Properties

Label 6004.2.a.h.1.20
Level $6004$
Weight $2$
Character 6004.1
Self dual yes
Analytic conductor $47.942$
Analytic rank $0$
Dimension $31$
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] = [6004,2,Mod(1,6004)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6004, 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("6004.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6004 = 2^{2} \cdot 19 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6004.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: \(47.9421813736\)
Analytic rank: \(0\)
Dimension: \(31\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.20
Character \(\chi\) \(=\) 6004.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.806301 q^{3} +1.48458 q^{5} +2.70067 q^{7} -2.34988 q^{9} +O(q^{10})\) \(q+0.806301 q^{3} +1.48458 q^{5} +2.70067 q^{7} -2.34988 q^{9} -5.03911 q^{11} +5.37759 q^{13} +1.19702 q^{15} +4.13412 q^{17} -1.00000 q^{19} +2.17755 q^{21} +7.14061 q^{23} -2.79603 q^{25} -4.31361 q^{27} +6.01825 q^{29} +5.76160 q^{31} -4.06304 q^{33} +4.00935 q^{35} +3.46239 q^{37} +4.33596 q^{39} +9.81340 q^{41} -8.87461 q^{43} -3.48857 q^{45} -12.7688 q^{47} +0.293600 q^{49} +3.33335 q^{51} -0.901754 q^{53} -7.48094 q^{55} -0.806301 q^{57} -2.24773 q^{59} -2.43729 q^{61} -6.34624 q^{63} +7.98345 q^{65} +1.66268 q^{67} +5.75749 q^{69} +0.0535198 q^{71} +1.80260 q^{73} -2.25444 q^{75} -13.6089 q^{77} -1.00000 q^{79} +3.57156 q^{81} +4.75299 q^{83} +6.13742 q^{85} +4.85252 q^{87} +16.5468 q^{89} +14.5231 q^{91} +4.64558 q^{93} -1.48458 q^{95} +7.23091 q^{97} +11.8413 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 31 q - 4 q^{3} + 10 q^{5} + 11 q^{7} + 47 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 31 q - 4 q^{3} + 10 q^{5} + 11 q^{7} + 47 q^{9} - 4 q^{11} + 11 q^{13} + 5 q^{15} + 14 q^{17} - 31 q^{19} + 22 q^{21} + 15 q^{23} + 59 q^{25} + 5 q^{27} + 34 q^{29} - 12 q^{31} + 10 q^{33} + 8 q^{35} + 16 q^{37} + 18 q^{39} + 27 q^{41} + 2 q^{43} + 22 q^{45} + 30 q^{47} + 62 q^{49} - 14 q^{51} + 35 q^{53} + 8 q^{55} + 4 q^{57} - 16 q^{59} + 37 q^{61} + 31 q^{63} + 80 q^{65} + 16 q^{67} + q^{69} + 19 q^{71} + 38 q^{73} + 21 q^{75} + 44 q^{77} - 31 q^{79} + 55 q^{81} - 12 q^{83} + 66 q^{85} + 58 q^{87} + 16 q^{89} - 42 q^{91} + 10 q^{93} - 10 q^{95} - 12 q^{97} + 12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.806301 0.465518 0.232759 0.972534i \(-0.425225\pi\)
0.232759 + 0.972534i \(0.425225\pi\)
\(4\) 0 0
\(5\) 1.48458 0.663923 0.331961 0.943293i \(-0.392290\pi\)
0.331961 + 0.943293i \(0.392290\pi\)
\(6\) 0 0
\(7\) 2.70067 1.02076 0.510378 0.859950i \(-0.329505\pi\)
0.510378 + 0.859950i \(0.329505\pi\)
\(8\) 0 0
\(9\) −2.34988 −0.783293
\(10\) 0 0
\(11\) −5.03911 −1.51935 −0.759674 0.650304i \(-0.774641\pi\)
−0.759674 + 0.650304i \(0.774641\pi\)
\(12\) 0 0
\(13\) 5.37759 1.49148 0.745738 0.666239i \(-0.232097\pi\)
0.745738 + 0.666239i \(0.232097\pi\)
\(14\) 0 0
\(15\) 1.19702 0.309068
\(16\) 0 0
\(17\) 4.13412 1.00267 0.501336 0.865253i \(-0.332842\pi\)
0.501336 + 0.865253i \(0.332842\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 2.17755 0.475181
\(22\) 0 0
\(23\) 7.14061 1.48892 0.744460 0.667667i \(-0.232707\pi\)
0.744460 + 0.667667i \(0.232707\pi\)
\(24\) 0 0
\(25\) −2.79603 −0.559207
\(26\) 0 0
\(27\) −4.31361 −0.830155
\(28\) 0 0
\(29\) 6.01825 1.11756 0.558780 0.829316i \(-0.311270\pi\)
0.558780 + 0.829316i \(0.311270\pi\)
\(30\) 0 0
\(31\) 5.76160 1.03481 0.517407 0.855740i \(-0.326897\pi\)
0.517407 + 0.855740i \(0.326897\pi\)
\(32\) 0 0
\(33\) −4.06304 −0.707284
\(34\) 0 0
\(35\) 4.00935 0.677703
\(36\) 0 0
\(37\) 3.46239 0.569213 0.284607 0.958644i \(-0.408137\pi\)
0.284607 + 0.958644i \(0.408137\pi\)
\(38\) 0 0
\(39\) 4.33596 0.694309
\(40\) 0 0
\(41\) 9.81340 1.53260 0.766298 0.642486i \(-0.222097\pi\)
0.766298 + 0.642486i \(0.222097\pi\)
\(42\) 0 0
\(43\) −8.87461 −1.35337 −0.676683 0.736275i \(-0.736583\pi\)
−0.676683 + 0.736275i \(0.736583\pi\)
\(44\) 0 0
\(45\) −3.48857 −0.520046
\(46\) 0 0
\(47\) −12.7688 −1.86253 −0.931264 0.364345i \(-0.881293\pi\)
−0.931264 + 0.364345i \(0.881293\pi\)
\(48\) 0 0
\(49\) 0.293600 0.0419428
\(50\) 0 0
\(51\) 3.33335 0.466762
\(52\) 0 0
