Properties

Label 76.3.j.a.21.1
Level $76$
Weight $3$
Character 76.21
Analytic conductor $2.071$
Analytic rank $0$
Dimension $18$
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] = [76,3,Mod(13,76)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(76, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("76.13");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 76.j (of order \(18\), degree \(6\), 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: \(2.07085000914\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{18})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 93 x^{16} + 3429 x^{14} + 64261 x^{12} + 647217 x^{10} + 3386277 x^{8} + 8232133 x^{6} + 8319228 x^{4} + 2467872 x^{2} + 69312 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 21.1
Root \(-1.57014i\) of defining polynomial
Character \(\chi\) \(=\) 76.21
Dual form 76.3.j.a.29.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.27531 + 2.71161i) q^{3} +(-5.68966 + 2.07087i) q^{5} +(-6.17078 - 10.6881i) q^{7} +(-0.612961 - 3.47627i) q^{9} +O(q^{10})\) \(q+(-2.27531 + 2.71161i) q^{3} +(-5.68966 + 2.07087i) q^{5} +(-6.17078 - 10.6881i) q^{7} +(-0.612961 - 3.47627i) q^{9} +(-8.21913 + 14.2359i) q^{11} +(12.0374 + 14.3456i) q^{13} +(7.33038 - 20.1400i) q^{15} +(-0.283090 + 1.60549i) q^{17} +(-17.9621 - 6.19392i) q^{19} +(43.0225 + 7.58602i) q^{21} +(21.0092 + 7.64671i) q^{23} +(8.93266 - 7.49539i) q^{25} +(-16.7687 - 9.68142i) q^{27} +(13.0677 - 2.30418i) q^{29} +(-21.3262 + 12.3127i) q^{31} +(-19.9013 - 54.6783i) q^{33} +(57.2433 + 48.0328i) q^{35} -29.5565i q^{37} -66.2886 q^{39} +(-10.7505 + 12.8120i) q^{41} +(19.3175 - 7.03101i) q^{43} +(10.6864 + 18.5095i) q^{45} +(3.01062 + 17.0741i) q^{47} +(-51.6570 + 89.4726i) q^{49} +(-3.70934 - 4.42062i) q^{51} +(-13.7549 + 37.7914i) q^{53} +(17.2833 - 98.0185i) q^{55} +(57.6648 - 34.6130i) q^{57} +(-86.1296 - 15.1870i) q^{59} +(-18.8722 - 6.86893i) q^{61} +(-33.3723 + 28.0027i) q^{63} +(-98.1966 - 56.6938i) q^{65} +(42.6487 - 7.52012i) q^{67} +(-68.5374 + 39.5701i) q^{69} +(-4.32434 - 11.8810i) q^{71} +(12.7881 + 10.7305i) q^{73} +41.2763i q^{75} +202.874 q^{77} +(13.3044 - 15.8555i) q^{79} +(94.2595 - 34.3077i) q^{81} +(61.6379 + 106.760i) q^{83} +(-1.71406 - 9.72091i) q^{85} +(-23.4850 + 40.6772i) q^{87} +(58.3959 + 69.5935i) q^{89} +(79.0472 - 217.180i) q^{91} +(15.1365 - 85.8435i) q^{93} +(115.025 - 1.95573i) q^{95} +(-111.640 - 19.6851i) q^{97} +(54.5260 + 19.8459i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 6 q^{3} + 9 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 6 q^{3} + 9 q^{7} + 6 q^{9} - 15 q^{11} + 51 q^{13} + 21 q^{15} - 45 q^{17} + 30 q^{19} - 63 q^{21} + 48 q^{23} - 54 q^{25} - 198 q^{27} - 39 q^{29} - 108 q^{31} - 105 q^{33} + 51 q^{35} + 48 q^{39} + 54 q^{41} + 75 q^{43} + 288 q^{45} + 339 q^{47} - 24 q^{49} + 360 q^{51} + 69 q^{53} - 51 q^{55} + 510 q^{57} - 483 q^{59} - 36 q^{61} - 267 q^{63} - 585 q^{65} - 87 q^{67} - 351 q^{69} - 234 q^{71} - 132 q^{73} + 108 q^{77} + 363 q^{79} + 258 q^{81} + 279 q^{83} + 666 q^{85} + 600 q^{89} + 270 q^{91} - 456 q^{93} - 39 q^{95} - 801 q^{97} - 267 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(39\)
\(\chi(n)\) \(e\left(\frac{1}{18}\right)\) \(1\)

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
\(3\) −2.27531 + 2.71161i −0.758438 + 0.903871i −0.997748 0.0670720i \(-0.978634\pi\)
0.239310 + 0.970943i \(0.423079\pi\)
\(4\) 0 0
\(5\) −5.68966 + 2.07087i −1.13793 + 0.414174i −0.841166 0.540777i \(-0.818130\pi\)
−0.296766 + 0.954950i \(0.595908\pi\)
\(6\) 0 0
\(7\) −6.17078 10.6881i −0.881540 1.52687i −0.849629 0.527381i \(-0.823174\pi\)
−0.0319111 0.999491i \(-0.510159\pi\)
\(8\) 0 0
\(9\) −0.612961 3.47627i −0.0681067 0.386253i
\(10\) 0 0
\(11\) −8.21913 + 14.2359i −0.747193 + 1.29418i 0.201970 + 0.979392i \(0.435266\pi\)
−0.949163 + 0.314785i \(0.898068\pi\)
\(12\) 0 0
\(13\) 12.0374 + 14.3456i 0.925953 + 1.10351i 0.994382 + 0.105852i \(0.0337571\pi\)
−0.0684284 + 0.997656i \(0.521798\pi\)
\(14\) 0 0
\(15\) 7.33038 20.1400i 0.488692 1.34267i
\(16\) 0 0
\(17\) −0.283090 + 1.60549i −0.0166524 + 0.0944403i −0.992001 0.126228i \(-0.959713\pi\)
0.975349 + 0.220669i \(0.0708239\pi\)
\(18\) 0 0
\(19\) −17.9621 6.19392i −0.945371 0.325996i
\(20\) 0 0
\(21\) 43.0225 + 7.58602i 2.04869 + 0.361239i
\(22\) 0 0
\(23\) 21.0092 + 7.64671i 0.913442 + 0.332466i 0.755626 0.655003i \(-0.227333\pi\)
0.157816 + 0.987469i \(0.449555\pi\)
\(24\) 0 0
\(25\) 8.93266 7.49539i 0.357306 0.299816i
\(26\) 0 0
\(27\) −16.7687 9.68142i −0.621063 0.358571i
\(28\) 0 0
\(29\) 13.0677 2.30418i 0.450609 0.0794545i 0.0562618 0.998416i \(-0.482082\pi\)
0.394347 + 0.918962i \(0.370971\pi\)
\(30\) 0 0
\(31\) −21.3262 + 12.3127i −0.687941 + 0.397183i −0.802840 0.596194i \(-0.796679\pi\)
0.114899 + 0.993377i \(0.463345\pi\)
\(32\) 0 0
\(33\) −19.9013 54.6783i −0.603069 1.65692i
\(34\) 0 0
\(35\) 57.2433 + 48.0328i 1.63552 + 1.37237i
\(36\) 0 0
\(37\) 29.5565i 0.798825i −0.916771 0.399412i \(-0.869214\pi\)
0.916771 0.399412i \(-0.130786\pi\)
\(38\) 0 0
\(39\) −66.2886 −1.69971
\(40\) 0 0
\(41\) −10.7505 + 12.8120i −0.262208 + 0.312487i −0.881045 0.473032i \(-0.843160\pi\)
0.618837 + 0.785519i \(0.287604\pi\)
\(42\) 0 0
\(43\) 19.3175 7.03101i 0.449245 0.163512i −0.107483 0.994207i \(-0.534279\pi\)
0.556728 + 0.830695i \(0.312057\pi\)
\(44\) 0 0
\(45\) 10.6864 + 18.5095i 0.237476 + 0.411321i
\(46\) 0 0
\(47\) 3.01062 + 17.0741i 0.0640558 + 0.363278i 0.999940 + 0.0109702i \(0.00349199\pi\)
−0.935884 + 0.352308i \(0.885397\pi\)
\(48\) 0 0
\(49\) −51.6570 + 89.4726i −1.05423 + 1.82597i
