Properties

Label 1143.2.a.f.1.3
Level $1143$
Weight $2$
Character 1143.1
Self dual yes
Analytic conductor $9.127$
Analytic rank $1$
Dimension $4$
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] = [1143,2,Mod(1,1143)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1143, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1143.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1143 = 3^{2} \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1143.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: \(9.12690095103\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.18890\) of defining polynomial
Character \(\chi\) \(=\) 1143.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.73205 q^{2} +1.00000 q^{4} -2.18890 q^{5} +0.791288 q^{7} -1.73205 q^{8} +O(q^{10})\) \(q+1.73205 q^{2} +1.00000 q^{4} -2.18890 q^{5} +0.791288 q^{7} -1.73205 q^{8} -3.79129 q^{10} +0.456850 q^{11} -5.79129 q^{13} +1.37055 q^{14} -5.00000 q^{16} +0.913701 q^{17} -5.58258 q^{19} -2.18890 q^{20} +0.791288 q^{22} -0.208712 q^{25} -10.0308 q^{26} +0.791288 q^{28} +4.83465 q^{29} -5.58258 q^{31} -5.19615 q^{32} +1.58258 q^{34} -1.73205 q^{35} +1.79129 q^{37} -9.66930 q^{38} +3.79129 q^{40} -6.20520 q^{41} +5.37386 q^{43} +0.456850 q^{44} +2.18890 q^{47} -6.37386 q^{49} -0.361500 q^{50} -5.79129 q^{52} +4.83465 q^{53} -1.00000 q^{55} -1.37055 q^{56} +8.37386 q^{58} -1.82740 q^{59} -2.79129 q^{61} -9.66930 q^{62} +1.00000 q^{64} +12.6766 q^{65} -5.79129 q^{67} +0.913701 q^{68} -3.00000 q^{70} -8.29875 q^{71} +0.373864 q^{73} +3.10260 q^{74} -5.58258 q^{76} +0.361500 q^{77} -3.58258 q^{79} +10.9445 q^{80} -10.7477 q^{82} +16.5975 q^{83} -2.00000 q^{85} +9.30780 q^{86} -0.791288 q^{88} -0.361500 q^{89} -4.58258 q^{91} +3.79129 q^{94} +12.2197 q^{95} -8.00000 q^{97} -11.0399 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} - 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{4} - 6 q^{7} - 6 q^{10} - 14 q^{13} - 20 q^{16} - 4 q^{19} - 6 q^{22} - 10 q^{25} - 6 q^{28} - 4 q^{31} - 12 q^{34} - 2 q^{37} + 6 q^{40} - 6 q^{43} + 2 q^{49} - 14 q^{52} - 4 q^{55} + 6 q^{58} - 2 q^{61} + 4 q^{64} - 14 q^{67} - 12 q^{70} - 26 q^{73} - 4 q^{76} + 4 q^{79} + 12 q^{82} - 8 q^{85} + 6 q^{88} + 6 q^{94} - 32 q^{97}+O(q^{100}) \) 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 \(0.290215\pi\)
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −2.18890 −0.978906 −0.489453 0.872030i \(-0.662804\pi\)
−0.489453 + 0.872030i \(0.662804\pi\)
\(6\) 0 0
\(7\) 0.791288 0.299079 0.149539 0.988756i \(-0.452221\pi\)
0.149539 + 0.988756i \(0.452221\pi\)
\(8\) −1.73205 −0.612372
\(9\) 0 0
\(10\) −3.79129 −1.19891
\(11\) 0.456850 0.137746 0.0688728 0.997625i \(-0.478060\pi\)
0.0688728 + 0.997625i \(0.478060\pi\)
\(12\) 0 0
\(13\) −5.79129 −1.60621 −0.803107 0.595835i \(-0.796821\pi\)
−0.803107 + 0.595835i \(0.796821\pi\)
\(14\) 1.37055 0.366295
\(15\) 0 0
\(16\) −5.00000 −1.25000
\(17\) 0.913701 0.221605 0.110802 0.993842i \(-0.464658\pi\)
0.110802 + 0.993842i \(0.464658\pi\)
\(18\) 0 0
\(19\) −5.58258 −1.28073 −0.640365 0.768070i \(-0.721217\pi\)
−0.640365 + 0.768070i \(0.721217\pi\)
\(20\) −2.18890 −0.489453
\(21\) 0 0
\(22\) 0.791288 0.168703
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −0.208712 −0.0417424
\(26\) −10.0308 −1.96720
\(27\) 0 0
\(28\) 0.791288 0.149539
\(29\) 4.83465 0.897772 0.448886 0.893589i \(-0.351821\pi\)
0.448886 + 0.893589i \(0.351821\pi\)
\(30\) 0 0
\(31\) −5.58258 −1.00266 −0.501330 0.865256i \(-0.667156\pi\)
−0.501330 + 0.865256i \(0.667156\pi\)
\(32\) −5.19615 −0.918559
\(33\) 0 0
\(34\) 1.58258 0.271409
\(35\) −1.73205 −0.292770
\(36\) 0 0
\(37\) 1.79129 0.294486 0.147243 0.989100i \(-0.452960\pi\)
0.147243 + 0.989100i \(0.452960\pi\)
\(38\) −9.66930 −1.56857
\(39\) 0 0
\(40\) 3.79129 0.599455
\(41\) −6.20520 −0.969090 −0.484545 0.874766i \(-0.661015\pi\)
−0.484545 + 0.874766i \(0.661015\pi\)
\(42\) 0 0
\(43\) 5.37386 0.819507 0.409753 0.912196i \(-0.365615\pi\)
0.409753 + 0.912196i \(0.365615\pi\)
\(44\) 0.456850 0.0688728
\(45\) 0 0
\(46\) 0 0
\(47\) 2.18890 0.319284 0.159642 0.987175i \(-0.448966\pi\)
0.159642 + 0.987175i \(0.448966\pi\)
\(48\) 0 0
\(49\) −6.37386 −0.910552
\(50\) −0.361500 −0.0511238
\(51\) 0 0
\(52\) −5.79129 −0.803107
\(53\) 4.83465 0.664091 0.332045 0.943263i \(-0.392261\pi\)
0.332045 + 0.943263i \(0.392261\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) −1.37055 −0.183148
\(57\) 0 0
\(58\) 8.37386 1.09954
\(59\) −1.82740 −0.237907 −0.118954 0.992900i \(-0.537954\pi\)
