Properties

Label 2738.2.a.x.1.4
Level $2738$
Weight $2$
Character 2738.1
Self dual yes
Analytic conductor $21.863$
Analytic rank $0$
Dimension $18$
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] = [2738,2,Mod(1,2738)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2738, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2738.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2738 = 2 \cdot 37^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2738.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: \(21.8630400734\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 8 x^{17} - 8 x^{16} + 209 x^{15} - 253 x^{14} - 1996 x^{13} + 4196 x^{12} + 8667 x^{11} - 24522 x^{10} - 18284 x^{9} + 68697 x^{8} + 20865 x^{7} - 97011 x^{6} + \cdots + 113 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(1.39715\) of defining polynomial
Character \(\chi\) \(=\) 2738.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.75172 q^{3} +1.00000 q^{4} -2.09131 q^{5} -1.75172 q^{6} +1.21116 q^{7} +1.00000 q^{8} +0.0685155 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.75172 q^{3} +1.00000 q^{4} -2.09131 q^{5} -1.75172 q^{6} +1.21116 q^{7} +1.00000 q^{8} +0.0685155 q^{9} -2.09131 q^{10} +5.93957 q^{11} -1.75172 q^{12} +1.68357 q^{13} +1.21116 q^{14} +3.66338 q^{15} +1.00000 q^{16} -5.38331 q^{17} +0.0685155 q^{18} -0.495716 q^{19} -2.09131 q^{20} -2.12161 q^{21} +5.93957 q^{22} -2.33910 q^{23} -1.75172 q^{24} -0.626440 q^{25} +1.68357 q^{26} +5.13513 q^{27} +1.21116 q^{28} -8.30651 q^{29} +3.66338 q^{30} +9.20267 q^{31} +1.00000 q^{32} -10.4044 q^{33} -5.38331 q^{34} -2.53291 q^{35} +0.0685155 q^{36} -0.495716 q^{38} -2.94915 q^{39} -2.09131 q^{40} +10.4036 q^{41} -2.12161 q^{42} +1.93690 q^{43} +5.93957 q^{44} -0.143287 q^{45} -2.33910 q^{46} +8.87715 q^{47} -1.75172 q^{48} -5.53309 q^{49} -0.626440 q^{50} +9.43004 q^{51} +1.68357 q^{52} -6.58882 q^{53} +5.13513 q^{54} -12.4215 q^{55} +1.21116 q^{56} +0.868354 q^{57} -8.30651 q^{58} +1.02706 q^{59} +3.66338 q^{60} -1.46885 q^{61} +9.20267 q^{62} +0.0829833 q^{63} +1.00000 q^{64} -3.52087 q^{65} -10.4044 q^{66} +12.2895 q^{67} -5.38331 q^{68} +4.09744 q^{69} -2.53291 q^{70} -13.0711 q^{71} +0.0685155 q^{72} +8.54942 q^{73} +1.09735 q^{75} -0.495716 q^{76} +7.19378 q^{77} -2.94915 q^{78} +6.28496 q^{79} -2.09131 q^{80} -9.20085 q^{81} +10.4036 q^{82} +3.91197 q^{83} -2.12161 q^{84} +11.2581 q^{85} +1.93690 q^{86} +14.5507 q^{87} +5.93957 q^{88} +18.0337 q^{89} -0.143287 q^{90} +2.03908 q^{91} -2.33910 q^{92} -16.1205 q^{93} +8.87715 q^{94} +1.03669 q^{95} -1.75172 q^{96} +8.12116 q^{97} -5.53309 q^{98} +0.406952 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 18 q^{2} + 8 q^{3} + 18 q^{4} + 9 q^{5} + 8 q^{6} + 18 q^{7} + 18 q^{8} + 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 18 q^{2} + 8 q^{3} + 18 q^{4} + 9 q^{5} + 8 q^{6} + 18 q^{7} + 18 q^{8} + 26 q^{9} + 9 q^{10} + 10 q^{11} + 8 q^{12} + 9 q^{13} + 18 q^{14} - 4 q^{15} + 18 q^{16} + 13 q^{17} + 26 q^{18} + 2 q^{19} + 9 q^{20} + 24 q^{21} + 10 q^{22} - 11 q^{23} + 8 q^{24} + 49 q^{25} + 9 q^{26} + 29 q^{27} + 18 q^{28} + 30 q^{29} - 4 q^{30} - 8 q^{31} + 18 q^{32} + 42 q^{33} + 13 q^{34} + 25 q^{35} + 26 q^{36} + 2 q^{38} - 45 q^{39} + 9 q^{40} + 5 q^{41} + 24 q^{42} - 3 q^{43} + 10 q^{44} - 30 q^{45} - 11 q^{46} + 37 q^{47} + 8 q^{48} + 50 q^{49} + 49 q^{50} + 10 q^{51} + 9 q^{52} + 25 q^{53} + 29 q^{54} - 44 q^{55} + 18 q^{56} + 22 q^{57} + 30 q^{58} + 26 q^{59} - 4 q^{60} + 27 q^{61} - 8 q^{62} + 74 q^{63} + 18 q^{64} - 16 q^{65} + 42 q^{66} + 23 q^{67} + 13 q^{68} - 2 q^{69} + 25 q^{70} - 25 q^{71} + 26 q^{72} + 77 q^{73} - q^{75} + 2 q^{76} - 6 q^{77} - 45 q^{78} - 13 q^{79} + 9 q^{80} + 38 q^{81} + 5 q^{82} - 10 q^{83} + 24 q^{84} + 16 q^{85} - 3 q^{86} - 55 q^{87} + 10 q^{88} + 55 q^{89} - 30 q^{90} + 12 q^{91} - 11 q^{92} - 58 q^{93} + 37 q^{94} - 18 q^{95} + 8 q^{96} - 59 q^{97} + 50 q^{98} + 11 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.75172 −1.01135 −0.505677 0.862723i \(-0.668757\pi\)
−0.505677 + 0.862723i \(0.668757\pi\)
\(4\) 1.00000 0.500000
\(5\) −2.09131 −0.935260 −0.467630 0.883924i \(-0.654892\pi\)
−0.467630 + 0.883924i \(0.654892\pi\)
\(6\) −1.75172 −0.715136
\(7\) 1.21116 0.457776 0.228888 0.973453i \(-0.426491\pi\)
0.228888 + 0.973453i \(0.426491\pi\)
\(8\) 1.00000 0.353553
\(9\) 0.0685155 0.0228385
\(10\) −2.09131 −0.661329
\(11\) 5.93957 1.79085 0.895423 0.445215i \(-0.146873\pi\)
0.895423 + 0.445215i \(0.146873\pi\)
\(12\) −1.75172 −0.505677
\(13\) 1.68357 0.466939 0.233470 0.972364i \(-0.424992\pi\)
0.233470 + 0.972364i \(0.424992\pi\)
\(14\) 1.21116 0.323697
\(15\) 3.66338 0.945880
\(16\) 1.00000 0.250000
\(17\) −5.38331 −1.30564 −0.652822 0.757511i \(-0.726415\pi\)
−0.652822 + 0.757511i \(0.726415\pi\)
\(18\) 0.0685155 0.0161492
\(19\) −0.495716 −0.113725 −0.0568625 0.998382i \(-0.518110\pi\)
−0.0568625 + 0.998382i \(0.518110\pi\)
\(20\) −2.09131 −0.467630