\(53\) −0.901754 −0.123865 −0.0619327 0.998080i \(-0.519726\pi\)
−0.0619327 + 0.998080i \(0.519726\pi\)
\(54\) 0 0
\(55\) −7.48094 −1.00873
\(56\) 0 0
\(57\) −0.806301 −0.106797
\(58\) 0 0
\(59\) −2.24773 −0.292630 −0.146315 0.989238i \(-0.546741\pi\)
−0.146315 + 0.989238i \(0.546741\pi\)
\(60\) 0 0
\(61\) −2.43729 −0.312063 −0.156032 0.987752i \(-0.549870\pi\)
−0.156032 + 0.987752i \(0.549870\pi\)
\(62\) 0 0
\(63\) −6.34624 −0.799551
\(64\) 0 0
\(65\) 7.98345 0.990225
\(66\) 0 0
\(67\) 1.66268 0.203129 0.101565 0.994829i \(-0.467615\pi\)
0.101565 + 0.994829i \(0.467615\pi\)
\(68\) 0 0
\(69\) 5.75749 0.693120
\(70\) 0 0
\(71\) 0.0535198 0.00635163 0.00317581 0.999995i \(-0.498989\pi\)
0.00317581 + 0.999995i \(0.498989\pi\)
\(72\) 0 0
\(73\) 1.80260 0.210978 0.105489 0.994420i \(-0.466359\pi\)
0.105489 + 0.994420i \(0.466359\pi\)
\(74\) 0 0
\(75\) −2.25444 −0.260321
\(76\) 0 0
\(77\) −13.6089 −1.55088
\(78\) 0 0
\(79\) −1.00000 −0.112509
\(80\) 0 0
\(81\) 3.57156 0.396840
\(82\) 0 0
\(83\) 4.75299 0.521709 0.260854 0.965378i \(-0.415996\pi\)
0.260854 + 0.965378i \(0.415996\pi\)
\(84\) 0 0
\(85\) 6.13742 0.665697
\(86\) 0 0
\(87\) 4.85252 0.520245
\(88\) 0 0
\(89\) 16.5468 1.75396 0.876979 0.480530i \(-0.159556\pi\)
0.876979 + 0.480530i \(0.159556\pi\)
\(90\) 0 0
\(91\) 14.5231 1.52243
\(92\) 0 0
\(93\) 4.64558 0.481725
\(94\) 0 0
\(95\) −1.48458 −0.152314
\(96\) 0 0
\(97\) 7.23091 0.734188 0.367094 0.930184i \(-0.380353\pi\)
0.367094 + 0.930184i \(0.380353\pi\)
\(98\) 0 0
\(99\) 11.8413 1.19009
\(100\) 0 0
\(101\) −15.8263 −1.57478 −0.787388 0.616458i \(-0.788567\pi\)
−0.787388 + 0.616458i \(0.788567\pi\)
\(102\) 0 0
\(103\) 10.9089 1.07489 0.537445 0.843299i \(-0.319390\pi\)
0.537445 + 0.843299i \(0.319390\pi\)
\(104\) 0 0
\(105\) 3.23274 0.315483
\(106\) 0 0
\(107\) 16.5069 1.59579 0.797893 0.602799i \(-0.205948\pi\)
0.797893 + 0.602799i \(0.205948\pi\)
\(108\) 0 0
\(109\) 7.23324 0.692819 0.346410 0.938083i \(-0.387401\pi\)
0.346410 + 0.938083i \(0.387401\pi\)
\(110\) 0 0
\(111\) 2.79173 0.264979
\(112\) 0 0
\(113\) −8.09683 −0.761686 −0.380843 0.924640i \(-0.624366\pi\)
−0.380843 + 0.924640i \(0.624366\pi\)
\(114\) 0 0
\(115\) 10.6008 0.988528
\(116\) 0 0
\(117\) −12.6367 −1.16826
\(118\) 0 0
\(119\) 11.1649 1.02348
\(120\) 0 0
\(121\) 14.3926 1.30842
\(122\) 0 0
\(123\) 7.91256 0.713451
\(124\) 0 0
\(125\) −11.5738 −1.03519
\(126\) 0 0
\(127\) 4.52754 0.401754 0.200877 0.979616i \(-0.435621\pi\)
0.200877 + 0.979616i \(0.435621\pi\)
\(128\) 0 0
\(129\) −7.15561 −0.630016
\(130\) 0 0
\(131\) −14.2696 −1.24674 −0.623371 0.781926i \(-0.714237\pi\)
−0.623371 + 0.781926i \(0.714237\pi\)
\(132\) 0 0
\(133\) −2.70067 −0.234177
\(134\) 0 0
\(135\) −6.40389 −0.551159
\(136\) 0 0
\(137\) 0.202571 0.0173068 0.00865342 0.999963i \(-0.497245\pi\)
0.00865342 + 0.999963i \(0.497245\pi\)
\(138\) 0 0
\(139\) −3.15719 −0.267789 −0.133895 0.990996i \(-0.542748\pi\)
−0.133895 + 0.990996i \(0.542748\pi\)
\(140\) 0 0
\(141\) −10.2955 −0.867041
\(142\) 0 0
\(143\) −27.0983 −2.26607
\(144\) 0 0
\(145\) 8.93455 0.741974
\(146\) 0 0
\(147\) 0.236730 0.0195251
\(148\) 0 0
\(149\) 17.3866 1.42437 0.712184 0.701993i \(-0.247706\pi\)
0.712184 + 0.701993i \(0.247706\pi\)
\(150\) 0 0
\(151\) 3.92205 0.319172 0.159586 0.987184i \(-0.448984\pi\)
0.159586 + 0.987184i \(0.448984\pi\)
\(152\) 0 0
\(153\) −9.71468 −0.785386
\(154\) 0 0
\(155\) 8.55353 0.687036
\(156\) 0 0
\(157\) 5.83804 0.465926 0.232963 0.972486i \(-0.425158\pi\)
0.232963 + 0.972486i \(0.425158\pi\)
\(158\) 0 0
\(159\) −0.727086 −0.0576617
\(160\) 0 0
\(161\) 19.2844 1.51982
\(162\) 0 0
\(163\) −0.771769 −0.0604496 −0.0302248 0.999543i \(-0.509622\pi\)
−0.0302248 + 0.999543i \(0.509622\pi\)
\(164\) 0 0
\(165\) −6.03189 −0.469582
\(166\) 0 0
\(167\) 17.6899 1.36888 0.684442 0.729067i \(-0.260046\pi\)
0.684442 + 0.729067i \(0.260046\pi\)
\(168\) 0 0
\(169\) 15.9185 1.22450
\(170\) 0 0
\(171\) 2.34988 0.179700
\(172\) 0 0
\(173\) 19.3702 1.47269 0.736345 0.676606i \(-0.236550\pi\)
0.736345 + 0.676606i \(0.236550\pi\)
\(174\) 0 0
\(175\) −7.55115 −0.570813
\(176\) 0 0
\(177\) −1.81235 −0.136225
\(178\) 0 0
\(179\) 0.802564 0.0599864 0.0299932 0.999550i \(-0.490451\pi\)
0.0299932 + 0.999550i \(0.490451\pi\)
