Properties

Label 2268.2.x.h.1889.1
Level $2268$
Weight $2$
Character 2268.1889
Analytic conductor $18.110$
Analytic rank $0$
Dimension $2$
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] = [2268,2,Mod(377,2268)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2268, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2268.377");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2268.x (of order \(6\), degree \(2\), 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: \(18.1100711784\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 756)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1889.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 2268.1889
Dual form 2268.2.x.h.377.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.50000 + 2.59808i) q^{5} +(0.500000 + 2.59808i) q^{7} +O(q^{10})\) \(q+(1.50000 + 2.59808i) q^{5} +(0.500000 + 2.59808i) q^{7} +(-4.50000 - 2.59808i) q^{11} +(-3.00000 + 1.73205i) q^{13} +6.00000 q^{17} -1.73205i q^{19} +(-4.50000 + 2.59808i) q^{23} +(-2.00000 + 3.46410i) q^{25} +(9.00000 + 5.19615i) q^{29} +(-4.50000 + 2.59808i) q^{31} +(-6.00000 + 5.19615i) q^{35} +1.00000 q^{37} +(-1.50000 - 2.59808i) q^{41} +(-5.00000 + 8.66025i) q^{43} +(-3.00000 + 5.19615i) q^{47} +(-6.50000 + 2.59808i) q^{49} -15.5885i q^{55} +(3.00000 + 5.19615i) q^{59} +(-12.0000 - 6.92820i) q^{61} +(-9.00000 - 5.19615i) q^{65} +(-1.00000 - 1.73205i) q^{67} -5.19615i q^{71} +3.46410i q^{73} +(4.50000 - 12.9904i) q^{77} +(-7.00000 + 12.1244i) q^{79} +(3.00000 - 5.19615i) q^{83} +(9.00000 + 15.5885i) q^{85} -9.00000 q^{89} +(-6.00000 - 6.92820i) q^{91} +(4.50000 - 2.59808i) q^{95} +(-6.00000 - 3.46410i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{5} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{5} + q^{7} - 9 q^{11} - 6 q^{13} + 12 q^{17} - 9 q^{23} - 4 q^{25} + 18 q^{29} - 9 q^{31} - 12 q^{35} + 2 q^{37} - 3 q^{41} - 10 q^{43} - 6 q^{47} - 13 q^{49} + 6 q^{59} - 24 q^{61} - 18 q^{65} - 2 q^{67} + 9 q^{77} - 14 q^{79} + 6 q^{83} + 18 q^{85} - 18 q^{89} - 12 q^{91} + 9 q^{95} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1135\) \(1541\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{1}{6}\right)\)

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 0
\(3\) 0 0
\(4\) 0 0
\(5\) 1.50000 + 2.59808i 0.670820 + 1.16190i 0.977672 + 0.210138i \(0.0673912\pi\)
−0.306851 + 0.951757i \(0.599275\pi\)
\(6\) 0 0
\(7\) 0.500000 + 2.59808i 0.188982 + 0.981981i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −4.50000 2.59808i −1.35680 0.783349i −0.367610 0.929980i \(-0.619824\pi\)
−0.989191 + 0.146631i \(0.953157\pi\)
\(12\) 0 0
\(13\) −3.00000 + 1.73205i −0.832050 + 0.480384i −0.854554 0.519362i \(-0.826170\pi\)
0.0225039 + 0.999747i \(0.492836\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) 1.73205i 0.397360i −0.980064 0.198680i \(-0.936335\pi\)
0.980064 0.198680i \(-0.0636654\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.50000 + 2.59808i −0.938315 + 0.541736i −0.889432 0.457068i \(-0.848900\pi\)
−0.0488832 + 0.998805i \(0.515566\pi\)
\(24\) 0 0
\(25\) −2.00000 + 3.46410i −0.400000 + 0.692820i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 9.00000 + 5.19615i 1.67126 + 0.964901i 0.966935 + 0.255021i \(0.0820825\pi\)
0.704323 + 0.709880i \(0.251251\pi\)
\(30\) 0 0
\(31\) −4.50000 + 2.59808i −0.808224 + 0.466628i −0.846339 0.532645i \(-0.821198\pi\)
0.0381148 + 0.999273i \(0.487865\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −6.00000 + 5.19615i −1.01419 + 0.878310i
\(36\) 0 0
\(37\) 1.00000 0.164399 0.0821995 0.996616i \(-0.473806\pi\)
0.0821995 + 0.996616i \(0.473806\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1.50000 2.59808i −0.234261 0.405751i 0.724797 0.688963i \(-0.241934\pi\)
−0.959058 + 0.283211i \(0.908600\pi\)
\(42\) 0 0
\(43\) −5.00000 + 8.66025i −0.762493 + 1.32068i 0.179069 + 0.983836i \(0.442691\pi\)
−0.941562 + 0.336840i \(0.890642\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.00000 + 5.19615i −0.437595 + 0.757937i −0.997503 0.0706177i \(-0.977503\pi\)
0.559908 + 0.828554i \(0.310836\pi\)
\(48\) 0 0
\(49\) −6.50000 + 2.59808i −0.928571 + 0.371154i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 15.5885i 2.10195i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.00000 + 5.19615i 0.390567 + 0.676481i 0.992524 0.122047i \(-0.0389457\pi\)
−0.601958 + 0.798528i \(0.705612\pi\)
\(60\) 0 0
\(61\) −12.0000 6.92820i −1.53644 0.887066i −0.999043 0.0437377i \(-0.986073\pi\)
−0.537400 0.843328i \(-0.680593\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −9.00000 5.19615i −1.11631 0.644503i
\(66\) 0 0
