Properties

Label 8030.2.a.bd.1.5
Level $8030$
Weight $2$
Character 8030.1
Self dual yes
Analytic conductor $64.120$
Analytic rank $0$
Dimension $14$
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] = [8030,2,Mod(1,8030)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8030, 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("8030.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8030 = 2 \cdot 5 \cdot 11 \cdot 73 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8030.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: \(64.1198728231\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 6 x^{13} - 15 x^{12} + 143 x^{11} - 13 x^{10} - 1176 x^{9} + 1018 x^{8} + 4076 x^{7} - 4865 x^{6} - 5483 x^{5} + 7607 x^{4} + 1210 x^{3} - 3153 x^{2} + 878 x - 54 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-0.856641\) of defining polynomial
Character \(\chi\) \(=\) 8030.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -0.856641 q^{3} +1.00000 q^{4} -1.00000 q^{5} -0.856641 q^{6} +1.42002 q^{7} +1.00000 q^{8} -2.26617 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -0.856641 q^{3} +1.00000 q^{4} -1.00000 q^{5} -0.856641 q^{6} +1.42002 q^{7} +1.00000 q^{8} -2.26617 q^{9} -1.00000 q^{10} +1.00000 q^{11} -0.856641 q^{12} +1.92336 q^{13} +1.42002 q^{14} +0.856641 q^{15} +1.00000 q^{16} +7.77749 q^{17} -2.26617 q^{18} -0.319182 q^{19} -1.00000 q^{20} -1.21645 q^{21} +1.00000 q^{22} -0.366337 q^{23} -0.856641 q^{24} +1.00000 q^{25} +1.92336 q^{26} +4.51121 q^{27} +1.42002 q^{28} -7.29502 q^{29} +0.856641 q^{30} -5.42821 q^{31} +1.00000 q^{32} -0.856641 q^{33} +7.77749 q^{34} -1.42002 q^{35} -2.26617 q^{36} +6.57006 q^{37} -0.319182 q^{38} -1.64763 q^{39} -1.00000 q^{40} +1.31661 q^{41} -1.21645 q^{42} +6.39077 q^{43} +1.00000 q^{44} +2.26617 q^{45} -0.366337 q^{46} -2.70055 q^{47} -0.856641 q^{48} -4.98354 q^{49} +1.00000 q^{50} -6.66251 q^{51} +1.92336 q^{52} +1.36622 q^{53} +4.51121 q^{54} -1.00000 q^{55} +1.42002 q^{56} +0.273424 q^{57} -7.29502 q^{58} +10.1268 q^{59} +0.856641 q^{60} +0.216728 q^{61} -5.42821 q^{62} -3.21801 q^{63} +1.00000 q^{64} -1.92336 q^{65} -0.856641 q^{66} +7.17786 q^{67} +7.77749 q^{68} +0.313819 q^{69} -1.42002 q^{70} -15.8861 q^{71} -2.26617 q^{72} +1.00000 q^{73} +6.57006 q^{74} -0.856641 q^{75} -0.319182 q^{76} +1.42002 q^{77} -1.64763 q^{78} +8.29940 q^{79} -1.00000 q^{80} +2.93401 q^{81} +1.31661 q^{82} -4.11020 q^{83} -1.21645 q^{84} -7.77749 q^{85} +6.39077 q^{86} +6.24921 q^{87} +1.00000 q^{88} -1.97389 q^{89} +2.26617 q^{90} +2.73121 q^{91} -0.366337 q^{92} +4.65003 q^{93} -2.70055 q^{94} +0.319182 q^{95} -0.856641 q^{96} -0.724943 q^{97} -4.98354 q^{98} -2.26617 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 14 q^{2} + 6 q^{3} + 14 q^{4} - 14 q^{5} + 6 q^{6} + 4 q^{7} + 14 q^{8} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 14 q^{2} + 6 q^{3} + 14 q^{4} - 14 q^{5} + 6 q^{6} + 4 q^{7} + 14 q^{8} + 24 q^{9} - 14 q^{10} + 14 q^{11} + 6 q^{12} + 4 q^{13} + 4 q^{14} - 6 q^{15} + 14 q^{16} + 20 q^{17} + 24 q^{18} - 14 q^{20} + 25 q^{21} + 14 q^{22} - 4 q^{23} + 6 q^{24} + 14 q^{25} + 4 q^{26} + 21 q^{27} + 4 q^{28} + 7 q^{29} - 6 q^{30} + 8 q^{31} + 14 q^{32} + 6 q^{33} + 20 q^{34} - 4 q^{35} + 24 q^{36} + 17 q^{37} + 7 q^{39} - 14 q^{40} - 14 q^{41} + 25 q^{42} + 12 q^{43} + 14 q^{44} - 24 q^{45} - 4 q^{46} + 28 q^{47} + 6 q^{48} + 20 q^{49} + 14 q^{50} - 13 q^{51} + 4 q^{52} + 13 q^{53} + 21 q^{54} - 14 q^{55} + 4 q^{56} - 23 q^{57} + 7 q^{58} + 36 q^{59} - 6 q^{60} + 25 q^{61} + 8 q^{62} + 45 q^{63} + 14 q^{64} - 4 q^{65} + 6 q^{66} + 20 q^{68} - 7 q^{69} - 4 q^{70} + 17 q^{71} + 24 q^{72} + 14 q^{73} + 17 q^{74} + 6 q^{75} + 4 q^{77} + 7 q^{78} + 10 q^{79} - 14 q^{80} + 58 q^{81} - 14 q^{82} - 6 q^{83} + 25 q^{84} - 20 q^{85} + 12 q^{86} + 44 q^{87} + 14 q^{88} + 36 q^{89} - 24 q^{90} - 15 q^{91} - 4 q^{92} - 2 q^{93} + 28 q^{94} + 6 q^{96} - 19 q^{97} + 20 q^{98} + 24 q^{99}+O(q^{100}) \) 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
\(3\) −0.856641 −0.494582 −0.247291 0.968941i \(-0.579540\pi\)
−0.247291 + 0.968941i \(0.579540\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −0.856641 −0.349722
\(7\) 1.42002 0.536718 0.268359 0.963319i \(-0.413519\pi\)
0.268359 + 0.963319i \(0.413519\pi\)
\(8\) 1.00000 0.353553
\(9\) −2.26617 −0.755389
\(10\) −1.00000 −0.316228
\(11\) 1.00000 0.301511
\(12\) −0.856641 −0.247291
\(13\) 1.92336 0.533444 0.266722 0.963774i \(-0.414060\pi\)
0.266722 + 0.963774i \(0.414060\pi\)
\(14\) 1.42002 0.379517
\(15\) 0.856641 0.221184
\(16\) 1.00000 0.250000
\(17\) 7.77749 1.88632 0.943159 0.332342i \(-0.107839\pi\)
0.943159 + 0.332342i \(0.107839\pi\)
\(18\) −2.26617 −0.534141
\(19\) −0.319182 −0.0732253 −0.0366127 0.999330i \(-0.511657\pi\)
−0.0366127 + 0.999330i \(0.511657\pi\)
\(20\) −1.00000 −0.223607
\(21\) −1.21645 −0.265451
\(22\) 1.00000 0.213201
\(23\) −0.366337 −0.0763865 −0.0381933 0.999270i \(-0.512160\pi\)
−0.0381933 + 0.999270i \(0.512160\pi\)
\(24\) −0.856641 −0.174861
\(25\) 1.00000 0.200000
\(26\) 1.92336 0.377202
\(27\) 4.51121 0.868183
\(28\) 1.42002 0.268359
\(29\) −7.29502 −1.35465 −0.677325 0.735684i \(-0.736861\pi\)
−0.677325 + 0.735684i \(0.736861\pi\)
\(30\) 0.856641 0.156400
\(31\) −5.42821 −0.974935 −0.487468 0.873141i \(-0.662079\pi\)
−0.487468 + 0.873141i \(0.662079\pi\)
