Properties

Label 4020.2.g.b.1609.16
Level $4020$
Weight $2$
Character 4020.1609
Analytic conductor $32.100$
Analytic rank $0$
Dimension $24$
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] = [4020,2,Mod(1609,4020)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4020, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4020.1609");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4020.g (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(32.0998616126\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1609.16
Character \(\chi\) \(=\) 4020.1609
Dual form 4020.2.g.b.1609.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{3} +(-1.63494 - 1.52544i) q^{5} +3.48382i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} +(-1.63494 - 1.52544i) q^{5} +3.48382i q^{7} -1.00000 q^{9} -1.37949 q^{11} +0.00615147i q^{13} +(1.52544 - 1.63494i) q^{15} +0.950372i q^{17} -5.93240 q^{19} -3.48382 q^{21} -5.79132i q^{23} +(0.346064 + 4.98801i) q^{25} -1.00000i q^{27} +1.84186 q^{29} -4.21197 q^{31} -1.37949i q^{33} +(5.31436 - 5.69584i) q^{35} -8.70804i q^{37} -0.00615147 q^{39} +5.62807 q^{41} +1.88362i q^{43} +(1.63494 + 1.52544i) q^{45} -5.91697i q^{47} -5.13700 q^{49} -0.950372 q^{51} +11.6894i q^{53} +(2.25538 + 2.10433i) q^{55} -5.93240i q^{57} +12.9173 q^{59} -8.73278 q^{61} -3.48382i q^{63} +(0.00938370 - 0.0100573i) q^{65} -1.00000i q^{67} +5.79132 q^{69} +5.85892 q^{71} -15.3918i q^{73} +(-4.98801 + 0.346064i) q^{75} -4.80589i q^{77} +7.34143 q^{79} +1.00000 q^{81} +17.1852i q^{83} +(1.44974 - 1.55380i) q^{85} +1.84186i q^{87} -0.0825605 q^{89} -0.0214306 q^{91} -4.21197i q^{93} +(9.69913 + 9.04953i) q^{95} -10.8586i q^{97} +1.37949 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 4 q^{5} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q - 4 q^{5} - 24 q^{9} - 8 q^{11} - 2 q^{15} + 16 q^{19} - 20 q^{21} + 10 q^{25} + 36 q^{29} - 2 q^{35} + 4 q^{39} - 24 q^{41} + 4 q^{45} - 4 q^{51} - 4 q^{55} + 24 q^{59} - 4 q^{61} - 20 q^{65} - 4 q^{69} + 20 q^{71} - 12 q^{75} - 28 q^{79} + 24 q^{81} - 16 q^{85} + 48 q^{89} - 20 q^{91} - 4 q^{95} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4020\mathbb{Z}\right)^\times\).

\(n\) \(1141\) \(2011\) \(2681\) \(3217\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(5\) −1.63494 1.52544i −0.731168 0.682198i
\(6\) 0 0
\(7\) 3.48382i 1.31676i 0.752686 + 0.658380i \(0.228758\pi\)
−0.752686 + 0.658380i \(0.771242\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −1.37949 −0.415932 −0.207966 0.978136i \(-0.566684\pi\)
−0.207966 + 0.978136i \(0.566684\pi\)
\(12\) 0 0
\(13\) 0.00615147i 0.00170611i 1.00000 0.000853055i \(0.000271536\pi\)
−1.00000 0.000853055i \(0.999728\pi\)
\(14\) 0 0
\(15\) 1.52544 1.63494i 0.393867 0.422140i
\(16\) 0 0
\(17\) 0.950372i 0.230499i 0.993337 + 0.115250i \(0.0367668\pi\)
−0.993337 + 0.115250i \(0.963233\pi\)
\(18\) 0 0
\(19\) −5.93240 −1.36099 −0.680493 0.732754i \(-0.738234\pi\)
−0.680493 + 0.732754i \(0.738234\pi\)
\(20\) 0 0
\(21\) −3.48382 −0.760232
\(22\) 0 0
\(23\) 5.79132i 1.20757i −0.797146 0.603787i \(-0.793658\pi\)
0.797146 0.603787i \(-0.206342\pi\)
\(24\) 0 0
\(25\) 0.346064 + 4.98801i 0.0692129 + 0.997602i
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 1.84186 0.342025 0.171013 0.985269i \(-0.445296\pi\)
0.171013 + 0.985269i \(0.445296\pi\)
\(30\) 0 0
\(31\) −4.21197 −0.756492 −0.378246 0.925705i \(-0.623473\pi\)
−0.378246 + 0.925705i \(0.623473\pi\)
\(32\) 0 0
\(33\) 1.37949i 0.240138i
\(34\) 0 0
\(35\) 5.31436 5.69584i 0.898290 0.962773i
\(36\) 0 0
\(37\) 8.70804i 1.43159i −0.698309 0.715796i \(-0.746064\pi\)
0.698309 0.715796i \(-0.253936\pi\)
\(38\) 0 0
\(39\) −0.00615147 −0.000985023
\(40\) 0 0
\(41\) 5.62807 0.878957 0.439478 0.898253i \(-0.355163\pi\)
0.439478 + 0.898253i \(0.355163\pi\)
\(42\) 0 0
\(43\) 1.88362i 0.287249i 0.989632 + 0.143625i \(0.0458758\pi\)
−0.989632 + 0.143625i \(0.954124\pi\)
\(44\) 0 0
\(45\) 1.63494 + 1.52544i 0.243723 + 0.227399i
\(46\) 0 0
\(47\) 5.91697i 0.863079i −0.902094 0.431540i \(-0.857971\pi\)
0.902094 0.431540i \(-0.142029\pi\)
\(48\) 0 0
\(49\) −5.13700 −0.733857
\(50\) 0 0
\(51\) −0.950372 −0.133079
\(52\) 0 0
\(53\) 11.6894i 1.60566i 0.596211 + 0.802828i \(0.296672\pi\)
−0.596211 + 0.802828i \(0.703328\pi\)
\(54\) 0 0
\(55\) 2.25538 + 2.10433i 0.304116 + 0.283748i
\(56\) 0 0
\(57\) 5.93240i 0.785766i
\(58\) 0 0
\(59\) 12.9173 1.68168 0.840842 0.541281i \(-0.182060\pi\)
0.840842 + 0.541281i \(0.182060\pi\)
\(60\) 0 0
\(61\) −8.73278 −1.11812 −0.559059 0.829128i \(-0.688837\pi\)
