Properties

Label 3072.2.d.j.1537.1
Level $3072$
Weight $2$
Character 3072.1537
Analytic conductor $24.530$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3072,2,Mod(1537,3072)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3072, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3072.1537");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3072 = 2^{10} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3072.d (of order \(2\), degree \(1\), 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: \(24.5300435009\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 1536)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1537.1
Root \(-0.382683 + 0.923880i\) of defining polynomial
Character \(\chi\) \(=\) 3072.1537
Dual form 3072.2.d.j.1537.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} -2.49661i q^{5} +0.917608 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} -2.49661i q^{5} +0.917608 q^{7} -1.00000 q^{9} +3.69552i q^{11} +5.81204i q^{13} -2.49661 q^{15} -0.867091 q^{17} -6.52395i q^{19} -0.917608i q^{21} +4.00000 q^{23} -1.23304 q^{25} +1.00000i q^{27} -7.72286i q^{29} +2.14386 q^{31} +3.69552 q^{33} -2.29090i q^{35} +2.47568i q^{37} +5.81204 q^{39} +9.58541 q^{41} -9.58541i q^{43} +2.49661i q^{45} +1.65685 q^{47} -6.15800 q^{49} +0.867091i q^{51} -3.39329i q^{53} +9.22625 q^{55} -6.52395 q^{57} -12.7183i q^{59} +0.0231773i q^{61} -0.917608 q^{63} +14.5104 q^{65} -5.32729i q^{67} -4.00000i q^{69} +11.8216 q^{71} +15.2809 q^{73} +1.23304i q^{75} +3.39104i q^{77} -8.40968 q^{79} +1.00000 q^{81} -1.96134i q^{83} +2.16478i q^{85} -7.72286 q^{87} -2.79565 q^{89} +5.33317i q^{91} -2.14386i q^{93} -16.2877 q^{95} -2.26582 q^{97} -3.69552i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{7} - 8 q^{9} + 32 q^{23} - 8 q^{25} - 16 q^{31} + 16 q^{39} - 32 q^{47} + 8 q^{49} + 32 q^{55} - 16 q^{63} - 16 q^{65} + 32 q^{71} + 16 q^{73} - 48 q^{79} + 8 q^{81} + 16 q^{89} - 64 q^{95} - 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3072\mathbb{Z}\right)^\times\).

\(n\) \(1025\) \(2047\) \(2053\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) − 1.00000i − 0.577350i
\(4\) 0 0
\(5\) − 2.49661i − 1.11652i −0.829667 0.558258i \(-0.811470\pi\)
0.829667 0.558258i \(-0.188530\pi\)
\(6\) 0 0
\(7\) 0.917608 0.346823 0.173412 0.984849i \(-0.444521\pi\)
0.173412 + 0.984849i \(0.444521\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 3.69552i 1.11424i 0.830432 + 0.557120i \(0.188094\pi\)
−0.830432 + 0.557120i \(0.811906\pi\)
\(12\) 0 0
\(13\) 5.81204i 1.61197i 0.591936 + 0.805985i \(0.298364\pi\)
−0.591936 + 0.805985i \(0.701636\pi\)
\(14\) 0 0
\(15\) −2.49661 −0.644621
\(16\) 0 0
\(17\) −0.867091 −0.210300 −0.105150 0.994456i \(-0.533532\pi\)
−0.105150 + 0.994456i \(0.533532\pi\)
\(18\) 0 0
\(19\) − 6.52395i − 1.49670i −0.663306 0.748348i \(-0.730847\pi\)
0.663306 0.748348i \(-0.269153\pi\)
\(20\) 0 0
\(21\) − 0.917608i − 0.200238i
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) −1.23304 −0.246608
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) − 7.72286i − 1.43410i −0.697022 0.717049i \(-0.745492\pi\)
0.697022 0.717049i \(-0.254508\pi\)
\(30\) 0 0
\(31\) 2.14386 0.385049 0.192524 0.981292i \(-0.438333\pi\)
0.192524 + 0.981292i \(0.438333\pi\)
\(32\) 0 0
\(33\) 3.69552 0.643307
\(34\) 0 0
\(35\) − 2.29090i − 0.387234i
\(36\) 0 0
\(37\) 2.47568i 0.406999i 0.979075 + 0.203500i \(0.0652316\pi\)
−0.979075 + 0.203500i \(0.934768\pi\)
\(38\) 0 0
\(39\) 5.81204 0.930671
\(40\) 0 0
\(41\) 9.58541 1.49699 0.748495 0.663140i \(-0.230777\pi\)
0.748495 + 0.663140i \(0.230777\pi\)
\(42\) 0 0
\(43\) − 9.58541i − 1.46176i −0.682505 0.730881i \(-0.739109\pi\)
0.682505 0.730881i \(-0.260891\pi\)
\(44\) 0 0
\(45\) 2.49661i 0.372172i
\(46\) 0 0
\(47\) 1.65685 0.241677 0.120839 0.992672i \(-0.461442\pi\)
0.120839 + 0.992672i \(0.461442\pi\)
\(48\) 0 0
\(49\) −6.15800 −0.879714
\(50\) 0 0
\(51\) 0.867091i 0.121417i
\(52\) 0 0
\(53\) − 3.39329i − 0.466104i −0.972464 0.233052i \(-0.925129\pi\)
0.972464 0.233052i \(-0.0748712\pi\)
\(54\) 0 0
\(55\) 9.22625 1.24407
\(56\) 0 0
\(57\) −6.52395 −0.864118
\(58\) 0 0
\(59\) − 12.7183i − 1.65578i −0.560887 0.827892i \(-0.689540\pi\)
0.560887 0.827892i \(-0.310460\pi\)
\(60\) 0 0
\(61\) 0.0231773i 0.00296755i 0.999999 + 0.00148377i \(0.000472300\pi\)
−0.999999 + 0.00148377i \(0.999528\pi\)
\(62\) 0 0
\(63\) −0.917608 −0.115608
\(64\) 0 0
\(65\) 14.5104 1.79979
\(66\) 0 0
\(67\) − 5.32729i − 0.650832i −0.945571 0.325416i \(-0.894496\pi\)
