Properties

Label 648.2.v.a.35.1
Level $648$
Weight $2$
Character 648.35
Analytic conductor $5.174$
Analytic rank $0$
Dimension $12$
CM discriminant -8
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [648,2,Mod(35,648)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(648, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 9, 13]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("648.35");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 648 = 2^{3} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 648.v (of order \(18\), degree \(6\), not 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: \(5.17430605098\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{18})\)
Coefficient field: 12.0.101559956668416.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 8x^{6} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 216)
Sato-Tate group: $\mathrm{U}(1)[D_{18}]$

Embedding invariants

Embedding label 35.1
Root \(0.483690 - 1.32893i\) of defining polynomial
Character \(\chi\) \(=\) 648.35
Dual form 648.2.v.a.611.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.909039 + 1.08335i) q^{2} +(-0.347296 - 1.96962i) q^{4} +(2.44949 + 1.41421i) q^{8} +O(q^{10})\) \(q+(-0.909039 + 1.08335i) q^{2} +(-0.347296 - 1.96962i) q^{4} +(2.44949 + 1.41421i) q^{8} +(-2.18893 - 6.01403i) q^{11} +(-3.75877 + 1.36808i) q^{16} +(-7.09286 + 4.09506i) q^{17} +(-0.511376 + 0.885729i) q^{19} +(8.50512 + 3.09561i) q^{22} +(-3.83022 - 3.21394i) q^{25} +(1.93476 - 5.31570i) q^{32} +(2.01130 - 11.4066i) q^{34} +(-0.494694 - 1.35916i) q^{38} +(-8.15282 - 9.71615i) q^{41} +(2.07316 - 0.754568i) q^{43} +(-11.0851 + 6.40000i) q^{44} +(-6.57785 - 2.39414i) q^{49} +(6.96364 - 1.22788i) q^{50} +(3.84333 - 10.5595i) q^{59} +(4.00000 + 6.92820i) q^{64} +(-4.53904 + 3.80871i) q^{67} +(10.5290 + 12.5480i) q^{68} +(4.68819 - 8.12018i) q^{73} +(1.92214 + 0.699604i) q^{76} +17.9372 q^{82} +(10.9291 - 13.0248i) q^{83} +(-1.06712 + 2.93189i) q^{86} +(3.14337 - 17.8269i) q^{88} +(11.0505 + 6.38002i) q^{89} +(-14.0731 + 5.12218i) q^{97} +(8.57321 - 4.94975i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 18 q^{11} + 12 q^{22} - 24 q^{34} - 72 q^{38} + 18 q^{41} + 30 q^{43} - 36 q^{59} + 48 q^{64} - 42 q^{67} + 72 q^{68} - 24 q^{76} - 36 q^{86} + 48 q^{88} + 162 q^{89} + 30 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/648\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(487\) \(569\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{13}{18}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.909039 + 1.08335i −0.642788 + 0.766044i
\(3\) 0 0
\(4\) −0.347296 1.96962i −0.173648 0.984808i
\(5\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(6\) 0 0
\(7\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(8\) 2.44949 + 1.41421i 0.866025 + 0.500000i
\(9\) 0 0
\(10\) 0 0
\(11\) −2.18893 6.01403i −0.659987 1.81330i −0.576994 0.816748i \(-0.695774\pi\)
−0.0829925 0.996550i \(-0.526448\pi\)
\(12\) 0 0
\(13\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −3.75877 + 1.36808i −0.939693 + 0.342020i
\(17\) −7.09286 + 4.09506i −1.72027 + 0.993199i −0.801920 + 0.597431i \(0.796188\pi\)
−0.918351 + 0.395768i \(0.870479\pi\)
\(18\) 0 0
\(19\) −0.511376 + 0.885729i −0.117318 + 0.203200i −0.918704 0.394947i \(-0.870763\pi\)
0.801386 + 0.598147i \(0.204096\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 8.50512 + 3.09561i 1.81330 + 0.659987i
