Properties

Label 1530.2.a.s.1.1
Level $1530$
Weight $2$
Character 1530.1
Self dual yes
Analytic conductor $12.217$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1530,2,Mod(1,1530)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1530, 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("1530.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1530 = 2 \cdot 3^{2} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1530.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: \(12.2171115093\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 510)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.44949\) of defining polynomial
Character \(\chi\) \(=\) 1530.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -4.89898 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -4.89898 q^{7} +1.00000 q^{8} -1.00000 q^{10} +6.89898 q^{13} -4.89898 q^{14} +1.00000 q^{16} +1.00000 q^{17} +4.00000 q^{19} -1.00000 q^{20} +4.00000 q^{23} +1.00000 q^{25} +6.89898 q^{26} -4.89898 q^{28} -6.00000 q^{29} +4.00000 q^{31} +1.00000 q^{32} +1.00000 q^{34} +4.89898 q^{35} +6.00000 q^{37} +4.00000 q^{38} -1.00000 q^{40} +2.89898 q^{41} +8.89898 q^{43} +4.00000 q^{46} -9.79796 q^{47} +17.0000 q^{49} +1.00000 q^{50} +6.89898 q^{52} +7.79796 q^{53} -4.89898 q^{56} -6.00000 q^{58} -4.89898 q^{59} +11.7980 q^{61} +4.00000 q^{62} +1.00000 q^{64} -6.89898 q^{65} +0.898979 q^{67} +1.00000 q^{68} +4.89898 q^{70} +8.89898 q^{71} -10.8990 q^{73} +6.00000 q^{74} +4.00000 q^{76} -5.79796 q^{79} -1.00000 q^{80} +2.89898 q^{82} -13.7980 q^{83} -1.00000 q^{85} +8.89898 q^{86} +7.79796 q^{89} -33.7980 q^{91} +4.00000 q^{92} -9.79796 q^{94} -4.00000 q^{95} -12.6969 q^{97} +17.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{8} - 2 q^{10} + 4 q^{13} + 2 q^{16} + 2 q^{17} + 8 q^{19} - 2 q^{20} + 8 q^{23} + 2 q^{25} + 4 q^{26} - 12 q^{29} + 8 q^{31} + 2 q^{32} + 2 q^{34} + 12 q^{37} + 8 q^{38} - 2 q^{40} - 4 q^{41} + 8 q^{43} + 8 q^{46} + 34 q^{49} + 2 q^{50} + 4 q^{52} - 4 q^{53} - 12 q^{58} + 4 q^{61} + 8 q^{62} + 2 q^{64} - 4 q^{65} - 8 q^{67} + 2 q^{68} + 8 q^{71} - 12 q^{73} + 12 q^{74} + 8 q^{76} + 8 q^{79} - 2 q^{80} - 4 q^{82} - 8 q^{83} - 2 q^{85} + 8 q^{86} - 4 q^{89} - 48 q^{91} + 8 q^{92} - 8 q^{95} + 4 q^{97} + 34 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −4.89898 −1.85164 −0.925820 0.377964i \(-0.876624\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 6.89898 1.91343 0.956716 0.291022i \(-0.0939953\pi\)
0.956716 + 0.291022i \(0.0939953\pi\)
\(14\) −4.89898 −1.30931
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 1.00000 0.242536
\(18\) 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
\(22\) 0 0
\(23\) 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
\(26\) 6.89898 1.35300
\(27\) 0 0
\(28\) −4.89898 −0.925820
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 1.00000 0.171499
\(35\) 4.89898 0.828079
\(36\) 0 0
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 4.00000 0.648886
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 2.89898 0.452745 0.226372 0.974041i \(-0.427313\pi\)
0.226372 + 0.974041i \(0.427313\pi\)
\(42\) 0 0
\(43\) 8.89898 1.35708 0.678541 0.734563i \(-0.262613\pi\)
0.678541 + 0.734563i \(0.262613\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) −9.79796 −1.42918 −0.714590 0.699544i \(-0.753387\pi\)
−0.714590 + 0.699544i \(0.753387\pi\)
\(48\) 0 0
\(49\) 17.0000 2.42857
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 6.89898 0.956716
\(53\) 7.79796 1.07113 0.535566 0.844493i \(-0.320098\pi\)
0.535566 + 0.844493i \(0.320098\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −4.89898 −0.654654
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) −4.89898 −0.637793 −0.318896 0.947790i \(-0.603312\pi\)
−0.318896 + 0.947790i \(0.603312\pi\)
\(60\) 0 0
\(61\) 11.7980 1.51057 0.755287 0.655394i \(-0.227498\pi\)
0.755287 + 0.655394i \(0.227498\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −6.89898 −0.855713
\(66\) 0 0
\(67\) 0.898979 0.109828 0.0549139 0.998491i \(-0.482512\pi\)
0.0549139 + 0.998491i \(0.482512\pi\)
\(68\) 1.00000 0.121268
\(69\) 0 0
\(70\) 4.89898 0.585540
\(71\) 8.89898 1.05611 0.528057 0.849209i \(-0.322921\pi\)
0.528057 + 0.849209i \(0.322921\pi\)
\(72\) 0 0
\(73\) −10.8990 −1.27563 −0.637815 0.770190i \(-0.720161\pi\)
−0.637815 + 0.770190i \(0.720161\pi\)
