Properties

Label 4334.2.a.a.1.12
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 1.12
Root \(-1.70683\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.70683 q^{3} +1.00000 q^{4} +1.14577 q^{5} -1.70683 q^{6} -1.19968 q^{7} -1.00000 q^{8} -0.0867259 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.70683 q^{3} +1.00000 q^{4} +1.14577 q^{5} -1.70683 q^{6} -1.19968 q^{7} -1.00000 q^{8} -0.0867259 q^{9} -1.14577 q^{10} +1.00000 q^{11} +1.70683 q^{12} +1.87419 q^{13} +1.19968 q^{14} +1.95564 q^{15} +1.00000 q^{16} -5.53362 q^{17} +0.0867259 q^{18} +1.61590 q^{19} +1.14577 q^{20} -2.04765 q^{21} -1.00000 q^{22} -5.50920 q^{23} -1.70683 q^{24} -3.68720 q^{25} -1.87419 q^{26} -5.26852 q^{27} -1.19968 q^{28} +8.88499 q^{29} -1.95564 q^{30} -7.44726 q^{31} -1.00000 q^{32} +1.70683 q^{33} +5.53362 q^{34} -1.37456 q^{35} -0.0867259 q^{36} +3.02478 q^{37} -1.61590 q^{38} +3.19893 q^{39} -1.14577 q^{40} -5.93132 q^{41} +2.04765 q^{42} -4.16246 q^{43} +1.00000 q^{44} -0.0993682 q^{45} +5.50920 q^{46} -6.81072 q^{47} +1.70683 q^{48} -5.56078 q^{49} +3.68720 q^{50} -9.44495 q^{51} +1.87419 q^{52} -6.71218 q^{53} +5.26852 q^{54} +1.14577 q^{55} +1.19968 q^{56} +2.75806 q^{57} -8.88499 q^{58} +8.03264 q^{59} +1.95564 q^{60} -8.11062 q^{61} +7.44726 q^{62} +0.104043 q^{63} +1.00000 q^{64} +2.14740 q^{65} -1.70683 q^{66} +8.40718 q^{67} -5.53362 q^{68} -9.40328 q^{69} +1.37456 q^{70} +1.08059 q^{71} +0.0867259 q^{72} -6.81363 q^{73} -3.02478 q^{74} -6.29344 q^{75} +1.61590 q^{76} -1.19968 q^{77} -3.19893 q^{78} -5.42875 q^{79} +1.14577 q^{80} -8.73230 q^{81} +5.93132 q^{82} +17.4686 q^{83} -2.04765 q^{84} -6.34027 q^{85} +4.16246 q^{86} +15.1652 q^{87} -1.00000 q^{88} -1.98437 q^{89} +0.0993682 q^{90} -2.24842 q^{91} -5.50920 q^{92} -12.7112 q^{93} +6.81072 q^{94} +1.85145 q^{95} -1.70683 q^{96} +16.3133 q^{97} +5.56078 q^{98} -0.0867259 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - 15 q^{2} - q^{3} + 15 q^{4} - 7 q^{5} + q^{6} + q^{7} - 15 q^{8} + 2 q^{9}+O(q^{10}) \) 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
\(3\) 1.70683 0.985440 0.492720 0.870188i \(-0.336003\pi\)
0.492720 + 0.870188i \(0.336003\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.14577 0.512405 0.256203 0.966623i \(-0.417529\pi\)
0.256203 + 0.966623i \(0.417529\pi\)
\(6\) −1.70683 −0.696811
\(7\) −1.19968 −0.453435 −0.226718 0.973961i \(-0.572799\pi\)
−0.226718 + 0.973961i \(0.572799\pi\)
\(8\) −1.00000 −0.353553
\(9\) −0.0867259 −0.0289086
\(10\) −1.14577 −0.362325
\(11\) 1.00000 0.301511
\(12\) 1.70683 0.492720
\(13\) 1.87419 0.519807 0.259903 0.965635i \(-0.416309\pi\)
0.259903 + 0.965635i \(0.416309\pi\)
\(14\) 1.19968 0.320627
\(15\) 1.95564 0.504944
\(16\) 1.00000 0.250000
\(17\) −5.53362 −1.34210 −0.671049 0.741413i \(-0.734156\pi\)
−0.671049 + 0.741413i \(0.734156\pi\)
\(18\) 0.0867259 0.0204415
\(19\) 1.61590 0.370712 0.185356 0.982671i \(-0.440656\pi\)
0.185356 + 0.982671i \(0.440656\pi\)
\(20\) 1.14577 0.256203
\(21\) −2.04765 −0.446833
\(22\) −1.00000 −0.213201
\(23\) −5.50920 −1.14875 −0.574374 0.818593i \(-0.694755\pi\)
−0.574374 + 0.818593i \(0.694755\pi\)
\(24\) −1.70683 −0.348406
\(25\) −3.68720 −0.737441
\(26\) −1.87419 −0.367559
\(27\) −5.26852 −1.01393
\(28\) −1.19968 −0.226718
\(29\) 8.88499 1.64990 0.824951 0.565205i \(-0.191203\pi\)
0.824951 + 0.565205i \(0.191203\pi\)
\(30\) −1.95564 −0.357050
\(31\) −7.44726 −1.33757 −0.668783 0.743457i \(-0.733185\pi\)
−0.668783 + 0.743457i \(0.733185\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.70683 0.297121
\(34\) 5.53362 0.949007
\(35\) −1.37456 −0.232343
\(36\) −0.0867259 −0.0144543
\(37\) 3.02478 0.497271 0.248635 0.968597i \(-0.420018\pi\)
0.248635 + 0.968597i \(0.420018\pi\)
\(38\) −1.61590 −0.262133
\(39\) 3.19893 0.512238
\(40\) −1.14577 −0.181163
\(41\) −5.93132 −0.926317 −0.463158 0.886276i \(-0.653284\pi\)
−0.463158 + 0.886276i \(0.653284\pi\)
\(42\) 2.04765 0.315959
\(43\) −4.16246 −0.634769 −0.317385 0.948297i \(-0.602805\pi\)
−0.317385 + 0.948297i \(0.602805\pi\)
\(44\) 1.00000 0.150756
\(45\) −0.0993682 −0.0148129
\(46\) 5.50920 0.812288
\(47\) −6.81072 −0.993446 −0.496723 0.867909i \(-0.665463\pi\)
−0.496723 + 0.867909i \(0.665463\pi\)
\(48\) 1.70683 0.246360
\(49\) −5.56078 −0.794396
\(50\) 3.68720 0.521449
\(51\) −9.44495 −1.32256
\(52\) 1.87419 0.259903
\(53\) −6.71218 −0.921989 −0.460994 0.887403i \(-0.652507\pi\)
−0.460994 + 0.887403i \(0.652507\pi\)
\(54\) 5.26852 0.716955
\(55\) 1.14577 0.154496
\(56\) 1.19968 0.160314
\(57\) 2.75806 0.365314
\(58\) −8.88499 −1.16666
\(59\) 8.03264 1.04576 0.522881 0.852406i \(-0.324857\pi\)
0.522881 + 0.852406i \(0.324857\pi\)
\(60\) 1.95564 0.252472
\(61\) −8.11062 −1.03846 −0.519229 0.854635i \(-0.673781\pi\)
−0.519229 + 0.854635i \(0.673781\pi\)
\(62\) 7.44726 0.945802
\(63\) 0.104043 0.0131082
\(64\) 1.00000 0.125000
\(65\) 2.14740 0.266352
\(66\) −1.70683 −0.210096
\(67\) 8.40718 1.02710 0.513550 0.858059i \(-0.328330\pi\)
0.513550 + 0.858059i \(0.328330\pi\)
\(68\) −5.53362 −0.671049
\(69\) −9.40328 −1.13202
\(70\) 1.37456 0.164291
