Properties

Label 6003.2.a.d.1.1
Level $6003$
Weight $2$
Character 6003.1
Self dual yes
Analytic conductor $47.934$
Analytic rank $1$
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] = [6003,2,Mod(1,6003)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6003, 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("6003.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6003.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: \(47.9341963334\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2001)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 6003.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{4} -0.561553 q^{5} +3.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{4} -0.561553 q^{5} +3.00000 q^{8} +0.561553 q^{10} -0.561553 q^{11} +0.561553 q^{13} -1.00000 q^{16} +3.12311 q^{19} +0.561553 q^{20} +0.561553 q^{22} -1.00000 q^{23} -4.68466 q^{25} -0.561553 q^{26} +1.00000 q^{29} -6.56155 q^{31} -5.00000 q^{32} +6.56155 q^{37} -3.12311 q^{38} -1.68466 q^{40} -3.43845 q^{41} +12.2462 q^{43} +0.561553 q^{44} +1.00000 q^{46} +1.12311 q^{47} -7.00000 q^{49} +4.68466 q^{50} -0.561553 q^{52} -12.2462 q^{53} +0.315342 q^{55} -1.00000 q^{58} -5.43845 q^{59} +8.80776 q^{61} +6.56155 q^{62} +7.00000 q^{64} -0.315342 q^{65} +10.5616 q^{67} -14.5616 q^{71} +3.12311 q^{73} -6.56155 q^{74} -3.12311 q^{76} -2.00000 q^{79} +0.561553 q^{80} +3.43845 q^{82} -1.12311 q^{83} -12.2462 q^{86} -1.68466 q^{88} +7.36932 q^{89} +1.00000 q^{92} -1.12311 q^{94} -1.75379 q^{95} +11.3693 q^{97} +7.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{4} + 3 q^{5} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 2 q^{4} + 3 q^{5} + 6 q^{8} - 3 q^{10} + 3 q^{11} - 3 q^{13} - 2 q^{16} - 2 q^{19} - 3 q^{20} - 3 q^{22} - 2 q^{23} + 3 q^{25} + 3 q^{26} + 2 q^{29} - 9 q^{31} - 10 q^{32} + 9 q^{37} + 2 q^{38} + 9 q^{40} - 11 q^{41} + 8 q^{43} - 3 q^{44} + 2 q^{46} - 6 q^{47} - 14 q^{49} - 3 q^{50} + 3 q^{52} - 8 q^{53} + 13 q^{55} - 2 q^{58} - 15 q^{59} - 3 q^{61} + 9 q^{62} + 14 q^{64} - 13 q^{65} + 17 q^{67} - 25 q^{71} - 2 q^{73} - 9 q^{74} + 2 q^{76} - 4 q^{79} - 3 q^{80} + 11 q^{82} + 6 q^{83} - 8 q^{86} + 9 q^{88} - 10 q^{89} + 2 q^{92} + 6 q^{94} - 20 q^{95} - 2 q^{97} + 14 q^{98}+O(q^{100}) \) 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 −0.353553 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) −0.561553 −0.251134 −0.125567 0.992085i \(-0.540075\pi\)
−0.125567 + 0.992085i \(0.540075\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 3.00000 1.06066
\(9\) 0 0
\(10\) 0.561553 0.177579
\(11\) −0.561553 −0.169315 −0.0846573 0.996410i \(-0.526980\pi\)
−0.0846573 + 0.996410i \(0.526980\pi\)
\(12\) 0 0
\(13\) 0.561553 0.155747 0.0778734 0.996963i \(-0.475187\pi\)
0.0778734 + 0.996963i \(0.475187\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 3.12311 0.716490 0.358245 0.933628i \(-0.383375\pi\)
0.358245 + 0.933628i \(0.383375\pi\)
\(20\) 0.561553 0.125567
\(21\) 0 0
\(22\) 0.561553 0.119723
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) −4.68466 −0.936932
\(26\) −0.561553 −0.110130
\(27\) 0 0
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −6.56155 −1.17849 −0.589245 0.807955i \(-0.700575\pi\)
−0.589245 + 0.807955i \(0.700575\pi\)
\(32\) −5.00000 −0.883883
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 6.56155 1.07871 0.539356 0.842078i \(-0.318668\pi\)
0.539356 + 0.842078i \(0.318668\pi\)
\(38\) −3.12311 −0.506635
\(39\) 0 0
\(40\) −1.68466 −0.266368
\(41\) −3.43845 −0.536995 −0.268498 0.963280i \(-0.586527\pi\)
−0.268498 + 0.963280i \(0.586527\pi\)
\(42\) 0 0
\(43\) 12.2462 1.86753 0.933765 0.357887i \(-0.116503\pi\)
0.933765 + 0.357887i \(0.116503\pi\)
\(44\) 0.561553 0.0846573
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 1.12311 0.163822 0.0819109 0.996640i \(-0.473898\pi\)
0.0819109 + 0.996640i \(0.473898\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 4.68466 0.662511
\(51\) 0 0
\(52\) −0.561553 −0.0778734
\(53\) −12.2462 −1.68215 −0.841073 0.540921i \(-0.818076\pi\)
−0.841073 + 0.540921i \(0.818076\pi\)
\(54\) 0 0
\(55\) 0.315342 0.0425206
\(56\) 0 0
\(57\) 0 0
\(58\) −1.00000 −0.131306
\(59\) −5.43845 −0.708026 −0.354013 0.935241i \(-0.615183\pi\)
−0.354013 + 0.935241i \(0.615183\pi\)
\(60\) 0 0
\(61\) 8.80776 1.12772 0.563859 0.825871i \(-0.309316\pi\)