\(180\) 0 0
\(181\) −6.95188 −0.516729 −0.258364 0.966048i \(-0.583184\pi\)
−0.258364 + 0.966048i \(0.583184\pi\)
\(182\) 0 0
\(183\) −1.96519 −0.145271
\(184\) 0 0
\(185\) 5.14018 0.377914
\(186\) 0 0
\(187\) −20.8323 −1.52341
\(188\) 0 0
\(189\) −11.6496 −0.847386
\(190\) 0 0
\(191\) 12.9890 0.939854 0.469927 0.882705i \(-0.344280\pi\)
0.469927 + 0.882705i \(0.344280\pi\)
\(192\) 0 0
\(193\) 0.0398276 0.00286686 0.00143343 0.999999i \(-0.499544\pi\)
0.00143343 + 0.999999i \(0.499544\pi\)
\(194\) 0 0
\(195\) 6.43706 0.460968
\(196\) 0 0
\(197\) 5.26601 0.375188 0.187594 0.982247i \(-0.439931\pi\)
0.187594 + 0.982247i \(0.439931\pi\)
\(198\) 0 0
\(199\) −15.5497 −1.10229 −0.551146 0.834409i \(-0.685809\pi\)
−0.551146 + 0.834409i \(0.685809\pi\)
\(200\) 0 0
\(201\) 1.34062 0.0945604
\(202\) 0 0
\(203\) 16.2533 1.14076
\(204\) 0 0
\(205\) 14.5687 1.01753
\(206\) 0 0
\(207\) −16.7796 −1.16626
\(208\) 0 0
\(209\) 5.03911 0.348562
\(210\) 0 0
\(211\) −17.8583 −1.22942 −0.614709 0.788754i \(-0.710727\pi\)
−0.614709 + 0.788754i \(0.710727\pi\)
\(212\) 0 0
\(213\) 0.0431531 0.00295680
\(214\) 0 0
\(215\) −13.1750 −0.898530
\(216\) 0 0
\(217\) 15.5602 1.05629
\(218\) 0 0
\(219\) 1.45344 0.0982141
\(220\) 0 0
\(221\) 22.2316 1.49546
\(222\) 0 0
\(223\) −4.86018 −0.325462 −0.162731 0.986671i \(-0.552030\pi\)
−0.162731 + 0.986671i \(0.552030\pi\)
\(224\) 0 0
\(225\) 6.57034 0.438022
\(226\) 0 0
\(227\) 17.9183 1.18928 0.594640 0.803992i \(-0.297294\pi\)
0.594640 + 0.803992i \(0.297294\pi\)
\(228\) 0 0
\(229\) −17.4287 −1.15172 −0.575860 0.817548i \(-0.695333\pi\)
−0.575860 + 0.817548i \(0.695333\pi\)
\(230\) 0 0
\(231\) −10.9729 −0.721965
\(232\) 0 0
\(233\) 7.49941 0.491303 0.245651 0.969358i \(-0.420998\pi\)
0.245651 + 0.969358i \(0.420998\pi\)
\(234\) 0 0
\(235\) −18.9563 −1.23657
\(236\) 0 0
\(237\) −0.806301 −0.0523749
\(238\) 0 0
\(239\) −21.9505 −1.41986 −0.709931 0.704271i \(-0.751274\pi\)
−0.709931 + 0.704271i \(0.751274\pi\)
\(240\) 0 0
\(241\) −0.728320 −0.0469152 −0.0234576 0.999725i \(-0.507467\pi\)
−0.0234576 + 0.999725i \(0.507467\pi\)
\(242\) 0 0
\(243\) 15.8206 1.01489
\(244\) 0 0
\(245\) 0.435871 0.0278468
\(246\) 0 0
\(247\) −5.37759 −0.342168
\(248\) 0 0
\(249\) 3.83234 0.242865
\(250\) 0 0
\(251\) 6.93064 0.437458 0.218729 0.975786i \(-0.429809\pi\)
0.218729 + 0.975786i \(0.429809\pi\)
\(252\) 0 0
\(253\) −35.9823 −2.26219
\(254\) 0 0
\(255\) 4.94861 0.309894
\(256\) 0 0
\(257\) 18.7320 1.16847 0.584236 0.811583i \(-0.301394\pi\)
0.584236 + 0.811583i \(0.301394\pi\)
\(258\) 0 0
\(259\) 9.35076 0.581028
\(260\) 0 0
\(261\) −14.1421 −0.875377
\(262\) 0 0
\(263\) 22.5991 1.39352 0.696762 0.717303i \(-0.254624\pi\)
0.696762 + 0.717303i \(0.254624\pi\)
\(264\) 0 0
\(265\) −1.33872 −0.0822371
\(266\) 0 0
\(267\) 13.3417 0.816499
\(268\) 0 0
\(269\) −10.2505 −0.624986 −0.312493 0.949920i \(-0.601164\pi\)
−0.312493 + 0.949920i \(0.601164\pi\)
\(270\) 0 0
\(271\) −20.6351 −1.25350 −0.626748 0.779222i \(-0.715614\pi\)
−0.626748 + 0.779222i \(0.715614\pi\)
\(272\) 0 0
\(273\) 11.7100 0.708720
\(274\) 0 0
\(275\) 14.0895 0.849629
\(276\) 0 0
\(277\) 17.9656 1.07945 0.539724 0.841842i \(-0.318529\pi\)
0.539724 + 0.841842i \(0.318529\pi\)
\(278\) 0 0
\(279\) −13.5391 −0.810562
\(280\) 0 0
\(281\) −8.00345 −0.477446 −0.238723 0.971088i \(-0.576729\pi\)
−0.238723 + 0.971088i \(0.576729\pi\)
\(282\) 0 0
\(283\) 24.3963 1.45021 0.725105 0.688638i \(-0.241791\pi\)
0.725105 + 0.688638i \(0.241791\pi\)
\(284\) 0 0
\(285\) −1.19702 −0.0709051
\(286\) 0 0
\(287\) 26.5027 1.56441
\(288\) 0 0
\(289\) 0.0909650 0.00535088
\(290\) 0 0
\(291\) 5.83029 0.341778
\(292\) 0 0
\(293\) −30.5340 −1.78382 −0.891908 0.452217i \(-0.850633\pi\)
−0.891908 + 0.452217i \(0.850633\pi\)
\(294\) 0 0
\(295\) −3.33693 −0.194284
\(296\) 0 0
\(297\) 21.7368 1.26129
\(298\) 0 0
\(299\) 38.3993 2.22069
\(300\) 0 0
\(301\) −23.9674 −1.38146
\(302\) 0 0
\(303\) −12.7608 −0.733087
\(304\) 0 0
\(305\) −3.61835 −0.207186
\(306\) 0 0
\(307\) −27.7976 −1.58649 −0.793247 0.608900i \(-0.791611\pi\)
−0.793247 + 0.608900i \(0.791611\pi\)
\(308\) 0 0
\(309\) 8.79589 0.500381
\(310\) 0 0
\(311\) 12.5560 0.711985 0.355992 0.934489i \(-0.384143\pi\)
