Properties

Label 4020.2.q.j.841.1
Level $4020$
Weight $2$
Character 4020.841
Analytic conductor $32.100$
Analytic rank $0$
Dimension $12$
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(841,4020)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4020, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4020.841");
 
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.q (of order \(3\), 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: \(32.0998616126\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + 30 x^{10} - 53 x^{9} + 798 x^{8} - 1096 x^{7} + 4060 x^{6} - 915 x^{5} + 10392 x^{4} - 7038 x^{3} + 4869 x^{2} - 675 x + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 841.1
Root \(-2.64313 + 4.57803i\) of defining polynomial
Character \(\chi\) \(=\) 4020.841
Dual form 4020.2.q.j.3781.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} +(-2.33998 + 4.05297i) q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.00000 q^{5} +(-2.33998 + 4.05297i) q^{7} +1.00000 q^{9} +(2.04957 - 3.54996i) q^{11} +(0.500000 + 0.866025i) q^{13} -1.00000 q^{15} +(0.405657 + 0.702619i) q^{17} +(-1.42209 - 2.46313i) q^{19} +(-2.33998 + 4.05297i) q^{21} +(2.18790 + 3.78956i) q^{23} +1.00000 q^{25} +1.00000 q^{27} +(-4.40150 + 7.62362i) q^{29} +(0.131676 - 0.228069i) q^{31} +(2.04957 - 3.54996i) q^{33} +(2.33998 - 4.05297i) q^{35} +(-0.572659 - 0.991874i) q^{37} +(0.500000 + 0.866025i) q^{39} +(-1.81958 + 3.15160i) q^{41} -0.155822 q^{43} -1.00000 q^{45} +(-1.69187 + 2.93041i) q^{47} +(-7.45103 - 12.9056i) q^{49} +(0.405657 + 0.702619i) q^{51} -0.736649 q^{53} +(-2.04957 + 3.54996i) q^{55} +(-1.42209 - 2.46313i) q^{57} +3.35790 q^{59} +(0.373303 + 0.646580i) q^{61} +(-2.33998 + 4.05297i) q^{63} +(-0.500000 - 0.866025i) q^{65} +(-8.03665 - 1.55314i) q^{67} +(2.18790 + 3.78956i) q^{69} +(1.60329 - 2.77699i) q^{71} +(4.76087 + 8.24608i) q^{73} +1.00000 q^{75} +(9.59191 + 16.6137i) q^{77} +(-5.82437 + 10.0881i) q^{79} +1.00000 q^{81} +(6.18900 + 10.7197i) q^{83} +(-0.405657 - 0.702619i) q^{85} +(-4.40150 + 7.62362i) q^{87} -13.1087 q^{89} -4.67996 q^{91} +(0.131676 - 0.228069i) q^{93} +(1.42209 + 2.46313i) q^{95} +(-9.60393 - 16.6345i) q^{97} +(2.04957 - 3.54996i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{3} - 12 q^{5} + 2 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{3} - 12 q^{5} + 2 q^{7} + 12 q^{9} - 5 q^{11} + 6 q^{13} - 12 q^{15} + 9 q^{17} - 11 q^{19} + 2 q^{21} + 19 q^{23} + 12 q^{25} + 12 q^{27} + 8 q^{29} - 4 q^{31} - 5 q^{33} - 2 q^{35} + 5 q^{37} + 6 q^{39} - 9 q^{41} - 14 q^{43} - 12 q^{45} - 6 q^{47} - 46 q^{49} + 9 q^{51} - 20 q^{53} + 5 q^{55} - 11 q^{57} + 6 q^{59} + 27 q^{61} + 2 q^{63} - 6 q^{65} + 3 q^{67} + 19 q^{69} + 25 q^{71} - 8 q^{73} + 12 q^{75} + 10 q^{77} - 2 q^{79} + 12 q^{81} + 20 q^{83} - 9 q^{85} + 8 q^{87} + 12 q^{89} + 4 q^{91} - 4 q^{93} + 11 q^{95} - q^{97} - 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(1141\) \(2011\) \(2681\) \(3217\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\) \(1\)

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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −2.33998 + 4.05297i −0.884430 + 1.53188i −0.0380644 + 0.999275i \(0.512119\pi\)
−0.846366 + 0.532602i \(0.821214\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.04957 3.54996i 0.617968 1.07035i −0.371888 0.928278i \(-0.621289\pi\)
0.989856 0.142075i \(-0.0453773\pi\)
\(12\) 0 0
\(13\) 0.500000 + 0.866025i 0.138675 + 0.240192i 0.926995 0.375073i \(-0.122382\pi\)
−0.788320 + 0.615265i \(0.789049\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 0.405657 + 0.702619i 0.0983864 + 0.170410i 0.911017 0.412369i \(-0.135299\pi\)
−0.812631 + 0.582779i \(0.801965\pi\)
\(18\) 0 0
\(19\) −1.42209 2.46313i −0.326250 0.565081i 0.655515 0.755182i \(-0.272452\pi\)
−0.981764 + 0.190101i \(0.939118\pi\)
\(20\) 0 0
\(21\) −2.33998 + 4.05297i −0.510626 + 0.884430i
\(22\) 0 0
\(23\) 2.18790 + 3.78956i 0.456209 + 0.790178i 0.998757 0.0498475i \(-0.0158735\pi\)
−0.542548 + 0.840025i \(0.682540\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −4.40150 + 7.62362i −0.817338 + 1.41567i 0.0902986 + 0.995915i \(0.471218\pi\)
−0.907637 + 0.419756i \(0.862115\pi\)
\(30\) 0 0
\(31\) 0.131676 0.228069i 0.0236496 0.0409624i −0.853958 0.520341i \(-0.825805\pi\)
0.877608 + 0.479379i \(0.159138\pi\)
\(32\) 0 0
\(33\) 2.04957 3.54996i 0.356784 0.617968i
\(34\) 0 0
\(35\) 2.33998 4.05297i 0.395529 0.685077i
\(36\) 0 0
\(37\) −0.572659 0.991874i −0.0941445 0.163063i 0.815107 0.579311i \(-0.196678\pi\)
−0.909251 + 0.416248i \(0.863345\pi\)
\(38\) 0 0
\(39\) 0.500000 + 0.866025i 0.0800641 + 0.138675i
\(40\) 0 0
\(41\) −1.81958 + 3.15160i −0.284170 + 0.492198i −0.972408 0.233288i \(-0.925051\pi\)
0.688237 + 0.725486i \(0.258385\pi\)
\(42\) 0 0
\(43\) −0.155822 −0.0237627 −0.0118813 0.999929i \(-0.503782\pi\)
−0.0118813 + 0.999929i \(0.503782\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) −1.69187 + 2.93041i −0.246785 + 0.427445i −0.962632 0.270813i \(-0.912708\pi\)
0.715847 + 0.698257i \(0.246041\pi\)
\(48\) 0 0
\(49\) −7.45103 12.9056i −1.06443 1.84365i
\(50\) 0 0
\(51\) 0.405657 + 0.702619i 0.0568034 + 0.0983864i
\(52\) 0 0