0.945571 0.325416i \(-0.105504\pi\)
\(68\) 0 0
\(69\) − 4.00000i − 0.481543i
\(70\) 0 0
\(71\) 11.8216 1.40297 0.701485 0.712684i \(-0.252521\pi\)
0.701485 + 0.712684i \(0.252521\pi\)
\(72\) 0 0
\(73\) 15.2809 1.78850 0.894249 0.447570i \(-0.147711\pi\)
0.894249 + 0.447570i \(0.147711\pi\)
\(74\) 0 0
\(75\) 1.23304i 0.142379i
\(76\) 0 0
\(77\) 3.39104i 0.386444i
\(78\) 0 0
\(79\) −8.40968 −0.946163 −0.473081 0.881019i \(-0.656858\pi\)
−0.473081 + 0.881019i \(0.656858\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 1.96134i − 0.215285i −0.994190 0.107642i \(-0.965670\pi\)
0.994190 0.107642i \(-0.0343301\pi\)
\(84\) 0 0
\(85\) 2.16478i 0.234804i
\(86\) 0 0
\(87\) −7.72286 −0.827977
\(88\) 0 0
\(89\) −2.79565 −0.296338 −0.148169 0.988962i \(-0.547338\pi\)
−0.148169 + 0.988962i \(0.547338\pi\)
\(90\) 0 0
\(91\) 5.33317i 0.559068i
\(92\) 0 0
\(93\) − 2.14386i − 0.222308i
\(94\) 0 0
\(95\) −16.2877 −1.67108
\(96\) 0 0
\(97\) −2.26582 −0.230059 −0.115029 0.993362i \(-0.536696\pi\)
−0.115029 + 0.993362i \(0.536696\pi\)
\(98\) 0 0
\(99\) − 3.69552i − 0.371414i
\(100\) 0 0
\(101\) 9.98868i 0.993910i 0.867776 + 0.496955i \(0.165549\pi\)
−0.867776 + 0.496955i \(0.834451\pi\)
\(102\) 0 0
\(103\) −16.4180 −1.61771 −0.808857 0.588006i \(-0.799913\pi\)
−0.808857 + 0.588006i \(0.799913\pi\)
\(104\) 0 0
\(105\) −2.29090 −0.223569
\(106\) 0 0
\(107\) − 12.2522i − 1.18447i −0.805766 0.592234i \(-0.798246\pi\)
0.805766 0.592234i \(-0.201754\pi\)
\(108\) 0 0
\(109\) 18.2973i 1.75257i 0.481797 + 0.876283i \(0.339984\pi\)
−0.481797 + 0.876283i \(0.660016\pi\)
\(110\) 0 0
\(111\) 2.47568 0.234981
\(112\) 0 0
\(113\) 3.65685 0.344008 0.172004 0.985096i \(-0.444976\pi\)
0.172004 + 0.985096i \(0.444976\pi\)
\(114\) 0 0
\(115\) − 9.98642i − 0.931239i
\(116\) 0 0
\(117\) − 5.81204i − 0.537323i
\(118\) 0 0
\(119\) −0.795649 −0.0729371
\(120\) 0 0
\(121\) −2.65685 −0.241532
\(122\) 0 0
\(123\) − 9.58541i − 0.864288i
\(124\) 0 0
\(125\) − 9.40461i − 0.841174i
\(126\) 0 0
\(127\) −20.6382 −1.83135 −0.915673 0.401925i \(-0.868341\pi\)
−0.915673 + 0.401925i \(0.868341\pi\)
\(128\) 0 0
\(129\) −9.58541 −0.843949
\(130\) 0 0
\(131\) − 17.5140i − 1.53020i −0.643910 0.765101i \(-0.722689\pi\)
0.643910 0.765101i \(-0.277311\pi\)
\(132\) 0 0
\(133\) − 5.98642i − 0.519089i
\(134\) 0 0
\(135\) 2.49661 0.214874
\(136\) 0 0
\(137\) −8.25813 −0.705539 −0.352770 0.935710i \(-0.614760\pi\)
−0.352770 + 0.935710i \(0.614760\pi\)
\(138\) 0 0
\(139\) − 14.7047i − 1.24724i −0.781728 0.623620i \(-0.785661\pi\)
0.781728 0.623620i \(-0.214339\pi\)
\(140\) 0 0
\(141\) − 1.65685i − 0.139532i
\(142\) 0 0
\(143\) −21.4785 −1.79612
\(144\) 0 0
\(145\) −19.2809 −1.60119
\(146\) 0 0
\(147\) 6.15800i 0.507903i
\(148\) 0 0
\(149\) 16.1535i 1.32334i 0.749794 + 0.661672i \(0.230153\pi\)
−0.749794 + 0.661672i \(0.769847\pi\)
\(150\) 0 0
\(151\) 19.2691 1.56810 0.784048 0.620701i \(-0.213152\pi\)
0.784048 + 0.620701i \(0.213152\pi\)
\(152\) 0 0
\(153\) 0.867091 0.0701002
\(154\) 0 0
\(155\) − 5.35237i − 0.429913i
\(156\) 0 0
\(157\) 0.818827i 0.0653495i 0.999466 + 0.0326747i \(0.0104025\pi\)
−0.999466 + 0.0326747i \(0.989597\pi\)
\(158\) 0 0
\(159\) −3.39329 −0.269105
\(160\) 0 0
\(161\) 3.67043 0.289271
\(162\) 0 0
\(163\) 23.6492i 1.85235i 0.377100 + 0.926173i \(0.376921\pi\)
−0.377100 + 0.926173i \(0.623079\pi\)
\(164\) 0 0
\(165\) − 9.22625i − 0.718263i
\(166\) 0 0
\(167\) 9.30358 0.719933 0.359966 0.932965i \(-0.382788\pi\)
0.359966 + 0.932965i \(0.382788\pi\)
\(168\) 0 0
\(169\) −20.7798 −1.59845
\(170\) 0 0
\(171\) 6.52395i 0.498899i
\(172\) 0 0
\(173\) − 7.10557i − 0.540226i −0.962829 0.270113i \(-0.912939\pi\)
0.962829 0.270113i \(-0.0870611\pi\)
\(174\) 0 0
\(175\) −1.13145 −0.0855294
\(176\) 0 0
\(177\) −12.7183 −0.955968
\(178\) 0 0
\(179\) 2.67271i 0.199768i 0.994999 + 0.0998840i \(0.0318472\pi\)
−0.994999 + 0.0998840i \(0.968153\pi\)
\(180\) 0 0
\(181\) 0.640465i 0.0476054i 0.999717 + 0.0238027i \(0.00757735\pi\)
−0.999717 + 0.0238027i \(0.992423\pi\)
\(182\) 0 0
\(183\) 0.0231773 0.00171332
\(184\) 0 0
\(185\) 6.18080 0.454421
\(186\) 0 0
\(187\) − 3.20435i − 0.234325i
\(188\) 0 0
\(189\) 0.917608i 0.0667461i
\(190\) 0 0
\(191\) 14.6037 1.05669 0.528344 0.849031i \(-0.322813\pi\)
0.528344 + 0.849031i \(0.322813\pi\)
\(192\) 0 0
\(193\) −8.81485 −0.634507 −0.317253 0.948341i \(-0.602761\pi\)
−0.317253 + 0.948341i \(0.602761\pi\)
\(194\) 0 0
\(195\) − 14.5104i − 1.03911i
\(196\) 0 0
\(197\) 3.74551i 0.266856i 0.991058 + 0.133428i \(0.0425985\pi\)
