Properties

Label 1008.2.cz.i.607.15
Level $1008$
Weight $2$
Character 1008.607
Analytic conductor $8.049$
Analytic rank $0$
Dimension $32$
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] = [1008,2,Mod(367,1008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1008, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 4, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.367");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.cz (of order \(6\), degree \(2\), 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: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 607.15
Character \(\chi\) \(=\) 1008.607
Dual form 1008.2.cz.i.367.15

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.69998 - 0.331769i) q^{3} +(-0.535741 + 0.309310i) q^{5} +(2.62638 + 0.319613i) q^{7} +(2.77986 - 1.12800i) q^{9} +O(q^{10})\) \(q+(1.69998 - 0.331769i) q^{3} +(-0.535741 + 0.309310i) q^{5} +(2.62638 + 0.319613i) q^{7} +(2.77986 - 1.12800i) q^{9} +(-2.58062 - 1.48992i) q^{11} +(1.64842 + 0.951716i) q^{13} +(-0.808129 + 0.703564i) q^{15} +(2.92194 - 1.68698i) q^{17} +(-1.13454 + 1.96509i) q^{19} +(4.57082 - 0.328016i) q^{21} +(5.84046 - 3.37199i) q^{23} +(-2.30865 + 3.99871i) q^{25} +(4.35146 - 2.83985i) q^{27} +(-1.74898 - 3.02933i) q^{29} +7.35193 q^{31} +(-4.88130 - 1.67666i) q^{33} +(-1.50592 + 0.641135i) q^{35} +(-3.25230 + 5.63315i) q^{37} +(3.11803 + 1.07100i) q^{39} +(-9.09363 - 5.25021i) q^{41} +(-2.48652 + 1.43559i) q^{43} +(-1.14038 + 1.46416i) q^{45} +5.60907 q^{47} +(6.79570 + 1.67885i) q^{49} +(4.40755 - 3.83725i) q^{51} +(2.50765 + 4.34338i) q^{53} +1.84339 q^{55} +(-1.27675 + 3.71702i) q^{57} +4.34188 q^{59} -3.63138i q^{61} +(7.66147 - 2.07408i) q^{63} -1.17750 q^{65} -5.01617i q^{67} +(8.80994 - 7.67000i) q^{69} +6.18857i q^{71} +(-12.5278 + 7.23291i) q^{73} +(-2.59802 + 7.56366i) q^{75} +(-6.30147 - 4.73789i) q^{77} -12.0606i q^{79} +(6.45522 - 6.27137i) q^{81} +(5.48427 + 9.49903i) q^{83} +(-1.04360 + 1.80757i) q^{85} +(-3.97828 - 4.56954i) q^{87} +(-6.03976 - 3.48706i) q^{89} +(4.02519 + 3.02642i) q^{91} +(12.4981 - 2.43915i) q^{93} -1.40371i q^{95} +(-2.71311 + 1.56642i) q^{97} +(-8.85438 - 1.23082i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 32 q - 4 q^{9} + 6 q^{13} - 18 q^{17} + 4 q^{21} + 16 q^{25} - 12 q^{29} + 2 q^{37} - 36 q^{41} + 12 q^{45} + 2 q^{49} - 12 q^{53} - 46 q^{57} - 36 q^{65} + 42 q^{69} + 42 q^{77} + 20 q^{81} - 12 q^{85} - 18 q^{89} - 38 q^{93} + 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(3\) 1.69998 0.331769i 0.981483 0.191547i
\(4\) 0 0
\(5\) −0.535741 + 0.309310i −0.239591 + 0.138328i −0.614989 0.788536i \(-0.710839\pi\)
0.375398 + 0.926864i \(0.377506\pi\)
\(6\) 0 0
\(7\) 2.62638 + 0.319613i 0.992677 + 0.120802i
\(8\) 0 0
\(9\) 2.77986 1.12800i 0.926619 0.376001i
\(10\) 0 0
\(11\) −2.58062 1.48992i −0.778085 0.449228i 0.0576660 0.998336i \(-0.481634\pi\)
−0.835751 + 0.549108i \(0.814967\pi\)
\(12\) 0 0
\(13\) 1.64842 + 0.951716i 0.457190 + 0.263959i 0.710862 0.703332i \(-0.248305\pi\)
−0.253672 + 0.967290i \(0.581638\pi\)
\(14\) 0 0
\(15\) −0.808129 + 0.703564i −0.208658 + 0.181659i
\(16\) 0 0
\(17\) 2.92194 1.68698i 0.708675 0.409154i −0.101895 0.994795i \(-0.532491\pi\)
0.810570 + 0.585641i \(0.199157\pi\)
\(18\) 0 0
\(19\) −1.13454 + 1.96509i −0.260282 + 0.450822i −0.966317 0.257355i \(-0.917149\pi\)
0.706034 + 0.708177i \(0.250482\pi\)
\(20\) 0 0
\(21\) 4.57082 0.328016i 0.997435 0.0715790i
\(22\) 0 0
\(23\) 5.84046 3.37199i 1.21782 0.703109i 0.253369 0.967370i \(-0.418461\pi\)
0.964451 + 0.264260i \(0.0851278\pi\)
\(24\) 0 0
\(25\) −2.30865 + 3.99871i −0.461731 + 0.799741i
\(26\) 0 0
\(27\) 4.35146 2.83985i 0.837440 0.546530i
\(28\) 0 0
\(29\) −1.74898 3.02933i −0.324778 0.562532i 0.656689 0.754161i \(-0.271956\pi\)
−0.981467 + 0.191629i \(0.938623\pi\)
\(30\) 0 0
\(31\) 7.35193 1.32045 0.660223 0.751070i \(-0.270462\pi\)
0.660223 + 0.751070i \(0.270462\pi\)
\(32\) 0 0
\(33\) −4.88130 1.67666i −0.849726 0.291870i
\(34\) 0 0
\(35\) −1.50592 + 0.641135i −0.254546 + 0.108372i
\(36\) 0 0
\(37\) −3.25230 + 5.63315i −0.534675 + 0.926085i 0.464504 + 0.885571i \(0.346233\pi\)
−0.999179 + 0.0405136i \(0.987101\pi\)
\(38\) 0 0
\(39\) 3.11803 + 1.07100i 0.499285 + 0.171498i
\(40\) 0 0
\(41\) −9.09363 5.25021i −1.42019 0.819945i −0.423872 0.905722i \(-0.639330\pi\)
−0.996314 + 0.0857766i \(0.972663\pi\)
\(42\) 0 0
\(43\) −2.48652 + 1.43559i −0.379191 + 0.218926i −0.677466 0.735554i \(-0.736922\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(44\) 0 0
\(45\) −1.14038 + 1.46416i −0.169998 + 0.218264i
\(46\) 0 0
\(47\) 5.60907 0.818167 0.409083 0.912497i \(-0.365848\pi\)
0.409083 + 0.912497i \(0.365848\pi\)
\(48\) 0 0
\(49\) 6.79570 + 1.67885i 0.970814 + 0.239835i
\(50\) 0 0
\(51\) 4.40755 3.83725i 0.617181 0.537322i
\(52\) 0 0
\(53\) 2.50765 + 4.34338i 0.344452 + 0.596609i 0.985254 0.171097i \(-0.0547313\pi\)
−0.640802 + 0.767706i \(0.721398\pi\)
\(54\) 0 0
\(55\) 1.84339 0.248563
\(56\) 0 0
\(57\) −1.27675 + 3.71702i −0.169109 + 0.492331i
\(58\) 0 0
\(59\) 4.34188 0.565265 0.282632 0.959228i \(-0.408792\pi\)
0.282632 + 0.959228i \(0.408792\pi\)
\(60\) 0 0
\(61\) 3.63138i 0.464951i −0.972602 0.232475i \(-0.925318\pi\)
0.972602 0.232475i \(-0.0746825\pi\)
\(62\) 0 0
\(63\) 7.66147 2.07408i 0.965255 0.261309i
\(64\) 0 0
\(65\) −1.17750 −0.146051
\(66\) 0 0
\(67\) 5.01617i 0.612822i −0.951899 0.306411i \(-0.900872\pi\)
0.951899 0.306411i \(-0.0991282\pi\)
\(68\) 0 0
\(69\) 8.80994 7.67000i 1.06059 0.923360i