0.563859 + 0.825871i \(0.309316\pi\)
\(62\) 6.56155 0.833318
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) −0.315342 −0.0391133
\(66\) 0 0
\(67\) 10.5616 1.29030 0.645150 0.764056i \(-0.276795\pi\)
0.645150 + 0.764056i \(0.276795\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −14.5616 −1.72814 −0.864069 0.503373i \(-0.832092\pi\)
−0.864069 + 0.503373i \(0.832092\pi\)
\(72\) 0 0
\(73\) 3.12311 0.365532 0.182766 0.983156i \(-0.441495\pi\)
0.182766 + 0.983156i \(0.441495\pi\)
\(74\) −6.56155 −0.762765
\(75\) 0 0
\(76\) −3.12311 −0.358245
\(77\) 0 0
\(78\) 0 0
\(79\) −2.00000 −0.225018 −0.112509 0.993651i \(-0.535889\pi\)
−0.112509 + 0.993651i \(0.535889\pi\)
\(80\) 0.561553 0.0627835
\(81\) 0 0
\(82\) 3.43845 0.379713
\(83\) −1.12311 −0.123277 −0.0616384 0.998099i \(-0.519633\pi\)
−0.0616384 + 0.998099i \(0.519633\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −12.2462 −1.32054
\(87\) 0 0
\(88\) −1.68466 −0.179585
\(89\) 7.36932 0.781146 0.390573 0.920572i \(-0.372277\pi\)
0.390573 + 0.920572i \(0.372277\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1.00000 0.104257
\(93\) 0 0
\(94\) −1.12311 −0.115840
\(95\) −1.75379 −0.179935
\(96\) 0 0
\(97\) 11.3693 1.15438 0.577190 0.816610i \(-0.304149\pi\)
0.577190 + 0.816610i \(0.304149\pi\)
\(98\) 7.00000 0.707107
\(99\) 0 0
\(100\) 4.68466 0.468466
\(101\) 3.43845 0.342138 0.171069 0.985259i \(-0.445278\pi\)
0.171069 + 0.985259i \(0.445278\pi\)
\(102\) 0 0
\(103\) −2.56155 −0.252397 −0.126199 0.992005i \(-0.540278\pi\)
−0.126199 + 0.992005i \(0.540278\pi\)
\(104\) 1.68466 0.165194
\(105\) 0 0
\(106\) 12.2462 1.18946
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) −0.315342 −0.0300666
\(111\) 0 0
\(112\) 0 0
\(113\) 11.3693 1.06954 0.534768 0.844999i \(-0.320399\pi\)
0.534768 + 0.844999i \(0.320399\pi\)
\(114\) 0 0
\(115\) 0.561553 0.0523651
\(116\) −1.00000 −0.0928477
\(117\) 0 0
\(118\) 5.43845 0.500650
\(119\) 0 0
\(120\) 0 0
\(121\) −10.6847 −0.971333
\(122\) −8.80776 −0.797417
\(123\) 0 0
\(124\) 6.56155 0.589245
\(125\) 5.43845 0.486430
\(126\) 0 0
\(127\) −8.80776 −0.781563 −0.390781 0.920484i \(-0.627795\pi\)
−0.390781 + 0.920484i \(0.627795\pi\)
\(128\) 3.00000 0.265165
\(129\) 0 0
\(130\) 0.315342 0.0276573
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −10.5616 −0.912379
\(135\) 0 0
\(136\) 0 0
\(137\) −18.2462 −1.55888 −0.779440 0.626477i \(-0.784496\pi\)
−0.779440 + 0.626477i \(0.784496\pi\)
\(138\) 0 0
\(139\) 21.6155 1.83341 0.916703 0.399570i \(-0.130841\pi\)
0.916703 + 0.399570i \(0.130841\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 14.5616 1.22198
\(143\) −0.315342 −0.0263702
\(144\) 0 0
\(145\) −0.561553 −0.0466344
\(146\) −3.12311 −0.258470
\(147\) 0 0
\(148\) −6.56155 −0.539356
\(149\) −11.9309 −0.977415 −0.488707 0.872448i \(-0.662531\pi\)
−0.488707 + 0.872448i \(0.662531\pi\)
\(150\) 0 0
\(151\) −10.2462 −0.833825 −0.416912 0.908947i \(-0.636888\pi\)
−0.416912 + 0.908947i \(0.636888\pi\)
\(152\) 9.36932 0.759952
\(153\) 0 0
\(154\) 0 0
\(155\) 3.68466 0.295959
\(156\) 0 0
\(157\) 10.8769 0.868071 0.434035 0.900896i \(-0.357089\pi\)
0.434035 + 0.900896i \(0.357089\pi\)
\(158\) 2.00000 0.159111
\(159\) 0 0
\(160\) 2.80776 0.221973
\(161\) 0 0
\(162\) 0 0
\(163\) 1.43845 0.112668 0.0563339 0.998412i \(-0.482059\pi\)
0.0563339 + 0.998412i \(0.482059\pi\)
\(164\) 3.43845 0.268498
\(165\) 0 0
\(166\) 1.12311 0.0871699
\(167\) −21.9309 −1.69706 −0.848531 0.529146i \(-0.822512\pi\)
−0.848531 + 0.529146i \(0.822512\pi\)
\(168\) 0 0
\(169\) −12.6847 −0.975743
\(170\) 0 0
\(171\) 0 0
\(172\) −12.2462 −0.933765
\(173\) 22.4924 1.71007 0.855034 0.518573i \(-0.173536\pi\)
0.855034 + 0.518573i \(0.173536\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0.561553 0.0423286
\(177\) 0 0
\(178\) −7.36932 −0.552354
\(179\) −16.4924 −1.23270 −0.616351 0.787472i \(-0.711390\pi\)
−0.616351 + 0.787472i \(0.711390\pi\)
\(180\) 0 0
\(181\) −14.4924 −1.07721 −0.538607 0.842557i \(-0.681049\pi\)
−0.538607 + 0.842557i \(0.681049\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −3.00000 −0.221163
\(185\) −3.68466 −0.270901
\(186\) 0 0
\(187\) 0 0
\(188\) −1.12311 −0.0819109
\(189\) 0 0
\(190\) 1.75379 0.127233
\(191\) 15.4384 1.11709 0.558543 0.829475i \(-0.311360\pi\)
0.558543 + 0.829475i \(0.311360\pi\)
\(192\) 0 0
