Properties

Label 4032.2.v.d.1583.18
Level $4032$
Weight $2$
Character 4032.1583
Analytic conductor $32.196$
Analytic rank $0$
Dimension $36$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4032,2,Mod(1583,4032)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4032, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4032.1583");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4032.v (of order \(4\), degree \(2\), not minimal)

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: no
Analytic conductor: \(32.1956820950\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 1008)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 1583.18
Character \(\chi\) \(=\) 4032.1583
Dual form 4032.2.v.d.3599.18

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.98923 + 2.98923i) q^{5} +1.00000 q^{7} +O(q^{10})\) \(q+(2.98923 + 2.98923i) q^{5} +1.00000 q^{7} +(-2.32631 + 2.32631i) q^{11} +(1.27678 + 1.27678i) q^{13} -3.56038i q^{17} +(0.796566 - 0.796566i) q^{19} -1.75549i q^{23} +12.8710i q^{25} +(1.87823 - 1.87823i) q^{29} +7.15500i q^{31} +(2.98923 + 2.98923i) q^{35} +(4.64081 - 4.64081i) q^{37} -8.98721 q^{41} +(6.04027 + 6.04027i) q^{43} +6.99236 q^{47} +1.00000 q^{49} +(-0.536812 - 0.536812i) q^{53} -13.9077 q^{55} +(-0.119264 + 0.119264i) q^{59} +(10.9735 + 10.9735i) q^{61} +7.63319i q^{65} +(-3.83581 + 3.83581i) q^{67} +9.96614i q^{71} +11.4452i q^{73} +(-2.32631 + 2.32631i) q^{77} -15.4311i q^{79} +(4.23544 + 4.23544i) q^{83} +(10.6428 - 10.6428i) q^{85} -2.38113 q^{89} +(1.27678 + 1.27678i) q^{91} +4.76225 q^{95} +6.00751 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 36 q + 36 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 36 q + 36 q^{7} - 16 q^{13} + 16 q^{19} + 20 q^{37} - 36 q^{43} + 36 q^{49} - 32 q^{55} + 112 q^{61} + 36 q^{67} - 96 q^{85} - 16 q^{91} + 56 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(1793\) \(3781\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(e\left(\frac{3}{4}\right)\)

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 0
\(4\) 0 0
\(5\) 2.98923 + 2.98923i 1.33683 + 1.33683i 0.899116 + 0.437710i \(0.144210\pi\)
0.437710 + 0.899116i \(0.355790\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.32631 + 2.32631i −0.701407 + 0.701407i −0.964713 0.263305i \(-0.915187\pi\)
0.263305 + 0.964713i \(0.415187\pi\)
\(12\) 0 0
\(13\) 1.27678 + 1.27678i 0.354115 + 0.354115i 0.861638 0.507523i \(-0.169439\pi\)
−0.507523 + 0.861638i \(0.669439\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.56038i 0.863519i −0.901989 0.431759i \(-0.857893\pi\)
0.901989 0.431759i \(-0.142107\pi\)
\(18\) 0 0
\(19\) 0.796566 0.796566i 0.182745 0.182745i −0.609806 0.792551i \(-0.708753\pi\)
0.792551 + 0.609806i \(0.208753\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.75549i 0.366045i −0.983109 0.183023i \(-0.941412\pi\)
0.983109 0.183023i \(-0.0585881\pi\)
\(24\) 0 0
\(25\) 12.8710i 2.57421i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.87823 1.87823i 0.348779 0.348779i −0.510876 0.859654i \(-0.670679\pi\)
0.859654 + 0.510876i \(0.170679\pi\)
\(30\) 0 0
\(31\) 7.15500i 1.28508i 0.766254 + 0.642538i \(0.222118\pi\)
−0.766254 + 0.642538i \(0.777882\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.98923 + 2.98923i 0.505273 + 0.505273i
\(36\) 0 0
\(37\) 4.64081 4.64081i 0.762945 0.762945i −0.213909 0.976854i \(-0.568619\pi\)
0.976854 + 0.213909i \(0.0686195\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −8.98721 −1.40357 −0.701783 0.712391i \(-0.747612\pi\)
−0.701783 + 0.712391i \(0.747612\pi\)
\(42\) 0 0
\(43\) 6.04027 + 6.04027i 0.921132 + 0.921132i 0.997110 0.0759775i \(-0.0242077\pi\)
−0.0759775 + 0.997110i \(0.524208\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.99236 1.01994 0.509970 0.860192i \(-0.329656\pi\)
0.509970 + 0.860192i \(0.329656\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −0.536812 0.536812i −0.0737369 0.0737369i 0.669277 0.743013i \(-0.266604\pi\)
−0.743013 + 0.669277i \(0.766604\pi\)
\(54\) 0 0
\(55\) −13.9077 −1.87532
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −0.119264 + 0.119264i −0.0155269 + 0.0155269i −0.714828 0.699301i \(-0.753495\pi\)
0.699301 + 0.714828i \(0.253495\pi\)
\(60\) 0 0
\(61\) 10.9735 + 10.9735i 1.40501 + 1.40501i 0.783055 + 0.621952i \(0.213660\pi\)
0.621952 + 0.783055i \(0.286340\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 7.63319i 0.946780i
\(66\) 0 0
\(67\) −3.83581 + 3.83581i −0.468618 + 0.468618i −0.901467 0.432848i \(-0.857509\pi\)
