Properties

Label 187.2.a.c.1.1
Level $187$
Weight $2$
Character 187.1
Self dual yes
Analytic conductor $1.493$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [187,2,Mod(1,187)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(187, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("187.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 187 = 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 187.a (trivial)

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: yes
Analytic conductor: \(1.49320251780\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 187.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.73205 q^{2} +1.73205 q^{3} +5.46410 q^{4} -3.73205 q^{5} -4.73205 q^{6} -2.00000 q^{7} -9.46410 q^{8} +O(q^{10})\) \(q-2.73205 q^{2} +1.73205 q^{3} +5.46410 q^{4} -3.73205 q^{5} -4.73205 q^{6} -2.00000 q^{7} -9.46410 q^{8} +10.1962 q^{10} +1.00000 q^{11} +9.46410 q^{12} -3.26795 q^{13} +5.46410 q^{14} -6.46410 q^{15} +14.9282 q^{16} +1.00000 q^{17} -6.19615 q^{19} -20.3923 q^{20} -3.46410 q^{21} -2.73205 q^{22} -3.73205 q^{23} -16.3923 q^{24} +8.92820 q^{25} +8.92820 q^{26} -5.19615 q^{27} -10.9282 q^{28} -1.26795 q^{29} +17.6603 q^{30} +2.26795 q^{31} -21.8564 q^{32} +1.73205 q^{33} -2.73205 q^{34} +7.46410 q^{35} -3.73205 q^{37} +16.9282 q^{38} -5.66025 q^{39} +35.3205 q^{40} +9.46410 q^{41} +9.46410 q^{42} -2.00000 q^{43} +5.46410 q^{44} +10.1962 q^{46} +0.928203 q^{47} +25.8564 q^{48} -3.00000 q^{49} -24.3923 q^{50} +1.73205 q^{51} -17.8564 q^{52} -2.53590 q^{53} +14.1962 q^{54} -3.73205 q^{55} +18.9282 q^{56} -10.7321 q^{57} +3.46410 q^{58} +3.00000 q^{59} -35.3205 q^{60} +2.92820 q^{61} -6.19615 q^{62} +29.8564 q^{64} +12.1962 q^{65} -4.73205 q^{66} +1.00000 q^{67} +5.46410 q^{68} -6.46410 q^{69} -20.3923 q^{70} +0.267949 q^{71} +3.80385 q^{73} +10.1962 q^{74} +15.4641 q^{75} -33.8564 q^{76} -2.00000 q^{77} +15.4641 q^{78} +2.19615 q^{79} -55.7128 q^{80} -9.00000 q^{81} -25.8564 q^{82} +15.8564 q^{83} -18.9282 q^{84} -3.73205 q^{85} +5.46410 q^{86} -2.19615 q^{87} -9.46410 q^{88} -11.9282 q^{89} +6.53590 q^{91} -20.3923 q^{92} +3.92820 q^{93} -2.53590 q^{94} +23.1244 q^{95} -37.8564 q^{96} +15.7321 q^{97} +8.19615 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 4 q^{4} - 4 q^{5} - 6 q^{6} - 4 q^{7} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 4 q^{4} - 4 q^{5} - 6 q^{6} - 4 q^{7} - 12 q^{8} + 10 q^{10} + 2 q^{11} + 12 q^{12} - 10 q^{13} + 4 q^{14} - 6 q^{15} + 16 q^{16} + 2 q^{17} - 2 q^{19} - 20 q^{20} - 2 q^{22} - 4 q^{23} - 12 q^{24} + 4 q^{25} + 4 q^{26} - 8 q^{28} - 6 q^{29} + 18 q^{30} + 8 q^{31} - 16 q^{32} - 2 q^{34} + 8 q^{35} - 4 q^{37} + 20 q^{38} + 6 q^{39} + 36 q^{40} + 12 q^{41} + 12 q^{42} - 4 q^{43} + 4 q^{44} + 10 q^{46} - 12 q^{47} + 24 q^{48} - 6 q^{49} - 28 q^{50} - 8 q^{52} - 12 q^{53} + 18 q^{54} - 4 q^{55} + 24 q^{56} - 18 q^{57} + 6 q^{59} - 36 q^{60} - 8 q^{61} - 2 q^{62} + 32 q^{64} + 14 q^{65} - 6 q^{66} + 2 q^{67} + 4 q^{68} - 6 q^{69} - 20 q^{70} + 4 q^{71} + 18 q^{73} + 10 q^{74} + 24 q^{75} - 40 q^{76} - 4 q^{77} + 24 q^{78} - 6 q^{79} - 56 q^{80} - 18 q^{81} - 24 q^{82} + 4 q^{83} - 24 q^{84} - 4 q^{85} + 4 q^{86} + 6 q^{87} - 12 q^{88} - 10 q^{89} + 20 q^{91} - 20 q^{92} - 6 q^{93} - 12 q^{94} + 22 q^{95} - 48 q^{96} + 28 q^{97} + 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.73205 −1.93185 −0.965926 0.258819i \(-0.916667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(3\) 1.73205 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(4\) 5.46410 2.73205
\(5\) −3.73205 −1.66902 −0.834512 0.550990i \(-0.814250\pi\)
−0.834512 + 0.550990i \(0.814250\pi\)
\(6\) −4.73205 −1.93185
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) −9.46410 −3.34607
\(9\) 0 0
\(10\) 10.1962 3.22431
\(11\) 1.00000 0.301511
\(12\) 9.46410 2.73205
\(13\) −3.26795 −0.906366 −0.453183 0.891417i \(-0.649712\pi\)
−0.453183 + 0.891417i \(0.649712\pi\)
\(14\) 5.46410 1.46034
\(15\) −6.46410 −1.66902
\(16\) 14.9282 3.73205
\(17\) 1.00000 0.242536
\(18\) 0 0
\(19\) −6.19615 −1.42149 −0.710747 0.703447i \(-0.751643\pi\)
−0.710747 + 0.703447i \(0.751643\pi\)
\(20\) −20.3923 −4.55986
\(21\) −3.46410 −0.755929
\(22\) −2.73205 −0.582475
\(23\) −3.73205 −0.778186 −0.389093 0.921198i \(-0.627212\pi\)
−0.389093 + 0.921198i \(0.627212\pi\)
\(24\) −16.3923 −3.34607
\(25\) 8.92820 1.78564
\(26\) 8.92820 1.75096
\(27\) −5.19615 −1.00000
\(28\) −10.9282 −2.06524
\(29\) −1.26795 −0.235452 −0.117726 0.993046i \(-0.537560\pi\)
−0.117726 + 0.993046i \(0.537560\pi\)
\(30\) 17.6603 3.22431
\(31\) 2.26795 0.407336 0.203668 0.979040i \(-0.434714\pi\)
0.203668 + 0.979040i \(0.434714\pi\)
\(32\) −21.8564 −3.86370
\(33\) 1.73205 0.301511
\(34\) −2.73205 −0.468543
\(35\) 7.46410 1.26166
\(36\) 0 0
\(37\) −3.73205 −0.613545 −0.306773 0.951783i \(-0.599249\pi\)
−0.306773 + 0.951783i \(0.599249\pi\)
\(38\) 16.9282 2.74612
\(39\) −5.66025 −0.906366
\(40\) 35.3205 5.58466
\(41\) 9.46410 1.47804 0.739022 0.673681i \(-0.235288\pi\)
0.739022 + 0.673681i \(0.235288\pi\)
\(42\) 9.46410 1.46034
