Properties

Label 1805.2.a.o.1.3
Level $1805$
Weight $2$
Character 1805.1
Self dual yes
Analytic conductor $14.413$
Analytic rank $0$
Dimension $4$
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] = [1805,2,Mod(1,1805)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1805, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1805.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1805 = 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1805.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: \(14.4129975648\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.7537.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + 4x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.37933\) of defining polynomial
Character \(\chi\) \(=\) 1805.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.09744 q^{2} -0.379334 q^{3} -0.795629 q^{4} +1.00000 q^{5} -0.416295 q^{6} +1.89307 q^{7} -3.06803 q^{8} -2.85611 q^{9} +O(q^{10})\) \(q+1.09744 q^{2} -0.379334 q^{3} -0.795629 q^{4} +1.00000 q^{5} -0.416295 q^{6} +1.89307 q^{7} -3.06803 q^{8} -2.85611 q^{9} +1.09744 q^{10} +0.134400 q^{11} +0.301809 q^{12} +3.51373 q^{13} +2.07752 q^{14} -0.379334 q^{15} -1.77572 q^{16} -1.66123 q^{17} -3.13440 q^{18} -0.795629 q^{20} -0.718104 q^{21} +0.147496 q^{22} +5.36984 q^{23} +1.16381 q^{24} +1.00000 q^{25} +3.85611 q^{26} +2.22142 q^{27} -1.50618 q^{28} +4.97059 q^{29} -0.416295 q^{30} +6.56472 q^{31} +4.18732 q^{32} -0.0509824 q^{33} -1.82310 q^{34} +1.89307 q^{35} +2.27240 q^{36} -1.69819 q^{37} -1.33288 q^{39} -3.06803 q^{40} +10.6327 q^{41} -0.788075 q^{42} +8.50784 q^{43} -0.106932 q^{44} -2.85611 q^{45} +5.89307 q^{46} -11.1154 q^{47} +0.673589 q^{48} -3.41630 q^{49} +1.09744 q^{50} +0.630160 q^{51} -2.79563 q^{52} -0.264847 q^{53} +2.43787 q^{54} +0.134400 q^{55} -5.80799 q^{56} +5.45492 q^{58} -6.89667 q^{59} +0.301809 q^{60} +9.17589 q^{61} +7.20437 q^{62} -5.40680 q^{63} +8.14676 q^{64} +3.51373 q^{65} -0.0559501 q^{66} -2.95354 q^{67} +1.32172 q^{68} -2.03696 q^{69} +2.07752 q^{70} +1.32835 q^{71} +8.76262 q^{72} -6.34237 q^{73} -1.86366 q^{74} -0.379334 q^{75} +0.254428 q^{77} -1.46275 q^{78} -1.46728 q^{79} -1.77572 q^{80} +7.72566 q^{81} +11.6688 q^{82} +7.44736 q^{83} +0.571345 q^{84} -1.66123 q^{85} +9.33683 q^{86} -1.88551 q^{87} -0.412343 q^{88} +9.73608 q^{89} -3.13440 q^{90} +6.65174 q^{91} -4.27240 q^{92} -2.49022 q^{93} -12.1985 q^{94} -1.58839 q^{96} +17.4689 q^{97} -3.74917 q^{98} -0.383860 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} + 3 q^{3} + 5 q^{4} + 4 q^{5} + 2 q^{6} - 4 q^{7} + 12 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} + 3 q^{3} + 5 q^{4} + 4 q^{5} + 2 q^{6} - 4 q^{7} + 12 q^{8} + q^{9} + q^{10} - 2 q^{11} + 6 q^{12} + 7 q^{13} - q^{14} + 3 q^{15} + 7 q^{16} - q^{17} - 10 q^{18} + 5 q^{20} - 4 q^{21} + 2 q^{22} + 2 q^{23} + 23 q^{24} + 4 q^{25} + 3 q^{26} + 12 q^{27} - 19 q^{28} - q^{29} + 2 q^{30} + 30 q^{32} + 19 q^{33} + 15 q^{34} - 4 q^{35} - 7 q^{36} - 2 q^{37} + 15 q^{39} + 12 q^{40} - 8 q^{41} - 15 q^{42} + q^{43} - 12 q^{44} + q^{45} + 12 q^{46} - 12 q^{47} + 23 q^{48} - 10 q^{49} + q^{50} + 22 q^{51} - 3 q^{52} - 5 q^{53} - 34 q^{54} - 2 q^{55} - 41 q^{56} - 27 q^{58} - 5 q^{59} + 6 q^{60} + 37 q^{62} - 3 q^{63} + 56 q^{64} + 7 q^{65} - 31 q^{66} + 4 q^{67} + 16 q^{68} - 9 q^{69} - q^{70} + 20 q^{71} + 17 q^{72} - 20 q^{73} + 25 q^{74} + 3 q^{75} + 14 q^{77} - 18 q^{78} + 17 q^{79} + 7 q^{80} + 12 q^{81} + 21 q^{82} + q^{83} - 20 q^{84} - q^{85} + 8 q^{86} - 16 q^{87} - 7 q^{88} + 11 q^{89} - 10 q^{90} + 6 q^{91} - q^{92} - 8 q^{93} - 31 q^{94} + 21 q^{96} + q^{97} + 9 q^{98} + 38 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.09744 0.776006 0.388003 0.921658i \(-0.373165\pi\)
0.388003 + 0.921658i \(0.373165\pi\)
\(3\) −0.379334 −0.219008 −0.109504 0.993986i \(-0.534926\pi\)
−0.109504 + 0.993986i \(0.534926\pi\)
\(4\) −0.795629 −0.397815
\(5\) 1.00000 0.447214
\(6\) −0.416295 −0.169952
\(7\) 1.89307 0.715512 0.357756 0.933815i \(-0.383542\pi\)
0.357756 + 0.933815i \(0.383542\pi\)
\(8\) −3.06803 −1.08471
\(9\) −2.85611 −0.952035
\(10\) 1.09744 0.347040
\(11\) 0.134400 0.0405231 0.0202615 0.999795i \(-0.493550\pi\)
0.0202615 + 0.999795i \(0.493550\pi\)
\(12\) 0.301809 0.0871248
\(13\) 3.51373 0.974534 0.487267 0.873253i \(-0.337994\pi\)
0.487267 + 0.873253i \(0.337994\pi\)
\(14\) 2.07752 0.555242
\(15\) −0.379334 −0.0979436
\(16\) −1.77572 −0.443929
\(17\) −1.66123 −0.402907 −0.201454 0.979498i \(-0.564567\pi\)
−0.201454 + 0.979498i \(0.564567\pi\)
\(18\) −3.13440 −0.738785
\(19\) 0 0
\(20\) −0.795629 −0.177908
\(21\) −0.718104 −0.156703
\(22\) 0.147496 0.0314462
\(23\) 5.36984 1.11969 0.559844 0.828598i \(-0.310861\pi\)
0.559844 + 0.828598i \(0.310861\pi\)
\(24\) 1.16381 0.237561
\(25\) 1.00000 0.200000
\(26\) 3.85611 0.756245
\(27\) 2.22142 0.427512
\(28\) −1.50618 −0.284641
\(29\) 4.97059 0.923016 0.461508 0.887136i \(-0.347309\pi\)
0.461508 + 0.887136i \(0.347309\pi\)
\(30\) −0.416295 −0.0760048
\(31\) 6.56472 1.17906 0.589529 0.807747i \(-0.299313\pi\)
0.589529 + 0.807747i \(0.299313\pi\)
\(32\) 4.18732 0.740221
\(33\) −0.0509824 −0.00887490
\(34\) −1.82310 −0.312658
\(35\) 1.89307 0.319987
\(36\) 2.27240 0.378734
\(37\) −1.69819 −0.279181 −0.139590 0.990209i \(-0.544579\pi\)
−0.139590 + 0.990209i \(0.544579\pi\)
\(38\) 0 0
\(39\) −1.33288 −0.213431