0.432848 + 0.901467i \(0.357509\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 9.96614i 1.18276i 0.806392 + 0.591382i \(0.201417\pi\)
−0.806392 + 0.591382i \(0.798583\pi\)
\(72\) 0 0
\(73\) 11.4452i 1.33956i 0.742562 + 0.669778i \(0.233611\pi\)
−0.742562 + 0.669778i \(0.766389\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.32631 + 2.32631i −0.265107 + 0.265107i
\(78\) 0 0
\(79\) 15.4311i 1.73613i −0.496449 0.868066i \(-0.665363\pi\)
0.496449 0.868066i \(-0.334637\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.23544 + 4.23544i 0.464900 + 0.464900i 0.900258 0.435358i \(-0.143378\pi\)
−0.435358 + 0.900258i \(0.643378\pi\)
\(84\) 0 0
\(85\) 10.6428 10.6428i 1.15437 1.15437i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −2.38113 −0.252400 −0.126200 0.992005i \(-0.540278\pi\)
−0.126200 + 0.992005i \(0.540278\pi\)
\(90\) 0 0
\(91\) 1.27678 + 1.27678i 0.133843 + 0.133843i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.76225 0.488596
\(96\) 0 0
\(97\) 6.00751 0.609970 0.304985 0.952357i \(-0.401348\pi\)
0.304985 + 0.952357i \(0.401348\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 6.15894 + 6.15894i 0.612837 + 0.612837i 0.943684 0.330847i \(-0.107334\pi\)
−0.330847 + 0.943684i \(0.607334\pi\)
\(102\) 0 0
\(103\) −17.7016 −1.74419 −0.872094 0.489338i \(-0.837238\pi\)
−0.872094 + 0.489338i \(0.837238\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −9.52746 + 9.52746i −0.921055 + 0.921055i −0.997104 0.0760494i \(-0.975769\pi\)
0.0760494 + 0.997104i \(0.475769\pi\)
\(108\) 0 0
\(109\) −11.3741 11.3741i −1.08945 1.08945i −0.995585 0.0938600i \(-0.970079\pi\)
−0.0938600 0.995585i \(-0.529921\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 9.65901i 0.908643i −0.890838 0.454321i \(-0.849882\pi\)
0.890838 0.454321i \(-0.150118\pi\)
\(114\) 0 0
\(115\) 5.24757 5.24757i 0.489339 0.489339i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 3.56038i 0.326379i
\(120\) 0 0
\(121\) 0.176606i 0.0160551i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −23.5284 + 23.5284i −2.10444 + 2.10444i
\(126\) 0 0
\(127\) 14.3501i 1.27336i 0.771127 + 0.636682i \(0.219693\pi\)
−0.771127 + 0.636682i \(0.780307\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −13.3814 13.3814i −1.16913 1.16913i −0.982413 0.186721i \(-0.940214\pi\)
−0.186721 0.982413i \(-0.559786\pi\)
\(132\) 0 0
\(133\) 0.796566 0.796566i 0.0690711 0.0690711i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −8.59574 −0.734384 −0.367192 0.930145i \(-0.619681\pi\)
−0.367192 + 0.930145i \(0.619681\pi\)
\(138\) 0 0
\(139\) −2.60098 2.60098i −0.220612 0.220612i 0.588144 0.808756i \(-0.299859\pi\)
−0.808756 + 0.588144i \(0.799859\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −5.94036 −0.496758
\(144\) 0 0
\(145\) 11.2289 0.932513
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −13.6833 13.6833i −1.12098 1.12098i −0.991594 0.129385i \(-0.958700\pi\)
−0.129385 0.991594i \(-0.541300\pi\)
\(150\) 0 0
\(151\) 7.21798 0.587391 0.293696 0.955899i \(-0.405115\pi\)
0.293696 + 0.955899i \(0.405115\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −21.3880 + 21.3880i −1.71792 + 1.71792i
\(156\) 0 0
\(157\) 2.28205 + 2.28205i 0.182127 + 0.182127i 0.792282 0.610155i \(-0.208893\pi\)
−0.610155 + 0.792282i \(0.708893\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 1.75549i 0.138352i
\(162\) 0 0
\(163\) 12.7454 12.7454i 0.998293 0.998293i −0.00170515 0.999999i \(-0.500543\pi\)
0.999999 + 0.00170515i \(0.000542768\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 7.86515i 0.608624i 0.952572 + 0.304312i \(0.0984265\pi\)
−0.952572 + 0.304312i \(0.901574\pi\)
\(168\) 0 0
\(169\) 9.73967i 0.749205i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 3.37538 3.37538i 0.256625 0.256625i −0.567055 0.823680i \(-0.691917\pi\)
0.823680 + 0.567055i \(0.191917\pi\)
\(174\) 0 0
\(175\) 12.8710i 0.972959i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −2.80060 2.80060i −0.209327 0.209327i 0.594655 0.803981i \(-0.297289\pi\)
−0.803981 + 0.594655i \(0.797289\pi\)
\(180\) 0 0
\(181\) 4.63652 4.63652i 0.344630 0.344630i −0.513475 0.858105i \(-0.671642\pi\)
0.858105 + 0.513475i \(0.171642\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 27.7450 2.03985
\(186\) 0 0
\(187\) 8.28253 + 8.28253i 0.605679 + 0.605679i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −14.1380 −1.02299 −0.511494 0.859287i \(-0.670908\pi\)
−0.511494 + 0.859287i \(0.670908\pi\)
\(192\) 0 0
\(193\) 22.2621 1.60246 0.801230 0.598356i \(-0.204179\pi\)
