Properties

Label 4334.2.a.a.1.15
Level $4334$
Weight $2$
Character 4334.1
Self dual yes
Analytic conductor $34.607$
Analytic rank $1$
Dimension $15$
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] = [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 1.15
Root \(-3.04439\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +3.04439 q^{3} +1.00000 q^{4} -2.75592 q^{5} -3.04439 q^{6} +0.383632 q^{7} -1.00000 q^{8} +6.26834 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +3.04439 q^{3} +1.00000 q^{4} -2.75592 q^{5} -3.04439 q^{6} +0.383632 q^{7} -1.00000 q^{8} +6.26834 q^{9} +2.75592 q^{10} +1.00000 q^{11} +3.04439 q^{12} +1.61809 q^{13} -0.383632 q^{14} -8.39009 q^{15} +1.00000 q^{16} -2.00124 q^{17} -6.26834 q^{18} -8.48996 q^{19} -2.75592 q^{20} +1.16793 q^{21} -1.00000 q^{22} -4.20484 q^{23} -3.04439 q^{24} +2.59507 q^{25} -1.61809 q^{26} +9.95012 q^{27} +0.383632 q^{28} +1.55492 q^{29} +8.39009 q^{30} -7.16290 q^{31} -1.00000 q^{32} +3.04439 q^{33} +2.00124 q^{34} -1.05726 q^{35} +6.26834 q^{36} -10.7757 q^{37} +8.48996 q^{38} +4.92611 q^{39} +2.75592 q^{40} +2.34341 q^{41} -1.16793 q^{42} -0.770372 q^{43} +1.00000 q^{44} -17.2750 q^{45} +4.20484 q^{46} +3.29424 q^{47} +3.04439 q^{48} -6.85283 q^{49} -2.59507 q^{50} -6.09256 q^{51} +1.61809 q^{52} +3.83128 q^{53} -9.95012 q^{54} -2.75592 q^{55} -0.383632 q^{56} -25.8468 q^{57} -1.55492 q^{58} -13.6272 q^{59} -8.39009 q^{60} +13.3093 q^{61} +7.16290 q^{62} +2.40474 q^{63} +1.00000 q^{64} -4.45932 q^{65} -3.04439 q^{66} -7.21647 q^{67} -2.00124 q^{68} -12.8012 q^{69} +1.05726 q^{70} +9.72559 q^{71} -6.26834 q^{72} -11.8569 q^{73} +10.7757 q^{74} +7.90042 q^{75} -8.48996 q^{76} +0.383632 q^{77} -4.92611 q^{78} -1.27011 q^{79} -2.75592 q^{80} +11.4871 q^{81} -2.34341 q^{82} +1.86918 q^{83} +1.16793 q^{84} +5.51525 q^{85} +0.770372 q^{86} +4.73378 q^{87} -1.00000 q^{88} -7.15795 q^{89} +17.2750 q^{90} +0.620752 q^{91} -4.20484 q^{92} -21.8067 q^{93} -3.29424 q^{94} +23.3976 q^{95} -3.04439 q^{96} +7.43357 q^{97} +6.85283 q^{98} +6.26834 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - 15 q^{2} - q^{3} + 15 q^{4} - 7 q^{5} + q^{6} + q^{7} - 15 q^{8} + 2 q^{9}+O(q^{10}) \) 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
\(3\) 3.04439 1.75768 0.878841 0.477115i \(-0.158317\pi\)
0.878841 + 0.477115i \(0.158317\pi\)
\(4\) 1.00000 0.500000
\(5\) −2.75592 −1.23248 −0.616241 0.787557i \(-0.711345\pi\)
−0.616241 + 0.787557i \(0.711345\pi\)
\(6\) −3.04439 −1.24287
\(7\) 0.383632 0.144999 0.0724997 0.997368i \(-0.476902\pi\)
0.0724997 + 0.997368i \(0.476902\pi\)
\(8\) −1.00000 −0.353553
\(9\) 6.26834 2.08945
\(10\) 2.75592 0.871497
\(11\) 1.00000 0.301511
\(12\) 3.04439 0.878841
\(13\) 1.61809 0.448778 0.224389 0.974500i \(-0.427961\pi\)
0.224389 + 0.974500i \(0.427961\pi\)
\(14\) −0.383632 −0.102530
\(15\) −8.39009 −2.16631
\(16\) 1.00000 0.250000
\(17\) −2.00124 −0.485372 −0.242686 0.970105i \(-0.578028\pi\)
−0.242686 + 0.970105i \(0.578028\pi\)
\(18\) −6.26834 −1.47746
\(19\) −8.48996 −1.94773 −0.973866 0.227125i \(-0.927067\pi\)
−0.973866 + 0.227125i \(0.927067\pi\)
\(20\) −2.75592 −0.616241
\(21\) 1.16793 0.254863
\(22\) −1.00000 −0.213201
\(23\) −4.20484 −0.876770 −0.438385 0.898787i \(-0.644449\pi\)
−0.438385 + 0.898787i \(0.644449\pi\)
\(24\) −3.04439 −0.621435
\(25\) 2.59507 0.519014
\(26\) −1.61809 −0.317334
\(27\) 9.95012 1.91490
\(28\) 0.383632 0.0724997
\(29\) 1.55492 0.288741 0.144370 0.989524i \(-0.453884\pi\)
0.144370 + 0.989524i \(0.453884\pi\)
\(30\) 8.39009 1.53181
\(31\) −7.16290 −1.28650 −0.643248 0.765658i \(-0.722413\pi\)
−0.643248 + 0.765658i \(0.722413\pi\)
\(32\) −1.00000 −0.176777
\(33\) 3.04439 0.529961
\(34\) 2.00124 0.343210
\(35\) −1.05726 −0.178709
\(36\) 6.26834 1.04472
\(37\) −10.7757 −1.77151 −0.885755 0.464154i \(-0.846359\pi\)
−0.885755 + 0.464154i \(0.846359\pi\)
\(38\) 8.48996 1.37725
\(39\) 4.92611 0.788808
\(40\) 2.75592 0.435748
\(41\) 2.34341 0.365979 0.182990 0.983115i \(-0.441422\pi\)
0.182990 + 0.983115i \(0.441422\pi\)
\(42\) −1.16793 −0.180215
\(43\) −0.770372 −0.117481 −0.0587403 0.998273i \(-0.518708\pi\)
−0.0587403 + 0.998273i \(0.518708\pi\)
\(44\) 1.00000 0.150756
\(45\) −17.2750 −2.57521
\(46\) 4.20484 0.619970
\(47\) 3.29424 0.480514 0.240257 0.970709i \(-0.422768\pi\)
0.240257 + 0.970709i \(0.422768\pi\)
\(48\) 3.04439 0.439421
\(49\) −6.85283 −0.978975
\(50\) −2.59507 −0.366998
\(51\) −6.09256 −0.853129
\(52\) 1.61809 0.224389
\(53\) 3.83128 0.526266 0.263133 0.964760i \(-0.415244\pi\)
0.263133 + 0.964760i \(0.415244\pi\)
\(54\) −9.95012 −1.35404
\(55\) −2.75592 −0.371608
\(56\) −0.383632 −0.0512650
\(57\) −25.8468 −3.42349
\(58\) −1.55492 −0.204170
\(59\) −13.6272 −1.77411 −0.887054 0.461665i \(-0.847252\pi\)
−0.887054 + 0.461665i \(0.847252\pi\)
\(60\) −8.39009 −1.08316
\(61\) 13.3093 1.70408 0.852039 0.523478i \(-0.175366\pi\)
0.852039 + 0.523478i \(0.175366\pi\)
\(62\) 7.16290 0.909690
\(63\) 2.40474 0.302968
\(64\) 1.00000 0.125000
\(65\) −4.45932 −0.553111
\(66\) −3.04439 −0.374739
\(67\) −7.21647 −0.881632 −0.440816 0.897598i \(-0.645311\pi\)