\(71\) 1.08059 0.128242 0.0641210 0.997942i \(-0.479576\pi\)
0.0641210 + 0.997942i \(0.479576\pi\)
\(72\) 0.0867259 0.0102207
\(73\) −6.81363 −0.797475 −0.398738 0.917065i \(-0.630552\pi\)
−0.398738 + 0.917065i \(0.630552\pi\)
\(74\) −3.02478 −0.351623
\(75\) −6.29344 −0.726703
\(76\) 1.61590 0.185356
\(77\) −1.19968 −0.136716
\(78\) −3.19893 −0.362207
\(79\) −5.42875 −0.610782 −0.305391 0.952227i \(-0.598787\pi\)
−0.305391 + 0.952227i \(0.598787\pi\)
\(80\) 1.14577 0.128101
\(81\) −8.73230 −0.970256
\(82\) 5.93132 0.655005
\(83\) 17.4686 1.91743 0.958715 0.284369i \(-0.0917840\pi\)
0.958715 + 0.284369i \(0.0917840\pi\)
\(84\) −2.04765 −0.223417
\(85\) −6.34027 −0.687699
\(86\) 4.16246 0.448850
\(87\) 15.1652 1.62588
\(88\) −1.00000 −0.106600
\(89\) −1.98437 −0.210343 −0.105171 0.994454i \(-0.533539\pi\)
−0.105171 + 0.994454i \(0.533539\pi\)
\(90\) 0.0993682 0.0104743
\(91\) −2.24842 −0.235699
\(92\) −5.50920 −0.574374
\(93\) −12.7112 −1.31809
\(94\) 6.81072 0.702472
\(95\) 1.85145 0.189955
\(96\) −1.70683 −0.174203
\(97\) 16.3133 1.65636 0.828181 0.560461i \(-0.189376\pi\)
0.828181 + 0.560461i \(0.189376\pi\)
\(98\) 5.56078 0.561723
\(99\) −0.0867259 −0.00871628
\(100\) −3.68720 −0.368720
\(101\) 10.2901 1.02391 0.511954 0.859013i \(-0.328922\pi\)
0.511954 + 0.859013i \(0.328922\pi\)
\(102\) 9.44495 0.935189
\(103\) −8.16069 −0.804097 −0.402048 0.915618i \(-0.631702\pi\)
−0.402048 + 0.915618i \(0.631702\pi\)
\(104\) −1.87419 −0.183779
\(105\) −2.34614 −0.228960
\(106\) 6.71218 0.651945
\(107\) 6.49869 0.628252 0.314126 0.949381i \(-0.398288\pi\)
0.314126 + 0.949381i \(0.398288\pi\)
\(108\) −5.26852 −0.506964
\(109\) −10.7274 −1.02750 −0.513748 0.857941i \(-0.671743\pi\)
−0.513748 + 0.857941i \(0.671743\pi\)
\(110\) −1.14577 −0.109245
\(111\) 5.16279 0.490030
\(112\) −1.19968 −0.113359
\(113\) −13.9026 −1.30785 −0.653924 0.756561i \(-0.726878\pi\)
−0.653924 + 0.756561i \(0.726878\pi\)
\(114\) −2.75806 −0.258316
\(115\) −6.31230 −0.588625
\(116\) 8.88499 0.824951
\(117\) −0.162541 −0.0150269
\(118\) −8.03264 −0.739465
\(119\) 6.63855 0.608555
\(120\) −1.95564 −0.178525
\(121\) 1.00000 0.0909091
\(122\) 8.11062 0.734301
\(123\) −10.1238 −0.912829
\(124\) −7.44726 −0.668783
\(125\) −9.95356 −0.890274
\(126\) −0.104043 −0.00926889
\(127\) 13.7068 1.21628 0.608141 0.793829i \(-0.291915\pi\)
0.608141 + 0.793829i \(0.291915\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −7.10462 −0.625527
\(130\) −2.14740 −0.188339
\(131\) −18.6812 −1.63218 −0.816091 0.577923i \(-0.803863\pi\)
−0.816091 + 0.577923i \(0.803863\pi\)
\(132\) 1.70683 0.148561
\(133\) −1.93855 −0.168094
\(134\) −8.40718 −0.726270
\(135\) −6.03653 −0.519542
\(136\) 5.53362 0.474504
\(137\) 7.98434 0.682148 0.341074 0.940036i \(-0.389209\pi\)
0.341074 + 0.940036i \(0.389209\pi\)
\(138\) 9.40328 0.800461
\(139\) −18.3920 −1.55999 −0.779993 0.625789i \(-0.784777\pi\)
−0.779993 + 0.625789i \(0.784777\pi\)
\(140\) −1.37456 −0.116171
\(141\) −11.6248 −0.978981
\(142\) −1.08059 −0.0906808
\(143\) 1.87419 0.156728
\(144\) −0.0867259 −0.00722716
\(145\) 10.1802 0.845418
\(146\) 6.81363 0.563900
\(147\) −9.49131 −0.782830
\(148\) 3.02478 0.248635
\(149\) −8.92237 −0.730949 −0.365474 0.930821i \(-0.619093\pi\)
−0.365474 + 0.930821i \(0.619093\pi\)
\(150\) 6.29344 0.513857
\(151\) 1.46432 0.119165 0.0595825 0.998223i \(-0.481023\pi\)
0.0595825 + 0.998223i \(0.481023\pi\)
\(152\) −1.61590 −0.131066
\(153\) 0.479908 0.0387982
\(154\) 1.19968 0.0966727
\(155\) −8.53286 −0.685376
\(156\) 3.19893 0.256119
\(157\) −13.7949 −1.10095 −0.550477 0.834850i \(-0.685554\pi\)
−0.550477 + 0.834850i \(0.685554\pi\)
\(158\) 5.42875 0.431888
\(159\) −11.4566 −0.908564
\(160\) −1.14577 −0.0905813
\(161\) 6.60926 0.520883
\(162\) 8.73230 0.686074
\(163\) 2.63145 0.206111 0.103056 0.994676i \(-0.467138\pi\)
0.103056 + 0.994676i \(0.467138\pi\)
\(164\) −5.93132 −0.463158
\(165\) 1.95564 0.152246
\(166\) −17.4686 −1.35583
\(167\) −1.78283 −0.137959 −0.0689797 0.997618i \(-0.521974\pi\)
−0.0689797 + 0.997618i \(0.521974\pi\)
\(168\) 2.04765 0.157979
\(169\) −9.48741 −0.729801
\(170\) 6.34027 0.486276
\(171\) −0.140140 −0.0107168
\(172\) −4.16246 −0.317385
\(173\) 12.1139 0.921003 0.460502 0.887659i \(-0.347670\pi\)
0.460502 + 0.887659i \(0.347670\pi\)
\(174\) −15.1652 −1.14967
\(175\) 4.42345 0.334382
\(176\) 1.00000 0.0753778
\(177\) 13.7104 1.03053
\(178\) 1.98437 0.148735
\(179\) −11.9199 −0.890933 −0.445467 0.895299i \(-0.646962\pi\)
−0.445467 + 0.895299i \(0.646962\pi\)
\(180\) −0.0993682 −0.00740647
\(181\) 1.58015 0.117452 0.0587258 0.998274i \(-0.481296\pi\)
0.0587258 + 0.998274i \(0.481296\pi\)
\(182\) 2.24842 0.166664
\(183\) −13.8435 −1.02334
\(184\) 5.50920 0.406144
\(185\) 3.46571 0.254804
\(186\) 12.7112 0.932031
\(187\) −5.53362 −0.404658
\(188\) −6.81072 −0.496723
\(189\) 6.32052 0.459750
\(190\) −1.85145 −0.134318
\(191\) −18.8839 −1.36639 −0.683196 0.730235i \(-0.739411\pi\)
−0.683196 + 0.730235i \(0.739411\pi\)
\(192\) 1.70683 0.123180
\(193\) 10.5416 0.758800 0.379400 0.925233i \(-0.376130\pi\)
0.379400 + 0.925233i \(0.376130\pi\)
\(194\) −16.3133 −1.17122