−0.118954 + 0.992900i \(0.537954\pi\)
\(60\) 0 0
\(61\) −2.79129 −0.357388 −0.178694 0.983905i \(-0.557187\pi\)
−0.178694 + 0.983905i \(0.557187\pi\)
\(62\) −9.66930 −1.22800
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 12.6766 1.57233
\(66\) 0 0
\(67\) −5.79129 −0.707518 −0.353759 0.935337i \(-0.615097\pi\)
−0.353759 + 0.935337i \(0.615097\pi\)
\(68\) 0.913701 0.110802
\(69\) 0 0
\(70\) −3.00000 −0.358569
\(71\) −8.29875 −0.984881 −0.492440 0.870346i \(-0.663895\pi\)
−0.492440 + 0.870346i \(0.663895\pi\)
\(72\) 0 0
\(73\) 0.373864 0.0437574 0.0218787 0.999761i \(-0.493035\pi\)
0.0218787 + 0.999761i \(0.493035\pi\)
\(74\) 3.10260 0.360670
\(75\) 0 0
\(76\) −5.58258 −0.640365
\(77\) 0.361500 0.0411968
\(78\) 0 0
\(79\) −3.58258 −0.403071 −0.201536 0.979481i \(-0.564593\pi\)
−0.201536 + 0.979481i \(0.564593\pi\)
\(80\) 10.9445 1.22363
\(81\) 0 0
\(82\) −10.7477 −1.18689
\(83\) 16.5975 1.82181 0.910907 0.412613i \(-0.135384\pi\)
0.910907 + 0.412613i \(0.135384\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) 9.30780 1.00369
\(87\) 0 0
\(88\) −0.791288 −0.0843516
\(89\) −0.361500 −0.0383189 −0.0191595 0.999816i \(-0.506099\pi\)
−0.0191595 + 0.999816i \(0.506099\pi\)
\(90\) 0 0
\(91\) −4.58258 −0.480384
\(92\) 0 0
\(93\) 0 0
\(94\) 3.79129 0.391041
\(95\) 12.2197 1.25372
\(96\) 0 0
\(97\) −8.00000 −0.812277 −0.406138 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) −11.0399 −1.11519
\(99\) 0 0
\(100\) −0.208712 −0.0208712
\(101\) −3.65480 −0.363666 −0.181833 0.983329i \(-0.558203\pi\)
−0.181833 + 0.983329i \(0.558203\pi\)
\(102\) 0 0
\(103\) 1.58258 0.155936 0.0779679 0.996956i \(-0.475157\pi\)
0.0779679 + 0.996956i \(0.475157\pi\)
\(104\) 10.0308 0.983601
\(105\) 0 0
\(106\) 8.37386 0.813342
\(107\) −12.7719 −1.23471 −0.617353 0.786686i \(-0.711795\pi\)
−0.617353 + 0.786686i \(0.711795\pi\)
\(108\) 0 0
\(109\) 20.7477 1.98727 0.993636 0.112640i \(-0.0359305\pi\)
0.993636 + 0.112640i \(0.0359305\pi\)
\(110\) −1.73205 −0.165145
\(111\) 0 0
\(112\) −3.95644 −0.373848
\(113\) 14.7701 1.38945 0.694727 0.719273i \(-0.255525\pi\)
0.694727 + 0.719273i \(0.255525\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 4.83465 0.448886
\(117\) 0 0
\(118\) −3.16515 −0.291376
\(119\) 0.723000 0.0662773
\(120\) 0 0
\(121\) −10.7913 −0.981026
\(122\) −4.83465 −0.437709
\(123\) 0 0
\(124\) −5.58258 −0.501330
\(125\) 11.4014 1.01977
\(126\) 0 0
\(127\) −1.00000 −0.0887357
\(128\) 12.1244 1.07165
\(129\) 0 0
\(130\) 21.9564 1.92571
\(131\) −4.73930 −0.414075 −0.207037 0.978333i \(-0.566382\pi\)
−0.207037 + 0.978333i \(0.566382\pi\)
\(132\) 0 0
\(133\) −4.41742 −0.383039
\(134\) −10.0308 −0.866530
\(135\) 0 0
\(136\) −1.58258 −0.135705
\(137\) 15.6838 1.33996 0.669979 0.742380i \(-0.266303\pi\)
0.669979 + 0.742380i \(0.266303\pi\)
\(138\) 0 0
\(139\) 7.16515 0.607740 0.303870 0.952713i \(-0.401721\pi\)
0.303870 + 0.952713i \(0.401721\pi\)
\(140\) −1.73205 −0.146385
\(141\) 0 0
\(142\) −14.3739 −1.20623
\(143\) −2.64575 −0.221249
\(144\) 0 0
\(145\) −10.5826 −0.878835
\(146\) 0.647551 0.0535917
\(147\) 0 0
\(148\) 1.79129 0.147243
\(149\) 14.9608 1.22564 0.612819 0.790224i \(-0.290036\pi\)
0.612819 + 0.790224i \(0.290036\pi\)
\(150\) 0 0
\(151\) −6.37386 −0.518698 −0.259349 0.965784i \(-0.583508\pi\)
−0.259349 + 0.965784i \(0.583508\pi\)
\(152\) 9.66930 0.784284
\(153\) 0 0
\(154\) 0.626136 0.0504555
\(155\) 12.2197 0.981510
\(156\) 0 0
\(157\) 0.373864 0.0298376 0.0149188 0.999889i \(-0.495251\pi\)
0.0149188 + 0.999889i \(0.495251\pi\)
\(158\) −6.20520 −0.493659
\(159\) 0 0
\(160\) 11.3739 0.899183
\(161\) 0 0
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) −6.20520 −0.484545
\(165\) 0 0
\(166\) 28.7477 2.23126
\(167\) −2.55040 −0.197356 −0.0986780 0.995119i \(-0.531461\pi\)
−0.0986780 + 0.995119i \(0.531461\pi\)
\(168\) 0 0
\(169\) 20.5390 1.57992
\(170\) −3.46410 −0.265684
\(171\) 0 0
\(172\) 5.37386 0.409753
\(173\) −5.84370 −0.444289 −0.222144 0.975014i \(-0.571306\pi\)
−0.222144 + 0.975014i \(0.571306\pi\)
\(174\) 0 0
\(175\) −0.165151 −0.0124843
\(176\) −2.28425 −0.172182
\(177\) 0 0
\(178\) −0.626136 −0.0469309
\(179\) −10.2215 −0.763991 −0.381996 0.924164i \(-0.624763\pi\)
−0.381996 + 0.924164i \(0.624763\pi\)
\(180\) 0 0
\(181\) −0.834849 −0.0620538 −0.0310269 0.999519i \(-0.509878\pi\)
−0.0310269 + 0.999519i \(0.509878\pi\)
\(182\) −7.93725 −0.588348
\(183\) 0 0
\(184\) 0 0
\(185\) −3.92095 −0.288274
\(186\) 0 0
\(187\) 0.417424 0.0305251
