Properties

Label 363.3.b.d.122.1
Level $363$
Weight $3$
Character 363.122
Analytic conductor $9.891$
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] = [363,3,Mod(122,363)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(363, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("363.122");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 363.b (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.89103359628\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-11}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 33)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 122.1
Root \(0.500000 + 1.65831i\) of defining polynomial
Character \(\chi\) \(=\) 363.122
Dual form 363.3.b.d.122.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.31662i q^{2} +3.00000 q^{3} -7.00000 q^{4} -6.63325i q^{5} -9.94987i q^{6} +8.00000 q^{7} +9.94987i q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-3.31662i q^{2} +3.00000 q^{3} -7.00000 q^{4} -6.63325i q^{5} -9.94987i q^{6} +8.00000 q^{7} +9.94987i q^{8} +9.00000 q^{9} -22.0000 q^{10} -21.0000 q^{12} -4.00000 q^{13} -26.5330i q^{14} -19.8997i q^{15} +5.00000 q^{16} -13.2665i q^{17} -29.8496i q^{18} +6.00000 q^{19} +46.4327i q^{20} +24.0000 q^{21} +6.63325i q^{23} +29.8496i q^{24} -19.0000 q^{25} +13.2665i q^{26} +27.0000 q^{27} -56.0000 q^{28} +39.7995i q^{29} -66.0000 q^{30} -26.0000 q^{31} +23.2164i q^{32} -44.0000 q^{34} -53.0660i q^{35} -63.0000 q^{36} +30.0000 q^{37} -19.8997i q^{38} -12.0000 q^{39} +66.0000 q^{40} -13.2665i q^{41} -79.5990i q^{42} -42.0000 q^{43} -59.6992i q^{45} +22.0000 q^{46} +86.2322i q^{47} +15.0000 q^{48} +15.0000 q^{49} +63.0159i q^{50} -39.7995i q^{51} +28.0000 q^{52} -59.6992i q^{53} -89.5489i q^{54} +79.5990i q^{56} +18.0000 q^{57} +132.000 q^{58} +66.3325i q^{59} +139.298i q^{60} -12.0000 q^{61} +86.2322i q^{62} +72.0000 q^{63} +97.0000 q^{64} +26.5330i q^{65} +2.00000 q^{67} +92.8655i q^{68} +19.8997i q^{69} -176.000 q^{70} -59.6992i q^{71} +89.5489i q^{72} +74.0000 q^{73} -99.4987i q^{74} -57.0000 q^{75} -42.0000 q^{76} +39.7995i q^{78} +40.0000 q^{79} -33.1662i q^{80} +81.0000 q^{81} -44.0000 q^{82} +39.7995i q^{83} -168.000 q^{84} -88.0000 q^{85} +139.298i q^{86} +119.398i q^{87} -119.398i q^{89} -198.000 q^{90} -32.0000 q^{91} -46.4327i q^{92} -78.0000 q^{93} +286.000 q^{94} -39.7995i q^{95} +69.6491i q^{96} +62.0000 q^{97} -49.7494i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} - 14 q^{4} + 16 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{3} - 14 q^{4} + 16 q^{7} + 18 q^{9} - 44 q^{10} - 42 q^{12} - 8 q^{13} + 10 q^{16} + 12 q^{19} + 48 q^{21} - 38 q^{25} + 54 q^{27} - 112 q^{28} - 132 q^{30} - 52 q^{31} - 88 q^{34} - 126 q^{36} + 60 q^{37} - 24 q^{39} + 132 q^{40} - 84 q^{43} + 44 q^{46} + 30 q^{48} + 30 q^{49} + 56 q^{52} + 36 q^{57} + 264 q^{58} - 24 q^{61} + 144 q^{63} + 194 q^{64} + 4 q^{67} - 352 q^{70} + 148 q^{73} - 114 q^{75} - 84 q^{76} + 80 q^{79} + 162 q^{81} - 88 q^{82} - 336 q^{84} - 176 q^{85} - 396 q^{90} - 64 q^{91} - 156 q^{93} + 572 q^{94} + 124 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/363\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(244\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 3.31662i − 1.65831i −0.559017 0.829156i \(-0.688821\pi\)
0.559017 0.829156i \(-0.311179\pi\)
\(3\) 3.00000 1.00000
\(4\) −7.00000 −1.75000
\(5\) − 6.63325i − 1.32665i −0.748331 0.663325i \(-0.769145\pi\)
0.748331 0.663325i \(-0.230855\pi\)
\(6\) − 9.94987i − 1.65831i
\(7\) 8.00000 1.14286 0.571429 0.820652i \(-0.306389\pi\)
0.571429 + 0.820652i \(0.306389\pi\)
\(8\) 9.94987i 1.24373i
\(9\) 9.00000 1.00000
\(10\) −22.0000 −2.20000
\(11\) 0 0
\(12\) −21.0000 −1.75000
\(13\) −4.00000 −0.307692 −0.153846 0.988095i \(-0.549166\pi\)
−0.153846 + 0.988095i \(0.549166\pi\)
\(14\) − 26.5330i − 1.89521i
\(15\) − 19.8997i − 1.32665i
\(16\) 5.00000 0.312500
\(17\) − 13.2665i − 0.780382i −0.920734 0.390191i \(-0.872409\pi\)
0.920734 0.390191i \(-0.127591\pi\)
\(18\) − 29.8496i − 1.65831i
\(19\) 6.00000 0.315789 0.157895 0.987456i \(-0.449529\pi\)
0.157895 + 0.987456i \(0.449529\pi\)
\(20\) 46.4327i 2.32164i
\(21\) 24.0000 1.14286
\(22\) 0 0
\(23\) 6.63325i 0.288402i 0.989548 + 0.144201i \(0.0460612\pi\)
−0.989548 + 0.144201i \(0.953939\pi\)
\(24\) 29.8496i 1.24373i
\(25\) −19.0000 −0.760000
\(26\) 13.2665i 0.510250i
\(27\) 27.0000 1.00000
\(28\) −56.0000 −2.00000
\(29\) 39.7995i 1.37240i 0.727415 + 0.686198i \(0.240722\pi\)
−0.727415 + 0.686198i \(0.759278\pi\)
\(30\) −66.0000 −2.20000
\(31\) −26.0000 −0.838710 −0.419355 0.907822i \(-0.637744\pi\)
−0.419355 + 0.907822i \(0.637744\pi\)
\(32\) 23.2164i 0.725512i
\(33\) 0 0
\(34\) −44.0000 −1.29412
\(35\) − 53.0660i − 1.51617i
\(36\) −63.0000 −1.75000
\(37\) 30.0000 0.810811 0.405405 0.914137i \(-0.367130\pi\)
0.405405 + 0.914137i \(0.367130\pi\)
\(38\) − 19.8997i − 0.523678i
\(39\) −12.0000 −0.307692
\(40\) 66.0000 1.65000
\(41\) − 13.2665i − 0.323573i −0.986826 0.161787i \(-0.948274\pi\)
0.986826 0.161787i \(-0.0517256\pi\)
\(42\) − 79.5990i − 1.89521i
\(43\) −42.0000 −0.976744 −0.488372 0.872635i \(-0.662409\pi\)
−0.488372 + 0.872635i \(0.662409\pi\)
\(44\) 0 0
\(45\) − 59.6992i − 1.32665i
\(46\) 22.0000 0.478261
\(47\) 86.2322i 1.83473i 0.398049 + 0.917364i \(0.369688\pi\)
−0.398049 + 0.917364i \(0.630312\pi\)
\(48\) 15.0000 0.312500