\(195\) 3.66524 0.262473
\(196\) −5.56078 −0.397198
\(197\) 1.00000 0.0712470
\(198\) 0.0867259 0.00616334
\(199\) −6.29150 −0.445992 −0.222996 0.974819i \(-0.571584\pi\)
−0.222996 + 0.974819i \(0.571584\pi\)
\(200\) 3.68720 0.260725
\(201\) 14.3496 1.01215
\(202\) −10.2901 −0.724012
\(203\) −10.6591 −0.748124
\(204\) −9.44495 −0.661279
\(205\) −6.79595 −0.474650
\(206\) 8.16069 0.568582
\(207\) 0.477791 0.0332087
\(208\) 1.87419 0.129952
\(209\) 1.61590 0.111774
\(210\) 2.34614 0.161899
\(211\) 1.80977 0.124589 0.0622947 0.998058i \(-0.480158\pi\)
0.0622947 + 0.998058i \(0.480158\pi\)
\(212\) −6.71218 −0.460994
\(213\) 1.84438 0.126375
\(214\) −6.49869 −0.444242
\(215\) −4.76924 −0.325259
\(216\) 5.26852 0.358477
\(217\) 8.93430 0.606500
\(218\) 10.7274 0.726549
\(219\) −11.6297 −0.785864
\(220\) 1.14577 0.0772480
\(221\) −10.3710 −0.697632
\(222\) −5.16279 −0.346504
\(223\) −7.58628 −0.508015 −0.254007 0.967202i \(-0.581749\pi\)
−0.254007 + 0.967202i \(0.581749\pi\)
\(224\) 1.19968 0.0801568
\(225\) 0.319776 0.0213184
\(226\) 13.9026 0.924788
\(227\) 12.3775 0.821521 0.410761 0.911743i \(-0.365263\pi\)
0.410761 + 0.911743i \(0.365263\pi\)
\(228\) 2.75806 0.182657
\(229\) −4.25397 −0.281110 −0.140555 0.990073i \(-0.544889\pi\)
−0.140555 + 0.990073i \(0.544889\pi\)
\(230\) 6.31230 0.416221
\(231\) −2.04765 −0.134725
\(232\) −8.88499 −0.583328
\(233\) −7.59700 −0.497696 −0.248848 0.968543i \(-0.580052\pi\)
−0.248848 + 0.968543i \(0.580052\pi\)
\(234\) 0.162541 0.0106256
\(235\) −7.80354 −0.509047
\(236\) 8.03264 0.522881
\(237\) −9.26595 −0.601888
\(238\) −6.63855 −0.430313
\(239\) 8.61541 0.557285 0.278642 0.960395i \(-0.410116\pi\)
0.278642 + 0.960395i \(0.410116\pi\)
\(240\) 1.95564 0.126236
\(241\) 21.9775 1.41569 0.707847 0.706366i \(-0.249667\pi\)
0.707847 + 0.706366i \(0.249667\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 0.900997 0.0577990
\(244\) −8.11062 −0.519229
\(245\) −6.37139 −0.407053
\(246\) 10.1238 0.645468
\(247\) 3.02849 0.192698
\(248\) 7.44726 0.472901
\(249\) 29.8160 1.88951
\(250\) 9.95356 0.629519
\(251\) 8.67911 0.547821 0.273910 0.961755i \(-0.411683\pi\)
0.273910 + 0.961755i \(0.411683\pi\)
\(252\) 0.104043 0.00655410
\(253\) −5.50920 −0.346361
\(254\) −13.7068 −0.860042
\(255\) −10.8218 −0.677685
\(256\) 1.00000 0.0625000
\(257\) 15.4743 0.965259 0.482629 0.875825i \(-0.339682\pi\)
0.482629 + 0.875825i \(0.339682\pi\)
\(258\) 7.10462 0.442314
\(259\) −3.62876 −0.225480
\(260\) 2.14740 0.133176
\(261\) −0.770559 −0.0476964
\(262\) 18.6812 1.15413
\(263\) 30.1966 1.86200 0.931001 0.365016i \(-0.118936\pi\)
0.931001 + 0.365016i \(0.118936\pi\)
\(264\) −1.70683 −0.105048
\(265\) −7.69063 −0.472432
\(266\) 1.93855 0.118860
\(267\) −3.38698 −0.207280
\(268\) 8.40718 0.513550
\(269\) −22.5400 −1.37429 −0.687145 0.726520i \(-0.741136\pi\)
−0.687145 + 0.726520i \(0.741136\pi\)
\(270\) 6.03653 0.367371
\(271\) 23.0806 1.40205 0.701024 0.713138i \(-0.252727\pi\)
0.701024 + 0.713138i \(0.252727\pi\)
\(272\) −5.53362 −0.335525
\(273\) −3.83768 −0.232267
\(274\) −7.98434 −0.482352
\(275\) −3.68720 −0.222347
\(276\) −9.40328 −0.566011
\(277\) 26.9413 1.61875 0.809373 0.587294i \(-0.199807\pi\)
0.809373 + 0.587294i \(0.199807\pi\)
\(278\) 18.3920 1.10308
\(279\) 0.645870 0.0386672
\(280\) 1.37456 0.0821455
\(281\) 5.19557 0.309942 0.154971 0.987919i \(-0.450472\pi\)
0.154971 + 0.987919i \(0.450472\pi\)
\(282\) 11.6248 0.692244
\(283\) −7.02171 −0.417397 −0.208699 0.977980i \(-0.566923\pi\)
−0.208699 + 0.977980i \(0.566923\pi\)
\(284\) 1.08059 0.0641210
\(285\) 3.16011 0.187189
\(286\) −1.87419 −0.110823
\(287\) 7.11567 0.420025
\(288\) 0.0867259 0.00511037
\(289\) 13.6209 0.801230
\(290\) −10.1802 −0.597801
\(291\) 27.8440 1.63224
\(292\) −6.81363 −0.398738
\(293\) 20.7718 1.21350 0.606752 0.794891i \(-0.292472\pi\)
0.606752 + 0.794891i \(0.292472\pi\)
\(294\) 9.49131 0.553544
\(295\) 9.20359 0.535854
\(296\) −3.02478 −0.175812
\(297\) −5.26852 −0.305711
\(298\) 8.92237 0.516859
\(299\) −10.3253 −0.597127
\(300\) −6.29344 −0.363352
\(301\) 4.99361 0.287827
\(302\) −1.46432 −0.0842623
\(303\) 17.5636 1.00900
\(304\) 1.61590 0.0926779
\(305\) −9.29292 −0.532111
\(306\) −0.479908 −0.0274345
\(307\) −27.0682 −1.54487 −0.772433 0.635097i \(-0.780960\pi\)
−0.772433 + 0.635097i \(0.780960\pi\)
\(308\) −1.19968 −0.0683579
\(309\) −13.9289 −0.792389
\(310\) 8.53286 0.484634
\(311\) −9.87629 −0.560033 −0.280016 0.959995i \(-0.590340\pi\)
−0.280016 + 0.959995i \(0.590340\pi\)
\(312\) −3.19893 −0.181103
\(313\) 2.58209 0.145949 0.0729743 0.997334i \(-0.476751\pi\)
0.0729743 + 0.997334i \(0.476751\pi\)
\(314\) 13.7949 0.778492
\(315\) 0.119210 0.00671671
\(316\) −5.42875 −0.305391
\(317\) −2.07068 −0.116301 −0.0581505 0.998308i \(-0.518520\pi\)
−0.0581505 + 0.998308i \(0.518520\pi\)
\(318\) 11.4566 0.642452
\(319\) 8.88499 0.497464
\(320\) 1.14577 0.0640507
\(321\) 11.0922 0.619105
\(322\) −6.60926 −0.368320
\(323\) −8.94174 −0.497532
\(324\) −8.73230 −0.485128
\(325\) −6.91052 −0.383327
\(326\) −2.63145 −0.145743
\(327\) −18.3098 −1.01253
