Properties

Label 304.2.i.d.49.1
Level $304$
Weight $2$
Character 304.49
Analytic conductor $2.427$
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] = [304,2,Mod(49,304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(304, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("304.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 304.i (of order \(3\), 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: \(2.42745222145\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 152)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 49.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 304.49
Dual form 304.2.i.d.273.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.500000 - 0.866025i) q^{3} +(2.00000 - 3.46410i) q^{5} +(1.00000 + 1.73205i) q^{9} +O(q^{10})\) \(q+(0.500000 - 0.866025i) q^{3} +(2.00000 - 3.46410i) q^{5} +(1.00000 + 1.73205i) q^{9} -3.00000 q^{11} +(-1.00000 - 1.73205i) q^{13} +(-2.00000 - 3.46410i) q^{15} +(-1.00000 + 1.73205i) q^{17} +(-0.500000 - 4.33013i) q^{19} +(3.00000 + 5.19615i) q^{23} +(-5.50000 - 9.52628i) q^{25} +5.00000 q^{27} +(2.00000 + 3.46410i) q^{29} +10.0000 q^{31} +(-1.50000 + 2.59808i) q^{33} +2.00000 q^{37} -2.00000 q^{39} +(-4.50000 + 7.79423i) q^{41} +(-2.00000 + 3.46410i) q^{43} +8.00000 q^{45} +(-6.00000 - 10.3923i) q^{47} -7.00000 q^{49} +(1.00000 + 1.73205i) q^{51} +(1.00000 + 1.73205i) q^{53} +(-6.00000 + 10.3923i) q^{55} +(-4.00000 - 1.73205i) q^{57} +(-0.500000 + 0.866025i) q^{59} +(4.00000 + 6.92820i) q^{61} -8.00000 q^{65} +(4.50000 + 7.79423i) q^{67} +6.00000 q^{69} +(-3.00000 + 5.19615i) q^{71} +(4.50000 - 7.79423i) q^{73} -11.0000 q^{75} +(-2.00000 + 3.46410i) q^{79} +(-0.500000 + 0.866025i) q^{81} +5.00000 q^{83} +(4.00000 + 6.92820i) q^{85} +4.00000 q^{87} +(9.00000 + 15.5885i) q^{89} +(5.00000 - 8.66025i) q^{93} +(-16.0000 - 6.92820i) q^{95} +(-0.500000 + 0.866025i) q^{97} +(-3.00000 - 5.19615i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + 4 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} + 4 q^{5} + 2 q^{9} - 6 q^{11} - 2 q^{13} - 4 q^{15} - 2 q^{17} - q^{19} + 6 q^{23} - 11 q^{25} + 10 q^{27} + 4 q^{29} + 20 q^{31} - 3 q^{33} + 4 q^{37} - 4 q^{39} - 9 q^{41} - 4 q^{43} + 16 q^{45} - 12 q^{47} - 14 q^{49} + 2 q^{51} + 2 q^{53} - 12 q^{55} - 8 q^{57} - q^{59} + 8 q^{61} - 16 q^{65} + 9 q^{67} + 12 q^{69} - 6 q^{71} + 9 q^{73} - 22 q^{75} - 4 q^{79} - q^{81} + 10 q^{83} + 8 q^{85} + 8 q^{87} + 18 q^{89} + 10 q^{93} - 32 q^{95} - q^{97} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/304\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(191\) \(229\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.500000 0.866025i 0.288675 0.500000i −0.684819 0.728714i \(-0.740119\pi\)
0.973494 + 0.228714i \(0.0734519\pi\)
\(4\) 0 0
\(5\) 2.00000 3.46410i 0.894427 1.54919i 0.0599153 0.998203i \(-0.480917\pi\)
0.834512 0.550990i \(-0.185750\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 1.00000 + 1.73205i 0.333333 + 0.577350i
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 0 0
\(13\) −1.00000 1.73205i −0.277350 0.480384i 0.693375 0.720577i \(-0.256123\pi\)
−0.970725 + 0.240192i \(0.922790\pi\)
\(14\) 0 0
\(15\) −2.00000 3.46410i −0.516398 0.894427i
\(16\) 0 0
\(17\) −1.00000 + 1.73205i −0.242536 + 0.420084i −0.961436 0.275029i \(-0.911312\pi\)
0.718900 + 0.695113i \(0.244646\pi\)
\(18\) 0 0
\(19\) −0.500000 4.33013i −0.114708 0.993399i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.00000 + 5.19615i 0.625543 + 1.08347i 0.988436 + 0.151642i \(0.0484560\pi\)
−0.362892 + 0.931831i \(0.618211\pi\)
\(24\) 0 0
\(25\) −5.50000 9.52628i −1.10000 1.90526i
\(26\) 0 0
\(27\) 5.00000 0.962250
\(28\) 0 0
\(29\) 2.00000 + 3.46410i 0.371391 + 0.643268i 0.989780 0.142605i \(-0.0455477\pi\)
−0.618389 + 0.785872i \(0.712214\pi\)
\(30\) 0 0
\(31\) 10.0000 1.79605 0.898027 0.439941i \(-0.145001\pi\)
0.898027 + 0.439941i \(0.145001\pi\)
\(32\) 0 0
\(33\) −1.50000 + 2.59808i −0.261116 + 0.452267i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) −4.50000 + 7.79423i −0.702782 + 1.21725i 0.264704 + 0.964330i \(0.414726\pi\)
−0.967486 + 0.252924i \(0.918608\pi\)
\(42\) 0 0
\(43\) −2.00000 + 3.46410i −0.304997 + 0.528271i −0.977261 0.212041i \(-0.931989\pi\)
0.672264 + 0.740312i \(0.265322\pi\)
\(44\) 0 0
\(45\) 8.00000 1.19257
\(46\) 0 0
\(47\) −6.00000 10.3923i −0.875190 1.51587i −0.856560 0.516047i \(-0.827403\pi\)
−0.0186297 0.999826i \(-0.505930\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 1.00000 + 1.73205i 0.140028 + 0.242536i
\(52\) 0 0
\(53\) 1.00000 + 1.73205i 0.137361 + 0.237915i 0.926497 0.376303i \(-0.122805\pi\)
−0.789136 + 0.614218i \(0.789471\pi\)
\(54\) 0 0
\(55\) −6.00000 + 10.3923i −0.809040 + 1.40130i
\(56\) 0 0
\(57\) −4.00000 1.73205i −0.529813 0.229416i
\(58\) 0 0
\(59\) −0.500000 + 0.866025i −0.0650945 + 0.112747i −0.896736 0.442566i \(-0.854068\pi\)
0.831641 + 0.555313i \(0.187402\pi\)
\(60\) 0 0
\(61\) 4.00000 + 6.92820i 0.512148 + 0.887066i 0.999901 + 0.0140840i \(0.00448323\pi\)