\(50\) 0 0
\(51\) −3.70934 4.42062i −0.0727321 0.0866787i
\(52\) 0 0
\(53\) −13.7549 + 37.7914i −0.259527 + 0.713045i 0.739670 + 0.672970i \(0.234982\pi\)
−0.999197 + 0.0400745i \(0.987240\pi\)
\(54\) 0 0
\(55\) 17.2833 98.0185i 0.314242 1.78215i
\(56\) 0 0
\(57\) 57.6648 34.6130i 1.01166 0.607246i
\(58\) 0 0
\(59\) −86.1296 15.1870i −1.45982 0.257406i −0.613339 0.789820i \(-0.710174\pi\)
−0.846484 + 0.532414i \(0.821285\pi\)
\(60\) 0 0
\(61\) −18.8722 6.86893i −0.309381 0.112605i 0.182664 0.983175i \(-0.441528\pi\)
−0.492045 + 0.870570i \(0.663750\pi\)
\(62\) 0 0
\(63\) −33.3723 + 28.0027i −0.529719 + 0.444487i
\(64\) 0 0
\(65\) −98.1966 56.6938i −1.51072 0.872213i
\(66\) 0 0
\(67\) 42.6487 7.52012i 0.636548 0.112241i 0.153944 0.988080i \(-0.450802\pi\)
0.482604 + 0.875839i \(0.339691\pi\)
\(68\) 0 0
\(69\) −68.5374 + 39.5701i −0.993295 + 0.573479i
\(70\) 0 0
\(71\) −4.32434 11.8810i −0.0609062 0.167338i 0.905507 0.424331i \(-0.139491\pi\)
−0.966413 + 0.256992i \(0.917268\pi\)
\(72\) 0 0
\(73\) 12.7881 + 10.7305i 0.175180 + 0.146993i 0.726162 0.687524i \(-0.241302\pi\)
−0.550982 + 0.834517i \(0.685747\pi\)
\(74\) 0 0
\(75\) 41.2763i 0.550351i
\(76\) 0 0
\(77\) 202.874 2.63472
\(78\) 0 0
\(79\) 13.3044 15.8555i 0.168410 0.200703i −0.675238 0.737600i \(-0.735959\pi\)
0.843648 + 0.536897i \(0.180404\pi\)
\(80\) 0 0
\(81\) 94.2595 34.3077i 1.16370 0.423551i
\(82\) 0 0
\(83\) 61.6379 + 106.760i 0.742626 + 1.28627i 0.951296 + 0.308280i \(0.0997532\pi\)
−0.208670 + 0.977986i \(0.566913\pi\)
\(84\) 0 0
\(85\) −1.71406 9.72091i −0.0201654 0.114364i
\(86\) 0 0
\(87\) −23.4850 + 40.6772i −0.269942 + 0.467554i
\(88\) 0 0
\(89\) 58.3959 + 69.5935i 0.656134 + 0.781950i 0.986825 0.161789i \(-0.0517264\pi\)
−0.330692 + 0.943739i \(0.607282\pi\)
\(90\) 0 0
\(91\) 79.0472 217.180i 0.868651 2.38660i
\(92\) 0 0
\(93\) 15.1365 85.8435i 0.162758 0.923048i
\(94\) 0 0
\(95\) 115.025 1.95573i 1.21079 0.0205867i
\(96\) 0 0
\(97\) −111.640 19.6851i −1.15093 0.202939i −0.434547 0.900649i \(-0.643091\pi\)
−0.716380 + 0.697710i \(0.754202\pi\)
\(98\) 0 0
\(99\) 54.5260 + 19.8459i 0.550768 + 0.200463i
\(100\) 0 0
\(101\) −144.471 + 121.225i −1.43040 + 1.20025i −0.484929 + 0.874554i \(0.661154\pi\)
−0.945474 + 0.325697i \(0.894401\pi\)
\(102\) 0 0
\(103\) 111.838 + 64.5696i 1.08580 + 0.626890i 0.932456 0.361283i \(-0.117661\pi\)
0.153348 + 0.988172i \(0.450994\pi\)
\(104\) 0 0
\(105\) −260.493 + 45.9319i −2.48089 + 0.437447i
\(106\) 0 0
\(107\) 9.92634 5.73098i 0.0927696 0.0535605i −0.452898 0.891563i \(-0.649610\pi\)
0.545667 + 0.838002i \(0.316276\pi\)
\(108\) 0 0
\(109\) −22.5020 61.8238i −0.206441 0.567191i 0.792657 0.609668i \(-0.208697\pi\)
−0.999097 + 0.0424773i \(0.986475\pi\)
\(110\) 0 0
\(111\) 80.1459 + 67.2504i 0.722035 + 0.605859i
\(112\) 0 0
\(113\) 140.102i 1.23984i −0.784665 0.619920i \(-0.787165\pi\)
0.784665 0.619920i \(-0.212835\pi\)
\(114\) 0 0
\(115\) −135.370 −1.17713
\(116\) 0 0
\(117\) 42.4908 50.6386i 0.363169 0.432808i
\(118\) 0 0
\(119\) 18.9065 6.88140i 0.158878 0.0578269i
\(120\) 0 0
\(121\) −74.6081 129.225i −0.616596 1.06798i
\(122\) 0 0
\(123\) −10.2803 58.3026i −0.0835798 0.474005i
\(124\) 0 0
\(125\) 40.3833 69.9459i 0.323066 0.559567i
\(126\) 0 0
\(127\) 50.9134 + 60.6763i 0.400893 + 0.477766i 0.928292 0.371852i \(-0.121277\pi\)
−0.527399 + 0.849618i \(0.676833\pi\)
\(128\) 0 0
\(129\) −24.8881 + 68.3794i −0.192931 + 0.530073i
\(130\) 0 0
\(131\) −33.5765 + 190.422i −0.256309 + 1.45360i 0.536381 + 0.843976i \(0.319791\pi\)
−0.792690 + 0.609625i \(0.791320\pi\)
\(132\) 0 0
\(133\) 44.6386 + 230.202i 0.335629 + 1.73084i
\(134\) 0 0
\(135\) 115.457 + 20.3582i 0.855239 + 0.150802i
\(136\) 0 0
\(137\) −68.5778 24.9603i −0.500568 0.182192i 0.0793814 0.996844i \(-0.474706\pi\)
−0.579949 + 0.814652i \(0.696928\pi\)
\(138\) 0 0
\(139\) −92.5208 + 77.6341i −0.665617 + 0.558519i −0.911764 0.410713i \(-0.865280\pi\)
0.246148 + 0.969232i \(0.420835\pi\)
\(140\) 0 0
\(141\) −53.1484 30.6853i −0.376939 0.217626i
\(142\) 0 0
\(143\) −303.160 + 53.4553i −2.12000 + 0.373813i
\(144\) 0 0
\(145\) −69.5789 + 40.1714i −0.479855 + 0.277044i
\(146\) 0 0
\(147\) −125.079 343.652i −0.850879 2.33777i
\(148\) 0 0
\(149\) 104.661 + 87.8213i 0.702425 + 0.589405i 0.922462 0.386087i \(-0.126173\pi\)
−0.220037 + 0.975491i \(0.570618\pi\)
\(150\) 0 0
\(151\) 226.087i 1.49727i 0.662985 + 0.748633i \(0.269289\pi\)
−0.662985 + 0.748633i \(0.730711\pi\)
\(152\) 0 0
\(153\) 5.75463 0.0376120
\(154\) 0 0
\(155\) 95.8408 114.219i 0.618328 0.736894i
\(156\) 0 0
\(157\) 208.620 75.9314i 1.32879 0.483639i 0.422524 0.906352i \(-0.361144\pi\)
0.906264 + 0.422712i \(0.138922\pi\)
\(158\) 0 0
\(159\) −71.1788 123.285i −0.447665 0.775379i
\(160\) 0 0
\(161\) −47.9141 271.734i −0.297603 1.68779i
\(162\) 0 0
\(163\) −51.5648 + 89.3128i −0.316348 + 0.547931i −0.979723 0.200356i \(-0.935790\pi\)
0.663375 + 0.748287i \(0.269123\pi\)
\(164\) 0 0
\(165\) 226.463 + 269.888i 1.37250 + 1.63569i
\(166\) 0 0
\(167\) 81.1500 222.958i 0.485928 1.33508i −0.418409 0.908259i \(-0.637412\pi\)
0.904337 0.426819i \(-0.140366\pi\)
\(168\) 0 0
\(169\) −31.5511 + 178.935i −0.186693 + 1.05879i
\(170\) 0 0
\(171\) −10.5217 + 66.2376i −0.0615305 + 0.387355i
\(172\) 0 0
\(173\) −246.700 43.4998i −1.42601 0.251444i −0.593223 0.805038i \(-0.702145\pi\)
−0.832786 + 0.553595i \(0.813256\pi\)
\(174\) 0 0
\(175\) −135.233 49.2208i −0.772760 0.281262i
\(176\) 0 0
\(177\) 237.153 198.995i 1.33985 1.12427i
\(178\) 0 0
\(179\) −29.4439 16.9994i −0.164491 0.0949689i 0.415495 0.909596i \(-0.363608\pi\)