\(43\) −2.00000 −0.304997 −0.152499 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 5.46410 0.823744
\(45\) 0 0
\(46\) 10.1962 1.50334
\(47\) 0.928203 0.135392 0.0676962 0.997706i \(-0.478435\pi\)
0.0676962 + 0.997706i \(0.478435\pi\)
\(48\) 25.8564 3.73205
\(49\) −3.00000 −0.428571
\(50\) −24.3923 −3.44959
\(51\) 1.73205 0.242536
\(52\) −17.8564 −2.47624
\(53\) −2.53590 −0.348332 −0.174166 0.984716i \(-0.555723\pi\)
−0.174166 + 0.984716i \(0.555723\pi\)
\(54\) 14.1962 1.93185
\(55\) −3.73205 −0.503230
\(56\) 18.9282 2.52939
\(57\) −10.7321 −1.42149
\(58\) 3.46410 0.454859
\(59\) 3.00000 0.390567 0.195283 0.980747i \(-0.437437\pi\)
0.195283 + 0.980747i \(0.437437\pi\)
\(60\) −35.3205 −4.55986
\(61\) 2.92820 0.374918 0.187459 0.982272i \(-0.439975\pi\)
0.187459 + 0.982272i \(0.439975\pi\)
\(62\) −6.19615 −0.786912
\(63\) 0 0
\(64\) 29.8564 3.73205
\(65\) 12.1962 1.51275
\(66\) −4.73205 −0.582475
\(67\) 1.00000 0.122169 0.0610847 0.998133i \(-0.480544\pi\)
0.0610847 + 0.998133i \(0.480544\pi\)
\(68\) 5.46410 0.662620
\(69\) −6.46410 −0.778186
\(70\) −20.3923 −2.43735
\(71\) 0.267949 0.0317997 0.0158999 0.999874i \(-0.494939\pi\)
0.0158999 + 0.999874i \(0.494939\pi\)
\(72\) 0 0
\(73\) 3.80385 0.445207 0.222603 0.974909i \(-0.428545\pi\)
0.222603 + 0.974909i \(0.428545\pi\)
\(74\) 10.1962 1.18528
\(75\) 15.4641 1.78564
\(76\) −33.8564 −3.88360
\(77\) −2.00000 −0.227921
\(78\) 15.4641 1.75096
\(79\) 2.19615 0.247086 0.123543 0.992339i \(-0.460574\pi\)
0.123543 + 0.992339i \(0.460574\pi\)
\(80\) −55.7128 −6.22888
\(81\) −9.00000 −1.00000
\(82\) −25.8564 −2.85536
\(83\) 15.8564 1.74047 0.870233 0.492640i \(-0.163968\pi\)
0.870233 + 0.492640i \(0.163968\pi\)
\(84\) −18.9282 −2.06524
\(85\) −3.73205 −0.404798
\(86\) 5.46410 0.589209
\(87\) −2.19615 −0.235452
\(88\) −9.46410 −1.00888
\(89\) −11.9282 −1.26439 −0.632194 0.774811i \(-0.717845\pi\)
−0.632194 + 0.774811i \(0.717845\pi\)
\(90\) 0 0
\(91\) 6.53590 0.685148
\(92\) −20.3923 −2.12604
\(93\) 3.92820 0.407336
\(94\) −2.53590 −0.261558
\(95\) 23.1244 2.37251
\(96\) −37.8564 −3.86370
\(97\) 15.7321 1.59735 0.798674 0.601764i \(-0.205535\pi\)
0.798674 + 0.601764i \(0.205535\pi\)
\(98\) 8.19615 0.827936
\(99\) 0 0
\(100\) 48.7846 4.87846
\(101\) 1.66025 0.165201 0.0826007 0.996583i \(-0.473677\pi\)
0.0826007 + 0.996583i \(0.473677\pi\)
\(102\) −4.73205 −0.468543
\(103\) −11.8564 −1.16825 −0.584123 0.811665i \(-0.698562\pi\)
−0.584123 + 0.811665i \(0.698562\pi\)
\(104\) 30.9282 3.03276
\(105\) 12.9282 1.26166
\(106\) 6.92820 0.672927
\(107\) −11.2679 −1.08931 −0.544657 0.838659i \(-0.683340\pi\)
−0.544657 + 0.838659i \(0.683340\pi\)
\(108\) −28.3923 −2.73205
\(109\) −2.19615 −0.210353 −0.105177 0.994454i \(-0.533541\pi\)
−0.105177 + 0.994454i \(0.533541\pi\)
\(110\) 10.1962 0.972165
\(111\) −6.46410 −0.613545
\(112\) −29.8564 −2.82117
\(113\) −6.26795 −0.589639 −0.294820 0.955553i \(-0.595260\pi\)
−0.294820 + 0.955553i \(0.595260\pi\)
\(114\) 29.3205 2.74612
\(115\) 13.9282 1.29881
\(116\) −6.92820 −0.643268
\(117\) 0 0
\(118\) −8.19615 −0.754517
\(119\) −2.00000 −0.183340
\(120\) 61.1769 5.58466
\(121\) 1.00000 0.0909091
\(122\) −8.00000 −0.724286
\(123\) 16.3923 1.47804
\(124\) 12.3923 1.11286
\(125\) −14.6603 −1.31125
\(126\) 0 0
\(127\) −4.00000 −0.354943 −0.177471 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) −37.8564 −3.34607
\(129\) −3.46410 −0.304997
\(130\) −33.3205 −2.92240
\(131\) −8.92820 −0.780061 −0.390030 0.920802i \(-0.627535\pi\)
−0.390030 + 0.920802i \(0.627535\pi\)
\(132\) 9.46410 0.823744
\(133\) 12.3923 1.07455
\(134\) −2.73205 −0.236013
\(135\) 19.3923 1.66902
\(136\) −9.46410 −0.811540
\(137\) −19.3923 −1.65680 −0.828398 0.560140i \(-0.810747\pi\)
−0.828398 + 0.560140i \(0.810747\pi\)
\(138\) 17.6603 1.50334
\(139\) 15.6603 1.32829 0.664143 0.747606i \(-0.268797\pi\)
0.664143 + 0.747606i \(0.268797\pi\)
\(140\) 40.7846 3.44693
\(141\) 1.60770 0.135392
\(142\) −0.732051 −0.0614323
\(143\) −3.26795 −0.273280
\(144\) 0 0
\(145\) 4.73205 0.392975
\(146\) −10.3923 −0.860073
\(147\) −5.19615 −0.428571
\(148\) −20.3923 −1.67624
\(149\) −19.1244 −1.56673 −0.783364 0.621563i \(-0.786498\pi\)
−0.783364 + 0.621563i \(0.786498\pi\)
\(150\) −42.2487 −3.44959
\(151\) −12.1962 −0.992509 −0.496254 0.868177i \(-0.665292\pi\)
−0.496254 + 0.868177i \(0.665292\pi\)
\(152\) 58.6410 4.75641
\(153\) 0 0
\(154\) 5.46410 0.440310
\(155\) −8.46410 −0.679853
\(156\) −30.9282 −2.47624
\(157\) −19.7846 −1.57898 −0.789492 0.613761i \(-0.789656\pi\)
−0.789492 + 0.613761i \(0.789656\pi\)
\(158\) −6.00000 −0.477334
\(159\) −4.39230 −0.348332
\(160\) 81.5692 6.44861
\(161\) 7.46410 0.588254
\(162\) 24.5885 1.93185
\(163\) −2.53590 −0.198627 −0.0993134 0.995056i \(-0.531665\pi\)
−0.0993134 + 0.995056i \(0.531665\pi\)
\(164\) 51.7128 4.03809
\(165\) −6.46410 −0.503230
\(166\) −43.3205 −3.36232
\(167\) 18.7321 1.44953 0.724765 0.688996i \(-0.241948\pi\)
0.724765 + 0.688996i \(0.241948\pi\)
\(168\) 32.7846 2.52939
\(169\) −2.32051 −0.178501
\(170\) 10.1962 0.782009
\(171\) 0 0
\(172\) −10.9282 −0.833268