\(70\) 0 0
\(71\) 6.18857i 0.734448i 0.930132 + 0.367224i \(0.119692\pi\)
−0.930132 + 0.367224i \(0.880308\pi\)
\(72\) 0 0
\(73\) −12.5278 + 7.23291i −1.46626 + 0.846548i −0.999288 0.0377215i \(-0.987990\pi\)
−0.466976 + 0.884270i \(0.654657\pi\)
\(74\) 0 0
\(75\) −2.59802 + 7.56366i −0.299993 + 0.873376i
\(76\) 0 0
\(77\) −6.30147 4.73789i −0.718119 0.539932i
\(78\) 0 0
\(79\) 12.0606i 1.35693i −0.734635 0.678463i \(-0.762646\pi\)
0.734635 0.678463i \(-0.237354\pi\)
\(80\) 0 0
\(81\) 6.45522 6.27137i 0.717247 0.696819i
\(82\) 0 0
\(83\) 5.48427 + 9.49903i 0.601977 + 1.04265i 0.992521 + 0.122070i \(0.0389533\pi\)
−0.390545 + 0.920584i \(0.627713\pi\)
\(84\) 0 0
\(85\) −1.04360 + 1.80757i −0.113195 + 0.196059i
\(86\) 0 0
\(87\) −3.97828 4.56954i −0.426516 0.489906i
\(88\) 0 0
\(89\) −6.03976 3.48706i −0.640213 0.369627i 0.144483 0.989507i \(-0.453848\pi\)
−0.784697 + 0.619880i \(0.787181\pi\)
\(90\) 0 0
\(91\) 4.02519 + 3.02642i 0.421955 + 0.317255i
\(92\) 0 0
\(93\) 12.4981 2.43915i 1.29600 0.252928i
\(94\) 0 0
\(95\) 1.40371i 0.144017i
\(96\) 0 0
\(97\) −2.71311 + 1.56642i −0.275475 + 0.159045i −0.631373 0.775479i \(-0.717508\pi\)
0.355898 + 0.934525i \(0.384175\pi\)
\(98\) 0 0
\(99\) −8.85438 1.23082i −0.889899 0.123702i
\(100\) 0 0
\(101\) −11.1962 6.46415i −1.11407 0.643207i −0.174187 0.984713i \(-0.555730\pi\)
−0.939880 + 0.341505i \(0.889063\pi\)
\(102\) 0 0
\(103\) −3.52663 6.10830i −0.347489 0.601869i 0.638314 0.769776i \(-0.279632\pi\)
−0.985803 + 0.167907i \(0.946299\pi\)
\(104\) 0 0
\(105\) −2.34732 + 1.58953i −0.229075 + 0.155123i
\(106\) 0 0
\(107\) −3.44421 1.98852i −0.332964 0.192237i 0.324192 0.945991i \(-0.394908\pi\)
−0.657156 + 0.753754i \(0.728241\pi\)
\(108\) 0 0
\(109\) 7.60818 + 13.1778i 0.728731 + 1.26220i 0.957420 + 0.288700i \(0.0932230\pi\)
−0.228688 + 0.973500i \(0.573444\pi\)
\(110\) 0 0
\(111\) −3.65994 + 10.6553i −0.347386 + 1.01135i
\(112\) 0 0
\(113\) −1.44699 + 2.50626i −0.136121 + 0.235769i −0.926025 0.377462i \(-0.876797\pi\)
0.789904 + 0.613231i \(0.210130\pi\)
\(114\) 0 0
\(115\) −2.08598 + 3.61303i −0.194519 + 0.336917i
\(116\) 0 0
\(117\) 5.65592 + 0.786214i 0.522890 + 0.0726855i
\(118\) 0 0
\(119\) 8.21330 3.49676i 0.752912 0.320548i
\(120\) 0 0
\(121\) −1.06028 1.83646i −0.0963889 0.166951i
\(122\) 0 0
\(123\) −17.2008 5.90826i −1.55095 0.532730i
\(124\) 0 0
\(125\) 5.94947i 0.532136i
\(126\) 0 0
\(127\) 2.01268i 0.178596i 0.996005 + 0.0892982i \(0.0284624\pi\)
−0.996005 + 0.0892982i \(0.971538\pi\)
\(128\) 0 0
\(129\) −3.75075 + 3.26543i −0.330235 + 0.287505i
\(130\) 0 0
\(131\) −9.03856 15.6553i −0.789703 1.36781i −0.926149 0.377158i \(-0.876901\pi\)
0.136446 0.990648i \(-0.456432\pi\)
\(132\) 0 0
\(133\) −3.60781 + 4.79845i −0.312837 + 0.416078i
\(134\) 0 0
\(135\) −1.45286 + 2.86738i −0.125043 + 0.246785i
\(136\) 0 0
\(137\) −10.3621 + 17.9477i −0.885297 + 1.53338i −0.0399235 + 0.999203i \(0.512711\pi\)
−0.845373 + 0.534176i \(0.820622\pi\)
\(138\) 0 0
\(139\) −3.98194 + 6.89693i −0.337744 + 0.584990i −0.984008 0.178124i \(-0.942997\pi\)
0.646264 + 0.763114i \(0.276330\pi\)
\(140\) 0 0
\(141\) 9.53530 1.86092i 0.803017 0.156718i
\(142\) 0 0
\(143\) −2.83596 4.91203i −0.237155 0.410765i
\(144\) 0 0
\(145\) 1.87401 + 1.08196i 0.155628 + 0.0898517i
\(146\) 0 0
\(147\) 12.1095 + 0.599400i 0.998777 + 0.0494377i
\(148\) 0 0
\(149\) −10.5029 18.1915i −0.860427 1.49030i −0.871517 0.490365i \(-0.836864\pi\)
0.0110903 0.999939i \(-0.496470\pi\)
\(150\) 0 0
\(151\) 12.8560 + 7.42241i 1.04621 + 0.604027i 0.921585 0.388177i \(-0.126895\pi\)
0.124621 + 0.992204i \(0.460228\pi\)
\(152\) 0 0
\(153\) 6.21966 7.98553i 0.502830 0.645592i
\(154\) 0 0
\(155\) −3.93873 + 2.27403i −0.316367 + 0.182654i
\(156\) 0 0
\(157\) 21.0499i 1.67996i 0.542614 + 0.839982i \(0.317435\pi\)
−0.542614 + 0.839982i \(0.682565\pi\)
\(158\) 0 0
\(159\) 5.70396 + 6.55169i 0.452353 + 0.519583i
\(160\) 0 0
\(161\) 16.4170 6.98943i 1.29384 0.550844i
\(162\) 0 0
\(163\) −1.53261 0.884855i −0.120044 0.0693072i 0.438776 0.898597i \(-0.355412\pi\)
−0.558819 + 0.829289i \(0.688746\pi\)
\(164\) 0 0
\(165\) 3.13373 0.611581i 0.243960 0.0476115i
\(166\) 0 0
\(167\) −2.50736 + 4.34288i −0.194025 + 0.336062i −0.946581 0.322467i \(-0.895488\pi\)
0.752555 + 0.658529i \(0.228821\pi\)
\(168\) 0 0
\(169\) −4.68847 8.12067i −0.360652 0.624667i
\(170\) 0 0
\(171\) −0.937249 + 6.74244i −0.0716732 + 0.515607i
\(172\) 0 0
\(173\) 3.13008i 0.237976i 0.992896 + 0.118988i \(0.0379650\pi\)
−0.992896 + 0.118988i \(0.962035\pi\)
\(174\) 0 0
\(175\) −7.34143 + 9.76423i −0.554960 + 0.738106i
\(176\) 0 0
\(177\) 7.38110 1.44050i 0.554798 0.108275i
\(178\) 0 0
\(179\) −15.1505 + 8.74714i −1.13240 + 0.653792i −0.944538 0.328404i \(-0.893489\pi\)
−0.187863 + 0.982195i \(0.560156\pi\)
\(180\) 0 0
\(181\) 4.76258i 0.354000i 0.984211 + 0.177000i \(0.0566392\pi\)
−0.984211 + 0.177000i \(0.943361\pi\)
\(182\) 0 0
\(183\) −1.20478 6.17327i −0.0890600 0.456341i
\(184\) 0 0
\(185\) 4.02388i 0.295842i
\(186\) 0 0
\(187\) −10.0539 −0.735213
\(188\) 0 0
\(189\) 12.3362 6.06773i 0.897329 0.441363i
\(190\) 0 0
\(191\) 9.60774i 0.695191i 0.937645 + 0.347596i \(0.113002\pi\)
−0.937645 + 0.347596i \(0.886998\pi\)
\(192\) 0 0
\(193\) 15.1521 1.09067 0.545336 0.838218i \(-0.316402\pi\)
0.545336 + 0.838218i \(0.316402\pi\)
\(194\) 0 0
\(195\) −2.00173 + 0.390660i −0.143347 + 0.0279757i
\(196\) 0 0
\(197\) −4.93656 −0.351715 −0.175858 0.984416i \(-0.556270\pi\)
−0.175858 + 0.984416i \(0.556270\pi\)
