Properties

Label 4334.2.a.a.1.9
Level $4334$
Weight $2$
Character 4334.1
Self dual yes
Analytic conductor $34.607$
Analytic rank $1$
Dimension $15$
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] = [4334,2,Mod(1,4334)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4334, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4334.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4334 = 2 \cdot 11 \cdot 197 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4334.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: \(34.6071642360\)
Analytic rank: \(1\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - x^{14} - 23 x^{13} + 19 x^{12} + 194 x^{11} - 124 x^{10} - 761 x^{9} + 353 x^{8} + 1417 x^{7} - 465 x^{6} - 1128 x^{5} + 288 x^{4} + 316 x^{3} - 79 x^{2} - 20 x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(-0.275495\) of defining polynomial
Character \(\chi\) \(=\) 4334.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +0.275495 q^{3} +1.00000 q^{4} -0.829257 q^{5} -0.275495 q^{6} +4.83738 q^{7} -1.00000 q^{8} -2.92410 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +0.275495 q^{3} +1.00000 q^{4} -0.829257 q^{5} -0.275495 q^{6} +4.83738 q^{7} -1.00000 q^{8} -2.92410 q^{9} +0.829257 q^{10} +1.00000 q^{11} +0.275495 q^{12} -4.69193 q^{13} -4.83738 q^{14} -0.228456 q^{15} +1.00000 q^{16} -3.61821 q^{17} +2.92410 q^{18} +3.96753 q^{19} -0.829257 q^{20} +1.33267 q^{21} -1.00000 q^{22} +3.89394 q^{23} -0.275495 q^{24} -4.31233 q^{25} +4.69193 q^{26} -1.63206 q^{27} +4.83738 q^{28} +7.12691 q^{29} +0.228456 q^{30} -5.82010 q^{31} -1.00000 q^{32} +0.275495 q^{33} +3.61821 q^{34} -4.01143 q^{35} -2.92410 q^{36} -7.26441 q^{37} -3.96753 q^{38} -1.29260 q^{39} +0.829257 q^{40} -2.08579 q^{41} -1.33267 q^{42} -10.3214 q^{43} +1.00000 q^{44} +2.42483 q^{45} -3.89394 q^{46} +5.67256 q^{47} +0.275495 q^{48} +16.4002 q^{49} +4.31233 q^{50} -0.996800 q^{51} -4.69193 q^{52} +8.95166 q^{53} +1.63206 q^{54} -0.829257 q^{55} -4.83738 q^{56} +1.09303 q^{57} -7.12691 q^{58} -9.73543 q^{59} -0.228456 q^{60} -12.1961 q^{61} +5.82010 q^{62} -14.1450 q^{63} +1.00000 q^{64} +3.89082 q^{65} -0.275495 q^{66} +12.1658 q^{67} -3.61821 q^{68} +1.07276 q^{69} +4.01143 q^{70} -7.84656 q^{71} +2.92410 q^{72} -0.146186 q^{73} +7.26441 q^{74} -1.18803 q^{75} +3.96753 q^{76} +4.83738 q^{77} +1.29260 q^{78} -6.17885 q^{79} -0.829257 q^{80} +8.32268 q^{81} +2.08579 q^{82} -13.7221 q^{83} +1.33267 q^{84} +3.00043 q^{85} +10.3214 q^{86} +1.96343 q^{87} -1.00000 q^{88} -5.85899 q^{89} -2.42483 q^{90} -22.6966 q^{91} +3.89394 q^{92} -1.60341 q^{93} -5.67256 q^{94} -3.29010 q^{95} -0.275495 q^{96} -12.1296 q^{97} -16.4002 q^{98} -2.92410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - 15 q^{2} - q^{3} + 15 q^{4} - 7 q^{5} + q^{6} + q^{7} - 15 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q - 15 q^{2} - q^{3} + 15 q^{4} - 7 q^{5} + q^{6} + q^{7} - 15 q^{8} + 2 q^{9} + 7 q^{10} + 15 q^{11} - q^{12} - q^{13} - q^{14} - 6 q^{15} + 15 q^{16} - 6 q^{17} - 2 q^{18} - 14 q^{19} - 7 q^{20} - 3 q^{21} - 15 q^{22} + 2 q^{23} + q^{24} - 10 q^{25} + q^{26} - 7 q^{27} + q^{28} + 8 q^{29} + 6 q^{30} - 33 q^{31} - 15 q^{32} - q^{33} + 6 q^{34} - 8 q^{35} + 2 q^{36} - 9 q^{37} + 14 q^{38} - 9 q^{39} + 7 q^{40} - 10 q^{41} + 3 q^{42} - 6 q^{43} + 15 q^{44} - 20 q^{45} - 2 q^{46} - q^{47} - q^{48} - 30 q^{49} + 10 q^{50} + 12 q^{51} - q^{52} + 6 q^{53} + 7 q^{54} - 7 q^{55} - q^{56} - 24 q^{57} - 8 q^{58} - 15 q^{59} - 6 q^{60} - 25 q^{61} + 33 q^{62} + 12 q^{63} + 15 q^{64} + 31 q^{65} + q^{66} - 13 q^{67} - 6 q^{68} - 43 q^{69} + 8 q^{70} - 4 q^{71} - 2 q^{72} - 4 q^{73} + 9 q^{74} - 5 q^{75} - 14 q^{76} + q^{77} + 9 q^{78} - 20 q^{79} - 7 q^{80} + 11 q^{81} + 10 q^{82} + q^{83} - 3 q^{84} - q^{85} + 6 q^{86} + 22 q^{87} - 15 q^{88} - 41 q^{89} + 20 q^{90} - 31 q^{91} + 2 q^{92} + 14 q^{93} + q^{94} + 41 q^{95} + q^{96} - 57 q^{97} + 30 q^{98} + 2 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0.275495 0.159057 0.0795285 0.996833i \(-0.474659\pi\)
0.0795285 + 0.996833i \(0.474659\pi\)
\(4\) 1.00000 0.500000
\(5\) −0.829257 −0.370855 −0.185428 0.982658i \(-0.559367\pi\)
−0.185428 + 0.982658i \(0.559367\pi\)
\(6\) −0.275495 −0.112470
\(7\) 4.83738 1.82836 0.914178 0.405313i \(-0.132837\pi\)
0.914178 + 0.405313i \(0.132837\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.92410 −0.974701
\(10\) 0.829257 0.262234
\(11\) 1.00000 0.301511
\(12\) 0.275495 0.0795285
\(13\) −4.69193 −1.30131 −0.650654 0.759374i \(-0.725505\pi\)
−0.650654 + 0.759374i \(0.725505\pi\)
\(14\) −4.83738 −1.29284
\(15\) −0.228456 −0.0589871
\(16\) 1.00000 0.250000
\(17\) −3.61821 −0.877546 −0.438773 0.898598i \(-0.644587\pi\)
−0.438773 + 0.898598i \(0.644587\pi\)
\(18\) 2.92410 0.689218
\(19\) 3.96753 0.910213 0.455107 0.890437i \(-0.349601\pi\)
0.455107 + 0.890437i \(0.349601\pi\)
\(20\) −0.829257 −0.185428
\(21\) 1.33267 0.290813
\(22\) −1.00000 −0.213201
\(23\) 3.89394 0.811942 0.405971 0.913886i \(-0.366933\pi\)
0.405971 + 0.913886i \(0.366933\pi\)
\(24\) −0.275495 −0.0562352
\(25\) −4.31233 −0.862466
\(26\) 4.69193 0.920163
\(27\) −1.63206 −0.314090
\(28\) 4.83738 0.914178
\(29\) 7.12691 1.32343 0.661717 0.749754i \(-0.269828\pi\)
0.661717 + 0.749754i \(0.269828\pi\)
\(30\) 0.228456 0.0417102
\(31\) −5.82010 −1.04532 −0.522661 0.852541i \(-0.675060\pi\)