−0.579986 + 0.814627i \(0.696942\pi\)
\(180\) 0 0
\(181\) 210.383 37.0962i 1.16234 0.204952i 0.440982 0.897516i \(-0.354630\pi\)
0.721356 + 0.692564i \(0.243519\pi\)
\(182\) 0 0
\(183\) 61.5661 35.5452i 0.336427 0.194236i
\(184\) 0 0
\(185\) 61.2077 + 168.167i 0.330852 + 0.909009i
\(186\) 0 0
\(187\) −20.5288 17.2257i −0.109780 0.0921163i
\(188\) 0 0
\(189\) 238.968i 1.26438i
\(190\) 0 0
\(191\) −28.5507 −0.149480 −0.0747399 0.997203i \(-0.523813\pi\)
−0.0747399 + 0.997203i \(0.523813\pi\)
\(192\) 0 0
\(193\) −1.67587 + 1.99723i −0.00868328 + 0.0103483i −0.770369 0.637599i \(-0.779928\pi\)
0.761685 + 0.647947i \(0.224372\pi\)
\(194\) 0 0
\(195\) 377.160 137.275i 1.93415 0.703974i
\(196\) 0 0
\(197\) −39.9438 69.1846i −0.202760 0.351191i 0.746657 0.665210i \(-0.231658\pi\)
−0.949417 + 0.314019i \(0.898325\pi\)
\(198\) 0 0
\(199\) 28.0487 + 159.072i 0.140948 + 0.799356i 0.970532 + 0.240974i \(0.0774667\pi\)
−0.829584 + 0.558383i \(0.811422\pi\)
\(200\) 0 0
\(201\) −76.6475 + 132.757i −0.381331 + 0.660485i
\(202\) 0 0
\(203\) −105.265 125.450i −0.518547 0.617980i
\(204\) 0 0
\(205\) 34.6350 95.1588i 0.168951 0.464189i
\(206\) 0 0
\(207\) 13.7043 77.7207i 0.0662042 0.375462i
\(208\) 0 0
\(209\) 235.809 204.798i 1.12827 0.979896i
\(210\) 0 0
\(211\) 231.472 + 40.8148i 1.09703 + 0.193435i 0.692733 0.721194i \(-0.256406\pi\)
0.404293 + 0.914630i \(0.367518\pi\)
\(212\) 0 0
\(213\) 42.0560 + 15.3071i 0.197446 + 0.0718645i
\(214\) 0 0
\(215\) −95.3500 + 80.0081i −0.443488 + 0.372131i
\(216\) 0 0
\(217\) 263.198 + 151.957i 1.21289 + 0.700265i
\(218\) 0 0
\(219\) −58.1940 + 10.2612i −0.265726 + 0.0468547i
\(220\) 0 0
\(221\) −26.4393 + 15.2648i −0.119635 + 0.0690713i
\(222\) 0 0
\(223\) −108.533 298.193i −0.486697 1.33719i −0.903654 0.428262i \(-0.859126\pi\)
0.416958 0.908926i \(-0.363096\pi\)
\(224\) 0 0
\(225\) −31.5314 26.4580i −0.140140 0.117591i
\(226\) 0 0
\(227\) 276.181i 1.21665i 0.793686 + 0.608327i \(0.208159\pi\)
−0.793686 + 0.608327i \(0.791841\pi\)
\(228\) 0 0
\(229\) 85.2800 0.372402 0.186201 0.982512i \(-0.440382\pi\)
0.186201 + 0.982512i \(0.440382\pi\)
\(230\) 0 0
\(231\) −461.601 + 550.115i −1.99827 + 2.38145i
\(232\) 0 0
\(233\) −156.992 + 57.1404i −0.673785 + 0.245238i −0.656177 0.754607i \(-0.727827\pi\)
−0.0176084 + 0.999845i \(0.505605\pi\)
\(234\) 0 0
\(235\) −52.4876 90.9112i −0.223351 0.386856i
\(236\) 0 0
\(237\) 12.7225 + 72.1527i 0.0536813 + 0.304442i
\(238\) 0 0
\(239\) 83.1245 143.976i 0.347801 0.602409i −0.638057 0.769989i \(-0.720262\pi\)
0.985859 + 0.167580i \(0.0535951\pi\)
\(240\) 0 0
\(241\) −98.2500 117.090i −0.407676 0.485850i 0.522668 0.852536i \(-0.324937\pi\)
−0.930345 + 0.366686i \(0.880492\pi\)
\(242\) 0 0
\(243\) −61.8386 + 169.900i −0.254480 + 0.699177i
\(244\) 0 0
\(245\) 108.625 616.044i 0.443368 2.51446i
\(246\) 0 0
\(247\) −127.361 332.235i −0.515631 1.34508i
\(248\) 0 0
\(249\) −429.738 75.7743i −1.72585 0.304315i
\(250\) 0 0
\(251\) −35.4203 12.8919i −0.141117 0.0513623i 0.270496 0.962721i \(-0.412812\pi\)
−0.411613 + 0.911359i \(0.635034\pi\)
\(252\) 0 0
\(253\) −281.535 + 236.236i −1.11279 + 0.933739i
\(254\) 0 0
\(255\) 30.2594 + 17.4703i 0.118664 + 0.0685108i
\(256\) 0 0
\(257\) −23.3796 + 4.12245i −0.0909712 + 0.0160407i −0.218948 0.975736i \(-0.570263\pi\)
0.127977 + 0.991777i \(0.459152\pi\)
\(258\) 0 0
\(259\) −315.903 + 182.387i −1.21970 + 0.704196i
\(260\) 0 0
\(261\) −16.0199 44.0144i −0.0613790 0.168638i
\(262\) 0 0
\(263\) 8.03448 + 6.74173i 0.0305494 + 0.0256339i 0.657934 0.753075i \(-0.271430\pi\)
−0.627385 + 0.778709i \(0.715875\pi\)
\(264\) 0 0
\(265\) 243.505i 0.918886i
\(266\) 0 0
\(267\) −321.580 −1.20442
\(268\) 0 0
\(269\) 279.542 333.145i 1.03919 1.23846i 0.0686169 0.997643i \(-0.478141\pi\)
0.970571 0.240813i \(-0.0774142\pi\)
\(270\) 0 0
\(271\) 89.5741 32.6023i 0.330532 0.120304i −0.171423 0.985197i \(-0.554837\pi\)
0.501955 + 0.864894i \(0.332614\pi\)
\(272\) 0 0
\(273\) 409.052 + 708.499i 1.49836 + 2.59524i
\(274\) 0 0
\(275\) 33.2853 + 188.770i 0.121038 + 0.686438i
\(276\) 0 0
\(277\) −134.671 + 233.256i −0.486175 + 0.842080i −0.999874 0.0158906i \(-0.994942\pi\)
0.513698 + 0.857971i \(0.328275\pi\)
\(278\) 0 0
\(279\) 55.8743 + 66.5884i 0.200266 + 0.238668i
\(280\) 0 0
\(281\) −33.9122 + 93.1731i −0.120684 + 0.331577i −0.985294 0.170867i \(-0.945343\pi\)
0.864610 + 0.502444i \(0.167566\pi\)
\(282\) 0 0
\(283\) −27.7556 + 157.410i −0.0980763 + 0.556218i 0.895685 + 0.444689i \(0.146686\pi\)
−0.993761 + 0.111529i \(0.964425\pi\)
\(284\) 0 0
\(285\) −256.414 + 316.353i −0.899700 + 1.11001i
\(286\) 0 0
\(287\) 203.275 + 35.8429i 0.708275 + 0.124888i
\(288\) 0 0
\(289\) 269.074 + 97.9348i 0.931051 + 0.338875i
\(290\) 0 0
\(291\) 307.394 257.934i 1.05634 0.886373i
\(292\) 0 0
\(293\) −443.330 255.957i −1.51307 0.873573i −0.999883 0.0152973i \(-0.995131\pi\)
−0.513189 0.858275i \(-0.671536\pi\)
\(294\) 0 0
\(295\) 521.498 91.9543i 1.76779 0.311709i
\(296\) 0 0
\(297\) 275.648 159.146i 0.928109 0.535844i
\(298\) 0 0
\(299\) 143.199 + 393.436i 0.478926 + 1.31584i
\(300\) 0 0
\(301\) −194.352 163.081i −0.645689 0.541797i
\(302\) 0 0
\(303\) 667.574i 2.20322i
\(304\) 0 0
\(305\) 121.601 0.398693
\(306\) 0 0
\(307\) 53.0718 63.2485i 0.172872 0.206021i −0.672651 0.739960i \(-0.734844\pi\)
0.845523 + 0.533939i \(0.179289\pi\)
\(308\) 0 0
\(309\) −429.554 + 156.345i −1.39014 + 0.505971i
\(310\) 0 0
\(311\) −98.5512 170.696i −0.316885 0.548861i 0.662951 0.748662i \(-0.269304\pi\)
−0.979836 + 0.199802i \(0.935970\pi\)
\(312\) 0 0
\(313\) 26.0879 + 147.952i 0.0833479 + 0.472689i 0.997701 + 0.0677713i \(0.0215888\pi\)
