Properties

Label 4024.2.a.g.1.10
Level $4024$
Weight $2$
Character 4024.1
Self dual yes
Analytic conductor $32.132$
Analytic rank $0$
Dimension $33$
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] = [4024,2,Mod(1,4024)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4024, 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("4024.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4024 = 2^{3} \cdot 503 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4024.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: \(32.1318017734\)
Analytic rank: \(0\)
Dimension: \(33\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Character \(\chi\) \(=\) 4024.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.02127 q^{3} +0.309055 q^{5} -3.45255 q^{7} -1.95700 q^{9} +O(q^{10})\) \(q-1.02127 q^{3} +0.309055 q^{5} -3.45255 q^{7} -1.95700 q^{9} +5.02189 q^{11} -5.26093 q^{13} -0.315630 q^{15} +5.38624 q^{17} -4.68014 q^{19} +3.52600 q^{21} -5.48438 q^{23} -4.90449 q^{25} +5.06246 q^{27} -3.54331 q^{29} +7.82064 q^{31} -5.12873 q^{33} -1.06703 q^{35} +2.44506 q^{37} +5.37285 q^{39} -1.26137 q^{41} -0.240872 q^{43} -0.604819 q^{45} -1.11939 q^{47} +4.92008 q^{49} -5.50083 q^{51} -6.70491 q^{53} +1.55204 q^{55} +4.77971 q^{57} -6.42730 q^{59} +4.92354 q^{61} +6.75663 q^{63} -1.62591 q^{65} +7.10771 q^{67} +5.60106 q^{69} -3.40052 q^{71} +2.77752 q^{73} +5.00883 q^{75} -17.3383 q^{77} +1.51015 q^{79} +0.700834 q^{81} +12.0362 q^{83} +1.66464 q^{85} +3.61870 q^{87} -4.39392 q^{89} +18.1636 q^{91} -7.98702 q^{93} -1.44642 q^{95} -9.39667 q^{97} -9.82783 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9} + 22 q^{11} - 17 q^{13} + 22 q^{15} + 9 q^{17} + 16 q^{19} + 6 q^{21} + 36 q^{23} + 47 q^{25} + 34 q^{27} + 13 q^{29} + 21 q^{31} + 14 q^{33} + 33 q^{35} - 55 q^{37} + 37 q^{39} + 42 q^{41} + 23 q^{43} + 5 q^{45} + 20 q^{47} + 55 q^{49} + 53 q^{51} - 32 q^{53} + 35 q^{55} + 21 q^{57} + 20 q^{59} - 15 q^{61} + 48 q^{63} + 34 q^{65} + 66 q^{67} - 4 q^{69} + 61 q^{71} + 19 q^{73} + 59 q^{75} + 2 q^{77} + 62 q^{79} + 77 q^{81} + 36 q^{83} - 14 q^{85} + 43 q^{87} + 34 q^{89} + 41 q^{91} - 11 q^{93} + 61 q^{95} - 8 q^{97} + 98 q^{99}+O(q^{100}) \) 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
\(3\) −1.02127 −0.589633 −0.294817 0.955554i \(-0.595259\pi\)
−0.294817 + 0.955554i \(0.595259\pi\)
\(4\) 0 0
\(5\) 0.309055 0.138213 0.0691067 0.997609i \(-0.477985\pi\)
0.0691067 + 0.997609i \(0.477985\pi\)
\(6\) 0 0
\(7\) −3.45255 −1.30494 −0.652470 0.757815i \(-0.726267\pi\)
−0.652470 + 0.757815i \(0.726267\pi\)
\(8\) 0 0
\(9\) −1.95700 −0.652333
\(10\) 0 0
\(11\) 5.02189 1.51416 0.757079 0.653324i \(-0.226626\pi\)
0.757079 + 0.653324i \(0.226626\pi\)
\(12\) 0 0
\(13\) −5.26093 −1.45912 −0.729560 0.683917i \(-0.760275\pi\)
−0.729560 + 0.683917i \(0.760275\pi\)
\(14\) 0 0
\(15\) −0.315630 −0.0814952
\(16\) 0 0
\(17\) 5.38624 1.30635 0.653177 0.757205i \(-0.273436\pi\)
0.653177 + 0.757205i \(0.273436\pi\)
\(18\) 0 0
\(19\) −4.68014 −1.07370 −0.536849 0.843678i \(-0.680386\pi\)
−0.536849 + 0.843678i \(0.680386\pi\)
\(20\) 0 0
\(21\) 3.52600 0.769436
\(22\) 0 0
\(23\) −5.48438 −1.14357 −0.571787 0.820402i \(-0.693749\pi\)
−0.571787 + 0.820402i \(0.693749\pi\)
\(24\) 0 0
\(25\) −4.90449 −0.980897
\(26\) 0 0
\(27\) 5.06246 0.974270
\(28\) 0 0
\(29\) −3.54331 −0.657977 −0.328988 0.944334i \(-0.606708\pi\)
−0.328988 + 0.944334i \(0.606708\pi\)
\(30\) 0 0
\(31\) 7.82064 1.40463 0.702314 0.711867i \(-0.252150\pi\)
0.702314 + 0.711867i \(0.252150\pi\)
\(32\) 0 0
\(33\) −5.12873 −0.892797
\(34\) 0 0
\(35\) −1.06703 −0.180360
\(36\) 0 0
\(37\) 2.44506 0.401965 0.200982 0.979595i \(-0.435587\pi\)
0.200982 + 0.979595i \(0.435587\pi\)
\(38\) 0 0
\(39\) 5.37285 0.860345
\(40\) 0 0
\(41\) −1.26137 −0.196992 −0.0984961 0.995137i \(-0.531403\pi\)
−0.0984961 + 0.995137i \(0.531403\pi\)
\(42\) 0 0
\(43\) −0.240872 −0.0367326 −0.0183663 0.999831i \(-0.505846\pi\)
−0.0183663 + 0.999831i \(0.505846\pi\)
\(44\) 0 0
\(45\) −0.604819 −0.0901611
\(46\) 0 0
\(47\) −1.11939 −0.163280 −0.0816402 0.996662i \(-0.526016\pi\)
−0.0816402 + 0.996662i \(0.526016\pi\)
\(48\) 0 0
\(49\) 4.92008 0.702868
\(50\) 0 0
\(51\) −5.50083 −0.770270
\(52\) 0 0