\(21\) −2.12161 −0.462974
\(22\) 5.93957 1.26632
\(23\) −2.33910 −0.487736 −0.243868 0.969808i \(-0.578416\pi\)
−0.243868 + 0.969808i \(0.578416\pi\)
\(24\) −1.75172 −0.357568
\(25\) −0.626440 −0.125288
\(26\) 1.68357 0.330176
\(27\) 5.13513 0.988257
\(28\) 1.21116 0.228888
\(29\) −8.30651 −1.54248 −0.771240 0.636544i \(-0.780363\pi\)
−0.771240 + 0.636544i \(0.780363\pi\)
\(30\) 3.66338 0.668838
\(31\) 9.20267 1.65285 0.826425 0.563047i \(-0.190371\pi\)
0.826425 + 0.563047i \(0.190371\pi\)
\(32\) 1.00000 0.176777
\(33\) −10.4044 −1.81118
\(34\) −5.38331 −0.923230
\(35\) −2.53291 −0.428140
\(36\) 0.0685155 0.0114192
\(37\) 0 0
\(38\) −0.495716 −0.0804157
\(39\) −2.94915 −0.472241
\(40\) −2.09131 −0.330664
\(41\) 10.4036 1.62477 0.812383 0.583124i \(-0.198170\pi\)
0.812383 + 0.583124i \(0.198170\pi\)
\(42\) −2.12161 −0.327372
\(43\) 1.93690 0.295375 0.147687 0.989034i \(-0.452817\pi\)
0.147687 + 0.989034i \(0.452817\pi\)
\(44\) 5.93957 0.895423
\(45\) −0.143287 −0.0213599
\(46\) −2.33910 −0.344881
\(47\) 8.87715 1.29487 0.647433 0.762122i \(-0.275843\pi\)
0.647433 + 0.762122i \(0.275843\pi\)
\(48\) −1.75172 −0.252839
\(49\) −5.53309 −0.790441
\(50\) −0.626440 −0.0885920
\(51\) 9.43004 1.32047
\(52\) 1.68357 0.233470
\(53\) −6.58882 −0.905044 −0.452522 0.891753i \(-0.649476\pi\)
−0.452522 + 0.891753i \(0.649476\pi\)
\(54\) 5.13513 0.698803
\(55\) −12.4215 −1.67491
\(56\) 1.21116 0.161848
\(57\) 0.868354 0.115016
\(58\) −8.30651 −1.09070
\(59\) 1.02706 0.133712 0.0668560 0.997763i \(-0.478703\pi\)
0.0668560 + 0.997763i \(0.478703\pi\)
\(60\) 3.66338 0.472940
\(61\) −1.46885 −0.188067 −0.0940337 0.995569i \(-0.529976\pi\)
−0.0940337 + 0.995569i \(0.529976\pi\)
\(62\) 9.20267 1.16874
\(63\) 0.0829833 0.0104549
\(64\) 1.00000 0.125000
\(65\) −3.52087 −0.436710
\(66\) −10.4044 −1.28070
\(67\) 12.2895 1.50140 0.750702 0.660642i \(-0.229716\pi\)
0.750702 + 0.660642i \(0.229716\pi\)
\(68\) −5.38331 −0.652822
\(69\) 4.09744 0.493274
\(70\) −2.53291 −0.302741
\(71\) −13.0711 −1.55125 −0.775627 0.631191i \(-0.782566\pi\)
−0.775627 + 0.631191i \(0.782566\pi\)
\(72\) 0.0685155 0.00807462
\(73\) 8.54942 1.00063 0.500317 0.865842i \(-0.333217\pi\)
0.500317 + 0.865842i \(0.333217\pi\)
\(74\) 0 0
\(75\) 1.09735 0.126711
\(76\) −0.495716 −0.0568625
\(77\) 7.19378 0.819807
\(78\) −2.94915 −0.333925
\(79\) 6.28496 0.707113 0.353556 0.935413i \(-0.384972\pi\)
0.353556 + 0.935413i \(0.384972\pi\)
\(80\) −2.09131 −0.233815
\(81\) −9.20085 −1.02232
\(82\) 10.4036 1.14888
\(83\) 3.91197 0.429394 0.214697 0.976681i \(-0.431124\pi\)
0.214697 + 0.976681i \(0.431124\pi\)
\(84\) −2.12161 −0.231487
\(85\) 11.2581 1.22112
\(86\) 1.93690 0.208861
\(87\) 14.5507 1.55999
\(88\) 5.93957 0.633160
\(89\) 18.0337 1.91157 0.955784 0.294070i \(-0.0950098\pi\)
0.955784 + 0.294070i \(0.0950098\pi\)
\(90\) −0.143287 −0.0151038
\(91\) 2.03908 0.213754
\(92\) −2.33910 −0.243868
\(93\) −16.1205 −1.67162
\(94\) 8.87715 0.915609
\(95\) 1.03669 0.106363
\(96\) −1.75172 −0.178784
\(97\) 8.12116 0.824579 0.412290 0.911053i \(-0.364729\pi\)
0.412290 + 0.911053i \(0.364729\pi\)
\(98\) −5.53309 −0.558926
\(99\) 0.406952 0.0409002
\(100\) −0.626440 −0.0626440
\(101\) −9.39363 −0.934701 −0.467351 0.884072i \(-0.654791\pi\)
−0.467351 + 0.884072i \(0.654791\pi\)
\(102\) 9.43004 0.933713
\(103\) −0.279274 −0.0275177 −0.0137588 0.999905i \(-0.504380\pi\)
−0.0137588 + 0.999905i \(0.504380\pi\)
\(104\) 1.68357 0.165088
\(105\) 4.43694 0.433001
\(106\) −6.58882 −0.639963
\(107\) 11.1237 1.07537 0.537685 0.843146i \(-0.319299\pi\)
0.537685 + 0.843146i \(0.319299\pi\)
\(108\) 5.13513 0.494128
\(109\) 4.40630 0.422047 0.211024 0.977481i \(-0.432320\pi\)
0.211024 + 0.977481i \(0.432320\pi\)
\(110\) −12.4215 −1.18434
\(111\) 0 0
\(112\) 1.21116 0.114444
\(113\) 13.3652 1.25729 0.628646 0.777691i \(-0.283609\pi\)
0.628646 + 0.777691i \(0.283609\pi\)
\(114\) 0.868354 0.0813288
\(115\) 4.89177 0.456160
\(116\) −8.30651 −0.771240
\(117\) 0.115351 0.0106642
\(118\) 1.02706 0.0945486
\(119\) −6.52006 −0.597692
\(120\) 3.66338 0.334419
\(121\) 24.2785 2.20713
\(122\) −1.46885 −0.132984
\(123\) −18.2241 −1.64321
\(124\) 9.20267 0.826425
\(125\) 11.7666 1.05244
\(126\) 0.0829833 0.00739274
\(127\) 19.0384 1.68938 0.844692 0.535253i \(-0.179784\pi\)
0.844692 + 0.535253i \(0.179784\pi\)
\(128\) 1.00000 0.0883883
\(129\) −3.39290 −0.298728
\(130\) −3.52087 −0.308801
\(131\) 2.32597 0.203221 0.101610 0.994824i \(-0.467600\pi\)
0.101610 + 0.994824i \(0.467600\pi\)
\(132\) −10.4044 −0.905591
\(133\) −0.600392 −0.0520606
\(134\) 12.2895 1.06165
\(135\) −10.7391 −0.924278
\(136\) −5.38331 −0.461615
\(137\) −9.64452 −0.823987 −0.411993 0.911187i \(-0.635167\pi\)
−0.411993 + 0.911187i \(0.635167\pi\)
\(138\) 4.09744 0.348797
\(139\) 6.05196 0.513321 0.256660 0.966502i \(-0.417378\pi\)
0.256660 + 0.966502i \(0.417378\pi\)
\(140\) −2.53291 −0.214070
\(141\) −15.5503 −1.30957
\(142\) −13.0711 −1.09690