\(32\) 1.00000 0.176777
\(33\) −0.856641 −0.149122
\(34\) 7.77749 1.33383
\(35\) −1.42002 −0.240028
\(36\) −2.26617 −0.377694
\(37\) 6.57006 1.08011 0.540056 0.841629i \(-0.318403\pi\)
0.540056 + 0.841629i \(0.318403\pi\)
\(38\) −0.319182 −0.0517781
\(39\) −1.64763 −0.263831
\(40\) −1.00000 −0.158114
\(41\) 1.31661 0.205620 0.102810 0.994701i \(-0.467217\pi\)
0.102810 + 0.994701i \(0.467217\pi\)
\(42\) −1.21645 −0.187702
\(43\) 6.39077 0.974584 0.487292 0.873239i \(-0.337985\pi\)
0.487292 + 0.873239i \(0.337985\pi\)
\(44\) 1.00000 0.150756
\(45\) 2.26617 0.337820
\(46\) −0.366337 −0.0540134
\(47\) −2.70055 −0.393915 −0.196958 0.980412i \(-0.563106\pi\)
−0.196958 + 0.980412i \(0.563106\pi\)
\(48\) −0.856641 −0.123645
\(49\) −4.98354 −0.711934
\(50\) 1.00000 0.141421
\(51\) −6.66251 −0.932938
\(52\) 1.92336 0.266722
\(53\) 1.36622 0.187665 0.0938326 0.995588i \(-0.470088\pi\)
0.0938326 + 0.995588i \(0.470088\pi\)
\(54\) 4.51121 0.613898
\(55\) −1.00000 −0.134840
\(56\) 1.42002 0.189758
\(57\) 0.273424 0.0362159
\(58\) −7.29502 −0.957883
\(59\) 10.1268 1.31840 0.659198 0.751969i \(-0.270896\pi\)
0.659198 + 0.751969i \(0.270896\pi\)
\(60\) 0.856641 0.110592
\(61\) 0.216728 0.0277492 0.0138746 0.999904i \(-0.495583\pi\)
0.0138746 + 0.999904i \(0.495583\pi\)
\(62\) −5.42821 −0.689383
\(63\) −3.21801 −0.405431
\(64\) 1.00000 0.125000
\(65\) −1.92336 −0.238563
\(66\) −0.856641 −0.105445
\(67\) 7.17786 0.876916 0.438458 0.898752i \(-0.355525\pi\)
0.438458 + 0.898752i \(0.355525\pi\)
\(68\) 7.77749 0.943159
\(69\) 0.313819 0.0377794
\(70\) −1.42002 −0.169725
\(71\) −15.8861 −1.88533 −0.942665 0.333739i \(-0.891689\pi\)
−0.942665 + 0.333739i \(0.891689\pi\)
\(72\) −2.26617 −0.267070
\(73\) 1.00000 0.117041
\(74\) 6.57006 0.763754
\(75\) −0.856641 −0.0989163
\(76\) −0.319182 −0.0366127
\(77\) 1.42002 0.161827
\(78\) −1.64763 −0.186557
\(79\) 8.29940 0.933756 0.466878 0.884322i \(-0.345379\pi\)
0.466878 + 0.884322i \(0.345379\pi\)
\(80\) −1.00000 −0.111803
\(81\) 2.93401 0.326001
\(82\) 1.31661 0.145396
\(83\) −4.11020 −0.451153 −0.225576 0.974226i \(-0.572426\pi\)
−0.225576 + 0.974226i \(0.572426\pi\)
\(84\) −1.21645 −0.132725
\(85\) −7.77749 −0.843587
\(86\) 6.39077 0.689135
\(87\) 6.24921 0.669985
\(88\) 1.00000 0.106600
\(89\) −1.97389 −0.209232 −0.104616 0.994513i \(-0.533361\pi\)
−0.104616 + 0.994513i \(0.533361\pi\)
\(90\) 2.26617 0.238875
\(91\) 2.73121 0.286309
\(92\) −0.366337 −0.0381933
\(93\) 4.65003 0.482185
\(94\) −2.70055 −0.278540
\(95\) 0.319182 0.0327474
\(96\) −0.856641 −0.0874305
\(97\) −0.724943 −0.0736068 −0.0368034 0.999323i \(-0.511718\pi\)
−0.0368034 + 0.999323i \(0.511718\pi\)
\(98\) −4.98354 −0.503413
\(99\) −2.26617 −0.227758
\(100\) 1.00000 0.100000
\(101\) −8.90533 −0.886114 −0.443057 0.896494i \(-0.646106\pi\)
−0.443057 + 0.896494i \(0.646106\pi\)
\(102\) −6.66251 −0.659687
\(103\) 14.8296 1.46120 0.730602 0.682804i \(-0.239240\pi\)
0.730602 + 0.682804i \(0.239240\pi\)
\(104\) 1.92336 0.188601
\(105\) 1.21645 0.118713
\(106\) 1.36622 0.132699
\(107\) −3.86024 −0.373184 −0.186592 0.982438i \(-0.559744\pi\)
−0.186592 + 0.982438i \(0.559744\pi\)
\(108\) 4.51121 0.434092
\(109\) −4.81249 −0.460953 −0.230476 0.973078i \(-0.574028\pi\)
−0.230476 + 0.973078i \(0.574028\pi\)
\(110\) −1.00000 −0.0953463
\(111\) −5.62818 −0.534203
\(112\) 1.42002 0.134179
\(113\) 1.53056 0.143983 0.0719917 0.997405i \(-0.477064\pi\)
0.0719917 + 0.997405i \(0.477064\pi\)
\(114\) 0.273424 0.0256085
\(115\) 0.366337 0.0341611
\(116\) −7.29502 −0.677325
\(117\) −4.35865 −0.402957
\(118\) 10.1268 0.932247
\(119\) 11.0442 1.01242
\(120\) 0.856641 0.0782002
\(121\) 1.00000 0.0909091
\(122\) 0.216728 0.0196217
\(123\) −1.12786 −0.101696
\(124\) −5.42821 −0.487468
\(125\) −1.00000 −0.0894427
\(126\) −3.21801 −0.286683
\(127\) 13.8567 1.22959 0.614793 0.788688i \(-0.289239\pi\)
0.614793 + 0.788688i \(0.289239\pi\)
\(128\) 1.00000 0.0883883
\(129\) −5.47460 −0.482011
\(130\) −1.92336 −0.168690
\(131\) −17.2098 −1.50363 −0.751814 0.659375i \(-0.770821\pi\)
−0.751814 + 0.659375i \(0.770821\pi\)
\(132\) −0.856641 −0.0745610
\(133\) −0.453245 −0.0393013
\(134\) 7.17786 0.620073
\(135\) −4.51121 −0.388263
\(136\) 7.77749 0.666914
\(137\) 17.0615 1.45766 0.728829 0.684696i \(-0.240065\pi\)
0.728829 + 0.684696i \(0.240065\pi\)
\(138\) 0.313819 0.0267141
\(139\) 11.8945 1.00887 0.504437 0.863448i \(-0.331700\pi\)
0.504437 + 0.863448i \(0.331700\pi\)
\(140\) −1.42002 −0.120014
\(141\) 2.31340 0.194823
\(142\) −15.8861 −1.33313
\(143\) 1.92336 0.160839
\(144\) −2.26617 −0.188847
\(145\) 7.29502 0.605818
\(146\) 1.00000 0.0827606
\(147\) 4.26910 0.352109
\(148\) 6.57006 0.540056
\(149\) 14.9784 1.22707 0.613537 0.789666i \(-0.289746\pi\)
0.613537 + 0.789666i \(0.289746\pi\)
\(150\) −0.856641 −0.0699444
\(151\) −0.304998 −0.0248204 −0.0124102 0.999923i \(-0.503950\pi\)
−0.0124102 + 0.999923i \(0.503950\pi\)
\(152\) −0.319182 −0.0258891
\(153\) −17.6251 −1.42490
\(154\) 1.42002 0.114429
\(155\) 5.42821 0.436004
\(156\) −1.64763 −0.131916
\(157\) −18.1757 −1.45058 −0.725290 0.688444i \(-0.758294\pi\)
−0.725290 + 0.688444i \(0.758294\pi\)
\(158\) 8.29940 0.660265
\(159\) −1.17036 −0.0928158
\(160\) −1.00000 −0.0790569
\(161\) −0.520206 −0.0409980
\(162\) 2.93401 0.230518
\(163\) 7.76954 0.608557 0.304279 0.952583i \(-0.401585\pi\)
0.304279 + 0.952583i \(0.401585\pi\)