\(53\) −6.70491 −0.920990 −0.460495 0.887662i \(-0.652328\pi\)
−0.460495 + 0.887662i \(0.652328\pi\)
\(54\) 0 0
\(55\) 1.55204 0.209277
\(56\) 0 0
\(57\) 4.77971 0.633088
\(58\) 0 0
\(59\) −6.42730 −0.836763 −0.418381 0.908271i \(-0.637402\pi\)
−0.418381 + 0.908271i \(0.637402\pi\)
\(60\) 0 0
\(61\) 4.92354 0.630394 0.315197 0.949026i \(-0.397929\pi\)
0.315197 + 0.949026i \(0.397929\pi\)
\(62\) 0 0
\(63\) 6.75663 0.851255
\(64\) 0 0
\(65\) −1.62591 −0.201670
\(66\) 0 0
\(67\) 7.10771 0.868345 0.434173 0.900830i \(-0.357041\pi\)
0.434173 + 0.900830i \(0.357041\pi\)
\(68\) 0 0
\(69\) 5.60106 0.674289
\(70\) 0 0
\(71\) −3.40052 −0.403567 −0.201784 0.979430i \(-0.564674\pi\)
−0.201784 + 0.979430i \(0.564674\pi\)
\(72\) 0 0
\(73\) 2.77752 0.325084 0.162542 0.986702i \(-0.448031\pi\)
0.162542 + 0.986702i \(0.448031\pi\)
\(74\) 0 0
\(75\) 5.00883 0.578370
\(76\) 0 0
\(77\) −17.3383 −1.97588
\(78\) 0 0
\(79\) 1.51015 0.169905 0.0849524 0.996385i \(-0.472926\pi\)
0.0849524 + 0.996385i \(0.472926\pi\)
\(80\) 0 0
\(81\) 0.700834 0.0778704
\(82\) 0 0
\(83\) 12.0362 1.32115 0.660573 0.750762i \(-0.270313\pi\)
0.660573 + 0.750762i \(0.270313\pi\)
\(84\) 0 0
\(85\) 1.66464 0.180556
\(86\) 0 0
\(87\) 3.61870 0.387965
\(88\) 0 0
\(89\) −4.39392 −0.465754 −0.232877 0.972506i \(-0.574814\pi\)
−0.232877 + 0.972506i \(0.574814\pi\)
\(90\) 0 0
\(91\) 18.1636 1.90406
\(92\) 0 0
\(93\) −7.98702 −0.828215
\(94\) 0 0
\(95\) −1.44642 −0.148400
\(96\) 0 0
\(97\) −9.39667 −0.954088 −0.477044 0.878880i \(-0.658292\pi\)
−0.477044 + 0.878880i \(0.658292\pi\)
\(98\) 0 0
\(99\) −9.82783 −0.987734
\(100\) 0 0
\(101\) −1.93173 −0.192214 −0.0961071 0.995371i \(-0.530639\pi\)
−0.0961071 + 0.995371i \(0.530639\pi\)
\(102\) 0 0
\(103\) 10.3218 1.01704 0.508518 0.861052i \(-0.330194\pi\)
0.508518 + 0.861052i \(0.330194\pi\)
\(104\) 0 0
\(105\) 1.08973 0.106346
\(106\) 0 0
\(107\) 13.6315 1.31780 0.658901 0.752229i \(-0.271022\pi\)
0.658901 + 0.752229i \(0.271022\pi\)
\(108\) 0 0
\(109\) 8.60551 0.824258 0.412129 0.911125i \(-0.364785\pi\)
0.412129 + 0.911125i \(0.364785\pi\)
\(110\) 0 0
\(111\) −2.49707 −0.237012
\(112\) 0 0
\(113\) 5.41649 0.509540 0.254770 0.967002i \(-0.418000\pi\)
0.254770 + 0.967002i \(0.418000\pi\)
\(114\) 0 0
\(115\) −1.69497 −0.158057
\(116\) 0 0
\(117\) 10.2956 0.951831
\(118\) 0 0
\(119\) −18.5962 −1.70471
\(120\) 0 0
\(121\) 14.2194 1.29267
\(122\) 0 0
\(123\) 1.28820 0.116153
\(124\) 0 0
\(125\) −3.06103 −0.273787
\(126\) 0 0
\(127\) −7.63382 −0.677392 −0.338696 0.940896i \(-0.609986\pi\)
−0.338696 + 0.940896i \(0.609986\pi\)
\(128\) 0 0
\(129\) 0.245996 0.0216587
\(130\) 0 0
\(131\) 18.9230 1.65331 0.826657 0.562706i \(-0.190240\pi\)
0.826657 + 0.562706i \(0.190240\pi\)
\(132\) 0 0
\(133\) 16.1584 1.40111
\(134\) 0 0
\(135\) 1.56458 0.134657
\(136\) 0 0
\(137\) −5.05935 −0.432250 −0.216125 0.976366i \(-0.569342\pi\)
−0.216125 + 0.976366i \(0.569342\pi\)
\(138\) 0 0
\(139\) 9.53038 0.808356 0.404178 0.914680i \(-0.367558\pi\)
0.404178 + 0.914680i \(0.367558\pi\)
\(140\) 0 0
\(141\) 1.14321 0.0962755
\(142\) 0 0
\(143\) −26.4198 −2.20934
\(144\) 0 0
\(145\) −1.09508 −0.0909412
\(146\) 0 0
\(147\) −5.02475 −0.414434
\(148\) 0 0
\(149\) −7.34906 −0.602058 −0.301029 0.953615i \(-0.597330\pi\)
−0.301029 + 0.953615i \(0.597330\pi\)
\(150\) 0 0
\(151\) 21.7971 1.77382 0.886910 0.461942i \(-0.152847\pi\)
0.886910 + 0.461942i \(0.152847\pi\)
\(152\) 0 0
\(153\) −10.5409 −0.852178
\(154\) 0 0
\(155\) 2.41700 0.194138
\(156\) 0 0
\(157\) 0.927317 0.0740079 0.0370040 0.999315i \(-0.488219\pi\)
0.0370040 + 0.999315i \(0.488219\pi\)
\(158\) 0 0
\(159\) 6.84755 0.543046
\(160\) 0 0
\(161\) 18.9351 1.49229
\(162\) 0 0
\(163\) 14.6883 1.15047 0.575237 0.817987i \(-0.304910\pi\)
0.575237 + 0.817987i \(0.304910\pi\)
\(164\) 0 0
\(165\) −1.58506 −0.123397
\(166\) 0 0
\(167\) 0.203800 0.0157705 0.00788526 0.999969i \(-0.497490\pi\)
0.00788526 + 0.999969i \(0.497490\pi\)
\(168\) 0 0
\(169\) 14.6774 1.12903
\(170\) 0 0
\(171\) 9.15903 0.700408
\(172\) 0 0
\(173\) −6.88045 −0.523111 −0.261555 0.965188i \(-0.584235\pi\)
−0.261555 + 0.965188i \(0.584235\pi\)
\(174\) 0 0
\(175\) 16.9330 1.28001
\(176\) 0 0
\(177\) 6.56404 0.493383
\(178\) 0 0
\(179\) 15.4125 1.15198 0.575991 0.817456i \(-0.304616\pi\)
