Properties

Label 4334.2.a.d.1.8
Level $4334$
Weight $2$
Character 4334.1
Self dual yes
Analytic conductor $34.607$
Analytic rank $1$
Dimension $17$
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: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{17} - 7 x^{16} - 7 x^{15} + 137 x^{14} - 98 x^{13} - 1048 x^{12} + 1313 x^{11} + 4085 x^{10} - 6021 x^{9} - 8879 x^{8} + 13530 x^{7} + 11150 x^{6} - 15676 x^{5} - 8037 x^{4} + \cdots - 400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 5 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(1.31815\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.31815 q^{3} +1.00000 q^{4} +1.46276 q^{5} -1.31815 q^{6} -4.53584 q^{7} +1.00000 q^{8} -1.26249 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.31815 q^{3} +1.00000 q^{4} +1.46276 q^{5} -1.31815 q^{6} -4.53584 q^{7} +1.00000 q^{8} -1.26249 q^{9} +1.46276 q^{10} -1.00000 q^{11} -1.31815 q^{12} +4.66998 q^{13} -4.53584 q^{14} -1.92814 q^{15} +1.00000 q^{16} +2.59877 q^{17} -1.26249 q^{18} -0.0248218 q^{19} +1.46276 q^{20} +5.97891 q^{21} -1.00000 q^{22} +3.39725 q^{23} -1.31815 q^{24} -2.86033 q^{25} +4.66998 q^{26} +5.61859 q^{27} -4.53584 q^{28} -3.84555 q^{29} -1.92814 q^{30} -3.79582 q^{31} +1.00000 q^{32} +1.31815 q^{33} +2.59877 q^{34} -6.63486 q^{35} -1.26249 q^{36} -2.61721 q^{37} -0.0248218 q^{38} -6.15572 q^{39} +1.46276 q^{40} +5.88709 q^{41} +5.97891 q^{42} +3.42657 q^{43} -1.00000 q^{44} -1.84672 q^{45} +3.39725 q^{46} -1.04331 q^{47} -1.31815 q^{48} +13.5739 q^{49} -2.86033 q^{50} -3.42557 q^{51} +4.66998 q^{52} -10.7672 q^{53} +5.61859 q^{54} -1.46276 q^{55} -4.53584 q^{56} +0.0327187 q^{57} -3.84555 q^{58} -7.61960 q^{59} -1.92814 q^{60} +4.98477 q^{61} -3.79582 q^{62} +5.72646 q^{63} +1.00000 q^{64} +6.83107 q^{65} +1.31815 q^{66} -13.9019 q^{67} +2.59877 q^{68} -4.47807 q^{69} -6.63486 q^{70} -8.51955 q^{71} -1.26249 q^{72} -6.22555 q^{73} -2.61721 q^{74} +3.77033 q^{75} -0.0248218 q^{76} +4.53584 q^{77} -6.15572 q^{78} -3.75592 q^{79} +1.46276 q^{80} -3.61865 q^{81} +5.88709 q^{82} -13.1972 q^{83} +5.97891 q^{84} +3.80139 q^{85} +3.42657 q^{86} +5.06900 q^{87} -1.00000 q^{88} +14.2555 q^{89} -1.84672 q^{90} -21.1823 q^{91} +3.39725 q^{92} +5.00345 q^{93} -1.04331 q^{94} -0.0363084 q^{95} -1.31815 q^{96} -17.1275 q^{97} +13.5739 q^{98} +1.26249 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17 q + 17 q^{2} - 7 q^{3} + 17 q^{4} - 4 q^{5} - 7 q^{6} - 5 q^{7} + 17 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 17 q + 17 q^{2} - 7 q^{3} + 17 q^{4} - 4 q^{5} - 7 q^{6} - 5 q^{7} + 17 q^{8} + 12 q^{9} - 4 q^{10} - 17 q^{11} - 7 q^{12} - 18 q^{13} - 5 q^{14} - 16 q^{15} + 17 q^{16} - 10 q^{17} + 12 q^{18} - 31 q^{19} - 4 q^{20} - 13 q^{21} - 17 q^{22} - 6 q^{23} - 7 q^{24} + 3 q^{25} - 18 q^{26} - 37 q^{27} - 5 q^{28} - 16 q^{29} - 16 q^{30} - 30 q^{31} + 17 q^{32} + 7 q^{33} - 10 q^{34} - 36 q^{35} + 12 q^{36} - 23 q^{37} - 31 q^{38} - 15 q^{39} - 4 q^{40} - 7 q^{41} - 13 q^{42} - 23 q^{43} - 17 q^{44} - 19 q^{45} - 6 q^{46} - 19 q^{47} - 7 q^{48} - 8 q^{49} + 3 q^{50} - 18 q^{51} - 18 q^{52} - 30 q^{53} - 37 q^{54} + 4 q^{55} - 5 q^{56} + 10 q^{57} - 16 q^{58} - 28 q^{59} - 16 q^{60} - 19 q^{61} - 30 q^{62} + 2 q^{63} + 17 q^{64} + 23 q^{65} + 7 q^{66} - 35 q^{67} - 10 q^{68} + q^{69} - 36 q^{70} + q^{71} + 12 q^{72} - 10 q^{73} - 23 q^{74} - 33 q^{75} - 31 q^{76} + 5 q^{77} - 15 q^{78} - 27 q^{79} - 4 q^{80} + 13 q^{81} - 7 q^{82} - 40 q^{83} - 13 q^{84} - 11 q^{85} - 23 q^{86} - 6 q^{87} - 17 q^{88} - 17 q^{89} - 19 q^{90} - 19 q^{91} - 6 q^{92} + 10 q^{93} - 19 q^{94} - 27 q^{95} - 7 q^{96} - 34 q^{97} - 8 q^{98} - 12 q^{99}+O(q^{100}) \) 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
\(3\) −1.31815 −0.761032 −0.380516 0.924774i \(-0.624254\pi\)
−0.380516 + 0.924774i \(0.624254\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.46276 0.654167 0.327084 0.944995i \(-0.393934\pi\)
0.327084 + 0.944995i \(0.393934\pi\)
\(6\) −1.31815 −0.538131
\(7\) −4.53584 −1.71439 −0.857194 0.514994i \(-0.827794\pi\)
−0.857194 + 0.514994i \(0.827794\pi\)
\(8\) 1.00000 0.353553
\(9\) −1.26249 −0.420830
\(10\) 1.46276 0.462566
\(11\) −1.00000 −0.301511
\(12\) −1.31815 −0.380516
\(13\) 4.66998 1.29522 0.647610 0.761972i \(-0.275769\pi\)
0.647610 + 0.761972i \(0.275769\pi\)
\(14\) −4.53584 −1.21226
\(15\) −1.92814 −0.497842
\(16\) 1.00000 0.250000
\(17\) 2.59877 0.630296 0.315148 0.949043i \(-0.397946\pi\)
0.315148 + 0.949043i \(0.397946\pi\)
\(18\) −1.26249 −0.297572
\(19\) −0.0248218 −0.00569451 −0.00284725 0.999996i \(-0.500906\pi\)
−0.00284725 + 0.999996i \(0.500906\pi\)
\(20\) 1.46276 0.327084
\(21\) 5.97891 1.30470
\(22\) −1.00000 −0.213201
\(23\) 3.39725 0.708375 0.354188 0.935174i \(-0.384757\pi\)
0.354188 + 0.935174i \(0.384757\pi\)
\(24\) −1.31815 −0.269066
\(25\) −2.86033 −0.572065
\(26\) 4.66998 0.915859
\(27\) 5.61859 1.08130
\(28\) −4.53584 −0.857194
\(29\) −3.84555 −0.714101 −0.357050 0.934085i \(-0.616218\pi\)
−0.357050 + 0.934085i \(0.616218\pi\)
\(30\) −1.92814 −0.352028
\(31\) −3.79582 −0.681750 −0.340875 0.940109i \(-0.610723\pi\)
−0.340875 + 0.940109i \(0.610723\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.31815 0.229460
\(34\) 2.59877 0.445686
\(35\) −6.63486 −1.12150
\(36\) −1.26249 −0.210415