−0.440816 + 0.897598i \(0.645311\pi\)
\(68\) −2.00124 −0.242686
\(69\) −12.8012 −1.54108
\(70\) 1.05726 0.126366
\(71\) 9.72559 1.15422 0.577108 0.816668i \(-0.304181\pi\)
0.577108 + 0.816668i \(0.304181\pi\)
\(72\) −6.26834 −0.738731
\(73\) −11.8569 −1.38775 −0.693874 0.720097i \(-0.744097\pi\)
−0.693874 + 0.720097i \(0.744097\pi\)
\(74\) 10.7757 1.25265
\(75\) 7.90042 0.912261
\(76\) −8.48996 −0.973866
\(77\) 0.383632 0.0437189
\(78\) −4.92611 −0.557772
\(79\) −1.27011 −0.142899 −0.0714493 0.997444i \(-0.522762\pi\)
−0.0714493 + 0.997444i \(0.522762\pi\)
\(80\) −2.75592 −0.308121
\(81\) 11.4871 1.27634
\(82\) −2.34341 −0.258787
\(83\) 1.86918 0.205170 0.102585 0.994724i \(-0.467289\pi\)
0.102585 + 0.994724i \(0.467289\pi\)
\(84\) 1.16793 0.127431
\(85\) 5.51525 0.598212
\(86\) 0.770372 0.0830713
\(87\) 4.73378 0.507514
\(88\) −1.00000 −0.106600
\(89\) −7.15795 −0.758742 −0.379371 0.925245i \(-0.623860\pi\)
−0.379371 + 0.925245i \(0.623860\pi\)
\(90\) 17.2750 1.82095
\(91\) 0.620752 0.0650724
\(92\) −4.20484 −0.438385
\(93\) −21.8067 −2.26125
\(94\) −3.29424 −0.339775
\(95\) 23.3976 2.40055
\(96\) −3.04439 −0.310717
\(97\) 7.43357 0.754764 0.377382 0.926058i \(-0.376824\pi\)
0.377382 + 0.926058i \(0.376824\pi\)
\(98\) 6.85283 0.692240
\(99\) 6.26834 0.629992
\(100\) 2.59507 0.259507
\(101\) −8.99486 −0.895022 −0.447511 0.894278i \(-0.647689\pi\)
−0.447511 + 0.894278i \(0.647689\pi\)
\(102\) 6.09256 0.603254
\(103\) 9.06428 0.893130 0.446565 0.894751i \(-0.352647\pi\)
0.446565 + 0.894751i \(0.352647\pi\)
\(104\) −1.61809 −0.158667
\(105\) −3.21871 −0.314114
\(106\) −3.83128 −0.372126
\(107\) 16.5612 1.60103 0.800516 0.599311i \(-0.204559\pi\)
0.800516 + 0.599311i \(0.204559\pi\)
\(108\) 9.95012 0.957451
\(109\) 5.52404 0.529107 0.264554 0.964371i \(-0.414775\pi\)
0.264554 + 0.964371i \(0.414775\pi\)
\(110\) 2.75592 0.262766
\(111\) −32.8054 −3.11375
\(112\) 0.383632 0.0362498
\(113\) −7.12104 −0.669891 −0.334945 0.942238i \(-0.608718\pi\)
−0.334945 + 0.942238i \(0.608718\pi\)
\(114\) 25.8468 2.42077
\(115\) 11.5882 1.08060
\(116\) 1.55492 0.144370
\(117\) 10.1427 0.937697
\(118\) 13.6272 1.25448
\(119\) −0.767740 −0.0703786
\(120\) 8.39009 0.765907
\(121\) 1.00000 0.0909091
\(122\) −13.3093 −1.20497
\(123\) 7.13427 0.643275
\(124\) −7.16290 −0.643248
\(125\) 6.62779 0.592807
\(126\) −2.40474 −0.214231
\(127\) 18.2064 1.61556 0.807779 0.589485i \(-0.200669\pi\)
0.807779 + 0.589485i \(0.200669\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −2.34532 −0.206494
\(130\) 4.45932 0.391108
\(131\) −2.98294 −0.260621 −0.130310 0.991473i \(-0.541597\pi\)
−0.130310 + 0.991473i \(0.541597\pi\)
\(132\) 3.04439 0.264981
\(133\) −3.25702 −0.282420
\(134\) 7.21647 0.623408
\(135\) −27.4217 −2.36008
\(136\) 2.00124 0.171605
\(137\) −4.94991 −0.422900 −0.211450 0.977389i \(-0.567819\pi\)
−0.211450 + 0.977389i \(0.567819\pi\)
\(138\) 12.8012 1.08971
\(139\) −12.6349 −1.07168 −0.535838 0.844321i \(-0.680004\pi\)
−0.535838 + 0.844321i \(0.680004\pi\)
\(140\) −1.05726 −0.0893546
\(141\) 10.0290 0.844591
\(142\) −9.72559 −0.816153
\(143\) 1.61809 0.135312
\(144\) 6.26834 0.522362
\(145\) −4.28522 −0.355868
\(146\) 11.8569 0.981286
\(147\) −20.8627 −1.72073
\(148\) −10.7757 −0.885755
\(149\) 23.3940 1.91651 0.958257 0.285907i \(-0.0922950\pi\)
0.958257 + 0.285907i \(0.0922950\pi\)
\(150\) −7.90042 −0.645066
\(151\) 3.38347 0.275343 0.137671 0.990478i \(-0.456038\pi\)
0.137671 + 0.990478i \(0.456038\pi\)
\(152\) 8.48996 0.688627
\(153\) −12.5444 −1.01416
\(154\) −0.383632 −0.0309140
\(155\) 19.7404 1.58558
\(156\) 4.92611 0.394404
\(157\) −18.2659 −1.45778 −0.728890 0.684631i \(-0.759963\pi\)
−0.728890 + 0.684631i \(0.759963\pi\)
\(158\) 1.27011 0.101045
\(159\) 11.6639 0.925009
\(160\) 2.75592 0.217874
\(161\) −1.61311 −0.127131
\(162\) −11.4871 −0.902509
\(163\) −13.1163 −1.02735 −0.513673 0.857986i \(-0.671716\pi\)
−0.513673 + 0.857986i \(0.671716\pi\)
\(164\) 2.34341 0.182990
\(165\) −8.39009 −0.653168
\(166\) −1.86918 −0.145077
\(167\) −14.1426 −1.09439 −0.547194 0.837006i \(-0.684304\pi\)
−0.547194 + 0.837006i \(0.684304\pi\)
\(168\) −1.16793 −0.0901076
\(169\) −10.3818 −0.798599
\(170\) −5.51525 −0.423000
\(171\) −53.2180 −4.06968
\(172\) −0.770372 −0.0587403
\(173\) −16.4029 −1.24709 −0.623543 0.781789i \(-0.714307\pi\)
−0.623543 + 0.781789i \(0.714307\pi\)
\(174\) −4.73378 −0.358867
\(175\) 0.995552 0.0752566
\(176\) 1.00000 0.0753778
\(177\) −41.4865 −3.11832
\(178\) 7.15795 0.536511
\(179\) 16.5328 1.23572 0.617860 0.786288i \(-0.288000\pi\)
0.617860 + 0.786288i \(0.288000\pi\)
\(180\) −17.2750 −1.28760
\(181\) −19.9372 −1.48192 −0.740962 0.671547i \(-0.765630\pi\)
−0.740962 + 0.671547i \(0.765630\pi\)
\(182\) −0.620752 −0.0460132
\(183\) 40.5187 2.99523
\(184\) 4.20484 0.309985
\(185\) 29.6968 2.18335
\(186\) 21.8067 1.59895
\(187\) −2.00124 −0.146345
\(188\) 3.29424 0.240257
\(189\) 3.81718 0.277659
\(190\) −23.3976 −1.69744
\(191\) 14.6512 1.06012 0.530061 0.847960i \(-0.322169\pi\)
0.530061 + 0.847960i \(0.322169\pi\)
\(192\) 3.04439 0.219710
