Properties

Label 1815.2.a.l.1.1
Level $1815$
Weight $2$
Character 1815.1
Self dual yes
Analytic conductor $14.493$
Analytic rank $0$
Dimension $3$
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] = [1815,2,Mod(1,1815)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1815, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1815.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1815.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: \(14.4928479669\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.469.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x + 4 \) 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 \(2.39138\) of defining polynomial
Character \(\chi\) \(=\) 1815.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.39138 q^{2} +1.00000 q^{3} +3.71871 q^{4} +1.00000 q^{5} -2.39138 q^{6} -5.11009 q^{7} -4.11009 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.39138 q^{2} +1.00000 q^{3} +3.71871 q^{4} +1.00000 q^{5} -2.39138 q^{6} -5.11009 q^{7} -4.11009 q^{8} +1.00000 q^{9} -2.39138 q^{10} +3.71871 q^{12} +5.43742 q^{13} +12.2202 q^{14} +1.00000 q^{15} +2.39138 q^{16} +1.32733 q^{17} -2.39138 q^{18} +1.67267 q^{19} +3.71871 q^{20} -5.11009 q^{21} -2.11009 q^{23} -4.11009 q^{24} +1.00000 q^{25} -13.0029 q^{26} +1.00000 q^{27} -19.0029 q^{28} +0.782765 q^{29} -2.39138 q^{30} -4.43742 q^{31} +2.50147 q^{32} -3.17415 q^{34} -5.11009 q^{35} +3.71871 q^{36} -11.3303 q^{37} -4.00000 q^{38} +5.43742 q^{39} -4.11009 q^{40} +11.5655 q^{41} +12.2202 q^{42} -4.78276 q^{43} +1.00000 q^{45} +5.04604 q^{46} +9.45544 q^{47} +2.39138 q^{48} +19.1130 q^{49} -2.39138 q^{50} +1.32733 q^{51} +20.2202 q^{52} -8.23820 q^{53} -2.39138 q^{54} +21.0029 q^{56} +1.67267 q^{57} -1.87189 q^{58} +10.0921 q^{59} +3.71871 q^{60} -0.779816 q^{61} +10.6116 q^{62} -5.11009 q^{63} -10.7647 q^{64} +5.43742 q^{65} -4.45544 q^{67} +4.93594 q^{68} -2.11009 q^{69} +12.2202 q^{70} +12.9109 q^{71} -4.11009 q^{72} +2.32733 q^{73} +27.0950 q^{74} +1.00000 q^{75} +6.22018 q^{76} -13.0029 q^{78} +1.12811 q^{79} +2.39138 q^{80} +1.00000 q^{81} -27.6576 q^{82} -4.00000 q^{83} -19.0029 q^{84} +1.32733 q^{85} +11.4374 q^{86} +0.782765 q^{87} +14.7828 q^{89} -2.39138 q^{90} -27.7857 q^{91} -7.84682 q^{92} -4.43742 q^{93} -22.6116 q^{94} +1.67267 q^{95} +2.50147 q^{96} +4.45544 q^{97} -45.7066 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 3 q^{3} + 5 q^{4} + 3 q^{5} - q^{6} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} + 3 q^{3} + 5 q^{4} + 3 q^{5} - q^{6} - 3 q^{7} + 3 q^{9} - q^{10} + 5 q^{12} + 4 q^{13} + 12 q^{14} + 3 q^{15} + q^{16} + 4 q^{17} - q^{18} + 5 q^{19} + 5 q^{20} - 3 q^{21} + 6 q^{23} + 3 q^{25} - 2 q^{26} + 3 q^{27} - 20 q^{28} - 10 q^{29} - q^{30} - q^{31} - 11 q^{32} + 9 q^{34} - 3 q^{35} + 5 q^{36} + 3 q^{37} - 12 q^{38} + 4 q^{39} + 10 q^{41} + 12 q^{42} - 2 q^{43} + 3 q^{45} + 9 q^{46} + 16 q^{47} + q^{48} + 8 q^{49} - q^{50} + 4 q^{51} + 36 q^{52} - q^{54} + 26 q^{56} + 5 q^{57} - 18 q^{58} + 18 q^{59} + 5 q^{60} - 27 q^{61} + q^{62} - 3 q^{63} - 20 q^{64} + 4 q^{65} - q^{67} + 21 q^{68} + 6 q^{69} + 12 q^{70} + 14 q^{71} + 7 q^{73} + 32 q^{74} + 3 q^{75} - 6 q^{76} - 2 q^{78} - 9 q^{79} + q^{80} + 3 q^{81} - 46 q^{82} - 12 q^{83} - 20 q^{84} + 4 q^{85} + 22 q^{86} - 10 q^{87} + 32 q^{89} - q^{90} - 34 q^{91} - 5 q^{92} - q^{93} - 37 q^{94} + 5 q^{95} - 11 q^{96} + q^{97} - 57 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.39138 −1.69096 −0.845481 0.534005i \(-0.820686\pi\)
−0.845481 + 0.534005i \(0.820686\pi\)
\(3\) 1.00000 0.577350
\(4\) 3.71871 1.85935
\(5\) 1.00000 0.447214
\(6\) −2.39138 −0.976278
\(7\) −5.11009 −1.93143 −0.965717 0.259599i \(-0.916410\pi\)
−0.965717 + 0.259599i \(0.916410\pi\)
\(8\) −4.11009 −1.45314
\(9\) 1.00000 0.333333
\(10\) −2.39138 −0.756222
\(11\) 0 0
\(12\) 3.71871 1.07350
\(13\) 5.43742 1.50807 0.754034 0.656835i \(-0.228105\pi\)
0.754034 + 0.656835i \(0.228105\pi\)
\(14\) 12.2202 3.26598
\(15\) 1.00000 0.258199
\(16\) 2.39138 0.597846
\(17\) 1.32733 0.321924 0.160962 0.986961i \(-0.448540\pi\)
0.160962 + 0.986961i \(0.448540\pi\)
\(18\) −2.39138 −0.563654
\(19\) 1.67267 0.383737 0.191869 0.981421i \(-0.438545\pi\)
0.191869 + 0.981421i \(0.438545\pi\)
\(20\) 3.71871 0.831529
\(21\) −5.11009 −1.11511
\(22\) 0 0
\(23\) −2.11009 −0.439985 −0.219992 0.975502i \(-0.570603\pi\)
−0.219992 + 0.975502i \(0.570603\pi\)
\(24\) −4.11009 −0.838969
\(25\) 1.00000 0.200000
\(26\) −13.0029 −2.55009
\(27\) 1.00000 0.192450
\(28\) −19.0029 −3.59122
\(29\) 0.782765 0.145356 0.0726779 0.997355i \(-0.476845\pi\)
0.0726779 + 0.997355i \(0.476845\pi\)
\(30\) −2.39138 −0.436605
\(31\) −4.43742 −0.796984 −0.398492 0.917172i \(-0.630466\pi\)
−0.398492 + 0.917172i \(0.630466\pi\)
\(32\) 2.50147 0.442202
\(33\) 0 0
\(34\) −3.17415 −0.544362
\(35\) −5.11009 −0.863763
\(36\) 3.71871 0.619785
\(37\) −11.3303 −1.86269 −0.931343 0.364143i \(-0.881362\pi\)
−0.931343 + 0.364143i \(0.881362\pi\)
\(38\) −4.00000 −0.648886
\(39\) 5.43742 0.870684
\(40\) −4.11009 −0.649863
\(41\) 11.5655 1.80623 0.903116 0.429396i \(-0.141274\pi\)