−0.991058 + 0.133428i \(0.957401\pi\)
\(198\) 0 0
\(199\) 14.8267 1.05104 0.525519 0.850782i \(-0.323871\pi\)
0.525519 + 0.850782i \(0.323871\pi\)
\(200\) 0 0
\(201\) −5.32729 −0.375758
\(202\) 0 0
\(203\) − 7.08655i − 0.497379i
\(204\) 0 0
\(205\) − 23.9310i − 1.67141i
\(206\) 0 0
\(207\) −4.00000 −0.278019
\(208\) 0 0
\(209\) 24.1094 1.66768
\(210\) 0 0
\(211\) − 11.3910i − 0.784191i −0.919924 0.392096i \(-0.871750\pi\)
0.919924 0.392096i \(-0.128250\pi\)
\(212\) 0 0
\(213\) − 11.8216i − 0.810005i
\(214\) 0 0
\(215\) −23.9310 −1.63208
\(216\) 0 0
\(217\) 1.96722 0.133544
\(218\) 0 0
\(219\) − 15.2809i − 1.03259i
\(220\) 0 0
\(221\) − 5.03957i − 0.338998i
\(222\) 0 0
\(223\) −4.78110 −0.320166 −0.160083 0.987104i \(-0.551176\pi\)
−0.160083 + 0.987104i \(0.551176\pi\)
\(224\) 0 0
\(225\) 1.23304 0.0822027
\(226\) 0 0
\(227\) − 3.83840i − 0.254764i −0.991854 0.127382i \(-0.959343\pi\)
0.991854 0.127382i \(-0.0406574\pi\)
\(228\) 0 0
\(229\) 1.34596i 0.0889434i 0.999011 + 0.0444717i \(0.0141604\pi\)
−0.999011 + 0.0444717i \(0.985840\pi\)
\(230\) 0 0
\(231\) 3.39104 0.223114
\(232\) 0 0
\(233\) 17.6433 1.15585 0.577925 0.816090i \(-0.303863\pi\)
0.577925 + 0.816090i \(0.303863\pi\)
\(234\) 0 0
\(235\) − 4.13651i − 0.269836i
\(236\) 0 0
\(237\) 8.40968i 0.546267i
\(238\) 0 0
\(239\) 9.61500 0.621943 0.310971 0.950419i \(-0.399346\pi\)
0.310971 + 0.950419i \(0.399346\pi\)
\(240\) 0 0
\(241\) 21.3583 1.37581 0.687903 0.725802i \(-0.258531\pi\)
0.687903 + 0.725802i \(0.258531\pi\)
\(242\) 0 0
\(243\) − 1.00000i − 0.0641500i
\(244\) 0 0
\(245\) 15.3741i 0.982214i
\(246\) 0 0
\(247\) 37.9174 2.41263
\(248\) 0 0
\(249\) −1.96134 −0.124295
\(250\) 0 0
\(251\) 4.30448i 0.271696i 0.990730 + 0.135848i \(0.0433760\pi\)
−0.990730 + 0.135848i \(0.956624\pi\)
\(252\) 0 0
\(253\) 14.7821i 0.929341i
\(254\) 0 0
\(255\) 2.16478 0.135564
\(256\) 0 0
\(257\) −4.67271 −0.291476 −0.145738 0.989323i \(-0.546556\pi\)
−0.145738 + 0.989323i \(0.546556\pi\)
\(258\) 0 0
\(259\) 2.27170i 0.141157i
\(260\) 0 0
\(261\) 7.72286i 0.478033i
\(262\) 0 0
\(263\) 15.4502 0.952701 0.476351 0.879255i \(-0.341959\pi\)
0.476351 + 0.879255i \(0.341959\pi\)
\(264\) 0 0
\(265\) −8.47170 −0.520413
\(266\) 0 0
\(267\) 2.79565i 0.171091i
\(268\) 0 0
\(269\) − 31.2459i − 1.90510i −0.304388 0.952548i \(-0.598452\pi\)
0.304388 0.952548i \(-0.401548\pi\)
\(270\) 0 0
\(271\) −9.88817 −0.600664 −0.300332 0.953835i \(-0.597097\pi\)
−0.300332 + 0.953835i \(0.597097\pi\)
\(272\) 0 0
\(273\) 5.33317 0.322778
\(274\) 0 0
\(275\) − 4.55672i − 0.274781i
\(276\) 0 0
\(277\) − 22.9565i − 1.37932i −0.724133 0.689660i \(-0.757760\pi\)
0.724133 0.689660i \(-0.242240\pi\)
\(278\) 0 0
\(279\) −2.14386 −0.128350
\(280\) 0 0
\(281\) 14.5754 0.869498 0.434749 0.900552i \(-0.356837\pi\)
0.434749 + 0.900552i \(0.356837\pi\)
\(282\) 0 0
\(283\) 4.71832i 0.280475i 0.990118 + 0.140238i \(0.0447866\pi\)
−0.990118 + 0.140238i \(0.955213\pi\)
\(284\) 0 0
\(285\) 16.2877i 0.964801i
\(286\) 0 0
\(287\) 8.79565 0.519191
\(288\) 0 0
\(289\) −16.2482 −0.955774
\(290\) 0 0
\(291\) 2.26582i 0.132825i
\(292\) 0 0
\(293\) 23.9204i 1.39745i 0.715392 + 0.698723i \(0.246248\pi\)
−0.715392 + 0.698723i \(0.753752\pi\)
\(294\) 0 0
\(295\) −31.7526 −1.84871
\(296\) 0 0
\(297\) −3.69552 −0.214436
\(298\) 0 0
\(299\) 23.2482i 1.34448i
\(300\) 0 0
\(301\) − 8.79565i − 0.506973i
\(302\) 0 0
\(303\) 9.98868 0.573834
\(304\) 0 0
\(305\) 0.0578646 0.00331332
\(306\) 0 0
\(307\) − 20.7685i − 1.18532i −0.805453 0.592660i \(-0.798078\pi\)
0.805453 0.592660i \(-0.201922\pi\)
\(308\) 0 0
\(309\) 16.4180i 0.933988i
\(310\) 0 0
\(311\) 11.7798 0.667971 0.333985 0.942578i \(-0.391606\pi\)
0.333985 + 0.942578i \(0.391606\pi\)
\(312\) 0 0
\(313\) −11.1580 −0.630687 −0.315344 0.948978i \(-0.602120\pi\)
−0.315344 + 0.948978i \(0.602120\pi\)
\(314\) 0 0
\(315\) 2.29090i 0.129078i
\(316\) 0 0
\(317\) 15.9140i 0.893822i 0.894578 + 0.446911i \(0.147476\pi\)
−0.894578 + 0.446911i \(0.852524\pi\)
\(318\) 0 0
\(319\) 28.5400 1.59793
\(320\) 0 0
\(321\) −12.2522 −0.683853
\(322\) 0 0
\(323\) 5.65685i 0.314756i
\(324\) 0 0
\(325\) − 7.16648i − 0.397525i
\(326\) 0 0
\(327\) 18.2973 1.01184
\(328\) 0 0
\(329\) 1.52034 0.0838192
\(330\) 0 0
\(331\) − 2.73190i − 0.150159i −0.997178 0.0750794i \(-0.976079\pi\)
0.997178 0.0750794i \(-0.0239210\pi\)
\(332\) 0 0
\(333\) − 2.47568i − 0.135666i
\(334\) 0 0
\(335\) −13.3001 −0.726664
\(336\) 0 0
\(337\) −1.49657 −0.0815236 −0.0407618 0.999169i \(-0.512978\pi\)
−0.0407618 + 0.999169i \(0.512978\pi\)