\(328\) 5.93132 0.327502
\(329\) 8.17066 0.450463
\(330\) −1.95564 −0.107655
\(331\) 7.21290 0.396457 0.198228 0.980156i \(-0.436481\pi\)
0.198228 + 0.980156i \(0.436481\pi\)
\(332\) 17.4686 0.958715
\(333\) −0.262327 −0.0143754
\(334\) 1.78283 0.0975520
\(335\) 9.63272 0.526292
\(336\) −2.04765 −0.111708
\(337\) −24.1101 −1.31336 −0.656682 0.754168i \(-0.728040\pi\)
−0.656682 + 0.754168i \(0.728040\pi\)
\(338\) 9.48741 0.516047
\(339\) −23.7294 −1.28880
\(340\) −6.34027 −0.343849
\(341\) −7.44726 −0.403291
\(342\) 0.140140 0.00757790
\(343\) 15.0689 0.813643
\(344\) 4.16246 0.224425
\(345\) −10.7740 −0.580054
\(346\) −12.1139 −0.651247
\(347\) 33.6719 1.80760 0.903802 0.427951i \(-0.140764\pi\)
0.903802 + 0.427951i \(0.140764\pi\)
\(348\) 15.1652 0.812939
\(349\) −10.9087 −0.583932 −0.291966 0.956429i \(-0.594309\pi\)
−0.291966 + 0.956429i \(0.594309\pi\)
\(350\) −4.42345 −0.236444
\(351\) −9.87421 −0.527046
\(352\) −1.00000 −0.0533002
\(353\) −34.0928 −1.81458 −0.907288 0.420510i \(-0.861851\pi\)
−0.907288 + 0.420510i \(0.861851\pi\)
\(354\) −13.7104 −0.728698
\(355\) 1.23811 0.0657119
\(356\) −1.98437 −0.105171
\(357\) 11.3309 0.599694
\(358\) 11.9199 0.629985
\(359\) −12.5651 −0.663162 −0.331581 0.943427i \(-0.607582\pi\)
−0.331581 + 0.943427i \(0.607582\pi\)
\(360\) 0.0993682 0.00523716
\(361\) −16.3889 −0.862573
\(362\) −1.58015 −0.0830509
\(363\) 1.70683 0.0895854
\(364\) −2.24842 −0.117849
\(365\) −7.80688 −0.408631
\(366\) 13.8435 0.723609
\(367\) −2.14332 −0.111880 −0.0559401 0.998434i \(-0.517816\pi\)
−0.0559401 + 0.998434i \(0.517816\pi\)
\(368\) −5.50920 −0.287187
\(369\) 0.514399 0.0267785
\(370\) −3.46571 −0.180174
\(371\) 8.05245 0.418062
\(372\) −12.7112 −0.659046
\(373\) −2.85050 −0.147593 −0.0737965 0.997273i \(-0.523512\pi\)
−0.0737965 + 0.997273i \(0.523512\pi\)
\(374\) 5.53362 0.286136
\(375\) −16.9891 −0.877311
\(376\) 6.81072 0.351236
\(377\) 16.6522 0.857630
\(378\) −6.32052 −0.325093
\(379\) −29.9256 −1.53717 −0.768586 0.639746i \(-0.779039\pi\)
−0.768586 + 0.639746i \(0.779039\pi\)
\(380\) 1.85145 0.0949773
\(381\) 23.3952 1.19857
\(382\) 18.8839 0.966185
\(383\) −13.3893 −0.684161 −0.342080 0.939671i \(-0.611131\pi\)
−0.342080 + 0.939671i \(0.611131\pi\)
\(384\) −1.70683 −0.0871014
\(385\) −1.37456 −0.0700539
\(386\) −10.5416 −0.536553
\(387\) 0.360993 0.0183503
\(388\) 16.3133 0.828181
\(389\) 14.9613 0.758570 0.379285 0.925280i \(-0.376170\pi\)
0.379285 + 0.925280i \(0.376170\pi\)
\(390\) −3.66524 −0.185597
\(391\) 30.4858 1.54173
\(392\) 5.56078 0.280862
\(393\) −31.8856 −1.60842
\(394\) −1.00000 −0.0503793
\(395\) −6.22011 −0.312968
\(396\) −0.0867259 −0.00435814
\(397\) −28.0241 −1.40649 −0.703244 0.710949i \(-0.748266\pi\)
−0.703244 + 0.710949i \(0.748266\pi\)
\(398\) 6.29150 0.315364
\(399\) −3.30878 −0.165646
\(400\) −3.68720 −0.184360
\(401\) −0.888495 −0.0443693 −0.0221847 0.999754i \(-0.507062\pi\)
−0.0221847 + 0.999754i \(0.507062\pi\)
\(402\) −14.3496 −0.715695
\(403\) −13.9576 −0.695276
\(404\) 10.2901 0.511954
\(405\) −10.0052 −0.497164
\(406\) 10.6591 0.529003
\(407\) 3.02478 0.149933
\(408\) 9.44495 0.467595
\(409\) −8.93554 −0.441834 −0.220917 0.975293i \(-0.570905\pi\)
−0.220917 + 0.975293i \(0.570905\pi\)
\(410\) 6.79595 0.335628
\(411\) 13.6279 0.672216
\(412\) −8.16069 −0.402048
\(413\) −9.63658 −0.474185
\(414\) −0.477791 −0.0234821
\(415\) 20.0151 0.982501
\(416\) −1.87419 −0.0918897
\(417\) −31.3920 −1.53727
\(418\) −1.61590 −0.0790360
\(419\) 8.77232 0.428556 0.214278 0.976773i \(-0.431260\pi\)
0.214278 + 0.976773i \(0.431260\pi\)
\(420\) −2.34614 −0.114480
\(421\) −34.4629 −1.67962 −0.839809 0.542882i \(-0.817333\pi\)
−0.839809 + 0.542882i \(0.817333\pi\)
\(422\) −1.80977 −0.0880981
\(423\) 0.590666 0.0287192
\(424\) 6.71218 0.325972
\(425\) 20.4036 0.989719
\(426\) −1.84438 −0.0893604
\(427\) 9.73012 0.470873
\(428\) 6.49869 0.314126
\(429\) 3.19893 0.154446
\(430\) 4.76924 0.229993
\(431\) −22.8149 −1.09896 −0.549479 0.835508i \(-0.685174\pi\)
−0.549479 + 0.835508i \(0.685174\pi\)
\(432\) −5.26852 −0.253482
\(433\) 32.0875 1.54203 0.771014 0.636818i \(-0.219750\pi\)
0.771014 + 0.636818i \(0.219750\pi\)
\(434\) −8.93430 −0.428860
\(435\) 17.3759 0.833109
\(436\) −10.7274 −0.513748
\(437\) −8.90230 −0.425855
\(438\) 11.6297 0.555690
\(439\) −32.1294 −1.53345 −0.766725 0.641975i \(-0.778115\pi\)
−0.766725 + 0.641975i \(0.778115\pi\)
\(440\) −1.14577 −0.0546226
\(441\) 0.482263 0.0229649
\(442\) 10.3710 0.493300
\(443\) 10.6860 0.507705 0.253853 0.967243i \(-0.418302\pi\)
0.253853 + 0.967243i \(0.418302\pi\)
\(444\) 5.16279 0.245015
\(445\) −2.27364 −0.107781
\(446\) 7.58628 0.359221
\(447\) −15.2290 −0.720306
\(448\) −1.19968 −0.0566794
\(449\) 18.0967 0.854037 0.427018 0.904243i \(-0.359564\pi\)
0.427018 + 0.904243i \(0.359564\pi\)
\(450\) −0.319776 −0.0150744
\(451\) −5.93132 −0.279295
\(452\) −13.9026 −0.653924
\(453\) 2.49935 0.117430
\(454\) −12.3775 −0.580903
\(455\) −2.57618 −0.120773
\(456\) −2.75806 −0.129158
\(457\) −10.3791 −0.485512 −0.242756 0.970087i \(-0.578051\pi\)
−0.242756 + 0.970087i \(0.578051\pi\)