−0.914353 + 0.404918i \(0.867300\pi\)
\(314\) 0 0
\(315\) 131.887 228.436i 0.418690 0.725192i
\(316\) 0 0
\(317\) −163.585 194.953i −0.516042 0.614995i 0.443598 0.896226i \(-0.353702\pi\)
−0.959640 + 0.281231i \(0.909257\pi\)
\(318\) 0 0
\(319\) −74.6026 + 204.969i −0.233864 + 0.642536i
\(320\) 0 0
\(321\) −7.04535 + 39.9562i −0.0219481 + 0.124474i
\(322\) 0 0
\(323\) 15.0291 27.0844i 0.0465298 0.0838526i
\(324\) 0 0
\(325\) 215.052 + 37.9195i 0.661698 + 0.116675i
\(326\) 0 0
\(327\) 218.841 + 79.6517i 0.669240 + 0.243583i
\(328\) 0 0
\(329\) 163.912 137.538i 0.498212 0.418049i
\(330\) 0 0
\(331\) −324.984 187.629i −0.981824 0.566856i −0.0790034 0.996874i \(-0.525174\pi\)
−0.902820 + 0.430018i \(0.858507\pi\)
\(332\) 0 0
\(333\) −102.747 + 18.1170i −0.308548 + 0.0544054i
\(334\) 0 0
\(335\) −227.084 + 131.107i −0.677862 + 0.391364i
\(336\) 0 0
\(337\) 70.8408 + 194.633i 0.210210 + 0.577547i 0.999326 0.0366965i \(-0.0116835\pi\)
−0.789117 + 0.614244i \(0.789461\pi\)
\(338\) 0 0
\(339\) 379.902 + 318.776i 1.12066 + 0.940342i
\(340\) 0 0
\(341\) 404.797i 1.18709i
\(342\) 0 0
\(343\) 670.320 1.95429
\(344\) 0 0
\(345\) 308.010 367.072i 0.892783 1.06398i
\(346\) 0 0
\(347\) −497.854 + 181.204i −1.43474 + 0.522201i −0.938285 0.345863i \(-0.887586\pi\)
−0.496452 + 0.868064i \(0.665364\pi\)
\(348\) 0 0
\(349\) −88.2775 152.901i −0.252944 0.438112i 0.711391 0.702796i \(-0.248066\pi\)
−0.964335 + 0.264684i \(0.914732\pi\)
\(350\) 0 0
\(351\) −62.9657 357.096i −0.179390 1.01737i
\(352\) 0 0
\(353\) −165.850 + 287.261i −0.469831 + 0.813771i −0.999405 0.0344927i \(-0.989018\pi\)
0.529574 + 0.848264i \(0.322352\pi\)
\(354\) 0 0
\(355\) 49.2081 + 58.6439i 0.138614 + 0.165194i
\(356\) 0 0
\(357\) −24.3585 + 66.9244i −0.0682311 + 0.187463i
\(358\) 0 0
\(359\) −46.5136 + 263.792i −0.129564 + 0.734796i 0.848927 + 0.528509i \(0.177249\pi\)
−0.978492 + 0.206286i \(0.933862\pi\)
\(360\) 0 0
\(361\) 284.271 + 222.511i 0.787454 + 0.616374i
\(362\) 0 0
\(363\) 520.165 + 91.7191i 1.43296 + 0.252670i
\(364\) 0 0
\(365\) −94.9817 34.5705i −0.260224 0.0947137i
\(366\) 0 0
\(367\) −31.9988 + 26.8502i −0.0871901 + 0.0731612i −0.685341 0.728222i \(-0.740347\pi\)
0.598151 + 0.801384i \(0.295902\pi\)
\(368\) 0 0
\(369\) 51.1276 + 29.5185i 0.138557 + 0.0799960i
\(370\) 0 0
\(371\) 488.797 86.1880i 1.31751 0.232313i
\(372\) 0 0
\(373\) 615.732 355.493i 1.65076 0.953064i 0.673994 0.738737i \(-0.264577\pi\)
0.976762 0.214327i \(-0.0687558\pi\)
\(374\) 0 0
\(375\) 97.7816 + 268.653i 0.260751 + 0.716408i
\(376\) 0 0
\(377\) 190.356 + 159.727i 0.504922 + 0.423680i
\(378\) 0 0
\(379\) 165.723i 0.437263i −0.975808 0.218631i \(-0.929841\pi\)
0.975808 0.218631i \(-0.0701592\pi\)
\(380\) 0 0
\(381\) −280.375 −0.735892
\(382\) 0 0
\(383\) −3.68599 + 4.39279i −0.00962399 + 0.0114694i −0.770835 0.637035i \(-0.780161\pi\)
0.761211 + 0.648504i \(0.224605\pi\)
\(384\) 0 0
\(385\) −1154.28 + 420.125i −2.99814 + 1.09123i
\(386\) 0 0
\(387\) −36.2826 62.8433i −0.0937535 0.162386i
\(388\) 0 0
\(389\) −13.0073 73.7682i −0.0334379 0.189636i 0.963514 0.267659i \(-0.0862501\pi\)
−0.996952 + 0.0780237i \(0.975139\pi\)
\(390\) 0 0
\(391\) −18.2242 + 31.5652i −0.0466092 + 0.0807294i
\(392\) 0 0
\(393\) −439.953 524.316i −1.11947 1.33414i
\(394\) 0 0
\(395\) −42.8627 + 117.764i −0.108513 + 0.298138i
\(396\) 0 0
\(397\) −43.4519 + 246.428i −0.109451 + 0.620726i 0.879898 + 0.475162i \(0.157611\pi\)
−0.989349 + 0.145564i \(0.953500\pi\)
\(398\) 0 0
\(399\) −725.785 402.738i −1.81901 1.00937i
\(400\) 0 0
\(401\) 178.013 + 31.3885i 0.443923 + 0.0782755i 0.391141 0.920331i \(-0.372080\pi\)
0.0527811 + 0.998606i \(0.483191\pi\)
\(402\) 0 0
\(403\) −433.344 157.724i −1.07530 0.391376i
\(404\) 0 0
\(405\) −465.258 + 390.398i −1.14879 + 0.963946i
\(406\) 0 0
\(407\) 420.765 + 242.929i 1.03382 + 0.596877i
\(408\) 0 0
\(409\) −211.559 + 37.3036i −0.517259 + 0.0912068i −0.426183 0.904637i \(-0.640142\pi\)
−0.0910766 + 0.995844i \(0.529031\pi\)
\(410\) 0 0
\(411\) 223.719 129.164i 0.544328 0.314268i
\(412\) 0 0
\(413\) 369.167 + 1014.28i 0.893866 + 2.45588i
\(414\) 0 0
\(415\) −571.785 479.785i −1.37780 1.15611i
\(416\) 0 0
\(417\) 427.523i 1.02523i
\(418\) 0 0
\(419\) −611.450 −1.45931 −0.729654 0.683816i \(-0.760319\pi\)
−0.729654 + 0.683816i \(0.760319\pi\)
\(420\) 0 0
\(421\) −521.010 + 620.916i −1.23755 + 1.47486i −0.411359 + 0.911473i \(0.634946\pi\)
−0.826194 + 0.563385i \(0.809499\pi\)
\(422\) 0 0
\(423\) 57.5088 20.9315i 0.135955 0.0494834i
\(424\) 0 0
\(425\) 9.50499 + 16.4631i 0.0223647 + 0.0387368i
\(426\) 0 0
\(427\) 43.0405 + 244.095i 0.100798 + 0.571651i
\(428\) 0 0
\(429\) 544.834 943.681i 1.27001 2.19972i
\(430\) 0 0
\(431\) 328.866 + 391.927i 0.763030 + 0.909344i 0.998036 0.0626502i \(-0.0199552\pi\)
−0.235005 + 0.971994i \(0.575511\pi\)
\(432\) 0 0
\(433\) 175.436 482.008i 0.405165 1.11318i −0.554536 0.832160i \(-0.687104\pi\)
0.959701 0.281022i \(-0.0906734\pi\)
\(434\) 0 0
\(435\) 49.3846 280.074i 0.113528 0.643848i
\(436\) 0 0
\(437\) −330.005 267.480i −0.755159 0.612082i
\(438\) 0 0
\(439\) 40.4326 + 7.12935i 0.0921015 + 0.0162400i 0.219509 0.975610i \(-0.429554\pi\)
−0.127407 + 0.991850i \(0.540666\pi\)
\(440\) 0 0
\(441\) 342.695 + 124.731i 0.777086 + 0.282836i
\(442\) 0 0
\(443\) 643.382 539.862i 1.45233 1.21865i 0.521472 0.853268i \(-0.325383\pi\)
0.930858 0.365382i \(-0.119062\pi\)
\(444\) 0 0
\(445\) −476.372 275.034i −1.07050 0.618053i
\(446\) 0 0
\(447\) −476.275 + 83.9801i −1.06549 + 0.187875i
\(448\) 0 0
\(449\) −417.559 + 241.078i −0.929975 + 0.536921i −0.886804 0.462146i \(-0.847080\pi\)
