Properties

Label 1536.2.d.f.769.3
Level $1536$
Weight $2$
Character 1536.769
Analytic conductor $12.265$
Analytic rank $0$
Dimension $4$
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] = [1536,2,Mod(769,1536)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1536, 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("1536.769");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1536 = 2^{9} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1536.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: \(12.2650217505\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 769.3
Root \(0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1536.769
Dual form 1536.2.d.f.769.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{3} +0.585786i q^{5} +3.41421 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} +0.585786i q^{5} +3.41421 q^{7} -1.00000 q^{9} -2.00000i q^{11} -2.82843i q^{13} -0.585786 q^{15} +3.65685 q^{17} -5.65685i q^{19} +3.41421i q^{21} +1.17157 q^{23} +4.65685 q^{25} -1.00000i q^{27} -0.585786i q^{29} -4.58579 q^{31} +2.00000 q^{33} +2.00000i q^{35} -9.65685i q^{37} +2.82843 q^{39} +11.6569 q^{41} -1.65685i q^{43} -0.585786i q^{45} -12.4853 q^{47} +4.65685 q^{49} +3.65685i q^{51} +11.8995i q^{53} +1.17157 q^{55} +5.65685 q^{57} +4.00000i q^{59} +9.65685i q^{61} -3.41421 q^{63} +1.65685 q^{65} -8.00000i q^{67} +1.17157i q^{69} +9.17157 q^{71} +1.65685 q^{73} +4.65685i q^{75} -6.82843i q^{77} -5.75736 q^{79} +1.00000 q^{81} +9.31371i q^{83} +2.14214i q^{85} +0.585786 q^{87} -2.00000 q^{89} -9.65685i q^{91} -4.58579i q^{93} +3.31371 q^{95} +13.3137 q^{97} +2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{7} - 4 q^{9} - 8 q^{15} - 8 q^{17} + 16 q^{23} - 4 q^{25} - 24 q^{31} + 8 q^{33} + 24 q^{41} - 16 q^{47} - 4 q^{49} + 16 q^{55} - 8 q^{63} - 16 q^{65} + 48 q^{71} - 16 q^{73} - 40 q^{79} + 4 q^{81} + 8 q^{87} - 8 q^{89} - 32 q^{95} + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1536\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(517\) \(1025\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(5\) 0.585786i 0.261972i 0.991384 + 0.130986i \(0.0418142\pi\)
−0.991384 + 0.130986i \(0.958186\pi\)
\(6\) 0 0
\(7\) 3.41421 1.29045 0.645226 0.763992i \(-0.276763\pi\)
0.645226 + 0.763992i \(0.276763\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) − 2.00000i − 0.603023i −0.953463 0.301511i \(-0.902509\pi\)
0.953463 0.301511i \(-0.0974911\pi\)
\(12\) 0 0
\(13\) − 2.82843i − 0.784465i −0.919866 0.392232i \(-0.871703\pi\)
0.919866 0.392232i \(-0.128297\pi\)
\(14\) 0 0
\(15\) −0.585786 −0.151249
\(16\) 0 0
\(17\) 3.65685 0.886917 0.443459 0.896295i \(-0.353751\pi\)
0.443459 + 0.896295i \(0.353751\pi\)
\(18\) 0 0
\(19\) − 5.65685i − 1.29777i −0.760886 0.648886i \(-0.775235\pi\)
0.760886 0.648886i \(-0.224765\pi\)
\(20\) 0 0
\(21\) 3.41421i 0.745042i
\(22\) 0 0
\(23\) 1.17157 0.244290 0.122145 0.992512i \(-0.461023\pi\)
0.122145 + 0.992512i \(0.461023\pi\)
\(24\) 0 0
\(25\) 4.65685 0.931371
\(26\) 0 0
\(27\) − 1.00000i − 0.192450i
\(28\) 0 0
\(29\) − 0.585786i − 0.108778i −0.998520 0.0543889i \(-0.982679\pi\)
0.998520 0.0543889i \(-0.0173211\pi\)
\(30\) 0 0
\(31\) −4.58579 −0.823632 −0.411816 0.911267i \(-0.635105\pi\)
−0.411816 + 0.911267i \(0.635105\pi\)
\(32\) 0 0
\(33\) 2.00000 0.348155
\(34\) 0 0
\(35\) 2.00000i 0.338062i
\(36\) 0 0
\(37\) − 9.65685i − 1.58758i −0.608194 0.793789i \(-0.708106\pi\)
0.608194 0.793789i \(-0.291894\pi\)
\(38\) 0 0
\(39\) 2.82843 0.452911
\(40\) 0 0
\(41\) 11.6569 1.82049 0.910247 0.414065i \(-0.135891\pi\)
0.910247 + 0.414065i \(0.135891\pi\)
\(42\) 0 0
\(43\) − 1.65685i − 0.252668i −0.991988 0.126334i \(-0.959679\pi\)
0.991988 0.126334i \(-0.0403211\pi\)
\(44\) 0 0
\(45\) − 0.585786i − 0.0873239i
\(46\) 0 0
\(47\) −12.4853 −1.82117 −0.910583 0.413327i \(-0.864367\pi\)
−0.910583 + 0.413327i \(0.864367\pi\)
\(48\) 0 0
\(49\) 4.65685 0.665265
\(50\) 0 0
\(51\) 3.65685i 0.512062i
\(52\) 0 0
\(53\) 11.8995i 1.63452i 0.576268 + 0.817261i \(0.304508\pi\)
−0.576268 + 0.817261i \(0.695492\pi\)
\(54\) 0 0
\(55\) 1.17157 0.157975
\(56\) 0 0
\(57\) 5.65685 0.749269
\(58\) 0 0
\(59\) 4.00000i 0.520756i 0.965507 + 0.260378i \(0.0838471\pi\)
−0.965507 + 0.260378i \(0.916153\pi\)
\(60\) 0 0
\(61\) 9.65685i 1.23643i 0.786008 + 0.618217i \(0.212145\pi\)
−0.786008 + 0.618217i \(0.787855\pi\)
\(62\) 0 0
\(63\) −3.41421 −0.430150
