Properties

Label 4029.2.a.e.1.11
Level $4029$
Weight $2$
Character 4029.1
Self dual yes
Analytic conductor $32.172$
Analytic rank $1$
Dimension $18$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4029,2,Mod(1,4029)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4029, 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("4029.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4029 = 3 \cdot 17 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4029.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: \(32.1717269744\)
Analytic rank: \(1\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 6 x^{17} - 10 x^{16} + 120 x^{15} - 56 x^{14} - 921 x^{13} + 1181 x^{12} + 3316 x^{11} - 6280 x^{10} - 5249 x^{9} + 15005 x^{8} + 1809 x^{7} - 16711 x^{6} + 2434 x^{5} + \cdots + 138 \) 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 1.11
Root \(-0.239152\) of defining polynomial
Character \(\chi\) \(=\) 4029.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.239152 q^{2} +1.00000 q^{3} -1.94281 q^{4} -0.863011 q^{5} +0.239152 q^{6} -3.18045 q^{7} -0.942930 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.239152 q^{2} +1.00000 q^{3} -1.94281 q^{4} -0.863011 q^{5} +0.239152 q^{6} -3.18045 q^{7} -0.942930 q^{8} +1.00000 q^{9} -0.206391 q^{10} +2.43892 q^{11} -1.94281 q^{12} +2.64738 q^{13} -0.760610 q^{14} -0.863011 q^{15} +3.66011 q^{16} +1.00000 q^{17} +0.239152 q^{18} -5.74248 q^{19} +1.67666 q^{20} -3.18045 q^{21} +0.583272 q^{22} +5.49541 q^{23} -0.942930 q^{24} -4.25521 q^{25} +0.633127 q^{26} +1.00000 q^{27} +6.17900 q^{28} +1.07170 q^{29} -0.206391 q^{30} +1.96792 q^{31} +2.76118 q^{32} +2.43892 q^{33} +0.239152 q^{34} +2.74476 q^{35} -1.94281 q^{36} +2.75352 q^{37} -1.37333 q^{38} +2.64738 q^{39} +0.813758 q^{40} +7.09994 q^{41} -0.760610 q^{42} -10.2572 q^{43} -4.73835 q^{44} -0.863011 q^{45} +1.31424 q^{46} -3.04303 q^{47} +3.66011 q^{48} +3.11526 q^{49} -1.01764 q^{50} +1.00000 q^{51} -5.14335 q^{52} +0.934787 q^{53} +0.239152 q^{54} -2.10481 q^{55} +2.99894 q^{56} -5.74248 q^{57} +0.256300 q^{58} -2.02087 q^{59} +1.67666 q^{60} -6.53682 q^{61} +0.470632 q^{62} -3.18045 q^{63} -6.65988 q^{64} -2.28472 q^{65} +0.583272 q^{66} -7.65642 q^{67} -1.94281 q^{68} +5.49541 q^{69} +0.656415 q^{70} -13.3580 q^{71} -0.942930 q^{72} +9.05559 q^{73} +0.658509 q^{74} -4.25521 q^{75} +11.1565 q^{76} -7.75686 q^{77} +0.633127 q^{78} -1.00000 q^{79} -3.15871 q^{80} +1.00000 q^{81} +1.69796 q^{82} -3.01545 q^{83} +6.17900 q^{84} -0.863011 q^{85} -2.45303 q^{86} +1.07170 q^{87} -2.29973 q^{88} +1.39929 q^{89} -0.206391 q^{90} -8.41987 q^{91} -10.6765 q^{92} +1.96792 q^{93} -0.727747 q^{94} +4.95583 q^{95} +2.76118 q^{96} +11.3288 q^{97} +0.745020 q^{98} +2.43892 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 6 q^{2} + 18 q^{3} + 20 q^{4} - 5 q^{5} - 6 q^{6} - 13 q^{7} - 12 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 6 q^{2} + 18 q^{3} + 20 q^{4} - 5 q^{5} - 6 q^{6} - 13 q^{7} - 12 q^{8} + 18 q^{9} - 15 q^{10} - 27 q^{11} + 20 q^{12} - 4 q^{13} - 5 q^{14} - 5 q^{15} + 16 q^{16} + 18 q^{17} - 6 q^{18} - 30 q^{19} - 16 q^{20} - 13 q^{21} + 13 q^{22} - 21 q^{23} - 12 q^{24} + 13 q^{25} - 20 q^{26} + 18 q^{27} - 33 q^{28} - 47 q^{29} - 15 q^{30} - 18 q^{31} - 45 q^{32} - 27 q^{33} - 6 q^{34} - 17 q^{35} + 20 q^{36} + q^{37} + 5 q^{38} - 4 q^{39} - 12 q^{40} - 18 q^{41} - 5 q^{42} - 39 q^{43} - 34 q^{44} - 5 q^{45} - 7 q^{46} + 16 q^{48} + 15 q^{49} - 23 q^{50} + 18 q^{51} + 5 q^{52} - 9 q^{53} - 6 q^{54} + q^{55} - 24 q^{56} - 30 q^{57} + 41 q^{58} - 42 q^{59} - 16 q^{60} - 43 q^{61} - 54 q^{62} - 13 q^{63} + 22 q^{64} - 25 q^{65} + 13 q^{66} + 20 q^{68} - 21 q^{69} + 17 q^{70} + 9 q^{71} - 12 q^{72} + 19 q^{73} - 30 q^{74} + 13 q^{75} - 17 q^{76} - 14 q^{77} - 20 q^{78} - 18 q^{79} + 36 q^{80} + 18 q^{81} - 3 q^{82} - 61 q^{83} - 33 q^{84} - 5 q^{85} - 24 q^{86} - 47 q^{87} - 25 q^{88} + 10 q^{89} - 15 q^{90} - 52 q^{91} - 74 q^{92} - 18 q^{93} + 31 q^{94} - 37 q^{95} - 45 q^{96} - 9 q^{97} + 27 q^{98} - 27 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.239152 0.169106 0.0845530 0.996419i \(-0.473054\pi\)
0.0845530 + 0.996419i \(0.473054\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.94281 −0.971403
\(5\) −0.863011 −0.385950 −0.192975 0.981204i \(-0.561814\pi\)
−0.192975 + 0.981204i \(0.561814\pi\)
\(6\) 0.239152 0.0976333
\(7\) −3.18045 −1.20210 −0.601048 0.799213i \(-0.705250\pi\)
−0.601048 + 0.799213i \(0.705250\pi\)
\(8\) −0.942930 −0.333376
\(9\) 1.00000 0.333333
\(10\) −0.206391 −0.0652665
\(11\) 2.43892 0.735362 0.367681 0.929952i \(-0.380152\pi\)
0.367681 + 0.929952i \(0.380152\pi\)
\(12\) −1.94281 −0.560840
\(13\) 2.64738 0.734252 0.367126 0.930171i \(-0.380342\pi\)
0.367126 + 0.930171i \(0.380342\pi\)
\(14\) −0.760610 −0.203282
\(15\) −0.863011 −0.222828
\(16\) 3.66011 0.915027
\(17\) 1.00000 0.242536
\(18\) 0.239152 0.0563686
\(19\) −5.74248 −1.31742 −0.658708 0.752399i \(-0.728897\pi\)
−0.658708 + 0.752399i \(0.728897\pi\)
\(20\) 1.67666 0.374913
\(21\) −3.18045 −0.694031
\(22\) 0.583272 0.124354
\(23\) 5.49541 1.14587 0.572936 0.819600i \(-0.305805\pi\)
0.572936 + 0.819600i \(0.305805\pi\)
\(24\) −0.942930 −0.192475
\(25\) −4.25521 −0.851042
\(26\) 0.633127 0.124166
\(27\) 1.00000 0.192450
\(28\) 6.17900 1.16772
\(29\) 1.07170 0.199010 0.0995052 0.995037i \(-0.468274\pi\)