\(164\) 1.31661 0.102810
\(165\) 0.856641 0.0666894
\(166\) −4.11020 −0.319013
\(167\) 18.1097 1.40137 0.700687 0.713469i \(-0.252877\pi\)
0.700687 + 0.713469i \(0.252877\pi\)
\(168\) −1.21645 −0.0938510
\(169\) −9.30069 −0.715438
\(170\) −7.77749 −0.596506
\(171\) 0.723319 0.0553136
\(172\) 6.39077 0.487292
\(173\) 15.6310 1.18840 0.594202 0.804316i \(-0.297468\pi\)
0.594202 + 0.804316i \(0.297468\pi\)
\(174\) 6.24921 0.473751
\(175\) 1.42002 0.107344
\(176\) 1.00000 0.0753778
\(177\) −8.67502 −0.652054
\(178\) −1.97389 −0.147949
\(179\) −12.6009 −0.941839 −0.470920 0.882176i \(-0.656078\pi\)
−0.470920 + 0.882176i \(0.656078\pi\)
\(180\) 2.26617 0.168910
\(181\) 0.664700 0.0494068 0.0247034 0.999695i \(-0.492136\pi\)
0.0247034 + 0.999695i \(0.492136\pi\)
\(182\) 2.73121 0.202451
\(183\) −0.185658 −0.0137243
\(184\) −0.366337 −0.0270067
\(185\) −6.57006 −0.483040
\(186\) 4.65003 0.340956
\(187\) 7.77749 0.568746
\(188\) −2.70055 −0.196958
\(189\) 6.40602 0.465969
\(190\) 0.319182 0.0231559
\(191\) 12.0626 0.872822 0.436411 0.899747i \(-0.356249\pi\)
0.436411 + 0.899747i \(0.356249\pi\)
\(192\) −0.856641 −0.0618227
\(193\) 6.55517 0.471852 0.235926 0.971771i \(-0.424188\pi\)
0.235926 + 0.971771i \(0.424188\pi\)
\(194\) −0.724943 −0.0520479
\(195\) 1.64763 0.117989
\(196\) −4.98354 −0.355967
\(197\) 25.0254 1.78299 0.891494 0.453033i \(-0.149658\pi\)
0.891494 + 0.453033i \(0.149658\pi\)
\(198\) −2.26617 −0.161049
\(199\) 5.56402 0.394423 0.197211 0.980361i \(-0.436811\pi\)
0.197211 + 0.980361i \(0.436811\pi\)
\(200\) 1.00000 0.0707107
\(201\) −6.14885 −0.433706
\(202\) −8.90533 −0.626577
\(203\) −10.3591 −0.727065
\(204\) −6.66251 −0.466469
\(205\) −1.31661 −0.0919563
\(206\) 14.8296 1.03323
\(207\) 0.830181 0.0577015
\(208\) 1.92336 0.133361
\(209\) −0.319182 −0.0220783
\(210\) 1.21645 0.0839429
\(211\) −13.8707 −0.954895 −0.477448 0.878660i \(-0.658438\pi\)
−0.477448 + 0.878660i \(0.658438\pi\)
\(212\) 1.36622 0.0938326
\(213\) 13.6087 0.932450
\(214\) −3.86024 −0.263881
\(215\) −6.39077 −0.435847
\(216\) 4.51121 0.306949
\(217\) −7.70818 −0.523265
\(218\) −4.81249 −0.325943
\(219\) −0.856641 −0.0578864
\(220\) −1.00000 −0.0674200
\(221\) 14.9589 1.00624
\(222\) −5.62818 −0.377739
\(223\) −2.04460 −0.136916 −0.0684582 0.997654i \(-0.521808\pi\)
−0.0684582 + 0.997654i \(0.521808\pi\)
\(224\) 1.42002 0.0948792
\(225\) −2.26617 −0.151078
\(226\) 1.53056 0.101812
\(227\) 9.45412 0.627492 0.313746 0.949507i \(-0.398416\pi\)
0.313746 + 0.949507i \(0.398416\pi\)
\(228\) 0.273424 0.0181080
\(229\) 13.6181 0.899912 0.449956 0.893051i \(-0.351440\pi\)
0.449956 + 0.893051i \(0.351440\pi\)
\(230\) 0.366337 0.0241555
\(231\) −1.21645 −0.0800364
\(232\) −7.29502 −0.478941
\(233\) 18.4436 1.20828 0.604140 0.796878i \(-0.293517\pi\)
0.604140 + 0.796878i \(0.293517\pi\)
\(234\) −4.35865 −0.284934
\(235\) 2.70055 0.176164
\(236\) 10.1268 0.659198
\(237\) −7.10961 −0.461819
\(238\) 11.0442 0.715889
\(239\) 20.7615 1.34295 0.671474 0.741028i \(-0.265662\pi\)
0.671474 + 0.741028i \(0.265662\pi\)
\(240\) 0.856641 0.0552959
\(241\) 14.9961 0.965983 0.482992 0.875625i \(-0.339550\pi\)
0.482992 + 0.875625i \(0.339550\pi\)
\(242\) 1.00000 0.0642824
\(243\) −16.0470 −1.02942
\(244\) 0.216728 0.0138746
\(245\) 4.98354 0.318387
\(246\) −1.12786 −0.0719100
\(247\) −0.613901 −0.0390616
\(248\) −5.42821 −0.344692
\(249\) 3.52096 0.223132
\(250\) −1.00000 −0.0632456
\(251\) −13.5759 −0.856906 −0.428453 0.903564i \(-0.640941\pi\)
−0.428453 + 0.903564i \(0.640941\pi\)
\(252\) −3.21801 −0.202715
\(253\) −0.366337 −0.0230314
\(254\) 13.8567 0.869449
\(255\) 6.66251 0.417223
\(256\) 1.00000 0.0625000
\(257\) −10.5016 −0.655071 −0.327535 0.944839i \(-0.606218\pi\)
−0.327535 + 0.944839i \(0.606218\pi\)
\(258\) −5.47460 −0.340834
\(259\) 9.32963 0.579715
\(260\) −1.92336 −0.119282
\(261\) 16.5317 1.02329
\(262\) −17.2098 −1.06323
\(263\) 4.29740 0.264989 0.132495 0.991184i \(-0.457701\pi\)
0.132495 + 0.991184i \(0.457701\pi\)
\(264\) −0.856641 −0.0527226
\(265\) −1.36622 −0.0839264
\(266\) −0.453245 −0.0277902
\(267\) 1.69092 0.103482
\(268\) 7.17786 0.438458
\(269\) −9.05841 −0.552301 −0.276151 0.961114i \(-0.589059\pi\)
−0.276151 + 0.961114i \(0.589059\pi\)
\(270\) −4.51121 −0.274544
\(271\) 12.1871 0.740314 0.370157 0.928969i \(-0.379304\pi\)
0.370157 + 0.928969i \(0.379304\pi\)
\(272\) 7.77749 0.471579
\(273\) −2.33967 −0.141603
\(274\) 17.0615 1.03072
\(275\) 1.00000 0.0603023
\(276\) 0.313819 0.0188897
\(277\) 17.0729 1.02581 0.512904 0.858446i \(-0.328570\pi\)
0.512904 + 0.858446i \(0.328570\pi\)
\(278\) 11.8945 0.713382
\(279\) 12.3012 0.736456
\(280\) −1.42002 −0.0848625
\(281\) −13.9544 −0.832451 −0.416226 0.909261i \(-0.636647\pi\)
−0.416226 + 0.909261i \(0.636647\pi\)
\(282\) 2.31340 0.137761
\(283\) −16.4666 −0.978836 −0.489418 0.872049i \(-0.662791\pi\)
−0.489418 + 0.872049i \(0.662791\pi\)
\(284\) −15.8861 −0.942665
\(285\) −0.273424 −0.0161962
\(286\) 1.92336 0.113731
\(287\) 1.86962 0.110360
\(288\) −2.26617 −0.133535
\(289\) 43.4893 2.55820
\(290\) 7.29502 0.428378
\(291\) 0.621016 0.0364046
\(292\) 1.00000 0.0585206
\(293\) −3.10799 −0.181571 −0.0907854 0.995870i \(-0.528938\pi\)
−0.0907854 + 0.995870i \(0.528938\pi\)
\(294\) 4.26910 0.248979
\(295\) −10.1268 −0.589605
\(296\) 6.57006 0.381877
\(297\) 4.51121 0.261767