\(40\) −3.06803 −0.485098
\(41\) 10.6327 1.66056 0.830278 0.557349i \(-0.188182\pi\)
0.830278 + 0.557349i \(0.188182\pi\)
\(42\) −0.788075 −0.121603
\(43\) 8.50784 1.29743 0.648717 0.761030i \(-0.275306\pi\)
0.648717 + 0.761030i \(0.275306\pi\)
\(44\) −0.106932 −0.0161207
\(45\) −2.85611 −0.425763
\(46\) 5.89307 0.868885
\(47\) −11.1154 −1.62135 −0.810675 0.585497i \(-0.800899\pi\)
−0.810675 + 0.585497i \(0.800899\pi\)
\(48\) 0.673589 0.0972242
\(49\) −3.41630 −0.488042
\(50\) 1.09744 0.155201
\(51\) 0.630160 0.0882401
\(52\) −2.79563 −0.387684
\(53\) −0.264847 −0.0363796 −0.0181898 0.999835i \(-0.505790\pi\)
−0.0181898 + 0.999835i \(0.505790\pi\)
\(54\) 2.43787 0.331752
\(55\) 0.134400 0.0181225
\(56\) −5.80799 −0.776125
\(57\) 0 0
\(58\) 5.45492 0.716266
\(59\) −6.89667 −0.897870 −0.448935 0.893564i \(-0.648196\pi\)
−0.448935 + 0.893564i \(0.648196\pi\)
\(60\) 0.301809 0.0389634
\(61\) 9.17589 1.17485 0.587426 0.809278i \(-0.300141\pi\)
0.587426 + 0.809278i \(0.300141\pi\)
\(62\) 7.20437 0.914956
\(63\) −5.40680 −0.681193
\(64\) 8.14676 1.01834
\(65\) 3.51373 0.435825
\(66\) −0.0559501 −0.00688698
\(67\) −2.95354 −0.360833 −0.180416 0.983590i \(-0.557745\pi\)
−0.180416 + 0.983590i \(0.557745\pi\)
\(68\) 1.32172 0.160282
\(69\) −2.03696 −0.245221
\(70\) 2.07752 0.248312
\(71\) 1.32835 0.157646 0.0788232 0.996889i \(-0.474884\pi\)
0.0788232 + 0.996889i \(0.474884\pi\)
\(72\) 8.76262 1.03268
\(73\) −6.34237 −0.742319 −0.371159 0.928569i \(-0.621040\pi\)
−0.371159 + 0.928569i \(0.621040\pi\)
\(74\) −1.86366 −0.216646
\(75\) −0.379334 −0.0438017
\(76\) 0 0
\(77\) 0.254428 0.0289948
\(78\) −1.46275 −0.165624
\(79\) −1.46728 −0.165082 −0.0825408 0.996588i \(-0.526303\pi\)
−0.0825408 + 0.996588i \(0.526303\pi\)
\(80\) −1.77572 −0.198531
\(81\) 7.72566 0.858406
\(82\) 11.6688 1.28860
\(83\) 7.44736 0.817454 0.408727 0.912657i \(-0.365973\pi\)
0.408727 + 0.912657i \(0.365973\pi\)
\(84\) 0.571345 0.0623388
\(85\) −1.66123 −0.180186
\(86\) 9.33683 1.00682
\(87\) −1.88551 −0.202148
\(88\) −0.412343 −0.0439559
\(89\) 9.73608 1.03202 0.516011 0.856582i \(-0.327416\pi\)
0.516011 + 0.856582i \(0.327416\pi\)
\(90\) −3.13440 −0.330395
\(91\) 6.65174 0.697291
\(92\) −4.27240 −0.445429
\(93\) −2.49022 −0.258224
\(94\) −12.1985 −1.25818
\(95\) 0 0
\(96\) −1.58839 −0.162115
\(97\) 17.4689 1.77370 0.886851 0.462055i \(-0.152888\pi\)
0.886851 + 0.462055i \(0.152888\pi\)
\(98\) −3.74917 −0.378724
\(99\) −0.383860 −0.0385794
\(100\) −0.795629 −0.0795629
\(101\) 5.39731 0.537052 0.268526 0.963272i \(-0.413463\pi\)
0.268526 + 0.963272i \(0.413463\pi\)
\(102\) 0.691562 0.0684749
\(103\) 2.14750 0.211599 0.105800 0.994387i \(-0.466260\pi\)
0.105800 + 0.994387i \(0.466260\pi\)
\(104\) −10.7802 −1.05709
\(105\) −0.718104 −0.0700798
\(106\) −0.290654 −0.0282308
\(107\) −1.00093 −0.0967631 −0.0483815 0.998829i \(-0.515406\pi\)
−0.0483815 + 0.998829i \(0.515406\pi\)
\(108\) −1.76743 −0.170071
\(109\) 16.2629 1.55770 0.778852 0.627208i \(-0.215802\pi\)
0.778852 + 0.627208i \(0.215802\pi\)
\(110\) 0.147496 0.0140632
\(111\) 0.644181 0.0611430
\(112\) −3.36155 −0.317637
\(113\) 0.843010 0.0793037 0.0396519 0.999214i \(-0.487375\pi\)
0.0396519 + 0.999214i \(0.487375\pi\)
\(114\) 0 0
\(115\) 5.36984 0.500740
\(116\) −3.95475 −0.367189
\(117\) −10.0356 −0.927791
\(118\) −7.56867 −0.696752
\(119\) −3.14482 −0.288285
\(120\) 1.16381 0.106241
\(121\) −10.9819 −0.998358
\(122\) 10.0700 0.911692
\(123\) −4.03336 −0.363676
\(124\) −5.22308 −0.469046
\(125\) 1.00000 0.0894427
\(126\) −5.93363 −0.528610
\(127\) 18.7397 1.66288 0.831439 0.555616i \(-0.187518\pi\)
0.831439 + 0.555616i \(0.187518\pi\)
\(128\) 0.565920 0.0500208
\(129\) −3.22731 −0.284149
\(130\) 3.85611 0.338203
\(131\) 2.88644 0.252189 0.126095 0.992018i \(-0.459756\pi\)
0.126095 + 0.992018i \(0.459756\pi\)
\(132\) 0.0405631 0.00353057
\(133\) 0 0
\(134\) −3.24133 −0.280008
\(135\) 2.22142 0.191189
\(136\) 5.09670 0.437039
\(137\) −18.8316 −1.60889 −0.804445 0.594027i \(-0.797537\pi\)
−0.804445 + 0.594027i \(0.797537\pi\)
\(138\) −2.23544 −0.190293
\(139\) −18.1795 −1.54196 −0.770982 0.636857i \(-0.780234\pi\)
−0.770982 + 0.636857i \(0.780234\pi\)
\(140\) −1.50618 −0.127295
\(141\) 4.21645 0.355089
\(142\) 1.45778 0.122334
\(143\) 0.472245 0.0394912
\(144\) 5.07163 0.422636
\(145\) 4.97059 0.412785
\(146\) −6.96036 −0.576044
\(147\) 1.29592 0.106885
\(148\) 1.35113 0.111062
\(149\) −22.2543 −1.82315 −0.911573 0.411138i \(-0.865131\pi\)
−0.911573 + 0.411138i \(0.865131\pi\)
\(150\) −0.416295 −0.0339904
\(151\) 3.33482 0.271384 0.135692 0.990751i \(-0.456674\pi\)
0.135692 + 0.990751i \(0.456674\pi\)
\(152\) 0 0
\(153\) 4.74465 0.383582
\(154\) 0.279219 0.0225001
\(155\) 6.56472 0.527291
\(156\) 1.06048 0.0849061
\(157\) 7.26291 0.579643 0.289822 0.957081i \(-0.406404\pi\)
0.289822 + 0.957081i \(0.406404\pi\)
\(158\) −1.61025 −0.128104
\(159\) 0.100466 0.00796744
\(160\) 4.18732 0.331037
\(161\) 10.1655 0.801151
\(162\) 8.47843 0.666129
\(163\) −19.7783 −1.54916 −0.774578 0.632478i \(-0.782038\pi\)
−0.774578 + 0.632478i \(0.782038\pi\)
\(164\) −8.45972 −0.660593
\(165\) −0.0509824 −0.00396898
\(166\) 8.17302 0.634350
\(167\) −3.40320 −0.263348 −0.131674 0.991293i \(-0.542035\pi\)
−0.131674 + 0.991293i \(0.542035\pi\)
\(168\) 2.20317 0.169978
\(169\) −0.653675 −0.0502827
