Properties

Label 1805.2.a.i.1.2
Level $1805$
Weight $2$
Character 1805.1
Self dual yes
Analytic conductor $14.413$
Analytic rank $1$
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: \(1\)
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 1.2
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 \(-0.626835\pi\)
−0.388003 + 0.921658i \(0.626835\pi\)
\(3\) 0.379334 0.219008 0.109504 0.993986i \(-0.465074\pi\)
0.109504 + 0.993986i \(0.465074\pi\)
\(4\) −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 \(-0.662006\pi\)
−0.487267 + 0.873253i \(0.662006\pi\)
\(14\) −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 \(-0.652691\pi\)
−0.461508 + 0.887136i \(0.652691\pi\)
\(30\) −0.416295 −0.0760048
\(31\) −6.56472 −1.17906 −0.589529 0.807747i \(-0.700687\pi\)
−0.589529 + 0.807747i \(0.700687\pi\)
\(32\) −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 \(-0.455421\pi\)
0.139590 + 0.990209i \(0.455421\pi\)
\(38\) 0 0
\(39\) −1.33288 −0.213431
\(40\) 3.06803 0.485098
\(41\) −10.6327 −1.66056 −0.830278 0.557349i \(-0.811818\pi\)
−0.830278 + 0.557349i \(0.811818\pi\)
\(42\) −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 \(-0.494210\pi\)
0.0181898 + 0.999835i \(0.494210\pi\)
\(54\) 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 \(-0.351804\pi\)
0.448935 + 0.893564i \(0.351804\pi\)
\(60\) −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 \(-0.442255\pi\)
0.180416 + 0.983590i \(0.442255\pi\)
\(68\) 1.32172 0.160282
\(69\) 2.03696 0.245221
\(70\) −2.07752 −0.248312
\(71\) −1.32835 −0.157646 −0.0788232 0.996889i \(-0.525116\pi\)
−0.0788232 + 0.996889i \(0.525116\pi\)
\(72\) −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 \(-0.473697\pi\)
0.0825408 + 0.996588i \(0.473697\pi\)
\(80\) −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 \(-0.672584\pi\)
−0.516011 + 0.856582i \(0.672584\pi\)
\(90\) 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 \(-0.847112\pi\)
−0.886851 + 0.462055i \(0.847112\pi\)
\(98\) 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 \(-0.533740\pi\)
−0.105800 + 0.994387i \(0.533740\pi\)
\(104\) −10.7802 −1.05709
\(105\) 0.718104 0.0700798
\(106\) −0.290654 −0.0282308
\(107\) 1.00093 0.0967631 0.0483815 0.998829i \(-0.484594\pi\)
0.0483815 + 0.998829i \(0.484594\pi\)
\(108\) 1.76743 0.170071
\(109\) −16.2629 −1.55770 −0.778852 0.627208i \(-0.784198\pi\)
−0.778852 + 0.627208i \(0.784198\pi\)
\(110\) −0.147496 −0.0140632
\(111\) 0.644181 0.0611430
\(112\) −3.36155 −0.317637
\(113\) −0.843010 −0.0793037 −0.0396519 0.999214i \(-0.512625\pi\)
−0.0396519 + 0.999214i \(0.512625\pi\)
\(114\) 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 \(-0.812482\pi\)
−0.831439 + 0.555616i \(0.812482\pi\)
\(128\) −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 \(-0.543326\pi\)
−0.135692 + 0.990751i \(0.543326\pi\)
\(152\) 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 \(-0.457965\pi\)
0.131674 + 0.991293i \(0.457965\pi\)
\(168\) 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 \(-0.368173\pi\)
0.402409 + 0.915460i \(0.368173\pi\)
\(174\) 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 \(-0.316257\pi\)
0.545718 + 0.837969i \(0.316257\pi\)
\(180\) 2.27240 0.169375
\(181\) 5.43261 0.403803 0.201901 0.979406i \(-0.435288\pi\)
0.201901 + 0.979406i \(0.435288\pi\)
\(182\) 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 \(-0.629860\pi\)
−0.396745 + 0.917929i \(0.629860\pi\)
\(194\) 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 \(-0.354454\pi\)
0.441478 + 0.897272i \(0.354454\pi\)
\(212\) −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 \(-0.739263\pi\)
−0.682856 + 0.730553i \(0.739263\pi\)
\(224\) −7.92688 −0.529637
\(225\) −2.85611 −0.190407
\(226\) 0.925152 0.0615402
\(227\) −25.4172 −1.68700 −0.843500 0.537129i \(-0.819509\pi\)
−0.843500 + 0.537129i \(0.819509\pi\)
\(228\) 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 \(-0.222305\pi\)