\(188\) 2.18890 0.159642
\(189\) 0 0
\(190\) 21.1652 1.53548
\(191\) 22.8027 1.64995 0.824973 0.565172i \(-0.191190\pi\)
0.824973 + 0.565172i \(0.191190\pi\)
\(192\) 0 0
\(193\) −16.0000 −1.15171 −0.575853 0.817554i \(-0.695330\pi\)
−0.575853 + 0.817554i \(0.695330\pi\)
\(194\) −13.8564 −0.994832
\(195\) 0 0
\(196\) −6.37386 −0.455276
\(197\) −19.8709 −1.41574 −0.707872 0.706341i \(-0.750345\pi\)
−0.707872 + 0.706341i \(0.750345\pi\)
\(198\) 0 0
\(199\) −12.7477 −0.903662 −0.451831 0.892103i \(-0.649229\pi\)
−0.451831 + 0.892103i \(0.649229\pi\)
\(200\) 0.361500 0.0255619
\(201\) 0 0
\(202\) −6.33030 −0.445399
\(203\) 3.82560 0.268505
\(204\) 0 0
\(205\) 13.5826 0.948648
\(206\) 2.74110 0.190982
\(207\) 0 0
\(208\) 28.9564 2.00777
\(209\) −2.55040 −0.176415
\(210\) 0 0
\(211\) 16.0000 1.10149 0.550743 0.834675i \(-0.314345\pi\)
0.550743 + 0.834675i \(0.314345\pi\)
\(212\) 4.83465 0.332045
\(213\) 0 0
\(214\) −22.1216 −1.51220
\(215\) −11.7629 −0.802220
\(216\) 0 0
\(217\) −4.41742 −0.299874
\(218\) 35.9361 2.43390
\(219\) 0 0
\(220\) −1.00000 −0.0674200
\(221\) −5.29150 −0.355945
\(222\) 0 0
\(223\) −12.5390 −0.839675 −0.419837 0.907599i \(-0.637913\pi\)
−0.419837 + 0.907599i \(0.637913\pi\)
\(224\) −4.11165 −0.274721
\(225\) 0 0
\(226\) 25.5826 1.70173
\(227\) 0.456850 0.0303222 0.0151611 0.999885i \(-0.495174\pi\)
0.0151611 + 0.999885i \(0.495174\pi\)
\(228\) 0 0
\(229\) 8.41742 0.556239 0.278120 0.960546i \(-0.410289\pi\)
0.278120 + 0.960546i \(0.410289\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −8.37386 −0.549771
\(233\) −14.5993 −0.956432 −0.478216 0.878242i \(-0.658716\pi\)
−0.478216 + 0.878242i \(0.658716\pi\)
\(234\) 0 0
\(235\) −4.79129 −0.312549
\(236\) −1.82740 −0.118954
\(237\) 0 0
\(238\) 1.25227 0.0811728
\(239\) −28.0942 −1.81726 −0.908632 0.417598i \(-0.862872\pi\)
−0.908632 + 0.417598i \(0.862872\pi\)
\(240\) 0 0
\(241\) −17.1652 −1.10570 −0.552852 0.833279i \(-0.686461\pi\)
−0.552852 + 0.833279i \(0.686461\pi\)
\(242\) −18.6911 −1.20151
\(243\) 0 0
\(244\) −2.79129 −0.178694
\(245\) 13.9518 0.891345
\(246\) 0 0
\(247\) 32.3303 2.05713
\(248\) 9.66930 0.614001
\(249\) 0 0
\(250\) 19.7477 1.24896
\(251\) −17.3205 −1.09326 −0.546630 0.837374i \(-0.684090\pi\)
−0.546630 + 0.837374i \(0.684090\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −1.73205 −0.108679
\(255\) 0 0
\(256\) 19.0000 1.18750
\(257\) −15.0363 −0.937936 −0.468968 0.883215i \(-0.655374\pi\)
−0.468968 + 0.883215i \(0.655374\pi\)
\(258\) 0 0
\(259\) 1.41742 0.0880745
\(260\) 12.6766 0.786167
\(261\) 0 0
\(262\) −8.20871 −0.507136
\(263\) −27.6374 −1.70419 −0.852096 0.523385i \(-0.824669\pi\)
−0.852096 + 0.523385i \(0.824669\pi\)
\(264\) 0 0
\(265\) −10.5826 −0.650082
\(266\) −7.65120 −0.469125
\(267\) 0 0
\(268\) −5.79129 −0.353759
\(269\) 19.3386 1.17910 0.589548 0.807733i \(-0.299306\pi\)
0.589548 + 0.807733i \(0.299306\pi\)
\(270\) 0 0
\(271\) 12.4174 0.754305 0.377152 0.926151i \(-0.376903\pi\)
0.377152 + 0.926151i \(0.376903\pi\)
\(272\) −4.56850 −0.277006
\(273\) 0 0
\(274\) 27.1652 1.64111
\(275\) −0.0953502 −0.00574983
\(276\) 0 0
\(277\) −13.1652 −0.791017 −0.395509 0.918462i \(-0.629432\pi\)
−0.395509 + 0.918462i \(0.629432\pi\)
\(278\) 12.4104 0.744327
\(279\) 0 0
\(280\) 3.00000 0.179284
\(281\) −16.0453 −0.957183 −0.478591 0.878038i \(-0.658852\pi\)
−0.478591 + 0.878038i \(0.658852\pi\)
\(282\) 0 0
\(283\) −12.7913 −0.760363 −0.380182 0.924912i \(-0.624138\pi\)
−0.380182 + 0.924912i \(0.624138\pi\)
\(284\) −8.29875 −0.492440
\(285\) 0 0
\(286\) −4.58258 −0.270973
\(287\) −4.91010 −0.289834
\(288\) 0 0
\(289\) −16.1652 −0.950891
\(290\) −18.3296 −1.07635
\(291\) 0 0
\(292\) 0.373864 0.0218787
\(293\) 20.5185 1.19870 0.599351 0.800487i \(-0.295426\pi\)
0.599351 + 0.800487i \(0.295426\pi\)
\(294\) 0 0
\(295\) 4.00000 0.232889
\(296\) −3.10260 −0.180335
\(297\) 0 0
\(298\) 25.9129 1.50109
\(299\) 0 0
\(300\) 0 0
\(301\) 4.25227 0.245097
\(302\) −11.0399 −0.635272
\(303\) 0 0
\(304\) 27.9129 1.60091
\(305\) 6.10985 0.349849
\(306\) 0 0
\(307\) 23.1216 1.31962 0.659809 0.751433i \(-0.270637\pi\)
0.659809 + 0.751433i \(0.270637\pi\)
\(308\) 0.361500 0.0205984
\(309\) 0 0
\(310\) 21.1652 1.20210
\(311\) −11.3060 −0.641105 −0.320552 0.947231i \(-0.603869\pi\)
−0.320552 + 0.947231i \(0.603869\pi\)
\(312\) 0 0
\(313\) −23.5826 −1.33297 −0.666483 0.745520i \(-0.732201\pi\)
−0.666483 + 0.745520i \(0.732201\pi\)
\(314\) 0.647551 0.0365434
\(315\) 0 0
\(316\) −3.58258 −0.201536