−0.559059 + 0.829128i \(0.688837\pi\)
\(62\) 0 0
\(63\) 3.48382i 0.438920i
\(64\) 0 0
\(65\) 0.00938370 0.0100573i 0.00116390 0.00124745i
\(66\) 0 0
\(67\) 1.00000i 0.122169i
\(68\) 0 0
\(69\) 5.79132 0.697193
\(70\) 0 0
\(71\) 5.85892 0.695325 0.347663 0.937620i \(-0.386975\pi\)
0.347663 + 0.937620i \(0.386975\pi\)
\(72\) 0 0
\(73\) 15.3918i 1.80148i −0.434364 0.900738i \(-0.643027\pi\)
0.434364 0.900738i \(-0.356973\pi\)
\(74\) 0 0
\(75\) −4.98801 + 0.346064i −0.575966 + 0.0399601i
\(76\) 0 0
\(77\) 4.80589i 0.547682i
\(78\) 0 0
\(79\) 7.34143 0.825976 0.412988 0.910737i \(-0.364485\pi\)
0.412988 + 0.910737i \(0.364485\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 17.1852i 1.88633i 0.332331 + 0.943163i \(0.392165\pi\)
−0.332331 + 0.943163i \(0.607835\pi\)
\(84\) 0 0
\(85\) 1.44974 1.55380i 0.157246 0.168533i
\(86\) 0 0
\(87\) 1.84186i 0.197468i
\(88\) 0 0
\(89\) −0.0825605 −0.00875139 −0.00437570 0.999990i \(-0.501393\pi\)
−0.00437570 + 0.999990i \(0.501393\pi\)
\(90\) 0 0
\(91\) −0.0214306 −0.00224654
\(92\) 0 0
\(93\) 4.21197i 0.436761i
\(94\) 0 0
\(95\) 9.69913 + 9.04953i 0.995110 + 0.928462i
\(96\) 0 0
\(97\) 10.8586i 1.10252i −0.834333 0.551260i \(-0.814147\pi\)
0.834333 0.551260i \(-0.185853\pi\)
\(98\) 0 0
\(99\) 1.37949 0.138644
\(100\) 0 0
\(101\) −12.2686 −1.22077 −0.610386 0.792104i \(-0.708986\pi\)
−0.610386 + 0.792104i \(0.708986\pi\)
\(102\) 0 0
\(103\) 8.37006i 0.824726i 0.911020 + 0.412363i \(0.135296\pi\)
−0.911020 + 0.412363i \(0.864704\pi\)
\(104\) 0 0
\(105\) 5.69584 + 5.31436i 0.555857 + 0.518628i
\(106\) 0 0
\(107\) 9.42903i 0.911538i −0.890098 0.455769i \(-0.849364\pi\)
0.890098 0.455769i \(-0.150636\pi\)
\(108\) 0 0
\(109\) −5.11856 −0.490270 −0.245135 0.969489i \(-0.578832\pi\)
−0.245135 + 0.969489i \(0.578832\pi\)
\(110\) 0 0
\(111\) 8.70804 0.826531
\(112\) 0 0
\(113\) 14.7880i 1.39114i −0.718458 0.695570i \(-0.755152\pi\)
0.718458 0.695570i \(-0.244848\pi\)
\(114\) 0 0
\(115\) −8.83432 + 9.46847i −0.823804 + 0.882939i
\(116\) 0 0
\(117\) 0.00615147i 0.000568703i
\(118\) 0 0
\(119\) −3.31092 −0.303512
\(120\) 0 0
\(121\) −9.09701 −0.827001
\(122\) 0 0
\(123\) 5.62807i 0.507466i
\(124\) 0 0
\(125\) 7.04312 8.68300i 0.629955 0.776631i
\(126\) 0 0
\(127\) 14.1851i 1.25873i −0.777111 0.629363i \(-0.783316\pi\)
0.777111 0.629363i \(-0.216684\pi\)
\(128\) 0 0
\(129\) −1.88362 −0.165843
\(130\) 0 0
\(131\) 10.3659 0.905675 0.452837 0.891593i \(-0.350412\pi\)
0.452837 + 0.891593i \(0.350412\pi\)
\(132\) 0 0
\(133\) 20.6674i 1.79209i
\(134\) 0 0
\(135\) −1.52544 + 1.63494i −0.131289 + 0.140713i
\(136\) 0 0
\(137\) 8.30628i 0.709653i 0.934932 + 0.354827i \(0.115460\pi\)
−0.934932 + 0.354827i \(0.884540\pi\)
\(138\) 0 0
\(139\) 13.9339 1.18186 0.590929 0.806723i \(-0.298761\pi\)
0.590929 + 0.806723i \(0.298761\pi\)
\(140\) 0 0
\(141\) 5.91697 0.498299
\(142\) 0 0
\(143\) 0.00848589i 0.000709626i
\(144\) 0 0
\(145\) −3.01133 2.80965i −0.250078 0.233329i
\(146\) 0 0
\(147\) 5.13700i 0.423692i
\(148\) 0 0
\(149\) 13.9868 1.14584 0.572921 0.819611i \(-0.305810\pi\)
0.572921 + 0.819611i \(0.305810\pi\)
\(150\) 0 0
\(151\) 19.2480 1.56638 0.783190 0.621783i \(-0.213591\pi\)
0.783190 + 0.621783i \(0.213591\pi\)
\(152\) 0 0
\(153\) 0.950372i 0.0768330i
\(154\) 0 0
\(155\) 6.88633 + 6.42511i 0.553123 + 0.516077i
\(156\) 0 0
\(157\) 10.4863i 0.836897i 0.908241 + 0.418449i \(0.137426\pi\)
−0.908241 + 0.418449i \(0.862574\pi\)
\(158\) 0 0
\(159\) −11.6894 −0.927026
\(160\) 0 0
\(161\) 20.1759 1.59009
\(162\) 0 0
\(163\) 7.57463i 0.593291i 0.954988 + 0.296645i \(0.0958680\pi\)
−0.954988 + 0.296645i \(0.904132\pi\)
\(164\) 0 0
\(165\) −2.10433 + 2.25538i −0.163822 + 0.175581i
\(166\) 0 0
\(167\) 17.1496i 1.32708i −0.748142 0.663539i \(-0.769054\pi\)
0.748142 0.663539i \(-0.230946\pi\)
\(168\) 0 0
\(169\) 13.0000 0.999997
\(170\) 0 0
\(171\) 5.93240 0.453662
\(172\) 0 0
\(173\) 10.7568i 0.817823i −0.912574 0.408912i \(-0.865908\pi\)
0.912574 0.408912i \(-0.134092\pi\)
\(174\) 0 0
\(175\) −17.3773 + 1.20563i −1.31360 + 0.0911368i
\(176\) 0 0
\(177\) 12.9173i 0.970921i
\(178\) 0 0
\(179\) 24.5022 1.83138 0.915691 0.401882i \(-0.131644\pi\)
0.915691 + 0.401882i \(0.131644\pi\)
\(180\) 0 0
\(181\) 0.467927 0.0347807 0.0173904 0.999849i \(-0.494464\pi\)
0.0173904 + 0.999849i \(0.494464\pi\)
\(182\) 0 0
\(183\) 8.73278i 0.645546i
\(184\) 0 0
\(185\) −13.2836 + 14.2371i −0.976629 + 1.04673i
\(186\) 0 0
\(187\) 1.31103i 0.0958719i
\(188\) 0 0
\(189\) 3.48382 0.253411
\(190\) 0 0