\(74\) 6.00000 0.697486
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) 0 0
\(79\) −5.79796 −0.652321 −0.326161 0.945314i \(-0.605755\pi\)
−0.326161 + 0.945314i \(0.605755\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) 2.89898 0.320139
\(83\) −13.7980 −1.51452 −0.757261 0.653112i \(-0.773463\pi\)
−0.757261 + 0.653112i \(0.773463\pi\)
\(84\) 0 0
\(85\) −1.00000 −0.108465
\(86\) 8.89898 0.959602
\(87\) 0 0
\(88\) 0 0
\(89\) 7.79796 0.826582 0.413291 0.910599i \(-0.364379\pi\)
0.413291 + 0.910599i \(0.364379\pi\)
\(90\) 0 0
\(91\) −33.7980 −3.54299
\(92\) 4.00000 0.417029
\(93\) 0 0
\(94\) −9.79796 −1.01058
\(95\) −4.00000 −0.410391
\(96\) 0 0
\(97\) −12.6969 −1.28918 −0.644589 0.764529i \(-0.722972\pi\)
−0.644589 + 0.764529i \(0.722972\pi\)
\(98\) 17.0000 1.71726
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 18.8990 1.88052 0.940259 0.340459i \(-0.110582\pi\)
0.940259 + 0.340459i \(0.110582\pi\)
\(102\) 0 0
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 6.89898 0.676501
\(105\) 0 0
\(106\) 7.79796 0.757405
\(107\) −5.79796 −0.560510 −0.280255 0.959926i \(-0.590419\pi\)
−0.280255 + 0.959926i \(0.590419\pi\)
\(108\) 0 0
\(109\) 11.7980 1.13004 0.565020 0.825077i \(-0.308869\pi\)
0.565020 + 0.825077i \(0.308869\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −4.89898 −0.462910
\(113\) −7.79796 −0.733570 −0.366785 0.930306i \(-0.619542\pi\)
−0.366785 + 0.930306i \(0.619542\pi\)
\(114\) 0 0
\(115\) −4.00000 −0.373002
\(116\) −6.00000 −0.557086
\(117\) 0 0
\(118\) −4.89898 −0.450988
\(119\) −4.89898 −0.449089
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 11.7980 1.06814
\(123\) 0 0
\(124\) 4.00000 0.359211
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −12.0000 −1.06483 −0.532414 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −6.89898 −0.605081
\(131\) −9.79796 −0.856052 −0.428026 0.903767i \(-0.640791\pi\)
−0.428026 + 0.903767i \(0.640791\pi\)
\(132\) 0 0
\(133\) −19.5959 −1.69918
\(134\) 0.898979 0.0776600
\(135\) 0 0
\(136\) 1.00000 0.0857493
\(137\) −14.0000 −1.19610 −0.598050 0.801459i \(-0.704058\pi\)
−0.598050 + 0.801459i \(0.704058\pi\)
\(138\) 0 0
\(139\) 13.7980 1.17033 0.585164 0.810915i \(-0.301030\pi\)
0.585164 + 0.810915i \(0.301030\pi\)
\(140\) 4.89898 0.414039
\(141\) 0 0
\(142\) 8.89898 0.746786
\(143\) 0 0
\(144\) 0 0
\(145\) 6.00000 0.498273
\(146\) −10.8990 −0.902006
\(147\) 0 0
\(148\) 6.00000 0.493197
\(149\) 18.8990 1.54826 0.774132 0.633024i \(-0.218186\pi\)
0.774132 + 0.633024i \(0.218186\pi\)
\(150\) 0 0
\(151\) −9.79796 −0.797347 −0.398673 0.917093i \(-0.630529\pi\)
−0.398673 + 0.917093i \(0.630529\pi\)
\(152\) 4.00000 0.324443
\(153\) 0 0
\(154\) 0 0
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) −1.10102 −0.0878710 −0.0439355 0.999034i \(-0.513990\pi\)
−0.0439355 + 0.999034i \(0.513990\pi\)
\(158\) −5.79796 −0.461261
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) −19.5959 −1.54437
\(162\) 0 0
\(163\) −2.20204 −0.172477 −0.0862386 0.996275i \(-0.527485\pi\)
−0.0862386 + 0.996275i \(0.527485\pi\)
\(164\) 2.89898 0.226372
\(165\) 0 0
\(166\) −13.7980 −1.07093
\(167\) −2.20204 −0.170399 −0.0851995 0.996364i \(-0.527153\pi\)
−0.0851995 + 0.996364i \(0.527153\pi\)
\(168\) 0 0
\(169\) 34.5959 2.66122
\(170\) −1.00000 −0.0766965
\(171\) 0 0
\(172\) 8.89898 0.678541
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) −4.89898 −0.370328
\(176\) 0 0
\(177\) 0 0
\(178\) 7.79796 0.584482
\(179\) 4.89898 0.366167 0.183083 0.983097i \(-0.441392\pi\)
0.183083 + 0.983097i \(0.441392\pi\)
\(180\) 0 0
\(181\) −4.20204 −0.312335 −0.156168 0.987731i \(-0.549914\pi\)
−0.156168 + 0.987731i \(0.549914\pi\)
\(182\) −33.7980 −2.50527
\(183\) 0 0
\(184\) 4.00000 0.294884
\(185\) −6.00000 −0.441129
\(186\) 0 0
\(187\) 0 0
\(188\) −9.79796 −0.714590
\(189\) 0 0
\(190\) −4.00000 −0.290191
\(191\) 9.79796 0.708955 0.354478 0.935064i \(-0.384659\pi\)
0.354478 + 0.935064i \(0.384659\pi\)
\(192\) 0 0
\(193\) −1.10102 −0.0792532 −0.0396266 0.999215i \(-0.512617\pi\)
−0.0396266 + 0.999215i \(0.512617\pi\)
\(194\) −12.6969 −0.911587
\(195\) 0 0
\(196\) 17.0000 1.21429
\(197\) 13.5959 0.968669 0.484335 0.874883i \(-0.339062\pi\)
0.484335 + 0.874883i \(0.339062\pi\)
\(198\) 0 0
\(199\) 15.5959 1.10557 0.552783 0.833325i \(-0.313566\pi\)