\(67\) −1.00000 1.73205i −0.122169 0.211604i 0.798454 0.602056i \(-0.205652\pi\)
−0.920623 + 0.390453i \(0.872318\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 5.19615i 0.616670i −0.951278 0.308335i \(-0.900228\pi\)
0.951278 0.308335i \(-0.0997717\pi\)
\(72\) 0 0
\(73\) 3.46410i 0.405442i 0.979236 + 0.202721i \(0.0649785\pi\)
−0.979236 + 0.202721i \(0.935021\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.50000 12.9904i 0.512823 1.48039i
\(78\) 0 0
\(79\) −7.00000 + 12.1244i −0.787562 + 1.36410i 0.139895 + 0.990166i \(0.455323\pi\)
−0.927457 + 0.373930i \(0.878010\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.00000 5.19615i 0.329293 0.570352i −0.653079 0.757290i \(-0.726523\pi\)
0.982372 + 0.186938i \(0.0598564\pi\)
\(84\) 0 0
\(85\) 9.00000 + 15.5885i 0.976187 + 1.69081i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −9.00000 −0.953998 −0.476999 0.878904i \(-0.658275\pi\)
−0.476999 + 0.878904i \(0.658275\pi\)
\(90\) 0 0
\(91\) −6.00000 6.92820i −0.628971 0.726273i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.50000 2.59808i 0.461690 0.266557i
\(96\) 0 0
\(97\) −6.00000 3.46410i −0.609208 0.351726i 0.163448 0.986552i \(-0.447739\pi\)
−0.772655 + 0.634826i \(0.781072\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 9.00000 15.5885i 0.895533 1.55111i 0.0623905 0.998052i \(-0.480128\pi\)
0.833143 0.553058i \(-0.186539\pi\)
\(102\) 0 0
\(103\) 7.50000 4.33013i 0.738997 0.426660i −0.0827075 0.996574i \(-0.526357\pi\)
0.821705 + 0.569914i \(0.193023\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 10.3923i 1.00466i −0.864675 0.502331i \(-0.832476\pi\)
0.864675 0.502331i \(-0.167524\pi\)
\(108\) 0 0
\(109\) −11.0000 −1.05361 −0.526804 0.849987i \(-0.676610\pi\)
−0.526804 + 0.849987i \(0.676610\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(114\) 0 0
\(115\) −13.5000 7.79423i −1.25888 0.726816i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 3.00000 + 15.5885i 0.275010 + 1.42899i
\(120\) 0 0
\(121\) 8.00000 + 13.8564i 0.727273 + 1.25967i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 3.00000 0.268328
\(126\) 0 0
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −9.00000 15.5885i −0.786334 1.36197i −0.928199 0.372084i \(-0.878643\pi\)
0.141865 0.989886i \(-0.454690\pi\)
\(132\) 0 0
\(133\) 4.50000 0.866025i 0.390199 0.0750939i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 9.00000 + 5.19615i 0.768922 + 0.443937i 0.832490 0.554040i \(-0.186915\pi\)
−0.0635680 + 0.997978i \(0.520248\pi\)
\(138\) 0 0
\(139\) −15.0000 + 8.66025i −1.27228 + 0.734553i −0.975417 0.220366i \(-0.929275\pi\)
−0.296866 + 0.954919i \(0.595942\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 18.0000 1.50524
\(144\) 0 0
\(145\) 31.1769i 2.58910i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(150\) 0 0
\(151\) −2.00000 + 3.46410i −0.162758 + 0.281905i −0.935857 0.352381i \(-0.885372\pi\)
0.773099 + 0.634285i \(0.218706\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −13.5000 7.79423i −1.08435 0.626048i
\(156\) 0 0
\(157\) 9.00000 5.19615i 0.718278 0.414698i −0.0958404 0.995397i \(-0.530554\pi\)
0.814119 + 0.580699i \(0.197221\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −9.00000 10.3923i −0.709299 0.819028i
\(162\) 0 0
\(163\) −10.0000 −0.783260 −0.391630 0.920123i \(-0.628089\pi\)
−0.391630 + 0.920123i \(0.628089\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 6.00000 + 10.3923i 0.464294 + 0.804181i 0.999169 0.0407502i \(-0.0129748\pi\)
−0.534875 + 0.844931i \(0.679641\pi\)
\(168\) 0 0
\(169\) −0.500000 + 0.866025i −0.0384615 + 0.0666173i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −4.50000 + 7.79423i −0.342129 + 0.592584i −0.984828 0.173534i \(-0.944481\pi\)
0.642699 + 0.766119i \(0.277815\pi\)
\(174\) 0 0
\(175\) −10.0000 3.46410i −0.755929 0.261861i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 10.3923i 0.776757i 0.921500 + 0.388379i \(0.126965\pi\)
−0.921500 + 0.388379i \(0.873035\pi\)
\(180\) 0 0
\(181\) 6.92820i 0.514969i 0.966282 + 0.257485i \(0.0828937\pi\)
−0.966282 + 0.257485i \(0.917106\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1.50000 + 2.59808i 0.110282 + 0.191014i
\(186\) 0 0
\(187\) −27.0000 15.5885i −1.97444 1.13994i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 4.50000 + 2.59808i 0.325609 + 0.187990i 0.653890 0.756590i \(-0.273136\pi\)
−0.328281 + 0.944580i \(0.606469\pi\)
\(192\) 0 0
\(193\) 1.00000 + 1.73205i 0.0719816 + 0.124676i 0.899770 0.436365i \(-0.143734\pi\)
−0.827788 + 0.561041i \(0.810401\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 20.7846i 1.48084i 0.672143 + 0.740421i \(0.265374\pi\)
−0.672143 + 0.740421i \(0.734626\pi\)
\(198\) 0 0