0.355992 + 0.934489i \(0.384143\pi\)
\(312\) 0 0
\(313\) −4.63902 −0.262213 −0.131106 0.991368i \(-0.541853\pi\)
−0.131106 + 0.991368i \(0.541853\pi\)
\(314\) 0 0
\(315\) −9.42148 −0.530840
\(316\) 0 0
\(317\) −12.6434 −0.710123 −0.355061 0.934843i \(-0.615540\pi\)
−0.355061 + 0.934843i \(0.615540\pi\)
\(318\) 0 0
\(319\) −30.3266 −1.69796
\(320\) 0 0
\(321\) 13.3096 0.742867
\(322\) 0 0
\(323\) −4.13412 −0.230029
\(324\) 0 0
\(325\) −15.0359 −0.834043
\(326\) 0 0
\(327\) 5.83217 0.322520
\(328\) 0 0
\(329\) −34.4844 −1.90119
\(330\) 0 0
\(331\) 0.893865 0.0491313 0.0245656 0.999698i \(-0.492180\pi\)
0.0245656 + 0.999698i \(0.492180\pi\)
\(332\) 0 0
\(333\) −8.13619 −0.445861
\(334\) 0 0
\(335\) 2.46838 0.134862
\(336\) 0 0
\(337\) 0.698634 0.0380570 0.0190285 0.999819i \(-0.493943\pi\)
0.0190285 + 0.999819i \(0.493943\pi\)
\(338\) 0 0
\(339\) −6.52848 −0.354579
\(340\) 0 0
\(341\) −29.0333 −1.57224
\(342\) 0 0
\(343\) −18.1118 −0.977943
\(344\) 0 0
\(345\) 8.54743 0.460178
\(346\) 0 0
\(347\) −22.3726 −1.20102 −0.600511 0.799616i \(-0.705036\pi\)
−0.600511 + 0.799616i \(0.705036\pi\)
\(348\) 0 0
\(349\) 16.3093 0.873017 0.436509 0.899700i \(-0.356215\pi\)
0.436509 + 0.899700i \(0.356215\pi\)
\(350\) 0 0
\(351\) −23.1969 −1.23816
\(352\) 0 0
\(353\) 26.6027 1.41592 0.707961 0.706252i \(-0.249615\pi\)
0.707961 + 0.706252i \(0.249615\pi\)
\(354\) 0 0
\(355\) 0.0794542 0.00421699
\(356\) 0 0
\(357\) 9.00226 0.476450
\(358\) 0 0
\(359\) 5.05161 0.266614 0.133307 0.991075i \(-0.457440\pi\)
0.133307 + 0.991075i \(0.457440\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 11.6048 0.609093
\(364\) 0 0
\(365\) 2.67609 0.140073
\(366\) 0 0
\(367\) 17.4420 0.910466 0.455233 0.890372i \(-0.349556\pi\)
0.455233 + 0.890372i \(0.349556\pi\)
\(368\) 0 0
\(369\) −23.0603 −1.20047
\(370\) 0 0
\(371\) −2.43534 −0.126436
\(372\) 0 0
\(373\) −13.6958 −0.709142 −0.354571 0.935029i \(-0.615373\pi\)
−0.354571 + 0.935029i \(0.615373\pi\)
\(374\) 0 0
\(375\) −9.33198 −0.481901
\(376\) 0 0
\(377\) 32.3637 1.66681
\(378\) 0 0
\(379\) 7.51884 0.386217 0.193108 0.981177i \(-0.438143\pi\)
0.193108 + 0.981177i \(0.438143\pi\)
\(380\) 0 0
\(381\) 3.65056 0.187024
\(382\) 0 0
\(383\) −6.00434 −0.306807 −0.153404 0.988164i \(-0.549023\pi\)
−0.153404 + 0.988164i \(0.549023\pi\)
\(384\) 0 0
\(385\) −20.2035 −1.02967
\(386\) 0 0
\(387\) 20.8543 1.06008
\(388\) 0 0
\(389\) −8.38695 −0.425235 −0.212618 0.977135i \(-0.568199\pi\)
−0.212618 + 0.977135i \(0.568199\pi\)
\(390\) 0 0
\(391\) 29.5202 1.49290
\(392\) 0 0
\(393\) −11.5056 −0.580381
\(394\) 0 0
\(395\) −1.48458 −0.0746972
\(396\) 0 0
\(397\) −10.4410 −0.524019 −0.262009 0.965065i \(-0.584385\pi\)
−0.262009 + 0.965065i \(0.584385\pi\)
\(398\) 0 0
\(399\) −2.17755 −0.109014
\(400\) 0 0
\(401\) 11.3891 0.568746 0.284373 0.958714i \(-0.408215\pi\)
0.284373 + 0.958714i \(0.408215\pi\)
\(402\) 0 0
\(403\) 30.9835 1.54340
\(404\) 0 0
\(405\) 5.30226 0.263471
\(406\) 0 0
\(407\) −17.4473 −0.864833
\(408\) 0 0
\(409\) −13.6248 −0.673704 −0.336852 0.941558i \(-0.609362\pi\)
−0.336852 + 0.941558i \(0.609362\pi\)
\(410\) 0 0
\(411\) 0.163334 0.00805665
\(412\) 0 0
\(413\) −6.07038 −0.298704
\(414\) 0 0
\(415\) 7.05618 0.346374
\(416\) 0 0
\(417\) −2.54564 −0.124661
\(418\) 0 0
\(419\) 5.77773 0.282260 0.141130 0.989991i \(-0.454926\pi\)
0.141130 + 0.989991i \(0.454926\pi\)
\(420\) 0 0
\(421\) −27.9535 −1.36237 −0.681186 0.732110i \(-0.738535\pi\)
−0.681186 + 0.732110i \(0.738535\pi\)
\(422\) 0 0
\(423\) 30.0052 1.45890
\(424\) 0 0
\(425\) −11.5591 −0.560701
\(426\) 0 0
\(427\) −6.58231 −0.318540
\(428\) 0 0
\(429\) −21.8494 −1.05490
\(430\) 0 0
\(431\) −20.7140 −0.997759 −0.498880 0.866671i \(-0.666255\pi\)
−0.498880 + 0.866671i \(0.666255\pi\)
\(432\) 0 0
\(433\) 5.33174 0.256227 0.128113 0.991760i \(-0.459108\pi\)
0.128113 + 0.991760i \(0.459108\pi\)
\(434\) 0 0
\(435\) 7.20394 0.345402
\(436\) 0 0
\(437\) −7.14061 −0.341582
\(438\) 0 0
\(439\) 39.4985 1.88516 0.942581 0.333978i \(-0.108391\pi\)
0.942581 + 0.333978i \(0.108391\pi\)
\(440\) 0 0
\(441\) −0.689923 −0.0328535
\(442\) 0 0
\(443\) 7.29356 0.346527 0.173264 0.984875i \(-0.444569\pi\)
0.173264 + 0.984875i \(0.444569\pi\)
\(444\) 0 0
\(445\) 24.5650 1.16449
