Properties

Label 1925.2.a.z.1.3
Level $1925$
Weight $2$
Character 1925.1
Self dual yes
Analytic conductor $15.371$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1925,2,Mod(1,1925)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1925, 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("1925.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1925 = 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1925.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: \(15.3712023891\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.9921856.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 7x^{4} + 11x^{2} - 2x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 385)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.226645\) of defining polynomial
Character \(\chi\) \(=\) 1925.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.226645 q^{2} +1.89494 q^{3} -1.94863 q^{4} -0.429477 q^{6} -1.00000 q^{7} +0.894936 q^{8} +0.590784 q^{9} +O(q^{10})\) \(q-0.226645 q^{2} +1.89494 q^{3} -1.94863 q^{4} -0.429477 q^{6} -1.00000 q^{7} +0.894936 q^{8} +0.590784 q^{9} -1.00000 q^{11} -3.69253 q^{12} -2.71469 q^{13} +0.226645 q^{14} +3.69443 q^{16} +6.92945 q^{17} -0.133898 q^{18} +6.38464 q^{19} -1.89494 q^{21} +0.226645 q^{22} -2.82331 q^{23} +1.69585 q^{24} +0.615271 q^{26} -4.56531 q^{27} +1.94863 q^{28} +4.88997 q^{29} -7.04479 q^{31} -2.62720 q^{32} -1.89494 q^{33} -1.57052 q^{34} -1.15122 q^{36} -1.27517 q^{37} -1.44704 q^{38} -5.14417 q^{39} +5.34306 q^{41} +0.429477 q^{42} +9.92780 q^{43} +1.94863 q^{44} +0.639888 q^{46} +7.64491 q^{47} +7.00071 q^{48} +1.00000 q^{49} +13.1309 q^{51} +5.28994 q^{52} +2.16500 q^{53} +1.03470 q^{54} -0.894936 q^{56} +12.0985 q^{57} -1.10829 q^{58} +12.0226 q^{59} -0.513231 q^{61} +1.59666 q^{62} -0.590784 q^{63} -6.79342 q^{64} +0.429477 q^{66} +13.7761 q^{67} -13.5030 q^{68} -5.34999 q^{69} +8.96893 q^{71} +0.528714 q^{72} +6.93424 q^{73} +0.289011 q^{74} -12.4413 q^{76} +1.00000 q^{77} +1.16590 q^{78} -10.8414 q^{79} -10.4233 q^{81} -1.21098 q^{82} -9.07907 q^{83} +3.69253 q^{84} -2.25008 q^{86} +9.26618 q^{87} -0.894936 q^{88} -5.69209 q^{89} +2.71469 q^{91} +5.50159 q^{92} -13.3494 q^{93} -1.73268 q^{94} -4.97837 q^{96} +6.05204 q^{97} -0.226645 q^{98} -0.590784 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{3} + 2 q^{4} - 2 q^{6} - 6 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{3} + 2 q^{4} - 2 q^{6} - 6 q^{7} + 10 q^{9} - 6 q^{11} + 12 q^{12} + 6 q^{13} - 6 q^{16} - 2 q^{17} + 14 q^{18} - 2 q^{19} - 6 q^{21} + 8 q^{23} + 22 q^{24} + 24 q^{27} - 2 q^{28} - 8 q^{29} + 4 q^{31} + 10 q^{32} - 6 q^{33} - 10 q^{34} + 26 q^{36} + 22 q^{37} - 10 q^{38} - 8 q^{39} + 4 q^{41} + 2 q^{42} + 30 q^{43} - 2 q^{44} - 8 q^{46} + 16 q^{47} - 8 q^{48} + 6 q^{49} - 4 q^{51} + 22 q^{52} + 6 q^{53} + 38 q^{54} + 18 q^{57} - 14 q^{58} + 14 q^{59} + 12 q^{61} - 14 q^{62} - 10 q^{63} - 22 q^{64} + 2 q^{66} + 46 q^{67} - 20 q^{68} + 12 q^{69} + 4 q^{71} + 32 q^{72} - 4 q^{73} - 8 q^{74} - 8 q^{76} + 6 q^{77} + 24 q^{78} - 8 q^{79} + 26 q^{81} - 10 q^{82} + 14 q^{83} - 12 q^{84} + 12 q^{86} - 2 q^{87} + 8 q^{89} - 6 q^{91} + 18 q^{92} + 12 q^{93} - 10 q^{94} - 16 q^{96} + 42 q^{97} - 10 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.226645 −0.160262 −0.0801310 0.996784i \(-0.525534\pi\)
−0.0801310 + 0.996784i \(0.525534\pi\)
\(3\) 1.89494 1.09404 0.547021 0.837119i \(-0.315762\pi\)
0.547021 + 0.837119i \(0.315762\pi\)
\(4\) −1.94863 −0.974316
\(5\) 0 0
\(6\) −0.429477 −0.175333
\(7\) −1.00000 −0.377964
\(8\) 0.894936 0.316408
\(9\) 0.590784 0.196928
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) −3.69253 −1.06594
\(13\) −2.71469 −0.752920 −0.376460 0.926433i \(-0.622859\pi\)
−0.376460 + 0.926433i \(0.622859\pi\)
\(14\) 0.226645 0.0605733
\(15\) 0 0
\(16\) 3.69443 0.923608
\(17\) 6.92945 1.68064 0.840319 0.542092i \(-0.182367\pi\)
0.840319 + 0.542092i \(0.182367\pi\)
\(18\) −0.133898 −0.0315601
\(19\) 6.38464 1.46474 0.732368 0.680909i \(-0.238415\pi\)
0.732368 + 0.680909i \(0.238415\pi\)
\(20\) 0 0
\(21\) −1.89494 −0.413509
\(22\) 0.226645 0.0483208
\(23\) −2.82331 −0.588700 −0.294350 0.955698i \(-0.595103\pi\)
−0.294350 + 0.955698i \(0.595103\pi\)
\(24\) 1.69585 0.346163
\(25\) 0 0
\(26\) 0.615271 0.120664
\(27\) −4.56531 −0.878595
\(28\) 1.94863 0.368257
\(29\) 4.88997 0.908045 0.454022 0.890990i \(-0.349989\pi\)
0.454022 + 0.890990i \(0.349989\pi\)
\(30\) 0 0
\(31\) −7.04479 −1.26528 −0.632640 0.774446i \(-0.718029\pi\)
−0.632640 + 0.774446i \(0.718029\pi\)
\(32\) −2.62720 −0.464427
\(33\) −1.89494 −0.329866
\(34\) −1.57052 −0.269342
\(35\) 0 0
\(36\) −1.15122 −0.191870
\(37\) −1.27517 −0.209637 −0.104819 0.994491i \(-0.533426\pi\)
−0.104819 + 0.994491i \(0.533426\pi\)
\(38\) −1.44704 −0.234742
\(39\) −5.14417 −0.823726
\(40\) 0 0
\(41\) 5.34306 0.834446 0.417223 0.908804i \(-0.363003\pi\)
0.417223 + 0.908804i \(0.363003\pi\)