\(37\) −2.61721 −0.430267 −0.215133 0.976585i \(-0.569019\pi\)
−0.215133 + 0.976585i \(0.569019\pi\)
\(38\) −0.0248218 −0.00402662
\(39\) −6.15572 −0.985704
\(40\) 1.46276 0.231283
\(41\) 5.88709 0.919409 0.459704 0.888072i \(-0.347955\pi\)
0.459704 + 0.888072i \(0.347955\pi\)
\(42\) 5.97891 0.922566
\(43\) 3.42657 0.522548 0.261274 0.965265i \(-0.415857\pi\)
0.261274 + 0.965265i \(0.415857\pi\)
\(44\) −1.00000 −0.150756
\(45\) −1.84672 −0.275293
\(46\) 3.39725 0.500897
\(47\) −1.04331 −0.152183 −0.0760913 0.997101i \(-0.524244\pi\)
−0.0760913 + 0.997101i \(0.524244\pi\)
\(48\) −1.31815 −0.190258
\(49\) 13.5739 1.93913
\(50\) −2.86033 −0.404511
\(51\) −3.42557 −0.479675
\(52\) 4.66998 0.647610
\(53\) −10.7672 −1.47899 −0.739494 0.673163i \(-0.764935\pi\)
−0.739494 + 0.673163i \(0.764935\pi\)
\(54\) 5.61859 0.764593
\(55\) −1.46276 −0.197239
\(56\) −4.53584 −0.606128
\(57\) 0.0327187 0.00433370
\(58\) −3.84555 −0.504946
\(59\) −7.61960 −0.991987 −0.495994 0.868326i \(-0.665196\pi\)
−0.495994 + 0.868326i \(0.665196\pi\)
\(60\) −1.92814 −0.248921
\(61\) 4.98477 0.638235 0.319117 0.947715i \(-0.396614\pi\)
0.319117 + 0.947715i \(0.396614\pi\)
\(62\) −3.79582 −0.482070
\(63\) 5.72646 0.721466
\(64\) 1.00000 0.125000
\(65\) 6.83107 0.847290
\(66\) 1.31815 0.162253
\(67\) −13.9019 −1.69839 −0.849196 0.528077i \(-0.822913\pi\)
−0.849196 + 0.528077i \(0.822913\pi\)
\(68\) 2.59877 0.315148
\(69\) −4.47807 −0.539096
\(70\) −6.63486 −0.793018
\(71\) −8.51955 −1.01109 −0.505543 0.862802i \(-0.668708\pi\)
−0.505543 + 0.862802i \(0.668708\pi\)
\(72\) −1.26249 −0.148786
\(73\) −6.22555 −0.728646 −0.364323 0.931273i \(-0.618700\pi\)
−0.364323 + 0.931273i \(0.618700\pi\)
\(74\) −2.61721 −0.304245
\(75\) 3.77033 0.435360
\(76\) −0.0248218 −0.00284725
\(77\) 4.53584 0.516907
\(78\) −6.15572 −0.696998
\(79\) −3.75592 −0.422574 −0.211287 0.977424i \(-0.567766\pi\)
−0.211287 + 0.977424i \(0.567766\pi\)
\(80\) 1.46276 0.163542
\(81\) −3.61865 −0.402072
\(82\) 5.88709 0.650120
\(83\) −13.1972 −1.44858 −0.724289 0.689497i \(-0.757832\pi\)
−0.724289 + 0.689497i \(0.757832\pi\)
\(84\) 5.97891 0.652352
\(85\) 3.80139 0.412319
\(86\) 3.42657 0.369497
\(87\) 5.06900 0.543454
\(88\) −1.00000 −0.106600
\(89\) 14.2555 1.51108 0.755542 0.655100i \(-0.227374\pi\)
0.755542 + 0.655100i \(0.227374\pi\)
\(90\) −1.84672 −0.194662
\(91\) −21.1823 −2.22051
\(92\) 3.39725 0.354188
\(93\) 5.00345 0.518834
\(94\) −1.04331 −0.107609
\(95\) −0.0363084 −0.00372516
\(96\) −1.31815 −0.134533
\(97\) −17.1275 −1.73904 −0.869518 0.493902i \(-0.835570\pi\)
−0.869518 + 0.493902i \(0.835570\pi\)
\(98\) 13.5739 1.37117
\(99\) 1.26249 0.126885
\(100\) −2.86033 −0.286033
\(101\) −3.10944 −0.309400 −0.154700 0.987961i \(-0.549441\pi\)
−0.154700 + 0.987961i \(0.549441\pi\)
\(102\) −3.42557 −0.339182
\(103\) 4.19477 0.413323 0.206661 0.978413i \(-0.433740\pi\)
0.206661 + 0.978413i \(0.433740\pi\)
\(104\) 4.66998 0.457929
\(105\) 8.74572 0.853495
\(106\) −10.7672 −1.04580
\(107\) −0.684451 −0.0661683 −0.0330842 0.999453i \(-0.510533\pi\)
−0.0330842 + 0.999453i \(0.510533\pi\)
\(108\) 5.61859 0.540649
\(109\) −18.5080 −1.77275 −0.886373 0.462971i \(-0.846783\pi\)
−0.886373 + 0.462971i \(0.846783\pi\)
\(110\) −1.46276 −0.139469
\(111\) 3.44987 0.327447
\(112\) −4.53584 −0.428597
\(113\) −3.74038 −0.351865 −0.175933 0.984402i \(-0.556294\pi\)
−0.175933 + 0.984402i \(0.556294\pi\)
\(114\) 0.0327187 0.00306439
\(115\) 4.96937 0.463396
\(116\) −3.84555 −0.357050
\(117\) −5.89580 −0.545067
\(118\) −7.61960 −0.701441
\(119\) −11.7876 −1.08057
\(120\) −1.92814 −0.176014
\(121\) 1.00000 0.0909091
\(122\) 4.98477 0.451300
\(123\) −7.76005 −0.699700
\(124\) −3.79582 −0.340875
\(125\) −11.4978 −1.02839
\(126\) 5.72646 0.510153
\(127\) −14.0597 −1.24759 −0.623796 0.781587i \(-0.714410\pi\)
−0.623796 + 0.781587i \(0.714410\pi\)
\(128\) 1.00000 0.0883883
\(129\) −4.51673 −0.397676
\(130\) 6.83107 0.599125
\(131\) −5.96454 −0.521124 −0.260562 0.965457i \(-0.583908\pi\)
−0.260562 + 0.965457i \(0.583908\pi\)
\(132\) 1.31815 0.114730
\(133\) 0.112588 0.00976259
\(134\) −13.9019 −1.20095
\(135\) 8.21866 0.707349
\(136\) 2.59877 0.222843
\(137\) −13.2498 −1.13201 −0.566003 0.824403i \(-0.691511\pi\)
−0.566003 + 0.824403i \(0.691511\pi\)
\(138\) −4.47807 −0.381199
\(139\) 9.15392 0.776425 0.388213 0.921570i \(-0.373093\pi\)
0.388213 + 0.921570i \(0.373093\pi\)
\(140\) −6.63486 −0.560748
\(141\) 1.37524 0.115816
\(142\) −8.51955 −0.714945
\(143\) −4.66998 −0.390523
\(144\) −1.26249 −0.105207
\(145\) −5.62513 −0.467141
\(146\) −6.22555 −0.515230
\(147\) −17.8924 −1.47574
\(148\) −2.61721 −0.215133
\(149\) 12.5013 1.02415 0.512073 0.858942i \(-0.328878\pi\)
0.512073 + 0.858942i \(0.328878\pi\)
\(150\) 3.77033 0.307846
\(151\) −11.5610 −0.940822 −0.470411 0.882447i \(-0.655894\pi\)
−0.470411 + 0.882447i \(0.655894\pi\)
\(152\) −0.0248218 −0.00201331
\(153\) −3.28093 −0.265247
\(154\) 4.53584 0.365509
\(155\) −5.55239 −0.445979
\(156\) −6.15572 −0.492852
\(157\) 10.4405 0.833242 0.416621 0.909080i \(-0.363214\pi\)
0.416621 + 0.909080i \(0.363214\pi\)
\(158\) −3.75592 −0.298805
\(159\) 14.1927 1.12556
\(160\) 1.46276 0.115642
\(161\) −15.4094 −1.21443
\(162\) −3.61865 −0.284308
\(163\) 18.9201 1.48194 0.740969 0.671540i \(-0.234367\pi\)
0.740969 + 0.671540i \(0.234367\pi\)
