Properties

Label 6042.2.a.bb.1.8
Level $6042$
Weight $2$
Character 6042.1
Self dual yes
Analytic conductor $48.246$
Analytic rank $1$
Dimension $9$
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] = [6042,2,Mod(1,6042)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6042, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6042.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6042 = 2 \cdot 3 \cdot 19 \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6042.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: \(48.2456129013\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 4x^{8} - 16x^{7} + 76x^{6} + 30x^{5} - 366x^{4} + 300x^{3} + 101x^{2} - 106x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(0.394407\) of defining polynomial
Character \(\chi\) \(=\) 6042.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.69036 q^{5} -1.00000 q^{6} -0.605593 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.69036 q^{5} -1.00000 q^{6} -0.605593 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.69036 q^{10} +5.02636 q^{11} -1.00000 q^{12} -2.72240 q^{13} -0.605593 q^{14} -1.69036 q^{15} +1.00000 q^{16} -5.51617 q^{17} +1.00000 q^{18} -1.00000 q^{19} +1.69036 q^{20} +0.605593 q^{21} +5.02636 q^{22} -7.58679 q^{23} -1.00000 q^{24} -2.14269 q^{25} -2.72240 q^{26} -1.00000 q^{27} -0.605593 q^{28} +3.16748 q^{29} -1.69036 q^{30} -9.54506 q^{31} +1.00000 q^{32} -5.02636 q^{33} -5.51617 q^{34} -1.02367 q^{35} +1.00000 q^{36} -9.34723 q^{37} -1.00000 q^{38} +2.72240 q^{39} +1.69036 q^{40} +0.396182 q^{41} +0.605593 q^{42} -3.81919 q^{43} +5.02636 q^{44} +1.69036 q^{45} -7.58679 q^{46} +4.86584 q^{47} -1.00000 q^{48} -6.63326 q^{49} -2.14269 q^{50} +5.51617 q^{51} -2.72240 q^{52} +1.00000 q^{53} -1.00000 q^{54} +8.49634 q^{55} -0.605593 q^{56} +1.00000 q^{57} +3.16748 q^{58} -1.33806 q^{59} -1.69036 q^{60} +9.23020 q^{61} -9.54506 q^{62} -0.605593 q^{63} +1.00000 q^{64} -4.60183 q^{65} -5.02636 q^{66} -10.6347 q^{67} -5.51617 q^{68} +7.58679 q^{69} -1.02367 q^{70} -6.43229 q^{71} +1.00000 q^{72} -11.5216 q^{73} -9.34723 q^{74} +2.14269 q^{75} -1.00000 q^{76} -3.04393 q^{77} +2.72240 q^{78} -1.45010 q^{79} +1.69036 q^{80} +1.00000 q^{81} +0.396182 q^{82} -11.7496 q^{83} +0.605593 q^{84} -9.32430 q^{85} -3.81919 q^{86} -3.16748 q^{87} +5.02636 q^{88} +11.5259 q^{89} +1.69036 q^{90} +1.64867 q^{91} -7.58679 q^{92} +9.54506 q^{93} +4.86584 q^{94} -1.69036 q^{95} -1.00000 q^{96} -6.47681 q^{97} -6.63326 q^{98} +5.02636 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 9 q^{2} - 9 q^{3} + 9 q^{4} - 5 q^{5} - 9 q^{6} - 5 q^{7} + 9 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 9 q^{2} - 9 q^{3} + 9 q^{4} - 5 q^{5} - 9 q^{6} - 5 q^{7} + 9 q^{8} + 9 q^{9} - 5 q^{10} + 3 q^{11} - 9 q^{12} - 4 q^{13} - 5 q^{14} + 5 q^{15} + 9 q^{16} - 16 q^{17} + 9 q^{18} - 9 q^{19} - 5 q^{20} + 5 q^{21} + 3 q^{22} - 9 q^{24} + 8 q^{25} - 4 q^{26} - 9 q^{27} - 5 q^{28} - 2 q^{29} + 5 q^{30} - 10 q^{31} + 9 q^{32} - 3 q^{33} - 16 q^{34} - q^{35} + 9 q^{36} - 14 q^{37} - 9 q^{38} + 4 q^{39} - 5 q^{40} - 21 q^{41} + 5 q^{42} - 4 q^{43} + 3 q^{44} - 5 q^{45} - 8 q^{47} - 9 q^{48} - 14 q^{49} + 8 q^{50} + 16 q^{51} - 4 q^{52} + 9 q^{53} - 9 q^{54} - 14 q^{55} - 5 q^{56} + 9 q^{57} - 2 q^{58} - 12 q^{59} + 5 q^{60} - 13 q^{61} - 10 q^{62} - 5 q^{63} + 9 q^{64} - 13 q^{65} - 3 q^{66} + 22 q^{67} - 16 q^{68} - q^{70} - 13 q^{71} + 9 q^{72} - 17 q^{73} - 14 q^{74} - 8 q^{75} - 9 q^{76} - 25 q^{77} + 4 q^{78} - 5 q^{80} + 9 q^{81} - 21 q^{82} - 24 q^{83} + 5 q^{84} - 16 q^{85} - 4 q^{86} + 2 q^{87} + 3 q^{88} - 23 q^{89} - 5 q^{90} - 11 q^{91} + 10 q^{93} - 8 q^{94} + 5 q^{95} - 9 q^{96} - 29 q^{97} - 14 q^{98} + 3 q^{99}+O(q^{100}) \) 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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 1.69036 0.755951 0.377975 0.925816i \(-0.376620\pi\)
0.377975 + 0.925816i \(0.376620\pi\)
\(6\) −1.00000 −0.408248
\(7\) −0.605593 −0.228893 −0.114446 0.993429i \(-0.536509\pi\)
−0.114446 + 0.993429i \(0.536509\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.69036 0.534538
\(11\) 5.02636 1.51550 0.757752 0.652543i \(-0.226298\pi\)
0.757752 + 0.652543i \(0.226298\pi\)
\(12\) −1.00000 −0.288675
\(13\) −2.72240 −0.755058 −0.377529 0.925998i \(-0.623226\pi\)
−0.377529 + 0.925998i \(0.623226\pi\)
\(14\) −0.605593 −0.161852
\(15\) −1.69036 −0.436448
\(16\) 1.00000 0.250000
\(17\) −5.51617 −1.33787 −0.668934 0.743322i \(-0.733249\pi\)
−0.668934 + 0.743322i \(0.733249\pi\)
\(18\) 1.00000 0.235702
\(19\) −1.00000 −0.229416
\(20\) 1.69036 0.377975
\(21\) 0.605593 0.132151
\(22\) 5.02636 1.07162
\(23\) −7.58679 −1.58196 −0.790978 0.611845i \(-0.790428\pi\)
−0.790978 + 0.611845i \(0.790428\pi\)
\(24\) −1.00000 −0.204124
\(25\) −2.14269 −0.428538
\(26\) −2.72240 −0.533907
\(27\) −1.00000 −0.192450
\(28\) −0.605593 −0.114446
\(29\) 3.16748 0.588186 0.294093 0.955777i \(-0.404983\pi\)
0.294093 + 0.955777i \(0.404983\pi\)
\(30\) −1.69036 −0.308616
\(31\) −9.54506 −1.71434 −0.857172 0.515031i \(-0.827781\pi\)
−0.857172 + 0.515031i \(0.827781\pi\)
\(32\) 1.00000 0.176777
\(33\) −5.02636 −0.874977
\(34\) −5.51617 −0.946016
\(35\) −1.02367 −0.173032
\(36\) 1.00000 0.166667
\(37\) −9.34723 −1.53667 −0.768337 0.640045i \(-0.778916\pi\)
−0.768337 + 0.640045i \(0.778916\pi\)