\(49\) 15.0000 0.306122
\(50\) 63.0159i 1.26032i
\(51\) − 39.7995i − 0.780382i
\(52\) 28.0000 0.538462
\(53\) − 59.6992i − 1.12640i −0.826320 0.563200i \(-0.809570\pi\)
0.826320 0.563200i \(-0.190430\pi\)
\(54\) − 89.5489i − 1.65831i
\(55\) 0 0
\(56\) 79.5990i 1.42141i
\(57\) 18.0000 0.315789
\(58\) 132.000 2.27586
\(59\) 66.3325i 1.12428i 0.827042 + 0.562140i \(0.190022\pi\)
−0.827042 + 0.562140i \(0.809978\pi\)
\(60\) 139.298i 2.32164i
\(61\) −12.0000 −0.196721 −0.0983607 0.995151i \(-0.531360\pi\)
−0.0983607 + 0.995151i \(0.531360\pi\)
\(62\) 86.2322i 1.39084i
\(63\) 72.0000 1.14286
\(64\) 97.0000 1.51562
\(65\) 26.5330i 0.408200i
\(66\) 0 0
\(67\) 2.00000 0.0298507 0.0149254 0.999889i \(-0.495249\pi\)
0.0149254 + 0.999889i \(0.495249\pi\)
\(68\) 92.8655i 1.36567i
\(69\) 19.8997i 0.288402i
\(70\) −176.000 −2.51429
\(71\) − 59.6992i − 0.840834i −0.907331 0.420417i \(-0.861884\pi\)
0.907331 0.420417i \(-0.138116\pi\)
\(72\) 89.5489i 1.24373i
\(73\) 74.0000 1.01370 0.506849 0.862035i \(-0.330810\pi\)
0.506849 + 0.862035i \(0.330810\pi\)
\(74\) − 99.4987i − 1.34458i
\(75\) −57.0000 −0.760000
\(76\) −42.0000 −0.552632
\(77\) 0 0
\(78\) 39.7995i 0.510250i
\(79\) 40.0000 0.506329 0.253165 0.967423i \(-0.418529\pi\)
0.253165 + 0.967423i \(0.418529\pi\)
\(80\) − 33.1662i − 0.414578i
\(81\) 81.0000 1.00000
\(82\) −44.0000 −0.536585
\(83\) 39.7995i 0.479512i 0.970833 + 0.239756i \(0.0770674\pi\)
−0.970833 + 0.239756i \(0.922933\pi\)
\(84\) −168.000 −2.00000
\(85\) −88.0000 −1.03529
\(86\) 139.298i 1.61975i
\(87\) 119.398i 1.37240i
\(88\) 0 0
\(89\) − 119.398i − 1.34156i −0.741658 0.670778i \(-0.765960\pi\)
0.741658 0.670778i \(-0.234040\pi\)
\(90\) −198.000 −2.20000
\(91\) −32.0000 −0.351648
\(92\) − 46.4327i − 0.504704i
\(93\) −78.0000 −0.838710
\(94\) 286.000 3.04255
\(95\) − 39.7995i − 0.418942i
\(96\) 69.6491i 0.725512i
\(97\) 62.0000 0.639175 0.319588 0.947557i \(-0.396456\pi\)
0.319588 + 0.947557i \(0.396456\pi\)
\(98\) − 49.7494i − 0.507647i
\(99\) 0 0
\(100\) 133.000 1.33000
\(101\) − 106.132i − 1.05081i −0.850852 0.525406i \(-0.823913\pi\)
0.850852 0.525406i \(-0.176087\pi\)
\(102\) −132.000 −1.29412
\(103\) 74.0000 0.718447 0.359223 0.933252i \(-0.383042\pi\)
0.359223 + 0.933252i \(0.383042\pi\)
\(104\) − 39.7995i − 0.382687i
\(105\) − 159.198i − 1.51617i
\(106\) −198.000 −1.86792
\(107\) 39.7995i 0.371958i 0.982554 + 0.185979i \(0.0595456\pi\)
−0.982554 + 0.185979i \(0.940454\pi\)
\(108\) −189.000 −1.75000
\(109\) 200.000 1.83486 0.917431 0.397894i \(-0.130259\pi\)
0.917431 + 0.397894i \(0.130259\pi\)
\(110\) 0 0
\(111\) 90.0000 0.810811
\(112\) 40.0000 0.357143
\(113\) − 39.7995i − 0.352208i −0.984372 0.176104i \(-0.943651\pi\)
0.984372 0.176104i \(-0.0563495\pi\)
\(114\) − 59.6992i − 0.523678i
\(115\) 44.0000 0.382609
\(116\) − 278.596i − 2.40169i
\(117\) −36.0000 −0.307692
\(118\) 220.000 1.86441
\(119\) − 106.132i − 0.891865i
\(120\) 198.000 1.65000
\(121\) 0 0
\(122\) 39.7995i 0.326225i
\(123\) − 39.7995i − 0.323573i
\(124\) 182.000 1.46774
\(125\) − 39.7995i − 0.318396i
\(126\) − 238.797i − 1.89521i
\(127\) −188.000 −1.48031 −0.740157 0.672434i \(-0.765249\pi\)
−0.740157 + 0.672434i \(0.765249\pi\)
\(128\) − 228.847i − 1.78787i
\(129\) −126.000 −0.976744
\(130\) 88.0000 0.676923
\(131\) − 39.7995i − 0.303813i −0.988395 0.151906i \(-0.951459\pi\)
0.988395 0.151906i \(-0.0485413\pi\)
\(132\) 0 0
\(133\) 48.0000 0.360902
\(134\) − 6.63325i − 0.0495019i
\(135\) − 179.098i − 1.32665i
\(136\) 132.000 0.970588
\(137\) 106.132i 0.774686i 0.921936 + 0.387343i \(0.126607\pi\)
−0.921936 + 0.387343i \(0.873393\pi\)
\(138\) 66.0000 0.478261
\(139\) 74.0000 0.532374 0.266187 0.963921i \(-0.414236\pi\)
0.266187 + 0.963921i \(0.414236\pi\)
\(140\) 371.462i 2.65330i
\(141\) 258.697i 1.83473i
\(142\) −198.000 −1.39437
\(143\) 0 0
\(144\) 45.0000 0.312500
\(145\) 264.000 1.82069
\(146\) − 245.430i − 1.68103i
\(147\) 45.0000 0.306122
\(148\) −210.000 −1.41892
\(149\) 92.8655i 0.623258i 0.950204 + 0.311629i \(0.100875\pi\)
−0.950204 + 0.311629i \(0.899125\pi\)
\(150\) 189.048i 1.26032i
\(151\) 160.000 1.05960 0.529801 0.848122i \(-0.322266\pi\)
0.529801 + 0.848122i \(0.322266\pi\)
\(152\) 59.6992i 0.392758i
\(153\) − 119.398i − 0.780382i
\(154\) 0 0
\(155\) 172.464i 1.11267i
\(156\) 84.0000 0.538462
\(157\) 182.000 1.15924 0.579618 0.814888i \(-0.303202\pi\)
0.579618 + 0.814888i \(0.303202\pi\)
\(158\) − 132.665i − 0.839652i
\(159\) − 179.098i − 1.12640i
\(160\) 154.000 0.962500
\(161\) 53.0660i 0.329602i
\(162\) − 268.647i − 1.65831i
\(163\) −290.000 −1.77914 −0.889571 0.456798i \(-0.848996\pi\)
−0.889571 + 0.456798i \(0.848996\pi\)
\(164\) 92.8655i 0.566253i
\(165\) 0 0
\(166\) 132.000 0.795181
\(167\) 238.797i 1.42992i 0.699164 + 0.714961i \(0.253556\pi\)
−0.699164 + 0.714961i \(0.746444\pi\)
\(168\) 238.797i 1.42141i
\(169\) −153.000 −0.905325
\(170\) 291.863i 1.71684i
\(171\) 54.0000 0.315789
\(172\) 294.000 1.70930
\(173\) − 198.997i − 1.15027i −0.818057 0.575137i \(-0.804949\pi\)
0.818057 0.575137i \(-0.195051\pi\)
\(174\) 396.000 2.27586
\(175\) −152.000 −0.868571
\(176\) 0 0
\(177\) 198.997i 1.12428i
\(178\) −396.000 −2.22472
\(179\) 198.997i 1.11172i 0.831277 + 0.555859i \(0.187611\pi\)
−0.831277 + 0.555859i \(0.812389\pi\)
\(180\) 417.895i 2.32164i
\(181\) 10.0000 0.0552486 0.0276243 0.999618i \(-0.491206\pi\)