−0.487753 + 0.872982i \(0.662183\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −8.00000 −0.992278
\(66\) 0 0
\(67\) 4.50000 + 7.79423i 0.549762 + 0.952217i 0.998290 + 0.0584478i \(0.0186151\pi\)
−0.448528 + 0.893769i \(0.648052\pi\)
\(68\) 0 0
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) −3.00000 + 5.19615i −0.356034 + 0.616670i −0.987294 0.158901i \(-0.949205\pi\)
0.631260 + 0.775571i \(0.282538\pi\)
\(72\) 0 0
\(73\) 4.50000 7.79423i 0.526685 0.912245i −0.472831 0.881153i \(-0.656768\pi\)
0.999517 0.0310925i \(-0.00989865\pi\)
\(74\) 0 0
\(75\) −11.0000 −1.27017
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −2.00000 + 3.46410i −0.225018 + 0.389742i −0.956325 0.292306i \(-0.905577\pi\)
0.731307 + 0.682048i \(0.238911\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) 5.00000 0.548821 0.274411 0.961613i \(-0.411517\pi\)
0.274411 + 0.961613i \(0.411517\pi\)
\(84\) 0 0
\(85\) 4.00000 + 6.92820i 0.433861 + 0.751469i
\(86\) 0 0
\(87\) 4.00000 0.428845
\(88\) 0 0
\(89\) 9.00000 + 15.5885i 0.953998 + 1.65237i 0.736644 + 0.676280i \(0.236409\pi\)
0.217354 + 0.976093i \(0.430258\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 5.00000 8.66025i 0.518476 0.898027i
\(94\) 0 0
\(95\) −16.0000 6.92820i −1.64157 0.710819i
\(96\) 0 0
\(97\) −0.500000 + 0.866025i −0.0507673 + 0.0879316i −0.890292 0.455389i \(-0.849500\pi\)
0.839525 + 0.543321i \(0.182833\pi\)
\(98\) 0 0
\(99\) −3.00000 5.19615i −0.301511 0.522233i
\(100\) 0 0
\(101\) −6.00000 10.3923i −0.597022 1.03407i −0.993258 0.115924i \(-0.963017\pi\)
0.396236 0.918149i \(-0.370316\pi\)
\(102\) 0 0
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −16.0000 −1.54678 −0.773389 0.633932i \(-0.781440\pi\)
−0.773389 + 0.633932i \(0.781440\pi\)
\(108\) 0 0
\(109\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(110\) 0 0
\(111\) 1.00000 1.73205i 0.0949158 0.164399i
\(112\) 0 0
\(113\) −1.00000 −0.0940721 −0.0470360 0.998893i \(-0.514978\pi\)
−0.0470360 + 0.998893i \(0.514978\pi\)
\(114\) 0 0
\(115\) 24.0000 2.23801
\(116\) 0 0
\(117\) 2.00000 3.46410i 0.184900 0.320256i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) 4.50000 + 7.79423i 0.405751 + 0.702782i
\(124\) 0 0
\(125\) −24.0000 −2.14663
\(126\) 0 0
\(127\) −3.00000 5.19615i −0.266207 0.461084i 0.701672 0.712500i \(-0.252437\pi\)
−0.967879 + 0.251416i \(0.919104\pi\)
\(128\) 0 0
\(129\) 2.00000 + 3.46410i 0.176090 + 0.304997i
\(130\) 0 0
\(131\) −7.50000 + 12.9904i −0.655278 + 1.13497i 0.326546 + 0.945181i \(0.394115\pi\)
−0.981824 + 0.189794i \(0.939218\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 10.0000 17.3205i 0.860663 1.49071i
\(136\) 0 0
\(137\) −4.50000 7.79423i −0.384461 0.665906i 0.607233 0.794524i \(-0.292279\pi\)
−0.991694 + 0.128618i \(0.958946\pi\)
\(138\) 0 0
\(139\) −6.50000 11.2583i −0.551323 0.954919i −0.998179 0.0603135i \(-0.980790\pi\)
0.446857 0.894606i \(-0.352543\pi\)
\(140\) 0 0
\(141\) −12.0000 −1.01058
\(142\) 0 0
\(143\) 3.00000 + 5.19615i 0.250873 + 0.434524i
\(144\) 0 0
\(145\) 16.0000 1.32873
\(146\) 0 0
\(147\) −3.50000 + 6.06218i −0.288675 + 0.500000i
\(148\) 0 0
\(149\) 5.00000 8.66025i 0.409616 0.709476i −0.585231 0.810867i \(-0.698996\pi\)
0.994847 + 0.101391i \(0.0323294\pi\)
\(150\) 0 0
\(151\) −2.00000 −0.162758 −0.0813788 0.996683i \(-0.525932\pi\)
−0.0813788 + 0.996683i \(0.525932\pi\)
\(152\) 0 0
\(153\) −4.00000 −0.323381
\(154\) 0 0
\(155\) 20.0000 34.6410i 1.60644 2.78243i
\(156\) 0 0
\(157\) −4.00000 + 6.92820i −0.319235 + 0.552931i −0.980329 0.197372i \(-0.936759\pi\)
0.661094 + 0.750303i \(0.270093\pi\)
\(158\) 0 0
\(159\) 2.00000 0.158610
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 11.0000 0.861586 0.430793 0.902451i \(-0.358234\pi\)
0.430793 + 0.902451i \(0.358234\pi\)
\(164\) 0 0
\(165\) 6.00000 + 10.3923i 0.467099 + 0.809040i
\(166\) 0 0
\(167\) 8.00000 + 13.8564i 0.619059 + 1.07224i 0.989658 + 0.143448i \(0.0458190\pi\)
−0.370599 + 0.928793i \(0.620848\pi\)
\(168\) 0 0
\(169\) 4.50000 7.79423i 0.346154 0.599556i
\(170\) 0 0
\(171\) 7.00000 5.19615i 0.535303 0.397360i
\(172\) 0 0
\(173\) 13.0000 22.5167i 0.988372 1.71191i 0.362500 0.931984i \(-0.381923\pi\)
0.625871 0.779926i \(-0.284744\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0.500000 + 0.866025i 0.0375823 + 0.0650945i
\(178\) 0 0
\(179\) −9.00000 −0.672692 −0.336346 0.941739i \(-0.609191\pi\)
−0.336346 + 0.941739i \(0.609191\pi\)
\(180\) 0 0
\(181\) −7.00000 12.1244i −0.520306 0.901196i −0.999721 0.0236082i \(-0.992485\pi\)
0.479415 0.877588i \(-0.340849\pi\)
\(182\) 0 0
\(183\) 8.00000 0.591377
\(184\) 0 0
\(185\) 4.00000 6.92820i 0.294086 0.509372i
\(186\) 0 0
\(187\) 3.00000 5.19615i 0.219382 0.379980i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 0 0
\(193\) 3.00000 5.19615i 0.215945 0.374027i −0.737620 0.675216i \(-0.764050\pi\)
0.953564 + 0.301189i \(0.0973836\pi\)
\(194\) 0 0
\(195\) −4.00000 + 6.92820i −0.286446 + 0.496139i