\(298\) 14.9784 0.867673
\(299\) −0.704597 −0.0407479
\(300\) −0.856641 −0.0494582
\(301\) 9.07504 0.523077
\(302\) −0.304998 −0.0175507
\(303\) 7.62867 0.438256
\(304\) −0.319182 −0.0183063
\(305\) −0.216728 −0.0124098
\(306\) −17.6251 −1.00756
\(307\) 11.7658 0.671510 0.335755 0.941949i \(-0.391009\pi\)
0.335755 + 0.941949i \(0.391009\pi\)
\(308\) 1.42002 0.0809133
\(309\) −12.7036 −0.722684
\(310\) 5.42821 0.308302
\(311\) 26.9845 1.53015 0.765076 0.643940i \(-0.222701\pi\)
0.765076 + 0.643940i \(0.222701\pi\)
\(312\) −1.64763 −0.0932785
\(313\) 1.88309 0.106439 0.0532194 0.998583i \(-0.483052\pi\)
0.0532194 + 0.998583i \(0.483052\pi\)
\(314\) −18.1757 −1.02571
\(315\) 3.21801 0.181314
\(316\) 8.29940 0.466878
\(317\) 20.8192 1.16932 0.584661 0.811278i \(-0.301228\pi\)
0.584661 + 0.811278i \(0.301228\pi\)
\(318\) −1.17036 −0.0656307
\(319\) −7.29502 −0.408443
\(320\) −1.00000 −0.0559017
\(321\) 3.30684 0.184570
\(322\) −0.520206 −0.0289900
\(323\) −2.48243 −0.138126
\(324\) 2.93401 0.163001
\(325\) 1.92336 0.106689
\(326\) 7.76954 0.430315
\(327\) 4.12257 0.227979
\(328\) 1.31661 0.0726978
\(329\) −3.83483 −0.211421
\(330\) 0.856641 0.0471565
\(331\) −4.28694 −0.235632 −0.117816 0.993035i \(-0.537589\pi\)
−0.117816 + 0.993035i \(0.537589\pi\)
\(332\) −4.11020 −0.225576
\(333\) −14.8889 −0.815904
\(334\) 18.1097 0.990921
\(335\) −7.17786 −0.392169
\(336\) −1.21645 −0.0663627
\(337\) −0.552780 −0.0301119 −0.0150559 0.999887i \(-0.504793\pi\)
−0.0150559 + 0.999887i \(0.504793\pi\)
\(338\) −9.30069 −0.505891
\(339\) −1.31114 −0.0712115
\(340\) −7.77749 −0.421794
\(341\) −5.42821 −0.293954
\(342\) 0.723319 0.0391126
\(343\) −17.0169 −0.918826
\(344\) 6.39077 0.344567
\(345\) −0.313819 −0.0168954
\(346\) 15.6310 0.840328
\(347\) −25.3958 −1.36332 −0.681659 0.731670i \(-0.738741\pi\)
−0.681659 + 0.731670i \(0.738741\pi\)
\(348\) 6.24921 0.334993
\(349\) −5.69642 −0.304922 −0.152461 0.988309i \(-0.548720\pi\)
−0.152461 + 0.988309i \(0.548720\pi\)
\(350\) 1.42002 0.0759034
\(351\) 8.67668 0.463127
\(352\) 1.00000 0.0533002
\(353\) 24.5096 1.30451 0.652256 0.757999i \(-0.273823\pi\)
0.652256 + 0.757999i \(0.273823\pi\)
\(354\) −8.67502 −0.461072
\(355\) 15.8861 0.843146
\(356\) −1.97389 −0.104616
\(357\) −9.46091 −0.500725
\(358\) −12.6009 −0.665981
\(359\) 15.1897 0.801680 0.400840 0.916148i \(-0.368718\pi\)
0.400840 + 0.916148i \(0.368718\pi\)
\(360\) 2.26617 0.119437
\(361\) −18.8981 −0.994638
\(362\) 0.664700 0.0349359
\(363\) −0.856641 −0.0449620
\(364\) 2.73121 0.143154
\(365\) −1.00000 −0.0523424
\(366\) −0.185658 −0.00970452
\(367\) 30.8165 1.60861 0.804304 0.594217i \(-0.202538\pi\)
0.804304 + 0.594217i \(0.202538\pi\)
\(368\) −0.366337 −0.0190966
\(369\) −2.98367 −0.155323
\(370\) −6.57006 −0.341561
\(371\) 1.94007 0.100723
\(372\) 4.65003 0.241093
\(373\) 21.6159 1.11923 0.559615 0.828752i \(-0.310949\pi\)
0.559615 + 0.828752i \(0.310949\pi\)
\(374\) 7.77749 0.402164
\(375\) 0.856641 0.0442367
\(376\) −2.70055 −0.139270
\(377\) −14.0309 −0.722630
\(378\) 6.40602 0.329490
\(379\) 21.1598 1.08691 0.543453 0.839440i \(-0.317117\pi\)
0.543453 + 0.839440i \(0.317117\pi\)
\(380\) 0.319182 0.0163737
\(381\) −11.8702 −0.608131
\(382\) 12.0626 0.617178
\(383\) −28.7204 −1.46755 −0.733773 0.679395i \(-0.762242\pi\)
−0.733773 + 0.679395i \(0.762242\pi\)
\(384\) −0.856641 −0.0437153
\(385\) −1.42002 −0.0723710
\(386\) 6.55517 0.333650
\(387\) −14.4826 −0.736190
\(388\) −0.724943 −0.0368034
\(389\) −12.2768 −0.622461 −0.311230 0.950335i \(-0.600741\pi\)
−0.311230 + 0.950335i \(0.600741\pi\)
\(390\) 1.64763 0.0834308
\(391\) −2.84918 −0.144089
\(392\) −4.98354 −0.251707
\(393\) 14.7426 0.743667
\(394\) 25.0254 1.26076
\(395\) −8.29940 −0.417588
\(396\) −2.26617 −0.113879
\(397\) 5.06869 0.254390 0.127195 0.991878i \(-0.459403\pi\)
0.127195 + 0.991878i \(0.459403\pi\)
\(398\) 5.56402 0.278899
\(399\) 0.388268 0.0194377
\(400\) 1.00000 0.0500000
\(401\) 11.7529 0.586911 0.293455 0.955973i \(-0.405195\pi\)
0.293455 + 0.955973i \(0.405195\pi\)
\(402\) −6.14885 −0.306677
\(403\) −10.4404 −0.520073
\(404\) −8.90533 −0.443057
\(405\) −2.93401 −0.145792
\(406\) −10.3591 −0.514113
\(407\) 6.57006 0.325666
\(408\) −6.66251 −0.329843
\(409\) −28.3149 −1.40008 −0.700041 0.714103i \(-0.746835\pi\)
−0.700041 + 0.714103i \(0.746835\pi\)
\(410\) −1.31661 −0.0650229
\(411\) −14.6155 −0.720931
\(412\) 14.8296 0.730602
\(413\) 14.3803 0.707607
\(414\) 0.830181 0.0408011
\(415\) 4.11020 0.201762
\(416\) 1.92336 0.0943004
\(417\) −10.1893 −0.498971
\(418\) −0.319182 −0.0156117
\(419\) 20.7058 1.01154 0.505771 0.862668i \(-0.331208\pi\)
0.505771 + 0.862668i \(0.331208\pi\)
\(420\) 1.21645 0.0593566
\(421\) 3.24741 0.158269 0.0791346 0.996864i \(-0.474784\pi\)
0.0791346 + 0.996864i \(0.474784\pi\)
\(422\) −13.8707 −0.675213
\(423\) 6.11989 0.297559
\(424\) 1.36622 0.0663497
\(425\) 7.77749 0.377264
\(426\) 13.6087 0.659342
\(427\) 0.307759 0.0148935
\(428\) −3.86024 −0.186592
\(429\) −1.64763 −0.0795482
\(430\) −6.39077 −0.308191
\(431\) −2.06655 −0.0995423 −0.0497711 0.998761i \(-0.515849\pi\)
−0.0497711 + 0.998761i \(0.515849\pi\)
\(432\) 4.51121 0.217046
\(433\) −33.3595 −1.60316 −0.801579 0.597889i \(-0.796006\pi\)
−0.801579 + 0.597889i \(0.796006\pi\)
\(434\) −7.70818 −0.370004
\(435\) −6.24921 −0.299627
\(436\) −4.81249 −0.230476