\(38\) −1.00000 −0.162221
\(39\) 2.72240 0.435933
\(40\) 1.69036 0.267269
\(41\) 0.396182 0.0618732 0.0309366 0.999521i \(-0.490151\pi\)
0.0309366 + 0.999521i \(0.490151\pi\)
\(42\) 0.605593 0.0934450
\(43\) −3.81919 −0.582421 −0.291211 0.956659i \(-0.594058\pi\)
−0.291211 + 0.956659i \(0.594058\pi\)
\(44\) 5.02636 0.757752
\(45\) 1.69036 0.251984
\(46\) −7.58679 −1.11861
\(47\) 4.86584 0.709756 0.354878 0.934913i \(-0.384522\pi\)
0.354878 + 0.934913i \(0.384522\pi\)
\(48\) −1.00000 −0.144338
\(49\) −6.63326 −0.947608
\(50\) −2.14269 −0.303022
\(51\) 5.51617 0.772419
\(52\) −2.72240 −0.377529
\(53\) 1.00000 0.137361
\(54\) −1.00000 −0.136083
\(55\) 8.49634 1.14565
\(56\) −0.605593 −0.0809258
\(57\) 1.00000 0.132453
\(58\) 3.16748 0.415910
\(59\) −1.33806 −0.174201 −0.0871004 0.996200i \(-0.527760\pi\)
−0.0871004 + 0.996200i \(0.527760\pi\)
\(60\) −1.69036 −0.218224
\(61\) 9.23020 1.18181 0.590903 0.806743i \(-0.298772\pi\)
0.590903 + 0.806743i \(0.298772\pi\)
\(62\) −9.54506 −1.21222
\(63\) −0.605593 −0.0762975
\(64\) 1.00000 0.125000
\(65\) −4.60183 −0.570787
\(66\) −5.02636 −0.618702
\(67\) −10.6347 −1.29923 −0.649616 0.760262i \(-0.725070\pi\)
−0.649616 + 0.760262i \(0.725070\pi\)
\(68\) −5.51617 −0.668934
\(69\) 7.58679 0.913342
\(70\) −1.02367 −0.122352
\(71\) −6.43229 −0.763373 −0.381686 0.924292i \(-0.624657\pi\)
−0.381686 + 0.924292i \(0.624657\pi\)
\(72\) 1.00000 0.117851
\(73\) −11.5216 −1.34850 −0.674250 0.738504i \(-0.735533\pi\)
−0.674250 + 0.738504i \(0.735533\pi\)
\(74\) −9.34723 −1.08659
\(75\) 2.14269 0.247417
\(76\) −1.00000 −0.114708
\(77\) −3.04393 −0.346888
\(78\) 2.72240 0.308251
\(79\) −1.45010 −0.163149 −0.0815746 0.996667i \(-0.525995\pi\)
−0.0815746 + 0.996667i \(0.525995\pi\)
\(80\) 1.69036 0.188988
\(81\) 1.00000 0.111111
\(82\) 0.396182 0.0437510
\(83\) −11.7496 −1.28969 −0.644844 0.764314i \(-0.723078\pi\)
−0.644844 + 0.764314i \(0.723078\pi\)
\(84\) 0.605593 0.0660756
\(85\) −9.32430 −1.01136
\(86\) −3.81919 −0.411834
\(87\) −3.16748 −0.339589
\(88\) 5.02636 0.535812
\(89\) 11.5259 1.22174 0.610872 0.791729i \(-0.290819\pi\)
0.610872 + 0.791729i \(0.290819\pi\)
\(90\) 1.69036 0.178179
\(91\) 1.64867 0.172827
\(92\) −7.58679 −0.790978
\(93\) 9.54506 0.989777
\(94\) 4.86584 0.501873
\(95\) −1.69036 −0.173427
\(96\) −1.00000 −0.102062
\(97\) −6.47681 −0.657621 −0.328810 0.944396i \(-0.606648\pi\)
−0.328810 + 0.944396i \(0.606648\pi\)
\(98\) −6.63326 −0.670060
\(99\) 5.02636 0.505168
\(100\) −2.14269 −0.214269
\(101\) 3.24349 0.322740 0.161370 0.986894i \(-0.448409\pi\)
0.161370 + 0.986894i \(0.448409\pi\)
\(102\) 5.51617 0.546182
\(103\) 20.1540 1.98583 0.992915 0.118824i \(-0.0379124\pi\)
0.992915 + 0.118824i \(0.0379124\pi\)
\(104\) −2.72240 −0.266953
\(105\) 1.02367 0.0998998
\(106\) 1.00000 0.0971286
\(107\) 2.85147 0.275662 0.137831 0.990456i \(-0.455987\pi\)
0.137831 + 0.990456i \(0.455987\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 4.79123 0.458916 0.229458 0.973319i \(-0.426305\pi\)
0.229458 + 0.973319i \(0.426305\pi\)
\(110\) 8.49634 0.810094
\(111\) 9.34723 0.887199
\(112\) −0.605593 −0.0572232
\(113\) 7.20098 0.677411 0.338706 0.940892i \(-0.390011\pi\)
0.338706 + 0.940892i \(0.390011\pi\)
\(114\) 1.00000 0.0936586
\(115\) −12.8244 −1.19588
\(116\) 3.16748 0.294093
\(117\) −2.72240 −0.251686
\(118\) −1.33806 −0.123179
\(119\) 3.34056 0.306228
\(120\) −1.69036 −0.154308
\(121\) 14.2643 1.29675
\(122\) 9.23020 0.835663
\(123\) −0.396182 −0.0357225
\(124\) −9.54506 −0.857172
\(125\) −12.0737 −1.07990
\(126\) −0.605593 −0.0539505
\(127\) 15.9360 1.41409 0.707046 0.707167i \(-0.250027\pi\)
0.707046 + 0.707167i \(0.250027\pi\)
\(128\) 1.00000 0.0883883
\(129\) 3.81919 0.336261
\(130\) −4.60183 −0.403607
\(131\) 14.5372 1.27012 0.635059 0.772464i \(-0.280976\pi\)
0.635059 + 0.772464i \(0.280976\pi\)
\(132\) −5.02636 −0.437488
\(133\) 0.605593 0.0525116
\(134\) −10.6347 −0.918696
\(135\) −1.69036 −0.145483
\(136\) −5.51617 −0.473008
\(137\) −15.6691 −1.33871 −0.669353 0.742945i \(-0.733428\pi\)
−0.669353 + 0.742945i \(0.733428\pi\)
\(138\) 7.58679 0.645831
\(139\) −0.653234 −0.0554066 −0.0277033 0.999616i \(-0.508819\pi\)
−0.0277033 + 0.999616i \(0.508819\pi\)
\(140\) −1.02367 −0.0865158
\(141\) −4.86584 −0.409778
\(142\) −6.43229 −0.539786
\(143\) −13.6838 −1.14429
\(144\) 1.00000 0.0833333
\(145\) 5.35417 0.444639
\(146\) −11.5216 −0.953533
\(147\) 6.63326 0.547102
\(148\) −9.34723 −0.768337
\(149\) 22.9327 1.87872 0.939360 0.342933i \(-0.111421\pi\)
0.939360 + 0.342933i \(0.111421\pi\)
\(150\) 2.14269 0.174950
\(151\) 19.3661 1.57599 0.787996 0.615680i \(-0.211119\pi\)
0.787996 + 0.615680i \(0.211119\pi\)
\(152\) −1.00000 −0.0811107
\(153\) −5.51617 −0.445956
\(154\) −3.04393 −0.245287
\(155\) −16.1346 −1.29596
\(156\) 2.72240 0.217967
\(157\) 0.0490296 0.00391299 0.00195649 0.999998i \(-0.499377\pi\)
0.00195649 + 0.999998i \(0.499377\pi\)
\(158\) −1.45010 −0.115364
\(159\) −1.00000 −0.0793052
\(160\) 1.69036 0.133635
\(161\) 4.59451 0.362098
\(162\) 1.00000 0.0785674
\(163\) −18.0273 −1.41201 −0.706005 0.708207i \(-0.749504\pi\)
−0.706005 + 0.708207i \(0.749504\pi\)
\(164\) 0.396182 0.0309366
\(165\) −8.49634 −0.661439
\(166\) −11.7496 −0.911947
\(167\) 23.0036 1.78007 0.890035 0.455892i \(-0.150680\pi\)