0.552783 + 0.833325i \(0.313566\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) 18.8990 1.32973
\(203\) 29.3939 2.06305
\(204\) 0 0
\(205\) −2.89898 −0.202474
\(206\) 4.00000 0.278693
\(207\) 0 0
\(208\) 6.89898 0.478358
\(209\) 0 0
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 7.79796 0.535566
\(213\) 0 0
\(214\) −5.79796 −0.396340
\(215\) −8.89898 −0.606905
\(216\) 0 0
\(217\) −19.5959 −1.33026
\(218\) 11.7980 0.799059
\(219\) 0 0
\(220\) 0 0
\(221\) 6.89898 0.464076
\(222\) 0 0
\(223\) −4.00000 −0.267860 −0.133930 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) −4.89898 −0.327327
\(225\) 0 0
\(226\) −7.79796 −0.518713
\(227\) 21.7980 1.44678 0.723391 0.690439i \(-0.242583\pi\)
0.723391 + 0.690439i \(0.242583\pi\)
\(228\) 0 0
\(229\) −13.5959 −0.898444 −0.449222 0.893420i \(-0.648299\pi\)
−0.449222 + 0.893420i \(0.648299\pi\)
\(230\) −4.00000 −0.263752
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) −14.0000 −0.917170 −0.458585 0.888650i \(-0.651644\pi\)
−0.458585 + 0.888650i \(0.651644\pi\)
\(234\) 0 0
\(235\) 9.79796 0.639148
\(236\) −4.89898 −0.318896
\(237\) 0 0
\(238\) −4.89898 −0.317554
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 13.5959 0.875790 0.437895 0.899026i \(-0.355724\pi\)
0.437895 + 0.899026i \(0.355724\pi\)
\(242\) −11.0000 −0.707107
\(243\) 0 0
\(244\) 11.7980 0.755287
\(245\) −17.0000 −1.08609
\(246\) 0 0
\(247\) 27.5959 1.75589
\(248\) 4.00000 0.254000
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) −4.89898 −0.309221 −0.154610 0.987976i \(-0.549412\pi\)
−0.154610 + 0.987976i \(0.549412\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −12.0000 −0.752947
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 2.00000 0.124757 0.0623783 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) 0 0
\(259\) −29.3939 −1.82645
\(260\) −6.89898 −0.427857
\(261\) 0 0
\(262\) −9.79796 −0.605320
\(263\) −8.00000 −0.493301 −0.246651 0.969104i \(-0.579330\pi\)
−0.246651 + 0.969104i \(0.579330\pi\)
\(264\) 0 0
\(265\) −7.79796 −0.479025
\(266\) −19.5959 −1.20150
\(267\) 0 0
\(268\) 0.898979 0.0549139
\(269\) −9.59592 −0.585073 −0.292537 0.956254i \(-0.594499\pi\)
−0.292537 + 0.956254i \(0.594499\pi\)
\(270\) 0 0
\(271\) −1.79796 −0.109218 −0.0546091 0.998508i \(-0.517391\pi\)
−0.0546091 + 0.998508i \(0.517391\pi\)
\(272\) 1.00000 0.0606339
\(273\) 0 0
\(274\) −14.0000 −0.845771
\(275\) 0 0
\(276\) 0 0
\(277\) 7.79796 0.468534 0.234267 0.972172i \(-0.424731\pi\)
0.234267 + 0.972172i \(0.424731\pi\)
\(278\) 13.7980 0.827547
\(279\) 0 0
\(280\) 4.89898 0.292770
\(281\) −27.7980 −1.65829 −0.829144 0.559036i \(-0.811171\pi\)
−0.829144 + 0.559036i \(0.811171\pi\)
\(282\) 0 0
\(283\) 23.5959 1.40263 0.701316 0.712851i \(-0.252596\pi\)
0.701316 + 0.712851i \(0.252596\pi\)
\(284\) 8.89898 0.528057
\(285\) 0 0
\(286\) 0 0
\(287\) −14.2020 −0.838320
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 6.00000 0.352332
\(291\) 0 0
\(292\) −10.8990 −0.637815
\(293\) −13.5959 −0.794282 −0.397141 0.917758i \(-0.629998\pi\)
−0.397141 + 0.917758i \(0.629998\pi\)
\(294\) 0 0
\(295\) 4.89898 0.285230
\(296\) 6.00000 0.348743
\(297\) 0 0
\(298\) 18.8990 1.09479
\(299\) 27.5959 1.59591
\(300\) 0 0
\(301\) −43.5959 −2.51283
\(302\) −9.79796 −0.563809
\(303\) 0 0
\(304\) 4.00000 0.229416
\(305\) −11.7980 −0.675549
\(306\) 0 0
\(307\) 0.898979 0.0513075 0.0256537 0.999671i \(-0.491833\pi\)
0.0256537 + 0.999671i \(0.491833\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −4.00000 −0.227185
\(311\) 7.10102 0.402662 0.201331 0.979523i \(-0.435473\pi\)
0.201331 + 0.979523i \(0.435473\pi\)
\(312\) 0 0
\(313\) 6.89898 0.389953 0.194977 0.980808i \(-0.437537\pi\)
0.194977 + 0.980808i \(0.437537\pi\)
\(314\) −1.10102 −0.0621342
\(315\) 0 0
\(316\) −5.79796 −0.326161
\(317\) −14.0000 −0.786318 −0.393159 0.919470i \(-0.628618\pi\)
−0.393159 + 0.919470i \(0.628618\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) −19.5959 −1.09204
\(323\) 4.00000 0.222566
\(324\) 0 0
\(325\) 6.89898 0.382687
\(326\) −2.20204 −0.121960
\(327\) 0 0
\(328\) 2.89898 0.160069
\(329\) 48.0000 2.64633
\(330\) 0 0
\(331\) −5.79796 −0.318685 −0.159342 0.987223i \(-0.550937\pi\)
−0.159342 + 0.987223i \(0.550937\pi\)
\(332\) −13.7980 −0.757261
\(333\) 0 0
\(334\) −2.20204 −0.120490
\(335\) −0.898979 −0.0491165
\(336\) 0 0
\(337\) 32.6969 1.78112 0.890558 0.454870i \(-0.150314\pi\)