\(170\) −1.82310 −0.139825
\(171\) 0 0
\(172\) −6.76909 −0.516138
\(173\) −10.5857 −0.804817 −0.402409 0.915460i \(-0.631827\pi\)
−0.402409 + 0.915460i \(0.631827\pi\)
\(174\) −2.06923 −0.156868
\(175\) 1.89307 0.143102
\(176\) −0.238656 −0.0179894
\(177\) 2.61614 0.196641
\(178\) 10.6847 0.800855
\(179\) −14.6024 −1.09144 −0.545718 0.837969i \(-0.683743\pi\)
−0.545718 + 0.837969i \(0.683743\pi\)
\(180\) 2.27240 0.169375
\(181\) −5.43261 −0.403803 −0.201901 0.979406i \(-0.564712\pi\)
−0.201901 + 0.979406i \(0.564712\pi\)
\(182\) 7.29987 0.541102
\(183\) −3.48072 −0.257303
\(184\) −16.4748 −1.21454
\(185\) −1.69819 −0.124853
\(186\) −2.73286 −0.200383
\(187\) −0.223269 −0.0163271
\(188\) 8.84375 0.644996
\(189\) 4.20530 0.305890
\(190\) 0 0
\(191\) 20.4758 1.48157 0.740787 0.671740i \(-0.234453\pi\)
0.740787 + 0.671740i \(0.234453\pi\)
\(192\) −3.09034 −0.223026
\(193\) 11.0235 0.793490 0.396745 0.917929i \(-0.370140\pi\)
0.396745 + 0.917929i \(0.370140\pi\)
\(194\) 19.1711 1.37640
\(195\) −1.33288 −0.0954494
\(196\) 2.71810 0.194150
\(197\) −19.8532 −1.41448 −0.707242 0.706971i \(-0.750061\pi\)
−0.707242 + 0.706971i \(0.750061\pi\)
\(198\) −0.421263 −0.0299379
\(199\) 21.0026 1.48883 0.744417 0.667715i \(-0.232728\pi\)
0.744417 + 0.667715i \(0.232728\pi\)
\(200\) −3.06803 −0.216943
\(201\) 1.12038 0.0790254
\(202\) 5.92321 0.416756
\(203\) 9.40967 0.660429
\(204\) −0.501374 −0.0351032
\(205\) 10.6327 0.742623
\(206\) 2.35674 0.164202
\(207\) −15.3368 −1.06598
\(208\) −6.23939 −0.432624
\(209\) 0 0
\(210\) −0.788075 −0.0543824
\(211\) −12.8257 −0.882956 −0.441478 0.897272i \(-0.645546\pi\)
−0.441478 + 0.897272i \(0.645546\pi\)
\(212\) 0.210720 0.0144723
\(213\) −0.503889 −0.0345259
\(214\) −1.09845 −0.0750887
\(215\) 8.50784 0.580230
\(216\) −6.81538 −0.463728
\(217\) 12.4275 0.843630
\(218\) 17.8475 1.20879
\(219\) 2.40588 0.162574
\(220\) −0.106932 −0.00720939
\(221\) −5.83712 −0.392647
\(222\) 0.706949 0.0474473
\(223\) 20.3944 1.36571 0.682856 0.730553i \(-0.260737\pi\)
0.682856 + 0.730553i \(0.260737\pi\)
\(224\) 7.92688 0.529637
\(225\) −2.85611 −0.190407
\(226\) 0.925152 0.0615402
\(227\) 25.4172 1.68700 0.843500 0.537129i \(-0.180491\pi\)
0.843500 + 0.537129i \(0.180491\pi\)
\(228\) 0 0
\(229\) 2.21553 0.146406 0.0732030 0.997317i \(-0.476678\pi\)
0.0732030 + 0.997317i \(0.476678\pi\)
\(230\) 5.89307 0.388577
\(231\) −0.0965132 −0.00635010
\(232\) −15.2499 −1.00121
\(233\) −14.1576 −0.927498 −0.463749 0.885967i \(-0.653496\pi\)
−0.463749 + 0.885967i \(0.653496\pi\)
\(234\) −11.0134 −0.719972
\(235\) −11.1154 −0.725089
\(236\) 5.48719 0.357186
\(237\) 0.556588 0.0361543
\(238\) −3.45125 −0.223711
\(239\) 3.01476 0.195008 0.0975042 0.995235i \(-0.468914\pi\)
0.0975042 + 0.995235i \(0.468914\pi\)
\(240\) 0.673589 0.0434800
\(241\) −23.7792 −1.53175 −0.765877 0.642987i \(-0.777695\pi\)
−0.765877 + 0.642987i \(0.777695\pi\)
\(242\) −12.0520 −0.774732
\(243\) −9.59486 −0.615511
\(244\) −7.30060 −0.467373
\(245\) −3.41630 −0.218259
\(246\) −4.42636 −0.282215
\(247\) 0 0
\(248\) −20.1407 −1.27894
\(249\) −2.82504 −0.179029
\(250\) 1.09744 0.0694081
\(251\) 17.1899 1.08502 0.542509 0.840050i \(-0.317475\pi\)
0.542509 + 0.840050i \(0.317475\pi\)
\(252\) 4.30181 0.270988
\(253\) 0.721706 0.0453733
\(254\) 20.5656 1.29040
\(255\) 0.630160 0.0394622
\(256\) −15.6725 −0.979529
\(257\) −19.5429 −1.21905 −0.609525 0.792767i \(-0.708640\pi\)
−0.609525 + 0.792767i \(0.708640\pi\)
\(258\) −3.54178 −0.220501
\(259\) −3.21479 −0.199757
\(260\) −2.79563 −0.173378
\(261\) −14.1965 −0.878744
\(262\) 3.16769 0.195700
\(263\) 8.81360 0.543470 0.271735 0.962372i \(-0.412403\pi\)
0.271735 + 0.962372i \(0.412403\pi\)
\(264\) 0.156416 0.00962672
\(265\) −0.264847 −0.0162694
\(266\) 0 0
\(267\) −3.69322 −0.226022
\(268\) 2.34993 0.143545
\(269\) 0.288362 0.0175818 0.00879088 0.999961i \(-0.497202\pi\)
0.00879088 + 0.999961i \(0.497202\pi\)
\(270\) 2.43787 0.148364
\(271\) −24.8712 −1.51082 −0.755409 0.655253i \(-0.772562\pi\)
−0.755409 + 0.655253i \(0.772562\pi\)
\(272\) 2.94987 0.178862
\(273\) −2.52323 −0.152713
\(274\) −20.6665 −1.24851
\(275\) 0.134400 0.00810462
\(276\) 1.62067 0.0975526
\(277\) −4.40486 −0.264662 −0.132331 0.991206i \(-0.542246\pi\)
−0.132331 + 0.991206i \(0.542246\pi\)
\(278\) −19.9509 −1.19657
\(279\) −18.7495 −1.12250
\(280\) −5.80799 −0.347094
\(281\) −32.7214 −1.95200 −0.975998 0.217778i \(-0.930119\pi\)
−0.975998 + 0.217778i \(0.930119\pi\)
\(282\) 4.62729 0.275551
\(283\) −1.32893 −0.0789964 −0.0394982 0.999220i \(-0.512576\pi\)
−0.0394982 + 0.999220i \(0.512576\pi\)
\(284\) −1.05688 −0.0627140
\(285\) 0 0
\(286\) 0.518260 0.0306454
\(287\) 20.1285 1.18815
\(288\) −11.9594 −0.704717
\(289\) −14.2403 −0.837666
\(290\) 5.45492 0.320324
\(291\) −6.62656 −0.388456
\(292\) 5.04618 0.295305
\(293\) −7.72365 −0.451220 −0.225610 0.974218i \(-0.572438\pi\)
−0.225610 + 0.974218i \(0.572438\pi\)
\(294\) 1.42219 0.0829437
\(295\) −6.89667 −0.401540
\(296\) 5.21010 0.302831
\(297\) 0.298558 0.0173241
\(298\) −24.4228 −1.41477
\(299\) 18.8682 1.09118
\(300\) 0.301809 0.0174250
\(301\) 16.1059 0.928330
\(302\) 3.65976 0.210595
\(303\) −2.04738 −0.117619
\(304\) 0 0
\(305\) 9.17589 0.525410
\(306\) 5.20696 0.297662
\(307\) −9.10002 −0.519366 −0.259683 0.965694i \(-0.583618\pi\)
−0.259683 + 0.965694i \(0.583618\pi\)