\(23\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(24\) 0 0
\(25\) −3.83022 3.21394i −0.766044 0.642788i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(30\) 0 0
\(31\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(32\) 1.93476 5.31570i 0.342020 0.939693i
\(33\) 0 0
\(34\) 2.01130 11.4066i 0.344934 1.95622i
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(38\) −0.494694 1.35916i −0.0802500 0.220485i
\(39\) 0 0
\(40\) 0 0
\(41\) −8.15282 9.71615i −1.27326 1.51741i −0.742424 0.669930i \(-0.766324\pi\)
−0.530831 0.847477i \(-0.678120\pi\)
\(42\) 0 0
\(43\) 2.07316 0.754568i 0.316154 0.115071i −0.179069 0.983836i \(-0.557309\pi\)
0.495223 + 0.868766i \(0.335086\pi\)
\(44\) −11.0851 + 6.40000i −1.67114 + 0.964836i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(48\) 0 0
\(49\) −6.57785 2.39414i −0.939693 0.342020i
\(50\) 6.96364 1.22788i 0.984808 0.173648i
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.84333 10.5595i 0.500359 1.37472i −0.390567 0.920575i \(-0.627721\pi\)
0.890925 0.454150i \(-0.150057\pi\)
\(60\) 0 0
\(61\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 4.00000 + 6.92820i 0.500000 + 0.866025i
\(65\) 0 0
\(66\) 0 0
\(67\) −4.53904 + 3.80871i −0.554532 + 0.465308i −0.876472 0.481452i \(-0.840109\pi\)
0.321940 + 0.946760i \(0.395665\pi\)
\(68\) 10.5290 + 12.5480i 1.27683 + 1.52167i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(72\) 0 0
\(73\) 4.68819 8.12018i 0.548711 0.950395i −0.449652 0.893204i \(-0.648452\pi\)
0.998363 0.0571917i \(-0.0182146\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 1.92214 + 0.699604i 0.220485 + 0.0802500i
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 17.9372 1.98083
\(83\) 10.9291 13.0248i 1.19963 1.42966i 0.324415 0.945915i \(-0.394833\pi\)
0.875210 0.483743i \(-0.160723\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1.06712 + 2.93189i −0.115071 + 0.316154i
\(87\) 0 0
\(88\) 3.14337 17.8269i 0.335084 1.90036i
\(89\) 11.0505 + 6.38002i 1.17135 + 0.676280i 0.953998 0.299813i \(-0.0969242\pi\)
0.217354 + 0.976093i \(0.430258\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −14.0731 + 5.12218i −1.42890 + 0.520079i −0.936617 0.350354i \(-0.886061\pi\)
−0.492287 + 0.870433i \(0.663839\pi\)
\(98\) 8.57321 4.94975i 0.866025 0.500000i
\(99\) 0 0
\(100\) −5.00000 + 8.66025i −0.500000 + 0.866025i
\(101\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(102\) 0 0
\(103\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 11.3102i 1.09340i 0.837330 + 0.546698i \(0.184115\pi\)
−0.837330 + 0.546698i \(0.815885\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −7.26632 + 19.9641i −0.683558 + 1.87806i −0.306510 + 0.951867i \(0.599161\pi\)
−0.377048 + 0.926194i \(0.623061\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 7.94586 + 13.7626i 0.731475 + 1.26695i
\(119\) 0 0
\(120\) 0 0
\(121\) −22.9507 + 19.2579i −2.08642 + 1.75072i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(128\) −11.1418 1.96460i −0.984808 0.173648i
\(129\) 0 0
\(130\) 0 0
\(131\) 8.38799 1.47903i 0.732862 0.129223i 0.205251 0.978709i \(-0.434199\pi\)
0.527611 + 0.849486i \(0.323088\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 8.37963i 0.723890i
\(135\) 0 0
\(136\) −23.1652 −1.98640
\(137\) −1.27249 + 1.51649i −0.108716 + 0.129562i −0.817658 0.575704i \(-0.804728\pi\)
0.708942 + 0.705266i \(0.249173\pi\)
\(138\) 0 0
\(139\) −2.11445 11.9916i −0.179345 1.01712i −0.933008 0.359856i \(-0.882826\pi\)