0.890035 + 0.455892i \(0.150680\pi\)
\(168\) 0.605593 0.0467225
\(169\) −5.58853 −0.429887
\(170\) −9.32430 −0.715142
\(171\) −1.00000 −0.0764719
\(172\) −3.81919 −0.291211
\(173\) −16.3622 −1.24400 −0.621998 0.783018i \(-0.713679\pi\)
−0.621998 + 0.783018i \(0.713679\pi\)
\(174\) −3.16748 −0.240126
\(175\) 1.29760 0.0980892
\(176\) 5.02636 0.378876
\(177\) 1.33806 0.100575
\(178\) 11.5259 0.863904
\(179\) 1.18880 0.0888552 0.0444276 0.999013i \(-0.485854\pi\)
0.0444276 + 0.999013i \(0.485854\pi\)
\(180\) 1.69036 0.125992
\(181\) −10.2668 −0.763127 −0.381564 0.924343i \(-0.624614\pi\)
−0.381564 + 0.924343i \(0.624614\pi\)
\(182\) 1.64867 0.122207
\(183\) −9.23020 −0.682316
\(184\) −7.58679 −0.559306
\(185\) −15.8002 −1.16165
\(186\) 9.54506 0.699878
\(187\) −27.7263 −2.02754
\(188\) 4.86584 0.354878
\(189\) 0.605593 0.0440504
\(190\) −1.69036 −0.122631
\(191\) 3.04006 0.219971 0.109985 0.993933i \(-0.464920\pi\)
0.109985 + 0.993933i \(0.464920\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −5.67822 −0.408728 −0.204364 0.978895i \(-0.565513\pi\)
−0.204364 + 0.978895i \(0.565513\pi\)
\(194\) −6.47681 −0.465008
\(195\) 4.60183 0.329544
\(196\) −6.63326 −0.473804
\(197\) −7.73625 −0.551185 −0.275593 0.961275i \(-0.588874\pi\)
−0.275593 + 0.961275i \(0.588874\pi\)
\(198\) 5.02636 0.357208
\(199\) −26.6283 −1.88763 −0.943815 0.330473i \(-0.892792\pi\)
−0.943815 + 0.330473i \(0.892792\pi\)
\(200\) −2.14269 −0.151511
\(201\) 10.6347 0.750112
\(202\) 3.24349 0.228211
\(203\) −1.91820 −0.134631
\(204\) 5.51617 0.386209
\(205\) 0.669689 0.0467731
\(206\) 20.1540 1.40419
\(207\) −7.58679 −0.527319
\(208\) −2.72240 −0.188765
\(209\) −5.02636 −0.347680
\(210\) 1.02367 0.0706399
\(211\) −25.9291 −1.78503 −0.892517 0.451013i \(-0.851063\pi\)
−0.892517 + 0.451013i \(0.851063\pi\)
\(212\) 1.00000 0.0686803
\(213\) 6.43229 0.440733
\(214\) 2.85147 0.194922
\(215\) −6.45580 −0.440282
\(216\) −1.00000 −0.0680414
\(217\) 5.78042 0.392401
\(218\) 4.79123 0.324503
\(219\) 11.5216 0.778556
\(220\) 8.49634 0.572823
\(221\) 15.0172 1.01017
\(222\) 9.34723 0.627345
\(223\) 25.3006 1.69425 0.847126 0.531392i \(-0.178331\pi\)
0.847126 + 0.531392i \(0.178331\pi\)
\(224\) −0.605593 −0.0404629
\(225\) −2.14269 −0.142846
\(226\) 7.20098 0.479002
\(227\) −8.58401 −0.569741 −0.284870 0.958566i \(-0.591951\pi\)
−0.284870 + 0.958566i \(0.591951\pi\)
\(228\) 1.00000 0.0662266
\(229\) −23.8247 −1.57438 −0.787190 0.616711i \(-0.788465\pi\)
−0.787190 + 0.616711i \(0.788465\pi\)
\(230\) −12.8244 −0.845615
\(231\) 3.04393 0.200276
\(232\) 3.16748 0.207955
\(233\) 20.2870 1.32904 0.664521 0.747269i \(-0.268636\pi\)
0.664521 + 0.747269i \(0.268636\pi\)
\(234\) −2.72240 −0.177969
\(235\) 8.22501 0.536540
\(236\) −1.33806 −0.0871004
\(237\) 1.45010 0.0941942
\(238\) 3.34056 0.216536
\(239\) 7.70613 0.498468 0.249234 0.968443i \(-0.419821\pi\)
0.249234 + 0.968443i \(0.419821\pi\)
\(240\) −1.69036 −0.109112
\(241\) 11.7300 0.755596 0.377798 0.925888i \(-0.376681\pi\)
0.377798 + 0.925888i \(0.376681\pi\)
\(242\) 14.2643 0.916942
\(243\) −1.00000 −0.0641500
\(244\) 9.23020 0.590903
\(245\) −11.2126 −0.716345
\(246\) −0.396182 −0.0252596
\(247\) 2.72240 0.173222
\(248\) −9.54506 −0.606112
\(249\) 11.7496 0.744602
\(250\) −12.0737 −0.763608
\(251\) 15.0680 0.951082 0.475541 0.879693i \(-0.342252\pi\)
0.475541 + 0.879693i \(0.342252\pi\)
\(252\) −0.605593 −0.0381488
\(253\) −38.1339 −2.39746
\(254\) 15.9360 0.999914
\(255\) 9.32430 0.583911
\(256\) 1.00000 0.0625000
\(257\) 11.0652 0.690231 0.345116 0.938560i \(-0.387840\pi\)
0.345116 + 0.938560i \(0.387840\pi\)
\(258\) 3.81919 0.237773
\(259\) 5.66061 0.351733
\(260\) −4.60183 −0.285393
\(261\) 3.16748 0.196062
\(262\) 14.5372 0.898109
\(263\) −7.24748 −0.446899 −0.223449 0.974716i \(-0.571732\pi\)
−0.223449 + 0.974716i \(0.571732\pi\)
\(264\) −5.02636 −0.309351
\(265\) 1.69036 0.103838
\(266\) 0.605593 0.0371313
\(267\) −11.5259 −0.705374
\(268\) −10.6347 −0.649616
\(269\) −13.0141 −0.793486 −0.396743 0.917930i \(-0.629860\pi\)
−0.396743 + 0.917930i \(0.629860\pi\)
\(270\) −1.69036 −0.102872
\(271\) −11.8664 −0.720831 −0.360416 0.932792i \(-0.617365\pi\)
−0.360416 + 0.932792i \(0.617365\pi\)
\(272\) −5.51617 −0.334467
\(273\) −1.64867 −0.0997819
\(274\) −15.6691 −0.946608
\(275\) −10.7699 −0.649451
\(276\) 7.58679 0.456671
\(277\) −23.5480 −1.41486 −0.707431 0.706783i \(-0.750146\pi\)
−0.707431 + 0.706783i \(0.750146\pi\)
\(278\) −0.653234 −0.0391784
\(279\) −9.54506 −0.571448
\(280\) −1.02367 −0.0611759
\(281\) −16.1100 −0.961041 −0.480521 0.876983i \(-0.659552\pi\)
−0.480521 + 0.876983i \(0.659552\pi\)
\(282\) −4.86584 −0.289757
\(283\) −8.17754 −0.486104 −0.243052 0.970013i \(-0.578149\pi\)
−0.243052 + 0.970013i \(0.578149\pi\)
\(284\) −6.43229 −0.381686
\(285\) 1.69036 0.100128
\(286\) −13.6838 −0.809138
\(287\) −0.239925 −0.0141623
\(288\) 1.00000 0.0589256
\(289\) 13.4282 0.789892
\(290\) 5.35417 0.314408
\(291\) 6.47681 0.379677
\(292\) −11.5216 −0.674250
\(293\) 16.7634 0.979328 0.489664 0.871911i \(-0.337119\pi\)
0.489664 + 0.871911i \(0.337119\pi\)
\(294\) 6.63326 0.386859
\(295\) −2.26180 −0.131687
\(296\) −9.34723 −0.543296
\(297\) −5.02636 −0.291659
\(298\) 22.9327 1.32846
\(299\) 20.6543 1.19447
\(300\) 2.14269 0.123708
\(301\) 2.31288 0.133312