\(42\) 0.429477 0.0662698
\(43\) 9.92780 1.51397 0.756987 0.653429i \(-0.226670\pi\)
0.756987 + 0.653429i \(0.226670\pi\)
\(44\) 1.94863 0.293767
\(45\) 0 0
\(46\) 0.639888 0.0943463
\(47\) 7.64491 1.11512 0.557562 0.830135i \(-0.311737\pi\)
0.557562 + 0.830135i \(0.311737\pi\)
\(48\) 7.00071 1.01047
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 13.1309 1.83869
\(52\) 5.28994 0.733582
\(53\) 2.16500 0.297386 0.148693 0.988883i \(-0.452493\pi\)
0.148693 + 0.988883i \(0.452493\pi\)
\(54\) 1.03470 0.140805
\(55\) 0 0
\(56\) −0.894936 −0.119591
\(57\) 12.0985 1.60248
\(58\) −1.10829 −0.145525
\(59\) 12.0226 1.56521 0.782606 0.622517i \(-0.213890\pi\)
0.782606 + 0.622517i \(0.213890\pi\)
\(60\) 0 0
\(61\) −0.513231 −0.0657124 −0.0328562 0.999460i \(-0.510460\pi\)
−0.0328562 + 0.999460i \(0.510460\pi\)
\(62\) 1.59666 0.202776
\(63\) −0.590784 −0.0744317
\(64\) −6.79342 −0.849178
\(65\) 0 0
\(66\) 0.429477 0.0528650
\(67\) 13.7761 1.68302 0.841509 0.540244i \(-0.181668\pi\)
0.841509 + 0.540244i \(0.181668\pi\)
\(68\) −13.5030 −1.63747
\(69\) −5.34999 −0.644063
\(70\) 0 0
\(71\) 8.96893 1.06442 0.532208 0.846614i \(-0.321362\pi\)
0.532208 + 0.846614i \(0.321362\pi\)
\(72\) 0.528714 0.0623095
\(73\) 6.93424 0.811592 0.405796 0.913964i \(-0.366994\pi\)
0.405796 + 0.913964i \(0.366994\pi\)
\(74\) 0.289011 0.0335969
\(75\) 0 0
\(76\) −12.4413 −1.42712
\(77\) 1.00000 0.113961
\(78\) 1.16590 0.132012
\(79\) −10.8414 −1.21975 −0.609875 0.792497i \(-0.708780\pi\)
−0.609875 + 0.792497i \(0.708780\pi\)
\(80\) 0 0
\(81\) −10.4233 −1.15815
\(82\) −1.21098 −0.133730
\(83\) −9.07907 −0.996557 −0.498279 0.867017i \(-0.666034\pi\)
−0.498279 + 0.867017i \(0.666034\pi\)
\(84\) 3.69253 0.402888
\(85\) 0 0
\(86\) −2.25008 −0.242633
\(87\) 9.26618 0.993439
\(88\) −0.894936 −0.0954005
\(89\) −5.69209 −0.603360 −0.301680 0.953409i \(-0.597547\pi\)
−0.301680 + 0.953409i \(0.597547\pi\)
\(90\) 0 0
\(91\) 2.71469 0.284577
\(92\) 5.50159 0.573580
\(93\) −13.3494 −1.38427
\(94\) −1.73268 −0.178712
\(95\) 0 0
\(96\) −4.97837 −0.508103
\(97\) 6.05204 0.614492 0.307246 0.951630i \(-0.400593\pi\)
0.307246 + 0.951630i \(0.400593\pi\)
\(98\) −0.226645 −0.0228946
\(99\) −0.590784 −0.0593760
\(100\) 0 0
\(101\) −8.71518 −0.867192 −0.433596 0.901107i \(-0.642756\pi\)
−0.433596 + 0.901107i \(0.642756\pi\)
\(102\) −2.97604 −0.294672
\(103\) 19.7813 1.94911 0.974553 0.224156i \(-0.0719625\pi\)
0.974553 + 0.224156i \(0.0719625\pi\)
\(104\) −2.42948 −0.238230
\(105\) 0 0
\(106\) −0.490687 −0.0476597
\(107\) 10.0523 0.971791 0.485895 0.874017i \(-0.338494\pi\)
0.485895 + 0.874017i \(0.338494\pi\)
\(108\) 8.89611 0.856029
\(109\) 2.52611 0.241958 0.120979 0.992655i \(-0.461397\pi\)
0.120979 + 0.992655i \(0.461397\pi\)
\(110\) 0 0
\(111\) −2.41637 −0.229352
\(112\) −3.69443 −0.349091
\(113\) 3.09105 0.290781 0.145391 0.989374i \(-0.453556\pi\)
0.145391 + 0.989374i \(0.453556\pi\)
\(114\) −2.74206 −0.256817
\(115\) 0 0
\(116\) −9.52875 −0.884722
\(117\) −1.60380 −0.148271
\(118\) −2.72486 −0.250844
\(119\) −6.92945 −0.635222
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0.116321 0.0105312
\(123\) 10.1248 0.912919
\(124\) 13.7277 1.23278
\(125\) 0 0
\(126\) 0.133898 0.0119286
\(127\) −13.1184 −1.16407 −0.582035 0.813164i \(-0.697743\pi\)
−0.582035 + 0.813164i \(0.697743\pi\)
\(128\) 6.79408 0.600518
\(129\) 18.8125 1.65635
\(130\) 0 0
\(131\) 0.919423 0.0803304 0.0401652 0.999193i \(-0.487212\pi\)
0.0401652 + 0.999193i \(0.487212\pi\)
\(132\) 3.69253 0.321394
\(133\) −6.38464 −0.553618
\(134\) −3.12228 −0.269724
\(135\) 0 0
\(136\) 6.20142 0.531767
\(137\) 13.1365 1.12233 0.561164 0.827704i \(-0.310354\pi\)
0.561164 + 0.827704i \(0.310354\pi\)
\(138\) 1.21255 0.103219
\(139\) −19.8836 −1.68651 −0.843254 0.537515i \(-0.819363\pi\)
−0.843254 + 0.537515i \(0.819363\pi\)
\(140\) 0 0
\(141\) 14.4866 1.21999
\(142\) −2.03276 −0.170585
\(143\) 2.71469 0.227014
\(144\) 2.18261 0.181884
\(145\) 0 0
\(146\) −1.57161 −0.130067
\(147\) 1.89494 0.156292
\(148\) 2.48485 0.204253
\(149\) −7.23587 −0.592785 −0.296393 0.955066i \(-0.595784\pi\)
−0.296393 + 0.955066i \(0.595784\pi\)
\(150\) 0 0
\(151\) −7.43840 −0.605329 −0.302664 0.953097i \(-0.597876\pi\)
−0.302664 + 0.953097i \(0.597876\pi\)
\(152\) 5.71384 0.463454
\(153\) 4.09381 0.330965
\(154\) −0.226645 −0.0182635
\(155\) 0 0
\(156\) 10.0241 0.802570
\(157\) −14.8104 −1.18200 −0.590999 0.806672i \(-0.701266\pi\)
−0.590999 + 0.806672i \(0.701266\pi\)
\(158\) 2.45714 0.195480
\(159\) 4.10255 0.325353
\(160\) 0 0
\(161\) 2.82331 0.222508
\(162\) 2.36239 0.185607
\(163\) 12.0213 0.941585 0.470792 0.882244i \(-0.343968\pi\)
0.470792 + 0.882244i \(0.343968\pi\)
\(164\) −10.4117 −0.813014
\(165\) 0 0
\(166\) 2.05772 0.159710
\(167\) 2.24124 0.173432 0.0867162 0.996233i \(-0.472363\pi\)
0.0867162 + 0.996233i \(0.472363\pi\)
\(168\) −1.69585 −0.130837
\(169\) −5.63044 −0.433111
\(170\) 0 0