−0.522661 + 0.852541i \(0.675060\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0.275495 0.0479575
\(34\) 3.61821 0.620519
\(35\) −4.01143 −0.678055
\(36\) −2.92410 −0.487350
\(37\) −7.26441 −1.19426 −0.597131 0.802144i \(-0.703693\pi\)
−0.597131 + 0.802144i \(0.703693\pi\)
\(38\) −3.96753 −0.643618
\(39\) −1.29260 −0.206982
\(40\) 0.829257 0.131117
\(41\) −2.08579 −0.325746 −0.162873 0.986647i \(-0.552076\pi\)
−0.162873 + 0.986647i \(0.552076\pi\)
\(42\) −1.33267 −0.205636
\(43\) −10.3214 −1.57399 −0.786996 0.616958i \(-0.788365\pi\)
−0.786996 + 0.616958i \(0.788365\pi\)
\(44\) 1.00000 0.150756
\(45\) 2.42483 0.361473
\(46\) −3.89394 −0.574130
\(47\) 5.67256 0.827428 0.413714 0.910407i \(-0.364231\pi\)
0.413714 + 0.910407i \(0.364231\pi\)
\(48\) 0.275495 0.0397643
\(49\) 16.4002 2.34289
\(50\) 4.31233 0.609856
\(51\) −0.996800 −0.139580
\(52\) −4.69193 −0.650654
\(53\) 8.95166 1.22961 0.614803 0.788681i \(-0.289236\pi\)
0.614803 + 0.788681i \(0.289236\pi\)
\(54\) 1.63206 0.222095
\(55\) −0.829257 −0.111817
\(56\) −4.83738 −0.646422
\(57\) 1.09303 0.144776
\(58\) −7.12691 −0.935809
\(59\) −9.73543 −1.26744 −0.633722 0.773560i \(-0.718474\pi\)
−0.633722 + 0.773560i \(0.718474\pi\)
\(60\) −0.228456 −0.0294936
\(61\) −12.1961 −1.56155 −0.780775 0.624812i \(-0.785176\pi\)
−0.780775 + 0.624812i \(0.785176\pi\)
\(62\) 5.82010 0.739154
\(63\) −14.1450 −1.78210
\(64\) 1.00000 0.125000
\(65\) 3.89082 0.482597
\(66\) −0.275495 −0.0339111
\(67\) 12.1658 1.48628 0.743142 0.669134i \(-0.233335\pi\)
0.743142 + 0.669134i \(0.233335\pi\)
\(68\) −3.61821 −0.438773
\(69\) 1.07276 0.129145
\(70\) 4.01143 0.479458
\(71\) −7.84656 −0.931216 −0.465608 0.884991i \(-0.654164\pi\)
−0.465608 + 0.884991i \(0.654164\pi\)
\(72\) 2.92410 0.344609
\(73\) −0.146186 −0.0171098 −0.00855491 0.999963i \(-0.502723\pi\)
−0.00855491 + 0.999963i \(0.502723\pi\)
\(74\) 7.26441 0.844470
\(75\) −1.18803 −0.137181
\(76\) 3.96753 0.455107
\(77\) 4.83738 0.551270
\(78\) 1.29260 0.146358
\(79\) −6.17885 −0.695174 −0.347587 0.937648i \(-0.612999\pi\)
−0.347587 + 0.937648i \(0.612999\pi\)
\(80\) −0.829257 −0.0927138
\(81\) 8.32268 0.924743
\(82\) 2.08579 0.230337
\(83\) −13.7221 −1.50620 −0.753100 0.657906i \(-0.771443\pi\)
−0.753100 + 0.657906i \(0.771443\pi\)
\(84\) 1.33267 0.145406
\(85\) 3.00043 0.325442
\(86\) 10.3214 1.11298
\(87\) 1.96343 0.210502
\(88\) −1.00000 −0.106600
\(89\) −5.85899 −0.621052 −0.310526 0.950565i \(-0.600505\pi\)
−0.310526 + 0.950565i \(0.600505\pi\)
\(90\) −2.42483 −0.255600
\(91\) −22.6966 −2.37925
\(92\) 3.89394 0.405971
\(93\) −1.60341 −0.166266
\(94\) −5.67256 −0.585080
\(95\) −3.29010 −0.337557
\(96\) −0.275495 −0.0281176
\(97\) −12.1296 −1.23157 −0.615786 0.787913i \(-0.711161\pi\)
−0.615786 + 0.787913i \(0.711161\pi\)
\(98\) −16.4002 −1.65667
\(99\) −2.92410 −0.293883
\(100\) −4.31233 −0.431233
\(101\) −10.1809 −1.01304 −0.506521 0.862228i \(-0.669069\pi\)
−0.506521 + 0.862228i \(0.669069\pi\)
\(102\) 0.996800 0.0986979
\(103\) −9.12868 −0.899475 −0.449738 0.893161i \(-0.648483\pi\)
−0.449738 + 0.893161i \(0.648483\pi\)
\(104\) 4.69193 0.460082
\(105\) −1.10513 −0.107849
\(106\) −8.95166 −0.869462
\(107\) 3.59450 0.347494 0.173747 0.984790i \(-0.444413\pi\)
0.173747 + 0.984790i \(0.444413\pi\)
\(108\) −1.63206 −0.157045
\(109\) 2.25058 0.215567 0.107783 0.994174i \(-0.465625\pi\)
0.107783 + 0.994174i \(0.465625\pi\)
\(110\) 0.829257 0.0790666
\(111\) −2.00131 −0.189956
\(112\) 4.83738 0.457089
\(113\) 0.169648 0.0159591 0.00797955 0.999968i \(-0.497460\pi\)
0.00797955 + 0.999968i \(0.497460\pi\)
\(114\) −1.09303 −0.102372
\(115\) −3.22908 −0.301113
\(116\) 7.12691 0.661717
\(117\) 13.7197 1.26839
\(118\) 9.73543 0.896219
\(119\) −17.5027 −1.60447
\(120\) 0.228456 0.0208551
\(121\) 1.00000 0.0909091
\(122\) 12.1961 1.10418
\(123\) −0.574625 −0.0518121
\(124\) −5.82010 −0.522661
\(125\) 7.72232 0.690705
\(126\) 14.1450 1.26014
\(127\) 9.99362 0.886790 0.443395 0.896326i \(-0.353774\pi\)
0.443395 + 0.896326i \(0.353774\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −2.84348 −0.250354
\(130\) −3.89082 −0.341247
\(131\) 12.0832 1.05572 0.527858 0.849333i \(-0.322995\pi\)
0.527858 + 0.849333i \(0.322995\pi\)
\(132\) 0.275495 0.0239788
\(133\) 19.1924 1.66419
\(134\) −12.1658 −1.05096
\(135\) 1.35340 0.116482
\(136\) 3.61821 0.310259
\(137\) 18.5604 1.58572 0.792860 0.609404i \(-0.208591\pi\)
0.792860 + 0.609404i \(0.208591\pi\)
\(138\) −1.07276 −0.0913193
\(139\) 7.49907 0.636063 0.318032 0.948080i \(-0.396978\pi\)
0.318032 + 0.948080i \(0.396978\pi\)
\(140\) −4.01143 −0.339028
\(141\) 1.56276 0.131608
\(142\) 7.84656 0.658469
\(143\) −4.69193 −0.392359
\(144\) −2.92410 −0.243675
\(145\) −5.91004 −0.490802
\(146\) 0.146186 0.0120985
\(147\) 4.51817 0.372653
\(148\) −7.26441 −0.597131
\(149\) 5.94309 0.486877 0.243438 0.969916i \(-0.421725\pi\)
0.243438 + 0.969916i \(0.421725\pi\)
\(150\) 1.18803 0.0970019
\(151\) −23.7015 −1.92880 −0.964400 0.264448i \(-0.914810\pi\)
−0.964400 + 0.264448i \(0.914810\pi\)
\(152\) −3.96753 −0.321809
\(153\) 10.5800 0.855345
\(154\) −4.83738 −0.389807
\(155\) 4.82636 0.387663
\(156\) −1.29260 −0.103491
\(157\) −9.43485 −0.752983 −0.376491 0.926420i \(-0.622870\pi\)
−0.376491 + 0.926420i \(0.622870\pi\)
\(158\) 6.17885 0.491563
\(159\) 2.46614 0.195577