\(53\) −0.736649 −0.101187 −0.0505933 0.998719i \(-0.516111\pi\)
−0.0505933 + 0.998719i \(0.516111\pi\)
\(54\) 0 0
\(55\) −2.04957 + 3.54996i −0.276364 + 0.478676i
\(56\) 0 0
\(57\) −1.42209 2.46313i −0.188360 0.326250i
\(58\) 0 0
\(59\) 3.35790 0.437162 0.218581 0.975819i \(-0.429857\pi\)
0.218581 + 0.975819i \(0.429857\pi\)
\(60\) 0 0
\(61\) 0.373303 + 0.646580i 0.0477966 + 0.0827861i 0.888934 0.458035i \(-0.151447\pi\)
−0.841137 + 0.540822i \(0.818113\pi\)
\(62\) 0 0
\(63\) −2.33998 + 4.05297i −0.294810 + 0.510626i
\(64\) 0 0
\(65\) −0.500000 0.866025i −0.0620174 0.107417i
\(66\) 0 0
\(67\) −8.03665 1.55314i −0.981833 0.189746i
\(68\) 0 0
\(69\) 2.18790 + 3.78956i 0.263393 + 0.456209i
\(70\) 0 0
\(71\) 1.60329 2.77699i 0.190276 0.329568i −0.755066 0.655649i \(-0.772395\pi\)
0.945342 + 0.326082i \(0.105728\pi\)
\(72\) 0 0
\(73\) 4.76087 + 8.24608i 0.557218 + 0.965130i 0.997727 + 0.0673821i \(0.0214646\pi\)
−0.440509 + 0.897748i \(0.645202\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 9.59191 + 16.6137i 1.09310 + 1.89330i
\(78\) 0 0
\(79\) −5.82437 + 10.0881i −0.655293 + 1.13500i 0.326527 + 0.945188i \(0.394122\pi\)
−0.981820 + 0.189813i \(0.939212\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 6.18900 + 10.7197i 0.679331 + 1.17664i 0.975183 + 0.221402i \(0.0710632\pi\)
−0.295852 + 0.955234i \(0.595603\pi\)
\(84\) 0 0
\(85\) −0.405657 0.702619i −0.0439997 0.0762097i
\(86\) 0 0
\(87\) −4.40150 + 7.62362i −0.471890 + 0.817338i
\(88\) 0 0
\(89\) −13.1087 −1.38952 −0.694761 0.719241i \(-0.744490\pi\)
−0.694761 + 0.719241i \(0.744490\pi\)
\(90\) 0 0
\(91\) −4.67996 −0.490593
\(92\) 0 0
\(93\) 0.131676 0.228069i 0.0136541 0.0236496i
\(94\) 0 0
\(95\) 1.42209 + 2.46313i 0.145903 + 0.252712i
\(96\) 0 0
\(97\) −9.60393 16.6345i −0.975131 1.68898i −0.679501 0.733675i \(-0.737804\pi\)
−0.295631 0.955302i \(-0.595530\pi\)
\(98\) 0 0
\(99\) 2.04957 3.54996i 0.205989 0.356784i
\(100\) 0 0
\(101\) −1.00397 + 1.73893i −0.0998989 + 0.173030i −0.911643 0.410984i \(-0.865185\pi\)
0.811744 + 0.584014i \(0.198519\pi\)
\(102\) 0 0
\(103\) 5.39205 9.33931i 0.531295 0.920229i −0.468038 0.883708i \(-0.655039\pi\)
0.999333 0.0365212i \(-0.0116276\pi\)
\(104\) 0 0
\(105\) 2.33998 4.05297i 0.228359 0.395529i
\(106\) 0 0
\(107\) −7.27630 −0.703426 −0.351713 0.936108i \(-0.614401\pi\)
−0.351713 + 0.936108i \(0.614401\pi\)
\(108\) 0 0
\(109\) −9.59298 −0.918840 −0.459420 0.888219i \(-0.651943\pi\)
−0.459420 + 0.888219i \(0.651943\pi\)
\(110\) 0 0
\(111\) −0.572659 0.991874i −0.0543544 0.0941445i
\(112\) 0 0
\(113\) 1.13168 1.96012i 0.106459 0.184393i −0.807874 0.589355i \(-0.799382\pi\)
0.914333 + 0.404962i \(0.132715\pi\)
\(114\) 0 0
\(115\) −2.18790 3.78956i −0.204023 0.353378i
\(116\) 0 0
\(117\) 0.500000 + 0.866025i 0.0462250 + 0.0800641i
\(118\) 0 0
\(119\) −3.79692 −0.348063
\(120\) 0 0
\(121\) −2.90146 5.02548i −0.263769 0.456862i
\(122\) 0 0
\(123\) −1.81958 + 3.15160i −0.164066 + 0.284170i
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −3.47244 + 6.01444i −0.308129 + 0.533695i −0.977953 0.208825i \(-0.933036\pi\)
0.669824 + 0.742520i \(0.266370\pi\)
\(128\) 0 0
\(129\) −0.155822 −0.0137194
\(130\) 0 0
\(131\) −17.3695 −1.51758 −0.758791 0.651334i \(-0.774210\pi\)
−0.758791 + 0.651334i \(0.774210\pi\)
\(132\) 0 0
\(133\) 13.3106 1.15418
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 12.9359 1.10519 0.552595 0.833450i \(-0.313638\pi\)
0.552595 + 0.833450i \(0.313638\pi\)
\(138\) 0 0
\(139\) −20.1973 −1.71311 −0.856555 0.516055i \(-0.827400\pi\)
−0.856555 + 0.516055i \(0.827400\pi\)
\(140\) 0 0
\(141\) −1.69187 + 2.93041i −0.142482 + 0.246785i
\(142\) 0 0
\(143\) 4.09914 0.342787
\(144\) 0 0
\(145\) 4.40150 7.62362i 0.365525 0.633107i
\(146\) 0 0
\(147\) −7.45103 12.9056i −0.614551 1.06443i
\(148\) 0 0
\(149\) −23.1440 −1.89603 −0.948013 0.318231i \(-0.896911\pi\)
−0.948013 + 0.318231i \(0.896911\pi\)
\(150\) 0 0
\(151\) 6.32638 + 10.9576i 0.514833 + 0.891717i 0.999852 + 0.0172135i \(0.00547950\pi\)
−0.485019 + 0.874504i \(0.661187\pi\)
\(152\) 0 0
\(153\) 0.405657 + 0.702619i 0.0327955 + 0.0568034i
\(154\) 0 0
\(155\) −0.131676 + 0.228069i −0.0105764 + 0.0183189i
\(156\) 0 0
\(157\) −1.72579 2.98916i −0.137733 0.238561i 0.788905 0.614515i \(-0.210648\pi\)
−0.926638 + 0.375954i \(0.877315\pi\)
\(158\) 0 0
\(159\) −0.736649 −0.0584201
\(160\) 0 0
\(161\) −20.4786 −1.61394
\(162\) 0 0
\(163\) −10.3321 + 17.8958i −0.809275 + 1.40170i 0.104092 + 0.994568i \(0.466806\pi\)
−0.913367 + 0.407137i \(0.866527\pi\)
\(164\) 0 0
\(165\) −2.04957 + 3.54996i −0.159559 + 0.276364i
\(166\) 0 0
\(167\) −12.4986 + 21.6483i −0.967173 + 1.67519i −0.263513 + 0.964656i \(0.584881\pi\)
−0.703660 + 0.710537i \(0.748452\pi\)
\(168\) 0 0
\(169\) 6.00000 10.3923i 0.461538 0.799408i
\(170\) 0 0
\(171\) −1.42209 2.46313i −0.108750 0.188360i
\(172\) 0 0
\(173\) 5.90917 + 10.2350i 0.449266 + 0.778151i 0.998338 0.0576233i \(-0.0183522\pi\)
−0.549072 + 0.835775i \(0.685019\pi\)
\(174\) 0 0
\(175\) −2.33998 + 4.05297i −0.176886 + 0.306376i
\(176\) 0 0
\(177\) 3.35790 0.252396
\(178\) 0 0
\(179\) −13.1659 −0.984063 −0.492032 0.870577i \(-0.663746\pi\)
−0.492032 + 0.870577i \(0.663746\pi\)
\(180\) 0 0
\(181\) −0.319852 + 0.553999i −0.0237744 + 0.0411785i −0.877668 0.479269i \(-0.840902\pi\)