\(198\) 0 0
\(199\) −11.3771 19.7058i −0.806503 1.39690i −0.915272 0.402837i \(-0.868024\pi\)
0.108769 0.994067i \(-0.465309\pi\)
\(200\) 0 0
\(201\) −1.66421 8.52738i −0.117384 0.601475i
\(202\) 0 0
\(203\) −3.62528 8.51516i −0.254445 0.597647i
\(204\) 0 0
\(205\) 6.49578 0.453685
\(206\) 0 0
\(207\) 12.4320 15.9617i 0.864087 1.10942i
\(208\) 0 0
\(209\) 5.85565 3.38076i 0.405044 0.233852i
\(210\) 0 0
\(211\) −3.10815 1.79449i −0.213974 0.123538i 0.389183 0.921160i \(-0.372757\pi\)
−0.603157 + 0.797623i \(0.706091\pi\)
\(212\) 0 0
\(213\) 2.05318 + 10.5204i 0.140682 + 0.720849i
\(214\) 0 0
\(215\) 0.888088 1.53821i 0.0605671 0.104905i
\(216\) 0 0
\(217\) 19.3089 + 2.34977i 1.31078 + 0.159513i
\(218\) 0 0
\(219\) −18.8973 + 16.4521i −1.27696 + 1.11173i
\(220\) 0 0
\(221\) 6.42212 0.431999
\(222\) 0 0
\(223\) 5.12744 + 8.88098i 0.343358 + 0.594714i 0.985054 0.172245i \(-0.0551021\pi\)
−0.641696 + 0.766959i \(0.721769\pi\)
\(224\) 0 0
\(225\) −1.90718 + 13.7200i −0.127145 + 0.914667i
\(226\) 0 0
\(227\) −7.19572 + 12.4634i −0.477597 + 0.827222i −0.999670 0.0256788i \(-0.991825\pi\)
0.522074 + 0.852900i \(0.325159\pi\)
\(228\) 0 0
\(229\) −3.35284 + 1.93576i −0.221562 + 0.127919i −0.606673 0.794951i \(-0.707496\pi\)
0.385111 + 0.922870i \(0.374163\pi\)
\(230\) 0 0
\(231\) −12.2843 5.96367i −0.808245 0.392381i
\(232\) 0 0
\(233\) 5.84465 10.1232i 0.382896 0.663195i −0.608579 0.793493i \(-0.708260\pi\)
0.991475 + 0.130298i \(0.0415935\pi\)
\(234\) 0 0
\(235\) −3.00501 + 1.73494i −0.196025 + 0.113175i
\(236\) 0 0
\(237\) −4.00134 20.5028i −0.259915 1.33180i
\(238\) 0 0
\(239\) −5.38726 3.11033i −0.348473 0.201191i 0.315540 0.948912i \(-0.397814\pi\)
−0.664012 + 0.747722i \(0.731148\pi\)
\(240\) 0 0
\(241\) −14.7081 8.49173i −0.947433 0.547001i −0.0551504 0.998478i \(-0.517564\pi\)
−0.892283 + 0.451477i \(0.850897\pi\)
\(242\) 0 0
\(243\) 8.89309 12.8028i 0.570492 0.821303i
\(244\) 0 0
\(245\) −4.16002 + 1.20255i −0.265774 + 0.0768282i
\(246\) 0 0
\(247\) −3.74042 + 2.15953i −0.237997 + 0.137408i
\(248\) 0 0
\(249\) 12.4746 + 14.3286i 0.790548 + 0.908041i
\(250\) 0 0
\(251\) −27.3406 −1.72572 −0.862861 0.505442i \(-0.831330\pi\)
−0.862861 + 0.505442i \(0.831330\pi\)
\(252\) 0 0
\(253\) −20.0960 −1.26342
\(254\) 0 0
\(255\) −1.17441 + 3.41907i −0.0735442 + 0.214111i
\(256\) 0 0
\(257\) −0.0373645 + 0.0215724i −0.00233074 + 0.00134565i −0.501165 0.865352i \(-0.667095\pi\)
0.498834 + 0.866697i \(0.333762\pi\)
\(258\) 0 0
\(259\) −10.3422 + 13.7553i −0.642633 + 0.854713i
\(260\) 0 0
\(261\) −8.27902 6.44825i −0.512458 0.399137i
\(262\) 0 0
\(263\) 13.4150 + 7.74514i 0.827203 + 0.477586i 0.852894 0.522084i \(-0.174845\pi\)
−0.0256913 + 0.999670i \(0.508179\pi\)
\(264\) 0 0
\(265\) −2.68690 1.55129i −0.165055 0.0952947i
\(266\) 0 0
\(267\) −11.4244 3.92412i −0.699160 0.240152i
\(268\) 0 0
\(269\) −24.9814 + 14.4230i −1.52314 + 0.879385i −0.523515 + 0.852017i \(0.675380\pi\)
−0.999625 + 0.0273690i \(0.991287\pi\)
\(270\) 0 0
\(271\) −7.16925 + 12.4175i −0.435501 + 0.754310i −0.997336 0.0729394i \(-0.976762\pi\)
0.561836 + 0.827249i \(0.310095\pi\)
\(272\) 0 0
\(273\) 7.84682 + 3.80942i 0.474911 + 0.230556i
\(274\) 0 0
\(275\) 11.9155 6.87942i 0.718532 0.414845i
\(276\) 0 0
\(277\) −7.30158 + 12.6467i −0.438709 + 0.759867i −0.997590 0.0693811i \(-0.977898\pi\)
0.558881 + 0.829248i \(0.311231\pi\)
\(278\) 0 0
\(279\) 20.4373 8.29299i 1.22355 0.496489i
\(280\) 0 0
\(281\) 1.04644 + 1.81248i 0.0624251 + 0.108123i 0.895549 0.444963i \(-0.146783\pi\)
−0.833124 + 0.553087i \(0.813450\pi\)
\(282\) 0 0
\(283\) −3.81783 −0.226946 −0.113473 0.993541i \(-0.536198\pi\)
−0.113473 + 0.993541i \(0.536198\pi\)
\(284\) 0 0
\(285\) −0.465707 2.38627i −0.0275861 0.141350i
\(286\) 0 0
\(287\) −22.2053 16.6955i −1.31073 0.985502i
\(288\) 0 0
\(289\) −2.80817 + 4.86389i −0.165186 + 0.286111i
\(290\) 0 0
\(291\) −4.09255 + 3.56300i −0.239909 + 0.208867i
\(292\) 0 0
\(293\) −11.7432 6.77992i −0.686043 0.396087i 0.116085 0.993239i \(-0.462966\pi\)
−0.802128 + 0.597152i \(0.796299\pi\)
\(294\) 0 0
\(295\) −2.32612 + 1.34299i −0.135432 + 0.0781918i
\(296\) 0 0
\(297\) −15.4606 + 0.845237i −0.897116 + 0.0490456i
\(298\) 0 0
\(299\) 12.8367 0.742367
\(300\) 0 0
\(301\) −6.98937 + 2.97568i −0.402861 + 0.171516i
\(302\) 0 0
\(303\) −21.1780 7.27435i −1.21664 0.417901i
\(304\) 0 0
\(305\) 1.12322 + 1.94548i 0.0643156 + 0.111398i
\(306\) 0 0
\(307\) 0.0932072 0.00531961 0.00265981 0.999996i \(-0.499153\pi\)
0.00265981 + 0.999996i \(0.499153\pi\)
\(308\) 0 0
\(309\) −8.02175 9.21396i −0.456341 0.524164i
\(310\) 0 0
\(311\) −0.00835538 −0.000473790 −0.000236895 1.00000i \(-0.500075\pi\)
−0.000236895 1.00000i \(0.500075\pi\)
\(312\) 0 0
\(313\) 0.0547355i 0.00309383i 0.999999 + 0.00154692i \(0.000492399\pi\)
−0.999999 + 0.00154692i \(0.999508\pi\)
\(314\) 0 0
\(315\) −3.46303 + 3.48094i −0.195120 + 0.196129i
\(316\) 0 0
\(317\) −10.5118 −0.590401 −0.295201 0.955435i \(-0.595386\pi\)
−0.295201 + 0.955435i \(0.595386\pi\)
\(318\) 0 0
\(319\) 10.4234i 0.583598i
\(320\) 0 0
\(321\) −6.51482 2.23775i −0.363622 0.124899i
\(322\) 0 0
\(323\) 7.65584i 0.425982i
\(324\) 0 0
\(325\) −7.61127 + 4.39437i −0.422197 + 0.243756i
\(326\) 0 0
\(327\) 17.3057 + 19.8777i 0.957008 + 1.09924i
\(328\) 0 0
\(329\) 14.7315 + 1.79273i 0.812175 + 0.0988364i
\(330\) 0 0
\(331\) 26.7610i 1.47092i −0.677569 0.735459i \(-0.736967\pi\)
0.677569 0.735459i \(-0.263033\pi\)
\(332\) 0 0
\(333\) −2.68673 + 19.3280i −0.147232 + 1.05917i