\(302\) 19.3661 1.11439
\(303\) −3.24349 −0.186334
\(304\) −1.00000 −0.0573539
\(305\) 15.6023 0.893388
\(306\) −5.51617 −0.315339
\(307\) 0.494661 0.0282318 0.0141159 0.999900i \(-0.495507\pi\)
0.0141159 + 0.999900i \(0.495507\pi\)
\(308\) −3.04393 −0.173444
\(309\) −20.1540 −1.14652
\(310\) −16.1346 −0.916382
\(311\) 0.669217 0.0379478 0.0189739 0.999820i \(-0.493960\pi\)
0.0189739 + 0.999820i \(0.493960\pi\)
\(312\) 2.72240 0.154126
\(313\) −21.8402 −1.23448 −0.617240 0.786775i \(-0.711749\pi\)
−0.617240 + 0.786775i \(0.711749\pi\)
\(314\) 0.0490296 0.00276690
\(315\) −1.02367 −0.0576772
\(316\) −1.45010 −0.0815746
\(317\) −12.3228 −0.692116 −0.346058 0.938213i \(-0.612480\pi\)
−0.346058 + 0.938213i \(0.612480\pi\)
\(318\) −1.00000 −0.0560772
\(319\) 15.9209 0.891397
\(320\) 1.69036 0.0944939
\(321\) −2.85147 −0.159153
\(322\) 4.59451 0.256042
\(323\) 5.51617 0.306928
\(324\) 1.00000 0.0555556
\(325\) 5.83326 0.323571
\(326\) −18.0273 −0.998442
\(327\) −4.79123 −0.264955
\(328\) 0.396182 0.0218755
\(329\) −2.94672 −0.162458
\(330\) −8.49634 −0.467708
\(331\) 15.9874 0.878745 0.439373 0.898305i \(-0.355201\pi\)
0.439373 + 0.898305i \(0.355201\pi\)
\(332\) −11.7496 −0.644844
\(333\) −9.34723 −0.512225
\(334\) 23.0036 1.25870
\(335\) −17.9764 −0.982156
\(336\) 0.605593 0.0330378
\(337\) 32.6872 1.78058 0.890292 0.455391i \(-0.150501\pi\)
0.890292 + 0.455391i \(0.150501\pi\)
\(338\) −5.58853 −0.303976
\(339\) −7.20098 −0.391104
\(340\) −9.32430 −0.505681
\(341\) −47.9769 −2.59809
\(342\) −1.00000 −0.0540738
\(343\) 8.25620 0.445793
\(344\) −3.81919 −0.205917
\(345\) 12.8244 0.690442
\(346\) −16.3622 −0.879639
\(347\) 24.6711 1.32441 0.662206 0.749321i \(-0.269620\pi\)
0.662206 + 0.749321i \(0.269620\pi\)
\(348\) −3.16748 −0.169795
\(349\) −15.1792 −0.812522 −0.406261 0.913757i \(-0.633168\pi\)
−0.406261 + 0.913757i \(0.633168\pi\)
\(350\) 1.29760 0.0693596
\(351\) 2.72240 0.145311
\(352\) 5.02636 0.267906
\(353\) −31.2775 −1.66473 −0.832367 0.554224i \(-0.813015\pi\)
−0.832367 + 0.554224i \(0.813015\pi\)
\(354\) 1.33806 0.0711172
\(355\) −10.8729 −0.577072
\(356\) 11.5259 0.610872
\(357\) −3.34056 −0.176801
\(358\) 1.18880 0.0628301
\(359\) 24.6442 1.30067 0.650335 0.759648i \(-0.274629\pi\)
0.650335 + 0.759648i \(0.274629\pi\)
\(360\) 1.69036 0.0890897
\(361\) 1.00000 0.0526316
\(362\) −10.2668 −0.539612
\(363\) −14.2643 −0.748680
\(364\) 1.64867 0.0864136
\(365\) −19.4756 −1.01940
\(366\) −9.23020 −0.482470
\(367\) −12.5851 −0.656938 −0.328469 0.944515i \(-0.606533\pi\)
−0.328469 + 0.944515i \(0.606533\pi\)
\(368\) −7.58679 −0.395489
\(369\) 0.396182 0.0206244
\(370\) −15.8002 −0.821411
\(371\) −0.605593 −0.0314408
\(372\) 9.54506 0.494888
\(373\) 30.5532 1.58198 0.790992 0.611827i \(-0.209565\pi\)
0.790992 + 0.611827i \(0.209565\pi\)
\(374\) −27.7263 −1.43369
\(375\) 12.0737 0.623483
\(376\) 4.86584 0.250937
\(377\) −8.62314 −0.444114
\(378\) 0.605593 0.0311483
\(379\) 5.55156 0.285164 0.142582 0.989783i \(-0.454459\pi\)
0.142582 + 0.989783i \(0.454459\pi\)
\(380\) −1.69036 −0.0867135
\(381\) −15.9360 −0.816427
\(382\) 3.04006 0.155543
\(383\) 6.04178 0.308721 0.154360 0.988015i \(-0.450668\pi\)
0.154360 + 0.988015i \(0.450668\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −5.14533 −0.262230
\(386\) −5.67822 −0.289014
\(387\) −3.81919 −0.194140
\(388\) −6.47681 −0.328810
\(389\) −7.97723 −0.404462 −0.202231 0.979338i \(-0.564819\pi\)
−0.202231 + 0.979338i \(0.564819\pi\)
\(390\) 4.60183 0.233023
\(391\) 41.8501 2.11645
\(392\) −6.63326 −0.335030
\(393\) −14.5372 −0.733303
\(394\) −7.73625 −0.389747
\(395\) −2.45119 −0.123333
\(396\) 5.02636 0.252584
\(397\) 30.2267 1.51703 0.758517 0.651653i \(-0.225924\pi\)
0.758517 + 0.651653i \(0.225924\pi\)
\(398\) −26.6283 −1.33476
\(399\) −0.605593 −0.0303176
\(400\) −2.14269 −0.107135
\(401\) 5.15637 0.257497 0.128748 0.991677i \(-0.458904\pi\)
0.128748 + 0.991677i \(0.458904\pi\)
\(402\) 10.6347 0.530409
\(403\) 25.9855 1.29443
\(404\) 3.24349 0.161370
\(405\) 1.69036 0.0839945
\(406\) −1.91820 −0.0951987
\(407\) −46.9825 −2.32884
\(408\) 5.51617 0.273091
\(409\) −9.56914 −0.473164 −0.236582 0.971612i \(-0.576027\pi\)
−0.236582 + 0.971612i \(0.576027\pi\)
\(410\) 0.669689 0.0330736
\(411\) 15.6691 0.772902
\(412\) 20.1540 0.992915
\(413\) 0.810321 0.0398733
\(414\) −7.58679 −0.372871
\(415\) −19.8611 −0.974941
\(416\) −2.72240 −0.133477
\(417\) 0.653234 0.0319890
\(418\) −5.02636 −0.245847
\(419\) −23.2252 −1.13463 −0.567313 0.823502i \(-0.692017\pi\)
−0.567313 + 0.823502i \(0.692017\pi\)
\(420\) 1.02367 0.0499499
\(421\) 5.67045 0.276361 0.138180 0.990407i \(-0.455875\pi\)
0.138180 + 0.990407i \(0.455875\pi\)
\(422\) −25.9291 −1.26221
\(423\) 4.86584 0.236585
\(424\) 1.00000 0.0485643
\(425\) 11.8195 0.573328
\(426\) 6.43229 0.311646
\(427\) −5.58975 −0.270507
\(428\) 2.85147 0.137831
\(429\) 13.6838 0.660658
\(430\) −6.45580 −0.311326
\(431\) −15.2246 −0.733342 −0.366671 0.930351i \(-0.619503\pi\)
−0.366671 + 0.930351i \(0.619503\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 6.37413 0.306321 0.153161 0.988201i \(-0.451055\pi\)
0.153161 + 0.988201i \(0.451055\pi\)
\(434\) 5.78042 0.277469
\(435\) −5.35417 −0.256713
\(436\) 4.79123 0.229458
\(437\) 7.58679 0.362926
\(438\) 11.5216 0.550522