\(173\) −14.0000 −1.06440 −0.532200 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) 6.00000 0.454859
\(175\) −17.8564 −1.34982
\(176\) 14.9282 1.12526
\(177\) 5.19615 0.390567
\(178\) 32.5885 2.44261
\(179\) −3.00000 −0.224231 −0.112115 0.993695i \(-0.535763\pi\)
−0.112115 + 0.993695i \(0.535763\pi\)
\(180\) 0 0
\(181\) 12.6603 0.941029 0.470515 0.882392i \(-0.344068\pi\)
0.470515 + 0.882392i \(0.344068\pi\)
\(182\) −17.8564 −1.32360
\(183\) 5.07180 0.374918
\(184\) 35.3205 2.60386
\(185\) 13.9282 1.02402
\(186\) −10.7321 −0.786912
\(187\) 1.00000 0.0731272
\(188\) 5.07180 0.369899
\(189\) 10.3923 0.755929
\(190\) −63.1769 −4.58334
\(191\) 26.4641 1.91488 0.957438 0.288640i \(-0.0932032\pi\)
0.957438 + 0.288640i \(0.0932032\pi\)
\(192\) 51.7128 3.73205
\(193\) −24.1962 −1.74168 −0.870839 0.491569i \(-0.836424\pi\)
−0.870839 + 0.491569i \(0.836424\pi\)
\(194\) −42.9808 −3.08584
\(195\) 21.1244 1.51275
\(196\) −16.3923 −1.17088
\(197\) −21.1244 −1.50505 −0.752524 0.658565i \(-0.771164\pi\)
−0.752524 + 0.658565i \(0.771164\pi\)
\(198\) 0 0
\(199\) −16.7846 −1.18983 −0.594915 0.803789i \(-0.702814\pi\)
−0.594915 + 0.803789i \(0.702814\pi\)
\(200\) −84.4974 −5.97487
\(201\) 1.73205 0.122169
\(202\) −4.53590 −0.319145
\(203\) 2.53590 0.177985
\(204\) 9.46410 0.662620
\(205\) −35.3205 −2.46689
\(206\) 32.3923 2.25688
\(207\) 0 0
\(208\) −48.7846 −3.38260
\(209\) −6.19615 −0.428597
\(210\) −35.3205 −2.43735
\(211\) −1.46410 −0.100793 −0.0503965 0.998729i \(-0.516048\pi\)
−0.0503965 + 0.998729i \(0.516048\pi\)
\(212\) −13.8564 −0.951662
\(213\) 0.464102 0.0317997
\(214\) 30.7846 2.10439
\(215\) 7.46410 0.509048
\(216\) 49.1769 3.34607
\(217\) −4.53590 −0.307917
\(218\) 6.00000 0.406371
\(219\) 6.58846 0.445207
\(220\) −20.3923 −1.37485
\(221\) −3.26795 −0.219826
\(222\) 17.6603 1.18528
\(223\) 20.3205 1.36076 0.680381 0.732859i \(-0.261814\pi\)
0.680381 + 0.732859i \(0.261814\pi\)
\(224\) 43.7128 2.92069
\(225\) 0 0
\(226\) 17.1244 1.13910
\(227\) 16.7321 1.11055 0.555273 0.831668i \(-0.312614\pi\)
0.555273 + 0.831668i \(0.312614\pi\)
\(228\) −58.6410 −3.88360
\(229\) −12.4641 −0.823651 −0.411826 0.911263i \(-0.635109\pi\)
−0.411826 + 0.911263i \(0.635109\pi\)
\(230\) −38.0526 −2.50911
\(231\) −3.46410 −0.227921
\(232\) 12.0000 0.787839
\(233\) −5.07180 −0.332264 −0.166132 0.986103i \(-0.553128\pi\)
−0.166132 + 0.986103i \(0.553128\pi\)
\(234\) 0 0
\(235\) −3.46410 −0.225973
\(236\) 16.3923 1.06705
\(237\) 3.80385 0.247086
\(238\) 5.46410 0.354185
\(239\) 4.19615 0.271427 0.135713 0.990748i \(-0.456667\pi\)
0.135713 + 0.990748i \(0.456667\pi\)
\(240\) −96.4974 −6.22888
\(241\) 18.9282 1.21927 0.609636 0.792681i \(-0.291315\pi\)
0.609636 + 0.792681i \(0.291315\pi\)
\(242\) −2.73205 −0.175623
\(243\) 0 0
\(244\) 16.0000 1.02430
\(245\) 11.1962 0.715296
\(246\) −44.7846 −2.85536
\(247\) 20.2487 1.28839
\(248\) −21.4641 −1.36297
\(249\) 27.4641 1.74047
\(250\) 40.0526 2.53315
\(251\) −26.3205 −1.66134 −0.830668 0.556768i \(-0.812041\pi\)
−0.830668 + 0.556768i \(0.812041\pi\)
\(252\) 0 0
\(253\) −3.73205 −0.234632
\(254\) 10.9282 0.685696
\(255\) −6.46410 −0.404798
\(256\) 43.7128 2.73205
\(257\) 20.7846 1.29651 0.648254 0.761424i \(-0.275499\pi\)
0.648254 + 0.761424i \(0.275499\pi\)
\(258\) 9.46410 0.589209
\(259\) 7.46410 0.463797
\(260\) 66.6410 4.13290
\(261\) 0 0
\(262\) 24.3923 1.50696
\(263\) 11.1244 0.685957 0.342979 0.939343i \(-0.388564\pi\)
0.342979 + 0.939343i \(0.388564\pi\)
\(264\) −16.3923 −1.00888
\(265\) 9.46410 0.581375
\(266\) −33.8564 −2.07587
\(267\) −20.6603 −1.26439
\(268\) 5.46410 0.333773
\(269\) −14.0000 −0.853595 −0.426798 0.904347i \(-0.640358\pi\)
−0.426798 + 0.904347i \(0.640358\pi\)
\(270\) −52.9808 −3.22431
\(271\) −9.07180 −0.551072 −0.275536 0.961291i \(-0.588855\pi\)
−0.275536 + 0.961291i \(0.588855\pi\)
\(272\) 14.9282 0.905155
\(273\) 11.3205 0.685148
\(274\) 52.9808 3.20068
\(275\) 8.92820 0.538391
\(276\) −35.3205 −2.12604
\(277\) 18.3923 1.10509 0.552543 0.833484i \(-0.313657\pi\)
0.552543 + 0.833484i \(0.313657\pi\)
\(278\) −42.7846 −2.56605
\(279\) 0 0
\(280\) −70.6410 −4.22161
\(281\) −27.1244 −1.61810 −0.809052 0.587737i \(-0.800019\pi\)
−0.809052 + 0.587737i \(0.800019\pi\)
\(282\) −4.39230 −0.261558
\(283\) −2.73205 −0.162404 −0.0812018 0.996698i \(-0.525876\pi\)
−0.0812018 + 0.996698i \(0.525876\pi\)
\(284\) 1.46410 0.0868784
\(285\) 40.0526 2.37251
\(286\) 8.92820 0.527936
\(287\) −18.9282 −1.11730
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) −12.9282 −0.759170
\(291\) 27.2487 1.59735
\(292\) 20.7846 1.21633
\(293\) −13.8564 −0.809500 −0.404750 0.914427i \(-0.632641\pi\)
−0.404750 + 0.914427i \(0.632641\pi\)
\(294\) 14.1962 0.827936
\(295\) −11.1962 −0.651865
\(296\) 35.3205 2.05296
\(297\) −5.19615 −0.301511
\(298\) 52.2487 3.02669
\(299\) 12.1962 0.705322
\(300\) 84.4974 4.87846
\(301\) 4.00000 0.230556
\(302\) 33.3205 1.91738
\(303\) 2.87564 0.165201
\(304\) −92.4974 −5.30509
\(305\) −10.9282 −0.625747
\(306\) 0 0
\(307\) −9.80385 −0.559535 −0.279768 0.960068i \(-0.590257\pi\)
−0.279768 + 0.960068i \(0.590257\pi\)
\(308\) −10.9282 −0.622692