−0.0431715 + 0.999068i \(0.513746\pi\)
\(450\) 0 0
\(451\) −94.0307 258.347i −0.208494 0.572832i
\(452\) 0 0
\(453\) −613.061 514.419i −1.35334 1.13558i
\(454\) 0 0
\(455\) 1399.38i 3.07556i
\(456\) 0 0
\(457\) 680.738 1.48958 0.744790 0.667299i \(-0.232550\pi\)
0.744790 + 0.667299i \(0.232550\pi\)
\(458\) 0 0
\(459\) 20.2904 24.1812i 0.0442058 0.0526824i
\(460\) 0 0
\(461\) 415.080 151.077i 0.900391 0.327716i 0.149982 0.988689i \(-0.452079\pi\)
0.750409 + 0.660973i \(0.229856\pi\)
\(462\) 0 0
\(463\) 33.8258 + 58.5880i 0.0730579 + 0.126540i 0.900240 0.435394i \(-0.143391\pi\)
−0.827182 + 0.561934i \(0.810058\pi\)
\(464\) 0 0
\(465\) 91.6488 + 519.766i 0.197094 + 1.11778i
\(466\) 0 0
\(467\) −78.4707 + 135.915i −0.168031 + 0.291039i −0.937728 0.347371i \(-0.887074\pi\)
0.769696 + 0.638410i \(0.220408\pi\)
\(468\) 0 0
\(469\) −343.552 409.429i −0.732519 0.872983i
\(470\) 0 0
\(471\) −268.779 + 738.464i −0.570656 + 1.56786i
\(472\) 0 0
\(473\) −58.6802 + 332.792i −0.124060 + 0.703577i
\(474\) 0 0
\(475\) −206.875 + 79.3045i −0.435526 + 0.166957i
\(476\) 0 0
\(477\) 139.804 + 24.6513i 0.293091 + 0.0516798i
\(478\) 0 0
\(479\) 421.235 + 153.317i 0.879405 + 0.320077i 0.741970 0.670434i \(-0.233892\pi\)
0.137435 + 0.990511i \(0.456114\pi\)
\(480\) 0 0
\(481\) 424.006 355.784i 0.881510 0.739675i
\(482\) 0 0
\(483\) 845.858 + 488.356i 1.75126 + 1.01109i
\(484\) 0 0
\(485\) 675.959 119.190i 1.39373 0.245752i
\(486\) 0 0
\(487\) 217.103 125.345i 0.445797 0.257381i −0.260256 0.965540i \(-0.583807\pi\)
0.706054 + 0.708158i \(0.250474\pi\)
\(488\) 0 0
\(489\) −124.856 343.038i −0.255329 0.701510i
\(490\) 0 0
\(491\) −413.096 346.629i −0.841337 0.705966i 0.116527 0.993188i \(-0.462824\pi\)
−0.957864 + 0.287222i \(0.907268\pi\)
\(492\) 0 0
\(493\) 21.6322i 0.0438788i
\(494\) 0 0
\(495\) −351.333 −0.709763
\(496\) 0 0
\(497\) −100.301 + 119.534i −0.201813 + 0.240512i
\(498\) 0 0
\(499\) 569.100 207.135i 1.14048 0.415101i 0.298393 0.954443i \(-0.403549\pi\)
0.842087 + 0.539342i \(0.181327\pi\)
\(500\) 0 0
\(501\) 419.934 + 727.347i 0.838191 + 1.45179i
\(502\) 0 0
\(503\) −56.2221 318.851i −0.111774 0.633899i −0.988297 0.152540i \(-0.951255\pi\)
0.876524 0.481359i \(-0.159857\pi\)
\(504\) 0 0
\(505\) 570.948 988.911i 1.13059 1.95824i
\(506\) 0 0
\(507\) −413.414 492.688i −0.815412 0.971770i
\(508\) 0 0
\(509\) −214.210 + 588.537i −0.420845 + 1.15626i 0.530380 + 0.847760i \(0.322049\pi\)
−0.951224 + 0.308501i \(0.900173\pi\)
\(510\) 0 0
\(511\) 35.7761 202.897i 0.0700120 0.397058i
\(512\) 0 0
\(513\) 241.235 + 277.762i 0.470243 + 0.541447i
\(514\) 0 0
\(515\) −770.035 135.778i −1.49521 0.263647i
\(516\) 0 0
\(517\) −267.810 97.4750i −0.518008 0.188540i
\(518\) 0 0
\(519\) 679.273 569.978i 1.30881 1.09822i
\(520\) 0 0
\(521\) 809.906 + 467.599i 1.55452 + 0.897503i 0.997765 + 0.0668280i \(0.0212879\pi\)
0.556757 + 0.830675i \(0.312045\pi\)
\(522\) 0 0
\(523\) 929.543 163.904i 1.77733 0.313391i 0.813830 0.581103i \(-0.197379\pi\)
0.963499 + 0.267712i \(0.0862674\pi\)
\(524\) 0 0
\(525\) 441.165 254.707i 0.840315 0.485156i
\(526\) 0 0
\(527\) −13.7306 37.7244i −0.0260542 0.0715834i
\(528\) 0 0
\(529\) −22.3246 18.7326i −0.0422015 0.0354113i
\(530\) 0 0
\(531\) 308.719i 0.581392i
\(532\) 0 0
\(533\) −313.204 −0.587625
\(534\) 0 0
\(535\) −44.6095 + 53.1635i −0.0833822 + 0.0993710i
\(536\) 0 0
\(537\) 113.090 41.1614i 0.210596 0.0766506i
\(538\) 0 0
\(539\) −849.151 1470.77i −1.57542 2.72871i
\(540\) 0 0
\(541\) 25.7223 + 145.878i 0.0475458 + 0.269646i 0.999308 0.0371903i \(-0.0118408\pi\)
−0.951762 + 0.306836i \(0.900730\pi\)
\(542\) 0 0
\(543\) −378.097 + 654.884i −0.696312 + 1.20605i
\(544\) 0 0
\(545\) 256.058 + 305.158i 0.469831 + 0.559922i
\(546\) 0 0
\(547\) −283.423 + 778.698i −0.518141 + 1.42358i 0.354425 + 0.935084i \(0.384676\pi\)
−0.872566 + 0.488496i \(0.837546\pi\)
\(548\) 0 0
\(549\) −12.3103 + 69.8154i −0.0224232 + 0.127168i
\(550\) 0 0
\(551\) −248.994 39.5522i −0.451895 0.0717826i
\(552\) 0 0
\(553\) −251.564 44.3576i −0.454908 0.0802126i
\(554\) 0 0
\(555\) −595.270 216.660i −1.07256 0.390379i
\(556\) 0 0
\(557\) −220.380 + 184.921i −0.395655 + 0.331994i −0.818812 0.574062i \(-0.805367\pi\)
0.423156 + 0.906057i \(0.360922\pi\)
\(558\) 0 0
\(559\) 333.397 + 192.487i 0.596417 + 0.344341i
\(560\) 0 0
\(561\) 93.4191 16.4723i 0.166523 0.0293624i
\(562\) 0 0
\(563\) 477.907 275.920i 0.848858 0.490089i −0.0114072 0.999935i \(-0.503631\pi\)
0.860265 + 0.509846i \(0.170298\pi\)
\(564\) 0 0
\(565\) 290.133 + 797.133i 0.513509 + 1.41085i
\(566\) 0 0
\(567\) −948.339 795.751i −1.67255 1.40344i
\(568\) 0 0
\(569\) 549.760i 0.966186i 0.875569 + 0.483093i \(0.160487\pi\)
−0.875569 + 0.483093i \(0.839513\pi\)
\(570\) 0 0
\(571\) −662.672 −1.16055 −0.580274 0.814422i \(-0.697054\pi\)
−0.580274 + 0.814422i \(0.697054\pi\)
\(572\) 0 0
\(573\) 64.9617 77.4183i 0.113371 0.135111i
\(574\) 0 0
\(575\) 244.983 89.1665i 0.426057 0.155072i
\(576\) 0 0
\(577\) 3.19265 + 5.52984i 0.00553320 + 0.00958378i 0.868779 0.495200i \(-0.164905\pi\)
−0.863246 + 0.504784i \(0.831572\pi\)
\(578\) 0 0
\(579\) −1.60257 9.08864i −0.00276783 0.0156971i
\(580\) 0 0
\(581\) 760.708 1317.59i 1.30931 2.26779i
\(582\) 0 0
\(583\) −424.942 506.426i −0.728889 0.868656i
\(584\) 0 0
\(585\) −136.893 + 376.109i −0.234004 + 0.642922i
\(586\) 0 0
\(587\) −177.108 + 1004.43i −0.301717 + 1.71112i 0.336851 + 0.941558i \(0.390638\pi\)
−0.638568 + 0.769565i \(0.720473\pi\)
\(588\) 0 0
\(589\) 459.325 89.0682i 0.779839 0.151219i
\(590\) 0 0