\(334\) 0 0
\(335\) 1.55155 + 2.68737i 0.0847703 + 0.146827i
\(336\) 0 0
\(337\) 5.90313 10.2245i 0.321564 0.556965i −0.659247 0.751927i \(-0.729125\pi\)
0.980811 + 0.194961i \(0.0624581\pi\)
\(338\) 0 0
\(339\) −1.62835 + 4.74066i −0.0884399 + 0.257477i
\(340\) 0 0
\(341\) −18.9725 10.9538i −1.02742 0.593181i
\(342\) 0 0
\(343\) 17.3115 + 6.58127i 0.934731 + 0.355355i
\(344\) 0 0
\(345\) −2.34744 + 6.83414i −0.126382 + 0.367938i
\(346\) 0 0
\(347\) 2.11386i 0.113478i −0.998389 0.0567390i \(-0.981930\pi\)
0.998389 0.0567390i \(-0.0180703\pi\)
\(348\) 0 0
\(349\) 18.9624 10.9480i 1.01504 0.586031i 0.102373 0.994746i \(-0.467356\pi\)
0.912662 + 0.408715i \(0.134023\pi\)
\(350\) 0 0
\(351\) 9.87578 0.539912i 0.527130 0.0288184i
\(352\) 0 0
\(353\) 7.95031 + 4.59011i 0.423152 + 0.244307i 0.696425 0.717630i \(-0.254773\pi\)
−0.273273 + 0.961937i \(0.588106\pi\)
\(354\) 0 0
\(355\) −1.91419 3.31547i −0.101595 0.175967i
\(356\) 0 0
\(357\) 12.8023 8.66935i 0.677571 0.458831i
\(358\) 0 0
\(359\) 26.1651 + 15.1064i 1.38094 + 0.797286i 0.992271 0.124091i \(-0.0396015\pi\)
0.388669 + 0.921377i \(0.372935\pi\)
\(360\) 0 0
\(361\) 6.92562 + 11.9955i 0.364506 + 0.631343i
\(362\) 0 0
\(363\) −2.41173 2.77017i −0.126583 0.145396i
\(364\) 0 0
\(365\) 4.47443 7.74994i 0.234202 0.405650i
\(366\) 0 0
\(367\) 2.42672 4.20320i 0.126674 0.219405i −0.795712 0.605675i \(-0.792903\pi\)
0.922386 + 0.386269i \(0.126237\pi\)
\(368\) 0 0
\(369\) −31.2013 4.33721i −1.62427 0.225786i
\(370\) 0 0
\(371\) 5.19783 + 12.2088i 0.269858 + 0.633850i
\(372\) 0 0
\(373\) 2.26478 + 3.92271i 0.117266 + 0.203110i 0.918683 0.394995i \(-0.129254\pi\)
−0.801418 + 0.598105i \(0.795920\pi\)
\(374\) 0 0
\(375\) −1.97385 10.1140i −0.101929 0.522283i
\(376\) 0 0
\(377\) 6.65815i 0.342912i
\(378\) 0 0
\(379\) 27.9107i 1.43368i −0.697239 0.716838i \(-0.745589\pi\)
0.697239 0.716838i \(-0.254411\pi\)
\(380\) 0 0
\(381\) 0.667745 + 3.42151i 0.0342096 + 0.175289i
\(382\) 0 0
\(383\) −10.7050 18.5416i −0.547000 0.947432i −0.998478 0.0551495i \(-0.982436\pi\)
0.451478 0.892282i \(-0.350897\pi\)
\(384\) 0 0
\(385\) 4.84144 + 0.589171i 0.246742 + 0.0300269i
\(386\) 0 0
\(387\) −5.29282 + 6.79555i −0.269049 + 0.345437i
\(388\) 0 0
\(389\) 7.13605 12.3600i 0.361812 0.626677i −0.626447 0.779464i \(-0.715492\pi\)
0.988259 + 0.152787i \(0.0488249\pi\)
\(390\) 0 0
\(391\) 11.3770 19.7055i 0.575360 0.996552i
\(392\) 0 0
\(393\) −20.5593 23.6149i −1.03708 1.19121i
\(394\) 0 0
\(395\) 3.73047 + 6.46137i 0.187701 + 0.325107i
\(396\) 0 0
\(397\) −13.2084 7.62586i −0.662909 0.382731i 0.130475 0.991452i \(-0.458350\pi\)
−0.793385 + 0.608721i \(0.791683\pi\)
\(398\) 0 0
\(399\) −4.54122 + 9.35422i −0.227345 + 0.468297i
\(400\) 0 0
\(401\) 17.3931 + 30.1257i 0.868569 + 1.50440i 0.863460 + 0.504418i \(0.168293\pi\)
0.00510899 + 0.999987i \(0.498374\pi\)
\(402\) 0 0
\(403\) 12.1191 + 6.99695i 0.603694 + 0.348543i
\(404\) 0 0
\(405\) −1.51853 + 5.35650i −0.0754563 + 0.266167i
\(406\) 0 0
\(407\) 16.7859 9.69134i 0.832046 0.480382i
\(408\) 0 0
\(409\) 27.7719i 1.37323i −0.727019 0.686617i \(-0.759095\pi\)
0.727019 0.686617i \(-0.240905\pi\)
\(410\) 0 0
\(411\) −11.6609 + 33.9486i −0.575190 + 1.67456i
\(412\) 0 0
\(413\) 11.4034 + 1.38772i 0.561125 + 0.0682852i
\(414\) 0 0
\(415\) −5.87630 3.39268i −0.288456 0.166540i
\(416\) 0 0
\(417\) −4.48103 + 13.0457i −0.219437 + 0.638852i
\(418\) 0 0
\(419\) 19.7876 34.2732i 0.966689 1.67436i 0.261684 0.965154i \(-0.415722\pi\)
0.705006 0.709202i \(-0.250944\pi\)
\(420\) 0 0
\(421\) 19.5960 + 33.9413i 0.955050 + 1.65420i 0.734253 + 0.678876i \(0.237533\pi\)
0.220798 + 0.975320i \(0.429134\pi\)
\(422\) 0 0
\(423\) 15.5924 6.32704i 0.758129 0.307631i
\(424\) 0 0
\(425\) 15.5787i 0.755676i
\(426\) 0 0
\(427\) 1.16064 9.53737i 0.0561671 0.461546i
\(428\) 0 0
\(429\) −6.45074 7.40946i −0.311445 0.357732i
\(430\) 0 0
\(431\) −31.3367 + 18.0923i −1.50944 + 0.871474i −0.509497 + 0.860472i \(0.670168\pi\)
−0.999939 + 0.0110011i \(0.996498\pi\)
\(432\) 0 0
\(433\) 36.6472i 1.76115i −0.473904 0.880577i \(-0.657156\pi\)
0.473904 0.880577i \(-0.342844\pi\)
\(434\) 0 0
\(435\) 3.54473 + 1.21757i 0.169957 + 0.0583779i
\(436\) 0 0
\(437\) 15.3027i 0.732028i
\(438\) 0 0
\(439\) −7.52279 −0.359043 −0.179522 0.983754i \(-0.557455\pi\)
−0.179522 + 0.983754i \(0.557455\pi\)
\(440\) 0 0
\(441\) 20.7848 2.99860i 0.989753 0.142791i
\(442\) 0 0
\(443\) 34.0670i 1.61857i 0.587416 + 0.809285i \(0.300145\pi\)
−0.587416 + 0.809285i \(0.699855\pi\)
\(444\) 0 0
\(445\) 4.31433 0.204519
\(446\) 0 0
\(447\) −23.8900 27.4406i −1.12996 1.29790i
\(448\) 0 0
\(449\) 23.4345 1.10594 0.552972 0.833200i \(-0.313494\pi\)
0.552972 + 0.833200i \(0.313494\pi\)
\(450\) 0 0
\(451\) 15.6448 + 27.0976i 0.736684 + 1.27597i
\(452\) 0 0
\(453\) 24.3175 + 8.35272i 1.14253 + 0.392445i
\(454\) 0 0
\(455\) −3.09257 0.376345i −0.144982 0.0176433i
\(456\) 0 0
\(457\) −4.96087 −0.232060 −0.116030 0.993246i \(-0.537017\pi\)
−0.116030 + 0.993246i \(0.537017\pi\)
\(458\) 0 0
\(459\) 7.92394 15.6387i 0.369858 0.729954i
\(460\) 0 0
\(461\) 18.2217 10.5203i 0.848671 0.489981i −0.0115310 0.999934i \(-0.503671\pi\)
0.860202 + 0.509953i \(0.170337\pi\)
\(462\) 0 0
\(463\) 24.0110 + 13.8627i 1.11589 + 0.644257i 0.940347 0.340216i \(-0.110500\pi\)
0.175538 + 0.984473i \(0.443834\pi\)
\(464\) 0 0
\(465\) −5.94131 + 5.17255i −0.275522 + 0.239871i
\(466\) 0 0
\(467\) 16.0302 27.7652i 0.741791 1.28482i −0.209887 0.977726i \(-0.567310\pi\)