\(199\) 1.73205i 0.122782i 0.998114 + 0.0613909i \(0.0195536\pi\)
−0.998114 + 0.0613909i \(0.980446\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −9.00000 + 25.9808i −0.631676 + 1.82349i
\(204\) 0 0
\(205\) 4.50000 7.79423i 0.314294 0.544373i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −4.50000 + 7.79423i −0.311272 + 0.539138i
\(210\) 0 0
\(211\) 10.0000 + 17.3205i 0.688428 + 1.19239i 0.972346 + 0.233544i \(0.0750324\pi\)
−0.283918 + 0.958849i \(0.591634\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −30.0000 −2.04598
\(216\) 0 0
\(217\) −9.00000 10.3923i −0.610960 0.705476i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −18.0000 + 10.3923i −1.21081 + 0.699062i
\(222\) 0 0
\(223\) 22.5000 + 12.9904i 1.50671 + 0.869900i 0.999970 + 0.00780243i \(0.00248362\pi\)
0.506742 + 0.862098i \(0.330850\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(228\) 0 0
\(229\) 18.0000 10.3923i 1.18947 0.686743i 0.231287 0.972886i \(-0.425707\pi\)
0.958187 + 0.286143i \(0.0923732\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 20.7846i 1.36165i 0.732448 + 0.680823i \(0.238378\pi\)
−0.732448 + 0.680823i \(0.761622\pi\)
\(234\) 0 0
\(235\) −18.0000 −1.17419
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 9.00000 5.19615i 0.582162 0.336111i −0.179830 0.983698i \(-0.557555\pi\)
0.761992 + 0.647586i \(0.224222\pi\)
\(240\) 0 0
\(241\) −3.00000 1.73205i −0.193247 0.111571i 0.400255 0.916404i \(-0.368922\pi\)
−0.593502 + 0.804833i \(0.702255\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −16.5000 12.9904i −1.05415 0.829925i
\(246\) 0 0
\(247\) 3.00000 + 5.19615i 0.190885 + 0.330623i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 24.0000 1.51487 0.757433 0.652913i \(-0.226453\pi\)
0.757433 + 0.652913i \(0.226453\pi\)
\(252\) 0 0
\(253\) 27.0000 1.69748
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 4.50000 + 7.79423i 0.280702 + 0.486191i 0.971558 0.236802i \(-0.0760993\pi\)
−0.690856 + 0.722993i \(0.742766\pi\)
\(258\) 0 0
\(259\) 0.500000 + 2.59808i 0.0310685 + 0.161437i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 4.50000 + 2.59808i 0.277482 + 0.160204i 0.632283 0.774738i \(-0.282118\pi\)
−0.354801 + 0.934942i \(0.615451\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 21.0000 1.28039 0.640196 0.768211i \(-0.278853\pi\)
0.640196 + 0.768211i \(0.278853\pi\)
\(270\) 0 0
\(271\) 3.46410i 0.210429i −0.994450 0.105215i \(-0.966447\pi\)
0.994450 0.105215i \(-0.0335529\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 18.0000 10.3923i 1.08544 0.626680i
\(276\) 0 0
\(277\) 9.50000 16.4545i 0.570800 0.988654i −0.425684 0.904872i \(-0.639967\pi\)
0.996484 0.0837823i \(-0.0267000\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −9.00000 5.19615i −0.536895 0.309976i 0.206925 0.978357i \(-0.433655\pi\)
−0.743820 + 0.668380i \(0.766988\pi\)
\(282\) 0 0
\(283\) 9.00000 5.19615i 0.534994 0.308879i −0.208053 0.978117i \(-0.566713\pi\)
0.743048 + 0.669238i \(0.233379\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 6.00000 5.19615i 0.354169 0.306719i
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 3.00000 + 5.19615i 0.175262 + 0.303562i 0.940252 0.340480i \(-0.110589\pi\)
−0.764990 + 0.644042i \(0.777256\pi\)
\(294\) 0 0
\(295\) −9.00000 + 15.5885i −0.524000 + 0.907595i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 9.00000 15.5885i 0.520483 0.901504i
\(300\) 0 0
\(301\) −25.0000 8.66025i −1.44098 0.499169i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 41.5692i 2.38025i
\(306\) 0 0
\(307\) 15.5885i 0.889680i −0.895610 0.444840i \(-0.853260\pi\)
0.895610 0.444840i \(-0.146740\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −3.00000 5.19615i −0.170114 0.294647i 0.768345 0.640036i \(-0.221080\pi\)
−0.938460 + 0.345389i \(0.887747\pi\)
\(312\) 0 0
\(313\) 3.00000 + 1.73205i 0.169570 + 0.0979013i 0.582383 0.812914i \(-0.302120\pi\)
−0.412813 + 0.910816i \(0.635454\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 27.0000 + 15.5885i 1.51647 + 0.875535i 0.999813 + 0.0193432i \(0.00615753\pi\)
0.516658 + 0.856192i \(0.327176\pi\)
\(318\) 0 0
\(319\) −27.0000 46.7654i −1.51171 2.61836i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 10.3923i 0.578243i
\(324\) 0 0
\(325\) 13.8564i 0.768615i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −15.0000 5.19615i −0.826977 0.286473i
\(330\) 0 0
\(331\) −2.00000 + 3.46410i −0.109930 + 0.190404i −0.915742 0.401768i \(-0.868396\pi\)
0.805812 + 0.592172i \(0.201729\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 3.00000 5.19615i 0.163908 0.283896i
\(336\) 0 0
\(337\) 14.5000 + 25.1147i 0.789865 + 1.36809i 0.926049 + 0.377403i \(0.123183\pi\)