0.0276243 + 0.999618i \(0.491206\pi\)
\(182\) 106.132i 0.583143i
\(183\) −36.0000 −0.196721
\(184\) −66.0000 −0.358696
\(185\) − 198.997i − 1.07566i
\(186\) 258.697i 1.39084i
\(187\) 0 0
\(188\) − 603.626i − 3.21078i
\(189\) 216.000 1.14286
\(190\) −132.000 −0.694737
\(191\) − 112.765i − 0.590394i −0.955436 0.295197i \(-0.904615\pi\)
0.955436 0.295197i \(-0.0953853\pi\)
\(192\) 291.000 1.51562
\(193\) −298.000 −1.54404 −0.772021 0.635597i \(-0.780754\pi\)
−0.772021 + 0.635597i \(0.780754\pi\)
\(194\) − 205.631i − 1.05995i
\(195\) 79.5990i 0.408200i
\(196\) −105.000 −0.535714
\(197\) − 132.665i − 0.673426i −0.941607 0.336713i \(-0.890685\pi\)
0.941607 0.336713i \(-0.109315\pi\)
\(198\) 0 0
\(199\) −42.0000 −0.211055 −0.105528 0.994416i \(-0.533653\pi\)
−0.105528 + 0.994416i \(0.533653\pi\)
\(200\) − 189.048i − 0.945238i
\(201\) 6.00000 0.0298507
\(202\) −352.000 −1.74257
\(203\) 318.396i 1.56845i
\(204\) 278.596i 1.36567i
\(205\) −88.0000 −0.429268
\(206\) − 245.430i − 1.19141i
\(207\) 59.6992i 0.288402i
\(208\) −20.0000 −0.0961538
\(209\) 0 0
\(210\) −528.000 −2.51429
\(211\) −246.000 −1.16588 −0.582938 0.812516i \(-0.698097\pi\)
−0.582938 + 0.812516i \(0.698097\pi\)
\(212\) 417.895i 1.97120i
\(213\) − 179.098i − 0.840834i
\(214\) 132.000 0.616822
\(215\) 278.596i 1.29580i
\(216\) 268.647i 1.24373i
\(217\) −208.000 −0.958525
\(218\) − 663.325i − 3.04278i
\(219\) 222.000 1.01370
\(220\) 0 0
\(221\) 53.0660i 0.240118i
\(222\) − 298.496i − 1.34458i
\(223\) −302.000 −1.35426 −0.677130 0.735863i \(-0.736777\pi\)
−0.677130 + 0.735863i \(0.736777\pi\)
\(224\) 185.731i 0.829156i
\(225\) −171.000 −0.760000
\(226\) −132.000 −0.584071
\(227\) 198.997i 0.876641i 0.898819 + 0.438320i \(0.144426\pi\)
−0.898819 + 0.438320i \(0.855574\pi\)
\(228\) −126.000 −0.552632
\(229\) 150.000 0.655022 0.327511 0.944847i \(-0.393790\pi\)
0.327511 + 0.944847i \(0.393790\pi\)
\(230\) − 145.931i − 0.634485i
\(231\) 0 0
\(232\) −396.000 −1.70690
\(233\) 437.794i 1.87895i 0.342623 + 0.939473i \(0.388685\pi\)
−0.342623 + 0.939473i \(0.611315\pi\)
\(234\) 119.398i 0.510250i
\(235\) 572.000 2.43404
\(236\) − 464.327i − 1.96749i
\(237\) 120.000 0.506329
\(238\) −352.000 −1.47899
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) − 99.4987i − 0.414578i
\(241\) −130.000 −0.539419 −0.269710 0.962942i \(-0.586928\pi\)
−0.269710 + 0.962942i \(0.586928\pi\)
\(242\) 0 0
\(243\) 243.000 1.00000
\(244\) 84.0000 0.344262
\(245\) − 99.4987i − 0.406117i
\(246\) −132.000 −0.536585
\(247\) −24.0000 −0.0971660
\(248\) − 258.697i − 1.04313i
\(249\) 119.398i 0.479512i
\(250\) −132.000 −0.528000
\(251\) 291.863i 1.16280i 0.813618 + 0.581400i \(0.197495\pi\)
−0.813618 + 0.581400i \(0.802505\pi\)
\(252\) −504.000 −2.00000
\(253\) 0 0
\(254\) 623.525i 2.45482i
\(255\) −264.000 −1.03529
\(256\) −371.000 −1.44922
\(257\) 490.860i 1.90996i 0.296665 + 0.954981i \(0.404125\pi\)
−0.296665 + 0.954981i \(0.595875\pi\)
\(258\) 417.895i 1.61975i
\(259\) 240.000 0.926641
\(260\) − 185.731i − 0.714350i
\(261\) 358.195i 1.37240i
\(262\) −132.000 −0.503817
\(263\) − 225.530i − 0.857530i −0.903416 0.428765i \(-0.858949\pi\)
0.903416 0.428765i \(-0.141051\pi\)
\(264\) 0 0
\(265\) −396.000 −1.49434
\(266\) − 159.198i − 0.598489i
\(267\) − 358.195i − 1.34156i
\(268\) −14.0000 −0.0522388
\(269\) 72.9657i 0.271248i 0.990760 + 0.135624i \(0.0433039\pi\)
−0.990760 + 0.135624i \(0.956696\pi\)
\(270\) −594.000 −2.20000
\(271\) 448.000 1.65314 0.826568 0.562836i \(-0.190290\pi\)
0.826568 + 0.562836i \(0.190290\pi\)
\(272\) − 66.3325i − 0.243869i
\(273\) −96.0000 −0.351648
\(274\) 352.000 1.28467
\(275\) 0 0
\(276\) − 139.298i − 0.504704i
\(277\) 260.000 0.938628 0.469314 0.883031i \(-0.344501\pi\)
0.469314 + 0.883031i \(0.344501\pi\)
\(278\) − 245.430i − 0.882843i
\(279\) −234.000 −0.838710
\(280\) 528.000 1.88571
\(281\) 198.997i 0.708176i 0.935212 + 0.354088i \(0.115209\pi\)
−0.935212 + 0.354088i \(0.884791\pi\)
\(282\) 858.000 3.04255
\(283\) 50.0000 0.176678 0.0883392 0.996090i \(-0.471844\pi\)
0.0883392 + 0.996090i \(0.471844\pi\)
\(284\) 417.895i 1.47146i
\(285\) − 119.398i − 0.418942i
\(286\) 0 0
\(287\) − 106.132i − 0.369798i
\(288\) 208.947i 0.725512i
\(289\) 113.000 0.391003
\(290\) − 875.589i − 3.01927i
\(291\) 186.000 0.639175
\(292\) −518.000 −1.77397
\(293\) − 477.594i − 1.63001i −0.579451 0.815007i \(-0.696733\pi\)
0.579451 0.815007i \(-0.303267\pi\)
\(294\) − 149.248i − 0.507647i
\(295\) 440.000 1.49153
\(296\) 298.496i 1.00843i
\(297\) 0 0
\(298\) 308.000 1.03356
\(299\) − 26.5330i − 0.0887391i
\(300\) 399.000 1.33000
\(301\) −336.000 −1.11628
\(302\) − 530.660i − 1.75715i
\(303\) − 318.396i − 1.05081i
\(304\) 30.0000 0.0986842
\(305\) 79.5990i 0.260980i
\(306\) −396.000 −1.29412
\(307\) −86.0000 −0.280130 −0.140065 0.990142i \(-0.544731\pi\)
−0.140065 + 0.990142i \(0.544731\pi\)
\(308\) 0 0
\(309\) 222.000 0.718447
\(310\) 572.000 1.84516
\(311\) 19.8997i 0.0639863i 0.999488 + 0.0319932i \(0.0101855\pi\)
−0.999488 + 0.0319932i \(0.989815\pi\)
\(312\) − 119.398i − 0.382687i
\(313\) 98.0000 0.313099 0.156550 0.987670i \(-0.449963\pi\)
0.156550 + 0.987670i \(0.449963\pi\)
\(314\) − 603.626i − 1.92237i
\(315\) − 477.594i − 1.51617i
\(316\) −280.000 −0.886076
\(317\) 311.763i 0.983479i 0.870742 + 0.491739i \(0.163639\pi\)
−0.870742 + 0.491739i \(0.836361\pi\)
\(318\) −594.000 −1.86792