0.575991 + 0.817456i \(0.304616\pi\)
\(180\) 0 0
\(181\) −0.574354 −0.0426914 −0.0213457 0.999772i \(-0.506795\pi\)
−0.0213457 + 0.999772i \(0.506795\pi\)
\(182\) 0 0
\(183\) −5.02828 −0.371701
\(184\) 0 0
\(185\) 0.755656 0.0555569
\(186\) 0 0
\(187\) 27.0491 1.97803
\(188\) 0 0
\(189\) −17.4784 −1.27136
\(190\) 0 0
\(191\) 14.9787 1.08382 0.541910 0.840437i \(-0.317701\pi\)
0.541910 + 0.840437i \(0.317701\pi\)
\(192\) 0 0
\(193\) −9.66032 −0.695365 −0.347682 0.937612i \(-0.613031\pi\)
−0.347682 + 0.937612i \(0.613031\pi\)
\(194\) 0 0
\(195\) 1.66051 0.118911
\(196\) 0 0
\(197\) −13.3503 −0.951172 −0.475586 0.879669i \(-0.657764\pi\)
−0.475586 + 0.879669i \(0.657764\pi\)
\(198\) 0 0
\(199\) −16.0422 −1.13720 −0.568600 0.822614i \(-0.692515\pi\)
−0.568600 + 0.822614i \(0.692515\pi\)
\(200\) 0 0
\(201\) −7.25893 −0.512005
\(202\) 0 0
\(203\) 12.2334 0.858620
\(204\) 0 0
\(205\) −0.389831 −0.0272270
\(206\) 0 0
\(207\) 10.7329 0.745990
\(208\) 0 0
\(209\) −23.5032 −1.62575
\(210\) 0 0
\(211\) −1.19766 −0.0824500 −0.0412250 0.999150i \(-0.513126\pi\)
−0.0412250 + 0.999150i \(0.513126\pi\)
\(212\) 0 0
\(213\) 3.47286 0.237957
\(214\) 0 0
\(215\) −0.0744425 −0.00507693
\(216\) 0 0
\(217\) −27.0011 −1.83295
\(218\) 0 0
\(219\) −2.83661 −0.191681
\(220\) 0 0
\(221\) −28.3366 −1.90613
\(222\) 0 0
\(223\) 11.9548 0.800553 0.400276 0.916394i \(-0.368914\pi\)
0.400276 + 0.916394i \(0.368914\pi\)
\(224\) 0 0
\(225\) 9.59807 0.639871
\(226\) 0 0
\(227\) −4.82722 −0.320394 −0.160197 0.987085i \(-0.551213\pi\)
−0.160197 + 0.987085i \(0.551213\pi\)
\(228\) 0 0
\(229\) 12.0946 0.799231 0.399616 0.916683i \(-0.369144\pi\)
0.399616 + 0.916683i \(0.369144\pi\)
\(230\) 0 0
\(231\) 17.7072 1.16505
\(232\) 0 0
\(233\) 15.1937 0.995372 0.497686 0.867357i \(-0.334183\pi\)
0.497686 + 0.867357i \(0.334183\pi\)
\(234\) 0 0
\(235\) −0.345954 −0.0225675
\(236\) 0 0
\(237\) −1.54228 −0.100182
\(238\) 0 0
\(239\) 4.48826 0.290321 0.145161 0.989408i \(-0.453630\pi\)
0.145161 + 0.989408i \(0.453630\pi\)
\(240\) 0 0
\(241\) 29.8757 1.92446 0.962231 0.272233i \(-0.0877621\pi\)
0.962231 + 0.272233i \(0.0877621\pi\)
\(242\) 0 0
\(243\) −15.9031 −1.02019
\(244\) 0 0
\(245\) 1.52057 0.0971458
\(246\) 0 0
\(247\) 24.6219 1.56665
\(248\) 0 0
\(249\) −12.2923 −0.778992
\(250\) 0 0
\(251\) 22.4988 1.42011 0.710055 0.704146i \(-0.248670\pi\)
0.710055 + 0.704146i \(0.248670\pi\)
\(252\) 0 0
\(253\) −27.5420 −1.73155
\(254\) 0 0
\(255\) −1.70006 −0.106462
\(256\) 0 0
\(257\) 1.62626 0.101443 0.0507215 0.998713i \(-0.483848\pi\)
0.0507215 + 0.998713i \(0.483848\pi\)
\(258\) 0 0
\(259\) −8.44167 −0.524540
\(260\) 0 0
\(261\) 6.93425 0.429220
\(262\) 0 0
\(263\) −22.8735 −1.41044 −0.705221 0.708988i \(-0.749152\pi\)
−0.705221 + 0.708988i \(0.749152\pi\)
\(264\) 0 0
\(265\) −2.07218 −0.127293
\(266\) 0 0
\(267\) 4.48740 0.274624
\(268\) 0 0
\(269\) −5.65236 −0.344631 −0.172315 0.985042i \(-0.555125\pi\)
−0.172315 + 0.985042i \(0.555125\pi\)
\(270\) 0 0
\(271\) −11.6860 −0.709877 −0.354938 0.934890i \(-0.615498\pi\)
−0.354938 + 0.934890i \(0.615498\pi\)
\(272\) 0 0
\(273\) −18.5500 −1.12270
\(274\) 0 0
\(275\) −24.6298 −1.48523
\(276\) 0 0
\(277\) −3.83467 −0.230403 −0.115202 0.993342i \(-0.536751\pi\)
−0.115202 + 0.993342i \(0.536751\pi\)
\(278\) 0 0
\(279\) −15.3050 −0.916284
\(280\) 0 0
\(281\) 10.9183 0.651332 0.325666 0.945485i \(-0.394411\pi\)
0.325666 + 0.945485i \(0.394411\pi\)
\(282\) 0 0
\(283\) −9.07660 −0.539548 −0.269774 0.962924i \(-0.586949\pi\)
−0.269774 + 0.962924i \(0.586949\pi\)
\(284\) 0 0
\(285\) 1.47719 0.0875013
\(286\) 0 0
\(287\) 4.35492 0.257063
\(288\) 0 0
\(289\) 12.0116 0.706563
\(290\) 0 0
\(291\) 9.59659 0.562562
\(292\) 0 0
\(293\) 25.6439 1.49813 0.749067 0.662494i \(-0.230502\pi\)
0.749067 + 0.662494i \(0.230502\pi\)
\(294\) 0 0
\(295\) −1.98639 −0.115652
\(296\) 0 0
\(297\) 25.4231 1.47520
\(298\) 0 0
\(299\) 28.8530 1.66861
\(300\) 0 0
\(301\) 0.831620 0.0479338
\(302\) 0 0
\(303\) 1.97283 0.113336
\(304\) 0 0
\(305\) 1.52164 0.0871289
\(306\) 0 0
\(307\) −13.5360 −0.772542 −0.386271 0.922385i \(-0.626237\pi\)
−0.386271 + 0.922385i \(0.626237\pi\)
\(308\) 0 0
\(309\) −10.5414 −0.599678
\(310\) 0 0