−0.136184 + 0.990684i \(0.543484\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 27.0000 1.46213
\(342\) 0 0
\(343\) −10.0000 15.5885i −0.539949 0.841698i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 4.50000 2.59808i 0.241573 0.139472i −0.374327 0.927297i \(-0.622126\pi\)
0.615899 + 0.787825i \(0.288793\pi\)
\(348\) 0 0
\(349\) 24.0000 + 13.8564i 1.28469 + 0.741716i 0.977702 0.209997i \(-0.0673454\pi\)
0.306988 + 0.951713i \(0.400679\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −7.50000 + 12.9904i −0.399185 + 0.691408i −0.993626 0.112731i \(-0.964040\pi\)
0.594441 + 0.804139i \(0.297373\pi\)
\(354\) 0 0
\(355\) 13.5000 7.79423i 0.716506 0.413675i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 10.3923i 0.548485i 0.961661 + 0.274242i \(0.0884271\pi\)
−0.961661 + 0.274242i \(0.911573\pi\)
\(360\) 0 0
\(361\) 16.0000 0.842105
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −9.00000 + 5.19615i −0.471082 + 0.271979i
\(366\) 0 0
\(367\) −1.50000 0.866025i −0.0782994 0.0452062i 0.460339 0.887743i \(-0.347728\pi\)
−0.538639 + 0.842537i \(0.681061\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −14.5000 25.1147i −0.750782 1.30039i −0.947444 0.319921i \(-0.896344\pi\)
0.196663 0.980471i \(-0.436990\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −36.0000 −1.85409
\(378\) 0 0
\(379\) 10.0000 0.513665 0.256833 0.966456i \(-0.417321\pi\)
0.256833 + 0.966456i \(0.417321\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −6.00000 10.3923i −0.306586 0.531022i 0.671027 0.741433i \(-0.265853\pi\)
−0.977613 + 0.210411i \(0.932520\pi\)
\(384\) 0 0
\(385\) 40.5000 7.79423i 2.06407 0.397231i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −18.0000 10.3923i −0.912636 0.526911i −0.0313578 0.999508i \(-0.509983\pi\)
−0.881278 + 0.472597i \(0.843316\pi\)
\(390\) 0 0
\(391\) −27.0000 + 15.5885i −1.36545 + 0.788342i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −42.0000 −2.11325
\(396\) 0 0
\(397\) 20.7846i 1.04315i 0.853206 + 0.521575i \(0.174655\pi\)
−0.853206 + 0.521575i \(0.825345\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 9.00000 5.19615i 0.449439 0.259483i −0.258154 0.966104i \(-0.583114\pi\)
0.707593 + 0.706620i \(0.249781\pi\)
\(402\) 0 0
\(403\) 9.00000 15.5885i 0.448322 0.776516i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −4.50000 2.59808i −0.223057 0.128782i
\(408\) 0 0
\(409\) −15.0000 + 8.66025i −0.741702 + 0.428222i −0.822688 0.568493i \(-0.807527\pi\)
0.0809857 + 0.996715i \(0.474193\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −12.0000 + 10.3923i −0.590481 + 0.511372i
\(414\) 0 0
\(415\) 18.0000 0.883585
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −12.0000 20.7846i −0.586238 1.01539i −0.994720 0.102628i \(-0.967275\pi\)
0.408481 0.912767i \(-0.366058\pi\)
\(420\) 0 0
\(421\) −2.50000 + 4.33013i −0.121843 + 0.211037i −0.920494 0.390756i \(-0.872214\pi\)
0.798652 + 0.601793i \(0.205547\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −12.0000 + 20.7846i −0.582086 + 1.00820i
\(426\) 0 0
\(427\) 12.0000 34.6410i 0.580721 1.67640i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 36.3731i 1.75203i 0.482285 + 0.876014i \(0.339807\pi\)
−0.482285 + 0.876014i \(0.660193\pi\)
\(432\) 0 0
\(433\) 10.3923i 0.499422i 0.968320 + 0.249711i \(0.0803357\pi\)
−0.968320 + 0.249711i \(0.919664\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 4.50000 + 7.79423i 0.215264 + 0.372849i
\(438\) 0 0
\(439\) 27.0000 + 15.5885i 1.28864 + 0.743996i 0.978412 0.206666i \(-0.0662612\pi\)
0.310228 + 0.950662i \(0.399595\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 13.5000 + 7.79423i 0.641404 + 0.370315i 0.785155 0.619299i \(-0.212583\pi\)
−0.143751 + 0.989614i \(0.545916\pi\)
\(444\) 0 0
\(445\) −13.5000 23.3827i −0.639961 1.10845i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 20.7846i 0.980886i −0.871473 0.490443i \(-0.836835\pi\)
0.871473 0.490443i \(-0.163165\pi\)
\(450\) 0 0
\(451\) 15.5885i 0.734032i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 9.00000 25.9808i 0.421927 1.21800i
\(456\) 0 0
\(457\) 8.50000 14.7224i 0.397613 0.688686i −0.595818 0.803120i \(-0.703172\pi\)
0.993431 + 0.114433i \(0.0365053\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 7.50000 12.9904i 0.349310 0.605022i −0.636817 0.771015i \(-0.719749\pi\)
0.986127 + 0.165992i \(0.0530827\pi\)
\(462\) 0 0
\(463\) 8.00000 + 13.8564i 0.371792 + 0.643962i 0.989841 0.142177i \(-0.0454103\pi\)
−0.618050 + 0.786139i \(0.712077\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 18.0000 0.832941 0.416470 0.909149i \(-0.363267\pi\)
0.416470 + 0.909149i \(0.363267\pi\)