\(446\) 0 0
\(447\) 14.0189 0.663069
\(448\) 0 0
\(449\) 7.19330 0.339473 0.169736 0.985489i \(-0.445708\pi\)
0.169736 + 0.985489i \(0.445708\pi\)
\(450\) 0 0
\(451\) −49.4508 −2.32855
\(452\) 0 0
\(453\) 3.16236 0.148580
\(454\) 0 0
\(455\) 21.5606 1.01078
\(456\) 0 0
\(457\) −9.34387 −0.437088 −0.218544 0.975827i \(-0.570131\pi\)
−0.218544 + 0.975827i \(0.570131\pi\)
\(458\) 0 0
\(459\) −17.8330 −0.832373
\(460\) 0 0
\(461\) 26.4003 1.22958 0.614791 0.788690i \(-0.289240\pi\)
0.614791 + 0.788690i \(0.289240\pi\)
\(462\) 0 0
\(463\) −24.8286 −1.15389 −0.576943 0.816785i \(-0.695754\pi\)
−0.576943 + 0.816785i \(0.695754\pi\)
\(464\) 0 0
\(465\) 6.89673 0.319828
\(466\) 0 0
\(467\) 24.0703 1.11384 0.556921 0.830566i \(-0.311983\pi\)
0.556921 + 0.830566i \(0.311983\pi\)
\(468\) 0 0
\(469\) 4.49036 0.207345
\(470\) 0 0
\(471\) 4.70722 0.216897
\(472\) 0 0
\(473\) 44.7201 2.05623
\(474\) 0 0
\(475\) 2.79603 0.128291
\(476\) 0 0
\(477\) 2.11901 0.0970229
\(478\) 0 0
\(479\) −12.4110 −0.567073 −0.283536 0.958961i \(-0.591508\pi\)
−0.283536 + 0.958961i \(0.591508\pi\)
\(480\) 0 0
\(481\) 18.6193 0.848968
\(482\) 0 0
\(483\) 15.5490 0.707506
\(484\) 0 0
\(485\) 10.7348 0.487444
\(486\) 0 0
\(487\) 38.9883 1.76673 0.883365 0.468686i \(-0.155273\pi\)
0.883365 + 0.468686i \(0.155273\pi\)
\(488\) 0 0
\(489\) −0.622278 −0.0281404
\(490\) 0 0
\(491\) −8.97418 −0.404999 −0.202499 0.979282i \(-0.564906\pi\)
−0.202499 + 0.979282i \(0.564906\pi\)
\(492\) 0 0
\(493\) 24.8802 1.12055
\(494\) 0 0
\(495\) 17.5793 0.790131
\(496\) 0 0
\(497\) 0.144539 0.00648346
\(498\) 0 0
\(499\) −13.3058 −0.595651 −0.297826 0.954620i \(-0.596261\pi\)
−0.297826 + 0.954620i \(0.596261\pi\)
\(500\) 0 0
\(501\) 14.2634 0.637241
\(502\) 0 0
\(503\) −40.0571 −1.78606 −0.893028 0.450001i \(-0.851424\pi\)
−0.893028 + 0.450001i \(0.851424\pi\)
\(504\) 0 0
\(505\) −23.4954 −1.04553
\(506\) 0 0
\(507\) 12.8351 0.570027
\(508\) 0 0
\(509\) −11.9829 −0.531134 −0.265567 0.964092i \(-0.585559\pi\)
−0.265567 + 0.964092i \(0.585559\pi\)
\(510\) 0 0
\(511\) 4.86821 0.215357
\(512\) 0 0
\(513\) 4.31361 0.190451
\(514\) 0 0
\(515\) 16.1952 0.713644
\(516\) 0 0
\(517\) 64.3436 2.82983
\(518\) 0 0
\(519\) 15.6182 0.685564
\(520\) 0 0
\(521\) 13.3214 0.583622 0.291811 0.956476i \(-0.405742\pi\)
0.291811 + 0.956476i \(0.405742\pi\)
\(522\) 0 0
\(523\) −19.0507 −0.833027 −0.416513 0.909130i \(-0.636748\pi\)
−0.416513 + 0.909130i \(0.636748\pi\)
\(524\) 0 0
\(525\) −6.08850 −0.265724
\(526\) 0 0
\(527\) 23.8192 1.03758
\(528\) 0 0
\(529\) 27.9884 1.21688
\(530\) 0 0
\(531\) 5.28190 0.229215
\(532\) 0 0
\(533\) 52.7725 2.28583
\(534\) 0 0
\(535\) 24.5058 1.05948
\(536\) 0 0
\(537\) 0.647108 0.0279248
\(538\) 0 0
\(539\) −1.47948 −0.0637257
\(540\) 0 0
\(541\) 9.78993 0.420902 0.210451 0.977604i \(-0.432507\pi\)
0.210451 + 0.977604i \(0.432507\pi\)
\(542\) 0 0
\(543\) −5.60531 −0.240547
\(544\) 0 0
\(545\) 10.7383 0.459978
\(546\) 0 0
\(547\) −35.2331 −1.50646 −0.753230 0.657757i \(-0.771505\pi\)
−0.753230 + 0.657757i \(0.771505\pi\)
\(548\) 0 0
\(549\) 5.72734 0.244437
\(550\) 0 0
\(551\) −6.01825 −0.256386
\(552\) 0 0
\(553\) −2.70067 −0.114844
\(554\) 0 0
\(555\) 4.14453 0.175926
\(556\) 0 0
\(557\) −35.3016 −1.49578 −0.747889 0.663824i \(-0.768933\pi\)
−0.747889 + 0.663824i \(0.768933\pi\)
\(558\) 0 0
\(559\) −47.7241 −2.01851
\(560\) 0 0
\(561\) −16.7971 −0.709174
\(562\) 0 0
\(563\) −20.7994 −0.876589 −0.438294 0.898831i \(-0.644417\pi\)
−0.438294 + 0.898831i \(0.644417\pi\)
\(564\) 0 0
\(565\) −12.0204 −0.505701
\(566\) 0 0
\(567\) 9.64560 0.405077
\(568\) 0 0
\(569\) −28.4346 −1.19204 −0.596020 0.802970i \(-0.703252\pi\)
−0.596020 + 0.802970i \(0.703252\pi\)
\(570\) 0 0
\(571\) 18.1303 0.758728 0.379364 0.925247i \(-0.376143\pi\)
0.379364 + 0.925247i \(0.376143\pi\)
\(572\) 0 0
\(573\) 10.4731 0.437519
\(574\) 0 0
\(575\) −19.9654 −0.832614
\(576\) 0 0
\(577\) −26.4651 −1.10176 −0.550879 0.834585i \(-0.685707\pi\)
−0.550879 + 0.834585i \(0.685707\pi\)
\(578\) 0 0
\(579\) 0.0321131 0.00133457
\(580\) 0 0
\(581\) 12.8362 0.532537
\(582\) 0 0
\(583\) 4.54404 0.188195
\(584\) 0 0
\(585\) −18.7601 −0.775636
\(586\) 0 0
\(587\) 44.8325 1.85043 0.925217 0.379439i \(-0.123883\pi\)
