Properties

Label 1764.2.b.j.1567.1
Level $1764$
Weight $2$
Character 1764.1567
Analytic conductor $14.086$
Analytic rank $0$
Dimension $8$
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] = [1764,2,Mod(1567,1764)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1764, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1764.1567");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1764.b (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: \(14.0856109166\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.562828176.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + x^{6} + 2x^{5} - 6x^{4} + 4x^{3} + 4x^{2} - 16x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1567.1
Root \(-1.33790 + 0.458297i\) of defining polynomial
Character \(\chi\) \(=\) 1764.1567
Dual form 1764.2.b.j.1567.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.33790 - 0.458297i) q^{2} +(1.57993 + 1.22631i) q^{4} -2.45262i q^{5} +(-1.55176 - 2.36475i) q^{8} +O(q^{10})\) \(q+(-1.33790 - 0.458297i) q^{2} +(1.57993 + 1.22631i) q^{4} -2.45262i q^{5} +(-1.55176 - 2.36475i) q^{8} +(-1.12403 + 3.28134i) q^{10} +1.26539i q^{11} +2.99744i q^{13} +(0.992338 + 3.87495i) q^{16} +1.83319i q^{17} -4.15985 q^{19} +(3.00766 - 3.87495i) q^{20} +(0.579927 - 1.69296i) q^{22} -6.73842i q^{23} -1.01532 q^{25} +(1.37372 - 4.01027i) q^{26} +9.42323 q^{29} +9.43978 q^{31} +(0.448237 - 5.63907i) q^{32} +(0.840146 - 2.45262i) q^{34} +7.51143 q^{37} +(5.56545 + 1.90645i) q^{38} +(-5.79982 + 3.80588i) q^{40} +1.08966i q^{41} -6.27176i q^{43} +(-1.55176 + 1.99923i) q^{44} +(-3.08820 + 9.01530i) q^{46} -7.35158 q^{47} +(1.35840 + 0.465320i) q^{50} +(-3.67579 + 4.73574i) q^{52} +0.0716524 q^{53} +3.10353 q^{55} +(-12.6073 - 4.31864i) q^{58} -3.36690 q^{59} -11.1024i q^{61} +(-12.6294 - 4.32623i) q^{62} +(-3.18406 + 7.33906i) q^{64} +7.35158 q^{65} -2.80766i q^{67} +(-2.24806 + 2.89631i) q^{68} +2.92285i q^{71} -8.10495i q^{73} +(-10.0495 - 3.44247i) q^{74} +(-6.57226 - 5.10126i) q^{76} +1.78368i q^{79} +(9.50377 - 2.43382i) q^{80} +(0.499388 - 1.45785i) q^{82} -5.33626 q^{83} +4.49611 q^{85} +(-2.87433 + 8.39096i) q^{86} +(2.99234 - 1.96359i) q^{88} -8.57161i q^{89} +(8.26338 - 10.6462i) q^{92} +(9.83564 + 3.36921i) q^{94} +10.2025i q^{95} -7.10394i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} + 2 q^{4} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{2} + 2 q^{4} - 4 q^{8} + 8 q^{10} + 10 q^{16} - 12 q^{19} + 22 q^{20} - 6 q^{22} - 4 q^{25} - 6 q^{26} + 16 q^{29} + 12 q^{31} + 12 q^{32} + 28 q^{34} - 12 q^{37} - 2 q^{38} - 4 q^{40} - 4 q^{44} - 12 q^{46} - 8 q^{47} - 2 q^{50} - 4 q^{52} - 8 q^{53} + 8 q^{55} - 14 q^{58} + 28 q^{59} - 48 q^{62} + 2 q^{64} + 8 q^{65} + 16 q^{68} - 38 q^{74} - 44 q^{76} + 6 q^{80} + 4 q^{82} + 4 q^{83} - 32 q^{85} - 6 q^{86} + 26 q^{88} + 28 q^{92} + 32 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.33790 0.458297i −0.946035 0.324065i
\(3\) 0 0
\(4\) 1.57993 + 1.22631i 0.789963 + 0.613154i
\(5\) 2.45262i 1.09684i −0.836202 0.548422i \(-0.815229\pi\)
0.836202 0.548422i \(-0.184771\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −1.55176 2.36475i −0.548631 0.836065i
\(9\) 0 0
\(10\) −1.12403 + 3.28134i −0.355449 + 1.03765i
\(11\) 1.26539i 0.381531i 0.981636 + 0.190765i \(0.0610970\pi\)
−0.981636 + 0.190765i \(0.938903\pi\)
\(12\) 0 0
\(13\) 2.99744i 0.831342i 0.909515 + 0.415671i \(0.136453\pi\)
−0.909515 + 0.415671i \(0.863547\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0.992338 + 3.87495i 0.248084 + 0.968738i
\(17\) 1.83319i 0.444614i 0.974977 + 0.222307i \(0.0713587\pi\)
−0.974977 + 0.222307i \(0.928641\pi\)
\(18\) 0 0
\(19\) −4.15985 −0.954336 −0.477168 0.878812i \(-0.658337\pi\)
−0.477168 + 0.878812i \(0.658337\pi\)
\(20\) 3.00766 3.87495i 0.672534 0.866466i
\(21\) 0 0
\(22\) 0.579927 1.69296i 0.123641 0.360941i
\(23\) 6.73842i 1.40506i −0.711655 0.702529i \(-0.752054\pi\)
0.711655 0.702529i \(-0.247946\pi\)
\(24\) 0 0
\(25\) −1.01532 −0.203065
\(26\) 1.37372 4.01027i 0.269409 0.786478i
\(27\) 0 0
\(28\) 0 0
\(29\) 9.42323 1.74985 0.874925 0.484258i \(-0.160910\pi\)
0.874925 + 0.484258i \(0.160910\pi\)
\(30\) 0 0
\(31\) 9.43978 1.69543 0.847717 0.530448i \(-0.177976\pi\)
0.847717 + 0.530448i \(0.177976\pi\)
\(32\) 0.448237 5.63907i 0.0792379 0.996856i
\(33\) 0 0
\(34\) 0.840146 2.45262i 0.144084 0.420620i
\(35\) 0 0
\(36\) 0 0
\(37\) 7.51143 1.23487 0.617436 0.786621i \(-0.288171\pi\)
0.617436 + 0.786621i \(0.288171\pi\)
\(38\) 5.56545 + 1.90645i 0.902835 + 0.309267i
\(39\) 0 0
\(40\) −5.79982 + 3.80588i −0.917032 + 0.601762i
\(41\) 1.08966i 0.170176i 0.996373 + 0.0850880i \(0.0271171\pi\)
−0.996373 + 0.0850880i \(0.972883\pi\)
\(42\) 0 0
\(43\) 6.27176i 0.956435i −0.878241 0.478218i \(-0.841283\pi\)
0.878241 0.478218i \(-0.158717\pi\)
\(44\) −1.55176 + 1.99923i −0.233937 + 0.301395i
\(45\) 0 0
\(46\) −3.08820 + 9.01530i −0.455330 + 1.32923i
\(47\) −7.35158 −1.07234 −0.536169 0.844111i \(-0.680129\pi\)
−0.536169 + 0.844111i \(0.680129\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 1.35840 + 0.465320i 0.192106 + 0.0658063i
\(51\) 0 0
\(52\) −3.67579 + 4.73574i −0.509740 + 0.656730i
\(53\) 0.0716524 0.00984222 0.00492111 0.999988i \(-0.498434\pi\)
0.00492111 + 0.999988i \(0.498434\pi\)
\(54\) 0 0
\(55\) 3.10353 0.418479
\(56\) 0 0
\(57\) 0 0
\(58\) −12.6073 4.31864i −1.65542 0.567066i
\(59\) −3.36690 −0.438334 −0.219167 0.975687i \(-0.570334\pi\)
−0.219167 + 0.975687i \(0.570334\pi\)