\(468\) 0 0
\(469\) 4.00000 3.46410i 0.184703 0.159957i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 45.0000 25.9808i 2.06910 1.19460i
\(474\) 0 0
\(475\) 6.00000 + 3.46410i 0.275299 + 0.158944i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 12.0000 20.7846i 0.548294 0.949673i −0.450098 0.892979i \(-0.648611\pi\)
0.998392 0.0566937i \(-0.0180558\pi\)
\(480\) 0 0
\(481\) −3.00000 + 1.73205i −0.136788 + 0.0789747i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 20.7846i 0.943781i
\(486\) 0 0
\(487\) −26.0000 −1.17817 −0.589086 0.808070i \(-0.700512\pi\)
−0.589086 + 0.808070i \(0.700512\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −13.5000 + 7.79423i −0.609246 + 0.351749i −0.772670 0.634807i \(-0.781079\pi\)
0.163424 + 0.986556i \(0.447746\pi\)
\(492\) 0 0
\(493\) 54.0000 + 31.1769i 2.43204 + 1.40414i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 13.5000 2.59808i 0.605558 0.116540i
\(498\) 0 0
\(499\) 7.00000 + 12.1244i 0.313363 + 0.542761i 0.979088 0.203436i \(-0.0652110\pi\)
−0.665725 + 0.746197i \(0.731878\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −6.00000 −0.267527 −0.133763 0.991013i \(-0.542706\pi\)
−0.133763 + 0.991013i \(0.542706\pi\)
\(504\) 0 0
\(505\) 54.0000 2.40297
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 21.0000 + 36.3731i 0.930809 + 1.61221i 0.781943 + 0.623350i \(0.214229\pi\)
0.148866 + 0.988857i \(0.452438\pi\)
\(510\) 0 0
\(511\) −9.00000 + 1.73205i −0.398137 + 0.0766214i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 22.5000 + 12.9904i 0.991468 + 0.572425i
\(516\) 0 0
\(517\) 27.0000 15.5885i 1.18746 0.685580i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −33.0000 −1.44576 −0.722878 0.690976i \(-0.757181\pi\)
−0.722878 + 0.690976i \(0.757181\pi\)
\(522\) 0 0
\(523\) 12.1244i 0.530161i 0.964226 + 0.265081i \(0.0853985\pi\)
−0.964226 + 0.265081i \(0.914601\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −27.0000 + 15.5885i −1.17614 + 0.679044i
\(528\) 0 0
\(529\) 2.00000 3.46410i 0.0869565 0.150613i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 9.00000 + 5.19615i 0.389833 + 0.225070i
\(534\) 0 0
\(535\) 27.0000 15.5885i 1.16731 0.673948i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 36.0000 + 5.19615i 1.55063 + 0.223814i
\(540\) 0 0
\(541\) 17.0000 0.730887 0.365444 0.930834i \(-0.380917\pi\)
0.365444 + 0.930834i \(0.380917\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −16.5000 28.5788i −0.706782 1.22418i
\(546\) 0 0
\(547\) 20.0000 34.6410i 0.855138 1.48114i −0.0213785 0.999771i \(-0.506805\pi\)
0.876517 0.481371i \(-0.159861\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 9.00000 15.5885i 0.383413 0.664091i
\(552\) 0 0
\(553\) −35.0000 12.1244i −1.48835 0.515580i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 20.7846i 0.880672i 0.897833 + 0.440336i \(0.145141\pi\)
−0.897833 + 0.440336i \(0.854859\pi\)
\(558\) 0 0
\(559\) 34.6410i 1.46516i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −9.00000 15.5885i −0.379305 0.656975i 0.611656 0.791123i \(-0.290503\pi\)
−0.990961 + 0.134148i \(0.957170\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −36.0000 20.7846i −1.50920 0.871336i −0.999943 0.0107211i \(-0.996587\pi\)
−0.509256 0.860615i \(-0.670079\pi\)
\(570\) 0 0
\(571\) −10.0000 17.3205i −0.418487 0.724841i 0.577301 0.816532i \(-0.304106\pi\)
−0.995788 + 0.0916910i \(0.970773\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 20.7846i 0.866778i
\(576\) 0 0
\(577\) 34.6410i 1.44212i 0.692870 + 0.721062i \(0.256346\pi\)
−0.692870 + 0.721062i \(0.743654\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 15.0000 + 5.19615i 0.622305 + 0.215573i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −18.0000 + 31.1769i −0.742940 + 1.28681i 0.208212 + 0.978084i \(0.433236\pi\)
−0.951151 + 0.308725i \(0.900098\pi\)
\(588\) 0 0
\(589\) 4.50000 + 7.79423i 0.185419 + 0.321156i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 39.0000 1.60154 0.800769 0.598973i \(-0.204424\pi\)
0.800769 + 0.598973i \(0.204424\pi\)
\(594\) 0 0
\(595\) −36.0000 + 31.1769i −1.47586 + 1.27813i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −22.5000 + 12.9904i −0.919325 + 0.530773i −0.883420 0.468583i \(-0.844765\pi\)
−0.0359054 + 0.999355i \(0.511431\pi\)
\(600\) 0 0
\(601\) −9.00000 5.19615i −0.367118 0.211955i 0.305081 0.952326i \(-0.401317\pi\)
−0.672198 + 0.740371i \(0.734650\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −24.0000 + 41.5692i −0.975739 + 1.69003i
\(606\) 0 0
\(607\) 15.0000 8.66025i 0.608831 0.351509i −0.163677 0.986514i \(-0.552335\pi\)