\(458\) 4.25397 0.198775
\(459\) 29.1540 1.36079
\(460\) −6.31230 −0.294312
\(461\) −13.4670 −0.627222 −0.313611 0.949552i \(-0.601539\pi\)
−0.313611 + 0.949552i \(0.601539\pi\)
\(462\) 2.04765 0.0952651
\(463\) 24.1611 1.12286 0.561430 0.827524i \(-0.310251\pi\)
0.561430 + 0.827524i \(0.310251\pi\)
\(464\) 8.88499 0.412475
\(465\) −14.5642 −0.675397
\(466\) 7.59700 0.351924
\(467\) −30.2544 −1.40000 −0.700002 0.714141i \(-0.746818\pi\)
−0.700002 + 0.714141i \(0.746818\pi\)
\(468\) −0.162541 −0.00751345
\(469\) −10.0859 −0.465724
\(470\) 7.80354 0.359950
\(471\) −23.5456 −1.08492
\(472\) −8.03264 −0.369732
\(473\) −4.16246 −0.191390
\(474\) 9.26595 0.425599
\(475\) −5.95814 −0.273378
\(476\) 6.63855 0.304277
\(477\) 0.582120 0.0266534
\(478\) −8.61541 −0.394060
\(479\) 15.8719 0.725204 0.362602 0.931944i \(-0.381888\pi\)
0.362602 + 0.931944i \(0.381888\pi\)
\(480\) −1.95564 −0.0892624
\(481\) 5.66901 0.258485
\(482\) −21.9775 −1.00105
\(483\) 11.2809 0.513299
\(484\) 1.00000 0.0454545
\(485\) 18.6913 0.848728
\(486\) −0.900997 −0.0408700
\(487\) 5.36705 0.243204 0.121602 0.992579i \(-0.461197\pi\)
0.121602 + 0.992579i \(0.461197\pi\)
\(488\) 8.11062 0.367150
\(489\) 4.49144 0.203110
\(490\) 6.37139 0.287830
\(491\) 38.8560 1.75355 0.876774 0.480902i \(-0.159691\pi\)
0.876774 + 0.480902i \(0.159691\pi\)
\(492\) −10.1238 −0.456415
\(493\) −49.1661 −2.21433
\(494\) −3.02849 −0.136258
\(495\) −0.0993682 −0.00446627
\(496\) −7.44726 −0.334392
\(497\) −1.29635 −0.0581494
\(498\) −29.8160 −1.33609
\(499\) 3.60886 0.161555 0.0807773 0.996732i \(-0.474260\pi\)
0.0807773 + 0.996732i \(0.474260\pi\)
\(500\) −9.95356 −0.445137
\(501\) −3.04299 −0.135951
\(502\) −8.67911 −0.387368
\(503\) 5.81700 0.259367 0.129684 0.991555i \(-0.458604\pi\)
0.129684 + 0.991555i \(0.458604\pi\)
\(504\) −0.104043 −0.00463445
\(505\) 11.7902 0.524656
\(506\) 5.50920 0.244914
\(507\) −16.1934 −0.719175
\(508\) 13.7068 0.608141
\(509\) −27.6151 −1.22402 −0.612009 0.790851i \(-0.709638\pi\)
−0.612009 + 0.790851i \(0.709638\pi\)
\(510\) 10.8218 0.479196
\(511\) 8.17416 0.361603
\(512\) −1.00000 −0.0441942
\(513\) −8.51338 −0.375875
\(514\) −15.4743 −0.682541
\(515\) −9.35030 −0.412023
\(516\) −7.10462 −0.312763
\(517\) −6.81072 −0.299535
\(518\) 3.62876 0.159438
\(519\) 20.6764 0.907593
\(520\) −2.14740 −0.0941695
\(521\) −36.4205 −1.59561 −0.797806 0.602914i \(-0.794006\pi\)
−0.797806 + 0.602914i \(0.794006\pi\)
\(522\) 0.770559 0.0337264
\(523\) −15.1819 −0.663857 −0.331928 0.943305i \(-0.607699\pi\)
−0.331928 + 0.943305i \(0.607699\pi\)
\(524\) −18.6812 −0.816091
\(525\) 7.55009 0.329513
\(526\) −30.1966 −1.31663
\(527\) 41.2102 1.79515
\(528\) 1.70683 0.0742803
\(529\) 7.35133 0.319623
\(530\) 7.69063 0.334060
\(531\) −0.696638 −0.0302315
\(532\) −1.93855 −0.0840469
\(533\) −11.1164 −0.481506
\(534\) 3.38698 0.146569
\(535\) 7.44603 0.321920
\(536\) −8.40718 −0.363135
\(537\) −20.3452 −0.877961
\(538\) 22.5400 0.971770
\(539\) −5.56078 −0.239520
\(540\) −6.03653 −0.259771
\(541\) 22.5318 0.968719 0.484359 0.874869i \(-0.339053\pi\)
0.484359 + 0.874869i \(0.339053\pi\)
\(542\) −23.0806 −0.991397
\(543\) 2.69705 0.115742
\(544\) 5.53362 0.237252
\(545\) −12.2911 −0.526494
\(546\) 3.83768 0.164237
\(547\) −26.8040 −1.14605 −0.573027 0.819536i \(-0.694231\pi\)
−0.573027 + 0.819536i \(0.694231\pi\)
\(548\) 7.98434 0.341074
\(549\) 0.703400 0.0300204
\(550\) 3.68720 0.157223
\(551\) 14.3572 0.611638
\(552\) 9.40328 0.400230
\(553\) 6.51274 0.276950
\(554\) −26.9413 −1.14463
\(555\) 5.91538 0.251094
\(556\) −18.3920 −0.779993
\(557\) 39.3675 1.66806 0.834028 0.551722i \(-0.186029\pi\)
0.834028 + 0.551722i \(0.186029\pi\)
\(558\) −0.645870 −0.0273418
\(559\) −7.80124 −0.329957
\(560\) −1.37456 −0.0580856
\(561\) −9.44495 −0.398766
\(562\) −5.19557 −0.219162
\(563\) −39.1964 −1.65193 −0.825966 0.563720i \(-0.809370\pi\)
−0.825966 + 0.563720i \(0.809370\pi\)
\(564\) −11.6248 −0.489490
\(565\) −15.9292 −0.670148
\(566\) 7.02171 0.295144
\(567\) 10.4759 0.439948
\(568\) −1.08059 −0.0453404
\(569\) 27.6319 1.15839 0.579195 0.815189i \(-0.303367\pi\)
0.579195 + 0.815189i \(0.303367\pi\)
\(570\) −3.16011 −0.132363
\(571\) 19.1761 0.802493 0.401246 0.915970i \(-0.368577\pi\)
0.401246 + 0.915970i \(0.368577\pi\)
\(572\) 1.87419 0.0783638
\(573\) −32.2317 −1.34650
\(574\) −7.11567 −0.297002
\(575\) 20.3136 0.847134
\(576\) −0.0867259 −0.00361358
\(577\) 6.82894 0.284292 0.142146 0.989846i \(-0.454600\pi\)
0.142146 + 0.989846i \(0.454600\pi\)
\(578\) −13.6209 −0.566555
\(579\) 17.9927 0.747752
\(580\) 10.1802 0.422709
\(581\) −20.9567 −0.869430
\(582\) −27.8440 −1.15417
\(583\) −6.71218 −0.277990
\(584\) 6.81363 0.281950
\(585\) −0.186235 −0.00769986
\(586\) −20.7718 −0.858077
\(587\) 27.1012 1.11859 0.559293 0.828970i \(-0.311073\pi\)
0.559293 + 0.828970i \(0.311073\pi\)
\(588\) −9.49131 −0.391415
\(589\) −12.0340 −0.495852
\(590\) −9.20359 −0.378906
\(591\) 1.70683 0.0702097
\(592\) 3.02478 0.124318
\(593\) 24.8088 1.01878 0.509388 0.860537i \(-0.329872\pi\)
0.509388 + 0.860537i \(0.329872\pi\)
\(594\) 5.26852 0.216170
\(595\) 7.60627 0.311827