\(64\) 0 0
\(65\) 1.65685 0.205507
\(66\) 0 0
\(67\) − 8.00000i − 0.977356i −0.872464 0.488678i \(-0.837479\pi\)
0.872464 0.488678i \(-0.162521\pi\)
\(68\) 0 0
\(69\) 1.17157i 0.141041i
\(70\) 0 0
\(71\) 9.17157 1.08847 0.544233 0.838934i \(-0.316821\pi\)
0.544233 + 0.838934i \(0.316821\pi\)
\(72\) 0 0
\(73\) 1.65685 0.193920 0.0969601 0.995288i \(-0.469088\pi\)
0.0969601 + 0.995288i \(0.469088\pi\)
\(74\) 0 0
\(75\) 4.65685i 0.537727i
\(76\) 0 0
\(77\) − 6.82843i − 0.778171i
\(78\) 0 0
\(79\) −5.75736 −0.647754 −0.323877 0.946099i \(-0.604986\pi\)
−0.323877 + 0.946099i \(0.604986\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 9.31371i 1.02231i 0.859488 + 0.511156i \(0.170783\pi\)
−0.859488 + 0.511156i \(0.829217\pi\)
\(84\) 0 0
\(85\) 2.14214i 0.232347i
\(86\) 0 0
\(87\) 0.585786 0.0628029
\(88\) 0 0
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) − 9.65685i − 1.01231i
\(92\) 0 0
\(93\) − 4.58579i − 0.475524i
\(94\) 0 0
\(95\) 3.31371 0.339979
\(96\) 0 0
\(97\) 13.3137 1.35180 0.675901 0.736992i \(-0.263755\pi\)
0.675901 + 0.736992i \(0.263755\pi\)
\(98\) 0 0
\(99\) 2.00000i 0.201008i
\(100\) 0 0
\(101\) − 17.5563i − 1.74692i −0.486894 0.873461i \(-0.661870\pi\)
0.486894 0.873461i \(-0.338130\pi\)
\(102\) 0 0
\(103\) 9.07107 0.893799 0.446899 0.894584i \(-0.352528\pi\)
0.446899 + 0.894584i \(0.352528\pi\)
\(104\) 0 0
\(105\) −2.00000 −0.195180
\(106\) 0 0
\(107\) 15.3137i 1.48043i 0.672369 + 0.740216i \(0.265277\pi\)
−0.672369 + 0.740216i \(0.734723\pi\)
\(108\) 0 0
\(109\) 13.1716i 1.26161i 0.775942 + 0.630804i \(0.217275\pi\)
−0.775942 + 0.630804i \(0.782725\pi\)
\(110\) 0 0
\(111\) 9.65685 0.916588
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 0.686292i 0.0639970i
\(116\) 0 0
\(117\) 2.82843i 0.261488i
\(118\) 0 0
\(119\) 12.4853 1.14452
\(120\) 0 0
\(121\) 7.00000 0.636364
\(122\) 0 0
\(123\) 11.6569i 1.05106i
\(124\) 0 0
\(125\) 5.65685i 0.505964i
\(126\) 0 0
\(127\) −6.72792 −0.597007 −0.298503 0.954409i \(-0.596487\pi\)
−0.298503 + 0.954409i \(0.596487\pi\)
\(128\) 0 0
\(129\) 1.65685 0.145878
\(130\) 0 0
\(131\) − 4.00000i − 0.349482i −0.984614 0.174741i \(-0.944091\pi\)
0.984614 0.174741i \(-0.0559088\pi\)
\(132\) 0 0
\(133\) − 19.3137i − 1.67471i
\(134\) 0 0
\(135\) 0.585786 0.0504165
\(136\) 0 0
\(137\) −4.34315 −0.371060 −0.185530 0.982639i \(-0.559400\pi\)
−0.185530 + 0.982639i \(0.559400\pi\)
\(138\) 0 0
\(139\) − 19.3137i − 1.63817i −0.573674 0.819084i \(-0.694482\pi\)
0.573674 0.819084i \(-0.305518\pi\)
\(140\) 0 0
\(141\) − 12.4853i − 1.05145i
\(142\) 0 0
\(143\) −5.65685 −0.473050
\(144\) 0 0
\(145\) 0.343146 0.0284967
\(146\) 0 0
\(147\) 4.65685i 0.384091i
\(148\) 0 0
\(149\) 14.2426i 1.16680i 0.812184 + 0.583401i \(0.198278\pi\)
−0.812184 + 0.583401i \(0.801722\pi\)
\(150\) 0 0
\(151\) 4.58579 0.373186 0.186593 0.982437i \(-0.440255\pi\)
0.186593 + 0.982437i \(0.440255\pi\)
\(152\) 0 0
\(153\) −3.65685 −0.295639
\(154\) 0 0
\(155\) − 2.68629i − 0.215768i
\(156\) 0 0
\(157\) 20.0000i 1.59617i 0.602542 + 0.798087i \(0.294154\pi\)
−0.602542 + 0.798087i \(0.705846\pi\)
\(158\) 0 0
\(159\) −11.8995 −0.943691
\(160\) 0 0
\(161\) 4.00000 0.315244
\(162\) 0 0
\(163\) 6.34315i 0.496834i 0.968653 + 0.248417i \(0.0799102\pi\)
−0.968653 + 0.248417i \(0.920090\pi\)
\(164\) 0 0
\(165\) 1.17157i 0.0912068i
\(166\) 0 0
\(167\) −21.6569 −1.67586 −0.837929 0.545779i \(-0.816234\pi\)
−0.837929 + 0.545779i \(0.816234\pi\)
\(168\) 0 0
\(169\) 5.00000 0.384615
\(170\) 0 0
\(171\) 5.65685i 0.432590i
\(172\) 0 0
\(173\) − 18.7279i − 1.42386i −0.702252 0.711929i \(-0.747822\pi\)
0.702252 0.711929i \(-0.252178\pi\)
\(174\) 0 0
\(175\) 15.8995 1.20189
\(176\) 0 0
\(177\) −4.00000 −0.300658
\(178\) 0 0
\(179\) − 4.00000i − 0.298974i −0.988764 0.149487i \(-0.952238\pi\)
0.988764 0.149487i \(-0.0477622\pi\)
\(180\) 0 0
\(181\) 8.48528i 0.630706i 0.948974 + 0.315353i \(0.102123\pi\)
−0.948974 + 0.315353i \(0.897877\pi\)
\(182\) 0 0
\(183\) −9.65685 −0.713855
\(184\) 0 0
\(185\) 5.65685 0.415900
\(186\) 0 0
\(187\) − 7.31371i − 0.534831i
\(188\) 0 0
\(189\) − 3.41421i − 0.248347i
\(190\) 0 0
\(191\) 16.9706 1.22795 0.613973 0.789327i \(-0.289570\pi\)
0.613973 + 0.789327i \(0.289570\pi\)
\(192\) 0 0
\(193\) −21.6569 −1.55889 −0.779447 0.626468i \(-0.784500\pi\)
−0.779447 + 0.626468i \(0.784500\pi\)
\(194\) 0 0
\(195\) 1.65685i 0.118650i
\(196\) 0 0