0.0995052 + 0.995037i \(0.468274\pi\)
\(30\) −0.206391 −0.0376816
\(31\) 1.96792 0.353449 0.176724 0.984260i \(-0.443450\pi\)
0.176724 + 0.984260i \(0.443450\pi\)
\(32\) 2.76118 0.488112
\(33\) 2.43892 0.424561
\(34\) 0.239152 0.0410142
\(35\) 2.74476 0.463950
\(36\) −1.94281 −0.323801
\(37\) 2.75352 0.452675 0.226338 0.974049i \(-0.427325\pi\)
0.226338 + 0.974049i \(0.427325\pi\)
\(38\) −1.37333 −0.222783
\(39\) 2.64738 0.423921
\(40\) 0.813758 0.128667
\(41\) 7.09994 1.10882 0.554412 0.832242i \(-0.312943\pi\)
0.554412 + 0.832242i \(0.312943\pi\)
\(42\) −0.760610 −0.117365
\(43\) −10.2572 −1.56421 −0.782105 0.623146i \(-0.785854\pi\)
−0.782105 + 0.623146i \(0.785854\pi\)
\(44\) −4.73835 −0.714333
\(45\) −0.863011 −0.128650
\(46\) 1.31424 0.193774
\(47\) −3.04303 −0.443872 −0.221936 0.975061i \(-0.571238\pi\)
−0.221936 + 0.975061i \(0.571238\pi\)
\(48\) 3.66011 0.528291
\(49\) 3.11526 0.445037
\(50\) −1.01764 −0.143916
\(51\) 1.00000 0.140028
\(52\) −5.14335 −0.713255
\(53\) 0.934787 0.128403 0.0642015 0.997937i \(-0.479550\pi\)
0.0642015 + 0.997937i \(0.479550\pi\)
\(54\) 0.239152 0.0325444
\(55\) −2.10481 −0.283813
\(56\) 2.99894 0.400750
\(57\) −5.74248 −0.760611
\(58\) 0.256300 0.0336538
\(59\) −2.02087 −0.263095 −0.131547 0.991310i \(-0.541995\pi\)
−0.131547 + 0.991310i \(0.541995\pi\)
\(60\) 1.67666 0.216456
\(61\) −6.53682 −0.836955 −0.418477 0.908227i \(-0.637436\pi\)
−0.418477 + 0.908227i \(0.637436\pi\)
\(62\) 0.470632 0.0597703
\(63\) −3.18045 −0.400699
\(64\) −6.65988 −0.832485
\(65\) −2.28472 −0.283385
\(66\) 0.583272 0.0717959
\(67\) −7.65642 −0.935380 −0.467690 0.883892i \(-0.654914\pi\)
−0.467690 + 0.883892i \(0.654914\pi\)
\(68\) −1.94281 −0.235600
\(69\) 5.49541 0.661569
\(70\) 0.656415 0.0784566
\(71\) −13.3580 −1.58530 −0.792652 0.609675i \(-0.791300\pi\)
−0.792652 + 0.609675i \(0.791300\pi\)
\(72\) −0.942930 −0.111125
\(73\) 9.05559 1.05988 0.529938 0.848036i \(-0.322215\pi\)
0.529938 + 0.848036i \(0.322215\pi\)
\(74\) 0.658509 0.0765501
\(75\) −4.25521 −0.491350
\(76\) 11.1565 1.27974
\(77\) −7.75686 −0.883976
\(78\) 0.633127 0.0716875
\(79\) −1.00000 −0.112509
\(80\) −3.15871 −0.353155
\(81\) 1.00000 0.111111
\(82\) 1.69796 0.187509
\(83\) −3.01545 −0.330988 −0.165494 0.986211i \(-0.552922\pi\)
−0.165494 + 0.986211i \(0.552922\pi\)
\(84\) 6.17900 0.674184
\(85\) −0.863011 −0.0936067
\(86\) −2.45303 −0.264517
\(87\) 1.07170 0.114899
\(88\) −2.29973 −0.245152
\(89\) 1.39929 0.148325 0.0741623 0.997246i \(-0.476372\pi\)
0.0741623 + 0.997246i \(0.476372\pi\)
\(90\) −0.206391 −0.0217555
\(91\) −8.41987 −0.882642
\(92\) −10.6765 −1.11310
\(93\) 1.96792 0.204064
\(94\) −0.727747 −0.0750614
\(95\) 4.95583 0.508457
\(96\) 2.76118 0.281812
\(97\) 11.3288 1.15027 0.575134 0.818059i \(-0.304950\pi\)
0.575134 + 0.818059i \(0.304950\pi\)
\(98\) 0.745020 0.0752584
\(99\) 2.43892 0.245121
\(100\) 8.26705 0.826705
\(101\) −10.4012 −1.03496 −0.517480 0.855695i \(-0.673130\pi\)
−0.517480 + 0.855695i \(0.673130\pi\)
\(102\) 0.239152 0.0236796
\(103\) −10.8386 −1.06796 −0.533979 0.845498i \(-0.679304\pi\)
−0.533979 + 0.845498i \(0.679304\pi\)
\(104\) −2.49630 −0.244782
\(105\) 2.74476 0.267861
\(106\) 0.223556 0.0217137
\(107\) 2.62365 0.253638 0.126819 0.991926i \(-0.459523\pi\)
0.126819 + 0.991926i \(0.459523\pi\)
\(108\) −1.94281 −0.186947
\(109\) −3.82176 −0.366058 −0.183029 0.983108i \(-0.558590\pi\)
−0.183029 + 0.983108i \(0.558590\pi\)
\(110\) −0.503370 −0.0479945
\(111\) 2.75352 0.261352
\(112\) −11.6408 −1.09995
\(113\) 6.14277 0.577864 0.288932 0.957350i \(-0.406700\pi\)
0.288932 + 0.957350i \(0.406700\pi\)
\(114\) −1.37333 −0.128624
\(115\) −4.74259 −0.442249
\(116\) −2.08211 −0.193319
\(117\) 2.64738 0.244751
\(118\) −0.483295 −0.0444909
\(119\) −3.18045 −0.291551
\(120\) 0.813758 0.0742856
\(121\) −5.05167 −0.459243
\(122\) −1.56329 −0.141534
\(123\) 7.09994 0.640180
\(124\) −3.82329 −0.343341
\(125\) 7.98735 0.714410
\(126\) −0.760610 −0.0677606
\(127\) −16.8085 −1.49151 −0.745754 0.666221i \(-0.767911\pi\)
−0.745754 + 0.666221i \(0.767911\pi\)
\(128\) −7.11508 −0.628891
\(129\) −10.2572 −0.903097
\(130\) −0.546395 −0.0479220
\(131\) 18.0054 1.57314 0.786570 0.617502i \(-0.211855\pi\)
0.786570 + 0.617502i \(0.211855\pi\)
\(132\) −4.73835 −0.412420
\(133\) 18.2637 1.58366
\(134\) −1.83105 −0.158178
\(135\) −0.863011 −0.0742762
\(136\) −0.942930 −0.0808555
\(137\) −12.8425 −1.09721 −0.548603 0.836083i \(-0.684840\pi\)
−0.548603 + 0.836083i \(0.684840\pi\)
\(138\) 1.31424 0.111875
\(139\) −13.7287 −1.16445 −0.582226 0.813027i \(-0.697818\pi\)
−0.582226 + 0.813027i \(0.697818\pi\)
\(140\) −5.33254 −0.450682
\(141\) −3.04303 −0.256270
\(142\) −3.19459 −0.268084
\(143\) 6.45676 0.539941
\(144\) 3.66011 0.305009
\(145\) −0.924892 −0.0768081
\(146\) 2.16566 0.179231
\(147\) 3.11526 0.256942
\(148\) −5.34955 −0.439730
\(149\) −14.2939 −1.17100 −0.585499 0.810673i \(-0.699102\pi\)
−0.585499 + 0.810673i \(0.699102\pi\)
\(150\) −1.01764 −0.0830901
\(151\) 2.98397 0.242832 0.121416 0.992602i \(-0.461257\pi\)
0.121416 + 0.992602i \(0.461257\pi\)
\(152\) 5.41476 0.439195
\(153\) 1.00000 0.0808452
\(154\) −1.85507 −0.149486
\(155\) −1.69834 −0.136414
\(156\) −5.14335 −0.411798
\(157\) −20.5319 −1.63862 −0.819310 0.573350i \(-0.805643\pi\)
−0.819310 + 0.573350i \(0.805643\pi\)
\(158\) −0.239152 −0.0190259
\(159\) 0.934787 0.0741335