\(591\) 278.486 + 49.1047i 0.471212 + 0.0830874i
\(592\) 0 0
\(593\) −729.961 265.684i −1.23096 0.448034i −0.357036 0.934091i \(-0.616213\pi\)
−0.873927 + 0.486057i \(0.838435\pi\)
\(594\) 0 0
\(595\) −93.3211 + 78.3057i −0.156842 + 0.131606i
\(596\) 0 0
\(597\) −495.161 285.881i −0.829415 0.478863i
\(598\) 0 0
\(599\) 644.567 113.654i 1.07607 0.189740i 0.392594 0.919712i \(-0.371578\pi\)
0.683477 + 0.729972i \(0.260467\pi\)
\(600\) 0 0
\(601\) −353.019 + 203.816i −0.587387 + 0.339128i −0.764063 0.645141i \(-0.776799\pi\)
0.176677 + 0.984269i \(0.443465\pi\)
\(602\) 0 0
\(603\) −52.2840 143.649i −0.0867064 0.238224i
\(604\) 0 0
\(605\) 692.103 + 580.743i 1.14397 + 0.959906i
\(606\) 0 0
\(607\) 342.676i 0.564540i 0.959335 + 0.282270i \(0.0910874\pi\)
−0.959335 + 0.282270i \(0.908913\pi\)
\(608\) 0 0
\(609\) 579.683 0.951860
\(610\) 0 0
\(611\) −208.698 + 248.717i −0.341568 + 0.407065i
\(612\) 0 0
\(613\) 119.947 43.6572i 0.195672 0.0712189i −0.242325 0.970195i \(-0.577910\pi\)
0.437997 + 0.898976i \(0.355688\pi\)
\(614\) 0 0
\(615\) 179.228 + 310.433i 0.291428 + 0.504769i
\(616\) 0 0
\(617\) 142.570 + 808.554i 0.231070 + 1.31046i 0.850735 + 0.525595i \(0.176157\pi\)
−0.619666 + 0.784866i \(0.712732\pi\)
\(618\) 0 0
\(619\) −181.415 + 314.220i −0.293077 + 0.507625i −0.974536 0.224232i \(-0.928013\pi\)
0.681458 + 0.731857i \(0.261346\pi\)
\(620\) 0 0
\(621\) −278.266 331.624i −0.448093 0.534016i
\(622\) 0 0
\(623\) 383.475 1053.59i 0.615529 1.69115i
\(624\) 0 0
\(625\) −135.540 + 768.687i −0.216865 + 1.22990i
\(626\) 0 0
\(627\) 18.7948 + 1105.40i 0.0299758 + 1.76300i
\(628\) 0 0
\(629\) 47.4526 + 8.36717i 0.0754413 + 0.0133023i
\(630\) 0 0
\(631\) −966.329 351.715i −1.53142 0.557393i −0.567455 0.823405i \(-0.692072\pi\)
−0.963970 + 0.266012i \(0.914294\pi\)
\(632\) 0 0
\(633\) −637.347 + 534.797i −1.00687 + 0.844861i
\(634\) 0 0
\(635\) −415.333 239.793i −0.654068 0.377626i
\(636\) 0 0
\(637\) −1905.35 + 335.965i −2.99114 + 0.527418i
\(638\) 0 0
\(639\) −38.6511 + 22.3152i −0.0604868 + 0.0349221i
\(640\) 0 0
\(641\) −152.065 417.795i −0.237231 0.651786i −0.999987 0.00510007i \(-0.998377\pi\)
0.762756 0.646686i \(-0.223846\pi\)
\(642\) 0 0
\(643\) 219.708 + 184.356i 0.341691 + 0.286713i 0.797443 0.603394i \(-0.206185\pi\)
−0.455752 + 0.890107i \(0.650630\pi\)
\(644\) 0 0
\(645\) 440.596i 0.683094i
\(646\) 0 0
\(647\) 116.031 0.179337 0.0896685 0.995972i \(-0.471419\pi\)
0.0896685 + 0.995972i \(0.471419\pi\)
\(648\) 0 0
\(649\) 924.111 1101.31i 1.42390 1.69694i
\(650\) 0 0
\(651\) −1010.91 + 367.940i −1.55285 + 0.565193i
\(652\) 0 0
\(653\) −306.370 530.649i −0.469173 0.812632i 0.530206 0.847869i \(-0.322115\pi\)
−0.999379 + 0.0352370i \(0.988781\pi\)
\(654\) 0 0
\(655\) −203.299 1152.97i −0.310381 1.76026i
\(656\) 0 0
\(657\) 29.4636 51.0324i 0.0448456 0.0776749i
\(658\) 0 0
\(659\) 190.151 + 226.613i 0.288545 + 0.343874i 0.890772 0.454451i \(-0.150164\pi\)
−0.602227 + 0.798325i \(0.705720\pi\)
\(660\) 0 0
\(661\) 267.124 733.918i 0.404121 1.11031i −0.556110 0.831109i \(-0.687707\pi\)
0.960231 0.279206i \(-0.0900711\pi\)
\(662\) 0 0
\(663\) 18.7657 106.425i 0.0283042 0.160521i
\(664\) 0 0
\(665\) −730.696 1217.33i −1.09879 1.83057i
\(666\) 0 0
\(667\) 292.160 + 51.5157i 0.438021 + 0.0772350i
\(668\) 0 0
\(669\) 1055.53 + 384.182i 1.57778 + 0.574263i
\(670\) 0 0
\(671\) 252.899 212.207i 0.376899 0.316256i
\(672\) 0 0
\(673\) 636.410 + 367.431i 0.945631 + 0.545960i 0.891721 0.452586i \(-0.149498\pi\)
0.0539099 + 0.998546i \(0.482832\pi\)
\(674\) 0 0
\(675\) −222.355 + 39.2072i −0.329415 + 0.0580848i
\(676\) 0 0
\(677\) −588.076 + 339.526i −0.868650 + 0.501515i −0.866899 0.498483i \(-0.833891\pi\)
−0.00175044 + 0.999998i \(0.500557\pi\)
\(678\) 0 0
\(679\) 478.509 + 1314.69i 0.704726 + 1.93622i
\(680\) 0 0
\(681\) −748.895 628.397i −1.09970 0.922757i
\(682\) 0 0
\(683\) 233.214i 0.341455i 0.985318 + 0.170727i \(0.0546118\pi\)
−0.985318 + 0.170727i \(0.945388\pi\)
\(684\) 0 0
\(685\) 441.874 0.645072
\(686\) 0 0
\(687\) −194.039 + 231.247i −0.282444 + 0.336603i
\(688\) 0 0
\(689\) −707.714 + 257.587i −1.02716 + 0.373856i
\(690\) 0 0
\(691\) 635.093 + 1100.01i 0.919092 + 1.59191i 0.800797 + 0.598936i \(0.204410\pi\)
0.118296 + 0.992978i \(0.462257\pi\)
\(692\) 0 0
\(693\) −124.354 705.244i −0.179442 1.01767i
\(694\) 0 0
\(695\) 365.642 633.310i 0.526103 0.911238i
\(696\) 0 0
\(697\) −17.5261 20.8868i −0.0251450 0.0299667i
\(698\) 0 0
\(699\) 202.263 555.714i 0.289361 0.795013i
\(700\) 0 0
\(701\) −39.0955 + 221.722i −0.0557711 + 0.316293i −0.999912 0.0132478i \(-0.995783\pi\)
0.944141 + 0.329541i \(0.106894\pi\)
\(702\) 0 0
\(703\) −183.071 + 530.896i −0.260413 + 0.755186i
\(704\) 0 0
\(705\) 365.942 + 64.5254i 0.519066 + 0.0915254i
\(706\) 0 0
\(707\) 2187.17 + 796.063i 3.09359 + 1.12597i
\(708\) 0 0
\(709\) −248.832 + 208.795i −0.350962 + 0.294492i −0.801176 0.598429i \(-0.795792\pi\)
0.450214 + 0.892921i \(0.351348\pi\)
\(710\) 0 0
\(711\) −63.2733 36.5308i −0.0889920 0.0513795i
\(712\) 0 0
\(713\) −542.196 + 95.6038i −0.760444 + 0.134087i
\(714\) 0 0
\(715\) 1614.18 931.947i 2.25759 1.30342i
\(716\) 0 0
\(717\) 201.272 + 552.992i 0.280715 + 0.771257i
\(718\) 0 0
\(719\) 341.999 + 286.972i 0.475660 + 0.399126i 0.848854 0.528627i \(-0.177293\pi\)
−0.373194 + 0.927753i \(0.621737\pi\)
\(720\) 0 0
\(721\) 1593.78i 2.21051i
\(722\) 0 0
\(723\) 541.052 0.748343
\(724\) 0 0
\(725\) 99.4582 118.530i 0.137184 0.163489i
\(726\) 0 0
\(727\) 262.192 95.4301i 0.360649 0.131266i −0.155339 0.987861i \(-0.549647\pi\)
0.515988 + 0.856596i \(0.327425\pi\)
\(728\) 0 0