0.925217 + 0.379439i \(0.123883\pi\)
\(588\) 0 0
\(589\) −5.76160 −0.237403
\(590\) 0 0
\(591\) 4.24599 0.174657
\(592\) 0 0
\(593\) 32.3793 1.32966 0.664829 0.746996i \(-0.268505\pi\)
0.664829 + 0.746996i \(0.268505\pi\)
\(594\) 0 0
\(595\) 16.5751 0.679514
\(596\) 0 0
\(597\) −12.5378 −0.513137
\(598\) 0 0
\(599\) 16.3023 0.666092 0.333046 0.942911i \(-0.391924\pi\)
0.333046 + 0.942911i \(0.391924\pi\)
\(600\) 0 0
\(601\) 9.18880 0.374819 0.187410 0.982282i \(-0.439991\pi\)
0.187410 + 0.982282i \(0.439991\pi\)
\(602\) 0 0
\(603\) −3.90711 −0.159110
\(604\) 0 0
\(605\) 21.3669 0.868689
\(606\) 0 0
\(607\) −20.9964 −0.852219 −0.426110 0.904672i \(-0.640116\pi\)
−0.426110 + 0.904672i \(0.640116\pi\)
\(608\) 0 0
\(609\) 13.1050 0.531043
\(610\) 0 0
\(611\) −68.6657 −2.77792
\(612\) 0 0
\(613\) 3.21357 0.129795 0.0648974 0.997892i \(-0.479328\pi\)
0.0648974 + 0.997892i \(0.479328\pi\)
\(614\) 0 0
\(615\) 11.7468 0.473677
\(616\) 0 0
\(617\) −30.6397 −1.23351 −0.616753 0.787157i \(-0.711552\pi\)
−0.616753 + 0.787157i \(0.711552\pi\)
\(618\) 0 0
\(619\) −2.35627 −0.0947064 −0.0473532 0.998878i \(-0.515079\pi\)
−0.0473532 + 0.998878i \(0.515079\pi\)
\(620\) 0 0
\(621\) −30.8018 −1.23604
\(622\) 0 0
\(623\) 44.6874 1.79036
\(624\) 0 0
\(625\) −3.20204 −0.128082
\(626\) 0 0
\(627\) 4.06304 0.162262
\(628\) 0 0
\(629\) 14.3139 0.570734
\(630\) 0 0
\(631\) −23.9916 −0.955089 −0.477545 0.878607i \(-0.658473\pi\)
−0.477545 + 0.878607i \(0.658473\pi\)
\(632\) 0 0
\(633\) −14.3992 −0.572317
\(634\) 0 0
\(635\) 6.72148 0.266734
\(636\) 0 0
\(637\) 1.57886 0.0625567
\(638\) 0 0
\(639\) −0.125765 −0.00497518
\(640\) 0 0
\(641\) −23.0351 −0.909833 −0.454916 0.890534i \(-0.650331\pi\)
−0.454916 + 0.890534i \(0.650331\pi\)
\(642\) 0 0
\(643\) −26.3109 −1.03760 −0.518801 0.854895i \(-0.673622\pi\)
−0.518801 + 0.854895i \(0.673622\pi\)
\(644\) 0 0
\(645\) −10.6231 −0.418282
\(646\) 0 0
\(647\) 5.51126 0.216670 0.108335 0.994114i \(-0.465448\pi\)
0.108335 + 0.994114i \(0.465448\pi\)
\(648\) 0 0
\(649\) 11.3266 0.444607
\(650\) 0 0
\(651\) 12.5462 0.491723
\(652\) 0 0
\(653\) −5.82044 −0.227772 −0.113886 0.993494i \(-0.536330\pi\)
−0.113886 + 0.993494i \(0.536330\pi\)
\(654\) 0 0
\(655\) −21.1843 −0.827741
\(656\) 0 0
\(657\) −4.23588 −0.165258
\(658\) 0 0
\(659\) 27.0908 1.05531 0.527654 0.849460i \(-0.323072\pi\)
0.527654 + 0.849460i \(0.323072\pi\)
\(660\) 0 0
\(661\) 11.9044 0.463029 0.231514 0.972831i \(-0.425632\pi\)
0.231514 + 0.972831i \(0.425632\pi\)
\(662\) 0 0
\(663\) 17.9254 0.696164
\(664\) 0 0
\(665\) −4.00935 −0.155476
\(666\) 0 0
\(667\) 42.9740 1.66396
\(668\) 0 0
\(669\) −3.91877 −0.151508
\(670\) 0 0
\(671\) 12.2818 0.474133
\(672\) 0 0
\(673\) 3.68644 0.142102 0.0710510 0.997473i \(-0.477365\pi\)
0.0710510 + 0.997473i \(0.477365\pi\)
\(674\) 0 0
\(675\) 12.0610 0.464228
\(676\) 0 0
\(677\) 45.7172 1.75705 0.878527 0.477693i \(-0.158527\pi\)
0.878527 + 0.477693i \(0.158527\pi\)
\(678\) 0 0
\(679\) 19.5283 0.749427
\(680\) 0 0
\(681\) 14.4476 0.553632
\(682\) 0 0
\(683\) 28.5355 1.09188 0.545939 0.837825i \(-0.316173\pi\)
0.545939 + 0.837825i \(0.316173\pi\)
\(684\) 0 0
\(685\) 0.300733 0.0114904
\(686\) 0 0
\(687\) −14.0528 −0.536147
\(688\) 0 0
\(689\) −4.84927 −0.184742
\(690\) 0 0
\(691\) −42.2935 −1.60892 −0.804460 0.594007i \(-0.797545\pi\)
−0.804460 + 0.594007i \(0.797545\pi\)
\(692\) 0 0
\(693\) 31.9794 1.21480
\(694\) 0 0
\(695\) −4.68709 −0.177791
\(696\) 0 0
\(697\) 40.5698 1.53669
\(698\) 0 0
\(699\) 6.04678 0.228710
\(700\) 0 0
\(701\) −38.2244 −1.44371 −0.721857 0.692042i \(-0.756711\pi\)
−0.721857 + 0.692042i \(0.756711\pi\)
\(702\) 0 0
\(703\) −3.46239 −0.130586
\(704\) 0 0
\(705\) −15.2845 −0.575648
\(706\) 0 0
\(707\) −42.7416 −1.60746
\(708\) 0 0
\(709\) −29.1321 −1.09408 −0.547039 0.837107i \(-0.684245\pi\)
−0.547039 + 0.837107i \(0.684245\pi\)
\(710\) 0 0
\(711\) 2.34988 0.0881273
\(712\) 0 0
\(713\) 41.1413 1.54076
\(714\) 0 0
\(715\) −40.2295 −1.50450
\(716\) 0 0
\(717\) −17.6987 −0.660972
\(718\) 0 0
\(719\) −7.13411 −0.266057 −0.133029 0.991112i \(-0.542470\pi\)
−0.133029 + 0.991112i \(0.542470\pi\)
\(720\) 0 0
\(721\) 29.4614 1.09720
\(722\) 0 0
\(723\) −0.587246 −0.0218399
\(724\) 0 0