\(160\) 0.829257 0.0655586
\(161\) 18.8364 1.48452
\(162\) −8.32268 −0.653892
\(163\) −9.82480 −0.769537 −0.384769 0.923013i \(-0.625719\pi\)
−0.384769 + 0.923013i \(0.625719\pi\)
\(164\) −2.08579 −0.162873
\(165\) −0.228456 −0.0177853
\(166\) 13.7221 1.06504
\(167\) −10.1212 −0.783206 −0.391603 0.920134i \(-0.628079\pi\)
−0.391603 + 0.920134i \(0.628079\pi\)
\(168\) −1.33267 −0.102818
\(169\) 9.01422 0.693402
\(170\) −3.00043 −0.230123
\(171\) −11.6015 −0.887185
\(172\) −10.3214 −0.786996
\(173\) 8.49931 0.646191 0.323095 0.946366i \(-0.395277\pi\)
0.323095 + 0.946366i \(0.395277\pi\)
\(174\) −1.96343 −0.148847
\(175\) −20.8604 −1.57690
\(176\) 1.00000 0.0753778
\(177\) −2.68206 −0.201596
\(178\) 5.85899 0.439150
\(179\) 9.56658 0.715040 0.357520 0.933906i \(-0.383622\pi\)
0.357520 + 0.933906i \(0.383622\pi\)
\(180\) 2.42483 0.180736
\(181\) −3.88330 −0.288644 −0.144322 0.989531i \(-0.546100\pi\)
−0.144322 + 0.989531i \(0.546100\pi\)
\(182\) 22.6966 1.68239
\(183\) −3.35996 −0.248376
\(184\) −3.89394 −0.287065
\(185\) 6.02406 0.442898
\(186\) 1.60341 0.117568
\(187\) −3.61821 −0.264590
\(188\) 5.67256 0.413714
\(189\) −7.89489 −0.574269
\(190\) 3.29010 0.238689
\(191\) 7.50821 0.543275 0.271637 0.962400i \(-0.412435\pi\)
0.271637 + 0.962400i \(0.412435\pi\)
\(192\) 0.275495 0.0198821
\(193\) −3.61012 −0.259862 −0.129931 0.991523i \(-0.541476\pi\)
−0.129931 + 0.991523i \(0.541476\pi\)
\(194\) 12.1296 0.870853
\(195\) 1.07190 0.0767604
\(196\) 16.4002 1.17144
\(197\) 1.00000 0.0712470
\(198\) 2.92410 0.207807
\(199\) −3.39694 −0.240803 −0.120401 0.992725i \(-0.538418\pi\)
−0.120401 + 0.992725i \(0.538418\pi\)
\(200\) 4.31233 0.304928
\(201\) 3.35160 0.236404
\(202\) 10.1809 0.716329
\(203\) 34.4755 2.41971
\(204\) −0.996800 −0.0697899
\(205\) 1.72966 0.120804
\(206\) 9.12868 0.636025
\(207\) −11.3863 −0.791400
\(208\) −4.69193 −0.325327
\(209\) 3.96753 0.274440
\(210\) 1.10513 0.0762611
\(211\) −6.08249 −0.418736 −0.209368 0.977837i \(-0.567141\pi\)
−0.209368 + 0.977837i \(0.567141\pi\)
\(212\) 8.95166 0.614803
\(213\) −2.16169 −0.148116
\(214\) −3.59450 −0.245715
\(215\) 8.55906 0.583723
\(216\) 1.63206 0.111048
\(217\) −28.1540 −1.91122
\(218\) −2.25058 −0.152429
\(219\) −0.0402736 −0.00272144
\(220\) −0.829257 −0.0559085
\(221\) 16.9764 1.14196
\(222\) 2.00131 0.134319
\(223\) −6.28035 −0.420563 −0.210282 0.977641i \(-0.567438\pi\)
−0.210282 + 0.977641i \(0.567438\pi\)
\(224\) −4.83738 −0.323211
\(225\) 12.6097 0.840647
\(226\) −0.169648 −0.0112848
\(227\) −20.0570 −1.33123 −0.665615 0.746295i \(-0.731831\pi\)
−0.665615 + 0.746295i \(0.731831\pi\)
\(228\) 1.09303 0.0723879
\(229\) −5.47928 −0.362081 −0.181040 0.983476i \(-0.557946\pi\)
−0.181040 + 0.983476i \(0.557946\pi\)
\(230\) 3.22908 0.212919
\(231\) 1.33267 0.0876834
\(232\) −7.12691 −0.467905
\(233\) 6.96448 0.456258 0.228129 0.973631i \(-0.426739\pi\)
0.228129 + 0.973631i \(0.426739\pi\)
\(234\) −13.7197 −0.896884
\(235\) −4.70401 −0.306856
\(236\) −9.73543 −0.633722
\(237\) −1.70224 −0.110572
\(238\) 17.5027 1.13453
\(239\) −4.60506 −0.297876 −0.148938 0.988847i \(-0.547586\pi\)
−0.148938 + 0.988847i \(0.547586\pi\)
\(240\) −0.228456 −0.0147468
\(241\) 7.16729 0.461686 0.230843 0.972991i \(-0.425852\pi\)
0.230843 + 0.972991i \(0.425852\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 7.18904 0.461177
\(244\) −12.1961 −0.780775
\(245\) −13.6000 −0.868872
\(246\) 0.574625 0.0366367
\(247\) −18.6154 −1.18447
\(248\) 5.82010 0.369577
\(249\) −3.78038 −0.239572
\(250\) −7.72232 −0.488402
\(251\) 0.901357 0.0568931 0.0284466 0.999595i \(-0.490944\pi\)
0.0284466 + 0.999595i \(0.490944\pi\)
\(252\) −14.1450 −0.891050
\(253\) 3.89394 0.244810
\(254\) −9.99362 −0.627055
\(255\) 0.826604 0.0517639
\(256\) 1.00000 0.0625000
\(257\) −26.9133 −1.67881 −0.839403 0.543509i \(-0.817095\pi\)
−0.839403 + 0.543509i \(0.817095\pi\)
\(258\) 2.84348 0.177027
\(259\) −35.1407 −2.18353
\(260\) 3.89082 0.241298
\(261\) −20.8398 −1.28995
\(262\) −12.0832 −0.746504
\(263\) −7.91989 −0.488361 −0.244181 0.969730i \(-0.578519\pi\)
−0.244181 + 0.969730i \(0.578519\pi\)
\(264\) −0.275495 −0.0169555
\(265\) −7.42323 −0.456006
\(266\) −19.1924 −1.17676
\(267\) −1.61412 −0.0987826
\(268\) 12.1658 0.743142
\(269\) −13.3200 −0.812137 −0.406069 0.913843i \(-0.633101\pi\)
−0.406069 + 0.913843i \(0.633101\pi\)
\(270\) −1.35340 −0.0823652
\(271\) 2.17709 0.132249 0.0661243 0.997811i \(-0.478937\pi\)
0.0661243 + 0.997811i \(0.478937\pi\)
\(272\) −3.61821 −0.219386
\(273\) −6.25281 −0.378437
\(274\) −18.5604 −1.12127
\(275\) −4.31233 −0.260043
\(276\) 1.07276 0.0645725
\(277\) 0.716770 0.0430666 0.0215333 0.999768i \(-0.493145\pi\)
0.0215333 + 0.999768i \(0.493145\pi\)
\(278\) −7.49907 −0.449765
\(279\) 17.0186 1.01888
\(280\) 4.01143 0.239729
\(281\) 4.61908 0.275551 0.137776 0.990463i \(-0.456005\pi\)
0.137776 + 0.990463i \(0.456005\pi\)
\(282\) −1.56276 −0.0930611
\(283\) −17.5616 −1.04393 −0.521965 0.852967i \(-0.674801\pi\)
−0.521965 + 0.852967i \(0.674801\pi\)
\(284\) −7.84656 −0.465608
\(285\) −0.906406 −0.0536909
\(286\) 4.69193 0.277440
\(287\) −10.0898 −0.595579
\(288\) 2.92410 0.172304
\(289\) −3.90852 −0.229913
\(290\) 5.91004 0.347050
\(291\) −3.34164 −0.195890
\(292\) −0.146186 −0.00855491
\(293\) 14.6631 0.856629 0.428314 0.903630i \(-0.359108\pi\)