\(191\) −2.48094 −0.179515 −0.0897574 0.995964i \(-0.528609\pi\)
−0.0897574 + 0.995964i \(0.528609\pi\)
\(192\) 0 0
\(193\) 24.4200i 1.75779i −0.477013 0.878896i \(-0.658281\pi\)
0.477013 0.878896i \(-0.341719\pi\)
\(194\) 0 0
\(195\) 0.0100573 + 0.00938370i 0.000720217 + 0.000671980i
\(196\) 0 0
\(197\) 24.5278i 1.74754i −0.486344 0.873768i \(-0.661670\pi\)
0.486344 0.873768i \(-0.338330\pi\)
\(198\) 0 0
\(199\) 20.9220 1.48312 0.741562 0.670884i \(-0.234085\pi\)
0.741562 + 0.670884i \(0.234085\pi\)
\(200\) 0 0
\(201\) 1.00000 0.0705346
\(202\) 0 0
\(203\) 6.41671i 0.450365i
\(204\) 0 0
\(205\) −9.20156 8.58528i −0.642665 0.599622i
\(206\) 0 0
\(207\) 5.79132i 0.402525i
\(208\) 0 0
\(209\) 8.18369 0.566078
\(210\) 0 0
\(211\) −8.45635 −0.582159 −0.291080 0.956699i \(-0.594014\pi\)
−0.291080 + 0.956699i \(0.594014\pi\)
\(212\) 0 0
\(213\) 5.85892i 0.401446i
\(214\) 0 0
\(215\) 2.87335 3.07961i 0.195961 0.210027i
\(216\) 0 0
\(217\) 14.6737i 0.996119i
\(218\) 0 0
\(219\) 15.3918 1.04008
\(220\) 0 0
\(221\) −0.00584618 −0.000393257
\(222\) 0 0
\(223\) 27.8779i 1.86684i −0.358783 0.933421i \(-0.616808\pi\)
0.358783 0.933421i \(-0.383192\pi\)
\(224\) 0 0
\(225\) −0.346064 4.98801i −0.0230710 0.332534i
\(226\) 0 0
\(227\) 12.0632i 0.800662i 0.916371 + 0.400331i \(0.131105\pi\)
−0.916371 + 0.400331i \(0.868895\pi\)
\(228\) 0 0
\(229\) −24.5262 −1.62074 −0.810370 0.585919i \(-0.800734\pi\)
−0.810370 + 0.585919i \(0.800734\pi\)
\(230\) 0 0
\(231\) 4.80589 0.316205
\(232\) 0 0
\(233\) 9.69308i 0.635015i −0.948256 0.317507i \(-0.897154\pi\)
0.948256 0.317507i \(-0.102846\pi\)
\(234\) 0 0
\(235\) −9.02599 + 9.67390i −0.588791 + 0.631056i
\(236\) 0 0
\(237\) 7.34143i 0.476877i
\(238\) 0 0
\(239\) −20.6671 −1.33684 −0.668420 0.743784i \(-0.733029\pi\)
−0.668420 + 0.743784i \(0.733029\pi\)
\(240\) 0 0
\(241\) −27.1359 −1.74798 −0.873989 0.485945i \(-0.838475\pi\)
−0.873989 + 0.485945i \(0.838475\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 8.39869 + 7.83618i 0.536572 + 0.500635i
\(246\) 0 0
\(247\) 0.0364930i 0.00232199i
\(248\) 0 0
\(249\) −17.1852 −1.08907
\(250\) 0 0
\(251\) −12.0570 −0.761029 −0.380515 0.924775i \(-0.624253\pi\)
−0.380515 + 0.924775i \(0.624253\pi\)
\(252\) 0 0
\(253\) 7.98907i 0.502269i
\(254\) 0 0
\(255\) 1.55380 + 1.44974i 0.0973029 + 0.0907860i
\(256\) 0 0
\(257\) 9.53277i 0.594638i −0.954778 0.297319i \(-0.903908\pi\)
0.954778 0.297319i \(-0.0960924\pi\)
\(258\) 0 0
\(259\) 30.3372 1.88506
\(260\) 0 0
\(261\) −1.84186 −0.114008
\(262\) 0 0
\(263\) 16.5536i 1.02074i −0.859955 0.510371i \(-0.829508\pi\)
0.859955 0.510371i \(-0.170492\pi\)
\(264\) 0 0
\(265\) 17.8314 19.1114i 1.09537 1.17400i
\(266\) 0 0
\(267\) 0.0825605i 0.00505262i
\(268\) 0 0
\(269\) 7.03378 0.428857 0.214429 0.976740i \(-0.431211\pi\)
0.214429 + 0.976740i \(0.431211\pi\)
\(270\) 0 0
\(271\) 21.8273 1.32591 0.662956 0.748658i \(-0.269302\pi\)
0.662956 + 0.748658i \(0.269302\pi\)
\(272\) 0 0
\(273\) 0.0214306i 0.00129704i
\(274\) 0 0
\(275\) −0.477392 6.88091i −0.0287878 0.414934i
\(276\) 0 0
\(277\) 26.8080i 1.61074i 0.592773 + 0.805370i \(0.298033\pi\)
−0.592773 + 0.805370i \(0.701967\pi\)
\(278\) 0 0
\(279\) 4.21197 0.252164
\(280\) 0 0
\(281\) −27.3824 −1.63350 −0.816750 0.576992i \(-0.804226\pi\)
−0.816750 + 0.576992i \(0.804226\pi\)
\(282\) 0 0
\(283\) 8.36562i 0.497284i −0.968595 0.248642i \(-0.920016\pi\)
0.968595 0.248642i \(-0.0799843\pi\)
\(284\) 0 0
\(285\) −9.04953 + 9.69913i −0.536048 + 0.574527i
\(286\) 0 0
\(287\) 19.6072i 1.15738i
\(288\) 0 0
\(289\) 16.0968 0.946870
\(290\) 0 0
\(291\) 10.8586 0.636540
\(292\) 0 0
\(293\) 13.9803i 0.816736i 0.912817 + 0.408368i \(0.133902\pi\)
−0.912817 + 0.408368i \(0.866098\pi\)
\(294\) 0 0
\(295\) −21.1190 19.7045i −1.22959 1.14724i
\(296\) 0 0
\(297\) 1.37949i 0.0800461i
\(298\) 0 0
\(299\) 0.0356251 0.00206025
\(300\) 0 0
\(301\) −6.56219 −0.378238
\(302\) 0 0
\(303\) 12.2686i 0.704813i
\(304\) 0 0
\(305\) 14.2776 + 13.3213i 0.817532 + 0.762777i
\(306\) 0 0
\(307\) 17.9185i 1.02266i 0.859384 + 0.511331i \(0.170847\pi\)
−0.859384 + 0.511331i \(0.829153\pi\)
\(308\) 0 0
\(309\) −8.37006 −0.476156
\(310\) 0 0
\(311\) 18.6785 1.05916 0.529580 0.848260i \(-0.322350\pi\)
0.529580 + 0.848260i \(0.322350\pi\)
\(312\) 0 0
\(313\) 9.32650i 0.527165i 0.964637 + 0.263583i \(0.0849041\pi\)
−0.964637 + 0.263583i \(0.915096\pi\)
\(314\) 0 0
\(315\) −5.31436 + 5.69584i −0.299430 + 0.320924i
\(316\) 0 0
\(317\) 10.6307i 0.597080i −0.954397 0.298540i \(-0.903500\pi\)
0.954397 0.298540i \(-0.0964996\pi\)
\(318\) 0 0