0.753663 0.657262i \(-0.228285\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 4.53526 + 12.4605i 0.375340 + 1.03124i
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(150\) 0 0
\(151\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(152\) −2.50522 + 1.44639i −0.203200 + 0.117318i
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 21.0454 1.64840 0.824202 0.566296i \(-0.191624\pi\)
0.824202 + 0.566296i \(0.191624\pi\)
\(164\) −16.3056 + 19.4323i −1.27326 + 1.51741i
\(165\) 0 0
\(166\) 4.17544 + 23.6801i 0.324077 + 1.83793i
\(167\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(168\) 0 0
\(169\) 2.25743 12.8025i 0.173648 0.984808i
\(170\) 0 0
\(171\) 0 0
\(172\) −2.20621 3.82127i −0.168222 0.291369i
\(173\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 16.4554 + 19.6107i 1.24037 + 1.47821i
\(177\) 0 0
\(178\) −16.9571 + 6.17189i −1.27099 + 0.462603i
\(179\) −22.0732 + 12.7440i −1.64983 + 0.952529i −0.672692 + 0.739923i \(0.734862\pi\)
−0.977138 + 0.212607i \(0.931805\pi\)
\(180\) 0 0
\(181\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 40.1536 + 33.6929i 2.93632 + 2.46387i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(192\) 0 0
\(193\) 2.58197 + 14.6431i 0.185854 + 1.05403i 0.924853 + 0.380325i \(0.124188\pi\)
−0.738999 + 0.673707i \(0.764701\pi\)
\(194\) 7.24386 19.9023i 0.520079 1.42890i
\(195\) 0 0
\(196\) −2.43107 + 13.7873i −0.173648 + 0.984808i
\(197\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(198\) 0 0
\(199\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(200\) −4.83690 13.2893i −0.342020 0.939693i
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 6.44617 + 1.13663i 0.445891 + 0.0786226i
\(210\) 0 0
\(211\) −14.1381 5.14583i −0.973304 0.354254i −0.194071 0.980988i \(-0.562169\pi\)
−0.779233 + 0.626734i \(0.784391\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) −12.2529 10.2814i −0.837590 0.702822i
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −15.0227 26.0201i −0.999295 1.73083i
\(227\) −7.33644 20.1567i −0.486937 1.33785i −0.903440 0.428714i \(-0.858967\pi\)
0.416503 0.909134i \(-0.363255\pi\)
\(228\) 0 0
\(229\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 13.4895 7.78815i 0.883725 0.510219i 0.0118403 0.999930i \(-0.496231\pi\)
0.871885 + 0.489711i \(0.162898\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −22.1328 3.90262i −1.44073 0.254039i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(240\) 0 0
\(241\) 16.2613 + 13.6448i 1.04748 + 0.878941i 0.992826 0.119564i \(-0.0381497\pi\)
0.0546547 + 0.998505i \(0.482594\pi\)
\(242\) 42.3698i 2.72363i
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 23.9806 + 13.8452i 1.51364 + 0.873902i 0.999872 + 0.0159750i \(0.00508522\pi\)
0.513771 + 0.857927i \(0.328248\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 12.2567 10.2846i 0.766044 0.642788i
\(257\) −17.7278 21.1272i −1.10583 1.31788i −0.943585 0.331130i \(-0.892570\pi\)
−0.162247 0.986750i \(-0.551874\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) −6.02270 + 10.4316i −0.372084 + 0.644468i
\(263\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 9.07808 + 7.61741i 0.554532 + 0.465308i
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 21.0580 25.0960i 1.27683 1.52167i
\(273\) 0 0
\(274\) −0.486151 2.75710i −0.0293694 0.166562i
\(275\) −10.9446 + 30.0702i −0.659987 + 1.81330i
\(276\) 0 0
\(277\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(278\) 14.9133 + 8.61018i 0.894438 + 0.516404i