\(596\) −8.92237 −0.365474
\(597\) −10.7385 −0.439499
\(598\) 10.3253 0.422233
\(599\) 21.2393 0.867816 0.433908 0.900957i \(-0.357134\pi\)
0.433908 + 0.900957i \(0.357134\pi\)
\(600\) 6.29344 0.256928
\(601\) −14.7481 −0.601586 −0.300793 0.953690i \(-0.597251\pi\)
−0.300793 + 0.953690i \(0.597251\pi\)
\(602\) −4.99361 −0.203524
\(603\) −0.729120 −0.0296921
\(604\) 1.46432 0.0595825
\(605\) 1.14577 0.0465823
\(606\) −17.5636 −0.713470
\(607\) −16.6528 −0.675915 −0.337958 0.941161i \(-0.609736\pi\)
−0.337958 + 0.941161i \(0.609736\pi\)
\(608\) −1.61590 −0.0655332
\(609\) −18.1933 −0.737231
\(610\) 9.29292 0.376259
\(611\) −12.7646 −0.516400
\(612\) 0.479908 0.0193991
\(613\) 13.8192 0.558150 0.279075 0.960269i \(-0.409972\pi\)
0.279075 + 0.960269i \(0.409972\pi\)
\(614\) 27.0682 1.09238
\(615\) −11.5995 −0.467739
\(616\) 1.19968 0.0483364
\(617\) −6.89996 −0.277782 −0.138891 0.990308i \(-0.544354\pi\)
−0.138891 + 0.990308i \(0.544354\pi\)
\(618\) 13.9289 0.560303
\(619\) 15.9224 0.639974 0.319987 0.947422i \(-0.396321\pi\)
0.319987 + 0.947422i \(0.396321\pi\)
\(620\) −8.53286 −0.342688
\(621\) 29.0254 1.16475
\(622\) 9.87629 0.396003
\(623\) 2.38060 0.0953767
\(624\) 3.19893 0.128060
\(625\) 7.03150 0.281260
\(626\) −2.58209 −0.103201
\(627\) 2.75806 0.110146
\(628\) −13.7949 −0.550477
\(629\) −16.7380 −0.667386
\(630\) −0.119210 −0.00474943
\(631\) 6.07520 0.241850 0.120925 0.992662i \(-0.461414\pi\)
0.120925 + 0.992662i \(0.461414\pi\)
\(632\) 5.42875 0.215944
\(633\) 3.08897 0.122775
\(634\) 2.07068 0.0822372
\(635\) 15.7049 0.623230
\(636\) −11.4566 −0.454282
\(637\) −10.4219 −0.412933
\(638\) −8.88499 −0.351760
\(639\) −0.0937148 −0.00370730
\(640\) −1.14577 −0.0452907
\(641\) −34.2875 −1.35427 −0.677137 0.735857i \(-0.736779\pi\)
−0.677137 + 0.735857i \(0.736779\pi\)
\(642\) −11.0922 −0.437773
\(643\) 29.4437 1.16115 0.580574 0.814208i \(-0.302828\pi\)
0.580574 + 0.814208i \(0.302828\pi\)
\(644\) 6.60926 0.260442
\(645\) −8.14028 −0.320523
\(646\) 8.94174 0.351808
\(647\) −1.67685 −0.0659237 −0.0329619 0.999457i \(-0.510494\pi\)
−0.0329619 + 0.999457i \(0.510494\pi\)
\(648\) 8.73230 0.343037
\(649\) 8.03264 0.315309
\(650\) 6.91052 0.271053
\(651\) 15.2493 0.597669
\(652\) 2.63145 0.103056
\(653\) 36.8803 1.44324 0.721619 0.692291i \(-0.243398\pi\)
0.721619 + 0.692291i \(0.243398\pi\)
\(654\) 18.3098 0.715970
\(655\) −21.4044 −0.836339
\(656\) −5.93132 −0.231579
\(657\) 0.590918 0.0230539
\(658\) −8.17066 −0.318526
\(659\) −17.0850 −0.665537 −0.332769 0.943009i \(-0.607983\pi\)
−0.332769 + 0.943009i \(0.607983\pi\)
\(660\) 1.95564 0.0761232
\(661\) 2.09579 0.0815169 0.0407584 0.999169i \(-0.487023\pi\)
0.0407584 + 0.999169i \(0.487023\pi\)
\(662\) −7.21290 −0.280337
\(663\) −17.7016 −0.687474
\(664\) −17.4686 −0.677914
\(665\) −2.22114 −0.0861321
\(666\) 0.262327 0.0101650
\(667\) −48.9492 −1.89532
\(668\) −1.78283 −0.0689797
\(669\) −12.9485 −0.500618
\(670\) −9.63272 −0.372145
\(671\) −8.11062 −0.313107
\(672\) 2.04765 0.0789897
\(673\) −17.6802 −0.681521 −0.340761 0.940150i \(-0.610685\pi\)
−0.340761 + 0.940150i \(0.610685\pi\)
\(674\) 24.1101 0.928688
\(675\) 19.4261 0.747711
\(676\) −9.48741 −0.364901
\(677\) 10.3928 0.399427 0.199713 0.979854i \(-0.435999\pi\)
0.199713 + 0.979854i \(0.435999\pi\)
\(678\) 23.7294 0.911323
\(679\) −19.5706 −0.751053
\(680\) 6.34027 0.243138
\(681\) 21.1262 0.809559
\(682\) 7.44726 0.285170
\(683\) −6.20800 −0.237542 −0.118771 0.992922i \(-0.537896\pi\)
−0.118771 + 0.992922i \(0.537896\pi\)
\(684\) −0.140140 −0.00535838
\(685\) 9.14824 0.349536
\(686\) −15.0689 −0.575332
\(687\) −7.26081 −0.277017
\(688\) −4.16246 −0.158692
\(689\) −12.5799 −0.479256
\(690\) 10.7740 0.410160
\(691\) −21.1760 −0.805572 −0.402786 0.915294i \(-0.631958\pi\)
−0.402786 + 0.915294i \(0.631958\pi\)
\(692\) 12.1139 0.460502
\(693\) 0.104043 0.00395227
\(694\) −33.6719 −1.27817
\(695\) −21.0730 −0.799345
\(696\) −15.1652 −0.574835
\(697\) 32.8217 1.24321
\(698\) 10.9087 0.412902
\(699\) −12.9668 −0.490449
\(700\) 4.42345 0.167191
\(701\) −12.9472 −0.489007 −0.244504 0.969648i \(-0.578625\pi\)
−0.244504 + 0.969648i \(0.578625\pi\)
\(702\) 9.87421 0.372678
\(703\) 4.88773 0.184344
\(704\) 1.00000 0.0376889
\(705\) −13.3193 −0.501635
\(706\) 34.0928 1.28310
\(707\) −12.3449 −0.464276
\(708\) 13.7104 0.515267
\(709\) −0.413937 −0.0155457 −0.00777287 0.999970i \(-0.502474\pi\)
−0.00777287 + 0.999970i \(0.502474\pi\)
\(710\) −1.23811 −0.0464653
\(711\) 0.470813 0.0176569
\(712\) 1.98437 0.0743673
\(713\) 41.0284 1.53653
\(714\) −11.3309 −0.424048
\(715\) 2.14740 0.0803080
\(716\) −11.9199 −0.445467
\(717\) 14.7051 0.549171
\(718\) 12.5651 0.468927
\(719\) 28.6745 1.06938 0.534690 0.845049i \(-0.320429\pi\)
0.534690 + 0.845049i \(0.320429\pi\)
\(720\) −0.0993682 −0.00370323
\(721\) 9.79019 0.364606
\(722\) 16.3889 0.609931
\(723\) 37.5119 1.39508
\(724\) 1.58015 0.0587258
\(725\) −32.7608 −1.21670
\(726\) −1.70683 −0.0633465
\(727\) −22.1091 −0.819980 −0.409990 0.912090i \(-0.634468\pi\)
−0.409990 + 0.912090i \(0.634468\pi\)
\(728\) 2.24842 0.0833321