\(196\) 0 0
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) −9.00000 15.5885i −0.637993 1.10504i −0.985873 0.167497i \(-0.946431\pi\)
0.347879 0.937539i \(-0.386902\pi\)
\(200\) 0 0
\(201\) 9.00000 0.634811
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 18.0000 + 31.1769i 1.25717 + 2.17749i
\(206\) 0 0
\(207\) −6.00000 + 10.3923i −0.417029 + 0.722315i
\(208\) 0 0
\(209\) 1.50000 + 12.9904i 0.103757 + 0.898563i
\(210\) 0 0
\(211\) −6.00000 + 10.3923i −0.413057 + 0.715436i −0.995222 0.0976347i \(-0.968872\pi\)
0.582165 + 0.813070i \(0.302206\pi\)
\(212\) 0 0
\(213\) 3.00000 + 5.19615i 0.205557 + 0.356034i
\(214\) 0 0
\(215\) 8.00000 + 13.8564i 0.545595 + 0.944999i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −4.50000 7.79423i −0.304082 0.526685i
\(220\) 0 0
\(221\) 4.00000 0.269069
\(222\) 0 0
\(223\) 5.00000 8.66025i 0.334825 0.579934i −0.648626 0.761107i \(-0.724656\pi\)
0.983451 + 0.181173i \(0.0579895\pi\)
\(224\) 0 0
\(225\) 11.0000 19.0526i 0.733333 1.27017i
\(226\) 0 0
\(227\) −19.0000 −1.26107 −0.630537 0.776159i \(-0.717165\pi\)
−0.630537 + 0.776159i \(0.717165\pi\)
\(228\) 0 0
\(229\) −8.00000 −0.528655 −0.264327 0.964433i \(-0.585150\pi\)
−0.264327 + 0.964433i \(0.585150\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −5.50000 + 9.52628i −0.360317 + 0.624087i −0.988013 0.154371i \(-0.950665\pi\)
0.627696 + 0.778459i \(0.283998\pi\)
\(234\) 0 0
\(235\) −48.0000 −3.13117
\(236\) 0 0
\(237\) 2.00000 + 3.46410i 0.129914 + 0.225018i
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) −10.5000 18.1865i −0.676364 1.17150i −0.976068 0.217465i \(-0.930221\pi\)
0.299704 0.954032i \(-0.403112\pi\)
\(242\) 0 0
\(243\) 8.00000 + 13.8564i 0.513200 + 0.888889i
\(244\) 0 0
\(245\) −14.0000 + 24.2487i −0.894427 + 1.54919i
\(246\) 0 0
\(247\) −7.00000 + 5.19615i −0.445399 + 0.330623i
\(248\) 0 0
\(249\) 2.50000 4.33013i 0.158431 0.274411i
\(250\) 0 0
\(251\) 2.50000 + 4.33013i 0.157799 + 0.273315i 0.934075 0.357078i \(-0.116227\pi\)
−0.776276 + 0.630393i \(0.782894\pi\)
\(252\) 0 0
\(253\) −9.00000 15.5885i −0.565825 0.980038i
\(254\) 0 0
\(255\) 8.00000 0.500979
\(256\) 0 0
\(257\) −1.50000 2.59808i −0.0935674 0.162064i 0.815442 0.578838i \(-0.196494\pi\)
−0.909010 + 0.416775i \(0.863160\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −4.00000 + 6.92820i −0.247594 + 0.428845i
\(262\) 0 0
\(263\) 8.00000 13.8564i 0.493301 0.854423i −0.506669 0.862141i \(-0.669123\pi\)
0.999970 + 0.00771799i \(0.00245674\pi\)
\(264\) 0 0
\(265\) 8.00000 0.491436
\(266\) 0 0
\(267\) 18.0000 1.10158
\(268\) 0 0
\(269\) −2.00000 + 3.46410i −0.121942 + 0.211210i −0.920534 0.390664i \(-0.872246\pi\)
0.798591 + 0.601874i \(0.205579\pi\)
\(270\) 0 0
\(271\) −10.0000 + 17.3205i −0.607457 + 1.05215i 0.384201 + 0.923249i \(0.374477\pi\)
−0.991658 + 0.128897i \(0.958856\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 16.5000 + 28.5788i 0.994987 + 1.72337i
\(276\) 0 0
\(277\) 12.0000 0.721010 0.360505 0.932757i \(-0.382604\pi\)
0.360505 + 0.932757i \(0.382604\pi\)
\(278\) 0 0
\(279\) 10.0000 + 17.3205i 0.598684 + 1.03695i
\(280\) 0 0
\(281\) 6.50000 + 11.2583i 0.387757 + 0.671616i 0.992148 0.125073i \(-0.0399165\pi\)
−0.604390 + 0.796689i \(0.706583\pi\)
\(282\) 0 0
\(283\) 6.50000 11.2583i 0.386385 0.669238i −0.605575 0.795788i \(-0.707057\pi\)
0.991960 + 0.126550i \(0.0403903\pi\)
\(284\) 0 0
\(285\) −14.0000 + 10.3923i −0.829288 + 0.615587i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 6.50000 + 11.2583i 0.382353 + 0.662255i
\(290\) 0 0
\(291\) 0.500000 + 0.866025i 0.0293105 + 0.0507673i
\(292\) 0 0
\(293\) −4.00000 −0.233682 −0.116841 0.993151i \(-0.537277\pi\)
−0.116841 + 0.993151i \(0.537277\pi\)
\(294\) 0 0
\(295\) 2.00000 + 3.46410i 0.116445 + 0.201688i
\(296\) 0 0
\(297\) −15.0000 −0.870388
\(298\) 0 0
\(299\) 6.00000 10.3923i 0.346989 0.601003i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −12.0000 −0.689382
\(304\) 0 0
\(305\) 32.0000 1.83231
\(306\) 0 0
\(307\) 12.5000 21.6506i 0.713413 1.23567i −0.250156 0.968206i \(-0.580482\pi\)
0.963569 0.267461i \(-0.0861848\pi\)
\(308\) 0 0
\(309\) 7.00000 12.1244i 0.398216 0.689730i
\(310\) 0 0
\(311\) 2.00000 0.113410 0.0567048 0.998391i \(-0.481941\pi\)
0.0567048 + 0.998391i \(0.481941\pi\)
\(312\) 0 0
\(313\) 9.50000 + 16.4545i 0.536972 + 0.930062i 0.999065 + 0.0432311i \(0.0137652\pi\)
−0.462093 + 0.886831i \(0.652902\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −3.00000 5.19615i −0.168497 0.291845i 0.769395 0.638774i \(-0.220558\pi\)
−0.937892 + 0.346929i \(0.887225\pi\)
\(318\) 0 0
\(319\) −6.00000 10.3923i −0.335936 0.581857i
\(320\) 0 0
\(321\) −8.00000 + 13.8564i −0.446516 + 0.773389i
\(322\) 0 0
\(323\) 8.00000 + 3.46410i 0.445132 + 0.192748i
\(324\) 0 0
\(325\) −11.0000 + 19.0526i −0.610170 + 1.05685i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −13.0000 −0.714545 −0.357272 0.934000i \(-0.616293\pi\)
−0.357272 + 0.934000i \(0.616293\pi\)
\(332\) 0 0