0.903116 + 0.429396i \(0.141274\pi\)
\(42\) 12.2202 1.88562
\(43\) −4.78276 −0.729365 −0.364682 0.931132i \(-0.618822\pi\)
−0.364682 + 0.931132i \(0.618822\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 5.04604 0.743998
\(47\) 9.45544 1.37922 0.689609 0.724182i \(-0.257783\pi\)
0.689609 + 0.724182i \(0.257783\pi\)
\(48\) 2.39138 0.345166
\(49\) 19.1130 2.73043
\(50\) −2.39138 −0.338193
\(51\) 1.32733 0.185863
\(52\) 20.2202 2.80404
\(53\) −8.23820 −1.13160 −0.565802 0.824541i \(-0.691433\pi\)
−0.565802 + 0.824541i \(0.691433\pi\)
\(54\) −2.39138 −0.325426
\(55\) 0 0
\(56\) 21.0029 2.80664
\(57\) 1.67267 0.221551
\(58\) −1.87189 −0.245791
\(59\) 10.0921 1.31388 0.656938 0.753945i \(-0.271851\pi\)
0.656938 + 0.753945i \(0.271851\pi\)
\(60\) 3.71871 0.480083
\(61\) −0.779816 −0.0998452 −0.0499226 0.998753i \(-0.515897\pi\)
−0.0499226 + 0.998753i \(0.515897\pi\)
\(62\) 10.6116 1.34767
\(63\) −5.11009 −0.643811
\(64\) −10.7647 −1.34559
\(65\) 5.43742 0.674429
\(66\) 0 0
\(67\) −4.45544 −0.544318 −0.272159 0.962252i \(-0.587738\pi\)
−0.272159 + 0.962252i \(0.587738\pi\)
\(68\) 4.93594 0.598571
\(69\) −2.11009 −0.254025
\(70\) 12.2202 1.46059
\(71\) 12.9109 1.53224 0.766119 0.642698i \(-0.222185\pi\)
0.766119 + 0.642698i \(0.222185\pi\)
\(72\) −4.11009 −0.484379
\(73\) 2.32733 0.272393 0.136197 0.990682i \(-0.456512\pi\)
0.136197 + 0.990682i \(0.456512\pi\)
\(74\) 27.0950 3.14973
\(75\) 1.00000 0.115470
\(76\) 6.22018 0.713504
\(77\) 0 0
\(78\) −13.0029 −1.47229
\(79\) 1.12811 0.126922 0.0634612 0.997984i \(-0.479786\pi\)
0.0634612 + 0.997984i \(0.479786\pi\)
\(80\) 2.39138 0.267365
\(81\) 1.00000 0.111111
\(82\) −27.6576 −3.05427
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) −19.0029 −2.07339
\(85\) 1.32733 0.143969
\(86\) 11.4374 1.23333
\(87\) 0.782765 0.0839212
\(88\) 0 0
\(89\) 14.7828 1.56697 0.783485 0.621411i \(-0.213440\pi\)
0.783485 + 0.621411i \(0.213440\pi\)
\(90\) −2.39138 −0.252074
\(91\) −27.7857 −2.91273
\(92\) −7.84682 −0.818088
\(93\) −4.43742 −0.460139
\(94\) −22.6116 −2.33221
\(95\) 1.67267 0.171613
\(96\) 2.50147 0.255306
\(97\) 4.45544 0.452381 0.226191 0.974083i \(-0.427373\pi\)
0.226191 + 0.974083i \(0.427373\pi\)
\(98\) −45.7066 −4.61706
\(99\) 0 0
\(100\) 3.71871 0.371871
\(101\) −6.87484 −0.684072 −0.342036 0.939687i \(-0.611116\pi\)
−0.342036 + 0.939687i \(0.611116\pi\)
\(102\) −3.17415 −0.314287
\(103\) 10.5835 1.04283 0.521414 0.853304i \(-0.325405\pi\)
0.521414 + 0.853304i \(0.325405\pi\)
\(104\) −22.3483 −2.19143
\(105\) −5.11009 −0.498694
\(106\) 19.7007 1.91350
\(107\) −1.41940 −0.137219 −0.0686094 0.997644i \(-0.521856\pi\)
−0.0686094 + 0.997644i \(0.521856\pi\)
\(108\) 3.71871 0.357833
\(109\) 15.2022 1.45610 0.728052 0.685522i \(-0.240426\pi\)
0.728052 + 0.685522i \(0.240426\pi\)
\(110\) 0 0
\(111\) −11.3303 −1.07542
\(112\) −12.2202 −1.15470
\(113\) 2.01802 0.189839 0.0949196 0.995485i \(-0.469741\pi\)
0.0949196 + 0.995485i \(0.469741\pi\)
\(114\) −4.00000 −0.374634
\(115\) −2.11009 −0.196767
\(116\) 2.91087 0.270268
\(117\) 5.43742 0.502690
\(118\) −24.1340 −2.22172
\(119\) −6.78276 −0.621775
\(120\) −4.11009 −0.375198
\(121\) 0 0
\(122\) 1.86484 0.168834
\(123\) 11.5655 1.04283
\(124\) −16.5015 −1.48188
\(125\) 1.00000 0.0894427
\(126\) 12.2202 1.08866
\(127\) −9.98493 −0.886019 −0.443010 0.896517i \(-0.646089\pi\)
−0.443010 + 0.896517i \(0.646089\pi\)
\(128\) 20.7397 1.83315
\(129\) −4.78276 −0.421099
\(130\) −13.0029 −1.14043
\(131\) −4.87484 −0.425917 −0.212958 0.977061i \(-0.568310\pi\)
−0.212958 + 0.977061i \(0.568310\pi\)
\(132\) 0 0
\(133\) −8.54751 −0.741163
\(134\) 10.6547 0.920422
\(135\) 1.00000 0.0860663
\(136\) −5.45544 −0.467800
\(137\) 19.7677 1.68887 0.844434 0.535659i \(-0.179937\pi\)
0.844434 + 0.535659i \(0.179937\pi\)
\(138\) 5.04604 0.429547
\(139\) −6.76475 −0.573778 −0.286889 0.957964i \(-0.592621\pi\)
−0.286889 + 0.957964i \(0.592621\pi\)
\(140\) −19.0029 −1.60604
\(141\) 9.45544 0.796291
\(142\) −30.8748 −2.59096
\(143\) 0 0
\(144\) 2.39138 0.199282
\(145\) 0.782765 0.0650051
\(146\) −5.56553 −0.460607
\(147\) 19.1130 1.57642
\(148\) −42.1340 −3.46339
\(149\) 0.128110 0.0104952 0.00524760 0.999986i \(-0.498330\pi\)
0.00524760 + 0.999986i \(0.498330\pi\)
\(150\) −2.39138 −0.195256
\(151\) 5.45544 0.443957 0.221979 0.975052i \(-0.428748\pi\)
0.221979 + 0.975052i \(0.428748\pi\)
\(152\) −6.87484 −0.557623
\(153\) 1.32733 0.107308
\(154\) 0 0
\(155\) −4.43742 −0.356422
\(156\) 20.2202 1.61891
\(157\) 11.9849 0.956502 0.478251 0.878223i \(-0.341271\pi\)
0.478251 + 0.878223i \(0.341271\pi\)
\(158\) −2.69774 −0.214621
\(159\) −8.23820 −0.653332
\(160\) 2.50147 0.197759
\(161\) 10.7828 0.849801
\(162\) −2.39138 −0.187885
\(163\) 9.98493 0.782080 0.391040 0.920374i \(-0.372115\pi\)
0.391040 + 0.920374i \(0.372115\pi\)
\(164\) 43.0088 3.35843
\(165\) 0 0
\(166\) 9.56553 0.742429
\(167\) 7.67562 0.593957 0.296979 0.954884i \(-0.404021\pi\)
0.296979 + 0.954884i \(0.404021\pi\)
\(168\) 21.0029 1.62041
\(169\) 16.5655 1.27427
\(170\) −3.17415 −0.243446
\(171\) 1.67267 0.127912
\(172\) −17.7857 −1.35615