0.853893 + 0.520448i \(0.174235\pi\)
\(182\) 0 0
\(183\) 0.373303 + 0.646580i 0.0275954 + 0.0477966i
\(184\) 0 0
\(185\) 0.572659 + 0.991874i 0.0421027 + 0.0729240i
\(186\) 0 0
\(187\) 3.32569 0.243199
\(188\) 0 0
\(189\) −2.33998 + 4.05297i −0.170209 + 0.294810i
\(190\) 0 0
\(191\) −8.19598 14.1959i −0.593040 1.02718i −0.993820 0.111002i \(-0.964594\pi\)
0.400780 0.916174i \(-0.368739\pi\)
\(192\) 0 0
\(193\) 9.88217 0.711334 0.355667 0.934613i \(-0.384254\pi\)
0.355667 + 0.934613i \(0.384254\pi\)
\(194\) 0 0
\(195\) −0.500000 0.866025i −0.0358057 0.0620174i
\(196\) 0 0
\(197\) −1.30338 + 2.25751i −0.0928617 + 0.160841i −0.908714 0.417419i \(-0.862935\pi\)
0.815852 + 0.578260i \(0.196268\pi\)
\(198\) 0 0
\(199\) −9.81017 16.9917i −0.695424 1.20451i −0.970037 0.242956i \(-0.921883\pi\)
0.274613 0.961555i \(-0.411450\pi\)
\(200\) 0 0
\(201\) −8.03665 1.55314i −0.566862 0.109550i
\(202\) 0 0
\(203\) −20.5989 35.6783i −1.44576 2.50412i
\(204\) 0 0
\(205\) 1.81958 3.15160i 0.127085 0.220117i
\(206\) 0 0
\(207\) 2.18790 + 3.78956i 0.152070 + 0.263393i
\(208\) 0 0
\(209\) −11.6587 −0.806447
\(210\) 0 0
\(211\) 5.30716 + 9.19227i 0.365360 + 0.632822i 0.988834 0.149022i \(-0.0476126\pi\)
−0.623474 + 0.781844i \(0.714279\pi\)
\(212\) 0 0
\(213\) 1.60329 2.77699i 0.109856 0.190276i
\(214\) 0 0
\(215\) 0.155822 0.0106270
\(216\) 0 0
\(217\) 0.616237 + 1.06735i 0.0418329 + 0.0724567i
\(218\) 0 0
\(219\) 4.76087 + 8.24608i 0.321710 + 0.557218i
\(220\) 0 0
\(221\) −0.405657 + 0.702619i −0.0272875 + 0.0472633i
\(222\) 0 0
\(223\) 7.92598 0.530763 0.265382 0.964143i \(-0.414502\pi\)
0.265382 + 0.964143i \(0.414502\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −5.84617 + 10.1259i −0.388024 + 0.672077i −0.992184 0.124786i \(-0.960176\pi\)
0.604160 + 0.796863i \(0.293509\pi\)
\(228\) 0 0
\(229\) 12.1270 + 21.0045i 0.801374 + 1.38802i 0.918712 + 0.394928i \(0.129230\pi\)
−0.117339 + 0.993092i \(0.537436\pi\)
\(230\) 0 0
\(231\) 9.59191 + 16.6137i 0.631101 + 1.09310i
\(232\) 0 0
\(233\) −1.42684 + 2.47137i −0.0934756 + 0.161905i −0.908971 0.416859i \(-0.863131\pi\)
0.815496 + 0.578763i \(0.196464\pi\)
\(234\) 0 0
\(235\) 1.69187 2.93041i 0.110366 0.191159i
\(236\) 0 0
\(237\) −5.82437 + 10.0881i −0.378334 + 0.655293i
\(238\) 0 0
\(239\) 1.17038 2.02715i 0.0757053 0.131125i −0.825687 0.564128i \(-0.809213\pi\)
0.901393 + 0.433002i \(0.142546\pi\)
\(240\) 0 0
\(241\) −0.784474 −0.0505324 −0.0252662 0.999681i \(-0.508043\pi\)
−0.0252662 + 0.999681i \(0.508043\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 7.45103 + 12.9056i 0.476029 + 0.824506i
\(246\) 0 0
\(247\) 1.42209 2.46313i 0.0904854 0.156725i
\(248\) 0 0
\(249\) 6.18900 + 10.7197i 0.392212 + 0.679331i
\(250\) 0 0
\(251\) 13.7408 + 23.7998i 0.867311 + 1.50223i 0.864734 + 0.502231i \(0.167487\pi\)
0.00257757 + 0.999997i \(0.499180\pi\)
\(252\) 0 0
\(253\) 17.9370 1.12769
\(254\) 0 0
\(255\) −0.405657 0.702619i −0.0254032 0.0439997i
\(256\) 0 0
\(257\) 11.3528 19.6637i 0.708169 1.22658i −0.257366 0.966314i \(-0.582855\pi\)
0.965536 0.260271i \(-0.0838119\pi\)
\(258\) 0 0
\(259\) 5.36004 0.333057
\(260\) 0 0
\(261\) −4.40150 + 7.62362i −0.272446 + 0.471890i
\(262\) 0 0
\(263\) 4.16533 0.256845 0.128423 0.991720i \(-0.459009\pi\)
0.128423 + 0.991720i \(0.459009\pi\)
\(264\) 0 0
\(265\) 0.736649 0.0452520
\(266\) 0 0
\(267\) −13.1087 −0.802241
\(268\) 0 0
\(269\) −15.9014 −0.969527 −0.484764 0.874645i \(-0.661094\pi\)
−0.484764 + 0.874645i \(0.661094\pi\)
\(270\) 0 0
\(271\) 27.3621 1.66213 0.831065 0.556175i \(-0.187732\pi\)
0.831065 + 0.556175i \(0.187732\pi\)
\(272\) 0 0
\(273\) −4.67996 −0.283244
\(274\) 0 0
\(275\) 2.04957 3.54996i 0.123594 0.214070i
\(276\) 0 0
\(277\) −13.3919 −0.804640 −0.402320 0.915499i \(-0.631796\pi\)
−0.402320 + 0.915499i \(0.631796\pi\)
\(278\) 0 0
\(279\) 0.131676 0.228069i 0.00788321 0.0136541i
\(280\) 0 0
\(281\) 12.4785 + 21.6133i 0.744402 + 1.28934i 0.950474 + 0.310806i \(0.100599\pi\)
−0.206071 + 0.978537i \(0.566068\pi\)
\(282\) 0 0
\(283\) −0.580906 −0.0345313 −0.0172656 0.999851i \(-0.505496\pi\)
−0.0172656 + 0.999851i \(0.505496\pi\)
\(284\) 0 0
\(285\) 1.42209 + 2.46313i 0.0842373 + 0.145903i
\(286\) 0 0
\(287\) −8.51556 14.7494i −0.502658 0.870629i
\(288\) 0 0
\(289\) 8.17088 14.1524i 0.480640 0.832493i
\(290\) 0 0
\(291\) −9.60393 16.6345i −0.562992 0.975131i
\(292\) 0 0
\(293\) −9.08517 −0.530761 −0.265381 0.964144i \(-0.585498\pi\)
−0.265381 + 0.964144i \(0.585498\pi\)
\(294\) 0 0
\(295\) −3.35790 −0.195505
\(296\) 0 0
\(297\) 2.04957 3.54996i 0.118928 0.205989i
\(298\) 0 0
\(299\) −2.18790 + 3.78956i −0.126530 + 0.219156i
\(300\) 0 0
\(301\) 0.364621 0.631542i 0.0210164 0.0364015i
\(302\) 0 0
\(303\) −1.00397 + 1.73893i −0.0576767 + 0.0998989i
\(304\) 0 0
\(305\) −0.373303 0.646580i −0.0213753 0.0370231i
\(306\) 0 0
\(307\) 6.85294 + 11.8696i 0.391118 + 0.677437i 0.992597 0.121452i \(-0.0387550\pi\)
−0.601479 + 0.798889i \(0.705422\pi\)
\(308\) 0 0
\(309\) 5.39205 9.33931i 0.306743 0.531295i
\(310\) 0 0
\(311\) 11.5669 0.655901 0.327950 0.944695i \(-0.393642\pi\)
0.327950 + 0.944695i \(0.393642\pi\)
\(312\) 0 0
\(313\) 14.6871 0.830163 0.415081 0.909784i \(-0.363753\pi\)