0.801230 + 0.598356i \(0.204179\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −6.84298 6.84298i −0.487542 0.487542i 0.419988 0.907530i \(-0.362034\pi\)
−0.907530 + 0.419988i \(0.862034\pi\)
\(198\) 0 0
\(199\) 1.39131 0.0986274 0.0493137 0.998783i \(-0.484297\pi\)
0.0493137 + 0.998783i \(0.484297\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.87823 1.87823i 0.131826 0.131826i
\(204\) 0 0
\(205\) −26.8649 26.8649i −1.87632 1.87632i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 3.70611i 0.256357i
\(210\) 0 0
\(211\) 6.57328 6.57328i 0.452523 0.452523i −0.443668 0.896191i \(-0.646323\pi\)
0.896191 + 0.443668i \(0.146323\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 36.1115i 2.46279i
\(216\) 0 0
\(217\) 7.15500i 0.485713i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 4.54582 4.54582i 0.305785 0.305785i
\(222\) 0 0
\(223\) 18.3598i 1.22946i −0.788737 0.614731i \(-0.789264\pi\)
0.788737 0.614731i \(-0.210736\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 14.0196 + 14.0196i 0.930514 + 0.930514i 0.997738 0.0672240i \(-0.0214142\pi\)
−0.0672240 + 0.997738i \(0.521414\pi\)
\(228\) 0 0
\(229\) 2.27105 2.27105i 0.150075 0.150075i −0.628076 0.778152i \(-0.716157\pi\)
0.778152 + 0.628076i \(0.216157\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −7.95841 −0.521373 −0.260687 0.965424i \(-0.583949\pi\)
−0.260687 + 0.965424i \(0.583949\pi\)
\(234\) 0 0
\(235\) 20.9018 + 20.9018i 1.36348 + 1.36348i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −23.4554 −1.51720 −0.758601 0.651556i \(-0.774117\pi\)
−0.758601 + 0.651556i \(0.774117\pi\)
\(240\) 0 0
\(241\) −15.8655 −1.02199 −0.510993 0.859585i \(-0.670722\pi\)
−0.510993 + 0.859585i \(0.670722\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.98923 + 2.98923i 0.190975 + 0.190975i
\(246\) 0 0
\(247\) 2.03408 0.129425
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 8.26193 8.26193i 0.521489 0.521489i −0.396532 0.918021i \(-0.629786\pi\)
0.918021 + 0.396532i \(0.129786\pi\)
\(252\) 0 0
\(253\) 4.08381 + 4.08381i 0.256747 + 0.256747i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 14.1727i 0.884070i −0.896998 0.442035i \(-0.854257\pi\)
0.896998 0.442035i \(-0.145743\pi\)
\(258\) 0 0
\(259\) 4.64081 4.64081i 0.288366 0.288366i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 26.0095i 1.60381i 0.597449 + 0.801907i \(0.296181\pi\)
−0.597449 + 0.801907i \(0.703819\pi\)
\(264\) 0 0
\(265\) 3.20932i 0.197147i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −6.51572 + 6.51572i −0.397270 + 0.397270i −0.877269 0.479999i \(-0.840637\pi\)
0.479999 + 0.877269i \(0.340637\pi\)
\(270\) 0 0
\(271\) 7.92154i 0.481199i 0.970624 + 0.240600i \(0.0773441\pi\)
−0.970624 + 0.240600i \(0.922656\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −29.9420 29.9420i −1.80557 1.80557i
\(276\) 0 0
\(277\) −5.35852 + 5.35852i −0.321962 + 0.321962i −0.849519 0.527557i \(-0.823108\pi\)
0.527557 + 0.849519i \(0.323108\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 20.2941 1.21065 0.605323 0.795980i \(-0.293044\pi\)
0.605323 + 0.795980i \(0.293044\pi\)
\(282\) 0 0
\(283\) 2.32260 + 2.32260i 0.138064 + 0.138064i 0.772761 0.634697i \(-0.218875\pi\)
−0.634697 + 0.772761i \(0.718875\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −8.98721 −0.530498
\(288\) 0 0
\(289\) 4.32369 0.254335
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 4.21418 + 4.21418i 0.246195 + 0.246195i 0.819407 0.573212i \(-0.194303\pi\)
−0.573212 + 0.819407i \(0.694303\pi\)
\(294\) 0 0
\(295\) −0.713017 −0.0415134
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 2.24137 2.24137i 0.129622 0.129622i
\(300\) 0 0
\(301\) 6.04027 + 6.04027i 0.348155 + 0.348155i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 65.6045i 3.75650i
\(306\) 0 0
\(307\) 18.6414 18.6414i 1.06392 1.06392i 0.0661106 0.997812i \(-0.478941\pi\)
0.997812 0.0661106i \(-0.0210590\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 19.0781i 1.08182i −0.841080 0.540910i \(-0.818080\pi\)
0.841080 0.540910i \(-0.181920\pi\)
\(312\) 0 0
\(313\) 32.3787i 1.83015i 0.403282 + 0.915076i \(0.367869\pi\)
−0.403282 + 0.915076i \(0.632131\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −1.55044 + 1.55044i −0.0870813 + 0.0870813i −0.749306 0.662224i \(-0.769613\pi\)
0.662224 + 0.749306i \(0.269613\pi\)
\(318\) 0 0
\(319\) 8.73867i 0.489272i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −2.83608 2.83608i −0.157804 0.157804i
\(324\) 0 0
\(325\) −16.4335 + 16.4335i −0.911565 + 0.911565i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 6.99236 0.385501