\(173\) −4.69069 −0.356627 −0.178313 0.983974i \(-0.557064\pi\)
−0.178313 + 0.983974i \(0.557064\pi\)
\(174\) −1.87189 −0.141908
\(175\) −5.11009 −0.386287
\(176\) 0 0
\(177\) 10.0921 0.758567
\(178\) −35.3512 −2.64969
\(179\) 11.4374 0.854873 0.427436 0.904045i \(-0.359417\pi\)
0.427436 + 0.904045i \(0.359417\pi\)
\(180\) 3.71871 0.277176
\(181\) −8.11304 −0.603038 −0.301519 0.953460i \(-0.597494\pi\)
−0.301519 + 0.953460i \(0.597494\pi\)
\(182\) 66.4463 4.92532
\(183\) −0.779816 −0.0576456
\(184\) 8.67267 0.639358
\(185\) −11.3303 −0.833018
\(186\) 10.6116 0.778078
\(187\) 0 0
\(188\) 35.1620 2.56445
\(189\) −5.11009 −0.371705
\(190\) −4.00000 −0.290191
\(191\) −1.00295 −0.0725708 −0.0362854 0.999341i \(-0.511553\pi\)
−0.0362854 + 0.999341i \(0.511553\pi\)
\(192\) −10.7647 −0.776879
\(193\) 15.9849 1.15062 0.575310 0.817935i \(-0.304881\pi\)
0.575310 + 0.817935i \(0.304881\pi\)
\(194\) −10.6547 −0.764960
\(195\) 5.43742 0.389382
\(196\) 71.0759 5.07685
\(197\) 16.2202 1.15564 0.577820 0.816164i \(-0.303904\pi\)
0.577820 + 0.816164i \(0.303904\pi\)
\(198\) 0 0
\(199\) −1.34240 −0.0951600 −0.0475800 0.998867i \(-0.515151\pi\)
−0.0475800 + 0.998867i \(0.515151\pi\)
\(200\) −4.11009 −0.290627
\(201\) −4.45544 −0.314262
\(202\) 16.4404 1.15674
\(203\) −4.00000 −0.280745
\(204\) 4.93594 0.345585
\(205\) 11.5655 0.807772
\(206\) −25.3093 −1.76338
\(207\) −2.11009 −0.146662
\(208\) 13.0029 0.901592
\(209\) 0 0
\(210\) 12.2202 0.843273
\(211\) 12.2231 0.841475 0.420738 0.907182i \(-0.361771\pi\)
0.420738 + 0.907182i \(0.361771\pi\)
\(212\) −30.6355 −2.10405
\(213\) 12.9109 0.884639
\(214\) 3.39433 0.232032
\(215\) −4.78276 −0.326182
\(216\) −4.11009 −0.279656
\(217\) 22.6756 1.53932
\(218\) −36.3542 −2.46222
\(219\) 2.32733 0.157266
\(220\) 0 0
\(221\) 7.21724 0.485484
\(222\) 27.0950 1.81850
\(223\) 11.2022 0.750153 0.375076 0.926994i \(-0.377617\pi\)
0.375076 + 0.926994i \(0.377617\pi\)
\(224\) −12.7828 −0.854084
\(225\) 1.00000 0.0666667
\(226\) −4.82585 −0.321011
\(227\) −14.5446 −0.965357 −0.482678 0.875798i \(-0.660336\pi\)
−0.482678 + 0.875798i \(0.660336\pi\)
\(228\) 6.22018 0.411942
\(229\) −14.9289 −0.986529 −0.493265 0.869879i \(-0.664197\pi\)
−0.493265 + 0.869879i \(0.664197\pi\)
\(230\) 5.04604 0.332726
\(231\) 0 0
\(232\) −3.21724 −0.211222
\(233\) 27.5475 1.80470 0.902349 0.431007i \(-0.141841\pi\)
0.902349 + 0.431007i \(0.141841\pi\)
\(234\) −13.0029 −0.850029
\(235\) 9.45544 0.616805
\(236\) 37.5295 2.44296
\(237\) 1.12811 0.0732786
\(238\) 16.2202 1.05140
\(239\) 3.21724 0.208106 0.104053 0.994572i \(-0.466819\pi\)
0.104053 + 0.994572i \(0.466819\pi\)
\(240\) 2.39138 0.154363
\(241\) 21.5835 1.39032 0.695159 0.718856i \(-0.255334\pi\)
0.695159 + 0.718856i \(0.255334\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) −2.89991 −0.185648
\(245\) 19.1130 1.22109
\(246\) −27.6576 −1.76338
\(247\) 9.09502 0.578702
\(248\) 18.2382 1.15813
\(249\) −4.00000 −0.253490
\(250\) −2.39138 −0.151244
\(251\) −5.43742 −0.343207 −0.171603 0.985166i \(-0.554895\pi\)
−0.171603 + 0.985166i \(0.554895\pi\)
\(252\) −19.0029 −1.19707
\(253\) 0 0
\(254\) 23.8778 1.49823
\(255\) 1.32733 0.0831205
\(256\) −28.0670 −1.75419
\(257\) −12.2022 −0.761150 −0.380575 0.924750i \(-0.624274\pi\)
−0.380575 + 0.924750i \(0.624274\pi\)
\(258\) 11.4374 0.712063
\(259\) 57.8988 3.59765
\(260\) 20.2202 1.25400
\(261\) 0.782765 0.0484519
\(262\) 11.6576 0.720209
\(263\) 1.85387 0.114315 0.0571573 0.998365i \(-0.481796\pi\)
0.0571573 + 0.998365i \(0.481796\pi\)
\(264\) 0 0
\(265\) −8.23820 −0.506069
\(266\) 20.4404 1.25328
\(267\) 14.7828 0.904691
\(268\) −16.5685 −1.01208
\(269\) −9.74968 −0.594448 −0.297224 0.954808i \(-0.596061\pi\)
−0.297224 + 0.954808i \(0.596061\pi\)
\(270\) −2.39138 −0.145535
\(271\) −15.2051 −0.923645 −0.461822 0.886972i \(-0.652804\pi\)
−0.461822 + 0.886972i \(0.652804\pi\)
\(272\) 3.17415 0.192461
\(273\) −27.7857 −1.68167
\(274\) −47.2721 −2.85581
\(275\) 0 0
\(276\) −7.84682 −0.472323
\(277\) 12.3332 0.741032 0.370516 0.928826i \(-0.379181\pi\)
0.370516 + 0.928826i \(0.379181\pi\)
\(278\) 16.1771 0.970238
\(279\) −4.43742 −0.265661
\(280\) 21.0029 1.25517
\(281\) −14.7828 −0.881866 −0.440933 0.897540i \(-0.645352\pi\)
−0.440933 + 0.897540i \(0.645352\pi\)
\(282\) −22.6116 −1.34650
\(283\) −29.0239 −1.72529 −0.862646 0.505808i \(-0.831195\pi\)
−0.862646 + 0.505808i \(0.831195\pi\)
\(284\) 48.0118 2.84898
\(285\) 1.67267 0.0990806
\(286\) 0 0
\(287\) −59.1009 −3.48862
\(288\) 2.50147 0.147401
\(289\) −15.2382 −0.896365
\(290\) −1.87189 −0.109921
\(291\) 4.45544 0.261182
\(292\) 8.65465 0.506475
\(293\) −9.32733 −0.544908 −0.272454 0.962169i \(-0.587835\pi\)
−0.272454 + 0.962169i \(0.587835\pi\)
\(294\) −45.7066 −2.66566
\(295\) 10.0921 0.587583
\(296\) 46.5685 2.70674
\(297\) 0 0
\(298\) −0.306360 −0.0177470
\(299\) −11.4735 −0.663527
\(300\) 3.71871 0.214700
\(301\) 24.4404 1.40872
\(302\) −13.0460 −0.750715
\(303\) −6.87484 −0.394949
\(304\) 4.00000 0.229416
\(305\) −0.779816 −0.0446521
\(306\) −3.17415 −0.181454
\(307\) 13.2382 0.755544 0.377772 0.925899i \(-0.376690\pi\)