\(729\) 131.389 + 227.572i 0.180232 + 0.312171i
\(730\) 0 0
\(731\) 5.81957 + 33.0044i 0.00796111 + 0.0451497i
\(732\) 0 0
\(733\) −610.557 + 1057.52i −0.832956 + 1.44272i 0.0627271 + 0.998031i \(0.480020\pi\)
−0.895683 + 0.444692i \(0.853313\pi\)
\(734\) 0 0
\(735\) 1423.32 + 1696.24i 1.93648 + 2.30781i
\(736\) 0 0
\(737\) −243.479 + 668.954i −0.330365 + 0.907671i
\(738\) 0 0
\(739\) 179.692 1019.08i 0.243155 1.37900i −0.581583 0.813487i \(-0.697566\pi\)
0.824738 0.565515i \(-0.191323\pi\)
\(740\) 0 0
\(741\) 1190.68 + 410.586i 1.60685 + 0.554097i
\(742\) 0 0
\(743\) −120.400 21.2298i −0.162046 0.0285731i 0.0920363 0.995756i \(-0.470662\pi\)
−0.254083 + 0.967183i \(0.581774\pi\)
\(744\) 0 0
\(745\) −777.354 282.934i −1.04343 0.379777i
\(746\) 0 0
\(747\) 333.345 279.710i 0.446246 0.374445i
\(748\) 0 0
\(749\) −122.507 70.7292i −0.163560 0.0944315i
\(750\) 0 0
\(751\) −745.826 + 131.509i −0.993111 + 0.175112i −0.646514 0.762902i \(-0.723774\pi\)
−0.346597 + 0.938014i \(0.612663\pi\)
\(752\) 0 0
\(753\) 115.550 66.7130i 0.153453 0.0885963i
\(754\) 0 0
\(755\) −468.197 1286.36i −0.620128 1.70379i
\(756\) 0 0
\(757\) 830.988 + 697.282i 1.09774 + 0.921112i 0.997271 0.0738286i \(-0.0235218\pi\)
0.100467 + 0.994940i \(0.467966\pi\)
\(758\) 0 0
\(759\) 1300.93i 1.71400i
\(760\) 0 0
\(761\) 125.987 0.165555 0.0827774 0.996568i \(-0.473621\pi\)
0.0827774 + 0.996568i \(0.473621\pi\)
\(762\) 0 0
\(763\) −521.924 + 622.005i −0.684042 + 0.815209i
\(764\) 0 0
\(765\) −32.7419 + 11.9171i −0.0427999 + 0.0155779i
\(766\) 0 0
\(767\) −818.909 1418.39i −1.06768 1.84927i
\(768\) 0 0
\(769\) −14.5819 82.6982i −0.0189622 0.107540i 0.973858 0.227159i \(-0.0729437\pi\)
−0.992820 + 0.119619i \(0.961833\pi\)
\(770\) 0 0
\(771\) 42.0174 72.7763i 0.0544973 0.0943921i
\(772\) 0 0
\(773\) −101.121 120.511i −0.130816 0.155900i 0.696660 0.717401i \(-0.254669\pi\)
−0.827476 + 0.561501i \(0.810224\pi\)
\(774\) 0 0
\(775\) −98.2111 + 269.833i −0.126724 + 0.348171i
\(776\) 0 0
\(777\) 224.216 1271.59i 0.288567 1.63654i
\(778\) 0 0
\(779\) 272.458 163.542i 0.349753 0.209938i
\(780\) 0 0
\(781\) 204.680 + 36.0906i 0.262074 + 0.0462108i
\(782\) 0 0
\(783\) −241.436 87.8754i −0.308347 0.112229i
\(784\) 0 0
\(785\) −1029.73 + 864.048i −1.31176 + 1.10070i
\(786\) 0 0
\(787\) −1094.56 631.944i −1.39080 0.802978i −0.397395 0.917648i \(-0.630086\pi\)
−0.993404 + 0.114670i \(0.963419\pi\)
\(788\) 0 0
\(789\) −36.5619 + 6.44685i −0.0463396 + 0.00817092i
\(790\) 0 0
\(791\) −1497.42 + 864.538i −1.89308 + 1.09297i
\(792\) 0 0
\(793\) −128.634 353.418i −0.162211 0.445672i
\(794\) 0 0
\(795\) 660.291 + 554.050i 0.830555 + 0.696918i
\(796\) 0 0
\(797\) 477.946i 0.599682i −0.953989 0.299841i \(-0.903066\pi\)
0.953989 0.299841i \(-0.0969335\pi\)
\(798\) 0 0
\(799\) −28.2645 −0.0353748
\(800\) 0 0
\(801\) 206.132 245.658i 0.257343 0.306689i
\(802\) 0 0
\(803\) −257.866 + 93.8557i −0.321129 + 0.116881i
\(804\) 0 0
\(805\) 835.341 + 1446.85i 1.03769 + 1.79733i
\(806\) 0 0
\(807\) 267.315 + 1516.02i 0.331245 + 1.87858i
\(808\) 0 0
\(809\) −232.123 + 402.050i −0.286926 + 0.496971i −0.973075 0.230491i \(-0.925967\pi\)
0.686148 + 0.727462i \(0.259300\pi\)
\(810\) 0 0
\(811\) 67.3090 + 80.2157i 0.0829951 + 0.0989097i 0.805944 0.591992i \(-0.201658\pi\)
−0.722949 + 0.690902i \(0.757214\pi\)
\(812\) 0 0
\(813\) −115.404 + 317.071i −0.141949 + 0.390001i
\(814\) 0 0
\(815\) 108.431 614.943i 0.133044 0.754532i
\(816\) 0 0
\(817\) −390.532 + 6.64010i −0.478007 + 0.00812742i
\(818\) 0 0
\(819\) −803.431 141.667i −0.980991 0.172975i
\(820\) 0 0
\(821\) 195.635 + 71.2054i 0.238289 + 0.0867300i 0.458404 0.888744i \(-0.348421\pi\)
−0.220115 + 0.975474i \(0.570643\pi\)
\(822\) 0 0
\(823\) 523.589 439.343i 0.636195 0.533831i −0.266652 0.963793i \(-0.585917\pi\)
0.902847 + 0.429962i \(0.141473\pi\)
\(824\) 0 0
\(825\) −587.607 339.255i −0.712251 0.411218i
\(826\) 0 0
\(827\) 831.241 146.570i 1.00513 0.177231i 0.353229 0.935537i \(-0.385084\pi\)
0.651899 + 0.758306i \(0.273973\pi\)
\(828\) 0 0
\(829\) −77.9552 + 45.0075i −0.0940353 + 0.0542913i −0.546280 0.837602i \(-0.683957\pi\)
0.452245 + 0.891894i \(0.350623\pi\)
\(830\) 0 0
\(831\) −326.083 895.906i −0.392398 1.07811i
\(832\) 0 0
\(833\) −129.023 108.263i −0.154890 0.129968i
\(834\) 0 0
\(835\) 1436.61i 1.72049i
\(836\) 0 0
\(837\) 476.816 0.569673
\(838\) 0 0
\(839\) 495.740 590.800i 0.590870 0.704171i −0.384903 0.922957i \(-0.625765\pi\)
0.975773 + 0.218786i \(0.0702096\pi\)
\(840\) 0 0
\(841\) −624.827 + 227.418i −0.742957 + 0.270414i
\(842\) 0 0
\(843\) −175.489 303.955i −0.208171 0.360563i
\(844\) 0 0
\(845\) −191.036 1083.42i −0.226078 1.28215i
\(846\) 0 0
\(847\) −920.780 + 1594.84i −1.08711 + 1.88293i
\(848\) 0 0
\(849\) −363.682 433.419i −0.428365 0.510505i
\(850\) 0 0
\(851\) 226.010 620.958i 0.265582 0.729680i
\(852\) 0 0
\(853\) 110.511 626.742i 0.129556 0.734750i −0.848941 0.528488i \(-0.822759\pi\)
0.978497 0.206262i \(-0.0661298\pi\)
\(854\) 0 0
\(855\) −77.3044 398.659i −0.0904145 0.466268i
\(856\) 0 0
\(857\) −861.767 151.953i −1.00556 0.177308i −0.353469 0.935446i \(-0.614998\pi\)
−0.652094 + 0.758138i \(0.726109\pi\)
\(858\) 0 0
\(859\) 328.931 + 119.721i 0.382923 + 0.139373i 0.526307 0.850294i \(-0.323576\pi\)
−0.143384 + 0.989667i \(0.545798\pi\)
\(860\) 0 0
\(861\) −559.706 + 469.649i −0.650065 + 0.545470i
\(862\) 0 0
\(863\) −573.859 331.317i −0.664958 0.383914i 0.129206 0.991618i \(-0.458757\pi\)
−0.794163 + 0.607704i \(0.792091\pi\)
\(864\) 0 0
\(865\) 1493.72 263.383i 1.72684 0.304489i
\(866\) 0 0
\(867\) −877.789 + 506.791i −1.01244 + 0.584535i