0.890558 + 0.454870i \(0.150314\pi\)
\(338\) 34.5959 1.88177
\(339\) 0 0
\(340\) −1.00000 −0.0542326
\(341\) 0 0
\(342\) 0 0
\(343\) −48.9898 −2.64520
\(344\) 8.89898 0.479801
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) −21.7980 −1.17018 −0.585088 0.810970i \(-0.698940\pi\)
−0.585088 + 0.810970i \(0.698940\pi\)
\(348\) 0 0
\(349\) 4.20204 0.224930 0.112465 0.993656i \(-0.464125\pi\)
0.112465 + 0.993656i \(0.464125\pi\)
\(350\) −4.89898 −0.261861
\(351\) 0 0
\(352\) 0 0
\(353\) 26.0000 1.38384 0.691920 0.721974i \(-0.256765\pi\)
0.691920 + 0.721974i \(0.256765\pi\)
\(354\) 0 0
\(355\) −8.89898 −0.472309
\(356\) 7.79796 0.413291
\(357\) 0 0
\(358\) 4.89898 0.258919
\(359\) −37.3939 −1.97357 −0.986787 0.162025i \(-0.948198\pi\)
−0.986787 + 0.162025i \(0.948198\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) −4.20204 −0.220854
\(363\) 0 0
\(364\) −33.7980 −1.77149
\(365\) 10.8990 0.570479
\(366\) 0 0
\(367\) −27.1010 −1.41466 −0.707331 0.706883i \(-0.750101\pi\)
−0.707331 + 0.706883i \(0.750101\pi\)
\(368\) 4.00000 0.208514
\(369\) 0 0
\(370\) −6.00000 −0.311925
\(371\) −38.2020 −1.98335
\(372\) 0 0
\(373\) 24.6969 1.27876 0.639380 0.768891i \(-0.279191\pi\)
0.639380 + 0.768891i \(0.279191\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −9.79796 −0.505291
\(377\) −41.3939 −2.13189
\(378\) 0 0
\(379\) −29.7980 −1.53062 −0.765309 0.643663i \(-0.777414\pi\)
−0.765309 + 0.643663i \(0.777414\pi\)
\(380\) −4.00000 −0.205196
\(381\) 0 0
\(382\) 9.79796 0.501307
\(383\) −8.00000 −0.408781 −0.204390 0.978889i \(-0.565521\pi\)
−0.204390 + 0.978889i \(0.565521\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −1.10102 −0.0560405
\(387\) 0 0
\(388\) −12.6969 −0.644589
\(389\) 38.4949 1.95177 0.975884 0.218288i \(-0.0700472\pi\)
0.975884 + 0.218288i \(0.0700472\pi\)
\(390\) 0 0
\(391\) 4.00000 0.202289
\(392\) 17.0000 0.858630
\(393\) 0 0
\(394\) 13.5959 0.684952
\(395\) 5.79796 0.291727
\(396\) 0 0
\(397\) 1.59592 0.0800968 0.0400484 0.999198i \(-0.487249\pi\)
0.0400484 + 0.999198i \(0.487249\pi\)
\(398\) 15.5959 0.781753
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −22.8990 −1.14352 −0.571760 0.820421i \(-0.693739\pi\)
−0.571760 + 0.820421i \(0.693739\pi\)
\(402\) 0 0
\(403\) 27.5959 1.37465
\(404\) 18.8990 0.940259
\(405\) 0 0
\(406\) 29.3939 1.45879
\(407\) 0 0
\(408\) 0 0
\(409\) −17.5959 −0.870062 −0.435031 0.900415i \(-0.643263\pi\)
−0.435031 + 0.900415i \(0.643263\pi\)
\(410\) −2.89898 −0.143170
\(411\) 0 0
\(412\) 4.00000 0.197066
\(413\) 24.0000 1.18096
\(414\) 0 0
\(415\) 13.7980 0.677315
\(416\) 6.89898 0.338250
\(417\) 0 0
\(418\) 0 0
\(419\) 17.7980 0.869487 0.434744 0.900554i \(-0.356839\pi\)
0.434744 + 0.900554i \(0.356839\pi\)
\(420\) 0 0
\(421\) 14.0000 0.682318 0.341159 0.940006i \(-0.389181\pi\)
0.341159 + 0.940006i \(0.389181\pi\)
\(422\) −12.0000 −0.584151
\(423\) 0 0
\(424\) 7.79796 0.378702
\(425\) 1.00000 0.0485071
\(426\) 0 0
\(427\) −57.7980 −2.79704
\(428\) −5.79796 −0.280255
\(429\) 0 0
\(430\) −8.89898 −0.429147
\(431\) 23.1010 1.11274 0.556369 0.830936i \(-0.312194\pi\)
0.556369 + 0.830936i \(0.312194\pi\)
\(432\) 0 0
\(433\) −35.3939 −1.70092 −0.850461 0.526039i \(-0.823677\pi\)
−0.850461 + 0.526039i \(0.823677\pi\)
\(434\) −19.5959 −0.940634
\(435\) 0 0
\(436\) 11.7980 0.565020
\(437\) 16.0000 0.765384
\(438\) 0 0
\(439\) 5.79796 0.276721 0.138361 0.990382i \(-0.455817\pi\)
0.138361 + 0.990382i \(0.455817\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 6.89898 0.328151
\(443\) −37.7980 −1.79584 −0.897918 0.440164i \(-0.854920\pi\)
−0.897918 + 0.440164i \(0.854920\pi\)
\(444\) 0 0
\(445\) −7.79796 −0.369659
\(446\) −4.00000 −0.189405
\(447\) 0 0
\(448\) −4.89898 −0.231455
\(449\) −6.89898 −0.325583 −0.162791 0.986660i \(-0.552050\pi\)
−0.162791 + 0.986660i \(0.552050\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −7.79796 −0.366785
\(453\) 0 0
\(454\) 21.7980 1.02303
\(455\) 33.7980 1.58447
\(456\) 0 0
\(457\) 16.2020 0.757900 0.378950 0.925417i \(-0.376285\pi\)
0.378950 + 0.925417i \(0.376285\pi\)
\(458\) −13.5959 −0.635296
\(459\) 0 0
\(460\) −4.00000 −0.186501
\(461\) 28.6969 1.33655 0.668275 0.743914i \(-0.267033\pi\)
0.668275 + 0.743914i \(0.267033\pi\)
\(462\) 0 0
\(463\) 7.59592 0.353012 0.176506 0.984300i \(-0.443520\pi\)