\(439\) −5.06870 −0.241916 −0.120958 0.992658i \(-0.538597\pi\)
−0.120958 + 0.992658i \(0.538597\pi\)
\(440\) 8.49634 0.405047
\(441\) −6.63326 −0.315869
\(442\) 15.0172 0.714297
\(443\) −22.9923 −1.09240 −0.546199 0.837656i \(-0.683926\pi\)
−0.546199 + 0.837656i \(0.683926\pi\)
\(444\) 9.34723 0.443600
\(445\) 19.4829 0.923579
\(446\) 25.3006 1.19802
\(447\) −22.9327 −1.08468
\(448\) −0.605593 −0.0286116
\(449\) 6.29872 0.297255 0.148627 0.988893i \(-0.452514\pi\)
0.148627 + 0.988893i \(0.452514\pi\)
\(450\) −2.14269 −0.101007
\(451\) 1.99135 0.0937692
\(452\) 7.20098 0.338706
\(453\) −19.3661 −0.909899
\(454\) −8.58401 −0.402868
\(455\) 2.78684 0.130649
\(456\) 1.00000 0.0468293
\(457\) 3.56508 0.166767 0.0833836 0.996518i \(-0.473427\pi\)
0.0833836 + 0.996518i \(0.473427\pi\)
\(458\) −23.8247 −1.11325
\(459\) 5.51617 0.257473
\(460\) −12.8244 −0.597940
\(461\) −19.6891 −0.917013 −0.458507 0.888691i \(-0.651615\pi\)
−0.458507 + 0.888691i \(0.651615\pi\)
\(462\) 3.04393 0.141616
\(463\) −19.2726 −0.895673 −0.447836 0.894116i \(-0.647805\pi\)
−0.447836 + 0.894116i \(0.647805\pi\)
\(464\) 3.16748 0.147046
\(465\) 16.1346 0.748223
\(466\) 20.2870 0.939775
\(467\) −9.21783 −0.426550 −0.213275 0.976992i \(-0.568413\pi\)
−0.213275 + 0.976992i \(0.568413\pi\)
\(468\) −2.72240 −0.125843
\(469\) 6.44028 0.297385
\(470\) 8.22501 0.379391
\(471\) −0.0490296 −0.00225917
\(472\) −1.33806 −0.0615893
\(473\) −19.1966 −0.882662
\(474\) 1.45010 0.0666054
\(475\) 2.14269 0.0983134
\(476\) 3.34056 0.153114
\(477\) 1.00000 0.0457869
\(478\) 7.70613 0.352470
\(479\) 26.8437 1.22652 0.613261 0.789880i \(-0.289857\pi\)
0.613261 + 0.789880i \(0.289857\pi\)
\(480\) −1.69036 −0.0771539
\(481\) 25.4469 1.16028
\(482\) 11.7300 0.534287
\(483\) −4.59451 −0.209057
\(484\) 14.2643 0.648376
\(485\) −10.9481 −0.497129
\(486\) −1.00000 −0.0453609
\(487\) −13.0977 −0.593515 −0.296757 0.954953i \(-0.595905\pi\)
−0.296757 + 0.954953i \(0.595905\pi\)
\(488\) 9.23020 0.417832
\(489\) 18.0273 0.815224
\(490\) −11.2126 −0.506533
\(491\) 33.6171 1.51712 0.758560 0.651603i \(-0.225903\pi\)
0.758560 + 0.651603i \(0.225903\pi\)
\(492\) −0.396182 −0.0178613
\(493\) −17.4723 −0.786915
\(494\) 2.72240 0.122487
\(495\) 8.49634 0.381882
\(496\) −9.54506 −0.428586
\(497\) 3.89535 0.174730
\(498\) 11.7496 0.526513
\(499\) 9.85886 0.441343 0.220672 0.975348i \(-0.429175\pi\)
0.220672 + 0.975348i \(0.429175\pi\)
\(500\) −12.0737 −0.539952
\(501\) −23.0036 −1.02772
\(502\) 15.0680 0.672517
\(503\) −38.9447 −1.73646 −0.868230 0.496162i \(-0.834742\pi\)
−0.868230 + 0.496162i \(0.834742\pi\)
\(504\) −0.605593 −0.0269753
\(505\) 5.48266 0.243975
\(506\) −38.1339 −1.69526
\(507\) 5.58853 0.248195
\(508\) 15.9360 0.707046
\(509\) −42.3073 −1.87524 −0.937620 0.347663i \(-0.886975\pi\)
−0.937620 + 0.347663i \(0.886975\pi\)
\(510\) 9.32430 0.412887
\(511\) 6.97739 0.308662
\(512\) 1.00000 0.0441942
\(513\) 1.00000 0.0441511
\(514\) 11.0652 0.488067
\(515\) 34.0674 1.50119
\(516\) 3.81919 0.168131
\(517\) 24.4575 1.07564
\(518\) 5.66061 0.248713
\(519\) 16.3622 0.718222
\(520\) −4.60183 −0.201804
\(521\) −17.9456 −0.786209 −0.393105 0.919494i \(-0.628599\pi\)
−0.393105 + 0.919494i \(0.628599\pi\)
\(522\) 3.16748 0.138637
\(523\) −25.5247 −1.11612 −0.558059 0.829801i \(-0.688454\pi\)
−0.558059 + 0.829801i \(0.688454\pi\)
\(524\) 14.5372 0.635059
\(525\) −1.29760 −0.0566318
\(526\) −7.24748 −0.316005
\(527\) 52.6522 2.29357
\(528\) −5.02636 −0.218744
\(529\) 34.5594 1.50258
\(530\) 1.69036 0.0734244
\(531\) −1.33806 −0.0580669
\(532\) 0.605593 0.0262558
\(533\) −1.07857 −0.0467179
\(534\) −11.5259 −0.498775
\(535\) 4.82000 0.208387
\(536\) −10.6347 −0.459348
\(537\) −1.18880 −0.0513006
\(538\) −13.0141 −0.561079
\(539\) −33.3411 −1.43610
\(540\) −1.69036 −0.0727414
\(541\) −34.6426 −1.48940 −0.744702 0.667397i \(-0.767408\pi\)
−0.744702 + 0.667397i \(0.767408\pi\)
\(542\) −11.8664 −0.509705
\(543\) 10.2668 0.440592
\(544\) −5.51617 −0.236504
\(545\) 8.09889 0.346918
\(546\) −1.64867 −0.0705564
\(547\) 6.22085 0.265984 0.132992 0.991117i \(-0.457541\pi\)
0.132992 + 0.991117i \(0.457541\pi\)
\(548\) −15.6691 −0.669353
\(549\) 9.23020 0.393935
\(550\) −10.7699 −0.459231
\(551\) −3.16748 −0.134939
\(552\) 7.58679 0.322915
\(553\) 0.878171 0.0373436
\(554\) −23.5480 −1.00046
\(555\) 15.8002 0.670679
\(556\) −0.653234 −0.0277033
\(557\) 10.0944 0.427712 0.213856 0.976865i \(-0.431398\pi\)
0.213856 + 0.976865i \(0.431398\pi\)
\(558\) −9.54506 −0.404075
\(559\) 10.3974 0.439762
\(560\) −1.02367 −0.0432579
\(561\) 27.7263 1.17060
\(562\) −16.1100 −0.679559
\(563\) 18.9732 0.799625 0.399813 0.916597i \(-0.369075\pi\)
0.399813 + 0.916597i \(0.369075\pi\)
\(564\) −4.86584 −0.204889
\(565\) 12.1722 0.512090
\(566\) −8.17754 −0.343728
\(567\) −0.605593 −0.0254325
\(568\) −6.43229 −0.269893
\(569\) 3.30430 0.138523 0.0692617 0.997599i \(-0.477936\pi\)
0.0692617 + 0.997599i \(0.477936\pi\)
\(570\) 1.69036 0.0708013
\(571\) 20.3094 0.849922 0.424961 0.905212i \(-0.360288\pi\)
0.424961 + 0.905212i \(0.360288\pi\)
\(572\) −13.6838 −0.572147
\(573\) −3.04006 −0.127000
\(574\) −0.239925 −0.0100143
\(575\) 16.2562 0.677928
\(576\) 1.00000 0.0416667
\(577\) 31.1092 1.29509 0.647547 0.762026i \(-0.275795\pi\)
0.647547 + 0.762026i \(0.275795\pi\)
\(578\) 13.4282 0.558538