\(164\) 5.88709 0.459704
\(165\) 1.92814 0.150105
\(166\) −13.1972 −1.02430
\(167\) 1.16959 0.0905053 0.0452526 0.998976i \(-0.485591\pi\)
0.0452526 + 0.998976i \(0.485591\pi\)
\(168\) 5.97891 0.461283
\(169\) 8.80872 0.677594
\(170\) 3.80139 0.291553
\(171\) 0.0313372 0.00239642
\(172\) 3.42657 0.261274
\(173\) 1.58761 0.120703 0.0603517 0.998177i \(-0.480778\pi\)
0.0603517 + 0.998177i \(0.480778\pi\)
\(174\) 5.06900 0.384280
\(175\) 12.9740 0.980742
\(176\) −1.00000 −0.0753778
\(177\) 10.0437 0.754934
\(178\) 14.2555 1.06850
\(179\) 2.59384 0.193873 0.0969363 0.995291i \(-0.469096\pi\)
0.0969363 + 0.995291i \(0.469096\pi\)
\(180\) −1.84672 −0.137647
\(181\) 10.5987 0.787798 0.393899 0.919154i \(-0.371126\pi\)
0.393899 + 0.919154i \(0.371126\pi\)
\(182\) −21.1823 −1.57014
\(183\) −6.57066 −0.485717
\(184\) 3.39725 0.250448
\(185\) −3.82836 −0.281467
\(186\) 5.00345 0.366871
\(187\) −2.59877 −0.190041
\(188\) −1.04331 −0.0760913
\(189\) −25.4850 −1.85376
\(190\) −0.0363084 −0.00263409
\(191\) −15.8379 −1.14599 −0.572997 0.819558i \(-0.694219\pi\)
−0.572997 + 0.819558i \(0.694219\pi\)
\(192\) −1.31815 −0.0951290
\(193\) 27.5855 1.98565 0.992826 0.119572i \(-0.0381523\pi\)
0.992826 + 0.119572i \(0.0381523\pi\)
\(194\) −17.1275 −1.22968
\(195\) −9.00435 −0.644815
\(196\) 13.5739 0.969563
\(197\) 1.00000 0.0712470
\(198\) 1.26249 0.0897212
\(199\) −9.49262 −0.672914 −0.336457 0.941699i \(-0.609229\pi\)
−0.336457 + 0.941699i \(0.609229\pi\)
\(200\) −2.86033 −0.202256
\(201\) 18.3248 1.29253
\(202\) −3.10944 −0.218779
\(203\) 17.4428 1.22425
\(204\) −3.42557 −0.239838
\(205\) 8.61141 0.601447
\(206\) 4.19477 0.292263
\(207\) −4.28899 −0.298105
\(208\) 4.66998 0.323805
\(209\) 0.0248218 0.00171696
\(210\) 8.74572 0.603512
\(211\) −4.54282 −0.312741 −0.156370 0.987698i \(-0.549979\pi\)
−0.156370 + 0.987698i \(0.549979\pi\)
\(212\) −10.7672 −0.739494
\(213\) 11.2300 0.769468
\(214\) −0.684451 −0.0467881
\(215\) 5.01226 0.341834
\(216\) 5.61859 0.382296
\(217\) 17.2173 1.16878
\(218\) −18.5080 −1.25352
\(219\) 8.20619 0.554523
\(220\) −1.46276 −0.0986194
\(221\) 12.1362 0.816371
\(222\) 3.44987 0.231540
\(223\) −21.7748 −1.45815 −0.729074 0.684435i \(-0.760049\pi\)
−0.729074 + 0.684435i \(0.760049\pi\)
\(224\) −4.53584 −0.303064
\(225\) 3.61113 0.240742
\(226\) −3.74038 −0.248806
\(227\) −4.31094 −0.286127 −0.143064 0.989714i \(-0.545695\pi\)
−0.143064 + 0.989714i \(0.545695\pi\)
\(228\) 0.0327187 0.00216685
\(229\) 19.2818 1.27418 0.637088 0.770791i \(-0.280139\pi\)
0.637088 + 0.770791i \(0.280139\pi\)
\(230\) 4.96937 0.327670
\(231\) −5.97891 −0.393383
\(232\) −3.84555 −0.252473
\(233\) −11.2994 −0.740251 −0.370126 0.928982i \(-0.620685\pi\)
−0.370126 + 0.928982i \(0.620685\pi\)
\(234\) −5.89580 −0.385421
\(235\) −1.52612 −0.0995528
\(236\) −7.61960 −0.495994
\(237\) 4.95086 0.321593
\(238\) −11.7876 −0.764079
\(239\) −3.22286 −0.208470 −0.104235 0.994553i \(-0.533239\pi\)
−0.104235 + 0.994553i \(0.533239\pi\)
\(240\) −1.92814 −0.124461
\(241\) 7.88943 0.508203 0.254101 0.967178i \(-0.418220\pi\)
0.254101 + 0.967178i \(0.418220\pi\)
\(242\) 1.00000 0.0642824
\(243\) −12.0858 −0.775307
\(244\) 4.98477 0.319117
\(245\) 19.8554 1.26851
\(246\) −7.76005 −0.494763
\(247\) −0.115917 −0.00737564
\(248\) −3.79582 −0.241035
\(249\) 17.3958 1.10241
\(250\) −11.4978 −0.727184
\(251\) −8.33299 −0.525974 −0.262987 0.964799i \(-0.584708\pi\)
−0.262987 + 0.964799i \(0.584708\pi\)
\(252\) 5.72646 0.360733
\(253\) −3.39725 −0.213583
\(254\) −14.0597 −0.882181
\(255\) −5.01079 −0.313788
\(256\) 1.00000 0.0625000
\(257\) 7.27548 0.453832 0.226916 0.973914i \(-0.427136\pi\)
0.226916 + 0.973914i \(0.427136\pi\)
\(258\) −4.51673 −0.281199
\(259\) 11.8713 0.737644
\(260\) 6.83107 0.423645
\(261\) 4.85497 0.300515
\(262\) −5.96454 −0.368490
\(263\) −20.7703 −1.28075 −0.640377 0.768061i \(-0.721222\pi\)
−0.640377 + 0.768061i \(0.721222\pi\)
\(264\) 1.31815 0.0811263
\(265\) −15.7499 −0.967506
\(266\) 0.112588 0.00690320
\(267\) −18.7909 −1.14998
\(268\) −13.9019 −0.849196
\(269\) 21.8670 1.33326 0.666628 0.745390i \(-0.267737\pi\)
0.666628 + 0.745390i \(0.267737\pi\)
\(270\) 8.21866 0.500172
\(271\) −12.8608 −0.781239 −0.390620 0.920552i \(-0.627739\pi\)
−0.390620 + 0.920552i \(0.627739\pi\)
\(272\) 2.59877 0.157574
\(273\) 27.9214 1.68988
\(274\) −13.2498 −0.800450
\(275\) 2.86033 0.172484
\(276\) −4.47807 −0.269548
\(277\) −7.99167 −0.480173 −0.240087 0.970751i \(-0.577176\pi\)
−0.240087 + 0.970751i \(0.577176\pi\)
\(278\) 9.15392 0.549016
\(279\) 4.79219 0.286901
\(280\) −6.63486 −0.396509
\(281\) −16.3440 −0.975000 −0.487500 0.873123i \(-0.662091\pi\)
−0.487500 + 0.873123i \(0.662091\pi\)
\(282\) 1.37524 0.0818941
\(283\) 6.23927 0.370886 0.185443 0.982655i \(-0.440628\pi\)
0.185443 + 0.982655i \(0.440628\pi\)
\(284\) −8.51955 −0.505543
\(285\) 0.0478597 0.00283497
\(286\) −4.66998 −0.276142
\(287\) −26.7029 −1.57622
\(288\) −1.26249 −0.0743929
\(289\) −10.2464 −0.602728
\(290\) −5.62513 −0.330319
\(291\) 22.5766 1.32346
\(292\) −6.22555 −0.364323
\(293\) 19.8379 1.15894 0.579470 0.814993i \(-0.303259\pi\)
0.579470 + 0.814993i \(0.303259\pi\)
\(294\) −17.8924 −1.04350
\(295\) −11.1457 −0.648926
\(296\) −2.61721 −0.152122
\(297\) −5.61859 −0.326023
\(298\) 12.5013 0.724180
\(299\) 15.8651 0.917501