\(317\) −3.27340 −0.183853 −0.0919263 0.995766i \(-0.529302\pi\)
−0.0919263 + 0.995766i \(0.529302\pi\)
\(318\) 0 0
\(319\) 2.20871 0.123664
\(320\) −2.18890 −0.122363
\(321\) 0 0
\(322\) 0 0
\(323\) −5.10080 −0.283816
\(324\) 0 0
\(325\) 1.20871 0.0670473
\(326\) −6.92820 −0.383718
\(327\) 0 0
\(328\) 10.7477 0.593444
\(329\) 1.73205 0.0954911
\(330\) 0 0
\(331\) 24.7042 1.35786 0.678932 0.734201i \(-0.262443\pi\)
0.678932 + 0.734201i \(0.262443\pi\)
\(332\) 16.5975 0.910907
\(333\) 0 0
\(334\) −4.41742 −0.241711
\(335\) 12.6766 0.692594
\(336\) 0 0
\(337\) −19.9129 −1.08472 −0.542362 0.840145i \(-0.682470\pi\)
−0.542362 + 0.840145i \(0.682470\pi\)
\(338\) 35.5746 1.93500
\(339\) 0 0
\(340\) −2.00000 −0.108465
\(341\) −2.55040 −0.138112
\(342\) 0 0
\(343\) −10.5826 −0.571405
\(344\) −9.30780 −0.501843
\(345\) 0 0
\(346\) −10.1216 −0.544140
\(347\) −15.8745 −0.852188 −0.426094 0.904679i \(-0.640111\pi\)
−0.426094 + 0.904679i \(0.640111\pi\)
\(348\) 0 0
\(349\) −19.5826 −1.04823 −0.524116 0.851647i \(-0.675604\pi\)
−0.524116 + 0.851647i \(0.675604\pi\)
\(350\) −0.286051 −0.0152900
\(351\) 0 0
\(352\) −2.37386 −0.126527
\(353\) −9.86001 −0.524795 −0.262398 0.964960i \(-0.584513\pi\)
−0.262398 + 0.964960i \(0.584513\pi\)
\(354\) 0 0
\(355\) 18.1652 0.964106
\(356\) −0.361500 −0.0191595
\(357\) 0 0
\(358\) −17.7042 −0.935694
\(359\) 19.3386 1.02065 0.510326 0.859981i \(-0.329525\pi\)
0.510326 + 0.859981i \(0.329525\pi\)
\(360\) 0 0
\(361\) 12.1652 0.640271
\(362\) −1.44600 −0.0760001
\(363\) 0 0
\(364\) −4.58258 −0.240192
\(365\) −0.818350 −0.0428344
\(366\) 0 0
\(367\) 2.00000 0.104399 0.0521996 0.998637i \(-0.483377\pi\)
0.0521996 + 0.998637i \(0.483377\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) −6.79129 −0.353062
\(371\) 3.82560 0.198615
\(372\) 0 0
\(373\) 4.00000 0.207112 0.103556 0.994624i \(-0.466978\pi\)
0.103556 + 0.994624i \(0.466978\pi\)
\(374\) 0.723000 0.0373854
\(375\) 0 0
\(376\) −3.79129 −0.195521
\(377\) −27.9989 −1.44201
\(378\) 0 0
\(379\) 12.1216 0.622644 0.311322 0.950304i \(-0.399228\pi\)
0.311322 + 0.950304i \(0.399228\pi\)
\(380\) 12.2197 0.626858
\(381\) 0 0
\(382\) 39.4955 2.02076
\(383\) 11.9536 0.610798 0.305399 0.952224i \(-0.401210\pi\)
0.305399 + 0.952224i \(0.401210\pi\)
\(384\) 0 0
\(385\) −0.791288 −0.0403278
\(386\) −27.7128 −1.41055
\(387\) 0 0
\(388\) −8.00000 −0.406138
\(389\) −16.5975 −0.841527 −0.420764 0.907170i \(-0.638238\pi\)
−0.420764 + 0.907170i \(0.638238\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 11.0399 0.557597
\(393\) 0 0
\(394\) −34.4174 −1.73392
\(395\) 7.84190 0.394569
\(396\) 0 0
\(397\) −34.0000 −1.70641 −0.853206 0.521575i \(-0.825345\pi\)
−0.853206 + 0.521575i \(0.825345\pi\)
\(398\) −22.0797 −1.10676
\(399\) 0 0
\(400\) 1.04356 0.0521780
\(401\) −7.30960 −0.365024 −0.182512 0.983204i \(-0.558423\pi\)
−0.182512 + 0.983204i \(0.558423\pi\)
\(402\) 0 0
\(403\) 32.3303 1.61049
\(404\) −3.65480 −0.181833
\(405\) 0 0
\(406\) 6.62614 0.328850
\(407\) 0.818350 0.0405641
\(408\) 0 0
\(409\) 7.16515 0.354294 0.177147 0.984184i \(-0.443313\pi\)
0.177147 + 0.984184i \(0.443313\pi\)
\(410\) 23.5257 1.16185
\(411\) 0 0
\(412\) 1.58258 0.0779679
\(413\) −1.44600 −0.0711530
\(414\) 0 0
\(415\) −36.3303 −1.78338
\(416\) 30.0924 1.47540
\(417\) 0 0
\(418\) −4.41742 −0.216063
\(419\) 20.8045 1.01637 0.508183 0.861249i \(-0.330317\pi\)
0.508183 + 0.861249i \(0.330317\pi\)
\(420\) 0 0
\(421\) 3.16515 0.154260 0.0771300 0.997021i \(-0.475424\pi\)
0.0771300 + 0.997021i \(0.475424\pi\)
\(422\) 27.7128 1.34904
\(423\) 0 0
\(424\) −8.37386 −0.406671
\(425\) −0.190700 −0.00925033
\(426\) 0 0
\(427\) −2.20871 −0.106887
\(428\) −12.7719 −0.617353
\(429\) 0 0
\(430\) −20.3739 −0.982515
\(431\) 12.0489 0.580375 0.290188 0.956970i \(-0.406282\pi\)
0.290188 + 0.956970i \(0.406282\pi\)
\(432\) 0 0
\(433\) −26.7042 −1.28332 −0.641660 0.766989i \(-0.721754\pi\)
−0.641660 + 0.766989i \(0.721754\pi\)
\(434\) −7.65120 −0.367270
\(435\) 0 0
\(436\) 20.7477 0.993636
\(437\) 0 0
\(438\) 0 0
\(439\) 18.2087 0.869054 0.434527 0.900659i \(-0.356915\pi\)
0.434527 + 0.900659i \(0.356915\pi\)
\(440\) 1.73205 0.0825723
\(441\) 0 0
\(442\) −9.16515 −0.435942
\(443\) 17.9681 0.853688 0.426844 0.904325i \(-0.359625\pi\)
0.426844 + 0.904325i \(0.359625\pi\)
\(444\) 0 0
\(445\) 0.791288 0.0375106
\(446\) −21.7182 −1.02839
\(447\) 0 0
\(448\) 0.791288 0.0373848
\(449\) −9.66930 −0.456323 −0.228161 0.973623i \(-0.573271\pi\)