0.377772 + 0.925899i \(0.376690\pi\)
\(308\) 0 0
\(309\) 10.5835 0.602077
\(310\) 10.6116 0.602696
\(311\) 2.78276 0.157796 0.0788981 0.996883i \(-0.474860\pi\)
0.0788981 + 0.996883i \(0.474860\pi\)
\(312\) −22.3483 −1.26522
\(313\) 3.87189 0.218852 0.109426 0.993995i \(-0.465099\pi\)
0.109426 + 0.993995i \(0.465099\pi\)
\(314\) −28.6606 −1.61741
\(315\) −5.11009 −0.287921
\(316\) 4.19511 0.235994
\(317\) −9.10714 −0.511508 −0.255754 0.966742i \(-0.582324\pi\)
−0.255754 + 0.966742i \(0.582324\pi\)
\(318\) 19.7007 1.10476
\(319\) 0 0
\(320\) −10.7647 −0.601768
\(321\) −1.41940 −0.0792233
\(322\) −25.7857 −1.43698
\(323\) 2.22018 0.123534
\(324\) 3.71871 0.206595
\(325\) 5.43742 0.301614
\(326\) −23.8778 −1.32247
\(327\) 15.2022 0.840682
\(328\) −47.5354 −2.62470
\(329\) −48.3182 −2.66387
\(330\) 0 0
\(331\) −6.21724 −0.341730 −0.170865 0.985294i \(-0.554656\pi\)
−0.170865 + 0.985294i \(0.554656\pi\)
\(332\) −14.8748 −0.816363
\(333\) −11.3303 −0.620895
\(334\) −18.3553 −1.00436
\(335\) −4.45544 −0.243427
\(336\) −12.2202 −0.666666
\(337\) 26.9879 1.47012 0.735062 0.678000i \(-0.237153\pi\)
0.735062 + 0.678000i \(0.237153\pi\)
\(338\) −39.6145 −2.15475
\(339\) 2.01802 0.109604
\(340\) 4.93594 0.267689
\(341\) 0 0
\(342\) −4.00000 −0.216295
\(343\) −61.8988 −3.34222
\(344\) 19.6576 1.05987
\(345\) −2.11009 −0.113604
\(346\) 11.2172 0.603042
\(347\) −12.1160 −0.650420 −0.325210 0.945642i \(-0.605435\pi\)
−0.325210 + 0.945642i \(0.605435\pi\)
\(348\) 2.91087 0.156039
\(349\) 23.4764 1.25666 0.628332 0.777946i \(-0.283738\pi\)
0.628332 + 0.777946i \(0.283738\pi\)
\(350\) 12.2202 0.653196
\(351\) 5.43742 0.290228
\(352\) 0 0
\(353\) 17.5115 0.932042 0.466021 0.884774i \(-0.345687\pi\)
0.466021 + 0.884774i \(0.345687\pi\)
\(354\) −24.1340 −1.28271
\(355\) 12.9109 0.685238
\(356\) 54.9728 2.91355
\(357\) −6.78276 −0.358982
\(358\) −27.3512 −1.44556
\(359\) 12.8388 0.677606 0.338803 0.940857i \(-0.389978\pi\)
0.338803 + 0.940857i \(0.389978\pi\)
\(360\) −4.11009 −0.216621
\(361\) −16.2022 −0.852746
\(362\) 19.4014 1.01971
\(363\) 0 0
\(364\) −103.327 −5.41581
\(365\) 2.32733 0.121818
\(366\) 1.86484 0.0974766
\(367\) −4.81880 −0.251539 −0.125770 0.992059i \(-0.540140\pi\)
−0.125770 + 0.992059i \(0.540140\pi\)
\(368\) −5.04604 −0.263043
\(369\) 11.5655 0.602077
\(370\) 27.0950 1.40860
\(371\) 42.0980 2.18562
\(372\) −16.5015 −0.855562
\(373\) 16.1071 0.833996 0.416998 0.908907i \(-0.363082\pi\)
0.416998 + 0.908907i \(0.363082\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) −38.8627 −2.00419
\(377\) 4.25622 0.219206
\(378\) 12.2202 0.628538
\(379\) 8.36631 0.429749 0.214874 0.976642i \(-0.431066\pi\)
0.214874 + 0.976642i \(0.431066\pi\)
\(380\) 6.22018 0.319089
\(381\) −9.98493 −0.511544
\(382\) 2.39843 0.122715
\(383\) −11.3453 −0.579720 −0.289860 0.957069i \(-0.593609\pi\)
−0.289860 + 0.957069i \(0.593609\pi\)
\(384\) 20.7397 1.05837
\(385\) 0 0
\(386\) −38.2261 −1.94566
\(387\) −4.78276 −0.243122
\(388\) 16.5685 0.841137
\(389\) 5.18120 0.262697 0.131349 0.991336i \(-0.458069\pi\)
0.131349 + 0.991336i \(0.458069\pi\)
\(390\) −13.0029 −0.658430
\(391\) −2.80078 −0.141642
\(392\) −78.5564 −3.96770
\(393\) −4.87484 −0.245903
\(394\) −38.7887 −1.95414
\(395\) 1.12811 0.0567614
\(396\) 0 0
\(397\) 0.767696 0.0385295 0.0192648 0.999814i \(-0.493867\pi\)
0.0192648 + 0.999814i \(0.493867\pi\)
\(398\) 3.21018 0.160912
\(399\) −8.54751 −0.427911
\(400\) 2.39138 0.119569
\(401\) 18.0360 0.900677 0.450338 0.892858i \(-0.351303\pi\)
0.450338 + 0.892858i \(0.351303\pi\)
\(402\) 10.6547 0.531406
\(403\) −24.1281 −1.20191
\(404\) −25.5655 −1.27193
\(405\) 1.00000 0.0496904
\(406\) 9.56553 0.474729
\(407\) 0 0
\(408\) −5.45544 −0.270084
\(409\) −2.55963 −0.126566 −0.0632828 0.997996i \(-0.520157\pi\)
−0.0632828 + 0.997996i \(0.520157\pi\)
\(410\) −27.6576 −1.36591
\(411\) 19.7677 0.975069
\(412\) 39.3571 1.93899
\(413\) −51.5714 −2.53766
\(414\) 5.04604 0.247999
\(415\) −4.00000 −0.196352
\(416\) 13.6016 0.666872
\(417\) −6.76475 −0.331271
\(418\) 0 0
\(419\) 11.0950 0.542027 0.271014 0.962575i \(-0.412641\pi\)
0.271014 + 0.962575i \(0.412641\pi\)
\(420\) −19.0029 −0.927249
\(421\) 15.3332 0.747296 0.373648 0.927571i \(-0.378107\pi\)
0.373648 + 0.927571i \(0.378107\pi\)
\(422\) −29.2302 −1.42290
\(423\) 9.45544 0.459739
\(424\) 33.8598 1.64438
\(425\) 1.32733 0.0643848
\(426\) −30.8748 −1.49589
\(427\) 3.98493 0.192844
\(428\) −5.27834 −0.255138
\(429\) 0 0
\(430\) 11.4374 0.551561
\(431\) −3.68774 −0.177632 −0.0888161 0.996048i \(-0.528308\pi\)
−0.0888161 + 0.996048i \(0.528308\pi\)
\(432\) 2.39138 0.115055
\(433\) 27.9489 1.34314 0.671569 0.740942i \(-0.265621\pi\)
0.671569 + 0.740942i \(0.265621\pi\)
\(434\) −54.2261 −2.60294
\(435\) 0.782765 0.0375307
\(436\) 56.5324 2.70741
\(437\) −3.52949 −0.168839
\(438\) −5.56553 −0.265931
\(439\) −24.0029 −1.14560 −0.572799 0.819696i \(-0.694142\pi\)
−0.572799 + 0.819696i \(0.694142\pi\)
\(440\) 0 0
\(441\) 19.1130 0.910145
\(442\) −17.2592 −0.820935
\(443\) −6.22018 −0.295530 −0.147765 0.989023i \(-0.547208\pi\)