\(729\) 27.7348 1.02721
\(730\) 7.80688 0.288945
\(731\) 23.0335 0.851923
\(732\) −13.8435 −0.511669
\(733\) −35.2251 −1.30107 −0.650535 0.759476i \(-0.725455\pi\)
−0.650535 + 0.759476i \(0.725455\pi\)
\(734\) 2.14332 0.0791113
\(735\) −10.8749 −0.401126
\(736\) 5.50920 0.203072
\(737\) 8.40718 0.309683
\(738\) −0.514399 −0.0189353
\(739\) 28.4707 1.04731 0.523655 0.851930i \(-0.324568\pi\)
0.523655 + 0.851930i \(0.324568\pi\)
\(740\) 3.46571 0.127402
\(741\) 5.16913 0.189893
\(742\) −8.05245 −0.295615
\(743\) −12.5455 −0.460249 −0.230125 0.973161i \(-0.573913\pi\)
−0.230125 + 0.973161i \(0.573913\pi\)
\(744\) 12.7112 0.466016
\(745\) −10.2230 −0.374542
\(746\) 2.85050 0.104364
\(747\) −1.51498 −0.0554303
\(748\) −5.53362 −0.202329
\(749\) −7.79633 −0.284872
\(750\) 16.9891 0.620353
\(751\) −8.96471 −0.327127 −0.163563 0.986533i \(-0.552299\pi\)
−0.163563 + 0.986533i \(0.552299\pi\)
\(752\) −6.81072 −0.248361
\(753\) 14.8138 0.539844
\(754\) −16.6522 −0.606436
\(755\) 1.67778 0.0610607
\(756\) 6.32052 0.229875
\(757\) 19.2191 0.698530 0.349265 0.937024i \(-0.386431\pi\)
0.349265 + 0.937024i \(0.386431\pi\)
\(758\) 29.9256 1.08694
\(759\) −9.40328 −0.341318
\(760\) −1.85145 −0.0671591
\(761\) −35.4948 −1.28669 −0.643343 0.765578i \(-0.722453\pi\)
−0.643343 + 0.765578i \(0.722453\pi\)
\(762\) −23.3952 −0.847519
\(763\) 12.8694 0.465903
\(764\) −18.8839 −0.683196
\(765\) 0.549865 0.0198804
\(766\) 13.3893 0.483775
\(767\) 15.0547 0.543594
\(768\) 1.70683 0.0615900
\(769\) −47.2942 −1.70547 −0.852736 0.522343i \(-0.825058\pi\)
−0.852736 + 0.522343i \(0.825058\pi\)
\(770\) 1.37456 0.0495356
\(771\) 26.4120 0.951204
\(772\) 10.5416 0.379400
\(773\) −0.0165424 −0.000594989 0 −0.000297495 1.00000i \(-0.500095\pi\)
−0.000297495 1.00000i \(0.500095\pi\)
\(774\) −0.360993 −0.0129756
\(775\) 27.4596 0.986376
\(776\) −16.3133 −0.585612
\(777\) −6.19368 −0.222197
\(778\) −14.9613 −0.536390
\(779\) −9.58439 −0.343397
\(780\) 3.66524 0.131237
\(781\) 1.08059 0.0386664
\(782\) −30.4858 −1.09017
\(783\) −46.8108 −1.67288
\(784\) −5.56078 −0.198599
\(785\) −15.8058 −0.564134
\(786\) 31.8856 1.13732
\(787\) −6.27120 −0.223544 −0.111772 0.993734i \(-0.535653\pi\)
−0.111772 + 0.993734i \(0.535653\pi\)
\(788\) 1.00000 0.0356235
\(789\) 51.5405 1.83489
\(790\) 6.22011 0.221302
\(791\) 16.6786 0.593024
\(792\) 0.0867259 0.00308167
\(793\) −15.2008 −0.539797
\(794\) 28.0241 0.994537
\(795\) −13.1266 −0.465553
\(796\) −6.29150 −0.222996
\(797\) 35.5928 1.26076 0.630381 0.776286i \(-0.282899\pi\)
0.630381 + 0.776286i \(0.282899\pi\)
\(798\) 3.30878 0.117130
\(799\) 37.6879 1.33330
\(800\) 3.68720 0.130362
\(801\) 0.172096 0.00608072
\(802\) 0.888495 0.0313738
\(803\) −6.81363 −0.240448
\(804\) 14.3496 0.506073
\(805\) 7.57272 0.266903
\(806\) 13.9576 0.491634
\(807\) −38.4720 −1.35428
\(808\) −10.2901 −0.362006
\(809\) 13.4244 0.471978 0.235989 0.971756i \(-0.424167\pi\)
0.235989 + 0.971756i \(0.424167\pi\)
\(810\) 10.0052 0.351548
\(811\) 22.7799 0.799911 0.399955 0.916535i \(-0.369026\pi\)
0.399955 + 0.916535i \(0.369026\pi\)
\(812\) −10.6591 −0.374062
\(813\) 39.3947 1.38163
\(814\) −3.02478 −0.106018
\(815\) 3.01504 0.105612
\(816\) −9.44495 −0.330639
\(817\) −6.72610 −0.235316
\(818\) 8.93554 0.312424
\(819\) 0.194996 0.00681372
\(820\) −6.79595 −0.237325
\(821\) 21.6675 0.756200 0.378100 0.925765i \(-0.376578\pi\)
0.378100 + 0.925765i \(0.376578\pi\)
\(822\) −13.6279 −0.475329
\(823\) −14.4015 −0.502003 −0.251002 0.967987i \(-0.580760\pi\)
−0.251002 + 0.967987i \(0.580760\pi\)
\(824\) 8.16069 0.284291
\(825\) −6.29344 −0.219109
\(826\) 9.63658 0.335299
\(827\) −36.1731 −1.25786 −0.628931 0.777461i \(-0.716507\pi\)
−0.628931 + 0.777461i \(0.716507\pi\)
\(828\) 0.477791 0.0166044
\(829\) 8.16328 0.283523 0.141761 0.989901i \(-0.454723\pi\)
0.141761 + 0.989901i \(0.454723\pi\)
\(830\) −20.0151 −0.694733
\(831\) 45.9843 1.59518
\(832\) 1.87419 0.0649758
\(833\) 30.7712 1.06616
\(834\) 31.3920 1.08702
\(835\) −2.04272 −0.0706911
\(836\) 1.61590 0.0558869
\(837\) 39.2360 1.35620
\(838\) −8.77232 −0.303035
\(839\) 20.6387 0.712527 0.356264 0.934386i \(-0.384051\pi\)
0.356264 + 0.934386i \(0.384051\pi\)
\(840\) 2.34614 0.0809494
\(841\) 49.9431 1.72218
\(842\) 34.4629 1.18767
\(843\) 8.86797 0.305429
\(844\) 1.80977 0.0622947
\(845\) −10.8704 −0.373954
\(846\) −0.590666 −0.0203075
\(847\) −1.19968 −0.0412214
\(848\) −6.71218 −0.230497
\(849\) −11.9849 −0.411320
\(850\) −20.4036 −0.699837
\(851\) −16.6641 −0.571239
\(852\) 1.84438 0.0631874
\(853\) 3.24788 0.111205 0.0556026 0.998453i \(-0.482292\pi\)
0.0556026 + 0.998453i \(0.482292\pi\)
\(854\) −9.73012 −0.332958
\(855\) −0.160569 −0.00549133
\(856\) −6.49869 −0.222121
\(857\) −3.52734 −0.120492 −0.0602458 0.998184i \(-0.519188\pi\)
−0.0602458 + 0.998184i \(0.519188\pi\)
\(858\) −3.19893 −0.109210
\(859\) 5.52256 0.188427 0.0942136 0.995552i \(-0.469966\pi\)
0.0942136 + 0.995552i \(0.469966\pi\)
\(860\) −4.76924 −0.162630
\(861\) 12.1452 0.413909
\(862\) 22.8149 0.777080
\(863\) 44.6795 1.52091 0.760454 0.649392i \(-0.224977\pi\)
0.760454 + 0.649392i \(0.224977\pi\)