\(319\) 0 0
\(320\) − 643.425i − 2.01070i
\(321\) 119.398i 0.371958i
\(322\) 176.000 0.546584
\(323\) − 79.5990i − 0.246437i
\(324\) −567.000 −1.75000
\(325\) 76.0000 0.233846
\(326\) 961.821i 2.95037i
\(327\) 600.000 1.83486
\(328\) 132.000 0.402439
\(329\) 689.858i 2.09683i
\(330\) 0 0
\(331\) −218.000 −0.658610 −0.329305 0.944224i \(-0.606814\pi\)
−0.329305 + 0.944224i \(0.606814\pi\)
\(332\) − 278.596i − 0.839146i
\(333\) 270.000 0.810811
\(334\) 792.000 2.37126
\(335\) − 13.2665i − 0.0396015i
\(336\) 120.000 0.357143
\(337\) −278.000 −0.824926 −0.412463 0.910974i \(-0.635331\pi\)
−0.412463 + 0.910974i \(0.635331\pi\)
\(338\) 507.444i 1.50131i
\(339\) − 119.398i − 0.352208i
\(340\) 616.000 1.81176
\(341\) 0 0
\(342\) − 179.098i − 0.523678i
\(343\) −272.000 −0.793003
\(344\) − 417.895i − 1.21481i
\(345\) 132.000 0.382609
\(346\) −660.000 −1.90751
\(347\) − 331.662i − 0.955800i −0.878414 0.477900i \(-0.841398\pi\)
0.878414 0.477900i \(-0.158602\pi\)
\(348\) − 835.789i − 2.40169i
\(349\) −324.000 −0.928367 −0.464183 0.885739i \(-0.653652\pi\)
−0.464183 + 0.885739i \(0.653652\pi\)
\(350\) 504.127i 1.44036i
\(351\) −108.000 −0.307692
\(352\) 0 0
\(353\) 397.995i 1.12746i 0.825958 + 0.563732i \(0.190635\pi\)
−0.825958 + 0.563732i \(0.809365\pi\)
\(354\) 660.000 1.86441
\(355\) −396.000 −1.11549
\(356\) 835.789i 2.34772i
\(357\) − 318.396i − 0.891865i
\(358\) 660.000 1.84358
\(359\) 305.129i 0.849943i 0.905207 + 0.424971i \(0.139716\pi\)
−0.905207 + 0.424971i \(0.860284\pi\)
\(360\) 594.000 1.65000
\(361\) −325.000 −0.900277
\(362\) − 33.1662i − 0.0916195i
\(363\) 0 0
\(364\) 224.000 0.615385
\(365\) − 490.860i − 1.34482i
\(366\) 119.398i 0.326225i
\(367\) −278.000 −0.757493 −0.378747 0.925500i \(-0.623645\pi\)
−0.378747 + 0.925500i \(0.623645\pi\)
\(368\) 33.1662i 0.0901257i
\(369\) − 119.398i − 0.323573i
\(370\) −660.000 −1.78378
\(371\) − 477.594i − 1.28732i
\(372\) 546.000 1.46774
\(373\) 68.0000 0.182306 0.0911528 0.995837i \(-0.470945\pi\)
0.0911528 + 0.995837i \(0.470945\pi\)
\(374\) 0 0
\(375\) − 119.398i − 0.318396i
\(376\) −858.000 −2.28191
\(377\) − 159.198i − 0.422276i
\(378\) − 716.391i − 1.89521i
\(379\) 670.000 1.76781 0.883905 0.467666i \(-0.154905\pi\)
0.883905 + 0.467666i \(0.154905\pi\)
\(380\) 278.596i 0.733149i
\(381\) −564.000 −1.48031
\(382\) −374.000 −0.979058
\(383\) − 33.1662i − 0.0865959i −0.999062 0.0432980i \(-0.986214\pi\)
0.999062 0.0432980i \(-0.0137865\pi\)
\(384\) − 686.541i − 1.78787i
\(385\) 0 0
\(386\) 988.354i 2.56050i
\(387\) −378.000 −0.976744
\(388\) −434.000 −1.11856
\(389\) − 417.895i − 1.07428i −0.843493 0.537140i \(-0.819505\pi\)
0.843493 0.537140i \(-0.180495\pi\)
\(390\) 264.000 0.676923
\(391\) 88.0000 0.225064
\(392\) 149.248i 0.380735i
\(393\) − 119.398i − 0.303813i
\(394\) −440.000 −1.11675
\(395\) − 265.330i − 0.671721i
\(396\) 0 0
\(397\) −86.0000 −0.216625 −0.108312 0.994117i \(-0.534545\pi\)
−0.108312 + 0.994117i \(0.534545\pi\)
\(398\) 139.298i 0.349996i
\(399\) 144.000 0.360902
\(400\) −95.0000 −0.237500
\(401\) − 252.063i − 0.628587i −0.949326 0.314294i \(-0.898232\pi\)
0.949326 0.314294i \(-0.101768\pi\)
\(402\) − 19.8997i − 0.0495019i
\(403\) 104.000 0.258065
\(404\) 742.924i 1.83892i
\(405\) − 537.293i − 1.32665i
\(406\) 1056.00 2.60099
\(407\) 0 0
\(408\) 396.000 0.970588
\(409\) −510.000 −1.24694 −0.623472 0.781846i \(-0.714278\pi\)
−0.623472 + 0.781846i \(0.714278\pi\)
\(410\) 291.863i 0.711861i
\(411\) 318.396i 0.774686i
\(412\) −518.000 −1.25728
\(413\) 530.660i 1.28489i
\(414\) 198.000 0.478261
\(415\) 264.000 0.636145
\(416\) − 92.8655i − 0.223234i
\(417\) 222.000 0.532374
\(418\) 0 0
\(419\) − 530.660i − 1.26649i −0.773951 0.633246i \(-0.781722\pi\)
0.773951 0.633246i \(-0.218278\pi\)
\(420\) 1114.39i 2.65330i
\(421\) −170.000 −0.403800 −0.201900 0.979406i \(-0.564712\pi\)
−0.201900 + 0.979406i \(0.564712\pi\)
\(422\) 815.890i 1.93339i
\(423\) 776.090i 1.83473i
\(424\) 594.000 1.40094
\(425\) 252.063i 0.593091i
\(426\) −594.000 −1.39437
\(427\) −96.0000 −0.224824
\(428\) − 278.596i − 0.650926i
\(429\) 0 0
\(430\) 924.000 2.14884
\(431\) − 278.596i − 0.646396i −0.946331 0.323198i \(-0.895242\pi\)
0.946331 0.323198i \(-0.104758\pi\)
\(432\) 135.000 0.312500
\(433\) −542.000 −1.25173 −0.625866 0.779931i \(-0.715254\pi\)
−0.625866 + 0.779931i \(0.715254\pi\)
\(434\) 689.858i 1.58953i
\(435\) 792.000 1.82069
\(436\) −1400.00 −3.21101
\(437\) 39.7995i 0.0910744i
\(438\) − 736.291i − 1.68103i
\(439\) −328.000 −0.747153 −0.373576 0.927599i \(-0.621869\pi\)
−0.373576 + 0.927599i \(0.621869\pi\)
\(440\) 0 0
\(441\) 135.000 0.306122
\(442\) 176.000 0.398190
\(443\) 132.665i 0.299470i 0.988726 + 0.149735i \(0.0478420\pi\)
−0.988726 + 0.149735i \(0.952158\pi\)
\(444\) −630.000 −1.41892
\(445\) −792.000 −1.77978
\(446\) 1001.62i 2.24579i
\(447\) 278.596i 0.623258i
\(448\) 776.000 1.73214
\(449\) − 451.061i − 1.00459i −0.864696 0.502295i \(-0.832489\pi\)
0.864696 0.502295i \(-0.167511\pi\)
\(450\) 567.143i 1.26032i
\(451\) 0 0
\(452\) 278.596i 0.616364i
\(453\) 480.000 1.05960
\(454\) 660.000 1.45374
\(455\) 212.264i 0.466514i
\(456\) 179.098i 0.392758i
\(457\) −342.000 −0.748359 −0.374179 0.927356i \(-0.622076\pi\)
−0.374179 + 0.927356i \(0.622076\pi\)
\(458\) − 497.494i − 1.08623i
\(459\) − 358.195i − 0.780382i
\(460\) −308.000 −0.669565