\(143\) 9.99970 0.836217
\(144\) 0.0685155 0.00570962
\(145\) 17.3715 1.44262
\(146\) 8.54942 0.707555
\(147\) 9.69241 0.799416
\(148\) 0 0
\(149\) 8.10585 0.664057 0.332029 0.943269i \(-0.392267\pi\)
0.332029 + 0.943269i \(0.392267\pi\)
\(150\) 1.09735 0.0895979
\(151\) 4.99063 0.406131 0.203066 0.979165i \(-0.434910\pi\)
0.203066 + 0.979165i \(0.434910\pi\)
\(152\) −0.495716 −0.0402079
\(153\) −0.368840 −0.0298189
\(154\) 7.19378 0.579691
\(155\) −19.2456 −1.54584
\(156\) −2.94915 −0.236121
\(157\) 4.80321 0.383338 0.191669 0.981460i \(-0.438610\pi\)
0.191669 + 0.981460i \(0.438610\pi\)
\(158\) 6.28496 0.500004
\(159\) 11.5418 0.915321
\(160\) −2.09131 −0.165332
\(161\) −2.83303 −0.223274
\(162\) −9.20085 −0.722887
\(163\) 5.92541 0.464114 0.232057 0.972702i \(-0.425454\pi\)
0.232057 + 0.972702i \(0.425454\pi\)
\(164\) 10.4036 0.812383
\(165\) 21.7589 1.69393
\(166\) 3.91197 0.303627
\(167\) −0.571625 −0.0442337 −0.0221168 0.999755i \(-0.507041\pi\)
−0.0221168 + 0.999755i \(0.507041\pi\)
\(168\) −2.12161 −0.163686
\(169\) −10.1656 −0.781968
\(170\) 11.2581 0.863460
\(171\) −0.0339642 −0.00259731
\(172\) 1.93690 0.147687
\(173\) 16.1829 1.23036 0.615180 0.788386i \(-0.289083\pi\)
0.615180 + 0.788386i \(0.289083\pi\)
\(174\) 14.5507 1.10308
\(175\) −0.758720 −0.0573538
\(176\) 5.93957 0.447712
\(177\) −1.79912 −0.135230
\(178\) 18.0337 1.35168
\(179\) −6.59689 −0.493075 −0.246538 0.969133i \(-0.579293\pi\)
−0.246538 + 0.969133i \(0.579293\pi\)
\(180\) −0.143287 −0.0106800
\(181\) 1.47280 0.109472 0.0547360 0.998501i \(-0.482568\pi\)
0.0547360 + 0.998501i \(0.482568\pi\)
\(182\) 2.03908 0.151147
\(183\) 2.57302 0.190203
\(184\) −2.33910 −0.172441
\(185\) 0 0
\(186\) −16.1205 −1.18201
\(187\) −31.9745 −2.33821
\(188\) 8.87715 0.647433
\(189\) 6.21948 0.452400
\(190\) 1.03669 0.0752097
\(191\) −23.8560 −1.72616 −0.863080 0.505067i \(-0.831468\pi\)
−0.863080 + 0.505067i \(0.831468\pi\)
\(192\) −1.75172 −0.126419
\(193\) −16.8415 −1.21228 −0.606138 0.795360i \(-0.707282\pi\)
−0.606138 + 0.795360i \(0.707282\pi\)
\(194\) 8.12116 0.583066
\(195\) 6.16757 0.441669
\(196\) −5.53309 −0.395221
\(197\) 18.0955 1.28925 0.644625 0.764499i \(-0.277014\pi\)
0.644625 + 0.764499i \(0.277014\pi\)
\(198\) 0.406952 0.0289208
\(199\) −15.9676 −1.13191 −0.565956 0.824435i \(-0.691493\pi\)
−0.565956 + 0.824435i \(0.691493\pi\)
\(200\) −0.626440 −0.0442960
\(201\) −21.5278 −1.51845
\(202\) −9.39363 −0.660934
\(203\) −10.0605 −0.706111
\(204\) 9.43004 0.660235
\(205\) −21.7571 −1.51958
\(206\) −0.279274 −0.0194579
\(207\) −0.160264 −0.0111391
\(208\) 1.68357 0.116735
\(209\) −2.94434 −0.203664
\(210\) 4.43694 0.306178
\(211\) −7.31223 −0.503395 −0.251697 0.967806i \(-0.580989\pi\)
−0.251697 + 0.967806i \(0.580989\pi\)
\(212\) −6.58882 −0.452522
\(213\) 22.8969 1.56887
\(214\) 11.1237 0.760402
\(215\) −4.05065 −0.276252
\(216\) 5.13513 0.349402
\(217\) 11.1459 0.756635
\(218\) 4.40630 0.298432
\(219\) −14.9762 −1.01200
\(220\) −12.4215 −0.837454
\(221\) −9.06320 −0.609657
\(222\) 0 0
\(223\) 15.9777 1.06995 0.534973 0.844869i \(-0.320322\pi\)
0.534973 + 0.844869i \(0.320322\pi\)
\(224\) 1.21116 0.0809241
\(225\) −0.0429208 −0.00286139
\(226\) 13.3652 0.889040
\(227\) 3.46336 0.229871 0.114936 0.993373i \(-0.463334\pi\)
0.114936 + 0.993373i \(0.463334\pi\)
\(228\) 0.868354 0.0575082
\(229\) −15.1822 −1.00327 −0.501635 0.865079i \(-0.667268\pi\)
−0.501635 + 0.865079i \(0.667268\pi\)
\(230\) 4.89177 0.322554
\(231\) −12.6015 −0.829116
\(232\) −8.30651 −0.545349
\(233\) −5.71207 −0.374210 −0.187105 0.982340i \(-0.559910\pi\)
−0.187105 + 0.982340i \(0.559910\pi\)
\(234\) 0.115351 0.00754072
\(235\) −18.5648 −1.21104
\(236\) 1.02706 0.0668560
\(237\) −11.0095 −0.715142
\(238\) −6.52006 −0.422632
\(239\) −7.10847 −0.459809 −0.229904 0.973213i \(-0.573841\pi\)
−0.229904 + 0.973213i \(0.573841\pi\)
\(240\) 3.66338 0.236470
\(241\) −10.1248 −0.652193 −0.326096 0.945337i \(-0.605733\pi\)
−0.326096 + 0.945337i \(0.605733\pi\)
\(242\) 24.2785 1.56068
\(243\) 0.711895 0.0456681
\(244\) −1.46885 −0.0940337
\(245\) 11.5714 0.739268
\(246\) −18.2241 −1.16193
\(247\) −0.834575 −0.0531027
\(248\) 9.20267 0.584370
\(249\) −6.85266 −0.434270
\(250\) 11.7666 0.744186
\(251\) 13.7490 0.867832 0.433916 0.900953i \(-0.357131\pi\)
0.433916 + 0.900953i \(0.357131\pi\)
\(252\) 0.0829833 0.00522746
\(253\) −13.8932 −0.873460
\(254\) 19.0384 1.19457
\(255\) −19.7211 −1.23498
\(256\) 1.00000 0.0625000
\(257\) −7.08203 −0.441765 −0.220883 0.975300i \(-0.570894\pi\)
−0.220883 + 0.975300i \(0.570894\pi\)
\(258\) −3.39290 −0.211233
\(259\) 0 0
\(260\) −3.52087 −0.218355
\(261\) −0.569124 −0.0352279
\(262\) 2.32597 0.143699
\(263\) 4.60769 0.284122 0.142061 0.989858i \(-0.454627\pi\)
0.142061 + 0.989858i \(0.454627\pi\)
\(264\) −10.4044 −0.640349
\(265\) 13.7792 0.846452
\(266\) −0.600392 −0.0368124
\(267\) −31.5899 −1.93327
\(268\) 12.2895 0.750702