−0.147765 + 0.989023i \(0.547208\pi\)
\(444\) −42.1340 −1.99959
\(445\) 14.7828 0.700770
\(446\) −26.7887 −1.26848
\(447\) 0.128110 0.00605940
\(448\) 55.0088 2.59892
\(449\) 22.0000 1.03824 0.519122 0.854700i \(-0.326259\pi\)
0.519122 + 0.854700i \(0.326259\pi\)
\(450\) −2.39138 −0.112731
\(451\) 0 0
\(452\) 7.50442 0.352978
\(453\) 5.45544 0.256319
\(454\) 34.7816 1.63238
\(455\) −27.7857 −1.30261
\(456\) −6.87484 −0.321944
\(457\) −16.8748 −0.789372 −0.394686 0.918816i \(-0.629147\pi\)
−0.394686 + 0.918816i \(0.629147\pi\)
\(458\) 35.7007 1.66818
\(459\) 1.32733 0.0619543
\(460\) −7.84682 −0.365860
\(461\) 23.8719 1.11182 0.555912 0.831241i \(-0.312369\pi\)
0.555912 + 0.831241i \(0.312369\pi\)
\(462\) 0 0
\(463\) 26.2261 1.21883 0.609415 0.792852i \(-0.291404\pi\)
0.609415 + 0.792852i \(0.291404\pi\)
\(464\) 1.87189 0.0869003
\(465\) −4.43742 −0.205780
\(466\) −65.8766 −3.05168
\(467\) −34.5505 −1.59880 −0.799402 0.600796i \(-0.794850\pi\)
−0.799402 + 0.600796i \(0.794850\pi\)
\(468\) 20.2202 0.934678
\(469\) 22.7677 1.05131
\(470\) −22.6116 −1.04299
\(471\) 11.9849 0.552236
\(472\) −41.4794 −1.90924
\(473\) 0 0
\(474\) −2.69774 −0.123911
\(475\) 1.67267 0.0767475
\(476\) −25.2231 −1.15610
\(477\) −8.23820 −0.377201
\(478\) −7.69364 −0.351899
\(479\) −35.2290 −1.60966 −0.804828 0.593508i \(-0.797742\pi\)
−0.804828 + 0.593508i \(0.797742\pi\)
\(480\) 2.50147 0.114176
\(481\) −61.6075 −2.80906
\(482\) −51.6145 −2.35098
\(483\) 10.7828 0.490633
\(484\) 0 0
\(485\) 4.45544 0.202311
\(486\) −2.39138 −0.108475
\(487\) −19.6576 −0.890771 −0.445386 0.895339i \(-0.646933\pi\)
−0.445386 + 0.895339i \(0.646933\pi\)
\(488\) 3.20511 0.145089
\(489\) 9.98493 0.451534
\(490\) −45.7066 −2.06481
\(491\) −0.470507 −0.0212337 −0.0106168 0.999944i \(-0.503380\pi\)
−0.0106168 + 0.999944i \(0.503380\pi\)
\(492\) 43.0088 1.93899
\(493\) 1.03899 0.0467935
\(494\) −21.7497 −0.978564
\(495\) 0 0
\(496\) −10.6116 −0.476473
\(497\) −65.9758 −2.95942
\(498\) 9.56553 0.428642
\(499\) −29.8568 −1.33657 −0.668287 0.743903i \(-0.732972\pi\)
−0.668287 + 0.743903i \(0.732972\pi\)
\(500\) 3.71871 0.166306
\(501\) 7.67562 0.342921
\(502\) 13.0029 0.580350
\(503\) 25.8598 1.15303 0.576515 0.817087i \(-0.304412\pi\)
0.576515 + 0.817087i \(0.304412\pi\)
\(504\) 21.0029 0.935546
\(505\) −6.87484 −0.305926
\(506\) 0 0
\(507\) 16.5655 0.735701
\(508\) −37.1311 −1.64742
\(509\) −25.4433 −1.12776 −0.563878 0.825858i \(-0.690691\pi\)
−0.563878 + 0.825858i \(0.690691\pi\)
\(510\) −3.17415 −0.140554
\(511\) −11.8929 −0.526109
\(512\) 25.6396 1.13312
\(513\) 1.67267 0.0738503
\(514\) 29.1800 1.28708
\(515\) 10.5835 0.466367
\(516\) −17.7857 −0.782972
\(517\) 0 0
\(518\) −138.458 −6.08350
\(519\) −4.69069 −0.205898
\(520\) −22.3483 −0.980038
\(521\) −40.5383 −1.77602 −0.888008 0.459827i \(-0.847911\pi\)
−0.888008 + 0.459827i \(0.847911\pi\)
\(522\) −1.87189 −0.0819304
\(523\) 35.3303 1.54489 0.772443 0.635085i \(-0.219035\pi\)
0.772443 + 0.635085i \(0.219035\pi\)
\(524\) −18.1281 −0.791930
\(525\) −5.11009 −0.223023
\(526\) −4.43332 −0.193302
\(527\) −5.88991 −0.256568
\(528\) 0 0
\(529\) −18.5475 −0.806414
\(530\) 19.7007 0.855743
\(531\) 10.0921 0.437959
\(532\) −31.7857 −1.37809
\(533\) 62.8866 2.72392
\(534\) −35.3512 −1.52980
\(535\) −1.41940 −0.0613661
\(536\) 18.3123 0.790969
\(537\) 11.4374 0.493561
\(538\) 23.3152 1.00519
\(539\) 0 0
\(540\) 3.71871 0.160028
\(541\) −22.6966 −0.975803 −0.487901 0.872899i \(-0.662237\pi\)
−0.487901 + 0.872899i \(0.662237\pi\)
\(542\) 36.3612 1.56185
\(543\) −8.11304 −0.348164
\(544\) 3.32028 0.142356
\(545\) 15.2022 0.651189
\(546\) 66.4463 2.84364
\(547\) 7.47346 0.319542 0.159771 0.987154i \(-0.448924\pi\)
0.159771 + 0.987154i \(0.448924\pi\)
\(548\) 73.5103 3.14021
\(549\) −0.779816 −0.0332817
\(550\) 0 0
\(551\) 1.30931 0.0557784
\(552\) 8.67267 0.369133
\(553\) −5.76475 −0.245142
\(554\) −29.4935 −1.25306
\(555\) −11.3303 −0.480943
\(556\) −25.1561 −1.06686
\(557\) −25.8037 −1.09334 −0.546670 0.837348i \(-0.684105\pi\)
−0.546670 + 0.837348i \(0.684105\pi\)
\(558\) 10.6116 0.449223
\(559\) −26.0059 −1.09993
\(560\) −12.2202 −0.516397
\(561\) 0 0
\(562\) 35.3512 1.49120
\(563\) −9.74968 −0.410900 −0.205450 0.978668i \(-0.565866\pi\)
−0.205450 + 0.978668i \(0.565866\pi\)
\(564\) 35.1620 1.48059
\(565\) 2.01802 0.0848987
\(566\) 69.4073 2.91741
\(567\) −5.11009 −0.214604
\(568\) −53.0649 −2.22655
\(569\) −21.2231 −0.889720 −0.444860 0.895600i \(-0.646747\pi\)
−0.444860 + 0.895600i \(0.646747\pi\)
\(570\) −4.00000 −0.167542
\(571\) −1.81880 −0.0761144 −0.0380572 0.999276i \(-0.512117\pi\)
−0.0380572 + 0.999276i \(0.512117\pi\)
\(572\) 0 0
\(573\) −1.00295 −0.0418988
\(574\) 141.333 5.89912
\(575\) −2.11009 −0.0879969
\(576\) −10.7647 −0.448531
\(577\) −39.4643 −1.64292 −0.821460 0.570266i \(-0.806840\pi\)
−0.821460 + 0.570266i \(0.806840\pi\)
\(578\) 36.4404 1.51572
\(579\) 15.9849 0.664311
\(580\) 2.91087 0.120868
\(581\) 20.4404 0.848009
\(582\) −10.6547 −0.441650
\(583\) 0 0
\(584\) −9.56553 −0.395824
\(585\) 5.43742 0.224810
\(586\) 22.3052 0.921420
\(587\) 29.6396 1.22336 0.611678 0.791107i \(-0.290495\pi\)