\(309\) −20.5359 −1.16825
\(310\) 23.1244 1.31338
\(311\) −0.392305 −0.0222456 −0.0111228 0.999938i \(-0.503541\pi\)
−0.0111228 + 0.999938i \(0.503541\pi\)
\(312\) 53.5692 3.03276
\(313\) 8.80385 0.497623 0.248811 0.968552i \(-0.419960\pi\)
0.248811 + 0.968552i \(0.419960\pi\)
\(314\) 54.0526 3.05036
\(315\) 0 0
\(316\) 12.0000 0.675053
\(317\) 25.4449 1.42913 0.714563 0.699571i \(-0.246626\pi\)
0.714563 + 0.699571i \(0.246626\pi\)
\(318\) 12.0000 0.672927
\(319\) −1.26795 −0.0709915
\(320\) −111.426 −6.22888
\(321\) −19.5167 −1.08931
\(322\) −20.3923 −1.13642
\(323\) −6.19615 −0.344763
\(324\) −49.1769 −2.73205
\(325\) −29.1769 −1.61844
\(326\) 6.92820 0.383718
\(327\) −3.80385 −0.210353
\(328\) −89.5692 −4.94563
\(329\) −1.85641 −0.102347
\(330\) 17.6603 0.972165
\(331\) −30.8564 −1.69602 −0.848011 0.529979i \(-0.822200\pi\)
−0.848011 + 0.529979i \(0.822200\pi\)
\(332\) 86.6410 4.75504
\(333\) 0 0
\(334\) −51.1769 −2.80028
\(335\) −3.73205 −0.203904
\(336\) −51.7128 −2.82117
\(337\) −9.60770 −0.523365 −0.261682 0.965154i \(-0.584277\pi\)
−0.261682 + 0.965154i \(0.584277\pi\)
\(338\) 6.33975 0.344837
\(339\) −10.8564 −0.589639
\(340\) −20.3923 −1.10593
\(341\) 2.26795 0.122816
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 18.9282 1.02054
\(345\) 24.1244 1.29881
\(346\) 38.2487 2.05626
\(347\) −18.5359 −0.995059 −0.497530 0.867447i \(-0.665759\pi\)
−0.497530 + 0.867447i \(0.665759\pi\)
\(348\) −12.0000 −0.643268
\(349\) −15.4641 −0.827774 −0.413887 0.910328i \(-0.635829\pi\)
−0.413887 + 0.910328i \(0.635829\pi\)
\(350\) 48.7846 2.60765
\(351\) 16.9808 0.906366
\(352\) −21.8564 −1.16495
\(353\) 21.0000 1.11772 0.558859 0.829263i \(-0.311239\pi\)
0.558859 + 0.829263i \(0.311239\pi\)
\(354\) −14.1962 −0.754517
\(355\) −1.00000 −0.0530745
\(356\) −65.1769 −3.45437
\(357\) −3.46410 −0.183340
\(358\) 8.19615 0.433180
\(359\) 2.19615 0.115908 0.0579542 0.998319i \(-0.481542\pi\)
0.0579542 + 0.998319i \(0.481542\pi\)
\(360\) 0 0
\(361\) 19.3923 1.02065
\(362\) −34.5885 −1.81793
\(363\) 1.73205 0.0909091
\(364\) 35.7128 1.87186
\(365\) −14.1962 −0.743061
\(366\) −13.8564 −0.724286
\(367\) 4.12436 0.215290 0.107645 0.994189i \(-0.465669\pi\)
0.107645 + 0.994189i \(0.465669\pi\)
\(368\) −55.7128 −2.90423
\(369\) 0 0
\(370\) −38.0526 −1.97826
\(371\) 5.07180 0.263315
\(372\) 21.4641 1.11286
\(373\) 8.19615 0.424381 0.212190 0.977228i \(-0.431940\pi\)
0.212190 + 0.977228i \(0.431940\pi\)
\(374\) −2.73205 −0.141271
\(375\) −25.3923 −1.31125
\(376\) −8.78461 −0.453032
\(377\) 4.14359 0.213406
\(378\) −28.3923 −1.46034
\(379\) −9.73205 −0.499902 −0.249951 0.968259i \(-0.580414\pi\)
−0.249951 + 0.968259i \(0.580414\pi\)
\(380\) 126.354 6.48181
\(381\) −6.92820 −0.354943
\(382\) −72.3013 −3.69925
\(383\) −20.3205 −1.03833 −0.519165 0.854674i \(-0.673757\pi\)
−0.519165 + 0.854674i \(0.673757\pi\)
\(384\) −65.5692 −3.34607
\(385\) 7.46410 0.380406
\(386\) 66.1051 3.36466
\(387\) 0 0
\(388\) 85.9615 4.36404
\(389\) 9.92820 0.503380 0.251690 0.967808i \(-0.419014\pi\)
0.251690 + 0.967808i \(0.419014\pi\)
\(390\) −57.7128 −2.92240
\(391\) −3.73205 −0.188738
\(392\) 28.3923 1.43403
\(393\) −15.4641 −0.780061
\(394\) 57.7128 2.90753
\(395\) −8.19615 −0.412393
\(396\) 0 0
\(397\) 27.7128 1.39087 0.695433 0.718591i \(-0.255213\pi\)
0.695433 + 0.718591i \(0.255213\pi\)
\(398\) 45.8564 2.29857
\(399\) 21.4641 1.07455
\(400\) 133.282 6.66410
\(401\) −28.7846 −1.43743 −0.718717 0.695302i \(-0.755270\pi\)
−0.718717 + 0.695302i \(0.755270\pi\)
\(402\) −4.73205 −0.236013
\(403\) −7.41154 −0.369195
\(404\) 9.07180 0.451339
\(405\) 33.5885 1.66902
\(406\) −6.92820 −0.343841
\(407\) −3.73205 −0.184991
\(408\) −16.3923 −0.811540
\(409\) −9.66025 −0.477669 −0.238834 0.971060i \(-0.576765\pi\)
−0.238834 + 0.971060i \(0.576765\pi\)
\(410\) 96.4974 4.76567
\(411\) −33.5885 −1.65680
\(412\) −64.7846 −3.19171
\(413\) −6.00000 −0.295241
\(414\) 0 0
\(415\) −59.1769 −2.90488
\(416\) 71.4256 3.50193
\(417\) 27.1244 1.32829
\(418\) 16.9282 0.827985
\(419\) 21.8564 1.06776 0.533878 0.845562i \(-0.320734\pi\)
0.533878 + 0.845562i \(0.320734\pi\)
\(420\) 70.6410 3.44693
\(421\) 17.8564 0.870268 0.435134 0.900366i \(-0.356701\pi\)
0.435134 + 0.900366i \(0.356701\pi\)
\(422\) 4.00000 0.194717
\(423\) 0 0
\(424\) 24.0000 1.16554
\(425\) 8.92820 0.433081
\(426\) −1.26795 −0.0614323
\(427\) −5.85641 −0.283411
\(428\) −61.5692 −2.97606
\(429\) −5.66025 −0.273280
\(430\) −20.3923 −0.983404
\(431\) 0.339746 0.0163650 0.00818249 0.999967i \(-0.497395\pi\)
0.00818249 + 0.999967i \(0.497395\pi\)
\(432\) −77.5692 −3.73205
\(433\) 18.6077 0.894229 0.447114 0.894477i \(-0.352452\pi\)
0.447114 + 0.894477i \(0.352452\pi\)
\(434\) 12.3923 0.594850
\(435\) 8.19615 0.392975
\(436\) −12.0000 −0.574696
\(437\) 23.1244 1.10619
\(438\) −18.0000 −0.860073
\(439\) −16.1962 −0.773000 −0.386500 0.922289i \(-0.626316\pi\)
−0.386500 + 0.922289i \(0.626316\pi\)
\(440\) 35.3205 1.68384
\(441\) 0 0
\(442\) 8.92820 0.424671
\(443\) 35.3923 1.68154 0.840770 0.541393i \(-0.182103\pi\)
0.840770 + 0.541393i \(0.182103\pi\)