\(60\) 0 0
\(61\) 11.1024i 1.42152i −0.703436 0.710758i \(-0.748352\pi\)
0.703436 0.710758i \(-0.251648\pi\)
\(62\) −12.6294 4.32623i −1.60394 0.549432i
\(63\) 0 0
\(64\) −3.18406 + 7.33906i −0.398008 + 0.917382i
\(65\) 7.35158 0.911851
\(66\) 0 0
\(67\) 2.80766i 0.343011i −0.985183 0.171505i \(-0.945137\pi\)
0.985183 0.171505i \(-0.0548631\pi\)
\(68\) −2.24806 + 2.89631i −0.272617 + 0.351229i
\(69\) 0 0
\(70\) 0 0
\(71\) 2.92285i 0.346878i 0.984845 + 0.173439i \(0.0554880\pi\)
−0.984845 + 0.173439i \(0.944512\pi\)
\(72\) 0 0
\(73\) 8.10495i 0.948613i −0.880360 0.474307i \(-0.842699\pi\)
0.880360 0.474307i \(-0.157301\pi\)
\(74\) −10.0495 3.44247i −1.16823 0.400179i
\(75\) 0 0
\(76\) −6.57226 5.10126i −0.753890 0.585155i
\(77\) 0 0
\(78\) 0 0
\(79\) 1.78368i 0.200680i 0.994953 + 0.100340i \(0.0319930\pi\)
−0.994953 + 0.100340i \(0.968007\pi\)
\(80\) 9.50377 2.43382i 1.06255 0.272110i
\(81\) 0 0
\(82\) 0.499388 1.45785i 0.0551481 0.160992i
\(83\) −5.33626 −0.585730 −0.292865 0.956154i \(-0.594609\pi\)
−0.292865 + 0.956154i \(0.594609\pi\)
\(84\) 0 0
\(85\) 4.49611 0.487672
\(86\) −2.87433 + 8.39096i −0.309947 + 0.904821i
\(87\) 0 0
\(88\) 2.99234 1.96359i 0.318984 0.209320i
\(89\) 8.57161i 0.908589i −0.890852 0.454294i \(-0.849891\pi\)
0.890852 0.454294i \(-0.150109\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 8.26338 10.6462i 0.861517 1.10994i
\(93\) 0 0
\(94\) 9.83564 + 3.36921i 1.01447 + 0.347508i
\(95\) 10.2025i 1.04676i
\(96\) 0 0
\(97\) 7.10394i 0.721296i −0.932702 0.360648i \(-0.882556\pi\)
0.932702 0.360648i \(-0.117444\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −1.60414 1.24510i −0.160414 0.124510i
\(101\) 0.933313i 0.0928682i 0.998921 + 0.0464341i \(0.0147857\pi\)
−0.998921 + 0.0464341i \(0.985214\pi\)
\(102\) 0 0
\(103\) 4.12921 0.406863 0.203431 0.979089i \(-0.434791\pi\)
0.203431 + 0.979089i \(0.434791\pi\)
\(104\) 7.08820 4.65132i 0.695055 0.456100i
\(105\) 0 0
\(106\) −0.0958634 0.0328381i −0.00931108 0.00318952i
\(107\) 13.8424i 1.33819i 0.743176 + 0.669096i \(0.233318\pi\)
−0.743176 + 0.669096i \(0.766682\pi\)
\(108\) 0 0
\(109\) −0.984676 −0.0943148 −0.0471574 0.998887i \(-0.515016\pi\)
−0.0471574 + 0.998887i \(0.515016\pi\)
\(110\) −4.15219 1.42234i −0.395896 0.135615i
\(111\) 0 0
\(112\) 0 0
\(113\) −5.03187 −0.473359 −0.236679 0.971588i \(-0.576059\pi\)
−0.236679 + 0.971588i \(0.576059\pi\)
\(114\) 0 0
\(115\) −16.5268 −1.54113
\(116\) 14.8880 + 11.5558i 1.38232 + 1.07293i
\(117\) 0 0
\(118\) 4.50457 + 1.54304i 0.414679 + 0.142049i
\(119\) 0 0
\(120\) 0 0
\(121\) 9.39878 0.854434
\(122\) −5.08820 + 14.8538i −0.460664 + 1.34480i
\(123\) 0 0
\(124\) 14.9142 + 11.5761i 1.33933 + 1.03956i
\(125\) 9.77288i 0.874113i
\(126\) 0 0
\(127\) 6.38337i 0.566433i −0.959056 0.283216i \(-0.908599\pi\)
0.959056 0.283216i \(-0.0914015\pi\)
\(128\) 7.62342 8.35964i 0.673821 0.738895i
\(129\) 0 0
\(130\) −9.83564 3.36921i −0.862643 0.295499i
\(131\) 3.87202 0.338300 0.169150 0.985590i \(-0.445898\pi\)
0.169150 + 0.985590i \(0.445898\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −1.28674 + 3.75636i −0.111158 + 0.324500i
\(135\) 0 0
\(136\) 4.33503 2.84468i 0.371726 0.243929i
\(137\) 14.7032 1.25618 0.628088 0.778142i \(-0.283838\pi\)
0.628088 + 0.778142i \(0.283838\pi\)
\(138\) 0 0
\(139\) −2.01655 −0.171041 −0.0855207 0.996336i \(-0.527255\pi\)
−0.0855207 + 0.996336i \(0.527255\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 1.33953 3.91046i 0.112411 0.328159i
\(143\) −3.79295 −0.317182
\(144\) 0 0
\(145\) 23.1116i 1.91931i
\(146\) −3.71448 + 10.8436i −0.307413 + 0.897421i
\(147\) 0 0
\(148\) 11.8675 + 9.21133i 0.975504 + 0.757167i
\(149\) 0.496110 0.0406429 0.0203215 0.999793i \(-0.493531\pi\)
0.0203215 + 0.999793i \(0.493531\pi\)
\(150\) 0 0
\(151\) 13.2370i 1.07721i −0.842558 0.538605i \(-0.818951\pi\)
0.842558 0.538605i \(-0.181049\pi\)
\(152\) 6.45511 + 9.83701i 0.523578 + 0.797886i
\(153\) 0 0
\(154\) 0 0
\(155\) 23.1522i 1.85963i
\(156\) 0 0
\(157\) 5.06158i 0.403958i −0.979390 0.201979i \(-0.935263\pi\)
0.979390 0.201979i \(-0.0647372\pi\)
\(158\) 0.817456 2.38638i 0.0650333 0.189850i
\(159\) 0 0
\(160\) −13.8305 1.09935i −1.09339 0.0869115i
\(161\) 0 0
\(162\) 0 0
\(163\) 12.0357i 0.942710i −0.881944 0.471355i \(-0.843765\pi\)
0.881944 0.471355i \(-0.156235\pi\)
\(164\) −1.33626 + 1.72158i −0.104344 + 0.134433i
\(165\) 0 0
\(166\) 7.13935 + 2.44559i 0.554121 + 0.189815i
\(167\) −7.46424 −0.577600 −0.288800 0.957389i \(-0.593256\pi\)
−0.288800 + 0.957389i \(0.593256\pi\)
\(168\) 0 0
\(169\) 4.01532 0.308871
\(170\) −6.01532 2.06056i −0.461354 0.158037i
\(171\) 0 0
\(172\) 7.69111 9.90893i 0.586442 0.755549i
\(173\) 4.36398i 0.331787i −0.986144 0.165894i \(-0.946949\pi\)
0.986144 0.165894i \(-0.0530508\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −4.90334 + 1.25570i −0.369603 + 0.0946518i
\(177\) 0 0
\(178\) −3.92835 + 11.4679i −0.294442 + 0.859557i
\(179\) 24.5667i 1.83620i −0.396343 0.918102i \(-0.629721\pi\)
0.396343 0.918102i \(-0.370279\pi\)
\(180\) 0 0
\(181\) 11.7182i 0.871011i 0.900186 + 0.435505i \(0.143430\pi\)
−0.900186 + 0.435505i \(0.856570\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −15.9347 + 10.4564i −1.17472 + 0.770858i
\(185\) 18.4227i 1.35446i
\(186\) 0 0
\(187\) −2.31971 −0.169634
\(188\) −11.6150 9.01530i −0.847108 0.657508i
\(189\) 0 0
\(190\) 4.67579 13.6499i 0.339217 0.990268i