\(461\) 79.5990i 0.172666i 0.996266 + 0.0863330i \(0.0275149\pi\)
−0.996266 + 0.0863330i \(0.972485\pi\)
\(462\) 0 0
\(463\) −86.0000 −0.185745 −0.0928726 0.995678i \(-0.529605\pi\)
−0.0928726 + 0.995678i \(0.529605\pi\)
\(464\) 198.997i 0.428874i
\(465\) 517.393i 1.11267i
\(466\) 1452.00 3.11588
\(467\) − 596.992i − 1.27836i −0.769059 0.639178i \(-0.779275\pi\)
0.769059 0.639178i \(-0.220725\pi\)
\(468\) 252.000 0.538462
\(469\) 16.0000 0.0341151
\(470\) − 1897.11i − 4.03640i
\(471\) 546.000 1.15924
\(472\) −660.000 −1.39831
\(473\) 0 0
\(474\) − 397.995i − 0.839652i
\(475\) −114.000 −0.240000
\(476\) 742.924i 1.56076i
\(477\) − 537.293i − 1.12640i
\(478\) 0 0
\(479\) − 225.530i − 0.470836i −0.971894 0.235418i \(-0.924354\pi\)
0.971894 0.235418i \(-0.0756459\pi\)
\(480\) 462.000 0.962500
\(481\) −120.000 −0.249480
\(482\) 431.161i 0.894525i
\(483\) 159.198i 0.329602i
\(484\) 0 0
\(485\) − 411.261i − 0.847962i
\(486\) − 805.940i − 1.65831i
\(487\) 446.000 0.915811 0.457906 0.889001i \(-0.348600\pi\)
0.457906 + 0.889001i \(0.348600\pi\)
\(488\) − 119.398i − 0.244669i
\(489\) −870.000 −1.77914
\(490\) −330.000 −0.673469
\(491\) 703.124i 1.43203i 0.698087 + 0.716013i \(0.254035\pi\)
−0.698087 + 0.716013i \(0.745965\pi\)
\(492\) 278.596i 0.566253i
\(493\) 528.000 1.07099
\(494\) 79.5990i 0.161132i
\(495\) 0 0
\(496\) −130.000 −0.262097
\(497\) − 477.594i − 0.960954i
\(498\) 396.000 0.795181
\(499\) −58.0000 −0.116232 −0.0581162 0.998310i \(-0.518509\pi\)
−0.0581162 + 0.998310i \(0.518509\pi\)
\(500\) 278.596i 0.557193i
\(501\) 716.391i 1.42992i
\(502\) 968.000 1.92829
\(503\) 504.127i 1.00224i 0.865378 + 0.501120i \(0.167079\pi\)
−0.865378 + 0.501120i \(0.832921\pi\)
\(504\) 716.391i 1.42141i
\(505\) −704.000 −1.39406
\(506\) 0 0
\(507\) −459.000 −0.905325
\(508\) 1316.00 2.59055
\(509\) 736.291i 1.44654i 0.690563 + 0.723272i \(0.257363\pi\)
−0.690563 + 0.723272i \(0.742637\pi\)
\(510\) 875.589i 1.71684i
\(511\) 592.000 1.15851
\(512\) 315.079i 0.615389i
\(513\) 162.000 0.315789
\(514\) 1628.00 3.16732
\(515\) − 490.860i − 0.953127i
\(516\) 882.000 1.70930
\(517\) 0 0
\(518\) − 795.990i − 1.53666i
\(519\) − 596.992i − 1.15027i
\(520\) −264.000 −0.507692
\(521\) 278.596i 0.534734i 0.963595 + 0.267367i \(0.0861536\pi\)
−0.963595 + 0.267367i \(0.913846\pi\)
\(522\) 1188.00 2.27586
\(523\) 142.000 0.271511 0.135755 0.990742i \(-0.456654\pi\)
0.135755 + 0.990742i \(0.456654\pi\)
\(524\) 278.596i 0.531673i
\(525\) −456.000 −0.868571
\(526\) −748.000 −1.42205
\(527\) 344.929i 0.654514i
\(528\) 0 0
\(529\) 485.000 0.916824
\(530\) 1313.38i 2.47808i
\(531\) 596.992i 1.12428i
\(532\) −336.000 −0.631579
\(533\) 53.0660i 0.0995610i
\(534\) −1188.00 −2.22472
\(535\) 264.000 0.493458
\(536\) 19.8997i 0.0371264i
\(537\) 596.992i 1.11172i
\(538\) 242.000 0.449814
\(539\) 0 0
\(540\) 1253.68i 2.32164i
\(541\) −400.000 −0.739372 −0.369686 0.929157i \(-0.620535\pi\)
−0.369686 + 0.929157i \(0.620535\pi\)
\(542\) − 1485.85i − 2.74142i
\(543\) 30.0000 0.0552486
\(544\) 308.000 0.566176
\(545\) − 1326.65i − 2.43422i
\(546\) 318.396i 0.583143i
\(547\) −170.000 −0.310786 −0.155393 0.987853i \(-0.549664\pi\)
−0.155393 + 0.987853i \(0.549664\pi\)
\(548\) − 742.924i − 1.35570i
\(549\) −108.000 −0.196721
\(550\) 0 0
\(551\) 238.797i 0.433388i
\(552\) −198.000 −0.358696
\(553\) 320.000 0.578662
\(554\) − 862.322i − 1.55654i
\(555\) − 596.992i − 1.07566i
\(556\) −518.000 −0.931655
\(557\) − 849.056i − 1.52434i −0.647378 0.762169i \(-0.724135\pi\)
0.647378 0.762169i \(-0.275865\pi\)
\(558\) 776.090i 1.39084i
\(559\) 168.000 0.300537
\(560\) − 265.330i − 0.473804i
\(561\) 0 0
\(562\) 660.000 1.17438
\(563\) 703.124i 1.24889i 0.781069 + 0.624444i \(0.214675\pi\)
−0.781069 + 0.624444i \(0.785325\pi\)
\(564\) − 1810.88i − 3.21078i
\(565\) −264.000 −0.467257
\(566\) − 165.831i − 0.292988i
\(567\) 648.000 1.14286
\(568\) 594.000 1.04577
\(569\) 119.398i 0.209839i 0.994481 + 0.104920i \(0.0334585\pi\)
−0.994481 + 0.104920i \(0.966541\pi\)
\(570\) −396.000 −0.694737
\(571\) 706.000 1.23643 0.618214 0.786010i \(-0.287857\pi\)
0.618214 + 0.786010i \(0.287857\pi\)
\(572\) 0 0
\(573\) − 338.296i − 0.590394i
\(574\) −352.000 −0.613240
\(575\) − 126.032i − 0.219186i
\(576\) 873.000 1.51562
\(577\) −738.000 −1.27903 −0.639515 0.768779i \(-0.720865\pi\)
−0.639515 + 0.768779i \(0.720865\pi\)
\(578\) − 374.779i − 0.648406i
\(579\) −894.000 −1.54404
\(580\) −1848.00 −3.18621
\(581\) 318.396i 0.548014i
\(582\) − 616.892i − 1.05995i
\(583\) 0 0
\(584\) 736.291i 1.26077i
\(585\) 238.797i 0.408200i
\(586\) −1584.00 −2.70307
\(587\) − 941.921i − 1.60464i −0.596897 0.802318i \(-0.703600\pi\)
0.596897 0.802318i \(-0.296400\pi\)
\(588\) −315.000 −0.535714
\(589\) −156.000 −0.264856
\(590\) − 1459.31i − 2.47342i
\(591\) − 397.995i − 0.673426i
\(592\) 150.000 0.253378
\(593\) − 543.926i − 0.917245i −0.888631 0.458623i \(-0.848343\pi\)
0.888631 0.458623i \(-0.151657\pi\)
\(594\) 0 0
\(595\) −704.000 −1.18319
\(596\) − 650.058i − 1.09070i
\(597\) −126.000 −0.211055
\(598\) −88.0000 −0.147157
\(599\) − 46.4327i − 0.0775171i −0.999249 0.0387586i \(-0.987660\pi\)
0.999249 0.0387586i \(-0.0123403\pi\)
\(600\) − 567.143i − 0.945238i
\(601\) −542.000 −0.901830 −0.450915 0.892567i \(-0.648902\pi\)
−0.450915 + 0.892567i \(0.648902\pi\)
\(602\) 1114.39i 1.85114i
\(603\) 18.0000 0.0298507