\(171\) 3.77194 0.288447
\(172\) −19.3456 −1.47509
\(173\) −5.43476 −0.413197 −0.206598 0.978426i \(-0.566239\pi\)
−0.206598 + 0.978426i \(0.566239\pi\)
\(174\) −2.10013 −0.159210
\(175\) 0 0
\(176\) −3.69443 −0.278478
\(177\) 22.7821 1.71241
\(178\) 1.29008 0.0966957
\(179\) 8.92059 0.666756 0.333378 0.942793i \(-0.391811\pi\)
0.333378 + 0.942793i \(0.391811\pi\)
\(180\) 0 0
\(181\) −5.84320 −0.434322 −0.217161 0.976136i \(-0.569680\pi\)
−0.217161 + 0.976136i \(0.569680\pi\)
\(182\) −0.615271 −0.0456069
\(183\) −0.972539 −0.0718922
\(184\) −2.52668 −0.186269
\(185\) 0 0
\(186\) 3.02557 0.221846
\(187\) −6.92945 −0.506732
\(188\) −14.8971 −1.08648
\(189\) 4.56531 0.332078
\(190\) 0 0
\(191\) 17.5949 1.27312 0.636560 0.771227i \(-0.280357\pi\)
0.636560 + 0.771227i \(0.280357\pi\)
\(192\) −12.8731 −0.929036
\(193\) 8.04491 0.579085 0.289543 0.957165i \(-0.406497\pi\)
0.289543 + 0.957165i \(0.406497\pi\)
\(194\) −1.37166 −0.0984797
\(195\) 0 0
\(196\) −1.94863 −0.139188
\(197\) 11.0440 0.786856 0.393428 0.919356i \(-0.371289\pi\)
0.393428 + 0.919356i \(0.371289\pi\)
\(198\) 0.133898 0.00951571
\(199\) 7.87899 0.558527 0.279264 0.960215i \(-0.409910\pi\)
0.279264 + 0.960215i \(0.409910\pi\)
\(200\) 0 0
\(201\) 26.1048 1.84129
\(202\) 1.97525 0.138978
\(203\) −4.88997 −0.343209
\(204\) −25.5872 −1.79146
\(205\) 0 0
\(206\) −4.48332 −0.312368
\(207\) −1.66796 −0.115932
\(208\) −10.0292 −0.695403
\(209\) −6.38464 −0.441635
\(210\) 0 0
\(211\) −21.0258 −1.44748 −0.723738 0.690074i \(-0.757578\pi\)
−0.723738 + 0.690074i \(0.757578\pi\)
\(212\) −4.21880 −0.289748
\(213\) 16.9956 1.16452
\(214\) −2.27830 −0.155741
\(215\) 0 0
\(216\) −4.08566 −0.277994
\(217\) 7.04479 0.478231
\(218\) −0.572530 −0.0387766
\(219\) 13.1399 0.887915
\(220\) 0 0
\(221\) −18.8113 −1.26539
\(222\) 0.547658 0.0367564
\(223\) −3.27301 −0.219177 −0.109588 0.993977i \(-0.534953\pi\)
−0.109588 + 0.993977i \(0.534953\pi\)
\(224\) 2.62720 0.175537
\(225\) 0 0
\(226\) −0.700570 −0.0466012
\(227\) −22.3224 −1.48159 −0.740794 0.671732i \(-0.765551\pi\)
−0.740794 + 0.671732i \(0.765551\pi\)
\(228\) −23.5755 −1.56133
\(229\) −2.14092 −0.141476 −0.0707378 0.997495i \(-0.522535\pi\)
−0.0707378 + 0.997495i \(0.522535\pi\)
\(230\) 0 0
\(231\) 1.89494 0.124678
\(232\) 4.37621 0.287312
\(233\) 10.3046 0.675075 0.337538 0.941312i \(-0.390406\pi\)
0.337538 + 0.941312i \(0.390406\pi\)
\(234\) 0.363492 0.0237622
\(235\) 0 0
\(236\) −23.4277 −1.52501
\(237\) −20.5437 −1.33446
\(238\) 1.57052 0.101802
\(239\) −10.4572 −0.676422 −0.338211 0.941070i \(-0.609822\pi\)
−0.338211 + 0.941070i \(0.609822\pi\)
\(240\) 0 0
\(241\) 23.4618 1.51131 0.755653 0.654973i \(-0.227320\pi\)
0.755653 + 0.654973i \(0.227320\pi\)
\(242\) −0.226645 −0.0145693
\(243\) −6.05560 −0.388467
\(244\) 1.00010 0.0640247
\(245\) 0 0
\(246\) −2.29472 −0.146306
\(247\) −17.3323 −1.10283
\(248\) −6.30464 −0.400345
\(249\) −17.2043 −1.09028
\(250\) 0 0
\(251\) −7.37954 −0.465793 −0.232896 0.972502i \(-0.574820\pi\)
−0.232896 + 0.972502i \(0.574820\pi\)
\(252\) 1.15122 0.0725201
\(253\) 2.82331 0.177500
\(254\) 2.97322 0.186556
\(255\) 0 0
\(256\) 12.0470 0.752938
\(257\) −29.3675 −1.83189 −0.915947 0.401300i \(-0.868559\pi\)
−0.915947 + 0.401300i \(0.868559\pi\)
\(258\) −4.26376 −0.265450
\(259\) 1.27517 0.0792355
\(260\) 0 0
\(261\) 2.88891 0.178819
\(262\) −0.208382 −0.0128739
\(263\) 15.2412 0.939814 0.469907 0.882716i \(-0.344287\pi\)
0.469907 + 0.882716i \(0.344287\pi\)
\(264\) −1.69585 −0.104372
\(265\) 0 0
\(266\) 1.44704 0.0887240
\(267\) −10.7861 −0.660102
\(268\) −26.8445 −1.63979
\(269\) 26.2583 1.60099 0.800497 0.599336i \(-0.204569\pi\)
0.800497 + 0.599336i \(0.204569\pi\)
\(270\) 0 0
\(271\) −14.5816 −0.885771 −0.442886 0.896578i \(-0.646045\pi\)
−0.442886 + 0.896578i \(0.646045\pi\)
\(272\) 25.6004 1.55225
\(273\) 5.14417 0.311339
\(274\) −2.97732 −0.179867
\(275\) 0 0
\(276\) 10.4252 0.627521
\(277\) 7.25963 0.436189 0.218095 0.975928i \(-0.430016\pi\)
0.218095 + 0.975928i \(0.430016\pi\)
\(278\) 4.50652 0.270283
\(279\) −4.16194 −0.249169
\(280\) 0 0
\(281\) −9.68302 −0.577640 −0.288820 0.957383i \(-0.593263\pi\)
−0.288820 + 0.957383i \(0.593263\pi\)
\(282\) −3.28331 −0.195519
\(283\) 3.97480 0.236277 0.118139 0.992997i \(-0.462307\pi\)
0.118139 + 0.992997i \(0.462307\pi\)
\(284\) −17.4771 −1.03708
\(285\) 0 0
\(286\) −0.615271 −0.0363817
\(287\) −5.34306 −0.315391
\(288\) −1.55210 −0.0914586
\(289\) 31.0173 1.82455
\(290\) 0 0
\(291\) 11.4682 0.672280
\(292\) −13.5123 −0.790747
\(293\) −18.9865 −1.10920 −0.554602 0.832116i \(-0.687130\pi\)
−0.554602 + 0.832116i \(0.687130\pi\)
\(294\) −0.429477 −0.0250476
\(295\) 0 0
\(296\) −1.14120 −0.0663309
\(297\) 4.56531 0.264906
\(298\) 1.63997 0.0950009
\(299\) 7.66441 0.443244
\(300\) 0 0
\(301\) −9.92780 −0.572229
\(302\) 1.68587 0.0970112
\(303\) −16.5147 −0.948745
\(304\) 23.5876 1.35284
\(305\) 0 0
\(306\) −0.927839 −0.0530410