\(437\) 0.116928 0.00559343
\(438\) −0.856641 −0.0409319
\(439\) −19.3908 −0.925473 −0.462736 0.886496i \(-0.653132\pi\)
−0.462736 + 0.886496i \(0.653132\pi\)
\(440\) −1.00000 −0.0476731
\(441\) 11.2935 0.537787
\(442\) 14.9589 0.711522
\(443\) −33.9145 −1.61132 −0.805662 0.592375i \(-0.798190\pi\)
−0.805662 + 0.592375i \(0.798190\pi\)
\(444\) −5.62818 −0.267102
\(445\) 1.97389 0.0935714
\(446\) −2.04460 −0.0968146
\(447\) −12.8311 −0.606889
\(448\) 1.42002 0.0670897
\(449\) −3.64868 −0.172192 −0.0860958 0.996287i \(-0.527439\pi\)
−0.0860958 + 0.996287i \(0.527439\pi\)
\(450\) −2.26617 −0.106828
\(451\) 1.31661 0.0619969
\(452\) 1.53056 0.0719917
\(453\) 0.261274 0.0122757
\(454\) 9.45412 0.443704
\(455\) −2.73121 −0.128041
\(456\) 0.273424 0.0128043
\(457\) −2.16910 −0.101466 −0.0507330 0.998712i \(-0.516156\pi\)
−0.0507330 + 0.998712i \(0.516156\pi\)
\(458\) 13.6181 0.636334
\(459\) 35.0859 1.63767
\(460\) 0.366337 0.0170805
\(461\) 23.1797 1.07958 0.539792 0.841798i \(-0.318503\pi\)
0.539792 + 0.841798i \(0.318503\pi\)
\(462\) −1.21645 −0.0565943
\(463\) −32.2635 −1.49941 −0.749707 0.661770i \(-0.769806\pi\)
−0.749707 + 0.661770i \(0.769806\pi\)
\(464\) −7.29502 −0.338663
\(465\) −4.65003 −0.215640
\(466\) 18.4436 0.854384
\(467\) 6.11798 0.283106 0.141553 0.989931i \(-0.454790\pi\)
0.141553 + 0.989931i \(0.454790\pi\)
\(468\) −4.35865 −0.201479
\(469\) 10.1927 0.470656
\(470\) 2.70055 0.124567
\(471\) 15.5700 0.717430
\(472\) 10.1268 0.466123
\(473\) 6.39077 0.293848
\(474\) −7.10961 −0.326555
\(475\) −0.319182 −0.0146451
\(476\) 11.0442 0.506210
\(477\) −3.09609 −0.141760
\(478\) 20.7615 0.949607
\(479\) 19.9231 0.910311 0.455156 0.890412i \(-0.349584\pi\)
0.455156 + 0.890412i \(0.349584\pi\)
\(480\) 0.856641 0.0391001
\(481\) 12.6366 0.576178
\(482\) 14.9961 0.683053
\(483\) 0.445630 0.0202769
\(484\) 1.00000 0.0454545
\(485\) 0.724943 0.0329180
\(486\) −16.0470 −0.727908
\(487\) −5.81713 −0.263599 −0.131800 0.991276i \(-0.542076\pi\)
−0.131800 + 0.991276i \(0.542076\pi\)
\(488\) 0.216728 0.00981084
\(489\) −6.65570 −0.300981
\(490\) 4.98354 0.225133
\(491\) 23.6181 1.06587 0.532936 0.846156i \(-0.321089\pi\)
0.532936 + 0.846156i \(0.321089\pi\)
\(492\) −1.12786 −0.0508481
\(493\) −56.7369 −2.55530
\(494\) −0.613901 −0.0276207
\(495\) 2.26617 0.101857
\(496\) −5.42821 −0.243734
\(497\) −22.5586 −1.01189
\(498\) 3.52096 0.157778
\(499\) −20.6660 −0.925137 −0.462568 0.886584i \(-0.653072\pi\)
−0.462568 + 0.886584i \(0.653072\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −15.5135 −0.693094
\(502\) −13.5759 −0.605924
\(503\) 17.0007 0.758023 0.379011 0.925392i \(-0.376264\pi\)
0.379011 + 0.925392i \(0.376264\pi\)
\(504\) −3.21801 −0.143341
\(505\) 8.90533 0.396282
\(506\) −0.366337 −0.0162857
\(507\) 7.96735 0.353842
\(508\) 13.8567 0.614793
\(509\) 2.52888 0.112091 0.0560454 0.998428i \(-0.482151\pi\)
0.0560454 + 0.998428i \(0.482151\pi\)
\(510\) 6.66251 0.295021
\(511\) 1.42002 0.0628181
\(512\) 1.00000 0.0441942
\(513\) −1.43990 −0.0635730
\(514\) −10.5016 −0.463205
\(515\) −14.8296 −0.653470
\(516\) −5.47460 −0.241006
\(517\) −2.70055 −0.118770
\(518\) 9.32963 0.409920
\(519\) −13.3902 −0.587763
\(520\) −1.92336 −0.0843449
\(521\) −42.7036 −1.87088 −0.935440 0.353487i \(-0.884996\pi\)
−0.935440 + 0.353487i \(0.884996\pi\)
\(522\) 16.5317 0.723574
\(523\) 18.1860 0.795217 0.397608 0.917555i \(-0.369840\pi\)
0.397608 + 0.917555i \(0.369840\pi\)
\(524\) −17.2098 −0.751814
\(525\) −1.21645 −0.0530902
\(526\) 4.29740 0.187376
\(527\) −42.2178 −1.83904
\(528\) −0.856641 −0.0372805
\(529\) −22.8658 −0.994165
\(530\) −1.36622 −0.0593449
\(531\) −22.9490 −0.995902
\(532\) −0.453245 −0.0196507
\(533\) 2.53232 0.109687
\(534\) 1.69092 0.0731731
\(535\) 3.86024 0.166893
\(536\) 7.17786 0.310037
\(537\) 10.7945 0.465816
\(538\) −9.05841 −0.390536
\(539\) −4.98354 −0.214656
\(540\) −4.51121 −0.194132
\(541\) 7.39842 0.318083 0.159041 0.987272i \(-0.449160\pi\)
0.159041 + 0.987272i \(0.449160\pi\)
\(542\) 12.1871 0.523481
\(543\) −0.569409 −0.0244357
\(544\) 7.77749 0.333457
\(545\) 4.81249 0.206144
\(546\) −2.33967 −0.100128
\(547\) −15.0443 −0.643248 −0.321624 0.946867i \(-0.604229\pi\)
−0.321624 + 0.946867i \(0.604229\pi\)
\(548\) 17.0615 0.728829
\(549\) −0.491143 −0.0209615
\(550\) 1.00000 0.0426401
\(551\) 2.32844 0.0991947
\(552\) 0.313819 0.0133570
\(553\) 11.7853 0.501164
\(554\) 17.0729 0.725356
\(555\) 5.62818 0.238903
\(556\) 11.8945 0.504437
\(557\) 31.9988 1.35583 0.677916 0.735139i \(-0.262883\pi\)
0.677916 + 0.735139i \(0.262883\pi\)
\(558\) 12.3012 0.520753
\(559\) 12.2918 0.519886
\(560\) −1.42002 −0.0600069
\(561\) −6.66251 −0.281291
\(562\) −13.9544 −0.588632
\(563\) 18.3938 0.775208 0.387604 0.921826i \(-0.373303\pi\)
0.387604 + 0.921826i \(0.373303\pi\)
\(564\) 2.31340 0.0974116
\(565\) −1.53056 −0.0643913
\(566\) −16.4666 −0.692141
\(567\) 4.16636 0.174971
\(568\) −15.8861 −0.666565
\(569\) −23.0568 −0.966593 −0.483296 0.875457i \(-0.660561\pi\)
−0.483296 + 0.875457i \(0.660561\pi\)
\(570\) −0.273424 −0.0114525
\(571\) 26.5044 1.10917 0.554587 0.832126i \(-0.312876\pi\)
0.554587 + 0.832126i \(0.312876\pi\)
\(572\) 1.92336 0.0804197
\(573\) −10.3333 −0.431682
\(574\) 1.86962 0.0780364
\(575\) −0.366337 −0.0152773
\(576\) −2.26617 −0.0944236
\(577\) −5.94907 −0.247663 −0.123832 0.992303i \(-0.539518\pi\)