\(193\) −15.1231 −1.08858 −0.544292 0.838896i \(-0.683202\pi\)
−0.544292 + 0.838896i \(0.683202\pi\)
\(194\) −11.3693 −0.816269
\(195\) 0 0
\(196\) 7.00000 0.500000
\(197\) 4.24621 0.302530 0.151265 0.988493i \(-0.451665\pi\)
0.151265 + 0.988493i \(0.451665\pi\)
\(198\) 0 0
\(199\) 15.6847 1.11186 0.555928 0.831230i \(-0.312363\pi\)
0.555928 + 0.831230i \(0.312363\pi\)
\(200\) −14.0540 −0.993766
\(201\) 0 0
\(202\) −3.43845 −0.241928
\(203\) 0 0
\(204\) 0 0
\(205\) 1.93087 0.134858
\(206\) 2.56155 0.178472
\(207\) 0 0
\(208\) −0.561553 −0.0389367
\(209\) −1.75379 −0.121312
\(210\) 0 0
\(211\) −24.1771 −1.66442 −0.832209 0.554461i \(-0.812924\pi\)
−0.832209 + 0.554461i \(0.812924\pi\)
\(212\) 12.2462 0.841073
\(213\) 0 0
\(214\) 0 0
\(215\) −6.87689 −0.469000
\(216\) 0 0
\(217\) 0 0
\(218\) −2.00000 −0.135457
\(219\) 0 0
\(220\) −0.315342 −0.0212603
\(221\) 0 0
\(222\) 0 0
\(223\) 4.49242 0.300835 0.150417 0.988623i \(-0.451938\pi\)
0.150417 + 0.988623i \(0.451938\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −11.3693 −0.756276
\(227\) 4.63068 0.307349 0.153675 0.988122i \(-0.450889\pi\)
0.153675 + 0.988122i \(0.450889\pi\)
\(228\) 0 0
\(229\) 20.8078 1.37502 0.687508 0.726177i \(-0.258705\pi\)
0.687508 + 0.726177i \(0.258705\pi\)
\(230\) −0.561553 −0.0370277
\(231\) 0 0
\(232\) 3.00000 0.196960
\(233\) −0.876894 −0.0574473 −0.0287236 0.999587i \(-0.509144\pi\)
−0.0287236 + 0.999587i \(0.509144\pi\)
\(234\) 0 0
\(235\) −0.630683 −0.0411412
\(236\) 5.43845 0.354013
\(237\) 0 0
\(238\) 0 0
\(239\) −14.5616 −0.941909 −0.470954 0.882158i \(-0.656090\pi\)
−0.470954 + 0.882158i \(0.656090\pi\)
\(240\) 0 0
\(241\) 23.6155 1.52121 0.760605 0.649215i \(-0.224902\pi\)
0.760605 + 0.649215i \(0.224902\pi\)
\(242\) 10.6847 0.686836
\(243\) 0 0
\(244\) −8.80776 −0.563859
\(245\) 3.93087 0.251134
\(246\) 0 0
\(247\) 1.75379 0.111591
\(248\) −19.6847 −1.24998
\(249\) 0 0
\(250\) −5.43845 −0.343958
\(251\) 1.68466 0.106335 0.0531673 0.998586i \(-0.483068\pi\)
0.0531673 + 0.998586i \(0.483068\pi\)
\(252\) 0 0
\(253\) 0.561553 0.0353045
\(254\) 8.80776 0.552648
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) −12.2462 −0.763898 −0.381949 0.924183i \(-0.624747\pi\)
−0.381949 + 0.924183i \(0.624747\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0.315342 0.0195567
\(261\) 0 0
\(262\) −4.00000 −0.247121
\(263\) −1.36932 −0.0844357 −0.0422178 0.999108i \(-0.513442\pi\)
−0.0422178 + 0.999108i \(0.513442\pi\)
\(264\) 0 0
\(265\) 6.87689 0.422444
\(266\) 0 0
\(267\) 0 0
\(268\) −10.5616 −0.645150
\(269\) −23.9309 −1.45909 −0.729545 0.683932i \(-0.760268\pi\)
−0.729545 + 0.683932i \(0.760268\pi\)
\(270\) 0 0
\(271\) −9.43845 −0.573345 −0.286672 0.958029i \(-0.592549\pi\)
−0.286672 + 0.958029i \(0.592549\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 18.2462 1.10229
\(275\) 2.63068 0.158636
\(276\) 0 0
\(277\) −15.9309 −0.957193 −0.478597 0.878035i \(-0.658854\pi\)
−0.478597 + 0.878035i \(0.658854\pi\)
\(278\) −21.6155 −1.29641
\(279\) 0 0
\(280\) 0 0
\(281\) −12.2462 −0.730548 −0.365274 0.930900i \(-0.619025\pi\)
−0.365274 + 0.930900i \(0.619025\pi\)
\(282\) 0 0
\(283\) −11.6847 −0.694581 −0.347290 0.937758i \(-0.612898\pi\)
−0.347290 + 0.937758i \(0.612898\pi\)
\(284\) 14.5616 0.864069
\(285\) 0 0
\(286\) 0.315342 0.0186465
\(287\) 0 0
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0.561553 0.0329755
\(291\) 0 0
\(292\) −3.12311 −0.182766
\(293\) 16.0000 0.934730 0.467365 0.884064i \(-0.345203\pi\)
0.467365 + 0.884064i \(0.345203\pi\)
\(294\) 0 0
\(295\) 3.05398 0.177809
\(296\) 19.6847 1.14415
\(297\) 0 0
\(298\) 11.9309 0.691137
\(299\) −0.561553 −0.0324754
\(300\) 0 0
\(301\) 0 0
\(302\) 10.2462 0.589603
\(303\) 0 0
\(304\) −3.12311 −0.179122
\(305\) −4.94602 −0.283209
\(306\) 0 0
\(307\) −19.6847 −1.12346 −0.561731 0.827320i \(-0.689865\pi\)
−0.561731 + 0.827320i \(0.689865\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −3.68466 −0.209275
\(311\) −25.1231 −1.42460 −0.712300 0.701875i \(-0.752347\pi\)
−0.712300 + 0.701875i \(0.752347\pi\)
\(312\) 0 0
\(313\) −20.2462 −1.14438 −0.572192 0.820120i \(-0.693907\pi\)
−0.572192 + 0.820120i \(0.693907\pi\)
\(314\) −10.8769 −0.613819
\(315\) 0 0
\(316\) 2.00000 0.112509
\(317\) 19.9309 1.11943 0.559715 0.828685i \(-0.310911\pi\)
0.559715 + 0.828685i \(0.310911\pi\)