\(604\) −1120.00 −1.85430
\(605\) 0 0
\(606\) −1056.00 −1.74257
\(607\) 700.000 1.15321 0.576606 0.817022i \(-0.304377\pi\)
0.576606 + 0.817022i \(0.304377\pi\)
\(608\) 139.298i 0.229109i
\(609\) 955.188i 1.56845i
\(610\) 264.000 0.432787
\(611\) − 344.929i − 0.564532i
\(612\) 835.789i 1.36567i
\(613\) −764.000 −1.24633 −0.623165 0.782091i \(-0.714153\pi\)
−0.623165 + 0.782091i \(0.714153\pi\)
\(614\) 285.230i 0.464544i
\(615\) −264.000 −0.429268
\(616\) 0 0
\(617\) 39.7995i 0.0645049i 0.999480 + 0.0322524i \(0.0102680\pi\)
−0.999480 + 0.0322524i \(0.989732\pi\)
\(618\) − 736.291i − 1.19141i
\(619\) −742.000 −1.19871 −0.599354 0.800484i \(-0.704576\pi\)
−0.599354 + 0.800484i \(0.704576\pi\)
\(620\) − 1207.25i − 1.94718i
\(621\) 179.098i 0.288402i
\(622\) 66.0000 0.106109
\(623\) − 955.188i − 1.53321i
\(624\) −60.0000 −0.0961538
\(625\) −739.000 −1.18240
\(626\) − 325.029i − 0.519216i
\(627\) 0 0
\(628\) −1274.00 −2.02866
\(629\) − 397.995i − 0.632742i
\(630\) −1584.00 −2.51429
\(631\) −410.000 −0.649762 −0.324881 0.945755i \(-0.605324\pi\)
−0.324881 + 0.945755i \(0.605324\pi\)
\(632\) 397.995i 0.629739i
\(633\) −738.000 −1.16588
\(634\) 1034.00 1.63091
\(635\) 1247.05i 1.96386i
\(636\) 1253.68i 1.97120i
\(637\) −60.0000 −0.0941915
\(638\) 0 0
\(639\) − 537.293i − 0.840834i
\(640\) −1518.00 −2.37188
\(641\) − 756.190i − 1.17970i −0.807511 0.589852i \(-0.799186\pi\)
0.807511 0.589852i \(-0.200814\pi\)
\(642\) 396.000 0.616822
\(643\) 890.000 1.38414 0.692068 0.721832i \(-0.256700\pi\)
0.692068 + 0.721832i \(0.256700\pi\)
\(644\) − 371.462i − 0.576804i
\(645\) 835.789i 1.29580i
\(646\) −264.000 −0.408669
\(647\) 484.227i 0.748419i 0.927344 + 0.374210i \(0.122086\pi\)
−0.927344 + 0.374210i \(0.877914\pi\)
\(648\) 805.940i 1.24373i
\(649\) 0 0
\(650\) − 252.063i − 0.387790i
\(651\) −624.000 −0.958525
\(652\) 2030.00 3.11350
\(653\) 391.362i 0.599329i 0.954045 + 0.299664i \(0.0968747\pi\)
−0.954045 + 0.299664i \(0.903125\pi\)
\(654\) − 1989.97i − 3.04278i
\(655\) −264.000 −0.403053
\(656\) − 66.3325i − 0.101117i
\(657\) 666.000 1.01370
\(658\) 2288.00 3.47720
\(659\) 384.728i 0.583806i 0.956448 + 0.291903i \(0.0942885\pi\)
−0.956448 + 0.291903i \(0.905711\pi\)
\(660\) 0 0
\(661\) −746.000 −1.12859 −0.564297 0.825572i \(-0.690853\pi\)
−0.564297 + 0.825572i \(0.690853\pi\)
\(662\) 723.024i 1.09218i
\(663\) 159.198i 0.240118i
\(664\) −396.000 −0.596386
\(665\) − 318.396i − 0.478791i
\(666\) − 895.489i − 1.34458i
\(667\) −264.000 −0.395802
\(668\) − 1671.58i − 2.50236i
\(669\) −906.000 −1.35426
\(670\) −44.0000 −0.0656716
\(671\) 0 0
\(672\) 557.193i 0.829156i
\(673\) 634.000 0.942051 0.471025 0.882120i \(-0.343884\pi\)
0.471025 + 0.882120i \(0.343884\pi\)
\(674\) 922.022i 1.36798i
\(675\) −513.000 −0.760000
\(676\) 1071.00 1.58432
\(677\) − 795.990i − 1.17576i −0.808948 0.587880i \(-0.799963\pi\)
0.808948 0.587880i \(-0.200037\pi\)
\(678\) −396.000 −0.584071
\(679\) 496.000 0.730486
\(680\) − 875.589i − 1.28763i
\(681\) 596.992i 0.876641i
\(682\) 0 0
\(683\) − 451.061i − 0.660411i −0.943909 0.330206i \(-0.892882\pi\)
0.943909 0.330206i \(-0.107118\pi\)
\(684\) −378.000 −0.552632
\(685\) 704.000 1.02774
\(686\) 902.122i 1.31505i
\(687\) 450.000 0.655022
\(688\) −210.000 −0.305233
\(689\) 238.797i 0.346585i
\(690\) − 437.794i − 0.634485i
\(691\) 458.000 0.662808 0.331404 0.943489i \(-0.392478\pi\)
0.331404 + 0.943489i \(0.392478\pi\)
\(692\) 1392.98i 2.01298i
\(693\) 0 0
\(694\) −1100.00 −1.58501
\(695\) − 490.860i − 0.706274i
\(696\) −1188.00 −1.70690
\(697\) −176.000 −0.252511
\(698\) 1074.59i 1.53952i
\(699\) 1313.38i 1.87895i
\(700\) 1064.00 1.52000
\(701\) − 504.127i − 0.719154i −0.933115 0.359577i \(-0.882921\pi\)
0.933115 0.359577i \(-0.117079\pi\)
\(702\) 358.195i 0.510250i
\(703\) 180.000 0.256046
\(704\) 0 0
\(705\) 1716.00 2.43404
\(706\) 1320.00 1.86969
\(707\) − 849.056i − 1.20093i
\(708\) − 1392.98i − 1.96749i
\(709\) −562.000 −0.792666 −0.396333 0.918107i \(-0.629717\pi\)
−0.396333 + 0.918107i \(0.629717\pi\)
\(710\) 1313.38i 1.84984i
\(711\) 360.000 0.506329
\(712\) 1188.00 1.66854
\(713\) − 172.464i − 0.241886i
\(714\) −1056.00 −1.47899
\(715\) 0 0
\(716\) − 1392.98i − 1.94551i
\(717\) 0 0
\(718\) 1012.00 1.40947
\(719\) 338.296i 0.470509i 0.971934 + 0.235254i \(0.0755923\pi\)
−0.971934 + 0.235254i \(0.924408\pi\)
\(720\) − 298.496i − 0.414578i
\(721\) 592.000 0.821082
\(722\) 1077.90i 1.49294i
\(723\) −390.000 −0.539419
\(724\) −70.0000 −0.0966851
\(725\) − 756.190i − 1.04302i
\(726\) 0 0
\(727\) −42.0000 −0.0577717 −0.0288858 0.999583i \(-0.509196\pi\)
−0.0288858 + 0.999583i \(0.509196\pi\)
\(728\) − 318.396i − 0.437357i
\(729\) 729.000 1.00000
\(730\) −1628.00 −2.23014
\(731\) 557.193i 0.762234i
\(732\) 252.000 0.344262
\(733\) 624.000 0.851296 0.425648 0.904889i \(-0.360046\pi\)
0.425648 + 0.904889i \(0.360046\pi\)
\(734\) 922.022i 1.25616i
\(735\) − 298.496i − 0.406117i
\(736\) −154.000 −0.209239
\(737\) 0 0
\(738\) −396.000 −0.536585
\(739\) −686.000 −0.928281 −0.464141 0.885761i \(-0.653637\pi\)
−0.464141 + 0.885761i \(0.653637\pi\)
\(740\) 1392.98i 1.88241i
\(741\) −72.0000 −0.0971660
\(742\) −1584.00 −2.13477
\(743\) 862.322i 1.16060i 0.814404 + 0.580298i \(0.197064\pi\)
−0.814404 + 0.580298i \(0.802936\pi\)
\(744\) − 776.090i − 1.04313i
\(745\) 616.000 0.826846
\(746\) − 225.530i − 0.302320i