\(269\) −5.21884 −0.318198 −0.159099 0.987263i \(-0.550859\pi\)
−0.159099 + 0.987263i \(0.550859\pi\)
\(270\) −10.7391 −0.653563
\(271\) 11.2863 0.685596 0.342798 0.939409i \(-0.388625\pi\)
0.342798 + 0.939409i \(0.388625\pi\)
\(272\) −5.38331 −0.326411
\(273\) −3.57189 −0.216181
\(274\) −9.64452 −0.582647
\(275\) −3.72078 −0.224372
\(276\) 4.09744 0.246637
\(277\) 0.522535 0.0313961 0.0156981 0.999877i \(-0.495003\pi\)
0.0156981 + 0.999877i \(0.495003\pi\)
\(278\) 6.05196 0.362973
\(279\) 0.630525 0.0377486
\(280\) −2.53291 −0.151370
\(281\) −14.8186 −0.884005 −0.442003 0.897014i \(-0.645732\pi\)
−0.442003 + 0.897014i \(0.645732\pi\)
\(282\) −15.5503 −0.926005
\(283\) 3.99925 0.237731 0.118865 0.992910i \(-0.462074\pi\)
0.118865 + 0.992910i \(0.462074\pi\)
\(284\) −13.0711 −0.775627
\(285\) −1.81599 −0.107570
\(286\) 9.99970 0.591295
\(287\) 12.6004 0.743779
\(288\) 0.0685155 0.00403731
\(289\) 11.9800 0.704706
\(290\) 17.3715 1.02009
\(291\) −14.2260 −0.833942
\(292\) 8.54942 0.500317
\(293\) −10.1604 −0.593579 −0.296789 0.954943i \(-0.595916\pi\)
−0.296789 + 0.954943i \(0.595916\pi\)
\(294\) 9.69241 0.565273
\(295\) −2.14790 −0.125056
\(296\) 0 0
\(297\) 30.5005 1.76982
\(298\) 8.10585 0.469559
\(299\) −3.93805 −0.227743
\(300\) 1.09735 0.0633553
\(301\) 2.34590 0.135215
\(302\) 4.99063 0.287178
\(303\) 16.4550 0.945314
\(304\) −0.495716 −0.0284313
\(305\) 3.07182 0.175892
\(306\) −0.368840 −0.0210852
\(307\) 22.4774 1.28285 0.641425 0.767186i \(-0.278344\pi\)
0.641425 + 0.767186i \(0.278344\pi\)
\(308\) 7.19378 0.409903
\(309\) 0.489209 0.0278301
\(310\) −19.2456 −1.09308
\(311\) −4.40563 −0.249820 −0.124910 0.992168i \(-0.539864\pi\)
−0.124910 + 0.992168i \(0.539864\pi\)
\(312\) −2.94915 −0.166963
\(313\) −0.741008 −0.0418842 −0.0209421 0.999781i \(-0.506667\pi\)
−0.0209421 + 0.999781i \(0.506667\pi\)
\(314\) 4.80321 0.271061
\(315\) −0.173543 −0.00977806
\(316\) 6.28496 0.353556
\(317\) 25.2386 1.41754 0.708772 0.705438i \(-0.249250\pi\)
0.708772 + 0.705438i \(0.249250\pi\)
\(318\) 11.5418 0.647229
\(319\) −49.3371 −2.76235
\(320\) −2.09131 −0.116908
\(321\) −19.4856 −1.08758
\(322\) −2.83303 −0.157878
\(323\) 2.66859 0.148484
\(324\) −9.20085 −0.511158
\(325\) −1.05466 −0.0585019
\(326\) 5.92541 0.328178
\(327\) −7.71859 −0.426839
\(328\) 10.4036 0.574442
\(329\) 10.7517 0.592759
\(330\) 21.7589 1.19779
\(331\) 4.58769 0.252162 0.126081 0.992020i \(-0.459760\pi\)
0.126081 + 0.992020i \(0.459760\pi\)
\(332\) 3.91197 0.214697
\(333\) 0 0
\(334\) −0.571625 −0.0312779
\(335\) −25.7011 −1.40420
\(336\) −2.12161 −0.115743
\(337\) −25.8343 −1.40729 −0.703643 0.710554i \(-0.748445\pi\)
−0.703643 + 0.710554i \(0.748445\pi\)
\(338\) −10.1656 −0.552935
\(339\) −23.4121 −1.27157
\(340\) 11.2581 0.610559
\(341\) 54.6599 2.96000
\(342\) −0.0339642 −0.00183657
\(343\) −15.1796 −0.819621
\(344\) 1.93690 0.104431
\(345\) −8.56900 −0.461340
\(346\) 16.1829 0.869996
\(347\) −4.89892 −0.262988 −0.131494 0.991317i \(-0.541977\pi\)
−0.131494 + 0.991317i \(0.541977\pi\)
\(348\) 14.5507 0.779997
\(349\) 29.0275 1.55381 0.776904 0.629619i \(-0.216789\pi\)
0.776904 + 0.629619i \(0.216789\pi\)
\(350\) −0.758720 −0.0405553
\(351\) 8.64538 0.461456
\(352\) 5.93957 0.316580
\(353\) −26.9530 −1.43456 −0.717282 0.696783i \(-0.754614\pi\)
−0.717282 + 0.696783i \(0.754614\pi\)
\(354\) −1.79912 −0.0956222
\(355\) 27.3357 1.45083
\(356\) 18.0337 0.955784
\(357\) 11.4213 0.604479
\(358\) −6.59689 −0.348657
\(359\) −2.33163 −0.123058 −0.0615292 0.998105i \(-0.519598\pi\)
−0.0615292 + 0.998105i \(0.519598\pi\)
\(360\) −0.143287 −0.00755188
\(361\) −18.7543 −0.987067
\(362\) 1.47280 0.0774084
\(363\) −42.5290 −2.23219
\(364\) 2.03908 0.106877
\(365\) −17.8795 −0.935854
\(366\) 2.57302 0.134494
\(367\) 1.11791 0.0583544 0.0291772 0.999574i \(-0.490711\pi\)
0.0291772 + 0.999574i \(0.490711\pi\)
\(368\) −2.33910 −0.121934
\(369\) 0.712806 0.0371072
\(370\) 0 0
\(371\) −7.98013 −0.414307
\(372\) −16.1205 −0.835808
\(373\) −7.12913 −0.369132 −0.184566 0.982820i \(-0.559088\pi\)
−0.184566 + 0.982820i \(0.559088\pi\)
\(374\) −31.9745 −1.65336
\(375\) −20.6118 −1.06439
\(376\) 8.87715 0.457804
\(377\) −13.9846 −0.720245
\(378\) 6.21948 0.319895
\(379\) 4.72138 0.242521 0.121260 0.992621i \(-0.461306\pi\)
0.121260 + 0.992621i \(0.461306\pi\)
\(380\) 1.03669 0.0531813
\(381\) −33.3499 −1.70857
\(382\) −23.8560 −1.22058
\(383\) −17.4079 −0.889505 −0.444752 0.895654i \(-0.646708\pi\)
−0.444752 + 0.895654i \(0.646708\pi\)
\(384\) −1.75172 −0.0893920
\(385\) −15.0444 −0.766733
\(386\) −16.8415 −0.857208
\(387\) 0.132708 0.00674591
\(388\) 8.12116 0.412290
\(389\) −35.7110 −1.81062 −0.905309 0.424754i \(-0.860361\pi\)
−0.905309 + 0.424754i \(0.860361\pi\)
\(390\) 6.16757 0.312307
\(391\) 12.5921 0.636809
\(392\) −5.53309 −0.279463
\(393\) −4.07444 −0.205528
\(394\) 18.0955 0.911637
\(395\) −13.1438 −0.661335
\(396\) 0.406952 0.0204501
\(397\) −11.6241 −0.583396 −0.291698 0.956510i \(-0.594220\pi\)