0.176506 + 0.984300i \(0.443520\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) −14.0000 −0.648537
\(467\) 7.59592 0.351497 0.175749 0.984435i \(-0.443765\pi\)
0.175749 + 0.984435i \(0.443765\pi\)
\(468\) 0 0
\(469\) −4.40408 −0.203362
\(470\) 9.79796 0.451946
\(471\) 0 0
\(472\) −4.89898 −0.225494
\(473\) 0 0
\(474\) 0 0
\(475\) 4.00000 0.183533
\(476\) −4.89898 −0.224544
\(477\) 0 0
\(478\) 0 0
\(479\) 32.8990 1.50319 0.751596 0.659623i \(-0.229284\pi\)
0.751596 + 0.659623i \(0.229284\pi\)
\(480\) 0 0
\(481\) 41.3939 1.88740
\(482\) 13.5959 0.619277
\(483\) 0 0
\(484\) −11.0000 −0.500000
\(485\) 12.6969 0.576538
\(486\) 0 0
\(487\) −20.8990 −0.947023 −0.473512 0.880788i \(-0.657014\pi\)
−0.473512 + 0.880788i \(0.657014\pi\)
\(488\) 11.7980 0.534069
\(489\) 0 0
\(490\) −17.0000 −0.767982
\(491\) 1.30306 0.0588063 0.0294032 0.999568i \(-0.490639\pi\)
0.0294032 + 0.999568i \(0.490639\pi\)
\(492\) 0 0
\(493\) −6.00000 −0.270226
\(494\) 27.5959 1.24160
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) −43.5959 −1.95554
\(498\) 0 0
\(499\) −39.5959 −1.77256 −0.886278 0.463153i \(-0.846718\pi\)
−0.886278 + 0.463153i \(0.846718\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) −4.89898 −0.218652
\(503\) −4.00000 −0.178351 −0.0891756 0.996016i \(-0.528423\pi\)
−0.0891756 + 0.996016i \(0.528423\pi\)
\(504\) 0 0
\(505\) −18.8990 −0.840994
\(506\) 0 0
\(507\) 0 0
\(508\) −12.0000 −0.532414
\(509\) 15.3031 0.678296 0.339148 0.940733i \(-0.389861\pi\)
0.339148 + 0.940733i \(0.389861\pi\)
\(510\) 0 0
\(511\) 53.3939 2.36201
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 2.00000 0.0882162
\(515\) −4.00000 −0.176261
\(516\) 0 0
\(517\) 0 0
\(518\) −29.3939 −1.29149
\(519\) 0 0
\(520\) −6.89898 −0.302540
\(521\) 40.2929 1.76526 0.882631 0.470066i \(-0.155770\pi\)
0.882631 + 0.470066i \(0.155770\pi\)
\(522\) 0 0
\(523\) 18.6969 0.817560 0.408780 0.912633i \(-0.365954\pi\)
0.408780 + 0.912633i \(0.365954\pi\)
\(524\) −9.79796 −0.428026
\(525\) 0 0
\(526\) −8.00000 −0.348817
\(527\) 4.00000 0.174243
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) −7.79796 −0.338722
\(531\) 0 0
\(532\) −19.5959 −0.849591
\(533\) 20.0000 0.866296
\(534\) 0 0
\(535\) 5.79796 0.250668
\(536\) 0.898979 0.0388300
\(537\) 0 0
\(538\) −9.59592 −0.413709
\(539\) 0 0
\(540\) 0 0
\(541\) −7.79796 −0.335260 −0.167630 0.985850i \(-0.553611\pi\)
−0.167630 + 0.985850i \(0.553611\pi\)
\(542\) −1.79796 −0.0772290
\(543\) 0 0
\(544\) 1.00000 0.0428746
\(545\) −11.7980 −0.505369
\(546\) 0 0
\(547\) 0.404082 0.0172773 0.00863865 0.999963i \(-0.497250\pi\)
0.00863865 + 0.999963i \(0.497250\pi\)
\(548\) −14.0000 −0.598050
\(549\) 0 0
\(550\) 0 0
\(551\) −24.0000 −1.02243
\(552\) 0 0
\(553\) 28.4041 1.20786
\(554\) 7.79796 0.331304
\(555\) 0 0
\(556\) 13.7980 0.585164
\(557\) −19.7980 −0.838866 −0.419433 0.907786i \(-0.637771\pi\)
−0.419433 + 0.907786i \(0.637771\pi\)
\(558\) 0 0
\(559\) 61.3939 2.59668
\(560\) 4.89898 0.207020
\(561\) 0 0
\(562\) −27.7980 −1.17259
\(563\) −21.7980 −0.918674 −0.459337 0.888262i \(-0.651913\pi\)
−0.459337 + 0.888262i \(0.651913\pi\)
\(564\) 0 0
\(565\) 7.79796 0.328063
\(566\) 23.5959 0.991810
\(567\) 0 0
\(568\) 8.89898 0.373393
\(569\) −34.0000 −1.42535 −0.712677 0.701492i \(-0.752517\pi\)
−0.712677 + 0.701492i \(0.752517\pi\)
\(570\) 0 0
\(571\) 29.7980 1.24701 0.623503 0.781821i \(-0.285709\pi\)
0.623503 + 0.781821i \(0.285709\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −14.2020 −0.592782
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) −27.3939 −1.14042 −0.570211 0.821498i \(-0.693139\pi\)
−0.570211 + 0.821498i \(0.693139\pi\)
\(578\) 1.00000 0.0415945
\(579\) 0 0
\(580\) 6.00000 0.249136
\(581\) 67.5959 2.80435
\(582\) 0 0
\(583\) 0 0
\(584\) −10.8990 −0.451003
\(585\) 0 0
\(586\) −13.5959 −0.561642
\(587\) 41.3939 1.70851 0.854254 0.519856i \(-0.174014\pi\)
0.854254 + 0.519856i \(0.174014\pi\)
\(588\) 0 0
\(589\) 16.0000 0.659269
\(590\) 4.89898 0.201688
\(591\) 0 0
\(592\) 6.00000 0.246598
\(593\) −1.59592 −0.0655365 −0.0327682 0.999463i \(-0.510432\pi\)
−0.0327682 + 0.999463i \(0.510432\pi\)
\(594\) 0 0
\(595\) 4.89898 0.200839
\(596\) 18.8990 0.774132
\(597\) 0 0
\(598\) 27.5959 1.12848
\(599\) −21.3939 −0.874130 −0.437065 0.899430i \(-0.643982\pi\)
−0.437065 + 0.899430i \(0.643982\pi\)
\(600\) 0 0