\(311\) 26.3152 1.49220 0.746099 0.665835i \(-0.231925\pi\)
0.746099 + 0.665835i \(0.231925\pi\)
\(312\) 0 0
\(313\) 24.8174 1.40277 0.701383 0.712785i \(-0.252566\pi\)
0.701383 + 0.712785i \(0.252566\pi\)
\(314\) 0 0
\(315\) 2.08817 0.117655
\(316\) 0 0
\(317\) −20.8273 −1.16977 −0.584887 0.811114i \(-0.698861\pi\)
−0.584887 + 0.811114i \(0.698861\pi\)
\(318\) 0 0
\(319\) −17.7941 −0.996280
\(320\) 0 0
\(321\) −13.9215 −0.777020
\(322\) 0 0
\(323\) −25.2084 −1.40263
\(324\) 0 0
\(325\) 25.8022 1.43125
\(326\) 0 0
\(327\) −8.78859 −0.486010
\(328\) 0 0
\(329\) 3.86476 0.213071
\(330\) 0 0
\(331\) 26.1051 1.43487 0.717434 0.696627i \(-0.245317\pi\)
0.717434 + 0.696627i \(0.245317\pi\)
\(332\) 0 0
\(333\) −4.78497 −0.262215
\(334\) 0 0
\(335\) 2.19667 0.120017
\(336\) 0 0
\(337\) −13.6273 −0.742327 −0.371163 0.928568i \(-0.621041\pi\)
−0.371163 + 0.928568i \(0.621041\pi\)
\(338\) 0 0
\(339\) −5.53172 −0.300442
\(340\) 0 0
\(341\) 39.2744 2.12683
\(342\) 0 0
\(343\) 7.18103 0.387739
\(344\) 0 0
\(345\) 1.73103 0.0931958
\(346\) 0 0
\(347\) −36.1034 −1.93813 −0.969065 0.246805i \(-0.920619\pi\)
−0.969065 + 0.246805i \(0.920619\pi\)
\(348\) 0 0
\(349\) −28.1385 −1.50622 −0.753110 0.657895i \(-0.771447\pi\)
−0.753110 + 0.657895i \(0.771447\pi\)
\(350\) 0 0
\(351\) −26.6332 −1.42158
\(352\) 0 0
\(353\) 15.8041 0.841168 0.420584 0.907254i \(-0.361825\pi\)
0.420584 + 0.907254i \(0.361825\pi\)
\(354\) 0 0
\(355\) −1.05095 −0.0557784
\(356\) 0 0
\(357\) 18.9919 1.00516
\(358\) 0 0
\(359\) 32.3544 1.70760 0.853800 0.520601i \(-0.174292\pi\)
0.853800 + 0.520601i \(0.174292\pi\)
\(360\) 0 0
\(361\) 2.90374 0.152828
\(362\) 0 0
\(363\) −14.5219 −0.762202
\(364\) 0 0
\(365\) 0.858406 0.0449310
\(366\) 0 0
\(367\) 28.4789 1.48659 0.743294 0.668965i \(-0.233262\pi\)
0.743294 + 0.668965i \(0.233262\pi\)
\(368\) 0 0
\(369\) 2.46849 0.128504
\(370\) 0 0
\(371\) 23.1490 1.20184
\(372\) 0 0
\(373\) −27.6287 −1.43056 −0.715280 0.698838i \(-0.753701\pi\)
−0.715280 + 0.698838i \(0.753701\pi\)
\(374\) 0 0
\(375\) 3.12615 0.161434
\(376\) 0 0
\(377\) 18.6411 0.960066
\(378\) 0 0
\(379\) −15.2850 −0.785139 −0.392570 0.919722i \(-0.628414\pi\)
−0.392570 + 0.919722i \(0.628414\pi\)
\(380\) 0 0
\(381\) 7.79623 0.399413
\(382\) 0 0
\(383\) −25.8035 −1.31850 −0.659250 0.751924i \(-0.729126\pi\)
−0.659250 + 0.751924i \(0.729126\pi\)
\(384\) 0 0
\(385\) −5.35848 −0.273094
\(386\) 0 0
\(387\) 0.471385 0.0239619
\(388\) 0 0
\(389\) 32.1863 1.63191 0.815955 0.578115i \(-0.196212\pi\)
0.815955 + 0.578115i \(0.196212\pi\)
\(390\) 0 0
\(391\) −29.5402 −1.49391
\(392\) 0 0
\(393\) −19.3256 −0.974849
\(394\) 0 0
\(395\) 0.466718 0.0234831
\(396\) 0 0
\(397\) 13.1315 0.659053 0.329526 0.944146i \(-0.393111\pi\)
0.329526 + 0.944146i \(0.393111\pi\)
\(398\) 0 0
\(399\) −16.5022 −0.826142
\(400\) 0 0
\(401\) 29.4650 1.47141 0.735707 0.677300i \(-0.236850\pi\)
0.735707 + 0.677300i \(0.236850\pi\)
\(402\) 0 0
\(403\) −41.1438 −2.04952
\(404\) 0 0
\(405\) 0.216596 0.0107627
\(406\) 0 0
\(407\) 12.2788 0.608638
\(408\) 0 0
\(409\) −21.9319 −1.08446 −0.542232 0.840229i \(-0.682421\pi\)
−0.542232 + 0.840229i \(0.682421\pi\)
\(410\) 0 0
\(411\) 5.16699 0.254869
\(412\) 0 0
\(413\) 22.1905 1.09193
\(414\) 0 0
\(415\) 3.71985 0.182600
\(416\) 0 0
\(417\) −9.73314 −0.476634
\(418\) 0 0
\(419\) −25.5104 −1.24626 −0.623131 0.782117i \(-0.714140\pi\)
−0.623131 + 0.782117i \(0.714140\pi\)
\(420\) 0 0
\(421\) 8.78072 0.427946 0.213973 0.976840i \(-0.431360\pi\)
0.213973 + 0.976840i \(0.431360\pi\)
\(422\) 0 0
\(423\) 2.19065 0.106513
\(424\) 0 0
\(425\) −26.4167 −1.28140
\(426\) 0 0
\(427\) −16.9987 −0.822627
\(428\) 0 0
\(429\) 26.9819 1.30270
\(430\) 0 0
\(431\) 6.43881 0.310146 0.155073 0.987903i \(-0.450439\pi\)
0.155073 + 0.987903i \(0.450439\pi\)
\(432\) 0 0
\(433\) 10.9900 0.528144 0.264072 0.964503i \(-0.414934\pi\)
0.264072 + 0.964503i \(0.414934\pi\)
\(434\) 0 0
\(435\) 1.11837 0.0536219
\(436\) 0 0
\(437\) 25.6677 1.22785
\(438\) 0 0
\(439\) 31.1090 1.48475 0.742375 0.669985i \(-0.233699\pi\)
0.742375 + 0.669985i \(0.233699\pi\)
\(440\) 0 0
\(441\) −9.62858 −0.458504
\(442\) 0 0
\(443\) −24.9901 −1.18731 −0.593657 0.804718i \(-0.702317\pi\)
−0.593657 + 0.804718i \(0.702317\pi\)
\(444\) 0 0