\(308\) −0.202430 −0.0115345
\(309\) −0.814618 −0.0463420
\(310\) 7.20437 0.409181
\(311\) −12.4569 −0.706364 −0.353182 0.935555i \(-0.614900\pi\)
−0.353182 + 0.935555i \(0.614900\pi\)
\(312\) 4.08931 0.231512
\(313\) 2.04553 0.115620 0.0578101 0.998328i \(-0.481588\pi\)
0.0578101 + 0.998328i \(0.481588\pi\)
\(314\) 7.97059 0.449807
\(315\) −5.40680 −0.304639
\(316\) 1.16741 0.0656719
\(317\) 23.5713 1.32389 0.661947 0.749551i \(-0.269730\pi\)
0.661947 + 0.749551i \(0.269730\pi\)
\(318\) 0.110255 0.00618278
\(319\) 0.668047 0.0374035
\(320\) 8.14676 0.455418
\(321\) 0.379685 0.0211919
\(322\) 11.1560 0.621698
\(323\) 0 0
\(324\) −6.14676 −0.341487
\(325\) 3.51373 0.194907
\(326\) −21.7055 −1.20215
\(327\) −6.16907 −0.341150
\(328\) −32.6216 −1.80123
\(329\) −21.0422 −1.16010
\(330\) −0.0559501 −0.00307995
\(331\) −18.7175 −1.02881 −0.514403 0.857549i \(-0.671986\pi\)
−0.514403 + 0.857549i \(0.671986\pi\)
\(332\) −5.92534 −0.325195
\(333\) 4.85021 0.265790
\(334\) −3.73480 −0.204359
\(335\) −2.95354 −0.161369
\(336\) 1.27515 0.0695651
\(337\) −32.6881 −1.78063 −0.890316 0.455343i \(-0.849517\pi\)
−0.890316 + 0.455343i \(0.849517\pi\)
\(338\) −0.717368 −0.0390197
\(339\) −0.319782 −0.0173682
\(340\) 1.32172 0.0716805
\(341\) 0.882297 0.0477791
\(342\) 0 0
\(343\) −19.7188 −1.06471
\(344\) −26.1023 −1.40734
\(345\) −2.03696 −0.109666
\(346\) −11.6172 −0.624543
\(347\) −2.56666 −0.137785 −0.0688927 0.997624i \(-0.521947\pi\)
−0.0688927 + 0.997624i \(0.521947\pi\)
\(348\) 1.50017 0.0804175
\(349\) 16.6195 0.889619 0.444810 0.895625i \(-0.353271\pi\)
0.444810 + 0.895625i \(0.353271\pi\)
\(350\) 2.07752 0.111048
\(351\) 7.80547 0.416625
\(352\) 0.562776 0.0299961
\(353\) 28.3629 1.50961 0.754803 0.655951i \(-0.227732\pi\)
0.754803 + 0.655951i \(0.227732\pi\)
\(354\) 2.87105 0.152595
\(355\) 1.32835 0.0705016
\(356\) −7.74631 −0.410553
\(357\) 1.19294 0.0631369
\(358\) −16.0252 −0.846961
\(359\) 17.3885 0.917733 0.458866 0.888505i \(-0.348256\pi\)
0.458866 + 0.888505i \(0.348256\pi\)
\(360\) 8.76262 0.461831
\(361\) 0 0
\(362\) −5.96195 −0.313353
\(363\) 4.16582 0.218649
\(364\) −5.29231 −0.277393
\(365\) −6.34237 −0.331975
\(366\) −3.81988 −0.199668
\(367\) 25.8048 1.34700 0.673501 0.739186i \(-0.264790\pi\)
0.673501 + 0.739186i \(0.264790\pi\)
\(368\) −9.53531 −0.497062
\(369\) −30.3683 −1.58091
\(370\) −1.86366 −0.0968871
\(371\) −0.501374 −0.0260300
\(372\) 1.98129 0.102725
\(373\) 27.0663 1.40144 0.700719 0.713437i \(-0.252862\pi\)
0.700719 + 0.713437i \(0.252862\pi\)
\(374\) −0.245024 −0.0126699
\(375\) −0.379334 −0.0195887
\(376\) 34.1024 1.75870
\(377\) 17.4653 0.899511
\(378\) 4.61505 0.237373
\(379\) 12.4028 0.637092 0.318546 0.947907i \(-0.396806\pi\)
0.318546 + 0.947907i \(0.396806\pi\)
\(380\) 0 0
\(381\) −7.10859 −0.364184
\(382\) 22.4709 1.14971
\(383\) −5.35942 −0.273854 −0.136927 0.990581i \(-0.543723\pi\)
−0.136927 + 0.990581i \(0.543723\pi\)
\(384\) −0.214673 −0.0109550
\(385\) 0.254428 0.0129669
\(386\) 12.0976 0.615753
\(387\) −24.2993 −1.23520
\(388\) −13.8988 −0.705605
\(389\) 8.56933 0.434482 0.217241 0.976118i \(-0.430294\pi\)
0.217241 + 0.976118i \(0.430294\pi\)
\(390\) −1.46275 −0.0740693
\(391\) −8.92053 −0.451131
\(392\) 10.4813 0.529386
\(393\) −1.09492 −0.0552316
\(394\) −21.7877 −1.09765
\(395\) −1.46728 −0.0738268
\(396\) 0.305411 0.0153475
\(397\) −10.6445 −0.534234 −0.267117 0.963664i \(-0.586071\pi\)
−0.267117 + 0.963664i \(0.586071\pi\)
\(398\) 23.0490 1.15534
\(399\) 0 0
\(400\) −1.77572 −0.0887858
\(401\) 7.65209 0.382127 0.191063 0.981578i \(-0.438806\pi\)
0.191063 + 0.981578i \(0.438806\pi\)
\(402\) 1.22955 0.0613242
\(403\) 23.0667 1.14903
\(404\) −4.29426 −0.213647
\(405\) 7.72566 0.383891
\(406\) 10.3265 0.512497
\(407\) −0.228237 −0.0113133
\(408\) −1.93335 −0.0957152
\(409\) −17.6887 −0.874650 −0.437325 0.899304i \(-0.644074\pi\)
−0.437325 + 0.899304i \(0.644074\pi\)
\(410\) 11.6688 0.576280
\(411\) 7.14345 0.352361
\(412\) −1.70861 −0.0841772
\(413\) −13.0559 −0.642437
\(414\) −16.8312 −0.827210
\(415\) 7.44736 0.365577
\(416\) 14.7131 0.721371
\(417\) 6.89609 0.337703
\(418\) 0 0
\(419\) 1.18732 0.0580045 0.0290023 0.999579i \(-0.490767\pi\)
0.0290023 + 0.999579i \(0.490767\pi\)
\(420\) 0.571345 0.0278788
\(421\) 33.3673 1.62622 0.813111 0.582109i \(-0.197772\pi\)
0.813111 + 0.582109i \(0.197772\pi\)
\(422\) −14.0754 −0.685180
\(423\) 31.7468 1.54358
\(424\) 0.812560 0.0394614
\(425\) −1.66123 −0.0805815
\(426\) −0.552987 −0.0267923
\(427\) 17.3706 0.840621
\(428\) 0.796365 0.0384938
\(429\) −0.179139 −0.00864890
\(430\) 9.33683 0.450262
\(431\) −6.17598 −0.297486 −0.148743 0.988876i \(-0.547523\pi\)
−0.148743 + 0.988876i \(0.547523\pi\)
\(432\) −3.94461 −0.189785
\(433\) −18.5552 −0.891707 −0.445854 0.895106i \(-0.647100\pi\)
−0.445854 + 0.895106i \(0.647100\pi\)
\(434\) 13.6384 0.654662
\(435\) −1.88551 −0.0904035
\(436\) −12.9392 −0.619677
\(437\) 0 0
\(438\) 2.64030 0.126158
\(439\) −0.227312 −0.0108490 −0.00542450 0.999985i \(-0.501727\pi\)
−0.00542450 + 0.999985i \(0.501727\pi\)
\(440\) −0.412343 −0.0196577
\(441\) 9.75730 0.464633
\(442\) −6.40588 −0.304696
\(443\) −34.9827 −1.66208 −0.831038 0.556215i \(-0.812253\pi\)
−0.831038 + 0.556215i \(0.812253\pi\)
\(444\) −0.512529 −0.0243236
\(445\) 9.73608 0.461534
\(446\) 22.3816 1.05980
\(447\) 8.44182 0.399285
\(448\) 15.4224 0.728638