\(279\) 0 0
\(280\) 0 0
\(281\) 10.1685 + 27.9376i 0.606600 + 1.66662i 0.737601 + 0.675236i \(0.235958\pi\)
−0.131002 + 0.991382i \(0.541819\pi\)
\(282\) 0 0
\(283\) 25.3143 21.2412i 1.50478 1.26266i 0.631557 0.775330i \(-0.282416\pi\)
0.873219 0.487327i \(-0.162028\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 25.0391 43.3690i 1.47289 2.55112i
\(290\) 0 0
\(291\) 0 0
\(292\) −17.6218 6.41382i −1.03124 0.375340i
\(293\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0.710396 4.02886i 0.0407440 0.231071i
\(305\) 0 0
\(306\) 0 0
\(307\) −17.4258 30.1824i −0.994545 1.72260i −0.587603 0.809149i \(-0.699928\pi\)
−0.406942 0.913454i \(-0.633405\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(312\) 0 0
\(313\) −19.7387 + 7.18430i −1.11570 + 0.406080i −0.833080 0.553153i \(-0.813425\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 8.37647i 0.466079i
\(324\) 0 0
\(325\) 0 0
\(326\) −19.1311 + 22.7996i −1.05957 + 1.26275i
\(327\) 0 0
\(328\) −6.22953 35.3294i −0.343968 1.95074i
\(329\) 0 0
\(330\) 0 0
\(331\) 1.57072 8.90799i 0.0863345 0.489627i −0.910726 0.413011i \(-0.864477\pi\)
0.997061 0.0766165i \(-0.0244117\pi\)
\(332\) −29.4495 17.0027i −1.61625 0.933143i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 24.9283 20.9173i 1.35793 1.13944i 0.381314 0.924445i \(-0.375472\pi\)
0.976616 0.214993i \(-0.0689729\pi\)
\(338\) 11.8175 + 14.0836i 0.642788 + 0.766044i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 6.14530 + 1.08358i 0.331333 + 0.0584229i
\(345\) 0 0
\(346\) 0 0
\(347\) 36.0481 6.35625i 1.93516 0.341222i 0.935245 0.354001i \(-0.115179\pi\)
0.999918 + 0.0127797i \(0.00406802\pi\)
\(348\) 0 0
\(349\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −36.2039 −1.92967
\(353\) −1.25345 + 1.49380i −0.0667144 + 0.0795071i −0.798369 0.602168i \(-0.794304\pi\)
0.731655 + 0.681675i \(0.238748\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 8.72837 23.9810i 0.462603 1.27099i
\(357\) 0 0
\(358\) 6.25922 35.4978i 0.330810 1.87612i
\(359\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(360\) 0 0
\(361\) 8.97699 + 15.5486i 0.472473 + 0.818347i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(374\) −73.0024 + 12.8723i −3.77486 + 0.665610i
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −32.8062 −1.68514 −0.842570 0.538587i \(-0.818958\pi\)
−0.842570 + 0.538587i \(0.818958\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −18.2107 10.5139i −0.926900 0.535146i
\(387\) 0 0
\(388\) 14.9763 + 25.9396i 0.760304 + 1.31689i
\(389\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −12.7265 15.1669i −0.642788 0.766044i
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 18.7939 + 6.84040i 0.939693 + 0.342020i
\(401\) −34.6237 + 6.10509i −1.72902 + 0.304874i −0.947679 0.319225i \(-0.896577\pi\)
−0.781345 + 0.624099i \(0.785466\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0.861982 + 4.88854i 0.0426223 + 0.241723i 0.998674 0.0514740i \(-0.0163919\pi\)
−0.956052 + 0.293197i \(0.905281\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) −7.09119 + 5.95021i −0.346841 + 0.291034i
\(419\) −21.8376 26.0250i −1.06684 1.27140i −0.960860 0.277036i \(-0.910648\pi\)
−0.105976 0.994369i \(-0.533797\pi\)
\(420\) 0 0
\(421\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(422\) 18.4268 10.6387i 0.897002 0.517884i
\(423\) 0 0
\(424\) 0 0
\(425\) 40.3285 + 7.11100i 1.95622 + 0.344934i