\(160\) −2.38293 −0.188387
\(161\) −17.4779 −1.37745
\(162\) 0.239152 0.0187895
\(163\) −19.9448 −1.56220 −0.781099 0.624407i \(-0.785341\pi\)
−0.781099 + 0.624407i \(0.785341\pi\)
\(164\) −13.7938 −1.07712
\(165\) −2.10481 −0.163860
\(166\) −0.721149 −0.0559721
\(167\) 3.27693 0.253576 0.126788 0.991930i \(-0.459533\pi\)
0.126788 + 0.991930i \(0.459533\pi\)
\(168\) 2.99894 0.231373
\(169\) −5.99136 −0.460874
\(170\) −0.206391 −0.0158294
\(171\) −5.74248 −0.439139
\(172\) 19.9278 1.51948
\(173\) −12.0891 −0.919116 −0.459558 0.888148i \(-0.651992\pi\)
−0.459558 + 0.888148i \(0.651992\pi\)
\(174\) 0.256300 0.0194300
\(175\) 13.5335 1.02304
\(176\) 8.92671 0.672876
\(177\) −2.02087 −0.151898
\(178\) 0.334643 0.0250826
\(179\) 6.13052 0.458217 0.229108 0.973401i \(-0.426419\pi\)
0.229108 + 0.973401i \(0.426419\pi\)
\(180\) 1.67666 0.124971
\(181\) 0.327510 0.0243437 0.0121718 0.999926i \(-0.496125\pi\)
0.0121718 + 0.999926i \(0.496125\pi\)
\(182\) −2.01363 −0.149260
\(183\) −6.53682 −0.483216
\(184\) −5.18178 −0.382006
\(185\) −2.37632 −0.174710
\(186\) 0.470632 0.0345084
\(187\) 2.43892 0.178351
\(188\) 5.91202 0.431179
\(189\) −3.18045 −0.231344
\(190\) 1.18520 0.0859831
\(191\) −1.51939 −0.109939 −0.0549695 0.998488i \(-0.517506\pi\)
−0.0549695 + 0.998488i \(0.517506\pi\)
\(192\) −6.65988 −0.480635
\(193\) −3.54344 −0.255062 −0.127531 0.991835i \(-0.540705\pi\)
−0.127531 + 0.991835i \(0.540705\pi\)
\(194\) 2.70931 0.194517
\(195\) −2.28472 −0.163612
\(196\) −6.05234 −0.432310
\(197\) −22.7150 −1.61838 −0.809188 0.587549i \(-0.800093\pi\)
−0.809188 + 0.587549i \(0.800093\pi\)
\(198\) 0.583272 0.0414514
\(199\) 8.31730 0.589597 0.294799 0.955559i \(-0.404747\pi\)
0.294799 + 0.955559i \(0.404747\pi\)
\(200\) 4.01237 0.283717
\(201\) −7.65642 −0.540042
\(202\) −2.48747 −0.175018
\(203\) −3.40850 −0.239230
\(204\) −1.94281 −0.136024
\(205\) −6.12733 −0.427951
\(206\) −2.59207 −0.180598
\(207\) 5.49541 0.381957
\(208\) 9.68971 0.671861
\(209\) −14.0055 −0.968778
\(210\) 0.656415 0.0452969
\(211\) −8.09013 −0.556948 −0.278474 0.960444i \(-0.589829\pi\)
−0.278474 + 0.960444i \(0.589829\pi\)
\(212\) −1.81611 −0.124731
\(213\) −13.3580 −0.915275
\(214\) 0.627451 0.0428917
\(215\) 8.85209 0.603707
\(216\) −0.942930 −0.0641582
\(217\) −6.25887 −0.424880
\(218\) −0.913980 −0.0619026
\(219\) 9.05559 0.611920
\(220\) 4.08925 0.275697
\(221\) 2.64738 0.178082
\(222\) 0.658509 0.0441962
\(223\) 13.9274 0.932646 0.466323 0.884614i \(-0.345578\pi\)
0.466323 + 0.884614i \(0.345578\pi\)
\(224\) −8.78180 −0.586758
\(225\) −4.25521 −0.283681
\(226\) 1.46906 0.0977201
\(227\) −16.9103 −1.12238 −0.561189 0.827688i \(-0.689656\pi\)
−0.561189 + 0.827688i \(0.689656\pi\)
\(228\) 11.1565 0.738860
\(229\) 9.20668 0.608395 0.304197 0.952609i \(-0.401612\pi\)
0.304197 + 0.952609i \(0.401612\pi\)
\(230\) −1.13420 −0.0747870
\(231\) −7.75686 −0.510364
\(232\) −1.01054 −0.0663452
\(233\) 5.70441 0.373709 0.186854 0.982388i \(-0.440171\pi\)
0.186854 + 0.982388i \(0.440171\pi\)
\(234\) 0.633127 0.0413888
\(235\) 2.62617 0.171312
\(236\) 3.92616 0.255571
\(237\) −1.00000 −0.0649570
\(238\) −0.760610 −0.0493030
\(239\) 8.86534 0.573451 0.286725 0.958013i \(-0.407433\pi\)
0.286725 + 0.958013i \(0.407433\pi\)
\(240\) −3.15871 −0.203894
\(241\) −22.9479 −1.47821 −0.739104 0.673592i \(-0.764751\pi\)
−0.739104 + 0.673592i \(0.764751\pi\)
\(242\) −1.20812 −0.0776606
\(243\) 1.00000 0.0641500
\(244\) 12.6998 0.813020
\(245\) −2.68850 −0.171762
\(246\) 1.69796 0.108258
\(247\) −15.2026 −0.967316
\(248\) −1.85561 −0.117831
\(249\) −3.01545 −0.191096
\(250\) 1.91019 0.120811
\(251\) −19.6612 −1.24100 −0.620501 0.784206i \(-0.713071\pi\)
−0.620501 + 0.784206i \(0.713071\pi\)
\(252\) 6.17900 0.389240
\(253\) 13.4029 0.842630
\(254\) −4.01977 −0.252223
\(255\) −0.863011 −0.0540438
\(256\) 11.6182 0.726136
\(257\) 6.41479 0.400144 0.200072 0.979781i \(-0.435882\pi\)
0.200072 + 0.979781i \(0.435882\pi\)
\(258\) −2.45303 −0.152719
\(259\) −8.75742 −0.544160
\(260\) 4.43877 0.275281
\(261\) 1.07170 0.0663368
\(262\) 4.30603 0.266027
\(263\) −0.0287981 −0.00177577 −0.000887883 1.00000i \(-0.500283\pi\)
−0.000887883 1.00000i \(0.500283\pi\)
\(264\) −2.29973 −0.141539
\(265\) −0.806732 −0.0495571
\(266\) 4.36779 0.267807
\(267\) 1.39929 0.0856352
\(268\) 14.8749 0.908632
\(269\) 21.0821 1.28540 0.642699 0.766119i \(-0.277815\pi\)
0.642699 + 0.766119i \(0.277815\pi\)
\(270\) −0.206391 −0.0125605
\(271\) 29.6658 1.80207 0.901034 0.433749i \(-0.142809\pi\)
0.901034 + 0.433749i \(0.142809\pi\)
\(272\) 3.66011 0.221927
\(273\) −8.41987 −0.509594
\(274\) −3.07130 −0.185544
\(275\) −10.3781 −0.625824
\(276\) −10.6765 −0.642650
\(277\) 13.6618 0.820856 0.410428 0.911893i \(-0.365379\pi\)
0.410428 + 0.911893i \(0.365379\pi\)
\(278\) −3.28324 −0.196916
\(279\) 1.96792 0.117816
\(280\) −2.58812 −0.154670
\(281\) 14.8601 0.886480 0.443240 0.896403i \(-0.353829\pi\)
0.443240 + 0.896403i \(0.353829\pi\)
\(282\) −0.727747 −0.0433367
\(283\) 1.34731 0.0800895 0.0400448 0.999198i \(-0.487250\pi\)
0.0400448 + 0.999198i \(0.487250\pi\)
\(284\) 25.9520 1.53997
\(285\) 4.95583 0.293558
\(286\) 1.54415 0.0913072
\(287\) −22.5810 −1.33291
\(288\) 2.76118 0.162704
\(289\) 1.00000 0.0588235
\(290\) −0.221190 −0.0129887
\(291\) 11.3288 0.664108
\(292\) −17.5933 −1.02957
\(293\) 22.8657 1.33583 0.667915 0.744237i \(-0.267187\pi\)