\(338\) 0 0
\(339\) − 3.65685i − 0.198613i
\(340\) 0 0
\(341\) 7.92267i 0.429037i
\(342\) 0 0
\(343\) −12.0739 −0.651928
\(344\) 0 0
\(345\) −9.98642 −0.537651
\(346\) 0 0
\(347\) 3.22944i 0.173365i 0.996236 + 0.0866826i \(0.0276266\pi\)
−0.996236 + 0.0866826i \(0.972373\pi\)
\(348\) 0 0
\(349\) 7.51074i 0.402041i 0.979587 + 0.201020i \(0.0644258\pi\)
−0.979587 + 0.201020i \(0.935574\pi\)
\(350\) 0 0
\(351\) −5.81204 −0.310224
\(352\) 0 0
\(353\) 2.17491 0.115759 0.0578795 0.998324i \(-0.481566\pi\)
0.0578795 + 0.998324i \(0.481566\pi\)
\(354\) 0 0
\(355\) − 29.5140i − 1.56644i
\(356\) 0 0
\(357\) 0.795649i 0.0421102i
\(358\) 0 0
\(359\) −12.6828 −0.669375 −0.334687 0.942329i \(-0.608631\pi\)
−0.334687 + 0.942329i \(0.608631\pi\)
\(360\) 0 0
\(361\) −23.5619 −1.24010
\(362\) 0 0
\(363\) 2.65685i 0.139449i
\(364\) 0 0
\(365\) − 38.1505i − 1.99689i
\(366\) 0 0
\(367\) 21.4576 1.12008 0.560038 0.828467i \(-0.310787\pi\)
0.560038 + 0.828467i \(0.310787\pi\)
\(368\) 0 0
\(369\) −9.58541 −0.498997
\(370\) 0 0
\(371\) − 3.11371i − 0.161656i
\(372\) 0 0
\(373\) 0.508459i 0.0263270i 0.999913 + 0.0131635i \(0.00419019\pi\)
−0.999913 + 0.0131635i \(0.995810\pi\)
\(374\) 0 0
\(375\) −9.40461 −0.485652
\(376\) 0 0
\(377\) 44.8855 2.31172
\(378\) 0 0
\(379\) 19.9286i 1.02366i 0.859086 + 0.511831i \(0.171032\pi\)
−0.859086 + 0.511831i \(0.828968\pi\)
\(380\) 0 0
\(381\) 20.6382i 1.05733i
\(382\) 0 0
\(383\) −29.3858 −1.50154 −0.750772 0.660562i \(-0.770318\pi\)
−0.750772 + 0.660562i \(0.770318\pi\)
\(384\) 0 0
\(385\) 8.46608 0.431471
\(386\) 0 0
\(387\) 9.58541i 0.487254i
\(388\) 0 0
\(389\) − 1.00379i − 0.0508941i −0.999676 0.0254470i \(-0.991899\pi\)
0.999676 0.0254470i \(-0.00810092\pi\)
\(390\) 0 0
\(391\) −3.46836 −0.175403
\(392\) 0 0
\(393\) −17.5140 −0.883463
\(394\) 0 0
\(395\) 20.9956i 1.05641i
\(396\) 0 0
\(397\) 7.02546i 0.352598i 0.984337 + 0.176299i \(0.0564125\pi\)
−0.984337 + 0.176299i \(0.943587\pi\)
\(398\) 0 0
\(399\) −5.98642 −0.299696
\(400\) 0 0
\(401\) 4.18080 0.208779 0.104390 0.994536i \(-0.466711\pi\)
0.104390 + 0.994536i \(0.466711\pi\)
\(402\) 0 0
\(403\) 12.4602i 0.620686i
\(404\) 0 0
\(405\) − 2.49661i − 0.124057i
\(406\) 0 0
\(407\) −9.14892 −0.453495
\(408\) 0 0
\(409\) −16.5818 −0.819918 −0.409959 0.912104i \(-0.634457\pi\)
−0.409959 + 0.912104i \(0.634457\pi\)
\(410\) 0 0
\(411\) 8.25813i 0.407343i
\(412\) 0 0
\(413\) − 11.6704i − 0.574264i
\(414\) 0 0
\(415\) −4.89668 −0.240369
\(416\) 0 0
\(417\) −14.7047 −0.720094
\(418\) 0 0
\(419\) − 22.2574i − 1.08734i −0.839298 0.543672i \(-0.817034\pi\)
0.839298 0.543672i \(-0.182966\pi\)
\(420\) 0 0
\(421\) − 28.4203i − 1.38512i −0.721361 0.692559i \(-0.756483\pi\)
0.721361 0.692559i \(-0.243517\pi\)
\(422\) 0 0
\(423\) −1.65685 −0.0805590
\(424\) 0 0
\(425\) 1.06916 0.0518618
\(426\) 0 0
\(427\) 0.0212677i 0.00102921i
\(428\) 0 0
\(429\) 21.4785i 1.03699i
\(430\) 0 0
\(431\) 36.6138 1.76363 0.881813 0.471599i \(-0.156323\pi\)
0.881813 + 0.471599i \(0.156323\pi\)
\(432\) 0 0
\(433\) 27.7526 1.33371 0.666853 0.745189i \(-0.267641\pi\)
0.666853 + 0.745189i \(0.267641\pi\)
\(434\) 0 0
\(435\) 19.2809i 0.924450i
\(436\) 0 0
\(437\) − 26.0958i − 1.24833i
\(438\) 0 0
\(439\) −21.1461 −1.00925 −0.504625 0.863339i \(-0.668369\pi\)
−0.504625 + 0.863339i \(0.668369\pi\)
\(440\) 0 0
\(441\) 6.15800 0.293238
\(442\) 0 0
\(443\) − 7.30677i − 0.347155i −0.984820 0.173577i \(-0.944467\pi\)
0.984820 0.173577i \(-0.0555327\pi\)
\(444\) 0 0
\(445\) 6.97963i 0.330866i
\(446\) 0 0
\(447\) 16.1535 0.764032
\(448\) 0 0
\(449\) −17.6356 −0.832275 −0.416137 0.909302i \(-0.636616\pi\)
−0.416137 + 0.909302i \(0.636616\pi\)
\(450\) 0 0
\(451\) 35.4231i 1.66801i
\(452\) 0 0
\(453\) − 19.2691i − 0.905340i
\(454\) 0 0
\(455\) 13.3148 0.624209
\(456\) 0 0
\(457\) −17.7070 −0.828300 −0.414150 0.910209i \(-0.635921\pi\)
−0.414150 + 0.910209i \(0.635921\pi\)
\(458\) 0 0
\(459\) − 0.867091i − 0.0404723i
\(460\) 0 0
\(461\) 36.9811i 1.72238i 0.508280 + 0.861192i \(0.330281\pi\)
−0.508280 + 0.861192i \(0.669719\pi\)
\(462\) 0 0
\(463\) −10.9632 −0.509504 −0.254752 0.967006i \(-0.581994\pi\)
−0.254752 + 0.967006i \(0.581994\pi\)
\(464\) 0 0
\(465\) −5.35237 −0.248210
\(466\) 0 0
\(467\) 12.6661i 0.586116i 0.956095 + 0.293058i \(0.0946730\pi\)
−0.956095 + 0.293058i \(0.905327\pi\)
\(468\) 0 0
\(469\) − 4.88836i − 0.225723i
\(470\) 0 0
\(471\) 0.818827 0.0377295
\(472\) 0 0
\(473\) 35.4231 1.62875
\(474\) 0 0
\(475\) 8.04429i 0.369097i
\(476\) 0 0
\(477\) 3.39329i 0.155368i