\(319\) −2.54083 −0.142259
\(320\) 0 0
\(321\) 9.42903 0.526277
\(322\) 0 0
\(323\) 5.63799i 0.313706i
\(324\) 0 0
\(325\) −0.0306836 + 0.00212880i −0.00170202 + 0.000118085i
\(326\) 0 0
\(327\) 5.11856i 0.283057i
\(328\) 0 0
\(329\) 20.6137 1.13647
\(330\) 0 0
\(331\) 0.0440326 0.00242025 0.00121013 0.999999i \(-0.499615\pi\)
0.00121013 + 0.999999i \(0.499615\pi\)
\(332\) 0 0
\(333\) 8.70804i 0.477198i
\(334\) 0 0
\(335\) −1.52544 + 1.63494i −0.0833437 + 0.0893264i
\(336\) 0 0
\(337\) 0.316789i 0.0172566i 0.999963 + 0.00862831i \(0.00274651\pi\)
−0.999963 + 0.00862831i \(0.997253\pi\)
\(338\) 0 0
\(339\) 14.7880 0.803175
\(340\) 0 0
\(341\) 5.81037 0.314649
\(342\) 0 0
\(343\) 6.49037i 0.350447i
\(344\) 0 0
\(345\) −9.46847 8.83432i −0.509765 0.475624i
\(346\) 0 0
\(347\) 6.85062i 0.367760i −0.982949 0.183880i \(-0.941134\pi\)
0.982949 0.183880i \(-0.0588658\pi\)
\(348\) 0 0
\(349\) −16.1577 −0.864903 −0.432452 0.901657i \(-0.642351\pi\)
−0.432452 + 0.901657i \(0.642351\pi\)
\(350\) 0 0
\(351\) 0.00615147 0.000328341
\(352\) 0 0
\(353\) 2.92185i 0.155514i −0.996972 0.0777571i \(-0.975224\pi\)
0.996972 0.0777571i \(-0.0247759\pi\)
\(354\) 0 0
\(355\) −9.57898 8.93742i −0.508400 0.474349i
\(356\) 0 0
\(357\) 3.31092i 0.175233i
\(358\) 0 0
\(359\) 17.8101 0.939980 0.469990 0.882672i \(-0.344257\pi\)
0.469990 + 0.882672i \(0.344257\pi\)
\(360\) 0 0
\(361\) 16.1934 0.852284
\(362\) 0 0
\(363\) 9.09701i 0.477469i
\(364\) 0 0
\(365\) −23.4793 + 25.1647i −1.22896 + 1.31718i
\(366\) 0 0
\(367\) 1.44265i 0.0753055i −0.999291 0.0376528i \(-0.988012\pi\)
0.999291 0.0376528i \(-0.0119881\pi\)
\(368\) 0 0
\(369\) −5.62807 −0.292986
\(370\) 0 0
\(371\) −40.7236 −2.11426
\(372\) 0 0
\(373\) 1.42957i 0.0740206i −0.999315 0.0370103i \(-0.988217\pi\)
0.999315 0.0370103i \(-0.0117834\pi\)
\(374\) 0 0
\(375\) 8.68300 + 7.04312i 0.448388 + 0.363705i
\(376\) 0 0
\(377\) 0.0113301i 0.000583532i
\(378\) 0 0
\(379\) −12.7653 −0.655712 −0.327856 0.944728i \(-0.606326\pi\)
−0.327856 + 0.944728i \(0.606326\pi\)
\(380\) 0 0
\(381\) 14.1851 0.726726
\(382\) 0 0
\(383\) 0.494136i 0.0252492i −0.999920 0.0126246i \(-0.995981\pi\)
0.999920 0.0126246i \(-0.00401864\pi\)
\(384\) 0 0
\(385\) −7.33110 + 7.85735i −0.373628 + 0.400448i
\(386\) 0 0
\(387\) 1.88362i 0.0957497i
\(388\) 0 0
\(389\) 6.28941 0.318886 0.159443 0.987207i \(-0.449030\pi\)
0.159443 + 0.987207i \(0.449030\pi\)
\(390\) 0 0
\(391\) 5.50391 0.278345
\(392\) 0 0
\(393\) 10.3659i 0.522892i
\(394\) 0 0
\(395\) −12.0028 11.1989i −0.603927 0.563479i
\(396\) 0 0
\(397\) 21.1867i 1.06333i −0.846954 0.531666i \(-0.821566\pi\)
0.846954 0.531666i \(-0.178434\pi\)
\(398\) 0 0
\(399\) 20.6674 1.03467
\(400\) 0 0
\(401\) 3.25488 0.162541 0.0812705 0.996692i \(-0.474102\pi\)
0.0812705 + 0.996692i \(0.474102\pi\)
\(402\) 0 0
\(403\) 0.0259098i 0.00129066i
\(404\) 0 0
\(405\) −1.63494 1.52544i −0.0812409 0.0757997i
\(406\) 0 0
\(407\) 12.0127i 0.595445i
\(408\) 0 0
\(409\) 15.8536 0.783913 0.391956 0.919984i \(-0.371798\pi\)
0.391956 + 0.919984i \(0.371798\pi\)
\(410\) 0 0
\(411\) −8.30628 −0.409718
\(412\) 0 0
\(413\) 45.0014i 2.21437i
\(414\) 0 0
\(415\) 26.2151 28.0969i 1.28685 1.37922i
\(416\) 0 0
\(417\) 13.9339i 0.682346i
\(418\) 0 0
\(419\) 3.42505 0.167325 0.0836624 0.996494i \(-0.473338\pi\)
0.0836624 + 0.996494i \(0.473338\pi\)
\(420\) 0 0
\(421\) 35.0605 1.70874 0.854371 0.519663i \(-0.173943\pi\)
0.854371 + 0.519663i \(0.173943\pi\)
\(422\) 0 0
\(423\) 5.91697i 0.287693i
\(424\) 0 0
\(425\) −4.74046 + 0.328890i −0.229946 + 0.0159535i
\(426\) 0 0
\(427\) 30.4234i 1.47229i
\(428\) 0 0
\(429\) 0.00848589 0.000409703
\(430\) 0 0
\(431\) −36.9897 −1.78173 −0.890865 0.454268i \(-0.849901\pi\)
−0.890865 + 0.454268i \(0.849901\pi\)
\(432\) 0 0
\(433\) 37.4130i 1.79795i −0.437996 0.898977i \(-0.644312\pi\)
0.437996 0.898977i \(-0.355688\pi\)
\(434\) 0 0
\(435\) 2.80965 3.01133i 0.134712 0.144382i
\(436\) 0 0
\(437\) 34.3565i 1.64349i
\(438\) 0 0
\(439\) −10.1011 −0.482098 −0.241049 0.970513i \(-0.577492\pi\)
−0.241049 + 0.970513i \(0.577492\pi\)
\(440\) 0 0
\(441\) 5.13700 0.244619
\(442\) 0 0
\(443\) 28.0156i 1.33106i 0.746370 + 0.665531i \(0.231795\pi\)
−0.746370 + 0.665531i \(0.768205\pi\)
\(444\) 0 0
\(445\) 0.134982 + 0.125941i 0.00639874 + 0.00597018i
\(446\) 0 0
\(447\) 13.9868i 0.661552i
\(448\) 0 0
\(449\) 23.8281 1.12452 0.562258 0.826962i \(-0.309933\pi\)
0.562258 + 0.826962i \(0.309933\pi\)
\(450\) 0 0
\(451\) −7.76386 −0.365586
\(452\) 0 0
\(453\) 19.2480i 0.904350i
\(454\) 0 0