0.428314 + 0.903630i \(0.359108\pi\)
\(294\) −4.51817 −0.263505
\(295\) 8.07318 0.470039
\(296\) 7.26441 0.422235
\(297\) −1.63206 −0.0947017
\(298\) −5.94309 −0.344274
\(299\) −18.2701 −1.05659
\(300\) −1.18803 −0.0685907
\(301\) −49.9283 −2.87782
\(302\) 23.7015 1.36387
\(303\) −2.80480 −0.161131
\(304\) 3.96753 0.227553
\(305\) 10.1137 0.579109
\(306\) −10.5800 −0.604820
\(307\) 13.5913 0.775695 0.387847 0.921724i \(-0.373219\pi\)
0.387847 + 0.921724i \(0.373219\pi\)
\(308\) 4.83738 0.275635
\(309\) −2.51490 −0.143068
\(310\) −4.82636 −0.274119
\(311\) −23.4225 −1.32817 −0.664084 0.747658i \(-0.731178\pi\)
−0.664084 + 0.747658i \(0.731178\pi\)
\(312\) 1.29260 0.0731792
\(313\) −16.2882 −0.920664 −0.460332 0.887747i \(-0.652270\pi\)
−0.460332 + 0.887747i \(0.652270\pi\)
\(314\) 9.43485 0.532439
\(315\) 11.7298 0.660901
\(316\) −6.17885 −0.347587
\(317\) 23.3554 1.31177 0.655884 0.754862i \(-0.272296\pi\)
0.655884 + 0.754862i \(0.272296\pi\)
\(318\) −2.46614 −0.138294
\(319\) 7.12691 0.399030
\(320\) −0.829257 −0.0463569
\(321\) 0.990267 0.0552713
\(322\) −18.8364 −1.04971
\(323\) −14.3554 −0.798754
\(324\) 8.32268 0.462371
\(325\) 20.2332 1.12233
\(326\) 9.82480 0.544145
\(327\) 0.620024 0.0342874
\(328\) 2.08579 0.115168
\(329\) 27.4403 1.51283
\(330\) 0.228456 0.0125761
\(331\) 34.2480 1.88244 0.941220 0.337793i \(-0.109680\pi\)
0.941220 + 0.337793i \(0.109680\pi\)
\(332\) −13.7221 −0.753100
\(333\) 21.2419 1.16405
\(334\) 10.1212 0.553810
\(335\) −10.0885 −0.551196
\(336\) 1.33267 0.0727032
\(337\) −25.7171 −1.40090 −0.700450 0.713701i \(-0.747017\pi\)
−0.700450 + 0.713701i \(0.747017\pi\)
\(338\) −9.01422 −0.490309
\(339\) 0.0467370 0.00253841
\(340\) 3.00043 0.162721
\(341\) −5.82010 −0.315176
\(342\) 11.6015 0.627335
\(343\) 45.4723 2.45527
\(344\) 10.3214 0.556490
\(345\) −0.889594 −0.0478941
\(346\) −8.49931 −0.456926
\(347\) 26.8693 1.44242 0.721210 0.692717i \(-0.243586\pi\)
0.721210 + 0.692717i \(0.243586\pi\)
\(348\) 1.96343 0.105251
\(349\) −19.8151 −1.06068 −0.530339 0.847786i \(-0.677935\pi\)
−0.530339 + 0.847786i \(0.677935\pi\)
\(350\) 20.8604 1.11503
\(351\) 7.65751 0.408728
\(352\) −1.00000 −0.0533002
\(353\) −23.1388 −1.23156 −0.615778 0.787920i \(-0.711158\pi\)
−0.615778 + 0.787920i \(0.711158\pi\)
\(354\) 2.68206 0.142550
\(355\) 6.50682 0.345346
\(356\) −5.85899 −0.310526
\(357\) −4.82189 −0.255202
\(358\) −9.56658 −0.505609
\(359\) −35.3470 −1.86554 −0.932771 0.360468i \(-0.882617\pi\)
−0.932771 + 0.360468i \(0.882617\pi\)
\(360\) −2.42483 −0.127800
\(361\) −3.25873 −0.171512
\(362\) 3.88330 0.204102
\(363\) 0.275495 0.0144597
\(364\) −22.6966 −1.18963
\(365\) 0.121226 0.00634526
\(366\) 3.35996 0.175628
\(367\) −1.43254 −0.0747779 −0.0373890 0.999301i \(-0.511904\pi\)
−0.0373890 + 0.999301i \(0.511904\pi\)
\(368\) 3.89394 0.202985
\(369\) 6.09906 0.317505
\(370\) −6.02406 −0.313176
\(371\) 43.3026 2.24816
\(372\) −1.60341 −0.0831328
\(373\) 21.8486 1.13128 0.565639 0.824653i \(-0.308630\pi\)
0.565639 + 0.824653i \(0.308630\pi\)
\(374\) 3.61821 0.187093
\(375\) 2.12746 0.109862
\(376\) −5.67256 −0.292540
\(377\) −33.4390 −1.72219
\(378\) 7.89489 0.406069
\(379\) −17.0176 −0.874135 −0.437067 0.899429i \(-0.643983\pi\)
−0.437067 + 0.899429i \(0.643983\pi\)
\(380\) −3.29010 −0.168779
\(381\) 2.75319 0.141050
\(382\) −7.50821 −0.384153
\(383\) 22.3816 1.14365 0.571823 0.820377i \(-0.306236\pi\)
0.571823 + 0.820377i \(0.306236\pi\)
\(384\) −0.275495 −0.0140588
\(385\) −4.01143 −0.204441
\(386\) 3.61012 0.183750
\(387\) 30.1807 1.53417
\(388\) −12.1296 −0.615786
\(389\) −17.5665 −0.890655 −0.445328 0.895368i \(-0.646913\pi\)
−0.445328 + 0.895368i \(0.646913\pi\)
\(390\) −1.07190 −0.0542778
\(391\) −14.0891 −0.712516
\(392\) −16.4002 −0.828335
\(393\) 3.32887 0.167919
\(394\) −1.00000 −0.0503793
\(395\) 5.12385 0.257809
\(396\) −2.92410 −0.146942
\(397\) −17.3428 −0.870411 −0.435205 0.900331i \(-0.643324\pi\)
−0.435205 + 0.900331i \(0.643324\pi\)
\(398\) 3.39694 0.170273
\(399\) 5.28741 0.264702
\(400\) −4.31233 −0.215617
\(401\) 9.02318 0.450596 0.225298 0.974290i \(-0.427664\pi\)
0.225298 + 0.974290i \(0.427664\pi\)
\(402\) −3.35160 −0.167163
\(403\) 27.3075 1.36028
\(404\) −10.1809 −0.506521
\(405\) −6.90165 −0.342946
\(406\) −34.4755 −1.71099
\(407\) −7.26441 −0.360083
\(408\) 0.996800 0.0493489
\(409\) 4.77298 0.236008 0.118004 0.993013i \(-0.462350\pi\)
0.118004 + 0.993013i \(0.462350\pi\)
\(410\) −1.72966 −0.0854217
\(411\) 5.11329 0.252220
\(412\) −9.12868 −0.449738
\(413\) −47.0939 −2.31734
\(414\) 11.3863 0.559605
\(415\) 11.3792 0.558582
\(416\) 4.69193 0.230041
\(417\) 2.06596 0.101170
\(418\) −3.96753 −0.194058
\(419\) −25.3146 −1.23670 −0.618350 0.785903i \(-0.712199\pi\)
−0.618350 + 0.785903i \(0.712199\pi\)
\(420\) −1.10513 −0.0539247
\(421\) −29.7629 −1.45056 −0.725279 0.688455i \(-0.758289\pi\)
−0.725279 + 0.688455i \(0.758289\pi\)
\(422\) 6.08249 0.296091
\(423\) −16.5871 −0.806495
\(424\) −8.95166 −0.434731
\(425\) 15.6029 0.756854
\(426\) 2.16169 0.104734
\(427\) −58.9971 −2.85507
\(428\) 3.59450 0.173747
\(429\) −1.29260 −0.0624075
\(430\) −8.55906 −0.412755
\(431\) −6.96059 −0.335280 −0.167640 0.985848i \(-0.553615\pi\)
−0.167640 + 0.985848i \(0.553615\pi\)
\(432\) −1.63206 −0.0785225
\(433\) 30.8500 1.48256 0.741278 0.671198i \(-0.234220\pi\)
0.741278 + 0.671198i \(0.234220\pi\)