\(725\) −16.8272 −0.624947
\(726\) 0 0
\(727\) −36.8284 −1.36589 −0.682945 0.730470i \(-0.739301\pi\)
−0.682945 + 0.730470i \(0.739301\pi\)
\(728\) 0 0
\(729\) 2.04148 0.0756104
\(730\) 0 0
\(731\) −36.6887 −1.35698
\(732\) 0 0
\(733\) 46.2173 1.70708 0.853538 0.521031i \(-0.174452\pi\)
0.853538 + 0.521031i \(0.174452\pi\)
\(734\) 0 0
\(735\) 0.351443 0.0129632
\(736\) 0 0
\(737\) −8.37845 −0.308624
\(738\) 0 0
\(739\) −45.8933 −1.68821 −0.844106 0.536176i \(-0.819868\pi\)
−0.844106 + 0.536176i \(0.819868\pi\)
\(740\) 0 0
\(741\) −4.33596 −0.159285
\(742\) 0 0
\(743\) 22.5619 0.827715 0.413858 0.910342i \(-0.364181\pi\)
0.413858 + 0.910342i \(0.364181\pi\)
\(744\) 0 0
\(745\) 25.8118 0.945670
\(746\) 0 0
\(747\) −11.1689 −0.408650
\(748\) 0 0
\(749\) 44.5797 1.62891
\(750\) 0 0
\(751\) 29.1881 1.06509 0.532544 0.846402i \(-0.321236\pi\)
0.532544 + 0.846402i \(0.321236\pi\)
\(752\) 0 0
\(753\) 5.58819 0.203645
\(754\) 0 0
\(755\) 5.82259 0.211906
\(756\) 0 0
\(757\) 24.1022 0.876010 0.438005 0.898973i \(-0.355685\pi\)
0.438005 + 0.898973i \(0.355685\pi\)
\(758\) 0 0
\(759\) −29.0126 −1.05309
\(760\) 0 0
\(761\) −25.6661 −0.930395 −0.465198 0.885207i \(-0.654017\pi\)
−0.465198 + 0.885207i \(0.654017\pi\)
\(762\) 0 0
\(763\) 19.5346 0.707199
\(764\) 0 0
\(765\) −14.4222 −0.521435
\(766\) 0 0
\(767\) −12.0874 −0.436450
\(768\) 0 0
\(769\) −30.1872 −1.08858 −0.544288 0.838898i \(-0.683200\pi\)
−0.544288 + 0.838898i \(0.683200\pi\)
\(770\) 0 0
\(771\) 15.1037 0.543945
\(772\) 0 0
\(773\) 46.7509 1.68151 0.840756 0.541414i \(-0.182111\pi\)
0.840756 + 0.541414i \(0.182111\pi\)
\(774\) 0 0
\(775\) −16.1096 −0.578675
\(776\) 0 0
\(777\) 7.53953 0.270479
\(778\) 0 0
\(779\) −9.81340 −0.351602
\(780\) 0 0
\(781\) −0.269692 −0.00965033
\(782\) 0 0
\(783\) −25.9604 −0.927749
\(784\) 0 0
\(785\) 8.66701 0.309339
\(786\) 0 0
\(787\) −34.7504 −1.23872 −0.619360 0.785107i \(-0.712608\pi\)
−0.619360 + 0.785107i \(0.712608\pi\)
\(788\) 0 0
\(789\) 18.2217 0.648710
\(790\) 0 0
\(791\) −21.8668 −0.777495
\(792\) 0 0
\(793\) −13.1068 −0.465435
\(794\) 0 0
\(795\) −1.07941 −0.0382829
\(796\) 0 0
\(797\) 14.7235 0.521534 0.260767 0.965402i \(-0.416025\pi\)
0.260767 + 0.965402i \(0.416025\pi\)
\(798\) 0 0
\(799\) −52.7880 −1.86750
\(800\) 0 0
\(801\) −38.8830 −1.37386
\(802\) 0 0
\(803\) −9.08348 −0.320549
\(804\) 0 0
\(805\) 28.6292 1.00905
\(806\) 0 0
\(807\) −8.26502 −0.290943
\(808\) 0 0
\(809\) −36.5316 −1.28438 −0.642191 0.766545i \(-0.721974\pi\)
−0.642191 + 0.766545i \(0.721974\pi\)
\(810\) 0 0
\(811\) −4.46592 −0.156820 −0.0784099 0.996921i \(-0.524984\pi\)
−0.0784099 + 0.996921i \(0.524984\pi\)
\(812\) 0 0
\(813\) −16.6381 −0.583525
\(814\) 0 0
\(815\) −1.14575 −0.0401339
\(816\) 0 0
\(817\) 8.87461 0.310483
\(818\) 0 0
\(819\) −34.1275 −1.19251
\(820\) 0 0
\(821\) −26.9756 −0.941453 −0.470727 0.882279i \(-0.656008\pi\)
−0.470727 + 0.882279i \(0.656008\pi\)
\(822\) 0 0
\(823\) −0.716195 −0.0249650 −0.0124825 0.999922i \(-0.503973\pi\)
−0.0124825 + 0.999922i \(0.503973\pi\)
\(824\) 0 0
\(825\) 11.3604 0.395518
\(826\) 0 0
\(827\) −4.21881 −0.146703 −0.0733513 0.997306i \(-0.523369\pi\)
−0.0733513 + 0.997306i \(0.523369\pi\)
\(828\) 0 0
\(829\) −10.3513 −0.359517 −0.179759 0.983711i \(-0.557532\pi\)
−0.179759 + 0.983711i \(0.557532\pi\)
\(830\) 0 0
\(831\) 14.4857 0.502503
\(832\) 0 0
\(833\) 1.21378 0.0420549
\(834\) 0 0
\(835\) 26.2620 0.908834
\(836\) 0 0
\(837\) −24.8533 −0.859056
\(838\) 0 0
\(839\) −43.1440 −1.48950 −0.744748 0.667345i \(-0.767430\pi\)
−0.744748 + 0.667345i \(0.767430\pi\)
\(840\) 0 0
\(841\) 7.21929 0.248941
\(842\) 0 0
\(843\) −6.45320 −0.222260
\(844\) 0 0
\(845\) 23.6322 0.812974
\(846\) 0 0
\(847\) 38.8696 1.33558
\(848\) 0 0
\(849\) 19.6708 0.675099
\(850\) 0 0
\(851\) 24.7236 0.847513
\(852\) 0 0
\(853\) −1.18391 −0.0405362 −0.0202681 0.999795i \(-0.506452\pi\)
−0.0202681 + 0.999795i \(0.506452\pi\)
\(854\) 0 0
\(855\) 3.48857 0.119307
\(856\) 0 0
\(857\) 7.08937 0.242168 0.121084 0.992642i \(-0.461363\pi\)
0.121084 + 0.992642i \(0.461363\pi\)
\(858\) 0 0
\(859\) −44.6682 −1.52406 −0.762030 0.647542i \(-0.775797\pi\)
−0.762030 + 0.647542i \(0.775797\pi\)
\(860\) 0 0
\(861\) 21.3692 0.728260
\(862\) 0 0