\(197\) 2.92893i 0.208678i 0.994542 + 0.104339i \(0.0332727\pi\)
−0.994542 + 0.104339i \(0.966727\pi\)
\(198\) 0 0
\(199\) −13.5563 −0.960984 −0.480492 0.876999i \(-0.659542\pi\)
−0.480492 + 0.876999i \(0.659542\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) 0 0
\(203\) − 2.00000i − 0.140372i
\(204\) 0 0
\(205\) 6.82843i 0.476918i
\(206\) 0 0
\(207\) −1.17157 −0.0814299
\(208\) 0 0
\(209\) −11.3137 −0.782586
\(210\) 0 0
\(211\) − 19.3137i − 1.32961i −0.747017 0.664805i \(-0.768515\pi\)
0.747017 0.664805i \(-0.231485\pi\)
\(212\) 0 0
\(213\) 9.17157i 0.628426i
\(214\) 0 0
\(215\) 0.970563 0.0661918
\(216\) 0 0
\(217\) −15.6569 −1.06286
\(218\) 0 0
\(219\) 1.65685i 0.111960i
\(220\) 0 0
\(221\) − 10.3431i − 0.695755i
\(222\) 0 0
\(223\) −7.89949 −0.528989 −0.264495 0.964387i \(-0.585205\pi\)
−0.264495 + 0.964387i \(0.585205\pi\)
\(224\) 0 0
\(225\) −4.65685 −0.310457
\(226\) 0 0
\(227\) 29.3137i 1.94562i 0.231606 + 0.972810i \(0.425602\pi\)
−0.231606 + 0.972810i \(0.574398\pi\)
\(228\) 0 0
\(229\) − 16.4853i − 1.08938i −0.838638 0.544689i \(-0.816648\pi\)
0.838638 0.544689i \(-0.183352\pi\)
\(230\) 0 0
\(231\) 6.82843 0.449278
\(232\) 0 0
\(233\) −9.31371 −0.610161 −0.305081 0.952327i \(-0.598683\pi\)
−0.305081 + 0.952327i \(0.598683\pi\)
\(234\) 0 0
\(235\) − 7.31371i − 0.477094i
\(236\) 0 0
\(237\) − 5.75736i − 0.373981i
\(238\) 0 0
\(239\) −16.9706 −1.09773 −0.548867 0.835910i \(-0.684941\pi\)
−0.548867 + 0.835910i \(0.684941\pi\)
\(240\) 0 0
\(241\) −17.6569 −1.13738 −0.568689 0.822553i \(-0.692549\pi\)
−0.568689 + 0.822553i \(0.692549\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 2.72792i 0.174281i
\(246\) 0 0
\(247\) −16.0000 −1.01806
\(248\) 0 0
\(249\) −9.31371 −0.590232
\(250\) 0 0
\(251\) 22.0000i 1.38863i 0.719672 + 0.694314i \(0.244292\pi\)
−0.719672 + 0.694314i \(0.755708\pi\)
\(252\) 0 0
\(253\) − 2.34315i − 0.147312i
\(254\) 0 0
\(255\) −2.14214 −0.134146
\(256\) 0 0
\(257\) −1.31371 −0.0819469 −0.0409734 0.999160i \(-0.513046\pi\)
−0.0409734 + 0.999160i \(0.513046\pi\)
\(258\) 0 0
\(259\) − 32.9706i − 2.04869i
\(260\) 0 0
\(261\) 0.585786i 0.0362593i
\(262\) 0 0
\(263\) −13.6569 −0.842118 −0.421059 0.907033i \(-0.638341\pi\)
−0.421059 + 0.907033i \(0.638341\pi\)
\(264\) 0 0
\(265\) −6.97056 −0.428198
\(266\) 0 0
\(267\) − 2.00000i − 0.122398i
\(268\) 0 0
\(269\) 6.24264i 0.380621i 0.981724 + 0.190310i \(0.0609494\pi\)
−0.981724 + 0.190310i \(0.939051\pi\)
\(270\) 0 0
\(271\) 19.4142 1.17933 0.589665 0.807648i \(-0.299260\pi\)
0.589665 + 0.807648i \(0.299260\pi\)
\(272\) 0 0
\(273\) 9.65685 0.584459
\(274\) 0 0
\(275\) − 9.31371i − 0.561638i
\(276\) 0 0
\(277\) − 7.51472i − 0.451516i −0.974183 0.225758i \(-0.927514\pi\)
0.974183 0.225758i \(-0.0724858\pi\)
\(278\) 0 0
\(279\) 4.58579 0.274544
\(280\) 0 0
\(281\) 2.00000 0.119310 0.0596550 0.998219i \(-0.481000\pi\)
0.0596550 + 0.998219i \(0.481000\pi\)
\(282\) 0 0
\(283\) − 15.3137i − 0.910305i −0.890413 0.455153i \(-0.849585\pi\)
0.890413 0.455153i \(-0.150415\pi\)
\(284\) 0 0
\(285\) 3.31371i 0.196287i
\(286\) 0 0
\(287\) 39.7990 2.34926
\(288\) 0 0
\(289\) −3.62742 −0.213377
\(290\) 0 0
\(291\) 13.3137i 0.780463i
\(292\) 0 0
\(293\) 27.8995i 1.62991i 0.579527 + 0.814953i \(0.303237\pi\)
−0.579527 + 0.814953i \(0.696763\pi\)
\(294\) 0 0
\(295\) −2.34315 −0.136423
\(296\) 0 0
\(297\) −2.00000 −0.116052
\(298\) 0 0
\(299\) − 3.31371i − 0.191637i
\(300\) 0 0
\(301\) − 5.65685i − 0.326056i
\(302\) 0 0
\(303\) 17.5563 1.00859
\(304\) 0 0
\(305\) −5.65685 −0.323911
\(306\) 0 0
\(307\) − 16.0000i − 0.913168i −0.889680 0.456584i \(-0.849073\pi\)
0.889680 0.456584i \(-0.150927\pi\)
\(308\) 0 0
\(309\) 9.07107i 0.516035i
\(310\) 0 0
\(311\) 11.3137 0.641542 0.320771 0.947157i \(-0.396058\pi\)
0.320771 + 0.947157i \(0.396058\pi\)
\(312\) 0 0
\(313\) −16.9706 −0.959233 −0.479616 0.877478i \(-0.659224\pi\)
−0.479616 + 0.877478i \(0.659224\pi\)
\(314\) 0 0
\(315\) − 2.00000i − 0.112687i
\(316\) 0 0
\(317\) − 5.07107i − 0.284820i −0.989808 0.142410i \(-0.954515\pi\)
0.989808 0.142410i \(-0.0454851\pi\)
\(318\) 0 0
\(319\) −1.17157 −0.0655955
\(320\) 0 0
\(321\) −15.3137 −0.854728
\(322\) 0 0
\(323\) − 20.6863i − 1.15102i
\(324\) 0 0
\(325\) − 13.1716i − 0.730627i
\(326\) 0 0
\(327\) −13.1716 −0.728390
\(328\) 0 0
\(329\) −42.6274 −2.35013
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 9.65685i 0.529192i