\(445\) −1.35796 −0.0643735
\(446\) 0 0
\(447\) 7.50541 0.354994
\(448\) 0 0
\(449\) 26.9545 1.27206 0.636031 0.771664i \(-0.280575\pi\)
0.636031 + 0.771664i \(0.280575\pi\)
\(450\) 0 0
\(451\) −6.33444 −0.298277
\(452\) 0 0
\(453\) −22.2608 −1.04590
\(454\) 0 0
\(455\) 5.61354 0.263167
\(456\) 0 0
\(457\) 18.8270 0.880690 0.440345 0.897829i \(-0.354856\pi\)
0.440345 + 0.897829i \(0.354856\pi\)
\(458\) 0 0
\(459\) 27.2676 1.27274
\(460\) 0 0
\(461\) −2.42542 −0.112963 −0.0564816 0.998404i \(-0.517988\pi\)
−0.0564816 + 0.998404i \(0.517988\pi\)
\(462\) 0 0
\(463\) 4.52321 0.210212 0.105106 0.994461i \(-0.466482\pi\)
0.105106 + 0.994461i \(0.466482\pi\)
\(464\) 0 0
\(465\) −2.46842 −0.114470
\(466\) 0 0
\(467\) −15.0113 −0.694641 −0.347320 0.937746i \(-0.612908\pi\)
−0.347320 + 0.937746i \(0.612908\pi\)
\(468\) 0 0
\(469\) −24.5397 −1.13314
\(470\) 0 0
\(471\) −0.947045 −0.0436375
\(472\) 0 0
\(473\) −1.20963 −0.0556189
\(474\) 0 0
\(475\) 22.9537 1.05319
\(476\) 0 0
\(477\) 13.1215 0.600792
\(478\) 0 0
\(479\) −31.8061 −1.45326 −0.726629 0.687031i \(-0.758914\pi\)
−0.726629 + 0.687031i \(0.758914\pi\)
\(480\) 0 0
\(481\) −12.8633 −0.586515
\(482\) 0 0
\(483\) −19.3379 −0.879906
\(484\) 0 0
\(485\) −2.90408 −0.131868
\(486\) 0 0
\(487\) 9.91432 0.449261 0.224630 0.974444i \(-0.427883\pi\)
0.224630 + 0.974444i \(0.427883\pi\)
\(488\) 0 0
\(489\) −15.0007 −0.678357
\(490\) 0 0
\(491\) 39.2384 1.77081 0.885403 0.464825i \(-0.153883\pi\)
0.885403 + 0.464825i \(0.153883\pi\)
\(492\) 0 0
\(493\) −19.0851 −0.859551
\(494\) 0 0
\(495\) −3.03734 −0.136518
\(496\) 0 0
\(497\) 11.7404 0.526631
\(498\) 0 0
\(499\) 27.3229 1.22314 0.611569 0.791191i \(-0.290538\pi\)
0.611569 + 0.791191i \(0.290538\pi\)
\(500\) 0 0
\(501\) −0.208136 −0.00929883
\(502\) 0 0
\(503\) 1.00000 0.0445878
\(504\) 0 0
\(505\) −0.597010 −0.0265666
\(506\) 0 0
\(507\) −14.9896 −0.665713
\(508\) 0 0
\(509\) −6.03664 −0.267569 −0.133785 0.991010i \(-0.542713\pi\)
−0.133785 + 0.991010i \(0.542713\pi\)
\(510\) 0 0
\(511\) −9.58953 −0.424216
\(512\) 0 0
\(513\) −23.6930 −1.04607
\(514\) 0 0
\(515\) 3.18999 0.140568
\(516\) 0 0
\(517\) −5.62147 −0.247232
\(518\) 0 0
\(519\) 7.02683 0.308443
\(520\) 0 0
\(521\) −16.5676 −0.725841 −0.362921 0.931820i \(-0.618220\pi\)
−0.362921 + 0.931820i \(0.618220\pi\)
\(522\) 0 0
\(523\) 13.3977 0.585840 0.292920 0.956137i \(-0.405373\pi\)
0.292920 + 0.956137i \(0.405373\pi\)
\(524\) 0 0
\(525\) −17.2932 −0.754737
\(526\) 0 0
\(527\) 42.1238 1.83494
\(528\) 0 0
\(529\) 7.07848 0.307760
\(530\) 0 0
\(531\) 12.5782 0.545848
\(532\) 0 0
\(533\) 6.63595 0.287435
\(534\) 0 0
\(535\) 4.21286 0.182138
\(536\) 0 0
\(537\) −15.7404 −0.679247
\(538\) 0 0
\(539\) 24.7081 1.06425
\(540\) 0 0
\(541\) −24.6329 −1.05905 −0.529525 0.848294i \(-0.677630\pi\)
−0.529525 + 0.848294i \(0.677630\pi\)
\(542\) 0 0
\(543\) 0.586573 0.0251723
\(544\) 0 0
\(545\) 2.65957 0.113924
\(546\) 0 0
\(547\) −21.0233 −0.898892 −0.449446 0.893307i \(-0.648379\pi\)
−0.449446 + 0.893307i \(0.648379\pi\)
\(548\) 0 0
\(549\) −9.63535 −0.411227
\(550\) 0 0
\(551\) 16.5832 0.706468
\(552\) 0 0
\(553\) −5.21385 −0.221716
\(554\) 0 0
\(555\) −0.771732 −0.0327582
\(556\) 0 0
\(557\) 31.8172 1.34814 0.674068 0.738669i \(-0.264545\pi\)
0.674068 + 0.738669i \(0.264545\pi\)
\(558\) 0 0
\(559\) 1.26721 0.0535972
\(560\) 0 0
\(561\) −27.6246 −1.16631
\(562\) 0 0
\(563\) −20.7675 −0.875246 −0.437623 0.899158i \(-0.644180\pi\)
−0.437623 + 0.899158i \(0.644180\pi\)
\(564\) 0 0
\(565\) 1.67399 0.0704253
\(566\) 0 0
\(567\) −2.41966 −0.101616
\(568\) 0 0
\(569\) 23.3842 0.980316 0.490158 0.871634i \(-0.336939\pi\)
0.490158 + 0.871634i \(0.336939\pi\)
\(570\) 0 0
\(571\) −18.2434 −0.763462 −0.381731 0.924273i \(-0.624672\pi\)
−0.381731 + 0.924273i \(0.624672\pi\)
\(572\) 0 0
\(573\) −15.2974 −0.639056
\(574\) 0 0
\(575\) 26.8981 1.12173
\(576\) 0 0
\(577\) −34.8760 −1.45191 −0.725953 0.687744i \(-0.758601\pi\)
−0.725953 + 0.687744i \(0.758601\pi\)
\(578\) 0 0
\(579\) 9.86584 0.410010
\(580\) 0 0
\(581\) −41.5556 −1.72402
\(582\) 0 0
\(583\) −33.6713 −1.39452
\(584\) 0 0
\(585\) 3.18191 0.131556
\(586\) 0 0
\(587\) 0.103791 0.00428390 0.00214195 0.999998i \(-0.499318\pi\)