\(478\) 0 0
\(479\) 13.3036 0.607856 0.303928 0.952695i \(-0.401702\pi\)
0.303928 + 0.952695i \(0.401702\pi\)
\(480\) 0 0
\(481\) −14.3888 −0.656071
\(482\) 0 0
\(483\) − 3.67043i − 0.167010i
\(484\) 0 0
\(485\) 5.65685i 0.256865i
\(486\) 0 0
\(487\) −5.44924 −0.246929 −0.123464 0.992349i \(-0.539400\pi\)
−0.123464 + 0.992349i \(0.539400\pi\)
\(488\) 0 0
\(489\) 23.6492 1.06945
\(490\) 0 0
\(491\) − 21.7844i − 0.983114i −0.870845 0.491557i \(-0.836428\pi\)
0.870845 0.491557i \(-0.163572\pi\)
\(492\) 0 0
\(493\) 6.69642i 0.301592i
\(494\) 0 0
\(495\) −9.22625 −0.414689
\(496\) 0 0
\(497\) 10.8476 0.486583
\(498\) 0 0
\(499\) 20.1595i 0.902465i 0.892407 + 0.451232i \(0.149015\pi\)
−0.892407 + 0.451232i \(0.850985\pi\)
\(500\) 0 0
\(501\) − 9.30358i − 0.415653i
\(502\) 0 0
\(503\) 33.5879 1.49761 0.748804 0.662791i \(-0.230628\pi\)
0.748804 + 0.662791i \(0.230628\pi\)
\(504\) 0 0
\(505\) 24.9378 1.10972
\(506\) 0 0
\(507\) 20.7798i 0.922863i
\(508\) 0 0
\(509\) − 11.1957i − 0.496242i −0.968729 0.248121i \(-0.920187\pi\)
0.968729 0.248121i \(-0.0798130\pi\)
\(510\) 0 0
\(511\) 14.0219 0.620292
\(512\) 0 0
\(513\) 6.52395 0.288039
\(514\) 0 0
\(515\) 40.9893i 1.80620i
\(516\) 0 0
\(517\) 6.12293i 0.269286i
\(518\) 0 0
\(519\) −7.10557 −0.311900
\(520\) 0 0
\(521\) 35.2151 1.54280 0.771401 0.636349i \(-0.219556\pi\)
0.771401 + 0.636349i \(0.219556\pi\)
\(522\) 0 0
\(523\) − 12.9444i − 0.566020i −0.959117 0.283010i \(-0.908667\pi\)
0.959117 0.283010i \(-0.0913329\pi\)
\(524\) 0 0
\(525\) 1.13145i 0.0493804i
\(526\) 0 0
\(527\) −1.85892 −0.0809759
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 12.7183i 0.551928i
\(532\) 0 0
\(533\) 55.7108i 2.41310i
\(534\) 0 0
\(535\) −30.5890 −1.32248
\(536\) 0 0
\(537\) 2.67271 0.115336
\(538\) 0 0
\(539\) − 22.7570i − 0.980213i
\(540\) 0 0
\(541\) 17.8584i 0.767792i 0.923376 + 0.383896i \(0.125418\pi\)
−0.923376 + 0.383896i \(0.874582\pi\)
\(542\) 0 0
\(543\) 0.640465 0.0274850
\(544\) 0 0
\(545\) 45.6812 1.95677
\(546\) 0 0
\(547\) 14.9191i 0.637893i 0.947773 + 0.318947i \(0.103329\pi\)
−0.947773 + 0.318947i \(0.896671\pi\)
\(548\) 0 0
\(549\) − 0.0231773i 0 0.000989183i
\(550\) 0 0
\(551\) −50.3835 −2.14641
\(552\) 0 0
\(553\) −7.71679 −0.328151
\(554\) 0 0
\(555\) − 6.18080i − 0.262360i
\(556\) 0 0
\(557\) − 24.4649i − 1.03661i −0.855196 0.518305i \(-0.826563\pi\)
0.855196 0.518305i \(-0.173437\pi\)
\(558\) 0 0
\(559\) 55.7108 2.35632
\(560\) 0 0
\(561\) −3.20435 −0.135288
\(562\) 0 0
\(563\) 40.4459i 1.70459i 0.523061 + 0.852295i \(0.324790\pi\)
−0.523061 + 0.852295i \(0.675210\pi\)
\(564\) 0 0
\(565\) − 9.12972i − 0.384090i
\(566\) 0 0
\(567\) 0.917608 0.0385359
\(568\) 0 0
\(569\) −42.1017 −1.76499 −0.882497 0.470318i \(-0.844139\pi\)
−0.882497 + 0.470318i \(0.844139\pi\)
\(570\) 0 0
\(571\) 1.24996i 0.0523091i 0.999658 + 0.0261546i \(0.00832620\pi\)
−0.999658 + 0.0261546i \(0.991674\pi\)
\(572\) 0 0
\(573\) − 14.6037i − 0.610079i
\(574\) 0 0
\(575\) −4.93216 −0.205685
\(576\) 0 0
\(577\) 29.3767 1.22297 0.611484 0.791257i \(-0.290573\pi\)
0.611484 + 0.791257i \(0.290573\pi\)
\(578\) 0 0
\(579\) 8.81485i 0.366333i
\(580\) 0 0
\(581\) − 1.79974i − 0.0746657i
\(582\) 0 0
\(583\) 12.5400 0.519352
\(584\) 0 0
\(585\) −14.5104 −0.599930
\(586\) 0 0
\(587\) 17.7070i 0.730847i 0.930841 + 0.365424i \(0.119076\pi\)
−0.930841 + 0.365424i \(0.880924\pi\)
\(588\) 0 0
\(589\) − 13.9864i − 0.576301i
\(590\) 0 0
\(591\) 3.74551 0.154070
\(592\) 0 0
\(593\) 17.1207 0.703061 0.351530 0.936176i \(-0.385661\pi\)
0.351530 + 0.936176i \(0.385661\pi\)
\(594\) 0 0
\(595\) 1.98642i 0.0814354i
\(596\) 0 0
\(597\) − 14.8267i − 0.606817i
\(598\) 0 0
\(599\) −25.1908 −1.02927 −0.514634 0.857410i \(-0.672072\pi\)
−0.514634 + 0.857410i \(0.672072\pi\)
\(600\) 0 0
\(601\) −22.5946 −0.921655 −0.460827 0.887490i \(-0.652447\pi\)
−0.460827 + 0.887490i \(0.652447\pi\)
\(602\) 0 0
\(603\) 5.32729i 0.216944i
\(604\) 0 0
\(605\) 6.63312i 0.269675i
\(606\) 0 0
\(607\) −8.88408 −0.360594 −0.180297 0.983612i \(-0.557706\pi\)
−0.180297 + 0.983612i \(0.557706\pi\)
\(608\) 0 0
\(609\) −7.08655 −0.287162
\(610\) 0 0
\(611\) 9.62970i 0.389576i
\(612\) 0 0
\(613\) − 15.9960i − 0.646073i −0.946386 0.323037i \(-0.895296\pi\)
0.946386 0.323037i \(-0.104704\pi\)
\(614\) 0 0
\(615\) −23.9310 −0.964991
\(616\) 0 0
\(617\) 2.99772 0.120684 0.0603418 0.998178i \(-0.480781\pi\)
0.0603418 + 0.998178i \(0.480781\pi\)
\(618\) 0 0
\(619\) 15.0279i 0.604024i 0.953304 + 0.302012i \(0.0976583\pi\)
−0.953304 + 0.302012i \(0.902342\pi\)