\(434\) 28.1540 1.35144
\(435\) −1.62819 −0.0780656
\(436\) 2.25058 0.107783
\(437\) 15.4493 0.739040
\(438\) 0.0402736 0.00192435
\(439\) 22.4922 1.07349 0.536747 0.843743i \(-0.319653\pi\)
0.536747 + 0.843743i \(0.319653\pi\)
\(440\) 0.829257 0.0395333
\(441\) −47.9559 −2.28361
\(442\) −16.9764 −0.807486
\(443\) −4.94585 −0.234984 −0.117492 0.993074i \(-0.537486\pi\)
−0.117492 + 0.993074i \(0.537486\pi\)
\(444\) −2.00131 −0.0949778
\(445\) 4.85861 0.230320
\(446\) 6.28035 0.297383
\(447\) 1.63729 0.0774412
\(448\) 4.83738 0.228545
\(449\) 29.3948 1.38723 0.693614 0.720347i \(-0.256017\pi\)
0.693614 + 0.720347i \(0.256017\pi\)
\(450\) −12.6097 −0.594427
\(451\) −2.08579 −0.0982160
\(452\) 0.169648 0.00797955
\(453\) −6.52964 −0.306789
\(454\) 20.0570 0.941322
\(455\) 18.8214 0.882359
\(456\) −1.09303 −0.0511860
\(457\) −31.6624 −1.48110 −0.740552 0.671999i \(-0.765436\pi\)
−0.740552 + 0.671999i \(0.765436\pi\)
\(458\) 5.47928 0.256030
\(459\) 5.90514 0.275628
\(460\) −3.22908 −0.150556
\(461\) 16.7107 0.778296 0.389148 0.921175i \(-0.372769\pi\)
0.389148 + 0.921175i \(0.372769\pi\)
\(462\) −1.33267 −0.0620015
\(463\) −26.5138 −1.23220 −0.616100 0.787668i \(-0.711288\pi\)
−0.616100 + 0.787668i \(0.711288\pi\)
\(464\) 7.12691 0.330859
\(465\) 1.32964 0.0616605
\(466\) −6.96448 −0.322623
\(467\) −15.2806 −0.707104 −0.353552 0.935415i \(-0.615026\pi\)
−0.353552 + 0.935415i \(0.615026\pi\)
\(468\) 13.7197 0.634193
\(469\) 58.8503 2.71746
\(470\) 4.70401 0.216980
\(471\) −2.59925 −0.119767
\(472\) 9.73543 0.448109
\(473\) −10.3214 −0.474576
\(474\) 1.70224 0.0781865
\(475\) −17.1093 −0.785028
\(476\) −17.5027 −0.802233
\(477\) −26.1756 −1.19850
\(478\) 4.60506 0.210630
\(479\) 37.3582 1.70694 0.853469 0.521143i \(-0.174494\pi\)
0.853469 + 0.521143i \(0.174494\pi\)
\(480\) 0.228456 0.0104276
\(481\) 34.0841 1.55410
\(482\) −7.16729 −0.326461
\(483\) 5.18934 0.236123
\(484\) 1.00000 0.0454545
\(485\) 10.0585 0.456735
\(486\) −7.18904 −0.326101
\(487\) −18.2127 −0.825296 −0.412648 0.910891i \(-0.635396\pi\)
−0.412648 + 0.910891i \(0.635396\pi\)
\(488\) 12.1961 0.552091
\(489\) −2.70668 −0.122400
\(490\) 13.6000 0.614385
\(491\) −40.6739 −1.83559 −0.917795 0.397055i \(-0.870032\pi\)
−0.917795 + 0.397055i \(0.870032\pi\)
\(492\) −0.574625 −0.0259061
\(493\) −25.7867 −1.16137
\(494\) 18.6154 0.837545
\(495\) 2.42483 0.108988
\(496\) −5.82010 −0.261330
\(497\) −37.9568 −1.70259
\(498\) 3.78038 0.169403
\(499\) 11.0473 0.494544 0.247272 0.968946i \(-0.420466\pi\)
0.247272 + 0.968946i \(0.420466\pi\)
\(500\) 7.72232 0.345353
\(501\) −2.78835 −0.124574
\(502\) −0.901357 −0.0402295
\(503\) −16.2803 −0.725904 −0.362952 0.931808i \(-0.618231\pi\)
−0.362952 + 0.931808i \(0.618231\pi\)
\(504\) 14.1450 0.630068
\(505\) 8.44262 0.375692
\(506\) −3.89394 −0.173107
\(507\) 2.48337 0.110290
\(508\) 9.99362 0.443395
\(509\) −4.58003 −0.203006 −0.101503 0.994835i \(-0.532365\pi\)
−0.101503 + 0.994835i \(0.532365\pi\)
\(510\) −0.826604 −0.0366026
\(511\) −0.707158 −0.0312828
\(512\) −1.00000 −0.0441942
\(513\) −6.47524 −0.285889
\(514\) 26.9133 1.18710
\(515\) 7.57002 0.333575
\(516\) −2.84348 −0.125177
\(517\) 5.67256 0.249479
\(518\) 35.1407 1.54399
\(519\) 2.34152 0.102781
\(520\) −3.89082 −0.170624
\(521\) 21.0983 0.924334 0.462167 0.886793i \(-0.347072\pi\)
0.462167 + 0.886793i \(0.347072\pi\)
\(522\) 20.8398 0.912134
\(523\) −26.1173 −1.14203 −0.571015 0.820939i \(-0.693450\pi\)
−0.571015 + 0.820939i \(0.693450\pi\)
\(524\) 12.0832 0.527858
\(525\) −5.74693 −0.250816
\(526\) 7.91989 0.345324
\(527\) 21.0584 0.917317
\(528\) 0.275495 0.0119894
\(529\) −7.83726 −0.340751
\(530\) 7.42323 0.322445
\(531\) 28.4674 1.23538
\(532\) 19.1924 0.832097
\(533\) 9.78638 0.423895
\(534\) 1.61412 0.0698499
\(535\) −2.98077 −0.128870
\(536\) −12.1658 −0.525481
\(537\) 2.63554 0.113732
\(538\) 13.3200 0.574268
\(539\) 16.4002 0.706407
\(540\) 1.35340 0.0582410
\(541\) −46.2299 −1.98758 −0.993788 0.111287i \(-0.964503\pi\)
−0.993788 + 0.111287i \(0.964503\pi\)
\(542\) −2.17709 −0.0935139
\(543\) −1.06983 −0.0459108
\(544\) 3.61821 0.155130
\(545\) −1.86631 −0.0799441
\(546\) 6.25281 0.267595
\(547\) 41.5825 1.77794 0.888969 0.457967i \(-0.151422\pi\)
0.888969 + 0.457967i \(0.151422\pi\)
\(548\) 18.5604 0.792860
\(549\) 35.6626 1.52204
\(550\) 4.31233 0.183878
\(551\) 28.2762 1.20461
\(552\) −1.07276 −0.0456597
\(553\) −29.8894 −1.27103
\(554\) −0.716770 −0.0304527
\(555\) 1.65960 0.0704460
\(556\) 7.49907 0.318032
\(557\) −40.7343 −1.72597 −0.862984 0.505232i \(-0.831407\pi\)
−0.862984 + 0.505232i \(0.831407\pi\)
\(558\) −17.0186 −0.720454
\(559\) 48.4271 2.04825
\(560\) −4.01143 −0.169514
\(561\) −0.996800 −0.0420849
\(562\) −4.61908 −0.194844
\(563\) 11.4817 0.483895 0.241947 0.970289i \(-0.422214\pi\)
0.241947 + 0.970289i \(0.422214\pi\)
\(564\) 1.56276 0.0658041
\(565\) −0.140681 −0.00591851
\(566\) 17.5616 0.738169
\(567\) 40.2599 1.69076
\(568\) 7.84656 0.329235
\(569\) 27.8058 1.16568 0.582841 0.812586i \(-0.301941\pi\)
0.582841 + 0.812586i \(0.301941\pi\)
\(570\) 0.906406 0.0379652
\(571\) −12.9931 −0.543746 −0.271873 0.962333i \(-0.587643\pi\)
−0.271873 + 0.962333i \(0.587643\pi\)
\(572\) −4.69193 −0.196180
\(573\) 2.06847 0.0864117
\(574\) 10.0898 0.421138
\(575\) −16.7919 −0.700272
\(576\) −2.92410 −0.121838