\(191\) 15.8919i 1.14990i −0.818190 0.574948i \(-0.805022\pi\)
0.818190 0.574948i \(-0.194978\pi\)
\(192\) 0 0
\(193\) 19.7338 1.42047 0.710235 0.703964i \(-0.248589\pi\)
0.710235 + 0.703964i \(0.248589\pi\)
\(194\) −3.25572 + 9.50433i −0.233747 + 0.682371i
\(195\) 0 0
\(196\) 0 0
\(197\) −0.998775 −0.0711598 −0.0355799 0.999367i \(-0.511328\pi\)
−0.0355799 + 0.999367i \(0.511328\pi\)
\(198\) 0 0
\(199\) −2.70316 −0.191622 −0.0958110 0.995400i \(-0.530544\pi\)
−0.0958110 + 0.995400i \(0.530544\pi\)
\(200\) 1.57554 + 2.40099i 0.111408 + 0.169775i
\(201\) 0 0
\(202\) 0.427735 1.24868i 0.0300953 0.0878565i
\(203\) 0 0
\(204\) 0 0
\(205\) 2.67251 0.186656
\(206\) −5.52444 1.89240i −0.384906 0.131850i
\(207\) 0 0
\(208\) −11.6150 + 2.97448i −0.805353 + 0.206243i
\(209\) 5.26385i 0.364108i
\(210\) 0 0
\(211\) 18.1798i 1.25155i 0.780004 + 0.625774i \(0.215217\pi\)
−0.780004 + 0.625774i \(0.784783\pi\)
\(212\) 0.113206 + 0.0878679i 0.00777499 + 0.00603479i
\(213\) 0 0
\(214\) 6.34392 18.5196i 0.433661 1.26598i
\(215\) −15.3822 −1.04906
\(216\) 0 0
\(217\) 0 0
\(218\) 1.31739 + 0.451274i 0.0892251 + 0.0305642i
\(219\) 0 0
\(220\) 4.90334 + 3.80588i 0.330583 + 0.256592i
\(221\) −5.49489 −0.369626
\(222\) 0 0
\(223\) −11.5996 −0.776769 −0.388385 0.921497i \(-0.626967\pi\)
−0.388385 + 0.921497i \(0.626967\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 6.73212 + 2.30609i 0.447814 + 0.153399i
\(227\) 10.1611 0.674414 0.337207 0.941430i \(-0.390518\pi\)
0.337207 + 0.941430i \(0.390518\pi\)
\(228\) 0 0
\(229\) 23.0970i 1.52629i 0.646228 + 0.763145i \(0.276346\pi\)
−0.646228 + 0.763145i \(0.723654\pi\)
\(230\) 22.1111 + 7.57417i 1.45796 + 0.499426i
\(231\) 0 0
\(232\) −14.6226 22.2836i −0.960022 1.46299i
\(233\) 6.85547 0.449117 0.224558 0.974461i \(-0.427906\pi\)
0.224558 + 0.974461i \(0.427906\pi\)
\(234\) 0 0
\(235\) 18.0306i 1.17619i
\(236\) −5.31946 4.12886i −0.346268 0.268766i
\(237\) 0 0
\(238\) 0 0
\(239\) 22.2257i 1.43766i 0.695184 + 0.718832i \(0.255323\pi\)
−0.695184 + 0.718832i \(0.744677\pi\)
\(240\) 0 0
\(241\) 15.4273i 0.993762i −0.867819 0.496881i \(-0.834478\pi\)
0.867819 0.496881i \(-0.165522\pi\)
\(242\) −12.5746 4.30744i −0.808325 0.276892i
\(243\) 0 0
\(244\) 13.6150 17.5410i 0.871608 1.12295i
\(245\) 0 0
\(246\) 0 0
\(247\) 12.4689i 0.793379i
\(248\) −14.6483 22.3227i −0.930168 1.41749i
\(249\) 0 0
\(250\) −4.47889 + 13.0751i −0.283270 + 0.826941i
\(251\) 22.2954 1.40727 0.703636 0.710561i \(-0.251559\pi\)
0.703636 + 0.710561i \(0.251559\pi\)
\(252\) 0 0
\(253\) 8.52676 0.536073
\(254\) −2.92548 + 8.54029i −0.183561 + 0.535865i
\(255\) 0 0
\(256\) −14.0305 + 7.69053i −0.876908 + 0.480658i
\(257\) 2.86976i 0.179011i −0.995986 0.0895055i \(-0.971471\pi\)
0.995986 0.0895055i \(-0.0285287\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 11.6150 + 9.01530i 0.720329 + 0.559105i
\(261\) 0 0
\(262\) −5.18036 1.77454i −0.320043 0.109631i
\(263\) 10.3714i 0.639526i −0.947498 0.319763i \(-0.896397\pi\)
0.947498 0.319763i \(-0.103603\pi\)
\(264\) 0 0
\(265\) 0.175736i 0.0107954i
\(266\) 0 0
\(267\) 0 0
\(268\) 3.44306 4.43590i 0.210318 0.270966i
\(269\) 5.17319i 0.315415i 0.987486 + 0.157707i \(0.0504103\pi\)
−0.987486 + 0.157707i \(0.949590\pi\)
\(270\) 0 0
\(271\) −24.2391 −1.47242 −0.736209 0.676755i \(-0.763386\pi\)
−0.736209 + 0.676755i \(0.763386\pi\)
\(272\) −7.10353 + 1.81914i −0.430714 + 0.110302i
\(273\) 0 0
\(274\) −19.6713 6.73842i −1.18839 0.407083i
\(275\) 1.28479i 0.0774755i
\(276\) 0 0
\(277\) −3.01532 −0.181173 −0.0905866 0.995889i \(-0.528874\pi\)
−0.0905866 + 0.995889i \(0.528874\pi\)
\(278\) 2.69793 + 0.924179i 0.161811 + 0.0554286i
\(279\) 0 0
\(280\) 0 0
\(281\) 6.91922 0.412766 0.206383 0.978471i \(-0.433831\pi\)
0.206383 + 0.978471i \(0.433831\pi\)
\(282\) 0 0
\(283\) −20.5740 −1.22299 −0.611497 0.791246i \(-0.709433\pi\)
−0.611497 + 0.791246i \(0.709433\pi\)
\(284\) −3.58431 + 4.61789i −0.212690 + 0.274021i
\(285\) 0 0
\(286\) 5.07457 + 1.73830i 0.300066 + 0.102788i
\(287\) 0 0
\(288\) 0 0
\(289\) 13.6394 0.802319
\(290\) −10.5920 + 30.9209i −0.621982 + 1.81574i
\(291\) 0 0
\(292\) 9.93917 12.8052i 0.581646 0.749370i
\(293\) 28.3113i 1.65396i 0.562229 + 0.826982i \(0.309944\pi\)
−0.562229 + 0.826982i \(0.690056\pi\)
\(294\) 0 0
\(295\) 8.25772i 0.480783i
\(296\) −11.6560 17.7626i −0.677489 1.03243i
\(297\) 0 0
\(298\) −0.663744 0.227366i −0.0384496 0.0131710i
\(299\) 20.1980 1.16808
\(300\) 0 0
\(301\) 0 0
\(302\) −6.06647 + 17.7097i −0.349086 + 1.01908i
\(303\) 0 0
\(304\) −4.12798 16.1192i −0.236756 0.924502i
\(305\) −27.2299 −1.55918
\(306\) 0 0
\(307\) 8.65596 0.494022 0.247011 0.969013i \(-0.420552\pi\)
0.247011 + 0.969013i \(0.420552\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −10.6106 + 30.9752i −0.602640 + 1.75927i
\(311\) 9.34258 0.529769 0.264884 0.964280i \(-0.414666\pi\)
0.264884 + 0.964280i \(0.414666\pi\)
\(312\) 0 0
\(313\) 7.37547i 0.416886i −0.978035 0.208443i \(-0.933160\pi\)
0.978035 0.208443i \(-0.0668396\pi\)
\(314\) −2.31971 + 6.77186i −0.130909 + 0.382158i
\(315\) 0 0
\(316\) −2.18734 + 2.81809i −0.123048 + 0.158530i
\(317\) 3.63028 0.203897 0.101949 0.994790i \(-0.467492\pi\)
0.101949 + 0.994790i \(0.467492\pi\)
\(318\) 0 0
\(319\) 11.9241i 0.667622i
\(320\) 17.9999 + 7.80929i 1.00622 + 0.436552i
\(321\) 0 0
\(322\) 0 0
\(323\) 7.62580i 0.424311i
\(324\) 0 0
\(325\) 3.04338i 0.168816i
\(326\) −5.51594 + 16.1025i −0.305499 + 0.891836i
\(327\) 0 0
\(328\) 2.57677 1.69089i 0.142278 0.0933638i