\(747\) 358.195i 0.479512i
\(748\) 0 0
\(749\) 318.396i 0.425095i
\(750\) −396.000 −0.528000
\(751\) 94.0000 0.125166 0.0625832 0.998040i \(-0.480066\pi\)
0.0625832 + 0.998040i \(0.480066\pi\)
\(752\) 431.161i 0.573353i
\(753\) 875.589i 1.16280i
\(754\) −528.000 −0.700265
\(755\) − 1061.32i − 1.40572i
\(756\) −1512.00 −2.00000
\(757\) 1118.00 1.47688 0.738441 0.674318i \(-0.235562\pi\)
0.738441 + 0.674318i \(0.235562\pi\)
\(758\) − 2222.14i − 2.93158i
\(759\) 0 0
\(760\) 396.000 0.521053
\(761\) 1154.19i 1.51667i 0.651865 + 0.758335i \(0.273987\pi\)
−0.651865 + 0.758335i \(0.726013\pi\)
\(762\) 1870.58i 2.45482i
\(763\) 1600.00 2.09699
\(764\) 789.357i 1.03319i
\(765\) −792.000 −1.03529
\(766\) −110.000 −0.143603
\(767\) − 265.330i − 0.345932i
\(768\) −1113.00 −1.44922
\(769\) −1274.00 −1.65670 −0.828349 0.560213i \(-0.810719\pi\)
−0.828349 + 0.560213i \(0.810719\pi\)
\(770\) 0 0
\(771\) 1472.58i 1.90996i
\(772\) 2086.00 2.70207
\(773\) − 935.288i − 1.20995i −0.796246 0.604973i \(-0.793184\pi\)
0.796246 0.604973i \(-0.206816\pi\)
\(774\) 1253.68i 1.61975i
\(775\) 494.000 0.637419
\(776\) 616.892i 0.794964i
\(777\) 720.000 0.926641
\(778\) −1386.00 −1.78149
\(779\) − 79.5990i − 0.102181i
\(780\) − 557.193i − 0.714350i
\(781\) 0 0
\(782\) − 291.863i − 0.373226i
\(783\) 1074.59i 1.37240i
\(784\) 75.0000 0.0956633
\(785\) − 1207.25i − 1.53790i
\(786\) −396.000 −0.503817
\(787\) −298.000 −0.378653 −0.189327 0.981914i \(-0.560630\pi\)
−0.189327 + 0.981914i \(0.560630\pi\)
\(788\) 928.655i 1.17850i
\(789\) − 676.591i − 0.857530i
\(790\) −880.000 −1.11392
\(791\) − 318.396i − 0.402523i
\(792\) 0 0
\(793\) 48.0000 0.0605296
\(794\) 285.230i 0.359231i
\(795\) −1188.00 −1.49434
\(796\) 294.000 0.369347
\(797\) 524.027i 0.657499i 0.944417 + 0.328750i \(0.106627\pi\)
−0.944417 + 0.328750i \(0.893373\pi\)
\(798\) − 477.594i − 0.598489i
\(799\) 1144.00 1.43179
\(800\) − 441.111i − 0.551389i
\(801\) − 1074.59i − 1.34156i
\(802\) −836.000 −1.04239
\(803\) 0 0
\(804\) −42.0000 −0.0522388
\(805\) 352.000 0.437267
\(806\) − 344.929i − 0.427952i
\(807\) 218.897i 0.271248i
\(808\) 1056.00 1.30693
\(809\) 915.388i 1.13151i 0.824575 + 0.565753i \(0.191414\pi\)
−0.824575 + 0.565753i \(0.808586\pi\)
\(810\) −1782.00 −2.20000
\(811\) 182.000 0.224414 0.112207 0.993685i \(-0.464208\pi\)
0.112207 + 0.993685i \(0.464208\pi\)
\(812\) − 2228.77i − 2.74479i
\(813\) 1344.00 1.65314
\(814\) 0 0
\(815\) 1923.64i 2.36030i
\(816\) − 198.997i − 0.243869i
\(817\) −252.000 −0.308446
\(818\) 1691.48i 2.06782i
\(819\) −288.000 −0.351648
\(820\) 616.000 0.751220
\(821\) 822.523i 1.00185i 0.865489 + 0.500927i \(0.167008\pi\)
−0.865489 + 0.500927i \(0.832992\pi\)
\(822\) 1056.00 1.28467
\(823\) −246.000 −0.298906 −0.149453 0.988769i \(-0.547751\pi\)
−0.149453 + 0.988769i \(0.547751\pi\)
\(824\) 736.291i 0.893557i
\(825\) 0 0
\(826\) 1760.00 2.13075
\(827\) − 543.926i − 0.657710i −0.944380 0.328855i \(-0.893337\pi\)
0.944380 0.328855i \(-0.106663\pi\)
\(828\) − 417.895i − 0.504704i
\(829\) 250.000 0.301568 0.150784 0.988567i \(-0.451820\pi\)
0.150784 + 0.988567i \(0.451820\pi\)
\(830\) − 875.589i − 1.05493i
\(831\) 780.000 0.938628
\(832\) −388.000 −0.466346
\(833\) − 198.997i − 0.238893i
\(834\) − 736.291i − 0.882843i
\(835\) 1584.00 1.89701
\(836\) 0 0
\(837\) −702.000 −0.838710
\(838\) −1760.00 −2.10024
\(839\) 470.961i 0.561336i 0.959805 + 0.280668i \(0.0905559\pi\)
−0.959805 + 0.280668i \(0.909444\pi\)
\(840\) 1584.00 1.88571
\(841\) −743.000 −0.883472
\(842\) 563.826i 0.669627i
\(843\) 596.992i 0.708176i
\(844\) 1722.00 2.04028
\(845\) 1014.89i 1.20105i
\(846\) 2574.00 3.04255
\(847\) 0 0
\(848\) − 298.496i − 0.352000i
\(849\) 150.000 0.176678
\(850\) 836.000 0.983529
\(851\) 198.997i 0.233840i
\(852\) 1253.68i 1.47146i
\(853\) −56.0000 −0.0656506 −0.0328253 0.999461i \(-0.510451\pi\)
−0.0328253 + 0.999461i \(0.510451\pi\)
\(854\) 318.396i 0.372829i
\(855\) − 358.195i − 0.418942i
\(856\) −396.000 −0.462617
\(857\) 252.063i 0.294123i 0.989127 + 0.147062i \(0.0469815\pi\)
−0.989127 + 0.147062i \(0.953018\pi\)
\(858\) 0 0
\(859\) 1278.00 1.48778 0.743888 0.668304i \(-0.232979\pi\)
0.743888 + 0.668304i \(0.232979\pi\)
\(860\) − 1950.18i − 2.26765i
\(861\) − 318.396i − 0.369798i
\(862\) −924.000 −1.07193
\(863\) − 1134.29i − 1.31435i −0.753737 0.657176i \(-0.771751\pi\)
0.753737 0.657176i \(-0.228249\pi\)
\(864\) 626.842i 0.725512i
\(865\) −1320.00 −1.52601
\(866\) 1797.61i 2.07576i
\(867\) 339.000 0.391003
\(868\) 1456.00 1.67742
\(869\) 0 0
\(870\) − 2626.77i − 3.01927i
\(871\) −8.00000 −0.00918485
\(872\) 1989.97i 2.28208i
\(873\) 558.000 0.639175
\(874\) 132.000 0.151030
\(875\) − 318.396i − 0.363881i
\(876\) −1554.00 −1.77397
\(877\) −456.000 −0.519954 −0.259977 0.965615i \(-0.583715\pi\)
−0.259977 + 0.965615i \(0.583715\pi\)
\(878\) 1087.85i 1.23901i
\(879\) − 1432.78i − 1.63001i
\(880\) 0 0
\(881\) − 610.259i − 0.692689i −0.938107 0.346344i \(-0.887423\pi\)
0.938107 0.346344i \(-0.112577\pi\)
\(882\) − 447.744i − 0.507647i
\(883\) −1094.00 −1.23896 −0.619479 0.785013i \(-0.712656\pi\)
−0.619479 + 0.785013i \(0.712656\pi\)
\(884\) − 371.462i − 0.420206i
\(885\) 1320.00 1.49153
\(886\) 440.000 0.496614
\(887\) − 145.931i − 0.164523i −0.996611 0.0822613i \(-0.973786\pi\)
0.996611 0.0822613i \(-0.0262142\pi\)
\(888\) 895.489i 1.00843i
\(889\) −1504.00 −1.69179