0.00214195 + 0.999998i \(0.499318\pi\)
\(588\) 0 0
\(589\) −36.6017 −1.50815
\(590\) 0 0
\(591\) 13.6344 0.560843
\(592\) 0 0
\(593\) 10.1581 0.417142 0.208571 0.978007i \(-0.433119\pi\)
0.208571 + 0.978007i \(0.433119\pi\)
\(594\) 0 0
\(595\) −5.74725 −0.235614
\(596\) 0 0
\(597\) 16.3835 0.670531
\(598\) 0 0
\(599\) −18.3660 −0.750416 −0.375208 0.926941i \(-0.622429\pi\)
−0.375208 + 0.926941i \(0.622429\pi\)
\(600\) 0 0
\(601\) −28.4392 −1.16006 −0.580030 0.814595i \(-0.696959\pi\)
−0.580030 + 0.814595i \(0.696959\pi\)
\(602\) 0 0
\(603\) −13.9098 −0.566450
\(604\) 0 0
\(605\) 4.39457 0.178665
\(606\) 0 0
\(607\) 47.6410 1.93369 0.966845 0.255364i \(-0.0821953\pi\)
0.966845 + 0.255364i \(0.0821953\pi\)
\(608\) 0 0
\(609\) −12.4937 −0.506271
\(610\) 0 0
\(611\) 5.88905 0.238245
\(612\) 0 0
\(613\) 18.6441 0.753029 0.376515 0.926411i \(-0.377122\pi\)
0.376515 + 0.926411i \(0.377122\pi\)
\(614\) 0 0
\(615\) 0.398124 0.0160539
\(616\) 0 0
\(617\) −20.3190 −0.818014 −0.409007 0.912531i \(-0.634125\pi\)
−0.409007 + 0.912531i \(0.634125\pi\)
\(618\) 0 0
\(619\) 14.0742 0.565689 0.282845 0.959166i \(-0.408722\pi\)
0.282845 + 0.959166i \(0.408722\pi\)
\(620\) 0 0
\(621\) −27.7645 −1.11415
\(622\) 0 0
\(623\) 15.1702 0.607782
\(624\) 0 0
\(625\) 23.5764 0.943056
\(626\) 0 0
\(627\) 24.0032 0.958595
\(628\) 0 0
\(629\) 13.1697 0.525109
\(630\) 0 0
\(631\) 14.9774 0.596240 0.298120 0.954528i \(-0.403641\pi\)
0.298120 + 0.954528i \(0.403641\pi\)
\(632\) 0 0
\(633\) 1.22314 0.0486153
\(634\) 0 0
\(635\) −2.35927 −0.0936247
\(636\) 0 0
\(637\) −25.8842 −1.02557
\(638\) 0 0
\(639\) 6.65480 0.263260
\(640\) 0 0
\(641\) 28.8102 1.13793 0.568967 0.822361i \(-0.307343\pi\)
0.568967 + 0.822361i \(0.307343\pi\)
\(642\) 0 0
\(643\) 13.4029 0.528560 0.264280 0.964446i \(-0.414866\pi\)
0.264280 + 0.964446i \(0.414866\pi\)
\(644\) 0 0
\(645\) 0.0760262 0.00299353
\(646\) 0 0
\(647\) −26.7597 −1.05203 −0.526016 0.850475i \(-0.676315\pi\)
−0.526016 + 0.850475i \(0.676315\pi\)
\(648\) 0 0
\(649\) −32.2772 −1.26699
\(650\) 0 0
\(651\) 27.5756 1.08077
\(652\) 0 0
\(653\) 3.60444 0.141053 0.0705263 0.997510i \(-0.477532\pi\)
0.0705263 + 0.997510i \(0.477532\pi\)
\(654\) 0 0
\(655\) 5.84825 0.228510
\(656\) 0 0
\(657\) −5.43561 −0.212063
\(658\) 0 0
\(659\) 28.6696 1.11681 0.558405 0.829568i \(-0.311413\pi\)
0.558405 + 0.829568i \(0.311413\pi\)
\(660\) 0 0
\(661\) −22.5484 −0.877030 −0.438515 0.898724i \(-0.644495\pi\)
−0.438515 + 0.898724i \(0.644495\pi\)
\(662\) 0 0
\(663\) 28.9395 1.12392
\(664\) 0 0
\(665\) 4.99383 0.193652
\(666\) 0 0
\(667\) 19.4329 0.752444
\(668\) 0 0
\(669\) −12.2091 −0.472033
\(670\) 0 0
\(671\) 24.7255 0.954516
\(672\) 0 0
\(673\) −9.07788 −0.349927 −0.174963 0.984575i \(-0.555981\pi\)
−0.174963 + 0.984575i \(0.555981\pi\)
\(674\) 0 0
\(675\) −24.8287 −0.955659
\(676\) 0 0
\(677\) −31.0614 −1.19379 −0.596893 0.802321i \(-0.703598\pi\)
−0.596893 + 0.802321i \(0.703598\pi\)
\(678\) 0 0
\(679\) 32.4425 1.24503
\(680\) 0 0
\(681\) 4.92992 0.188915
\(682\) 0 0
\(683\) 21.7817 0.833453 0.416727 0.909032i \(-0.363177\pi\)
0.416727 + 0.909032i \(0.363177\pi\)
\(684\) 0 0
\(685\) −1.56362 −0.0597427
\(686\) 0 0
\(687\) −12.3519 −0.471253
\(688\) 0 0
\(689\) 35.2740 1.34383
\(690\) 0 0
\(691\) −46.4717 −1.76787 −0.883934 0.467611i \(-0.845115\pi\)
−0.883934 + 0.467611i \(0.845115\pi\)
\(692\) 0 0
\(693\) 33.9310 1.28893
\(694\) 0 0
\(695\) 2.94541 0.111726
\(696\) 0 0
\(697\) −6.79401 −0.257342
\(698\) 0 0
\(699\) −15.5169 −0.586905
\(700\) 0 0
\(701\) 11.5534 0.436368 0.218184 0.975908i \(-0.429987\pi\)
0.218184 + 0.975908i \(0.429987\pi\)
\(702\) 0 0
\(703\) −11.4432 −0.431589
\(704\) 0 0
\(705\) 0.353314 0.0133066
\(706\) 0 0
\(707\) 6.66938 0.250828
\(708\) 0 0
\(709\) −21.6178 −0.811873 −0.405937 0.913901i \(-0.633055\pi\)
−0.405937 + 0.913901i \(0.633055\pi\)
\(710\) 0 0
\(711\) −2.95535 −0.110834
\(712\) 0 0
\(713\) −42.8914 −1.60629
\(714\) 0 0
\(715\) −8.16516 −0.305360
\(716\) 0 0
\(717\) −4.58374 −0.171183
\(718\) 0 0
\(719\) −14.4493 −0.538867 −0.269433 0.963019i \(-0.586836\pi\)
−0.269433 + 0.963019i \(0.586836\pi\)
\(720\) 0 0
\(721\) −35.6364 −1.32717
\(722\) 0 0
\(723\) −30.5113 −1.13473
\(724\) 0 0
\(725\) 17.3781 0.645407
\(726\) 0 0