\(577\) −7.55341 −0.314453 −0.157226 0.987563i \(-0.550255\pi\)
−0.157226 + 0.987563i \(0.550255\pi\)
\(578\) 3.90852 0.162573
\(579\) −0.994569 −0.0413329
\(580\) −5.91004 −0.245401
\(581\) −66.3792 −2.75387
\(582\) 3.34164 0.138515
\(583\) 8.95166 0.370740
\(584\) 0.146186 0.00604923
\(585\) −11.3772 −0.470387
\(586\) −14.6631 −0.605728
\(587\) −27.0191 −1.11520 −0.557600 0.830110i \(-0.688278\pi\)
−0.557600 + 0.830110i \(0.688278\pi\)
\(588\) 4.51817 0.186326
\(589\) −23.0914 −0.951465
\(590\) −8.07318 −0.332367
\(591\) 0.275495 0.0113323
\(592\) −7.26441 −0.298565
\(593\) −38.9055 −1.59766 −0.798829 0.601559i \(-0.794547\pi\)
−0.798829 + 0.601559i \(0.794547\pi\)
\(594\) 1.63206 0.0669642
\(595\) 14.5142 0.595025
\(596\) 5.94309 0.243438
\(597\) −0.935839 −0.0383013
\(598\) 18.2701 0.747119
\(599\) −32.3648 −1.32239 −0.661195 0.750214i \(-0.729950\pi\)
−0.661195 + 0.750214i \(0.729950\pi\)
\(600\) 1.18803 0.0485009
\(601\) 13.9539 0.569189 0.284595 0.958648i \(-0.408141\pi\)
0.284595 + 0.958648i \(0.408141\pi\)
\(602\) 49.9283 2.03492
\(603\) −35.5739 −1.44868
\(604\) −23.7015 −0.964400
\(605\) −0.829257 −0.0337141
\(606\) 2.80480 0.113937
\(607\) 25.2293 1.02403 0.512013 0.858978i \(-0.328900\pi\)
0.512013 + 0.858978i \(0.328900\pi\)
\(608\) −3.96753 −0.160904
\(609\) 9.49784 0.384872
\(610\) −10.1137 −0.409492
\(611\) −26.6153 −1.07674
\(612\) 10.5800 0.427672
\(613\) 42.7405 1.72627 0.863135 0.504973i \(-0.168497\pi\)
0.863135 + 0.504973i \(0.168497\pi\)
\(614\) −13.5913 −0.548499
\(615\) 0.476512 0.0192148
\(616\) −4.83738 −0.194903
\(617\) −16.1662 −0.650828 −0.325414 0.945572i \(-0.605504\pi\)
−0.325414 + 0.945572i \(0.605504\pi\)
\(618\) 2.51490 0.101164
\(619\) 23.6337 0.949917 0.474958 0.880008i \(-0.342463\pi\)
0.474958 + 0.880008i \(0.342463\pi\)
\(620\) 4.82636 0.193831
\(621\) −6.35514 −0.255023
\(622\) 23.4225 0.939156
\(623\) −28.3421 −1.13550
\(624\) −1.29260 −0.0517455
\(625\) 15.1579 0.606315
\(626\) 16.2882 0.651008
\(627\) 1.09303 0.0436515
\(628\) −9.43485 −0.376491
\(629\) 26.2842 1.04802
\(630\) −11.7298 −0.467328
\(631\) −15.9302 −0.634171 −0.317085 0.948397i \(-0.602704\pi\)
−0.317085 + 0.948397i \(0.602704\pi\)
\(632\) 6.17885 0.245781
\(633\) −1.67569 −0.0666029
\(634\) −23.3554 −0.927560
\(635\) −8.28728 −0.328871
\(636\) 2.46614 0.0977887
\(637\) −76.9486 −3.04882
\(638\) −7.12691 −0.282157
\(639\) 22.9442 0.907657
\(640\) 0.829257 0.0327793
\(641\) 44.2360 1.74722 0.873608 0.486630i \(-0.161774\pi\)
0.873608 + 0.486630i \(0.161774\pi\)
\(642\) −0.990267 −0.0390827
\(643\) 40.0020 1.57752 0.788762 0.614698i \(-0.210722\pi\)
0.788762 + 0.614698i \(0.210722\pi\)
\(644\) 18.8364 0.742259
\(645\) 2.35798 0.0928453
\(646\) 14.3554 0.564804
\(647\) 11.3381 0.445748 0.222874 0.974847i \(-0.428456\pi\)
0.222874 + 0.974847i \(0.428456\pi\)
\(648\) −8.32268 −0.326946
\(649\) −9.73543 −0.382149
\(650\) −20.2332 −0.793610
\(651\) −7.75629 −0.303993
\(652\) −9.82480 −0.384769
\(653\) 6.92242 0.270895 0.135448 0.990785i \(-0.456753\pi\)
0.135448 + 0.990785i \(0.456753\pi\)
\(654\) −0.620024 −0.0242449
\(655\) −10.0201 −0.391518
\(656\) −2.08579 −0.0814364
\(657\) 0.427464 0.0166769
\(658\) −27.4403 −1.06973
\(659\) 15.1395 0.589750 0.294875 0.955536i \(-0.404722\pi\)
0.294875 + 0.955536i \(0.404722\pi\)
\(660\) −0.228456 −0.00889265
\(661\) −34.8415 −1.35518 −0.677588 0.735442i \(-0.736975\pi\)
−0.677588 + 0.735442i \(0.736975\pi\)
\(662\) −34.2480 −1.33109
\(663\) 4.67692 0.181636
\(664\) 13.7221 0.532522
\(665\) −15.9155 −0.617175
\(666\) −21.2419 −0.823106
\(667\) 27.7517 1.07455
\(668\) −10.1212 −0.391603
\(669\) −1.73020 −0.0668936
\(670\) 10.0885 0.389754
\(671\) −12.1961 −0.470825
\(672\) −1.33267 −0.0514090
\(673\) 42.6829 1.64531 0.822653 0.568544i \(-0.192493\pi\)
0.822653 + 0.568544i \(0.192493\pi\)
\(674\) 25.7171 0.990586
\(675\) 7.03798 0.270892
\(676\) 9.01422 0.346701
\(677\) 23.8986 0.918500 0.459250 0.888307i \(-0.348118\pi\)
0.459250 + 0.888307i \(0.348118\pi\)
\(678\) −0.0467370 −0.00179492
\(679\) −58.6753 −2.25175
\(680\) −3.00043 −0.115061
\(681\) −5.52560 −0.211741
\(682\) 5.82010 0.222863
\(683\) −14.9459 −0.571887 −0.285944 0.958246i \(-0.592307\pi\)
−0.285944 + 0.958246i \(0.592307\pi\)
\(684\) −11.6015 −0.443593
\(685\) −15.3913 −0.588072
\(686\) −45.4723 −1.73614
\(687\) −1.50951 −0.0575915
\(688\) −10.3214 −0.393498
\(689\) −42.0006 −1.60009
\(690\) 0.889594 0.0338663
\(691\) 21.4833 0.817263 0.408632 0.912699i \(-0.366006\pi\)
0.408632 + 0.912699i \(0.366006\pi\)
\(692\) 8.49931 0.323095
\(693\) −14.1450 −0.537323
\(694\) −26.8693 −1.01994
\(695\) −6.21866 −0.235887
\(696\) −1.96343 −0.0744235
\(697\) 7.54684 0.285857
\(698\) 19.8151 0.750012
\(699\) 1.91868 0.0725711
\(700\) −20.8604 −0.788448
\(701\) 35.2764 1.33237 0.666186 0.745785i \(-0.267926\pi\)
0.666186 + 0.745785i \(0.267926\pi\)
\(702\) −7.65751 −0.289014
\(703\) −28.8217 −1.08703
\(704\) 1.00000 0.0376889
\(705\) −1.29593 −0.0488076
\(706\) 23.1388 0.870842
\(707\) −49.2491 −1.85220
\(708\) −2.68206 −0.100798
\(709\) 24.3875 0.915893 0.457947 0.888980i \(-0.348585\pi\)
0.457947 + 0.888980i \(0.348585\pi\)
\(710\) −6.50682 −0.244197
\(711\) 18.0676 0.677587
\(712\) 5.85899 0.219575
\(713\) −22.6631 −0.848740
\(714\) 4.82189 0.180455
\(715\) 3.89082 0.145508
\(716\) 9.56658 0.357520