−0.228161 + 0.973623i \(0.573271\pi\)
\(450\) 0 0
\(451\) −2.83485 −0.133488
\(452\) 14.7701 0.694727
\(453\) 0 0
\(454\) 0.791288 0.0371370
\(455\) 10.0308 0.470251
\(456\) 0 0
\(457\) −24.3739 −1.14016 −0.570081 0.821589i \(-0.693088\pi\)
−0.570081 + 0.821589i \(0.693088\pi\)
\(458\) 14.5794 0.681251
\(459\) 0 0
\(460\) 0 0
\(461\) 5.74835 0.267727 0.133864 0.991000i \(-0.457262\pi\)
0.133864 + 0.991000i \(0.457262\pi\)
\(462\) 0 0
\(463\) −4.00000 −0.185896 −0.0929479 0.995671i \(-0.529629\pi\)
−0.0929479 + 0.995671i \(0.529629\pi\)
\(464\) −24.1733 −1.12222
\(465\) 0 0
\(466\) −25.2867 −1.17139
\(467\) −20.9753 −0.970622 −0.485311 0.874342i \(-0.661294\pi\)
−0.485311 + 0.874342i \(0.661294\pi\)
\(468\) 0 0
\(469\) −4.58258 −0.211604
\(470\) −8.29875 −0.382793
\(471\) 0 0
\(472\) 3.16515 0.145688
\(473\) 2.45505 0.112883
\(474\) 0 0
\(475\) 1.16515 0.0534608
\(476\) 0.723000 0.0331387
\(477\) 0 0
\(478\) −48.6606 −2.22568
\(479\) 39.0387 1.78372 0.891862 0.452307i \(-0.149399\pi\)
0.891862 + 0.452307i \(0.149399\pi\)
\(480\) 0 0
\(481\) −10.3739 −0.473007
\(482\) −29.7309 −1.35421
\(483\) 0 0
\(484\) −10.7913 −0.490513
\(485\) 17.5112 0.795143
\(486\) 0 0
\(487\) 8.95644 0.405855 0.202928 0.979194i \(-0.434954\pi\)
0.202928 + 0.979194i \(0.434954\pi\)
\(488\) 4.83465 0.218854
\(489\) 0 0
\(490\) 24.1652 1.09167
\(491\) −25.1624 −1.13556 −0.567782 0.823179i \(-0.692198\pi\)
−0.567782 + 0.823179i \(0.692198\pi\)
\(492\) 0 0
\(493\) 4.41742 0.198951
\(494\) 55.9977 2.51946
\(495\) 0 0
\(496\) 27.9129 1.25333
\(497\) −6.56670 −0.294557
\(498\) 0 0
\(499\) 38.3739 1.71785 0.858925 0.512101i \(-0.171133\pi\)
0.858925 + 0.512101i \(0.171133\pi\)
\(500\) 11.4014 0.509884
\(501\) 0 0
\(502\) −30.0000 −1.33897
\(503\) −3.84550 −0.171462 −0.0857312 0.996318i \(-0.527323\pi\)
−0.0857312 + 0.996318i \(0.527323\pi\)
\(504\) 0 0
\(505\) 8.00000 0.355995
\(506\) 0 0
\(507\) 0 0
\(508\) −1.00000 −0.0443678
\(509\) −4.75920 −0.210948 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(510\) 0 0
\(511\) 0.295834 0.0130869
\(512\) 8.66025 0.382733
\(513\) 0 0
\(514\) −26.0436 −1.14873
\(515\) −3.46410 −0.152647
\(516\) 0 0
\(517\) 1.00000 0.0439799
\(518\) 2.45505 0.107869
\(519\) 0 0
\(520\) −21.9564 −0.962854
\(521\) −29.5402 −1.29418 −0.647090 0.762414i \(-0.724014\pi\)
−0.647090 + 0.762414i \(0.724014\pi\)
\(522\) 0 0
\(523\) −43.1652 −1.88748 −0.943740 0.330688i \(-0.892719\pi\)
−0.943740 + 0.330688i \(0.892719\pi\)
\(524\) −4.73930 −0.207037
\(525\) 0 0
\(526\) −47.8693 −2.08720
\(527\) −5.10080 −0.222194
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) −18.3296 −0.796185
\(531\) 0 0
\(532\) −4.41742 −0.191520
\(533\) 35.9361 1.55657
\(534\) 0 0
\(535\) 27.9564 1.20866
\(536\) 10.0308 0.433265
\(537\) 0 0
\(538\) 33.4955 1.44409
\(539\) −2.91190 −0.125424
\(540\) 0 0
\(541\) 40.3303 1.73394 0.866968 0.498365i \(-0.166066\pi\)
0.866968 + 0.498365i \(0.166066\pi\)
\(542\) 21.5076 0.923831
\(543\) 0 0
\(544\) −4.74773 −0.203557
\(545\) −45.4147 −1.94535
\(546\) 0 0
\(547\) 31.9564 1.36636 0.683179 0.730251i \(-0.260597\pi\)
0.683179 + 0.730251i \(0.260597\pi\)
\(548\) 15.6838 0.669979
\(549\) 0 0
\(550\) −0.165151 −0.00704208
\(551\) −26.9898 −1.14980
\(552\) 0 0
\(553\) −2.83485 −0.120550
\(554\) −22.8027 −0.968794
\(555\) 0 0
\(556\) 7.16515 0.303870
\(557\) 14.0471 0.595195 0.297598 0.954691i \(-0.403815\pi\)
0.297598 + 0.954691i \(0.403815\pi\)
\(558\) 0 0
\(559\) −31.1216 −1.31630
\(560\) 8.66025 0.365963
\(561\) 0 0
\(562\) −27.7913 −1.17230
\(563\) 14.7701 0.622486 0.311243 0.950330i \(-0.399255\pi\)
0.311243 + 0.950330i \(0.399255\pi\)
\(564\) 0 0
\(565\) −32.3303 −1.36015
\(566\) −22.1552 −0.931251
\(567\) 0 0
\(568\) 14.3739 0.603114
\(569\) 13.8564 0.580891 0.290445 0.956892i \(-0.406197\pi\)
0.290445 + 0.956892i \(0.406197\pi\)
\(570\) 0 0
\(571\) −34.8693 −1.45924 −0.729618 0.683855i \(-0.760302\pi\)
−0.729618 + 0.683855i \(0.760302\pi\)
\(572\) −2.64575 −0.110624
\(573\) 0 0
\(574\) −8.50455 −0.354973
\(575\) 0 0
\(576\) 0 0
\(577\) −18.0000 −0.749350 −0.374675 0.927156i \(-0.622246\pi\)
−0.374675 + 0.927156i \(0.622246\pi\)
\(578\) −27.9989 −1.16460
\(579\) 0 0
\(580\) −10.5826 −0.439418
\(581\) 13.1334 0.544866
\(582\) 0 0
\(583\) 2.20871 0.0914755
\(584\) −0.647551 −0.0267958
\(585\) 0 0
\(586\) 35.5390 1.46810
\(587\) 15.2270 0.628484 0.314242 0.949343i \(-0.398250\pi\)
0.314242 + 0.949343i \(0.398250\pi\)
\(588\) 0 0
\(589\) 31.1652 1.28414
\(590\) 6.92820 0.285230
\(591\) 0 0