0.951679 0.307095i \(-0.0993569\pi\)
\(468\) 0 0
\(469\) 1.60323 13.1743i 0.0740303 0.608334i
\(470\) 0 0
\(471\) 6.98371 + 35.7844i 0.321792 + 1.64886i
\(472\) 0 0
\(473\) 8.55568 0.393390
\(474\) 0 0
\(475\) −5.23854 9.07342i −0.240361 0.416317i
\(476\) 0 0
\(477\) 11.8703 + 9.24534i 0.543502 + 0.423315i
\(478\) 0 0
\(479\) 7.30666 12.6555i 0.333850 0.578245i −0.649414 0.760435i \(-0.724986\pi\)
0.983263 + 0.182191i \(0.0583189\pi\)
\(480\) 0 0
\(481\) −10.7223 + 6.19054i −0.488896 + 0.282264i
\(482\) 0 0
\(483\) 25.5896 17.3285i 1.16437 0.788476i
\(484\) 0 0
\(485\) 0.969018 1.67839i 0.0440008 0.0762117i
\(486\) 0 0
\(487\) 28.9705 16.7261i 1.31278 0.757932i 0.330222 0.943903i \(-0.392877\pi\)
0.982555 + 0.185971i \(0.0595432\pi\)
\(488\) 0 0
\(489\) −2.89898 0.995761i −0.131096 0.0450299i
\(490\) 0 0
\(491\) 35.7574 + 20.6445i 1.61371 + 0.931675i 0.988501 + 0.151217i \(0.0483192\pi\)
0.625208 + 0.780458i \(0.285014\pi\)
\(492\) 0 0
\(493\) −10.2209 5.90102i −0.460325 0.265769i
\(494\) 0 0
\(495\) 5.12436 2.07935i 0.230323 0.0934598i
\(496\) 0 0
\(497\) −1.97795 + 16.2535i −0.0887230 + 0.729070i
\(498\) 0 0
\(499\) 16.0288 9.25425i 0.717549 0.414277i −0.0963011 0.995352i \(-0.530701\pi\)
0.813850 + 0.581075i \(0.197368\pi\)
\(500\) 0 0
\(501\) −2.82163 + 8.21466i −0.126061 + 0.367004i
\(502\) 0 0
\(503\) −16.4055 −0.731483 −0.365742 0.930716i \(-0.619185\pi\)
−0.365742 + 0.930716i \(0.619185\pi\)
\(504\) 0 0
\(505\) 7.99772 0.355894
\(506\) 0 0
\(507\) −10.6645 12.2495i −0.473627 0.544018i
\(508\) 0 0
\(509\) 19.5541 11.2896i 0.866721 0.500401i 0.000463610 1.00000i \(-0.499852\pi\)
0.866257 + 0.499598i \(0.166519\pi\)
\(510\) 0 0
\(511\) −35.2144 + 14.9923i −1.55779 + 0.663221i
\(512\) 0 0
\(513\) 0.643632 + 11.7730i 0.0284170 + 0.519789i
\(514\) 0 0
\(515\) 3.77872 + 2.18165i 0.166510 + 0.0961348i
\(516\) 0 0
\(517\) −14.4749 8.35706i −0.636604 0.367543i
\(518\) 0 0
\(519\) 1.03847 + 5.32107i 0.0455836 + 0.233569i
\(520\) 0 0
\(521\) −24.6511 + 14.2323i −1.07998 + 0.623529i −0.930892 0.365294i \(-0.880968\pi\)
−0.149092 + 0.988823i \(0.547635\pi\)
\(522\) 0 0
\(523\) −12.2249 + 21.1742i −0.534558 + 0.925882i 0.464627 + 0.885507i \(0.346189\pi\)
−0.999185 + 0.0403749i \(0.987145\pi\)
\(524\) 0 0
\(525\) −9.24081 + 19.0346i −0.403302 + 0.830740i
\(526\) 0 0
\(527\) 21.4819 12.4026i 0.935767 0.540265i
\(528\) 0 0
\(529\) 11.2407 19.4694i 0.488725 0.846496i
\(530\) 0 0
\(531\) 12.0698 4.89765i 0.523785 0.212540i
\(532\) 0 0
\(533\) −9.99343 17.3091i −0.432863 0.749741i
\(534\) 0 0
\(535\) 2.46027 0.106367
\(536\) 0 0
\(537\) −22.8535 + 19.8964i −0.986200 + 0.858594i
\(538\) 0 0
\(539\) −15.0357 14.4575i −0.647635 0.622729i
\(540\) 0 0
\(541\) 21.2501 36.8062i 0.913613 1.58242i 0.104693 0.994505i \(-0.466614\pi\)
0.808920 0.587919i \(-0.200053\pi\)
\(542\) 0 0
\(543\) 1.58008 + 8.09628i 0.0678076 + 0.347445i
\(544\) 0 0
\(545\) −8.15203 4.70658i −0.349195 0.201608i
\(546\) 0 0
\(547\) 40.0427 23.1187i 1.71210 0.988483i 0.780386 0.625298i \(-0.215022\pi\)
0.931717 0.363185i \(-0.118311\pi\)
\(548\) 0 0
\(549\) −4.09621 10.0947i −0.174822 0.430832i
\(550\) 0 0
\(551\) 7.93721 0.338136
\(552\) 0 0
\(553\) 3.85473 31.6757i 0.163920 1.34699i
\(554\) 0 0
\(555\) −1.33500 6.84052i −0.0566677 0.290364i
\(556\) 0 0
\(557\) 14.8064 + 25.6454i 0.627365 + 1.08663i 0.988078 + 0.153952i \(0.0492000\pi\)
−0.360713 + 0.932677i \(0.617467\pi\)
\(558\) 0 0
\(559\) −5.46511 −0.231150
\(560\) 0 0
\(561\) −17.0914 + 3.33557i −0.721599 + 0.140828i
\(562\) 0 0
\(563\) 10.6407 0.448453 0.224226 0.974537i \(-0.428015\pi\)
0.224226 + 0.974537i \(0.428015\pi\)
\(564\) 0 0
\(565\) 1.79028i 0.0753175i
\(566\) 0 0
\(567\) 18.9582 14.4078i 0.796172 0.605071i
\(568\) 0 0
\(569\) 40.1154 1.68173 0.840863 0.541249i \(-0.182048\pi\)
0.840863 + 0.541249i \(0.182048\pi\)
\(570\) 0 0
\(571\) 27.9281i 1.16875i 0.811483 + 0.584376i \(0.198661\pi\)
−0.811483 + 0.584376i \(0.801339\pi\)
\(572\) 0 0
\(573\) 3.18755 + 16.3330i 0.133162 + 0.682319i
\(574\) 0 0
\(575\) 31.1391i 1.29859i
\(576\) 0 0
\(577\) 20.5370 11.8571i 0.854969 0.493616i −0.00735564 0.999973i \(-0.502341\pi\)
0.862324 + 0.506357i \(0.169008\pi\)
\(578\) 0 0
\(579\) 25.7582 5.02700i 1.07048 0.208915i
\(580\) 0 0
\(581\) 11.3677 + 26.7009i 0.471613 + 1.10774i
\(582\) 0 0
\(583\) 14.9448i 0.618950i
\(584\) 0 0
\(585\) −3.27329 + 1.32823i −0.135334 + 0.0549154i
\(586\) 0 0
\(587\) −2.00010 3.46427i −0.0825530 0.142986i 0.821793 0.569786i \(-0.192974\pi\)
−0.904346 + 0.426800i \(0.859641\pi\)
\(588\) 0 0
\(589\) −8.34110 + 14.4472i −0.343689 + 0.595287i
\(590\) 0 0
\(591\) −8.39205 + 1.63780i −0.345203 + 0.0673701i
\(592\) 0 0
\(593\) 13.2297 + 7.63817i 0.543279 + 0.313662i 0.746407 0.665490i \(-0.231777\pi\)
−0.203128 + 0.979152i \(0.565111\pi\)
\(594\) 0 0
\(595\) −3.31862 + 4.41382i −0.136050 + 0.180949i
\(596\) 0 0
\(597\) −25.8786 29.7248i −1.05914 1.21655i
\(598\) 0 0
\(599\) 18.2542i 0.745848i −0.927862 0.372924i \(-0.878355\pi\)
0.927862 0.372924i \(-0.121645\pi\)
\(600\) 0 0
\(601\) −0.833186 + 0.481040i −0.0339863 + 0.0196220i −0.516897 0.856048i \(-0.672913\pi\)
0.482911 + 0.875670i \(0.339580\pi\)
\(602\) 0 0
\(603\) −5.65825 13.9442i −0.230422 0.567853i
\(604\) 0 0
\(605\) 1.13607 + 0.655910i 0.0461878 + 0.0266665i
\(606\) 0 0
\(607\) −10.1758 17.6249i −0.413021 0.715374i 0.582197 0.813048i \(-0.302193\pi\)
−0.995218 + 0.0976738i \(0.968860\pi\)
\(608\) 0 0
\(609\) −8.98796 13.2728i −0.364211 0.537842i
\(610\) 0 0