\(449\) −16.9509 −0.799961 −0.399980 0.916524i \(-0.630983\pi\)
−0.399980 + 0.916524i \(0.630983\pi\)
\(450\) −3.13440 −0.147757
\(451\) 1.42904 0.0672909
\(452\) −0.670724 −0.0315482
\(453\) −1.26501 −0.0594353
\(454\) 27.8938 1.30912
\(455\) 6.65174 0.311838
\(456\) 0 0
\(457\) 1.60241 0.0749578 0.0374789 0.999297i \(-0.488067\pi\)
0.0374789 + 0.999297i \(0.488067\pi\)
\(458\) 2.43140 0.113612
\(459\) −3.69029 −0.172248
\(460\) −4.27240 −0.199202
\(461\) −8.74162 −0.407138 −0.203569 0.979061i \(-0.565254\pi\)
−0.203569 + 0.979061i \(0.565254\pi\)
\(462\) −0.105917 −0.00492772
\(463\) 21.1886 0.984718 0.492359 0.870392i \(-0.336135\pi\)
0.492359 + 0.870392i \(0.336135\pi\)
\(464\) −8.82636 −0.409753
\(465\) −2.49022 −0.115481
\(466\) −15.5371 −0.719744
\(467\) 20.4516 0.946388 0.473194 0.880958i \(-0.343101\pi\)
0.473194 + 0.880958i \(0.343101\pi\)
\(468\) 7.98461 0.369089
\(469\) −5.59126 −0.258180
\(470\) −12.1985 −0.562674
\(471\) −2.75507 −0.126947
\(472\) 21.1592 0.973931
\(473\) 1.14345 0.0525760
\(474\) 0.610821 0.0280559
\(475\) 0 0
\(476\) 2.50211 0.114684
\(477\) 0.756432 0.0346347
\(478\) 3.30851 0.151328
\(479\) 23.5491 1.07599 0.537994 0.842949i \(-0.319182\pi\)
0.537994 + 0.842949i \(0.319182\pi\)
\(480\) −1.58839 −0.0724999
\(481\) −5.96699 −0.272071
\(482\) −26.0962 −1.18865
\(483\) −3.85611 −0.175459
\(484\) 8.73755 0.397161
\(485\) 17.4689 0.793224
\(486\) −10.5298 −0.477640
\(487\) 36.0392 1.63309 0.816546 0.577280i \(-0.195886\pi\)
0.816546 + 0.577280i \(0.195886\pi\)
\(488\) −28.1519 −1.27438
\(489\) 7.50258 0.339278
\(490\) −3.74917 −0.169370
\(491\) 20.0595 0.905271 0.452635 0.891696i \(-0.350484\pi\)
0.452635 + 0.891696i \(0.350484\pi\)
\(492\) 3.20906 0.144676
\(493\) −8.25729 −0.371890
\(494\) 0 0
\(495\) −0.383860 −0.0172532
\(496\) −11.6571 −0.523418
\(497\) 2.51466 0.112798
\(498\) −3.10030 −0.138928
\(499\) −36.8729 −1.65066 −0.825328 0.564653i \(-0.809010\pi\)
−0.825328 + 0.564653i \(0.809010\pi\)
\(500\) −0.795629 −0.0355816
\(501\) 1.29095 0.0576753
\(502\) 18.8649 0.841980
\(503\) −21.6487 −0.965268 −0.482634 0.875822i \(-0.660320\pi\)
−0.482634 + 0.875822i \(0.660320\pi\)
\(504\) 16.5882 0.738899
\(505\) 5.39731 0.240177
\(506\) 0.792028 0.0352099
\(507\) 0.247961 0.0110123
\(508\) −14.9098 −0.661517
\(509\) −36.4558 −1.61588 −0.807938 0.589267i \(-0.799417\pi\)
−0.807938 + 0.589267i \(0.799417\pi\)
\(510\) 0.691562 0.0306229
\(511\) −12.0065 −0.531138
\(512\) −18.3314 −0.810141
\(513\) 0 0
\(514\) −21.4471 −0.945990
\(515\) 2.14750 0.0946300
\(516\) 2.56774 0.113039
\(517\) −1.49391 −0.0657021
\(518\) −3.52803 −0.155013
\(519\) 4.01552 0.176262
\(520\) −10.7802 −0.472745
\(521\) −22.6092 −0.990528 −0.495264 0.868742i \(-0.664929\pi\)
−0.495264 + 0.868742i \(0.664929\pi\)
\(522\) −15.5798 −0.681910
\(523\) 0.532911 0.0233026 0.0116513 0.999932i \(-0.496291\pi\)
0.0116513 + 0.999932i \(0.496291\pi\)
\(524\) −2.29654 −0.100325
\(525\) −0.718104 −0.0313406
\(526\) 9.67238 0.421736
\(527\) −10.9055 −0.475051
\(528\) 0.0905303 0.00393983
\(529\) 5.83518 0.253703
\(530\) −0.290654 −0.0126252
\(531\) 19.6976 0.854804
\(532\) 0 0
\(533\) 37.3606 1.61827
\(534\) −4.05308 −0.175394
\(535\) −1.00093 −0.0432738
\(536\) 9.06156 0.391400
\(537\) 5.53919 0.239034
\(538\) 0.316460 0.0136436
\(539\) −0.459150 −0.0197770
\(540\) −1.76743 −0.0760579
\(541\) 5.01640 0.215672 0.107836 0.994169i \(-0.465608\pi\)
0.107836 + 0.994169i \(0.465608\pi\)
\(542\) −27.2946 −1.17240
\(543\) 2.06077 0.0884362
\(544\) −6.95610 −0.298240
\(545\) 16.2629 0.696626
\(546\) −2.76909 −0.118506
\(547\) 22.6299 0.967584 0.483792 0.875183i \(-0.339259\pi\)
0.483792 + 0.875183i \(0.339259\pi\)
\(548\) 14.9830 0.640040
\(549\) −26.2073 −1.11850
\(550\) 0.147496 0.00628923
\(551\) 0 0
\(552\) 6.24946 0.265995
\(553\) −2.77766 −0.118118
\(554\) −4.83406 −0.205380
\(555\) 0.644181 0.0273440
\(556\) 14.4641 0.613416
\(557\) 35.2554 1.49382 0.746910 0.664925i \(-0.231537\pi\)
0.746910 + 0.664925i \(0.231537\pi\)
\(558\) −20.5764 −0.871070
\(559\) 29.8943 1.26439
\(560\) −3.36155 −0.142051
\(561\) 0.0846935 0.00357576
\(562\) −35.9097 −1.51476
\(563\) −24.6295 −1.03801 −0.519005 0.854771i \(-0.673698\pi\)
−0.519005 + 0.854771i \(0.673698\pi\)
\(564\) −3.35473 −0.141260
\(565\) 0.843010 0.0354657
\(566\) −1.45841 −0.0613017
\(567\) 14.6252 0.614200
\(568\) −4.07542 −0.171001
\(569\) −20.0193 −0.839252 −0.419626 0.907697i \(-0.637839\pi\)
−0.419626 + 0.907697i \(0.637839\pi\)
\(570\) 0 0
\(571\) −16.6121 −0.695195 −0.347597 0.937644i \(-0.613002\pi\)
−0.347597 + 0.937644i \(0.613002\pi\)
\(572\) −0.375732 −0.0157102
\(573\) −7.76715 −0.324477
\(574\) 22.0898 0.922010
\(575\) 5.36984 0.223938
\(576\) −23.2680 −0.969500
\(577\) 12.4486 0.518244 0.259122 0.965845i \(-0.416567\pi\)
0.259122 + 0.965845i \(0.416567\pi\)
\(578\) −15.6279 −0.650034
\(579\) −4.18159 −0.173781
\(580\) −3.95475 −0.164212
\(581\) 14.0984 0.584899
\(582\) −7.27224 −0.301444
\(583\) −0.0355955 −0.00147421
\(584\) 19.4586 0.805202
\(585\) −10.0356 −0.414921
\(586\) −8.47623 −0.350150
\(587\) 4.51144 0.186207 0.0931036 0.995656i \(-0.470321\pi\)
0.0931036 + 0.995656i \(0.470321\pi\)
\(588\) −1.03107 −0.0425206
\(589\) 0 0
\(590\) −7.56867 −0.311597
\(591\) 7.53101 0.309784
\(592\) 3.01550 0.123936
\(593\) 19.6082 0.805213 0.402606 0.915373i \(-0.368104\pi\)