\(329\) 0 0
\(330\) 0 0
\(331\) 0.628347i 0.0345371i 0.999851 + 0.0172685i \(0.00549702\pi\)
−0.999851 + 0.0172685i \(0.994503\pi\)
\(332\) −8.43090 6.54389i −0.462705 0.359143i
\(333\) 0 0
\(334\) 9.98637 + 3.42084i 0.546430 + 0.187180i
\(335\) −6.88612 −0.376229
\(336\) 0 0
\(337\) −22.3119 −1.21541 −0.607704 0.794164i \(-0.707909\pi\)
−0.607704 + 0.794164i \(0.707909\pi\)
\(338\) −5.37208 1.84021i −0.292203 0.100094i
\(339\) 0 0
\(340\) 7.10353 + 5.51362i 0.385243 + 0.299018i
\(341\) 11.9450i 0.646860i
\(342\) 0 0
\(343\) 0 0
\(344\) −14.8311 + 9.73229i −0.799642 + 0.524730i
\(345\) 0 0
\(346\) −2.00000 + 5.83854i −0.107521 + 0.313882i
\(347\) 6.89477i 0.370130i 0.982726 + 0.185065i \(0.0592496\pi\)
−0.982726 + 0.185065i \(0.940750\pi\)
\(348\) 0 0
\(349\) 13.4768i 0.721399i 0.932682 + 0.360699i \(0.117462\pi\)
−0.932682 + 0.360699i \(0.882538\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 7.13564 + 0.567197i 0.380331 + 0.0302317i
\(353\) 28.5846i 1.52140i −0.649101 0.760702i \(-0.724855\pi\)
0.649101 0.760702i \(-0.275145\pi\)
\(354\) 0 0
\(355\) 7.16862 0.380471
\(356\) 10.5114 13.5425i 0.557105 0.717752i
\(357\) 0 0
\(358\) −11.2589 + 32.8677i −0.595050 + 1.73711i
\(359\) 6.92820i 0.365657i 0.983145 + 0.182828i \(0.0585252\pi\)
−0.983145 + 0.182828i \(0.941475\pi\)
\(360\) 0 0
\(361\) −1.69562 −0.0892430
\(362\) 5.37044 15.6778i 0.282264 0.824006i
\(363\) 0 0
\(364\) 0 0
\(365\) −19.8783 −1.04048
\(366\) 0 0
\(367\) 12.9437 0.675654 0.337827 0.941208i \(-0.390308\pi\)
0.337827 + 0.941208i \(0.390308\pi\)
\(368\) 26.1111 6.68679i 1.36113 0.348573i
\(369\) 0 0
\(370\) −8.44306 + 24.6476i −0.438934 + 1.28137i
\(371\) 0 0
\(372\) 0 0
\(373\) 3.07907 0.159428 0.0797141 0.996818i \(-0.474599\pi\)
0.0797141 + 0.996818i \(0.474599\pi\)
\(374\) 3.10353 + 1.06312i 0.160479 + 0.0549724i
\(375\) 0 0
\(376\) 11.4079 + 17.3846i 0.588318 + 0.896544i
\(377\) 28.2456i 1.45472i
\(378\) 0 0
\(379\) 5.21020i 0.267630i 0.991006 + 0.133815i \(0.0427228\pi\)
−0.991006 + 0.133815i \(0.957277\pi\)
\(380\) −12.5114 + 16.1192i −0.641823 + 0.826900i
\(381\) 0 0
\(382\) −7.28321 + 21.2617i −0.372641 + 1.08784i
\(383\) 38.8706 1.98619 0.993096 0.117301i \(-0.0374242\pi\)
0.993096 + 0.117301i \(0.0374242\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −26.4018 9.04395i −1.34381 0.460325i
\(387\) 0 0
\(388\) 8.71162 11.2237i 0.442265 0.569797i
\(389\) 3.73503 0.189374 0.0946869 0.995507i \(-0.469815\pi\)
0.0946869 + 0.995507i \(0.469815\pi\)
\(390\) 0 0
\(391\) 12.3528 0.624708
\(392\) 0 0
\(393\) 0 0
\(394\) 1.33626 + 0.457736i 0.0673196 + 0.0230604i
\(395\) 4.37468 0.220114
\(396\) 0 0
\(397\) 6.70976i 0.336753i −0.985723 0.168377i \(-0.946148\pi\)
0.985723 0.168377i \(-0.0538525\pi\)
\(398\) 3.61655 + 1.23885i 0.181281 + 0.0620980i
\(399\) 0 0
\(400\) −1.00754 3.93433i −0.0503772 0.196717i
\(401\) −5.84769 −0.292020 −0.146010 0.989283i \(-0.546643\pi\)
−0.146010 + 0.989283i \(0.546643\pi\)
\(402\) 0 0
\(403\) 28.2952i 1.40949i
\(404\) −1.14453 + 1.47457i −0.0569425 + 0.0733625i
\(405\) 0 0
\(406\) 0 0
\(407\) 9.50492i 0.471142i
\(408\) 0 0
\(409\) 30.8651i 1.52618i 0.646293 + 0.763089i \(0.276318\pi\)
−0.646293 + 0.763089i \(0.723682\pi\)
\(410\) −3.57554 1.22481i −0.176583 0.0604889i
\(411\) 0 0
\(412\) 6.52384 + 5.06368i 0.321407 + 0.249469i
\(413\) 0 0
\(414\) 0 0
\(415\) 13.0878i 0.642454i
\(416\) 16.9028 + 1.34357i 0.828728 + 0.0658738i
\(417\) 0 0
\(418\) −2.41241 + 7.04249i −0.117995 + 0.344459i
\(419\) 29.0866 1.42097 0.710487 0.703710i \(-0.248475\pi\)
0.710487 + 0.703710i \(0.248475\pi\)
\(420\) 0 0
\(421\) 13.8642 0.675702 0.337851 0.941200i \(-0.390300\pi\)
0.337851 + 0.941200i \(0.390300\pi\)
\(422\) 8.33175 24.3227i 0.405583 1.18401i
\(423\) 0 0
\(424\) −0.111188 0.169440i −0.00539974 0.00822873i
\(425\) 1.86128i 0.0902854i
\(426\) 0 0
\(427\) 0 0
\(428\) −16.9750 + 21.8699i −0.820517 + 1.05712i
\(429\) 0 0
\(430\) 20.5798 + 7.04964i 0.992447 + 0.339964i
\(431\) 31.8995i 1.53655i 0.640123 + 0.768273i \(0.278883\pi\)
−0.640123 + 0.768273i \(0.721117\pi\)
\(432\) 0 0
\(433\) 9.82239i 0.472034i 0.971749 + 0.236017i \(0.0758421\pi\)
−0.971749 + 0.236017i \(0.924158\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −1.55572 1.20752i −0.0745053 0.0578295i
\(437\) 28.0308i 1.34090i
\(438\) 0 0
\(439\) 17.0398 0.813264 0.406632 0.913592i \(-0.366703\pi\)
0.406632 + 0.913592i \(0.366703\pi\)
\(440\) −4.81594 7.33906i −0.229591 0.349876i
\(441\) 0 0
\(442\) 7.35158 + 2.51829i 0.349679 + 0.119783i
\(443\) 8.42931i 0.400489i 0.979746 + 0.200244i \(0.0641736\pi\)
−0.979746 + 0.200244i \(0.935826\pi\)
\(444\) 0 0
\(445\) −21.0229 −0.996580
\(446\) 15.5191 + 5.31608i 0.734851 + 0.251724i
\(447\) 0 0
\(448\) 0 0
\(449\) −9.64064 −0.454970 −0.227485 0.973782i \(-0.573050\pi\)
−0.227485 + 0.973782i \(0.573050\pi\)
\(450\) 0 0
\(451\) −1.37885 −0.0649274
\(452\) −7.94999 6.17063i −0.373936 0.290242i
\(453\) 0 0
\(454\) −13.5945 4.65680i −0.638020 0.218554i
\(455\) 0 0
\(456\) 0 0
\(457\) −29.0457 −1.35870 −0.679351 0.733813i \(-0.737739\pi\)
−0.679351 + 0.733813i \(0.737739\pi\)
\(458\) 10.5853 30.9013i 0.494617 1.44392i
\(459\) 0 0
\(460\) −26.1111 20.2669i −1.21743 0.944949i
\(461\) 23.9796i 1.11684i −0.829559 0.558420i \(-0.811408\pi\)
0.829559 0.558420i \(-0.188592\pi\)
\(462\) 0 0
\(463\) 28.4975i 1.32439i −0.749331 0.662196i \(-0.769625\pi\)
0.749331 0.662196i \(-0.230375\pi\)