0.772508 + 0.635005i \(0.219002\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 20.7846i 0.840855i
\(612\) 0 0
\(613\) −19.0000 −0.767403 −0.383701 0.923457i \(-0.625351\pi\)
−0.383701 + 0.923457i \(0.625351\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −27.0000 + 15.5885i −1.08698 + 0.627568i −0.932771 0.360471i \(-0.882616\pi\)
−0.154209 + 0.988038i \(0.549283\pi\)
\(618\) 0 0
\(619\) −28.5000 16.4545i −1.14551 0.661361i −0.197722 0.980258i \(-0.563354\pi\)
−0.947790 + 0.318897i \(0.896688\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −4.50000 23.3827i −0.180289 0.936808i
\(624\) 0 0
\(625\) 14.5000 + 25.1147i 0.580000 + 1.00459i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 6.00000 0.239236
\(630\) 0 0
\(631\) 26.0000 1.03504 0.517522 0.855670i \(-0.326855\pi\)
0.517522 + 0.855670i \(0.326855\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 3.00000 + 5.19615i 0.119051 + 0.206203i
\(636\) 0 0
\(637\) 15.0000 19.0526i 0.594322 0.754890i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −18.0000 10.3923i −0.710957 0.410471i 0.100458 0.994941i \(-0.467969\pi\)
−0.811415 + 0.584470i \(0.801302\pi\)
\(642\) 0 0
\(643\) 40.5000 23.3827i 1.59716 0.922123i 0.605134 0.796124i \(-0.293120\pi\)
0.992030 0.125999i \(-0.0402137\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 18.0000 0.707653 0.353827 0.935311i \(-0.384880\pi\)
0.353827 + 0.935311i \(0.384880\pi\)
\(648\) 0 0
\(649\) 31.1769i 1.22380i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −9.00000 + 5.19615i −0.352197 + 0.203341i −0.665653 0.746262i \(-0.731847\pi\)
0.313455 + 0.949603i \(0.398513\pi\)
\(654\) 0 0
\(655\) 27.0000 46.7654i 1.05498 1.82727i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 31.5000 + 18.1865i 1.22707 + 0.708447i 0.966415 0.256986i \(-0.0827296\pi\)
0.260651 + 0.965433i \(0.416063\pi\)
\(660\) 0 0
\(661\) −21.0000 + 12.1244i −0.816805 + 0.471583i −0.849314 0.527889i \(-0.822984\pi\)
0.0325082 + 0.999471i \(0.489650\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 9.00000 + 10.3923i 0.349005 + 0.402996i
\(666\) 0 0
\(667\) −54.0000 −2.09089
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 36.0000 + 62.3538i 1.38976 + 2.40714i
\(672\) 0 0
\(673\) −11.0000 + 19.0526i −0.424019 + 0.734422i −0.996328 0.0856156i \(-0.972714\pi\)
0.572309 + 0.820038i \(0.306048\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 13.5000 23.3827i 0.518847 0.898670i −0.480913 0.876768i \(-0.659695\pi\)
0.999760 0.0219013i \(-0.00697196\pi\)
\(678\) 0 0
\(679\) 6.00000 17.3205i 0.230259 0.664700i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 15.5885i 0.596476i −0.954492 0.298238i \(-0.903601\pi\)
0.954492 0.298238i \(-0.0963989\pi\)
\(684\) 0 0
\(685\) 31.1769i 1.19121i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −9.00000 5.19615i −0.342376 0.197671i 0.318946 0.947773i \(-0.396671\pi\)
−0.661322 + 0.750102i \(0.730004\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −45.0000 25.9808i −1.70695 0.985506i
\(696\) 0 0
\(697\) −9.00000 15.5885i −0.340899 0.590455i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 31.1769i 1.17754i 0.808302 + 0.588768i \(0.200387\pi\)
−0.808302 + 0.588768i \(0.799613\pi\)
\(702\) 0 0
\(703\) 1.73205i 0.0653255i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 45.0000 + 15.5885i 1.69240 + 0.586264i
\(708\) 0 0
\(709\) 11.5000 19.9186i 0.431892 0.748058i −0.565145 0.824992i \(-0.691180\pi\)
0.997036 + 0.0769337i \(0.0245130\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 13.5000 23.3827i 0.505579 0.875688i
\(714\) 0 0
\(715\) 27.0000 + 46.7654i 1.00974 + 1.74893i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −6.00000 −0.223762 −0.111881 0.993722i \(-0.535688\pi\)
−0.111881 + 0.993722i \(0.535688\pi\)
\(720\) 0 0
\(721\) 15.0000 + 17.3205i 0.558629 + 0.645049i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −36.0000 + 20.7846i −1.33701 + 0.771921i
\(726\) 0 0
\(727\) −27.0000 15.5885i −1.00137 0.578144i −0.0927199 0.995692i \(-0.529556\pi\)
−0.908655 + 0.417548i \(0.862889\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −30.0000 + 51.9615i −1.10959 + 1.92187i
\(732\) 0 0
\(733\) −9.00000 + 5.19615i −0.332423 + 0.191924i −0.656916 0.753964i \(-0.728139\pi\)
0.324494 + 0.945888i \(0.394806\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 10.3923i 0.382805i
\(738\) 0 0
\(739\) −16.0000 −0.588570 −0.294285 0.955718i \(-0.595081\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 31.5000 18.1865i 1.15562 0.667199i 0.205372 0.978684i \(-0.434160\pi\)
0.950251 + 0.311485i \(0.100826\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 27.0000 5.19615i 0.986559 0.189863i
\(750\) 0 0