\(307\) 2.29041 0.130721 0.0653604 0.997862i \(-0.479180\pi\)
0.0653604 + 0.997862i \(0.479180\pi\)
\(308\) −1.94863 −0.111034
\(309\) 37.4843 2.13240
\(310\) 0 0
\(311\) −3.70629 −0.210165 −0.105082 0.994464i \(-0.533511\pi\)
−0.105082 + 0.994464i \(0.533511\pi\)
\(312\) −4.60370 −0.260633
\(313\) −10.4251 −0.589263 −0.294631 0.955611i \(-0.595197\pi\)
−0.294631 + 0.955611i \(0.595197\pi\)
\(314\) 3.35670 0.189429
\(315\) 0 0
\(316\) 21.1259 1.18842
\(317\) −8.16477 −0.458579 −0.229289 0.973358i \(-0.573640\pi\)
−0.229289 + 0.973358i \(0.573640\pi\)
\(318\) −0.929820 −0.0521417
\(319\) −4.88997 −0.273786
\(320\) 0 0
\(321\) 19.0484 1.06318
\(322\) −0.639888 −0.0356595
\(323\) 44.2420 2.46169
\(324\) 20.3112 1.12840
\(325\) 0 0
\(326\) −2.72457 −0.150900
\(327\) 4.78682 0.264712
\(328\) 4.78170 0.264025
\(329\) −7.64491 −0.421478
\(330\) 0 0
\(331\) 11.2519 0.618459 0.309230 0.950987i \(-0.399929\pi\)
0.309230 + 0.950987i \(0.399929\pi\)
\(332\) 17.6918 0.970962
\(333\) −0.753352 −0.0412834
\(334\) −0.507965 −0.0277946
\(335\) 0 0
\(336\) −7.00071 −0.381920
\(337\) −23.5677 −1.28381 −0.641907 0.766782i \(-0.721857\pi\)
−0.641907 + 0.766782i \(0.721857\pi\)
\(338\) 1.27611 0.0694112
\(339\) 5.85734 0.318127
\(340\) 0 0
\(341\) 7.04479 0.381497
\(342\) −0.854890 −0.0462272
\(343\) −1.00000 −0.0539949
\(344\) 8.88475 0.479033
\(345\) 0 0
\(346\) 1.23176 0.0662197
\(347\) 26.8229 1.43993 0.719965 0.694011i \(-0.244158\pi\)
0.719965 + 0.694011i \(0.244158\pi\)
\(348\) −18.0564 −0.967924
\(349\) 20.5516 1.10010 0.550052 0.835131i \(-0.314608\pi\)
0.550052 + 0.835131i \(0.314608\pi\)
\(350\) 0 0
\(351\) 12.3934 0.661512
\(352\) 2.62720 0.140030
\(353\) −34.1433 −1.81726 −0.908631 0.417599i \(-0.862872\pi\)
−0.908631 + 0.417599i \(0.862872\pi\)
\(354\) −5.16344 −0.274434
\(355\) 0 0
\(356\) 11.0918 0.587864
\(357\) −13.1309 −0.694959
\(358\) −2.02180 −0.106856
\(359\) −21.2017 −1.11898 −0.559491 0.828836i \(-0.689003\pi\)
−0.559491 + 0.828836i \(0.689003\pi\)
\(360\) 0 0
\(361\) 21.7636 1.14545
\(362\) 1.32433 0.0696053
\(363\) 1.89494 0.0994584
\(364\) −5.28994 −0.277268
\(365\) 0 0
\(366\) 0.220421 0.0115216
\(367\) 32.1138 1.67633 0.838163 0.545421i \(-0.183630\pi\)
0.838163 + 0.545421i \(0.183630\pi\)
\(368\) −10.4305 −0.543728
\(369\) 3.15659 0.164326
\(370\) 0 0
\(371\) −2.16500 −0.112401
\(372\) 26.0131 1.34872
\(373\) 10.5291 0.545177 0.272589 0.962131i \(-0.412120\pi\)
0.272589 + 0.962131i \(0.412120\pi\)
\(374\) 1.57052 0.0812098
\(375\) 0 0
\(376\) 6.84171 0.352834
\(377\) −13.2748 −0.683685
\(378\) −1.03470 −0.0532194
\(379\) −25.1566 −1.29221 −0.646105 0.763249i \(-0.723603\pi\)
−0.646105 + 0.763249i \(0.723603\pi\)
\(380\) 0 0
\(381\) −24.8585 −1.27354
\(382\) −3.98778 −0.204033
\(383\) −20.8031 −1.06299 −0.531494 0.847062i \(-0.678369\pi\)
−0.531494 + 0.847062i \(0.678369\pi\)
\(384\) 12.8744 0.656992
\(385\) 0 0
\(386\) −1.82334 −0.0928053
\(387\) 5.86518 0.298144
\(388\) −11.7932 −0.598709
\(389\) 4.17853 0.211860 0.105930 0.994374i \(-0.466218\pi\)
0.105930 + 0.994374i \(0.466218\pi\)
\(390\) 0 0
\(391\) −19.5640 −0.989393
\(392\) 0.894936 0.0452011
\(393\) 1.74225 0.0878848
\(394\) −2.50307 −0.126103
\(395\) 0 0
\(396\) 1.15122 0.0578510
\(397\) 20.1701 1.01231 0.506155 0.862442i \(-0.331066\pi\)
0.506155 + 0.862442i \(0.331066\pi\)
\(398\) −1.78573 −0.0895106
\(399\) −12.0985 −0.605682
\(400\) 0 0
\(401\) −9.05971 −0.452421 −0.226210 0.974078i \(-0.572634\pi\)
−0.226210 + 0.974078i \(0.572634\pi\)
\(402\) −5.91652 −0.295089
\(403\) 19.1244 0.952656
\(404\) 16.9827 0.844919
\(405\) 0 0
\(406\) 1.10829 0.0550033
\(407\) 1.27517 0.0632080
\(408\) 11.7513 0.581776
\(409\) 5.19698 0.256974 0.128487 0.991711i \(-0.458988\pi\)
0.128487 + 0.991711i \(0.458988\pi\)
\(410\) 0 0
\(411\) 24.8929 1.22787
\(412\) −38.5464 −1.89905
\(413\) −12.0226 −0.591595
\(414\) 0.378035 0.0185794
\(415\) 0 0
\(416\) 7.13203 0.349677
\(417\) −37.6782 −1.84511
\(418\) 1.44704 0.0707772
\(419\) −34.0187 −1.66192 −0.830962 0.556329i \(-0.812210\pi\)
−0.830962 + 0.556329i \(0.812210\pi\)
\(420\) 0 0
\(421\) −34.3577 −1.67449 −0.837245 0.546828i \(-0.815835\pi\)
−0.837245 + 0.546828i \(0.815835\pi\)
\(422\) 4.76539 0.231975
\(423\) 4.51649 0.219599
\(424\) 1.93754 0.0940953
\(425\) 0 0
\(426\) −3.85195 −0.186628
\(427\) 0.513231 0.0248370
\(428\) −19.5882 −0.946831
\(429\) 5.14417 0.248363
\(430\) 0 0
\(431\) −25.5131 −1.22892 −0.614462 0.788947i \(-0.710627\pi\)
−0.614462 + 0.788947i \(0.710627\pi\)
\(432\) −16.8662 −0.811477
\(433\) −1.07504 −0.0516629 −0.0258315 0.999666i \(-0.508223\pi\)
−0.0258315 + 0.999666i \(0.508223\pi\)
\(434\) −1.59666 −0.0766423
\(435\) 0 0
\(436\) −4.92247 −0.235743
\(437\) −18.0258 −0.862291
\(438\) −2.97810 −0.142299
\(439\) −11.6914 −0.558002 −0.279001 0.960291i \(-0.590003\pi\)
−0.279001 + 0.960291i \(0.590003\pi\)
\(440\) 0 0
\(441\) 0.590784 0.0281326