−0.291698 + 0.956510i \(0.594220\pi\)
\(398\) −15.9676 −0.800383
\(399\) 1.05172 0.0526517
\(400\) −0.626440 −0.0313220
\(401\) 29.4827 1.47229 0.736147 0.676822i \(-0.236643\pi\)
0.736147 + 0.676822i \(0.236643\pi\)
\(402\) −21.5278 −1.07371
\(403\) 15.4934 0.771781
\(404\) −9.39363 −0.467351
\(405\) 19.2418 0.956133
\(406\) −10.0605 −0.499296
\(407\) 0 0
\(408\) 9.43004 0.466856
\(409\) 32.8533 1.62449 0.812245 0.583317i \(-0.198245\pi\)
0.812245 + 0.583317i \(0.198245\pi\)
\(410\) −21.7571 −1.07450
\(411\) 16.8945 0.833343
\(412\) −0.279274 −0.0137588
\(413\) 1.24394 0.0612101
\(414\) −0.160264 −0.00787657
\(415\) −8.18112 −0.401595
\(416\) 1.68357 0.0825440
\(417\) −10.6013 −0.519149
\(418\) −2.94434 −0.144012
\(419\) 19.2145 0.938692 0.469346 0.883014i \(-0.344490\pi\)
0.469346 + 0.883014i \(0.344490\pi\)
\(420\) 4.43694 0.216501
\(421\) 36.4583 1.77687 0.888435 0.459002i \(-0.151793\pi\)
0.888435 + 0.459002i \(0.151793\pi\)
\(422\) −7.31223 −0.355954
\(423\) 0.608222 0.0295728
\(424\) −6.58882 −0.319981
\(425\) 3.37232 0.163581
\(426\) 22.8969 1.10936
\(427\) −1.77902 −0.0860928
\(428\) 11.1237 0.537685
\(429\) −17.5167 −0.845712
\(430\) −4.05065 −0.195340
\(431\) −9.26831 −0.446439 −0.223219 0.974768i \(-0.571657\pi\)
−0.223219 + 0.974768i \(0.571657\pi\)
\(432\) 5.13513 0.247064
\(433\) −24.0022 −1.15347 −0.576735 0.816931i \(-0.695674\pi\)
−0.576735 + 0.816931i \(0.695674\pi\)
\(434\) 11.1459 0.535022
\(435\) −30.4299 −1.45900
\(436\) 4.40630 0.211024
\(437\) 1.15953 0.0554678
\(438\) −14.9762 −0.715590
\(439\) −6.86675 −0.327732 −0.163866 0.986483i \(-0.552396\pi\)
−0.163866 + 0.986483i \(0.552396\pi\)
\(440\) −12.4215 −0.592169
\(441\) −0.379102 −0.0180525
\(442\) −9.06320 −0.431092
\(443\) 0.332029 0.0157752 0.00788759 0.999969i \(-0.497489\pi\)
0.00788759 + 0.999969i \(0.497489\pi\)
\(444\) 0 0
\(445\) −37.7140 −1.78781
\(446\) 15.9777 0.756566
\(447\) −14.1992 −0.671597
\(448\) 1.21116 0.0572220
\(449\) 9.94564 0.469364 0.234682 0.972072i \(-0.424595\pi\)
0.234682 + 0.972072i \(0.424595\pi\)
\(450\) −0.0429208 −0.00202331
\(451\) 61.7928 2.90971
\(452\) 13.3652 0.628646
\(453\) −8.74217 −0.410743
\(454\) 3.46336 0.162544
\(455\) −4.26434 −0.199915
\(456\) 0.868354 0.0406644
\(457\) 28.3766 1.32740 0.663701 0.747998i \(-0.268985\pi\)
0.663701 + 0.747998i \(0.268985\pi\)
\(458\) −15.1822 −0.709420
\(459\) −27.6440 −1.29031
\(460\) 4.89177 0.228080
\(461\) 4.57081 0.212884 0.106442 0.994319i \(-0.466054\pi\)
0.106442 + 0.994319i \(0.466054\pi\)
\(462\) −12.6015 −0.586273
\(463\) 13.6661 0.635116 0.317558 0.948239i \(-0.397137\pi\)
0.317558 + 0.948239i \(0.397137\pi\)
\(464\) −8.30651 −0.385620
\(465\) 33.7129 1.56340
\(466\) −5.71207 −0.264607
\(467\) 5.16486 0.239001 0.119501 0.992834i \(-0.461871\pi\)
0.119501 + 0.992834i \(0.461871\pi\)
\(468\) 0.115351 0.00533210
\(469\) 14.8846 0.687306
\(470\) −18.5648 −0.856332
\(471\) −8.41386 −0.387690
\(472\) 1.02706 0.0472743
\(473\) 11.5044 0.528971
\(474\) −11.0095 −0.505682
\(475\) 0.310536 0.0142484
\(476\) −6.52006 −0.298846
\(477\) −0.451436 −0.0206698
\(478\) −7.10847 −0.325134
\(479\) −33.7622 −1.54264 −0.771318 0.636450i \(-0.780402\pi\)
−0.771318 + 0.636450i \(0.780402\pi\)
\(480\) 3.66338 0.167210
\(481\) 0 0
\(482\) −10.1248 −0.461170
\(483\) 4.96266 0.225809
\(484\) 24.2785 1.10357
\(485\) −16.9838 −0.771196
\(486\) 0.711895 0.0322922
\(487\) 32.6970 1.48164 0.740821 0.671703i \(-0.234437\pi\)
0.740821 + 0.671703i \(0.234437\pi\)
\(488\) −1.46885 −0.0664919
\(489\) −10.3796 −0.469384
\(490\) 11.5714 0.522742
\(491\) −24.2572 −1.09471 −0.547356 0.836900i \(-0.684366\pi\)
−0.547356 + 0.836900i \(0.684366\pi\)
\(492\) −18.2241 −0.821607
\(493\) 44.7165 2.01393
\(494\) −0.834575 −0.0375493
\(495\) −0.851061 −0.0382524
\(496\) 9.20267 0.413212
\(497\) −15.8312 −0.710127
\(498\) −6.85266 −0.307075
\(499\) −18.6640 −0.835514 −0.417757 0.908559i \(-0.637184\pi\)
−0.417757 + 0.908559i \(0.637184\pi\)
\(500\) 11.7666 0.526219
\(501\) 1.00133 0.0447359
\(502\) 13.7490 0.613650
\(503\) −8.58318 −0.382705 −0.191353 0.981521i \(-0.561287\pi\)
−0.191353 + 0.981521i \(0.561287\pi\)
\(504\) 0.0829833 0.00369637
\(505\) 19.6450 0.874189
\(506\) −13.8932 −0.617630
\(507\) 17.8072 0.790847
\(508\) 19.0384 0.844692
\(509\) −24.8965 −1.10352 −0.551759 0.834004i \(-0.686043\pi\)
−0.551759 + 0.834004i \(0.686043\pi\)
\(510\) −19.7211 −0.873265
\(511\) 10.3547 0.458066
\(512\) 1.00000 0.0441942
\(513\) −2.54557 −0.112390
\(514\) −7.08203 −0.312375
\(515\) 0.584047 0.0257362
\(516\) −3.39290 −0.149364
\(517\) 52.7265 2.31891
\(518\) 0 0
\(519\) −28.3478 −1.24433
\(520\) −3.52087 −0.154400
\(521\) −14.2132 −0.622691 −0.311345 0.950297i \(-0.600780\pi\)
−0.311345 + 0.950297i \(0.600780\pi\)
\(522\) −0.569124 −0.0249099
\(523\) 33.4160 1.46118 0.730591 0.682816i \(-0.239245\pi\)
0.730591 + 0.682816i \(0.239245\pi\)
\(524\) 2.32597 0.101610
\(525\) 1.32906 0.0580051