\(318\) 0 0
\(319\) −0.561553 −0.0314409
\(320\) −3.93087 −0.219742
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −2.63068 −0.145924
\(326\) −1.43845 −0.0796682
\(327\) 0 0
\(328\) −10.3153 −0.569569
\(329\) 0 0
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 1.12311 0.0616384
\(333\) 0 0
\(334\) 21.9309 1.20000
\(335\) −5.93087 −0.324038
\(336\) 0 0
\(337\) −3.19224 −0.173892 −0.0869461 0.996213i \(-0.527711\pi\)
−0.0869461 + 0.996213i \(0.527711\pi\)
\(338\) 12.6847 0.689954
\(339\) 0 0
\(340\) 0 0
\(341\) 3.68466 0.199535
\(342\) 0 0
\(343\) 0 0
\(344\) 36.7386 1.98081
\(345\) 0 0
\(346\) −22.4924 −1.20920
\(347\) 8.49242 0.455897 0.227949 0.973673i \(-0.426798\pi\)
0.227949 + 0.973673i \(0.426798\pi\)
\(348\) 0 0
\(349\) −31.3002 −1.67546 −0.837730 0.546084i \(-0.816118\pi\)
−0.837730 + 0.546084i \(0.816118\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 2.80776 0.149654
\(353\) 22.4924 1.19715 0.598575 0.801066i \(-0.295734\pi\)
0.598575 + 0.801066i \(0.295734\pi\)
\(354\) 0 0
\(355\) 8.17708 0.433994
\(356\) −7.36932 −0.390573
\(357\) 0 0
\(358\) 16.4924 0.871652
\(359\) −3.12311 −0.164831 −0.0824156 0.996598i \(-0.526263\pi\)
−0.0824156 + 0.996598i \(0.526263\pi\)
\(360\) 0 0
\(361\) −9.24621 −0.486643
\(362\) 14.4924 0.761705
\(363\) 0 0
\(364\) 0 0
\(365\) −1.75379 −0.0917975
\(366\) 0 0
\(367\) −24.7386 −1.29135 −0.645673 0.763614i \(-0.723423\pi\)
−0.645673 + 0.763614i \(0.723423\pi\)
\(368\) 1.00000 0.0521286
\(369\) 0 0
\(370\) 3.68466 0.191556
\(371\) 0 0
\(372\) 0 0
\(373\) −30.4924 −1.57884 −0.789419 0.613855i \(-0.789618\pi\)
−0.789419 + 0.613855i \(0.789618\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 3.36932 0.173759
\(377\) 0.561553 0.0289214
\(378\) 0 0
\(379\) 30.0000 1.54100 0.770498 0.637442i \(-0.220007\pi\)
0.770498 + 0.637442i \(0.220007\pi\)
\(380\) 1.75379 0.0899675
\(381\) 0 0
\(382\) −15.4384 −0.789900
\(383\) −17.1231 −0.874950 −0.437475 0.899231i \(-0.644127\pi\)
−0.437475 + 0.899231i \(0.644127\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 15.1231 0.769746
\(387\) 0 0
\(388\) −11.3693 −0.577190
\(389\) −21.7538 −1.10296 −0.551480 0.834188i \(-0.685937\pi\)
−0.551480 + 0.834188i \(0.685937\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −21.0000 −1.06066
\(393\) 0 0
\(394\) −4.24621 −0.213921
\(395\) 1.12311 0.0565096
\(396\) 0 0
\(397\) −32.2462 −1.61839 −0.809195 0.587540i \(-0.800097\pi\)
−0.809195 + 0.587540i \(0.800097\pi\)
\(398\) −15.6847 −0.786201
\(399\) 0 0
\(400\) 4.68466 0.234233
\(401\) 22.8078 1.13897 0.569483 0.822003i \(-0.307144\pi\)
0.569483 + 0.822003i \(0.307144\pi\)
\(402\) 0 0
\(403\) −3.68466 −0.183546
\(404\) −3.43845 −0.171069
\(405\) 0 0
\(406\) 0 0
\(407\) −3.68466 −0.182642
\(408\) 0 0
\(409\) 3.75379 0.185613 0.0928065 0.995684i \(-0.470416\pi\)
0.0928065 + 0.995684i \(0.470416\pi\)
\(410\) −1.93087 −0.0953589
\(411\) 0 0
\(412\) 2.56155 0.126199
\(413\) 0 0
\(414\) 0 0
\(415\) 0.630683 0.0309590
\(416\) −2.80776 −0.137662
\(417\) 0 0
\(418\) 1.75379 0.0857806
\(419\) −14.8769 −0.726784 −0.363392 0.931636i \(-0.618381\pi\)
−0.363392 + 0.931636i \(0.618381\pi\)
\(420\) 0 0
\(421\) −0.807764 −0.0393680 −0.0196840 0.999806i \(-0.506266\pi\)
−0.0196840 + 0.999806i \(0.506266\pi\)
\(422\) 24.1771 1.17692
\(423\) 0 0
\(424\) −36.7386 −1.78419
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 6.87689 0.331633
\(431\) −29.1231 −1.40281 −0.701405 0.712763i \(-0.747444\pi\)
−0.701405 + 0.712763i \(0.747444\pi\)
\(432\) 0 0
\(433\) −29.1231 −1.39957 −0.699784 0.714355i \(-0.746720\pi\)
−0.699784 + 0.714355i \(0.746720\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −2.00000 −0.0957826
\(437\) −3.12311 −0.149398
\(438\) 0 0
\(439\) −27.8617 −1.32977 −0.664884 0.746947i \(-0.731519\pi\)
−0.664884 + 0.746947i \(0.731519\pi\)
\(440\) 0.946025 0.0451000
\(441\) 0 0
\(442\) 0 0
\(443\) −13.1231 −0.623498 −0.311749 0.950165i \(-0.600915\pi\)
−0.311749 + 0.950165i \(0.600915\pi\)
\(444\) 0 0
\(445\) −4.13826 −0.196172
\(446\) −4.49242 −0.212722
\(447\) 0 0
\(448\) 0 0
\(449\) −29.6847 −1.40091 −0.700453 0.713699i \(-0.747019\pi\)
−0.700453 + 0.713699i \(0.747019\pi\)
\(450\) 0 0
\(451\) 1.93087 0.0909211
\(452\) −11.3693 −0.534768
\(453\) 0 0
\(454\) −4.63068 −0.217329
\(455\) 0 0
\(456\) 0 0