\(442\) 4.26349 0.202793
\(443\) 6.92747 0.329134 0.164567 0.986366i \(-0.447377\pi\)
0.164567 + 0.986366i \(0.447377\pi\)
\(444\) 4.70862 0.223461
\(445\) 0 0
\(446\) 0.741809 0.0351257
\(447\) −13.7115 −0.648532
\(448\) 6.79342 0.320959
\(449\) −6.65819 −0.314220 −0.157110 0.987581i \(-0.550218\pi\)
−0.157110 + 0.987581i \(0.550218\pi\)
\(450\) 0 0
\(451\) −5.34306 −0.251595
\(452\) −6.02332 −0.283313
\(453\) −14.0953 −0.662255
\(454\) 5.05925 0.237442
\(455\) 0 0
\(456\) 10.8274 0.507038
\(457\) 9.78442 0.457696 0.228848 0.973462i \(-0.426504\pi\)
0.228848 + 0.973462i \(0.426504\pi\)
\(458\) 0.485227 0.0226732
\(459\) −31.6351 −1.47660
\(460\) 0 0
\(461\) −9.87760 −0.460046 −0.230023 0.973185i \(-0.573880\pi\)
−0.230023 + 0.973185i \(0.573880\pi\)
\(462\) −0.429477 −0.0199811
\(463\) 14.5142 0.674531 0.337266 0.941410i \(-0.390498\pi\)
0.337266 + 0.941410i \(0.390498\pi\)
\(464\) 18.0657 0.838677
\(465\) 0 0
\(466\) −2.33548 −0.108189
\(467\) −24.3690 −1.12766 −0.563832 0.825889i \(-0.690674\pi\)
−0.563832 + 0.825889i \(0.690674\pi\)
\(468\) 3.12521 0.144463
\(469\) −13.7761 −0.636121
\(470\) 0 0
\(471\) −28.0647 −1.29316
\(472\) 10.7595 0.495245
\(473\) −9.92780 −0.456481
\(474\) 4.65612 0.213863
\(475\) 0 0
\(476\) 13.5030 0.618907
\(477\) 1.27905 0.0585637
\(478\) 2.37007 0.108405
\(479\) −6.49628 −0.296823 −0.148411 0.988926i \(-0.547416\pi\)
−0.148411 + 0.988926i \(0.547416\pi\)
\(480\) 0 0
\(481\) 3.46171 0.157840
\(482\) −5.31748 −0.242205
\(483\) 5.34999 0.243433
\(484\) −1.94863 −0.0885742
\(485\) 0 0
\(486\) 1.37247 0.0622565
\(487\) 30.2243 1.36960 0.684798 0.728733i \(-0.259891\pi\)
0.684798 + 0.728733i \(0.259891\pi\)
\(488\) −0.459309 −0.0207919
\(489\) 22.7797 1.03013
\(490\) 0 0
\(491\) −15.3260 −0.691653 −0.345826 0.938298i \(-0.612401\pi\)
−0.345826 + 0.938298i \(0.612401\pi\)
\(492\) −19.7294 −0.889471
\(493\) 33.8848 1.52609
\(494\) 3.92828 0.176742
\(495\) 0 0
\(496\) −26.0265 −1.16862
\(497\) −8.96893 −0.402312
\(498\) 3.89925 0.174730
\(499\) 23.5020 1.05209 0.526046 0.850456i \(-0.323674\pi\)
0.526046 + 0.850456i \(0.323674\pi\)
\(500\) 0 0
\(501\) 4.24701 0.189742
\(502\) 1.67253 0.0746488
\(503\) 36.5297 1.62878 0.814390 0.580318i \(-0.197072\pi\)
0.814390 + 0.580318i \(0.197072\pi\)
\(504\) −0.528714 −0.0235508
\(505\) 0 0
\(506\) −0.639888 −0.0284465
\(507\) −10.6693 −0.473842
\(508\) 25.5629 1.13417
\(509\) −9.57002 −0.424184 −0.212092 0.977250i \(-0.568028\pi\)
−0.212092 + 0.977250i \(0.568028\pi\)
\(510\) 0 0
\(511\) −6.93424 −0.306753
\(512\) −16.3186 −0.721185
\(513\) −29.1479 −1.28691
\(514\) 6.65598 0.293583
\(515\) 0 0
\(516\) −36.6587 −1.61381
\(517\) −7.64491 −0.336223
\(518\) −0.289011 −0.0126984
\(519\) −10.2985 −0.452055
\(520\) 0 0
\(521\) 17.6884 0.774943 0.387471 0.921882i \(-0.373349\pi\)
0.387471 + 0.921882i \(0.373349\pi\)
\(522\) −0.654757 −0.0286579
\(523\) 37.8115 1.65338 0.826691 0.562656i \(-0.190220\pi\)
0.826691 + 0.562656i \(0.190220\pi\)
\(524\) −1.79162 −0.0782672
\(525\) 0 0
\(526\) −3.45434 −0.150616
\(527\) −48.8165 −2.12648
\(528\) −7.00071 −0.304667
\(529\) −15.0289 −0.653432
\(530\) 0 0
\(531\) 7.10277 0.308234
\(532\) 12.4413 0.539399
\(533\) −14.5048 −0.628271
\(534\) 2.44462 0.105789
\(535\) 0 0
\(536\) 12.3287 0.532520
\(537\) 16.9039 0.729459
\(538\) −5.95129 −0.256579
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) −4.25551 −0.182959 −0.0914794 0.995807i \(-0.529160\pi\)
−0.0914794 + 0.995807i \(0.529160\pi\)
\(542\) 3.30485 0.141955
\(543\) −11.0725 −0.475166
\(544\) −18.2050 −0.780534
\(545\) 0 0
\(546\) −1.16590 −0.0498959
\(547\) 0.773105 0.0330556 0.0165278 0.999863i \(-0.494739\pi\)
0.0165278 + 0.999863i \(0.494739\pi\)
\(548\) −25.5982 −1.09350
\(549\) −0.303208 −0.0129406
\(550\) 0 0
\(551\) 31.2207 1.33005
\(552\) −4.78790 −0.203787
\(553\) 10.8414 0.461022
\(554\) −1.64536 −0.0699045
\(555\) 0 0
\(556\) 38.7459 1.64319
\(557\) −24.5625 −1.04074 −0.520372 0.853940i \(-0.674207\pi\)
−0.520372 + 0.853940i \(0.674207\pi\)
\(558\) 0.943282 0.0399323
\(559\) −26.9509 −1.13990
\(560\) 0 0
\(561\) −13.1309 −0.554386
\(562\) 2.19460 0.0925738
\(563\) −3.70616 −0.156196 −0.0780981 0.996946i \(-0.524885\pi\)
−0.0780981 + 0.996946i \(0.524885\pi\)
\(564\) −28.2291 −1.18866
\(565\) 0 0
\(566\) −0.900867 −0.0378663
\(567\) 10.4233 0.437739
\(568\) 8.02662 0.336790
\(569\) 14.2498 0.597381 0.298691 0.954350i \(-0.403450\pi\)
0.298691 + 0.954350i \(0.403450\pi\)
\(570\) 0 0
\(571\) 8.31485 0.347966 0.173983 0.984749i \(-0.444336\pi\)
0.173983 + 0.984749i \(0.444336\pi\)
\(572\) −5.28994 −0.221183
\(573\) 33.3412 1.39285
\(574\) 1.21098 0.0505452
\(575\) 0 0
\(576\) −4.01344 −0.167227
\(577\) 9.01917 0.375473 0.187736 0.982219i \(-0.439885\pi\)
0.187736 + 0.982219i \(0.439885\pi\)
\(578\) −7.02990 −0.292405
\(579\) 15.2446 0.633543
\(580\) 0 0
\(581\) 9.07907 0.376663
\(582\) −2.59921 −0.107741
\(583\) −2.16500 −0.0896653