0.415081 + 0.909784i \(0.363753\pi\)
\(314\) 0 0
\(315\) 2.33998 4.05297i 0.131843 0.228359i
\(316\) 0 0
\(317\) 2.18470 + 3.78401i 0.122705 + 0.212531i 0.920834 0.389956i \(-0.127510\pi\)
−0.798128 + 0.602487i \(0.794176\pi\)
\(318\) 0 0
\(319\) 18.0424 + 31.2503i 1.01018 + 1.74968i
\(320\) 0 0
\(321\) −7.27630 −0.406123
\(322\) 0 0
\(323\) 1.15376 1.99837i 0.0641970 0.111192i
\(324\) 0 0
\(325\) 0.500000 + 0.866025i 0.0277350 + 0.0480384i
\(326\) 0 0
\(327\) −9.59298 −0.530493
\(328\) 0 0
\(329\) −7.91791 13.7142i −0.436529 0.756090i
\(330\) 0 0
\(331\) 5.15199 8.92351i 0.283179 0.490480i −0.688987 0.724774i \(-0.741944\pi\)
0.972166 + 0.234293i \(0.0752776\pi\)
\(332\) 0 0
\(333\) −0.572659 0.991874i −0.0313815 0.0543544i
\(334\) 0 0
\(335\) 8.03665 + 1.55314i 0.439089 + 0.0848572i
\(336\) 0 0
\(337\) 1.87626 + 3.24978i 0.102207 + 0.177027i 0.912593 0.408868i \(-0.134076\pi\)
−0.810387 + 0.585895i \(0.800743\pi\)
\(338\) 0 0
\(339\) 1.13168 1.96012i 0.0614642 0.106459i
\(340\) 0 0
\(341\) −0.539756 0.934885i −0.0292294 0.0506269i
\(342\) 0 0
\(343\) 36.9813 1.99681
\(344\) 0 0
\(345\) −2.18790 3.78956i −0.117793 0.204023i
\(346\) 0 0
\(347\) −0.308623 + 0.534551i −0.0165678 + 0.0286962i −0.874190 0.485583i \(-0.838607\pi\)
0.857623 + 0.514279i \(0.171941\pi\)
\(348\) 0 0
\(349\) 19.1107 1.02297 0.511485 0.859292i \(-0.329095\pi\)
0.511485 + 0.859292i \(0.329095\pi\)
\(350\) 0 0
\(351\) 0.500000 + 0.866025i 0.0266880 + 0.0462250i
\(352\) 0 0
\(353\) −5.37455 9.30900i −0.286059 0.495468i 0.686807 0.726840i \(-0.259012\pi\)
−0.972865 + 0.231372i \(0.925679\pi\)
\(354\) 0 0
\(355\) −1.60329 + 2.77699i −0.0850940 + 0.147387i
\(356\) 0 0
\(357\) −3.79692 −0.200954
\(358\) 0 0
\(359\) 10.3998 0.548880 0.274440 0.961604i \(-0.411508\pi\)
0.274440 + 0.961604i \(0.411508\pi\)
\(360\) 0 0
\(361\) 5.45533 9.44890i 0.287122 0.497311i
\(362\) 0 0
\(363\) −2.90146 5.02548i −0.152287 0.263769i
\(364\) 0 0
\(365\) −4.76087 8.24608i −0.249196 0.431619i
\(366\) 0 0
\(367\) −8.99567 + 15.5810i −0.469570 + 0.813319i −0.999395 0.0347879i \(-0.988924\pi\)
0.529825 + 0.848107i \(0.322258\pi\)
\(368\) 0 0
\(369\) −1.81958 + 3.15160i −0.0947235 + 0.164066i
\(370\) 0 0
\(371\) 1.72375 2.98561i 0.0894924 0.155005i
\(372\) 0 0
\(373\) 4.92391 8.52846i 0.254950 0.441587i −0.709932 0.704271i \(-0.751274\pi\)
0.964882 + 0.262683i \(0.0846075\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −8.80300 −0.453378
\(378\) 0 0
\(379\) 16.7497 + 29.0113i 0.860374 + 1.49021i 0.871568 + 0.490275i \(0.163104\pi\)
−0.0111935 + 0.999937i \(0.503563\pi\)
\(380\) 0 0
\(381\) −3.47244 + 6.01444i −0.177898 + 0.308129i
\(382\) 0 0
\(383\) −6.55000 11.3449i −0.334690 0.579699i 0.648736 0.761014i \(-0.275298\pi\)
−0.983425 + 0.181315i \(0.941965\pi\)
\(384\) 0 0
\(385\) −9.59191 16.6137i −0.488849 0.846711i
\(386\) 0 0
\(387\) −0.155822 −0.00792088
\(388\) 0 0
\(389\) 2.20799 + 3.82436i 0.111950 + 0.193903i 0.916556 0.399906i \(-0.130957\pi\)
−0.804607 + 0.593808i \(0.797624\pi\)
\(390\) 0 0
\(391\) −1.77508 + 3.07452i −0.0897695 + 0.155485i
\(392\) 0 0
\(393\) −17.3695 −0.876176
\(394\) 0 0
\(395\) 5.82437 10.0881i 0.293056 0.507588i
\(396\) 0 0
\(397\) 21.9026 1.09926 0.549631 0.835408i \(-0.314768\pi\)
0.549631 + 0.835408i \(0.314768\pi\)
\(398\) 0 0
\(399\) 13.3106 0.666366
\(400\) 0 0
\(401\) −17.5467 −0.876239 −0.438120 0.898917i \(-0.644355\pi\)
−0.438120 + 0.898917i \(0.644355\pi\)
\(402\) 0 0
\(403\) 0.263351 0.0131185
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −4.69481 −0.232713
\(408\) 0 0
\(409\) 5.41748 9.38335i 0.267877 0.463977i −0.700436 0.713715i \(-0.747011\pi\)
0.968313 + 0.249738i \(0.0803445\pi\)
\(410\) 0 0
\(411\) 12.9359 0.638082
\(412\) 0 0
\(413\) −7.85743 + 13.6095i −0.386639 + 0.669678i
\(414\) 0 0
\(415\) −6.18900 10.7197i −0.303806 0.526207i
\(416\) 0 0
\(417\) −20.1973 −0.989065
\(418\) 0 0
\(419\) 18.9375 + 32.8007i 0.925158 + 1.60242i 0.791308 + 0.611418i \(0.209401\pi\)
0.133850 + 0.991002i \(0.457266\pi\)
\(420\) 0 0
\(421\) 3.30128 + 5.71799i 0.160895 + 0.278678i 0.935190 0.354147i \(-0.115229\pi\)
−0.774295 + 0.632825i \(0.781895\pi\)
\(422\) 0 0
\(423\) −1.69187 + 2.93041i −0.0822618 + 0.142482i
\(424\) 0 0
\(425\) 0.405657 + 0.702619i 0.0196773 + 0.0340820i
\(426\) 0 0
\(427\) −3.49409 −0.169091
\(428\) 0 0
\(429\) 4.09914 0.197908
\(430\) 0 0
\(431\) 7.84252 13.5836i 0.377761 0.654301i −0.612975 0.790102i \(-0.710027\pi\)
0.990736 + 0.135801i \(0.0433607\pi\)
\(432\) 0 0
\(433\) 6.41965 11.1192i 0.308509 0.534353i −0.669528 0.742787i \(-0.733503\pi\)
0.978036 + 0.208434i \(0.0668367\pi\)
\(434\) 0 0
\(435\) 4.40150 7.62362i 0.211036 0.365525i
\(436\) 0 0
\(437\) 6.22278 10.7782i 0.297676 0.515590i
\(438\) 0 0
\(439\) −17.7785 30.7932i −0.848519 1.46968i −0.882530 0.470257i \(-0.844161\pi\)
0.0340103 0.999421i \(-0.489172\pi\)
\(440\) 0 0
\(441\) −7.45103 12.9056i −0.354811 0.614551i
\(442\) 0 0
\(443\) 8.25994 14.3066i 0.392442 0.679729i −0.600329 0.799753i \(-0.704964\pi\)
0.992771 + 0.120024i \(0.0382971\pi\)
\(444\) 0 0
\(445\) 13.1087 0.621413
\(446\) 0 0
\(447\) −23.1440 −1.09467
\(448\) 0 0
\(449\) −3.80900 + 6.59738i −0.179758 + 0.311349i −0.941797 0.336181i \(-0.890865\pi\)