\(601\) 16.2020 0.660895 0.330448 0.943824i \(-0.392800\pi\)
0.330448 + 0.943824i \(0.392800\pi\)
\(602\) −43.5959 −1.77684
\(603\) 0 0
\(604\) −9.79796 −0.398673
\(605\) 11.0000 0.447214
\(606\) 0 0
\(607\) −32.4949 −1.31893 −0.659464 0.751736i \(-0.729217\pi\)
−0.659464 + 0.751736i \(0.729217\pi\)
\(608\) 4.00000 0.162221
\(609\) 0 0
\(610\) −11.7980 −0.477685
\(611\) −67.5959 −2.73464
\(612\) 0 0
\(613\) −14.4949 −0.585443 −0.292722 0.956198i \(-0.594561\pi\)
−0.292722 + 0.956198i \(0.594561\pi\)
\(614\) 0.898979 0.0362799
\(615\) 0 0
\(616\) 0 0
\(617\) 37.5959 1.51355 0.756777 0.653673i \(-0.226773\pi\)
0.756777 + 0.653673i \(0.226773\pi\)
\(618\) 0 0
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) −4.00000 −0.160644
\(621\) 0 0
\(622\) 7.10102 0.284725
\(623\) −38.2020 −1.53053
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 6.89898 0.275739
\(627\) 0 0
\(628\) −1.10102 −0.0439355
\(629\) 6.00000 0.239236
\(630\) 0 0
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) −5.79796 −0.230630
\(633\) 0 0
\(634\) −14.0000 −0.556011
\(635\) 12.0000 0.476205
\(636\) 0 0
\(637\) 117.283 4.64691
\(638\) 0 0
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) 20.6969 0.817480 0.408740 0.912651i \(-0.365968\pi\)
0.408740 + 0.912651i \(0.365968\pi\)
\(642\) 0 0
\(643\) 29.7980 1.17512 0.587558 0.809182i \(-0.300089\pi\)
0.587558 + 0.809182i \(0.300089\pi\)
\(644\) −19.5959 −0.772187
\(645\) 0 0
\(646\) 4.00000 0.157378
\(647\) 32.0000 1.25805 0.629025 0.777385i \(-0.283454\pi\)
0.629025 + 0.777385i \(0.283454\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 6.89898 0.270600
\(651\) 0 0
\(652\) −2.20204 −0.0862386
\(653\) −30.0000 −1.17399 −0.586995 0.809590i \(-0.699689\pi\)
−0.586995 + 0.809590i \(0.699689\pi\)
\(654\) 0 0
\(655\) 9.79796 0.382838
\(656\) 2.89898 0.113186
\(657\) 0 0
\(658\) 48.0000 1.87123
\(659\) −30.6969 −1.19578 −0.597891 0.801577i \(-0.703995\pi\)
−0.597891 + 0.801577i \(0.703995\pi\)
\(660\) 0 0
\(661\) −19.7980 −0.770051 −0.385026 0.922906i \(-0.625807\pi\)
−0.385026 + 0.922906i \(0.625807\pi\)
\(662\) −5.79796 −0.225344
\(663\) 0 0
\(664\) −13.7980 −0.535465
\(665\) 19.5959 0.759897
\(666\) 0 0
\(667\) −24.0000 −0.929284
\(668\) −2.20204 −0.0851995
\(669\) 0 0
\(670\) −0.898979 −0.0347306
\(671\) 0 0
\(672\) 0 0
\(673\) −33.1010 −1.27595 −0.637975 0.770057i \(-0.720228\pi\)
−0.637975 + 0.770057i \(0.720228\pi\)
\(674\) 32.6969 1.25944
\(675\) 0 0
\(676\) 34.5959 1.33061
\(677\) 13.5959 0.522534 0.261267 0.965267i \(-0.415860\pi\)
0.261267 + 0.965267i \(0.415860\pi\)
\(678\) 0 0
\(679\) 62.2020 2.38710
\(680\) −1.00000 −0.0383482
\(681\) 0 0
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 0 0
\(685\) 14.0000 0.534913
\(686\) −48.9898 −1.87044
\(687\) 0 0
\(688\) 8.89898 0.339270
\(689\) 53.7980 2.04954
\(690\) 0 0
\(691\) −27.1918 −1.03443 −0.517213 0.855857i \(-0.673031\pi\)
−0.517213 + 0.855857i \(0.673031\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) −21.7980 −0.827439
\(695\) −13.7980 −0.523386
\(696\) 0 0
\(697\) 2.89898 0.109807
\(698\) 4.20204 0.159050
\(699\) 0 0
\(700\) −4.89898 −0.185164
\(701\) −14.8990 −0.562727 −0.281363 0.959601i \(-0.590787\pi\)
−0.281363 + 0.959601i \(0.590787\pi\)
\(702\) 0 0
\(703\) 24.0000 0.905177
\(704\) 0 0
\(705\) 0 0
\(706\) 26.0000 0.978523
\(707\) −92.5857 −3.48204
\(708\) 0 0
\(709\) −31.7980 −1.19420 −0.597099 0.802168i \(-0.703680\pi\)
−0.597099 + 0.802168i \(0.703680\pi\)
\(710\) −8.89898 −0.333973
\(711\) 0 0
\(712\) 7.79796 0.292241
\(713\) 16.0000 0.599205
\(714\) 0 0
\(715\) 0 0
\(716\) 4.89898 0.183083
\(717\) 0 0
\(718\) −37.3939 −1.39553
\(719\) 12.4949 0.465981 0.232991 0.972479i \(-0.425149\pi\)
0.232991 + 0.972479i \(0.425149\pi\)
\(720\) 0 0
\(721\) −19.5959 −0.729790
\(722\) −3.00000 −0.111648
\(723\) 0 0
\(724\) −4.20204 −0.156168
\(725\) −6.00000 −0.222834
\(726\) 0 0
\(727\) 0.404082 0.0149866 0.00749329 0.999972i \(-0.497615\pi\)
0.00749329 + 0.999972i \(0.497615\pi\)
\(728\) −33.7980 −1.25264
\(729\) 0 0
\(730\) 10.8990 0.403389
\(731\) 8.89898 0.329141
\(732\) 0 0
\(733\) −46.4949 −1.71733 −0.858664 0.512539i \(-0.828705\pi\)
−0.858664 + 0.512539i \(0.828705\pi\)
\(734\) −27.1010 −1.00032
\(735\) 0 0
\(736\) 4.00000 0.147442
\(737\) 0 0
\(738\) 0 0
\(739\) −9.39388 −0.345559 −0.172780 0.984960i \(-0.555275\pi\)
−0.172780 + 0.984960i \(0.555275\pi\)