\(426\) 0 0
\(427\) 0 0
\(428\) 22.2767 3.92798i 1.07679 0.189866i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 18.2012 0.874695 0.437347 0.899293i \(-0.355918\pi\)
0.437347 + 0.899293i \(0.355918\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 5.82211 + 15.9961i 0.276617 + 0.759998i 0.997740 + 0.0671913i \(0.0214038\pi\)
−0.721124 + 0.692807i \(0.756374\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 34.9698 20.1898i 1.65033 0.952816i 0.673389 0.739288i \(-0.264838\pi\)
0.976937 0.213528i \(-0.0684954\pi\)
\(450\) 0 0
\(451\) −40.5873 + 70.2992i −1.91118 + 3.31026i
\(452\) 41.8451 + 7.37842i 1.96823 + 0.347052i
\(453\) 0 0
\(454\) 28.5059 + 10.3753i 1.33785 + 0.486937i
\(455\) 0 0
\(456\) 0 0
\(457\) −9.82334 8.24276i −0.459516 0.385580i 0.383437 0.923567i \(-0.374740\pi\)
−0.842953 + 0.537987i \(0.819185\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(462\) 0 0
\(463\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −3.82516 + 21.6936i −0.177197 + 1.00494i
\(467\) −34.2054 19.7485i −1.58284 0.913851i −0.994443 0.105276i \(-0.966427\pi\)
−0.588393 0.808575i \(-0.700239\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 24.3475 20.4300i 1.12069 0.940367i
\(473\) −9.07600 10.8164i −0.417315 0.497336i
\(474\) 0 0
\(475\) 4.80536 1.74901i 0.220485 0.0802500i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −29.5643 + 5.21298i −1.34662 + 0.237445i
\(483\) 0 0
\(484\) 45.9013 + 38.5158i 2.08642 + 1.75072i
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −12.9388 + 35.5491i −0.583921 + 1.60431i 0.197499 + 0.980303i \(0.436718\pi\)
−0.781419 + 0.624006i \(0.785504\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −12.6754 + 10.6359i −0.567428 + 0.476129i −0.880791 0.473504i \(-0.842989\pi\)
0.313363 + 0.949633i \(0.398544\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −36.7985 + 13.3936i −1.64240 + 0.597784i
\(503\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 22.6274i 1.00000i
\(513\) 0 0
\(514\) 39.0035 1.72037
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −26.6827 15.4052i −1.16899 0.674916i −0.215547 0.976493i \(-0.569153\pi\)
−0.953442 + 0.301577i \(0.902487\pi\)
\(522\) 0 0
\(523\) −1.52270 2.63740i −0.0665832 0.115325i 0.830812 0.556553i \(-0.187876\pi\)
−0.897395 + 0.441228i \(0.854543\pi\)
\(524\) −5.82624 16.0075i −0.254520 0.699289i
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 21.6129 7.86646i 0.939693 0.342020i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) −16.5047 + 2.91022i −0.712893 + 0.125702i
\(537\) 0 0
\(538\) 0 0
\(539\) 44.8000i 1.92967i
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 8.04518 + 45.6265i 0.344934 + 1.95622i
\(545\) 0 0
\(546\) 0 0
\(547\) −5.17190 + 29.3313i −0.221135 + 1.25412i 0.648803 + 0.760956i \(0.275270\pi\)
−0.869938 + 0.493161i \(0.835841\pi\)
\(548\) 3.42883 + 1.97964i 0.146472 + 0.0845659i
\(549\) 0 0
\(550\) −22.6274 39.1918i −0.964836 1.67114i
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −22.8846 + 8.32931i −0.970522 + 0.353241i
\(557\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −39.5098 14.3804i −1.66662 0.606600i
\(563\) 22.8074 4.02156i 0.961218 0.169489i 0.329044 0.944315i \(-0.393274\pi\)
0.632175 + 0.774826i \(0.282163\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 46.7333i 1.96435i
\(567\) 0 0
\(568\) 0 0
\(569\) −4.43393 + 5.28415i −0.185880 + 0.221523i −0.850935 0.525271i \(-0.823964\pi\)