\(334\) 0 0
\(335\) 4.68629 0.256039
\(336\) 0 0
\(337\) 20.9706 1.14234 0.571170 0.820832i \(-0.306490\pi\)
0.571170 + 0.820832i \(0.306490\pi\)
\(338\) 0 0
\(339\) 6.00000i 0.325875i
\(340\) 0 0
\(341\) 9.17157i 0.496669i
\(342\) 0 0
\(343\) −8.00000 −0.431959
\(344\) 0 0
\(345\) −0.686292 −0.0369487
\(346\) 0 0
\(347\) − 25.3137i − 1.35891i −0.733717 0.679456i \(-0.762216\pi\)
0.733717 0.679456i \(-0.237784\pi\)
\(348\) 0 0
\(349\) 0.686292i 0.0367363i 0.999831 + 0.0183682i \(0.00584710\pi\)
−0.999831 + 0.0183682i \(0.994153\pi\)
\(350\) 0 0
\(351\) −2.82843 −0.150970
\(352\) 0 0
\(353\) −13.3137 −0.708617 −0.354309 0.935129i \(-0.615284\pi\)
−0.354309 + 0.935129i \(0.615284\pi\)
\(354\) 0 0
\(355\) 5.37258i 0.285147i
\(356\) 0 0
\(357\) 12.4853i 0.660791i
\(358\) 0 0
\(359\) −15.7990 −0.833839 −0.416919 0.908943i \(-0.636890\pi\)
−0.416919 + 0.908943i \(0.636890\pi\)
\(360\) 0 0
\(361\) −13.0000 −0.684211
\(362\) 0 0
\(363\) 7.00000i 0.367405i
\(364\) 0 0
\(365\) 0.970563i 0.0508016i
\(366\) 0 0
\(367\) 22.7279 1.18639 0.593194 0.805060i \(-0.297867\pi\)
0.593194 + 0.805060i \(0.297867\pi\)
\(368\) 0 0
\(369\) −11.6569 −0.606832
\(370\) 0 0
\(371\) 40.6274i 2.10927i
\(372\) 0 0
\(373\) − 24.2843i − 1.25739i −0.777651 0.628696i \(-0.783589\pi\)
0.777651 0.628696i \(-0.216411\pi\)
\(374\) 0 0
\(375\) −5.65685 −0.292119
\(376\) 0 0
\(377\) −1.65685 −0.0853323
\(378\) 0 0
\(379\) 18.3431i 0.942224i 0.882073 + 0.471112i \(0.156147\pi\)
−0.882073 + 0.471112i \(0.843853\pi\)
\(380\) 0 0
\(381\) − 6.72792i − 0.344682i
\(382\) 0 0
\(383\) 19.3137 0.986884 0.493442 0.869779i \(-0.335738\pi\)
0.493442 + 0.869779i \(0.335738\pi\)
\(384\) 0 0
\(385\) 4.00000 0.203859
\(386\) 0 0
\(387\) 1.65685i 0.0842226i
\(388\) 0 0
\(389\) − 22.2426i − 1.12775i −0.825861 0.563873i \(-0.809311\pi\)
0.825861 0.563873i \(-0.190689\pi\)
\(390\) 0 0
\(391\) 4.28427 0.216665
\(392\) 0 0
\(393\) 4.00000 0.201773
\(394\) 0 0
\(395\) − 3.37258i − 0.169693i
\(396\) 0 0
\(397\) 4.00000i 0.200754i 0.994949 + 0.100377i \(0.0320049\pi\)
−0.994949 + 0.100377i \(0.967995\pi\)
\(398\) 0 0
\(399\) 19.3137 0.966895
\(400\) 0 0
\(401\) −31.6569 −1.58087 −0.790434 0.612547i \(-0.790145\pi\)
−0.790434 + 0.612547i \(0.790145\pi\)
\(402\) 0 0
\(403\) 12.9706i 0.646110i
\(404\) 0 0
\(405\) 0.585786i 0.0291080i
\(406\) 0 0
\(407\) −19.3137 −0.957345
\(408\) 0 0
\(409\) −1.31371 −0.0649587 −0.0324794 0.999472i \(-0.510340\pi\)
−0.0324794 + 0.999472i \(0.510340\pi\)
\(410\) 0 0
\(411\) − 4.34315i − 0.214232i
\(412\) 0 0
\(413\) 13.6569i 0.672010i
\(414\) 0 0
\(415\) −5.45584 −0.267817
\(416\) 0 0
\(417\) 19.3137 0.945796
\(418\) 0 0
\(419\) − 1.31371i − 0.0641789i −0.999485 0.0320894i \(-0.989784\pi\)
0.999485 0.0320894i \(-0.0102161\pi\)
\(420\) 0 0
\(421\) 22.1421i 1.07914i 0.841940 + 0.539571i \(0.181413\pi\)
−0.841940 + 0.539571i \(0.818587\pi\)
\(422\) 0 0
\(423\) 12.4853 0.607055
\(424\) 0 0
\(425\) 17.0294 0.826049
\(426\) 0 0
\(427\) 32.9706i 1.59556i
\(428\) 0 0
\(429\) − 5.65685i − 0.273115i
\(430\) 0 0
\(431\) −35.1127 −1.69132 −0.845660 0.533723i \(-0.820793\pi\)
−0.845660 + 0.533723i \(0.820793\pi\)
\(432\) 0 0
\(433\) −16.6274 −0.799063 −0.399531 0.916720i \(-0.630827\pi\)
−0.399531 + 0.916720i \(0.630827\pi\)
\(434\) 0 0
\(435\) 0.343146i 0.0164526i
\(436\) 0 0
\(437\) − 6.62742i − 0.317032i
\(438\) 0 0
\(439\) −15.8995 −0.758841 −0.379421 0.925224i \(-0.623877\pi\)
−0.379421 + 0.925224i \(0.623877\pi\)
\(440\) 0 0
\(441\) −4.65685 −0.221755
\(442\) 0 0
\(443\) 27.9411i 1.32752i 0.747944 + 0.663761i \(0.231041\pi\)
−0.747944 + 0.663761i \(0.768959\pi\)
\(444\) 0 0
\(445\) − 1.17157i − 0.0555379i
\(446\) 0 0
\(447\) −14.2426 −0.673654
\(448\) 0 0
\(449\) −7.65685 −0.361349 −0.180675 0.983543i \(-0.557828\pi\)
−0.180675 + 0.983543i \(0.557828\pi\)
\(450\) 0 0
\(451\) − 23.3137i − 1.09780i
\(452\) 0 0
\(453\) 4.58579i 0.215459i
\(454\) 0 0
\(455\) 5.65685 0.265197
\(456\) 0 0
\(457\) 17.3137 0.809901 0.404951 0.914339i \(-0.367289\pi\)
0.404951 + 0.914339i \(0.367289\pi\)
\(458\) 0 0
\(459\) − 3.65685i − 0.170687i
\(460\) 0 0
\(461\) 10.9289i 0.509011i 0.967071 + 0.254506i \(0.0819127\pi\)
−0.967071 + 0.254506i \(0.918087\pi\)
\(462\) 0 0
\(463\) −2.44365 −0.113566 −0.0567830 0.998387i \(-0.518084\pi\)
−0.0567830 + 0.998387i \(0.518084\pi\)
\(464\) 0 0
\(465\) 2.68629 0.124574
\(466\) 0 0