0.765877 + 0.642987i \(0.222305\pi\)
\(242\) 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 \(-0.291360\pi\)
0.609525 + 0.792767i \(0.291360\pi\)
\(258\) −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 \(-0.502798\pi\)
−0.00879088 + 0.999961i \(0.502798\pi\)
\(270\) 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 \(-0.0698809\pi\)
0.975998 + 0.217778i \(0.0698809\pi\)
\(282\) 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 \(-0.427562\pi\)
0.225610 + 0.974218i \(0.427562\pi\)
\(294\) 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 \(-0.416382\pi\)
0.259683 + 0.965694i \(0.416382\pi\)
\(308\) −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 \(-0.730270\pi\)
−0.661947 + 0.749551i \(0.730270\pi\)
\(318\) −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 \(-0.328014\pi\)
0.514403 + 0.857549i \(0.328014\pi\)
\(332\) −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 \(-0.150483\pi\)
0.890316 + 0.455343i \(0.150483\pi\)
\(338\) 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 \(-0.747138\pi\)
−0.700719 + 0.713437i \(0.747138\pi\)
\(374\) 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 \(-0.603194\pi\)
−0.318546 + 0.947907i \(0.603194\pi\)
\(380\) 0 0
\(381\) −7.10859 −0.364184
\(382\) −22.4709 −1.14971
\(383\) 5.35942 0.273854 0.136927 0.990581i \(-0.456277\pi\)
0.136927 + 0.990581i \(0.456277\pi\)
\(384\) −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 \(-0.561194\pi\)
−0.191063 + 0.981578i \(0.561194\pi\)
\(402\) −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 \(-0.355926\pi\)
0.437325 + 0.899304i \(0.355926\pi\)
\(410\) 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 \(-0.802228\pi\)
−0.813111 + 0.582109i \(0.802228\pi\)
\(422\) −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 \(-0.452477\pi\)
0.148743 + 0.988876i \(0.452477\pi\)
\(432\) 3.94461 0.189785
\(433\) 18.5552 0.891707 0.445854 0.895106i \(-0.352900\pi\)
0.445854 + 0.895106i \(0.352900\pi\)
\(434\) 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 \(-0.498273\pi\)
0.00542450 + 0.999985i \(0.498273\pi\)
\(440\) 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 \(-0.369017\pi\)
0.399980 + 0.916524i \(0.369017\pi\)
\(450\) 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 \(-0.804114\pi\)
−0.816546 + 0.577280i \(0.804114\pi\)
\(488\) 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 \(-0.200583\pi\)
0.807938 + 0.589267i \(0.200583\pi\)
\(510\) 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 \(-0.335071\pi\)
0.495264 + 0.868742i \(0.335071\pi\)
\(522\) −15.5798 −0.681910
\(523\) −0.532911 −0.0233026 −0.0116513 0.999932i \(-0.503709\pi\)
−0.0116513 + 0.999932i \(0.503709\pi\)
\(524\) −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 \(-0.660741\pi\)
−0.483792 + 0.875183i \(0.660741\pi\)
\(548\) 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 \(-0.326302\pi\)
0.519005 + 0.854771i \(0.326302\pi\)
\(564\) 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 \(-0.362161\pi\)
0.419626 + 0.907697i \(0.362161\pi\)
\(570\) 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 \(-0.570653\pi\)
−0.220146 + 0.975467i \(0.570653\pi\)
\(600\) 1.16381 0.0475122
\(601\) −15.0244 −0.612860 −0.306430 0.951893i \(-0.599134\pi\)
−0.306430 + 0.951893i \(0.599134\pi\)
\(602\) −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 \(-0.670805\pi\)
−0.511217 + 0.859452i \(0.670805\pi\)
\(608\) 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 \(-0.546764\pi\)
−0.146386 + 0.989227i \(0.546764\pi\)
\(642\) −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 \(-0.328544\pi\)
0.512972 + 0.858405i \(0.328544\pi\)
\(660\) −0.0405631 −0.00157892
\(661\) −3.78419 −0.147188 −0.0735941 0.997288i \(-0.523447\pi\)
−0.0735941 + 0.997288i \(0.523447\pi\)
\(662\) −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 \(-0.367071\pi\)
0.405575 + 0.914062i \(0.367071\pi\)