\(193\) −25.8325 −1.85947 −0.929733 0.368235i \(-0.879962\pi\)
−0.929733 + 0.368235i \(0.879962\pi\)
\(194\) −7.43357 −0.533699
\(195\) −13.5759 −0.972193
\(196\) −6.85283 −0.489488
\(197\) 1.00000 0.0712470
\(198\) −6.26834 −0.445472
\(199\) −17.5504 −1.24411 −0.622056 0.782973i \(-0.713702\pi\)
−0.622056 + 0.782973i \(0.713702\pi\)
\(200\) −2.59507 −0.183499
\(201\) −21.9698 −1.54963
\(202\) 8.99486 0.632876
\(203\) 0.596516 0.0418672
\(204\) −6.09256 −0.426565
\(205\) −6.45824 −0.451063
\(206\) −9.06428 −0.631538
\(207\) −26.3574 −1.83196
\(208\) 1.61809 0.112194
\(209\) −8.48996 −0.587263
\(210\) 3.21871 0.222112
\(211\) −8.00255 −0.550918 −0.275459 0.961313i \(-0.588830\pi\)
−0.275459 + 0.961313i \(0.588830\pi\)
\(212\) 3.83128 0.263133
\(213\) 29.6085 2.02874
\(214\) −16.5612 −1.13210
\(215\) 2.12308 0.144793
\(216\) −9.95012 −0.677020
\(217\) −2.74792 −0.186541
\(218\) −5.52404 −0.374135
\(219\) −36.0971 −2.43922
\(220\) −2.75592 −0.185804
\(221\) −3.23819 −0.217824
\(222\) 32.8054 2.20175
\(223\) −5.24703 −0.351367 −0.175684 0.984447i \(-0.556214\pi\)
−0.175684 + 0.984447i \(0.556214\pi\)
\(224\) −0.383632 −0.0256325
\(225\) 16.2668 1.08445
\(226\) 7.12104 0.473684
\(227\) 20.9474 1.39033 0.695164 0.718851i \(-0.255332\pi\)
0.695164 + 0.718851i \(0.255332\pi\)
\(228\) −25.8468 −1.71175
\(229\) 4.66063 0.307983 0.153992 0.988072i \(-0.450787\pi\)
0.153992 + 0.988072i \(0.450787\pi\)
\(230\) −11.5882 −0.764103
\(231\) 1.16793 0.0768440
\(232\) −1.55492 −0.102085
\(233\) 20.3782 1.33502 0.667510 0.744601i \(-0.267360\pi\)
0.667510 + 0.744601i \(0.267360\pi\)
\(234\) −10.1427 −0.663052
\(235\) −9.07864 −0.592225
\(236\) −13.6272 −0.887054
\(237\) −3.86672 −0.251170
\(238\) 0.767740 0.0497652
\(239\) −27.7949 −1.79790 −0.898950 0.438051i \(-0.855669\pi\)
−0.898950 + 0.438051i \(0.855669\pi\)
\(240\) −8.39009 −0.541578
\(241\) −7.74842 −0.499120 −0.249560 0.968359i \(-0.580286\pi\)
−0.249560 + 0.968359i \(0.580286\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 5.12082 0.328501
\(244\) 13.3093 0.852039
\(245\) 18.8858 1.20657
\(246\) −7.13427 −0.454864
\(247\) −13.7375 −0.874098
\(248\) 7.16290 0.454845
\(249\) 5.69054 0.360623
\(250\) −6.62779 −0.419178
\(251\) 7.34619 0.463688 0.231844 0.972753i \(-0.425524\pi\)
0.231844 + 0.972753i \(0.425524\pi\)
\(252\) 2.40474 0.151484
\(253\) −4.20484 −0.264356
\(254\) −18.2064 −1.14237
\(255\) 16.7906 1.05147
\(256\) 1.00000 0.0625000
\(257\) −7.10183 −0.443000 −0.221500 0.975160i \(-0.571095\pi\)
−0.221500 + 0.975160i \(0.571095\pi\)
\(258\) 2.34532 0.146013
\(259\) −4.13389 −0.256868
\(260\) −4.45932 −0.276555
\(261\) 9.74674 0.603308
\(262\) 2.98294 0.184287
\(263\) −2.12702 −0.131158 −0.0655788 0.997847i \(-0.520889\pi\)
−0.0655788 + 0.997847i \(0.520889\pi\)
\(264\) −3.04439 −0.187370
\(265\) −10.5587 −0.648614
\(266\) 3.25702 0.199701
\(267\) −21.7916 −1.33363
\(268\) −7.21647 −0.440816
\(269\) 10.0081 0.610203 0.305102 0.952320i \(-0.401310\pi\)
0.305102 + 0.952320i \(0.401310\pi\)
\(270\) 27.4217 1.66883
\(271\) 16.3640 0.994043 0.497022 0.867738i \(-0.334427\pi\)
0.497022 + 0.867738i \(0.334427\pi\)
\(272\) −2.00124 −0.121343
\(273\) 1.88981 0.114377
\(274\) 4.94991 0.299035
\(275\) 2.59507 0.156489
\(276\) −12.8012 −0.770542
\(277\) 9.46792 0.568872 0.284436 0.958695i \(-0.408194\pi\)
0.284436 + 0.958695i \(0.408194\pi\)
\(278\) 12.6349 0.757789
\(279\) −44.8995 −2.68806
\(280\) 1.05726 0.0631832
\(281\) −0.209325 −0.0124873 −0.00624363 0.999981i \(-0.501987\pi\)
−0.00624363 + 0.999981i \(0.501987\pi\)
\(282\) −10.0290 −0.597216
\(283\) −22.3914 −1.33103 −0.665514 0.746385i \(-0.731788\pi\)
−0.665514 + 0.746385i \(0.731788\pi\)
\(284\) 9.72559 0.577108
\(285\) 71.2316 4.21940
\(286\) −1.61809 −0.0956797
\(287\) 0.899008 0.0530668
\(288\) −6.26834 −0.369365
\(289\) −12.9950 −0.764414
\(290\) 4.28522 0.251637
\(291\) 22.6307 1.32664
\(292\) −11.8569 −0.693874
\(293\) −19.3539 −1.13067 −0.565335 0.824862i \(-0.691253\pi\)
−0.565335 + 0.824862i \(0.691253\pi\)
\(294\) 20.8627 1.21674
\(295\) 37.5554 2.18656
\(296\) 10.7757 0.626323
\(297\) 9.95012 0.577364
\(298\) −23.3940 −1.35518
\(299\) −6.80382 −0.393475
\(300\) 7.90042 0.456131
\(301\) −0.295539 −0.0170346
\(302\) −3.38347 −0.194697
\(303\) −27.3839 −1.57316
\(304\) −8.48996 −0.486933
\(305\) −36.6792 −2.10025
\(306\) 12.5444 0.717118
\(307\) 12.6575 0.722403 0.361201 0.932488i \(-0.382367\pi\)
0.361201 + 0.932488i \(0.382367\pi\)
\(308\) 0.383632 0.0218595
\(309\) 27.5952 1.56984
\(310\) −19.7404 −1.12118
\(311\) −7.07558 −0.401219 −0.200610 0.979671i \(-0.564292\pi\)
−0.200610 + 0.979671i \(0.564292\pi\)
\(312\) −4.92611 −0.278886
\(313\) 21.3478 1.20665 0.603324 0.797496i \(-0.293843\pi\)
0.603324 + 0.797496i \(0.293843\pi\)
\(314\) 18.2659 1.03081
\(315\) −6.62725 −0.373403
\(316\) −1.27011 −0.0714493
\(317\) 3.77787 0.212186 0.106093 0.994356i \(-0.466166\pi\)
0.106093 + 0.994356i \(0.466166\pi\)
\(318\) −11.6639 −0.654080
\(319\) 1.55492 0.0870586
\(320\) −2.75592 −0.154060
\(321\) 50.4189 2.81411
\(322\) 1.61311 0.0898952
\(323\) 16.9904 0.945374