\(751\) 1.00000 + 1.73205i 0.0364905 + 0.0632034i 0.883694 0.468065i \(-0.155049\pi\)
−0.847203 + 0.531269i \(0.821715\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −12.0000 −0.436725
\(756\) 0 0
\(757\) −26.0000 −0.944986 −0.472493 0.881334i \(-0.656646\pi\)
−0.472493 + 0.881334i \(0.656646\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 3.00000 + 5.19615i 0.108750 + 0.188360i 0.915264 0.402854i \(-0.131982\pi\)
−0.806514 + 0.591215i \(0.798649\pi\)
\(762\) 0 0
\(763\) −5.50000 28.5788i −0.199113 1.03462i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −18.0000 10.3923i −0.649942 0.375244i
\(768\) 0 0
\(769\) 3.00000 1.73205i 0.108183 0.0624593i −0.444932 0.895564i \(-0.646772\pi\)
0.553115 + 0.833105i \(0.313439\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −39.0000 −1.40273 −0.701366 0.712801i \(-0.747426\pi\)
−0.701366 + 0.712801i \(0.747426\pi\)
\(774\) 0 0
\(775\) 20.7846i 0.746605i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −4.50000 + 2.59808i −0.161229 + 0.0930857i
\(780\) 0 0
\(781\) −13.5000 + 23.3827i −0.483068 + 0.836698i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 27.0000 + 15.5885i 0.963671 + 0.556376i
\(786\) 0 0
\(787\) 33.0000 19.0526i 1.17632 0.679150i 0.221162 0.975237i \(-0.429015\pi\)
0.955161 + 0.296087i \(0.0956817\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 48.0000 1.70453
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −22.5000 38.9711i −0.796991 1.38043i −0.921567 0.388219i \(-0.873091\pi\)
0.124576 0.992210i \(-0.460243\pi\)
\(798\) 0 0
\(799\) −18.0000 + 31.1769i −0.636794 + 1.10296i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 9.00000 15.5885i 0.317603 0.550105i
\(804\) 0 0
\(805\) 13.5000 38.9711i 0.475812 1.37355i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 31.1769i 1.09612i 0.836438 + 0.548061i \(0.184634\pi\)
−0.836438 + 0.548061i \(0.815366\pi\)
\(810\) 0 0
\(811\) 5.19615i 0.182462i 0.995830 + 0.0912308i \(0.0290801\pi\)
−0.995830 + 0.0912308i \(0.970920\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −15.0000 25.9808i −0.525427 0.910066i
\(816\) 0 0
\(817\) 15.0000 + 8.66025i 0.524784 + 0.302984i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 27.0000 + 15.5885i 0.942306 + 0.544041i 0.890683 0.454626i \(-0.150227\pi\)
0.0516239 + 0.998667i \(0.483560\pi\)
\(822\) 0 0
\(823\) −10.0000 17.3205i −0.348578 0.603755i 0.637419 0.770517i \(-0.280002\pi\)
−0.985997 + 0.166762i \(0.946669\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 46.7654i 1.62619i 0.582130 + 0.813096i \(0.302219\pi\)
−0.582130 + 0.813096i \(0.697781\pi\)
\(828\) 0 0
\(829\) 45.0333i 1.56407i −0.623233 0.782036i \(-0.714181\pi\)
0.623233 0.782036i \(-0.285819\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −39.0000 + 15.5885i −1.35127 + 0.540108i
\(834\) 0 0
\(835\) −18.0000 + 31.1769i −0.622916 + 1.07892i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −21.0000 + 36.3731i −0.725001 + 1.25574i 0.233973 + 0.972243i \(0.424827\pi\)
−0.958974 + 0.283495i \(0.908506\pi\)
\(840\) 0 0
\(841\) 39.5000 + 68.4160i 1.36207 + 2.35917i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −3.00000 −0.103203
\(846\) 0 0
\(847\) −32.0000 + 27.7128i −1.09953 + 0.952224i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −4.50000 + 2.59808i −0.154258 + 0.0890609i
\(852\) 0 0
\(853\) −42.0000 24.2487i −1.43805 0.830260i −0.440338 0.897832i \(-0.645141\pi\)
−0.997714 + 0.0675719i \(0.978475\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 10.5000 18.1865i 0.358673 0.621240i −0.629066 0.777352i \(-0.716563\pi\)
0.987739 + 0.156112i \(0.0498959\pi\)
\(858\) 0 0
\(859\) −25.5000 + 14.7224i −0.870049 + 0.502323i −0.867364 0.497674i \(-0.834188\pi\)
−0.00268433 + 0.999996i \(0.500854\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 10.3923i 0.353758i 0.984233 + 0.176879i \(0.0566002\pi\)
−0.984233 + 0.176879i \(0.943400\pi\)
\(864\) 0 0
\(865\) −27.0000 −0.918028
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 63.0000 36.3731i 2.13713 1.23387i
\(870\) 0 0
\(871\) 6.00000 + 3.46410i 0.203302 + 0.117377i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 1.50000 + 7.79423i 0.0507093 + 0.263493i
\(876\) 0 0
\(877\) −1.00000 1.73205i −0.0337676 0.0584872i 0.848648 0.528958i \(-0.177417\pi\)
−0.882415 + 0.470471i \(0.844084\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 27.0000 0.909653 0.454827 0.890580i \(-0.349701\pi\)
0.454827 + 0.890580i \(0.349701\pi\)
\(882\) 0 0
\(883\) 28.0000 0.942275 0.471138 0.882060i \(-0.343844\pi\)
0.471138 + 0.882060i \(0.343844\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −21.0000 36.3731i −0.705111 1.22129i −0.966651 0.256096i \(-0.917564\pi\)