0.611678 + 0.791107i \(0.290495\pi\)
\(588\) 71.0759 2.93112
\(589\) −7.42235 −0.305833
\(590\) −24.1340 −0.993581
\(591\) 16.2202 0.667209
\(592\) −27.0950 −1.11360
\(593\) −29.1311 −1.19627 −0.598135 0.801396i \(-0.704091\pi\)
−0.598135 + 0.801396i \(0.704091\pi\)
\(594\) 0 0
\(595\) −6.78276 −0.278066
\(596\) 0.476404 0.0195143
\(597\) −1.34240 −0.0549406
\(598\) 27.4374 1.12200
\(599\) 44.1039 1.80204 0.901018 0.433782i \(-0.142821\pi\)
0.901018 + 0.433782i \(0.142821\pi\)
\(600\) −4.11009 −0.167794
\(601\) −34.5174 −1.40799 −0.703997 0.710203i \(-0.748603\pi\)
−0.703997 + 0.710203i \(0.748603\pi\)
\(602\) −58.4463 −2.38209
\(603\) −4.45544 −0.181439
\(604\) 20.2872 0.825474
\(605\) 0 0
\(606\) 16.4404 0.667844
\(607\) 2.99705 0.121647 0.0608233 0.998149i \(-0.480627\pi\)
0.0608233 + 0.998149i \(0.480627\pi\)
\(608\) 4.18415 0.169690
\(609\) −4.00000 −0.162088
\(610\) 1.86484 0.0755051
\(611\) 51.4132 2.07995
\(612\) 4.93594 0.199524
\(613\) −0.761798 −0.0307687 −0.0153844 0.999882i \(-0.504897\pi\)
−0.0153844 + 0.999882i \(0.504897\pi\)
\(614\) −31.6576 −1.27760
\(615\) 11.5655 0.466367
\(616\) 0 0
\(617\) −23.8217 −0.959028 −0.479514 0.877534i \(-0.659187\pi\)
−0.479514 + 0.877534i \(0.659187\pi\)
\(618\) −25.3093 −1.01809
\(619\) −30.4463 −1.22374 −0.611869 0.790959i \(-0.709582\pi\)
−0.611869 + 0.790959i \(0.709582\pi\)
\(620\) −16.5015 −0.662715
\(621\) −2.11009 −0.0846751
\(622\) −6.65465 −0.266827
\(623\) −75.5413 −3.02650
\(624\) 13.0029 0.520535
\(625\) 1.00000 0.0400000
\(626\) −9.25917 −0.370071
\(627\) 0 0
\(628\) 44.5685 1.77848
\(629\) −15.0390 −0.599644
\(630\) 12.2202 0.486864
\(631\) −10.5446 −0.419772 −0.209886 0.977726i \(-0.567309\pi\)
−0.209886 + 0.977726i \(0.567309\pi\)
\(632\) −4.63664 −0.184435
\(633\) 12.2231 0.485826
\(634\) 21.7787 0.864941
\(635\) −9.98493 −0.396240
\(636\) −30.6355 −1.21478
\(637\) 103.926 4.11768
\(638\) 0 0
\(639\) 12.9109 0.510746
\(640\) 20.7397 0.819808
\(641\) 17.4014 0.687313 0.343657 0.939095i \(-0.388334\pi\)
0.343657 + 0.939095i \(0.388334\pi\)
\(642\) 3.39433 0.133964
\(643\) −32.3693 −1.27652 −0.638260 0.769821i \(-0.720345\pi\)
−0.638260 + 0.769821i \(0.720345\pi\)
\(644\) 40.0980 1.58008
\(645\) −4.78276 −0.188321
\(646\) −5.30931 −0.208892
\(647\) 38.3362 1.50715 0.753575 0.657362i \(-0.228328\pi\)
0.753575 + 0.657362i \(0.228328\pi\)
\(648\) −4.11009 −0.161460
\(649\) 0 0
\(650\) −13.0029 −0.510018
\(651\) 22.6756 0.888728
\(652\) 37.1311 1.45416
\(653\) 3.74968 0.146736 0.0733681 0.997305i \(-0.476625\pi\)
0.0733681 + 0.997305i \(0.476625\pi\)
\(654\) −36.3542 −1.42156
\(655\) −4.87484 −0.190476
\(656\) 27.6576 1.07985
\(657\) 2.32733 0.0907977
\(658\) 115.547 4.50450
\(659\) −15.9079 −0.619685 −0.309842 0.950788i \(-0.600276\pi\)
−0.309842 + 0.950788i \(0.600276\pi\)
\(660\) 0 0
\(661\) 17.2382 0.670488 0.335244 0.942131i \(-0.391181\pi\)
0.335244 + 0.942131i \(0.391181\pi\)
\(662\) 14.8678 0.577853
\(663\) 7.21724 0.280294
\(664\) 16.4404 0.638010
\(665\) −8.54751 −0.331458
\(666\) 27.0950 1.04991
\(667\) −1.65171 −0.0639543
\(668\) 28.5434 1.10438
\(669\) 11.2022 0.433101
\(670\) 10.6547 0.411625
\(671\) 0 0
\(672\) −12.7828 −0.493106
\(673\) 21.2942 0.820833 0.410416 0.911898i \(-0.365383\pi\)
0.410416 + 0.911898i \(0.365383\pi\)
\(674\) −64.5383 −2.48592
\(675\) 1.00000 0.0384900
\(676\) 61.6024 2.36932
\(677\) −38.0419 −1.46207 −0.731035 0.682340i \(-0.760962\pi\)
−0.731035 + 0.682340i \(0.760962\pi\)
\(678\) −4.82585 −0.185336
\(679\) −22.7677 −0.873744
\(680\) −5.45544 −0.209206
\(681\) −14.5446 −0.557349
\(682\) 0 0
\(683\) 33.0950 1.26635 0.633173 0.774010i \(-0.281752\pi\)
0.633173 + 0.774010i \(0.281752\pi\)
\(684\) 6.22018 0.237835
\(685\) 19.7677 0.755285
\(686\) 148.024 5.65157
\(687\) −14.9289 −0.569573
\(688\) −11.4374 −0.436048
\(689\) −44.7946 −1.70654
\(690\) 5.04604 0.192099
\(691\) −32.8778 −1.25073 −0.625365 0.780332i \(-0.715050\pi\)
−0.625365 + 0.780332i \(0.715050\pi\)
\(692\) −17.4433 −0.663095
\(693\) 0 0
\(694\) 28.9740 1.09984
\(695\) −6.76475 −0.256601
\(696\) −3.21724 −0.121949
\(697\) 15.3512 0.581470
\(698\) −56.1411 −2.12497
\(699\) 27.5475 1.04194
\(700\) −19.0029 −0.718244
\(701\) −35.2231 −1.33036 −0.665180 0.746683i \(-0.731645\pi\)
−0.665180 + 0.746683i \(0.731645\pi\)
\(702\) −13.0029 −0.490765
\(703\) −18.9518 −0.714782
\(704\) 0 0
\(705\) 9.45544 0.356112
\(706\) −41.8766 −1.57605
\(707\) 35.1311 1.32124
\(708\) 37.5295 1.41044
\(709\) 1.14318 0.0429330 0.0214665 0.999770i \(-0.493166\pi\)
0.0214665 + 0.999770i \(0.493166\pi\)
\(710\) −30.8748 −1.15871
\(711\) 1.12811 0.0423074
\(712\) −60.7585 −2.27702
\(713\) 9.36336 0.350661
\(714\) 16.2202 0.607025
\(715\) 0 0
\(716\) 42.5324 1.58951
\(717\) 3.21724 0.120150
\(718\) −30.7025 −1.14581
\(719\) 1.16120 0.0433054 0.0216527 0.999766i \(-0.493107\pi\)
0.0216527 + 0.999766i \(0.493107\pi\)
\(720\) 2.39138 0.0891216
\(721\) −54.0829 −2.01415
\(722\) 38.7456 1.44196
\(723\) 21.5835 0.802701
\(724\) −30.1700 −1.12126
\(725\) 0.782765 0.0290712
\(726\) 0 0