\(324\) 11.4871 0.638170
\(325\) 4.19906 0.232922
\(326\) 13.1163 0.726444
\(327\) 16.8174 0.930002
\(328\) −2.34341 −0.129393
\(329\) 1.26378 0.0696742
\(330\) 8.39009 0.461859
\(331\) −4.50178 −0.247440 −0.123720 0.992317i \(-0.539483\pi\)
−0.123720 + 0.992317i \(0.539483\pi\)
\(332\) 1.86918 0.102585
\(333\) −67.5456 −3.70147
\(334\) 14.1426 0.773850
\(335\) 19.8880 1.08660
\(336\) 1.16793 0.0637157
\(337\) 2.06321 0.112390 0.0561950 0.998420i \(-0.482103\pi\)
0.0561950 + 0.998420i \(0.482103\pi\)
\(338\) 10.3818 0.564695
\(339\) −21.6792 −1.17746
\(340\) 5.51525 0.299106
\(341\) −7.16290 −0.387893
\(342\) 53.2180 2.87770
\(343\) −5.31439 −0.286950
\(344\) 0.770372 0.0415357
\(345\) 35.2790 1.89936
\(346\) 16.4029 0.881823
\(347\) −5.80849 −0.311816 −0.155908 0.987772i \(-0.549830\pi\)
−0.155908 + 0.987772i \(0.549830\pi\)
\(348\) 4.73378 0.253757
\(349\) 14.1619 0.758071 0.379035 0.925382i \(-0.376256\pi\)
0.379035 + 0.925382i \(0.376256\pi\)
\(350\) −0.995552 −0.0532145
\(351\) 16.1002 0.859365
\(352\) −1.00000 −0.0533002
\(353\) −36.4956 −1.94246 −0.971232 0.238137i \(-0.923463\pi\)
−0.971232 + 0.238137i \(0.923463\pi\)
\(354\) 41.4865 2.20498
\(355\) −26.8029 −1.42255
\(356\) −7.15795 −0.379371
\(357\) −2.33730 −0.123703
\(358\) −16.5328 −0.873786
\(359\) 31.8263 1.67973 0.839864 0.542797i \(-0.182635\pi\)
0.839864 + 0.542797i \(0.182635\pi\)
\(360\) 17.2750 0.910473
\(361\) 53.0795 2.79366
\(362\) 19.9372 1.04788
\(363\) 3.04439 0.159789
\(364\) 0.620752 0.0325362
\(365\) 32.6767 1.71037
\(366\) −40.5187 −2.11795
\(367\) 27.4423 1.43248 0.716238 0.697856i \(-0.245862\pi\)
0.716238 + 0.697856i \(0.245862\pi\)
\(368\) −4.20484 −0.219193
\(369\) 14.6893 0.764694
\(370\) −29.6968 −1.54387
\(371\) 1.46980 0.0763082
\(372\) −21.8067 −1.13063
\(373\) 26.4484 1.36945 0.684724 0.728802i \(-0.259923\pi\)
0.684724 + 0.728802i \(0.259923\pi\)
\(374\) 2.00124 0.103482
\(375\) 20.1776 1.04197
\(376\) −3.29424 −0.169887
\(377\) 2.51600 0.129580
\(378\) −3.81718 −0.196335
\(379\) −14.8445 −0.762511 −0.381256 0.924470i \(-0.624508\pi\)
−0.381256 + 0.924470i \(0.624508\pi\)
\(380\) 23.3976 1.20027
\(381\) 55.4275 2.83964
\(382\) −14.6512 −0.749619
\(383\) −19.6478 −1.00396 −0.501978 0.864880i \(-0.667394\pi\)
−0.501978 + 0.864880i \(0.667394\pi\)
\(384\) −3.04439 −0.155359
\(385\) −1.05726 −0.0538828
\(386\) 25.8325 1.31484
\(387\) −4.82895 −0.245469
\(388\) 7.43357 0.377382
\(389\) −33.9620 −1.72194 −0.860970 0.508656i \(-0.830142\pi\)
−0.860970 + 0.508656i \(0.830142\pi\)
\(390\) 13.5759 0.687444
\(391\) 8.41490 0.425560
\(392\) 6.85283 0.346120
\(393\) −9.08124 −0.458088
\(394\) −1.00000 −0.0503793
\(395\) 3.50032 0.176120
\(396\) 6.26834 0.314996
\(397\) 25.6886 1.28927 0.644637 0.764489i \(-0.277009\pi\)
0.644637 + 0.764489i \(0.277009\pi\)
\(398\) 17.5504 0.879720
\(399\) −9.91566 −0.496404
\(400\) 2.59507 0.129753
\(401\) 20.5313 1.02528 0.512642 0.858603i \(-0.328667\pi\)
0.512642 + 0.858603i \(0.328667\pi\)
\(402\) 21.9698 1.09575
\(403\) −11.5902 −0.577350
\(404\) −8.99486 −0.447511
\(405\) −31.6574 −1.57307
\(406\) −0.596516 −0.0296046
\(407\) −10.7757 −0.534130
\(408\) 6.09256 0.301627
\(409\) 8.39367 0.415040 0.207520 0.978231i \(-0.433461\pi\)
0.207520 + 0.978231i \(0.433461\pi\)
\(410\) 6.45824 0.318950
\(411\) −15.0695 −0.743323
\(412\) 9.06428 0.446565
\(413\) −5.22783 −0.257244
\(414\) 26.3574 1.29539
\(415\) −5.15131 −0.252868
\(416\) −1.61809 −0.0793334
\(417\) −38.4655 −1.88367
\(418\) 8.48996 0.415258
\(419\) −28.1301 −1.37425 −0.687123 0.726541i \(-0.741127\pi\)
−0.687123 + 0.726541i \(0.741127\pi\)
\(420\) −3.21871 −0.157057
\(421\) −6.66048 −0.324612 −0.162306 0.986740i \(-0.551893\pi\)
−0.162306 + 0.986740i \(0.551893\pi\)
\(422\) 8.00255 0.389558
\(423\) 20.6494 1.00401
\(424\) −3.83128 −0.186063
\(425\) −5.19335 −0.251915
\(426\) −29.6085 −1.43454
\(427\) 5.10586 0.247090
\(428\) 16.5612 0.800516
\(429\) 4.92611 0.237835
\(430\) −2.12308 −0.102384
\(431\) 26.5395 1.27836 0.639181 0.769057i \(-0.279274\pi\)
0.639181 + 0.769057i \(0.279274\pi\)
\(432\) 9.95012 0.478725
\(433\) −5.66324 −0.272158 −0.136079 0.990698i \(-0.543450\pi\)
−0.136079 + 0.990698i \(0.543450\pi\)
\(434\) 2.74792 0.131904
\(435\) −13.0459 −0.625503
\(436\) 5.52404 0.264554
\(437\) 35.6990 1.70771
\(438\) 36.0971 1.72479
\(439\) −26.4140 −1.26067 −0.630336 0.776322i \(-0.717083\pi\)
−0.630336 + 0.776322i \(0.717083\pi\)
\(440\) 2.75592 0.131383
\(441\) −42.9558 −2.04552
\(442\) 3.23819 0.154025
\(443\) 26.5681 1.26229 0.631145 0.775665i \(-0.282585\pi\)
0.631145 + 0.775665i \(0.282585\pi\)
\(444\) −32.8054 −1.55688
\(445\) 19.7267 0.935136
\(446\) 5.24703 0.248454
\(447\) 71.2207 3.36862
\(448\) 0.383632 0.0181249
\(449\) −0.0968120 −0.00456884 −0.00228442 0.999997i \(-0.500727\pi\)
−0.00228442 + 0.999997i \(0.500727\pi\)
\(450\) −16.2668 −0.766823
\(451\) 2.34341 0.110347
\(452\) −7.12104 −0.334945
\(453\) 10.3006 0.483965
\(454\) −20.9474 −0.983111
\(455\) −1.71074 −0.0802007
\(456\) 25.8468 1.21039
\(457\) −0.866576 −0.0405367 −0.0202684 0.999795i \(-0.506452\pi\)