\(717\) −1.26867 −0.0473793
\(718\) 35.3470 1.31914
\(719\) 6.56128 0.244695 0.122347 0.992487i \(-0.460958\pi\)
0.122347 + 0.992487i \(0.460958\pi\)
\(720\) 2.42483 0.0903682
\(721\) −44.1588 −1.64456
\(722\) 3.25873 0.121277
\(723\) 1.97455 0.0734344
\(724\) −3.88330 −0.144322
\(725\) −30.7336 −1.14142
\(726\) −0.275495 −0.0102246
\(727\) 24.5434 0.910264 0.455132 0.890424i \(-0.349592\pi\)
0.455132 + 0.890424i \(0.349592\pi\)
\(728\) 22.6966 0.841193
\(729\) −22.9875 −0.851389
\(730\) −0.121226 −0.00448678
\(731\) 37.3449 1.38125
\(732\) −3.35996 −0.124188
\(733\) −3.91980 −0.144781 −0.0723906 0.997376i \(-0.523063\pi\)
−0.0723906 + 0.997376i \(0.523063\pi\)
\(734\) 1.43254 0.0528760
\(735\) −3.74673 −0.138200
\(736\) −3.89394 −0.143532
\(737\) 12.1658 0.448131
\(738\) −6.09906 −0.224510
\(739\) 25.0985 0.923263 0.461632 0.887072i \(-0.347264\pi\)
0.461632 + 0.887072i \(0.347264\pi\)
\(740\) 6.02406 0.221449
\(741\) −5.12844 −0.188398
\(742\) −43.3026 −1.58969
\(743\) 23.4706 0.861051 0.430526 0.902578i \(-0.358328\pi\)
0.430526 + 0.902578i \(0.358328\pi\)
\(744\) 1.60341 0.0587838
\(745\) −4.92835 −0.180561
\(746\) −21.8486 −0.799934
\(747\) 40.1250 1.46810
\(748\) −3.61821 −0.132295
\(749\) 17.3880 0.635342
\(750\) −2.12746 −0.0776839
\(751\) 10.1705 0.371127 0.185564 0.982632i \(-0.440589\pi\)
0.185564 + 0.982632i \(0.440589\pi\)
\(752\) 5.67256 0.206857
\(753\) 0.248319 0.00904925
\(754\) 33.4390 1.21778
\(755\) 19.6546 0.715305
\(756\) −7.89489 −0.287134
\(757\) −37.6292 −1.36766 −0.683828 0.729643i \(-0.739686\pi\)
−0.683828 + 0.729643i \(0.739686\pi\)
\(758\) 17.0176 0.618106
\(759\) 1.07276 0.0389387
\(760\) 3.29010 0.119345
\(761\) 0.0484792 0.00175737 0.000878684 1.00000i \(-0.499720\pi\)
0.000878684 1.00000i \(0.499720\pi\)
\(762\) −2.75319 −0.0997376
\(763\) 10.8869 0.394133
\(764\) 7.50821 0.271637
\(765\) −8.77357 −0.317209
\(766\) −22.3816 −0.808680
\(767\) 45.6780 1.64934
\(768\) 0.275495 0.00994107
\(769\) 38.9130 1.40324 0.701620 0.712551i \(-0.252460\pi\)
0.701620 + 0.712551i \(0.252460\pi\)
\(770\) 4.01143 0.144562
\(771\) −7.41448 −0.267026
\(772\) −3.61012 −0.129931
\(773\) 50.5900 1.81960 0.909798 0.415051i \(-0.136236\pi\)
0.909798 + 0.415051i \(0.136236\pi\)
\(774\) −30.1807 −1.08482
\(775\) 25.0982 0.901554
\(776\) 12.1296 0.435426
\(777\) −9.68107 −0.347307
\(778\) 17.5665 0.629788
\(779\) −8.27543 −0.296498
\(780\) 1.07190 0.0383802
\(781\) −7.84656 −0.280772
\(782\) 14.0891 0.503825
\(783\) −11.6315 −0.415678
\(784\) 16.4002 0.585722
\(785\) 7.82392 0.279248
\(786\) −3.32887 −0.118737
\(787\) −14.1567 −0.504632 −0.252316 0.967645i \(-0.581192\pi\)
−0.252316 + 0.967645i \(0.581192\pi\)
\(788\) 1.00000 0.0356235
\(789\) −2.18189 −0.0776773
\(790\) −5.12385 −0.182299
\(791\) 0.820649 0.0291789
\(792\) 2.92410 0.103903
\(793\) 57.2233 2.03206
\(794\) 17.3428 0.615473
\(795\) −2.04506 −0.0725309
\(796\) −3.39694 −0.120401
\(797\) 14.8443 0.525811 0.262905 0.964822i \(-0.415319\pi\)
0.262905 + 0.964822i \(0.415319\pi\)
\(798\) −5.28741 −0.187172
\(799\) −20.5245 −0.726106
\(800\) 4.31233 0.152464
\(801\) 17.1323 0.605340
\(802\) −9.02318 −0.318620
\(803\) −0.146186 −0.00515880
\(804\) 3.35160 0.118202
\(805\) −15.6203 −0.550542
\(806\) −27.3075 −0.961866
\(807\) −3.66960 −0.129176
\(808\) 10.1809 0.358164
\(809\) −2.79386 −0.0982271 −0.0491135 0.998793i \(-0.515640\pi\)
−0.0491135 + 0.998793i \(0.515640\pi\)
\(810\) 6.90165 0.242499
\(811\) −8.96871 −0.314934 −0.157467 0.987524i \(-0.550333\pi\)
−0.157467 + 0.987524i \(0.550333\pi\)
\(812\) 34.4755 1.20985
\(813\) 0.599777 0.0210351
\(814\) 7.26441 0.254617
\(815\) 8.14729 0.285387
\(816\) −0.996800 −0.0348950
\(817\) −40.9503 −1.43267
\(818\) −4.77298 −0.166883
\(819\) 66.3673 2.31906
\(820\) 1.72966 0.0604022
\(821\) 28.1486 0.982392 0.491196 0.871049i \(-0.336560\pi\)
0.491196 + 0.871049i \(0.336560\pi\)
\(822\) −5.11329 −0.178346
\(823\) −24.6879 −0.860566 −0.430283 0.902694i \(-0.641586\pi\)
−0.430283 + 0.902694i \(0.641586\pi\)
\(824\) 9.12868 0.318013
\(825\) −1.18803 −0.0413617
\(826\) 47.0939 1.63861
\(827\) 44.8396 1.55923 0.779613 0.626261i \(-0.215416\pi\)
0.779613 + 0.626261i \(0.215416\pi\)
\(828\) −11.3863 −0.395700
\(829\) −12.0773 −0.419463 −0.209732 0.977759i \(-0.567259\pi\)
−0.209732 + 0.977759i \(0.567259\pi\)
\(830\) −11.3792 −0.394977
\(831\) 0.197467 0.00685004
\(832\) −4.69193 −0.162663
\(833\) −59.3395 −2.05599
\(834\) −2.06596 −0.0715382
\(835\) 8.39312 0.290456
\(836\) 3.96753 0.137220
\(837\) 9.49875 0.328325
\(838\) 25.3146 0.874479
\(839\) −53.8304 −1.85843 −0.929216 0.369537i \(-0.879516\pi\)
−0.929216 + 0.369537i \(0.879516\pi\)
\(840\) 1.10513 0.0381306
\(841\) 21.7929 0.751478
\(842\) 29.7629 1.02570
\(843\) 1.27253 0.0438283
\(844\) −6.08249 −0.209368
\(845\) −7.47511 −0.257152
\(846\) 16.5871 0.570278
\(847\) 4.83738 0.166214
\(848\) 8.95166 0.307401
\(849\) −4.83813 −0.166044
\(850\) −15.6029 −0.535177
\(851\) −28.2871 −0.969670
\(852\) −2.16169 −0.0740582
\(853\) 23.6977 0.811393 0.405697 0.914008i \(-0.367029\pi\)
0.405697 + 0.914008i \(0.367029\pi\)
\(854\) 58.9971 2.01884
\(855\) 9.62059 0.329017
\(856\) −3.59450 −0.122858
\(857\) −34.2047 −1.16841 −0.584204 0.811607i \(-0.698593\pi\)
−0.584204 + 0.811607i \(0.698593\pi\)
\(858\) 1.29260 0.0441287
\(859\) −10.7322 −0.366180 −0.183090 0.983096i \(-0.558610\pi\)