\(890\) 2626.77i 2.95142i
\(891\) 0 0
\(892\) 2114.00 2.36996
\(893\) 517.393i 0.579388i
\(894\) 924.000 1.03356
\(895\) 1320.00 1.47486
\(896\) − 1830.78i − 2.04328i
\(897\) − 79.5990i − 0.0887391i
\(898\) −1496.00 −1.66592
\(899\) − 1034.79i − 1.15104i
\(900\) 1197.00 1.33000
\(901\) −792.000 −0.879023
\(902\) 0 0
\(903\) −1008.00 −1.11628
\(904\) 396.000 0.438053
\(905\) − 66.3325i − 0.0732956i
\(906\) − 1591.98i − 1.75715i
\(907\) 186.000 0.205072 0.102536 0.994729i \(-0.467304\pi\)
0.102536 + 0.994729i \(0.467304\pi\)
\(908\) − 1392.98i − 1.53412i
\(909\) − 955.188i − 1.05081i
\(910\) 704.000 0.773626
\(911\) 855.689i 0.939286i 0.882857 + 0.469643i \(0.155617\pi\)
−0.882857 + 0.469643i \(0.844383\pi\)
\(912\) 90.0000 0.0986842
\(913\) 0 0
\(914\) 1134.29i 1.24101i
\(915\) 238.797i 0.260980i
\(916\) −1050.00 −1.14629
\(917\) − 318.396i − 0.347215i
\(918\) −1188.00 −1.29412
\(919\) 428.000 0.465724 0.232862 0.972510i \(-0.425191\pi\)
0.232862 + 0.972510i \(0.425191\pi\)
\(920\) 437.794i 0.475864i
\(921\) −258.000 −0.280130
\(922\) 264.000 0.286334
\(923\) 238.797i 0.258718i
\(924\) 0 0
\(925\) −570.000 −0.616216
\(926\) 285.230i 0.308023i
\(927\) 666.000 0.718447
\(928\) −924.000 −0.995690
\(929\) 636.792i 0.685460i 0.939434 + 0.342730i \(0.111352\pi\)
−0.939434 + 0.342730i \(0.888648\pi\)
\(930\) 1716.00 1.84516
\(931\) 90.0000 0.0966702
\(932\) − 3064.56i − 3.28816i
\(933\) 59.6992i 0.0639863i
\(934\) −1980.00 −2.11991
\(935\) 0 0
\(936\) − 358.195i − 0.382687i
\(937\) −290.000 −0.309498 −0.154749 0.987954i \(-0.549457\pi\)
−0.154749 + 0.987954i \(0.549457\pi\)
\(938\) − 53.0660i − 0.0565736i
\(939\) 294.000 0.313099
\(940\) −4004.00 −4.25957
\(941\) − 875.589i − 0.930488i −0.885183 0.465244i \(-0.845967\pi\)
0.885183 0.465244i \(-0.154033\pi\)
\(942\) − 1810.88i − 1.92237i
\(943\) 88.0000 0.0933192
\(944\) 331.662i 0.351337i
\(945\) − 1432.78i − 1.51617i
\(946\) 0 0
\(947\) − 79.5990i − 0.0840538i −0.999116 0.0420269i \(-0.986618\pi\)
0.999116 0.0420269i \(-0.0133815\pi\)
\(948\) −840.000 −0.886076
\(949\) −296.000 −0.311907
\(950\) 378.095i 0.397995i
\(951\) 935.288i 0.983479i
\(952\) 1056.00 1.10924
\(953\) 782.723i 0.821326i 0.911787 + 0.410663i \(0.134703\pi\)
−0.911787 + 0.410663i \(0.865297\pi\)
\(954\) −1782.00 −1.86792
\(955\) −748.000 −0.783246
\(956\) 0 0
\(957\) 0 0
\(958\) −748.000 −0.780793
\(959\) 849.056i 0.885356i
\(960\) − 1930.28i − 2.01070i
\(961\) −285.000 −0.296566
\(962\) 397.995i 0.413716i
\(963\) 358.195i 0.371958i
\(964\) 910.000 0.943983
\(965\) 1976.71i 2.04840i
\(966\) 528.000 0.546584
\(967\) −460.000 −0.475698 −0.237849 0.971302i \(-0.576442\pi\)
−0.237849 + 0.971302i \(0.576442\pi\)
\(968\) 0 0
\(969\) − 238.797i − 0.246437i
\(970\) −1364.00 −1.40619
\(971\) − 1167.45i − 1.20232i −0.799129 0.601160i \(-0.794706\pi\)
0.799129 0.601160i \(-0.205294\pi\)
\(972\) −1701.00 −1.75000
\(973\) 592.000 0.608428
\(974\) − 1479.21i − 1.51870i
\(975\) 228.000 0.233846
\(976\) −60.0000 −0.0614754
\(977\) 1313.38i 1.34430i 0.740414 + 0.672151i \(0.234630\pi\)
−0.740414 + 0.672151i \(0.765370\pi\)
\(978\) 2885.46i 2.95037i
\(979\) 0 0
\(980\) 696.491i 0.710705i
\(981\) 1800.00 1.83486
\(982\) 2332.00 2.37475
\(983\) 417.895i 0.425122i 0.977148 + 0.212561i \(0.0681804\pi\)
−0.977148 + 0.212561i \(0.931820\pi\)
\(984\) 396.000 0.402439
\(985\) −880.000 −0.893401
\(986\) − 1751.18i − 1.77604i
\(987\) 2069.57i 2.09683i
\(988\) 168.000 0.170040
\(989\) − 278.596i − 0.281695i
\(990\) 0 0
\(991\) 838.000 0.845610 0.422805 0.906221i \(-0.361045\pi\)
0.422805 + 0.906221i \(0.361045\pi\)
\(992\) − 603.626i − 0.608494i
\(993\) −654.000 −0.658610
\(994\) −1584.00 −1.59356
\(995\) 278.596i 0.279996i
\(996\) − 835.789i − 0.839146i
\(997\) 52.0000 0.0521565 0.0260782 0.999660i \(-0.491698\pi\)
0.0260782 + 0.999660i \(0.491698\pi\)
\(998\) 192.364i 0.192750i
\(999\) 810.000 0.810811
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 363.3.b.d.122.1 2
3.2 odd 2 inner 363.3.b.d.122.2 2
11.2 odd 10 363.3.h.e.323.2 8
11.3 even 5 363.3.h.d.251.1 8
11.4 even 5 363.3.h.d.269.2 8
11.5 even 5 363.3.h.d.245.2 8
11.6 odd 10 363.3.h.e.245.1 8
11.7 odd 10 363.3.h.e.269.1 8
11.8 odd 10 363.3.h.e.251.2 8
11.9 even 5 363.3.h.d.323.1 8
11.10 odd 2 33.3.b.a.23.2 yes 2
33.2 even 10 363.3.h.e.323.1 8
33.5 odd 10 363.3.h.d.245.1 8
33.8 even 10 363.3.h.e.251.1 8
33.14 odd 10 363.3.h.d.251.2 8
33.17 even 10 363.3.h.e.245.2 8
33.20 odd 10 363.3.h.d.323.2 8
33.26 odd 10 363.3.h.d.269.1 8
33.29 even 10 363.3.h.e.269.2 8
33.32 even 2 33.3.b.a.23.1 2
44.43 even 2 528.3.i.a.353.1 2
132.131 odd 2 528.3.i.a.353.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.3.b.a.23.1 2 33.32 even 2
33.3.b.a.23.2 yes 2 11.10 odd 2
363.3.b.d.122.1 2 1.1 even 1 trivial
363.3.b.d.122.2 2 3.2 odd 2 inner
363.3.h.d.245.1 8 33.5 odd 10
363.3.h.d.245.2 8 11.5 even 5
363.3.h.d.251.1 8 11.3 even 5
363.3.h.d.251.2 8 33.14 odd 10
363.3.h.d.269.1 8 33.26 odd 10
363.3.h.d.269.2 8 11.4 even 5
363.3.h.d.323.1 8 11.9 even 5
363.3.h.d.323.2 8 33.20 odd 10
363.3.h.e.245.1 8 11.6 odd 10
363.3.h.e.245.2 8 33.17 even 10
363.3.h.e.251.1 8 33.8 even 10
363.3.h.e.251.2 8 11.8 odd 10
363.3.h.e.269.1 8 11.7 odd 10
363.3.h.e.269.2 8 33.29 even 10
363.3.h.e.323.1 8 33.2 even 10
363.3.h.e.323.2 8 11.2 odd 10
528.3.i.a.353.1 2 44.43 even 2
528.3.i.a.353.2 2 132.131 odd 2