0.762040 + 0.647530i \(0.224198\pi\)
\(450\) 0 0
\(451\) 7.45870 + 12.9188i 0.351216 + 0.608325i
\(452\) 0 0
\(453\) 6.32638 + 10.9576i 0.297239 + 0.514833i
\(454\) 0 0
\(455\) 4.67996 0.219400
\(456\) 0 0
\(457\) −1.66587 + 2.88537i −0.0779260 + 0.134972i −0.902355 0.430994i \(-0.858163\pi\)
0.824429 + 0.565966i \(0.191496\pi\)
\(458\) 0 0
\(459\) 0.405657 + 0.702619i 0.0189345 + 0.0327955i
\(460\) 0 0
\(461\) 13.8604 0.645544 0.322772 0.946477i \(-0.395385\pi\)
0.322772 + 0.946477i \(0.395385\pi\)
\(462\) 0 0
\(463\) −7.46192 12.9244i −0.346785 0.600649i 0.638891 0.769297i \(-0.279393\pi\)
−0.985676 + 0.168648i \(0.946060\pi\)
\(464\) 0 0
\(465\) −0.131676 + 0.228069i −0.00610631 + 0.0105764i
\(466\) 0 0
\(467\) 3.86736 + 6.69846i 0.178960 + 0.309968i 0.941525 0.336944i \(-0.109393\pi\)
−0.762565 + 0.646912i \(0.776060\pi\)
\(468\) 0 0
\(469\) 25.1004 28.9380i 1.15903 1.33623i
\(470\) 0 0
\(471\) −1.72579 2.98916i −0.0795204 0.137733i
\(472\) 0 0
\(473\) −0.319368 + 0.553162i −0.0146846 + 0.0254344i
\(474\) 0 0
\(475\) −1.42209 2.46313i −0.0652499 0.113016i
\(476\) 0 0
\(477\) −0.736649 −0.0337288
\(478\) 0 0
\(479\) −0.447058 0.774327i −0.0204266 0.0353799i 0.855631 0.517586i \(-0.173169\pi\)
−0.876058 + 0.482206i \(0.839836\pi\)
\(480\) 0 0
\(481\) 0.572659 0.991874i 0.0261110 0.0452256i
\(482\) 0 0
\(483\) −20.4786 −0.931809
\(484\) 0 0
\(485\) 9.60393 + 16.6345i 0.436092 + 0.755334i
\(486\) 0 0
\(487\) 3.67507 + 6.36541i 0.166533 + 0.288444i 0.937199 0.348796i \(-0.113409\pi\)
−0.770665 + 0.637240i \(0.780076\pi\)
\(488\) 0 0
\(489\) −10.3321 + 17.8958i −0.467235 + 0.809275i
\(490\) 0 0
\(491\) −19.0899 −0.861516 −0.430758 0.902467i \(-0.641754\pi\)
−0.430758 + 0.902467i \(0.641754\pi\)
\(492\) 0 0
\(493\) −7.14200 −0.321660
\(494\) 0 0
\(495\) −2.04957 + 3.54996i −0.0921212 + 0.159559i
\(496\) 0 0
\(497\) 7.50335 + 12.9962i 0.336571 + 0.582959i
\(498\) 0 0
\(499\) 3.47567 + 6.02004i 0.155593 + 0.269494i 0.933275 0.359164i \(-0.116938\pi\)
−0.777682 + 0.628658i \(0.783605\pi\)
\(500\) 0 0
\(501\) −12.4986 + 21.6483i −0.558398 + 0.967173i
\(502\) 0 0
\(503\) 3.11221 5.39050i 0.138767 0.240351i −0.788263 0.615338i \(-0.789020\pi\)
0.927030 + 0.374987i \(0.122353\pi\)
\(504\) 0 0
\(505\) 1.00397 1.73893i 0.0446762 0.0773814i
\(506\) 0 0
\(507\) 6.00000 10.3923i 0.266469 0.461538i
\(508\) 0 0
\(509\) 8.49202 0.376402 0.188201 0.982131i \(-0.439734\pi\)
0.188201 + 0.982131i \(0.439734\pi\)
\(510\) 0 0
\(511\) −44.5614 −1.97128
\(512\) 0 0
\(513\) −1.42209 2.46313i −0.0627868 0.108750i
\(514\) 0 0
\(515\) −5.39205 + 9.33931i −0.237602 + 0.411539i
\(516\) 0 0
\(517\) 6.93523 + 12.0122i 0.305011 + 0.528294i
\(518\) 0 0
\(519\) 5.90917 + 10.2350i 0.259384 + 0.449266i
\(520\) 0 0
\(521\) −13.7233 −0.601229 −0.300615 0.953746i \(-0.597192\pi\)
−0.300615 + 0.953746i \(0.597192\pi\)
\(522\) 0 0
\(523\) −11.0672 19.1690i −0.483935 0.838201i 0.515894 0.856652i \(-0.327460\pi\)
−0.999830 + 0.0184516i \(0.994126\pi\)
\(524\) 0 0
\(525\) −2.33998 + 4.05297i −0.102125 + 0.176886i
\(526\) 0 0
\(527\) 0.213661 0.00930720
\(528\) 0 0
\(529\) 1.92616 3.33621i 0.0837463 0.145053i
\(530\) 0 0
\(531\) 3.35790 0.145721
\(532\) 0 0
\(533\) −3.63916 −0.157629
\(534\) 0 0
\(535\) 7.27630 0.314582
\(536\) 0 0
\(537\) −13.1659 −0.568149
\(538\) 0 0
\(539\) −61.0856 −2.63114
\(540\) 0 0
\(541\) −12.9506 −0.556790 −0.278395 0.960467i \(-0.589802\pi\)
−0.278395 + 0.960467i \(0.589802\pi\)
\(542\) 0 0
\(543\) −0.319852 + 0.553999i −0.0137262 + 0.0237744i
\(544\) 0 0
\(545\) 9.59298 0.410918
\(546\) 0 0
\(547\) 3.02003 5.23084i 0.129127 0.223655i −0.794212 0.607641i \(-0.792116\pi\)
0.923339 + 0.383987i \(0.125449\pi\)
\(548\) 0 0
\(549\) 0.373303 + 0.646580i 0.0159322 + 0.0275954i
\(550\) 0 0
\(551\) 25.0373 1.06662
\(552\) 0 0
\(553\) −27.2579 47.2120i −1.15912 2.00766i
\(554\) 0 0
\(555\) 0.572659 + 0.991874i 0.0243080 + 0.0421027i
\(556\) 0 0
\(557\) −8.77286 + 15.1950i −0.371718 + 0.643835i −0.989830 0.142256i \(-0.954565\pi\)
0.618112 + 0.786090i \(0.287898\pi\)
\(558\) 0 0
\(559\) −0.0779111 0.134946i −0.00329529 0.00570760i
\(560\) 0 0
\(561\) 3.32569 0.140411
\(562\) 0 0
\(563\) 6.35200 0.267705 0.133853 0.991001i \(-0.457265\pi\)
0.133853 + 0.991001i \(0.457265\pi\)
\(564\) 0 0
\(565\) −1.13168 + 1.96012i −0.0476099 + 0.0824628i
\(566\) 0 0
\(567\) −2.33998 + 4.05297i −0.0982700 + 0.170209i
\(568\) 0 0
\(569\) 10.8063 18.7171i 0.453026 0.784663i −0.545547 0.838080i \(-0.683678\pi\)
0.998572 + 0.0534170i \(0.0170113\pi\)
\(570\) 0 0
\(571\) 11.1088 19.2410i 0.464889 0.805212i −0.534307 0.845290i \(-0.679427\pi\)
0.999197 + 0.0400783i \(0.0127608\pi\)
\(572\) 0 0
\(573\) −8.19598 14.1959i −0.342392 0.593040i
\(574\) 0 0
\(575\) 2.18790 + 3.78956i 0.0912418 + 0.158036i
\(576\) 0 0
\(577\) 3.02464 5.23883i 0.125917 0.218095i −0.796174 0.605068i \(-0.793146\pi\)
0.922091 + 0.386973i \(0.126479\pi\)
\(578\) 0 0
\(579\) 9.88217 0.410689
\(580\) 0 0
\(581\) −57.9286 −2.40328
\(582\) 0 0
\(583\) −1.50981 + 2.61507i −0.0625300 + 0.108305i
\(584\) 0 0
\(585\) −0.500000 0.866025i −0.0206725 0.0358057i
\(586\) 0 0
\(587\) 22.1727 + 38.4042i 0.915164 + 1.58511i 0.806661 + 0.591014i \(0.201272\pi\)
0.108502 + 0.994096i \(0.465395\pi\)