\(455\) 0.0350378 + 0.0326911i 0.00164260 + 0.00153258i
\(456\) 0 0
\(457\) 7.67425i 0.358986i 0.983759 + 0.179493i \(0.0574457\pi\)
−0.983759 + 0.179493i \(0.942554\pi\)
\(458\) 0 0
\(459\) 0.950372 0.0443596
\(460\) 0 0
\(461\) −5.39545 −0.251291 −0.125646 0.992075i \(-0.540100\pi\)
−0.125646 + 0.992075i \(0.540100\pi\)
\(462\) 0 0
\(463\) 8.15948i 0.379203i −0.981861 0.189602i \(-0.939280\pi\)
0.981861 0.189602i \(-0.0607197\pi\)
\(464\) 0 0
\(465\) −6.42511 + 6.88633i −0.297957 + 0.319346i
\(466\) 0 0
\(467\) 33.9518i 1.57110i −0.618797 0.785551i \(-0.712380\pi\)
0.618797 0.785551i \(-0.287620\pi\)
\(468\) 0 0
\(469\) 3.48382 0.160868
\(470\) 0 0
\(471\) −10.4863 −0.483183
\(472\) 0 0
\(473\) 2.59843i 0.119476i
\(474\) 0 0
\(475\) −2.05299 29.5909i −0.0941978 1.35772i
\(476\) 0 0
\(477\) 11.6894i 0.535219i
\(478\) 0 0
\(479\) 7.97687 0.364472 0.182236 0.983255i \(-0.441666\pi\)
0.182236 + 0.983255i \(0.441666\pi\)
\(480\) 0 0
\(481\) 0.0535672 0.00244246
\(482\) 0 0
\(483\) 20.1759i 0.918036i
\(484\) 0 0
\(485\) −16.5641 + 17.7531i −0.752137 + 0.806128i
\(486\) 0 0
\(487\) 3.77376i 0.171005i −0.996338 0.0855027i \(-0.972750\pi\)
0.996338 0.0855027i \(-0.0272496\pi\)
\(488\) 0 0
\(489\) −7.57463 −0.342537
\(490\) 0 0
\(491\) 34.0282 1.53567 0.767836 0.640647i \(-0.221334\pi\)
0.767836 + 0.640647i \(0.221334\pi\)
\(492\) 0 0
\(493\) 1.75045i 0.0788364i
\(494\) 0 0
\(495\) −2.25538 2.10433i −0.101372 0.0945826i
\(496\) 0 0
\(497\) 20.4114i 0.915577i
\(498\) 0 0
\(499\) −36.3624 −1.62781 −0.813903 0.581001i \(-0.802661\pi\)
−0.813903 + 0.581001i \(0.802661\pi\)
\(500\) 0 0
\(501\) 17.1496 0.766189
\(502\) 0 0
\(503\) 24.5726i 1.09564i −0.836596 0.547820i \(-0.815458\pi\)
0.836596 0.547820i \(-0.184542\pi\)
\(504\) 0 0
\(505\) 20.0584 + 18.7150i 0.892589 + 0.832808i
\(506\) 0 0
\(507\) 13.0000i 0.577349i
\(508\) 0 0
\(509\) 6.04776 0.268062 0.134031 0.990977i \(-0.457208\pi\)
0.134031 + 0.990977i \(0.457208\pi\)
\(510\) 0 0
\(511\) 53.6223 2.37211
\(512\) 0 0
\(513\) 5.93240i 0.261922i
\(514\) 0 0
\(515\) 12.7680 13.6845i 0.562626 0.603013i
\(516\) 0 0
\(517\) 8.16240i 0.358982i
\(518\) 0 0
\(519\) 10.7568 0.472170
\(520\) 0 0
\(521\) −43.6035 −1.91030 −0.955152 0.296116i \(-0.904308\pi\)
−0.955152 + 0.296116i \(0.904308\pi\)
\(522\) 0 0
\(523\) 4.21345i 0.184241i −0.995748 0.0921207i \(-0.970635\pi\)
0.995748 0.0921207i \(-0.0293646\pi\)
\(524\) 0 0
\(525\) −1.20563 17.3773i −0.0526178 0.758409i
\(526\) 0 0
\(527\) 4.00294i 0.174371i
\(528\) 0 0
\(529\) −10.5394 −0.458235
\(530\) 0 0
\(531\) −12.9173 −0.560561
\(532\) 0 0
\(533\) 0.0346209i 0.00149960i
\(534\) 0 0
\(535\) −14.3834 + 15.4159i −0.621849 + 0.666487i
\(536\) 0 0
\(537\) 24.5022i 1.05735i
\(538\) 0 0
\(539\) 7.08643 0.305234
\(540\) 0 0
\(541\) −10.1030 −0.434362 −0.217181 0.976131i \(-0.569686\pi\)
−0.217181 + 0.976131i \(0.569686\pi\)
\(542\) 0 0
\(543\) 0.467927i 0.0200807i
\(544\) 0 0
\(545\) 8.36855 + 7.80806i 0.358469 + 0.334461i
\(546\) 0 0
\(547\) 39.1148i 1.67243i 0.548404 + 0.836214i \(0.315236\pi\)
−0.548404 + 0.836214i \(0.684764\pi\)
\(548\) 0 0
\(549\) 8.73278 0.372706
\(550\) 0 0
\(551\) −10.9267 −0.465491
\(552\) 0 0
\(553\) 25.5762i 1.08761i
\(554\) 0 0
\(555\) −14.2371 13.2836i −0.604333 0.563857i
\(556\) 0 0
\(557\) 7.54268i 0.319594i −0.987150 0.159797i \(-0.948916\pi\)
0.987150 0.159797i \(-0.0510839\pi\)
\(558\) 0 0
\(559\) −0.0115870 −0.000490079
\(560\) 0 0
\(561\) 1.31103 0.0553517
\(562\) 0 0
\(563\) 25.0314i 1.05495i 0.849572 + 0.527473i \(0.176860\pi\)
−0.849572 + 0.527473i \(0.823140\pi\)
\(564\) 0 0
\(565\) −22.5582 + 24.1775i −0.949032 + 1.01716i
\(566\) 0 0
\(567\) 3.48382i 0.146307i
\(568\) 0 0
\(569\) 14.0066 0.587188 0.293594 0.955930i \(-0.405149\pi\)
0.293594 + 0.955930i \(0.405149\pi\)
\(570\) 0 0
\(571\) −3.22997 −0.135170 −0.0675849 0.997714i \(-0.521529\pi\)
−0.0675849 + 0.997714i \(0.521529\pi\)
\(572\) 0 0
\(573\) 2.48094i 0.103643i
\(574\) 0 0
\(575\) 28.8872 2.00417i 1.20468 0.0835797i
\(576\) 0 0
\(577\) 5.24717i 0.218443i 0.994017 + 0.109221i \(0.0348357\pi\)
−0.994017 + 0.109221i \(0.965164\pi\)
\(578\) 0 0
\(579\) 24.4200 1.01486
\(580\) 0 0
\(581\) −59.8703 −2.48384
\(582\) 0 0
\(583\) 16.1253i 0.667843i
\(584\) 0 0
\(585\) −0.00938370 + 0.0100573i −0.000387968 + 0.000415818i
\(586\) 0 0
\(587\) 36.7849i 1.51827i 0.650931 + 0.759137i \(0.274379\pi\)
−0.650931 + 0.759137i \(0.725621\pi\)
\(588\) 0 0
\(589\) 24.9871 1.02958
\(590\) 0 0
\(591\) 24.5278 1.00894
\(592\) 0 0
\(593\) 5.23589i 0.215012i 0.994204 + 0.107506i \(0.0342865\pi\)