\(464\) 9.35103 + 36.5146i 0.434111 + 1.69515i
\(465\) 0 0
\(466\) −9.17190 3.14184i −0.424880 0.145543i
\(467\) −18.5815 −0.859849 −0.429925 0.902865i \(-0.641460\pi\)
−0.429925 + 0.902865i \(0.641460\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 8.26338 24.1231i 0.381161 1.11271i
\(471\) 0 0
\(472\) 5.22464 + 7.96188i 0.240483 + 0.366475i
\(473\) 7.93625 0.364909
\(474\) 0 0
\(475\) 4.22360 0.193792
\(476\) 0 0
\(477\) 0 0
\(478\) 10.1860 29.7357i 0.465897 1.36008i
\(479\) −28.3414 −1.29495 −0.647475 0.762087i \(-0.724175\pi\)
−0.647475 + 0.762087i \(0.724175\pi\)
\(480\) 0 0
\(481\) 22.5151i 1.02660i
\(482\) −7.07031 + 20.6402i −0.322044 + 0.940134i
\(483\) 0 0
\(484\) 14.8494 + 11.5258i 0.674972 + 0.523900i
\(485\) −17.4232 −0.791148
\(486\) 0 0
\(487\) 41.5113i 1.88106i −0.339718 0.940528i \(-0.610331\pi\)
0.339718 0.940528i \(-0.389669\pi\)
\(488\) −26.2544 + 17.2283i −1.18848 + 0.779888i
\(489\) 0 0
\(490\) 0 0
\(491\) 1.72728i 0.0779509i 0.999240 + 0.0389755i \(0.0124094\pi\)
−0.999240 + 0.0389755i \(0.987591\pi\)
\(492\) 0 0
\(493\) 17.2746i 0.778008i
\(494\) −5.71448 + 16.6821i −0.257107 + 0.750564i
\(495\) 0 0
\(496\) 9.36745 + 36.5787i 0.420611 + 1.64243i
\(497\) 0 0
\(498\) 0 0
\(499\) 42.2870i 1.89303i 0.322663 + 0.946514i \(0.395422\pi\)
−0.322663 + 0.946514i \(0.604578\pi\)
\(500\) 11.9846 15.4404i 0.535966 0.690517i
\(501\) 0 0
\(502\) −29.8289 10.2179i −1.33133 0.456048i
\(503\) 4.23770 0.188950 0.0944748 0.995527i \(-0.469883\pi\)
0.0944748 + 0.995527i \(0.469883\pi\)
\(504\) 0 0
\(505\) 2.28906 0.101862
\(506\) −11.4079 3.90779i −0.507143 0.173723i
\(507\) 0 0
\(508\) 7.82798 10.0853i 0.347311 0.447461i
\(509\) 41.7756i 1.85167i 0.377924 + 0.925836i \(0.376638\pi\)
−0.377924 + 0.925836i \(0.623362\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 22.2959 3.85896i 0.985350 0.170544i
\(513\) 0 0
\(514\) −1.31521 + 3.83944i −0.0580112 + 0.169351i
\(515\) 10.1274i 0.446265i
\(516\) 0 0
\(517\) 9.30265i 0.409130i
\(518\) 0 0
\(519\) 0 0
\(520\) −11.4079 17.3846i −0.500270 0.762367i
\(521\) 34.9506i 1.53121i 0.643309 + 0.765607i \(0.277561\pi\)
−0.643309 + 0.765607i \(0.722439\pi\)
\(522\) 0 0
\(523\) 12.2701 0.536532 0.268266 0.963345i \(-0.413549\pi\)
0.268266 + 0.963345i \(0.413549\pi\)
\(524\) 6.11751 + 4.74829i 0.267245 + 0.207430i
\(525\) 0 0
\(526\) −4.75317 + 13.8758i −0.207248 + 0.605014i
\(527\) 17.3049i 0.753814i
\(528\) 0 0
\(529\) −22.4063 −0.974188
\(530\) −0.0805393 + 0.235116i −0.00349840 + 0.0102128i
\(531\) 0 0
\(532\) 0 0
\(533\) −3.26619 −0.141474
\(534\) 0 0
\(535\) 33.9500 1.46779
\(536\) −6.63942 + 4.35683i −0.286779 + 0.188186i
\(537\) 0 0
\(538\) 2.37086 6.92118i 0.102215 0.298393i
\(539\) 0 0
\(540\) 0 0
\(541\) −0.935765 −0.0402317 −0.0201158 0.999798i \(-0.506404\pi\)
−0.0201158 + 0.999798i \(0.506404\pi\)
\(542\) 32.4293 + 11.1087i 1.39296 + 0.477159i
\(543\) 0 0
\(544\) 10.3375 + 0.821704i 0.443216 + 0.0352303i
\(545\) 2.41503i 0.103449i
\(546\) 0 0
\(547\) 7.13048i 0.304877i 0.988313 + 0.152439i \(0.0487127\pi\)
−0.988313 + 0.152439i \(0.951287\pi\)
\(548\) 23.2299 + 18.0306i 0.992333 + 0.770229i
\(549\) 0 0
\(550\) −0.588814 + 1.71891i −0.0251071 + 0.0732945i
\(551\) −39.1993 −1.66995
\(552\) 0 0
\(553\) 0 0
\(554\) 4.03419 + 1.38192i 0.171396 + 0.0587120i
\(555\) 0 0
\(556\) −3.18600 2.47291i −0.135116 0.104875i
\(557\) −9.95244 −0.421698 −0.210849 0.977519i \(-0.567623\pi\)
−0.210849 + 0.977519i \(0.567623\pi\)
\(558\) 0 0
\(559\) 18.7993 0.795124
\(560\) 0 0
\(561\) 0 0
\(562\) −9.25719 3.17106i −0.390491 0.133763i
\(563\) −1.68906 −0.0711855 −0.0355927 0.999366i \(-0.511332\pi\)
−0.0355927 + 0.999366i \(0.511332\pi\)
\(564\) 0 0
\(565\) 12.3413i 0.519200i
\(566\) 27.5258 + 9.42899i 1.15700 + 0.396330i
\(567\) 0 0
\(568\) 6.91180 4.53557i 0.290013 0.190308i
\(569\) 13.9387 0.584341 0.292170 0.956366i \(-0.405623\pi\)
0.292170 + 0.956366i \(0.405623\pi\)
\(570\) 0 0
\(571\) 18.6589i 0.780853i −0.920634 0.390426i \(-0.872328\pi\)
0.920634 0.390426i \(-0.127672\pi\)
\(572\) −5.99258 4.65132i −0.250562 0.194482i
\(573\) 0 0
\(574\) 0 0
\(575\) 6.84168i 0.285318i
\(576\) 0 0
\(577\) 34.0165i 1.41612i −0.706150 0.708062i \(-0.749570\pi\)
0.706150 0.708062i \(-0.250430\pi\)
\(578\) −18.2481 6.25091i −0.759021 0.260004i
\(579\) 0 0
\(580\) 28.3419 36.5146i 1.17683 1.51619i
\(581\) 0 0
\(582\) 0 0
\(583\) 0.0906685i 0.00375511i
\(584\) −19.1662 + 12.5770i −0.793102 + 0.520439i
\(585\) 0 0
\(586\) 12.9750 37.8775i 0.535992 1.56471i
\(587\) −41.9153 −1.73003 −0.865015 0.501746i \(-0.832691\pi\)
−0.865015 + 0.501746i \(0.832691\pi\)
\(588\) 0 0
\(589\) −39.2681 −1.61801
\(590\) 3.78449 11.0480i 0.155805 0.454838i
\(591\) 0 0
\(592\) 7.45388 + 29.1065i 0.306353 + 1.19627i
\(593\) 24.4051i 1.00220i −0.865391 0.501098i \(-0.832930\pi\)
0.865391 0.501098i \(-0.167070\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0.783818 + 0.608384i 0.0321064 + 0.0249204i
\(597\) 0 0
\(598\) −27.0229 9.25671i −1.10505 0.378535i
\(599\) 20.7846i 0.849236i 0.905373 + 0.424618i \(0.139592\pi\)
−0.905373 + 0.424618i \(0.860408\pi\)
\(600\) 0 0
\(601\) 10.6623i 0.434924i −0.976069 0.217462i \(-0.930222\pi\)
0.976069 0.217462i \(-0.0697778\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 16.2326 20.9135i 0.660496 0.850957i
\(605\) 23.0516i 0.937180i
\(606\) 0 0
\(607\) 40.0429 1.62529 0.812646 0.582757i \(-0.198026\pi\)
0.812646 + 0.582757i \(0.198026\pi\)