0.667915 + 0.744237i \(0.267187\pi\)
\(294\) 0.745020 0.0434504
\(295\) 1.74403 0.101542
\(296\) −2.59637 −0.150911
\(297\) 2.43892 0.141520
\(298\) −3.41840 −0.198023
\(299\) 14.5484 0.841358
\(300\) 8.26705 0.477299
\(301\) 32.6225 1.88033
\(302\) 0.713621 0.0410643
\(303\) −10.4012 −0.597534
\(304\) −21.0181 −1.20547
\(305\) 5.64135 0.323023
\(306\) 0.239152 0.0136714
\(307\) −31.9557 −1.82381 −0.911905 0.410402i \(-0.865388\pi\)
−0.911905 + 0.410402i \(0.865388\pi\)
\(308\) 15.0701 0.858697
\(309\) −10.8386 −0.616586
\(310\) −0.406160 −0.0230684
\(311\) 0.969169 0.0549565 0.0274783 0.999622i \(-0.491252\pi\)
0.0274783 + 0.999622i \(0.491252\pi\)
\(312\) −2.49630 −0.141325
\(313\) −11.2022 −0.633183 −0.316592 0.948562i \(-0.602538\pi\)
−0.316592 + 0.948562i \(0.602538\pi\)
\(314\) −4.91023 −0.277100
\(315\) 2.74476 0.154650
\(316\) 1.94281 0.109291
\(317\) 4.44816 0.249833 0.124917 0.992167i \(-0.460134\pi\)
0.124917 + 0.992167i \(0.460134\pi\)
\(318\) 0.223556 0.0125364
\(319\) 2.61380 0.146345
\(320\) 5.74755 0.321298
\(321\) 2.62365 0.146438
\(322\) −4.17986 −0.232935
\(323\) −5.74248 −0.319520
\(324\) −1.94281 −0.107934
\(325\) −11.2652 −0.624880
\(326\) −4.76984 −0.264177
\(327\) −3.82176 −0.211344
\(328\) −6.69475 −0.369655
\(329\) 9.67821 0.533577
\(330\) −0.503370 −0.0277096
\(331\) 26.2905 1.44506 0.722530 0.691340i \(-0.242979\pi\)
0.722530 + 0.691340i \(0.242979\pi\)
\(332\) 5.85843 0.321523
\(333\) 2.75352 0.150892
\(334\) 0.783683 0.0428812
\(335\) 6.60757 0.361010
\(336\) −11.6408 −0.635057
\(337\) 8.13050 0.442896 0.221448 0.975172i \(-0.428922\pi\)
0.221448 + 0.975172i \(0.428922\pi\)
\(338\) −1.43285 −0.0779365
\(339\) 6.14277 0.333630
\(340\) 1.67666 0.0909298
\(341\) 4.79960 0.259913
\(342\) −1.37333 −0.0742610
\(343\) 12.3552 0.667119
\(344\) 9.67183 0.521470
\(345\) −4.74259 −0.255333
\(346\) −2.89113 −0.155428
\(347\) −35.3063 −1.89534 −0.947672 0.319247i \(-0.896570\pi\)
−0.947672 + 0.319247i \(0.896570\pi\)
\(348\) −2.08211 −0.111613
\(349\) 32.4168 1.73523 0.867615 0.497237i \(-0.165652\pi\)
0.867615 + 0.497237i \(0.165652\pi\)
\(350\) 3.23656 0.173001
\(351\) 2.64738 0.141307
\(352\) 6.73430 0.358939
\(353\) 14.7374 0.784395 0.392198 0.919881i \(-0.371715\pi\)
0.392198 + 0.919881i \(0.371715\pi\)
\(354\) −0.483295 −0.0256868
\(355\) 11.5281 0.611848
\(356\) −2.71855 −0.144083
\(357\) −3.18045 −0.168327
\(358\) 1.46613 0.0774872
\(359\) −28.0424 −1.48002 −0.740011 0.672595i \(-0.765180\pi\)
−0.740011 + 0.672595i \(0.765180\pi\)
\(360\) 0.813758 0.0428888
\(361\) 13.9761 0.735586
\(362\) 0.0783247 0.00411666
\(363\) −5.05167 −0.265144
\(364\) 16.3582 0.857401
\(365\) −7.81507 −0.409060
\(366\) −1.56329 −0.0817147
\(367\) 7.99865 0.417526 0.208763 0.977966i \(-0.433056\pi\)
0.208763 + 0.977966i \(0.433056\pi\)
\(368\) 20.1138 1.04850
\(369\) 7.09994 0.369608
\(370\) −0.568300 −0.0295445
\(371\) −2.97304 −0.154353
\(372\) −3.82329 −0.198228
\(373\) 16.4571 0.852116 0.426058 0.904696i \(-0.359902\pi\)
0.426058 + 0.904696i \(0.359902\pi\)
\(374\) 0.583272 0.0301603
\(375\) 7.98735 0.412465
\(376\) 2.86937 0.147976
\(377\) 2.83721 0.146124
\(378\) −0.760610 −0.0391216
\(379\) −30.8578 −1.58506 −0.792529 0.609834i \(-0.791236\pi\)
−0.792529 + 0.609834i \(0.791236\pi\)
\(380\) −9.62821 −0.493917
\(381\) −16.8085 −0.861123
\(382\) −0.363364 −0.0185913
\(383\) −32.9059 −1.68141 −0.840705 0.541493i \(-0.817859\pi\)
−0.840705 + 0.541493i \(0.817859\pi\)
\(384\) −7.11508 −0.363090
\(385\) 6.69426 0.341171
\(386\) −0.847420 −0.0431325
\(387\) −10.2572 −0.521403
\(388\) −22.0097 −1.11737
\(389\) −8.82733 −0.447563 −0.223782 0.974639i \(-0.571840\pi\)
−0.223782 + 0.974639i \(0.571840\pi\)
\(390\) −0.546395 −0.0276678
\(391\) 5.49541 0.277915
\(392\) −2.93747 −0.148365
\(393\) 18.0054 0.908252
\(394\) −5.43233 −0.273677
\(395\) 0.863011 0.0434228
\(396\) −4.73835 −0.238111
\(397\) −23.1481 −1.16177 −0.580885 0.813985i \(-0.697294\pi\)
−0.580885 + 0.813985i \(0.697294\pi\)
\(398\) 1.98910 0.0997044
\(399\) 18.2637 0.914328
\(400\) −15.5745 −0.778727
\(401\) −3.34254 −0.166919 −0.0834593 0.996511i \(-0.526597\pi\)
−0.0834593 + 0.996511i \(0.526597\pi\)
\(402\) −1.83105 −0.0913243
\(403\) 5.20984 0.259521
\(404\) 20.2075 1.00536
\(405\) −0.863011 −0.0428834
\(406\) −0.815149 −0.0404551
\(407\) 6.71561 0.332880
\(408\) −0.942930 −0.0466820
\(409\) 10.8048 0.534261 0.267130 0.963660i \(-0.413925\pi\)
0.267130 + 0.963660i \(0.413925\pi\)
\(410\) −1.46536 −0.0723691
\(411\) −12.8425 −0.633472
\(412\) 21.0573 1.03742
\(413\) 6.42727 0.316266
\(414\) 1.31424 0.0645912
\(415\) 2.60236 0.127745
\(416\) 7.30990 0.358398
\(417\) −13.7287 −0.672297
\(418\) −3.34943 −0.163826
\(419\) −22.9673 −1.12203 −0.561014 0.827806i \(-0.689589\pi\)
−0.561014 + 0.827806i \(0.689589\pi\)
\(420\) −5.33254 −0.260201
\(421\) −28.1847 −1.37364 −0.686818 0.726829i \(-0.740993\pi\)
−0.686818 + 0.726829i \(0.740993\pi\)
\(422\) −1.93477 −0.0941831
\(423\) −3.04303 −0.147957
\(424\) −0.881439 −0.0428064
\(425\) −4.25521 −0.206408
\(426\) −3.19459 −0.154778
\(427\) 20.7900 1.00610
\(428\) −5.09725 −0.246385
\(429\) 6.45676 0.311735
\(430\) 2.11699 0.102090
\(431\) 0.0906693 0.00436739 0.00218369 0.999998i \(-0.499305\pi\)
0.00218369 + 0.999998i \(0.499305\pi\)
\(432\) 3.66011 0.176097
\(433\) 32.5020 1.56195 0.780973 0.624564i \(-0.214723\pi\)