0.402606 + 0.915373i \(0.368104\pi\)
\(594\) 0.327650 0.0134436
\(595\) −3.14482 −0.128925
\(596\) 17.7062 0.725274
\(597\) −7.96699 −0.326067
\(598\) 20.7067 0.846759
\(599\) 10.7759 0.440292 0.220146 0.975467i \(-0.429347\pi\)
0.220146 + 0.975467i \(0.429347\pi\)
\(600\) 1.16381 0.0475122
\(601\) 15.0244 0.612860 0.306430 0.951893i \(-0.400866\pi\)
0.306430 + 0.951893i \(0.400866\pi\)
\(602\) 17.6753 0.720389
\(603\) 8.43563 0.343526
\(604\) −2.65328 −0.107960
\(605\) −10.9819 −0.446479
\(606\) −2.24687 −0.0912730
\(607\) 25.1901 1.02243 0.511217 0.859452i \(-0.329195\pi\)
0.511217 + 0.859452i \(0.329195\pi\)
\(608\) 0 0
\(609\) −3.56940 −0.144640
\(610\) 10.0700 0.407721
\(611\) −39.0566 −1.58006
\(612\) −3.77498 −0.152595
\(613\) 16.2351 0.655728 0.327864 0.944725i \(-0.393671\pi\)
0.327864 + 0.944725i \(0.393671\pi\)
\(614\) −9.98672 −0.403031
\(615\) −4.03336 −0.162641
\(616\) −0.780593 −0.0314510
\(617\) 6.51826 0.262415 0.131208 0.991355i \(-0.458115\pi\)
0.131208 + 0.991355i \(0.458115\pi\)
\(618\) −0.893993 −0.0359617
\(619\) −4.39112 −0.176494 −0.0882470 0.996099i \(-0.528126\pi\)
−0.0882470 + 0.996099i \(0.528126\pi\)
\(620\) −5.22308 −0.209764
\(621\) 11.9287 0.478681
\(622\) −13.6706 −0.548142
\(623\) 18.4311 0.738425
\(624\) 2.36681 0.0947483
\(625\) 1.00000 0.0400000
\(626\) 2.24484 0.0897220
\(627\) 0 0
\(628\) −5.77858 −0.230590
\(629\) 2.82108 0.112484
\(630\) −5.93363 −0.236402
\(631\) −34.6209 −1.37824 −0.689118 0.724650i \(-0.742002\pi\)
−0.689118 + 0.724650i \(0.742002\pi\)
\(632\) 4.50165 0.179066
\(633\) 4.86521 0.193375
\(634\) 25.8680 1.02735
\(635\) 18.7397 0.743661
\(636\) −0.0799333 −0.00316956
\(637\) −12.0040 −0.475614
\(638\) 0.733141 0.0290253
\(639\) −3.79391 −0.150085
\(640\) 0.565920 0.0223700
\(641\) 7.41241 0.292773 0.146386 0.989227i \(-0.453236\pi\)
0.146386 + 0.989227i \(0.453236\pi\)
\(642\) 0.416681 0.0164451
\(643\) −16.5459 −0.652506 −0.326253 0.945282i \(-0.605786\pi\)
−0.326253 + 0.945282i \(0.605786\pi\)
\(644\) −8.08794 −0.318710
\(645\) −3.22731 −0.127075
\(646\) 0 0
\(647\) 29.4822 1.15907 0.579533 0.814949i \(-0.303235\pi\)
0.579533 + 0.814949i \(0.303235\pi\)
\(648\) −23.7026 −0.931124
\(649\) −0.926912 −0.0363845
\(650\) 3.85611 0.151249
\(651\) −4.71415 −0.184762
\(652\) 15.7362 0.616277
\(653\) 6.57421 0.257269 0.128634 0.991692i \(-0.458941\pi\)
0.128634 + 0.991692i \(0.458941\pi\)
\(654\) −6.77017 −0.264735
\(655\) 2.88644 0.112782
\(656\) −18.8807 −0.737169
\(657\) 18.1145 0.706713
\(658\) −23.0925 −0.900241
\(659\) −26.3370 −1.02594 −0.512972 0.858405i \(-0.671456\pi\)
−0.512972 + 0.858405i \(0.671456\pi\)
\(660\) 0.0405631 0.00157892
\(661\) 3.78419 0.147188 0.0735941 0.997288i \(-0.476553\pi\)
0.0735941 + 0.997288i \(0.476553\pi\)
\(662\) −20.5413 −0.798359
\(663\) 2.21422 0.0859930
\(664\) −22.8487 −0.886703
\(665\) 0 0
\(666\) 5.32281 0.206255
\(667\) 26.6913 1.03349
\(668\) 2.70769 0.104763
\(669\) −7.73630 −0.299103
\(670\) −3.24133 −0.125224
\(671\) 1.23324 0.0476086
\(672\) −3.00694 −0.115995
\(673\) −21.0431 −0.811150 −0.405575 0.914062i \(-0.632929\pi\)
−0.405575 + 0.914062i \(0.632929\pi\)
\(674\) −35.8731 −1.38178
\(675\) 2.22142 0.0855025
\(676\) 0.520083 0.0200032
\(677\) −15.2744 −0.587043 −0.293522 0.955952i \(-0.594827\pi\)
−0.293522 + 0.955952i \(0.594827\pi\)
\(678\) −0.350941 −0.0134778
\(679\) 33.0699 1.26911
\(680\) 5.09670 0.195450
\(681\) −9.64161 −0.369467
\(682\) 0.968267 0.0370769
\(683\) −8.60225 −0.329156 −0.164578 0.986364i \(-0.552626\pi\)
−0.164578 + 0.986364i \(0.552626\pi\)
\(684\) 0 0
\(685\) −18.8316 −0.719518
\(686\) −21.6401 −0.826223
\(687\) −0.840424 −0.0320642
\(688\) −15.1075 −0.575968
\(689\) −0.930603 −0.0354532
\(690\) −2.23544 −0.0851017
\(691\) 34.1079 1.29753 0.648763 0.760990i \(-0.275286\pi\)
0.648763 + 0.760990i \(0.275286\pi\)
\(692\) 8.42231 0.320168
\(693\) −0.726674 −0.0276040
\(694\) −2.81675 −0.106922
\(695\) −18.1795 −0.689587
\(696\) 5.78481 0.219273
\(697\) −17.6634 −0.669050
\(698\) 18.2388 0.690350
\(699\) 5.37047 0.203130
\(700\) −1.50618 −0.0569282
\(701\) 10.7293 0.405239 0.202619 0.979258i \(-0.435055\pi\)
0.202619 + 0.979258i \(0.435055\pi\)
\(702\) 8.56603 0.323304
\(703\) 0 0
\(704\) 1.09492 0.0412665
\(705\) 4.21645 0.158801
\(706\) 31.1266 1.17146
\(707\) 10.2175 0.384267
\(708\) −2.08148 −0.0782267
\(709\) −28.8476 −1.08339 −0.541697 0.840574i \(-0.682218\pi\)
−0.541697 + 0.840574i \(0.682218\pi\)
\(710\) 1.45778 0.0547096
\(711\) 4.19070 0.157164
\(712\) −29.8706 −1.11945
\(713\) 35.2515 1.32018
\(714\) 1.30917 0.0489946
\(715\) 0.472245 0.0176610
\(716\) 11.6181 0.434189
\(717\) −1.14360 −0.0427085
\(718\) 19.0829 0.712166
\(719\) −20.0555 −0.747944 −0.373972 0.927440i \(-0.622004\pi\)
−0.373972 + 0.927440i \(0.622004\pi\)
\(720\) 5.07163 0.189009
\(721\) 4.06535 0.151402
\(722\) 0 0
\(723\) 9.02026 0.335467
\(724\) 4.32234 0.160639
\(725\) 4.97059 0.184603
\(726\) 4.57173 0.169673
\(727\) −0.780521 −0.0289479 −0.0144740 0.999895i \(-0.504607\pi\)
−0.0144740 + 0.999895i \(0.504607\pi\)
\(728\) −20.4077 −0.756361
\(729\) −19.5373 −0.723604
\(730\) −6.96036 −0.257615
\(731\) −14.1335 −0.522745
\(732\) 2.76937 0.102359
\(733\) −26.8391 −0.991326 −0.495663 0.868515i \(-0.665075\pi\)
−0.495663 + 0.868515i \(0.665075\pi\)
\(734\) 28.3192 1.04528
\(735\) 1.29592 0.0478006