\(457\) 7.12311 0.333205 0.166602 0.986024i \(-0.446720\pi\)
0.166602 + 0.986024i \(0.446720\pi\)
\(458\) −20.8078 −0.972283
\(459\) 0 0
\(460\) −0.561553 −0.0261825
\(461\) −15.3002 −0.712601 −0.356300 0.934371i \(-0.615962\pi\)
−0.356300 + 0.934371i \(0.615962\pi\)
\(462\) 0 0
\(463\) −12.4924 −0.580572 −0.290286 0.956940i \(-0.593750\pi\)
−0.290286 + 0.956940i \(0.593750\pi\)
\(464\) −1.00000 −0.0464238
\(465\) 0 0
\(466\) 0.876894 0.0406213
\(467\) 36.4233 1.68547 0.842734 0.538329i \(-0.180944\pi\)
0.842734 + 0.538329i \(0.180944\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0.630683 0.0290912
\(471\) 0 0
\(472\) −16.3153 −0.750974
\(473\) −6.87689 −0.316200
\(474\) 0 0
\(475\) −14.6307 −0.671302
\(476\) 0 0
\(477\) 0 0
\(478\) 14.5616 0.666030
\(479\) −33.6847 −1.53909 −0.769546 0.638592i \(-0.779517\pi\)
−0.769546 + 0.638592i \(0.779517\pi\)
\(480\) 0 0
\(481\) 3.68466 0.168006
\(482\) −23.6155 −1.07566
\(483\) 0 0
\(484\) 10.6847 0.485666
\(485\) −6.38447 −0.289904
\(486\) 0 0
\(487\) 38.7386 1.75542 0.877708 0.479197i \(-0.159072\pi\)
0.877708 + 0.479197i \(0.159072\pi\)
\(488\) 26.4233 1.19613
\(489\) 0 0
\(490\) −3.93087 −0.177579
\(491\) 27.3693 1.23516 0.617580 0.786508i \(-0.288113\pi\)
0.617580 + 0.786508i \(0.288113\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) −1.75379 −0.0789067
\(495\) 0 0
\(496\) 6.56155 0.294622
\(497\) 0 0
\(498\) 0 0
\(499\) −19.3693 −0.867090 −0.433545 0.901132i \(-0.642737\pi\)
−0.433545 + 0.901132i \(0.642737\pi\)
\(500\) −5.43845 −0.243215
\(501\) 0 0
\(502\) −1.68466 −0.0751900
\(503\) −41.3693 −1.84457 −0.922283 0.386514i \(-0.873679\pi\)
−0.922283 + 0.386514i \(0.873679\pi\)
\(504\) 0 0
\(505\) −1.93087 −0.0859226
\(506\) −0.561553 −0.0249641
\(507\) 0 0
\(508\) 8.80776 0.390781
\(509\) 10.0000 0.443242 0.221621 0.975133i \(-0.428865\pi\)
0.221621 + 0.975133i \(0.428865\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 11.0000 0.486136
\(513\) 0 0
\(514\) 12.2462 0.540157
\(515\) 1.43845 0.0633856
\(516\) 0 0
\(517\) −0.630683 −0.0277374
\(518\) 0 0
\(519\) 0 0
\(520\) −0.946025 −0.0414859
\(521\) 11.7538 0.514943 0.257471 0.966286i \(-0.417111\pi\)
0.257471 + 0.966286i \(0.417111\pi\)
\(522\) 0 0
\(523\) 12.8078 0.560044 0.280022 0.959994i \(-0.409658\pi\)
0.280022 + 0.959994i \(0.409658\pi\)
\(524\) −4.00000 −0.174741
\(525\) 0 0
\(526\) 1.36932 0.0597051
\(527\) 0 0
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) −6.87689 −0.298713
\(531\) 0 0
\(532\) 0 0
\(533\) −1.93087 −0.0836353
\(534\) 0 0
\(535\) 0 0
\(536\) 31.6847 1.36857
\(537\) 0 0
\(538\) 23.9309 1.03173
\(539\) 3.93087 0.169315
\(540\) 0 0
\(541\) 12.2462 0.526506 0.263253 0.964727i \(-0.415205\pi\)
0.263253 + 0.964727i \(0.415205\pi\)
\(542\) 9.43845 0.405416
\(543\) 0 0
\(544\) 0 0
\(545\) −1.12311 −0.0481086
\(546\) 0 0
\(547\) 25.7538 1.10115 0.550576 0.834785i \(-0.314408\pi\)
0.550576 + 0.834785i \(0.314408\pi\)
\(548\) 18.2462 0.779440
\(549\) 0 0
\(550\) −2.63068 −0.112173
\(551\) 3.12311 0.133049
\(552\) 0 0
\(553\) 0 0
\(554\) 15.9309 0.676838
\(555\) 0 0
\(556\) −21.6155 −0.916703
\(557\) −42.1771 −1.78710 −0.893550 0.448963i \(-0.851793\pi\)
−0.893550 + 0.448963i \(0.851793\pi\)
\(558\) 0 0
\(559\) 6.87689 0.290862
\(560\) 0 0
\(561\) 0 0
\(562\) 12.2462 0.516575
\(563\) −43.9309 −1.85147 −0.925733 0.378178i \(-0.876551\pi\)
−0.925733 + 0.378178i \(0.876551\pi\)
\(564\) 0 0
\(565\) −6.38447 −0.268597
\(566\) 11.6847 0.491143
\(567\) 0 0
\(568\) −43.6847 −1.83297
\(569\) −18.7386 −0.785564 −0.392782 0.919632i \(-0.628487\pi\)
−0.392782 + 0.919632i \(0.628487\pi\)
\(570\) 0 0
\(571\) 32.1771 1.34657 0.673284 0.739384i \(-0.264883\pi\)
0.673284 + 0.739384i \(0.264883\pi\)
\(572\) 0.315342 0.0131851
\(573\) 0 0
\(574\) 0 0
\(575\) 4.68466 0.195364
\(576\) 0 0
\(577\) 3.12311 0.130017 0.0650083 0.997885i \(-0.479293\pi\)
0.0650083 + 0.997885i \(0.479293\pi\)
\(578\) 17.0000 0.707107
\(579\) 0 0
\(580\) 0.561553 0.0233172
\(581\) 0 0
\(582\) 0 0
\(583\) 6.87689 0.284812
\(584\) 9.36932 0.387705
\(585\) 0 0
\(586\) −16.0000 −0.660954
\(587\) 47.2311 1.94943 0.974717 0.223442i \(-0.0717294\pi\)
0.974717 + 0.223442i \(0.0717294\pi\)
\(588\) 0 0
\(589\) −20.4924 −0.844376
\(590\) −3.05398 −0.125730
\(591\) 0 0
\(592\) −6.56155 −0.269678
\(593\) −29.3693 −1.20605 −0.603027 0.797721i \(-0.706039\pi\)