0.780973 + 0.624564i \(0.214723\pi\)
\(434\) −1.49682 −0.0718497
\(435\) −0.924892 −0.0443452
\(436\) 7.42494 0.355590
\(437\) −31.5573 −1.50959
\(438\) 2.16566 0.103479
\(439\) 11.3014 0.539386 0.269693 0.962946i \(-0.413078\pi\)
0.269693 + 0.962946i \(0.413078\pi\)
\(440\) 1.98469 0.0946165
\(441\) 3.11526 0.148346
\(442\) 0.633127 0.0301148
\(443\) 40.8366 1.94021 0.970103 0.242695i \(-0.0780314\pi\)
0.970103 + 0.242695i \(0.0780314\pi\)
\(444\) −5.34955 −0.253878
\(445\) −1.20760 −0.0572459
\(446\) 3.33076 0.157716
\(447\) −14.2939 −0.676077
\(448\) 21.1814 1.00073
\(449\) 5.87709 0.277357 0.138678 0.990337i \(-0.455715\pi\)
0.138678 + 0.990337i \(0.455715\pi\)
\(450\) −1.01764 −0.0479721
\(451\) 17.3162 0.815388
\(452\) −11.9342 −0.561338
\(453\) 2.98397 0.140199
\(454\) −4.04413 −0.189801
\(455\) 7.26644 0.340656
\(456\) 5.41476 0.253569
\(457\) 35.7045 1.67019 0.835093 0.550108i \(-0.185414\pi\)
0.835093 + 0.550108i \(0.185414\pi\)
\(458\) 2.20180 0.102883
\(459\) 1.00000 0.0466760
\(460\) 9.21394 0.429602
\(461\) −35.7072 −1.66305 −0.831524 0.555488i \(-0.812531\pi\)
−0.831524 + 0.555488i \(0.812531\pi\)
\(462\) −1.85507 −0.0863056
\(463\) 14.5370 0.675593 0.337796 0.941219i \(-0.390319\pi\)
0.337796 + 0.941219i \(0.390319\pi\)
\(464\) 3.92255 0.182100
\(465\) −1.69834 −0.0787585
\(466\) 1.36422 0.0631963
\(467\) −10.0292 −0.464094 −0.232047 0.972705i \(-0.574542\pi\)
−0.232047 + 0.972705i \(0.574542\pi\)
\(468\) −5.14335 −0.237752
\(469\) 24.3509 1.12442
\(470\) 0.628054 0.0289700
\(471\) −20.5319 −0.946058
\(472\) 1.90554 0.0877095
\(473\) −25.0165 −1.15026
\(474\) −0.239152 −0.0109846
\(475\) 24.4355 1.12118
\(476\) 6.17900 0.283214
\(477\) 0.934787 0.0428010
\(478\) 2.12016 0.0969739
\(479\) 5.86372 0.267920 0.133960 0.990987i \(-0.457231\pi\)
0.133960 + 0.990987i \(0.457231\pi\)
\(480\) −2.38293 −0.108765
\(481\) 7.28962 0.332378
\(482\) −5.48804 −0.249974
\(483\) −17.4779 −0.795270
\(484\) 9.81442 0.446110
\(485\) −9.77690 −0.443946
\(486\) 0.239152 0.0108481
\(487\) 7.90286 0.358113 0.179056 0.983839i \(-0.442696\pi\)
0.179056 + 0.983839i \(0.442696\pi\)
\(488\) 6.16376 0.279020
\(489\) −19.9448 −0.901936
\(490\) −0.642960 −0.0290460
\(491\) 35.6953 1.61091 0.805454 0.592659i \(-0.201922\pi\)
0.805454 + 0.592659i \(0.201922\pi\)
\(492\) −13.7938 −0.621873
\(493\) 1.07170 0.0482671
\(494\) −3.63572 −0.163579
\(495\) −2.10481 −0.0946044
\(496\) 7.20280 0.323415
\(497\) 42.4844 1.90569
\(498\) −0.721149 −0.0323155
\(499\) −22.8901 −1.02470 −0.512352 0.858776i \(-0.671226\pi\)
−0.512352 + 0.858776i \(0.671226\pi\)
\(500\) −15.5179 −0.693980
\(501\) 3.27693 0.146402
\(502\) −4.70200 −0.209861
\(503\) −15.5920 −0.695211 −0.347606 0.937641i \(-0.613005\pi\)
−0.347606 + 0.937641i \(0.613005\pi\)
\(504\) 2.99894 0.133583
\(505\) 8.97636 0.399443
\(506\) 3.20532 0.142494
\(507\) −5.99136 −0.266086
\(508\) 32.6556 1.44886
\(509\) −42.0078 −1.86196 −0.930982 0.365064i \(-0.881047\pi\)
−0.930982 + 0.365064i \(0.881047\pi\)
\(510\) −0.206391 −0.00913913
\(511\) −28.8008 −1.27407
\(512\) 17.0087 0.751684
\(513\) −5.74248 −0.253537
\(514\) 1.53411 0.0676666
\(515\) 9.35383 0.412179
\(516\) 19.9278 0.877272
\(517\) −7.42171 −0.326407
\(518\) −2.09435 −0.0920206
\(519\) −12.0891 −0.530652
\(520\) 2.15433 0.0944736
\(521\) −24.8991 −1.09085 −0.545424 0.838160i \(-0.683631\pi\)
−0.545424 + 0.838160i \(0.683631\pi\)
\(522\) 0.256300 0.0112179
\(523\) 13.3217 0.582518 0.291259 0.956644i \(-0.405926\pi\)
0.291259 + 0.956644i \(0.405926\pi\)
\(524\) −34.9810 −1.52815
\(525\) 13.5335 0.590650
\(526\) −0.00688711 −0.000300292 0
\(527\) 1.96792 0.0857240
\(528\) 8.92671 0.388485
\(529\) 7.19948 0.313021
\(530\) −0.192931 −0.00838040
\(531\) −2.02087 −0.0876983
\(532\) −35.4828 −1.53837
\(533\) 18.7963 0.814157
\(534\) 0.334643 0.0144814
\(535\) −2.26424 −0.0978917
\(536\) 7.21946 0.311833
\(537\) 6.13052 0.264552
\(538\) 5.04182 0.217368
\(539\) 7.59786 0.327263
\(540\) 1.67666 0.0721521
\(541\) 18.7724 0.807088 0.403544 0.914960i \(-0.367778\pi\)
0.403544 + 0.914960i \(0.367778\pi\)
\(542\) 7.09463 0.304740
\(543\) 0.327510 0.0140548
\(544\) 2.76118 0.118385
\(545\) 3.29822 0.141280
\(546\) −2.01363 −0.0861753
\(547\) −12.8576 −0.549751 −0.274875 0.961480i \(-0.588637\pi\)
−0.274875 + 0.961480i \(0.588637\pi\)
\(548\) 24.9504 1.06583
\(549\) −6.53682 −0.278985
\(550\) −2.48195 −0.105831
\(551\) −6.15424 −0.262179
\(552\) −5.18178 −0.220551
\(553\) 3.18045 0.135246
\(554\) 3.26724 0.138812
\(555\) −2.37632 −0.100869
\(556\) 26.6722 1.13115
\(557\) −3.20201 −0.135673 −0.0678367 0.997696i \(-0.521610\pi\)
−0.0678367 + 0.997696i \(0.521610\pi\)
\(558\) 0.470632 0.0199234
\(559\) −27.1548 −1.14852
\(560\) 10.0461 0.424527
\(561\) 2.43892 0.102971
\(562\) 3.55382 0.149909
\(563\) 3.76069 0.158494 0.0792472 0.996855i \(-0.474748\pi\)
0.0792472 + 0.996855i \(0.474748\pi\)
\(564\) 5.91202 0.248941
\(565\) −5.30128 −0.223027
\(566\) 0.322213 0.0135436
\(567\) −3.18045 −0.133566
\(568\) 12.5957 0.528502
\(569\) −12.8179 −0.537356 −0.268678 0.963230i \(-0.586587\pi\)
−0.268678 + 0.963230i \(0.586587\pi\)
\(570\) 1.18520 0.0496424
\(571\) −7.95308 −0.332826 −0.166413 0.986056i \(-0.553219\pi\)
−0.166413 + 0.986056i \(0.553219\pi\)
\(572\) −12.5442 −0.524500
\(573\) −1.51939 −0.0634733
\(574\) −5.40029 −0.225404
\(575\) −23.3841 −0.975185