\(330\) 0 0
\(331\) −6.20654 6.20654i −0.341142 0.341142i 0.515654 0.856797i \(-0.327549\pi\)
−0.856797 + 0.515654i \(0.827549\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −22.9322 −1.25292
\(336\) 0 0
\(337\) −11.6890 −0.636739 −0.318369 0.947967i \(-0.603135\pi\)
−0.318369 + 0.947967i \(0.603135\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −16.6447 16.6447i −0.901362 0.901362i
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 18.2790 18.2790i 0.981268 0.981268i −0.0185600 0.999828i \(-0.505908\pi\)
0.999828 + 0.0185600i \(0.00590816\pi\)
\(348\) 0 0
\(349\) 15.5945 + 15.5945i 0.834753 + 0.834753i 0.988163 0.153409i \(-0.0490253\pi\)
−0.153409 + 0.988163i \(0.549025\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 22.5053i 1.19784i −0.800810 0.598918i \(-0.795598\pi\)
0.800810 0.598918i \(-0.204402\pi\)
\(354\) 0 0
\(355\) −29.7911 + 29.7911i −1.58115 + 1.58115i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 13.4623i 0.710512i 0.934769 + 0.355256i \(0.115606\pi\)
−0.934769 + 0.355256i \(0.884394\pi\)
\(360\) 0 0
\(361\) 17.7310i 0.933209i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −34.2123 + 34.2123i −1.79075 + 1.79075i
\(366\) 0 0
\(367\) 4.96018i 0.258919i 0.991585 + 0.129460i \(0.0413242\pi\)
−0.991585 + 0.129460i \(0.958676\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −0.536812 0.536812i −0.0278699 0.0278699i
\(372\) 0 0
\(373\) −3.46439 + 3.46439i −0.179379 + 0.179379i −0.791085 0.611706i \(-0.790484\pi\)
0.611706 + 0.791085i \(0.290484\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 4.79617 0.247015
\(378\) 0 0
\(379\) −12.5198 12.5198i −0.643097 0.643097i 0.308218 0.951316i \(-0.400267\pi\)
−0.951316 + 0.308218i \(0.900267\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 32.5201 1.66170 0.830849 0.556497i \(-0.187855\pi\)
0.830849 + 0.556497i \(0.187855\pi\)
\(384\) 0 0
\(385\) −13.9077 −0.708804
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −6.73683 6.73683i −0.341571 0.341571i 0.515387 0.856958i \(-0.327648\pi\)
−0.856958 + 0.515387i \(0.827648\pi\)
\(390\) 0 0
\(391\) −6.25021 −0.316087
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 46.1271 46.1271i 2.32091 2.32091i
\(396\) 0 0
\(397\) −17.6040 17.6040i −0.883520 0.883520i 0.110370 0.993891i \(-0.464796\pi\)
−0.993891 + 0.110370i \(0.964796\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 17.9384i 0.895803i 0.894083 + 0.447902i \(0.147829\pi\)
−0.894083 + 0.447902i \(0.852171\pi\)
\(402\) 0 0
\(403\) −9.13536 + 9.13536i −0.455064 + 0.455064i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 21.5919i 1.07027i
\(408\) 0 0
\(409\) 13.5200i 0.668521i 0.942481 + 0.334261i \(0.108487\pi\)
−0.942481 + 0.334261i \(0.891513\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −0.119264 + 0.119264i −0.00586861 + 0.00586861i
\(414\) 0 0
\(415\) 25.3214i 1.24298i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 8.57126 + 8.57126i 0.418733 + 0.418733i 0.884767 0.466034i \(-0.154317\pi\)
−0.466034 + 0.884767i \(0.654317\pi\)
\(420\) 0 0
\(421\) 6.96735 6.96735i 0.339568 0.339568i −0.516637 0.856205i \(-0.672816\pi\)
0.856205 + 0.516637i \(0.172816\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 45.8258 2.22288
\(426\) 0 0
\(427\) 10.9735 + 10.9735i 0.531043 + 0.531043i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 37.2769 1.79557 0.897783 0.440438i \(-0.145177\pi\)
0.897783 + 0.440438i \(0.145177\pi\)
\(432\) 0 0
\(433\) 25.5067 1.22577 0.612886 0.790171i \(-0.290008\pi\)
0.612886 + 0.790171i \(0.290008\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −1.39837 1.39837i −0.0668929 0.0668929i
\(438\) 0 0
\(439\) 38.7171 1.84786 0.923932 0.382556i \(-0.124956\pi\)
0.923932 + 0.382556i \(0.124956\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 10.8179 10.8179i 0.513974 0.513974i −0.401767 0.915742i \(-0.631604\pi\)
0.915742 + 0.401767i \(0.131604\pi\)
\(444\) 0 0
\(445\) −7.11777 7.11777i −0.337415 0.337415i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 6.74427i 0.318282i 0.987256 + 0.159141i \(0.0508724\pi\)
−0.987256 + 0.159141i \(0.949128\pi\)
\(450\) 0 0
\(451\) 20.9070 20.9070i 0.984472 0.984472i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 7.63319i 0.357849i
\(456\) 0 0
\(457\) 31.1453i 1.45692i −0.685091 0.728458i \(-0.740238\pi\)
0.685091 0.728458i \(-0.259762\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −27.6054 + 27.6054i −1.28571 + 1.28571i −0.348346 + 0.937366i \(0.613256\pi\)
−0.937366 + 0.348346i \(0.886744\pi\)
\(462\) 0 0