\(584\) 6.20571 0.256794
\(585\) 0 0
\(586\) 4.30319 0.177763
\(587\) 5.17843 0.213737 0.106868 0.994273i \(-0.465918\pi\)
0.106868 + 0.994273i \(0.465918\pi\)
\(588\) −3.69253 −0.152278
\(589\) −44.9784 −1.85330
\(590\) 0 0
\(591\) 20.9278 0.860853
\(592\) −4.71104 −0.193623
\(593\) 17.3248 0.711443 0.355722 0.934592i \(-0.384235\pi\)
0.355722 + 0.934592i \(0.384235\pi\)
\(594\) −1.03470 −0.0424544
\(595\) 0 0
\(596\) 14.1000 0.577560
\(597\) 14.9302 0.611052
\(598\) −1.73710 −0.0710352
\(599\) 24.7227 1.01014 0.505070 0.863078i \(-0.331467\pi\)
0.505070 + 0.863078i \(0.331467\pi\)
\(600\) 0 0
\(601\) 46.3237 1.88958 0.944792 0.327670i \(-0.106263\pi\)
0.944792 + 0.327670i \(0.106263\pi\)
\(602\) 2.25008 0.0917065
\(603\) 8.13869 0.331433
\(604\) 14.4947 0.589782
\(605\) 0 0
\(606\) 3.74297 0.152048
\(607\) 32.6404 1.32483 0.662416 0.749136i \(-0.269531\pi\)
0.662416 + 0.749136i \(0.269531\pi\)
\(608\) −16.7737 −0.680263
\(609\) −9.26618 −0.375485
\(610\) 0 0
\(611\) −20.7536 −0.839600
\(612\) −7.97732 −0.322464
\(613\) −17.5678 −0.709555 −0.354778 0.934951i \(-0.615443\pi\)
−0.354778 + 0.934951i \(0.615443\pi\)
\(614\) −0.519110 −0.0209496
\(615\) 0 0
\(616\) 0.894936 0.0360580
\(617\) −41.9900 −1.69045 −0.845226 0.534409i \(-0.820534\pi\)
−0.845226 + 0.534409i \(0.820534\pi\)
\(618\) −8.49560 −0.341743
\(619\) 9.54044 0.383463 0.191731 0.981447i \(-0.438590\pi\)
0.191731 + 0.981447i \(0.438590\pi\)
\(620\) 0 0
\(621\) 12.8893 0.517229
\(622\) 0.840011 0.0336814
\(623\) 5.69209 0.228049
\(624\) −19.0048 −0.760800
\(625\) 0 0
\(626\) 2.36280 0.0944364
\(627\) −12.0985 −0.483167
\(628\) 28.8600 1.15164
\(629\) −8.83626 −0.352325
\(630\) 0 0
\(631\) −29.9289 −1.19145 −0.595726 0.803188i \(-0.703135\pi\)
−0.595726 + 0.803188i \(0.703135\pi\)
\(632\) −9.70234 −0.385939
\(633\) −39.8426 −1.58360
\(634\) 1.85050 0.0734928
\(635\) 0 0
\(636\) −7.99435 −0.316997
\(637\) −2.71469 −0.107560
\(638\) 1.10829 0.0438774
\(639\) 5.29870 0.209613
\(640\) 0 0
\(641\) −29.4319 −1.16249 −0.581246 0.813728i \(-0.697434\pi\)
−0.581246 + 0.813728i \(0.697434\pi\)
\(642\) −4.31722 −0.170387
\(643\) −13.3683 −0.527196 −0.263598 0.964633i \(-0.584909\pi\)
−0.263598 + 0.964633i \(0.584909\pi\)
\(644\) −5.50159 −0.216793
\(645\) 0 0
\(646\) −10.0272 −0.394516
\(647\) 39.5103 1.55331 0.776655 0.629926i \(-0.216915\pi\)
0.776655 + 0.629926i \(0.216915\pi\)
\(648\) −9.32821 −0.366447
\(649\) −12.0226 −0.471929
\(650\) 0 0
\(651\) 13.3494 0.523205
\(652\) −23.4252 −0.917401
\(653\) 12.4194 0.486009 0.243004 0.970025i \(-0.421867\pi\)
0.243004 + 0.970025i \(0.421867\pi\)
\(654\) −1.08491 −0.0424233
\(655\) 0 0
\(656\) 19.7396 0.770701
\(657\) 4.09664 0.159825
\(658\) 1.73268 0.0675468
\(659\) −22.6922 −0.883962 −0.441981 0.897024i \(-0.645724\pi\)
−0.441981 + 0.897024i \(0.645724\pi\)
\(660\) 0 0
\(661\) 0.980639 0.0381424 0.0190712 0.999818i \(-0.493929\pi\)
0.0190712 + 0.999818i \(0.493929\pi\)
\(662\) −2.55018 −0.0991155
\(663\) −35.6463 −1.38439
\(664\) −8.12519 −0.315318
\(665\) 0 0
\(666\) 0.170743 0.00661617
\(667\) −13.8059 −0.534566
\(668\) −4.36736 −0.168978
\(669\) −6.20214 −0.239789
\(670\) 0 0
\(671\) 0.513231 0.0198130
\(672\) 4.97837 0.192045
\(673\) −14.7455 −0.568397 −0.284199 0.958765i \(-0.591728\pi\)
−0.284199 + 0.958765i \(0.591728\pi\)
\(674\) 5.34149 0.205747
\(675\) 0 0
\(676\) 10.9717 0.421987
\(677\) 9.42359 0.362178 0.181089 0.983467i \(-0.442038\pi\)
0.181089 + 0.983467i \(0.442038\pi\)
\(678\) −1.32753 −0.0509837
\(679\) −6.05204 −0.232256
\(680\) 0 0
\(681\) −42.2995 −1.62092
\(682\) −1.59666 −0.0611394
\(683\) 8.86005 0.339020 0.169510 0.985528i \(-0.445781\pi\)
0.169510 + 0.985528i \(0.445781\pi\)
\(684\) −7.35012 −0.281039
\(685\) 0 0
\(686\) 0.226645 0.00865333
\(687\) −4.05690 −0.154780
\(688\) 36.6776 1.39832
\(689\) −5.87732 −0.223908
\(690\) 0 0
\(691\) −7.69667 −0.292795 −0.146398 0.989226i \(-0.546768\pi\)
−0.146398 + 0.989226i \(0.546768\pi\)
\(692\) 10.5903 0.402584
\(693\) 0.590784 0.0224420
\(694\) −6.07927 −0.230766
\(695\) 0 0
\(696\) 8.29264 0.314332
\(697\) 37.0245 1.40240
\(698\) −4.65792 −0.176305
\(699\) 19.5265 0.738561
\(700\) 0 0
\(701\) −8.21711 −0.310356 −0.155178 0.987887i \(-0.549595\pi\)
−0.155178 + 0.987887i \(0.549595\pi\)
\(702\) −2.80890 −0.106015
\(703\) −8.14153 −0.307063
\(704\) 6.79342 0.256037
\(705\) 0 0
\(706\) 7.73839 0.291238
\(707\) 8.71518 0.327768
\(708\) −44.3940 −1.66843
\(709\) −8.20752 −0.308240 −0.154120 0.988052i \(-0.549254\pi\)
−0.154120 + 0.988052i \(0.549254\pi\)
\(710\) 0 0
\(711\) −6.40491 −0.240203
\(712\) −5.09406 −0.190908
\(713\) 19.8896 0.744871
\(714\) 2.97604 0.111376
\(715\) 0 0
\(716\) −17.3829 −0.649631
\(717\) −19.8158 −0.740034
\(718\) 4.80525 0.179330
\(719\) 2.38303 0.0888720 0.0444360 0.999012i \(-0.485851\pi\)
0.0444360 + 0.999012i \(0.485851\pi\)
\(720\) 0 0
\(721\) −19.7813 −0.736693
\(722\) −4.93260 −0.183572