\(608\) −1.86460 + 23.4577i −0.0756196 + 0.951335i
\(609\) 0 0
\(610\) 36.4308 + 12.4794i 1.47504 + 0.505276i
\(611\) 22.0360i 0.891479i
\(612\) 0 0
\(613\) 22.4961 0.908609 0.454305 0.890846i \(-0.349888\pi\)
0.454305 + 0.890846i \(0.349888\pi\)
\(614\) −11.5808 3.96701i −0.467362 0.160095i
\(615\) 0 0
\(616\) 0 0
\(617\) 18.0820 0.727954 0.363977 0.931408i \(-0.381419\pi\)
0.363977 + 0.931408i \(0.381419\pi\)
\(618\) 0 0
\(619\) 32.0929 1.28992 0.644962 0.764215i \(-0.276873\pi\)
0.644962 + 0.764215i \(0.276873\pi\)
\(620\) 28.3917 36.5787i 1.14024 1.46904i
\(621\) 0 0
\(622\) −12.4994 4.28168i −0.501180 0.171680i
\(623\) 0 0
\(624\) 0 0
\(625\) −29.0457 −1.16183
\(626\) −3.38016 + 9.86760i −0.135098 + 0.394389i
\(627\) 0 0
\(628\) 6.20705 7.99692i 0.247688 0.319112i
\(629\) 13.7699i 0.549041i
\(630\) 0 0
\(631\) 2.95509i 0.117640i −0.998269 0.0588201i \(-0.981266\pi\)
0.998269 0.0588201i \(-0.0187338\pi\)
\(632\) 4.21796 2.76785i 0.167781 0.110099i
\(633\) 0 0
\(634\) −4.85694 1.66375i −0.192894 0.0660760i
\(635\) −15.6560 −0.621288
\(636\) 0 0
\(637\) 0 0
\(638\) 5.46479 15.9532i 0.216353 0.631593i
\(639\) 0 0
\(640\) −20.5030 18.6973i −0.810451 0.739076i
\(641\) 41.4917 1.63882 0.819412 0.573205i \(-0.194300\pi\)
0.819412 + 0.573205i \(0.194300\pi\)
\(642\) 0 0
\(643\) −16.7686 −0.661290 −0.330645 0.943755i \(-0.607266\pi\)
−0.330645 + 0.943755i \(0.607266\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −3.49489 + 10.2025i −0.137504 + 0.401413i
\(647\) −18.6236 −0.732169 −0.366085 0.930582i \(-0.619302\pi\)
−0.366085 + 0.930582i \(0.619302\pi\)
\(648\) 0 0
\(649\) 4.26046i 0.167238i
\(650\) −1.39477 + 4.07172i −0.0547075 + 0.159706i
\(651\) 0 0
\(652\) 14.7595 19.0155i 0.578026 0.744706i
\(653\) −25.6609 −1.00419 −0.502095 0.864813i \(-0.667437\pi\)
−0.502095 + 0.864813i \(0.667437\pi\)
\(654\) 0 0
\(655\) 9.49658i 0.371062i
\(656\) −4.22238 + 1.08131i −0.164856 + 0.0422180i
\(657\) 0 0
\(658\) 0 0
\(659\) 27.7044i 1.07921i 0.841919 + 0.539604i \(0.181426\pi\)
−0.841919 + 0.539604i \(0.818574\pi\)
\(660\) 0 0
\(661\) 34.1182i 1.32704i 0.748157 + 0.663522i \(0.230939\pi\)
−0.748157 + 0.663522i \(0.769061\pi\)
\(662\) 0.287970 0.840662i 0.0111923 0.0326732i
\(663\) 0 0
\(664\) 8.28060 + 12.6189i 0.321350 + 0.489708i
\(665\) 0 0
\(666\) 0 0
\(667\) 63.4977i 2.45864i
\(668\) −11.7929 9.15345i −0.456283 0.354158i
\(669\) 0 0
\(670\) 9.21290 + 3.15589i 0.355926 + 0.121923i
\(671\) 14.0489 0.542352
\(672\) 0 0
\(673\) −17.7032 −0.682407 −0.341203 0.939990i \(-0.610834\pi\)
−0.341203 + 0.939990i \(0.610834\pi\)
\(674\) 29.8510 + 10.2255i 1.14982 + 0.393872i
\(675\) 0 0
\(676\) 6.34392 + 4.92402i 0.243997 + 0.189386i
\(677\) 41.0852i 1.57903i 0.613730 + 0.789516i \(0.289668\pi\)
−0.613730 + 0.789516i \(0.710332\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −6.97690 10.6322i −0.267552 0.407725i
\(681\) 0 0
\(682\) 5.47438 15.9812i 0.209625 0.611952i
\(683\) 21.2282i 0.812275i 0.913812 + 0.406137i \(0.133125\pi\)
−0.913812 + 0.406137i \(0.866875\pi\)
\(684\) 0 0
\(685\) 36.0612i 1.37783i
\(686\) 0 0
\(687\) 0 0
\(688\) 24.3028 6.22371i 0.926535 0.237277i
\(689\) 0.214774i 0.00818224i
\(690\) 0 0
\(691\) −38.9757 −1.48270 −0.741352 0.671116i \(-0.765815\pi\)
−0.741352 + 0.671116i \(0.765815\pi\)
\(692\) 5.35158 6.89477i 0.203437 0.262100i
\(693\) 0 0
\(694\) 3.15985 9.22447i 0.119946 0.350156i
\(695\) 4.94582i 0.187606i
\(696\) 0 0
\(697\) −1.99755 −0.0756626
\(698\) 6.17640 18.0306i 0.233780 0.682468i
\(699\) 0 0
\(700\) 0 0
\(701\) −4.28115 −0.161697 −0.0808485 0.996726i \(-0.525763\pi\)
−0.0808485 + 0.996726i \(0.525763\pi\)
\(702\) 0 0
\(703\) −31.2465 −1.17848
\(704\) −9.28680 4.02910i −0.350009 0.151852i
\(705\) 0 0
\(706\) −13.1002 + 38.2432i −0.493034 + 1.43930i
\(707\) 0 0
\(708\) 0 0
\(709\) −43.7591 −1.64341 −0.821704 0.569914i \(-0.806976\pi\)
−0.821704 + 0.569914i \(0.806976\pi\)
\(710\) −9.59087 3.28536i −0.359939 0.123297i
\(711\) 0 0
\(712\) −20.2697 + 13.3011i −0.759639 + 0.498480i
\(713\) 63.6092i 2.38218i
\(714\) 0 0
\(715\) 9.30265i 0.347899i
\(716\) 30.1264 38.8137i 1.12588 1.45053i
\(717\) 0 0
\(718\) 3.17518 9.26921i 0.118497 0.345924i
\(719\) 41.5314 1.54886 0.774429 0.632660i \(-0.218037\pi\)
0.774429 + 0.632660i \(0.218037\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 2.26856 + 0.777097i 0.0844270 + 0.0289205i
\(723\) 0 0
\(724\) −14.3702 + 18.5140i −0.534064 + 0.688067i
\(725\) −9.56764 −0.355333
\(726\) 0 0
\(727\) 7.19963 0.267020 0.133510 0.991047i \(-0.457375\pi\)
0.133510 + 0.991047i \(0.457375\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 26.5951 + 9.11019i 0.984330 + 0.337183i
\(731\) 11.4973 0.425244
\(732\) 0 0
\(733\) 24.1585i 0.892315i 0.894954 + 0.446158i \(0.147208\pi\)
−0.894954 + 0.446158i \(0.852792\pi\)
\(734\) −17.3173 5.93205i −0.639192 0.218956i
\(735\) 0 0
\(736\) −37.9984 3.02041i −1.40064 0.111334i
\(737\) 3.55280 0.130869
\(738\) 0 0
\(739\) 9.43033i 0.346900i 0.984843 + 0.173450i \(0.0554915\pi\)
−0.984843 + 0.173450i \(0.944508\pi\)
\(740\) 22.5919 29.1065i 0.830493 1.06997i
\(741\) 0 0
\(742\) 0 0
\(743\) 2.32851i 0.0854248i −0.999087 0.0427124i \(-0.986400\pi\)
0.999087 0.0427124i \(-0.0135999\pi\)
\(744\) 0 0
\(745\) 1.21677i 0.0445789i
\(746\) −4.11947 1.41113i −0.150825 0.0516651i
\(747\) 0 0
\(748\) −3.66497 2.84468i −0.134005 0.104012i
\(749\) 0 0
\(750\) 0 0
\(751\) 31.1577i 1.13696i 0.822696 + 0.568481i \(0.192469\pi\)
−0.822696 + 0.568481i \(0.807531\pi\)