\(463\) 20.2024i 0.938884i 0.882963 + 0.469442i \(0.155545\pi\)
−0.882963 + 0.469442i \(0.844455\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 14.5889 + 14.5889i 0.675095 + 0.675095i 0.958886 0.283791i \(-0.0915924\pi\)
−0.283791 + 0.958886i \(0.591592\pi\)
\(468\) 0 0
\(469\) −3.83581 + 3.83581i −0.177121 + 0.177121i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −28.1030 −1.29218
\(474\) 0 0
\(475\) 10.2526 + 10.2526i 0.470423 + 0.470423i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −3.50499 −0.160147 −0.0800735 0.996789i \(-0.525515\pi\)
−0.0800735 + 0.996789i \(0.525515\pi\)
\(480\) 0 0
\(481\) 11.8506 0.540340
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 17.9579 + 17.9579i 0.815424 + 0.815424i
\(486\) 0 0
\(487\) 11.8906 0.538813 0.269407 0.963027i \(-0.413172\pi\)
0.269407 + 0.963027i \(0.413172\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 8.36496 8.36496i 0.377505 0.377505i −0.492696 0.870201i \(-0.663989\pi\)
0.870201 + 0.492696i \(0.163989\pi\)
\(492\) 0 0
\(493\) −6.68721 6.68721i −0.301177 0.301177i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 9.96614i 0.447043i
\(498\) 0 0
\(499\) 23.4014 23.4014i 1.04759 1.04759i 0.0487824 0.998809i \(-0.484466\pi\)
0.998809 0.0487824i \(-0.0155341\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 11.4682i 0.511342i −0.966764 0.255671i \(-0.917704\pi\)
0.966764 0.255671i \(-0.0822963\pi\)
\(504\) 0 0
\(505\) 36.8210i 1.63851i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 27.9963 27.9963i 1.24091 1.24091i 0.281292 0.959622i \(-0.409237\pi\)
0.959622 0.281292i \(-0.0907631\pi\)
\(510\) 0 0
\(511\) 11.4452i 0.506304i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −52.9142 52.9142i −2.33168 2.33168i
\(516\) 0 0
\(517\) −16.2664 + 16.2664i −0.715394 + 0.715394i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 11.7733 0.515799 0.257900 0.966172i \(-0.416970\pi\)
0.257900 + 0.966172i \(0.416970\pi\)
\(522\) 0 0
\(523\) −20.3443 20.3443i −0.889593 0.889593i 0.104891 0.994484i \(-0.466551\pi\)
−0.994484 + 0.104891i \(0.966551\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 25.4745 1.10969
\(528\) 0 0
\(529\) 19.9183 0.866011
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −11.4747 11.4747i −0.497024 0.497024i
\(534\) 0 0
\(535\) −56.9596 −2.46258
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −2.32631 + 2.32631i −0.100201 + 0.100201i
\(540\) 0 0
\(541\) −15.0824 15.0824i −0.648445 0.648445i 0.304172 0.952617i \(-0.401620\pi\)
−0.952617 + 0.304172i \(0.901620\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 68.0000i 2.91280i
\(546\) 0 0
\(547\) 10.2321 10.2321i 0.437492 0.437492i −0.453675 0.891167i \(-0.649887\pi\)
0.891167 + 0.453675i \(0.149887\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 2.99227i 0.127475i
\(552\) 0 0
\(553\) 15.4311i 0.656196i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0.823599 0.823599i 0.0348970 0.0348970i −0.689443 0.724340i \(-0.742145\pi\)
0.724340 + 0.689443i \(0.242145\pi\)
\(558\) 0 0
\(559\) 15.4242i 0.652373i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −11.9474 11.9474i −0.503523 0.503523i 0.409008 0.912531i \(-0.365875\pi\)
−0.912531 + 0.409008i \(0.865875\pi\)
\(564\) 0 0
\(565\) 28.8730 28.8730i 1.21470 1.21470i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 14.9392 0.626285 0.313142 0.949706i \(-0.398618\pi\)
0.313142 + 0.949706i \(0.398618\pi\)
\(570\) 0 0
\(571\) 21.1630 + 21.1630i 0.885642 + 0.885642i 0.994101 0.108459i \(-0.0345916\pi\)
−0.108459 + 0.994101i \(0.534592\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 22.5950 0.942276
\(576\) 0 0
\(577\) −13.7022 −0.570430 −0.285215 0.958463i \(-0.592065\pi\)
−0.285215 + 0.958463i \(0.592065\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 4.23544 + 4.23544i 0.175716 + 0.175716i
\(582\) 0 0
\(583\) 2.49758 0.103439
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 14.9582 14.9582i 0.617390 0.617390i −0.327471 0.944861i \(-0.606196\pi\)
0.944861 + 0.327471i \(0.106196\pi\)
\(588\) 0 0
\(589\) 5.69943 + 5.69943i 0.234841 + 0.234841i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 6.84205i 0.280969i 0.990083 + 0.140485i \(0.0448661\pi\)
−0.990083 + 0.140485i \(0.955134\pi\)
\(594\) 0 0
\(595\) 10.6428 10.6428i 0.436313 0.436313i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 7.02266i 0.286938i −0.989655 0.143469i \(-0.954174\pi\)
0.989655 0.143469i \(-0.0458257\pi\)
\(600\) 0 0
\(601\) 9.23315i 0.376628i 0.982109 + 0.188314i \(0.0603023\pi\)
