0000 −0.586207
\(842\) 38.1051 1.31319
\(843\) −3.46410 −0.119310
\(844\) −3.46410 −0.119239
\(845\) 1.00000 0.0344010
\(846\) 0 0
\(847\) 0 0
\(848\) 30.0000 1.03020
\(849\) −20.7846 −0.713326
\(850\) 0 0
\(851\) 0 0
\(852\) −12.0000 −0.411113
\(853\) −51.9615 −1.77913 −0.889564 0.456810i \(-0.848992\pi\)
−0.889564 + 0.456810i \(0.848992\pi\)
\(854\) 0 0
\(855\) 3.46410 0.118470
\(856\) 18.0000 0.615227
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) −4.00000 −0.136478 −0.0682391 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) 6.92820 0.236250
\(861\) 0 0
\(862\) 36.0000 1.22616
\(863\) −24.0000 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) 5.19615 0.176777
\(865\) 0 0
\(866\) −58.8897 −2.00115
\(867\) 17.0000 0.577350
\(868\) 0 0
\(869\) 0 0
\(870\) −6.00000 −0.203419
\(871\) −27.7128 −0.939013
\(872\) −24.0000 −0.812743
\(873\) −10.0000 −0.338449
\(874\) 0 0
\(875\) 0 0
\(876\) −3.46410 −0.117041
\(877\) 31.1769 1.05277 0.526385 0.850246i \(-0.323547\pi\)
0.526385 + 0.850246i \(0.323547\pi\)
\(878\) 42.0000 1.41743
\(879\) −13.8564 −0.467365
\(880\) 0 0
\(881\) −42.0000 −1.41502 −0.707508 0.706705i \(-0.750181\pi\)
−0.707508 + 0.706705i \(0.750181\pi\)
\(882\) −12.1244 −0.408248
\(883\) 28.0000 0.942275 0.471138 0.882060i \(-0.343844\pi\)
0.471138 + 0.882060i \(0.343844\pi\)
\(884\) 0 0
\(885\) 12.0000 0.403376
\(886\) 62.3538 2.09482
\(887\) 17.3205 0.581566 0.290783 0.956789i \(-0.406084\pi\)
0.290783 + 0.956789i \(0.406084\pi\)
\(888\) −3.46410 −0.116248
\(889\) 0 0
\(890\) 10.3923 0.348351
\(891\) 0 0
\(892\) −16.0000 −0.535720
\(893\) 0 0
\(894\) −18.0000 −0.602010
\(895\) 12.0000 0.401116
\(896\) 0 0
\(897\) 0 0
\(898\) −10.3923 −0.346796
\(899\) 27.7128 0.924274
\(900\) 1.00000 0.0333333
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 10.3923 0.345643
\(905\) −2.00000 −0.0664822
\(906\) 6.00000 0.199337
\(907\) 20.0000 0.664089 0.332045 0.943264i \(-0.392262\pi\)
0.332045 + 0.943264i \(0.392262\pi\)
\(908\) 10.3923 0.344881
\(909\) −10.3923 −0.344691
\(910\) 0 0
\(911\) −24.0000 −0.795155 −0.397578 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) −17.3205 −0.573539
\(913\) 0 0
\(914\) −30.0000 −0.992312
\(915\) −13.8564 −0.458079
\(916\) 10.0000 0.330409
\(917\) 0 0
\(918\) 0 0
\(919\) 38.1051 1.25697 0.628486 0.777821i \(-0.283675\pi\)
0.628486 + 0.777821i \(0.283675\pi\)
\(920\) 0 0
\(921\) 13.8564 0.456584
\(922\) −6.00000 −0.197599
\(923\) 41.5692 1.36827
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) 13.8564 0.455350
\(927\) −4.00000 −0.131377
\(928\) 18.0000 0.590879
\(929\) −42.0000 −1.37798 −0.688988 0.724773i \(-0.741945\pi\)
−0.688988 + 0.724773i \(0.741945\pi\)
\(930\) −13.8564 −0.454369
\(931\) 24.2487 0.794719
\(932\) 0 0
\(933\) 0 0
\(934\) 62.3538 2.04028
\(935\) 0 0
\(936\) −6.00000 −0.196116
\(937\) −10.3923 −0.339502 −0.169751 0.985487i \(-0.554296\pi\)
−0.169751 + 0.985487i \(0.554296\pi\)
\(938\) 0 0
\(939\) 14.0000 0.456873
\(940\) 0 0
\(941\) 31.1769 1.01634 0.508169 0.861257i \(-0.330322\pi\)
0.508169 + 0.861257i \(0.330322\pi\)
\(942\) 24.2487 0.790066
\(943\) 0 0
\(944\) −60.0000 −1.95283
\(945\) 0 0
\(946\) 0 0
\(947\) −12.0000 −0.389948 −0.194974 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) −10.3923 −0.337526
\(949\) 12.0000 0.389536
\(950\) −6.00000 −0.194666
\(951\) −30.0000 −0.972817
\(952\) 0 0
\(953\) −34.6410 −1.12213 −0.561066 0.827771i \(-0.689609\pi\)
−0.561066 + 0.827771i \(0.689609\pi\)
\(954\) −10.3923 −0.336463
\(955\) 0 0
\(956\) 13.8564 0.448148
\(957\) 0 0
\(958\) −24.0000 −0.775405
\(959\) 0 0
\(960\) 1.00000 0.0322749
\(961\) 33.0000 1.06452
\(962\) −12.0000 −0.386896
\(963\) −10.3923 −0.334887
\(964\) 13.8564 0.446285
\(965\) −24.2487 −0.780594
\(966\) 0 0
\(967\) −48.4974 −1.55957 −0.779786 0.626046i \(-0.784672\pi\)
−0.779786 + 0.626046i \(0.784672\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 17.3205 0.556128
\(971\) 24.0000 0.770197 0.385098 0.922876i \(-0.374168\pi\)
0.385098 + 0.922876i \(0.374168\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −34.6410 −1.10997
\(975\) −3.46410 −0.110940
\(976\) 69.2820 2.21766
\(977\) −18.0000 −0.575871 −0.287936 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) 27.7128 0.886158
\(979\) 0 0
\(980\) 7.00000 0.223607
\(981\) 13.8564 0.442401
\(982\) 12.0000 0.382935
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) −18.0000 −0.573819
\(985\) −13.8564 −0.441502
\(986\) 0 0
\(987\) 0 0
\(988\) −12.0000 −0.381771
\(989\) 0 0
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 41.5692 1.31982
\(993\) −20.0000 −0.634681
\(994\) 0 0
\(995\) −16.0000 −0.507234
\(996\) −17.3205 −0.548821
\(997\) 31.1769 0.987383 0.493691 0.869637i \(-0.335647\pi\)
0.493691 + 0.869637i \(0.335647\pi\)
\(998\) 6.92820 0.219308
\(999\) 2.00000 0.0632772
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 1815.2.a.h.1.2 yes 2
3.2 odd 2 5445.2.a.t.1.1 2
5.4 even 2 9075.2.a.bp.1.1 2
11.10 odd 2 inner 1815.2.a.h.1.1 2
33.32 even 2 5445.2.a.t.1.2 2
55.54 odd 2 9075.2.a.bp.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1815.2.a.h.1.1 2 11.10 odd 2 inner
1815.2.a.h.1.2 yes 2 1.1 even 1 trivial
5445.2.a.t.1.1 2 3.2 odd 2
5445.2.a.t.1.2 2 33.32 even 2
9075.2.a.bp.1.1 2 5.4 even 2
9075.2.a.bp.1.2 2 55.54 odd 2