\(300\) 3.77033 0.217680
\(301\) −15.5424 −0.895849
\(302\) −11.5610 −0.665262
\(303\) 4.09869 0.235464
\(304\) −0.0248218 −0.00142363
\(305\) 7.29154 0.417512
\(306\) −3.28093 −0.187558
\(307\) 30.5600 1.74415 0.872076 0.489371i \(-0.162774\pi\)
0.872076 + 0.489371i \(0.162774\pi\)
\(308\) 4.53584 0.258454
\(309\) −5.52932 −0.314552
\(310\) −5.55239 −0.315355
\(311\) −15.7508 −0.893148 −0.446574 0.894747i \(-0.647356\pi\)
−0.446574 + 0.894747i \(0.647356\pi\)
\(312\) −6.15572 −0.348499
\(313\) 27.8952 1.57673 0.788364 0.615209i \(-0.210928\pi\)
0.788364 + 0.615209i \(0.210928\pi\)
\(314\) 10.4405 0.589191
\(315\) 8.37645 0.471959
\(316\) −3.75592 −0.211287
\(317\) −16.8081 −0.944040 −0.472020 0.881588i \(-0.656475\pi\)
−0.472020 + 0.881588i \(0.656475\pi\)
\(318\) 14.1927 0.795890
\(319\) 3.84555 0.215310
\(320\) 1.46276 0.0817709
\(321\) 0.902206 0.0503562
\(322\) −15.4094 −0.858732
\(323\) −0.0645062 −0.00358922
\(324\) −3.61865 −0.201036
\(325\) −13.3577 −0.740950
\(326\) 18.9201 1.04789
\(327\) 24.3963 1.34912
\(328\) 5.88709 0.325060
\(329\) 4.73230 0.260900
\(330\) 1.92814 0.106140
\(331\) −5.46500 −0.300384 −0.150192 0.988657i \(-0.547989\pi\)
−0.150192 + 0.988657i \(0.547989\pi\)
\(332\) −13.1972 −0.724289
\(333\) 3.30420 0.181069
\(334\) 1.16959 0.0639969
\(335\) −20.3352 −1.11103
\(336\) 5.97891 0.326176
\(337\) −9.06460 −0.493780 −0.246890 0.969044i \(-0.579409\pi\)
−0.246890 + 0.969044i \(0.579409\pi\)
\(338\) 8.80872 0.479131
\(339\) 4.93037 0.267781
\(340\) 3.80139 0.206159
\(341\) 3.79582 0.205555
\(342\) 0.0313372 0.00169452
\(343\) −29.8181 −1.61003
\(344\) 3.42657 0.184748
\(345\) −6.55035 −0.352659
\(346\) 1.58761 0.0853502
\(347\) −20.4406 −1.09731 −0.548655 0.836049i \(-0.684860\pi\)
−0.548655 + 0.836049i \(0.684860\pi\)
\(348\) 5.06900 0.271727
\(349\) −21.3694 −1.14388 −0.571939 0.820296i \(-0.693809\pi\)
−0.571939 + 0.820296i \(0.693809\pi\)
\(350\) 12.9740 0.693489
\(351\) 26.2387 1.40052
\(352\) −1.00000 −0.0533002
\(353\) 14.5443 0.774117 0.387058 0.922055i \(-0.373491\pi\)
0.387058 + 0.922055i \(0.373491\pi\)
\(354\) 10.0437 0.533819
\(355\) −12.4621 −0.661419
\(356\) 14.2555 0.755542
\(357\) 15.5378 0.822350
\(358\) 2.59384 0.137089
\(359\) −5.04096 −0.266052 −0.133026 0.991113i \(-0.542469\pi\)
−0.133026 + 0.991113i \(0.542469\pi\)
\(360\) −1.84672 −0.0973308
\(361\) −18.9994 −0.999968
\(362\) 10.5987 0.557057
\(363\) −1.31815 −0.0691848
\(364\) −21.1823 −1.11025
\(365\) −9.10651 −0.476656
\(366\) −6.57066 −0.343454
\(367\) 0.931053 0.0486006 0.0243003 0.999705i \(-0.492264\pi\)
0.0243003 + 0.999705i \(0.492264\pi\)
\(368\) 3.39725 0.177094
\(369\) −7.43239 −0.386915
\(370\) −3.82836 −0.199027
\(371\) 48.8383 2.53556
\(372\) 5.00345 0.259417
\(373\) 9.53258 0.493578 0.246789 0.969069i \(-0.420624\pi\)
0.246789 + 0.969069i \(0.420624\pi\)
\(374\) −2.59877 −0.134379
\(375\) 15.1558 0.782641
\(376\) −1.04331 −0.0538046
\(377\) −17.9586 −0.924917
\(378\) −25.4850 −1.31081
\(379\) −25.5474 −1.31228 −0.656141 0.754638i \(-0.727812\pi\)
−0.656141 + 0.754638i \(0.727812\pi\)
\(380\) −0.0363084 −0.00186258
\(381\) 18.5327 0.949458
\(382\) −15.8379 −0.810340
\(383\) 4.42112 0.225909 0.112954 0.993600i \(-0.463969\pi\)
0.112954 + 0.993600i \(0.463969\pi\)
\(384\) −1.31815 −0.0672664
\(385\) 6.63486 0.338144
\(386\) 27.5855 1.40407
\(387\) −4.32601 −0.219904
\(388\) −17.1275 −0.869518
\(389\) 26.6493 1.35117 0.675586 0.737281i \(-0.263891\pi\)
0.675586 + 0.737281i \(0.263891\pi\)
\(390\) −9.00435 −0.455953
\(391\) 8.82868 0.446486
\(392\) 13.5739 0.685585
\(393\) 7.86213 0.396592
\(394\) 1.00000 0.0503793
\(395\) −5.49402 −0.276434
\(396\) 1.26249 0.0634425
\(397\) 33.9654 1.70468 0.852338 0.522992i \(-0.175184\pi\)
0.852338 + 0.522992i \(0.175184\pi\)
\(398\) −9.49262 −0.475822
\(399\) −0.148407 −0.00742965
\(400\) −2.86033 −0.143016
\(401\) 19.9873 0.998119 0.499059 0.866568i \(-0.333679\pi\)
0.499059 + 0.866568i \(0.333679\pi\)
\(402\) 18.3248 0.913958
\(403\) −17.7264 −0.883016
\(404\) −3.10944 −0.154700
\(405\) −5.29323 −0.263023
\(406\) 17.4428 0.865673
\(407\) 2.61721 0.129730
\(408\) −3.42557 −0.169591
\(409\) −13.2776 −0.656533 −0.328266 0.944585i \(-0.606464\pi\)
−0.328266 + 0.944585i \(0.606464\pi\)
\(410\) 8.61141 0.425287
\(411\) 17.4652 0.861494
\(412\) 4.19477 0.206661
\(413\) 34.5613 1.70065
\(414\) −4.28899 −0.210792
\(415\) −19.3043 −0.947612
\(416\) 4.66998 0.228965
\(417\) −12.0662 −0.590885
\(418\) 0.0248218 0.00121407
\(419\) 12.0765 0.589977 0.294989 0.955501i \(-0.404684\pi\)
0.294989 + 0.955501i \(0.404684\pi\)
\(420\) 8.74572 0.426748
\(421\) 5.00005 0.243687 0.121844 0.992549i \(-0.461119\pi\)
0.121844 + 0.992549i \(0.461119\pi\)
\(422\) −4.54282 −0.221141
\(423\) 1.31717 0.0640429
\(424\) −10.7672 −0.522901
\(425\) −7.43334 −0.360570
\(426\) 11.2300 0.544096
\(427\) −22.6102 −1.09418
\(428\) −0.684451 −0.0330842
\(429\) 6.15572 0.297201
\(430\) 5.01226 0.241713
\(431\) 21.5480 1.03793 0.518964 0.854796i \(-0.326318\pi\)
0.518964 + 0.854796i \(0.326318\pi\)
\(432\) 5.61859 0.270324
\(433\) 28.9405 1.39079 0.695397 0.718626i \(-0.255229\pi\)
0.695397 + 0.718626i \(0.255229\pi\)
\(434\) 17.2173 0.826455
\(435\) 7.41474 0.355510
\(436\) −18.5080 −0.886373
\(437\) −0.0843257 −0.00403385
\(438\) 8.20619 0.392107