\(727\) 6.10057 0.226258 0.113129 0.993580i \(-0.463913\pi\)
0.113129 + 0.993580i \(0.463913\pi\)
\(728\) 0 0
\(729\) 14.1389 0.523665
\(730\) 0 0
\(731\) −1.29739 −0.0479858
\(732\) 0 0
\(733\) −18.5571 −0.685423 −0.342711 0.939441i \(-0.611345\pi\)
−0.342711 + 0.939441i \(0.611345\pi\)
\(734\) 0 0
\(735\) −1.55292 −0.0572804
\(736\) 0 0
\(737\) 35.6942 1.31481
\(738\) 0 0
\(739\) −5.85643 −0.215432 −0.107716 0.994182i \(-0.534354\pi\)
−0.107716 + 0.994182i \(0.534354\pi\)
\(740\) 0 0
\(741\) −25.1457 −0.923751
\(742\) 0 0
\(743\) −52.1008 −1.91139 −0.955696 0.294354i \(-0.904895\pi\)
−0.955696 + 0.294354i \(0.904895\pi\)
\(744\) 0 0
\(745\) −2.27126 −0.0832125
\(746\) 0 0
\(747\) −23.5548 −0.861827
\(748\) 0 0
\(749\) −47.0632 −1.71965
\(750\) 0 0
\(751\) −5.47538 −0.199799 −0.0998997 0.994998i \(-0.531852\pi\)
−0.0998997 + 0.994998i \(0.531852\pi\)
\(752\) 0 0
\(753\) −22.9774 −0.837344
\(754\) 0 0
\(755\) 6.73648 0.245166
\(756\) 0 0
\(757\) −27.4211 −0.996638 −0.498319 0.866994i \(-0.666049\pi\)
−0.498319 + 0.866994i \(0.666049\pi\)
\(758\) 0 0
\(759\) 28.1279 1.02098
\(760\) 0 0
\(761\) −41.9789 −1.52173 −0.760867 0.648907i \(-0.775226\pi\)
−0.760867 + 0.648907i \(0.775226\pi\)
\(762\) 0 0
\(763\) −29.7109 −1.07561
\(764\) 0 0
\(765\) −3.25770 −0.117782
\(766\) 0 0
\(767\) 33.8136 1.22094
\(768\) 0 0
\(769\) −33.6416 −1.21315 −0.606574 0.795027i \(-0.707457\pi\)
−0.606574 + 0.795027i \(0.707457\pi\)
\(770\) 0 0
\(771\) −1.66085 −0.0598142
\(772\) 0 0
\(773\) 35.8072 1.28790 0.643949 0.765069i \(-0.277295\pi\)
0.643949 + 0.765069i \(0.277295\pi\)
\(774\) 0 0
\(775\) −38.3562 −1.37780
\(776\) 0 0
\(777\) 8.62126 0.309286
\(778\) 0 0
\(779\) 5.90337 0.211510
\(780\) 0 0
\(781\) −17.0770 −0.611064
\(782\) 0 0
\(783\) −17.9379 −0.641047
\(784\) 0 0
\(785\) 0.286591 0.0102289
\(786\) 0 0
\(787\) −29.9460 −1.06746 −0.533729 0.845655i \(-0.679210\pi\)
−0.533729 + 0.845655i \(0.679210\pi\)
\(788\) 0 0
\(789\) 23.3601 0.831643
\(790\) 0 0
\(791\) −18.7007 −0.664920
\(792\) 0 0
\(793\) −25.9024 −0.919820
\(794\) 0 0
\(795\) 2.11627 0.0750563
\(796\) 0 0
\(797\) −38.0445 −1.34761 −0.673803 0.738911i \(-0.735340\pi\)
−0.673803 + 0.738911i \(0.735340\pi\)
\(798\) 0 0
\(799\) −6.02932 −0.213302
\(800\) 0 0
\(801\) 8.59889 0.303827
\(802\) 0 0
\(803\) 13.9484 0.492229
\(804\) 0 0
\(805\) 5.85198 0.206255
\(806\) 0 0
\(807\) 5.77262 0.203206
\(808\) 0 0
\(809\) 43.6315 1.53400 0.767000 0.641647i \(-0.221749\pi\)
0.767000 + 0.641647i \(0.221749\pi\)
\(810\) 0 0
\(811\) 0.874118 0.0306944 0.0153472 0.999882i \(-0.495115\pi\)
0.0153472 + 0.999882i \(0.495115\pi\)
\(812\) 0 0
\(813\) 11.9347 0.418567
\(814\) 0 0
\(815\) 4.53947 0.159011
\(816\) 0 0
\(817\) 1.12731 0.0394397
\(818\) 0 0
\(819\) −35.5461 −1.24208
\(820\) 0 0
\(821\) 14.7684 0.515419 0.257710 0.966222i \(-0.417032\pi\)
0.257710 + 0.966222i \(0.417032\pi\)
\(822\) 0 0
\(823\) −4.18690 −0.145946 −0.0729731 0.997334i \(-0.523249\pi\)
−0.0729731 + 0.997334i \(0.523249\pi\)
\(824\) 0 0
\(825\) 25.1538 0.875742
\(826\) 0 0
\(827\) −52.0721 −1.81072 −0.905362 0.424641i \(-0.860400\pi\)
−0.905362 + 0.424641i \(0.860400\pi\)
\(828\) 0 0
\(829\) −18.6676 −0.648352 −0.324176 0.945997i \(-0.605087\pi\)
−0.324176 + 0.945997i \(0.605087\pi\)
\(830\) 0 0
\(831\) 3.91625 0.135853
\(832\) 0 0
\(833\) 26.5007 0.918195
\(834\) 0 0
\(835\) 0.0629854 0.00217970
\(836\) 0 0
\(837\) 39.5916 1.36849
\(838\) 0 0
\(839\) 37.7079 1.30182 0.650910 0.759155i \(-0.274388\pi\)
0.650910 + 0.759155i \(0.274388\pi\)
\(840\) 0 0
\(841\) −16.4449 −0.567067
\(842\) 0 0
\(843\) −11.1506 −0.384047
\(844\) 0 0
\(845\) 4.53611 0.156047
\(846\) 0 0
\(847\) −49.0931 −1.68686
\(848\) 0 0
\(849\) 9.26971 0.318136
\(850\) 0 0
\(851\) −13.4096 −0.459676
\(852\) 0 0
\(853\) −50.5660 −1.73135 −0.865674 0.500608i \(-0.833110\pi\)
−0.865674 + 0.500608i \(0.833110\pi\)
\(854\) 0 0
\(855\) 2.83064 0.0968058
\(856\) 0 0
\(857\) 20.3814 0.696216 0.348108 0.937454i \(-0.386824\pi\)
0.348108 + 0.937454i \(0.386824\pi\)
\(858\) 0 0
\(859\) 1.18317 0.0403693 0.0201847 0.999796i \(-0.493575\pi\)
0.0201847 + 0.999796i \(0.493575\pi\)
\(860\) 0 0
\(861\) −4.44757 −0.151573
\(862\) 0 0
\(863\) −39.8418 −1.35623 −0.678116 0.734955i \(-0.737203\pi\)