−0.0202684 + 0.999795i \(0.506452\pi\)
\(458\) −4.66063 −0.217777
\(459\) −19.9126 −0.929439
\(460\) 11.5882 0.540302
\(461\) −17.0541 −0.794287 −0.397143 0.917757i \(-0.629998\pi\)
−0.397143 + 0.917757i \(0.629998\pi\)
\(462\) −1.16793 −0.0543369
\(463\) −34.0789 −1.58378 −0.791890 0.610664i \(-0.790903\pi\)
−0.791890 + 0.610664i \(0.790903\pi\)
\(464\) 1.55492 0.0721852
\(465\) 60.0974 2.78695
\(466\) −20.3782 −0.944001
\(467\) 11.9629 0.553576 0.276788 0.960931i \(-0.410730\pi\)
0.276788 + 0.960931i \(0.410730\pi\)
\(468\) 10.1427 0.468848
\(469\) −2.76847 −0.127836
\(470\) 9.07864 0.418766
\(471\) −55.6087 −2.56231
\(472\) 13.6272 0.627242
\(473\) −0.770372 −0.0354217
\(474\) 3.86672 0.177604
\(475\) −22.0320 −1.01090
\(476\) −0.767740 −0.0351893
\(477\) 24.0157 1.09961
\(478\) 27.7949 1.27131
\(479\) 27.8579 1.27286 0.636431 0.771334i \(-0.280410\pi\)
0.636431 + 0.771334i \(0.280410\pi\)
\(480\) 8.39009 0.382954
\(481\) −17.4360 −0.795014
\(482\) 7.74842 0.352931
\(483\) −4.91095 −0.223456
\(484\) 1.00000 0.0454545
\(485\) −20.4863 −0.930234
\(486\) −5.12082 −0.232285
\(487\) −18.4547 −0.836261 −0.418131 0.908387i \(-0.637315\pi\)
−0.418131 + 0.908387i \(0.637315\pi\)
\(488\) −13.3093 −0.602483
\(489\) −39.9311 −1.80575
\(490\) −18.8858 −0.853174
\(491\) 0.669667 0.0302217 0.0151108 0.999886i \(-0.495190\pi\)
0.0151108 + 0.999886i \(0.495190\pi\)
\(492\) 7.13427 0.321638
\(493\) −3.11176 −0.140147
\(494\) 13.7375 0.618081
\(495\) −17.2750 −0.776454
\(496\) −7.16290 −0.321624
\(497\) 3.73105 0.167360
\(498\) −5.69054 −0.254999
\(499\) −2.87546 −0.128723 −0.0643615 0.997927i \(-0.520501\pi\)
−0.0643615 + 0.997927i \(0.520501\pi\)
\(500\) 6.62779 0.296404
\(501\) −43.0557 −1.92359
\(502\) −7.34619 −0.327877
\(503\) 10.8054 0.481791 0.240895 0.970551i \(-0.422559\pi\)
0.240895 + 0.970551i \(0.422559\pi\)
\(504\) −2.40474 −0.107115
\(505\) 24.7891 1.10310
\(506\) 4.20484 0.186928
\(507\) −31.6062 −1.40368
\(508\) 18.2064 0.807779
\(509\) 33.9202 1.50349 0.751744 0.659455i \(-0.229213\pi\)
0.751744 + 0.659455i \(0.229213\pi\)
\(510\) −16.7906 −0.743500
\(511\) −4.54870 −0.201222
\(512\) −1.00000 −0.0441942
\(513\) −84.4761 −3.72971
\(514\) 7.10183 0.313248
\(515\) −24.9804 −1.10077
\(516\) −2.34532 −0.103247
\(517\) 3.29424 0.144880
\(518\) 4.13389 0.181633
\(519\) −49.9368 −2.19198
\(520\) 4.45932 0.195554
\(521\) −0.0387645 −0.00169830 −0.000849152 1.00000i \(-0.500270\pi\)
−0.000849152 1.00000i \(0.500270\pi\)
\(522\) −9.74674 −0.426603
\(523\) −35.7567 −1.56353 −0.781766 0.623572i \(-0.785681\pi\)
−0.781766 + 0.623572i \(0.785681\pi\)
\(524\) −2.98294 −0.130310
\(525\) 3.03085 0.132277
\(526\) 2.12702 0.0927425
\(527\) 14.3347 0.624429
\(528\) 3.04439 0.132490
\(529\) −5.31930 −0.231274
\(530\) 10.5587 0.458639
\(531\) −85.4198 −3.70690
\(532\) −3.25702 −0.141210
\(533\) 3.79185 0.164243
\(534\) 21.7916 0.943017
\(535\) −45.6413 −1.97325
\(536\) 7.21647 0.311704
\(537\) 50.3324 2.17200
\(538\) −10.0081 −0.431479
\(539\) −6.85283 −0.295172
\(540\) −27.4217 −1.18004
\(541\) −25.9239 −1.11456 −0.557278 0.830326i \(-0.688154\pi\)
−0.557278 + 0.830326i \(0.688154\pi\)
\(542\) −16.3640 −0.702895
\(543\) −60.6968 −2.60475
\(544\) 2.00124 0.0858024
\(545\) −15.2238 −0.652115
\(546\) −1.88981 −0.0808765
\(547\) 29.8991 1.27839 0.639197 0.769043i \(-0.279267\pi\)
0.639197 + 0.769043i \(0.279267\pi\)
\(548\) −4.94991 −0.211450
\(549\) 83.4271 3.56058
\(550\) −2.59507 −0.110654
\(551\) −13.2012 −0.562389
\(552\) 12.8012 0.544855
\(553\) −0.487255 −0.0207202
\(554\) −9.46792 −0.402253
\(555\) 90.4089 3.83764
\(556\) −12.6349 −0.535838
\(557\) 30.9023 1.30937 0.654686 0.755901i \(-0.272801\pi\)
0.654686 + 0.755901i \(0.272801\pi\)
\(558\) 44.8995 1.90075
\(559\) −1.24653 −0.0527227
\(560\) −1.05726 −0.0446773
\(561\) −6.09256 −0.257228
\(562\) 0.209325 0.00882983
\(563\) −7.18778 −0.302929 −0.151464 0.988463i \(-0.548399\pi\)
−0.151464 + 0.988463i \(0.548399\pi\)
\(564\) 10.0290 0.422295
\(565\) 19.6250 0.825629
\(566\) 22.3914 0.941179
\(567\) 4.40681 0.185069
\(568\) −9.72559 −0.408077
\(569\) 6.15835 0.258172 0.129086 0.991633i \(-0.458796\pi\)
0.129086 + 0.991633i \(0.458796\pi\)
\(570\) −71.2316 −2.98356
\(571\) −7.81281 −0.326956 −0.163478 0.986547i \(-0.552271\pi\)
−0.163478 + 0.986547i \(0.552271\pi\)
\(572\) 1.61809 0.0676558
\(573\) 44.6039 1.86336
\(574\) −0.899008 −0.0375239
\(575\) −10.9119 −0.455056
\(576\) 6.26834 0.261181
\(577\) 31.3260 1.30412 0.652059 0.758169i \(-0.273906\pi\)
0.652059 + 0.758169i \(0.273906\pi\)
\(578\) 12.9950 0.540522
\(579\) −78.6444 −3.26835
\(580\) −4.28522 −0.177934
\(581\) 0.717079 0.0297495
\(582\) −22.6307 −0.938073
\(583\) 3.83128 0.158675
\(584\) 11.8569 0.490643
\(585\) −27.9525 −1.15570
\(586\) 19.3539 0.799504
\(587\) 24.7550 1.02175 0.510874 0.859656i \(-0.329322\pi\)
0.510874 + 0.859656i \(0.329322\pi\)
\(588\) −20.8627 −0.860364
\(589\) 60.8128 2.50575
\(590\) −37.5554 −1.54613
\(591\) 3.04439 0.125230
\(592\) −10.7757 −0.442877
\(593\) 4.44248 0.182431 0.0912154 0.995831i \(-0.470925\pi\)
0.0912154 + 0.995831i \(0.470925\pi\)