\(588\) 0 0
\(589\) −0.749017 −0.0308627
\(590\) 0 0
\(591\) −1.30338 + 2.25751i −0.0536137 + 0.0928617i
\(592\) 0 0
\(593\) −18.1509 31.4383i −0.745369 1.29102i −0.950022 0.312183i \(-0.898940\pi\)
0.204653 0.978835i \(-0.434393\pi\)
\(594\) 0 0
\(595\) 3.79692 0.155659
\(596\) 0 0
\(597\) −9.81017 16.9917i −0.401504 0.695424i
\(598\) 0 0
\(599\) −11.1631 + 19.3350i −0.456110 + 0.790006i −0.998751 0.0499585i \(-0.984091\pi\)
0.542641 + 0.839965i \(0.317424\pi\)
\(600\) 0 0
\(601\) −11.3273 19.6195i −0.462052 0.800297i 0.537011 0.843575i \(-0.319553\pi\)
−0.999063 + 0.0432779i \(0.986220\pi\)
\(602\) 0 0
\(603\) −8.03665 1.55314i −0.327278 0.0632488i
\(604\) 0 0
\(605\) 2.90146 + 5.02548i 0.117961 + 0.204315i
\(606\) 0 0
\(607\) 0.378766 0.656041i 0.0153736 0.0266279i −0.858236 0.513255i \(-0.828440\pi\)
0.873610 + 0.486627i \(0.161773\pi\)
\(608\) 0 0
\(609\) −20.5989 35.6783i −0.834708 1.44576i
\(610\) 0 0
\(611\) −3.38375 −0.136892
\(612\) 0 0
\(613\) −13.3625 23.1445i −0.539706 0.934799i −0.998920 0.0464728i \(-0.985202\pi\)
0.459213 0.888326i \(-0.348131\pi\)
\(614\) 0 0
\(615\) 1.81958 3.15160i 0.0733725 0.127085i
\(616\) 0 0
\(617\) 2.14925 0.0865256 0.0432628 0.999064i \(-0.486225\pi\)
0.0432628 + 0.999064i \(0.486225\pi\)
\(618\) 0 0
\(619\) −0.592777 1.02672i −0.0238257 0.0412674i 0.853867 0.520492i \(-0.174251\pi\)
−0.877692 + 0.479224i \(0.840918\pi\)
\(620\) 0 0
\(621\) 2.18790 + 3.78956i 0.0877975 + 0.152070i
\(622\) 0 0
\(623\) 30.6742 53.1292i 1.22893 2.12858i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −11.6587 −0.465603
\(628\) 0 0
\(629\) 0.464606 0.804722i 0.0185251 0.0320864i
\(630\) 0 0
\(631\) 3.64522 + 6.31371i 0.145114 + 0.251345i 0.929415 0.369035i \(-0.120312\pi\)
−0.784301 + 0.620380i \(0.786978\pi\)
\(632\) 0 0
\(633\) 5.30716 + 9.19227i 0.210941 + 0.365360i
\(634\) 0 0
\(635\) 3.47244 6.01444i 0.137800 0.238676i
\(636\) 0 0
\(637\) 7.45103 12.9056i 0.295221 0.511337i
\(638\) 0 0
\(639\) 1.60329 2.77699i 0.0634253 0.109856i
\(640\) 0 0
\(641\) 14.0975 24.4176i 0.556817 0.964435i −0.440943 0.897535i \(-0.645356\pi\)
0.997760 0.0669002i \(-0.0213109\pi\)
\(642\) 0 0
\(643\) 41.0318 1.61814 0.809069 0.587714i \(-0.199972\pi\)
0.809069 + 0.587714i \(0.199972\pi\)
\(644\) 0 0
\(645\) 0.155822 0.00613549
\(646\) 0 0
\(647\) 24.6239 + 42.6498i 0.968064 + 1.67674i 0.701149 + 0.713015i \(0.252671\pi\)
0.266915 + 0.963720i \(0.413996\pi\)
\(648\) 0 0
\(649\) 6.88225 11.9204i 0.270152 0.467917i
\(650\) 0 0
\(651\) 0.616237 + 1.06735i 0.0241522 + 0.0418329i
\(652\) 0 0
\(653\) −9.95225 17.2378i −0.389462 0.674567i 0.602915 0.797805i \(-0.294006\pi\)
−0.992377 + 0.123238i \(0.960672\pi\)
\(654\) 0 0
\(655\) 17.3695 0.678683
\(656\) 0 0
\(657\) 4.76087 + 8.24608i 0.185739 + 0.321710i
\(658\) 0 0
\(659\) −11.6665 + 20.2070i −0.454462 + 0.787151i −0.998657 0.0518074i \(-0.983502\pi\)
0.544195 + 0.838959i \(0.316835\pi\)
\(660\) 0 0
\(661\) −3.67541 −0.142957 −0.0714784 0.997442i \(-0.522772\pi\)
−0.0714784 + 0.997442i \(0.522772\pi\)
\(662\) 0 0
\(663\) −0.405657 + 0.702619i −0.0157544 + 0.0272875i
\(664\) 0 0
\(665\) −13.3106 −0.516165
\(666\) 0 0
\(667\) −38.5202 −1.49151
\(668\) 0 0
\(669\) 7.92598 0.306436
\(670\) 0 0
\(671\) 3.06044 0.118147
\(672\) 0 0
\(673\) −22.0122 −0.848509 −0.424254 0.905543i \(-0.639464\pi\)
−0.424254 + 0.905543i \(0.639464\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 4.43919 7.68890i 0.170612 0.295508i −0.768022 0.640423i \(-0.778759\pi\)
0.938634 + 0.344915i \(0.112092\pi\)
\(678\) 0 0
\(679\) 89.8921 3.44974
\(680\) 0 0
\(681\) −5.84617 + 10.1259i −0.224026 + 0.388024i
\(682\) 0 0
\(683\) 11.5061 + 19.9292i 0.440269 + 0.762569i 0.997709 0.0676485i \(-0.0215496\pi\)
−0.557440 + 0.830217i \(0.688216\pi\)
\(684\) 0 0
\(685\) −12.9359 −0.494256
\(686\) 0 0
\(687\) 12.1270 + 21.0045i 0.462673 + 0.801374i
\(688\) 0 0
\(689\) −0.368324 0.637957i −0.0140320 0.0243042i
\(690\) 0 0
\(691\) −19.9060 + 34.4783i −0.757261 + 1.31162i 0.186981 + 0.982364i \(0.440130\pi\)
−0.944242 + 0.329252i \(0.893204\pi\)
\(692\) 0 0
\(693\) 9.59191 + 16.6137i 0.364366 + 0.631101i
\(694\) 0 0
\(695\) 20.1973 0.766126
\(696\) 0 0
\(697\) −2.95250 −0.111834
\(698\) 0 0
\(699\) −1.42684 + 2.47137i −0.0539682 + 0.0934756i
\(700\) 0 0
\(701\) 6.60794 11.4453i 0.249579 0.432283i −0.713830 0.700319i \(-0.753041\pi\)
0.963409 + 0.268036i \(0.0863746\pi\)
\(702\) 0 0
\(703\) −1.62874 + 2.82107i −0.0614292 + 0.106399i
\(704\) 0 0
\(705\) 1.69187 2.93041i 0.0637197 0.110366i
\(706\) 0 0
\(707\) −4.69855 8.13813i −0.176707 0.306066i
\(708\) 0 0
\(709\) −1.34170 2.32389i −0.0503885 0.0872754i 0.839731 0.543003i \(-0.182713\pi\)
−0.890120 + 0.455727i \(0.849379\pi\)
\(710\) 0 0
\(711\) −5.82437 + 10.0881i −0.218431 + 0.378334i
\(712\) 0 0
\(713\) 1.15237 0.0431567
\(714\) 0 0
\(715\) −4.09914 −0.153299
\(716\) 0 0
\(717\) 1.17038 2.02715i 0.0437085 0.0757053i
\(718\) 0 0
\(719\) −15.9485 27.6236i −0.594777 1.03018i −0.993578 0.113147i \(-0.963907\pi\)
0.398801 0.917038i \(-0.369426\pi\)
\(720\) 0 0
\(721\) 25.2346 + 43.7076i 0.939786 + 1.62776i
\(722\) 0 0
\(723\) −0.784474 −0.0291749
\(724\) 0 0
\(725\) −4.40150 + 7.62362i −0.163468 + 0.283134i
\(726\) 0 0