\(727\) 6.65465 0.246807 0.123404 0.992357i \(-0.460619\pi\)
0.123404 + 0.992357i \(0.460619\pi\)
\(728\) 114.202 4.23260
\(729\) 1.00000 0.0370370
\(730\) −5.56553 −0.205989
\(731\) −6.34829 −0.234800
\(732\) −2.89991 −0.107184
\(733\) −22.0000 −0.812589 −0.406294 0.913742i \(-0.633179\pi\)
−0.406294 + 0.913742i \(0.633179\pi\)
\(734\) 11.5236 0.425344
\(735\) 19.1130 0.704995
\(736\) −5.27834 −0.194562
\(737\) 0 0
\(738\) −27.6576 −1.01809
\(739\) 4.47346 0.164559 0.0822794 0.996609i \(-0.473780\pi\)
0.0822794 + 0.996609i \(0.473780\pi\)
\(740\) −42.1340 −1.54888
\(741\) 9.09502 0.334114
\(742\) −100.672 −3.69580
\(743\) 24.3303 0.892591 0.446296 0.894886i \(-0.352743\pi\)
0.446296 + 0.894886i \(0.352743\pi\)
\(744\) 18.2382 0.668645
\(745\) 0.128110 0.00469359
\(746\) −38.5183 −1.41026
\(747\) −4.00000 −0.146352
\(748\) 0 0
\(749\) 7.25327 0.265029
\(750\) −2.39138 −0.0873209
\(751\) 32.6635 1.19191 0.595954 0.803019i \(-0.296774\pi\)
0.595954 + 0.803019i \(0.296774\pi\)
\(752\) 22.6116 0.824559
\(753\) −5.43742 −0.198151
\(754\) −10.1783 −0.370670
\(755\) 5.45544 0.198544
\(756\) −19.0029 −0.691131
\(757\) −12.0711 −0.438732 −0.219366 0.975643i \(-0.570399\pi\)
−0.219366 + 0.975643i \(0.570399\pi\)
\(758\) −20.0071 −0.726689
\(759\) 0 0
\(760\) −6.87484 −0.249377
\(761\) 16.3123 0.591319 0.295659 0.955293i \(-0.404461\pi\)
0.295659 + 0.955293i \(0.404461\pi\)
\(762\) 23.8778 0.865001
\(763\) −77.6845 −2.81237
\(764\) −3.72968 −0.134935
\(765\) 1.32733 0.0479896
\(766\) 27.1311 0.980285
\(767\) 54.8748 1.98142
\(768\) −28.0670 −1.01278
\(769\) −14.6016 −0.526546 −0.263273 0.964721i \(-0.584802\pi\)
−0.263273 + 0.964721i \(0.584802\pi\)
\(770\) 0 0
\(771\) −12.2022 −0.439450
\(772\) 59.4433 2.13941
\(773\) −26.2742 −0.945019 −0.472509 0.881326i \(-0.656652\pi\)
−0.472509 + 0.881326i \(0.656652\pi\)
\(774\) 11.4374 0.411110
\(775\) −4.43742 −0.159397
\(776\) −18.3123 −0.657372
\(777\) 57.8988 2.07711
\(778\) −12.3902 −0.444211
\(779\) 19.3453 0.693119
\(780\) 20.2202 0.723999
\(781\) 0 0
\(782\) 6.69774 0.239511
\(783\) 0.782765 0.0279737
\(784\) 45.7066 1.63238
\(785\) 11.9849 0.427761
\(786\) 11.6576 0.415813
\(787\) 44.8807 1.59983 0.799913 0.600116i \(-0.204879\pi\)
0.799913 + 0.600116i \(0.204879\pi\)
\(788\) 60.3182 2.14875
\(789\) 1.85387 0.0659996
\(790\) −2.69774 −0.0959814
\(791\) −10.3123 −0.366662
\(792\) 0 0
\(793\) −4.24019 −0.150573
\(794\) −1.83585 −0.0651520
\(795\) −8.23820 −0.292179
\(796\) −4.99198 −0.176936
\(797\) −38.5124 −1.36418 −0.682090 0.731268i \(-0.738929\pi\)
−0.682090 + 0.731268i \(0.738929\pi\)
\(798\) 20.4404 0.723581
\(799\) 12.5505 0.444003
\(800\) 2.50147 0.0884405
\(801\) 14.7828 0.522323
\(802\) −43.1311 −1.52301
\(803\) 0 0
\(804\) −16.5685 −0.584325
\(805\) 10.7828 0.380043
\(806\) 57.6995 2.03238
\(807\) −9.74968 −0.343205
\(808\) 28.2562 0.994050
\(809\) 9.09502 0.319764 0.159882 0.987136i \(-0.448889\pi\)
0.159882 + 0.987136i \(0.448889\pi\)
\(810\) −2.39138 −0.0840246
\(811\) −16.7297 −0.587458 −0.293729 0.955889i \(-0.594896\pi\)
−0.293729 + 0.955889i \(0.594896\pi\)
\(812\) −14.8748 −0.522005
\(813\) −15.2051 −0.533267
\(814\) 0 0
\(815\) 9.98493 0.349757
\(816\) 3.17415 0.111117
\(817\) −8.00000 −0.279885
\(818\) 6.12106 0.214018
\(819\) −27.7857 −0.970911
\(820\) 43.0088 1.50193
\(821\) −7.38138 −0.257612 −0.128806 0.991670i \(-0.541114\pi\)
−0.128806 + 0.991670i \(0.541114\pi\)
\(822\) −47.2721 −1.64880
\(823\) −38.0410 −1.32602 −0.663012 0.748608i \(-0.730722\pi\)
−0.663012 + 0.748608i \(0.730722\pi\)
\(824\) −43.4994 −1.51537
\(825\) 0 0
\(826\) 123.327 4.29110
\(827\) 46.1900 1.60619 0.803093 0.595854i \(-0.203186\pi\)
0.803093 + 0.595854i \(0.203186\pi\)
\(828\) −7.84682 −0.272696
\(829\) 8.55963 0.297288 0.148644 0.988891i \(-0.452509\pi\)
0.148644 + 0.988891i \(0.452509\pi\)
\(830\) 9.56553 0.332024
\(831\) 12.3332 0.427835
\(832\) −58.5324 −2.02925
\(833\) 25.3693 0.878993
\(834\) 16.1771 0.560167
\(835\) 7.67562 0.265626
\(836\) 0 0
\(837\) −4.43742 −0.153380
\(838\) −26.5324 −0.916548
\(839\) 55.7556 1.92490 0.962448 0.271466i \(-0.0875084\pi\)
0.962448 + 0.271466i \(0.0875084\pi\)
\(840\) 21.0029 0.724671
\(841\) −28.3873 −0.978872
\(842\) −36.6676 −1.26365
\(843\) −14.7828 −0.509145
\(844\) 45.4543 1.56460
\(845\) 16.5655 0.569872
\(846\) −22.6116 −0.777402
\(847\) 0 0
\(848\) −19.7007 −0.676525
\(849\) −29.0239 −0.996098
\(850\) −3.17415 −0.108872
\(851\) 23.9079 0.819553
\(852\) 48.0118 1.64486
\(853\) −12.7258 −0.435722 −0.217861 0.975980i \(-0.569908\pi\)
−0.217861 + 0.975980i \(0.569908\pi\)
\(854\) −9.52949 −0.326093
\(855\) 1.67267 0.0572042
\(856\) 5.83387 0.199398
\(857\) 13.5835 0.464005 0.232003 0.972715i \(-0.425472\pi\)
0.232003 + 0.972715i \(0.425472\pi\)
\(858\) 0 0
\(859\) 11.0239 0.376131 0.188066 0.982156i \(-0.439778\pi\)
0.188066 + 0.982156i \(0.439778\pi\)
\(860\) −17.7857 −0.606488
\(861\) −59.1009 −2.01415
\(862\) 8.81880 0.300370
\(863\) −46.1900 −1.57233 −0.786164 0.618018i \(-0.787936\pi\)
−0.786164 + 0.618018i \(0.787936\pi\)
\(864\) 2.50147 0.0851019
\(865\) −4.69069 −0.159488
\(866\) −66.8365 −2.27120