\(611\) 9.24611 + 5.33824i 0.374058 + 0.215962i
\(612\) 0 0
\(613\) −0.545611 0.945026i −0.0220370 0.0381692i 0.854797 0.518963i \(-0.173682\pi\)
−0.876834 + 0.480794i \(0.840348\pi\)
\(614\) 0 0
\(615\) 11.0427 2.15510i 0.445284 0.0869021i
\(616\) 0 0
\(617\) −19.9190 + 34.5008i −0.801911 + 1.38895i 0.116446 + 0.993197i \(0.462850\pi\)
−0.918357 + 0.395753i \(0.870484\pi\)
\(618\) 0 0
\(619\) 4.72902 8.19091i 0.190075 0.329220i −0.755200 0.655495i \(-0.772460\pi\)
0.945275 + 0.326275i \(0.105793\pi\)
\(620\) 0 0
\(621\) 15.8386 31.2592i 0.635581 1.25439i
\(622\) 0 0
\(623\) −14.7482 11.0887i −0.590873 0.444260i
\(624\) 0 0
\(625\) −9.70304 16.8062i −0.388122 0.672246i
\(626\) 0 0
\(627\) 8.83285 7.68995i 0.352750 0.307107i
\(628\) 0 0
\(629\) 21.9463i 0.875058i
\(630\) 0 0
\(631\) 21.9236i 0.872765i 0.899761 + 0.436382i \(0.143741\pi\)
−0.899761 + 0.436382i \(0.856259\pi\)
\(632\) 0 0
\(633\) −5.87915 2.01941i −0.233675 0.0802643i
\(634\) 0 0
\(635\) −0.622543 1.07828i −0.0247049 0.0427901i
\(636\) 0 0
\(637\) 9.60438 + 9.23502i 0.380540 + 0.365905i
\(638\) 0 0
\(639\) 6.98072 + 17.2034i 0.276153 + 0.680554i
\(640\) 0 0
\(641\) −0.830084 + 1.43775i −0.0327863 + 0.0567876i −0.881953 0.471337i \(-0.843771\pi\)
0.849167 + 0.528125i \(0.177105\pi\)
\(642\) 0 0
\(643\) −7.32745 + 12.6915i −0.288966 + 0.500504i −0.973563 0.228418i \(-0.926645\pi\)
0.684597 + 0.728922i \(0.259978\pi\)
\(644\) 0 0
\(645\) 0.999399 2.90957i 0.0393513 0.114564i
\(646\) 0 0
\(647\) 7.70627 + 13.3477i 0.302965 + 0.524750i 0.976806 0.214126i \(-0.0686902\pi\)
−0.673841 + 0.738876i \(0.735357\pi\)
\(648\) 0 0
\(649\) −11.2047 6.46905i −0.439824 0.253933i
\(650\) 0 0
\(651\) 33.6044 2.41155i 1.31706 0.0945161i
\(652\) 0 0
\(653\) −0.694964 1.20371i −0.0271960 0.0471049i 0.852107 0.523368i \(-0.175324\pi\)
−0.879303 + 0.476263i \(0.841991\pi\)
\(654\) 0 0
\(655\) 9.68466 + 5.59144i 0.378411 + 0.218476i
\(656\) 0 0
\(657\) −26.6667 + 34.2378i −1.04037 + 1.33574i
\(658\) 0 0
\(659\) 6.58496 3.80183i 0.256514 0.148098i −0.366230 0.930525i \(-0.619352\pi\)
0.622743 + 0.782426i \(0.286018\pi\)
\(660\) 0 0
\(661\) 36.3951i 1.41561i 0.706410 + 0.707803i \(0.250314\pi\)
−0.706410 + 0.707803i \(0.749686\pi\)
\(662\) 0 0
\(663\) 10.9175 2.13066i 0.424000 0.0827481i
\(664\) 0 0
\(665\) 0.448642 3.68666i 0.0173976 0.142962i
\(666\) 0 0
\(667\) −20.4298 11.7951i −0.791043 0.456709i
\(668\) 0 0
\(669\) 11.6630 + 13.3964i 0.450916 + 0.517933i
\(670\) 0 0
\(671\) −5.41047 + 9.37120i −0.208869 + 0.361771i
\(672\) 0 0
\(673\) −13.1537 22.7829i −0.507038 0.878216i −0.999967 0.00814606i \(-0.997407\pi\)
0.492929 0.870070i \(-0.335926\pi\)
\(674\) 0 0
\(675\) 1.30971 + 23.9565i 0.0504107 + 0.922085i
\(676\) 0 0
\(677\) 36.4345i 1.40029i −0.714001 0.700145i \(-0.753119\pi\)
0.714001 0.700145i \(-0.246881\pi\)
\(678\) 0 0
\(679\) −7.62630 + 3.24685i −0.292670 + 0.124603i
\(680\) 0 0
\(681\) −8.09761 + 23.5748i −0.310301 + 0.903387i
\(682\) 0 0
\(683\) −42.8648 + 24.7480i −1.64017 + 0.946955i −0.659405 + 0.751788i \(0.729192\pi\)
−0.980770 + 0.195167i \(0.937475\pi\)
\(684\) 0 0
\(685\) 12.8205i 0.489845i
\(686\) 0 0
\(687\) −5.05753 + 4.40313i −0.192957 + 0.167990i
\(688\) 0 0
\(689\) 9.54629i 0.363685i
\(690\) 0 0
\(691\) −38.5566 −1.46676 −0.733381 0.679818i \(-0.762059\pi\)
−0.733381 + 0.679818i \(0.762059\pi\)
\(692\) 0 0
\(693\) −22.8615 6.06258i −0.868438 0.230298i
\(694\) 0 0
\(695\) 4.92662i 0.186877i
\(696\) 0 0
\(697\) −35.4281 −1.34193
\(698\) 0 0
\(699\) 6.57720 19.1484i 0.248773 0.724257i
\(700\) 0 0
\(701\) −10.2585 −0.387458 −0.193729 0.981055i \(-0.562058\pi\)
−0.193729 + 0.981055i \(0.562058\pi\)
\(702\) 0 0
\(703\) −7.37977 12.7821i −0.278333 0.482087i
\(704\) 0 0
\(705\) −4.53285 + 3.94634i −0.170717 + 0.148628i
\(706\) 0 0
\(707\) −27.3395 20.5557i −1.02821 0.773078i
\(708\) 0 0
\(709\) 11.1248 0.417799 0.208900 0.977937i \(-0.433012\pi\)
0.208900 + 0.977937i \(0.433012\pi\)
\(710\) 0 0
\(711\) −13.6044 33.5268i −0.510205 1.25735i
\(712\) 0 0
\(713\) 42.9387 24.7907i 1.60807 0.928417i
\(714\) 0 0
\(715\) 3.03868 + 1.75439i 0.113640 + 0.0656103i
\(716\) 0 0
\(717\) −10.1901 3.50018i −0.380558 0.130716i
\(718\) 0 0
\(719\) 9.64721 16.7095i 0.359780 0.623158i −0.628144 0.778097i \(-0.716185\pi\)
0.987924 + 0.154940i \(0.0495183\pi\)
\(720\) 0 0
\(721\) −7.30996 17.1699i −0.272237 0.639439i
\(722\) 0 0
\(723\) −27.8208 9.55607i −1.03467 0.355394i
\(724\) 0 0
\(725\) 16.1512 0.599841
\(726\) 0 0
\(727\) 13.0702 + 22.6382i 0.484747 + 0.839606i 0.999846 0.0175243i \(-0.00557846\pi\)
−0.515100 + 0.857130i \(0.672245\pi\)
\(728\) 0 0
\(729\) 10.8705 24.7150i 0.402610 0.915371i
\(730\) 0 0
\(731\) −4.84365 + 8.38944i −0.179149 + 0.310295i
\(732\) 0 0
\(733\) 17.1392 9.89531i 0.633050 0.365492i −0.148882 0.988855i \(-0.547567\pi\)
0.781932 + 0.623363i \(0.214234\pi\)
\(734\) 0 0
\(735\) −6.67298 + 3.42448i −0.246136 + 0.126314i
\(736\) 0 0
\(737\) −7.47368 + 12.9448i −0.275297 + 0.476828i
\(738\) 0 0
\(739\) 38.3346 22.1325i 1.41016 0.814158i 0.414759 0.909931i \(-0.363866\pi\)
0.995403 + 0.0957737i \(0.0305325\pi\)
\(740\) 0 0
\(741\) −5.64216 + 4.91211i −0.207270 + 0.180451i
\(742\) 0 0
\(743\) −32.3816 18.6955i −1.18797 0.685873i −0.230123 0.973162i \(-0.573913\pi\)
−0.957844 + 0.287289i \(0.907246\pi\)
\(744\) 0 0
\(745\) 11.2536 + 6.49728i 0.412301 + 0.238042i
\(746\) 0 0
\(747\) 25.9604 + 20.2197i 0.949842 + 0.739800i
\(748\) 0 0
\(749\) −8.41024 6.32340i −0.307303 0.231052i
\(750\) 0 0