\(752\) −7.29525 28.4870i −0.266030 1.03882i
\(753\) 0 0
\(754\) 12.9449 37.7897i 0.471425 1.37622i
\(755\) −32.4652 −1.18153
\(756\) 0 0
\(757\) −0.559856 −0.0203483 −0.0101742 0.999948i \(-0.503239\pi\)
−0.0101742 + 0.999948i \(0.503239\pi\)
\(758\) 2.38782 6.97070i 0.0867296 0.253187i
\(759\) 0 0
\(760\) 24.1264 15.8319i 0.875156 0.574283i
\(761\) 45.7488i 1.65839i −0.558958 0.829196i \(-0.688799\pi\)
0.558958 0.829196i \(-0.311201\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 19.4883 25.1080i 0.705063 0.908376i
\(765\) 0 0
\(766\) −52.0047 17.8143i −1.87901 0.643656i
\(767\) 10.0921i 0.364405i
\(768\) 0 0
\(769\) 8.33377i 0.300524i −0.988646 0.150262i \(-0.951988\pi\)
0.988646 0.150262i \(-0.0480117\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 31.1780 + 24.1997i 1.12212 + 0.870967i
\(773\) 30.6774i 1.10339i 0.834046 + 0.551695i \(0.186019\pi\)
−0.834046 + 0.551695i \(0.813981\pi\)
\(774\) 0 0
\(775\) −9.58444 −0.344283
\(776\) −16.7990 + 11.0236i −0.603050 + 0.395725i
\(777\) 0 0
\(778\) −4.99708 1.71176i −0.179154 0.0613695i
\(779\) 4.53282i 0.162405i
\(780\) 0 0
\(781\) −3.69855 −0.132345
\(782\) −16.5268 5.66126i −0.590996 0.202446i
\(783\) 0 0
\(784\) 0 0
\(785\) −12.4141 −0.443078
\(786\) 0 0
\(787\) 22.1584 0.789861 0.394931 0.918711i \(-0.370769\pi\)
0.394931 + 0.918711i \(0.370769\pi\)
\(788\) −1.57799 1.22481i −0.0562136 0.0436319i
\(789\) 0 0
\(790\) −5.85287 2.00491i −0.208236 0.0713314i
\(791\) 0 0
\(792\) 0 0
\(793\) 33.2788 1.18177
\(794\) −3.07506 + 8.97695i −0.109130 + 0.318580i
\(795\) 0 0
\(796\) −4.27080 3.31491i −0.151374 0.117494i
\(797\) 24.1705i 0.856164i −0.903740 0.428082i \(-0.859189\pi\)
0.903740 0.428082i \(-0.140811\pi\)
\(798\) 0 0
\(799\) 13.4768i 0.476776i
\(800\) −0.455106 + 5.72548i −0.0160904 + 0.202426i
\(801\) 0 0
\(802\) 7.82360 + 2.67998i 0.276261 + 0.0946334i
\(803\) 10.2560 0.361925
\(804\) 0 0
\(805\) 0 0
\(806\) 12.9676 37.8560i 0.456765 1.33342i
\(807\) 0 0
\(808\) 2.20705 1.44828i 0.0776438 0.0509503i
\(809\) −15.2209 −0.535139 −0.267569 0.963539i \(-0.586220\pi\)
−0.267569 + 0.963539i \(0.586220\pi\)
\(810\) 0 0
\(811\) −48.4574 −1.70157 −0.850785 0.525515i \(-0.823873\pi\)
−0.850785 + 0.525515i \(0.823873\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 4.35608 12.7166i 0.152681 0.445716i
\(815\) −29.5190 −1.03400
\(816\) 0 0
\(817\) 26.0896i 0.912760i
\(818\) 14.1454 41.2942i 0.494581 1.44382i
\(819\) 0 0
\(820\) 4.22238 + 3.27732i 0.147452 + 0.114449i
\(821\) −51.5293 −1.79839 −0.899193 0.437552i \(-0.855846\pi\)
−0.899193 + 0.437552i \(0.855846\pi\)
\(822\) 0 0
\(823\) 2.19716i 0.0765881i 0.999267 + 0.0382941i \(0.0121924\pi\)
−0.999267 + 0.0382941i \(0.987808\pi\)
\(824\) −6.40755 9.76453i −0.223217 0.340164i
\(825\) 0 0
\(826\) 0 0
\(827\) 10.2864i 0.357693i −0.983877 0.178846i \(-0.942763\pi\)
0.983877 0.178846i \(-0.0572365\pi\)
\(828\) 0 0
\(829\) 0.765044i 0.0265711i −0.999912 0.0132855i \(-0.995771\pi\)
0.999912 0.0132855i \(-0.00422904\pi\)
\(830\) 5.99810 17.5101i 0.208197 0.607784i
\(831\) 0 0
\(832\) −21.9984 9.54406i −0.762658 0.330881i
\(833\) 0 0
\(834\) 0 0
\(835\) 18.3069i 0.633537i
\(836\) 6.45511 8.31651i 0.223255 0.287632i
\(837\) 0 0
\(838\) −38.9148 13.3303i −1.34429 0.460488i
\(839\) −3.50389 −0.120968 −0.0604839 0.998169i \(-0.519264\pi\)
−0.0604839 + 0.998169i \(0.519264\pi\)
\(840\) 0 0
\(841\) 59.7973 2.06198
\(842\) −18.5489 6.35395i −0.639237 0.218971i
\(843\) 0 0
\(844\) −22.2940 + 28.7227i −0.767392 + 0.988678i
\(845\) 9.84805i 0.338783i
\(846\) 0 0
\(847\) 0 0
\(848\) 0.0711034 + 0.277650i 0.00244170 + 0.00953453i
\(849\) 0 0
\(850\) −0.853021 + 2.49020i −0.0292584 + 0.0854132i
\(851\) 50.6152i 1.73507i
\(852\) 0 0
\(853\) 25.3974i 0.869589i −0.900530 0.434795i \(-0.856821\pi\)
0.900530 0.434795i \(-0.143179\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 32.7337 21.4801i 1.11881 0.734173i
\(857\) 28.9379i 0.988499i −0.869320 0.494249i \(-0.835443\pi\)
0.869320 0.494249i \(-0.164557\pi\)
\(858\) 0 0
\(859\) 4.68807 0.159955 0.0799775 0.996797i \(-0.474515\pi\)
0.0799775 + 0.996797i \(0.474515\pi\)
\(860\) −24.3028 18.8633i −0.828718 0.643235i
\(861\) 0 0
\(862\) 14.6195 42.6782i 0.497941 1.45363i
\(863\) 43.4651i 1.47957i −0.672844 0.739784i \(-0.734928\pi\)
0.672844 0.739784i \(-0.265072\pi\)
\(864\) 0 0
\(865\) −10.7032 −0.363918
\(866\) 4.50158 13.1413i 0.152970 0.446560i
\(867\) 0 0
\(868\) 0 0
\(869\) −2.25706 −0.0765655
\(870\) 0 0
\(871\) 8.41582 0.285159
\(872\) 1.52798 + 2.32851i 0.0517440 + 0.0788533i
\(873\) 0 0
\(874\) 12.8465 37.5023i 0.434538 1.26854i
\(875\) 0 0
\(876\) 0 0
\(877\) −46.0126 −1.55374 −0.776868 0.629663i \(-0.783193\pi\)
−0.776868 + 0.629663i \(0.783193\pi\)
\(878\) −22.7974 7.80929i −0.769376 0.263551i
\(879\) 0 0
\(880\) 3.07975 + 12.0260i 0.103818 + 0.405397i
\(881\) 40.8047i 1.37475i 0.726304 + 0.687373i \(0.241236\pi\)
−0.726304 + 0.687373i \(0.758764\pi\)
\(882\) 0 0
\(883\) 6.06234i 0.204014i −0.994784 0.102007i \(-0.967474\pi\)
0.994784 0.102007i \(-0.0325264\pi\)
\(884\) −8.68152 6.73842i −0.291991 0.226638i
\(885\) 0 0
\(886\) 3.86313 11.2775i 0.129784 0.378876i
\(887\) −44.2552 −1.48595 −0.742973 0.669322i \(-0.766585\pi\)
−0.742973 + 0.669322i \(0.766585\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 28.1264 + 9.63473i 0.942799 + 0.322957i
\(891\) 0 0
\(892\) −18.3266 14.2247i −0.613619 0.476279i
\(893\) 30.5815 1.02337
\(894\) 0 0