\(579\) 5.67822 0.235979
\(580\) 5.35417 0.222320
\(581\) 7.11549 0.295200
\(582\) 6.47681 0.268473
\(583\) 5.02636 0.208170
\(584\) −11.5216 −0.476766
\(585\) −4.60183 −0.190262
\(586\) 16.7634 0.692490
\(587\) 20.2031 0.833870 0.416935 0.908936i \(-0.363104\pi\)
0.416935 + 0.908936i \(0.363104\pi\)
\(588\) 6.63326 0.273551
\(589\) 9.54506 0.393297
\(590\) −2.26180 −0.0931170
\(591\) 7.73625 0.318227
\(592\) −9.34723 −0.384169
\(593\) −36.3982 −1.49469 −0.747347 0.664434i \(-0.768673\pi\)
−0.747347 + 0.664434i \(0.768673\pi\)
\(594\) −5.02636 −0.206234
\(595\) 5.64673 0.231493
\(596\) 22.9327 0.939360
\(597\) 26.6283 1.08982
\(598\) 20.6543 0.844617
\(599\) 17.7160 0.723855 0.361927 0.932206i \(-0.382119\pi\)
0.361927 + 0.932206i \(0.382119\pi\)
\(600\) 2.14269 0.0874750
\(601\) −46.5989 −1.90081 −0.950405 0.311015i \(-0.899331\pi\)
−0.950405 + 0.311015i \(0.899331\pi\)
\(602\) 2.31288 0.0942658
\(603\) −10.6347 −0.433077
\(604\) 19.3661 0.787996
\(605\) 24.1117 0.980281
\(606\) −3.24349 −0.131758
\(607\) −36.0480 −1.46314 −0.731572 0.681765i \(-0.761213\pi\)
−0.731572 + 0.681765i \(0.761213\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 1.91820 0.0777294
\(610\) 15.6023 0.631720
\(611\) −13.2468 −0.535907
\(612\) −5.51617 −0.222978
\(613\) 5.92023 0.239116 0.119558 0.992827i \(-0.461852\pi\)
0.119558 + 0.992827i \(0.461852\pi\)
\(614\) 0.494661 0.0199629
\(615\) −0.669689 −0.0270045
\(616\) −3.04393 −0.122643
\(617\) −1.51772 −0.0611013 −0.0305506 0.999533i \(-0.509726\pi\)
−0.0305506 + 0.999533i \(0.509726\pi\)
\(618\) −20.1540 −0.810712
\(619\) 37.9119 1.52381 0.761904 0.647690i \(-0.224265\pi\)
0.761904 + 0.647690i \(0.224265\pi\)
\(620\) −16.1346 −0.647980
\(621\) 7.58679 0.304447
\(622\) 0.669217 0.0268332
\(623\) −6.98001 −0.279648
\(624\) 2.72240 0.108983
\(625\) −9.69542 −0.387817
\(626\) −21.8402 −0.872909
\(627\) 5.02636 0.200733
\(628\) 0.0490296 0.00195649
\(629\) 51.5609 2.05587
\(630\) −1.02367 −0.0407839
\(631\) 40.5797 1.61545 0.807726 0.589559i \(-0.200698\pi\)
0.807726 + 0.589559i \(0.200698\pi\)
\(632\) −1.45010 −0.0576819
\(633\) 25.9291 1.03059
\(634\) −12.3228 −0.489400
\(635\) 26.9376 1.06898
\(636\) −1.00000 −0.0396526
\(637\) 18.0584 0.715499
\(638\) 15.9209 0.630313
\(639\) −6.43229 −0.254458
\(640\) 1.69036 0.0668173
\(641\) 2.95517 0.116722 0.0583610 0.998296i \(-0.481413\pi\)
0.0583610 + 0.998296i \(0.481413\pi\)
\(642\) −2.85147 −0.112538
\(643\) 8.38579 0.330703 0.165352 0.986235i \(-0.447124\pi\)
0.165352 + 0.986235i \(0.447124\pi\)
\(644\) 4.59451 0.181049
\(645\) 6.45580 0.254197
\(646\) 5.51617 0.217031
\(647\) −39.1819 −1.54040 −0.770200 0.637803i \(-0.779843\pi\)
−0.770200 + 0.637803i \(0.779843\pi\)
\(648\) 1.00000 0.0392837
\(649\) −6.72558 −0.264002
\(650\) 5.83326 0.228799
\(651\) −5.78042 −0.226553
\(652\) −18.0273 −0.706005
\(653\) 8.24952 0.322829 0.161414 0.986887i \(-0.448394\pi\)
0.161414 + 0.986887i \(0.448394\pi\)
\(654\) −4.79123 −0.187352
\(655\) 24.5730 0.960147
\(656\) 0.396182 0.0154683
\(657\) −11.5216 −0.449500
\(658\) −2.94672 −0.114875
\(659\) −30.7268 −1.19695 −0.598473 0.801143i \(-0.704226\pi\)
−0.598473 + 0.801143i \(0.704226\pi\)
\(660\) −8.49634 −0.330720
\(661\) −30.8161 −1.19861 −0.599303 0.800522i \(-0.704556\pi\)
−0.599303 + 0.800522i \(0.704556\pi\)
\(662\) 15.9874 0.621367
\(663\) −15.0172 −0.583221
\(664\) −11.7496 −0.455974
\(665\) 1.02367 0.0396962
\(666\) −9.34723 −0.362198
\(667\) −24.0310 −0.930483
\(668\) 23.0036 0.890035
\(669\) −25.3006 −0.978177
\(670\) −17.9764 −0.694489
\(671\) 46.3943 1.79103
\(672\) 0.605593 0.0233613
\(673\) 40.7427 1.57051 0.785257 0.619170i \(-0.212531\pi\)
0.785257 + 0.619170i \(0.212531\pi\)
\(674\) 32.6872 1.25906
\(675\) 2.14269 0.0824722
\(676\) −5.58853 −0.214944
\(677\) −20.6647 −0.794208 −0.397104 0.917774i \(-0.629985\pi\)
−0.397104 + 0.917774i \(0.629985\pi\)
\(678\) −7.20098 −0.276552
\(679\) 3.92231 0.150525
\(680\) −9.32430 −0.357571
\(681\) 8.58401 0.328940
\(682\) −47.9769 −1.83713
\(683\) 27.0666 1.03567 0.517837 0.855479i \(-0.326737\pi\)
0.517837 + 0.855479i \(0.326737\pi\)
\(684\) −1.00000 −0.0382360
\(685\) −26.4865 −1.01200
\(686\) 8.25620 0.315223
\(687\) 23.8247 0.908969
\(688\) −3.81919 −0.145605
\(689\) −2.72240 −0.103715
\(690\) 12.8244 0.488216
\(691\) 34.8261 1.32485 0.662423 0.749130i \(-0.269528\pi\)
0.662423 + 0.749130i \(0.269528\pi\)
\(692\) −16.3622 −0.621998
\(693\) −3.04393 −0.115629
\(694\) 24.6711 0.936501
\(695\) −1.10420 −0.0418847
\(696\) −3.16748 −0.120063
\(697\) −2.18541 −0.0827783
\(698\) −15.1792 −0.574540
\(699\) −20.2870 −0.767323
\(700\) 1.29760 0.0490446
\(701\) 41.5066 1.56768 0.783842 0.620960i \(-0.213257\pi\)
0.783842 + 0.620960i \(0.213257\pi\)
\(702\) 2.72240 0.102750
\(703\) 9.34723 0.352537
\(704\) 5.02636 0.189438
\(705\) −8.22501 −0.309772
\(706\) −31.2775 −1.17715
\(707\) −1.96424 −0.0738727
\(708\) 1.33806 0.0502874
\(709\) −8.49344 −0.318978 −0.159489 0.987200i \(-0.550985\pi\)
−0.159489 + 0.987200i \(0.550985\pi\)
\(710\) −10.8729 −0.408052
\(711\) −1.45010 −0.0543830
\(712\) 11.5259 0.431952
\(713\) 72.4164 2.71202
\(714\) −3.34056 −0.125017
\(715\) −23.1304 −0.865030
\(716\) 1.18880 0.0444276
\(717\) −7.70613 −0.287791
\(718\) 24.6442 0.919712
\(719\) −15.2099 −0.567235 −0.283617 0.958938i \(-0.591535\pi\)