\(674\) −35.8731 −1.38178
\(675\) −2.22142 −0.0855025
\(676\) 0.520083 0.0200032
\(677\) 15.2744 0.587043 0.293522 0.955952i \(-0.405173\pi\)
0.293522 + 0.955952i \(0.405173\pi\)
\(678\) 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 \(-0.447374\pi\)
0.164578 + 0.986364i \(0.447374\pi\)
\(684\) 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 \(-0.508603\pi\)
−0.0270227 + 0.999635i \(0.508603\pi\)
\(744\) −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 \(-0.591642\pi\)
−0.283940 + 0.958842i \(0.591642\pi\)
\(752\) 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 \(-0.682452\pi\)
−0.542314 + 0.840176i \(0.682452\pi\)
\(774\) 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 \(-0.778807\pi\)
−0.768119 + 0.640307i \(0.778807\pi\)
\(788\) 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 \(-0.147525\pi\)
0.894510 + 0.447048i \(0.147525\pi\)
\(798\) 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 \(-0.632018\pi\)
−0.402958 + 0.915218i \(0.632018\pi\)
\(812\) 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 \(-0.690542\pi\)
−0.563490 + 0.826123i \(0.690542\pi\)
\(828\) 12.2024 0.424064
\(829\) 18.5305 0.643592 0.321796 0.946809i \(-0.395713\pi\)
0.321796 + 0.946809i \(0.395713\pi\)
\(830\) −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 \(-0.504542\pi\)
−0.0142688 + 0.999898i \(0.504542\pi\)
\(840\) 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 \(-0.217974\pi\)
0.774556 + 0.632506i \(0.217974\pi\)
\(858\) 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 \(-0.746461\pi\)
−0.699201 + 0.714925i \(0.746461\pi\)
\(864\) 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 \(-0.873754\pi\)
−0.922374 + 0.386298i \(0.873754\pi\)
\(878\) −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 \(-0.324892\pi\)
0.522788 + 0.852463i \(0.324892\pi\)
\(888\) 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 \(-0.757185\pi\)
−0.722886 + 0.690967i \(0.757185\pi\)
\(908\) 20.2227 0.671113
\(909\) −15.4153 −0.511293
\(910\) 7.29987 0.241988
\(911\) −5.72789 −0.189774 −0.0948868 0.995488i \(-0.530249\pi\)
−0.0948868 + 0.995488i \(0.530249\pi\)
\(912\) 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 \(-0.499197\pi\)
0.00252243 + 0.999997i \(0.499197\pi\)
\(942\) −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 \(-0.619700\pi\)
−0.367247 + 0.930124i \(0.619700\pi\)
\(954\) 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 \(-0.639046\pi\)
−0.423065 + 0.906099i \(0.639046\pi\)
\(972\) −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 \(-0.523404\pi\)
−0.0734585 + 0.997298i \(0.523404\pi\)
\(978\) 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 \(-0.535172\pi\)
−0.110273 + 0.993901i \(0.535172\pi\)
\(984\) −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 \(-0.293706\pi\)
0.603666 + 0.797237i \(0.293706\pi\)
\(992\) 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.i.1.2 4
5.4 even 2 9025.2.a.bp.1.3 4
19.8 odd 6 95.2.e.c.26.2 yes 8
19.12 odd 6 95.2.e.c.11.2 8
19.18 odd 2 1805.2.a.o.1.3 4
57.8 even 6 855.2.k.h.406.3 8
57.50 even 6 855.2.k.h.676.3 8
76.27 even 6 1520.2.q.o.881.2 8
76.31 even 6 1520.2.q.o.961.2 8
95.8 even 12 475.2.j.c.349.4 16
95.12 even 12 475.2.j.c.49.4 16
95.27 even 12 475.2.j.c.349.5 16
95.69 odd 6 475.2.e.e.201.3 8
95.84 odd 6 475.2.e.e.26.3 8
95.88 even 12 475.2.j.c.49.5 16
95.94 odd 2 9025.2.a.bg.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.e.c.11.2 8 19.12 odd 6
95.2.e.c.26.2 yes 8 19.8 odd 6
475.2.e.e.26.3 8 95.84 odd 6
475.2.e.e.201.3 8 95.69 odd 6
475.2.j.c.49.4 16 95.12 even 12
475.2.j.c.49.5 16 95.88 even 12
475.2.j.c.349.4 16 95.8 even 12
475.2.j.c.349.5 16 95.27 even 12
855.2.k.h.406.3 8 57.8 even 6
855.2.k.h.676.3 8 57.50 even 6
1520.2.q.o.881.2 8 76.27 even 6
1520.2.q.o.961.2 8 76.31 even 6
1805.2.a.i.1.2 4 1.1 even 1 trivial
1805.2.a.o.1.3 4 19.18 odd 2
9025.2.a.bg.1.2 4 95.94 odd 2
9025.2.a.bp.1.3 4 5.4 even 2