\(444\) −35.3205 −1.67624
\(445\) 44.5167 2.11029
\(446\) −55.5167 −2.62879
\(447\) −33.1244 −1.56673
\(448\) −59.7128 −2.82117
\(449\) 14.5167 0.685084 0.342542 0.939503i \(-0.388712\pi\)
0.342542 + 0.939503i \(0.388712\pi\)
\(450\) 0 0
\(451\) 9.46410 0.445647
\(452\) −34.2487 −1.61092
\(453\) −21.1244 −0.992509
\(454\) −45.7128 −2.14541
\(455\) −24.3923 −1.14353
\(456\) 101.569 4.75641
\(457\) −4.73205 −0.221356 −0.110678 0.993856i \(-0.535302\pi\)
−0.110678 + 0.993856i \(0.535302\pi\)
\(458\) 34.0526 1.59117
\(459\) −5.19615 −0.242536
\(460\) 76.1051 3.54842
\(461\) −13.0718 −0.608814 −0.304407 0.952542i \(-0.598458\pi\)
−0.304407 + 0.952542i \(0.598458\pi\)
\(462\) 9.46410 0.440310
\(463\) 41.9282 1.94857 0.974284 0.225322i \(-0.0723433\pi\)
0.974284 + 0.225322i \(0.0723433\pi\)
\(464\) −18.9282 −0.878720
\(465\) −14.6603 −0.679853
\(466\) 13.8564 0.641886
\(467\) −5.78461 −0.267680 −0.133840 0.991003i \(-0.542731\pi\)
−0.133840 + 0.991003i \(0.542731\pi\)
\(468\) 0 0
\(469\) −2.00000 −0.0923514
\(470\) 9.46410 0.436546
\(471\) −34.2679 −1.57898
\(472\) −28.3923 −1.30686
\(473\) −2.00000 −0.0919601
\(474\) −10.3923 −0.477334
\(475\) −55.3205 −2.53828
\(476\) −10.9282 −0.500893
\(477\) 0 0
\(478\) −11.4641 −0.524356
\(479\) −8.78461 −0.401379 −0.200690 0.979655i \(-0.564318\pi\)
−0.200690 + 0.979655i \(0.564318\pi\)
\(480\) 141.282 6.44861
\(481\) 12.1962 0.556097
\(482\) −51.7128 −2.35545
\(483\) 12.9282 0.588254
\(484\) 5.46410 0.248368
\(485\) −58.7128 −2.66601
\(486\) 0 0
\(487\) 0.947441 0.0429327 0.0214663 0.999770i \(-0.493167\pi\)
0.0214663 + 0.999770i \(0.493167\pi\)
\(488\) −27.7128 −1.25450
\(489\) −4.39230 −0.198627
\(490\) −30.5885 −1.38185
\(491\) −1.12436 −0.0507415 −0.0253707 0.999678i \(-0.508077\pi\)
−0.0253707 + 0.999678i \(0.508077\pi\)
\(492\) 89.5692 4.03809
\(493\) −1.26795 −0.0571056
\(494\) −55.3205 −2.48899
\(495\) 0 0
\(496\) 33.8564 1.52020
\(497\) −0.535898 −0.0240383
\(498\) −75.0333 −3.36232
\(499\) 6.39230 0.286159 0.143079 0.989711i \(-0.454300\pi\)
0.143079 + 0.989711i \(0.454300\pi\)
\(500\) −80.1051 −3.58241
\(501\) 32.4449 1.44953
\(502\) 71.9090 3.20945
\(503\) 39.3731 1.75556 0.877779 0.479066i \(-0.159024\pi\)
0.877779 + 0.479066i \(0.159024\pi\)
\(504\) 0 0
\(505\) −6.19615 −0.275725
\(506\) 10.1962 0.453274
\(507\) −4.01924 −0.178501
\(508\) −21.8564 −0.969721
\(509\) 3.78461 0.167750 0.0838749 0.996476i \(-0.473270\pi\)
0.0838749 + 0.996476i \(0.473270\pi\)
\(510\) 17.6603 0.782009
\(511\) −7.60770 −0.336545
\(512\) −43.7128 −1.93185
\(513\) 32.1962 1.42149
\(514\) −56.7846 −2.50466
\(515\) 44.2487 1.94983
\(516\) −18.9282 −0.833268
\(517\) 0.928203 0.0408223
\(518\) −20.3923 −0.895986
\(519\) −24.2487 −1.06440
\(520\) −115.426 −5.06175
\(521\) 0.947441 0.0415081 0.0207541 0.999785i \(-0.493393\pi\)
0.0207541 + 0.999785i \(0.493393\pi\)
\(522\) 0 0
\(523\) 6.24871 0.273237 0.136619 0.990624i \(-0.456377\pi\)
0.136619 + 0.990624i \(0.456377\pi\)
\(524\) −48.7846 −2.13117
\(525\) −30.9282 −1.34982
\(526\) −30.3923 −1.32517
\(527\) 2.26795 0.0987934
\(528\) 25.8564 1.12526
\(529\) −9.07180 −0.394426
\(530\) −25.8564 −1.12313
\(531\) 0 0
\(532\) 67.7128 2.93572
\(533\) −30.9282 −1.33965
\(534\) 56.4449 2.44261
\(535\) 42.0526 1.81809
\(536\) −9.46410 −0.408787
\(537\) −5.19615 −0.224231
\(538\) 38.2487 1.64902
\(539\) −3.00000 −0.129219
\(540\) 105.962 4.55986
\(541\) 6.33975 0.272567 0.136283 0.990670i \(-0.456484\pi\)
0.136283 + 0.990670i \(0.456484\pi\)
\(542\) 24.7846 1.06459
\(543\) 21.9282 0.941029
\(544\) −21.8564 −0.937086
\(545\) 8.19615 0.351085
\(546\) −30.9282 −1.32360
\(547\) 38.7321 1.65606 0.828031 0.560682i \(-0.189461\pi\)
0.828031 + 0.560682i \(0.189461\pi\)
\(548\) −105.962 −4.52645
\(549\) 0 0
\(550\) −24.3923 −1.04009
\(551\) 7.85641 0.334694
\(552\) 61.1769 2.60386
\(553\) −4.39230 −0.186780
\(554\) −50.2487 −2.13486
\(555\) 24.1244 1.02402
\(556\) 85.5692 3.62894
\(557\) −12.9282 −0.547786 −0.273893 0.961760i \(-0.588311\pi\)
−0.273893 + 0.961760i \(0.588311\pi\)
\(558\) 0 0
\(559\) 6.53590 0.276439
\(560\) 111.426 4.70859
\(561\) 1.73205 0.0731272
\(562\) 74.1051 3.12594
\(563\) −11.2679 −0.474887 −0.237444 0.971401i \(-0.576310\pi\)
−0.237444 + 0.971401i \(0.576310\pi\)
\(564\) 8.78461 0.369899
\(565\) 23.3923 0.984122
\(566\) 7.46410 0.313740
\(567\) 18.0000 0.755929
\(568\) −2.53590 −0.106404
\(569\) 4.00000 0.167689 0.0838444 0.996479i \(-0.473280\pi\)
0.0838444 + 0.996479i \(0.473280\pi\)
\(570\) −109.426 −4.58334
\(571\) 8.98076 0.375833 0.187917 0.982185i \(-0.439827\pi\)
0.187917 + 0.982185i \(0.439827\pi\)
\(572\) −17.8564 −0.746614
\(573\) 45.8372 1.91488
\(574\) 51.7128 2.15845
\(575\) −33.3205 −1.38956
\(576\) 0 0
\(577\) −3.39230 −0.141223 −0.0706117 0.997504i \(-0.522495\pi\)
−0.0706117 + 0.997504i \(0.522495\pi\)
\(578\) −2.73205 −0.113638
\(579\) −41.9090 −1.74168
\(580\) 25.8564 1.07363
\(581\) −31.7128 −1.31567
\(582\) −74.4449 −3.08584
\(583\) −2.53590 −0.105026
\(584\) −36.0000 −1.48969
\(585\) 0 0
\(586\) 37.8564 1.56383
\(587\) −44.9282 −1.85439 −0.927193 0.374585i \(-0.877785\pi\)