−0.982109 + 0.188314i \(0.939698\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −0.527918 + 0.527918i −0.0214629 + 0.0214629i
\(606\) 0 0
\(607\) 18.3413i 0.744450i −0.928142 0.372225i \(-0.878595\pi\)
0.928142 0.372225i \(-0.121405\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 8.92770 + 8.92770i 0.361176 + 0.361176i
\(612\) 0 0
\(613\) −30.5942 + 30.5942i −1.23569 + 1.23569i −0.273943 + 0.961746i \(0.588328\pi\)
−0.961746 + 0.273943i \(0.911672\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −11.9418 −0.480757 −0.240379 0.970679i \(-0.577272\pi\)
−0.240379 + 0.970679i \(0.577272\pi\)
\(618\) 0 0
\(619\) 7.31388 + 7.31388i 0.293970 + 0.293970i 0.838646 0.544676i \(-0.183348\pi\)
−0.544676 + 0.838646i \(0.683348\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −2.38113 −0.0953982
\(624\) 0 0
\(625\) −76.3084 −3.05234
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −16.5231 16.5231i −0.658817 0.658817i
\(630\) 0 0
\(631\) −26.9094 −1.07125 −0.535623 0.844457i \(-0.679923\pi\)
−0.535623 + 0.844457i \(0.679923\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −42.8957 + 42.8957i −1.70226 + 1.70226i
\(636\) 0 0
\(637\) 1.27678 + 1.27678i 0.0505878 + 0.0505878i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 8.10097i 0.319969i 0.987120 + 0.159984i \(0.0511444\pi\)
−0.987120 + 0.159984i \(0.948856\pi\)
\(642\) 0 0
\(643\) −8.56333 + 8.56333i −0.337705 + 0.337705i −0.855503 0.517798i \(-0.826752\pi\)
0.517798 + 0.855503i \(0.326752\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 47.0564i 1.84998i −0.379994 0.924989i \(-0.624074\pi\)
0.379994 0.924989i \(-0.375926\pi\)
\(648\) 0 0
\(649\) 0.554890i 0.0217813i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 32.5577 32.5577i 1.27408 1.27408i 0.330153 0.943927i \(-0.392900\pi\)
0.943927 0.330153i \(-0.107100\pi\)
\(654\) 0 0
\(655\) 80.0000i 3.12586i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 19.5376 + 19.5376i 0.761077 + 0.761077i 0.976517 0.215440i \(-0.0691186\pi\)
−0.215440 + 0.976517i \(0.569119\pi\)
\(660\) 0 0
\(661\) 20.8137 20.8137i 0.809559 0.809559i −0.175008 0.984567i \(-0.555995\pi\)
0.984567 + 0.175008i \(0.0559951\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 4.76225 0.184672
\(666\) 0 0
\(667\) −3.29722 3.29722i −0.127669 0.127669i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −51.0552 −1.97097
\(672\) 0 0
\(673\) 12.0918 0.466103 0.233052 0.972464i \(-0.425129\pi\)
0.233052 + 0.972464i \(0.425129\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −9.11293 9.11293i −0.350238 0.350238i 0.509960 0.860198i \(-0.329660\pi\)
−0.860198 + 0.509960i \(0.829660\pi\)
\(678\) 0 0
\(679\) 6.00751 0.230547
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −27.2646 + 27.2646i −1.04325 + 1.04325i −0.0442295 + 0.999021i \(0.514083\pi\)
−0.999021 + 0.0442295i \(0.985917\pi\)
\(684\) 0 0
\(685\) −25.6947 25.6947i −0.981743 0.981743i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 1.37078i 0.0522226i
\(690\) 0 0
\(691\) −0.651287 + 0.651287i −0.0247761 + 0.0247761i −0.719386 0.694610i \(-0.755577\pi\)
0.694610 + 0.719386i \(0.255577\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 15.5499i 0.589840i
\(696\) 0 0
\(697\) 31.9979i 1.21201i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −3.49984 + 3.49984i −0.132187 + 0.132187i −0.770105 0.637918i \(-0.779796\pi\)
0.637918 + 0.770105i \(0.279796\pi\)
\(702\) 0 0
\(703\) 7.39343i 0.278849i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 6.15894 + 6.15894i 0.231631 + 0.231631i
\(708\) 0 0
\(709\) 24.4219 24.4219i 0.917184 0.917184i −0.0796402 0.996824i \(-0.525377\pi\)
0.996824 + 0.0796402i \(0.0253771\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 12.5605 0.470396
\(714\) 0 0
\(715\) −17.7571 17.7571i −0.664079 0.664079i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −14.6847 −0.547648 −0.273824 0.961780i \(-0.588289\pi\)
−0.273824 + 0.961780i \(0.588289\pi\)
\(720\) 0 0
\(721\) −17.7016 −0.659241
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 24.1748 + 24.1748i 0.897828 + 0.897828i
\(726\) 0 0
\(727\) −18.8567 −0.699355 −0.349678 0.936870i \(-0.613709\pi\)
−0.349678 + 0.936870i \(0.613709\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 21.5056 21.5056i 0.795415 0.795415i
\(732\) 0 0
\(733\) −7.31258 7.31258i −0.270096 0.270096i 0.559043 0.829139i \(-0.311169\pi\)
−0.829139 + 0.559043i \(0.811169\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 17.8465i 0.657385i
\(738\) 0 0
\(739\) 32.5266 32.5266i 1.19651 1.19651i 0.221305 0.975205i \(-0.428968\pi\)