\(439\) 5.31799 0.253814 0.126907 0.991915i \(-0.459495\pi\)
0.126907 + 0.991915i \(0.459495\pi\)
\(440\) −1.46276 −0.0697345
\(441\) −17.1369 −0.816042
\(442\) 12.1362 0.577262
\(443\) −4.59279 −0.218210 −0.109105 0.994030i \(-0.534798\pi\)
−0.109105 + 0.994030i \(0.534798\pi\)
\(444\) 3.44987 0.163723
\(445\) 20.8525 0.988502
\(446\) −21.7748 −1.03107
\(447\) −16.4785 −0.779408
\(448\) −4.53584 −0.214299
\(449\) −31.4568 −1.48454 −0.742269 0.670102i \(-0.766250\pi\)
−0.742269 + 0.670102i \(0.766250\pi\)
\(450\) 3.61113 0.170230
\(451\) −5.88709 −0.277212
\(452\) −3.74038 −0.175933
\(453\) 15.2391 0.715996
\(454\) −4.31094 −0.202322
\(455\) −30.9847 −1.45258
\(456\) 0.0327187 0.00153220
\(457\) −42.5459 −1.99021 −0.995106 0.0988167i \(-0.968494\pi\)
−0.995106 + 0.0988167i \(0.968494\pi\)
\(458\) 19.2818 0.900979
\(459\) 14.6014 0.681537
\(460\) 4.96937 0.231698
\(461\) −33.8876 −1.57830 −0.789152 0.614198i \(-0.789480\pi\)
−0.789152 + 0.614198i \(0.789480\pi\)
\(462\) −5.97891 −0.278164
\(463\) −17.2341 −0.800936 −0.400468 0.916311i \(-0.631152\pi\)
−0.400468 + 0.916311i \(0.631152\pi\)
\(464\) −3.84555 −0.178525
\(465\) 7.31886 0.339404
\(466\) −11.2994 −0.523437
\(467\) −26.4443 −1.22370 −0.611849 0.790975i \(-0.709574\pi\)
−0.611849 + 0.790975i \(0.709574\pi\)
\(468\) −5.89580 −0.272534
\(469\) 63.0571 2.91170
\(470\) −1.52612 −0.0703945
\(471\) −13.7621 −0.634124
\(472\) −7.61960 −0.350720
\(473\) −3.42657 −0.157554
\(474\) 4.95086 0.227400
\(475\) 0.0709984 0.00325763
\(476\) −11.7876 −0.540286
\(477\) 13.5935 0.622402
\(478\) −3.22286 −0.147410
\(479\) −18.2359 −0.833220 −0.416610 0.909085i \(-0.636782\pi\)
−0.416610 + 0.909085i \(0.636782\pi\)
\(480\) −1.92814 −0.0880069
\(481\) −12.2223 −0.557290
\(482\) 7.88943 0.359354
\(483\) 20.3118 0.924220
\(484\) 1.00000 0.0454545
\(485\) −25.0535 −1.13762
\(486\) −12.0858 −0.548225
\(487\) 39.2794 1.77992 0.889960 0.456038i \(-0.150732\pi\)
0.889960 + 0.456038i \(0.150732\pi\)
\(488\) 4.98477 0.225650
\(489\) −24.9395 −1.12780
\(490\) 19.8554 0.896974
\(491\) 28.5260 1.28736 0.643680 0.765295i \(-0.277407\pi\)
0.643680 + 0.765295i \(0.277407\pi\)
\(492\) −7.76005 −0.349850
\(493\) −9.99372 −0.450095
\(494\) −0.115917 −0.00521536
\(495\) 1.84672 0.0830040
\(496\) −3.79582 −0.170438
\(497\) 38.6434 1.73339
\(498\) 17.3958 0.779524
\(499\) 33.4415 1.49705 0.748525 0.663107i \(-0.230763\pi\)
0.748525 + 0.663107i \(0.230763\pi\)
\(500\) −11.4978 −0.514197
\(501\) −1.54169 −0.0688774
\(502\) −8.33299 −0.371920
\(503\) −34.0731 −1.51924 −0.759621 0.650366i \(-0.774616\pi\)
−0.759621 + 0.650366i \(0.774616\pi\)
\(504\) 5.72646 0.255077
\(505\) −4.54837 −0.202400
\(506\) −3.39725 −0.151026
\(507\) −11.6112 −0.515671
\(508\) −14.0597 −0.623796
\(509\) −33.7263 −1.49489 −0.747446 0.664322i \(-0.768720\pi\)
−0.747446 + 0.664322i \(0.768720\pi\)
\(510\) −5.01079 −0.221882
\(511\) 28.2381 1.24918
\(512\) 1.00000 0.0441942
\(513\) −0.139463 −0.00615745
\(514\) 7.27548 0.320908
\(515\) 6.13595 0.270382
\(516\) −4.51673 −0.198838
\(517\) 1.04331 0.0458848
\(518\) 11.8713 0.521593
\(519\) −2.09270 −0.0918592
\(520\) 6.83107 0.299562
\(521\) 19.2060 0.841430 0.420715 0.907193i \(-0.361779\pi\)
0.420715 + 0.907193i \(0.361779\pi\)
\(522\) 4.85497 0.212496
\(523\) 8.98858 0.393043 0.196522 0.980499i \(-0.437035\pi\)
0.196522 + 0.980499i \(0.437035\pi\)
\(524\) −5.96454 −0.260562
\(525\) −17.1016 −0.746376
\(526\) −20.7703 −0.905629
\(527\) −9.86449 −0.429704
\(528\) 1.31815 0.0573650
\(529\) −11.4587 −0.498205
\(530\) −15.7499 −0.684130
\(531\) 9.61966 0.417458
\(532\) 0.112588 0.00488130
\(533\) 27.4926 1.19084
\(534\) −18.7909 −0.813161
\(535\) −1.00119 −0.0432852
\(536\) −13.9019 −0.600473
\(537\) −3.41906 −0.147543
\(538\) 21.8670 0.942755
\(539\) −13.5739 −0.584669
\(540\) 8.21866 0.353675
\(541\) −24.9312 −1.07188 −0.535938 0.844257i \(-0.680042\pi\)
−0.535938 + 0.844257i \(0.680042\pi\)
\(542\) −12.8608 −0.552420
\(543\) −13.9707 −0.599540
\(544\) 2.59877 0.111422
\(545\) −27.0728 −1.15967
\(546\) 27.9214 1.19492
\(547\) 4.51435 0.193020 0.0965098 0.995332i \(-0.469232\pi\)
0.0965098 + 0.995332i \(0.469232\pi\)
\(548\) −13.2498 −0.566003
\(549\) −6.29323 −0.268588
\(550\) 2.86033 0.121965
\(551\) 0.0954534 0.00406645
\(552\) −4.47807 −0.190599
\(553\) 17.0363 0.724457
\(554\) −7.99167 −0.339534
\(555\) 5.04634 0.214205
\(556\) 9.15392 0.388213
\(557\) 0.450981 0.0191087 0.00955433 0.999954i \(-0.496959\pi\)
0.00955433 + 0.999954i \(0.496959\pi\)
\(558\) 4.79219 0.202870
\(559\) 16.0020 0.676814
\(560\) −6.63486 −0.280374
\(561\) 3.42557 0.144628
\(562\) −16.3440 −0.689429
\(563\) −12.6290 −0.532247 −0.266124 0.963939i \(-0.585743\pi\)
−0.266124 + 0.963939i \(0.585743\pi\)
\(564\) 1.37524 0.0579079
\(565\) −5.47128 −0.230179
\(566\) 6.23927 0.262256
\(567\) 16.4136 0.689308
\(568\) −8.51955 −0.357473
\(569\) −10.4797 −0.439331 −0.219665 0.975575i \(-0.570497\pi\)
−0.219665 + 0.975575i \(0.570497\pi\)
\(570\) 0.0478597 0.00200462
\(571\) −12.0789 −0.505487 −0.252744 0.967533i \(-0.581333\pi\)
−0.252744 + 0.967533i \(0.581333\pi\)
\(572\) −4.66998 −0.195262
\(573\) 20.8767 0.872138
\(574\) −26.7029 −1.11456
\(575\) −9.71724 −0.405237
\(576\) −1.26249 −0.0526037
\(577\) −33.6728 −1.40182 −0.700908 0.713251i \(-0.747222\pi\)