\(333\) 2.00000 + 3.46410i 0.109599 + 0.189832i
\(334\) 0 0
\(335\) 36.0000 1.96689
\(336\) 0 0
\(337\) −1.50000 + 2.59808i −0.0817102 + 0.141526i −0.903985 0.427565i \(-0.859372\pi\)
0.822274 + 0.569091i \(0.192705\pi\)
\(338\) 0 0
\(339\) −0.500000 + 0.866025i −0.0271563 + 0.0470360i
\(340\) 0 0
\(341\) −30.0000 −1.62459
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 12.0000 20.7846i 0.646058 1.11901i
\(346\) 0 0
\(347\) −4.50000 + 7.79423i −0.241573 + 0.418416i −0.961162 0.275983i \(-0.910997\pi\)
0.719590 + 0.694399i \(0.244330\pi\)
\(348\) 0 0
\(349\) −28.0000 −1.49881 −0.749403 0.662114i \(-0.769659\pi\)
−0.749403 + 0.662114i \(0.769659\pi\)
\(350\) 0 0
\(351\) −5.00000 8.66025i −0.266880 0.462250i
\(352\) 0 0
\(353\) 11.0000 0.585471 0.292735 0.956193i \(-0.405434\pi\)
0.292735 + 0.956193i \(0.405434\pi\)
\(354\) 0 0
\(355\) 12.0000 + 20.7846i 0.636894 + 1.10313i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −1.00000 + 1.73205i −0.0527780 + 0.0914141i −0.891207 0.453596i \(-0.850141\pi\)
0.838429 + 0.545010i \(0.183474\pi\)
\(360\) 0 0
\(361\) −18.5000 + 4.33013i −0.973684 + 0.227901i
\(362\) 0 0
\(363\) −1.00000 + 1.73205i −0.0524864 + 0.0909091i
\(364\) 0 0
\(365\) −18.0000 31.1769i −0.942163 1.63187i
\(366\) 0 0
\(367\) −13.0000 22.5167i −0.678594 1.17536i −0.975404 0.220423i \(-0.929256\pi\)
0.296810 0.954937i \(-0.404077\pi\)
\(368\) 0 0
\(369\) −18.0000 −0.937043
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 4.00000 0.207112 0.103556 0.994624i \(-0.466978\pi\)
0.103556 + 0.994624i \(0.466978\pi\)
\(374\) 0 0
\(375\) −12.0000 + 20.7846i −0.619677 + 1.07331i
\(376\) 0 0
\(377\) 4.00000 6.92820i 0.206010 0.356821i
\(378\) 0 0
\(379\) 12.0000 0.616399 0.308199 0.951322i \(-0.400274\pi\)
0.308199 + 0.951322i \(0.400274\pi\)
\(380\) 0 0
\(381\) −6.00000 −0.307389
\(382\) 0 0
\(383\) −4.00000 + 6.92820i −0.204390 + 0.354015i −0.949938 0.312437i \(-0.898855\pi\)
0.745548 + 0.666452i \(0.232188\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −8.00000 −0.406663
\(388\) 0 0
\(389\) −2.00000 3.46410i −0.101404 0.175637i 0.810859 0.585241i \(-0.199000\pi\)
−0.912263 + 0.409604i \(0.865667\pi\)
\(390\) 0 0
\(391\) −12.0000 −0.606866
\(392\) 0 0
\(393\) 7.50000 + 12.9904i 0.378325 + 0.655278i
\(394\) 0 0
\(395\) 8.00000 + 13.8564i 0.402524 + 0.697191i
\(396\) 0 0
\(397\) 7.00000 12.1244i 0.351320 0.608504i −0.635161 0.772380i \(-0.719066\pi\)
0.986481 + 0.163876i \(0.0523996\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1.50000 2.59808i 0.0749064 0.129742i −0.826139 0.563466i \(-0.809468\pi\)
0.901046 + 0.433724i \(0.142801\pi\)
\(402\) 0 0
\(403\) −10.0000 17.3205i −0.498135 0.862796i
\(404\) 0 0
\(405\) 2.00000 + 3.46410i 0.0993808 + 0.172133i
\(406\) 0 0
\(407\) −6.00000 −0.297409
\(408\) 0 0
\(409\) 17.5000 + 30.3109i 0.865319 + 1.49878i 0.866730 + 0.498778i \(0.166218\pi\)
−0.00141047 + 0.999999i \(0.500449\pi\)
\(410\) 0 0
\(411\) −9.00000 −0.443937
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 10.0000 17.3205i 0.490881 0.850230i
\(416\) 0 0
\(417\) −13.0000 −0.636613
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) 11.0000 19.0526i 0.536107 0.928565i −0.463002 0.886357i \(-0.653228\pi\)
0.999109 0.0422075i \(-0.0134391\pi\)
\(422\) 0 0
\(423\) 12.0000 20.7846i 0.583460 1.01058i
\(424\) 0 0
\(425\) 22.0000 1.06716
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 6.00000 0.289683
\(430\) 0 0
\(431\) −15.0000 25.9808i −0.722525 1.25145i −0.959985 0.280052i \(-0.909648\pi\)
0.237460 0.971397i \(-0.423685\pi\)
\(432\) 0 0
\(433\) −5.00000 8.66025i −0.240285 0.416185i 0.720511 0.693444i \(-0.243907\pi\)
−0.960795 + 0.277259i \(0.910574\pi\)
\(434\) 0 0
\(435\) 8.00000 13.8564i 0.383571 0.664364i
\(436\) 0 0
\(437\) 21.0000 15.5885i 1.00457 0.745697i
\(438\) 0 0
\(439\) −7.00000 + 12.1244i −0.334092 + 0.578664i −0.983310 0.181938i \(-0.941763\pi\)
0.649218 + 0.760602i \(0.275096\pi\)
\(440\) 0 0
\(441\) −7.00000 12.1244i −0.333333 0.577350i
\(442\) 0 0
\(443\) 0.500000 + 0.866025i 0.0237557 + 0.0411461i 0.877659 0.479286i \(-0.159104\pi\)
−0.853903 + 0.520432i \(0.825771\pi\)
\(444\) 0 0
\(445\) 72.0000 3.41313
\(446\) 0 0
\(447\) −5.00000 8.66025i −0.236492 0.409616i
\(448\) 0 0
\(449\) 33.0000 1.55737 0.778683 0.627417i \(-0.215888\pi\)
0.778683 + 0.627417i \(0.215888\pi\)
\(450\) 0 0
\(451\) 13.5000 23.3827i 0.635690 1.10105i
\(452\) 0 0
\(453\) −1.00000 + 1.73205i −0.0469841 + 0.0813788i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −11.0000 −0.514558 −0.257279 0.966337i \(-0.582826\pi\)
−0.257279 + 0.966337i \(0.582826\pi\)
\(458\) 0 0
\(459\) −5.00000 + 8.66025i −0.233380 + 0.404226i
\(460\) 0 0
\(461\) −3.00000 + 5.19615i −0.139724 + 0.242009i −0.927392 0.374091i \(-0.877955\pi\)
0.787668 + 0.616100i \(0.211288\pi\)
\(462\) 0 0
\(463\) 34.0000 1.58011 0.790057 0.613033i \(-0.210051\pi\)
0.790057 + 0.613033i \(0.210051\pi\)
\(464\) 0 0
\(465\) −20.0000 34.6410i −0.927478 1.60644i
\(466\) 0 0