\(467\) − 9.31371i − 0.430987i −0.976505 0.215494i \(-0.930864\pi\)
0.976505 0.215494i \(-0.0691360\pi\)
\(468\) 0 0
\(469\) − 27.3137i − 1.26123i
\(470\) 0 0
\(471\) −20.0000 −0.921551
\(472\) 0 0
\(473\) −3.31371 −0.152364
\(474\) 0 0
\(475\) − 26.3431i − 1.20871i
\(476\) 0 0
\(477\) − 11.8995i − 0.544840i
\(478\) 0 0
\(479\) 4.48528 0.204938 0.102469 0.994736i \(-0.467326\pi\)
0.102469 + 0.994736i \(0.467326\pi\)
\(480\) 0 0
\(481\) −27.3137 −1.24540
\(482\) 0 0
\(483\) 4.00000i 0.182006i
\(484\) 0 0
\(485\) 7.79899i 0.354134i
\(486\) 0 0
\(487\) −5.55635 −0.251782 −0.125891 0.992044i \(-0.540179\pi\)
−0.125891 + 0.992044i \(0.540179\pi\)
\(488\) 0 0
\(489\) −6.34315 −0.286847
\(490\) 0 0
\(491\) 0.686292i 0.0309719i 0.999880 + 0.0154860i \(0.00492953\pi\)
−0.999880 + 0.0154860i \(0.995070\pi\)
\(492\) 0 0
\(493\) − 2.14214i − 0.0964769i
\(494\) 0 0
\(495\) −1.17157 −0.0526583
\(496\) 0 0
\(497\) 31.3137 1.40461
\(498\) 0 0
\(499\) 23.3137i 1.04366i 0.853048 + 0.521832i \(0.174751\pi\)
−0.853048 + 0.521832i \(0.825249\pi\)
\(500\) 0 0
\(501\) − 21.6569i − 0.967557i
\(502\) 0 0
\(503\) −4.48528 −0.199989 −0.0999944 0.994988i \(-0.531882\pi\)
−0.0999944 + 0.994988i \(0.531882\pi\)
\(504\) 0 0
\(505\) 10.2843 0.457644
\(506\) 0 0
\(507\) 5.00000i 0.222058i
\(508\) 0 0
\(509\) − 23.2132i − 1.02891i −0.857518 0.514454i \(-0.827995\pi\)
0.857518 0.514454i \(-0.172005\pi\)
\(510\) 0 0
\(511\) 5.65685 0.250244
\(512\) 0 0
\(513\) −5.65685 −0.249756
\(514\) 0 0
\(515\) 5.31371i 0.234150i
\(516\) 0 0
\(517\) 24.9706i 1.09820i
\(518\) 0 0
\(519\) 18.7279 0.822065
\(520\) 0 0
\(521\) 10.9706 0.480629 0.240315 0.970695i \(-0.422749\pi\)
0.240315 + 0.970695i \(0.422749\pi\)
\(522\) 0 0
\(523\) − 26.3431i − 1.15191i −0.817483 0.575953i \(-0.804631\pi\)
0.817483 0.575953i \(-0.195369\pi\)
\(524\) 0 0
\(525\) 15.8995i 0.693911i
\(526\) 0 0
\(527\) −16.7696 −0.730493
\(528\) 0 0
\(529\) −21.6274 −0.940322
\(530\) 0 0
\(531\) − 4.00000i − 0.173585i
\(532\) 0 0
\(533\) − 32.9706i − 1.42811i
\(534\) 0 0
\(535\) −8.97056 −0.387831
\(536\) 0 0
\(537\) 4.00000 0.172613
\(538\) 0 0
\(539\) − 9.31371i − 0.401170i
\(540\) 0 0
\(541\) 5.17157i 0.222343i 0.993801 + 0.111172i \(0.0354603\pi\)
−0.993801 + 0.111172i \(0.964540\pi\)
\(542\) 0 0
\(543\) −8.48528 −0.364138
\(544\) 0 0
\(545\) −7.71573 −0.330506
\(546\) 0 0
\(547\) 28.9706i 1.23869i 0.785118 + 0.619346i \(0.212602\pi\)
−0.785118 + 0.619346i \(0.787398\pi\)
\(548\) 0 0
\(549\) − 9.65685i − 0.412144i
\(550\) 0 0
\(551\) −3.31371 −0.141169
\(552\) 0 0
\(553\) −19.6569 −0.835894
\(554\) 0 0
\(555\) 5.65685i 0.240120i
\(556\) 0 0
\(557\) 13.0711i 0.553839i 0.960893 + 0.276919i \(0.0893135\pi\)
−0.960893 + 0.276919i \(0.910686\pi\)
\(558\) 0 0
\(559\) −4.68629 −0.198209
\(560\) 0 0
\(561\) 7.31371 0.308785
\(562\) 0 0
\(563\) − 28.6274i − 1.20650i −0.797551 0.603251i \(-0.793872\pi\)
0.797551 0.603251i \(-0.206128\pi\)
\(564\) 0 0
\(565\) 3.51472i 0.147865i
\(566\) 0 0
\(567\) 3.41421 0.143383
\(568\) 0 0
\(569\) −18.9706 −0.795287 −0.397644 0.917540i \(-0.630172\pi\)
−0.397644 + 0.917540i \(0.630172\pi\)
\(570\) 0 0
\(571\) 20.0000i 0.836974i 0.908223 + 0.418487i \(0.137439\pi\)
−0.908223 + 0.418487i \(0.862561\pi\)
\(572\) 0 0
\(573\) 16.9706i 0.708955i
\(574\) 0 0
\(575\) 5.45584 0.227524
\(576\) 0 0
\(577\) 28.2843 1.17749 0.588745 0.808319i \(-0.299622\pi\)
0.588745 + 0.808319i \(0.299622\pi\)
\(578\) 0 0
\(579\) − 21.6569i − 0.900028i
\(580\) 0 0
\(581\) 31.7990i 1.31924i
\(582\) 0 0
\(583\) 23.7990 0.985653
\(584\) 0 0
\(585\) −1.65685 −0.0685025
\(586\) 0 0
\(587\) 8.68629i 0.358522i 0.983802 + 0.179261i \(0.0573706\pi\)
−0.983802 + 0.179261i \(0.942629\pi\)
\(588\) 0 0
\(589\) 25.9411i 1.06889i
\(590\) 0 0
\(591\) −2.92893 −0.120480
\(592\) 0 0
\(593\) −2.00000 −0.0821302 −0.0410651 0.999156i \(-0.513075\pi\)
−0.0410651 + 0.999156i \(0.513075\pi\)
\(594\) 0 0
\(595\) 7.31371i 0.299833i
\(596\) 0 0
\(597\) − 13.5563i − 0.554824i
\(598\) 0 0
\(599\) 28.4853 1.16388 0.581939 0.813233i \(-0.302294\pi\)
0.581939 + 0.813233i \(0.302294\pi\)
\(600\) 0 0
\(601\) −5.65685 −0.230748 −0.115374 0.993322i \(-0.536807\pi\)
−0.115374 + 0.993322i \(0.536807\pi\)
\(602\) 0 0
\(603\) 8.00000i 0.325785i
\(604\) 0 0
\(605\) 4.10051i 0.166709i
\(606\) 0 0
\(607\) 15.8995 0.645341 0.322670 0.946511i \(-0.395419\pi\)
0.322670 + 0.946511i \(0.395419\pi\)
\(608\) 0 0