0.261540 0.965193i \(-0.415770\pi\)
\(888\) 0 0
\(889\) 1.00000 + 5.19615i 0.0335389 + 0.174273i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 9.00000 + 5.19615i 0.301174 + 0.173883i
\(894\) 0 0
\(895\) −27.0000 + 15.5885i −0.902510 + 0.521065i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −54.0000 −1.80100
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −18.0000 + 10.3923i −0.598340 + 0.345452i
\(906\) 0 0
\(907\) 16.0000 27.7128i 0.531271 0.920189i −0.468063 0.883695i \(-0.655048\pi\)
0.999334 0.0364935i \(-0.0116188\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 9.00000 + 5.19615i 0.298183 + 0.172156i 0.641626 0.767017i \(-0.278260\pi\)
−0.343443 + 0.939173i \(0.611593\pi\)
\(912\) 0 0
\(913\) −27.0000 + 15.5885i −0.893570 + 0.515903i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 36.0000 31.1769i 1.18882 1.02955i
\(918\) 0 0
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 9.00000 + 15.5885i 0.296239 + 0.513100i
\(924\) 0 0
\(925\) −2.00000 + 3.46410i −0.0657596 + 0.113899i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 15.0000 25.9808i 0.492134 0.852401i −0.507825 0.861460i \(-0.669550\pi\)
0.999959 + 0.00905914i \(0.00288365\pi\)
\(930\) 0 0
\(931\) 4.50000 + 11.2583i 0.147482 + 0.368977i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 93.5307i 3.05878i
\(936\) 0 0
\(937\) 55.4256i 1.81068i 0.424691 + 0.905338i \(0.360383\pi\)
−0.424691 + 0.905338i \(0.639617\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −10.5000 18.1865i −0.342290 0.592864i 0.642567 0.766229i \(-0.277869\pi\)
−0.984858 + 0.173365i \(0.944536\pi\)
\(942\) 0 0
\(943\) 13.5000 + 7.79423i 0.439620 + 0.253815i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −22.5000 12.9904i −0.731152 0.422131i 0.0876916 0.996148i \(-0.472051\pi\)
−0.818843 + 0.574017i \(0.805384\pi\)
\(948\) 0 0
\(949\) −6.00000 10.3923i −0.194768 0.337348i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 31.1769i 1.00992i −0.863143 0.504960i \(-0.831507\pi\)
0.863143 0.504960i \(-0.168493\pi\)
\(954\) 0 0
\(955\) 15.5885i 0.504431i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −9.00000 + 25.9808i −0.290625 + 0.838963i
\(960\) 0 0
\(961\) −2.00000 + 3.46410i −0.0645161 + 0.111745i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −3.00000 + 5.19615i −0.0965734 + 0.167270i
\(966\) 0 0
\(967\) 8.00000 + 13.8564i 0.257263 + 0.445592i 0.965508 0.260375i \(-0.0838461\pi\)
−0.708245 + 0.705967i \(0.750513\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −6.00000 −0.192549 −0.0962746 0.995355i \(-0.530693\pi\)
−0.0962746 + 0.995355i \(0.530693\pi\)
\(972\) 0 0
\(973\) −30.0000 34.6410i −0.961756 1.11054i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 36.0000 20.7846i 1.15174 0.664959i 0.202431 0.979297i \(-0.435116\pi\)
0.949311 + 0.314338i \(0.101783\pi\)
\(978\) 0 0
\(979\) 40.5000 + 23.3827i 1.29439 + 0.747314i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 21.0000 36.3731i 0.669796 1.16012i −0.308165 0.951333i \(-0.599715\pi\)
0.977961 0.208788i \(-0.0669518\pi\)
\(984\) 0 0
\(985\) −54.0000 + 31.1769i −1.72058 + 0.993379i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 51.9615i 1.65228i
\(990\) 0 0
\(991\) 10.0000 0.317660 0.158830 0.987306i \(-0.449228\pi\)
0.158830 + 0.987306i \(0.449228\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −4.50000 + 2.59808i −0.142660 + 0.0823646i
\(996\) 0 0
\(997\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(998\) 0 0
\(999\) 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 2268.2.x.h.1889.1 2
3.2 odd 2 2268.2.x.b.1889.1 2
7.6 odd 2 2268.2.x.a.1889.1 2
9.2 odd 6 756.2.f.c.377.1 yes 2
9.4 even 3 2268.2.x.g.377.1 2
9.5 odd 6 2268.2.x.a.377.1 2
9.7 even 3 756.2.f.a.377.1 2
21.20 even 2 2268.2.x.g.1889.1 2
36.7 odd 6 3024.2.k.a.1889.2 2
36.11 even 6 3024.2.k.d.1889.2 2
63.13 odd 6 2268.2.x.b.377.1 2
63.20 even 6 756.2.f.a.377.2 yes 2
63.34 odd 6 756.2.f.c.377.2 yes 2
63.41 even 6 inner 2268.2.x.h.377.1 2
252.83 odd 6 3024.2.k.a.1889.1 2
252.223 even 6 3024.2.k.d.1889.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
756.2.f.a.377.1 2 9.7 even 3
756.2.f.a.377.2 yes 2 63.20 even 6
756.2.f.c.377.1 yes 2 9.2 odd 6
756.2.f.c.377.2 yes 2 63.34 odd 6
2268.2.x.a.377.1 2 9.5 odd 6
2268.2.x.a.1889.1 2 7.6 odd 2
2268.2.x.b.377.1 2 63.13 odd 6
2268.2.x.b.1889.1 2 3.2 odd 2
2268.2.x.g.377.1 2 9.4 even 3
2268.2.x.g.1889.1 2 21.20 even 2
2268.2.x.h.377.1 2 63.41 even 6 inner
2268.2.x.h.1889.1 2 1.1 even 1 trivial
3024.2.k.a.1889.1 2 252.83 odd 6
3024.2.k.a.1889.2 2 36.7 odd 6
3024.2.k.d.1889.1 2 252.223 even 6
3024.2.k.d.1889.2 2 36.11 even 6