\(467\) −5.00000 −0.231372 −0.115686 0.993286i \(-0.536907\pi\)
−0.115686 + 0.993286i \(0.536907\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 4.00000 + 6.92820i 0.184310 + 0.319235i
\(472\) 0 0
\(473\) 6.00000 10.3923i 0.275880 0.477839i
\(474\) 0 0
\(475\) −38.5000 + 28.5788i −1.76650 + 1.31129i
\(476\) 0 0
\(477\) −2.00000 + 3.46410i −0.0915737 + 0.158610i
\(478\) 0 0
\(479\) 10.0000 + 17.3205i 0.456912 + 0.791394i 0.998796 0.0490589i \(-0.0156222\pi\)
−0.541884 + 0.840453i \(0.682289\pi\)
\(480\) 0 0
\(481\) −2.00000 3.46410i −0.0911922 0.157949i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 2.00000 + 3.46410i 0.0908153 + 0.157297i
\(486\) 0 0
\(487\) 26.0000 1.17817 0.589086 0.808070i \(-0.299488\pi\)
0.589086 + 0.808070i \(0.299488\pi\)
\(488\) 0 0
\(489\) 5.50000 9.52628i 0.248719 0.430793i
\(490\) 0 0
\(491\) 8.00000 13.8564i 0.361035 0.625331i −0.627096 0.778942i \(-0.715757\pi\)
0.988131 + 0.153611i \(0.0490902\pi\)
\(492\) 0 0
\(493\) −8.00000 −0.360302
\(494\) 0 0
\(495\) −24.0000 −1.07872
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 3.50000 6.06218i 0.156682 0.271380i −0.776989 0.629515i \(-0.783254\pi\)
0.933670 + 0.358134i \(0.116587\pi\)
\(500\) 0 0
\(501\) 16.0000 0.714827
\(502\) 0 0
\(503\) −3.00000 5.19615i −0.133763 0.231685i 0.791361 0.611349i \(-0.209373\pi\)
−0.925124 + 0.379664i \(0.876040\pi\)
\(504\) 0 0
\(505\) −48.0000 −2.13597
\(506\) 0 0
\(507\) −4.50000 7.79423i −0.199852 0.346154i
\(508\) 0 0
\(509\) −8.00000 13.8564i −0.354594 0.614174i 0.632455 0.774597i \(-0.282047\pi\)
−0.987048 + 0.160423i \(0.948714\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −2.50000 21.6506i −0.110378 0.955899i
\(514\) 0 0
\(515\) 28.0000 48.4974i 1.23383 2.13705i
\(516\) 0 0
\(517\) 18.0000 + 31.1769i 0.791639 + 1.37116i
\(518\) 0 0
\(519\) −13.0000 22.5167i −0.570637 0.988372i
\(520\) 0 0
\(521\) 1.00000 0.0438108 0.0219054 0.999760i \(-0.493027\pi\)
0.0219054 + 0.999760i \(0.493027\pi\)
\(522\) 0 0
\(523\) 10.0000 + 17.3205i 0.437269 + 0.757373i 0.997478 0.0709788i \(-0.0226123\pi\)
−0.560208 + 0.828352i \(0.689279\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −10.0000 + 17.3205i −0.435607 + 0.754493i
\(528\) 0 0
\(529\) −6.50000 + 11.2583i −0.282609 + 0.489493i
\(530\) 0 0
\(531\) −2.00000 −0.0867926
\(532\) 0 0
\(533\) 18.0000 0.779667
\(534\) 0 0
\(535\) −32.0000 + 55.4256i −1.38348 + 2.39626i
\(536\) 0 0
\(537\) −4.50000 + 7.79423i −0.194189 + 0.336346i
\(538\) 0 0
\(539\) 21.0000 0.904534
\(540\) 0 0
\(541\) −2.00000 3.46410i −0.0859867 0.148933i 0.819825 0.572615i \(-0.194071\pi\)
−0.905811 + 0.423681i \(0.860738\pi\)
\(542\) 0 0
\(543\) −14.0000 −0.600798
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 14.0000 + 24.2487i 0.598597 + 1.03680i 0.993028 + 0.117875i \(0.0376081\pi\)
−0.394432 + 0.918925i \(0.629059\pi\)
\(548\) 0 0
\(549\) −8.00000 + 13.8564i −0.341432 + 0.591377i
\(550\) 0 0
\(551\) 14.0000 10.3923i 0.596420 0.442727i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −4.00000 6.92820i −0.169791 0.294086i
\(556\) 0 0
\(557\) 20.0000 + 34.6410i 0.847427 + 1.46779i 0.883497 + 0.468438i \(0.155183\pi\)
−0.0360693 + 0.999349i \(0.511484\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) −3.00000 5.19615i −0.126660 0.219382i
\(562\) 0 0
\(563\) −11.0000 −0.463595 −0.231797 0.972764i \(-0.574461\pi\)
−0.231797 + 0.972764i \(0.574461\pi\)
\(564\) 0 0
\(565\) −2.00000 + 3.46410i −0.0841406 + 0.145736i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −46.0000 −1.92842 −0.964210 0.265139i \(-0.914582\pi\)
−0.964210 + 0.265139i \(0.914582\pi\)
\(570\) 0 0
\(571\) −17.0000 −0.711428 −0.355714 0.934595i \(-0.615762\pi\)
−0.355714 + 0.934595i \(0.615762\pi\)
\(572\) 0 0
\(573\) −6.00000 + 10.3923i −0.250654 + 0.434145i
\(574\) 0 0
\(575\) 33.0000 57.1577i 1.37620 2.38364i
\(576\) 0 0
\(577\) −37.0000 −1.54033 −0.770165 0.637845i \(-0.779826\pi\)
−0.770165 + 0.637845i \(0.779826\pi\)
\(578\) 0 0
\(579\) −3.00000 5.19615i −0.124676 0.215945i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −3.00000 5.19615i −0.124247 0.215203i
\(584\) 0 0
\(585\) −8.00000 13.8564i −0.330759 0.572892i
\(586\) 0 0
\(587\) 2.00000 3.46410i 0.0825488 0.142979i −0.821795 0.569783i \(-0.807027\pi\)
0.904344 + 0.426804i \(0.140361\pi\)
\(588\) 0 0
\(589\) −5.00000 43.3013i −0.206021 1.78420i
\(590\) 0 0
\(591\) −9.00000 + 15.5885i −0.370211 + 0.641223i
\(592\) 0 0
\(593\) 10.5000 + 18.1865i 0.431183 + 0.746831i 0.996976 0.0777165i \(-0.0247629\pi\)
−0.565792 + 0.824548i \(0.691430\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −18.0000 −0.736691
\(598\) 0 0
\(599\) 19.0000 + 32.9090i 0.776319 + 1.34462i 0.934050 + 0.357142i \(0.116249\pi\)
−0.157731 + 0.987482i \(0.550418\pi\)
\(600\) 0 0
\(601\) −13.0000 −0.530281 −0.265141 0.964210i \(-0.585418\pi\)
−0.265141 + 0.964210i \(0.585418\pi\)
\(602\) 0 0
\(603\) −9.00000 + 15.5885i −0.366508 + 0.634811i
\(604\) 0 0
\(605\) −4.00000 + 6.92820i −0.162623 + 0.281672i