−0.994204 + 0.107506i \(0.965713\pi\)
\(594\) 0 0
\(595\) 5.41317 + 5.05062i 0.221918 + 0.207055i
\(596\) 0 0
\(597\) 20.9220i 0.856282i
\(598\) 0 0
\(599\) −26.2714 −1.07342 −0.536711 0.843766i \(-0.680333\pi\)
−0.536711 + 0.843766i \(0.680333\pi\)
\(600\) 0 0
\(601\) −46.1863 −1.88398 −0.941990 0.335641i \(-0.891047\pi\)
−0.941990 + 0.335641i \(0.891047\pi\)
\(602\) 0 0
\(603\) 1.00000i 0.0407231i
\(604\) 0 0
\(605\) 14.8731 + 13.8769i 0.604676 + 0.564178i
\(606\) 0 0
\(607\) 14.4091i 0.584849i −0.956289 0.292425i \(-0.905538\pi\)
0.956289 0.292425i \(-0.0944621\pi\)
\(608\) 0 0
\(609\) −6.41671 −0.260018
\(610\) 0 0
\(611\) 0.0363981 0.00147251
\(612\) 0 0
\(613\) 31.3208i 1.26504i −0.774545 0.632518i \(-0.782021\pi\)
0.774545 0.632518i \(-0.217979\pi\)
\(614\) 0 0
\(615\) 8.58528 9.20156i 0.346192 0.371043i
\(616\) 0 0
\(617\) 17.0998i 0.688411i 0.938894 + 0.344206i \(0.111852\pi\)
−0.938894 + 0.344206i \(0.888148\pi\)
\(618\) 0 0
\(619\) −32.1087 −1.29056 −0.645278 0.763948i \(-0.723259\pi\)
−0.645278 + 0.763948i \(0.723259\pi\)
\(620\) 0 0
\(621\) −5.79132 −0.232398
\(622\) 0 0
\(623\) 0.287626i 0.0115235i
\(624\) 0 0
\(625\) −24.7605 + 3.45235i −0.990419 + 0.138094i
\(626\) 0 0
\(627\) 8.18369i 0.326825i
\(628\) 0 0
\(629\) 8.27588 0.329981
\(630\) 0 0
\(631\) −25.4447 −1.01294 −0.506469 0.862258i \(-0.669049\pi\)
−0.506469 + 0.862258i \(0.669049\pi\)
\(632\) 0 0
\(633\) 8.45635i 0.336110i
\(634\) 0 0
\(635\) −21.6386 + 23.1919i −0.858701 + 0.920341i
\(636\) 0 0
\(637\) 0.0316001i 0.00125204i
\(638\) 0 0
\(639\) −5.85892 −0.231775
\(640\) 0 0
\(641\) −50.1762 −1.98184 −0.990920 0.134455i \(-0.957071\pi\)
−0.990920 + 0.134455i \(0.957071\pi\)
\(642\) 0 0
\(643\) 18.6849i 0.736860i −0.929655 0.368430i \(-0.879895\pi\)
0.929655 0.368430i \(-0.120105\pi\)
\(644\) 0 0
\(645\) 3.07961 + 2.87335i 0.121259 + 0.113138i
\(646\) 0 0
\(647\) 27.6483i 1.08697i 0.839420 + 0.543483i \(0.182895\pi\)
−0.839420 + 0.543483i \(0.817105\pi\)
\(648\) 0 0
\(649\) −17.8192 −0.699466
\(650\) 0 0
\(651\) 14.6737 0.575110
\(652\) 0 0
\(653\) 32.8195i 1.28433i −0.766568 0.642163i \(-0.778037\pi\)
0.766568 0.642163i \(-0.221963\pi\)
\(654\) 0 0
\(655\) −16.9477 15.8126i −0.662200 0.617849i
\(656\) 0 0
\(657\) 15.3918i 0.600492i
\(658\) 0 0
\(659\) 21.9909 0.856642 0.428321 0.903627i \(-0.359105\pi\)
0.428321 + 0.903627i \(0.359105\pi\)
\(660\) 0 0
\(661\) −43.8094 −1.70399 −0.851995 0.523550i \(-0.824607\pi\)
−0.851995 + 0.523550i \(0.824607\pi\)
\(662\) 0 0
\(663\) 0.00584618i 0.000227047i
\(664\) 0 0
\(665\) −31.5269 + 33.7900i −1.22256 + 1.31032i
\(666\) 0 0
\(667\) 10.6668i 0.413021i
\(668\) 0 0
\(669\) 27.8779 1.07782
\(670\) 0 0
\(671\) 12.0468 0.465061
\(672\) 0 0
\(673\) 15.5926i 0.601051i −0.953774 0.300526i \(-0.902838\pi\)
0.953774 0.300526i \(-0.0971621\pi\)
\(674\) 0 0
\(675\) 4.98801 0.346064i 0.191989 0.0133200i
\(676\) 0 0
\(677\) 6.15504i 0.236558i 0.992980 + 0.118279i \(0.0377376\pi\)
−0.992980 + 0.118279i \(0.962262\pi\)
\(678\) 0 0
\(679\) 37.8293 1.45175
\(680\) 0 0
\(681\) −12.0632 −0.462263
\(682\) 0 0
\(683\) 46.6388i 1.78459i −0.451457 0.892293i \(-0.649096\pi\)
0.451457 0.892293i \(-0.350904\pi\)
\(684\) 0 0
\(685\) 12.6707 13.5803i 0.484124 0.518876i
\(686\) 0 0
\(687\) 24.5262i 0.935735i
\(688\) 0 0
\(689\) −0.0719067 −0.00273943
\(690\) 0 0
\(691\) −26.8639 −1.02195 −0.510975 0.859595i \(-0.670716\pi\)
−0.510975 + 0.859595i \(0.670716\pi\)
\(692\) 0 0
\(693\) 4.80589i 0.182561i
\(694\) 0 0
\(695\) −22.7811 21.2553i −0.864137 0.806261i
\(696\) 0 0
\(697\) 5.34876i 0.202599i
\(698\) 0 0
\(699\) 9.69308 0.366626
\(700\) 0 0
\(701\) 14.9337 0.564037 0.282019 0.959409i \(-0.408996\pi\)
0.282019 + 0.959409i \(0.408996\pi\)
\(702\) 0 0
\(703\) 51.6596i 1.94838i
\(704\) 0 0
\(705\) −9.67390 9.02599i −0.364340 0.339938i
\(706\) 0 0
\(707\) 42.7416i 1.60746i
\(708\) 0 0
\(709\) −4.23386 −0.159006 −0.0795030 0.996835i \(-0.525333\pi\)
−0.0795030 + 0.996835i \(0.525333\pi\)
\(710\) 0 0
\(711\) −7.34143 −0.275325
\(712\) 0 0
\(713\) 24.3929i 0.913521i
\(714\) 0 0
\(715\) −0.0129447 + 0.0138739i −0.000484105 + 0.000518855i
\(716\) 0 0
\(717\) 20.6671i 0.771825i
\(718\) 0 0
\(719\) −4.71665 −0.175901 −0.0879507 0.996125i \(-0.528032\pi\)
−0.0879507 + 0.996125i \(0.528032\pi\)
\(720\) 0 0
\(721\) −29.1598 −1.08597
\(722\) 0 0
\(723\) 27.1359i 1.00920i
\(724\) 0 0
\(725\) 0.637403 + 9.18722i 0.0236725 + 0.341205i
\(726\) 0 0
\(727\) 4.54853i 0.168695i 0.996436 + 0.0843477i \(0.0268807\pi\)