\(727\) −10.3166 17.8689i −0.382622 0.662721i 0.608814 0.793313i \(-0.291646\pi\)
−0.991436 + 0.130592i \(0.958312\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −0.0632104 0.109484i −0.00233792 0.00404940i
\(732\) 0 0
\(733\) −4.89664 + 8.48123i −0.180862 + 0.313262i −0.942174 0.335123i \(-0.891222\pi\)
0.761313 + 0.648385i \(0.224555\pi\)
\(734\) 0 0
\(735\) 7.45103 + 12.9056i 0.274835 + 0.476029i
\(736\) 0 0
\(737\) −21.9852 + 25.3465i −0.809837 + 0.933650i
\(738\) 0 0
\(739\) 12.2101 + 21.1486i 0.449157 + 0.777962i 0.998331 0.0577452i \(-0.0183911\pi\)
−0.549175 + 0.835708i \(0.685058\pi\)
\(740\) 0 0
\(741\) 1.42209 2.46313i 0.0522417 0.0904854i
\(742\) 0 0
\(743\) 19.0989 + 33.0803i 0.700671 + 1.21360i 0.968231 + 0.250057i \(0.0804493\pi\)
−0.267560 + 0.963541i \(0.586217\pi\)
\(744\) 0 0
\(745\) 23.1440 0.847929
\(746\) 0 0
\(747\) 6.18900 + 10.7197i 0.226444 + 0.392212i
\(748\) 0 0
\(749\) 17.0264 29.4906i 0.622131 1.07756i
\(750\) 0 0
\(751\) −40.2739 −1.46962 −0.734808 0.678276i \(-0.762728\pi\)
−0.734808 + 0.678276i \(0.762728\pi\)
\(752\) 0 0
\(753\) 13.7408 + 23.7998i 0.500742 + 0.867311i
\(754\) 0 0
\(755\) −6.32638 10.9576i −0.230240 0.398788i
\(756\) 0 0
\(757\) 8.86526 15.3551i 0.322213 0.558090i −0.658731 0.752379i \(-0.728906\pi\)
0.980944 + 0.194288i \(0.0622398\pi\)
\(758\) 0 0
\(759\) 17.9370 0.651073
\(760\) 0 0
\(761\) −5.44137 −0.197250 −0.0986248 0.995125i \(-0.531444\pi\)
−0.0986248 + 0.995125i \(0.531444\pi\)
\(762\) 0 0
\(763\) 22.4474 38.8800i 0.812650 1.40755i
\(764\) 0 0
\(765\) −0.405657 0.702619i −0.0146666 0.0254032i
\(766\) 0 0
\(767\) 1.67895 + 2.90803i 0.0606234 + 0.105003i
\(768\) 0 0
\(769\) −3.36334 + 5.82547i −0.121285 + 0.210072i −0.920275 0.391273i \(-0.872035\pi\)
0.798990 + 0.601345i \(0.205368\pi\)
\(770\) 0 0
\(771\) 11.3528 19.6637i 0.408862 0.708169i
\(772\) 0 0
\(773\) 4.87223 8.43895i 0.175242 0.303528i −0.765003 0.644027i \(-0.777263\pi\)
0.940245 + 0.340499i \(0.110596\pi\)
\(774\) 0 0
\(775\) 0.131676 0.228069i 0.00472993 0.00819247i
\(776\) 0 0
\(777\) 5.36004 0.192291
\(778\) 0 0
\(779\) 10.3504 0.370842
\(780\) 0 0
\(781\) −6.57212 11.3832i −0.235169 0.407324i
\(782\) 0 0
\(783\) −4.40150 + 7.62362i −0.157297 + 0.272446i
\(784\) 0 0
\(785\) 1.72579 + 2.98916i 0.0615962 + 0.106688i
\(786\) 0 0
\(787\) −0.519118 0.899139i −0.0185046 0.0320509i 0.856625 0.515940i \(-0.172557\pi\)
−0.875129 + 0.483889i \(0.839224\pi\)
\(788\) 0 0
\(789\) 4.16533 0.148290
\(790\) 0 0
\(791\) 5.29620 + 9.17329i 0.188311 + 0.326165i
\(792\) 0 0
\(793\) −0.373303 + 0.646580i −0.0132564 + 0.0229607i
\(794\) 0 0
\(795\) 0.736649 0.0261262
\(796\) 0 0
\(797\) −5.17411 + 8.96183i −0.183276 + 0.317444i −0.942994 0.332809i \(-0.892004\pi\)
0.759718 + 0.650253i \(0.225337\pi\)
\(798\) 0 0
\(799\) −2.74529 −0.0971212
\(800\) 0 0
\(801\) −13.1087 −0.463174
\(802\) 0 0
\(803\) 39.0310 1.37737
\(804\) 0 0
\(805\) 20.4786 0.721776
\(806\) 0 0
\(807\) −15.9014 −0.559757
\(808\) 0 0
\(809\) −18.6926 −0.657198 −0.328599 0.944470i \(-0.606576\pi\)
−0.328599 + 0.944470i \(0.606576\pi\)
\(810\) 0 0
\(811\) 1.10449 1.91303i 0.0387839 0.0671757i −0.845982 0.533212i \(-0.820985\pi\)
0.884766 + 0.466036i \(0.154318\pi\)
\(812\) 0 0
\(813\) 27.3621 0.959631
\(814\) 0 0
\(815\) 10.3321 17.8958i 0.361919 0.626861i
\(816\) 0 0
\(817\) 0.221593 + 0.383810i 0.00775255 + 0.0134278i
\(818\) 0 0
\(819\) −4.67996 −0.163531
\(820\) 0 0
\(821\) 18.9988 + 32.9069i 0.663063 + 1.14846i 0.979806 + 0.199948i \(0.0640774\pi\)
−0.316743 + 0.948511i \(0.602589\pi\)
\(822\) 0 0
\(823\) 21.8384 + 37.8252i 0.761239 + 1.31851i 0.942212 + 0.335017i \(0.108742\pi\)
−0.180973 + 0.983488i \(0.557925\pi\)
\(824\) 0 0
\(825\) 2.04957 3.54996i 0.0713568 0.123594i
\(826\) 0 0
\(827\) −26.5786 46.0355i −0.924230 1.60081i −0.792796 0.609488i \(-0.791375\pi\)
−0.131434 0.991325i \(-0.541958\pi\)
\(828\) 0 0
\(829\) 33.1302 1.15066 0.575329 0.817922i \(-0.304874\pi\)
0.575329 + 0.817922i \(0.304874\pi\)
\(830\) 0 0
\(831\) −13.3919 −0.464559
\(832\) 0 0
\(833\) 6.04513 10.4705i 0.209451 0.362780i
\(834\) 0 0
\(835\) 12.4986 21.6483i 0.432533 0.749169i
\(836\) 0 0
\(837\) 0.131676 0.228069i 0.00455137 0.00788321i
\(838\) 0 0
\(839\) −6.42532 + 11.1290i −0.221827 + 0.384215i −0.955363 0.295436i \(-0.904535\pi\)
0.733536 + 0.679651i \(0.237869\pi\)
\(840\) 0 0
\(841\) −24.2464 41.9960i −0.836083 1.44814i
\(842\) 0 0
\(843\) 12.4785 + 21.6133i 0.429781 + 0.744402i
\(844\) 0 0
\(845\) −6.00000 + 10.3923i −0.206406 + 0.357506i
\(846\) 0 0
\(847\) 27.1575 0.933142
\(848\) 0 0
\(849\) −0.580906 −0.0199366
\(850\) 0 0
\(851\) 2.50584 4.34025i 0.0858992 0.148782i
\(852\) 0 0
\(853\) 4.53454 + 7.85406i 0.155260 + 0.268918i 0.933154 0.359478i \(-0.117045\pi\)
−0.777894 + 0.628396i \(0.783712\pi\)
\(854\) 0 0
\(855\) 1.42209 + 2.46313i 0.0486344 + 0.0842373i
\(856\) 0 0
\(857\) 39.5338 1.35045 0.675225 0.737612i \(-0.264047\pi\)
0.675225 + 0.737612i \(0.264047\pi\)
\(858\) 0 0
\(859\) −25.1474 + 43.5566i −0.858019 + 1.48613i 0.0157966 + 0.999875i \(0.494972\pi\)
−0.873816 + 0.486257i \(0.838362\pi\)
\(860\) 0 0
\(861\) −8.51556 14.7494i −0.290210 0.502658i
\(862\) 0 0
\(863\) 22.5929 0.769071 0.384535 0.923110i \(-0.374362\pi\)