−0.183090 + 0.983096i \(0.558610\pi\)
\(860\) 8.55906 0.291862
\(861\) −2.77967 −0.0947311
\(862\) 6.96059 0.237079
\(863\) 12.8060 0.435922 0.217961 0.975957i \(-0.430059\pi\)
0.217961 + 0.975957i \(0.430059\pi\)
\(864\) 1.63206 0.0555238
\(865\) −7.04812 −0.239643
\(866\) −30.8500 −1.04833
\(867\) −1.07678 −0.0365693
\(868\) −28.1540 −0.955610
\(869\) −6.17885 −0.209603
\(870\) 1.62819 0.0552007
\(871\) −57.0809 −1.93411
\(872\) −2.25058 −0.0762144
\(873\) 35.4681 1.20041
\(874\) −15.4493 −0.522580
\(875\) 37.3558 1.26286
\(876\) −0.0402736 −0.00136072
\(877\) −40.5951 −1.37080 −0.685399 0.728168i \(-0.740372\pi\)
−0.685399 + 0.728168i \(0.740372\pi\)
\(878\) −22.4922 −0.759075
\(879\) 4.03962 0.136253
\(880\) −0.829257 −0.0279543
\(881\) −26.1642 −0.881496 −0.440748 0.897631i \(-0.645287\pi\)
−0.440748 + 0.897631i \(0.645287\pi\)
\(882\) 47.9559 1.61476
\(883\) 2.99051 0.100639 0.0503194 0.998733i \(-0.483976\pi\)
0.0503194 + 0.998733i \(0.483976\pi\)
\(884\) 16.9764 0.570979
\(885\) 2.22412 0.0747629
\(886\) 4.94585 0.166159
\(887\) −24.8922 −0.835797 −0.417899 0.908494i \(-0.637233\pi\)
−0.417899 + 0.908494i \(0.637233\pi\)
\(888\) 2.00131 0.0671595
\(889\) 48.3429 1.62137
\(890\) −4.85861 −0.162861
\(891\) 8.32268 0.278820
\(892\) −6.28035 −0.210282
\(893\) 22.5060 0.753136
\(894\) −1.63729 −0.0547592
\(895\) −7.93316 −0.265176
\(896\) −4.83738 −0.161605
\(897\) −5.03331 −0.168057
\(898\) −29.3948 −0.980918
\(899\) −41.4793 −1.38341
\(900\) 12.6097 0.420323
\(901\) −32.3890 −1.07904
\(902\) 2.08579 0.0694492
\(903\) −13.7550 −0.457737
\(904\) −0.169648 −0.00564239
\(905\) 3.22026 0.107045
\(906\) 6.52964 0.216933
\(907\) −1.84829 −0.0613714 −0.0306857 0.999529i \(-0.509769\pi\)
−0.0306857 + 0.999529i \(0.509769\pi\)
\(908\) −20.0570 −0.665615
\(909\) 29.7701 0.987413
\(910\) −18.8214 −0.623922
\(911\) −2.63136 −0.0871808 −0.0435904 0.999049i \(-0.513880\pi\)
−0.0435904 + 0.999049i \(0.513880\pi\)
\(912\) 1.09303 0.0361940
\(913\) −13.7221 −0.454137
\(914\) 31.6624 1.04730
\(915\) 2.78627 0.0921114
\(916\) −5.47928 −0.181040
\(917\) 58.4511 1.93023
\(918\) −5.90514 −0.194899
\(919\) −42.5543 −1.40374 −0.701868 0.712307i \(-0.747650\pi\)
−0.701868 + 0.712307i \(0.747650\pi\)
\(920\) 3.22908 0.106459
\(921\) 3.74432 0.123380
\(922\) −16.7107 −0.550339
\(923\) 36.8155 1.21180
\(924\) 1.33267 0.0438417
\(925\) 31.3265 1.03001
\(926\) 26.5138 0.871298
\(927\) 26.6932 0.876719
\(928\) −7.12691 −0.233952
\(929\) −19.6061 −0.643255 −0.321628 0.946866i \(-0.604230\pi\)
−0.321628 + 0.946866i \(0.604230\pi\)
\(930\) −1.32964 −0.0436006
\(931\) 65.0682 2.13253
\(932\) 6.96448 0.228129
\(933\) −6.45277 −0.211254
\(934\) 15.2806 0.499998
\(935\) 3.00043 0.0981246
\(936\) −13.7197 −0.448442
\(937\) 27.8111 0.908550 0.454275 0.890861i \(-0.349898\pi\)
0.454275 + 0.890861i \(0.349898\pi\)
\(938\) −58.8503 −1.92153
\(939\) −4.48732 −0.146438
\(940\) −4.70401 −0.153428
\(941\) −22.8584 −0.745161 −0.372581 0.928000i \(-0.621527\pi\)
−0.372581 + 0.928000i \(0.621527\pi\)
\(942\) 2.59925 0.0846882
\(943\) −8.12193 −0.264487
\(944\) −9.73543 −0.316861
\(945\) 6.54689 0.212970
\(946\) 10.3214 0.335576
\(947\) −5.64337 −0.183385 −0.0916925 0.995787i \(-0.529228\pi\)
−0.0916925 + 0.995787i \(0.529228\pi\)
\(948\) −1.70224 −0.0552862
\(949\) 0.685896 0.0222651
\(950\) 17.1093 0.555099
\(951\) 6.43428 0.208646
\(952\) 17.5027 0.567265
\(953\) −3.87642 −0.125570 −0.0627848 0.998027i \(-0.519998\pi\)
−0.0627848 + 0.998027i \(0.519998\pi\)
\(954\) 26.1756 0.847466
\(955\) −6.22624 −0.201476
\(956\) −4.60506 −0.148938
\(957\) 1.96343 0.0634686
\(958\) −37.3582 −1.20699
\(959\) 89.7835 2.89926
\(960\) −0.228456 −0.00737339
\(961\) 2.87358 0.0926961
\(962\) −34.0841 −1.09892
\(963\) −10.5107 −0.338702
\(964\) 7.16729 0.230843
\(965\) 2.99372 0.0963711
\(966\) −5.18934 −0.166964
\(967\) 42.7324 1.37418 0.687091 0.726572i \(-0.258888\pi\)
0.687091 + 0.726572i \(0.258888\pi\)
\(968\) −1.00000 −0.0321412
\(969\) −3.95483 −0.127047
\(970\) −10.0585 −0.322960
\(971\) −23.2887 −0.747371 −0.373685 0.927556i \(-0.621906\pi\)
−0.373685 + 0.927556i \(0.621906\pi\)
\(972\) 7.18904 0.230588
\(973\) 36.2758 1.16295
\(974\) 18.2127 0.583572
\(975\) 5.57413 0.178515
\(976\) −12.1961 −0.390388
\(977\) −22.7720 −0.728541 −0.364270 0.931293i \(-0.618682\pi\)
−0.364270 + 0.931293i \(0.618682\pi\)
\(978\) 2.70668 0.0865501
\(979\) −5.85899 −0.187254
\(980\) −13.6000 −0.434436
\(981\) −6.58094 −0.210113
\(982\) 40.6739 1.29796
\(983\) 51.6985 1.64892 0.824462 0.565917i \(-0.191478\pi\)
0.824462 + 0.565917i \(0.191478\pi\)
\(984\) 0.574625 0.0183184
\(985\) −0.829257 −0.0264223
\(986\) 25.7867 0.821216
\(987\) 7.55966 0.240627
\(988\) −18.6154 −0.592234
\(989\) −40.1907 −1.27799
\(990\) −2.42483 −0.0770663
\(991\) −28.4828 −0.904787 −0.452393 0.891819i \(-0.649430\pi\)
−0.452393 + 0.891819i \(0.649430\pi\)
\(992\) 5.82010 0.184788
\(993\) 9.43515 0.299415
\(994\) 37.9568 1.20392
\(995\) 2.81694 0.0893029
\(996\) −3.78038 −0.119786
\(997\) 48.3041 1.52980 0.764902 0.644146i \(-0.222787\pi\)
0.764902 + 0.644146i \(0.222787\pi\)
\(998\) −11.0473 −0.349696
\(999\) 11.8559 0.375106
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 4334.2.a.a.1.9 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4334.2.a.a.1.9 15 1.1 even 1 trivial