0.975205 0.221305i \(-0.0710316\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 24.8397i 0.911279i −0.890164 0.455640i \(-0.849411\pi\)
0.890164 0.455640i \(-0.150589\pi\)
\(744\) 0 0
\(745\) 81.8052i 2.99711i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −9.52746 + 9.52746i −0.348126 + 0.348126i
\(750\) 0 0
\(751\) 11.9769i 0.437042i −0.975832 0.218521i \(-0.929877\pi\)
0.975832 0.218521i \(-0.0701233\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 21.5762 + 21.5762i 0.785240 + 0.785240i
\(756\) 0 0
\(757\) 21.1197 21.1197i 0.767609 0.767609i −0.210076 0.977685i \(-0.567371\pi\)
0.977685 + 0.210076i \(0.0673710\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −22.2727 −0.807383 −0.403692 0.914895i \(-0.632273\pi\)
−0.403692 + 0.914895i \(0.632273\pi\)
\(762\) 0 0
\(763\) −11.3741 11.3741i −0.411772 0.411772i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −0.304548 −0.0109966
\(768\) 0 0
\(769\) 1.60282 0.0577992 0.0288996 0.999582i \(-0.490800\pi\)
0.0288996 + 0.999582i \(0.490800\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −1.88678 1.88678i −0.0678627 0.0678627i 0.672361 0.740224i \(-0.265280\pi\)
−0.740224 + 0.672361i \(0.765280\pi\)
\(774\) 0 0
\(775\) −92.0923 −3.30805
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −7.15891 + 7.15891i −0.256495 + 0.256495i
\(780\) 0 0
\(781\) −23.1843 23.1843i −0.829599 0.829599i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 13.6432i 0.486945i
\(786\) 0 0
\(787\) −14.9704 + 14.9704i −0.533636 + 0.533636i −0.921652 0.388016i \(-0.873160\pi\)
0.388016 + 0.921652i \(0.373160\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 9.65901i 0.343435i
\(792\) 0 0
\(793\) 28.0214i 0.995068i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −11.5079 + 11.5079i −0.407631 + 0.407631i −0.880912 0.473281i \(-0.843070\pi\)
0.473281 + 0.880912i \(0.343070\pi\)
\(798\) 0 0
\(799\) 24.8955i 0.880738i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −26.6250 26.6250i −0.939574 0.939574i
\(804\) 0 0
\(805\) 5.24757 5.24757i 0.184953 0.184953i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −18.4422 −0.648393 −0.324197 0.945990i \(-0.605094\pi\)
−0.324197 + 0.945990i \(0.605094\pi\)
\(810\) 0 0
\(811\) 20.1494 + 20.1494i 0.707541 + 0.707541i 0.966018 0.258476i \(-0.0832205\pi\)
−0.258476 + 0.966018i \(0.583220\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 76.1977 2.66909
\(816\) 0 0
\(817\) 9.62295 0.336664
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −16.2511 16.2511i −0.567167 0.567167i 0.364167 0.931334i \(-0.381354\pi\)
−0.931334 + 0.364167i \(0.881354\pi\)
\(822\) 0 0
\(823\) 29.9919 1.04545 0.522725 0.852501i \(-0.324915\pi\)
0.522725 + 0.852501i \(0.324915\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −3.98308 + 3.98308i −0.138505 + 0.138505i −0.772960 0.634455i \(-0.781225\pi\)
0.634455 + 0.772960i \(0.281225\pi\)
\(828\) 0 0
\(829\) −35.3406 35.3406i −1.22743 1.22743i −0.964933 0.262495i \(-0.915455\pi\)
−0.262495 0.964933i \(-0.584545\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 3.56038i 0.123360i
\(834\) 0 0
\(835\) −23.5108 + 23.5108i −0.813624 + 0.813624i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 9.84103i 0.339750i −0.985466 0.169875i \(-0.945664\pi\)
0.985466 0.169875i \(-0.0543364\pi\)
\(840\) 0 0
\(841\) 21.9445i 0.756707i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 29.1141 29.1141i 1.00156 1.00156i
\(846\) 0 0
\(847\) 0.176606i 0.00606827i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −8.14691 8.14691i −0.279272 0.279272i
\(852\) 0 0
\(853\) 15.4376 15.4376i 0.528575 0.528575i −0.391573 0.920147i \(-0.628069\pi\)
0.920147 + 0.391573i \(0.128069\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −22.5480 −0.770225 −0.385113 0.922870i \(-0.625837\pi\)
−0.385113 + 0.922870i \(0.625837\pi\)
\(858\) 0 0
\(859\) −11.5198 11.5198i −0.393051 0.393051i 0.482722 0.875774i \(-0.339648\pi\)
−0.875774 + 0.482722i \(0.839648\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 39.0019 1.32764 0.663821 0.747892i \(-0.268934\pi\)
0.663821 + 0.747892i \(0.268934\pi\)
\(864\) 0 0
\(865\) 20.1796 0.686127
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 35.8974 + 35.8974i 1.21774 + 1.21774i
\(870\) 0 0
\(871\) −9.79495 −0.331889
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −23.5284 + 23.5284i −0.795404 + 0.795404i
\(876\) 0 0
\(877\) 10.4024 + 10.4024i 0.351264 + 0.351264i 0.860580 0.509316i \(-0.170101\pi\)