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −12.0000 + 20.7846i −0.485468 + 0.840855i
\(612\) 0 0
\(613\) 11.0000 19.0526i 0.444286 0.769526i −0.553716 0.832705i \(-0.686791\pi\)
0.998002 + 0.0631797i \(0.0201241\pi\)
\(614\) 0 0
\(615\) 36.0000 1.45166
\(616\) 0 0
\(617\) −1.50000 2.59808i −0.0603877 0.104595i 0.834251 0.551385i \(-0.185900\pi\)
−0.894639 + 0.446790i \(0.852567\pi\)
\(618\) 0 0
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 0 0
\(621\) 15.0000 + 25.9808i 0.601929 + 1.04257i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −20.5000 + 35.5070i −0.820000 + 1.42028i
\(626\) 0 0
\(627\) 12.0000 + 5.19615i 0.479234 + 0.207514i
\(628\) 0 0
\(629\) −2.00000 + 3.46410i −0.0797452 + 0.138123i
\(630\) 0 0
\(631\) −4.00000 6.92820i −0.159237 0.275807i 0.775356 0.631524i \(-0.217570\pi\)
−0.934594 + 0.355716i \(0.884237\pi\)
\(632\) 0 0
\(633\) 6.00000 + 10.3923i 0.238479 + 0.413057i
\(634\) 0 0
\(635\) −24.0000 −0.952411
\(636\) 0 0
\(637\) 7.00000 + 12.1244i 0.277350 + 0.480384i
\(638\) 0 0
\(639\) −12.0000 −0.474713
\(640\) 0 0
\(641\) −16.5000 + 28.5788i −0.651711 + 1.12880i 0.330997 + 0.943632i \(0.392615\pi\)
−0.982708 + 0.185164i \(0.940718\pi\)
\(642\) 0 0
\(643\) 2.50000 4.33013i 0.0985904 0.170764i −0.812511 0.582946i \(-0.801900\pi\)
0.911101 + 0.412182i \(0.135233\pi\)
\(644\) 0 0
\(645\) 16.0000 0.629999
\(646\) 0 0
\(647\) 14.0000 0.550397 0.275198 0.961387i \(-0.411256\pi\)
0.275198 + 0.961387i \(0.411256\pi\)
\(648\) 0 0
\(649\) 1.50000 2.59808i 0.0588802 0.101983i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 36.0000 1.40879 0.704394 0.709809i \(-0.251219\pi\)
0.704394 + 0.709809i \(0.251219\pi\)
\(654\) 0 0
\(655\) 30.0000 + 51.9615i 1.17220 + 2.03030i
\(656\) 0 0
\(657\) 18.0000 0.702247
\(658\) 0 0
\(659\) 6.00000 + 10.3923i 0.233727 + 0.404827i 0.958902 0.283738i \(-0.0915745\pi\)
−0.725175 + 0.688565i \(0.758241\pi\)
\(660\) 0 0
\(661\) −10.0000 17.3205i −0.388955 0.673690i 0.603354 0.797473i \(-0.293830\pi\)
−0.992309 + 0.123784i \(0.960497\pi\)
\(662\) 0 0
\(663\) 2.00000 3.46410i 0.0776736 0.134535i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −12.0000 + 20.7846i −0.464642 + 0.804783i
\(668\) 0 0
\(669\) −5.00000 8.66025i −0.193311 0.334825i
\(670\) 0 0
\(671\) −12.0000 20.7846i −0.463255 0.802381i
\(672\) 0 0
\(673\) 30.0000 1.15642 0.578208 0.815890i \(-0.303752\pi\)
0.578208 + 0.815890i \(0.303752\pi\)
\(674\) 0 0
\(675\) −27.5000 47.6314i −1.05848 1.83333i
\(676\) 0 0
\(677\) −2.00000 −0.0768662 −0.0384331 0.999261i \(-0.512237\pi\)
−0.0384331 + 0.999261i \(0.512237\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −9.50000 + 16.4545i −0.364041 + 0.630537i
\(682\) 0 0
\(683\) 28.0000 1.07139 0.535695 0.844411i \(-0.320050\pi\)
0.535695 + 0.844411i \(0.320050\pi\)
\(684\) 0 0
\(685\) −36.0000 −1.37549
\(686\) 0 0
\(687\) −4.00000 + 6.92820i −0.152610 + 0.264327i
\(688\) 0 0
\(689\) 2.00000 3.46410i 0.0761939 0.131972i
\(690\) 0 0
\(691\) −36.0000 −1.36950 −0.684752 0.728776i \(-0.740090\pi\)
−0.684752 + 0.728776i \(0.740090\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −52.0000 −1.97247
\(696\) 0 0
\(697\) −9.00000 15.5885i −0.340899 0.590455i
\(698\) 0 0
\(699\) 5.50000 + 9.52628i 0.208029 + 0.360317i
\(700\) 0 0
\(701\) 6.00000 10.3923i 0.226617 0.392512i −0.730186 0.683248i \(-0.760567\pi\)
0.956803 + 0.290736i \(0.0939001\pi\)
\(702\) 0 0
\(703\) −1.00000 8.66025i −0.0377157 0.326628i
\(704\) 0 0
\(705\) −24.0000 + 41.5692i −0.903892 + 1.56559i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −9.00000 15.5885i −0.338002 0.585437i 0.646055 0.763291i \(-0.276418\pi\)
−0.984057 + 0.177854i \(0.943084\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 0 0
\(713\) 30.0000 + 51.9615i 1.12351 + 1.94597i
\(714\) 0 0
\(715\) 24.0000 0.897549
\(716\) 0 0
\(717\) −6.00000 + 10.3923i −0.224074 + 0.388108i
\(718\) 0 0
\(719\) −17.0000 + 29.4449i −0.633993 + 1.09811i 0.352735 + 0.935723i \(0.385252\pi\)
−0.986728 + 0.162385i \(0.948081\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −21.0000 −0.780998
\(724\) 0 0
\(725\) 22.0000 38.1051i 0.817059 1.41519i
\(726\) 0 0
\(727\) −4.00000 + 6.92820i −0.148352 + 0.256953i −0.930618 0.365991i \(-0.880730\pi\)
0.782267 + 0.622944i \(0.214063\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −4.00000 6.92820i −0.147945 0.256249i
\(732\) 0 0
\(733\) 22.0000 0.812589 0.406294 0.913742i \(-0.366821\pi\)
0.406294 + 0.913742i \(0.366821\pi\)
\(734\) 0 0
\(735\) 14.0000 + 24.2487i 0.516398 + 0.894427i
\(736\) 0 0
\(737\) −13.5000 23.3827i −0.497279 0.861312i
\(738\) 0 0
\(739\) −2.50000 + 4.33013i −0.0919640 + 0.159286i −0.908337 0.418238i \(-0.862648\pi\)
0.816373 + 0.577524i \(0.195981\pi\)
\(740\) 0 0
\(741\) 1.00000 + 8.66025i 0.0367359 + 0.318142i
\(742\) 0 0
\(743\) −3.00000 + 5.19615i −0.110059 + 0.190628i −0.915794 0.401648i \(-0.868437\pi\)
0.805735 + 0.592277i \(0.201771\pi\)
\(744\) 0 0
\(745\) −20.0000 34.6410i −0.732743 1.26915i