\(592\) −8.95644 −0.368107
\(593\) 7.57575 0.311099 0.155549 0.987828i \(-0.450285\pi\)
0.155549 + 0.987828i \(0.450285\pi\)
\(594\) 0 0
\(595\) −1.58258 −0.0648793
\(596\) 14.9608 0.612819
\(597\) 0 0
\(598\) 0 0
\(599\) −3.84550 −0.157123 −0.0785615 0.996909i \(-0.525033\pi\)
−0.0785615 + 0.996909i \(0.525033\pi\)
\(600\) 0 0
\(601\) 6.74773 0.275246 0.137623 0.990485i \(-0.456054\pi\)
0.137623 + 0.990485i \(0.456054\pi\)
\(602\) 7.36515 0.300181
\(603\) 0 0
\(604\) −6.37386 −0.259349
\(605\) 23.6211 0.960333
\(606\) 0 0
\(607\) 33.4955 1.35954 0.679769 0.733426i \(-0.262080\pi\)
0.679769 + 0.733426i \(0.262080\pi\)
\(608\) 29.0079 1.17643
\(609\) 0 0
\(610\) 10.5826 0.428476
\(611\) −12.6766 −0.512839
\(612\) 0 0
\(613\) 2.74773 0.110980 0.0554898 0.998459i \(-0.482328\pi\)
0.0554898 + 0.998459i \(0.482328\pi\)
\(614\) 40.0478 1.61620
\(615\) 0 0
\(616\) −0.626136 −0.0252278
\(617\) 44.5209 1.79234 0.896172 0.443706i \(-0.146337\pi\)
0.896172 + 0.443706i \(0.146337\pi\)
\(618\) 0 0
\(619\) 0.791288 0.0318045 0.0159023 0.999874i \(-0.494938\pi\)
0.0159023 + 0.999874i \(0.494938\pi\)
\(620\) 12.2197 0.490755
\(621\) 0 0
\(622\) −19.5826 −0.785190
\(623\) −0.286051 −0.0114604
\(624\) 0 0
\(625\) −23.9129 −0.956515
\(626\) −40.8462 −1.63254
\(627\) 0 0
\(628\) 0.373864 0.0149188
\(629\) 1.63670 0.0652595
\(630\) 0 0
\(631\) 15.7042 0.625173 0.312586 0.949889i \(-0.398805\pi\)
0.312586 + 0.949889i \(0.398805\pi\)
\(632\) 6.20520 0.246830
\(633\) 0 0
\(634\) −5.66970 −0.225172
\(635\) 2.18890 0.0868639
\(636\) 0 0
\(637\) 36.9129 1.46254
\(638\) 3.82560 0.151457
\(639\) 0 0
\(640\) −26.5390 −1.04905
\(641\) −7.67110 −0.302990 −0.151495 0.988458i \(-0.548409\pi\)
−0.151495 + 0.988458i \(0.548409\pi\)
\(642\) 0 0
\(643\) −6.00000 −0.236617 −0.118308 0.992977i \(-0.537747\pi\)
−0.118308 + 0.992977i \(0.537747\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −8.83485 −0.347602
\(647\) 24.2487 0.953315 0.476658 0.879089i \(-0.341848\pi\)
0.476658 + 0.879089i \(0.341848\pi\)
\(648\) 0 0
\(649\) −0.834849 −0.0327707
\(650\) 2.09355 0.0821158
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) 23.3350 0.913170 0.456585 0.889680i \(-0.349073\pi\)
0.456585 + 0.889680i \(0.349073\pi\)
\(654\) 0 0
\(655\) 10.3739 0.405340
\(656\) 31.0260 1.21136
\(657\) 0 0
\(658\) 3.00000 0.116952
\(659\) −45.0333 −1.75425 −0.877125 0.480263i \(-0.840541\pi\)
−0.877125 + 0.480263i \(0.840541\pi\)
\(660\) 0 0
\(661\) 34.7042 1.34984 0.674918 0.737893i \(-0.264179\pi\)
0.674918 + 0.737893i \(0.264179\pi\)
\(662\) 42.7889 1.66304
\(663\) 0 0
\(664\) −28.7477 −1.11563
\(665\) 9.66930 0.374960
\(666\) 0 0
\(667\) 0 0
\(668\) −2.55040 −0.0986780
\(669\) 0 0
\(670\) 21.9564 0.848251
\(671\) −1.27520 −0.0492286
\(672\) 0 0
\(673\) 23.7042 0.913728 0.456864 0.889536i \(-0.348973\pi\)
0.456864 + 0.889536i \(0.348973\pi\)
\(674\) −34.4901 −1.32851
\(675\) 0 0
\(676\) 20.5390 0.789962
\(677\) −36.6591 −1.40892 −0.704462 0.709742i \(-0.748812\pi\)
−0.704462 + 0.709742i \(0.748812\pi\)
\(678\) 0 0
\(679\) −6.33030 −0.242935
\(680\) 3.46410 0.132842
\(681\) 0 0
\(682\) −4.41742 −0.169152
\(683\) 20.5939 0.788004 0.394002 0.919110i \(-0.371090\pi\)
0.394002 + 0.919110i \(0.371090\pi\)
\(684\) 0 0
\(685\) −34.3303 −1.31169
\(686\) −18.3296 −0.699826
\(687\) 0 0
\(688\) −26.8693 −1.02438
\(689\) −27.9989 −1.06667
\(690\) 0 0
\(691\) −18.2087 −0.692692 −0.346346 0.938107i \(-0.612578\pi\)
−0.346346 + 0.938107i \(0.612578\pi\)
\(692\) −5.84370 −0.222144
\(693\) 0 0
\(694\) −27.4955 −1.04371
\(695\) −15.6838 −0.594921
\(696\) 0 0
\(697\) −5.66970 −0.214755
\(698\) −33.9180 −1.28382
\(699\) 0 0
\(700\) −0.165151 −0.00624214
\(701\) 7.57575 0.286132 0.143066 0.989713i \(-0.454304\pi\)
0.143066 + 0.989713i \(0.454304\pi\)
\(702\) 0 0
\(703\) −10.0000 −0.377157
\(704\) 0.456850 0.0172182
\(705\) 0 0
\(706\) −17.0780 −0.642740
\(707\) −2.89200 −0.108765
\(708\) 0 0
\(709\) −0.121591 −0.00456643 −0.00228322 0.999997i \(-0.500727\pi\)
−0.00228322 + 0.999997i \(0.500727\pi\)
\(710\) 31.4630 1.18078
\(711\) 0 0
\(712\) 0.626136 0.0234655
\(713\) 0 0
\(714\) 0 0
\(715\) 5.79129 0.216582
\(716\) −10.2215 −0.381996
\(717\) 0 0
\(718\) 33.4955 1.25004
\(719\) −27.7327 −1.03426 −0.517128 0.855908i \(-0.672999\pi\)
−0.517128 + 0.855908i \(0.672999\pi\)
\(720\) 0 0
\(721\) 1.25227 0.0466371
\(722\) 21.0707 0.784169
\(723\) 0 0
\(724\) −0.834849 −0.0310269
\(725\) −1.00905 −0.0374752
\(726\) 0 0
\(727\) −17.7913 −0.659842 −0.329921 0.944008i \(-0.607022\pi\)