\(740\) −6.00000 −0.220564
\(741\) 0 0
\(742\) −38.2020 −1.40244
\(743\) 9.39388 0.344628 0.172314 0.985042i \(-0.444876\pi\)
0.172314 + 0.985042i \(0.444876\pi\)
\(744\) 0 0
\(745\) −18.8990 −0.692405
\(746\) 24.6969 0.904219
\(747\) 0 0
\(748\) 0 0
\(749\) 28.4041 1.03786
\(750\) 0 0
\(751\) 21.7980 0.795419 0.397709 0.917511i \(-0.369805\pi\)
0.397709 + 0.917511i \(0.369805\pi\)
\(752\) −9.79796 −0.357295
\(753\) 0 0
\(754\) −41.3939 −1.50748
\(755\) 9.79796 0.356584
\(756\) 0 0
\(757\) −4.69694 −0.170713 −0.0853566 0.996350i \(-0.527203\pi\)
−0.0853566 + 0.996350i \(0.527203\pi\)
\(758\) −29.7980 −1.08231
\(759\) 0 0
\(760\) −4.00000 −0.145095
\(761\) 1.59592 0.0578520 0.0289260 0.999582i \(-0.490791\pi\)
0.0289260 + 0.999582i \(0.490791\pi\)
\(762\) 0 0
\(763\) −57.7980 −2.09243
\(764\) 9.79796 0.354478
\(765\) 0 0
\(766\) −8.00000 −0.289052
\(767\) −33.7980 −1.22037
\(768\) 0 0
\(769\) 26.0000 0.937584 0.468792 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −1.10102 −0.0396266
\(773\) 35.3939 1.27303 0.636515 0.771265i \(-0.280375\pi\)
0.636515 + 0.771265i \(0.280375\pi\)
\(774\) 0 0
\(775\) 4.00000 0.143684
\(776\) −12.6969 −0.455794
\(777\) 0 0
\(778\) 38.4949 1.38011
\(779\) 11.5959 0.415467
\(780\) 0 0
\(781\) 0 0
\(782\) 4.00000 0.143040
\(783\) 0 0
\(784\) 17.0000 0.607143
\(785\) 1.10102 0.0392971
\(786\) 0 0
\(787\) −45.7980 −1.63252 −0.816260 0.577684i \(-0.803957\pi\)
−0.816260 + 0.577684i \(0.803957\pi\)
\(788\) 13.5959 0.484335
\(789\) 0 0
\(790\) 5.79796 0.206282
\(791\) 38.2020 1.35831
\(792\) 0 0
\(793\) 81.3939 2.89038
\(794\) 1.59592 0.0566370
\(795\) 0 0
\(796\) 15.5959 0.552783
\(797\) −26.0000 −0.920967 −0.460484 0.887668i \(-0.652324\pi\)
−0.460484 + 0.887668i \(0.652324\pi\)
\(798\) 0 0
\(799\) −9.79796 −0.346627
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) −22.8990 −0.808591
\(803\) 0 0
\(804\) 0 0
\(805\) 19.5959 0.690665
\(806\) 27.5959 0.972025
\(807\) 0 0
\(808\) 18.8990 0.664864
\(809\) −32.6969 −1.14956 −0.574782 0.818307i \(-0.694913\pi\)
−0.574782 + 0.818307i \(0.694913\pi\)
\(810\) 0 0
\(811\) 25.3939 0.891700 0.445850 0.895108i \(-0.352902\pi\)
0.445850 + 0.895108i \(0.352902\pi\)
\(812\) 29.3939 1.03152
\(813\) 0 0
\(814\) 0 0
\(815\) 2.20204 0.0771341
\(816\) 0 0
\(817\) 35.5959 1.24534
\(818\) −17.5959 −0.615227
\(819\) 0 0
\(820\) −2.89898 −0.101237
\(821\) −15.7980 −0.551353 −0.275676 0.961251i \(-0.588902\pi\)
−0.275676 + 0.961251i \(0.588902\pi\)
\(822\) 0 0
\(823\) 20.8990 0.728493 0.364246 0.931303i \(-0.381327\pi\)
0.364246 + 0.931303i \(0.381327\pi\)
\(824\) 4.00000 0.139347
\(825\) 0 0
\(826\) 24.0000 0.835067
\(827\) 4.00000 0.139094 0.0695468 0.997579i \(-0.477845\pi\)
0.0695468 + 0.997579i \(0.477845\pi\)
\(828\) 0 0
\(829\) 3.39388 0.117874 0.0589371 0.998262i \(-0.481229\pi\)
0.0589371 + 0.998262i \(0.481229\pi\)
\(830\) 13.7980 0.478934
\(831\) 0 0
\(832\) 6.89898 0.239179
\(833\) 17.0000 0.589015
\(834\) 0 0
\(835\) 2.20204 0.0762048
\(836\) 0 0
\(837\) 0 0
\(838\) 17.7980 0.614820
\(839\) 7.10102 0.245154 0.122577 0.992459i \(-0.460884\pi\)
0.122577 + 0.992459i \(0.460884\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 14.0000 0.482472
\(843\) 0 0
\(844\) −12.0000 −0.413057
\(845\) −34.5959 −1.19014
\(846\) 0 0
\(847\) 53.8888 1.85164
\(848\) 7.79796 0.267783
\(849\) 0 0
\(850\) 1.00000 0.0342997
\(851\) 24.0000 0.822709
\(852\) 0 0
\(853\) 11.3939 0.390119 0.195059 0.980791i \(-0.437510\pi\)
0.195059 + 0.980791i \(0.437510\pi\)
\(854\) −57.7980 −1.97781
\(855\) 0 0
\(856\) −5.79796 −0.198170
\(857\) 10.0000 0.341593 0.170797 0.985306i \(-0.445366\pi\)
0.170797 + 0.985306i \(0.445366\pi\)
\(858\) 0 0
\(859\) 5.79796 0.197824 0.0989119 0.995096i \(-0.468464\pi\)
0.0989119 + 0.995096i \(0.468464\pi\)
\(860\) −8.89898 −0.303453
\(861\) 0 0
\(862\) 23.1010 0.786824
\(863\) 45.3939 1.54523 0.772613 0.634878i \(-0.218949\pi\)
0.772613 + 0.634878i \(0.218949\pi\)
\(864\) 0 0
\(865\) 6.00000 0.204006
\(866\) −35.3939 −1.20273
\(867\) 0 0
\(868\) −19.5959 −0.665129
\(869\) 0 0
\(870\) 0 0
\(871\) 6.20204 0.210148
\(872\) 11.7980 0.399529
\(873\) 0 0
\(874\) 16.0000 0.541208
\(875\) 4.89898 0.165616
\(876\) 0 0
\(877\) −31.3939 −1.06010 −0.530048 0.847968i \(-0.677826\pi\)
−0.530048 + 0.847968i \(0.677826\pi\)
\(878\) 5.79796 0.195672
\(879\) 0 0
\(880\) 0 0