\(576\) −6.65988 −0.277495
\(577\) 24.2094 1.00785 0.503925 0.863747i \(-0.331889\pi\)
0.503925 + 0.863747i \(0.331889\pi\)
\(578\) 0.239152 0.00994741
\(579\) −3.54344 −0.147260
\(580\) 1.79689 0.0746116
\(581\) 9.59047 0.397880
\(582\) 2.70931 0.112305
\(583\) 2.27987 0.0944226
\(584\) −8.53878 −0.353337
\(585\) −2.28472 −0.0944616
\(586\) 5.46838 0.225897
\(587\) −5.53028 −0.228259 −0.114130 0.993466i \(-0.536408\pi\)
−0.114130 + 0.993466i \(0.536408\pi\)
\(588\) −6.05234 −0.249594
\(589\) −11.3008 −0.465639
\(590\) 0.417089 0.0171713
\(591\) −22.7150 −0.934370
\(592\) 10.0782 0.414210
\(593\) −44.8582 −1.84211 −0.921053 0.389438i \(-0.872669\pi\)
−0.921053 + 0.389438i \(0.872669\pi\)
\(594\) 0.583272 0.0239320
\(595\) 2.74476 0.112524
\(596\) 27.7702 1.13751
\(597\) 8.31730 0.340404
\(598\) 3.47929 0.142279
\(599\) 42.7788 1.74790 0.873948 0.486019i \(-0.161551\pi\)
0.873948 + 0.486019i \(0.161551\pi\)
\(600\) 4.01237 0.163804
\(601\) 23.2667 0.949069 0.474535 0.880237i \(-0.342616\pi\)
0.474535 + 0.880237i \(0.342616\pi\)
\(602\) 7.80174 0.317975
\(603\) −7.65642 −0.311793
\(604\) −5.79727 −0.235888
\(605\) 4.35965 0.177245
\(606\) −2.48747 −0.101047
\(607\) −25.7282 −1.04428 −0.522138 0.852861i \(-0.674865\pi\)
−0.522138 + 0.852861i \(0.674865\pi\)
\(608\) −15.8560 −0.643047
\(609\) −3.40850 −0.138119
\(610\) 1.34914 0.0546251
\(611\) −8.05608 −0.325914
\(612\) −1.94281 −0.0785333
\(613\) 2.00650 0.0810416 0.0405208 0.999179i \(-0.487098\pi\)
0.0405208 + 0.999179i \(0.487098\pi\)
\(614\) −7.64227 −0.308417
\(615\) −6.12733 −0.247078
\(616\) 7.31417 0.294696
\(617\) −34.1639 −1.37539 −0.687694 0.726000i \(-0.741377\pi\)
−0.687694 + 0.726000i \(0.741377\pi\)
\(618\) −2.59207 −0.104268
\(619\) −2.06276 −0.0829093 −0.0414547 0.999140i \(-0.513199\pi\)
−0.0414547 + 0.999140i \(0.513199\pi\)
\(620\) 3.29954 0.132513
\(621\) 5.49541 0.220523
\(622\) 0.231779 0.00929348
\(623\) −4.45038 −0.178301
\(624\) 9.68971 0.387899
\(625\) 14.3829 0.575316
\(626\) −2.67902 −0.107075
\(627\) −14.0055 −0.559324
\(628\) 39.8894 1.59176
\(629\) 2.75352 0.109790
\(630\) 0.656415 0.0261522
\(631\) 15.8528 0.631089 0.315545 0.948911i \(-0.397813\pi\)
0.315545 + 0.948911i \(0.397813\pi\)
\(632\) 0.942930 0.0375077
\(633\) −8.09013 −0.321554
\(634\) 1.06378 0.0422483
\(635\) 14.5059 0.575648
\(636\) −1.81611 −0.0720135
\(637\) 8.24728 0.326769
\(638\) 0.625095 0.0247477
\(639\) −13.3580 −0.528434
\(640\) 6.14040 0.242720
\(641\) −37.8179 −1.49372 −0.746859 0.664982i \(-0.768439\pi\)
−0.746859 + 0.664982i \(0.768439\pi\)
\(642\) 0.627451 0.0247635
\(643\) 3.96886 0.156517 0.0782583 0.996933i \(-0.475064\pi\)
0.0782583 + 0.996933i \(0.475064\pi\)
\(644\) 33.9561 1.33806
\(645\) 8.85209 0.348551
\(646\) −1.37333 −0.0540328
\(647\) −8.04118 −0.316131 −0.158066 0.987429i \(-0.550526\pi\)
−0.158066 + 0.987429i \(0.550526\pi\)
\(648\) −0.942930 −0.0370418
\(649\) −4.92874 −0.193470
\(650\) −2.69409 −0.105671
\(651\) −6.25887 −0.245304
\(652\) 38.7489 1.51752
\(653\) 30.2921 1.18542 0.592710 0.805416i \(-0.298058\pi\)
0.592710 + 0.805416i \(0.298058\pi\)
\(654\) −0.913980 −0.0357395
\(655\) −15.5389 −0.607153
\(656\) 25.9866 1.01461
\(657\) 9.05559 0.353292
\(658\) 2.31456 0.0902310
\(659\) 33.5567 1.30718 0.653591 0.756848i \(-0.273262\pi\)
0.653591 + 0.756848i \(0.273262\pi\)
\(660\) 4.08925 0.159174
\(661\) −5.23536 −0.203632 −0.101816 0.994803i \(-0.532465\pi\)
−0.101816 + 0.994803i \(0.532465\pi\)
\(662\) 6.28743 0.244368
\(663\) 2.64738 0.102816
\(664\) 2.84335 0.110343
\(665\) −15.7618 −0.611215
\(666\) 0.658509 0.0255167
\(667\) 5.88944 0.228040
\(668\) −6.36643 −0.246325
\(669\) 13.9274 0.538464
\(670\) 1.58021 0.0610490
\(671\) −15.9428 −0.615465
\(672\) −8.78180 −0.338765
\(673\) −20.5506 −0.792169 −0.396084 0.918214i \(-0.629631\pi\)
−0.396084 + 0.918214i \(0.629631\pi\)
\(674\) 1.94442 0.0748964
\(675\) −4.25521 −0.163783
\(676\) 11.6401 0.447694
\(677\) 19.0877 0.733601 0.366801 0.930300i \(-0.380453\pi\)
0.366801 + 0.930300i \(0.380453\pi\)
\(678\) 1.46906 0.0564187
\(679\) −36.0308 −1.38273
\(680\) 0.813758 0.0312062
\(681\) −16.9103 −0.648005
\(682\) 1.14783 0.0439528
\(683\) 25.1210 0.961229 0.480615 0.876932i \(-0.340414\pi\)
0.480615 + 0.876932i \(0.340414\pi\)
\(684\) 11.1565 0.426581
\(685\) 11.0832 0.423467
\(686\) 2.95478 0.112814
\(687\) 9.20668 0.351257
\(688\) −37.5425 −1.43130
\(689\) 2.47474 0.0942801
\(690\) −1.13420 −0.0431783
\(691\) 24.7036 0.939771 0.469885 0.882728i \(-0.344295\pi\)
0.469885 + 0.882728i \(0.344295\pi\)
\(692\) 23.4868 0.892832
\(693\) −7.75686 −0.294659
\(694\) −8.44357 −0.320514
\(695\) 11.8480 0.449420
\(696\) −1.01054 −0.0383044
\(697\) 7.09994 0.268930
\(698\) 7.75253 0.293438
\(699\) 5.70441 0.215761
\(700\) −26.2929 −0.993780
\(701\) −39.8419 −1.50481 −0.752403 0.658703i \(-0.771105\pi\)
−0.752403 + 0.658703i \(0.771105\pi\)
\(702\) 0.633127 0.0238958
\(703\) −15.8120 −0.596362
\(704\) −16.2429 −0.612178
\(705\) 2.62617 0.0989073
\(706\) 3.52449 0.132646
\(707\) 33.0805 1.24412
\(708\) 3.92616 0.147554
\(709\) 38.5043 1.44606 0.723030 0.690817i \(-0.242749\pi\)
0.723030 + 0.690817i \(0.242749\pi\)
\(710\) 2.75697 0.103467
\(711\) −1.00000 −0.0375029
\(712\) −1.31943 −0.0494478
\(713\) 10.8145 0.405007
\(714\) −0.760610 −0.0284651
\(715\) −5.57225 −0.208390
\(716\) −11.9104 −0.445113