\(746\) 0 0
\(747\) 5.00000 + 8.66025i 0.182940 + 0.316862i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 19.0000 + 32.9090i 0.693320 + 1.20087i 0.970744 + 0.240118i \(0.0771860\pi\)
−0.277424 + 0.960748i \(0.589481\pi\)
\(752\) 0 0
\(753\) 5.00000 0.182210
\(754\) 0 0
\(755\) −4.00000 + 6.92820i −0.145575 + 0.252143i
\(756\) 0 0
\(757\) −7.00000 + 12.1244i −0.254419 + 0.440667i −0.964738 0.263213i \(-0.915218\pi\)
0.710318 + 0.703881i \(0.248551\pi\)
\(758\) 0 0
\(759\) −18.0000 −0.653359
\(760\) 0 0
\(761\) 17.0000 0.616250 0.308125 0.951346i \(-0.400299\pi\)
0.308125 + 0.951346i \(0.400299\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −8.00000 + 13.8564i −0.289241 + 0.500979i
\(766\) 0 0
\(767\) 2.00000 0.0722158
\(768\) 0 0
\(769\) −1.00000 1.73205i −0.0360609 0.0624593i 0.847432 0.530904i \(-0.178148\pi\)
−0.883493 + 0.468445i \(0.844814\pi\)
\(770\) 0 0
\(771\) −3.00000 −0.108042
\(772\) 0 0
\(773\) −26.0000 45.0333i −0.935155 1.61974i −0.774357 0.632749i \(-0.781927\pi\)
−0.160798 0.986987i \(-0.551407\pi\)
\(774\) 0 0
\(775\) −55.0000 95.2628i −1.97566 3.42194i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 36.0000 + 15.5885i 1.28983 + 0.558514i
\(780\) 0 0
\(781\) 9.00000 15.5885i 0.322045 0.557799i
\(782\) 0 0
\(783\) 10.0000 + 17.3205i 0.357371 + 0.618984i
\(784\) 0 0
\(785\) 16.0000 + 27.7128i 0.571064 + 0.989113i
\(786\) 0 0
\(787\) −41.0000 −1.46149 −0.730746 0.682649i \(-0.760828\pi\)
−0.730746 + 0.682649i \(0.760828\pi\)
\(788\) 0 0
\(789\) −8.00000 13.8564i −0.284808 0.493301i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 8.00000 13.8564i 0.284088 0.492055i
\(794\) 0 0
\(795\) 4.00000 6.92820i 0.141865 0.245718i
\(796\) 0 0
\(797\) 42.0000 1.48772 0.743858 0.668338i \(-0.232994\pi\)
0.743858 + 0.668338i \(0.232994\pi\)
\(798\) 0 0
\(799\) 24.0000 0.849059
\(800\) 0 0
\(801\) −18.0000 + 31.1769i −0.635999 + 1.10158i
\(802\) 0 0
\(803\) −13.5000 + 23.3827i −0.476405 + 0.825157i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 2.00000 + 3.46410i 0.0704033 + 0.121942i
\(808\) 0 0
\(809\) 21.0000 0.738321 0.369160 0.929366i \(-0.379645\pi\)
0.369160 + 0.929366i \(0.379645\pi\)
\(810\) 0 0
\(811\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(812\) 0 0
\(813\) 10.0000 + 17.3205i 0.350715 + 0.607457i
\(814\) 0 0
\(815\) 22.0000 38.1051i 0.770626 1.33476i
\(816\) 0 0
\(817\) 16.0000 + 6.92820i 0.559769 + 0.242387i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 19.0000 + 32.9090i 0.663105 + 1.14853i 0.979795 + 0.200002i \(0.0640949\pi\)
−0.316691 + 0.948529i \(0.602572\pi\)
\(822\) 0 0
\(823\) −13.0000 22.5167i −0.453152 0.784881i 0.545428 0.838157i \(-0.316367\pi\)
−0.998580 + 0.0532760i \(0.983034\pi\)
\(824\) 0 0
\(825\) 33.0000 1.14891
\(826\) 0 0
\(827\) −27.5000 47.6314i −0.956269 1.65631i −0.731438 0.681908i \(-0.761150\pi\)
−0.224831 0.974398i \(-0.572183\pi\)
\(828\) 0 0
\(829\) −12.0000 −0.416777 −0.208389 0.978046i \(-0.566822\pi\)
−0.208389 + 0.978046i \(0.566822\pi\)
\(830\) 0 0
\(831\) 6.00000 10.3923i 0.208138 0.360505i
\(832\) 0 0
\(833\) 7.00000 12.1244i 0.242536 0.420084i
\(834\) 0 0
\(835\) 64.0000 2.21481
\(836\) 0 0
\(837\) 50.0000 1.72825
\(838\) 0 0
\(839\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(840\) 0 0
\(841\) 6.50000 11.2583i 0.224138 0.388218i
\(842\) 0 0
\(843\) 13.0000 0.447744
\(844\) 0 0
\(845\) −18.0000 31.1769i −0.619219 1.07252i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −6.50000 11.2583i −0.223079 0.386385i
\(850\) 0 0
\(851\) 6.00000 + 10.3923i 0.205677 + 0.356244i
\(852\) 0 0
\(853\) −21.0000 + 36.3731i −0.719026 + 1.24539i 0.242360 + 0.970186i \(0.422079\pi\)
−0.961386 + 0.275204i \(0.911255\pi\)
\(854\) 0 0
\(855\) −4.00000 34.6410i −0.136797 1.18470i
\(856\) 0 0
\(857\) 17.5000 30.3109i 0.597789 1.03540i −0.395358 0.918527i \(-0.629380\pi\)
0.993147 0.116873i \(-0.0372871\pi\)
\(858\) 0 0
\(859\) 14.5000 + 25.1147i 0.494734 + 0.856904i 0.999982 0.00607046i \(-0.00193230\pi\)
−0.505248 + 0.862974i \(0.668599\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −6.00000 −0.204242 −0.102121 0.994772i \(-0.532563\pi\)
−0.102121 + 0.994772i \(0.532563\pi\)
\(864\) 0 0
\(865\) −52.0000 90.0666i −1.76805 3.06236i
\(866\) 0 0
\(867\) 13.0000 0.441503
\(868\) 0 0
\(869\) 6.00000 10.3923i 0.203536 0.352535i
\(870\) 0 0
\(871\) 9.00000 15.5885i 0.304953 0.528195i
\(872\) 0 0
\(873\) −2.00000 −0.0676897
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −20.0000 + 34.6410i −0.675352 + 1.16974i 0.301014 + 0.953620i \(0.402675\pi\)
−0.976366 + 0.216124i \(0.930658\pi\)
\(878\) 0 0
\(879\) −2.00000 + 3.46410i −0.0674583 + 0.116841i
\(880\) 0 0
\(881\) 7.00000 0.235836 0.117918 0.993023i \(-0.462378\pi\)
0.117918 + 0.993023i \(0.462378\pi\)
\(882\) 0 0
\(883\) −21.5000 37.2391i −0.723533 1.25320i −0.959575 0.281453i \(-0.909184\pi\)
0.236043 0.971743i \(-0.424150\pi\)
\(884\) 0 0
\(885\) 4.00000 0.134459
\(886\) 0 0