−0.283617 + 0.958938i \(0.591535\pi\)
\(720\) 1.69036 0.0629959
\(721\) −12.2051 −0.454542
\(722\) 1.00000 0.0372161
\(723\) −11.7300 −0.436244
\(724\) −10.2668 −0.381564
\(725\) −6.78692 −0.252060
\(726\) −14.2643 −0.529397
\(727\) −8.57395 −0.317990 −0.158995 0.987279i \(-0.550825\pi\)
−0.158995 + 0.987279i \(0.550825\pi\)
\(728\) 1.64867 0.0611037
\(729\) 1.00000 0.0370370
\(730\) −19.4756 −0.720824
\(731\) 21.0673 0.779203
\(732\) −9.23020 −0.341158
\(733\) −35.1291 −1.29752 −0.648762 0.760991i \(-0.724713\pi\)
−0.648762 + 0.760991i \(0.724713\pi\)
\(734\) −12.5851 −0.464525
\(735\) 11.2126 0.413582
\(736\) −7.58679 −0.279653
\(737\) −53.4537 −1.96899
\(738\) 0.396182 0.0145837
\(739\) 35.2246 1.29576 0.647879 0.761743i \(-0.275656\pi\)
0.647879 + 0.761743i \(0.275656\pi\)
\(740\) −15.8002 −0.580825
\(741\) −2.72240 −0.100010
\(742\) −0.605593 −0.0222320
\(743\) 17.4349 0.639625 0.319813 0.947481i \(-0.396380\pi\)
0.319813 + 0.947481i \(0.396380\pi\)
\(744\) 9.54506 0.349939
\(745\) 38.7645 1.42022
\(746\) 30.5532 1.11863
\(747\) −11.7496 −0.429896
\(748\) −27.7263 −1.01377
\(749\) −1.72683 −0.0630969
\(750\) 12.0737 0.440869
\(751\) −15.6510 −0.571112 −0.285556 0.958362i \(-0.592178\pi\)
−0.285556 + 0.958362i \(0.592178\pi\)
\(752\) 4.86584 0.177439
\(753\) −15.0680 −0.549108
\(754\) −8.62314 −0.314036
\(755\) 32.7357 1.19137
\(756\) 0.605593 0.0220252
\(757\) 22.8933 0.832072 0.416036 0.909348i \(-0.363419\pi\)
0.416036 + 0.909348i \(0.363419\pi\)
\(758\) 5.55156 0.201642
\(759\) 38.1339 1.38417
\(760\) −1.69036 −0.0613157
\(761\) −21.4767 −0.778528 −0.389264 0.921126i \(-0.627271\pi\)
−0.389264 + 0.921126i \(0.627271\pi\)
\(762\) −15.9360 −0.577301
\(763\) −2.90153 −0.105043
\(764\) 3.04006 0.109985
\(765\) −9.32430 −0.337121
\(766\) 6.04178 0.218299
\(767\) 3.64274 0.131532
\(768\) −1.00000 −0.0360844
\(769\) −19.3350 −0.697236 −0.348618 0.937265i \(-0.613349\pi\)
−0.348618 + 0.937265i \(0.613349\pi\)
\(770\) −5.14533 −0.185425
\(771\) −11.0652 −0.398505
\(772\) −5.67822 −0.204364
\(773\) −53.2953 −1.91690 −0.958449 0.285263i \(-0.907919\pi\)
−0.958449 + 0.285263i \(0.907919\pi\)
\(774\) −3.81919 −0.137278
\(775\) 20.4521 0.734662
\(776\) −6.47681 −0.232504
\(777\) −5.66061 −0.203073
\(778\) −7.97723 −0.285998
\(779\) −0.396182 −0.0141947
\(780\) 4.60183 0.164772
\(781\) −32.3310 −1.15689
\(782\) 41.8501 1.49655
\(783\) −3.16748 −0.113196
\(784\) −6.63326 −0.236902
\(785\) 0.0828776 0.00295803
\(786\) −14.5372 −0.518523
\(787\) 7.91503 0.282140 0.141070 0.990000i \(-0.454946\pi\)
0.141070 + 0.990000i \(0.454946\pi\)
\(788\) −7.73625 −0.275593
\(789\) 7.24748 0.258017
\(790\) −2.45119 −0.0872094
\(791\) −4.36086 −0.155054
\(792\) 5.02636 0.178604
\(793\) −25.1283 −0.892332
\(794\) 30.2267 1.07270
\(795\) −1.69036 −0.0599508
\(796\) −26.6283 −0.943815
\(797\) −39.3451 −1.39368 −0.696838 0.717229i \(-0.745410\pi\)
−0.696838 + 0.717229i \(0.745410\pi\)
\(798\) −0.605593 −0.0214378
\(799\) −26.8408 −0.949560
\(800\) −2.14269 −0.0757556
\(801\) 11.5259 0.407248
\(802\) 5.15637 0.182078
\(803\) −57.9116 −2.04366
\(804\) 10.6347 0.375056
\(805\) 7.76636 0.273728
\(806\) 25.9855 0.915300
\(807\) 13.0141 0.458119
\(808\) 3.24349 0.114106
\(809\) −31.7853 −1.11751 −0.558756 0.829332i \(-0.688721\pi\)
−0.558756 + 0.829332i \(0.688721\pi\)
\(810\) 1.69036 0.0593931
\(811\) 40.2364 1.41289 0.706446 0.707767i \(-0.250297\pi\)
0.706446 + 0.707767i \(0.250297\pi\)
\(812\) −1.91820 −0.0673157
\(813\) 11.8664 0.416172
\(814\) −46.9825 −1.64674
\(815\) −30.4726 −1.06741
\(816\) 5.51617 0.193105
\(817\) 3.81919 0.133617
\(818\) −9.56914 −0.334577
\(819\) 1.64867 0.0576091
\(820\) 0.669689 0.0233866
\(821\) −33.7454 −1.17772 −0.588862 0.808234i \(-0.700424\pi\)
−0.588862 + 0.808234i \(0.700424\pi\)
\(822\) 15.6691 0.546524
\(823\) 39.5858 1.37987 0.689936 0.723870i \(-0.257638\pi\)
0.689936 + 0.723870i \(0.257638\pi\)
\(824\) 20.1540 0.702097
\(825\) 10.7699 0.374961
\(826\) 0.810321 0.0281947
\(827\) 5.55246 0.193078 0.0965390 0.995329i \(-0.469223\pi\)
0.0965390 + 0.995329i \(0.469223\pi\)
\(828\) −7.58679 −0.263659
\(829\) 3.98995 0.138577 0.0692884 0.997597i \(-0.477927\pi\)
0.0692884 + 0.997597i \(0.477927\pi\)
\(830\) −19.8611 −0.689388
\(831\) 23.5480 0.816871
\(832\) −2.72240 −0.0943823
\(833\) 36.5902 1.26778
\(834\) 0.653234 0.0226197
\(835\) 38.8843 1.34565
\(836\) −5.02636 −0.173840
\(837\) 9.54506 0.329926
\(838\) −23.2252 −0.802302
\(839\) 9.30463 0.321231 0.160616 0.987017i \(-0.448652\pi\)
0.160616 + 0.987017i \(0.448652\pi\)
\(840\) 1.02367 0.0353199
\(841\) −18.9671 −0.654038
\(842\) 5.67045 0.195416
\(843\) 16.1100 0.554857
\(844\) −25.9291 −0.892517
\(845\) −9.44662 −0.324974
\(846\) 4.86584 0.167291
\(847\) −8.63834 −0.296817
\(848\) 1.00000 0.0343401
\(849\) 8.17754 0.280652
\(850\) 11.8195 0.405404
\(851\) 70.9155 2.43095
\(852\) 6.43229 0.220367
\(853\) −21.6029 −0.739668 −0.369834 0.929098i \(-0.620586\pi\)
−0.369834 + 0.929098i \(0.620586\pi\)
\(854\) −5.58975 −0.191277
\(855\) −1.69036 −0.0578090
\(856\) 2.85147 0.0974611
\(857\) −38.0281 −1.29901 −0.649507 0.760356i \(-0.725025\pi\)
−0.649507 + 0.760356i \(0.725025\pi\)
\(858\) 13.6838 0.467156
\(859\) 3.42251 0.116774 0.0583872 0.998294i \(-0.481404\pi\)
0.0583872 + 0.998294i \(0.481404\pi\)