−0.700908 + 0.713251i \(0.747222\pi\)
\(578\) −10.2464 −0.426193
\(579\) −36.3618 −1.51114
\(580\) −5.62513 −0.233571
\(581\) 59.8603 2.48342
\(582\) 22.5766 0.935829
\(583\) 10.7672 0.445932
\(584\) −6.22555 −0.257615
\(585\) −8.62416 −0.356565
\(586\) 19.8379 0.819495
\(587\) 22.6738 0.935848 0.467924 0.883769i \(-0.345002\pi\)
0.467924 + 0.883769i \(0.345002\pi\)
\(588\) −17.8924 −0.737869
\(589\) 0.0942191 0.00388223
\(590\) −11.1457 −0.458860
\(591\) −1.31815 −0.0542213
\(592\) −2.61721 −0.107567
\(593\) 2.49682 0.102532 0.0512660 0.998685i \(-0.483674\pi\)
0.0512660 + 0.998685i \(0.483674\pi\)
\(594\) −5.61859 −0.230533
\(595\) −17.2425 −0.706874
\(596\) 12.5013 0.512073
\(597\) 12.5127 0.512109
\(598\) 15.8651 0.648771
\(599\) 22.7518 0.929615 0.464808 0.885412i \(-0.346123\pi\)
0.464808 + 0.885412i \(0.346123\pi\)
\(600\) 3.77033 0.153923
\(601\) −1.79248 −0.0731170 −0.0365585 0.999332i \(-0.511640\pi\)
−0.0365585 + 0.999332i \(0.511640\pi\)
\(602\) −15.5424 −0.633461
\(603\) 17.5511 0.714734
\(604\) −11.5610 −0.470411
\(605\) 1.46276 0.0594698
\(606\) 4.09869 0.166498
\(607\) 34.7467 1.41033 0.705163 0.709045i \(-0.250874\pi\)
0.705163 + 0.709045i \(0.250874\pi\)
\(608\) −0.0248218 −0.00100666
\(609\) −22.9922 −0.931691
\(610\) 7.29154 0.295226
\(611\) −4.87224 −0.197110
\(612\) −3.28093 −0.132624
\(613\) 4.84273 0.195596 0.0977980 0.995206i \(-0.468820\pi\)
0.0977980 + 0.995206i \(0.468820\pi\)
\(614\) 30.5600 1.23330
\(615\) −11.3511 −0.457721
\(616\) 4.53584 0.182754
\(617\) −5.70514 −0.229680 −0.114840 0.993384i \(-0.536636\pi\)
−0.114840 + 0.993384i \(0.536636\pi\)
\(618\) −5.52932 −0.222422
\(619\) 41.3552 1.66220 0.831102 0.556119i \(-0.187710\pi\)
0.831102 + 0.556119i \(0.187710\pi\)
\(620\) −5.55239 −0.222989
\(621\) 19.0877 0.765964
\(622\) −15.7508 −0.631551
\(623\) −64.6609 −2.59058
\(624\) −6.15572 −0.246426
\(625\) −2.51691 −0.100676
\(626\) 27.8952 1.11491
\(627\) −0.0327187 −0.00130666
\(628\) 10.4405 0.416621
\(629\) −6.80154 −0.271195
\(630\) 8.37645 0.333726
\(631\) 23.4081 0.931861 0.465930 0.884821i \(-0.345720\pi\)
0.465930 + 0.884821i \(0.345720\pi\)
\(632\) −3.75592 −0.149403
\(633\) 5.98811 0.238006
\(634\) −16.8081 −0.667537
\(635\) −20.5659 −0.816134
\(636\) 14.1927 0.562779
\(637\) 63.3898 2.51159
\(638\) 3.84555 0.152247
\(639\) 10.7558 0.425495
\(640\) 1.46276 0.0578208
\(641\) 35.8595 1.41636 0.708182 0.706030i \(-0.249516\pi\)
0.708182 + 0.706030i \(0.249516\pi\)
\(642\) 0.902206 0.0356072
\(643\) −18.6741 −0.736434 −0.368217 0.929740i \(-0.620032\pi\)
−0.368217 + 0.929740i \(0.620032\pi\)
\(644\) −15.4094 −0.607215
\(645\) −6.60690 −0.260146
\(646\) −0.0645062 −0.00253796
\(647\) 39.1856 1.54054 0.770272 0.637715i \(-0.220120\pi\)
0.770272 + 0.637715i \(0.220120\pi\)
\(648\) −3.61865 −0.142154
\(649\) 7.61960 0.299095
\(650\) −13.3577 −0.523931
\(651\) −22.6949 −0.889483
\(652\) 18.9201 0.740969
\(653\) −19.3221 −0.756132 −0.378066 0.925779i \(-0.623411\pi\)
−0.378066 + 0.925779i \(0.623411\pi\)
\(654\) 24.3963 0.953970
\(655\) −8.72470 −0.340902
\(656\) 5.88709 0.229852
\(657\) 7.85970 0.306636
\(658\) 4.73230 0.184484
\(659\) −1.52151 −0.0592697 −0.0296348 0.999561i \(-0.509434\pi\)
−0.0296348 + 0.999561i \(0.509434\pi\)
\(660\) 1.92814 0.0750526
\(661\) −29.3132 −1.14015 −0.570075 0.821592i \(-0.693086\pi\)
−0.570075 + 0.821592i \(0.693086\pi\)
\(662\) −5.46500 −0.212403
\(663\) −15.9973 −0.621285
\(664\) −13.1972 −0.512149
\(665\) 0.164689 0.00638637
\(666\) 3.30420 0.128035
\(667\) −13.0643 −0.505851
\(668\) 1.16959 0.0452526
\(669\) 28.7024 1.10970
\(670\) −20.3352 −0.785619
\(671\) −4.98477 −0.192435
\(672\) 5.97891 0.230641
\(673\) 15.0588 0.580472 0.290236 0.956955i \(-0.406266\pi\)
0.290236 + 0.956955i \(0.406266\pi\)
\(674\) −9.06460 −0.349155
\(675\) −16.0710 −0.618573
\(676\) 8.80872 0.338797
\(677\) −35.3887 −1.36010 −0.680048 0.733167i \(-0.738041\pi\)
−0.680048 + 0.733167i \(0.738041\pi\)
\(678\) 4.93037 0.189350
\(679\) 77.6877 2.98138
\(680\) 3.80139 0.145777
\(681\) 5.68245 0.217752
\(682\) 3.79582 0.145350
\(683\) 14.3648 0.549652 0.274826 0.961494i \(-0.411380\pi\)
0.274826 + 0.961494i \(0.411380\pi\)
\(684\) 0.0313372 0.00119821
\(685\) −19.3813 −0.740522
\(686\) −29.8181 −1.13846
\(687\) −25.4162 −0.969689
\(688\) 3.42657 0.130637
\(689\) −50.2826 −1.91561
\(690\) −6.55035 −0.249368
\(691\) −42.9587 −1.63423 −0.817113 0.576478i \(-0.804427\pi\)
−0.817113 + 0.576478i \(0.804427\pi\)
\(692\) 1.58761 0.0603517
\(693\) −5.72646 −0.217530
\(694\) −20.4406 −0.775915
\(695\) 13.3900 0.507912
\(696\) 5.06900 0.192140
\(697\) 15.2992 0.579499
\(698\) −21.3694 −0.808844
\(699\) 14.8943 0.563355
\(700\) 12.9740 0.490371
\(701\) −12.2510 −0.462716 −0.231358 0.972869i \(-0.574317\pi\)
−0.231358 + 0.972869i \(0.574317\pi\)
\(702\) 26.2387 0.990315
\(703\) 0.0649638 0.00245016
\(704\) −1.00000 −0.0376889
\(705\) 2.01164 0.0757629
\(706\) 14.5443 0.547383
\(707\) 14.1039 0.530432
\(708\) 10.0437 0.377467
\(709\) 29.5882 1.11121 0.555605 0.831446i \(-0.312487\pi\)
0.555605 + 0.831446i \(0.312487\pi\)
\(710\) −12.4621 −0.467694
\(711\) 4.74181 0.177832
\(712\) 14.2555 0.534249
\(713\) −12.8954 −0.482935
\(714\) 15.5378 0.581489
\(715\) −6.83107 −0.255468
\(716\) 2.59384 0.0969363
\(717\) 4.24820 0.158652
\(718\) −5.04096 −0.188127