−0.996436 + 0.0843477i \(0.973119\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −1.79014 −0.0662106
\(732\) 0 0
\(733\) 11.8615i 0.438114i −0.975712 0.219057i \(-0.929702\pi\)
0.975712 0.219057i \(-0.0702980\pi\)
\(734\) 0 0
\(735\) −7.83618 + 8.39869i −0.289042 + 0.309790i
\(736\) 0 0
\(737\) 1.37949i 0.0508142i
\(738\) 0 0
\(739\) 22.2083 0.816946 0.408473 0.912770i \(-0.366061\pi\)
0.408473 + 0.912770i \(0.366061\pi\)
\(740\) 0 0
\(741\) 0.0364930 0.00134060
\(742\) 0 0
\(743\) 29.7672i 1.09205i −0.837768 0.546027i \(-0.816140\pi\)
0.837768 0.546027i \(-0.183860\pi\)
\(744\) 0 0
\(745\) −22.8676 21.3360i −0.837802 0.781690i
\(746\) 0 0
\(747\) 17.1852i 0.628775i
\(748\) 0 0
\(749\) 32.8490 1.20028
\(750\) 0 0
\(751\) −18.9738 −0.692363 −0.346182 0.938168i \(-0.612522\pi\)
−0.346182 + 0.938168i \(0.612522\pi\)
\(752\) 0 0
\(753\) 12.0570i 0.439380i
\(754\) 0 0
\(755\) −31.4693 29.3617i −1.14529 1.06858i
\(756\) 0 0
\(757\) 3.59503i 0.130664i 0.997864 + 0.0653319i \(0.0208106\pi\)
−0.997864 + 0.0653319i \(0.979189\pi\)
\(758\) 0 0
\(759\) −7.98907 −0.289985
\(760\) 0 0
\(761\) 43.3749 1.57234 0.786169 0.618012i \(-0.212062\pi\)
0.786169 + 0.618012i \(0.212062\pi\)
\(762\) 0 0
\(763\) 17.8322i 0.645567i
\(764\) 0 0
\(765\) −1.44974 + 1.55380i −0.0524153 + 0.0561778i
\(766\) 0 0
\(767\) 0.0794601i 0.00286914i
\(768\) 0 0
\(769\) −33.2888 −1.20042 −0.600212 0.799841i \(-0.704917\pi\)
−0.600212 + 0.799841i \(0.704917\pi\)
\(770\) 0 0
\(771\) 9.53277 0.343314
\(772\) 0 0
\(773\) 8.75564i 0.314918i 0.987525 + 0.157459i \(0.0503303\pi\)
−0.987525 + 0.157459i \(0.949670\pi\)
\(774\) 0 0
\(775\) −1.45761 21.0094i −0.0523590 0.754678i
\(776\) 0 0
\(777\) 30.3372i 1.08834i
\(778\) 0 0
\(779\) −33.3880 −1.19625
\(780\) 0 0
\(781\) −8.08231 −0.289208
\(782\) 0 0
\(783\) 1.84186i 0.0658227i
\(784\) 0 0
\(785\) 15.9962 17.1445i 0.570929 0.611912i
\(786\) 0 0
\(787\) 35.5156i 1.26600i 0.774154 + 0.632998i \(0.218176\pi\)
−0.774154 + 0.632998i \(0.781824\pi\)
\(788\) 0 0
\(789\) 16.5536 0.589325
\(790\) 0 0
\(791\) 51.5188 1.83180
\(792\) 0 0
\(793\) 0.0537194i 0.00190763i
\(794\) 0 0
\(795\) 19.1114 + 17.8314i 0.677812 + 0.632415i
\(796\) 0 0
\(797\) 19.8606i 0.703500i −0.936094 0.351750i \(-0.885587\pi\)
0.936094 0.351750i \(-0.114413\pi\)
\(798\) 0 0
\(799\) 5.62332 0.198939
\(800\) 0 0
\(801\) 0.0825605 0.00291713
\(802\) 0 0
\(803\) 21.2328i 0.749291i
\(804\) 0 0
\(805\) −32.9864 30.7772i −1.16262 1.08475i
\(806\) 0 0
\(807\) 7.03378i 0.247601i
\(808\) 0 0
\(809\) 53.1079 1.86717 0.933587 0.358352i \(-0.116661\pi\)
0.933587 + 0.358352i \(0.116661\pi\)
\(810\) 0 0
\(811\) 3.45248 0.121233 0.0606166 0.998161i \(-0.480693\pi\)
0.0606166 + 0.998161i \(0.480693\pi\)
\(812\) 0 0
\(813\) 21.8273i 0.765516i
\(814\) 0 0
\(815\) 11.5546 12.3841i 0.404742 0.433795i
\(816\) 0 0
\(817\) 11.1744i 0.390942i
\(818\) 0 0
\(819\) 0.0214306 0.000748846
\(820\) 0 0
\(821\) 47.4174 1.65488 0.827439 0.561556i \(-0.189797\pi\)
0.827439 + 0.561556i \(0.189797\pi\)
\(822\) 0 0
\(823\) 1.81465i 0.0632546i 0.999500 + 0.0316273i \(0.0100690\pi\)
−0.999500 + 0.0316273i \(0.989931\pi\)
\(824\) 0 0
\(825\) 6.88091 0.477392i 0.239562 0.0166207i
\(826\) 0 0
\(827\) 26.7154i 0.928985i −0.885577 0.464492i \(-0.846237\pi\)
0.885577 0.464492i \(-0.153763\pi\)
\(828\) 0 0
\(829\) −36.0126 −1.25077 −0.625384 0.780317i \(-0.715058\pi\)
−0.625384 + 0.780317i \(0.715058\pi\)
\(830\) 0 0
\(831\) −26.8080 −0.929961
\(832\) 0 0
\(833\) 4.88206i 0.169153i
\(834\) 0 0
\(835\) −26.1607 + 28.0386i −0.905329 + 0.970317i
\(836\) 0 0
\(837\) 4.21197i 0.145587i
\(838\) 0 0
\(839\) −2.88016 −0.0994342 −0.0497171 0.998763i \(-0.515832\pi\)
−0.0497171 + 0.998763i \(0.515832\pi\)
\(840\) 0 0
\(841\) −25.6075 −0.883019
\(842\) 0 0
\(843\) 27.3824i 0.943101i
\(844\) 0 0
\(845\) −21.2542 19.8307i −0.731166 0.682196i
\(846\) 0 0
\(847\) 31.6923i 1.08896i
\(848\) 0 0
\(849\) 8.36562 0.287107
\(850\) 0 0
\(851\) −50.4311 −1.72875
\(852\) 0 0
\(853\) 27.2580i 0.933297i 0.884443 + 0.466648i \(0.154539\pi\)
−0.884443 + 0.466648i \(0.845461\pi\)
\(854\) 0 0
\(855\) −9.69913 9.04953i −0.331703 0.309487i
\(856\) 0 0
\(857\) 26.1104i 0.891916i −0.895054 0.445958i \(-0.852863\pi\)
0.895054 0.445958i \(-0.147137\pi\)
\(858\) 0 0
\(859\) 5.32404 0.181654 0.0908270 0.995867i \(-0.471049\pi\)
0.0908270 + 0.995867i \(0.471049\pi\)
\(860\) 0 0
\(861\) −19.6072 −0.668211
\(862\) 0 0
\(863\) 12.0033i 0.408596i −0.978909 0.204298i \(-0.934509\pi\)
0.978909 0.204298i \(-0.0654911\pi\)
\(864\) 0 0