−0.123832 + 0.992303i \(0.539518\pi\)
\(578\) 43.4893 1.80892
\(579\) −5.61543 −0.233369
\(580\) 7.29502 0.302909
\(581\) −5.83657 −0.242142
\(582\) 0.621016 0.0257419
\(583\) 1.36622 0.0565832
\(584\) 1.00000 0.0413803
\(585\) 4.35865 0.180208
\(586\) −3.10799 −0.128390
\(587\) −29.6858 −1.22527 −0.612633 0.790368i \(-0.709890\pi\)
−0.612633 + 0.790368i \(0.709890\pi\)
\(588\) 4.26910 0.176055
\(589\) 1.73259 0.0713900
\(590\) −10.1268 −0.416913
\(591\) −21.4378 −0.881833
\(592\) 6.57006 0.270028
\(593\) 36.1342 1.48385 0.741926 0.670482i \(-0.233913\pi\)
0.741926 + 0.670482i \(0.233913\pi\)
\(594\) 4.51121 0.185097
\(595\) −11.0442 −0.452768
\(596\) 14.9784 0.613537
\(597\) −4.76636 −0.195074
\(598\) −0.704597 −0.0288131
\(599\) 6.73300 0.275103 0.137552 0.990495i \(-0.456077\pi\)
0.137552 + 0.990495i \(0.456077\pi\)
\(600\) −0.856641 −0.0349722
\(601\) −23.3603 −0.952886 −0.476443 0.879205i \(-0.658074\pi\)
−0.476443 + 0.879205i \(0.658074\pi\)
\(602\) 9.07504 0.369871
\(603\) −16.2662 −0.662412
\(604\) −0.304998 −0.0124102
\(605\) −1.00000 −0.0406558
\(606\) 7.62867 0.309893
\(607\) 12.3739 0.502241 0.251120 0.967956i \(-0.419201\pi\)
0.251120 + 0.967956i \(0.419201\pi\)
\(608\) −0.319182 −0.0129445
\(609\) 8.87401 0.359593
\(610\) −0.216728 −0.00877508
\(611\) −5.19412 −0.210132
\(612\) −17.6251 −0.712452
\(613\) −5.74470 −0.232026 −0.116013 0.993248i \(-0.537011\pi\)
−0.116013 + 0.993248i \(0.537011\pi\)
\(614\) 11.7658 0.474829
\(615\) 1.12786 0.0454799
\(616\) 1.42002 0.0572143
\(617\) −0.237263 −0.00955185 −0.00477593 0.999989i \(-0.501520\pi\)
−0.00477593 + 0.999989i \(0.501520\pi\)
\(618\) −12.7036 −0.511015
\(619\) −26.1137 −1.04960 −0.524799 0.851226i \(-0.675860\pi\)
−0.524799 + 0.851226i \(0.675860\pi\)
\(620\) 5.42821 0.218002
\(621\) −1.65262 −0.0663175
\(622\) 26.9845 1.08198
\(623\) −2.80297 −0.112299
\(624\) −1.64763 −0.0659579
\(625\) 1.00000 0.0400000
\(626\) 1.88309 0.0752636
\(627\) 0.273424 0.0109195
\(628\) −18.1757 −0.725290
\(629\) 51.0986 2.03743
\(630\) 3.21801 0.128208
\(631\) −36.3962 −1.44891 −0.724455 0.689322i \(-0.757908\pi\)
−0.724455 + 0.689322i \(0.757908\pi\)
\(632\) 8.29940 0.330133
\(633\) 11.8822 0.472274
\(634\) 20.8192 0.826835
\(635\) −13.8567 −0.549888
\(636\) −1.17036 −0.0464079
\(637\) −9.58513 −0.379777
\(638\) −7.29502 −0.288812
\(639\) 36.0005 1.42416
\(640\) −1.00000 −0.0395285
\(641\) −4.27457 −0.168835 −0.0844176 0.996430i \(-0.526903\pi\)
−0.0844176 + 0.996430i \(0.526903\pi\)
\(642\) 3.30684 0.130511
\(643\) 45.3473 1.78832 0.894161 0.447745i \(-0.147773\pi\)
0.894161 + 0.447745i \(0.147773\pi\)
\(644\) −0.520206 −0.0204990
\(645\) 5.47460 0.215562
\(646\) −2.48243 −0.0976700
\(647\) −38.8178 −1.52608 −0.763042 0.646349i \(-0.776295\pi\)
−0.763042 + 0.646349i \(0.776295\pi\)
\(648\) 2.93401 0.115259
\(649\) 10.1268 0.397511
\(650\) 1.92336 0.0754403
\(651\) 6.60314 0.258797
\(652\) 7.76954 0.304279
\(653\) −29.6602 −1.16069 −0.580347 0.814369i \(-0.697083\pi\)
−0.580347 + 0.814369i \(0.697083\pi\)
\(654\) 4.12257 0.161205
\(655\) 17.2098 0.672443
\(656\) 1.31661 0.0514051
\(657\) −2.26617 −0.0884116
\(658\) −3.83483 −0.149497
\(659\) 19.1253 0.745018 0.372509 0.928029i \(-0.378498\pi\)
0.372509 + 0.928029i \(0.378498\pi\)
\(660\) 0.856641 0.0333447
\(661\) 26.9202 1.04707 0.523536 0.852003i \(-0.324612\pi\)
0.523536 + 0.852003i \(0.324612\pi\)
\(662\) −4.28694 −0.166617
\(663\) −12.8144 −0.497670
\(664\) −4.11020 −0.159507
\(665\) 0.453245 0.0175761
\(666\) −14.8889 −0.576931
\(667\) 2.67243 0.103477
\(668\) 18.1097 0.700687
\(669\) 1.75149 0.0677164
\(670\) −7.17786 −0.277305
\(671\) 0.216728 0.00836671
\(672\) −1.21645 −0.0469255
\(673\) 29.7766 1.14780 0.573902 0.818924i \(-0.305429\pi\)
0.573902 + 0.818924i \(0.305429\pi\)
\(674\) −0.552780 −0.0212923
\(675\) 4.51121 0.173637
\(676\) −9.30069 −0.357719
\(677\) 13.1342 0.504789 0.252395 0.967624i \(-0.418782\pi\)
0.252395 + 0.967624i \(0.418782\pi\)
\(678\) −1.31114 −0.0503542
\(679\) −1.02944 −0.0395061
\(680\) −7.77749 −0.298253
\(681\) −8.09878 −0.310346
\(682\) −5.42821 −0.207857
\(683\) 5.19913 0.198939 0.0994697 0.995041i \(-0.468285\pi\)
0.0994697 + 0.995041i \(0.468285\pi\)
\(684\) 0.723319 0.0276568
\(685\) −17.0615 −0.651885
\(686\) −17.0169 −0.649708
\(687\) −11.6659 −0.445080
\(688\) 6.39077 0.243646
\(689\) 2.62774 0.100109
\(690\) −0.313819 −0.0119469
\(691\) −24.2108 −0.921021 −0.460511 0.887654i \(-0.652334\pi\)
−0.460511 + 0.887654i \(0.652334\pi\)
\(692\) 15.6310 0.594202
\(693\) −3.21801 −0.122242
\(694\) −25.3958 −0.964012
\(695\) −11.8945 −0.451183
\(696\) 6.24921 0.236876
\(697\) 10.2399 0.387866
\(698\) −5.69642 −0.215613
\(699\) −15.7995 −0.597594
\(700\) 1.42002 0.0536718
\(701\) −38.1565 −1.44115 −0.720575 0.693377i \(-0.756122\pi\)
−0.720575 + 0.693377i \(0.756122\pi\)
\(702\) 8.67668 0.327480
\(703\) −2.09704 −0.0790915
\(704\) 1.00000 0.0376889
\(705\) −2.31340 −0.0871276
\(706\) 24.5096 0.922429
\(707\) −12.6458 −0.475593
\(708\) −8.67502 −0.326027
\(709\) 9.03220 0.339211 0.169606 0.985512i \(-0.445751\pi\)
0.169606 + 0.985512i \(0.445751\pi\)
\(710\) 15.8861 0.596194
\(711\) −18.8078 −0.705349
\(712\) −1.97389 −0.0739747
\(713\) 1.98855 0.0744719
\(714\) −9.46091 −0.354066
\(715\) −1.92336 −0.0719295
\(716\) −12.6009 −0.470920
\(717\) −17.7851 −0.664197