0.665055 + 0.746795i \(0.268408\pi\)
\(570\) 0 0
\(571\) −7.66208 43.4538i −0.320648 1.81848i −0.538642 0.842535i \(-0.681062\pi\)
0.217994 0.975950i \(-0.430049\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 19.6648 + 34.0604i 0.818655 + 1.41795i 0.906674 + 0.421833i \(0.138613\pi\)
−0.0880190 + 0.996119i \(0.528054\pi\)
\(578\) 24.2223 + 66.5502i 1.00751 + 2.76812i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 22.9673 13.2602i 0.950395 0.548711i
\(585\) 0 0
\(586\) 0 0
\(587\) −2.40364 0.423827i −0.0992090 0.0174932i 0.123823 0.992304i \(-0.460484\pi\)
−0.223032 + 0.974811i \(0.571596\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 38.2159i 1.56934i −0.619915 0.784669i \(-0.712833\pi\)
0.619915 0.784669i \(-0.287167\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(600\) 0 0
\(601\) 1.51507 8.59237i 0.0618009 0.350490i −0.938190 0.346122i \(-0.887498\pi\)
0.999990 0.00436841i \(-0.00139051\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(608\) 3.71889 + 4.43199i 0.150821 + 0.179741i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(614\) 48.5389 + 8.55872i 1.95887 + 0.345402i
\(615\) 0 0
\(616\) 0 0
\(617\) 35.8670 6.32433i 1.44395 0.254608i 0.603877 0.797077i \(-0.293622\pi\)
0.840076 + 0.542469i \(0.182511\pi\)
\(618\) 0 0
\(619\) −29.8318 25.0319i −1.19904 1.00612i −0.999657 0.0261952i \(-0.991661\pi\)
−0.199386 0.979921i \(-0.563895\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 4.34120 + 24.6202i 0.173648 + 0.984808i
\(626\) 10.1601 27.9147i 0.406080 1.11570i
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −40.3213 7.10974i −1.59260 0.280818i −0.694127 0.719852i \(-0.744209\pi\)
−0.898470 + 0.439034i \(0.855321\pi\)
\(642\) 0 0
\(643\) −0.306376 0.111512i −0.0120823 0.00439759i 0.335972 0.941872i \(-0.390935\pi\)
−0.348054 + 0.937474i \(0.613157\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 9.07465 + 7.61454i 0.357037 + 0.299590i
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) −71.9177 −2.82302
\(650\) 0 0
\(651\) 0 0
\(652\) −7.30899 41.4514i −0.286242 1.62336i
\(653\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 43.9370 + 25.3671i 1.71545 + 0.990417i
\(657\) 0 0
\(658\) 0 0
\(659\) −2.89116 7.94338i −0.112623 0.309430i 0.870557 0.492068i \(-0.163759\pi\)
−0.983180 + 0.182637i \(0.941537\pi\)
\(660\) 0 0
\(661\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(662\) 8.22263 + 9.79935i 0.319582 + 0.380862i
\(663\) 0 0
\(664\) 45.1906 16.4480i 1.75373 0.638307i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −29.9453 25.1271i −1.15431 0.968578i −0.154495 0.987994i \(-0.549375\pi\)
−0.999812 + 0.0194154i \(0.993820\pi\)
\(674\) 46.0207i 1.77265i
\(675\) 0 0
\(676\) −26.0000 −1.00000
\(677\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −31.1416 17.9796i −1.19160 0.687971i −0.232932 0.972493i \(-0.574832\pi\)
−0.958670 + 0.284522i \(0.908165\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −6.76022 + 5.67250i −0.257731 + 0.216262i
\(689\) 0 0
\(690\) 0 0
\(691\) −42.3288 + 15.4064i −1.61026 + 0.586088i −0.981492 0.191501i \(-0.938665\pi\)
−0.628772 + 0.777589i \(0.716442\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −25.8831 + 44.8308i −0.982508 + 1.70175i
\(695\) 0 0
\(696\) 0 0
\(697\) 97.6150 + 35.5290i 3.69743 + 1.34576i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 32.9107 39.2215i 1.24037 1.47821i
\(705\) 0 0
\(706\) −0.478878 2.71585i −0.0180228 0.102212i