\(751\) 2.04602 1.18127i 0.0746603 0.0431052i −0.462205 0.886773i \(-0.652942\pi\)
0.536865 + 0.843668i \(0.319608\pi\)
\(752\) 0 0
\(753\) −46.4784 + 9.07076i −1.69377 + 0.330557i
\(754\) 0 0
\(755\) −9.18332 −0.334215
\(756\) 0 0
\(757\) 13.6132 0.494779 0.247389 0.968916i \(-0.420427\pi\)
0.247389 + 0.968916i \(0.420427\pi\)
\(758\) 0 0
\(759\) −34.1628 + 6.66724i −1.24003 + 0.242005i
\(760\) 0 0
\(761\) −18.0200 + 10.4039i −0.653225 + 0.377140i −0.789691 0.613505i \(-0.789759\pi\)
0.136466 + 0.990645i \(0.456426\pi\)
\(762\) 0 0
\(763\) 15.7702 + 37.0414i 0.570918 + 1.34099i
\(764\) 0 0
\(765\) −0.862122 + 6.20199i −0.0311701 + 0.224233i
\(766\) 0 0
\(767\) 7.15725 + 4.13224i 0.258433 + 0.149206i
\(768\) 0 0
\(769\) −10.5703 6.10279i −0.381176 0.220072i 0.297154 0.954830i \(-0.403963\pi\)
−0.678330 + 0.734757i \(0.737296\pi\)
\(770\) 0 0
\(771\) −0.0563618 + 0.0490691i −0.00202982 + 0.00176718i
\(772\) 0 0
\(773\) 3.16232 1.82577i 0.113741 0.0656683i −0.442050 0.896990i \(-0.645749\pi\)
0.555791 + 0.831322i \(0.312415\pi\)
\(774\) 0 0
\(775\) −16.9731 + 29.3982i −0.609691 + 1.05601i
\(776\) 0 0
\(777\) −13.0179 + 26.8149i −0.467016 + 0.961981i
\(778\) 0 0
\(779\) 20.6343 11.9132i 0.739299 0.426835i
\(780\) 0 0
\(781\) 9.22048 15.9703i 0.329935 0.571463i
\(782\) 0 0
\(783\) −16.2135 8.21516i −0.579423 0.293586i
\(784\) 0 0
\(785\) −6.51095 11.2773i −0.232386 0.402504i
\(786\) 0 0
\(787\) 8.32993 0.296930 0.148465 0.988918i \(-0.452567\pi\)
0.148465 + 0.988918i \(0.452567\pi\)
\(788\) 0 0
\(789\) 25.3748 + 8.71589i 0.903366 + 0.310294i
\(790\) 0 0
\(791\) −4.60137 + 6.11990i −0.163606 + 0.217599i
\(792\) 0 0
\(793\) 3.45604 5.98605i 0.122728 0.212571i
\(794\) 0 0
\(795\) −5.08235 1.74572i −0.180252 0.0619143i
\(796\) 0 0
\(797\) 27.3707 + 15.8025i 0.969520 + 0.559753i 0.899090 0.437764i \(-0.144229\pi\)
0.0704305 + 0.997517i \(0.477563\pi\)
\(798\) 0 0
\(799\) 16.3894 9.46241i 0.579815 0.334756i
\(800\) 0 0
\(801\) −20.7231 2.88066i −0.732214 0.101783i
\(802\) 0 0
\(803\) 43.1058 1.52117
\(804\) 0 0
\(805\) −6.63335 + 8.82247i −0.233795 + 0.310951i
\(806\) 0 0
\(807\) −37.6827 + 32.8068i −1.32649 + 1.15486i
\(808\) 0 0
\(809\) −6.15525 10.6612i −0.216407 0.374828i 0.737300 0.675566i \(-0.236100\pi\)
−0.953707 + 0.300738i \(0.902767\pi\)
\(810\) 0 0
\(811\) −0.741828 −0.0260491 −0.0130245 0.999915i \(-0.504146\pi\)
−0.0130245 + 0.999915i \(0.504146\pi\)
\(812\) 0 0
\(813\) −8.06783 + 23.4880i −0.282951 + 0.823761i
\(814\) 0 0
\(815\) 1.09478 0.0383485
\(816\) 0 0
\(817\) 6.51498i 0.227930i
\(818\) 0 0
\(819\) 14.6033 + 3.87260i 0.510280 + 0.135319i
\(820\) 0 0
\(821\) −12.5572 −0.438250 −0.219125 0.975697i \(-0.570320\pi\)
−0.219125 + 0.975697i \(0.570320\pi\)
\(822\) 0 0
\(823\) 1.95398i 0.0681113i −0.999420 0.0340557i \(-0.989158\pi\)
0.999420 0.0340557i \(-0.0108424\pi\)
\(824\) 0 0
\(825\) 17.9737 15.6481i 0.625765 0.544796i
\(826\) 0 0
\(827\) 26.2468i 0.912692i −0.889802 0.456346i \(-0.849158\pi\)
0.889802 0.456346i \(-0.150842\pi\)
\(828\) 0 0
\(829\) −0.405249 + 0.233970i −0.0140749 + 0.00812613i −0.507021 0.861934i \(-0.669253\pi\)
0.492946 + 0.870060i \(0.335920\pi\)
\(830\) 0 0
\(831\) −8.21674 + 23.9216i −0.285036 + 0.829830i
\(832\) 0 0
\(833\) 22.6888 6.55874i 0.786121 0.227247i
\(834\) 0 0
\(835\) 3.10221i 0.107356i
\(836\) 0 0
\(837\) 31.9917 20.8784i 1.10579 0.721663i
\(838\) 0 0
\(839\) 6.54979 + 11.3446i 0.226124 + 0.391658i 0.956656 0.291220i \(-0.0940613\pi\)
−0.730532 + 0.682878i \(0.760728\pi\)
\(840\) 0 0
\(841\) 8.38211 14.5182i 0.289038 0.500629i
\(842\) 0 0
\(843\) 2.38024 + 2.73400i 0.0819799 + 0.0941640i
\(844\) 0 0
\(845\) 5.02362 + 2.90039i 0.172818 + 0.0997763i
\(846\) 0 0
\(847\) −2.19773 5.16210i −0.0755150 0.177372i
\(848\) 0 0
\(849\) −6.49023 + 1.26664i −0.222744 + 0.0434709i
\(850\) 0 0
\(851\) 43.8670i 1.50374i
\(852\) 0 0
\(853\) −12.5601 + 7.25157i −0.430049 + 0.248289i −0.699368 0.714762i \(-0.746535\pi\)
0.269318 + 0.963051i \(0.413202\pi\)
\(854\) 0 0
\(855\) −1.58338 3.90210i −0.0541506 0.133449i
\(856\) 0 0
\(857\) 26.9380 + 15.5527i 0.920186 + 0.531269i 0.883694 0.468065i \(-0.155049\pi\)
0.0364913 + 0.999334i \(0.488382\pi\)
\(858\) 0 0
\(859\) −13.0019 22.5200i −0.443619 0.768371i 0.554335 0.832293i \(-0.312972\pi\)
−0.997955 + 0.0639219i \(0.979639\pi\)
\(860\) 0 0
\(861\) −43.2875 21.0149i −1.47523 0.716187i
\(862\) 0 0
\(863\) 26.1288 + 15.0854i 0.889433 + 0.513515i 0.873757 0.486363i \(-0.161677\pi\)
0.0156763 + 0.999877i \(0.495010\pi\)
\(864\) 0 0
\(865\) −0.968167 1.67691i −0.0329187 0.0570168i
\(866\) 0 0
\(867\) −3.16014 + 9.20017i −0.107324 + 0.312454i
\(868\) 0 0
\(869\) −17.9694 + 31.1238i −0.609569 + 1.05580i
\(870\) 0 0
\(871\) 4.77397 8.26875i 0.161760 0.280176i
\(872\) 0 0
\(873\) −5.77515 + 7.41481i −0.195459 + 0.250953i
\(874\) 0 0
\(875\) 1.90153 15.6255i 0.0642833 0.528239i
\(876\) 0 0
\(877\) −7.57800 13.1255i −0.255891 0.443216i 0.709246 0.704961i \(-0.249035\pi\)
−0.965137 + 0.261745i \(0.915702\pi\)
\(878\) 0 0
\(879\) −22.2125 7.62970i −0.749210 0.257344i
\(880\) 0 0
\(881\) 46.0172i 1.55036i 0.631741 + 0.775180i \(0.282341\pi\)
−0.631741 + 0.775180i \(0.717659\pi\)
\(882\) 0 0
\(883\) 20.3035i 0.683267i −0.939833 0.341633i \(-0.889020\pi\)
0.939833 0.341633i \(-0.110980\pi\)
\(884\) 0 0
\(885\) −3.50880 + 3.05479i −0.117947 + 0.102686i
\(886\) 0 0
\(887\) −19.5332 33.8325i −0.655861 1.13598i −0.981677 0.190551i \(-0.938973\pi\)
0.325817 0.945433i \(-0.394361\pi\)
\(888\) 0 0