−0.678116 + 0.734955i \(0.737203\pi\)
\(864\) 0 0
\(865\) −2.12643 −0.0723009
\(866\) 0 0
\(867\) −12.2671 −0.416613
\(868\) 0 0
\(869\) 7.58380 0.257263
\(870\) 0 0
\(871\) −37.3932 −1.26702
\(872\) 0 0
\(873\) 18.3893 0.622382
\(874\) 0 0
\(875\) 10.5683 0.357275
\(876\) 0 0
\(877\) −33.5913 −1.13430 −0.567149 0.823615i \(-0.691954\pi\)
−0.567149 + 0.823615i \(0.691954\pi\)
\(878\) 0 0
\(879\) −26.1895 −0.883349
\(880\) 0 0
\(881\) −37.1675 −1.25221 −0.626103 0.779741i \(-0.715351\pi\)
−0.626103 + 0.779741i \(0.715351\pi\)
\(882\) 0 0
\(883\) −9.78604 −0.329326 −0.164663 0.986350i \(-0.552654\pi\)
−0.164663 + 0.986350i \(0.552654\pi\)
\(884\) 0 0
\(885\) 2.02865 0.0681922
\(886\) 0 0
\(887\) −34.1694 −1.14730 −0.573648 0.819102i \(-0.694472\pi\)
−0.573648 + 0.819102i \(0.694472\pi\)
\(888\) 0 0
\(889\) 26.3561 0.883956
\(890\) 0 0
\(891\) 3.51951 0.117908
\(892\) 0 0
\(893\) 5.23892 0.175314
\(894\) 0 0
\(895\) 4.76329 0.159219
\(896\) 0 0
\(897\) −29.4668 −0.983868
\(898\) 0 0
\(899\) −27.7110 −0.924212
\(900\) 0 0
\(901\) −36.1142 −1.20314
\(902\) 0 0
\(903\) −0.849313 −0.0282634
\(904\) 0 0
\(905\) −0.177507 −0.00590052
\(906\) 0 0
\(907\) −33.1793 −1.10170 −0.550850 0.834604i \(-0.685696\pi\)
−0.550850 + 0.834604i \(0.685696\pi\)
\(908\) 0 0
\(909\) 3.78039 0.125388
\(910\) 0 0
\(911\) −53.0164 −1.75651 −0.878257 0.478190i \(-0.841293\pi\)
−0.878257 + 0.478190i \(0.841293\pi\)
\(912\) 0 0
\(913\) 60.4446 2.00042
\(914\) 0 0
\(915\) −1.55401 −0.0513741
\(916\) 0 0
\(917\) −65.3327 −2.15748
\(918\) 0 0
\(919\) 50.8988 1.67900 0.839498 0.543363i \(-0.182849\pi\)
0.839498 + 0.543363i \(0.182849\pi\)
\(920\) 0 0
\(921\) 13.8240 0.455516
\(922\) 0 0
\(923\) 17.8899 0.588853
\(924\) 0 0
\(925\) −11.9917 −0.394286
\(926\) 0 0
\(927\) −20.1997 −0.663445
\(928\) 0 0
\(929\) 2.45820 0.0806510 0.0403255 0.999187i \(-0.487161\pi\)
0.0403255 + 0.999187i \(0.487161\pi\)
\(930\) 0 0
\(931\) −23.0267 −0.754668
\(932\) 0 0
\(933\) −26.8750 −0.879849
\(934\) 0 0
\(935\) 8.35965 0.273390
\(936\) 0 0
\(937\) 33.7157 1.10145 0.550723 0.834688i \(-0.314352\pi\)
0.550723 + 0.834688i \(0.314352\pi\)
\(938\) 0 0
\(939\) −25.3454 −0.827117
\(940\) 0 0
\(941\) −34.8370 −1.13565 −0.567826 0.823148i \(-0.692215\pi\)
−0.567826 + 0.823148i \(0.692215\pi\)
\(942\) 0 0
\(943\) 6.91781 0.225275
\(944\) 0 0
\(945\) −5.40177 −0.175720
\(946\) 0 0
\(947\) −54.3894 −1.76742 −0.883709 0.468038i \(-0.844961\pi\)
−0.883709 + 0.468038i \(0.844961\pi\)
\(948\) 0 0
\(949\) −14.6124 −0.474337
\(950\) 0 0
\(951\) 21.2703 0.689738
\(952\) 0 0
\(953\) −17.4651 −0.565749 −0.282874 0.959157i \(-0.591288\pi\)
−0.282874 + 0.959157i \(0.591288\pi\)
\(954\) 0 0
\(955\) 4.62923 0.149798
\(956\) 0 0
\(957\) 18.1727 0.587440
\(958\) 0 0
\(959\) 17.4677 0.564060
\(960\) 0 0
\(961\) 30.1623 0.972979
\(962\) 0 0
\(963\) −26.6767 −0.859646
\(964\) 0 0
\(965\) −2.98557 −0.0961087
\(966\) 0 0
\(967\) 54.2305 1.74394 0.871968 0.489563i \(-0.162844\pi\)
0.871968 + 0.489563i \(0.162844\pi\)
\(968\) 0 0
\(969\) 25.7447 0.827038
\(970\) 0 0
\(971\) −20.8932 −0.670495 −0.335247 0.942130i \(-0.608820\pi\)
−0.335247 + 0.942130i \(0.608820\pi\)
\(972\) 0 0
\(973\) −32.9041 −1.05486
\(974\) 0 0
\(975\) −26.3511 −0.843910
\(976\) 0 0
\(977\) 17.5287 0.560793 0.280396 0.959884i \(-0.409534\pi\)
0.280396 + 0.959884i \(0.409534\pi\)
\(978\) 0 0
\(979\) −22.0658 −0.705225
\(980\) 0 0
\(981\) −16.8410 −0.537690
\(982\) 0 0
\(983\) −23.8464 −0.760582 −0.380291 0.924867i \(-0.624176\pi\)
−0.380291 + 0.924867i \(0.624176\pi\)
\(984\) 0 0
\(985\) −4.12598 −0.131465
\(986\) 0 0
\(987\) −3.94698 −0.125634
\(988\) 0 0
\(989\) 1.32103 0.0420064
\(990\) 0 0
\(991\) 22.8232 0.725004 0.362502 0.931983i \(-0.381923\pi\)
0.362502 + 0.931983i \(0.381923\pi\)
\(992\) 0 0
\(993\) −26.6605 −0.846046
\(994\) 0 0
\(995\) −4.95791 −0.157176
\(996\) 0 0
\(997\) 39.3974 1.24773 0.623864 0.781533i \(-0.285562\pi\)
0.623864 + 0.781533i \(0.285562\pi\)
\(998\) 0 0
\(999\) 12.3780 0.391622
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 4024.2.a.g.1.10 33
4.3 odd 2 8048.2.a.x.1.24 33
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4024.2.a.g.1.10 33 1.1 even 1 trivial
8048.2.a.x.1.24 33 4.3 odd 2