\(620\) 0 0
\(621\) 4.00000i 0.160514i
\(622\) 0 0
\(623\) −2.56531 −0.102777
\(624\) 0 0
\(625\) −29.6448 −1.18579
\(626\) 0 0
\(627\) − 24.1094i − 0.962835i
\(628\) 0 0
\(629\) − 2.14664i − 0.0855922i
\(630\) 0 0
\(631\) −19.3701 −0.771112 −0.385556 0.922684i \(-0.625990\pi\)
−0.385556 + 0.922684i \(0.625990\pi\)
\(632\) 0 0
\(633\) −11.3910 −0.452753
\(634\) 0 0
\(635\) 51.5255i 2.04473i
\(636\) 0 0
\(637\) − 35.7905i − 1.41807i
\(638\) 0 0
\(639\) −11.8216 −0.467657
\(640\) 0 0
\(641\) 13.4489 0.531200 0.265600 0.964083i \(-0.414430\pi\)
0.265600 + 0.964083i \(0.414430\pi\)
\(642\) 0 0
\(643\) 2.44662i 0.0964852i 0.998836 + 0.0482426i \(0.0153621\pi\)
−0.998836 + 0.0482426i \(0.984638\pi\)
\(644\) 0 0
\(645\) 23.9310i 0.942282i
\(646\) 0 0
\(647\) −16.1275 −0.634038 −0.317019 0.948419i \(-0.602682\pi\)
−0.317019 + 0.948419i \(0.602682\pi\)
\(648\) 0 0
\(649\) 47.0008 1.84494
\(650\) 0 0
\(651\) − 1.96722i − 0.0771015i
\(652\) 0 0
\(653\) 28.6244i 1.12016i 0.828439 + 0.560080i \(0.189230\pi\)
−0.828439 + 0.560080i \(0.810770\pi\)
\(654\) 0 0
\(655\) −43.7255 −1.70850
\(656\) 0 0
\(657\) −15.2809 −0.596166
\(658\) 0 0
\(659\) 4.75185i 0.185106i 0.995708 + 0.0925528i \(0.0295027\pi\)
−0.995708 + 0.0925528i \(0.970497\pi\)
\(660\) 0 0
\(661\) − 21.8086i − 0.848256i −0.905602 0.424128i \(-0.860581\pi\)
0.905602 0.424128i \(-0.139419\pi\)
\(662\) 0 0
\(663\) −5.03957 −0.195721
\(664\) 0 0
\(665\) −14.9457 −0.579571
\(666\) 0 0
\(667\) − 30.8914i − 1.19612i
\(668\) 0 0
\(669\) 4.78110i 0.184848i
\(670\) 0 0
\(671\) −0.0856521 −0.00330656
\(672\) 0 0
\(673\) 6.34315 0.244510 0.122255 0.992499i \(-0.460987\pi\)
0.122255 + 0.992499i \(0.460987\pi\)
\(674\) 0 0
\(675\) − 1.23304i − 0.0474597i
\(676\) 0 0
\(677\) 45.6339i 1.75385i 0.480624 + 0.876927i \(0.340410\pi\)
−0.480624 + 0.876927i \(0.659590\pi\)
\(678\) 0 0
\(679\) −2.07913 −0.0797898
\(680\) 0 0
\(681\) −3.83840 −0.147088
\(682\) 0 0
\(683\) 37.9342i 1.45151i 0.687953 + 0.725756i \(0.258510\pi\)
−0.687953 + 0.725756i \(0.741490\pi\)
\(684\) 0 0
\(685\) 20.6173i 0.787746i
\(686\) 0 0
\(687\) 1.34596 0.0513515
\(688\) 0 0
\(689\) 19.7219 0.751345
\(690\) 0 0
\(691\) 33.3716i 1.26951i 0.772712 + 0.634757i \(0.218900\pi\)
−0.772712 + 0.634757i \(0.781100\pi\)
\(692\) 0 0
\(693\) − 3.39104i − 0.128815i
\(694\) 0 0
\(695\) −36.7120 −1.39256
\(696\) 0 0
\(697\) −8.31143 −0.314818
\(698\) 0 0
\(699\) − 17.6433i − 0.667330i
\(700\) 0 0
\(701\) − 23.4480i − 0.885618i −0.896616 0.442809i \(-0.853982\pi\)
0.896616 0.442809i \(-0.146018\pi\)
\(702\) 0 0
\(703\) 16.1512 0.609154
\(704\) 0 0
\(705\) −4.13651 −0.155790
\(706\) 0 0
\(707\) 9.16569i 0.344711i
\(708\) 0 0
\(709\) 12.1562i 0.456537i 0.973598 + 0.228269i \(0.0733064\pi\)
−0.973598 + 0.228269i \(0.926694\pi\)
\(710\) 0 0
\(711\) 8.40968 0.315388
\(712\) 0 0
\(713\) 8.57544 0.321153
\(714\) 0 0
\(715\) 53.6233i 2.00540i
\(716\) 0 0
\(717\) − 9.61500i − 0.359079i
\(718\) 0 0
\(719\) 4.06900 0.151748 0.0758741 0.997117i \(-0.475825\pi\)
0.0758741 + 0.997117i \(0.475825\pi\)
\(720\) 0 0
\(721\) −15.0653 −0.561061
\(722\) 0 0
\(723\) − 21.3583i − 0.794322i
\(724\) 0 0
\(725\) 9.52259i 0.353660i
\(726\) 0 0
\(727\) −17.3211 −0.642402 −0.321201 0.947011i \(-0.604087\pi\)
−0.321201 + 0.947011i \(0.604087\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 8.31143i 0.307409i
\(732\) 0 0
\(733\) − 21.4361i − 0.791761i −0.918302 0.395880i \(-0.870439\pi\)
0.918302 0.395880i \(-0.129561\pi\)
\(734\) 0 0
\(735\) 15.3741 0.567082
\(736\) 0 0
\(737\) 19.6871 0.725183
\(738\) 0 0
\(739\) − 31.2545i − 1.14972i −0.818253 0.574858i \(-0.805057\pi\)
0.818253 0.574858i \(-0.194943\pi\)
\(740\) 0 0
\(741\) − 37.9174i − 1.39293i
\(742\) 0 0
\(743\) −17.7244 −0.650244 −0.325122 0.945672i \(-0.605405\pi\)
−0.325122 + 0.945672i \(0.605405\pi\)
\(744\) 0 0
\(745\) 40.3288 1.47753
\(746\) 0 0
\(747\) 1.96134i 0.0717615i
\(748\) 0 0
\(749\) − 11.2428i − 0.410801i
\(750\) 0 0
\(751\) −7.38826 −0.269601 −0.134801 0.990873i \(-0.543039\pi\)
−0.134801 + 0.990873i \(0.543039\pi\)
\(752\) 0 0
\(753\) 4.30448 0.156864
\(754\) 0 0
\(755\) − 48.1073i − 1.75080i
\(756\) 0 0
\(757\) − 13.2804i − 0.482684i −0.970440 0.241342i \(-0.922412\pi\)
0.970440 0.241342i \(-0.0775876\pi\)
\(758\) 0 0
\(759\) 14.7821 0.536555
\(760\) 0 0
\(761\) −28.1473 −1.02034 −0.510169 0.860074i \(-0.670417\pi\)
−0.510169 + 0.860074i \(0.670417\pi\)
\(762\) 0 0
\(763\) 16.7898i 0.607830i
\(764\) 0 0
\(765\) − 2.16478i − 0.0782679i
\(766\) 0 0
\(767\) 73.9194 2.66907