\(736\) 22.4853 0.828817
\(737\) −0.396956 −0.0146221
\(738\) −33.3273 −1.22679
\(739\) 20.6530 0.759735 0.379867 0.925041i \(-0.375970\pi\)
0.379867 + 0.925041i \(0.375970\pi\)
\(740\) 1.35113 0.0496685
\(741\) 0 0
\(742\) −0.550227 −0.0201995
\(743\) 1.47317 0.0540454 0.0270227 0.999635i \(-0.491397\pi\)
0.0270227 + 0.999635i \(0.491397\pi\)
\(744\) 7.64007 0.280098
\(745\) −22.2543 −0.815336
\(746\) 29.7036 1.08752
\(747\) −21.2705 −0.778245
\(748\) 0.177639 0.00649514
\(749\) −1.89482 −0.0692352
\(750\) −0.416295 −0.0152010
\(751\) 15.5624 0.567881 0.283940 0.958842i \(-0.408358\pi\)
0.283940 + 0.958842i \(0.408358\pi\)
\(752\) 19.7378 0.719764
\(753\) −6.52071 −0.237628
\(754\) 19.1671 0.698026
\(755\) 3.33482 0.121366
\(756\) −3.34586 −0.121688
\(757\) 20.1756 0.733295 0.366647 0.930360i \(-0.380506\pi\)
0.366647 + 0.930360i \(0.380506\pi\)
\(758\) 13.6114 0.494387
\(759\) −0.273767 −0.00993713
\(760\) 0 0
\(761\) −15.1076 −0.547649 −0.273825 0.961780i \(-0.588289\pi\)
−0.273825 + 0.961780i \(0.588289\pi\)
\(762\) −7.80124 −0.282609
\(763\) 30.7868 1.11456
\(764\) −16.2911 −0.589392
\(765\) 4.74465 0.171543
\(766\) −5.88163 −0.212512
\(767\) −24.2331 −0.875005
\(768\) 5.94509 0.214525
\(769\) 51.6580 1.86284 0.931418 0.363951i \(-0.118573\pi\)
0.931418 + 0.363951i \(0.118573\pi\)
\(770\) 0.279219 0.0100624
\(771\) 7.41327 0.266982
\(772\) −8.77063 −0.315662
\(773\) 30.1558 1.08463 0.542314 0.840176i \(-0.317548\pi\)
0.542314 + 0.840176i \(0.317548\pi\)
\(774\) −26.6670 −0.958525
\(775\) 6.56472 0.235812
\(776\) −53.5952 −1.92396
\(777\) 1.21948 0.0437485
\(778\) 9.40431 0.337161
\(779\) 0 0
\(780\) 1.06048 0.0379712
\(781\) 0.178530 0.00638832
\(782\) −9.78974 −0.350080
\(783\) 11.0418 0.394601
\(784\) 6.06637 0.216656
\(785\) 7.26291 0.259224
\(786\) −1.20161 −0.0428601
\(787\) 43.0969 1.53624 0.768119 0.640307i \(-0.221193\pi\)
0.768119 + 0.640307i \(0.221193\pi\)
\(788\) 15.7958 0.562703
\(789\) −3.34330 −0.119025
\(790\) −1.61025 −0.0572900
\(791\) 1.59588 0.0567428
\(792\) 1.17770 0.0418476
\(793\) 32.2416 1.14493
\(794\) −11.6817 −0.414569
\(795\) 0.100466 0.00356315
\(796\) −16.7103 −0.592280
\(797\) −50.5062 −1.78902 −0.894510 0.447048i \(-0.852475\pi\)
−0.894510 + 0.447048i \(0.852475\pi\)
\(798\) 0 0
\(799\) 18.4652 0.653253
\(800\) 4.18732 0.148044
\(801\) −27.8073 −0.982522
\(802\) 8.39769 0.296533
\(803\) −0.852414 −0.0300810
\(804\) −0.891406 −0.0314375
\(805\) 10.1655 0.358286
\(806\) 25.3142 0.891656
\(807\) −0.109386 −0.00385056
\(808\) −16.5591 −0.582547
\(809\) −30.0872 −1.05781 −0.528905 0.848681i \(-0.677397\pi\)
−0.528905 + 0.848681i \(0.677397\pi\)
\(810\) 8.47843 0.297902
\(811\) 22.9509 0.805916 0.402958 0.915218i \(-0.367982\pi\)
0.402958 + 0.915218i \(0.367982\pi\)
\(812\) −7.48661 −0.262728
\(813\) 9.43449 0.330882
\(814\) −0.250476 −0.00877917
\(815\) −19.7783 −0.692804
\(816\) −1.11899 −0.0391723
\(817\) 0 0
\(818\) −19.4123 −0.678734
\(819\) −18.9981 −0.663846
\(820\) −8.45972 −0.295426
\(821\) −38.0807 −1.32903 −0.664513 0.747277i \(-0.731361\pi\)
−0.664513 + 0.747277i \(0.731361\pi\)
\(822\) 7.83950 0.273434
\(823\) 53.7932 1.87511 0.937556 0.347835i \(-0.113083\pi\)
0.937556 + 0.347835i \(0.113083\pi\)
\(824\) −6.58858 −0.229524
\(825\) −0.0509824 −0.00177498
\(826\) −14.3280 −0.498535
\(827\) 32.4092 1.12698 0.563490 0.826123i \(-0.309458\pi\)
0.563490 + 0.826123i \(0.309458\pi\)
\(828\) 12.2024 0.424064
\(829\) −18.5305 −0.643592 −0.321796 0.946809i \(-0.604287\pi\)
−0.321796 + 0.946809i \(0.604287\pi\)
\(830\) 8.17302 0.283690
\(831\) 1.67091 0.0579633
\(832\) 28.6255 0.992412
\(833\) 5.67525 0.196636
\(834\) 7.56804 0.262060
\(835\) −3.40320 −0.117773
\(836\) 0 0
\(837\) 14.5830 0.504062
\(838\) 1.30301 0.0450118
\(839\) 0.826608 0.0285377 0.0142688 0.999898i \(-0.495458\pi\)
0.0142688 + 0.999898i \(0.495458\pi\)
\(840\) 2.20317 0.0760165
\(841\) −4.29321 −0.148042
\(842\) 36.6185 1.26196
\(843\) 12.4123 0.427504
\(844\) 10.2045 0.351253
\(845\) −0.653675 −0.0224871
\(846\) 34.8401 1.19783
\(847\) −20.7895 −0.714337
\(848\) 0.470294 0.0161500
\(849\) 0.504106 0.0173009
\(850\) −1.82310 −0.0625317
\(851\) −9.11901 −0.312596
\(852\) 0.400908 0.0137349
\(853\) −7.34938 −0.251638 −0.125819 0.992053i \(-0.540156\pi\)
−0.125819 + 0.992053i \(0.540156\pi\)
\(854\) 19.0631 0.652327
\(855\) 0 0
\(856\) 3.07087 0.104960
\(857\) −45.3496 −1.54911 −0.774556 0.632506i \(-0.782026\pi\)
−0.774556 + 0.632506i \(0.782026\pi\)
\(858\) −0.196594 −0.00671160
\(859\) 29.6285 1.01091 0.505456 0.862852i \(-0.331324\pi\)
0.505456 + 0.862852i \(0.331324\pi\)
\(860\) −6.76909 −0.230824
\(861\) −7.63542 −0.260215
\(862\) −6.77775 −0.230851
\(863\) 41.0807 1.39840 0.699201 0.714925i \(-0.253539\pi\)
0.699201 + 0.714925i \(0.253539\pi\)
\(864\) 9.30180 0.316454
\(865\) −10.5857 −0.359925
\(866\) −20.3632 −0.691970
\(867\) 5.40183 0.183456
\(868\) −9.88764 −0.335608
\(869\) −0.197202 −0.00668962
\(870\) −2.06923 −0.0701536
\(871\) −10.3780 −0.351644
\(872\) −49.8951 −1.68966
\(873\) −49.8931 −1.68863
\(874\) 0 0
\(875\) 1.89307 0.0639974
\(876\) −1.91419 −0.0646743
\(877\) 54.6307 1.84475 0.922374 0.386298i \(-0.126246\pi\)
0.922374 + 0.386298i \(0.126246\pi\)
\(878\) −0.249460 −0.00841888
\(879\) 2.92984 0.0988211
\(880\) −0.238656 −0.00804509
\(881\) −15.4805 −0.521552 −0.260776 0.965399i \(-0.583978\pi\)