−0.603027 + 0.797721i \(0.706039\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 11.9309 0.488707
\(597\) 0 0
\(598\) 0.561553 0.0229636
\(599\) 44.9848 1.83803 0.919015 0.394221i \(-0.128986\pi\)
0.919015 + 0.394221i \(0.128986\pi\)
\(600\) 0 0
\(601\) −4.24621 −0.173207 −0.0866033 0.996243i \(-0.527601\pi\)
−0.0866033 + 0.996243i \(0.527601\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 10.2462 0.416912
\(605\) 6.00000 0.243935
\(606\) 0 0
\(607\) 47.2311 1.91705 0.958525 0.285009i \(-0.0919967\pi\)
0.958525 + 0.285009i \(0.0919967\pi\)
\(608\) −15.6155 −0.633293
\(609\) 0 0
\(610\) 4.94602 0.200259
\(611\) 0.630683 0.0255147
\(612\) 0 0
\(613\) 22.4924 0.908460 0.454230 0.890884i \(-0.349914\pi\)
0.454230 + 0.890884i \(0.349914\pi\)
\(614\) 19.6847 0.794408
\(615\) 0 0
\(616\) 0 0
\(617\) −2.38447 −0.0959952 −0.0479976 0.998847i \(-0.515284\pi\)
−0.0479976 + 0.998847i \(0.515284\pi\)
\(618\) 0 0
\(619\) 8.24621 0.331443 0.165722 0.986173i \(-0.447005\pi\)
0.165722 + 0.986173i \(0.447005\pi\)
\(620\) −3.68466 −0.147979
\(621\) 0 0
\(622\) 25.1231 1.00734
\(623\) 0 0
\(624\) 0 0
\(625\) 20.3693 0.814773
\(626\) 20.2462 0.809201
\(627\) 0 0
\(628\) −10.8769 −0.434035
\(629\) 0 0
\(630\) 0 0
\(631\) 34.5616 1.37587 0.687937 0.725771i \(-0.258517\pi\)
0.687937 + 0.725771i \(0.258517\pi\)
\(632\) −6.00000 −0.238667
\(633\) 0 0
\(634\) −19.9309 −0.791556
\(635\) 4.94602 0.196277
\(636\) 0 0
\(637\) −3.93087 −0.155747
\(638\) 0.561553 0.0222321
\(639\) 0 0
\(640\) −1.68466 −0.0665920
\(641\) −10.7386 −0.424151 −0.212075 0.977253i \(-0.568022\pi\)
−0.212075 + 0.977253i \(0.568022\pi\)
\(642\) 0 0
\(643\) −16.4924 −0.650398 −0.325199 0.945646i \(-0.605431\pi\)
−0.325199 + 0.945646i \(0.605431\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −12.3153 −0.484166 −0.242083 0.970256i \(-0.577831\pi\)
−0.242083 + 0.970256i \(0.577831\pi\)
\(648\) 0 0
\(649\) 3.05398 0.119879
\(650\) 2.63068 0.103184
\(651\) 0 0
\(652\) −1.43845 −0.0563339
\(653\) −39.4384 −1.54335 −0.771673 0.636020i \(-0.780580\pi\)
−0.771673 + 0.636020i \(0.780580\pi\)
\(654\) 0 0
\(655\) −2.24621 −0.0877667
\(656\) 3.43845 0.134249
\(657\) 0 0
\(658\) 0 0
\(659\) −24.8769 −0.969066 −0.484533 0.874773i \(-0.661010\pi\)
−0.484533 + 0.874773i \(0.661010\pi\)
\(660\) 0 0
\(661\) −23.6155 −0.918538 −0.459269 0.888297i \(-0.651889\pi\)
−0.459269 + 0.888297i \(0.651889\pi\)
\(662\) −20.0000 −0.777322
\(663\) 0 0
\(664\) −3.36932 −0.130755
\(665\) 0 0
\(666\) 0 0
\(667\) −1.00000 −0.0387202
\(668\) 21.9309 0.848531
\(669\) 0 0
\(670\) 5.93087 0.229129
\(671\) −4.94602 −0.190939
\(672\) 0 0
\(673\) −30.0000 −1.15642 −0.578208 0.815890i \(-0.696248\pi\)
−0.578208 + 0.815890i \(0.696248\pi\)
\(674\) 3.19224 0.122960
\(675\) 0 0
\(676\) 12.6847 0.487871
\(677\) 29.7538 1.14353 0.571765 0.820417i \(-0.306259\pi\)
0.571765 + 0.820417i \(0.306259\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) −3.68466 −0.141093
\(683\) −5.43845 −0.208096 −0.104048 0.994572i \(-0.533180\pi\)
−0.104048 + 0.994572i \(0.533180\pi\)
\(684\) 0 0
\(685\) 10.2462 0.391488
\(686\) 0 0
\(687\) 0 0
\(688\) −12.2462 −0.466882
\(689\) −6.87689 −0.261989
\(690\) 0 0
\(691\) −50.1080 −1.90620 −0.953098 0.302661i \(-0.902125\pi\)
−0.953098 + 0.302661i \(0.902125\pi\)
\(692\) −22.4924 −0.855034
\(693\) 0 0
\(694\) −8.49242 −0.322368
\(695\) −12.1383 −0.460430
\(696\) 0 0
\(697\) 0 0
\(698\) 31.3002 1.18473
\(699\) 0 0
\(700\) 0 0
\(701\) 21.0540 0.795198 0.397599 0.917559i \(-0.369844\pi\)
0.397599 + 0.917559i \(0.369844\pi\)
\(702\) 0 0
\(703\) 20.4924 0.772886
\(704\) −3.93087 −0.148150
\(705\) 0 0
\(706\) −22.4924 −0.846513
\(707\) 0 0
\(708\) 0 0
\(709\) −7.12311 −0.267514 −0.133757 0.991014i \(-0.542704\pi\)
−0.133757 + 0.991014i \(0.542704\pi\)
\(710\) −8.17708 −0.306880
\(711\) 0 0
\(712\) 22.1080 0.828530
\(713\) 6.56155 0.245732
\(714\) 0 0
\(715\) 0.177081 0.00662245
\(716\) 16.4924 0.616351
\(717\) 0 0
\(718\) 3.12311 0.116553
\(719\) 13.3002 0.496013 0.248007 0.968758i \(-0.420225\pi\)
0.248007 + 0.968758i \(0.420225\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 9.24621 0.344108
\(723\) 0 0
\(724\) 14.4924 0.538607
\(725\) −4.68466 −0.173984
\(726\) 0 0
\(727\) 0.738634 0.0273944 0.0136972 0.999906i \(-0.495640\pi\)