−0.329921 + 0.944008i \(0.607022\pi\)
\(728\) 7.93725 0.294174
\(729\) 0 0
\(730\) −1.41742 −0.0524612
\(731\) 4.91010 0.181607
\(732\) 0 0
\(733\) −38.0345 −1.40484 −0.702418 0.711765i \(-0.747896\pi\)
−0.702418 + 0.711765i \(0.747896\pi\)
\(734\) 3.46410 0.127862
\(735\) 0 0
\(736\) 0 0
\(737\) −2.64575 −0.0974575
\(738\) 0 0
\(739\) −13.9129 −0.511794 −0.255897 0.966704i \(-0.582371\pi\)
−0.255897 + 0.966704i \(0.582371\pi\)
\(740\) −3.92095 −0.144137
\(741\) 0 0
\(742\) 6.62614 0.243253
\(743\) 10.5830 0.388253 0.194126 0.980977i \(-0.437813\pi\)
0.194126 + 0.980977i \(0.437813\pi\)
\(744\) 0 0
\(745\) −32.7477 −1.19978
\(746\) 6.92820 0.253660
\(747\) 0 0
\(748\) 0.417424 0.0152625
\(749\) −10.1063 −0.369274
\(750\) 0 0
\(751\) 22.3303 0.814844 0.407422 0.913240i \(-0.366428\pi\)
0.407422 + 0.913240i \(0.366428\pi\)
\(752\) −10.9445 −0.399105
\(753\) 0 0
\(754\) −48.4955 −1.76610
\(755\) 13.9518 0.507756
\(756\) 0 0
\(757\) 5.20871 0.189314 0.0946569 0.995510i \(-0.469825\pi\)
0.0946569 + 0.995510i \(0.469825\pi\)
\(758\) 20.9952 0.762580
\(759\) 0 0
\(760\) −21.1652 −0.767741
\(761\) −43.6628 −1.58277 −0.791387 0.611315i \(-0.790641\pi\)
−0.791387 + 0.611315i \(0.790641\pi\)
\(762\) 0 0
\(763\) 16.4174 0.594351
\(764\) 22.8027 0.824973
\(765\) 0 0
\(766\) 20.7042 0.748072
\(767\) 10.5830 0.382130
\(768\) 0 0
\(769\) 9.58258 0.345557 0.172778 0.984961i \(-0.444726\pi\)
0.172778 + 0.984961i \(0.444726\pi\)
\(770\) −1.37055 −0.0493912
\(771\) 0 0
\(772\) −16.0000 −0.575853
\(773\) −41.4183 −1.48971 −0.744857 0.667224i \(-0.767482\pi\)
−0.744857 + 0.667224i \(0.767482\pi\)
\(774\) 0 0
\(775\) 1.16515 0.0418535
\(776\) 13.8564 0.497416
\(777\) 0 0
\(778\) −28.7477 −1.03066
\(779\) 34.6410 1.24114
\(780\) 0 0
\(781\) −3.79129 −0.135663
\(782\) 0 0
\(783\) 0 0
\(784\) 31.8693 1.13819
\(785\) −0.818350 −0.0292082
\(786\) 0 0
\(787\) −5.58258 −0.198997 −0.0994987 0.995038i \(-0.531724\pi\)
−0.0994987 + 0.995038i \(0.531724\pi\)
\(788\) −19.8709 −0.707872
\(789\) 0 0
\(790\) 13.5826 0.483246
\(791\) 11.6874 0.415556
\(792\) 0 0
\(793\) 16.1652 0.574041
\(794\) −58.8897 −2.08992
\(795\) 0 0
\(796\) −12.7477 −0.451831
\(797\) −19.3386 −0.685009 −0.342504 0.939516i \(-0.611275\pi\)
−0.342504 + 0.939516i \(0.611275\pi\)
\(798\) 0 0
\(799\) 2.00000 0.0707549
\(800\) 1.08450 0.0383429
\(801\) 0 0
\(802\) −12.6606 −0.447062
\(803\) 0.170800 0.00602739
\(804\) 0 0
\(805\) 0 0
\(806\) 55.9977 1.97244
\(807\) 0 0
\(808\) 6.33030 0.222699
\(809\) 35.2131 1.23803 0.619014 0.785380i \(-0.287533\pi\)
0.619014 + 0.785380i \(0.287533\pi\)
\(810\) 0 0
\(811\) 32.0000 1.12367 0.561836 0.827249i \(-0.310095\pi\)
0.561836 + 0.827249i \(0.310095\pi\)
\(812\) 3.82560 0.134252
\(813\) 0 0
\(814\) 1.41742 0.0496807
\(815\) 8.75560 0.306695
\(816\) 0 0
\(817\) −30.0000 −1.04957
\(818\) 12.4104 0.433920
\(819\) 0 0
\(820\) 13.5826 0.474324
\(821\) −47.0713 −1.64280 −0.821400 0.570352i \(-0.806807\pi\)
−0.821400 + 0.570352i \(0.806807\pi\)
\(822\) 0 0
\(823\) 12.4174 0.432844 0.216422 0.976300i \(-0.430561\pi\)
0.216422 + 0.976300i \(0.430561\pi\)
\(824\) −2.74110 −0.0954908
\(825\) 0 0
\(826\) −2.50455 −0.0871443
\(827\) −34.1087 −1.18608 −0.593038 0.805174i \(-0.702072\pi\)
−0.593038 + 0.805174i \(0.702072\pi\)
\(828\) 0 0
\(829\) −0.834849 −0.0289955 −0.0144977 0.999895i \(-0.504615\pi\)
−0.0144977 + 0.999895i \(0.504615\pi\)
\(830\) −62.9259 −2.18419
\(831\) 0 0
\(832\) −5.79129 −0.200777
\(833\) −5.82380 −0.201783
\(834\) 0 0
\(835\) 5.58258 0.193193
\(836\) −2.55040 −0.0882075
\(837\) 0 0
\(838\) 36.0345 1.24479
\(839\) 3.46410 0.119594 0.0597970 0.998211i \(-0.480955\pi\)
0.0597970 + 0.998211i \(0.480955\pi\)
\(840\) 0 0
\(841\) −5.62614 −0.194005
\(842\) 5.48220 0.188929
\(843\) 0 0
\(844\) 16.0000 0.550743
\(845\) −44.9579 −1.54660
\(846\) 0 0
\(847\) −8.53901 −0.293404
\(848\) −24.1733 −0.830113
\(849\) 0 0
\(850\) −0.330303 −0.0113293
\(851\) 0 0
\(852\) 0 0
\(853\) −8.74773 −0.299516 −0.149758 0.988723i \(-0.547850\pi\)
−0.149758 + 0.988723i \(0.547850\pi\)
\(854\) −3.82560 −0.130909
\(855\) 0 0
\(856\) 22.1216 0.756100
\(857\) −27.7128 −0.946652 −0.473326 0.880887i \(-0.656947\pi\)
−0.473326 + 0.880887i \(0.656947\pi\)
\(858\) 0 0
\(859\) 18.3739 0.626908 0.313454 0.949603i \(-0.398514\pi\)
0.313454 + 0.949603i \(0.398514\pi\)
\(860\) −11.7629 −0.401110
\(861\) 0 0
\(862\) 20.8693 0.710812
\(863\) 15.1515 0.515763 0.257882 0.966177i \(-0.416976\pi\)