\(717\) 8.86534 0.331082
\(718\) −6.70640 −0.250280
\(719\) −33.8862 −1.26374 −0.631872 0.775073i \(-0.717713\pi\)
−0.631872 + 0.775073i \(0.717713\pi\)
\(720\) −3.15871 −0.117718
\(721\) 34.4716 1.28379
\(722\) 3.34242 0.124392
\(723\) −22.9479 −0.853443
\(724\) −0.636289 −0.0236475
\(725\) −4.56032 −0.169366
\(726\) −1.20812 −0.0448374
\(727\) 1.02338 0.0379549 0.0189775 0.999820i \(-0.493959\pi\)
0.0189775 + 0.999820i \(0.493959\pi\)
\(728\) 7.93934 0.294252
\(729\) 1.00000 0.0370370
\(730\) −1.86899 −0.0691744
\(731\) −10.2572 −0.379377
\(732\) 12.6998 0.469397
\(733\) −14.6930 −0.542697 −0.271349 0.962481i \(-0.587470\pi\)
−0.271349 + 0.962481i \(0.587470\pi\)
\(734\) 1.91289 0.0706061
\(735\) −2.68850 −0.0991669
\(736\) 15.1738 0.559314
\(737\) −18.6734 −0.687843
\(738\) 1.69796 0.0625029
\(739\) −52.7776 −1.94145 −0.970727 0.240185i \(-0.922792\pi\)
−0.970727 + 0.240185i \(0.922792\pi\)
\(740\) 4.61672 0.169714
\(741\) −15.2026 −0.558480
\(742\) −0.711009 −0.0261020
\(743\) −10.7061 −0.392769 −0.196384 0.980527i \(-0.562920\pi\)
−0.196384 + 0.980527i \(0.562920\pi\)
\(744\) −1.85561 −0.0680300
\(745\) 12.3358 0.451947
\(746\) 3.93575 0.144098
\(747\) −3.01545 −0.110329
\(748\) −4.73835 −0.173251
\(749\) −8.34439 −0.304898
\(750\) 1.91019 0.0697503
\(751\) 47.8336 1.74547 0.872736 0.488193i \(-0.162344\pi\)
0.872736 + 0.488193i \(0.162344\pi\)
\(752\) −11.1378 −0.406155
\(753\) −19.6612 −0.716492
\(754\) 0.678524 0.0247104
\(755\) −2.57520 −0.0937210
\(756\) 6.17900 0.224728
\(757\) 10.5151 0.382176 0.191088 0.981573i \(-0.438798\pi\)
0.191088 + 0.981573i \(0.438798\pi\)
\(758\) −7.37970 −0.268043
\(759\) 13.4029 0.486493
\(760\) −4.67300 −0.169507
\(761\) 16.1834 0.586647 0.293323 0.956013i \(-0.405239\pi\)
0.293323 + 0.956013i \(0.405239\pi\)
\(762\) −4.01977 −0.145621
\(763\) 12.1549 0.440037
\(764\) 2.95187 0.106795
\(765\) −0.863011 −0.0312022
\(766\) −7.86950 −0.284337
\(767\) −5.35002 −0.193178
\(768\) 11.6182 0.419235
\(769\) −4.82718 −0.174073 −0.0870363 0.996205i \(-0.527740\pi\)
−0.0870363 + 0.996205i \(0.527740\pi\)
\(770\) 1.60094 0.0576940
\(771\) 6.41479 0.231023
\(772\) 6.88421 0.247768
\(773\) −31.7089 −1.14049 −0.570245 0.821475i \(-0.693152\pi\)
−0.570245 + 0.821475i \(0.693152\pi\)
\(774\) −2.45303 −0.0881724
\(775\) −8.37392 −0.300800
\(776\) −10.6823 −0.383472
\(777\) −8.75742 −0.314171
\(778\) −2.11107 −0.0756856
\(779\) −40.7713 −1.46078
\(780\) 4.43877 0.158933
\(781\) −32.5791 −1.16577
\(782\) 1.31424 0.0469970
\(783\) 1.07170 0.0382996
\(784\) 11.4022 0.407221
\(785\) 17.7192 0.632426
\(786\) 4.30603 0.153591
\(787\) −2.74964 −0.0980140 −0.0490070 0.998798i \(-0.515606\pi\)
−0.0490070 + 0.998798i \(0.515606\pi\)
\(788\) 44.1308 1.57210
\(789\) −0.0287981 −0.00102524
\(790\) 0.206391 0.00734305
\(791\) −19.5368 −0.694648
\(792\) −2.29973 −0.0817173
\(793\) −17.3055 −0.614536
\(794\) −5.53591 −0.196462
\(795\) −0.806732 −0.0286118
\(796\) −16.1589 −0.572737
\(797\) 44.0207 1.55929 0.779647 0.626220i \(-0.215399\pi\)
0.779647 + 0.626220i \(0.215399\pi\)
\(798\) 4.36779 0.154618
\(799\) −3.04303 −0.107655
\(800\) −11.7494 −0.415404
\(801\) 1.39929 0.0494415
\(802\) −0.799375 −0.0282269
\(803\) 22.0859 0.779393
\(804\) 14.8749 0.524599
\(805\) 15.0836 0.531626
\(806\) 1.24594 0.0438865
\(807\) 21.0821 0.742125
\(808\) 9.80761 0.345031
\(809\) −3.47782 −0.122274 −0.0611369 0.998129i \(-0.519473\pi\)
−0.0611369 + 0.998129i \(0.519473\pi\)
\(810\) −0.206391 −0.00725183
\(811\) −4.82488 −0.169424 −0.0847122 0.996405i \(-0.526997\pi\)
−0.0847122 + 0.996405i \(0.526997\pi\)
\(812\) 6.62205 0.232388
\(813\) 29.6658 1.04042
\(814\) 1.60605 0.0562920
\(815\) 17.2126 0.602931
\(816\) 3.66011 0.128129
\(817\) 58.9019 2.06072
\(818\) 2.58398 0.0903467
\(819\) −8.41987 −0.294214
\(820\) 11.9042 0.415713
\(821\) −16.2032 −0.565497 −0.282748 0.959194i \(-0.591246\pi\)
−0.282748 + 0.959194i \(0.591246\pi\)
\(822\) −3.07130 −0.107124
\(823\) −1.17277 −0.0408803 −0.0204401 0.999791i \(-0.506507\pi\)
−0.0204401 + 0.999791i \(0.506507\pi\)
\(824\) 10.2200 0.356032
\(825\) −10.3781 −0.361320
\(826\) 1.53709 0.0534824
\(827\) −20.0839 −0.698386 −0.349193 0.937051i \(-0.613544\pi\)
−0.349193 + 0.937051i \(0.613544\pi\)
\(828\) −10.6765 −0.371034
\(829\) −40.0111 −1.38964 −0.694821 0.719182i \(-0.744517\pi\)
−0.694821 + 0.719182i \(0.744517\pi\)
\(830\) 0.622360 0.0216024
\(831\) 13.6618 0.473921
\(832\) −17.6312 −0.611254
\(833\) 3.11526 0.107937
\(834\) −3.28324 −0.113689
\(835\) −2.82802 −0.0978678
\(836\) 27.2099 0.941074
\(837\) 1.96792 0.0680213
\(838\) −5.49268 −0.189742
\(839\) −4.32613 −0.149355 −0.0746773 0.997208i \(-0.523793\pi\)
−0.0746773 + 0.997208i \(0.523793\pi\)
\(840\) −2.58812 −0.0892985
\(841\) −27.8515 −0.960395
\(842\) −6.74041 −0.232290
\(843\) 14.8601 0.511809
\(844\) 15.7176 0.541021
\(845\) 5.17061 0.177874
\(846\) −0.727747 −0.0250205
\(847\) 16.0666 0.552054
\(848\) 3.42142 0.117492
\(849\) 1.34731 0.0462397
\(850\) −1.01764 −0.0349048
\(851\) 15.1317 0.518708
\(852\) 25.9520 0.889101
\(853\) 10.1701 0.348217 0.174109 0.984726i \(-0.444296\pi\)
0.174109 + 0.984726i \(0.444296\pi\)
\(854\) 4.97198 0.170138
\(855\) 4.95583 0.169486
\(856\) −2.47392 −0.0845568
\(857\) −50.1986 −1.71475 −0.857375 0.514692i \(-0.827906\pi\)
−0.857375 + 0.514692i \(0.827906\pi\)
\(858\) 1.54415 0.0527163