\(868\) 0 0
\(869\) 116.368 + 319.719i 0.133911 + 0.367916i
\(870\) 0 0
\(871\) 621.260 + 521.299i 0.713272 + 0.598506i
\(872\) 0 0
\(873\) 400.157i 0.458370i
\(874\) 0 0
\(875\) −996.786 −1.13918
\(876\) 0 0
\(877\) −361.550 + 430.879i −0.412258 + 0.491310i −0.931717 0.363186i \(-0.881689\pi\)
0.519459 + 0.854495i \(0.326134\pi\)
\(878\) 0 0
\(879\) 1702.77 619.758i 1.93717 0.705072i
\(880\) 0 0
\(881\) 638.016 + 1105.08i 0.724195 + 1.25434i 0.959305 + 0.282373i \(0.0911217\pi\)
−0.235110 + 0.971969i \(0.575545\pi\)
\(882\) 0 0
\(883\) 129.225 + 732.873i 0.146348 + 0.829981i 0.966275 + 0.257513i \(0.0829030\pi\)
−0.819927 + 0.572468i \(0.805986\pi\)
\(884\) 0 0
\(885\) −937.228 + 1623.33i −1.05902 + 1.83427i
\(886\) 0 0
\(887\) −630.328 751.196i −0.710629 0.846895i 0.283055 0.959104i \(-0.408652\pi\)
−0.993685 + 0.112208i \(0.964208\pi\)
\(888\) 0 0
\(889\) 334.339 918.588i 0.376084 1.03328i
\(890\) 0 0
\(891\) −286.329 + 1623.85i −0.321357 + 1.82251i
\(892\) 0 0
\(893\) 51.6785 325.333i 0.0578707 0.364315i
\(894\) 0 0
\(895\) 202.729 + 35.7466i 0.226513 + 0.0399404i
\(896\) 0 0
\(897\) −1392.67 506.890i −1.55258 0.565094i
\(898\) 0 0
\(899\) −250.312 + 210.037i −0.278434 + 0.233634i
\(900\) 0 0
\(901\) −56.7796 32.7817i −0.0630184 0.0363837i
\(902\) 0 0
\(903\) 884.425 155.948i 0.979430 0.172700i
\(904\) 0 0
\(905\) −1120.19 + 646.741i −1.23778 + 0.714631i
\(906\) 0 0
\(907\) −342.939 942.216i −0.378102 1.03883i −0.972142 0.234392i \(-0.924690\pi\)
0.594040 0.804435i \(-0.297532\pi\)
\(908\) 0 0
\(909\) 509.967 + 427.913i 0.561020 + 0.470752i
\(910\) 0 0
\(911\) 420.233i 0.461287i −0.973038 0.230644i \(-0.925917\pi\)
0.973038 0.230644i \(-0.0740832\pi\)
\(912\) 0 0
\(913\) −2026.44 −2.21954
\(914\) 0 0
\(915\) −276.681 + 329.736i −0.302384 + 0.360367i
\(916\) 0 0
\(917\) 2242.44 816.181i 2.44541 0.890056i
\(918\) 0 0
\(919\) 143.066 + 247.798i 0.155676 + 0.269639i 0.933305 0.359085i \(-0.116911\pi\)
−0.777629 + 0.628724i \(0.783578\pi\)
\(920\) 0 0
\(921\) 50.7505 + 287.820i 0.0551036 + 0.312508i
\(922\) 0 0
\(923\) 118.387 205.052i 0.128263 0.222158i
\(924\) 0 0
\(925\) −221.538 264.018i −0.239500 0.285425i
\(926\) 0 0
\(927\) 155.909 428.358i 0.168187 0.462090i
\(928\) 0 0
\(929\) 133.744 758.499i 0.143965 0.816468i −0.824227 0.566260i \(-0.808390\pi\)
0.968192 0.250208i \(-0.0804991\pi\)
\(930\) 0 0
\(931\) 1482.05 1287.15i 1.59189 1.38255i
\(932\) 0 0
\(933\) 687.096 + 121.153i 0.736437 + 0.129854i
\(934\) 0 0
\(935\) 152.474 + 55.4962i 0.163074 + 0.0593542i
\(936\) 0 0
\(937\) 73.1662 61.3938i 0.0780856 0.0655216i −0.602909 0.797810i \(-0.705992\pi\)
0.680995 + 0.732288i \(0.261548\pi\)
\(938\) 0 0
\(939\) −460.546 265.896i −0.490464 0.283170i
\(940\) 0 0
\(941\) 980.482 172.885i 1.04196 0.183725i 0.373619 0.927582i \(-0.378117\pi\)
0.668339 + 0.743857i \(0.267006\pi\)
\(942\) 0 0
\(943\) −323.829 + 186.963i −0.343403 + 0.198264i
\(944\) 0 0
\(945\) −494.870 1359.65i −0.523672 1.43878i
\(946\) 0 0
\(947\) 537.232 + 450.791i 0.567299 + 0.476021i 0.880748 0.473584i \(-0.157040\pi\)
−0.313449 + 0.949605i \(0.601485\pi\)
\(948\) 0 0
\(949\) 312.621i 0.329422i
\(950\) 0 0
\(951\) 900.846 0.947262
\(952\) 0 0
\(953\) 441.334 525.961i 0.463099 0.551900i −0.483066 0.875584i \(-0.660477\pi\)
0.946165 + 0.323683i \(0.104921\pi\)
\(954\) 0 0
\(955\) 162.444 59.1246i 0.170098 0.0619106i
\(956\) 0 0
\(957\) −386.052 668.662i −0.403398 0.698706i
\(958\) 0 0
\(959\) 156.401 + 886.991i 0.163087 + 0.924913i
\(960\) 0 0
\(961\) −177.297 + 307.087i −0.184492 + 0.319549i
\(962\) 0 0
\(963\) −26.0069 30.9938i −0.0270061 0.0321846i
\(964\) 0 0
\(965\) 5.39916 14.8341i 0.00559498 0.0153721i
\(966\) 0 0
\(967\) 140.913 799.159i 0.145722 0.826431i −0.821062 0.570838i \(-0.806618\pi\)
0.966785 0.255593i \(-0.0822706\pi\)
\(968\) 0 0
\(969\) 39.2464 + 102.379i 0.0405019 + 0.105654i
\(970\) 0 0
\(971\) −130.384 22.9902i −0.134278 0.0236769i 0.106105 0.994355i \(-0.466162\pi\)
−0.240383 + 0.970678i \(0.577273\pi\)
\(972\) 0 0
\(973\) 1400.69 + 509.808i 1.43955 + 0.523955i
\(974\) 0 0
\(975\) −592.133 + 496.859i −0.607316 + 0.509599i
\(976\) 0 0
\(977\) 233.290 + 134.690i 0.238782 + 0.137861i 0.614617 0.788826i \(-0.289311\pi\)
−0.375835 + 0.926687i \(0.622644\pi\)
\(978\) 0 0
\(979\) −1470.69 + 259.323i −1.50224 + 0.264885i
\(980\) 0 0
\(981\) −201.123 + 116.119i −0.205019 + 0.118368i
\(982\) 0 0
\(983\) −228.444 627.646i −0.232395 0.638501i 0.767602 0.640927i \(-0.221450\pi\)
−0.999997 + 0.00242643i \(0.999228\pi\)
\(984\) 0 0
\(985\) 370.539 + 310.919i 0.376181 + 0.315654i
\(986\) 0 0
\(987\) 757.408i 0.767384i
\(988\) 0 0
\(989\) 459.609 0.464721
\(990\) 0 0
\(991\) −678.867 + 809.043i −0.685033 + 0.816390i −0.990745 0.135734i \(-0.956661\pi\)
0.305713 + 0.952124i \(0.401105\pi\)
\(992\) 0 0
\(993\) 1248.22 454.314i 1.25702 0.457517i
\(994\) 0 0
\(995\) −489.004 846.980i −0.491462 0.851237i
\(996\) 0 0
\(997\) −252.072 1429.57i −0.252830 1.43387i −0.801581 0.597886i \(-0.796008\pi\)
0.548751 0.835986i \(-0.315104\pi\)
\(998\) 0 0
\(999\) −286.149 + 495.625i −0.286436 + 0.496121i
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 76.3.j.a.21.1 18
4.3 odd 2 304.3.z.b.97.3 18
19.3 odd 18 1444.3.c.c.721.3 18
19.10 odd 18 inner 76.3.j.a.29.1 yes 18
19.16 even 9 1444.3.c.c.721.16 18
76.67 even 18 304.3.z.b.257.3 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
76.3.j.a.21.1 18 1.1 even 1 trivial
76.3.j.a.29.1 yes 18 19.10 odd 18 inner
304.3.z.b.97.3 18 4.3 odd 2
304.3.z.b.257.3 18 76.67 even 18
1444.3.c.c.721.3 18 19.3 odd 18
1444.3.c.c.721.16 18 19.16 even 9