\(863\) −43.5011 −1.48079 −0.740397 0.672170i \(-0.765362\pi\)
−0.740397 + 0.672170i \(0.765362\pi\)
\(864\) 0 0
\(865\) 28.7566 0.977752
\(866\) 0 0
\(867\) 0.0733452 0.00249093
\(868\) 0 0
\(869\) 5.03911 0.170940
\(870\) 0 0
\(871\) 8.94124 0.302962
\(872\) 0 0
\(873\) −16.9918 −0.575084
\(874\) 0 0
\(875\) −31.2570 −1.05668
\(876\) 0 0
\(877\) 32.9131 1.11140 0.555698 0.831384i \(-0.312451\pi\)
0.555698 + 0.831384i \(0.312451\pi\)
\(878\) 0 0
\(879\) −24.6196 −0.830399
\(880\) 0 0
\(881\) −22.2390 −0.749252 −0.374626 0.927176i \(-0.622229\pi\)
−0.374626 + 0.927176i \(0.622229\pi\)
\(882\) 0 0
\(883\) −12.0056 −0.404022 −0.202011 0.979383i \(-0.564748\pi\)
−0.202011 + 0.979383i \(0.564748\pi\)
\(884\) 0 0
\(885\) −2.69057 −0.0904426
\(886\) 0 0
\(887\) −35.3233 −1.18604 −0.593020 0.805188i \(-0.702065\pi\)
−0.593020 + 0.805188i \(0.702065\pi\)
\(888\) 0 0
\(889\) 12.2274 0.410093
\(890\) 0 0
\(891\) −17.9975 −0.602938
\(892\) 0 0
\(893\) 12.7688 0.427293
\(894\) 0 0
\(895\) 1.19147 0.0398264
\(896\) 0 0
\(897\) 30.9614 1.03377
\(898\) 0 0
\(899\) 34.6747 1.15647
\(900\) 0 0
\(901\) −3.72796 −0.124196
\(902\) 0 0
\(903\) −19.3249 −0.643093
\(904\) 0 0
\(905\) −10.3206 −0.343068
\(906\) 0 0
\(907\) 1.15934 0.0384954 0.0192477 0.999815i \(-0.493873\pi\)
0.0192477 + 0.999815i \(0.493873\pi\)
\(908\) 0 0
\(909\) 37.1899 1.23351
\(910\) 0 0
\(911\) −36.5361 −1.21050 −0.605248 0.796037i \(-0.706926\pi\)
−0.605248 + 0.796037i \(0.706926\pi\)
\(912\) 0 0
\(913\) −23.9508 −0.792657
\(914\) 0 0
\(915\) −2.91748 −0.0964488
\(916\) 0 0
\(917\) −38.5375 −1.27262
\(918\) 0 0
\(919\) −11.6007 −0.382672 −0.191336 0.981525i \(-0.561282\pi\)
−0.191336 + 0.981525i \(0.561282\pi\)
\(920\) 0 0
\(921\) −22.4133 −0.738542
\(922\) 0 0
\(923\) 0.287808 0.00947330
\(924\) 0 0
\(925\) −9.68095 −0.318308
\(926\) 0 0
\(927\) −25.6347 −0.841953
\(928\) 0 0
\(929\) 30.6131 1.00438 0.502192 0.864756i \(-0.332527\pi\)
0.502192 + 0.864756i \(0.332527\pi\)
\(930\) 0 0
\(931\) −0.293600 −0.00962234
\(932\) 0 0
\(933\) 10.1239 0.331442
\(934\) 0 0
\(935\) −30.9271 −1.01142
\(936\) 0 0
\(937\) 25.3848 0.829284 0.414642 0.909985i \(-0.363907\pi\)
0.414642 + 0.909985i \(0.363907\pi\)
\(938\) 0 0
\(939\) −3.74045 −0.122065
\(940\) 0 0
\(941\) −32.1381 −1.04767 −0.523835 0.851820i \(-0.675499\pi\)
−0.523835 + 0.851820i \(0.675499\pi\)
\(942\) 0 0
\(943\) 70.0737 2.28191
\(944\) 0 0
\(945\) −17.2948 −0.562599
\(946\) 0 0
\(947\) 37.3415 1.21344 0.606718 0.794917i \(-0.292486\pi\)
0.606718 + 0.794917i \(0.292486\pi\)
\(948\) 0 0
\(949\) 9.69364 0.314669
\(950\) 0 0
\(951\) −10.1944 −0.330575
\(952\) 0 0
\(953\) −18.4110 −0.596389 −0.298195 0.954505i \(-0.596384\pi\)
−0.298195 + 0.954505i \(0.596384\pi\)
\(954\) 0 0
\(955\) 19.2832 0.623990
\(956\) 0 0
\(957\) −24.4524 −0.790433
\(958\) 0 0
\(959\) 0.547078 0.0176661
\(960\) 0 0
\(961\) 2.19602 0.0708395
\(962\) 0 0
\(963\) −38.7893 −1.24997
\(964\) 0 0
\(965\) 0.0591272 0.00190337
\(966\) 0 0
\(967\) 38.8004 1.24774 0.623868 0.781530i \(-0.285560\pi\)
0.623868 + 0.781530i \(0.285560\pi\)
\(968\) 0 0
\(969\) −3.33335 −0.107083
\(970\) 0 0
\(971\) −18.8198 −0.603956 −0.301978 0.953315i \(-0.597647\pi\)
−0.301978 + 0.953315i \(0.597647\pi\)
\(972\) 0 0
\(973\) −8.52651 −0.273347
\(974\) 0 0
\(975\) −12.1235 −0.388262
\(976\) 0 0
\(977\) 42.3647 1.35537 0.677683 0.735354i \(-0.262984\pi\)
0.677683 + 0.735354i \(0.262984\pi\)
\(978\) 0 0
\(979\) −83.3811 −2.66487
\(980\) 0 0
\(981\) −16.9972 −0.542680
\(982\) 0 0
\(983\) 26.2497 0.837235 0.418617 0.908163i \(-0.362515\pi\)
0.418617 + 0.908163i \(0.362515\pi\)
\(984\) 0 0
\(985\) 7.81779 0.249096
\(986\) 0 0
\(987\) −27.8048 −0.885037
\(988\) 0 0
\(989\) −63.3702 −2.01505
\(990\) 0 0
\(991\) 48.8280 1.55107 0.775536 0.631304i \(-0.217480\pi\)
0.775536 + 0.631304i \(0.217480\pi\)
\(992\) 0 0
\(993\) 0.720724 0.0228715
\(994\) 0 0
\(995\) −23.0848 −0.731837
\(996\) 0 0
\(997\) 36.5535 1.15766 0.578831 0.815448i \(-0.303509\pi\)
0.578831 + 0.815448i \(0.303509\pi\)
\(998\) 0 0
\(999\) −14.9354 −0.472535
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 6004.2.a.h.1.20 31
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6004.2.a.h.1.20 31 1.1 even 1 trivial