\(859\) −51.8503 −1.76911 −0.884555 0.466437i \(-0.845538\pi\)
−0.884555 + 0.466437i \(0.845538\pi\)
\(860\) −17.1979 −0.586443
\(861\) −22.5810 −0.769559
\(862\) 0.0216837 0.000738551 0
\(863\) 41.6870 1.41904 0.709521 0.704684i \(-0.248911\pi\)
0.709521 + 0.704684i \(0.248911\pi\)
\(864\) 2.76118 0.0939373
\(865\) 10.4330 0.354733
\(866\) 7.77292 0.264134
\(867\) 1.00000 0.0339618
\(868\) 12.1598 0.412730
\(869\) −2.43892 −0.0827347
\(870\) −0.221190 −0.00749903
\(871\) −20.2695 −0.686805
\(872\) 3.60365 0.122035
\(873\) 11.3288 0.383423
\(874\) −7.54698 −0.255280
\(875\) −25.4034 −0.858790
\(876\) −17.5933 −0.594421
\(877\) 39.6264 1.33809 0.669045 0.743222i \(-0.266703\pi\)
0.669045 + 0.743222i \(0.266703\pi\)
\(878\) 2.70275 0.0912134
\(879\) 22.8657 0.771242
\(880\) −7.70385 −0.259697
\(881\) 12.5505 0.422838 0.211419 0.977396i \(-0.432192\pi\)
0.211419 + 0.977396i \(0.432192\pi\)
\(882\) 0.745020 0.0250861
\(883\) −29.9924 −1.00933 −0.504663 0.863317i \(-0.668383\pi\)
−0.504663 + 0.863317i \(0.668383\pi\)
\(884\) −5.14335 −0.172990
\(885\) 1.74403 0.0586250
\(886\) 9.76615 0.328100
\(887\) −6.05781 −0.203401 −0.101701 0.994815i \(-0.532428\pi\)
−0.101701 + 0.994815i \(0.532428\pi\)
\(888\) −2.59637 −0.0871286
\(889\) 53.4584 1.79294
\(890\) −0.288801 −0.00968062
\(891\) 2.43892 0.0817069
\(892\) −27.0582 −0.905975
\(893\) 17.4746 0.584764
\(894\) −3.41840 −0.114329
\(895\) −5.29071 −0.176849
\(896\) 22.6292 0.755987
\(897\) 14.5484 0.485758
\(898\) 1.40552 0.0469027
\(899\) 2.10903 0.0703400
\(900\) 8.26705 0.275568
\(901\) 0.934787 0.0311423
\(902\) 4.14120 0.137887
\(903\) 32.6225 1.08561
\(904\) −5.79220 −0.192646
\(905\) −0.282645 −0.00939544
\(906\) 0.713621 0.0237085
\(907\) −15.0098 −0.498391 −0.249196 0.968453i \(-0.580166\pi\)
−0.249196 + 0.968453i \(0.580166\pi\)
\(908\) 32.8535 1.09028
\(909\) −10.4012 −0.344987
\(910\) 1.73778 0.0576069
\(911\) 2.98742 0.0989777 0.0494888 0.998775i \(-0.484241\pi\)
0.0494888 + 0.998775i \(0.484241\pi\)
\(912\) −21.0181 −0.695980
\(913\) −7.35443 −0.243396
\(914\) 8.53880 0.282438
\(915\) 5.64135 0.186497
\(916\) −17.8868 −0.590997
\(917\) −57.2653 −1.89107
\(918\) 0.239152 0.00789319
\(919\) 39.7421 1.31097 0.655486 0.755207i \(-0.272464\pi\)
0.655486 + 0.755207i \(0.272464\pi\)
\(920\) 4.47193 0.147435
\(921\) −31.9557 −1.05298
\(922\) −8.53944 −0.281231
\(923\) −35.3638 −1.16401
\(924\) 15.0701 0.495769
\(925\) −11.7168 −0.385246
\(926\) 3.47655 0.114247
\(927\) −10.8386 −0.355986
\(928\) 2.95917 0.0971394
\(929\) 5.97867 0.196154 0.0980769 0.995179i \(-0.468731\pi\)
0.0980769 + 0.995179i \(0.468731\pi\)
\(930\) −0.406160 −0.0133185
\(931\) −17.8893 −0.586299
\(932\) −11.0826 −0.363022
\(933\) 0.969169 0.0317292
\(934\) −2.39849 −0.0784810
\(935\) −2.10481 −0.0688348
\(936\) −2.49630 −0.0815940
\(937\) −32.5487 −1.06332 −0.531659 0.846958i \(-0.678431\pi\)
−0.531659 + 0.846958i \(0.678431\pi\)
\(938\) 5.82355 0.190146
\(939\) −11.2022 −0.365568
\(940\) −5.10214 −0.166414
\(941\) 11.0582 0.360488 0.180244 0.983622i \(-0.442311\pi\)
0.180244 + 0.983622i \(0.442311\pi\)
\(942\) −4.91023 −0.159984
\(943\) 39.0171 1.27057
\(944\) −7.39661 −0.240739
\(945\) 2.74476 0.0892871
\(946\) −5.98275 −0.194516
\(947\) 6.89840 0.224168 0.112084 0.993699i \(-0.464247\pi\)
0.112084 + 0.993699i \(0.464247\pi\)
\(948\) 1.94281 0.0630994
\(949\) 23.9736 0.778216
\(950\) 5.84379 0.189598
\(951\) 4.44816 0.144241
\(952\) 2.99894 0.0971962
\(953\) 49.5728 1.60582 0.802911 0.596099i \(-0.203284\pi\)
0.802911 + 0.596099i \(0.203284\pi\)
\(954\) 0.223556 0.00723790
\(955\) 1.31125 0.0424309
\(956\) −17.2236 −0.557052
\(957\) 2.61380 0.0844921
\(958\) 1.40232 0.0453069
\(959\) 40.8448 1.31895
\(960\) 5.74755 0.185501
\(961\) −27.1273 −0.875074
\(962\) 1.74333 0.0562071
\(963\) 2.62365 0.0845460
\(964\) 44.5834 1.43594
\(965\) 3.05802 0.0984413
\(966\) −4.17986 −0.134485
\(967\) 2.22239 0.0714671 0.0357336 0.999361i \(-0.488623\pi\)
0.0357336 + 0.999361i \(0.488623\pi\)
\(968\) 4.76337 0.153100
\(969\) −5.74248 −0.184475
\(970\) −2.33816 −0.0750739
\(971\) −3.76483 −0.120819 −0.0604096 0.998174i \(-0.519241\pi\)
−0.0604096 + 0.998174i \(0.519241\pi\)
\(972\) −1.94281 −0.0623155
\(973\) 43.6634 1.39978
\(974\) 1.88998 0.0605590
\(975\) −11.2652 −0.360774
\(976\) −23.9255 −0.765836
\(977\) 32.9125 1.05296 0.526482 0.850186i \(-0.323511\pi\)
0.526482 + 0.850186i \(0.323511\pi\)
\(978\) −4.76984 −0.152523
\(979\) 3.41276 0.109072
\(980\) 5.22324 0.166850
\(981\) −3.82176 −0.122019
\(982\) 8.53660 0.272414
\(983\) 59.7889 1.90697 0.953484 0.301442i \(-0.0974680\pi\)
0.953484 + 0.301442i \(0.0974680\pi\)
\(984\) −6.69475 −0.213421
\(985\) 19.6033 0.624613
\(986\) 0.256300 0.00816225
\(987\) 9.67821 0.308061
\(988\) 29.5356 0.939654
\(989\) −56.3675 −1.79238
\(990\) −0.503370 −0.0159982
\(991\) −8.30891 −0.263941 −0.131971 0.991254i \(-0.542130\pi\)
−0.131971 + 0.991254i \(0.542130\pi\)
\(992\) 5.43378 0.172523
\(993\) 26.2905 0.834305
\(994\) 10.1602 0.322263
\(995\) −7.17792 −0.227555
\(996\) 5.85843 0.185631
\(997\) 5.07254 0.160649 0.0803245 0.996769i \(-0.474404\pi\)
0.0803245 + 0.996769i \(0.474404\pi\)
\(998\) −5.47422 −0.173283
\(999\) 2.75352 0.0871174
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 4029.2.a.e.1.11 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4029.2.a.e.1.11 18 1.1 even 1 trivial