−0.509316 + 0.860580i \(0.670101\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 19.3973i 0.653511i −0.945109 0.326755i \(-0.894045\pi\)
0.945109 0.326755i \(-0.105955\pi\)
\(882\) 0 0
\(883\) 0.435961 0.435961i 0.0146712 0.0146712i −0.699733 0.714404i \(-0.746698\pi\)
0.714404 + 0.699733i \(0.246698\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 14.2161i 0.477331i −0.971102 0.238665i \(-0.923290\pi\)
0.971102 0.238665i \(-0.0767099\pi\)
\(888\) 0 0
\(889\) 14.3501i 0.481286i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 5.56988 5.56988i 0.186389 0.186389i
\(894\) 0 0
\(895\) 16.7433i 0.559667i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 13.4387 + 13.4387i 0.448207 + 0.448207i
\(900\) 0 0
\(901\) −1.91126 + 1.91126i −0.0636732 + 0.0636732i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 27.7193 0.921420
\(906\) 0 0
\(907\) 1.15583 + 1.15583i 0.0383788 + 0.0383788i 0.726036 0.687657i \(-0.241361\pi\)
−0.687657 + 0.726036i \(0.741361\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −34.5370 −1.14426 −0.572132 0.820162i \(-0.693883\pi\)
−0.572132 + 0.820162i \(0.693883\pi\)
\(912\) 0 0
\(913\) −19.7058 −0.652168
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −13.3814 13.3814i −0.441891 0.441891i
\(918\) 0 0
\(919\) −15.0003 −0.494816 −0.247408 0.968911i \(-0.579579\pi\)
−0.247408 + 0.968911i \(0.579579\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −12.7246 + 12.7246i −0.418834 + 0.418834i
\(924\) 0 0
\(925\) 59.7321 + 59.7321i 1.96398 + 1.96398i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 6.94212i 0.227764i −0.993494 0.113882i \(-0.963671\pi\)
0.993494 0.113882i \(-0.0363285\pi\)
\(930\) 0 0
\(931\) 0.796566 0.796566i 0.0261064 0.0261064i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 49.5168i 1.61937i
\(936\) 0 0
\(937\) 8.96161i 0.292763i −0.989228 0.146382i \(-0.953237\pi\)
0.989228 0.146382i \(-0.0467627\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 21.6536 21.6536i 0.705886 0.705886i −0.259782 0.965667i \(-0.583651\pi\)
0.965667 + 0.259782i \(0.0836506\pi\)
\(942\) 0 0
\(943\) 15.7770i 0.513769i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −5.14094 5.14094i −0.167058 0.167058i 0.618627 0.785685i \(-0.287689\pi\)
−0.785685 + 0.618627i \(0.787689\pi\)
\(948\) 0 0
\(949\) −14.6130 + 14.6130i −0.474357 + 0.474357i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −29.5695 −0.957851 −0.478926 0.877856i \(-0.658974\pi\)
−0.478926 + 0.877856i \(0.658974\pi\)
\(954\) 0 0
\(955\) −42.2617 42.2617i −1.36756 1.36756i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −8.59574 −0.277571
\(960\) 0 0
\(961\) −20.1940 −0.651420
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 66.5466 + 66.5466i 2.14221 + 2.14221i
\(966\) 0 0
\(967\) 19.3751 0.623063 0.311531 0.950236i \(-0.399158\pi\)
0.311531 + 0.950236i \(0.399158\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 9.45788 9.45788i 0.303518 0.303518i −0.538871 0.842388i \(-0.681149\pi\)
0.842388 + 0.538871i \(0.181149\pi\)
\(972\) 0 0
\(973\) −2.60098 2.60098i −0.0833836 0.0833836i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 32.4796i 1.03912i −0.854435 0.519558i \(-0.826097\pi\)
0.854435 0.519558i \(-0.173903\pi\)
\(978\) 0 0
\(979\) 5.53925 5.53925i 0.177035 0.177035i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 12.8424i 0.409609i −0.978803 0.204804i \(-0.934344\pi\)
0.978803 0.204804i \(-0.0656558\pi\)
\(984\) 0 0
\(985\) 40.9106i 1.30352i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 10.6036 10.6036i 0.337176 0.337176i
\(990\) 0 0
\(991\) 9.67971i 0.307486i −0.988111 0.153743i \(-0.950867\pi\)
0.988111 0.153743i \(-0.0491328\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 4.15895 + 4.15895i 0.131848 + 0.131848i
\(996\) 0 0
\(997\) 1.70473 1.70473i 0.0539893 0.0539893i −0.679597 0.733586i \(-0.737845\pi\)
0.733586 + 0.679597i \(0.237845\pi\)
\(998\) 0 0
\(999\) 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 4032.2.v.d.1583.18 36
3.2 odd 2 inner 4032.2.v.d.1583.1 36
4.3 odd 2 1008.2.v.d.323.16 yes 36
12.11 even 2 1008.2.v.d.323.3 36
16.5 even 4 1008.2.v.d.827.3 yes 36
16.11 odd 4 inner 4032.2.v.d.3599.1 36
48.5 odd 4 1008.2.v.d.827.16 yes 36
48.11 even 4 inner 4032.2.v.d.3599.18 36
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1008.2.v.d.323.3 36 12.11 even 2
1008.2.v.d.323.16 yes 36 4.3 odd 2
1008.2.v.d.827.3 yes 36 16.5 even 4
1008.2.v.d.827.16 yes 36 48.5 odd 4
4032.2.v.d.1583.1 36 3.2 odd 2 inner
4032.2.v.d.1583.18 36 1.1 even 1 trivial
4032.2.v.d.3599.1 36 16.11 odd 4 inner
4032.2.v.d.3599.18 36 48.11 even 4 inner