−0.927193 + 0.374585i \(0.877785\pi\)
\(588\) −28.3923 −1.17088
\(589\) −14.0526 −0.579026
\(590\) 30.5885 1.25931
\(591\) −36.5885 −1.50505
\(592\) −55.7128 −2.28978
\(593\) 2.87564 0.118089 0.0590443 0.998255i \(-0.481195\pi\)
0.0590443 + 0.998255i \(0.481195\pi\)
\(594\) 14.1962 0.582475
\(595\) 7.46410 0.305998
\(596\) −104.497 −4.28038
\(597\) −29.0718 −1.18983
\(598\) −33.3205 −1.36258
\(599\) 12.2487 0.500469 0.250234 0.968185i \(-0.419492\pi\)
0.250234 + 0.968185i \(0.419492\pi\)
\(600\) −146.354 −5.97487
\(601\) −16.5359 −0.674513 −0.337257 0.941413i \(-0.609499\pi\)
−0.337257 + 0.941413i \(0.609499\pi\)
\(602\) −10.9282 −0.445400
\(603\) 0 0
\(604\) −66.6410 −2.71158
\(605\) −3.73205 −0.151729
\(606\) −7.85641 −0.319145
\(607\) 22.7846 0.924799 0.462399 0.886672i \(-0.346989\pi\)
0.462399 + 0.886672i \(0.346989\pi\)
\(608\) 135.426 5.49223
\(609\) 4.39230 0.177985
\(610\) 29.8564 1.20885
\(611\) −3.03332 −0.122715
\(612\) 0 0
\(613\) −30.6410 −1.23758 −0.618789 0.785557i \(-0.712377\pi\)
−0.618789 + 0.785557i \(0.712377\pi\)
\(614\) 26.7846 1.08094
\(615\) −61.1769 −2.46689
\(616\) 18.9282 0.762639
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 56.1051 2.25688
\(619\) −13.0526 −0.524627 −0.262313 0.964983i \(-0.584485\pi\)
−0.262313 + 0.964983i \(0.584485\pi\)
\(620\) −46.2487 −1.85739
\(621\) 19.3923 0.778186
\(622\) 1.07180 0.0429751
\(623\) 23.8564 0.955787
\(624\) −84.4974 −3.38260
\(625\) 10.0718 0.402872
\(626\) −24.0526 −0.961334
\(627\) −10.7321 −0.428597
\(628\) −108.105 −4.31386
\(629\) −3.73205 −0.148807
\(630\) 0 0
\(631\) 47.3923 1.88666 0.943329 0.331859i \(-0.107676\pi\)
0.943329 + 0.331859i \(0.107676\pi\)
\(632\) −20.7846 −0.826767
\(633\) −2.53590 −0.100793
\(634\) −69.5167 −2.76086
\(635\) 14.9282 0.592408
\(636\) −24.0000 −0.951662
\(637\) 9.80385 0.388443
\(638\) 3.46410 0.137145
\(639\) 0 0
\(640\) 141.282 5.58466
\(641\) −39.8372 −1.57347 −0.786737 0.617289i \(-0.788231\pi\)
−0.786737 + 0.617289i \(0.788231\pi\)
\(642\) 53.3205 2.10439
\(643\) −23.9808 −0.945709 −0.472854 0.881141i \(-0.656776\pi\)
−0.472854 + 0.881141i \(0.656776\pi\)
\(644\) 40.7846 1.60714
\(645\) 12.9282 0.509048
\(646\) 16.9282 0.666031
\(647\) 29.2487 1.14989 0.574943 0.818194i \(-0.305024\pi\)
0.574943 + 0.818194i \(0.305024\pi\)
\(648\) 85.1769 3.34607
\(649\) 3.00000 0.117760
\(650\) 79.7128 3.12659
\(651\) −7.85641 −0.307917
\(652\) −13.8564 −0.542659
\(653\) 42.3731 1.65819 0.829093 0.559111i \(-0.188857\pi\)
0.829093 + 0.559111i \(0.188857\pi\)
\(654\) 10.3923 0.406371
\(655\) 33.3205 1.30194
\(656\) 141.282 5.51614
\(657\) 0 0
\(658\) 5.07180 0.197719
\(659\) −35.1244 −1.36825 −0.684125 0.729364i \(-0.739816\pi\)
−0.684125 + 0.729364i \(0.739816\pi\)
\(660\) −35.3205 −1.37485
\(661\) −34.4641 −1.34050 −0.670249 0.742136i \(-0.733813\pi\)
−0.670249 + 0.742136i \(0.733813\pi\)
\(662\) 84.3013 3.27646
\(663\) −5.66025 −0.219826
\(664\) −150.067 −5.82372
\(665\) −46.2487 −1.79345
\(666\) 0 0
\(667\) 4.73205 0.183226
\(668\) 102.354 3.96019
\(669\) 35.1962 1.36076
\(670\) 10.1962 0.393912
\(671\) 2.92820 0.113042
\(672\) 75.7128 2.92069
\(673\) 10.1962 0.393033 0.196516 0.980501i \(-0.437037\pi\)
0.196516 + 0.980501i \(0.437037\pi\)
\(674\) 26.2487 1.01106
\(675\) −46.3923 −1.78564
\(676\) −12.6795 −0.487673
\(677\) −28.0526 −1.07815 −0.539074 0.842259i \(-0.681226\pi\)
−0.539074 + 0.842259i \(0.681226\pi\)
\(678\) 29.6603 1.13910
\(679\) −31.4641 −1.20748
\(680\) 35.3205 1.35448
\(681\) 28.9808 1.11055
\(682\) −6.19615 −0.237263
\(683\) −1.46410 −0.0560223 −0.0280111 0.999608i \(-0.508917\pi\)
−0.0280111 + 0.999608i \(0.508917\pi\)
\(684\) 0 0
\(685\) 72.3731 2.76523
\(686\) −54.6410 −2.08620
\(687\) −21.5885 −0.823651
\(688\) −29.8564 −1.13826
\(689\) 8.28719 0.315717
\(690\) −65.9090 −2.50911
\(691\) −24.5167 −0.932658 −0.466329 0.884611i \(-0.654424\pi\)
−0.466329 + 0.884611i \(0.654424\pi\)
\(692\) −76.4974 −2.90800
\(693\) 0 0
\(694\) 50.6410 1.92231
\(695\) −58.4449 −2.21694
\(696\) 20.7846 0.787839
\(697\) 9.46410 0.358478
\(698\) 42.2487 1.59914
\(699\) −8.78461 −0.332264
\(700\) −97.5692 −3.68777
\(701\) 28.6410 1.08176 0.540878 0.841101i \(-0.318092\pi\)
0.540878 + 0.841101i \(0.318092\pi\)
\(702\) −46.3923 −1.75096
\(703\) 23.1244 0.872152
\(704\) 29.8564 1.12526
\(705\) −6.00000 −0.225973
\(706\) −57.3731 −2.15926
\(707\) −3.32051 −0.124881
\(708\) 28.3923 1.06705
\(709\) 17.3397 0.651208 0.325604 0.945506i \(-0.394432\pi\)
0.325604 + 0.945506i \(0.394432\pi\)
\(710\) 2.73205 0.102532
\(711\) 0 0
\(712\) 112.890 4.23072
\(713\) −8.46410 −0.316983
\(714\) 9.46410 0.354185
\(715\) 12.1962 0.456110
\(716\) −16.3923 −0.612609
\(717\) 7.26795 0.271427
\(718\) −6.00000 −0.223918
\(719\) 28.3731 1.05814 0.529068 0.848579i \(-0.322541\pi\)
0.529068 + 0.848579i \(0.322541\pi\)
\(720\) 0 0
\(721\) 23.7128 0.883111
\(722\) −52.9808 −1.97174
\(723\) 32.7846 1.21927
\(724\) 69.1769 2.57094
\(725\) −11.3205 −0.420433
\(726\) −4.73205 −0.175623
\(727\) −23.9282 −0.887448 −0.443724 0.896164i \(-0.646343\pi\)