−0.260776 + 0.965399i \(0.583978\pi\)
\(882\) 10.7080 0.360558
\(883\) −45.3495 −1.52613 −0.763066 0.646321i \(-0.776307\pi\)
−0.763066 + 0.646321i \(0.776307\pi\)
\(884\) 4.64418 0.156201
\(885\) 2.61614 0.0879406
\(886\) −38.3913 −1.28978
\(887\) −31.1399 −1.04558 −0.522788 0.852463i \(-0.675108\pi\)
−0.522788 + 0.852463i \(0.675108\pi\)
\(888\) −1.97637 −0.0663226
\(889\) 35.4755 1.18981
\(890\) 10.6847 0.358153
\(891\) 1.03833 0.0347853
\(892\) −16.2264 −0.543301
\(893\) 0 0
\(894\) 9.26438 0.309847
\(895\) −14.6024 −0.488105
\(896\) 1.07133 0.0357905
\(897\) −7.15734 −0.238977
\(898\) −18.6025 −0.620775
\(899\) 32.6305 1.08829
\(900\) 2.27240 0.0757467
\(901\) 0.439972 0.0146576
\(902\) 1.56828 0.0522181
\(903\) −6.10952 −0.203312
\(904\) −2.58638 −0.0860218
\(905\) −5.43261 −0.180586
\(906\) −1.38827 −0.0461222
\(907\) 43.5415 1.44577 0.722886 0.690967i \(-0.242815\pi\)
0.722886 + 0.690967i \(0.242815\pi\)
\(908\) −20.2227 −0.671113
\(909\) −15.4153 −0.511293
\(910\) 7.29987 0.241988
\(911\) 5.72789 0.189774 0.0948868 0.995488i \(-0.469751\pi\)
0.0948868 + 0.995488i \(0.469751\pi\)
\(912\) 0 0
\(913\) 1.00093 0.0331258
\(914\) 1.75855 0.0581677
\(915\) −3.48072 −0.115069
\(916\) −1.76274 −0.0582425
\(917\) 5.46422 0.180445
\(918\) −4.04986 −0.133665
\(919\) 7.93860 0.261870 0.130935 0.991391i \(-0.458202\pi\)
0.130935 + 0.991391i \(0.458202\pi\)
\(920\) −16.4748 −0.543159
\(921\) 3.45195 0.113746
\(922\) −9.59339 −0.315941
\(923\) 4.66747 0.153632
\(924\) 0.0767887 0.00252616
\(925\) −1.69819 −0.0558362
\(926\) 23.2532 0.764147
\(927\) −6.13347 −0.201450
\(928\) 20.8135 0.683236
\(929\) −28.9567 −0.950039 −0.475019 0.879975i \(-0.657559\pi\)
−0.475019 + 0.879975i \(0.657559\pi\)
\(930\) −2.73286 −0.0896141
\(931\) 0 0
\(932\) 11.2642 0.368972
\(933\) 4.72531 0.154700
\(934\) 22.4444 0.734403
\(935\) −0.223269 −0.00730168
\(936\) 30.7895 1.00639
\(937\) −14.6816 −0.479627 −0.239813 0.970819i \(-0.577086\pi\)
−0.239813 + 0.970819i \(0.577086\pi\)
\(938\) −6.13606 −0.200349
\(939\) −0.775939 −0.0253218
\(940\) 8.84375 0.288451
\(941\) −0.154755 −0.00504485 −0.00252243 0.999997i \(-0.500803\pi\)
−0.00252243 + 0.999997i \(0.500803\pi\)
\(942\) −3.02352 −0.0985114
\(943\) 57.0961 1.85931
\(944\) 12.2465 0.398590
\(945\) 4.20530 0.136798
\(946\) 1.25487 0.0407993
\(947\) −7.51807 −0.244304 −0.122152 0.992511i \(-0.538980\pi\)
−0.122152 + 0.992511i \(0.538980\pi\)
\(948\) −0.442838 −0.0143827
\(949\) −22.2854 −0.723415
\(950\) 0 0
\(951\) −8.94137 −0.289944
\(952\) 9.64840 0.312706
\(953\) 22.6743 0.734493 0.367247 0.930124i \(-0.380300\pi\)
0.367247 + 0.930124i \(0.380300\pi\)
\(954\) 0.830138 0.0268767
\(955\) 20.4758 0.662580
\(956\) −2.39863 −0.0775772
\(957\) −0.253413 −0.00819167
\(958\) 25.8437 0.834973
\(959\) −35.6494 −1.15118
\(960\) −3.09034 −0.0997403
\(961\) 12.0955 0.390177
\(962\) −6.54840 −0.211129
\(963\) 2.85875 0.0921219
\(964\) 18.9194 0.609354
\(965\) 11.0235 0.354860
\(966\) −4.23184 −0.136157
\(967\) 28.8344 0.927253 0.463627 0.886031i \(-0.346548\pi\)
0.463627 + 0.886031i \(0.346548\pi\)
\(968\) 33.6929 1.08293
\(969\) 0 0
\(970\) 19.1711 0.615546
\(971\) 26.3661 0.846130 0.423065 0.906099i \(-0.360954\pi\)
0.423065 + 0.906099i \(0.360954\pi\)
\(972\) 7.63395 0.244859
\(973\) −34.4150 −1.10329
\(974\) 39.5508 1.26729
\(975\) −1.33288 −0.0426863
\(976\) −16.2938 −0.521551
\(977\) 4.59218 0.146917 0.0734585 0.997298i \(-0.476596\pi\)
0.0734585 + 0.997298i \(0.476596\pi\)
\(978\) 8.23362 0.263282
\(979\) 1.30853 0.0418207
\(980\) 2.71810 0.0868267
\(981\) −46.4486 −1.48299
\(982\) 22.0140 0.702496
\(983\) 6.91473 0.220546 0.110273 0.993901i \(-0.464828\pi\)
0.110273 + 0.993901i \(0.464828\pi\)
\(984\) 12.3745 0.394484
\(985\) −19.8532 −0.632577
\(986\) −9.06187 −0.288589
\(987\) 7.98203 0.254071
\(988\) 0 0
\(989\) 45.6857 1.45272
\(990\) −0.421263 −0.0133886
\(991\) −38.0070 −1.20733 −0.603666 0.797237i \(-0.706294\pi\)
−0.603666 + 0.797237i \(0.706294\pi\)
\(992\) 27.4886 0.872763
\(993\) 7.10017 0.225317
\(994\) 2.75968 0.0875318
\(995\) 21.0026 0.665827
\(996\) 2.24768 0.0712205
\(997\) 16.2265 0.513899 0.256950 0.966425i \(-0.417283\pi\)
0.256950 + 0.966425i \(0.417283\pi\)
\(998\) −40.4657 −1.28092
\(999\) −3.77239 −0.119353
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 1805.2.a.o.1.3 4
5.4 even 2 9025.2.a.bg.1.2 4
19.7 even 3 95.2.e.c.11.2 8
19.11 even 3 95.2.e.c.26.2 yes 8
19.18 odd 2 1805.2.a.i.1.2 4
57.11 odd 6 855.2.k.h.406.3 8
57.26 odd 6 855.2.k.h.676.3 8
76.7 odd 6 1520.2.q.o.961.2 8
76.11 odd 6 1520.2.q.o.881.2 8
95.7 odd 12 475.2.j.c.49.4 16
95.49 even 6 475.2.e.e.26.3 8
95.64 even 6 475.2.e.e.201.3 8
95.68 odd 12 475.2.j.c.349.4 16
95.83 odd 12 475.2.j.c.49.5 16
95.87 odd 12 475.2.j.c.349.5 16
95.94 odd 2 9025.2.a.bp.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.e.c.11.2 8 19.7 even 3
95.2.e.c.26.2 yes 8 19.11 even 3
475.2.e.e.26.3 8 95.49 even 6
475.2.e.e.201.3 8 95.64 even 6
475.2.j.c.49.4 16 95.7 odd 12
475.2.j.c.49.5 16 95.83 odd 12
475.2.j.c.349.4 16 95.68 odd 12
475.2.j.c.349.5 16 95.87 odd 12
855.2.k.h.406.3 8 57.11 odd 6
855.2.k.h.676.3 8 57.26 odd 6
1520.2.q.o.881.2 8 76.11 odd 6
1520.2.q.o.961.2 8 76.7 odd 6
1805.2.a.i.1.2 4 19.18 odd 2
1805.2.a.o.1.3 4 1.1 even 1 trivial
9025.2.a.bg.1.2 4 5.4 even 2
9025.2.a.bp.1.3 4 95.94 odd 2