\(609\) 2.00000 0.0810441
\(610\) 0 0
\(611\) 35.3137i 1.42864i
\(612\) 0 0
\(613\) 34.6274i 1.39859i 0.714834 + 0.699294i \(0.246502\pi\)
−0.714834 + 0.699294i \(0.753498\pi\)
\(614\) 0 0
\(615\) −6.82843 −0.275349
\(616\) 0 0
\(617\) 43.9411 1.76900 0.884502 0.466537i \(-0.154499\pi\)
0.884502 + 0.466537i \(0.154499\pi\)
\(618\) 0 0
\(619\) − 4.00000i − 0.160774i −0.996764 0.0803868i \(-0.974384\pi\)
0.996764 0.0803868i \(-0.0256155\pi\)
\(620\) 0 0
\(621\) − 1.17157i − 0.0470136i
\(622\) 0 0
\(623\) −6.82843 −0.273575
\(624\) 0 0
\(625\) 19.9706 0.798823
\(626\) 0 0
\(627\) − 11.3137i − 0.451826i
\(628\) 0 0
\(629\) − 35.3137i − 1.40805i
\(630\) 0 0
\(631\) 33.0711 1.31654 0.658269 0.752783i \(-0.271289\pi\)
0.658269 + 0.752783i \(0.271289\pi\)
\(632\) 0 0
\(633\) 19.3137 0.767651
\(634\) 0 0
\(635\) − 3.94113i − 0.156399i
\(636\) 0 0
\(637\) − 13.1716i − 0.521877i
\(638\) 0 0
\(639\) −9.17157 −0.362822
\(640\) 0 0
\(641\) 42.9706 1.69724 0.848618 0.529007i \(-0.177435\pi\)
0.848618 + 0.529007i \(0.177435\pi\)
\(642\) 0 0
\(643\) 24.2843i 0.957678i 0.877903 + 0.478839i \(0.158942\pi\)
−0.877903 + 0.478839i \(0.841058\pi\)
\(644\) 0 0
\(645\) 0.970563i 0.0382159i
\(646\) 0 0
\(647\) −17.1716 −0.675084 −0.337542 0.941310i \(-0.609596\pi\)
−0.337542 + 0.941310i \(0.609596\pi\)
\(648\) 0 0
\(649\) 8.00000 0.314027
\(650\) 0 0
\(651\) − 15.6569i − 0.613640i
\(652\) 0 0
\(653\) − 23.4142i − 0.916269i −0.888883 0.458134i \(-0.848518\pi\)
0.888883 0.458134i \(-0.151482\pi\)
\(654\) 0 0
\(655\) 2.34315 0.0915543
\(656\) 0 0
\(657\) −1.65685 −0.0646400
\(658\) 0 0
\(659\) − 16.6863i − 0.650006i −0.945713 0.325003i \(-0.894635\pi\)
0.945713 0.325003i \(-0.105365\pi\)
\(660\) 0 0
\(661\) 16.6863i 0.649022i 0.945882 + 0.324511i \(0.105200\pi\)
−0.945882 + 0.324511i \(0.894800\pi\)
\(662\) 0 0
\(663\) 10.3431 0.401694
\(664\) 0 0
\(665\) 11.3137 0.438727
\(666\) 0 0
\(667\) − 0.686292i − 0.0265733i
\(668\) 0 0
\(669\) − 7.89949i − 0.305412i
\(670\) 0 0
\(671\) 19.3137 0.745597
\(672\) 0 0
\(673\) 36.6274 1.41188 0.705942 0.708270i \(-0.250524\pi\)
0.705942 + 0.708270i \(0.250524\pi\)
\(674\) 0 0
\(675\) − 4.65685i − 0.179242i
\(676\) 0 0
\(677\) 7.21320i 0.277226i 0.990347 + 0.138613i \(0.0442644\pi\)
−0.990347 + 0.138613i \(0.955736\pi\)
\(678\) 0 0
\(679\) 45.4558 1.74444
\(680\) 0 0
\(681\) −29.3137 −1.12330
\(682\) 0 0
\(683\) 25.3137i 0.968602i 0.874901 + 0.484301i \(0.160926\pi\)
−0.874901 + 0.484301i \(0.839074\pi\)
\(684\) 0 0
\(685\) − 2.54416i − 0.0972072i
\(686\) 0 0
\(687\) 16.4853 0.628953
\(688\) 0 0
\(689\) 33.6569 1.28222
\(690\) 0 0
\(691\) − 17.6569i − 0.671698i −0.941916 0.335849i \(-0.890977\pi\)
0.941916 0.335849i \(-0.109023\pi\)
\(692\) 0 0
\(693\) 6.82843i 0.259390i
\(694\) 0 0
\(695\) 11.3137 0.429153
\(696\) 0 0
\(697\) 42.6274 1.61463
\(698\) 0 0
\(699\) − 9.31371i − 0.352277i
\(700\) 0 0
\(701\) − 7.41421i − 0.280031i −0.990149 0.140015i \(-0.955285\pi\)
0.990149 0.140015i \(-0.0447152\pi\)
\(702\) 0 0
\(703\) −54.6274 −2.06031
\(704\) 0 0
\(705\) 7.31371 0.275450
\(706\) 0 0
\(707\) − 59.9411i − 2.25432i
\(708\) 0 0
\(709\) 31.1127i 1.16846i 0.811587 + 0.584231i \(0.198604\pi\)
−0.811587 + 0.584231i \(0.801396\pi\)
\(710\) 0 0
\(711\) 5.75736 0.215918
\(712\) 0 0
\(713\) −5.37258 −0.201205
\(714\) 0 0
\(715\) − 3.31371i − 0.123926i
\(716\) 0 0
\(717\) − 16.9706i − 0.633777i
\(718\) 0 0
\(719\) −25.1716 −0.938741 −0.469371 0.883001i \(-0.655519\pi\)
−0.469371 + 0.883001i \(0.655519\pi\)
\(720\) 0 0
\(721\) 30.9706 1.15340
\(722\) 0 0
\(723\) − 17.6569i − 0.656665i
\(724\) 0 0
\(725\) − 2.72792i − 0.101312i
\(726\) 0 0
\(727\) −13.7574 −0.510232 −0.255116 0.966910i \(-0.582114\pi\)
−0.255116 + 0.966910i \(0.582114\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) − 6.05887i − 0.224096i
\(732\) 0 0
\(733\) − 40.4853i − 1.49536i −0.664060 0.747679i \(-0.731168\pi\)
0.664060 0.747679i \(-0.268832\pi\)
\(734\) 0 0
\(735\) −2.72792 −0.100621
\(736\) 0 0
\(737\) −16.0000 −0.589368
\(738\) 0 0
\(739\) 34.6274i 1.27379i 0.770951 + 0.636895i \(0.219782\pi\)
−0.770951 + 0.636895i \(0.780218\pi\)
\(740\) 0 0
\(741\) − 16.0000i − 0.587775i
\(742\) 0 0
\(743\) 40.0000 1.46746 0.733729 0.679442i \(-0.237778\pi\)
0.733729 + 0.679442i \(0.237778\pi\)
\(744\) 0 0
\(745\) −8.34315 −0.305669
\(746\) 0 0
\(747\) − 9.31371i − 0.340771i
\(748\) 0 0
\(749\) 52.2843i 1.91043i