\(867\) −15.2382 −0.517516
\(868\) 84.3241 2.86214
\(869\) 0 0
\(870\) −1.87189 −0.0634630
\(871\) −24.2261 −0.820869
\(872\) −62.4823 −2.11592
\(873\) 4.45544 0.150794
\(874\) 8.44037 0.285500
\(875\) −5.11009 −0.172753
\(876\) 8.65465 0.292414
\(877\) −22.3634 −0.755157 −0.377579 0.925978i \(-0.623243\pi\)
−0.377579 + 0.925978i \(0.623243\pi\)
\(878\) 57.4002 1.93716
\(879\) −9.32733 −0.314603
\(880\) 0 0
\(881\) 36.5124 1.23014 0.615068 0.788474i \(-0.289129\pi\)
0.615068 + 0.788474i \(0.289129\pi\)
\(882\) −45.7066 −1.53902
\(883\) −42.4672 −1.42914 −0.714568 0.699566i \(-0.753377\pi\)
−0.714568 + 0.699566i \(0.753377\pi\)
\(884\) 26.8388 0.902687
\(885\) 10.0921 0.339241
\(886\) 14.8748 0.499730
\(887\) 17.5354 0.588781 0.294390 0.955685i \(-0.404883\pi\)
0.294390 + 0.955685i \(0.404883\pi\)
\(888\) 46.5685 1.56274
\(889\) 51.0239 1.71129
\(890\) −35.3512 −1.18498
\(891\) 0 0
\(892\) 41.6576 1.39480
\(893\) 15.8159 0.529257
\(894\) −0.306360 −0.0102462
\(895\) 11.4374 0.382311
\(896\) −105.982 −3.54060
\(897\) −11.4735 −0.383088
\(898\) −52.6104 −1.75563
\(899\) −3.47346 −0.115846
\(900\) 3.71871 0.123957
\(901\) −10.9348 −0.364291
\(902\) 0 0
\(903\) 24.4404 0.813325
\(904\) −8.29424 −0.275862
\(905\) −8.11304 −0.269687
\(906\) −13.0460 −0.433426
\(907\) −24.5835 −0.816283 −0.408142 0.912919i \(-0.633823\pi\)
−0.408142 + 0.912919i \(0.633823\pi\)
\(908\) −54.0870 −1.79494
\(909\) −6.87484 −0.228024
\(910\) 66.4463 2.20267
\(911\) −42.5685 −1.41036 −0.705178 0.709030i \(-0.749133\pi\)
−0.705178 + 0.709030i \(0.749133\pi\)
\(912\) 4.00000 0.132453
\(913\) 0 0
\(914\) 40.3542 1.33480
\(915\) −0.779816 −0.0257799
\(916\) −55.5162 −1.83431
\(917\) 24.9109 0.822630
\(918\) −3.17415 −0.104762
\(919\) 32.1491 1.06050 0.530250 0.847841i \(-0.322098\pi\)
0.530250 + 0.847841i \(0.322098\pi\)
\(920\) 8.67267 0.285930
\(921\) 13.2382 0.436214
\(922\) −57.0868 −1.88005
\(923\) 70.2018 2.31072
\(924\) 0 0
\(925\) −11.3303 −0.372537
\(926\) −62.7166 −2.06100
\(927\) 10.5835 0.347609
\(928\) 1.95807 0.0642767
\(929\) 8.17825 0.268320 0.134160 0.990960i \(-0.457166\pi\)
0.134160 + 0.990960i \(0.457166\pi\)
\(930\) 10.6116 0.347967
\(931\) 31.9699 1.04777
\(932\) 102.441 3.35557
\(933\) 2.78276 0.0911036
\(934\) 82.6234 2.70352
\(935\) 0 0
\(936\) −22.3483 −0.730477
\(937\) −19.0741 −0.623122 −0.311561 0.950226i \(-0.600852\pi\)
−0.311561 + 0.950226i \(0.600852\pi\)
\(938\) −54.4463 −1.77773
\(939\) 3.87189 0.126354
\(940\) 35.1620 1.14686
\(941\) −31.0029 −1.01067 −0.505334 0.862924i \(-0.668631\pi\)
−0.505334 + 0.862924i \(0.668631\pi\)
\(942\) −28.6606 −0.933811
\(943\) −24.4043 −0.794714
\(944\) 24.1340 0.785495
\(945\) −5.11009 −0.166231
\(946\) 0 0
\(947\) 14.3604 0.466651 0.233325 0.972399i \(-0.425039\pi\)
0.233325 + 0.972399i \(0.425039\pi\)
\(948\) 4.19511 0.136251
\(949\) 12.6547 0.410787
\(950\) −4.00000 −0.129777
\(951\) −9.10714 −0.295319
\(952\) 27.8778 0.903524
\(953\) 50.0000 1.61966 0.809829 0.586665i \(-0.199560\pi\)
0.809829 + 0.586665i \(0.199560\pi\)
\(954\) 19.7007 0.637833
\(955\) −1.00295 −0.0324547
\(956\) 11.9640 0.386942
\(957\) 0 0
\(958\) 84.2461 2.72187
\(959\) −101.015 −3.26194
\(960\) −10.7647 −0.347431
\(961\) −11.3093 −0.364816
\(962\) 147.327 4.75001
\(963\) −1.41940 −0.0457396
\(964\) 80.2629 2.58510
\(965\) 15.9849 0.514573
\(966\) −25.7857 −0.829642
\(967\) −21.4165 −0.688707 −0.344353 0.938840i \(-0.611902\pi\)
−0.344353 + 0.938840i \(0.611902\pi\)
\(968\) 0 0
\(969\) 2.22018 0.0713226
\(970\) −10.6547 −0.342100
\(971\) −3.61567 −0.116032 −0.0580162 0.998316i \(-0.518477\pi\)
−0.0580162 + 0.998316i \(0.518477\pi\)
\(972\) 3.71871 0.119278
\(973\) 34.5685 1.10821
\(974\) 47.0088 1.50626
\(975\) 5.43742 0.174137
\(976\) −1.86484 −0.0596920
\(977\) 20.2022 0.646325 0.323162 0.946344i \(-0.395254\pi\)
0.323162 + 0.946344i \(0.395254\pi\)
\(978\) −23.8778 −0.763527
\(979\) 0 0
\(980\) 71.0759 2.27043
\(981\) 15.2022 0.485368
\(982\) 1.12516 0.0359053
\(983\) −46.8067 −1.49290 −0.746451 0.665441i \(-0.768244\pi\)
−0.746451 + 0.665441i \(0.768244\pi\)
\(984\) −47.5354 −1.51537
\(985\) 16.2202 0.516818
\(986\) −2.48461 −0.0791261
\(987\) −48.3182 −1.53798
\(988\) 33.8217 1.07601
\(989\) 10.0921 0.320909
\(990\) 0 0
\(991\) −47.3893 −1.50537 −0.752685 0.658381i \(-0.771242\pi\)
−0.752685 + 0.658381i \(0.771242\pi\)
\(992\) −11.1001 −0.352428
\(993\) −6.21724 −0.197298
\(994\) 157.773 5.00426
\(995\) −1.34240 −0.0425568
\(996\) −14.8748 −0.471327
\(997\) 30.4554 0.964533 0.482267 0.876024i \(-0.339814\pi\)
0.482267 + 0.876024i \(0.339814\pi\)
\(998\) 71.3991 2.26010
\(999\) −11.3303 −0.358474
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 1815.2.a.l.1.1 3
3.2 odd 2 5445.2.a.bc.1.3 3
5.4 even 2 9075.2.a.ci.1.3 3
11.10 odd 2 1815.2.a.n.1.3 yes 3
33.32 even 2 5445.2.a.ba.1.1 3
55.54 odd 2 9075.2.a.ce.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1815.2.a.l.1.1 3 1.1 even 1 trivial
1815.2.a.n.1.3 yes 3 11.10 odd 2
5445.2.a.ba.1.1 3 33.32 even 2
5445.2.a.bc.1.3 3 3.2 odd 2
9075.2.a.ce.1.1 3 55.54 odd 2
9075.2.a.ci.1.3 3 5.4 even 2