0.384535 + 0.923110i \(0.374362\pi\)
\(864\) 0 0
\(865\) −5.90917 10.2350i −0.200918 0.348000i
\(866\) 0 0
\(867\) 8.17088 14.1524i 0.277498 0.480640i
\(868\) 0 0
\(869\) 23.8749 + 41.3525i 0.809900 + 1.40279i
\(870\) 0 0
\(871\) −2.67327 7.73651i −0.0905801 0.262142i
\(872\) 0 0
\(873\) −9.60393 16.6345i −0.325044 0.562992i
\(874\) 0 0
\(875\) 2.33998 4.05297i 0.0791058 0.137015i
\(876\) 0 0
\(877\) −22.3198 38.6590i −0.753686 1.30542i −0.946025 0.324093i \(-0.894941\pi\)
0.192340 0.981328i \(-0.438392\pi\)
\(878\) 0 0
\(879\) −9.08517 −0.306435
\(880\) 0 0
\(881\) 14.0804 + 24.3880i 0.474381 + 0.821652i 0.999570 0.0293338i \(-0.00933858\pi\)
−0.525189 + 0.850986i \(0.676005\pi\)
\(882\) 0 0
\(883\) −8.97646 + 15.5477i −0.302082 + 0.523221i −0.976607 0.215030i \(-0.931015\pi\)
0.674526 + 0.738252i \(0.264348\pi\)
\(884\) 0 0
\(885\) −3.35790 −0.112875
\(886\) 0 0
\(887\) −2.68575 4.65186i −0.0901787 0.156194i 0.817408 0.576060i \(-0.195410\pi\)
−0.907586 + 0.419866i \(0.862077\pi\)
\(888\) 0 0
\(889\) −16.2509 28.1474i −0.545037 0.944033i
\(890\) 0 0
\(891\) 2.04957 3.54996i 0.0686631 0.118928i
\(892\) 0 0
\(893\) 9.62398 0.322054
\(894\) 0 0
\(895\) 13.1659 0.440086
\(896\) 0 0
\(897\) −2.18790 + 3.78956i −0.0730519 + 0.126530i
\(898\) 0 0
\(899\) 1.15914 + 2.00769i 0.0386595 + 0.0669602i
\(900\) 0 0
\(901\) −0.298827 0.517584i −0.00995537 0.0172432i
\(902\) 0 0
\(903\) 0.364621 0.631542i 0.0121338 0.0210164i
\(904\) 0 0
\(905\) 0.319852 0.553999i 0.0106322 0.0184156i
\(906\) 0 0
\(907\) −4.52227 + 7.83279i −0.150159 + 0.260084i −0.931286 0.364289i \(-0.881312\pi\)
0.781127 + 0.624373i \(0.214645\pi\)
\(908\) 0 0
\(909\) −1.00397 + 1.73893i −0.0332996 + 0.0576767i
\(910\) 0 0
\(911\) −47.3846 −1.56992 −0.784960 0.619546i \(-0.787317\pi\)
−0.784960 + 0.619546i \(0.787317\pi\)
\(912\) 0 0
\(913\) 50.7391 1.67922
\(914\) 0 0
\(915\) −0.373303 0.646580i −0.0123410 0.0213753i
\(916\) 0 0
\(917\) 40.6443 70.3981i 1.34219 2.32475i
\(918\) 0 0
\(919\) 12.9197 + 22.3775i 0.426180 + 0.738166i 0.996530 0.0832362i \(-0.0265256\pi\)
−0.570350 + 0.821402i \(0.693192\pi\)
\(920\) 0 0
\(921\) 6.85294 + 11.8696i 0.225812 + 0.391118i
\(922\) 0 0
\(923\) 3.20659 0.105546
\(924\) 0 0
\(925\) −0.572659 0.991874i −0.0188289 0.0326126i
\(926\) 0 0
\(927\) 5.39205 9.33931i 0.177098 0.306743i
\(928\) 0 0
\(929\) 49.2162 1.61473 0.807366 0.590051i \(-0.200892\pi\)
0.807366 + 0.590051i \(0.200892\pi\)
\(930\) 0 0
\(931\) −21.1921 + 36.7057i −0.694542 + 1.20298i
\(932\) 0 0
\(933\) 11.5669 0.378684
\(934\) 0 0
\(935\) −3.32569 −0.108762
\(936\) 0 0
\(937\) 5.17439 0.169040 0.0845200 0.996422i \(-0.473064\pi\)
0.0845200 + 0.996422i \(0.473064\pi\)
\(938\) 0 0
\(939\) 14.6871 0.479295
\(940\) 0 0
\(941\) −24.6020 −0.802003 −0.401002 0.916077i \(-0.631338\pi\)
−0.401002 + 0.916077i \(0.631338\pi\)
\(942\) 0 0
\(943\) −15.9242 −0.518565
\(944\) 0 0
\(945\) 2.33998 4.05297i 0.0761196 0.131843i
\(946\) 0 0
\(947\) −18.3406 −0.595990 −0.297995 0.954567i \(-0.596318\pi\)
−0.297995 + 0.954567i \(0.596318\pi\)
\(948\) 0 0
\(949\) −4.76087 + 8.24608i −0.154545 + 0.267679i
\(950\) 0 0
\(951\) 2.18470 + 3.78401i 0.0708438 + 0.122705i
\(952\) 0 0
\(953\) 18.0193 0.583702 0.291851 0.956464i \(-0.405729\pi\)
0.291851 + 0.956464i \(0.405729\pi\)
\(954\) 0 0
\(955\) 8.19598 + 14.1959i 0.265216 + 0.459367i
\(956\) 0 0
\(957\) 18.0424 + 31.2503i 0.583226 + 1.01018i
\(958\) 0 0
\(959\) −30.2698 + 52.4289i −0.977463 + 1.69302i
\(960\) 0 0
\(961\) 15.4653 + 26.7867i 0.498881 + 0.864088i
\(962\) 0 0
\(963\) −7.27630 −0.234475
\(964\) 0 0
\(965\) −9.88217 −0.318118
\(966\) 0 0
\(967\) 26.2202 45.4148i 0.843186 1.46044i −0.0440006 0.999032i \(-0.514010\pi\)
0.887187 0.461410i \(-0.152656\pi\)
\(968\) 0 0
\(969\) 1.15376 1.99837i 0.0370642 0.0641970i
\(970\) 0 0
\(971\) −0.268358 + 0.464809i −0.00861201 + 0.0149164i −0.870299 0.492523i \(-0.836075\pi\)
0.861687 + 0.507440i \(0.169408\pi\)
\(972\) 0 0
\(973\) 47.2613 81.8589i 1.51513 2.62428i
\(974\) 0 0
\(975\) 0.500000 + 0.866025i 0.0160128 + 0.0277350i
\(976\) 0 0
\(977\) 2.05815 + 3.56482i 0.0658460 + 0.114049i 0.897069 0.441891i \(-0.145692\pi\)
−0.831223 + 0.555939i \(0.812359\pi\)
\(978\) 0 0
\(979\) −26.8672 + 46.5354i −0.858680 + 1.48728i
\(980\) 0 0
\(981\) −9.59298 −0.306280
\(982\) 0 0
\(983\) 34.1266 1.08847 0.544235 0.838933i \(-0.316820\pi\)
0.544235 + 0.838933i \(0.316820\pi\)
\(984\) 0 0
\(985\) 1.30338 2.25751i 0.0415290 0.0719303i
\(986\) 0 0
\(987\) −7.91791 13.7142i −0.252030 0.436529i
\(988\) 0 0
\(989\) −0.340924 0.590497i −0.0108407 0.0187767i
\(990\) 0 0
\(991\) 38.8833 1.23517 0.617584 0.786505i \(-0.288112\pi\)
0.617584 + 0.786505i \(0.288112\pi\)
\(992\) 0 0
\(993\) 5.15199 8.92351i 0.163493 0.283179i
\(994\) 0 0
\(995\) 9.81017 + 16.9917i 0.311003 + 0.538673i
\(996\) 0 0
\(997\) −33.2131 −1.05187 −0.525935 0.850525i \(-0.676284\pi\)
−0.525935 + 0.850525i \(0.676284\pi\)
\(998\) 0 0
\(999\) −0.572659 0.991874i −0.0181181 0.0313815i
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.q.j.841.1 12
67.29 even 3 inner 4020.2.q.j.3781.1 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4020.2.q.j.841.1 12 1.1 even 1 trivial
4020.2.q.j.3781.1 yes 12 67.29 even 3 inner