−0.443724 + 0.896164i \(0.646343\pi\)
\(728\) −61.8564 −2.29255
\(729\) 27.0000 1.00000
\(730\) 38.7846 1.43548
\(731\) −2.00000 −0.0739727
\(732\) 27.7128 1.02430
\(733\) 20.3923 0.753207 0.376603 0.926375i \(-0.377092\pi\)
0.376603 + 0.926375i \(0.377092\pi\)
\(734\) −11.2679 −0.415908
\(735\) 19.3923 0.715296
\(736\) 81.5692 3.00668
\(737\) 1.00000 0.0368355
\(738\) 0 0
\(739\) 26.0526 0.958359 0.479179 0.877717i \(-0.340934\pi\)
0.479179 + 0.877717i \(0.340934\pi\)
\(740\) 76.1051 2.79768
\(741\) 35.0718 1.28839
\(742\) −13.8564 −0.508685
\(743\) 4.39230 0.161138 0.0805690 0.996749i \(-0.474326\pi\)
0.0805690 + 0.996749i \(0.474326\pi\)
\(744\) −37.1769 −1.36297
\(745\) 71.3731 2.61491
\(746\) −22.3923 −0.819841
\(747\) 0 0
\(748\) 5.46410 0.199787
\(749\) 22.5359 0.823444
\(750\) 69.3731 2.53315
\(751\) 33.0526 1.20611 0.603053 0.797701i \(-0.293951\pi\)
0.603053 + 0.797701i \(0.293951\pi\)
\(752\) 13.8564 0.505291
\(753\) −45.5885 −1.66134
\(754\) −11.3205 −0.412269
\(755\) 45.5167 1.65652
\(756\) 56.7846 2.06524
\(757\) −43.5692 −1.58355 −0.791775 0.610813i \(-0.790843\pi\)
−0.791775 + 0.610813i \(0.790843\pi\)
\(758\) 26.5885 0.965736
\(759\) −6.46410 −0.234632
\(760\) −218.851 −7.93857
\(761\) 3.94744 0.143095 0.0715473 0.997437i \(-0.477206\pi\)
0.0715473 + 0.997437i \(0.477206\pi\)
\(762\) 18.9282 0.685696
\(763\) 4.39230 0.159012
\(764\) 144.603 5.23154
\(765\) 0 0
\(766\) 55.5167 2.00590
\(767\) −9.80385 −0.353996
\(768\) 75.7128 2.73205
\(769\) 33.5692 1.21054 0.605269 0.796021i \(-0.293066\pi\)
0.605269 + 0.796021i \(0.293066\pi\)
\(770\) −20.3923 −0.734888
\(771\) 36.0000 1.29651
\(772\) −132.210 −4.75835
\(773\) −43.3205 −1.55813 −0.779065 0.626943i \(-0.784306\pi\)
−0.779065 + 0.626943i \(0.784306\pi\)
\(774\) 0 0
\(775\) 20.2487 0.727355
\(776\) −148.890 −5.34483
\(777\) 12.9282 0.463797
\(778\) −27.1244 −0.972455
\(779\) −58.6410 −2.10103
\(780\) 115.426 4.13290
\(781\) 0.267949 0.00958798
\(782\) 10.1962 0.364614
\(783\) 6.58846 0.235452
\(784\) −44.7846 −1.59945
\(785\) 73.8372 2.63536
\(786\) 42.2487 1.50696
\(787\) 7.12436 0.253956 0.126978 0.991906i \(-0.459472\pi\)
0.126978 + 0.991906i \(0.459472\pi\)
\(788\) −115.426 −4.11187
\(789\) 19.2679 0.685957
\(790\) 22.3923 0.796682
\(791\) 12.5359 0.445725
\(792\) 0 0
\(793\) −9.56922 −0.339813
\(794\) −75.7128 −2.68695
\(795\) 16.3923 0.581375
\(796\) −91.7128 −3.25067
\(797\) 5.39230 0.191005 0.0955026 0.995429i \(-0.469554\pi\)
0.0955026 + 0.995429i \(0.469554\pi\)
\(798\) −58.6410 −2.07587
\(799\) 0.928203 0.0328375
\(800\) −195.138 −6.89919
\(801\) 0 0
\(802\) 78.6410 2.77691
\(803\) 3.80385 0.134235
\(804\) 9.46410 0.333773
\(805\) −27.8564 −0.981809
\(806\) 20.2487 0.713230
\(807\) −24.2487 −0.853595
\(808\) −15.7128 −0.552775
\(809\) 31.9090 1.12186 0.560930 0.827863i \(-0.310444\pi\)
0.560930 + 0.827863i \(0.310444\pi\)
\(810\) −91.7654 −3.22431
\(811\) −0.732051 −0.0257058 −0.0128529 0.999917i \(-0.504091\pi\)
−0.0128529 + 0.999917i \(0.504091\pi\)
\(812\) 13.8564 0.486265
\(813\) −15.7128 −0.551072
\(814\) 10.1962 0.357375
\(815\) 9.46410 0.331513
\(816\) 25.8564 0.905155
\(817\) 12.3923 0.433552
\(818\) 26.3923 0.922785
\(819\) 0 0
\(820\) −192.995 −6.73967
\(821\) −4.05256 −0.141435 −0.0707176 0.997496i \(-0.522529\pi\)
−0.0707176 + 0.997496i \(0.522529\pi\)
\(822\) 91.7654 3.20068
\(823\) 39.7321 1.38497 0.692486 0.721431i \(-0.256515\pi\)
0.692486 + 0.721431i \(0.256515\pi\)
\(824\) 112.210 3.90903
\(825\) 15.4641 0.538391
\(826\) 16.3923 0.570361
\(827\) −29.6603 −1.03139 −0.515694 0.856773i \(-0.672466\pi\)
−0.515694 + 0.856773i \(0.672466\pi\)
\(828\) 0 0
\(829\) −13.9282 −0.483746 −0.241873 0.970308i \(-0.577762\pi\)
−0.241873 + 0.970308i \(0.577762\pi\)
\(830\) 161.674 5.61180
\(831\) 31.8564 1.10509
\(832\) −97.5692 −3.38260
\(833\) −3.00000 −0.103944
\(834\) −74.1051 −2.56605
\(835\) −69.9090 −2.41930
\(836\) −33.8564 −1.17095
\(837\) −11.7846 −0.407336
\(838\) −59.7128 −2.06274
\(839\) −5.48334 −0.189306 −0.0946530 0.995510i \(-0.530174\pi\)
−0.0946530 + 0.995510i \(0.530174\pi\)
\(840\) −122.354 −4.22161
\(841\) −27.3923 −0.944562
\(842\) −48.7846 −1.68123
\(843\) −46.9808 −1.61810
\(844\) −8.00000 −0.275371
\(845\) 8.66025 0.297922
\(846\) 0 0
\(847\) −2.00000 −0.0687208
\(848\) −37.8564 −1.29999
\(849\) −4.73205 −0.162404
\(850\) −24.3923 −0.836649
\(851\) 13.9282 0.477453
\(852\) 2.53590 0.0868784
\(853\) −13.1244 −0.449369 −0.224685 0.974432i \(-0.572135\pi\)
−0.224685 + 0.974432i \(0.572135\pi\)
\(854\) 16.0000 0.547509
\(855\) 0 0
\(856\) 106.641 3.64491
\(857\) −10.4449 −0.356790 −0.178395 0.983959i \(-0.557090\pi\)
−0.178395 + 0.983959i \(0.557090\pi\)
\(858\) 15.4641 0.527936
\(859\) 17.1436 0.584932 0.292466 0.956276i \(-0.405524\pi\)
0.292466 + 0.956276i \(0.405524\pi\)
\(860\) 40.7846 1.39074
\(861\) −32.7846 −1.11730
\(862\) −0.928203 −0.0316147
\(863\) −36.6410 −1.24727 −0.623637 0.781714i \(-0.714346\pi\)
−0.623637 + 0.781714i \(0.714346\pi\)
\(864\) 113.569 3.86370
\(865\) 52.2487 1.77651
\(866\) −50.8372 −1.72752
\(867\) 1.73205 0.0588235
\(868\) −24.7846 −0.841244