\(594\) −9.95012 −0.408258
\(595\) 2.11583 0.0867404
\(596\) 23.3940 0.958257
\(597\) −53.4302 −2.18675
\(598\) 6.80382 0.278229
\(599\) −32.4237 −1.32480 −0.662398 0.749152i \(-0.730461\pi\)
−0.662398 + 0.749152i \(0.730461\pi\)
\(600\) −7.90042 −0.322533
\(601\) 13.2488 0.540431 0.270215 0.962800i \(-0.412905\pi\)
0.270215 + 0.962800i \(0.412905\pi\)
\(602\) 0.295539 0.0120453
\(603\) −45.2353 −1.84212
\(604\) 3.38347 0.137671
\(605\) −2.75592 −0.112044
\(606\) 27.3839 1.11239
\(607\) 14.5637 0.591124 0.295562 0.955324i \(-0.404493\pi\)
0.295562 + 0.955324i \(0.404493\pi\)
\(608\) 8.48996 0.344313
\(609\) 1.81603 0.0735892
\(610\) 36.6792 1.48510
\(611\) 5.33038 0.215644
\(612\) −12.5444 −0.507079
\(613\) −39.1195 −1.58002 −0.790010 0.613094i \(-0.789925\pi\)
−0.790010 + 0.613094i \(0.789925\pi\)
\(614\) −12.6575 −0.510816
\(615\) −19.6614 −0.792826
\(616\) −0.383632 −0.0154570
\(617\) 27.2597 1.09743 0.548717 0.836008i \(-0.315116\pi\)
0.548717 + 0.836008i \(0.315116\pi\)
\(618\) −27.5952 −1.11004
\(619\) 21.7957 0.876042 0.438021 0.898965i \(-0.355680\pi\)
0.438021 + 0.898965i \(0.355680\pi\)
\(620\) 19.7404 0.792792
\(621\) −41.8387 −1.67893
\(622\) 7.07558 0.283705
\(623\) −2.74602 −0.110017
\(624\) 4.92611 0.197202
\(625\) −31.2410 −1.24964
\(626\) −21.3478 −0.853229
\(627\) −25.8468 −1.03222
\(628\) −18.2659 −0.728890
\(629\) 21.5647 0.859841
\(630\) 6.62725 0.264036
\(631\) 44.3795 1.76672 0.883360 0.468696i \(-0.155276\pi\)
0.883360 + 0.468696i \(0.155276\pi\)
\(632\) 1.27011 0.0505223
\(633\) −24.3629 −0.968339
\(634\) −3.77787 −0.150038
\(635\) −50.1754 −1.99115
\(636\) 11.6639 0.462504
\(637\) −11.0885 −0.439342
\(638\) −1.55492 −0.0615597
\(639\) 60.9633 2.41167
\(640\) 2.75592 0.108937
\(641\) −21.2491 −0.839291 −0.419645 0.907688i \(-0.637846\pi\)
−0.419645 + 0.907688i \(0.637846\pi\)
\(642\) −50.4189 −1.98987
\(643\) −14.1784 −0.559142 −0.279571 0.960125i \(-0.590192\pi\)
−0.279571 + 0.960125i \(0.590192\pi\)
\(644\) −1.61311 −0.0635655
\(645\) 6.46349 0.254500
\(646\) −16.9904 −0.668480
\(647\) 3.99270 0.156969 0.0784847 0.996915i \(-0.474992\pi\)
0.0784847 + 0.996915i \(0.474992\pi\)
\(648\) −11.4871 −0.451255
\(649\) −13.6272 −0.534914
\(650\) −4.19906 −0.164701
\(651\) −8.36575 −0.327880
\(652\) −13.1163 −0.513673
\(653\) 13.7833 0.539382 0.269691 0.962947i \(-0.413078\pi\)
0.269691 + 0.962947i \(0.413078\pi\)
\(654\) −16.8174 −0.657611
\(655\) 8.22073 0.321210
\(656\) 2.34341 0.0914949
\(657\) −74.3232 −2.89962
\(658\) −1.26378 −0.0492671
\(659\) −21.7022 −0.845398 −0.422699 0.906270i \(-0.638917\pi\)
−0.422699 + 0.906270i \(0.638917\pi\)
\(660\) −8.39009 −0.326584
\(661\) 4.50144 0.175086 0.0875429 0.996161i \(-0.472099\pi\)
0.0875429 + 0.996161i \(0.472099\pi\)
\(662\) 4.50178 0.174967
\(663\) −9.85832 −0.382865
\(664\) −1.86918 −0.0725384
\(665\) 8.97608 0.348077
\(666\) 67.5456 2.61734
\(667\) −6.53818 −0.253159
\(668\) −14.1426 −0.547194
\(669\) −15.9740 −0.617592
\(670\) −19.8880 −0.768339
\(671\) 13.3093 0.513799
\(672\) −1.16793 −0.0450538
\(673\) −5.41207 −0.208620 −0.104310 0.994545i \(-0.533263\pi\)
−0.104310 + 0.994545i \(0.533263\pi\)
\(674\) −2.06321 −0.0794717
\(675\) 25.8212 0.993860
\(676\) −10.3818 −0.399299
\(677\) −8.07849 −0.310481 −0.155241 0.987877i \(-0.549615\pi\)
−0.155241 + 0.987877i \(0.549615\pi\)
\(678\) 21.6792 0.832586
\(679\) 2.85175 0.109440
\(680\) −5.51525 −0.211500
\(681\) 63.7722 2.44376
\(682\) 7.16290 0.274282
\(683\) −37.2188 −1.42414 −0.712069 0.702110i \(-0.752242\pi\)
−0.712069 + 0.702110i \(0.752242\pi\)
\(684\) −53.2180 −2.03484
\(685\) 13.6415 0.521217
\(686\) 5.31439 0.202904
\(687\) 14.1888 0.541337
\(688\) −0.770372 −0.0293701
\(689\) 6.19935 0.236177
\(690\) −35.2790 −1.34305
\(691\) 18.0478 0.686569 0.343285 0.939231i \(-0.388460\pi\)
0.343285 + 0.939231i \(0.388460\pi\)
\(692\) −16.4029 −0.623543
\(693\) 2.40474 0.0913484
\(694\) 5.80849 0.220487
\(695\) 34.8206 1.32082
\(696\) −4.73378 −0.179433
\(697\) −4.68973 −0.177636
\(698\) −14.1619 −0.536037
\(699\) 62.0393 2.34654
\(700\) 0.995552 0.0376283
\(701\) 48.4483 1.82987 0.914933 0.403605i \(-0.132243\pi\)
0.914933 + 0.403605i \(0.132243\pi\)
\(702\) −16.1002 −0.607663
\(703\) 91.4851 3.45042
\(704\) 1.00000 0.0376889
\(705\) −27.6390 −1.04094
\(706\) 36.4956 1.37353
\(707\) −3.45072 −0.129778
\(708\) −41.4865 −1.55916
\(709\) 43.5804 1.63670 0.818348 0.574723i \(-0.194890\pi\)
0.818348 + 0.574723i \(0.194890\pi\)
\(710\) 26.8029 1.00589
\(711\) −7.96149 −0.298579
\(712\) 7.15795 0.268256
\(713\) 30.1189 1.12796
\(714\) 2.33730 0.0874713
\(715\) −4.45932 −0.166769
\(716\) 16.5328 0.617860
\(717\) −84.6185 −3.16014
\(718\) −31.8263 −1.18775
\(719\) −15.4904 −0.577696 −0.288848 0.957375i \(-0.593272\pi\)
−0.288848 + 0.957375i \(0.593272\pi\)
\(720\) −17.2750 −0.643802
\(721\) 3.47735 0.129503
\(722\) −53.0795 −1.97541
\(723\) −23.5893 −0.877294
\(724\) −19.9372 −0.740962
\(725\) 4.03511 0.149860
\(726\) −3.04439 −0.112988
\(727\) 26.2426 0.973284 0.486642 0.873602i \(-0.338222\pi\)
0.486642 + 0.873602i \(0.338222\pi\)
\(728\) −0.620752 −0.0230066