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 18.0454 + 31.2556i 0.676280 + 1.17135i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 32.7667 + 39.0498i 1.22455 + 1.45936i
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −25.0050 4.40906i −0.930590 0.164088i
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −11.6146 + 13.8418i −0.429582 + 0.511956i
\(732\) 0 0
\(733\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 32.8413 + 18.9609i 1.20973 + 0.698435i
\(738\) 0 0
\(739\) −17.9389 31.0711i −0.659893 1.14297i −0.980643 0.195805i \(-0.937268\pi\)
0.320749 0.947164i \(-0.396065\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 52.4168 90.7886i 1.91655 3.31956i
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 29.8221 35.5406i 1.08319 1.29089i
\(759\) 0 0
\(760\) 0 0
\(761\) 3.86952 10.6314i 0.140270 0.385388i −0.849589 0.527446i \(-0.823150\pi\)
0.989858 + 0.142058i \(0.0453719\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −25.3490 + 21.2704i −0.914110 + 0.767029i −0.972896 0.231242i \(-0.925721\pi\)
0.0587868 + 0.998271i \(0.481277\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 27.9445 10.1710i 1.00574 0.366061i
\(773\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −41.7157 7.35561i −1.49751 0.264051i
\(777\) 0 0
\(778\) 0 0
\(779\) 12.7750 2.25258i 0.457713 0.0807071i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 28.0000 1.00000
\(785\) 0 0
\(786\) 0 0
\(787\) −0.513060 2.90971i −0.0182886 0.103720i 0.974297 0.225267i \(-0.0723255\pi\)
−0.992586 + 0.121547i \(0.961214\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −24.4949 + 14.1421i −0.866025 + 0.500000i
\(801\) 0 0
\(802\) 24.8603 43.0593i 0.877849 1.52048i
\(803\) −59.0971 10.4204i −2.08549 0.367729i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 47.1907i 1.65914i −0.558404 0.829569i \(-0.688586\pi\)
0.558404 0.829569i \(-0.311414\pi\)
\(810\) 0 0
\(811\) 48.3805 1.69887 0.849434 0.527694i \(-0.176943\pi\)
0.849434 + 0.527694i \(0.176943\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −0.391821 + 2.22213i −0.0137081 + 0.0777424i
\(818\) −6.07958 3.51005i −0.212568 0.122726i
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(822\) 0 0
\(823\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 17.1464 9.89949i 0.596240 0.344239i −0.171321 0.985215i \(-0.554804\pi\)
0.767561 + 0.640976i \(0.221470\pi\)
\(828\) 0 0
\(829\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 56.4599 9.95540i 1.95622 0.344934i
\(834\) 0 0
\(835\) 0 0
\(836\) 13.0912i 0.452769i
\(837\) 0 0
\(838\) 48.0454 1.65970
\(839\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(840\) 0 0
\(841\) −5.03580 28.5594i −0.173648 0.984808i
\(842\) 0 0
\(843\) 0 0
\(844\) −5.22522 + 29.6337i −0.179859 + 1.02003i
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) −44.3639 + 37.2257i −1.52167 + 1.27683i
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −15.9950 + 27.7042i −0.546698 + 0.946909i
\(857\) −22.2837 3.92921i −0.761195 0.134219i −0.220441 0.975400i \(-0.570750\pi\)
−0.540754 + 0.841181i \(0.681861\pi\)
\(858\) 0 0
\(859\) 17.3166 + 6.30274i 0.590836 + 0.215047i 0.620097 0.784525i \(-0.287093\pi\)
−0.0292613 + 0.999572i \(0.509316\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −16.5456 + 19.7183i −0.562243 + 0.670055i
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 10.9949 6.34791i 0.370428 0.213866i −0.303218 0.952921i \(-0.598061\pi\)
0.673645 + 0.739055i \(0.264728\pi\)