\(526\) 4.60769 0.200905
\(527\) −49.5408 −2.15803
\(528\) −10.4044 −0.452795
\(529\) −17.5286 −0.762114
\(530\) 13.7792 0.598532
\(531\) 0.0703696 0.00305378
\(532\) −0.600392 −0.0260303
\(533\) 17.5152 0.758667
\(534\) −31.5899 −1.36703
\(535\) −23.2631 −1.00575
\(536\) 12.2895 0.530826
\(537\) 11.5559 0.498674
\(538\) −5.21884 −0.225000
\(539\) −32.8641 −1.41556
\(540\) −10.7391 −0.462139
\(541\) −26.5225 −1.14029 −0.570146 0.821544i \(-0.693113\pi\)
−0.570146 + 0.821544i \(0.693113\pi\)
\(542\) 11.2863 0.484790
\(543\) −2.57992 −0.110715
\(544\) −5.38331 −0.230807
\(545\) −9.21492 −0.394724
\(546\) −3.57189 −0.152863
\(547\) 7.21041 0.308295 0.154147 0.988048i \(-0.450737\pi\)
0.154147 + 0.988048i \(0.450737\pi\)
\(548\) −9.64452 −0.411993
\(549\) −0.100639 −0.00429518
\(550\) −3.72078 −0.158655
\(551\) 4.11767 0.175419
\(552\) 4.09744 0.174399
\(553\) 7.61210 0.323699
\(554\) 0.522535 0.0222004
\(555\) 0 0
\(556\) 6.05196 0.256660
\(557\) 12.5459 0.531585 0.265793 0.964030i \(-0.414366\pi\)
0.265793 + 0.964030i \(0.414366\pi\)
\(558\) 0.630525 0.0266923
\(559\) 3.26092 0.137922
\(560\) −2.53291 −0.107035
\(561\) 56.0103 2.36476
\(562\) −14.8186 −0.625086
\(563\) 19.9469 0.840660 0.420330 0.907371i \(-0.361914\pi\)
0.420330 + 0.907371i \(0.361914\pi\)
\(564\) −15.5503 −0.654784
\(565\) −27.9507 −1.17590
\(566\) 3.99925 0.168101
\(567\) −11.1437 −0.467992
\(568\) −13.0711 −0.548451
\(569\) −2.55565 −0.107138 −0.0535692 0.998564i \(-0.517060\pi\)
−0.0535692 + 0.998564i \(0.517060\pi\)
\(570\) −1.81599 −0.0760636
\(571\) −11.1405 −0.466216 −0.233108 0.972451i \(-0.574889\pi\)
−0.233108 + 0.972451i \(0.574889\pi\)
\(572\) 9.99970 0.418109
\(573\) 41.7890 1.74576
\(574\) 12.6004 0.525931
\(575\) 1.46531 0.0611074
\(576\) 0.0685155 0.00285481
\(577\) −16.6278 −0.692226 −0.346113 0.938193i \(-0.612499\pi\)
−0.346113 + 0.938193i \(0.612499\pi\)
\(578\) 11.9800 0.498302
\(579\) 29.5015 1.22604
\(580\) 17.3715 0.721310
\(581\) 4.73802 0.196566
\(582\) −14.2260 −0.589686
\(583\) −39.1347 −1.62080
\(584\) 8.54942 0.353778
\(585\) −0.241234 −0.00997380
\(586\) −10.1604 −0.419724
\(587\) −26.0244 −1.07414 −0.537070 0.843538i \(-0.680469\pi\)
−0.537070 + 0.843538i \(0.680469\pi\)
\(588\) 9.69241 0.399708
\(589\) −4.56191 −0.187970
\(590\) −2.14790 −0.0884276
\(591\) −31.6982 −1.30389
\(592\) 0 0
\(593\) −17.1659 −0.704917 −0.352459 0.935827i \(-0.614654\pi\)
−0.352459 + 0.935827i \(0.614654\pi\)
\(594\) 30.5005 1.25145
\(595\) 13.6354 0.558998
\(596\) 8.10585 0.332029
\(597\) 27.9707 1.14477
\(598\) −3.93805 −0.161039
\(599\) −18.8295 −0.769351 −0.384675 0.923052i \(-0.625687\pi\)
−0.384675 + 0.923052i \(0.625687\pi\)
\(600\) 1.09735 0.0447990
\(601\) 28.5102 1.16295 0.581477 0.813563i \(-0.302475\pi\)
0.581477 + 0.813563i \(0.302475\pi\)
\(602\) 2.34590 0.0956117
\(603\) 0.842022 0.0342898
\(604\) 4.99063 0.203066
\(605\) −50.7737 −2.06424
\(606\) 16.4550 0.668438
\(607\) −35.6551 −1.44720 −0.723598 0.690222i \(-0.757513\pi\)
−0.723598 + 0.690222i \(0.757513\pi\)
\(608\) −0.495716 −0.0201039
\(609\) 17.6232 0.714128
\(610\) 3.07182 0.124374
\(611\) 14.9453 0.604624
\(612\) −0.368840 −0.0149095
\(613\) −21.3524 −0.862416 −0.431208 0.902253i \(-0.641912\pi\)
−0.431208 + 0.902253i \(0.641912\pi\)
\(614\) 22.4774 0.907112
\(615\) 38.1122 1.53683
\(616\) 7.19378 0.289845
\(617\) 4.49896 0.181121 0.0905606 0.995891i \(-0.471134\pi\)
0.0905606 + 0.995891i \(0.471134\pi\)
\(618\) 0.489209 0.0196789
\(619\) 28.2607 1.13589 0.567946 0.823066i \(-0.307738\pi\)
0.567946 + 0.823066i \(0.307738\pi\)
\(620\) −19.2456 −0.772922
\(621\) −12.0116 −0.482008
\(622\) −4.40563 −0.176650
\(623\) 21.8417 0.875070
\(624\) −2.94915 −0.118060
\(625\) −21.4754 −0.859015
\(626\) −0.741008 −0.0296166
\(627\) 5.15765 0.205977
\(628\) 4.80321 0.191669
\(629\) 0 0
\(630\) −0.173543 −0.00691414
\(631\) −0.688404 −0.0274049 −0.0137025 0.999906i \(-0.504362\pi\)
−0.0137025 + 0.999906i \(0.504362\pi\)
\(632\) 6.28496 0.250002
\(633\) 12.8090 0.509111
\(634\) 25.2386 1.00235
\(635\) −39.8151 −1.58001
\(636\) 11.5418 0.457660
\(637\) −9.31536 −0.369088
\(638\) −49.3371 −1.95327
\(639\) −0.895572 −0.0354283
\(640\) −2.09131 −0.0826661
\(641\) −23.6488 −0.934073 −0.467036 0.884238i \(-0.654678\pi\)
−0.467036 + 0.884238i \(0.654678\pi\)
\(642\) −19.4856 −0.769036
\(643\) −43.3182 −1.70830 −0.854151 0.520026i \(-0.825922\pi\)
−0.854151 + 0.520026i \(0.825922\pi\)
\(644\) −2.83303 −0.111637
\(645\) 7.09560 0.279389
\(646\) 2.66859 0.104994
\(647\) 34.6124 1.36075 0.680376 0.732863i \(-0.261816\pi\)
0.680376 + 0.732863i \(0.261816\pi\)
\(648\) −9.20085 −0.361444
\(649\) 6.10030 0.239458
\(650\) −1.05466 −0.0413671
\(651\) −19.5245 −0.765226
\(652\) 5.92541 0.232057
\(653\) 29.5930 1.15806 0.579032 0.815305i \(-0.303431\pi\)
0.579032 + 0.815305i \(0.303431\pi\)
\(654\) −7.71859 −0.301821
\(655\) −4.86431 −0.190064
\(656\) 10.4036 0.406192
\(657\) 0.585768 0.0228530