\(719\) −14.0402 −0.523612 −0.261806 0.965120i \(-0.584318\pi\)
−0.261806 + 0.965120i \(0.584318\pi\)
\(720\) −1.84672 −0.0688233
\(721\) −19.0268 −0.708596
\(722\) −18.9994 −0.707084
\(723\) −10.3994 −0.386759
\(724\) 10.5987 0.393899
\(725\) 10.9995 0.408512
\(726\) −1.31815 −0.0489210
\(727\) −25.3104 −0.938711 −0.469355 0.883009i \(-0.655514\pi\)
−0.469355 + 0.883009i \(0.655514\pi\)
\(728\) −21.1823 −0.785069
\(729\) 26.7869 0.992106
\(730\) −9.10651 −0.337047
\(731\) 8.90489 0.329359
\(732\) −6.57066 −0.242859
\(733\) −17.0730 −0.630607 −0.315304 0.948991i \(-0.602106\pi\)
−0.315304 + 0.948991i \(0.602106\pi\)
\(734\) 0.931053 0.0343658
\(735\) −26.1723 −0.965380
\(736\) 3.39725 0.125224
\(737\) 13.9019 0.512085
\(738\) −7.43239 −0.273590
\(739\) −23.0350 −0.847357 −0.423679 0.905813i \(-0.639261\pi\)
−0.423679 + 0.905813i \(0.639261\pi\)
\(740\) −3.82836 −0.140733
\(741\) 0.152796 0.00561310
\(742\) 48.8383 1.79291
\(743\) 17.8764 0.655820 0.327910 0.944709i \(-0.393656\pi\)
0.327910 + 0.944709i \(0.393656\pi\)
\(744\) 5.00345 0.183435
\(745\) 18.2864 0.669962
\(746\) 9.53258 0.349013
\(747\) 16.6613 0.609605
\(748\) −2.59877 −0.0950206
\(749\) 3.10456 0.113438
\(750\) 15.1558 0.553411
\(751\) 44.7023 1.63121 0.815605 0.578609i \(-0.196404\pi\)
0.815605 + 0.578609i \(0.196404\pi\)
\(752\) −1.04331 −0.0380456
\(753\) 10.9841 0.400283
\(754\) −17.9586 −0.654015
\(755\) −16.9110 −0.615455
\(756\) −25.4850 −0.926882
\(757\) 27.1075 0.985239 0.492619 0.870245i \(-0.336040\pi\)
0.492619 + 0.870245i \(0.336040\pi\)
\(758\) −25.5474 −0.927924
\(759\) 4.47807 0.162544
\(760\) −0.0363084 −0.00131704
\(761\) −4.64957 −0.168547 −0.0842734 0.996443i \(-0.526857\pi\)
−0.0842734 + 0.996443i \(0.526857\pi\)
\(762\) 18.5327 0.671368
\(763\) 83.9495 3.03918
\(764\) −15.8379 −0.572997
\(765\) −4.79922 −0.173516
\(766\) 4.42112 0.159742
\(767\) −35.5834 −1.28484
\(768\) −1.31815 −0.0475645
\(769\) −30.3365 −1.09396 −0.546981 0.837145i \(-0.684223\pi\)
−0.546981 + 0.837145i \(0.684223\pi\)
\(770\) 6.63486 0.239104
\(771\) −9.59015 −0.345381
\(772\) 27.5855 0.992826
\(773\) 11.2227 0.403651 0.201825 0.979422i \(-0.435313\pi\)
0.201825 + 0.979422i \(0.435313\pi\)
\(774\) −4.32601 −0.155495
\(775\) 10.8573 0.390006
\(776\) −17.1275 −0.614842
\(777\) −15.6481 −0.561371
\(778\) 26.6493 0.955423
\(779\) −0.146128 −0.00523558
\(780\) −9.00435 −0.322408
\(781\) 8.51955 0.304854
\(782\) 8.82868 0.315713
\(783\) −21.6066 −0.772155
\(784\) 13.5739 0.484782
\(785\) 15.2720 0.545079
\(786\) 7.86213 0.280433
\(787\) 29.4930 1.05131 0.525657 0.850697i \(-0.323820\pi\)
0.525657 + 0.850697i \(0.323820\pi\)
\(788\) 1.00000 0.0356235
\(789\) 27.3783 0.974695
\(790\) −5.49402 −0.195469
\(791\) 16.9658 0.603233
\(792\) 1.26249 0.0448606
\(793\) 23.2788 0.826654
\(794\) 33.9654 1.20539
\(795\) 20.7606 0.736303
\(796\) −9.49262 −0.336457
\(797\) 52.7756 1.86941 0.934703 0.355429i \(-0.115665\pi\)
0.934703 + 0.355429i \(0.115665\pi\)
\(798\) −0.148407 −0.00525355
\(799\) −2.71133 −0.0959200
\(800\) −2.86033 −0.101128
\(801\) −17.9975 −0.635909
\(802\) 19.9873 0.705776
\(803\) 6.22555 0.219695
\(804\) 18.3248 0.646266
\(805\) −22.5403 −0.794440
\(806\) −17.7264 −0.624387
\(807\) −28.8240 −1.01465
\(808\) −3.10944 −0.109390
\(809\) −7.72464 −0.271584 −0.135792 0.990737i \(-0.543358\pi\)
−0.135792 + 0.990737i \(0.543358\pi\)
\(810\) −5.29323 −0.185985
\(811\) −33.7411 −1.18481 −0.592404 0.805641i \(-0.701821\pi\)
−0.592404 + 0.805641i \(0.701821\pi\)
\(812\) 17.4428 0.612123
\(813\) 16.9525 0.594548
\(814\) 2.61721 0.0917332
\(815\) 27.6756 0.969435
\(816\) −3.42557 −0.119919
\(817\) −0.0850536 −0.00297565
\(818\) −13.2776 −0.464239
\(819\) 26.7424 0.934457
\(820\) 8.61141 0.300724
\(821\) 4.30254 0.150160 0.0750798 0.997178i \(-0.476079\pi\)
0.0750798 + 0.997178i \(0.476079\pi\)
\(822\) 17.4652 0.609168
\(823\) 18.1369 0.632214 0.316107 0.948724i \(-0.397624\pi\)
0.316107 + 0.948724i \(0.397624\pi\)
\(824\) 4.19477 0.146132
\(825\) −3.77033 −0.131266
\(826\) 34.5613 1.20254
\(827\) 24.1572 0.840028 0.420014 0.907518i \(-0.362025\pi\)
0.420014 + 0.907518i \(0.362025\pi\)
\(828\) −4.28899 −0.149053
\(829\) −27.8647 −0.967781 −0.483890 0.875129i \(-0.660777\pi\)
−0.483890 + 0.875129i \(0.660777\pi\)
\(830\) −19.3043 −0.670063
\(831\) 10.5342 0.365427
\(832\) 4.66998 0.161902
\(833\) 35.2755 1.22222
\(834\) −12.0662 −0.417819
\(835\) 1.71083 0.0592056
\(836\) 0.0248218 0.000858479 0
\(837\) −21.3272 −0.737175
\(838\) 12.0765 0.417177
\(839\) −3.57511 −0.123427 −0.0617133 0.998094i \(-0.519656\pi\)
−0.0617133 + 0.998094i \(0.519656\pi\)
\(840\) 8.74572 0.301756
\(841\) −14.2117 −0.490060
\(842\) 5.00005 0.172313
\(843\) 21.5438 0.742006
\(844\) −4.54282 −0.156370
\(845\) 12.8851 0.443260
\(846\) 1.31717 0.0452852
\(847\) −4.53584 −0.155853
\(848\) −10.7672 −0.369747
\(849\) −8.22427 −0.282256
\(850\) −7.43334 −0.254962
\(851\) −8.89132 −0.304790
\(852\) 11.2300 0.384734
\(853\) 2.95659 0.101232 0.0506159 0.998718i \(-0.483882\pi\)
0.0506159 + 0.998718i \(0.483882\pi\)
\(854\) −22.6102 −0.773704
\(855\) 0.0458389 0.00156766
\(856\) −0.684451 −0.0233940
\(857\) 12.6615 0.432509 0.216254 0.976337i \(-0.430616\pi\)
0.216254 + 0.976337i \(0.430616\pi\)
\(858\) 6.15572 0.210153
\(859\) 14.0078 0.477940 0.238970 0.971027i \(-0.423190\pi\)