\(718\) 15.1897 0.566873
\(719\) 21.4938 0.801584 0.400792 0.916169i \(-0.368735\pi\)
0.400792 + 0.916169i \(0.368735\pi\)
\(720\) 2.26617 0.0844551
\(721\) 21.0583 0.784254
\(722\) −18.8981 −0.703315
\(723\) −12.8463 −0.477758
\(724\) 0.664700 0.0247034
\(725\) −7.29502 −0.270930
\(726\) −0.856641 −0.0317929
\(727\) 45.9739 1.70508 0.852538 0.522665i \(-0.175062\pi\)
0.852538 + 0.522665i \(0.175062\pi\)
\(728\) 2.73121 0.101225
\(729\) 4.94450 0.183130
\(730\) −1.00000 −0.0370117
\(731\) 49.7042 1.83838
\(732\) −0.185658 −0.00686213
\(733\) −37.1150 −1.37087 −0.685437 0.728132i \(-0.740389\pi\)
−0.685437 + 0.728132i \(0.740389\pi\)
\(734\) 30.8165 1.13746
\(735\) −4.26910 −0.157468
\(736\) −0.366337 −0.0135034
\(737\) 7.17786 0.264400
\(738\) −2.98367 −0.109830
\(739\) 35.3924 1.30193 0.650965 0.759107i \(-0.274364\pi\)
0.650965 + 0.759107i \(0.274364\pi\)
\(740\) −6.57006 −0.241520
\(741\) 0.525893 0.0193191
\(742\) 1.94007 0.0712221
\(743\) 15.9911 0.586658 0.293329 0.956012i \(-0.405237\pi\)
0.293329 + 0.956012i \(0.405237\pi\)
\(744\) 4.65003 0.170478
\(745\) −14.9784 −0.548765
\(746\) 21.6159 0.791416
\(747\) 9.31439 0.340796
\(748\) 7.77749 0.284373
\(749\) −5.48163 −0.200294
\(750\) 0.856641 0.0312801
\(751\) 12.6936 0.463195 0.231597 0.972812i \(-0.425605\pi\)
0.231597 + 0.972812i \(0.425605\pi\)
\(752\) −2.70055 −0.0984788
\(753\) 11.6297 0.423810
\(754\) −14.0309 −0.510976
\(755\) 0.304998 0.0111000
\(756\) 6.40602 0.232985
\(757\) 5.74200 0.208696 0.104348 0.994541i \(-0.466724\pi\)
0.104348 + 0.994541i \(0.466724\pi\)
\(758\) 21.1598 0.768558
\(759\) 0.313819 0.0113909
\(760\) 0.319182 0.0115779
\(761\) −45.3644 −1.64446 −0.822229 0.569157i \(-0.807270\pi\)
−0.822229 + 0.569157i \(0.807270\pi\)
\(762\) −11.8702 −0.430013
\(763\) −6.83384 −0.247401
\(764\) 12.0626 0.436411
\(765\) 17.6251 0.637236
\(766\) −28.7204 −1.03771
\(767\) 19.4774 0.703290
\(768\) −0.856641 −0.0309114
\(769\) 37.8677 1.36554 0.682772 0.730632i \(-0.260774\pi\)
0.682772 + 0.730632i \(0.260774\pi\)
\(770\) −1.42002 −0.0511740
\(771\) 8.99608 0.323986
\(772\) 6.55517 0.235926
\(773\) −46.7882 −1.68286 −0.841428 0.540369i \(-0.818285\pi\)
−0.841428 + 0.540369i \(0.818285\pi\)
\(774\) −14.4826 −0.520565
\(775\) −5.42821 −0.194987
\(776\) −0.724943 −0.0260239
\(777\) −7.99214 −0.286716
\(778\) −12.2768 −0.440146
\(779\) −0.420239 −0.0150566
\(780\) 1.64763 0.0589945
\(781\) −15.8861 −0.568449
\(782\) −2.84918 −0.101887
\(783\) −32.9094 −1.17608
\(784\) −4.98354 −0.177983
\(785\) 18.1757 0.648719
\(786\) 14.7426 0.525852
\(787\) 36.1280 1.28783 0.643913 0.765099i \(-0.277310\pi\)
0.643913 + 0.765099i \(0.277310\pi\)
\(788\) 25.0254 0.891494
\(789\) −3.68133 −0.131059
\(790\) −8.29940 −0.295280
\(791\) 2.17344 0.0772785
\(792\) −2.26617 −0.0805247
\(793\) 0.416847 0.0148027
\(794\) 5.06869 0.179881
\(795\) 1.17036 0.0415085
\(796\) 5.56402 0.197211
\(797\) −25.5167 −0.903847 −0.451924 0.892057i \(-0.649262\pi\)
−0.451924 + 0.892057i \(0.649262\pi\)
\(798\) 0.388268 0.0137445
\(799\) −21.0035 −0.743049
\(800\) 1.00000 0.0353553
\(801\) 4.47317 0.158052
\(802\) 11.7529 0.415009
\(803\) 1.00000 0.0352892
\(804\) −6.14885 −0.216853
\(805\) 0.520206 0.0183349
\(806\) −10.4404 −0.367747
\(807\) 7.75980 0.273158
\(808\) −8.90533 −0.313288
\(809\) −24.4471 −0.859513 −0.429757 0.902945i \(-0.641401\pi\)
−0.429757 + 0.902945i \(0.641401\pi\)
\(810\) −2.93401 −0.103091
\(811\) 43.4252 1.52486 0.762432 0.647069i \(-0.224005\pi\)
0.762432 + 0.647069i \(0.224005\pi\)
\(812\) −10.3591 −0.363533
\(813\) −10.4400 −0.366146
\(814\) 6.57006 0.230280
\(815\) −7.76954 −0.272155
\(816\) −6.66251 −0.233235
\(817\) −2.03982 −0.0713642
\(818\) −28.3149 −0.990007
\(819\) −6.18938 −0.216274
\(820\) −1.31661 −0.0459781
\(821\) −7.50953 −0.262084 −0.131042 0.991377i \(-0.541832\pi\)
−0.131042 + 0.991377i \(0.541832\pi\)
\(822\) −14.6155 −0.509775
\(823\) −39.8884 −1.39042 −0.695211 0.718806i \(-0.744689\pi\)
−0.695211 + 0.718806i \(0.744689\pi\)
\(824\) 14.8296 0.516613
\(825\) −0.856641 −0.0298244
\(826\) 14.3803 0.500353
\(827\) 12.1798 0.423534 0.211767 0.977320i \(-0.432078\pi\)
0.211767 + 0.977320i \(0.432078\pi\)
\(828\) 0.830181 0.0288508
\(829\) 5.50080 0.191051 0.0955254 0.995427i \(-0.469547\pi\)
0.0955254 + 0.995427i \(0.469547\pi\)
\(830\) 4.11020 0.142667
\(831\) −14.6253 −0.507346
\(832\) 1.92336 0.0666805
\(833\) −38.7594 −1.34293
\(834\) −10.1893 −0.352826
\(835\) −18.1097 −0.626714
\(836\) −0.319182 −0.0110391
\(837\) −24.4878 −0.846423
\(838\) 20.7058 0.715269
\(839\) 34.0175 1.17441 0.587207 0.809437i \(-0.300228\pi\)
0.587207 + 0.809437i \(0.300228\pi\)
\(840\) 1.21645 0.0419715
\(841\) 24.2173 0.835078
\(842\) 3.24741 0.111913
\(843\) 11.9539 0.411715
\(844\) −13.8707 −0.477448
\(845\) 9.30069 0.319954
\(846\) 6.11989 0.210406
\(847\) 1.42002 0.0487925
\(848\) 1.36622 0.0469163
\(849\) 14.1059 0.484114
\(850\) 7.77749 0.266766
\(851\) −2.40686 −0.0825059
\(852\) 13.6087 0.466225
\(853\) 55.0578 1.88514 0.942571 0.334007i \(-0.108401\pi\)
0.942571 + 0.334007i \(0.108401\pi\)
\(854\) 0.307759 0.0105313
\(855\) −0.723319 −0.0247370
\(856\) −3.86024 −0.131940
\(857\) −39.5466 −1.35089 −0.675444 0.737412i \(-0.736048\pi\)
−0.675444 + 0.737412i \(0.736048\pi\)
\(858\) −1.64763 −0.0562491
\(859\) −6.67313 −0.227684 −0.113842 0.993499i \(-0.536316\pi\)