\(658\) 10.7517 0.419144
\(659\) −44.4752 −1.73251 −0.866253 0.499606i \(-0.833478\pi\)
−0.866253 + 0.499606i \(0.833478\pi\)
\(660\) 21.7589 0.846963
\(661\) 1.68358 0.0654838 0.0327419 0.999464i \(-0.489576\pi\)
0.0327419 + 0.999464i \(0.489576\pi\)
\(662\) 4.58769 0.178306
\(663\) 15.8762 0.616579
\(664\) 3.91197 0.151814
\(665\) 1.25560 0.0486902
\(666\) 0 0
\(667\) 19.4298 0.752323
\(668\) −0.571625 −0.0221168
\(669\) −27.9884 −1.08209
\(670\) −25.7011 −0.992921
\(671\) −8.72436 −0.336800
\(672\) −2.12161 −0.0818430
\(673\) 26.6133 1.02587 0.512933 0.858429i \(-0.328559\pi\)
0.512933 + 0.858429i \(0.328559\pi\)
\(674\) −25.8343 −0.995102
\(675\) −3.21685 −0.123817
\(676\) −10.1656 −0.390984
\(677\) 6.85446 0.263438 0.131719 0.991287i \(-0.457950\pi\)
0.131719 + 0.991287i \(0.457950\pi\)
\(678\) −23.4121 −0.899135
\(679\) 9.83604 0.377473
\(680\) 11.2581 0.431730
\(681\) −6.06683 −0.232482
\(682\) 54.6599 2.09304
\(683\) −41.2133 −1.57698 −0.788491 0.615046i \(-0.789137\pi\)
−0.788491 + 0.615046i \(0.789137\pi\)
\(684\) −0.0339642 −0.00129865
\(685\) 20.1696 0.770642
\(686\) −15.1796 −0.579560
\(687\) 26.5950 1.01466
\(688\) 1.93690 0.0738436
\(689\) −11.0928 −0.422601
\(690\) −8.56900 −0.326216
\(691\) −4.89261 −0.186123 −0.0930617 0.995660i \(-0.529665\pi\)
−0.0930617 + 0.995660i \(0.529665\pi\)
\(692\) 16.1829 0.615180
\(693\) 0.492885 0.0187231
\(694\) −4.89892 −0.185961
\(695\) −12.6565 −0.480089
\(696\) 14.5507 0.551541
\(697\) −56.0057 −2.12137
\(698\) 29.0275 1.09871
\(699\) 10.0059 0.378459
\(700\) −0.758720 −0.0286769
\(701\) 17.3040 0.653562 0.326781 0.945100i \(-0.394036\pi\)
0.326781 + 0.945100i \(0.394036\pi\)
\(702\) 8.64538 0.326299
\(703\) 0 0
\(704\) 5.93957 0.223856
\(705\) 32.5204 1.22479
\(706\) −26.9530 −1.01439
\(707\) −11.3772 −0.427884
\(708\) −1.79912 −0.0676151
\(709\) −5.07624 −0.190642 −0.0953212 0.995447i \(-0.530388\pi\)
−0.0953212 + 0.995447i \(0.530388\pi\)
\(710\) 27.3357 1.02589
\(711\) 0.430617 0.0161494
\(712\) 18.0337 0.675841
\(713\) −21.5260 −0.806154
\(714\) 11.4213 0.427431
\(715\) −20.9124 −0.782081
\(716\) −6.59689 −0.246538
\(717\) 12.4520 0.465030
\(718\) −2.33163 −0.0870155
\(719\) 24.7862 0.924371 0.462185 0.886783i \(-0.347066\pi\)
0.462185 + 0.886783i \(0.347066\pi\)
\(720\) −0.143287 −0.00533998
\(721\) −0.338246 −0.0125969
\(722\) −18.7543 −0.697961
\(723\) 17.7357 0.659598
\(724\) 1.47280 0.0547360
\(725\) 5.20353 0.193254
\(726\) −42.5290 −1.57840
\(727\) 3.10240 0.115062 0.0575308 0.998344i \(-0.481677\pi\)
0.0575308 + 0.998344i \(0.481677\pi\)
\(728\) 2.03908 0.0755733
\(729\) 26.3555 0.976130
\(730\) −17.8795 −0.661749
\(731\) −10.4269 −0.385654
\(732\) 2.57302 0.0951015
\(733\) 17.7619 0.656050 0.328025 0.944669i \(-0.393617\pi\)
0.328025 + 0.944669i \(0.393617\pi\)
\(734\) 1.11791 0.0412628
\(735\) −20.2698 −0.747663
\(736\) −2.33910 −0.0862203
\(737\) 72.9944 2.68878
\(738\) 0.712806 0.0262387
\(739\) −20.8171 −0.765771 −0.382886 0.923796i \(-0.625070\pi\)
−0.382886 + 0.923796i \(0.625070\pi\)
\(740\) 0 0
\(741\) 1.46194 0.0537057
\(742\) −7.98013 −0.292960
\(743\) 47.5877 1.74583 0.872913 0.487877i \(-0.162228\pi\)
0.872913 + 0.487877i \(0.162228\pi\)
\(744\) −16.1205 −0.591006
\(745\) −16.9518 −0.621066
\(746\) −7.12913 −0.261016
\(747\) 0.268030 0.00980671
\(748\) −31.9745 −1.16910
\(749\) 13.4726 0.492279
\(750\) −20.6118 −0.752636
\(751\) 31.6665 1.15553 0.577764 0.816204i \(-0.303925\pi\)
0.577764 + 0.816204i \(0.303925\pi\)
\(752\) 8.87715 0.323717
\(753\) −24.0844 −0.877686
\(754\) −13.9846 −0.509290
\(755\) −10.4369 −0.379839
\(756\) 6.21948 0.226200
\(757\) −33.1698 −1.20558 −0.602789 0.797900i \(-0.705944\pi\)
−0.602789 + 0.797900i \(0.705944\pi\)
\(758\) 4.72138 0.171488
\(759\) 24.3370 0.883378
\(760\) 1.03669 0.0376048
\(761\) −5.42177 −0.196539 −0.0982696 0.995160i \(-0.531331\pi\)
−0.0982696 + 0.995160i \(0.531331\pi\)
\(762\) −33.3499 −1.20814
\(763\) 5.33674 0.193203
\(764\) −23.8560 −0.863080
\(765\) 0.771357 0.0278885
\(766\) −17.4079 −0.628975
\(767\) 1.72913 0.0624354
\(768\) −1.75172 −0.0632097
\(769\) 38.1392 1.37533 0.687666 0.726027i \(-0.258635\pi\)
0.687666 + 0.726027i \(0.258635\pi\)
\(770\) −15.0444 −0.542162
\(771\) 12.4057 0.446781
\(772\) −16.8415 −0.606138
\(773\) −16.5982 −0.596995 −0.298497 0.954410i \(-0.596485\pi\)
−0.298497 + 0.954410i \(0.596485\pi\)
\(774\) 0.132708 0.00477008
\(775\) −5.76492 −0.207082
\(776\) 8.12116 0.291533
\(777\) 0 0
\(778\) −35.7110 −1.28030
\(779\) −5.15722 −0.184777
\(780\) 6.16757 0.220834
\(781\) −77.6367 −2.77806
\(782\) 12.5921 0.450292
\(783\) −42.6550 −1.52437
\(784\) −5.53309 −0.197610
\(785\) −10.0450 −0.358521
\(786\) −4.07444 −0.145330
\(787\) 36.6308 1.30575 0.652873 0.757467i \(-0.273564\pi\)
0.652873 + 0.757467i \(0.273564\pi\)
\(788\) 18.0955 0.644625
\(789\) −8.07137 −0.287348
\(790\) −13.1438 −0.467634
\(791\) 16.1874 0.575558
\(792\) 0.406952 0.0144604
\(793\) −2.47292 −0.0878161