0.0136972 + 0.999906i \(0.495640\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 1.75379 0.0649106
\(731\) 0 0
\(732\) 0 0
\(733\) −29.4384 −1.08733 −0.543667 0.839301i \(-0.682965\pi\)
−0.543667 + 0.839301i \(0.682965\pi\)
\(734\) 24.7386 0.913120
\(735\) 0 0
\(736\) 5.00000 0.184302
\(737\) −5.93087 −0.218466
\(738\) 0 0
\(739\) 39.5464 1.45474 0.727369 0.686247i \(-0.240743\pi\)
0.727369 + 0.686247i \(0.240743\pi\)
\(740\) 3.68466 0.135451
\(741\) 0 0
\(742\) 0 0
\(743\) −31.3002 −1.14829 −0.574146 0.818753i \(-0.694666\pi\)
−0.574146 + 0.818753i \(0.694666\pi\)
\(744\) 0 0
\(745\) 6.69981 0.245462
\(746\) 30.4924 1.11641
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −37.8617 −1.38159 −0.690797 0.723049i \(-0.742740\pi\)
−0.690797 + 0.723049i \(0.742740\pi\)
\(752\) −1.12311 −0.0409554
\(753\) 0 0
\(754\) −0.561553 −0.0204505
\(755\) 5.75379 0.209402
\(756\) 0 0
\(757\) 8.31534 0.302226 0.151113 0.988516i \(-0.451714\pi\)
0.151113 + 0.988516i \(0.451714\pi\)
\(758\) −30.0000 −1.08965
\(759\) 0 0
\(760\) −5.26137 −0.190850
\(761\) −46.9848 −1.70320 −0.851600 0.524193i \(-0.824367\pi\)
−0.851600 + 0.524193i \(0.824367\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −15.4384 −0.558543
\(765\) 0 0
\(766\) 17.1231 0.618683
\(767\) −3.05398 −0.110273
\(768\) 0 0
\(769\) −1.43845 −0.0518717 −0.0259359 0.999664i \(-0.508257\pi\)
−0.0259359 + 0.999664i \(0.508257\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 15.1231 0.544292
\(773\) 44.0000 1.58257 0.791285 0.611448i \(-0.209412\pi\)
0.791285 + 0.611448i \(0.209412\pi\)
\(774\) 0 0
\(775\) 30.7386 1.10416
\(776\) 34.1080 1.22440
\(777\) 0 0
\(778\) 21.7538 0.779911
\(779\) −10.7386 −0.384751
\(780\) 0 0
\(781\) 8.17708 0.292599
\(782\) 0 0
\(783\) 0 0
\(784\) 7.00000 0.250000
\(785\) −6.10795 −0.218002
\(786\) 0 0
\(787\) −55.0540 −1.96246 −0.981231 0.192835i \(-0.938232\pi\)
−0.981231 + 0.192835i \(0.938232\pi\)
\(788\) −4.24621 −0.151265
\(789\) 0 0
\(790\) −1.12311 −0.0399583
\(791\) 0 0
\(792\) 0 0
\(793\) 4.94602 0.175638
\(794\) 32.2462 1.14438
\(795\) 0 0
\(796\) −15.6847 −0.555928
\(797\) 42.1080 1.49154 0.745770 0.666203i \(-0.232082\pi\)
0.745770 + 0.666203i \(0.232082\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 23.4233 0.828138
\(801\) 0 0
\(802\) −22.8078 −0.805370
\(803\) −1.75379 −0.0618899
\(804\) 0 0
\(805\) 0 0
\(806\) 3.68466 0.129787
\(807\) 0 0
\(808\) 10.3153 0.362892
\(809\) −51.3002 −1.80362 −0.901809 0.432134i \(-0.857761\pi\)
−0.901809 + 0.432134i \(0.857761\pi\)
\(810\) 0 0
\(811\) −1.75379 −0.0615839 −0.0307919 0.999526i \(-0.509803\pi\)
−0.0307919 + 0.999526i \(0.509803\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 3.68466 0.129147
\(815\) −0.807764 −0.0282947
\(816\) 0 0
\(817\) 38.2462 1.33807
\(818\) −3.75379 −0.131248
\(819\) 0 0
\(820\) −1.93087 −0.0674289
\(821\) 21.3693 0.745794 0.372897 0.927873i \(-0.378364\pi\)
0.372897 + 0.927873i \(0.378364\pi\)
\(822\) 0 0
\(823\) −25.3002 −0.881909 −0.440955 0.897529i \(-0.645360\pi\)
−0.440955 + 0.897529i \(0.645360\pi\)
\(824\) −7.68466 −0.267708
\(825\) 0 0
\(826\) 0 0
\(827\) −3.93087 −0.136690 −0.0683449 0.997662i \(-0.521772\pi\)
−0.0683449 + 0.997662i \(0.521772\pi\)
\(828\) 0 0
\(829\) 50.3542 1.74887 0.874436 0.485141i \(-0.161232\pi\)
0.874436 + 0.485141i \(0.161232\pi\)
\(830\) −0.630683 −0.0218913
\(831\) 0 0
\(832\) 3.93087 0.136278
\(833\) 0 0
\(834\) 0 0
\(835\) 12.3153 0.426190
\(836\) 1.75379 0.0606561
\(837\) 0 0
\(838\) 14.8769 0.513914
\(839\) −29.1922 −1.00783 −0.503914 0.863754i \(-0.668107\pi\)
−0.503914 + 0.863754i \(0.668107\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0.807764 0.0278374
\(843\) 0 0
\(844\) 24.1771 0.832209
\(845\) 7.12311 0.245042
\(846\) 0 0
\(847\) 0 0
\(848\) 12.2462 0.420537
\(849\) 0 0
\(850\) 0 0
\(851\) −6.56155 −0.224927
\(852\) 0 0
\(853\) 5.36932 0.183842 0.0919210 0.995766i \(-0.470699\pi\)
0.0919210 + 0.995766i \(0.470699\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −32.2462 −1.10151 −0.550755 0.834667i \(-0.685660\pi\)
−0.550755 + 0.834667i \(0.685660\pi\)
\(858\) 0 0
\(859\) −52.9848 −1.80782 −0.903910 0.427723i \(-0.859316\pi\)
−0.903910 + 0.427723i \(0.859316\pi\)
\(860\) 6.87689 0.234500
\(861\) 0 0
\(862\) 29.1231 0.991937
\(863\) −5.93087 −0.201889 −0.100945 0.994892i \(-0.532186\pi\)
−0.100945 + 0.994892i \(0.532186\pi\)