\(889\) −0.643278 + 5.28605i −0.0215749 + 0.177288i
\(890\) 0 0
\(891\) −26.0023 + 6.56624i −0.871110 + 0.219977i
\(892\) 0 0
\(893\) −6.36374 + 11.0223i −0.212954 + 0.368848i
\(894\) 0 0
\(895\) 5.41116 9.37240i 0.180875 0.313285i
\(896\) 0 0
\(897\) 21.8222 4.25883i 0.728621 0.142198i
\(898\) 0 0
\(899\) −12.8584 22.2714i −0.428852 0.742794i
\(900\) 0 0
\(901\) 14.6544 + 8.46074i 0.488210 + 0.281868i
\(902\) 0 0
\(903\) −10.8945 + 7.37746i −0.362548 + 0.245506i
\(904\) 0 0
\(905\) −1.47311 2.55151i −0.0489680 0.0848150i
\(906\) 0 0
\(907\) 25.0468 + 14.4608i 0.831667 + 0.480163i 0.854423 0.519578i \(-0.173911\pi\)
−0.0227563 + 0.999741i \(0.507244\pi\)
\(908\) 0 0
\(909\) −38.4155 5.34004i −1.27416 0.177118i
\(910\) 0 0
\(911\) −50.0777 + 28.9124i −1.65915 + 0.957910i −0.686039 + 0.727565i \(0.740652\pi\)
−0.973109 + 0.230345i \(0.926015\pi\)
\(912\) 0 0
\(913\) 32.6845i 1.08170i
\(914\) 0 0
\(915\) 2.55491 + 2.93463i 0.0844626 + 0.0970157i
\(916\) 0 0
\(917\) −18.7350 44.0054i −0.618686 1.45319i
\(918\) 0 0
\(919\) −37.0466 21.3889i −1.22205 0.705554i −0.256699 0.966491i \(-0.582635\pi\)
−0.965356 + 0.260938i \(0.915968\pi\)
\(920\) 0 0
\(921\) 0.158450 0.0309233i 0.00522111 0.00101896i
\(922\) 0 0
\(923\) −5.88977 + 10.2014i −0.193864 + 0.335782i
\(924\) 0 0
\(925\) −15.0169 26.0100i −0.493752 0.855204i
\(926\) 0 0
\(927\) −16.6937 13.0022i −0.548293 0.427047i
\(928\) 0 0
\(929\) 31.2840i 1.02639i 0.858271 + 0.513197i \(0.171539\pi\)
−0.858271 + 0.513197i \(0.828461\pi\)
\(930\) 0 0
\(931\) −11.0091 + 11.4494i −0.360809 + 0.375240i
\(932\) 0 0
\(933\) −0.0142040 + 0.00277206i −0.000465017 + 9.07531e-5i
\(934\) 0 0
\(935\) 5.38628 3.10977i 0.176150 0.101700i
\(936\) 0 0
\(937\) 12.2887i 0.401454i 0.979647 + 0.200727i \(0.0643305\pi\)
−0.979647 + 0.200727i \(0.935670\pi\)
\(938\) 0 0
\(939\) 0.0181596 + 0.0930492i 0.000592615 + 0.00303655i
\(940\) 0 0
\(941\) 15.5977i 0.508469i −0.967143 0.254235i \(-0.918176\pi\)
0.967143 0.254235i \(-0.0818236\pi\)
\(942\) 0 0
\(943\) −70.8147 −2.30604
\(944\) 0 0
\(945\) −4.73222 + 7.06646i −0.153939 + 0.229872i
\(946\) 0 0
\(947\) 40.4755i 1.31528i 0.753334 + 0.657639i \(0.228445\pi\)
−0.753334 + 0.657639i \(0.771555\pi\)
\(948\) 0 0
\(949\) −27.5347 −0.893815
\(950\) 0 0
\(951\) −17.8698 + 3.48749i −0.579469 + 0.113090i
\(952\) 0 0
\(953\) 47.9759 1.55409 0.777046 0.629444i \(-0.216717\pi\)
0.777046 + 0.629444i \(0.216717\pi\)
\(954\) 0 0
\(955\) −2.97177 5.14726i −0.0961643 0.166561i
\(956\) 0 0
\(957\) 3.45816 + 17.7195i 0.111786 + 0.572791i
\(958\) 0 0
\(959\) −32.9512 + 43.8256i −1.06405 + 1.41520i
\(960\) 0 0
\(961\) 23.0509 0.743577
\(962\) 0 0
\(963\) −11.8175 1.64272i −0.380813 0.0529358i
\(964\) 0 0
\(965\) −8.11760 + 4.68670i −0.261315 + 0.150870i
\(966\) 0 0
\(967\) −41.0636 23.7081i −1.32052 0.762401i −0.336707 0.941610i \(-0.609313\pi\)
−0.983811 + 0.179208i \(0.942646\pi\)
\(968\) 0 0
\(969\) 2.53997 + 13.0148i 0.0815957 + 0.418094i
\(970\) 0 0
\(971\) 13.0930 22.6777i 0.420173 0.727761i −0.575783 0.817602i \(-0.695303\pi\)
0.995956 + 0.0898415i \(0.0286361\pi\)
\(972\) 0 0
\(973\) −12.6624 + 16.8412i −0.405939 + 0.539905i
\(974\) 0 0
\(975\) −11.4811 + 9.99552i −0.367689 + 0.320113i
\(976\) 0 0
\(977\) 26.8970 0.860513 0.430256 0.902707i \(-0.358423\pi\)
0.430256 + 0.902707i \(0.358423\pi\)
\(978\) 0 0
\(979\) 10.3909 + 17.9975i 0.332094 + 0.575203i
\(980\) 0 0
\(981\) 36.0142 + 28.0502i 1.14984 + 0.895575i
\(982\) 0 0
\(983\) −7.46289 + 12.9261i −0.238029 + 0.412279i −0.960149 0.279490i \(-0.909835\pi\)
0.722119 + 0.691768i \(0.243168\pi\)
\(984\) 0 0
\(985\) 2.64472 1.52693i 0.0842677 0.0486520i
\(986\) 0 0
\(987\) 25.6381 1.83986i 0.816068 0.0585635i
\(988\) 0 0
\(989\) −9.68162 + 16.7691i −0.307858 + 0.533225i
\(990\) 0 0
\(991\) −33.6509 + 19.4284i −1.06896 + 0.617163i −0.927896 0.372838i \(-0.878385\pi\)
−0.141061 + 0.990001i \(0.545051\pi\)
\(992\) 0 0
\(993\) −8.87849 45.4932i −0.281750 1.44368i
\(994\) 0 0
\(995\) 12.1904 + 7.03812i 0.386461 + 0.223124i
\(996\) 0 0
\(997\) 32.5019 + 18.7650i 1.02935 + 0.594293i 0.916797 0.399353i \(-0.130765\pi\)
0.112549 + 0.993646i \(0.464099\pi\)
\(998\) 0 0
\(999\) 1.84504 + 33.7485i 0.0583746 + 1.06776i
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 1008.2.cz.i.607.15 yes 32
3.2 odd 2 3024.2.cz.i.1279.9 32
4.3 odd 2 inner 1008.2.cz.i.607.2 yes 32
7.3 odd 6 1008.2.bf.i.31.12 yes 32
9.2 odd 6 3024.2.bf.i.2287.9 32
9.7 even 3 1008.2.bf.i.943.5 yes 32
12.11 even 2 3024.2.cz.i.1279.10 32
21.17 even 6 3024.2.bf.i.1711.7 32
28.3 even 6 1008.2.bf.i.31.5 32
36.7 odd 6 1008.2.bf.i.943.12 yes 32
36.11 even 6 3024.2.bf.i.2287.10 32
63.38 even 6 3024.2.cz.i.2719.10 32
63.52 odd 6 inner 1008.2.cz.i.367.2 yes 32
84.59 odd 6 3024.2.bf.i.1711.8 32
252.115 even 6 inner 1008.2.cz.i.367.15 yes 32
252.227 odd 6 3024.2.cz.i.2719.9 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1008.2.bf.i.31.5 32 28.3 even 6
1008.2.bf.i.31.12 yes 32 7.3 odd 6
1008.2.bf.i.943.5 yes 32 9.7 even 3
1008.2.bf.i.943.12 yes 32 36.7 odd 6
1008.2.cz.i.367.2 yes 32 63.52 odd 6 inner
1008.2.cz.i.367.15 yes 32 252.115 even 6 inner
1008.2.cz.i.607.2 yes 32 4.3 odd 2 inner
1008.2.cz.i.607.15 yes 32 1.1 even 1 trivial
3024.2.bf.i.1711.7 32 21.17 even 6
3024.2.bf.i.1711.8 32 84.59 odd 6
3024.2.bf.i.2287.9 32 9.2 odd 6
3024.2.bf.i.2287.10 32 36.11 even 6
3024.2.cz.i.1279.9 32 3.2 odd 2
3024.2.cz.i.1279.10 32 12.11 even 2
3024.2.cz.i.2719.9 32 252.227 odd 6
3024.2.cz.i.2719.10 32 63.38 even 6