\(768\) 0 0
\(769\) −11.9355 −0.430405 −0.215203 0.976569i \(-0.569041\pi\)
−0.215203 + 0.976569i \(0.569041\pi\)
\(770\) 0 0
\(771\) 4.67271i 0.168284i
\(772\) 0 0
\(773\) − 14.5868i − 0.524649i −0.964980 0.262325i \(-0.915511\pi\)
0.964980 0.262325i \(-0.0844891\pi\)
\(774\) 0 0
\(775\) −2.64347 −0.0949561
\(776\) 0 0
\(777\) 2.27170 0.0814969
\(778\) 0 0
\(779\) − 62.5347i − 2.24054i
\(780\) 0 0
\(781\) 43.6871i 1.56325i
\(782\) 0 0
\(783\) 7.72286 0.275992
\(784\) 0 0
\(785\) 2.04429 0.0729638
\(786\) 0 0
\(787\) 38.6835i 1.37892i 0.724325 + 0.689459i \(0.242151\pi\)
−0.724325 + 0.689459i \(0.757849\pi\)
\(788\) 0 0
\(789\) − 15.4502i − 0.550042i
\(790\) 0 0
\(791\) 3.35556 0.119310
\(792\) 0 0
\(793\) −0.134707 −0.00478360
\(794\) 0 0
\(795\) 8.47170i 0.300460i
\(796\) 0 0
\(797\) − 29.4107i − 1.04178i −0.853624 0.520890i \(-0.825600\pi\)
0.853624 0.520890i \(-0.174400\pi\)
\(798\) 0 0
\(799\) −1.43664 −0.0508248
\(800\) 0 0
\(801\) 2.79565 0.0987794
\(802\) 0 0
\(803\) 56.4710i 1.99282i
\(804\) 0 0
\(805\) − 9.16362i − 0.322975i
\(806\) 0 0
\(807\) −31.2459 −1.09991
\(808\) 0 0
\(809\) −2.80334 −0.0985602 −0.0492801 0.998785i \(-0.515693\pi\)
−0.0492801 + 0.998785i \(0.515693\pi\)
\(810\) 0 0
\(811\) 51.0722i 1.79339i 0.442650 + 0.896694i \(0.354038\pi\)
−0.442650 + 0.896694i \(0.645962\pi\)
\(812\) 0 0
\(813\) 9.88817i 0.346793i
\(814\) 0 0
\(815\) 59.0426 2.06817
\(816\) 0 0
\(817\) −62.5347 −2.18781
\(818\) 0 0
\(819\) − 5.33317i − 0.186356i
\(820\) 0 0
\(821\) − 17.4389i − 0.608622i −0.952573 0.304311i \(-0.901574\pi\)
0.952573 0.304311i \(-0.0984261\pi\)
\(822\) 0 0
\(823\) 22.0665 0.769191 0.384595 0.923085i \(-0.374341\pi\)
0.384595 + 0.923085i \(0.374341\pi\)
\(824\) 0 0
\(825\) −4.55672 −0.158645
\(826\) 0 0
\(827\) 1.76621i 0.0614172i 0.999528 + 0.0307086i \(0.00977639\pi\)
−0.999528 + 0.0307086i \(0.990224\pi\)
\(828\) 0 0
\(829\) 22.0514i 0.765879i 0.923774 + 0.382939i \(0.125088\pi\)
−0.923774 + 0.382939i \(0.874912\pi\)
\(830\) 0 0
\(831\) −22.9565 −0.796351
\(832\) 0 0
\(833\) 5.33954 0.185004
\(834\) 0 0
\(835\) − 23.2274i − 0.803816i
\(836\) 0 0
\(837\) 2.14386i 0.0741026i
\(838\) 0 0
\(839\) 22.1904 0.766099 0.383050 0.923728i \(-0.374874\pi\)
0.383050 + 0.923728i \(0.374874\pi\)
\(840\) 0 0
\(841\) −30.6425 −1.05664
\(842\) 0 0
\(843\) − 14.5754i − 0.502005i
\(844\) 0 0
\(845\) 51.8789i 1.78469i
\(846\) 0 0
\(847\) −2.43795 −0.0837690
\(848\) 0 0
\(849\) 4.71832 0.161932
\(850\) 0 0
\(851\) 9.90272i 0.339461i
\(852\) 0 0
\(853\) 37.4972i 1.28388i 0.766756 + 0.641939i \(0.221870\pi\)
−0.766756 + 0.641939i \(0.778130\pi\)
\(854\) 0 0
\(855\) 16.2877 0.557028
\(856\) 0 0
\(857\) 17.4761 0.596971 0.298485 0.954414i \(-0.403519\pi\)
0.298485 + 0.954414i \(0.403519\pi\)
\(858\) 0 0
\(859\) 10.7442i 0.366586i 0.983058 + 0.183293i \(0.0586757\pi\)
−0.983058 + 0.183293i \(0.941324\pi\)
\(860\) 0 0
\(861\) − 8.79565i − 0.299755i
\(862\) 0 0
\(863\) 23.1791 0.789027 0.394514 0.918890i \(-0.370913\pi\)
0.394514 + 0.918890i \(0.370913\pi\)
\(864\) 0 0
\(865\) −17.7398 −0.603171
\(866\) 0 0
\(867\) 16.2482i 0.551816i
\(868\) 0 0
\(869\) − 31.0781i − 1.05425i
\(870\) 0 0
\(871\) 30.9624 1.04912
\(872\) 0 0
\(873\) 2.26582 0.0766863
\(874\) 0 0
\(875\) − 8.62975i − 0.291739i
\(876\) 0 0
\(877\) 28.3694i 0.957966i 0.877824 + 0.478983i \(0.158995\pi\)
−0.877824 + 0.478983i \(0.841005\pi\)
\(878\) 0 0
\(879\) 23.9204 0.806816
\(880\) 0 0
\(881\) −58.0822 −1.95684 −0.978420 0.206628i \(-0.933751\pi\)
−0.978420 + 0.206628i \(0.933751\pi\)
\(882\) 0 0
\(883\) 5.55826i 0.187050i 0.995617 + 0.0935252i \(0.0298136\pi\)
−0.995617 + 0.0935252i \(0.970186\pi\)
\(884\) 0 0
\(885\) 31.7526i 1.06735i
\(886\) 0 0
\(887\) 9.24359 0.310369 0.155185 0.987885i \(-0.450403\pi\)
0.155185 + 0.987885i \(0.450403\pi\)
\(888\) 0 0
\(889\) −18.9378 −0.635153
\(890\) 0 0
\(891\) 3.69552i 0.123805i
\(892\) 0 0
\(893\) − 10.8092i − 0.361717i
\(894\) 0 0
\(895\) 6.67271 0.223044
\(896\) 0 0
\(897\) 23.2482 0.776233
\(898\) 0 0
\(899\) − 16.5567i − 0.552198i
\(900\) 0 0
\(901\) 2.94229i 0.0980219i
\(902\) 0 0
\(903\) −8.79565 −0.292701
\(904\) 0 0
\(905\) 1.59899 0.0531522
\(906\) 0 0
\(907\) 15.4945i 0.514487i 0.966347 + 0.257243i \(0.0828142\pi\)
−0.966347 + 0.257243i \(0.917186\pi\)
\(908\) 0 0
\(909\) − 9.98868i − 0.331303i
\(910\) 0 0
\(911\) −43.7108 −1.44820 −0.724101 0.689693i \(-0.757745\pi\)
−0.724101 + 0.689693i \(0.757745\pi\)
\(912\) 0 0
\(913\) 7.24815 0.239879
\(914\) 0 0
\(915\) − 0.0578646i − 0.00191294i