\(881\) −34.4949 −1.16216 −0.581081 0.813846i \(-0.697370\pi\)
−0.581081 + 0.813846i \(0.697370\pi\)
\(882\) 0 0
\(883\) 26.6969 0.898424 0.449212 0.893425i \(-0.351705\pi\)
0.449212 + 0.893425i \(0.351705\pi\)
\(884\) 6.89898 0.232038
\(885\) 0 0
\(886\) −37.7980 −1.26985
\(887\) 12.0000 0.402921 0.201460 0.979497i \(-0.435431\pi\)
0.201460 + 0.979497i \(0.435431\pi\)
\(888\) 0 0
\(889\) 58.7878 1.97168
\(890\) −7.79796 −0.261388
\(891\) 0 0
\(892\) −4.00000 −0.133930
\(893\) −39.1918 −1.31150
\(894\) 0 0
\(895\) −4.89898 −0.163755
\(896\) −4.89898 −0.163663
\(897\) 0 0
\(898\) −6.89898 −0.230222
\(899\) −24.0000 −0.800445
\(900\) 0 0
\(901\) 7.79796 0.259788
\(902\) 0 0
\(903\) 0 0
\(904\) −7.79796 −0.259356
\(905\) 4.20204 0.139681
\(906\) 0 0
\(907\) −33.3939 −1.10883 −0.554413 0.832242i \(-0.687057\pi\)
−0.554413 + 0.832242i \(0.687057\pi\)
\(908\) 21.7980 0.723391
\(909\) 0 0
\(910\) 33.7980 1.12039
\(911\) −23.1010 −0.765371 −0.382685 0.923879i \(-0.625001\pi\)
−0.382685 + 0.923879i \(0.625001\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 16.2020 0.535916
\(915\) 0 0
\(916\) −13.5959 −0.449222
\(917\) 48.0000 1.58510
\(918\) 0 0
\(919\) 1.79796 0.0593092 0.0296546 0.999560i \(-0.490559\pi\)
0.0296546 + 0.999560i \(0.490559\pi\)
\(920\) −4.00000 −0.131876
\(921\) 0 0
\(922\) 28.6969 0.945083
\(923\) 61.3939 2.02080
\(924\) 0 0
\(925\) 6.00000 0.197279
\(926\) 7.59592 0.249617
\(927\) 0 0
\(928\) −6.00000 −0.196960
\(929\) −36.2929 −1.19073 −0.595365 0.803455i \(-0.702993\pi\)
−0.595365 + 0.803455i \(0.702993\pi\)
\(930\) 0 0
\(931\) 68.0000 2.22861
\(932\) −14.0000 −0.458585
\(933\) 0 0
\(934\) 7.59592 0.248546
\(935\) 0 0
\(936\) 0 0
\(937\) 39.3939 1.28694 0.643471 0.765471i \(-0.277494\pi\)
0.643471 + 0.765471i \(0.277494\pi\)
\(938\) −4.40408 −0.143798
\(939\) 0 0
\(940\) 9.79796 0.319574
\(941\) 16.2020 0.528171 0.264086 0.964499i \(-0.414930\pi\)
0.264086 + 0.964499i \(0.414930\pi\)
\(942\) 0 0
\(943\) 11.5959 0.377615
\(944\) −4.89898 −0.159448
\(945\) 0 0
\(946\) 0 0
\(947\) −21.7980 −0.708338 −0.354169 0.935181i \(-0.615236\pi\)
−0.354169 + 0.935181i \(0.615236\pi\)
\(948\) 0 0
\(949\) −75.1918 −2.44083
\(950\) 4.00000 0.129777
\(951\) 0 0
\(952\) −4.89898 −0.158777
\(953\) 41.1918 1.33433 0.667167 0.744908i \(-0.267507\pi\)
0.667167 + 0.744908i \(0.267507\pi\)
\(954\) 0 0
\(955\) −9.79796 −0.317055
\(956\) 0 0
\(957\) 0 0
\(958\) 32.8990 1.06292
\(959\) 68.5857 2.21475
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 41.3939 1.33459
\(963\) 0 0
\(964\) 13.5959 0.437895
\(965\) 1.10102 0.0354431
\(966\) 0 0
\(967\) −41.3939 −1.33114 −0.665569 0.746337i \(-0.731811\pi\)
−0.665569 + 0.746337i \(0.731811\pi\)
\(968\) −11.0000 −0.353553
\(969\) 0 0
\(970\) 12.6969 0.407674
\(971\) −38.6969 −1.24184 −0.620922 0.783872i \(-0.713242\pi\)
−0.620922 + 0.783872i \(0.713242\pi\)
\(972\) 0 0
\(973\) −67.5959 −2.16703
\(974\) −20.8990 −0.669646
\(975\) 0 0
\(976\) 11.7980 0.377643
\(977\) 29.5959 0.946857 0.473429 0.880832i \(-0.343016\pi\)
0.473429 + 0.880832i \(0.343016\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −17.0000 −0.543045
\(981\) 0 0
\(982\) 1.30306 0.0415824
\(983\) −9.39388 −0.299618 −0.149809 0.988715i \(-0.547866\pi\)
−0.149809 + 0.988715i \(0.547866\pi\)
\(984\) 0 0
\(985\) −13.5959 −0.433202
\(986\) −6.00000 −0.191079
\(987\) 0 0
\(988\) 27.5959 0.877943
\(989\) 35.5959 1.13188
\(990\) 0 0
\(991\) −15.5959 −0.495421 −0.247710 0.968834i \(-0.579678\pi\)
−0.247710 + 0.968834i \(0.579678\pi\)
\(992\) 4.00000 0.127000
\(993\) 0 0
\(994\) −43.5959 −1.38278
\(995\) −15.5959 −0.494424
\(996\) 0 0
\(997\) −21.5959 −0.683950 −0.341975 0.939709i \(-0.611096\pi\)
−0.341975 + 0.939709i \(0.611096\pi\)
\(998\) −39.5959 −1.25339
\(999\) 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 1530.2.a.s.1.1 2
3.2 odd 2 510.2.a.h.1.1 2
5.4 even 2 7650.2.a.cu.1.2 2
12.11 even 2 4080.2.a.bq.1.2 2
15.2 even 4 2550.2.d.u.2449.1 4
15.8 even 4 2550.2.d.u.2449.4 4
15.14 odd 2 2550.2.a.bl.1.2 2
51.50 odd 2 8670.2.a.be.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
510.2.a.h.1.1 2 3.2 odd 2
1530.2.a.s.1.1 2 1.1 even 1 trivial
2550.2.a.bl.1.2 2 15.14 odd 2
2550.2.d.u.2449.1 4 15.2 even 4
2550.2.d.u.2449.4 4 15.8 even 4
4080.2.a.bq.1.2 2 12.11 even 2
7650.2.a.cu.1.2 2 5.4 even 2
8670.2.a.be.1.2 2 51.50 odd 2