−0.113842 + 0.993499i \(0.536316\pi\)
\(860\) −6.39077 −0.217924
\(861\) −1.60159 −0.0545821
\(862\) −2.06655 −0.0703870
\(863\) −27.1979 −0.925826 −0.462913 0.886404i \(-0.653196\pi\)
−0.462913 + 0.886404i \(0.653196\pi\)
\(864\) 4.51121 0.153475
\(865\) −15.6310 −0.531470
\(866\) −33.3595 −1.13360
\(867\) −37.2547 −1.26524
\(868\) −7.70818 −0.261633
\(869\) 8.29940 0.281538
\(870\) −6.24921 −0.211868
\(871\) 13.8056 0.467785
\(872\) −4.81249 −0.162971
\(873\) 1.64284 0.0556018
\(874\) 0.116928 0.00395515
\(875\) −1.42002 −0.0480055
\(876\) −0.856641 −0.0289432
\(877\) 51.7917 1.74888 0.874440 0.485134i \(-0.161229\pi\)
0.874440 + 0.485134i \(0.161229\pi\)
\(878\) −19.3908 −0.654408
\(879\) 2.66243 0.0898016
\(880\) −1.00000 −0.0337100
\(881\) 6.86060 0.231140 0.115570 0.993299i \(-0.463131\pi\)
0.115570 + 0.993299i \(0.463131\pi\)
\(882\) 11.2935 0.380273
\(883\) −30.4470 −1.02462 −0.512312 0.858799i \(-0.671211\pi\)
−0.512312 + 0.858799i \(0.671211\pi\)
\(884\) 14.9589 0.503122
\(885\) 8.67502 0.291608
\(886\) −33.9145 −1.13938
\(887\) −12.3277 −0.413924 −0.206962 0.978349i \(-0.566358\pi\)
−0.206962 + 0.978349i \(0.566358\pi\)
\(888\) −5.62818 −0.188869
\(889\) 19.6769 0.659941
\(890\) 1.97389 0.0661650
\(891\) 2.93401 0.0982931
\(892\) −2.04460 −0.0684582
\(893\) 0.861965 0.0288446
\(894\) −12.8311 −0.429135
\(895\) 12.6009 0.421203
\(896\) 1.42002 0.0474396
\(897\) 0.603587 0.0201532
\(898\) −3.64868 −0.121758
\(899\) 39.5989 1.32070
\(900\) −2.26617 −0.0755389
\(901\) 10.6258 0.353996
\(902\) 1.31661 0.0438384
\(903\) −7.77405 −0.258704
\(904\) 1.53056 0.0509058
\(905\) −0.664700 −0.0220954
\(906\) 0.261274 0.00868023
\(907\) −28.9561 −0.961472 −0.480736 0.876865i \(-0.659630\pi\)
−0.480736 + 0.876865i \(0.659630\pi\)
\(908\) 9.45412 0.313746
\(909\) 20.1810 0.669361
\(910\) −2.73121 −0.0905388
\(911\) −20.1064 −0.666153 −0.333077 0.942900i \(-0.608087\pi\)
−0.333077 + 0.942900i \(0.608087\pi\)
\(912\) 0.273424 0.00905398
\(913\) −4.11020 −0.136028
\(914\) −2.16910 −0.0717473
\(915\) 0.185658 0.00613768
\(916\) 13.6181 0.449956
\(917\) −24.4383 −0.807024
\(918\) 35.0859 1.15801
\(919\) 6.73924 0.222307 0.111154 0.993803i \(-0.464545\pi\)
0.111154 + 0.993803i \(0.464545\pi\)
\(920\) 0.366337 0.0120778
\(921\) −10.0791 −0.332117
\(922\) 23.1797 0.763381
\(923\) −30.5546 −1.00572
\(924\) −1.21645 −0.0400182
\(925\) 6.57006 0.216022
\(926\) −32.2635 −1.06025
\(927\) −33.6063 −1.10378
\(928\) −7.29502 −0.239471
\(929\) 59.6218 1.95613 0.978065 0.208302i \(-0.0667936\pi\)
0.978065 + 0.208302i \(0.0667936\pi\)
\(930\) −4.65003 −0.152480
\(931\) 1.59065 0.0521316
\(932\) 18.4436 0.604140
\(933\) −23.1160 −0.756785
\(934\) 6.11798 0.200186
\(935\) −7.77749 −0.254351
\(936\) −4.35865 −0.142467
\(937\) −9.94544 −0.324903 −0.162452 0.986717i \(-0.551940\pi\)
−0.162452 + 0.986717i \(0.551940\pi\)
\(938\) 10.1927 0.332804
\(939\) −1.61313 −0.0526427
\(940\) 2.70055 0.0880821
\(941\) −27.2194 −0.887328 −0.443664 0.896193i \(-0.646322\pi\)
−0.443664 + 0.896193i \(0.646322\pi\)
\(942\) 15.5700 0.507300
\(943\) −0.482324 −0.0157066
\(944\) 10.1268 0.329599
\(945\) −6.40602 −0.208388
\(946\) 6.39077 0.207782
\(947\) 41.5656 1.35070 0.675350 0.737497i \(-0.263993\pi\)
0.675350 + 0.737497i \(0.263993\pi\)
\(948\) −7.10961 −0.230909
\(949\) 1.92336 0.0624349
\(950\) −0.319182 −0.0103556
\(951\) −17.8346 −0.578325
\(952\) 11.0442 0.357945
\(953\) −48.4306 −1.56882 −0.784410 0.620243i \(-0.787034\pi\)
−0.784410 + 0.620243i \(0.787034\pi\)
\(954\) −3.09609 −0.100240
\(955\) −12.0626 −0.390338
\(956\) 20.7615 0.671474
\(957\) 6.24921 0.202008
\(958\) 19.9231 0.643687
\(959\) 24.2276 0.782351
\(960\) 0.856641 0.0276480
\(961\) −1.53453 −0.0495008
\(962\) 12.6366 0.407420
\(963\) 8.74795 0.281899
\(964\) 14.9961 0.482992
\(965\) −6.55517 −0.211019
\(966\) 0.445630 0.0143379
\(967\) −10.8619 −0.349295 −0.174648 0.984631i \(-0.555879\pi\)
−0.174648 + 0.984631i \(0.555879\pi\)
\(968\) 1.00000 0.0321412
\(969\) 2.12655 0.0683147
\(970\) 0.724943 0.0232765
\(971\) 9.71748 0.311849 0.155924 0.987769i \(-0.450164\pi\)
0.155924 + 0.987769i \(0.450164\pi\)
\(972\) −16.0470 −0.514709
\(973\) 16.8904 0.541481
\(974\) −5.81713 −0.186393
\(975\) −1.64763 −0.0527663
\(976\) 0.216728 0.00693731
\(977\) −31.9230 −1.02131 −0.510654 0.859786i \(-0.670597\pi\)
−0.510654 + 0.859786i \(0.670597\pi\)
\(978\) −6.65570 −0.212826
\(979\) −1.97389 −0.0630858
\(980\) 4.98354 0.159193
\(981\) 10.9059 0.348199
\(982\) 23.6181 0.753685
\(983\) 29.0080 0.925213 0.462606 0.886564i \(-0.346914\pi\)
0.462606 + 0.886564i \(0.346914\pi\)
\(984\) −1.12786 −0.0359550
\(985\) −25.0254 −0.797376
\(986\) −56.7369 −1.80687
\(987\) 3.28508 0.104565
\(988\) −0.613901 −0.0195308
\(989\) −2.34118 −0.0744451
\(990\) 2.26617 0.0720235
\(991\) −15.5813 −0.494955 −0.247478 0.968894i \(-0.579602\pi\)
−0.247478 + 0.968894i \(0.579602\pi\)
\(992\) −5.42821 −0.172346
\(993\) 3.67237 0.116539
\(994\) −22.5586 −0.715515
\(995\) −5.56402 −0.176391
\(996\) 3.52096 0.111566
\(997\) 62.8524 1.99055 0.995277 0.0970708i \(-0.0309473\pi\)
0.995277 + 0.0970708i \(0.0309473\pi\)
\(998\) −20.6660 −0.654170
\(999\) 29.6389 0.937734
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 8030.2.a.bd.1.5 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8030.2.a.bd.1.5 14 1.1 even 1 trivial