\(864\) 0 0
\(865\) −12.6307 −0.429456
\(866\) 29.1231 0.989643
\(867\) 0 0
\(868\) 0 0
\(869\) 1.12311 0.0380987
\(870\) 0 0
\(871\) 5.93087 0.200960
\(872\) 6.00000 0.203186
\(873\) 0 0
\(874\) 3.12311 0.105641
\(875\) 0 0
\(876\) 0 0
\(877\) −49.2311 −1.66241 −0.831207 0.555963i \(-0.812350\pi\)
−0.831207 + 0.555963i \(0.812350\pi\)
\(878\) 27.8617 0.940288
\(879\) 0 0
\(880\) −0.315342 −0.0106302
\(881\) 38.1080 1.28389 0.641945 0.766751i \(-0.278128\pi\)
0.641945 + 0.766751i \(0.278128\pi\)
\(882\) 0 0
\(883\) 5.26137 0.177059 0.0885295 0.996074i \(-0.471783\pi\)
0.0885295 + 0.996074i \(0.471783\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 13.1231 0.440879
\(887\) 28.4924 0.956682 0.478341 0.878174i \(-0.341238\pi\)
0.478341 + 0.878174i \(0.341238\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 4.13826 0.138715
\(891\) 0 0
\(892\) −4.49242 −0.150417
\(893\) 3.50758 0.117377
\(894\) 0 0
\(895\) 9.26137 0.309573
\(896\) 0 0
\(897\) 0 0
\(898\) 29.6847 0.990590
\(899\) −6.56155 −0.218840
\(900\) 0 0
\(901\) 0 0
\(902\) −1.93087 −0.0642909
\(903\) 0 0
\(904\) 34.1080 1.13441
\(905\) 8.13826 0.270525
\(906\) 0 0
\(907\) 2.00000 0.0664089 0.0332045 0.999449i \(-0.489429\pi\)
0.0332045 + 0.999449i \(0.489429\pi\)
\(908\) −4.63068 −0.153675
\(909\) 0 0
\(910\) 0 0
\(911\) 41.6847 1.38107 0.690537 0.723297i \(-0.257374\pi\)
0.690537 + 0.723297i \(0.257374\pi\)
\(912\) 0 0
\(913\) 0.630683 0.0208726
\(914\) −7.12311 −0.235611
\(915\) 0 0
\(916\) −20.8078 −0.687508
\(917\) 0 0
\(918\) 0 0
\(919\) −8.80776 −0.290541 −0.145271 0.989392i \(-0.546405\pi\)
−0.145271 + 0.989392i \(0.546405\pi\)
\(920\) 1.68466 0.0555415
\(921\) 0 0
\(922\) 15.3002 0.503885
\(923\) −8.17708 −0.269152
\(924\) 0 0
\(925\) −30.7386 −1.01068
\(926\) 12.4924 0.410526
\(927\) 0 0
\(928\) −5.00000 −0.164133
\(929\) −14.0000 −0.459325 −0.229663 0.973270i \(-0.573762\pi\)
−0.229663 + 0.973270i \(0.573762\pi\)
\(930\) 0 0
\(931\) −21.8617 −0.716490
\(932\) 0.876894 0.0287236
\(933\) 0 0
\(934\) −36.4233 −1.19181
\(935\) 0 0
\(936\) 0 0
\(937\) −46.9848 −1.53493 −0.767464 0.641092i \(-0.778482\pi\)
−0.767464 + 0.641092i \(0.778482\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0.630683 0.0205706
\(941\) −25.5464 −0.832789 −0.416394 0.909184i \(-0.636706\pi\)
−0.416394 + 0.909184i \(0.636706\pi\)
\(942\) 0 0
\(943\) 3.43845 0.111971
\(944\) 5.43845 0.177006
\(945\) 0 0
\(946\) 6.87689 0.223587
\(947\) −49.4773 −1.60780 −0.803898 0.594768i \(-0.797244\pi\)
−0.803898 + 0.594768i \(0.797244\pi\)
\(948\) 0 0
\(949\) 1.75379 0.0569304
\(950\) 14.6307 0.474682
\(951\) 0 0
\(952\) 0 0
\(953\) 38.9848 1.26284 0.631422 0.775440i \(-0.282472\pi\)
0.631422 + 0.775440i \(0.282472\pi\)
\(954\) 0 0
\(955\) −8.66950 −0.280539
\(956\) 14.5616 0.470954
\(957\) 0 0
\(958\) 33.6847 1.08830
\(959\) 0 0
\(960\) 0 0
\(961\) 12.0540 0.388838
\(962\) −3.68466 −0.118798
\(963\) 0 0
\(964\) −23.6155 −0.760605
\(965\) 8.49242 0.273381
\(966\) 0 0
\(967\) 54.2462 1.74444 0.872220 0.489113i \(-0.162680\pi\)
0.872220 + 0.489113i \(0.162680\pi\)
\(968\) −32.0540 −1.03025
\(969\) 0 0
\(970\) 6.38447 0.204993
\(971\) 2.80776 0.0901054 0.0450527 0.998985i \(-0.485654\pi\)
0.0450527 + 0.998985i \(0.485654\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −38.7386 −1.24127
\(975\) 0 0
\(976\) −8.80776 −0.281930
\(977\) −30.1771 −0.965450 −0.482725 0.875772i \(-0.660353\pi\)
−0.482725 + 0.875772i \(0.660353\pi\)
\(978\) 0 0
\(979\) −4.13826 −0.132259
\(980\) −3.93087 −0.125567
\(981\) 0 0
\(982\) −27.3693 −0.873390
\(983\) 3.43845 0.109669 0.0548347 0.998495i \(-0.482537\pi\)
0.0548347 + 0.998495i \(0.482537\pi\)
\(984\) 0 0
\(985\) −2.38447 −0.0759756
\(986\) 0 0
\(987\) 0 0
\(988\) −1.75379 −0.0557955
\(989\) −12.2462 −0.389407
\(990\) 0 0
\(991\) −44.4924 −1.41335 −0.706674 0.707539i \(-0.749805\pi\)
−0.706674 + 0.707539i \(0.749805\pi\)
\(992\) 32.8078 1.04165
\(993\) 0 0
\(994\) 0 0
\(995\) −8.80776 −0.279225
\(996\) 0 0
\(997\) 24.7386 0.783480 0.391740 0.920076i \(-0.371873\pi\)
0.391740 + 0.920076i \(0.371873\pi\)
\(998\) 19.3693 0.613125
\(999\) 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 6003.2.a.d.1.1 2
3.2 odd 2 2001.2.a.f.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2001.2.a.f.1.2 2 3.2 odd 2
6003.2.a.d.1.1 2 1.1 even 1 trivial