\(882\) 0 0
\(883\) −19.8559 + 34.3914i −0.668203 + 1.15736i 0.310203 + 0.950670i \(0.399603\pi\)
−0.978406 + 0.206691i \(0.933730\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −22.6219 8.23370i −0.759998 0.276617i
\(887\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −9.91625 + 56.2379i −0.330910 + 1.87668i
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) −39.2633 107.875i −1.30732 3.59184i
\(903\) 0 0
\(904\) −46.0322 + 38.6256i −1.53101 + 1.28467i
\(905\) 0 0
\(906\) 0 0
\(907\) 56.5987 20.6002i 1.87933 0.684020i 0.937018 0.349281i \(-0.113574\pi\)
0.942311 0.334738i \(-0.108648\pi\)
\(908\) −37.1531 + 21.4503i −1.23297 + 0.711854i
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(912\) 0 0
\(913\) −102.255 37.2176i −3.38413 1.23172i
\(914\) 17.8596 3.14913i 0.590743 0.104164i
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −9.67379 26.5785i −0.317387 0.872013i −0.991112 0.133031i \(-0.957529\pi\)
0.673725 0.738982i \(-0.264693\pi\)
\(930\) 0 0
\(931\) 5.48431 4.60189i 0.179741 0.150821i
\(932\) −20.0245 23.8643i −0.655925 0.781701i
\(933\) 0 0
\(934\) 52.4886 19.1043i 1.71748 0.625111i
\(935\) 0 0
\(936\) 0 0
\(937\) 13.5454 23.4613i 0.442509 0.766448i −0.555366 0.831606i \(-0.687422\pi\)
0.997875 + 0.0651578i \(0.0207551\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 44.9486i 1.46295i
\(945\) 0 0
\(946\) 19.9683 0.649226
\(947\) −38.8571 + 46.3081i −1.26269 + 1.50481i −0.487435 + 0.873160i \(0.662067\pi\)
−0.775252 + 0.631652i \(0.782377\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −2.47347 + 6.79581i −0.0802500 + 0.220485i
\(951\) 0 0
\(952\) 0 0
\(953\) −42.6261 24.6102i −1.38079 0.797202i −0.388540 0.921432i \(-0.627021\pi\)
−0.992253 + 0.124230i \(0.960354\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −29.1305 + 10.6026i −0.939693 + 0.342020i
\(962\) 0 0
\(963\) 0 0
\(964\) 21.2276 36.7673i 0.683695 1.18419i
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(968\) −83.4522 + 14.7149i −2.68226 + 0.472954i
\(969\) 0 0
\(970\) 0 0
\(971\) 31.1127i 0.998454i −0.866471 0.499227i \(-0.833617\pi\)
0.866471 0.499227i \(-0.166383\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −20.7007 + 56.8748i −0.662275 + 1.81959i −0.0959785 + 0.995383i \(0.530598\pi\)
−0.566296 + 0.824202i \(0.691624\pi\)
\(978\) 0 0
\(979\) 14.1808 80.4235i 0.453221 2.57035i
\(980\) 0 0
\(981\) 0 0
\(982\) −26.7503 46.3328i −0.853635 1.47854i
\(983\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(998\) 23.4004i 0.740725i
\(999\) 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 648.2.v.a.35.1 12
3.2 odd 2 216.2.v.a.11.2 12
8.3 odd 2 CM 648.2.v.a.35.1 12
12.11 even 2 864.2.bh.a.335.2 12
24.5 odd 2 864.2.bh.a.335.2 12
24.11 even 2 216.2.v.a.11.2 12
27.5 odd 18 inner 648.2.v.a.611.1 12
27.22 even 9 216.2.v.a.59.2 yes 12
108.103 odd 18 864.2.bh.a.815.2 12
216.59 even 18 inner 648.2.v.a.611.1 12
216.157 even 18 864.2.bh.a.815.2 12
216.211 odd 18 216.2.v.a.59.2 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
216.2.v.a.11.2 12 3.2 odd 2
216.2.v.a.11.2 12 24.11 even 2
216.2.v.a.59.2 yes 12 27.22 even 9
216.2.v.a.59.2 yes 12 216.211 odd 18
648.2.v.a.35.1 12 1.1 even 1 trivial
648.2.v.a.35.1 12 8.3 odd 2 CM
648.2.v.a.611.1 12 27.5 odd 18 inner
648.2.v.a.611.1 12 216.59 even 18 inner
864.2.bh.a.335.2 12 12.11 even 2
864.2.bh.a.335.2 12 24.5 odd 2
864.2.bh.a.815.2 12 108.103 odd 18
864.2.bh.a.815.2 12 216.157 even 18