\(723\) 44.4585 1.65343
\(724\) 11.3863 0.423167
\(725\) 0 0
\(726\) −0.429477 −0.0159394
\(727\) 24.8734 0.922504 0.461252 0.887269i \(-0.347400\pi\)
0.461252 + 0.887269i \(0.347400\pi\)
\(728\) 2.42948 0.0900424
\(729\) 19.7950 0.733148
\(730\) 0 0
\(731\) 68.7942 2.54444
\(732\) 1.89512 0.0700457
\(733\) 3.23427 0.119461 0.0597303 0.998215i \(-0.480976\pi\)
0.0597303 + 0.998215i \(0.480976\pi\)
\(734\) −7.27841 −0.268651
\(735\) 0 0
\(736\) 7.41738 0.273408
\(737\) −13.7761 −0.507449
\(738\) −0.715425 −0.0263351
\(739\) 2.59092 0.0953084 0.0476542 0.998864i \(-0.484825\pi\)
0.0476542 + 0.998864i \(0.484825\pi\)
\(740\) 0 0
\(741\) −32.8437 −1.20654
\(742\) 0.490687 0.0180137
\(743\) −35.2841 −1.29445 −0.647225 0.762299i \(-0.724070\pi\)
−0.647225 + 0.762299i \(0.724070\pi\)
\(744\) −11.9469 −0.437994
\(745\) 0 0
\(746\) −2.38637 −0.0873712
\(747\) −5.36377 −0.196250
\(748\) 13.5030 0.493717
\(749\) −10.0523 −0.367302
\(750\) 0 0
\(751\) −7.09221 −0.258798 −0.129399 0.991593i \(-0.541305\pi\)
−0.129399 + 0.991593i \(0.541305\pi\)
\(752\) 28.2436 1.02994
\(753\) −13.9838 −0.509597
\(754\) 3.00865 0.109569
\(755\) 0 0
\(756\) −8.89611 −0.323549
\(757\) 35.6937 1.29731 0.648654 0.761083i \(-0.275332\pi\)
0.648654 + 0.761083i \(0.275332\pi\)
\(758\) 5.70162 0.207092
\(759\) 5.34999 0.194192
\(760\) 0 0
\(761\) −33.8388 −1.22666 −0.613328 0.789828i \(-0.710170\pi\)
−0.613328 + 0.789828i \(0.710170\pi\)
\(762\) 5.63406 0.204100
\(763\) −2.52611 −0.0914515
\(764\) −34.2859 −1.24042
\(765\) 0 0
\(766\) 4.71491 0.170356
\(767\) −32.6377 −1.17848
\(768\) 22.8283 0.823746
\(769\) 33.1746 1.19631 0.598153 0.801382i \(-0.295902\pi\)
0.598153 + 0.801382i \(0.295902\pi\)
\(770\) 0 0
\(771\) −55.6495 −2.00417
\(772\) −15.6766 −0.564212
\(773\) 38.6688 1.39082 0.695411 0.718613i \(-0.255223\pi\)
0.695411 + 0.718613i \(0.255223\pi\)
\(774\) −1.32931 −0.0477811
\(775\) 0 0
\(776\) 5.41619 0.194430
\(777\) 2.41637 0.0866869
\(778\) −0.947041 −0.0339531
\(779\) 34.1135 1.22224
\(780\) 0 0
\(781\) −8.96893 −0.320934
\(782\) 4.43407 0.158562
\(783\) −22.3242 −0.797803
\(784\) 3.69443 0.131944
\(785\) 0 0
\(786\) −0.394871 −0.0140846
\(787\) −35.4748 −1.26454 −0.632269 0.774749i \(-0.717876\pi\)
−0.632269 + 0.774749i \(0.717876\pi\)
\(788\) −21.5208 −0.766646
\(789\) 28.8812 1.02820
\(790\) 0 0
\(791\) −3.09105 −0.109905
\(792\) −0.528714 −0.0187870
\(793\) 1.39326 0.0494762
\(794\) −4.57146 −0.162235
\(795\) 0 0
\(796\) −15.3533 −0.544182
\(797\) −10.5336 −0.373120 −0.186560 0.982444i \(-0.559734\pi\)
−0.186560 + 0.982444i \(0.559734\pi\)
\(798\) 2.74206 0.0970677
\(799\) 52.9750 1.87412
\(800\) 0 0
\(801\) −3.36279 −0.118818
\(802\) 2.05334 0.0725058
\(803\) −6.93424 −0.244704
\(804\) −50.8687 −1.79400
\(805\) 0 0
\(806\) −4.33445 −0.152674
\(807\) 49.7577 1.75156
\(808\) −7.79953 −0.274386
\(809\) −51.6743 −1.81677 −0.908385 0.418135i \(-0.862684\pi\)
−0.908385 + 0.418135i \(0.862684\pi\)
\(810\) 0 0
\(811\) 20.1386 0.707161 0.353581 0.935404i \(-0.384964\pi\)
0.353581 + 0.935404i \(0.384964\pi\)
\(812\) 9.52875 0.334394
\(813\) −27.6313 −0.969071
\(814\) −0.289011 −0.0101298
\(815\) 0 0
\(816\) 48.5111 1.69823
\(817\) 63.3854 2.21757
\(818\) −1.17787 −0.0411832
\(819\) 1.60380 0.0560412
\(820\) 0 0
\(821\) −3.72653 −0.130057 −0.0650284 0.997883i \(-0.520714\pi\)
−0.0650284 + 0.997883i \(0.520714\pi\)
\(822\) −5.64184 −0.196782
\(823\) −45.3334 −1.58022 −0.790111 0.612964i \(-0.789977\pi\)
−0.790111 + 0.612964i \(0.789977\pi\)
\(824\) 17.7030 0.616712
\(825\) 0 0
\(826\) 2.72486 0.0948101
\(827\) 23.2710 0.809214 0.404607 0.914491i \(-0.367408\pi\)
0.404607 + 0.914491i \(0.367408\pi\)
\(828\) 3.25025 0.112954
\(829\) 49.0037 1.70197 0.850985 0.525190i \(-0.176006\pi\)
0.850985 + 0.525190i \(0.176006\pi\)
\(830\) 0 0
\(831\) 13.7565 0.477209
\(832\) 18.4421 0.639363
\(833\) 6.92945 0.240091
\(834\) 8.53957 0.295701
\(835\) 0 0
\(836\) 12.4413 0.430292
\(837\) 32.1616 1.11167
\(838\) 7.71017 0.266343
\(839\) −6.44771 −0.222600 −0.111300 0.993787i \(-0.535501\pi\)
−0.111300 + 0.993787i \(0.535501\pi\)
\(840\) 0 0
\(841\) −5.08820 −0.175455
\(842\) 7.78698 0.268357
\(843\) −18.3487 −0.631963
\(844\) 40.9716 1.41030
\(845\) 0 0
\(846\) −1.02364 −0.0351934
\(847\) −1.00000 −0.0343604
\(848\) 7.99846 0.274668
\(849\) 7.53199 0.258497
\(850\) 0 0
\(851\) 3.60021 0.123414
\(852\) −33.1181 −1.13461
\(853\) 3.55016 0.121555 0.0607777 0.998151i \(-0.480642\pi\)
0.0607777 + 0.998151i \(0.480642\pi\)
\(854\) −0.116321 −0.00398042
\(855\) 0 0
\(856\) 8.99615 0.307482
\(857\) −51.9633 −1.77503 −0.887516 0.460777i \(-0.847571\pi\)
−0.887516 + 0.460777i \(0.847571\pi\)
\(858\) −1.16590 −0.0398031
\(859\) −36.2287 −1.23611 −0.618054 0.786136i \(-0.712079\pi\)
−0.618054 + 0.786136i \(0.712079\pi\)
\(860\) 0 0
\(861\) −10.1248 −0.345051
\(862\) 5.78241 0.196950
\(863\) −1.69567 −0.0577213 −0.0288607 0.999583i \(-0.509188\pi\)