\(729\) −18.8714 −0.698941
\(730\) −32.6767 −1.20942
\(731\) 1.54170 0.0570218
\(732\) 40.5187 1.49761
\(733\) −23.7212 −0.876162 −0.438081 0.898935i \(-0.644342\pi\)
−0.438081 + 0.898935i \(0.644342\pi\)
\(734\) −27.4423 −1.01291
\(735\) 57.4959 2.12077
\(736\) 4.20484 0.154993
\(737\) −7.21647 −0.265822
\(738\) −14.6893 −0.540721
\(739\) 2.03317 0.0747915 0.0373957 0.999301i \(-0.488094\pi\)
0.0373957 + 0.999301i \(0.488094\pi\)
\(740\) 29.6968 1.09168
\(741\) −41.8225 −1.53639
\(742\) −1.46980 −0.0539581
\(743\) 3.07793 0.112918 0.0564592 0.998405i \(-0.482019\pi\)
0.0564592 + 0.998405i \(0.482019\pi\)
\(744\) 21.8067 0.799473
\(745\) −64.4720 −2.36207
\(746\) −26.4484 −0.968346
\(747\) 11.7167 0.428691
\(748\) −2.00124 −0.0731725
\(749\) 6.35341 0.232149
\(750\) −20.1776 −0.736782
\(751\) −20.3855 −0.743876 −0.371938 0.928258i \(-0.621307\pi\)
−0.371938 + 0.928258i \(0.621307\pi\)
\(752\) 3.29424 0.120129
\(753\) 22.3647 0.815015
\(754\) −2.51600 −0.0916271
\(755\) −9.32455 −0.339355
\(756\) 3.81718 0.138830
\(757\) 31.8420 1.15732 0.578659 0.815569i \(-0.303576\pi\)
0.578659 + 0.815569i \(0.303576\pi\)
\(758\) 14.8445 0.539177
\(759\) −12.8012 −0.464654
\(760\) −23.3976 −0.848721
\(761\) −25.6419 −0.929517 −0.464758 0.885438i \(-0.653859\pi\)
−0.464758 + 0.885438i \(0.653859\pi\)
\(762\) −55.4275 −2.00793
\(763\) 2.11920 0.0767202
\(764\) 14.6512 0.530061
\(765\) 34.5714 1.24993
\(766\) 19.6478 0.709904
\(767\) −22.0500 −0.796180
\(768\) 3.04439 0.109855
\(769\) −30.6970 −1.10696 −0.553481 0.832862i \(-0.686701\pi\)
−0.553481 + 0.832862i \(0.686701\pi\)
\(770\) 1.05726 0.0381009
\(771\) −21.6208 −0.778653
\(772\) −25.8325 −0.929733
\(773\) −14.8616 −0.534533 −0.267267 0.963623i \(-0.586120\pi\)
−0.267267 + 0.963623i \(0.586120\pi\)
\(774\) 4.82895 0.173573
\(775\) −18.5882 −0.667709
\(776\) −7.43357 −0.266849
\(777\) −12.5852 −0.451492
\(778\) 33.9620 1.21760
\(779\) −19.8955 −0.712829
\(780\) −13.5759 −0.486096
\(781\) 9.72559 0.348009
\(782\) −8.41490 −0.300916
\(783\) 15.4716 0.552910
\(784\) −6.85283 −0.244744
\(785\) 50.3394 1.79669
\(786\) 9.08124 0.323917
\(787\) 37.6561 1.34229 0.671147 0.741324i \(-0.265802\pi\)
0.671147 + 0.741324i \(0.265802\pi\)
\(788\) 1.00000 0.0356235
\(789\) −6.47549 −0.230534
\(790\) −3.50032 −0.124536
\(791\) −2.73186 −0.0971337
\(792\) −6.26834 −0.222736
\(793\) 21.5356 0.764752
\(794\) −25.6886 −0.911654
\(795\) −32.1448 −1.14006
\(796\) −17.5504 −0.622056
\(797\) −41.2260 −1.46030 −0.730150 0.683287i \(-0.760550\pi\)
−0.730150 + 0.683287i \(0.760550\pi\)
\(798\) 9.91566 0.351011
\(799\) −6.59256 −0.233228
\(800\) −2.59507 −0.0917496
\(801\) −44.8685 −1.58535
\(802\) −20.5313 −0.724985
\(803\) −11.8569 −0.418422
\(804\) −21.9698 −0.774814
\(805\) 4.44560 0.156687
\(806\) 11.5902 0.408248
\(807\) 30.4685 1.07254
\(808\) 8.99486 0.316438
\(809\) 27.4592 0.965414 0.482707 0.875782i \(-0.339654\pi\)
0.482707 + 0.875782i \(0.339654\pi\)
\(810\) 31.6574 1.11233
\(811\) 16.5611 0.581540 0.290770 0.956793i \(-0.406089\pi\)
0.290770 + 0.956793i \(0.406089\pi\)
\(812\) 0.596516 0.0209336
\(813\) 49.8185 1.74721
\(814\) 10.7757 0.377687
\(815\) 36.1474 1.26619
\(816\) −6.09256 −0.213282
\(817\) 6.54043 0.228821
\(818\) −8.39367 −0.293478
\(819\) 3.89108 0.135965
\(820\) −6.45824 −0.225532
\(821\) −3.82753 −0.133582 −0.0667908 0.997767i \(-0.521276\pi\)
−0.0667908 + 0.997767i \(0.521276\pi\)
\(822\) 15.0695 0.525609
\(823\) 0.146999 0.00512405 0.00256202 0.999997i \(-0.499184\pi\)
0.00256202 + 0.999997i \(0.499184\pi\)
\(824\) −9.06428 −0.315769
\(825\) 7.90042 0.275057
\(826\) 5.22783 0.181899
\(827\) 28.4139 0.988049 0.494025 0.869448i \(-0.335525\pi\)
0.494025 + 0.869448i \(0.335525\pi\)
\(828\) −26.3574 −0.915982
\(829\) −16.4109 −0.569974 −0.284987 0.958531i \(-0.591989\pi\)
−0.284987 + 0.958531i \(0.591989\pi\)
\(830\) 5.15131 0.178805
\(831\) 28.8241 0.999896
\(832\) 1.61809 0.0560972
\(833\) 13.7141 0.475167
\(834\) 38.4655 1.33195
\(835\) 38.9759 1.34882
\(836\) −8.48996 −0.293632
\(837\) −71.2717 −2.46351
\(838\) 28.1301 0.971739
\(839\) 28.3737 0.979568 0.489784 0.871844i \(-0.337076\pi\)
0.489784 + 0.871844i \(0.337076\pi\)
\(840\) 3.21871 0.111056
\(841\) −26.5822 −0.916629
\(842\) 6.66048 0.229535
\(843\) −0.637267 −0.0219487
\(844\) −8.00255 −0.275459
\(845\) 28.6113 0.984259
\(846\) −20.6494 −0.709941
\(847\) 0.383632 0.0131818
\(848\) 3.83128 0.131567
\(849\) −68.1681 −2.33952
\(850\) 5.19335 0.178131
\(851\) 45.3100 1.55321
\(852\) 29.6085 1.01437
\(853\) 45.8296 1.56917 0.784587 0.620018i \(-0.212875\pi\)
0.784587 + 0.620018i \(0.212875\pi\)
\(854\) −5.10586 −0.174719
\(855\) 146.664 5.01581
\(856\) −16.5612 −0.566050
\(857\) −11.5742 −0.395367 −0.197683 0.980266i \(-0.563342\pi\)
−0.197683 + 0.980266i \(0.563342\pi\)
\(858\) −4.92611 −0.168175
\(859\) −14.2960 −0.487773 −0.243886 0.969804i \(-0.578422\pi\)
−0.243886 + 0.969804i \(0.578422\pi\)
\(860\) 2.12308 0.0723964
\(861\) 2.73694 0.0932745
\(862\) −26.5395 −0.903938
\(863\) 26.3281 0.896220 0.448110 0.893979i \(-0.352097\pi\)
0.448110 + 0.893979i \(0.352097\pi\)