\(869\) 2.19615 0.0744994
\(870\) −22.3923 −0.759170
\(871\) −3.26795 −0.110730
\(872\) 20.7846 0.703856
\(873\) 0 0
\(874\) −63.1769 −2.13699
\(875\) 29.3205 0.991214
\(876\) 36.0000 1.21633
\(877\) 12.5885 0.425082 0.212541 0.977152i \(-0.431826\pi\)
0.212541 + 0.977152i \(0.431826\pi\)
\(878\) 44.2487 1.49332
\(879\) −24.0000 −0.809500
\(880\) −55.7128 −1.87808
\(881\) 22.9090 0.771823 0.385911 0.922536i \(-0.373887\pi\)
0.385911 + 0.922536i \(0.373887\pi\)
\(882\) 0 0
\(883\) 6.92820 0.233153 0.116576 0.993182i \(-0.462808\pi\)
0.116576 + 0.993182i \(0.462808\pi\)
\(884\) −17.8564 −0.600576
\(885\) −19.3923 −0.651865
\(886\) −96.6936 −3.24848
\(887\) 21.7128 0.729045 0.364522 0.931195i \(-0.381232\pi\)
0.364522 + 0.931195i \(0.381232\pi\)
\(888\) 61.1769 2.05296
\(889\) 8.00000 0.268311
\(890\) −121.622 −4.07677
\(891\) −9.00000 −0.301511
\(892\) 111.033 3.71767
\(893\) −5.75129 −0.192460
\(894\) 90.4974 3.02669
\(895\) 11.1962 0.374246
\(896\) 75.7128 2.52939
\(897\) 21.1244 0.705322
\(898\) −39.6603 −1.32348
\(899\) −2.87564 −0.0959081
\(900\) 0 0
\(901\) −2.53590 −0.0844830
\(902\) −25.8564 −0.860924
\(903\) 6.92820 0.230556
\(904\) 59.3205 1.97297
\(905\) −47.2487 −1.57060
\(906\) 57.7128 1.91738
\(907\) 1.07180 0.0355884 0.0177942 0.999842i \(-0.494336\pi\)
0.0177942 + 0.999842i \(0.494336\pi\)
\(908\) 91.4256 3.03407
\(909\) 0 0
\(910\) 66.6410 2.20913
\(911\) −5.60770 −0.185791 −0.0928956 0.995676i \(-0.529612\pi\)
−0.0928956 + 0.995676i \(0.529612\pi\)
\(912\) −160.210 −5.30509
\(913\) 15.8564 0.524770
\(914\) 12.9282 0.427627
\(915\) −18.9282 −0.625747
\(916\) −68.1051 −2.25026
\(917\) 17.8564 0.589670
\(918\) 14.1962 0.468543
\(919\) −6.67949 −0.220336 −0.110168 0.993913i \(-0.535139\pi\)
−0.110168 + 0.993913i \(0.535139\pi\)
\(920\) −131.818 −4.34591
\(921\) −16.9808 −0.559535
\(922\) 35.7128 1.17614
\(923\) −0.875644 −0.0288222
\(924\) −18.9282 −0.622692
\(925\) −33.3205 −1.09557
\(926\) −114.550 −3.76435
\(927\) 0 0
\(928\) 27.7128 0.909718
\(929\) 45.0333 1.47750 0.738748 0.673982i \(-0.235418\pi\)
0.738748 + 0.673982i \(0.235418\pi\)
\(930\) 40.0526 1.31338
\(931\) 18.5885 0.609212
\(932\) −27.7128 −0.907763
\(933\) −0.679492 −0.0222456
\(934\) 15.8038 0.517118
\(935\) −3.73205 −0.122051
\(936\) 0 0
\(937\) 14.9808 0.489400 0.244700 0.969599i \(-0.421310\pi\)
0.244700 + 0.969599i \(0.421310\pi\)
\(938\) 5.46410 0.178409
\(939\) 15.2487 0.497623
\(940\) −18.9282 −0.617370
\(941\) 24.2487 0.790485 0.395243 0.918577i \(-0.370660\pi\)
0.395243 + 0.918577i \(0.370660\pi\)
\(942\) 93.6218 3.05036
\(943\) −35.3205 −1.15019
\(944\) 44.7846 1.45761
\(945\) −38.7846 −1.26166
\(946\) 5.46410 0.177653
\(947\) −2.26795 −0.0736984 −0.0368492 0.999321i \(-0.511732\pi\)
−0.0368492 + 0.999321i \(0.511732\pi\)
\(948\) 20.7846 0.675053
\(949\) −12.4308 −0.403520
\(950\) 151.138 4.90358
\(951\) 44.0718 1.42913
\(952\) 18.9282 0.613467
\(953\) −54.7846 −1.77465 −0.887324 0.461147i \(-0.847438\pi\)
−0.887324 + 0.461147i \(0.847438\pi\)
\(954\) 0 0
\(955\) −98.7654 −3.19597
\(956\) 22.9282 0.741551
\(957\) −2.19615 −0.0709915
\(958\) 24.0000 0.775405
\(959\) 38.7846 1.25242
\(960\) −192.995 −6.22888
\(961\) −25.8564 −0.834078
\(962\) −33.3205 −1.07430
\(963\) 0 0
\(964\) 103.426 3.33112
\(965\) 90.3013 2.90690
\(966\) −35.3205 −1.13642
\(967\) 41.9090 1.34770 0.673851 0.738868i \(-0.264639\pi\)
0.673851 + 0.738868i \(0.264639\pi\)
\(968\) −9.46410 −0.304188
\(969\) −10.7321 −0.344763
\(970\) 160.406 5.15034
\(971\) −28.8564 −0.926046 −0.463023 0.886346i \(-0.653235\pi\)
−0.463023 + 0.886346i \(0.653235\pi\)
\(972\) 0 0
\(973\) −31.3205 −1.00409
\(974\) −2.58846 −0.0829395
\(975\) −50.5359 −1.61844
\(976\) 43.7128 1.39921
\(977\) −53.7846 −1.72072 −0.860361 0.509685i \(-0.829762\pi\)
−0.860361 + 0.509685i \(0.829762\pi\)
\(978\) 12.0000 0.383718
\(979\) −11.9282 −0.381227
\(980\) 61.1769 1.95422
\(981\) 0 0
\(982\) 3.07180 0.0980250
\(983\) 2.66025 0.0848489 0.0424245 0.999100i \(-0.486492\pi\)
0.0424245 + 0.999100i \(0.486492\pi\)
\(984\) −155.138 −4.94563
\(985\) 78.8372 2.51196
\(986\) 3.46410 0.110319
\(987\) −3.21539 −0.102347
\(988\) 110.641 3.51996
\(989\) 7.46410 0.237345
\(990\) 0 0
\(991\) −22.6410 −0.719216 −0.359608 0.933104i \(-0.617090\pi\)
−0.359608 + 0.933104i \(0.617090\pi\)
\(992\) −49.5692 −1.57382
\(993\) −53.4449 −1.69602
\(994\) 1.46410 0.0464385
\(995\) 62.6410 1.98585
\(996\) 150.067 4.75504
\(997\) −61.2295 −1.93916 −0.969578 0.244781i \(-0.921284\pi\)
−0.969578 + 0.244781i \(0.921284\pi\)
\(998\) −17.4641 −0.552816
\(999\) 19.3923 0.613545
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 187.2.a.c.1.1 2
3.2 odd 2 1683.2.a.r.1.2 2
4.3 odd 2 2992.2.a.j.1.1 2
5.4 even 2 4675.2.a.v.1.2 2
7.6 odd 2 9163.2.a.h.1.1 2
11.10 odd 2 2057.2.a.o.1.2 2
17.16 even 2 3179.2.a.k.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
187.2.a.c.1.1 2 1.1 even 1 trivial
1683.2.a.r.1.2 2 3.2 odd 2
2057.2.a.o.1.2 2 11.10 odd 2
2992.2.a.j.1.1 2 4.3 odd 2
3179.2.a.k.1.1 2 17.16 even 2
4675.2.a.v.1.2 2 5.4 even 2
9163.2.a.h.1.1 2 7.6 odd 2