0.257882 + 0.966177i \(0.416976\pi\)
\(864\) 0 0
\(865\) 12.7913 0.434917
\(866\) −46.2530 −1.57174
\(867\) 0 0
\(868\) −4.41742 −0.149937
\(869\) −1.63670 −0.0555213
\(870\) 0 0
\(871\) 33.5390 1.13643
\(872\) −35.9361 −1.21695
\(873\) 0 0
\(874\) 0 0
\(875\) 9.02175 0.304991
\(876\) 0 0
\(877\) −46.6170 −1.57415 −0.787073 0.616860i \(-0.788404\pi\)
−0.787073 + 0.616860i \(0.788404\pi\)
\(878\) 31.5384 1.06437
\(879\) 0 0
\(880\) 5.00000 0.168550
\(881\) −32.9289 −1.10940 −0.554701 0.832050i \(-0.687167\pi\)
−0.554701 + 0.832050i \(0.687167\pi\)
\(882\) 0 0
\(883\) −9.58258 −0.322479 −0.161240 0.986915i \(-0.551549\pi\)
−0.161240 + 0.986915i \(0.551549\pi\)
\(884\) −5.29150 −0.177972
\(885\) 0 0
\(886\) 31.1216 1.04555
\(887\) −35.7454 −1.20021 −0.600107 0.799920i \(-0.704875\pi\)
−0.600107 + 0.799920i \(0.704875\pi\)
\(888\) 0 0
\(889\) −0.791288 −0.0265389
\(890\) 1.37055 0.0459410
\(891\) 0 0
\(892\) −12.5390 −0.419837
\(893\) −12.2197 −0.408917
\(894\) 0 0
\(895\) 22.3739 0.747876
\(896\) 9.59386 0.320508
\(897\) 0 0
\(898\) −16.7477 −0.558879
\(899\) −26.9898 −0.900161
\(900\) 0 0
\(901\) 4.41742 0.147166
\(902\) −4.91010 −0.163489
\(903\) 0 0
\(904\) −25.5826 −0.850864
\(905\) 1.82740 0.0607449
\(906\) 0 0
\(907\) −22.3303 −0.741466 −0.370733 0.928740i \(-0.620893\pi\)
−0.370733 + 0.928740i \(0.620893\pi\)
\(908\) 0.456850 0.0151611
\(909\) 0 0
\(910\) 17.3739 0.575938
\(911\) −46.6899 −1.54691 −0.773453 0.633854i \(-0.781472\pi\)
−0.773453 + 0.633854i \(0.781472\pi\)
\(912\) 0 0
\(913\) 7.58258 0.250947
\(914\) −42.2168 −1.39641
\(915\) 0 0
\(916\) 8.41742 0.278120
\(917\) −3.75015 −0.123841
\(918\) 0 0
\(919\) 9.25227 0.305204 0.152602 0.988288i \(-0.451235\pi\)
0.152602 + 0.988288i \(0.451235\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 9.95644 0.327898
\(923\) 48.0605 1.58193
\(924\) 0 0
\(925\) −0.373864 −0.0122926
\(926\) −6.92820 −0.227675
\(927\) 0 0
\(928\) −25.1216 −0.824657
\(929\) −50.0587 −1.64237 −0.821186 0.570661i \(-0.806687\pi\)
−0.821186 + 0.570661i \(0.806687\pi\)
\(930\) 0 0
\(931\) 35.5826 1.16617
\(932\) −14.5993 −0.478216
\(933\) 0 0
\(934\) −36.3303 −1.18876
\(935\) −0.913701 −0.0298812
\(936\) 0 0
\(937\) 49.4083 1.61410 0.807050 0.590483i \(-0.201063\pi\)
0.807050 + 0.590483i \(0.201063\pi\)
\(938\) −7.93725 −0.259161
\(939\) 0 0
\(940\) −4.79129 −0.156275
\(941\) 45.2240 1.47426 0.737130 0.675750i \(-0.236180\pi\)
0.737130 + 0.675750i \(0.236180\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 9.13701 0.297384
\(945\) 0 0
\(946\) 4.25227 0.138253
\(947\) −24.2487 −0.787977 −0.393989 0.919115i \(-0.628905\pi\)
−0.393989 + 0.919115i \(0.628905\pi\)
\(948\) 0 0
\(949\) −2.16515 −0.0702838
\(950\) 2.01810 0.0654759
\(951\) 0 0
\(952\) −1.25227 −0.0405864
\(953\) 24.2487 0.785493 0.392746 0.919647i \(-0.371525\pi\)
0.392746 + 0.919647i \(0.371525\pi\)
\(954\) 0 0
\(955\) −49.9129 −1.61514
\(956\) −28.0942 −0.908632
\(957\) 0 0
\(958\) 67.6170 2.18461
\(959\) 12.4104 0.400753
\(960\) 0 0
\(961\) 0.165151 0.00532746
\(962\) −17.9681 −0.579313
\(963\) 0 0
\(964\) −17.1652 −0.552852
\(965\) 35.0224 1.12741
\(966\) 0 0
\(967\) 46.2087 1.48597 0.742986 0.669307i \(-0.233409\pi\)
0.742986 + 0.669307i \(0.233409\pi\)
\(968\) 18.6911 0.600753
\(969\) 0 0
\(970\) 30.3303 0.973847
\(971\) −0.0754495 −0.00242129 −0.00121064 0.999999i \(-0.500385\pi\)
−0.00121064 + 0.999999i \(0.500385\pi\)
\(972\) 0 0
\(973\) 5.66970 0.181762
\(974\) 15.5130 0.497069
\(975\) 0 0
\(976\) 13.9564 0.446735
\(977\) 38.2958 1.22519 0.612596 0.790396i \(-0.290125\pi\)
0.612596 + 0.790396i \(0.290125\pi\)
\(978\) 0 0
\(979\) −0.165151 −0.00527826
\(980\) 13.9518 0.445673
\(981\) 0 0
\(982\) −43.5826 −1.39078
\(983\) 54.7225 1.74538 0.872689 0.488277i \(-0.162374\pi\)
0.872689 + 0.488277i \(0.162374\pi\)
\(984\) 0 0
\(985\) 43.4955 1.38588
\(986\) 7.65120 0.243664
\(987\) 0 0
\(988\) 32.3303 1.02856
\(989\) 0 0
\(990\) 0 0
\(991\) 4.04356 0.128448 0.0642240 0.997936i \(-0.479543\pi\)
0.0642240 + 0.997936i \(0.479543\pi\)
\(992\) 29.0079 0.921002
\(993\) 0 0
\(994\) −11.3739 −0.360757
\(995\) 27.9035 0.884601
\(996\) 0 0
\(997\) −2.41742 −0.0765606 −0.0382803 0.999267i \(-0.512188\pi\)
−0.0382803 + 0.999267i \(0.512188\pi\)
\(998\) 66.4655 2.10393
\(999\) 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 1143.2.a.f.1.3 yes 4
3.2 odd 2 inner 1143.2.a.f.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1143.2.a.f.1.2 4 3.2 odd 2 inner
1143.2.a.f.1.3 yes 4 1.1 even 1 trivial