\(864\) −9.95012 −0.338510
\(865\) 45.2049 1.53701
\(866\) 5.66324 0.192445
\(867\) −39.5620 −1.34360
\(868\) −2.74792 −0.0932705
\(869\) −1.27011 −0.0430856
\(870\) 13.0459 0.442297
\(871\) −11.6769 −0.395657
\(872\) −5.52404 −0.187068
\(873\) 46.5961 1.57704
\(874\) −35.6990 −1.20754
\(875\) 2.54263 0.0859566
\(876\) −36.0971 −1.21961
\(877\) −16.6055 −0.560727 −0.280364 0.959894i \(-0.590455\pi\)
−0.280364 + 0.959894i \(0.590455\pi\)
\(878\) 26.4140 0.891430
\(879\) −58.9210 −1.98736
\(880\) −2.75592 −0.0929019
\(881\) −41.0974 −1.38461 −0.692304 0.721606i \(-0.743404\pi\)
−0.692304 + 0.721606i \(0.743404\pi\)
\(882\) 42.9558 1.44640
\(883\) −20.2997 −0.683138 −0.341569 0.939857i \(-0.610958\pi\)
−0.341569 + 0.939857i \(0.610958\pi\)
\(884\) −3.23819 −0.108912
\(885\) 114.333 3.84327
\(886\) −26.5681 −0.892573
\(887\) 26.0819 0.875743 0.437872 0.899037i \(-0.355732\pi\)
0.437872 + 0.899037i \(0.355732\pi\)
\(888\) 32.8054 1.10088
\(889\) 6.98457 0.234255
\(890\) −19.7267 −0.661241
\(891\) 11.4871 0.384831
\(892\) −5.24703 −0.175684
\(893\) −27.9680 −0.935912
\(894\) −71.2207 −2.38198
\(895\) −45.5630 −1.52300
\(896\) −0.383632 −0.0128162
\(897\) −20.7135 −0.691604
\(898\) 0.0968120 0.00323066
\(899\) −11.1377 −0.371464
\(900\) 16.2668 0.542226
\(901\) −7.66730 −0.255435
\(902\) −2.34341 −0.0780271
\(903\) −0.899738 −0.0299414
\(904\) 7.12104 0.236842
\(905\) 54.9453 1.82645
\(906\) −10.3006 −0.342215
\(907\) 5.18562 0.172186 0.0860929 0.996287i \(-0.472562\pi\)
0.0860929 + 0.996287i \(0.472562\pi\)
\(908\) 20.9474 0.695164
\(909\) −56.3828 −1.87010
\(910\) 1.71074 0.0567104
\(911\) −29.0767 −0.963355 −0.481678 0.876348i \(-0.659972\pi\)
−0.481678 + 0.876348i \(0.659972\pi\)
\(912\) −25.8468 −0.855873
\(913\) 1.86918 0.0618610
\(914\) 0.866576 0.0286638
\(915\) −111.666 −3.69157
\(916\) 4.66063 0.153992
\(917\) −1.14435 −0.0377898
\(918\) 19.9126 0.657213
\(919\) −2.50468 −0.0826219 −0.0413110 0.999146i \(-0.513153\pi\)
−0.0413110 + 0.999146i \(0.513153\pi\)
\(920\) −11.5882 −0.382051
\(921\) 38.5345 1.26975
\(922\) 17.0541 0.561645
\(923\) 15.7369 0.517986
\(924\) 1.16793 0.0384220
\(925\) −27.9636 −0.919438
\(926\) 34.0789 1.11990
\(927\) 56.8180 1.86615
\(928\) −1.55492 −0.0510426
\(929\) −33.1956 −1.08911 −0.544555 0.838725i \(-0.683302\pi\)
−0.544555 + 0.838725i \(0.683302\pi\)
\(930\) −60.0974 −1.97067
\(931\) 58.1802 1.90678
\(932\) 20.3782 0.667510
\(933\) −21.5409 −0.705216
\(934\) −11.9629 −0.391438
\(935\) 5.51525 0.180368
\(936\) −10.1427 −0.331526
\(937\) 39.3783 1.28643 0.643216 0.765685i \(-0.277600\pi\)
0.643216 + 0.765685i \(0.277600\pi\)
\(938\) 2.76847 0.0903937
\(939\) 64.9911 2.12090
\(940\) −9.07864 −0.296113
\(941\) −11.4503 −0.373270 −0.186635 0.982429i \(-0.559758\pi\)
−0.186635 + 0.982429i \(0.559758\pi\)
\(942\) 55.6087 1.81183
\(943\) −9.85368 −0.320880
\(944\) −13.6272 −0.443527
\(945\) −10.5198 −0.342210
\(946\) 0.770372 0.0250469
\(947\) 18.8501 0.612547 0.306274 0.951943i \(-0.400918\pi\)
0.306274 + 0.951943i \(0.400918\pi\)
\(948\) −3.86672 −0.125585
\(949\) −19.1856 −0.622790
\(950\) 22.0320 0.714814
\(951\) 11.5013 0.372956
\(952\) 0.767740 0.0248826
\(953\) −6.47359 −0.209700 −0.104850 0.994488i \(-0.533436\pi\)
−0.104850 + 0.994488i \(0.533436\pi\)
\(954\) −24.0157 −0.777538
\(955\) −40.3774 −1.30658
\(956\) −27.7949 −0.898950
\(957\) 4.73378 0.153021
\(958\) −27.8579 −0.900049
\(959\) −1.89895 −0.0613202
\(960\) −8.39009 −0.270789
\(961\) 20.3072 0.655070
\(962\) 17.4360 0.562160
\(963\) 103.811 3.34527
\(964\) −7.74842 −0.249560
\(965\) 71.1922 2.29176
\(966\) 4.91095 0.158007
\(967\) −45.1682 −1.45251 −0.726256 0.687424i \(-0.758741\pi\)
−0.726256 + 0.687424i \(0.758741\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 51.7256 1.66167
\(970\) 20.4863 0.657775
\(971\) 48.1671 1.54576 0.772878 0.634554i \(-0.218816\pi\)
0.772878 + 0.634554i \(0.218816\pi\)
\(972\) 5.12082 0.164250
\(973\) −4.84714 −0.155392
\(974\) 18.4547 0.591326
\(975\) 12.7836 0.409403
\(976\) 13.3093 0.426019
\(977\) 26.7116 0.854581 0.427290 0.904114i \(-0.359468\pi\)
0.427290 + 0.904114i \(0.359468\pi\)
\(978\) 39.9311 1.27686
\(979\) −7.15795 −0.228769
\(980\) 18.8858 0.603285
\(981\) 34.6266 1.10554
\(982\) −0.669667 −0.0213699
\(983\) −0.674008 −0.0214975 −0.0107488 0.999942i \(-0.503421\pi\)
−0.0107488 + 0.999942i \(0.503421\pi\)
\(984\) −7.13427 −0.227432
\(985\) −2.75592 −0.0878108
\(986\) 3.11176 0.0990986
\(987\) 3.84743 0.122465
\(988\) −13.7375 −0.437049
\(989\) 3.23929 0.103003
\(990\) 17.2750 0.549036
\(991\) −1.94468 −0.0617750 −0.0308875 0.999523i \(-0.509833\pi\)
−0.0308875 + 0.999523i \(0.509833\pi\)
\(992\) 7.16290 0.227422
\(993\) −13.7052 −0.434921
\(994\) −3.73105 −0.118342
\(995\) 48.3673 1.53335
\(996\) 5.69054 0.180312
\(997\) 15.2599 0.483285 0.241643 0.970365i \(-0.422314\pi\)
0.241643 + 0.970365i \(0.422314\pi\)
\(998\) 2.87546 0.0910209
\(999\) −107.219 −3.39227
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.1.15 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4334.2.a.a.1.15 15 1.1 even 1 trivial