\(750\) 0 0
\(751\) −30.7279 −1.12128 −0.560639 0.828060i \(-0.689444\pi\)
−0.560639 + 0.828060i \(0.689444\pi\)
\(752\) 0 0
\(753\) −22.0000 −0.801725
\(754\) 0 0
\(755\) 2.68629i 0.0977642i
\(756\) 0 0
\(757\) 15.5147i 0.563892i 0.959430 + 0.281946i \(0.0909799\pi\)
−0.959430 + 0.281946i \(0.909020\pi\)
\(758\) 0 0
\(759\) 2.34315 0.0850508
\(760\) 0 0
\(761\) −24.3431 −0.882438 −0.441219 0.897399i \(-0.645454\pi\)
−0.441219 + 0.897399i \(0.645454\pi\)
\(762\) 0 0
\(763\) 44.9706i 1.62804i
\(764\) 0 0
\(765\) − 2.14214i − 0.0774491i
\(766\) 0 0
\(767\) 11.3137 0.408514
\(768\) 0 0
\(769\) −13.6569 −0.492479 −0.246239 0.969209i \(-0.579195\pi\)
−0.246239 + 0.969209i \(0.579195\pi\)
\(770\) 0 0
\(771\) − 1.31371i − 0.0473121i
\(772\) 0 0
\(773\) 18.7279i 0.673597i 0.941577 + 0.336798i \(0.109344\pi\)
−0.941577 + 0.336798i \(0.890656\pi\)
\(774\) 0 0
\(775\) −21.3553 −0.767106
\(776\) 0 0
\(777\) 32.9706 1.18281
\(778\) 0 0
\(779\) − 65.9411i − 2.36259i
\(780\) 0 0
\(781\) − 18.3431i − 0.656369i
\(782\) 0 0
\(783\) −0.585786 −0.0209343
\(784\) 0 0
\(785\) −11.7157 −0.418152
\(786\) 0 0
\(787\) 10.3431i 0.368693i 0.982861 + 0.184347i \(0.0590169\pi\)
−0.982861 + 0.184347i \(0.940983\pi\)
\(788\) 0 0
\(789\) − 13.6569i − 0.486197i
\(790\) 0 0
\(791\) 20.4853 0.728373
\(792\) 0 0
\(793\) 27.3137 0.969938
\(794\) 0 0
\(795\) − 6.97056i − 0.247220i
\(796\) 0 0
\(797\) − 32.5858i − 1.15425i −0.816657 0.577124i \(-0.804175\pi\)
0.816657 0.577124i \(-0.195825\pi\)
\(798\) 0 0
\(799\) −45.6569 −1.61522
\(800\) 0 0
\(801\) 2.00000 0.0706665
\(802\) 0 0
\(803\) − 3.31371i − 0.116938i
\(804\) 0 0
\(805\) 2.34315i 0.0825850i
\(806\) 0 0
\(807\) −6.24264 −0.219751
\(808\) 0 0
\(809\) 38.9706 1.37013 0.685066 0.728481i \(-0.259773\pi\)
0.685066 + 0.728481i \(0.259773\pi\)
\(810\) 0 0
\(811\) 29.6569i 1.04139i 0.853742 + 0.520697i \(0.174328\pi\)
−0.853742 + 0.520697i \(0.825672\pi\)
\(812\) 0 0
\(813\) 19.4142i 0.680886i
\(814\) 0 0
\(815\) −3.71573 −0.130156
\(816\) 0 0
\(817\) −9.37258 −0.327905
\(818\) 0 0
\(819\) 9.65685i 0.337438i
\(820\) 0 0
\(821\) − 44.8701i − 1.56598i −0.622037 0.782988i \(-0.713695\pi\)
0.622037 0.782988i \(-0.286305\pi\)
\(822\) 0 0
\(823\) −3.21320 −0.112005 −0.0560026 0.998431i \(-0.517836\pi\)
−0.0560026 + 0.998431i \(0.517836\pi\)
\(824\) 0 0
\(825\) 9.31371 0.324262
\(826\) 0 0
\(827\) − 37.9411i − 1.31934i −0.751554 0.659671i \(-0.770696\pi\)
0.751554 0.659671i \(-0.229304\pi\)
\(828\) 0 0
\(829\) − 20.7696i − 0.721356i −0.932690 0.360678i \(-0.882545\pi\)
0.932690 0.360678i \(-0.117455\pi\)
\(830\) 0 0
\(831\) 7.51472 0.260683
\(832\) 0 0
\(833\) 17.0294 0.590035
\(834\) 0 0
\(835\) − 12.6863i − 0.439027i
\(836\) 0 0
\(837\) 4.58579i 0.158508i
\(838\) 0 0
\(839\) 11.5147 0.397532 0.198766 0.980047i \(-0.436307\pi\)
0.198766 + 0.980047i \(0.436307\pi\)
\(840\) 0 0
\(841\) 28.6569 0.988167
\(842\) 0 0
\(843\) 2.00000i 0.0688837i
\(844\) 0 0
\(845\) 2.92893i 0.100758i
\(846\) 0 0
\(847\) 23.8995 0.821196
\(848\) 0 0
\(849\) 15.3137 0.525565
\(850\) 0 0
\(851\) − 11.3137i − 0.387829i
\(852\) 0 0
\(853\) − 10.6274i − 0.363876i −0.983310 0.181938i \(-0.941763\pi\)
0.983310 0.181938i \(-0.0582370\pi\)
\(854\) 0 0
\(855\) −3.31371 −0.113326
\(856\) 0 0
\(857\) 37.5980 1.28432 0.642161 0.766570i \(-0.278038\pi\)
0.642161 + 0.766570i \(0.278038\pi\)
\(858\) 0 0
\(859\) − 11.0294i − 0.376320i −0.982138 0.188160i \(-0.939748\pi\)
0.982138 0.188160i \(-0.0602523\pi\)
\(860\) 0 0
\(861\) 39.7990i 1.35635i
\(862\) 0 0
\(863\) −48.9706 −1.66698 −0.833489 0.552537i \(-0.813660\pi\)
−0.833489 + 0.552537i \(0.813660\pi\)
\(864\) 0 0
\(865\) 10.9706 0.373010
\(866\) 0 0
\(867\) − 3.62742i − 0.123194i
\(868\) 0 0
\(869\) 11.5147i 0.390610i
\(870\) 0 0
\(871\) −22.6274 −0.766701
\(872\) 0 0
\(873\) −13.3137 −0.450601
\(874\) 0 0
\(875\) 19.3137i 0.652923i
\(876\) 0 0
\(877\) 0.686292i 0.0231744i 0.999933 + 0.0115872i \(0.00368841\pi\)
−0.999933 + 0.0115872i \(0.996312\pi\)
\(878\) 0 0
\(879\) −27.8995 −0.941027
\(880\) 0 0
\(881\) 22.6863 0.764321 0.382160 0.924096i \(-0.375180\pi\)
0.382160 + 0.924096i \(0.375180\pi\)
\(882\) 0 0
\(883\) 41.6569i 1.40186i 0.713228 + 0.700932i \(0.247233\pi\)
−0.713228 + 0.700932i \(0.752767\pi\)
\(884\) 0 0
\(885\) − 2.34315i − 0.0787640i
\(886\) 0 0
\(887\) −18.3431 −0.615903 −0.307951 0.951402i \(-0.599643\pi\)
−0.307951 + 0.951402i \(0.599643\pi\)