\(887\) 16.0000 + 27.7128i 0.537227 + 0.930505i 0.999052 + 0.0435339i \(0.0138616\pi\)
−0.461825 + 0.886971i \(0.652805\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 1.50000 2.59808i 0.0502519 0.0870388i
\(892\) 0 0
\(893\) −42.0000 + 31.1769i −1.40548 + 1.04330i
\(894\) 0 0
\(895\) −18.0000 + 31.1769i −0.601674 + 1.04213i
\(896\) 0 0
\(897\) −6.00000 10.3923i −0.200334 0.346989i
\(898\) 0 0
\(899\) 20.0000 + 34.6410i 0.667037 + 1.15534i
\(900\) 0 0
\(901\) −4.00000 −0.133259
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −56.0000 −1.86150
\(906\) 0 0
\(907\) 24.5000 42.4352i 0.813509 1.40904i −0.0968843 0.995296i \(-0.530888\pi\)
0.910393 0.413744i \(-0.135779\pi\)
\(908\) 0 0
\(909\) 12.0000 20.7846i 0.398015 0.689382i
\(910\) 0 0
\(911\) 14.0000 0.463841 0.231920 0.972735i \(-0.425499\pi\)
0.231920 + 0.972735i \(0.425499\pi\)
\(912\) 0 0
\(913\) −15.0000 −0.496428
\(914\) 0 0
\(915\) 16.0000 27.7128i 0.528944 0.916157i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 26.0000 0.857661 0.428830 0.903385i \(-0.358926\pi\)
0.428830 + 0.903385i \(0.358926\pi\)
\(920\) 0 0
\(921\) −12.5000 21.6506i −0.411889 0.713413i
\(922\) 0 0
\(923\) 12.0000 0.394985
\(924\) 0 0
\(925\) −11.0000 19.0526i −0.361678 0.626444i
\(926\) 0 0
\(927\) 14.0000 + 24.2487i 0.459820 + 0.796432i
\(928\) 0 0
\(929\) −10.5000 + 18.1865i −0.344494 + 0.596681i −0.985262 0.171054i \(-0.945283\pi\)
0.640768 + 0.767735i \(0.278616\pi\)
\(930\) 0 0
\(931\) 3.50000 + 30.3109i 0.114708 + 0.993399i
\(932\) 0 0
\(933\) 1.00000 1.73205i 0.0327385 0.0567048i
\(934\) 0 0
\(935\) −12.0000 20.7846i −0.392442 0.679729i
\(936\) 0 0
\(937\) −5.50000 9.52628i −0.179677 0.311210i 0.762093 0.647468i \(-0.224172\pi\)
−0.941770 + 0.336258i \(0.890839\pi\)
\(938\) 0 0
\(939\) 19.0000 0.620042
\(940\) 0 0
\(941\) 5.00000 + 8.66025i 0.162995 + 0.282316i 0.935942 0.352155i \(-0.114551\pi\)
−0.772946 + 0.634472i \(0.781218\pi\)
\(942\) 0 0
\(943\) −54.0000 −1.75848
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −14.0000 + 24.2487i −0.454939 + 0.787977i −0.998685 0.0512727i \(-0.983672\pi\)
0.543746 + 0.839250i \(0.317006\pi\)
\(948\) 0 0
\(949\) −18.0000 −0.584305
\(950\) 0 0
\(951\) −6.00000 −0.194563
\(952\) 0 0
\(953\) −0.500000 + 0.866025i −0.0161966 + 0.0280533i −0.874010 0.485908i \(-0.838489\pi\)
0.857814 + 0.513961i \(0.171822\pi\)
\(954\) 0 0
\(955\) −24.0000 + 41.5692i −0.776622 + 1.34515i
\(956\) 0 0
\(957\) −12.0000 −0.387905
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) 0 0
\(963\) −16.0000 27.7128i −0.515593 0.893033i
\(964\) 0 0
\(965\) −12.0000 20.7846i −0.386294 0.669080i
\(966\) 0 0
\(967\) 27.0000 46.7654i 0.868261 1.50387i 0.00448958 0.999990i \(-0.498571\pi\)
0.863772 0.503883i \(-0.168096\pi\)
\(968\) 0 0
\(969\) 7.00000 5.19615i 0.224872 0.166924i
\(970\) 0 0
\(971\) 26.5000 45.8993i 0.850425 1.47298i −0.0303994 0.999538i \(-0.509678\pi\)
0.880825 0.473442i \(-0.156989\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 11.0000 + 19.0526i 0.352282 + 0.610170i
\(976\) 0 0
\(977\) −47.0000 −1.50366 −0.751832 0.659355i \(-0.770829\pi\)
−0.751832 + 0.659355i \(0.770829\pi\)
\(978\) 0 0
\(979\) −27.0000 46.7654i −0.862924 1.49463i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 16.0000 27.7128i 0.510321 0.883901i −0.489608 0.871943i \(-0.662860\pi\)
0.999928 0.0119587i \(-0.00380665\pi\)
\(984\) 0 0
\(985\) −36.0000 + 62.3538i −1.14706 + 1.98676i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −24.0000 −0.763156
\(990\) 0 0
\(991\) 4.00000 6.92820i 0.127064 0.220082i −0.795474 0.605988i \(-0.792778\pi\)
0.922538 + 0.385906i \(0.126111\pi\)
\(992\) 0 0
\(993\) −6.50000 + 11.2583i −0.206271 + 0.357272i
\(994\) 0 0
\(995\) −72.0000 −2.28255
\(996\) 0 0
\(997\) 16.0000 + 27.7128i 0.506725 + 0.877674i 0.999970 + 0.00778294i \(0.00247741\pi\)
−0.493245 + 0.869891i \(0.664189\pi\)
\(998\) 0 0
\(999\) 10.0000 0.316386
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 304.2.i.d.49.1 2
3.2 odd 2 2736.2.s.a.1873.1 2
4.3 odd 2 152.2.i.b.49.1 2
8.3 odd 2 1216.2.i.f.961.1 2
8.5 even 2 1216.2.i.b.961.1 2
12.11 even 2 1368.2.s.a.505.1 2
19.7 even 3 inner 304.2.i.d.273.1 2
19.8 odd 6 5776.2.a.j.1.1 1
19.11 even 3 5776.2.a.e.1.1 1
57.26 odd 6 2736.2.s.a.577.1 2
76.7 odd 6 152.2.i.b.121.1 yes 2
76.11 odd 6 2888.2.a.d.1.1 1
76.27 even 6 2888.2.a.a.1.1 1
152.45 even 6 1216.2.i.b.577.1 2
152.83 odd 6 1216.2.i.f.577.1 2
228.83 even 6 1368.2.s.a.577.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.2.i.b.49.1 2 4.3 odd 2
152.2.i.b.121.1 yes 2 76.7 odd 6
304.2.i.d.49.1 2 1.1 even 1 trivial
304.2.i.d.273.1 2 19.7 even 3 inner
1216.2.i.b.577.1 2 152.45 even 6
1216.2.i.b.961.1 2 8.5 even 2
1216.2.i.f.577.1 2 152.83 odd 6
1216.2.i.f.961.1 2 8.3 odd 2
1368.2.s.a.505.1 2 12.11 even 2
1368.2.s.a.577.1 2 228.83 even 6
2736.2.s.a.577.1 2 57.26 odd 6
2736.2.s.a.1873.1 2 3.2 odd 2
2888.2.a.a.1.1 1 76.27 even 6
2888.2.a.d.1.1 1 76.11 odd 6
5776.2.a.e.1.1 1 19.11 even 3
5776.2.a.j.1.1 1 19.8 odd 6