\(864\) 5.26852 0.179239
\(865\) 13.8798 0.471927
\(866\) −32.0875 −1.09038
\(867\) 23.2486 0.789564
\(868\) 8.93430 0.303250
\(869\) −5.42875 −0.184158
\(870\) −17.3759 −0.589097
\(871\) 15.7567 0.533894
\(872\) 10.7274 0.363275
\(873\) −1.41478 −0.0478831
\(874\) 8.90230 0.301125
\(875\) 11.9411 0.403682
\(876\) −11.6297 −0.392932
\(877\) −11.8032 −0.398566 −0.199283 0.979942i \(-0.563861\pi\)
−0.199283 + 0.979942i \(0.563861\pi\)
\(878\) 32.1294 1.08431
\(879\) 35.4540 1.19584
\(880\) 1.14577 0.0386240
\(881\) −2.57876 −0.0868807 −0.0434403 0.999056i \(-0.513832\pi\)
−0.0434403 + 0.999056i \(0.513832\pi\)
\(882\) −0.482263 −0.0162386
\(883\) −19.5155 −0.656748 −0.328374 0.944548i \(-0.606501\pi\)
−0.328374 + 0.944548i \(0.606501\pi\)
\(884\) −10.3710 −0.348816
\(885\) 15.7090 0.528051
\(886\) −10.6860 −0.359002
\(887\) 54.2383 1.82114 0.910572 0.413350i \(-0.135642\pi\)
0.910572 + 0.413350i \(0.135642\pi\)
\(888\) −5.16279 −0.173252
\(889\) −16.4437 −0.551505
\(890\) 2.27364 0.0762124
\(891\) −8.73230 −0.292543
\(892\) −7.58628 −0.254007
\(893\) −11.0054 −0.368282
\(894\) 15.2290 0.509333
\(895\) −13.6575 −0.456519
\(896\) 1.19968 0.0400784
\(897\) −17.6235 −0.588433
\(898\) −18.0967 −0.603895
\(899\) −66.1688 −2.20685
\(900\) 0.319776 0.0106592
\(901\) 37.1426 1.23740
\(902\) 5.93132 0.197491
\(903\) 8.52325 0.283636
\(904\) 13.9026 0.462394
\(905\) 1.81049 0.0601829
\(906\) −2.49935 −0.0830355
\(907\) 11.8694 0.394118 0.197059 0.980392i \(-0.436861\pi\)
0.197059 + 0.980392i \(0.436861\pi\)
\(908\) 12.3775 0.410761
\(909\) −0.892422 −0.0295998
\(910\) 2.57618 0.0853996
\(911\) 43.6223 1.44527 0.722636 0.691229i \(-0.242930\pi\)
0.722636 + 0.691229i \(0.242930\pi\)
\(912\) 2.75806 0.0913285
\(913\) 17.4686 0.578127
\(914\) 10.3791 0.343309
\(915\) −15.8615 −0.524364
\(916\) −4.25397 −0.140555
\(917\) 22.4114 0.740089
\(918\) −29.1540 −0.962224
\(919\) −13.0653 −0.430985 −0.215492 0.976506i \(-0.569136\pi\)
−0.215492 + 0.976506i \(0.569136\pi\)
\(920\) 6.31230 0.208110
\(921\) −46.2009 −1.52237
\(922\) 13.4670 0.443513
\(923\) 2.02522 0.0666610
\(924\) −2.04765 −0.0673626
\(925\) −11.1530 −0.366708
\(926\) −24.1611 −0.793982
\(927\) 0.707743 0.0232453
\(928\) −8.88499 −0.291664
\(929\) 14.3779 0.471724 0.235862 0.971787i \(-0.424209\pi\)
0.235862 + 0.971787i \(0.424209\pi\)
\(930\) 14.5642 0.477578
\(931\) −8.98563 −0.294492
\(932\) −7.59700 −0.248848
\(933\) −16.8572 −0.551879
\(934\) 30.2544 0.989953
\(935\) −6.34027 −0.207349
\(936\) 0.162541 0.00531281
\(937\) −56.7189 −1.85292 −0.926462 0.376388i \(-0.877166\pi\)
−0.926462 + 0.376388i \(0.877166\pi\)
\(938\) 10.0859 0.329316
\(939\) 4.40720 0.143824
\(940\) −7.80354 −0.254523
\(941\) 36.4087 1.18689 0.593444 0.804875i \(-0.297768\pi\)
0.593444 + 0.804875i \(0.297768\pi\)
\(942\) 23.5456 0.767157
\(943\) 32.6769 1.06410
\(944\) 8.03264 0.261440
\(945\) 7.24188 0.235579
\(946\) 4.16246 0.135333
\(947\) −10.8371 −0.352159 −0.176079 0.984376i \(-0.556342\pi\)
−0.176079 + 0.984376i \(0.556342\pi\)
\(948\) −9.26595 −0.300944
\(949\) −12.7700 −0.414533
\(950\) 5.95814 0.193307
\(951\) −3.53430 −0.114608
\(952\) −6.63855 −0.215157
\(953\) −25.8897 −0.838651 −0.419325 0.907836i \(-0.637733\pi\)
−0.419325 + 0.907836i \(0.637733\pi\)
\(954\) −0.582120 −0.0188468
\(955\) −21.6367 −0.700146
\(956\) 8.61541 0.278642
\(957\) 15.1652 0.490221
\(958\) −15.8719 −0.512797
\(959\) −9.57863 −0.309310
\(960\) 1.95564 0.0631181
\(961\) 24.4616 0.789084
\(962\) −5.66901 −0.182776
\(963\) −0.563605 −0.0181619
\(964\) 21.9775 0.707847
\(965\) 12.0783 0.388813
\(966\) −11.2809 −0.362957
\(967\) −3.72928 −0.119926 −0.0599628 0.998201i \(-0.519098\pi\)
−0.0599628 + 0.998201i \(0.519098\pi\)
\(968\) −1.00000 −0.0321412
\(969\) −15.2620 −0.490288
\(970\) −18.6913 −0.600141
\(971\) 2.01530 0.0646740 0.0323370 0.999477i \(-0.489705\pi\)
0.0323370 + 0.999477i \(0.489705\pi\)
\(972\) 0.900997 0.0288995
\(973\) 22.0644 0.707352
\(974\) −5.36705 −0.171971
\(975\) −11.7951 −0.377745
\(976\) −8.11062 −0.259614
\(977\) −28.2612 −0.904156 −0.452078 0.891978i \(-0.649317\pi\)
−0.452078 + 0.891978i \(0.649317\pi\)
\(978\) −4.49144 −0.143620
\(979\) −1.98437 −0.0634207
\(980\) −6.37139 −0.203526
\(981\) 0.930341 0.0297035
\(982\) −38.8560 −1.23995
\(983\) 40.5240 1.29252 0.646258 0.763119i \(-0.276333\pi\)
0.646258 + 0.763119i \(0.276333\pi\)
\(984\) 10.1238 0.322734
\(985\) 1.14577 0.0365074
\(986\) 49.1661 1.56577
\(987\) 13.9459 0.443904
\(988\) 3.02849 0.0963492
\(989\) 22.9318 0.729190
\(990\) 0.0993682 0.00315813
\(991\) 30.4052 0.965852 0.482926 0.875661i \(-0.339574\pi\)
0.482926 + 0.875661i \(0.339574\pi\)
\(992\) 7.44726 0.236451
\(993\) 12.3112 0.390684
\(994\) 1.29635 0.0411179
\(995\) −7.20863 −0.228529
\(996\) 29.8160 0.944756
\(997\) −20.2150 −0.640216 −0.320108 0.947381i \(-0.603719\pi\)
−0.320108 + 0.947381i \(0.603719\pi\)
\(998\) −3.60886 −0.114236
\(999\) −15.9361 −0.504196
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.1.12 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4334.2.a.a.1.12 15 1.1 even 1 trivial