−0.0288607 + 0.999583i \(0.509188\pi\)
\(864\) 11.9940 0.408043
\(865\) 0 0
\(866\) 0.243651 0.00827960
\(867\) 58.7758 1.99613
\(868\) −13.7277 −0.465948
\(869\) 10.8414 0.367769
\(870\) 0 0
\(871\) −37.3978 −1.26718
\(872\) 2.26071 0.0765573
\(873\) 3.57545 0.121011
\(874\) 4.08545 0.138192
\(875\) 0 0
\(876\) −25.6049 −0.865110
\(877\) 22.4861 0.759303 0.379652 0.925130i \(-0.376044\pi\)
0.379652 + 0.925130i \(0.376044\pi\)
\(878\) 2.64980 0.0894265
\(879\) −35.9783 −1.21352
\(880\) 0 0
\(881\) 54.4829 1.83558 0.917788 0.397071i \(-0.129973\pi\)
0.917788 + 0.397071i \(0.129973\pi\)
\(882\) −0.133898 −0.00450858
\(883\) −48.5342 −1.63331 −0.816654 0.577128i \(-0.804173\pi\)
−0.816654 + 0.577128i \(0.804173\pi\)
\(884\) 36.6564 1.23289
\(885\) 0 0
\(886\) −1.57007 −0.0527476
\(887\) 5.91327 0.198548 0.0992742 0.995060i \(-0.468348\pi\)
0.0992742 + 0.995060i \(0.468348\pi\)
\(888\) −2.16250 −0.0725688
\(889\) 13.1184 0.439977
\(890\) 0 0
\(891\) 10.4233 0.349195
\(892\) 6.37789 0.213547
\(893\) 48.8100 1.63336
\(894\) 3.10764 0.103935
\(895\) 0 0
\(896\) −6.79408 −0.226974
\(897\) 14.5236 0.484928
\(898\) 1.50904 0.0503574
\(899\) −34.4488 −1.14893
\(900\) 0 0
\(901\) 15.0023 0.499799
\(902\) 1.21098 0.0403211
\(903\) −18.8125 −0.626042
\(904\) 2.76629 0.0920055
\(905\) 0 0
\(906\) 3.19462 0.106134
\(907\) 18.1745 0.603474 0.301737 0.953391i \(-0.402434\pi\)
0.301737 + 0.953391i \(0.402434\pi\)
\(908\) 43.4981 1.44354
\(909\) −5.14878 −0.170774
\(910\) 0 0
\(911\) 32.9701 1.09235 0.546174 0.837671i \(-0.316084\pi\)
0.546174 + 0.837671i \(0.316084\pi\)
\(912\) 44.6970 1.48007
\(913\) 9.07907 0.300473
\(914\) −2.21759 −0.0733512
\(915\) 0 0
\(916\) 4.17186 0.137842
\(917\) −0.919423 −0.0303620
\(918\) 7.16993 0.236643
\(919\) 32.4023 1.06885 0.534426 0.845215i \(-0.320528\pi\)
0.534426 + 0.845215i \(0.320528\pi\)
\(920\) 0 0
\(921\) 4.34019 0.143014
\(922\) 2.23871 0.0737279
\(923\) −24.3479 −0.801421
\(924\) −3.69253 −0.121475
\(925\) 0 0
\(926\) −3.28956 −0.108102
\(927\) 11.6865 0.383833
\(928\) −12.8469 −0.421720
\(929\) 15.6805 0.514462 0.257231 0.966350i \(-0.417190\pi\)
0.257231 + 0.966350i \(0.417190\pi\)
\(930\) 0 0
\(931\) 6.38464 0.209248
\(932\) −20.0798 −0.657737
\(933\) −7.02319 −0.229929
\(934\) 5.52311 0.180722
\(935\) 0 0
\(936\) −1.43530 −0.0469141
\(937\) 13.6128 0.444710 0.222355 0.974966i \(-0.428626\pi\)
0.222355 + 0.974966i \(0.428626\pi\)
\(938\) 3.12228 0.101946
\(939\) −19.7549 −0.644678
\(940\) 0 0
\(941\) 28.5175 0.929644 0.464822 0.885404i \(-0.346118\pi\)
0.464822 + 0.885404i \(0.346118\pi\)
\(942\) 6.36072 0.207244
\(943\) −15.0851 −0.491238
\(944\) 44.4168 1.44564
\(945\) 0 0
\(946\) 2.25008 0.0731565
\(947\) 4.38617 0.142531 0.0712657 0.997457i \(-0.477296\pi\)
0.0712657 + 0.997457i \(0.477296\pi\)
\(948\) 40.0322 1.30018
\(949\) −18.8243 −0.611064
\(950\) 0 0
\(951\) −15.4717 −0.501705
\(952\) −6.20142 −0.200989
\(953\) −56.7069 −1.83692 −0.918458 0.395519i \(-0.870565\pi\)
−0.918458 + 0.395519i \(0.870565\pi\)
\(954\) −0.289890 −0.00938553
\(955\) 0 0
\(956\) 20.3773 0.659048
\(957\) −9.26618 −0.299533
\(958\) 1.47235 0.0475694
\(959\) −13.1365 −0.424200
\(960\) 0 0
\(961\) 18.6290 0.600936
\(962\) −0.784577 −0.0252958
\(963\) 5.93872 0.191373
\(964\) −45.7183 −1.47249
\(965\) 0 0
\(966\) −1.21255 −0.0390130
\(967\) −23.3022 −0.749349 −0.374675 0.927156i \(-0.622246\pi\)
−0.374675 + 0.927156i \(0.622246\pi\)
\(968\) 0.894936 0.0287643
\(969\) 83.8358 2.69319
\(970\) 0 0
\(971\) 5.11862 0.164264 0.0821322 0.996621i \(-0.473827\pi\)
0.0821322 + 0.996621i \(0.473827\pi\)
\(972\) 11.8001 0.378490
\(973\) 19.8836 0.637440
\(974\) −6.85018 −0.219494
\(975\) 0 0
\(976\) −1.89610 −0.0606925
\(977\) 0.477556 0.0152784 0.00763918 0.999971i \(-0.497568\pi\)
0.00763918 + 0.999971i \(0.497568\pi\)
\(978\) −5.16289 −0.165091
\(979\) 5.69209 0.181920
\(980\) 0 0
\(981\) 1.49239 0.0476482
\(982\) 3.47356 0.110846
\(983\) −35.0509 −1.11795 −0.558975 0.829185i \(-0.688805\pi\)
−0.558975 + 0.829185i \(0.688805\pi\)
\(984\) 9.06101 0.288855
\(985\) 0 0
\(986\) −7.67981 −0.244575
\(987\) −14.4866 −0.461114
\(988\) 33.7743 1.07450
\(989\) −28.0292 −0.891278
\(990\) 0 0
\(991\) −12.7459 −0.404887 −0.202443 0.979294i \(-0.564888\pi\)
−0.202443 + 0.979294i \(0.564888\pi\)
\(992\) 18.5080 0.587631
\(993\) 21.3216 0.676621
\(994\) 2.03276 0.0644752
\(995\) 0 0
\(996\) 33.5248 1.06227
\(997\) −0.547098 −0.0173268 −0.00866338 0.999962i \(-0.502758\pi\)
−0.00866338 + 0.999962i \(0.502758\pi\)
\(998\) −5.32659 −0.168610
\(999\) 5.82157 0.184186
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 1925.2.a.z.1.3 6
5.2 odd 4 385.2.b.c.309.6 12
5.3 odd 4 385.2.b.c.309.7 yes 12
5.4 even 2 1925.2.a.y.1.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
385.2.b.c.309.6 12 5.2 odd 4
385.2.b.c.309.7 yes 12 5.3 odd 4
1925.2.a.y.1.4 6 5.4 even 2
1925.2.a.z.1.3 6 1.1 even 1 trivial