\(916\) 0 0
\(917\) − 16.0710i − 0.530710i
\(918\) 0 0
\(919\) −41.7955 −1.37871 −0.689353 0.724426i \(-0.742105\pi\)
−0.689353 + 0.724426i \(0.742105\pi\)
\(920\) 0 0
\(921\) −20.7685 −0.684345
\(922\) 0 0
\(923\) 68.7078i 2.26155i
\(924\) 0 0
\(925\) − 3.05261i − 0.100369i
\(926\) 0 0
\(927\) 16.4180 0.539238
\(928\) 0 0
\(929\) 24.1989 0.793942 0.396971 0.917831i \(-0.370061\pi\)
0.396971 + 0.917831i \(0.370061\pi\)
\(930\) 0 0
\(931\) 40.1744i 1.31666i
\(932\) 0 0
\(933\) − 11.7798i − 0.385653i
\(934\) 0 0
\(935\) −8.00000 −0.261628
\(936\) 0 0
\(937\) −15.3183 −0.500426 −0.250213 0.968191i \(-0.580501\pi\)
−0.250213 + 0.968191i \(0.580501\pi\)
\(938\) 0 0
\(939\) 11.1580i 0.364127i
\(940\) 0 0
\(941\) − 35.3016i − 1.15080i −0.817872 0.575400i \(-0.804846\pi\)
0.817872 0.575400i \(-0.195154\pi\)
\(942\) 0 0
\(943\) 38.3417 1.24858
\(944\) 0 0
\(945\) 2.29090 0.0745231
\(946\) 0 0
\(947\) − 14.2051i − 0.461605i −0.973001 0.230802i \(-0.925865\pi\)
0.973001 0.230802i \(-0.0741351\pi\)
\(948\) 0 0
\(949\) 88.8134i 2.88300i
\(950\) 0 0
\(951\) 15.9140 0.516048
\(952\) 0 0
\(953\) 19.6428 0.636292 0.318146 0.948042i \(-0.396940\pi\)
0.318146 + 0.948042i \(0.396940\pi\)
\(954\) 0 0
\(955\) − 36.4597i − 1.17981i
\(956\) 0 0
\(957\) − 28.5400i − 0.922566i
\(958\) 0 0
\(959\) −7.57772 −0.244697
\(960\) 0 0
\(961\) −26.4039 −0.851738
\(962\) 0 0
\(963\) 12.2522i 0.394823i
\(964\) 0 0
\(965\) 22.0072i 0.708437i
\(966\) 0 0
\(967\) 44.5337 1.43211 0.716054 0.698045i \(-0.245946\pi\)
0.716054 + 0.698045i \(0.245946\pi\)
\(968\) 0 0
\(969\) 5.65685 0.181724
\(970\) 0 0
\(971\) − 1.40255i − 0.0450099i −0.999747 0.0225049i \(-0.992836\pi\)
0.999747 0.0225049i \(-0.00716415\pi\)
\(972\) 0 0
\(973\) − 13.4932i − 0.432572i
\(974\) 0 0
\(975\) −7.16648 −0.229511
\(976\) 0 0
\(977\) −37.0794 −1.18628 −0.593138 0.805101i \(-0.702111\pi\)
−0.593138 + 0.805101i \(0.702111\pi\)
\(978\) 0 0
\(979\) − 10.3314i − 0.330192i
\(980\) 0 0
\(981\) − 18.2973i − 0.584188i
\(982\) 0 0
\(983\) 46.6894 1.48916 0.744580 0.667534i \(-0.232650\pi\)
0.744580 + 0.667534i \(0.232650\pi\)
\(984\) 0 0
\(985\) 9.35105 0.297949
\(986\) 0 0
\(987\) − 1.52034i − 0.0483930i
\(988\) 0 0
\(989\) − 38.3417i − 1.21919i
\(990\) 0 0
\(991\) 11.8780 0.377318 0.188659 0.982043i \(-0.439586\pi\)
0.188659 + 0.982043i \(0.439586\pi\)
\(992\) 0 0
\(993\) −2.73190 −0.0866942
\(994\) 0 0
\(995\) − 37.0164i − 1.17350i
\(996\) 0 0
\(997\) 51.1515i 1.61998i 0.586441 + 0.809992i \(0.300528\pi\)
−0.586441 + 0.809992i \(0.699472\pi\)
\(998\) 0 0
\(999\) −2.47568 −0.0783271
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 3072.2.d.j.1537.1 8
4.3 odd 2 3072.2.d.e.1537.5 8
8.3 odd 2 3072.2.d.e.1537.4 8
8.5 even 2 inner 3072.2.d.j.1537.8 8
16.3 odd 4 3072.2.a.m.1.1 4
16.5 even 4 3072.2.a.j.1.4 4
16.11 odd 4 3072.2.a.s.1.4 4
16.13 even 4 3072.2.a.p.1.1 4
32.3 odd 8 1536.2.j.j.385.2 yes 8
32.5 even 8 1536.2.j.i.1153.4 yes 8
32.11 odd 8 1536.2.j.e.1153.3 yes 8
32.13 even 8 1536.2.j.f.385.1 yes 8
32.19 odd 8 1536.2.j.e.385.3 8
32.21 even 8 1536.2.j.f.1153.1 yes 8
32.27 odd 8 1536.2.j.j.1153.2 yes 8
32.29 even 8 1536.2.j.i.385.4 yes 8
48.5 odd 4 9216.2.a.ba.1.1 4
48.11 even 4 9216.2.a.bm.1.1 4
48.29 odd 4 9216.2.a.z.1.4 4
48.35 even 4 9216.2.a.bl.1.4 4
96.5 odd 8 4608.2.k.bc.1153.2 8
96.11 even 8 4608.2.k.bh.1153.3 8
96.29 odd 8 4608.2.k.bc.3457.2 8
96.35 even 8 4608.2.k.be.3457.2 8
96.53 odd 8 4608.2.k.bj.1153.3 8
96.59 even 8 4608.2.k.be.1153.2 8
96.77 odd 8 4608.2.k.bj.3457.3 8
96.83 even 8 4608.2.k.bh.3457.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1536.2.j.e.385.3 8 32.19 odd 8
1536.2.j.e.1153.3 yes 8 32.11 odd 8
1536.2.j.f.385.1 yes 8 32.13 even 8
1536.2.j.f.1153.1 yes 8 32.21 even 8
1536.2.j.i.385.4 yes 8 32.29 even 8
1536.2.j.i.1153.4 yes 8 32.5 even 8
1536.2.j.j.385.2 yes 8 32.3 odd 8
1536.2.j.j.1153.2 yes 8 32.27 odd 8
3072.2.a.j.1.4 4 16.5 even 4
3072.2.a.m.1.1 4 16.3 odd 4
3072.2.a.p.1.1 4 16.13 even 4
3072.2.a.s.1.4 4 16.11 odd 4
3072.2.d.e.1537.4 8 8.3 odd 2
3072.2.d.e.1537.5 8 4.3 odd 2
3072.2.d.j.1537.1 8 1.1 even 1 trivial
3072.2.d.j.1537.8 8 8.5 even 2 inner
4608.2.k.bc.1153.2 8 96.5 odd 8
4608.2.k.bc.3457.2 8 96.29 odd 8
4608.2.k.be.1153.2 8 96.59 even 8
4608.2.k.be.3457.2 8 96.35 even 8
4608.2.k.bh.1153.3 8 96.11 even 8
4608.2.k.bh.3457.3 8 96.83 even 8
4608.2.k.bj.1153.3 8 96.53 odd 8
4608.2.k.bj.3457.3 8 96.77 odd 8
9216.2.a.z.1.4 4 48.29 odd 4
9216.2.a.ba.1.1 4 48.5 odd 4
9216.2.a.bl.1.4 4 48.35 even 4
9216.2.a.bm.1.1 4 48.11 even 4