\(888\) 0 0
\(889\) −22.9706 −0.770408
\(890\) 0 0
\(891\) − 2.00000i − 0.0670025i
\(892\) 0 0
\(893\) 70.6274i 2.36346i
\(894\) 0 0
\(895\) 2.34315 0.0783227
\(896\) 0 0
\(897\) 3.31371 0.110642
\(898\) 0 0
\(899\) 2.68629i 0.0895928i
\(900\) 0 0
\(901\) 43.5147i 1.44969i
\(902\) 0 0
\(903\) 5.65685 0.188248
\(904\) 0 0
\(905\) −4.97056 −0.165227
\(906\) 0 0
\(907\) 27.5980i 0.916376i 0.888855 + 0.458188i \(0.151501\pi\)
−0.888855 + 0.458188i \(0.848499\pi\)
\(908\) 0 0
\(909\) 17.5563i 0.582307i
\(910\) 0 0
\(911\) −51.3137 −1.70010 −0.850050 0.526703i \(-0.823428\pi\)
−0.850050 + 0.526703i \(0.823428\pi\)
\(912\) 0 0
\(913\) 18.6274 0.616478
\(914\) 0 0
\(915\) − 5.65685i − 0.187010i
\(916\) 0 0
\(917\) − 13.6569i − 0.450989i
\(918\) 0 0
\(919\) 53.3553 1.76003 0.880015 0.474946i \(-0.157532\pi\)
0.880015 + 0.474946i \(0.157532\pi\)
\(920\) 0 0
\(921\) 16.0000 0.527218
\(922\) 0 0
\(923\) − 25.9411i − 0.853863i
\(924\) 0 0
\(925\) − 44.9706i − 1.47862i
\(926\) 0 0
\(927\) −9.07107 −0.297933
\(928\) 0 0
\(929\) −49.5980 −1.62726 −0.813628 0.581385i \(-0.802511\pi\)
−0.813628 + 0.581385i \(0.802511\pi\)
\(930\) 0 0
\(931\) − 26.3431i − 0.863362i
\(932\) 0 0
\(933\) 11.3137i 0.370394i
\(934\) 0 0
\(935\) 4.28427 0.140111
\(936\) 0 0
\(937\) 8.62742 0.281845 0.140923 0.990021i \(-0.454993\pi\)
0.140923 + 0.990021i \(0.454993\pi\)
\(938\) 0 0
\(939\) − 16.9706i − 0.553813i
\(940\) 0 0
\(941\) − 16.3848i − 0.534128i −0.963679 0.267064i \(-0.913946\pi\)
0.963679 0.267064i \(-0.0860536\pi\)
\(942\) 0 0
\(943\) 13.6569 0.444728
\(944\) 0 0
\(945\) 2.00000 0.0650600
\(946\) 0 0
\(947\) 10.6274i 0.345345i 0.984979 + 0.172672i \(0.0552402\pi\)
−0.984979 + 0.172672i \(0.944760\pi\)
\(948\) 0 0
\(949\) − 4.68629i − 0.152123i
\(950\) 0 0
\(951\) 5.07107 0.164441
\(952\) 0 0
\(953\) 2.97056 0.0962260 0.0481130 0.998842i \(-0.484679\pi\)
0.0481130 + 0.998842i \(0.484679\pi\)
\(954\) 0 0
\(955\) 9.94113i 0.321687i
\(956\) 0 0
\(957\) − 1.17157i − 0.0378716i
\(958\) 0 0
\(959\) −14.8284 −0.478835
\(960\) 0 0
\(961\) −9.97056 −0.321631
\(962\) 0 0
\(963\) − 15.3137i − 0.493477i
\(964\) 0 0
\(965\) − 12.6863i − 0.408386i
\(966\) 0 0
\(967\) 10.2426 0.329381 0.164691 0.986345i \(-0.447337\pi\)
0.164691 + 0.986345i \(0.447337\pi\)
\(968\) 0 0
\(969\) 20.6863 0.664539
\(970\) 0 0
\(971\) − 6.68629i − 0.214573i −0.994228 0.107287i \(-0.965784\pi\)
0.994228 0.107287i \(-0.0342163\pi\)
\(972\) 0 0
\(973\) − 65.9411i − 2.11398i
\(974\) 0 0
\(975\) 13.1716 0.421828
\(976\) 0 0
\(977\) 26.2843 0.840908 0.420454 0.907314i \(-0.361871\pi\)
0.420454 + 0.907314i \(0.361871\pi\)
\(978\) 0 0
\(979\) 4.00000i 0.127841i
\(980\) 0 0
\(981\) − 13.1716i − 0.420536i
\(982\) 0 0
\(983\) 52.2843 1.66761 0.833805 0.552060i \(-0.186158\pi\)
0.833805 + 0.552060i \(0.186158\pi\)
\(984\) 0 0
\(985\) −1.71573 −0.0546677
\(986\) 0 0
\(987\) − 42.6274i − 1.35685i
\(988\) 0 0
\(989\) − 1.94113i − 0.0617242i
\(990\) 0 0
\(991\) 7.89949 0.250936 0.125468 0.992098i \(-0.459957\pi\)
0.125468 + 0.992098i \(0.459957\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 7.94113i − 0.251751i
\(996\) 0 0
\(997\) − 20.9706i − 0.664144i −0.943254 0.332072i \(-0.892252\pi\)
0.943254 0.332072i \(-0.107748\pi\)
\(998\) 0 0
\(999\) −9.65685 −0.305529
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 1536.2.d.f.769.3 4
3.2 odd 2 4608.2.d.o.2305.2 4
4.3 odd 2 1536.2.d.a.769.1 4
8.3 odd 2 1536.2.d.a.769.4 4
8.5 even 2 inner 1536.2.d.f.769.2 4
12.11 even 2 4608.2.d.c.2305.2 4
16.3 odd 4 1536.2.a.l.1.1 yes 2
16.5 even 4 1536.2.a.g.1.2 yes 2
16.11 odd 4 1536.2.a.b.1.2 2
16.13 even 4 1536.2.a.e.1.1 yes 2
24.5 odd 2 4608.2.d.o.2305.3 4
24.11 even 2 4608.2.d.c.2305.3 4
48.5 odd 4 4608.2.a.n.1.1 2
48.11 even 4 4608.2.a.r.1.1 2
48.29 odd 4 4608.2.a.a.1.2 2
48.35 even 4 4608.2.a.e.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1536.2.a.b.1.2 2 16.11 odd 4
1536.2.a.e.1.1 yes 2 16.13 even 4
1536.2.a.g.1.2 yes 2 16.5 even 4
1536.2.a.l.1.1 yes 2 16.3 odd 4
1536.2.d.a.769.1 4 4.3 odd 2
1536.2.d.a.769.4 4 8.3 odd 2
1536.2.d.f.769.2 4 8.5 even 2 inner
1536.2.d.f.769.3 4 1.1 even 1 trivial
4608.2.a.a.1.2 2 48.29 odd 4
4608.2.a.e.1.2 2 48.35 even 4
4608.2.a.n.1.1 2 48.5 odd 4
4608.2.a.r.1.1 2 48.11 even 4
4608.2.d.c.2305.2 4 12.11 even 2
4608.2.d.c.2305.3 4 24.11 even 2
4608.2.d.o.2305.2 4 3.2 odd 2
4608.2.d.o.2305.3 4 24.5 odd 2