0.238970 + 0.971027i \(0.423190\pi\)
\(860\) 5.01226 0.170917
\(861\) 35.1984 1.19956
\(862\) 21.5480 0.733926
\(863\) 12.8050 0.435888 0.217944 0.975961i \(-0.430065\pi\)
0.217944 + 0.975961i \(0.430065\pi\)
\(864\) 5.61859 0.191148
\(865\) 2.32229 0.0789603
\(866\) 28.9405 0.983440
\(867\) 13.5062 0.458695
\(868\) 17.2173 0.584392
\(869\) 3.75592 0.127411
\(870\) 7.41474 0.251383
\(871\) −64.9218 −2.19979
\(872\) −18.5080 −0.626761
\(873\) 21.6233 0.731838
\(874\) −0.0843257 −0.00285236
\(875\) 52.1522 1.76307
\(876\) 8.20619 0.277262
\(877\) −33.5833 −1.13403 −0.567014 0.823708i \(-0.691902\pi\)
−0.567014 + 0.823708i \(0.691902\pi\)
\(878\) 5.31799 0.179474
\(879\) −26.1492 −0.881991
\(880\) −1.46276 −0.0493097
\(881\) 19.2099 0.647199 0.323599 0.946194i \(-0.395107\pi\)
0.323599 + 0.946194i \(0.395107\pi\)
\(882\) −17.1369 −0.577029
\(883\) 54.6291 1.83841 0.919207 0.393774i \(-0.128831\pi\)
0.919207 + 0.393774i \(0.128831\pi\)
\(884\) 12.1362 0.408186
\(885\) 14.6916 0.493853
\(886\) −4.59279 −0.154298
\(887\) 42.3730 1.42275 0.711374 0.702814i \(-0.248073\pi\)
0.711374 + 0.702814i \(0.248073\pi\)
\(888\) 3.44987 0.115770
\(889\) 63.7724 2.13886
\(890\) 20.8525 0.698976
\(891\) 3.61865 0.121229
\(892\) −21.7748 −0.729074
\(893\) 0.0258968 0.000866604 0
\(894\) −16.4785 −0.551124
\(895\) 3.79417 0.126825
\(896\) −4.53584 −0.151532
\(897\) −20.9125 −0.698248
\(898\) −31.4568 −1.04973
\(899\) 14.5970 0.486838
\(900\) 3.61113 0.120371
\(901\) −27.9815 −0.932200
\(902\) −5.88709 −0.196019
\(903\) 20.4872 0.681770
\(904\) −3.74038 −0.124403
\(905\) 15.5034 0.515352
\(906\) 15.2391 0.506286
\(907\) 7.93844 0.263592 0.131796 0.991277i \(-0.457926\pi\)
0.131796 + 0.991277i \(0.457926\pi\)
\(908\) −4.31094 −0.143064
\(909\) 3.92563 0.130205
\(910\) −30.9847 −1.02713
\(911\) −8.36949 −0.277294 −0.138647 0.990342i \(-0.544275\pi\)
−0.138647 + 0.990342i \(0.544275\pi\)
\(912\) 0.0327187 0.00108343
\(913\) 13.1972 0.436762
\(914\) −42.5459 −1.40729
\(915\) −9.61132 −0.317740
\(916\) 19.2818 0.637088
\(917\) 27.0542 0.893409
\(918\) 14.6014 0.481919
\(919\) 32.5862 1.07492 0.537459 0.843290i \(-0.319384\pi\)
0.537459 + 0.843290i \(0.319384\pi\)
\(920\) 4.96937 0.163835
\(921\) −40.2826 −1.32736
\(922\) −33.8876 −1.11603
\(923\) −39.7862 −1.30958
\(924\) −5.97891 −0.196692
\(925\) 7.48608 0.246141
\(926\) −17.2341 −0.566348
\(927\) −5.29585 −0.173939
\(928\) −3.84555 −0.126236
\(929\) −14.7296 −0.483262 −0.241631 0.970368i \(-0.577682\pi\)
−0.241631 + 0.970368i \(0.577682\pi\)
\(930\) 7.31886 0.239995
\(931\) −0.336928 −0.0110424
\(932\) −11.2994 −0.370126
\(933\) 20.7619 0.679714
\(934\) −26.4443 −0.865285
\(935\) −3.80139 −0.124319
\(936\) −5.89580 −0.192710
\(937\) 33.2345 1.08572 0.542861 0.839822i \(-0.317341\pi\)
0.542861 + 0.839822i \(0.317341\pi\)
\(938\) 63.0571 2.05889
\(939\) −36.7699 −1.19994
\(940\) −1.52612 −0.0497764
\(941\) 49.7545 1.62195 0.810975 0.585081i \(-0.198937\pi\)
0.810975 + 0.585081i \(0.198937\pi\)
\(942\) −13.7621 −0.448393
\(943\) 19.9999 0.651286
\(944\) −7.61960 −0.247997
\(945\) −37.2786 −1.21267
\(946\) −3.42657 −0.111408
\(947\) −15.4538 −0.502180 −0.251090 0.967964i \(-0.580789\pi\)
−0.251090 + 0.967964i \(0.580789\pi\)
\(948\) 4.95086 0.160796
\(949\) −29.0732 −0.943756
\(950\) 0.0709984 0.00230349
\(951\) 22.1556 0.718445
\(952\) −11.7876 −0.382040
\(953\) −58.8609 −1.90669 −0.953346 0.301879i \(-0.902386\pi\)
−0.953346 + 0.301879i \(0.902386\pi\)
\(954\) 13.5935 0.440105
\(955\) −23.1672 −0.749671
\(956\) −3.22286 −0.104235
\(957\) −5.06900 −0.163857
\(958\) −18.2359 −0.589176
\(959\) 60.0990 1.94070
\(960\) −1.92814 −0.0622303
\(961\) −16.5917 −0.535217
\(962\) −12.2223 −0.394064
\(963\) 0.864112 0.0278456
\(964\) 7.88943 0.254101
\(965\) 40.3511 1.29895
\(966\) 20.3118 0.653523
\(967\) 30.4623 0.979600 0.489800 0.871835i \(-0.337070\pi\)
0.489800 + 0.871835i \(0.337070\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0.0850286 0.00273151
\(970\) −25.0535 −0.804419
\(971\) 4.88016 0.156612 0.0783058 0.996929i \(-0.475049\pi\)
0.0783058 + 0.996929i \(0.475049\pi\)
\(972\) −12.0858 −0.387654
\(973\) −41.5208 −1.33109
\(974\) 39.2794 1.25859
\(975\) 17.6074 0.563887
\(976\) 4.98477 0.159559
\(977\) 11.7518 0.375972 0.187986 0.982172i \(-0.439804\pi\)
0.187986 + 0.982172i \(0.439804\pi\)
\(978\) −24.9395 −0.797476
\(979\) −14.2555 −0.455609
\(980\) 19.8554 0.634257
\(981\) 23.3662 0.746025
\(982\) 28.5260 0.910301
\(983\) −16.3167 −0.520422 −0.260211 0.965552i \(-0.583792\pi\)
−0.260211 + 0.965552i \(0.583792\pi\)
\(984\) −7.76005 −0.247381
\(985\) 1.46276 0.0466075
\(986\) −9.99372 −0.318265
\(987\) −6.23786 −0.198553
\(988\) −0.115917 −0.00368782
\(989\) 11.6409 0.370160
\(990\) 1.84672 0.0586927
\(991\) 46.3234 1.47151 0.735755 0.677248i \(-0.236827\pi\)
0.735755 + 0.677248i \(0.236827\pi\)
\(992\) −3.79582 −0.120518
\(993\) 7.20368 0.228602
\(994\) 38.6434 1.22569
\(995\) −13.8855 −0.440198
\(996\) 17.3958 0.551207
\(997\) −27.4837 −0.870419 −0.435210 0.900329i \(-0.643326\pi\)
−0.435210 + 0.900329i \(0.643326\pi\)
\(998\) 33.4415 1.05857
\(999\) −14.7050 −0.465246
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.d.1.8 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4334.2.a.d.1.8 17 1.1 even 1 trivial