\(865\) −16.4088 + 17.5867i −0.557917 + 0.597966i
\(866\) 0 0
\(867\) 16.0968i 0.546676i
\(868\) 0 0
\(869\) −10.1274 −0.343550
\(870\) 0 0
\(871\) 0.00615147 0.000208435
\(872\) 0 0
\(873\) 10.8586i 0.367507i
\(874\) 0 0
\(875\) 30.2500 + 24.5369i 1.02264 + 0.829500i
\(876\) 0 0
\(877\) 6.18961i 0.209008i 0.994524 + 0.104504i \(0.0333255\pi\)
−0.994524 + 0.104504i \(0.966674\pi\)
\(878\) 0 0
\(879\) −13.9803 −0.471543
\(880\) 0 0
\(881\) −12.6288 −0.425474 −0.212737 0.977110i \(-0.568238\pi\)
−0.212737 + 0.977110i \(0.568238\pi\)
\(882\) 0 0
\(883\) 46.1877i 1.55434i 0.629291 + 0.777170i \(0.283346\pi\)
−0.629291 + 0.777170i \(0.716654\pi\)
\(884\) 0 0
\(885\) 19.7045 21.1190i 0.662360 0.709906i
\(886\) 0 0
\(887\) 22.1189i 0.742682i 0.928497 + 0.371341i \(0.121102\pi\)
−0.928497 + 0.371341i \(0.878898\pi\)
\(888\) 0 0
\(889\) 49.4184 1.65744
\(890\) 0 0
\(891\) −1.37949 −0.0462147
\(892\) 0 0
\(893\) 35.1019i 1.17464i
\(894\) 0 0
\(895\) −40.0597 37.3767i −1.33905 1.24936i
\(896\) 0 0
\(897\) 0.0356251i 0.00118949i
\(898\) 0 0
\(899\) −7.75787 −0.258739
\(900\) 0 0
\(901\) −11.1092 −0.370102
\(902\) 0 0
\(903\) 6.56219i 0.218376i
\(904\) 0 0
\(905\) −0.765033 0.713794i −0.0254305 0.0237273i
\(906\) 0 0
\(907\) 12.7759i 0.424217i 0.977246 + 0.212108i \(0.0680330\pi\)
−0.977246 + 0.212108i \(0.931967\pi\)
\(908\) 0 0
\(909\) 12.2686 0.406924
\(910\) 0 0
\(911\) −3.19007 −0.105692 −0.0528459 0.998603i \(-0.516829\pi\)
−0.0528459 + 0.998603i \(0.516829\pi\)
\(912\) 0 0
\(913\) 23.7069i 0.784583i
\(914\) 0 0
\(915\) −13.3213 + 14.2776i −0.440390 + 0.472002i
\(916\) 0 0
\(917\) 36.1130i 1.19256i
\(918\) 0 0
\(919\) 27.0087 0.890935 0.445468 0.895298i \(-0.353037\pi\)
0.445468 + 0.895298i \(0.353037\pi\)
\(920\) 0 0
\(921\) −17.9185 −0.590434
\(922\) 0 0
\(923\) 0.0360409i 0.00118630i
\(924\) 0 0
\(925\) 43.4358 3.01354i 1.42816 0.0990847i
\(926\) 0 0
\(927\) 8.37006i 0.274909i
\(928\) 0 0
\(929\) −42.4873 −1.39396 −0.696981 0.717089i \(-0.745474\pi\)
−0.696981 + 0.717089i \(0.745474\pi\)
\(930\) 0 0
\(931\) 30.4747 0.998769
\(932\) 0 0
\(933\) 18.6785i 0.611506i
\(934\) 0 0
\(935\) −1.99990 + 2.14345i −0.0654036 + 0.0700984i
\(936\) 0 0
\(937\) 38.3512i 1.25288i 0.779470 + 0.626440i \(0.215489\pi\)
−0.779470 + 0.626440i \(0.784511\pi\)
\(938\) 0 0
\(939\) −9.32650 −0.304359
\(940\) 0 0
\(941\) −1.24423 −0.0405607 −0.0202803 0.999794i \(-0.506456\pi\)
−0.0202803 + 0.999794i \(0.506456\pi\)
\(942\) 0 0
\(943\) 32.5940i 1.06141i
\(944\) 0 0
\(945\) −5.69584 5.31436i −0.185286 0.172876i
\(946\) 0 0
\(947\) 12.1202i 0.393853i 0.980418 + 0.196926i \(0.0630960\pi\)
−0.980418 + 0.196926i \(0.936904\pi\)
\(948\) 0 0
\(949\) 0.0946822 0.00307352
\(950\) 0 0
\(951\) 10.6307 0.344724
\(952\) 0 0
\(953\) 13.9713i 0.452576i 0.974060 + 0.226288i \(0.0726591\pi\)
−0.974060 + 0.226288i \(0.927341\pi\)
\(954\) 0 0
\(955\) 4.05619 + 3.78453i 0.131255 + 0.122465i
\(956\) 0 0
\(957\) 2.54083i 0.0821333i
\(958\) 0 0
\(959\) −28.9376 −0.934443
\(960\) 0 0
\(961\) −13.2593 −0.427719
\(962\) 0 0
\(963\) 9.42903i 0.303846i
\(964\) 0 0
\(965\) −37.2513 + 39.9253i −1.19916 + 1.28524i
\(966\) 0 0
\(967\) 7.52975i 0.242140i −0.992644 0.121070i \(-0.961367\pi\)
0.992644 0.121070i \(-0.0386326\pi\)
\(968\) 0 0
\(969\) 5.63799 0.181118
\(970\) 0 0
\(971\) 7.60699 0.244120 0.122060 0.992523i \(-0.461050\pi\)
0.122060 + 0.992523i \(0.461050\pi\)
\(972\) 0 0
\(973\) 48.5432i 1.55622i
\(974\) 0 0
\(975\) −0.00212880 0.0306836i −6.81763e−5 0.000982661i
\(976\) 0 0
\(977\) 16.8460i 0.538952i −0.963007 0.269476i \(-0.913149\pi\)
0.963007 0.269476i \(-0.0868506\pi\)
\(978\) 0 0
\(979\) 0.113891 0.00363998
\(980\) 0 0
\(981\) 5.11856 0.163423
\(982\) 0 0
\(983\) 12.3613i 0.394263i −0.980377 0.197132i \(-0.936837\pi\)
0.980377 0.197132i \(-0.0631627\pi\)
\(984\) 0 0
\(985\) −37.4157 + 40.1015i −1.19216 + 1.27774i
\(986\) 0 0
\(987\) 20.6137i 0.656140i
\(988\) 0 0
\(989\) 10.9086 0.346875
\(990\) 0 0
\(991\) 41.4738 1.31746 0.658729 0.752380i \(-0.271094\pi\)
0.658729 + 0.752380i \(0.271094\pi\)
\(992\) 0 0
\(993\) 0.0440326i 0.00139733i
\(994\) 0 0
\(995\) −34.2063 31.9153i −1.08441 1.01178i
\(996\) 0 0
\(997\) 17.8188i 0.564328i −0.959366 0.282164i \(-0.908948\pi\)
0.959366 0.282164i \(-0.0910523\pi\)
\(998\) 0 0
\(999\) −8.70804 −0.275510
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 4020.2.g.b.1609.16 yes 24
5.4 even 2 inner 4020.2.g.b.1609.4 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4020.2.g.b.1609.4 24 5.4 even 2 inner
4020.2.g.b.1609.16 yes 24 1.1 even 1 trivial