\(860\) −6.45580 −0.220141
\(861\) 0.239925 0.00817663
\(862\) −15.2246 −0.518551
\(863\) 16.8323 0.572978 0.286489 0.958083i \(-0.407512\pi\)
0.286489 + 0.958083i \(0.407512\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −27.6580 −0.940401
\(866\) 6.37413 0.216602
\(867\) −13.4282 −0.456044
\(868\) 5.78042 0.196200
\(869\) −7.28873 −0.247253
\(870\) −5.35417 −0.181523
\(871\) 28.9518 0.980996
\(872\) 4.79123 0.162251
\(873\) −6.47681 −0.219207
\(874\) 7.58679 0.256627
\(875\) 7.31175 0.247182
\(876\) 11.5216 0.389278
\(877\) 46.1626 1.55880 0.779400 0.626527i \(-0.215524\pi\)
0.779400 + 0.626527i \(0.215524\pi\)
\(878\) −5.06870 −0.171060
\(879\) −16.7634 −0.565415
\(880\) 8.49634 0.286412
\(881\) −11.0089 −0.370899 −0.185449 0.982654i \(-0.559374\pi\)
−0.185449 + 0.982654i \(0.559374\pi\)
\(882\) −6.63326 −0.223353
\(883\) 5.16240 0.173729 0.0868644 0.996220i \(-0.472315\pi\)
0.0868644 + 0.996220i \(0.472315\pi\)
\(884\) 15.0172 0.505084
\(885\) 2.26180 0.0760297
\(886\) −22.9923 −0.772442
\(887\) 28.7966 0.966896 0.483448 0.875373i \(-0.339384\pi\)
0.483448 + 0.875373i \(0.339384\pi\)
\(888\) 9.34723 0.313672
\(889\) −9.65074 −0.323675
\(890\) 19.4829 0.653069
\(891\) 5.02636 0.168389
\(892\) 25.3006 0.847126
\(893\) −4.86584 −0.162829
\(894\) −22.9327 −0.766984
\(895\) 2.00950 0.0671702
\(896\) −0.605593 −0.0202314
\(897\) −20.6543 −0.689627
\(898\) 6.29872 0.210191
\(899\) −30.2338 −1.00835
\(900\) −2.14269 −0.0714230
\(901\) −5.51617 −0.183770
\(902\) 1.99135 0.0663048
\(903\) −2.31288 −0.0769677
\(904\) 7.20098 0.239501
\(905\) −17.3546 −0.576887
\(906\) −19.3661 −0.643396
\(907\) 34.1769 1.13483 0.567413 0.823433i \(-0.307944\pi\)
0.567413 + 0.823433i \(0.307944\pi\)
\(908\) −8.58401 −0.284870
\(909\) 3.24349 0.107580
\(910\) 2.78684 0.0923827
\(911\) −23.1219 −0.766061 −0.383031 0.923736i \(-0.625120\pi\)
−0.383031 + 0.923736i \(0.625120\pi\)
\(912\) 1.00000 0.0331133
\(913\) −59.0578 −1.95453
\(914\) 3.56508 0.117922
\(915\) −15.6023 −0.515798
\(916\) −23.8247 −0.787190
\(917\) −8.80360 −0.290721
\(918\) 5.51617 0.182061
\(919\) −19.1746 −0.632512 −0.316256 0.948674i \(-0.602426\pi\)
−0.316256 + 0.948674i \(0.602426\pi\)
\(920\) −12.8244 −0.422808
\(921\) −0.494661 −0.0162996
\(922\) −19.6891 −0.648426
\(923\) 17.5113 0.576391
\(924\) 3.04393 0.100138
\(925\) 20.0282 0.658524
\(926\) −19.2726 −0.633336
\(927\) 20.1540 0.661944
\(928\) 3.16748 0.103977
\(929\) 53.5219 1.75600 0.877998 0.478665i \(-0.158879\pi\)
0.877998 + 0.478665i \(0.158879\pi\)
\(930\) 16.1346 0.529073
\(931\) 6.63326 0.217396
\(932\) 20.2870 0.664521
\(933\) −0.669217 −0.0219092
\(934\) −9.21783 −0.301617
\(935\) −46.8673 −1.53272
\(936\) −2.72240 −0.0889845
\(937\) 30.7200 1.00358 0.501789 0.864990i \(-0.332675\pi\)
0.501789 + 0.864990i \(0.332675\pi\)
\(938\) 6.44028 0.210283
\(939\) 21.8402 0.712727
\(940\) 8.22501 0.268270
\(941\) −44.0264 −1.43522 −0.717610 0.696445i \(-0.754764\pi\)
−0.717610 + 0.696445i \(0.754764\pi\)
\(942\) −0.0490296 −0.00159747
\(943\) −3.00575 −0.0978807
\(944\) −1.33806 −0.0435502
\(945\) 1.02367 0.0332999
\(946\) −19.1966 −0.624136
\(947\) −14.3997 −0.467928 −0.233964 0.972245i \(-0.575170\pi\)
−0.233964 + 0.972245i \(0.575170\pi\)
\(948\) 1.45010 0.0470971
\(949\) 31.3664 1.01820
\(950\) 2.14269 0.0695181
\(951\) 12.3228 0.399593
\(952\) 3.34056 0.108268
\(953\) −9.44941 −0.306096 −0.153048 0.988219i \(-0.548909\pi\)
−0.153048 + 0.988219i \(0.548909\pi\)
\(954\) 1.00000 0.0323762
\(955\) 5.13878 0.166287
\(956\) 7.70613 0.249234
\(957\) −15.9209 −0.514649
\(958\) 26.8437 0.867282
\(959\) 9.48912 0.306420
\(960\) −1.69036 −0.0545561
\(961\) 60.1082 1.93897
\(962\) 25.4469 0.820441
\(963\) 2.85147 0.0918872
\(964\) 11.7300 0.377798
\(965\) −9.59823 −0.308978
\(966\) −4.59451 −0.147826
\(967\) −18.6951 −0.601194 −0.300597 0.953751i \(-0.597186\pi\)
−0.300597 + 0.953751i \(0.597186\pi\)
\(968\) 14.2643 0.458471
\(969\) −5.51617 −0.177205
\(970\) −10.9481 −0.351523
\(971\) 34.4053 1.10412 0.552060 0.833805i \(-0.313842\pi\)
0.552060 + 0.833805i \(0.313842\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0.395594 0.0126822
\(974\) −13.0977 −0.419678
\(975\) −5.83326 −0.186814
\(976\) 9.23020 0.295452
\(977\) −5.03680 −0.161142 −0.0805708 0.996749i \(-0.525674\pi\)
−0.0805708 + 0.996749i \(0.525674\pi\)
\(978\) 18.0273 0.576450
\(979\) 57.9334 1.85156
\(980\) −11.2126 −0.358173
\(981\) 4.79123 0.152972
\(982\) 33.6171 1.07277
\(983\) −16.7963 −0.535718 −0.267859 0.963458i \(-0.586316\pi\)
−0.267859 + 0.963458i \(0.586316\pi\)
\(984\) −0.396182 −0.0126298
\(985\) −13.0770 −0.416669
\(986\) −17.4723 −0.556433
\(987\) 2.94672 0.0937951
\(988\) 2.72240 0.0866111
\(989\) 28.9754 0.921365
\(990\) 8.49634 0.270031
\(991\) 55.7358 1.77051 0.885254 0.465109i \(-0.153985\pi\)
0.885254 + 0.465109i \(0.153985\pi\)
\(992\) −9.54506 −0.303056
\(993\) −15.9874 −0.507344
\(994\) 3.89535 0.123553
\(995\) −45.0114 −1.42696
\(996\) 11.7496 0.372301
\(997\) −37.5350 −1.18874 −0.594372 0.804190i \(-0.702599\pi\)
−0.594372 + 0.804190i \(0.702599\pi\)
\(998\) 9.85886 0.312077
\(999\) 9.34723 0.295733
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 6042.2.a.bb.1.8 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6042.2.a.bb.1.8 9 1.1 even 1 trivial