\(895\) −60.2528 −2.01403
\(896\) 0 0
\(897\) 0 0
\(898\) 12.8982 + 4.41828i 0.430417 + 0.147440i
\(899\) 88.9533 2.96676
\(900\) 0 0
\(901\) 0.131352i 0.00437599i
\(902\) 1.84475 + 0.631922i 0.0614236 + 0.0210407i
\(903\) 0 0
\(904\) 7.80827 + 11.8991i 0.259699 + 0.395759i
\(905\) 28.7404 0.955362
\(906\) 0 0
\(907\) 6.40312i 0.212612i −0.994333 0.106306i \(-0.966098\pi\)
0.994333 0.106306i \(-0.0339023\pi\)
\(908\) 16.0538 + 12.4606i 0.532763 + 0.413520i
\(909\) 0 0
\(910\) 0 0
\(911\) 48.9687i 1.62241i −0.584765 0.811203i \(-0.698813\pi\)
0.584765 0.811203i \(-0.301187\pi\)
\(912\) 0 0
\(913\) 6.75247i 0.223474i
\(914\) 38.8602 + 13.3116i 1.28538 + 0.440308i
\(915\) 0 0
\(916\) −28.3240 + 36.4915i −0.935850 + 1.20571i
\(917\) 0 0
\(918\) 0 0
\(919\) 24.5577i 0.810085i 0.914298 + 0.405042i \(0.132743\pi\)
−0.914298 + 0.405042i \(0.867257\pi\)
\(920\) 25.6456 + 39.0816i 0.845511 + 1.28848i
\(921\) 0 0
\(922\) −10.9898 + 32.0821i −0.361929 + 1.05657i
\(923\) −8.76108 −0.288374
\(924\) 0 0
\(925\) −7.62654 −0.250759
\(926\) −13.0603 + 38.1267i −0.429189 + 1.25292i
\(927\) 0 0
\(928\) 4.22384 53.1382i 0.138654 1.74435i
\(929\) 9.83435i 0.322655i 0.986901 + 0.161327i \(0.0515775\pi\)
−0.986901 + 0.161327i \(0.948423\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 10.8311 + 8.40692i 0.354786 + 0.275378i
\(933\) 0 0
\(934\) 24.8601 + 8.51585i 0.813447 + 0.278647i
\(935\) 5.68935i 0.186062i
\(936\) 0 0
\(937\) 38.1447i 1.24613i 0.782168 + 0.623067i \(0.214114\pi\)
−0.782168 + 0.623067i \(0.785886\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −22.1111 + 28.4870i −0.721184 + 0.929145i
\(941\) 34.3856i 1.12094i 0.828176 + 0.560469i \(0.189379\pi\)
−0.828176 + 0.560469i \(0.810621\pi\)
\(942\) 0 0
\(943\) 7.34258 0.239107
\(944\) −3.34111 13.0466i −0.108744 0.424631i
\(945\) 0 0
\(946\) −10.6179 3.63716i −0.345217 0.118254i
\(947\) 16.4695i 0.535187i 0.963532 + 0.267594i \(0.0862285\pi\)
−0.963532 + 0.267594i \(0.913772\pi\)
\(948\) 0 0
\(949\) 24.2942 0.788622
\(950\) −5.65073 1.93567i −0.183334 0.0628013i
\(951\) 0 0
\(952\) 0 0
\(953\) 52.4540 1.69915 0.849576 0.527466i \(-0.176858\pi\)
0.849576 + 0.527466i \(0.176858\pi\)
\(954\) 0 0
\(955\) −38.9767 −1.26126
\(956\) −27.2556 + 35.1150i −0.881509 + 1.13570i
\(957\) 0 0
\(958\) 37.9178 + 12.9888i 1.22507 + 0.419648i
\(959\) 0 0
\(960\) 0 0
\(961\) 58.1095 1.87450
\(962\) 10.3186 30.1229i 0.332686 0.971200i
\(963\) 0 0
\(964\) 18.9187 24.3741i 0.609329 0.785036i
\(965\) 48.3995i 1.55803i
\(966\) 0 0
\(967\) 16.3573i 0.526016i 0.964794 + 0.263008i \(0.0847146\pi\)
−0.964794 + 0.263008i \(0.915285\pi\)
\(968\) −14.5847 22.2257i −0.468769 0.714362i
\(969\) 0 0
\(970\) 23.3105 + 7.98502i 0.748454 + 0.256384i
\(971\) 23.3182 0.748318 0.374159 0.927365i \(-0.377931\pi\)
0.374159 + 0.927365i \(0.377931\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −19.0245 + 55.5377i −0.609585 + 1.77954i
\(975\) 0 0
\(976\) 43.0213 11.0173i 1.37708 0.352656i
\(977\) −39.9844 −1.27922 −0.639608 0.768701i \(-0.720903\pi\)
−0.639608 + 0.768701i \(0.720903\pi\)
\(978\) 0 0
\(979\) 10.8465 0.346655
\(980\) 0 0
\(981\) 0 0
\(982\) 0.791607 2.31092i 0.0252612 0.0737443i
\(983\) −21.5255 −0.686558 −0.343279 0.939233i \(-0.611538\pi\)
−0.343279 + 0.939233i \(0.611538\pi\)
\(984\) 0 0
\(985\) 2.44961i 0.0780511i
\(986\) 7.91689 23.1116i 0.252125 0.736022i
\(987\) 0 0
\(988\) 15.2908 19.7000i 0.486464 0.626741i
\(989\) −42.2618 −1.34385
\(990\) 0 0
\(991\) 11.1967i 0.355676i 0.984060 + 0.177838i \(0.0569103\pi\)
−0.984060 + 0.177838i \(0.943090\pi\)
\(992\) 4.23126 53.2316i 0.134343 1.69010i
\(993\) 0 0
\(994\) 0 0
\(995\) 6.62982i 0.210179i
\(996\) 0 0
\(997\) 56.7766i 1.79813i 0.437813 + 0.899066i \(0.355753\pi\)
−0.437813 + 0.899066i \(0.644247\pi\)
\(998\) 19.3800 56.5756i 0.613464 1.79087i
\(999\) 0 0
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 1764.2.b.j.1567.1 8
3.2 odd 2 588.2.b.a.391.8 8
4.3 odd 2 1764.2.b.i.1567.2 8
7.2 even 3 252.2.bf.f.199.3 8
7.3 odd 6 252.2.bf.g.19.4 8
7.6 odd 2 1764.2.b.i.1567.1 8
12.11 even 2 588.2.b.b.391.7 8
21.2 odd 6 84.2.o.b.31.2 yes 8
21.5 even 6 588.2.o.b.31.2 8
21.11 odd 6 588.2.o.d.19.1 8
21.17 even 6 84.2.o.a.19.1 8
21.20 even 2 588.2.b.b.391.8 8
28.3 even 6 252.2.bf.f.19.3 8
28.23 odd 6 252.2.bf.g.199.4 8
28.27 even 2 inner 1764.2.b.j.1567.2 8
84.11 even 6 588.2.o.b.19.2 8
84.23 even 6 84.2.o.a.31.1 yes 8
84.47 odd 6 588.2.o.d.31.1 8
84.59 odd 6 84.2.o.b.19.2 yes 8
84.83 odd 2 588.2.b.a.391.7 8
168.59 odd 6 1344.2.bl.i.1279.1 8
168.101 even 6 1344.2.bl.j.1279.1 8
168.107 even 6 1344.2.bl.j.703.1 8
168.149 odd 6 1344.2.bl.i.703.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.2.o.a.19.1 8 21.17 even 6
84.2.o.a.31.1 yes 8 84.23 even 6
84.2.o.b.19.2 yes 8 84.59 odd 6
84.2.o.b.31.2 yes 8 21.2 odd 6
252.2.bf.f.19.3 8 28.3 even 6
252.2.bf.f.199.3 8 7.2 even 3
252.2.bf.g.19.4 8 7.3 odd 6
252.2.bf.g.199.4 8 28.23 odd 6
588.2.b.a.391.7 8 84.83 odd 2
588.2.b.a.391.8 8 3.2 odd 2
588.2.b.b.391.7 8 12.11 even 2
588.2.b.b.391.8 8 21.20 even 2
588.2.o.b.19.2 8 84.11 even 6
588.2.o.b.31.2 8 21.5 even 6
588.2.o.d.19.1 8 21.11 odd 6
588.2.o.d.31.1 8 84.47 odd 6
1344.2.bl.i.703.1 8 168.149 odd 6
1344.2.bl.i.1279.1 8 168.59 odd 6
1344.2.bl.j.703.1 8 168.107 even 6
1344.2.bl.j.1279.1 8 168.101 even 6
1764.2.b.i.1567.1 8 7.6 odd 2
1764.2.b.i.1567.2 8 4.3 odd 2
1764.2.b.j.1567.1 8 1.1 even 1 trivial
1764.2.b.j.1567.2 8 28.27 even 2 inner