\(794\) −11.6241 −0.412523
\(795\) −24.1373 −0.856063
\(796\) −15.9676 −0.565956
\(797\) 25.4663 0.902063 0.451031 0.892508i \(-0.351056\pi\)
0.451031 + 0.892508i \(0.351056\pi\)
\(798\) 1.05172 0.0372304
\(799\) −47.7885 −1.69063
\(800\) −0.626440 −0.0221480
\(801\) 1.23559 0.0436573
\(802\) 29.4827 1.04107
\(803\) 50.7799 1.79198
\(804\) −21.5278 −0.759226
\(805\) 5.92473 0.208819
\(806\) 15.4934 0.545731
\(807\) 9.14194 0.321811
\(808\) −9.39363 −0.330467
\(809\) 25.7424 0.905055 0.452528 0.891750i \(-0.350522\pi\)
0.452528 + 0.891750i \(0.350522\pi\)
\(810\) 19.2418 0.676088
\(811\) −24.9867 −0.877403 −0.438702 0.898633i \(-0.644561\pi\)
−0.438702 + 0.898633i \(0.644561\pi\)
\(812\) −10.0605 −0.353055
\(813\) −19.7705 −0.693381
\(814\) 0 0
\(815\) −12.3918 −0.434067
\(816\) 9.43004 0.330117
\(817\) −0.960152 −0.0335915
\(818\) 32.8533 1.14869
\(819\) 0.139709 0.00488181
\(820\) −21.7571 −0.759790
\(821\) −27.6049 −0.963417 −0.481708 0.876332i \(-0.659984\pi\)
−0.481708 + 0.876332i \(0.659984\pi\)
\(822\) 16.8945 0.589262
\(823\) −47.0065 −1.63855 −0.819273 0.573404i \(-0.805622\pi\)
−0.819273 + 0.573404i \(0.805622\pi\)
\(824\) −0.279274 −0.00972897
\(825\) 6.51776 0.226919
\(826\) 1.24394 0.0432821
\(827\) −20.9656 −0.729044 −0.364522 0.931195i \(-0.618768\pi\)
−0.364522 + 0.931195i \(0.618768\pi\)
\(828\) −0.160264 −0.00556957
\(829\) 47.6008 1.65324 0.826622 0.562758i \(-0.190260\pi\)
0.826622 + 0.562758i \(0.190260\pi\)
\(830\) −8.18112 −0.283971
\(831\) −0.915335 −0.0317526
\(832\) 1.68357 0.0583674
\(833\) 29.7863 1.03203
\(834\) −10.6013 −0.367094
\(835\) 1.19544 0.0413700
\(836\) −2.94434 −0.101832
\(837\) 47.2570 1.63344
\(838\) 19.2145 0.663755
\(839\) −18.2819 −0.631162 −0.315581 0.948899i \(-0.602199\pi\)
−0.315581 + 0.948899i \(0.602199\pi\)
\(840\) 4.43694 0.153089
\(841\) 39.9981 1.37925
\(842\) 36.4583 1.25644
\(843\) 25.9580 0.894043
\(844\) −7.31223 −0.251697
\(845\) 21.2593 0.731343
\(846\) 0.608222 0.0209111
\(847\) 29.4051 1.01037
\(848\) −6.58882 −0.226261
\(849\) −7.00556 −0.240430
\(850\) 3.37232 0.115670
\(851\) 0 0
\(852\) 22.8969 0.784434
\(853\) −18.3882 −0.629601 −0.314801 0.949158i \(-0.601938\pi\)
−0.314801 + 0.949158i \(0.601938\pi\)
\(854\) −1.77902 −0.0608768
\(855\) 0.0710295 0.00242916
\(856\) 11.1237 0.380201
\(857\) −5.01983 −0.171474 −0.0857371 0.996318i \(-0.527324\pi\)
−0.0857371 + 0.996318i \(0.527324\pi\)
\(858\) −17.5167 −0.598009
\(859\) 7.42957 0.253494 0.126747 0.991935i \(-0.459546\pi\)
0.126747 + 0.991935i \(0.459546\pi\)
\(860\) −4.05065 −0.138126
\(861\) −22.0724 −0.752224
\(862\) −9.26831 −0.315680
\(863\) −12.2648 −0.417498 −0.208749 0.977969i \(-0.566939\pi\)
−0.208749 + 0.977969i \(0.566939\pi\)
\(864\) 5.13513 0.174701
\(865\) −33.8433 −1.15071
\(866\) −24.0022 −0.815627
\(867\) −20.9856 −0.712708
\(868\) 11.1459 0.378317
\(869\) 37.3299 1.26633
\(870\) −30.4299 −1.03167
\(871\) 20.6903 0.701064
\(872\) 4.40630 0.149216
\(873\) 0.556425 0.0188321
\(874\) 1.15953 0.0392216
\(875\) 14.2513 0.481781
\(876\) −14.9762 −0.505998
\(877\) 32.9639 1.11311 0.556557 0.830810i \(-0.312122\pi\)
0.556557 + 0.830810i \(0.312122\pi\)
\(878\) −6.86675 −0.231742
\(879\) 17.7982 0.600319
\(880\) −12.4215 −0.418727
\(881\) 9.29779 0.313250 0.156625 0.987658i \(-0.449939\pi\)
0.156625 + 0.987658i \(0.449939\pi\)
\(882\) −0.379102 −0.0127650
\(883\) 20.6007 0.693268 0.346634 0.938000i \(-0.387325\pi\)
0.346634 + 0.938000i \(0.387325\pi\)
\(884\) −9.06320 −0.304828
\(885\) 3.76251 0.126475
\(886\) 0.332029 0.0111547
\(887\) −29.9457 −1.00548 −0.502740 0.864438i \(-0.667675\pi\)
−0.502740 + 0.864438i \(0.667675\pi\)
\(888\) 0 0
\(889\) 23.0586 0.773359
\(890\) −37.7140 −1.26418
\(891\) −54.6491 −1.83081
\(892\) 15.9777 0.534973
\(893\) −4.40055 −0.147259
\(894\) −14.1992 −0.474891
\(895\) 13.7961 0.461154
\(896\) 1.21116 0.0404621
\(897\) 6.89835 0.230329
\(898\) 9.94564 0.331890
\(899\) −76.4421 −2.54949
\(900\) −0.0429208 −0.00143069
\(901\) 35.4696 1.18167
\(902\) 61.7928 2.05747
\(903\) −4.10935 −0.136751
\(904\) 13.3652 0.444520
\(905\) −3.08007 −0.102385
\(906\) −8.74217 −0.290439
\(907\) 51.2156 1.70059 0.850294 0.526308i \(-0.176424\pi\)
0.850294 + 0.526308i \(0.176424\pi\)
\(908\) 3.46336 0.114936
\(909\) −0.643609 −0.0213472
\(910\) −4.26434 −0.141362
\(911\) −26.5131 −0.878417 −0.439208 0.898385i \(-0.644741\pi\)
−0.439208 + 0.898385i \(0.644741\pi\)
\(912\) 0.868354 0.0287541
\(913\) 23.2354 0.768979
\(914\) 28.3766 0.938614
\(915\) −5.38097 −0.177889
\(916\) −15.1822 −0.501635
\(917\) 2.81712 0.0930296
\(918\) −27.6440 −0.912388
\(919\) −27.6594 −0.912400 −0.456200 0.889877i \(-0.650790\pi\)
−0.456200 + 0.889877i \(0.650790\pi\)
\(920\) 4.89177 0.161277
\(921\) −39.3740 −1.29742
\(922\) 4.57081 0.150532
\(923\) −22.0062 −0.724342
\(924\) −12.6015 −0.414558
\(925\) 0 0
\(926\) 13.6661 0.449095
\(927\) −0.0191346 −0.000628462 0
\(928\) −8.30651 −0.272675
\(929\) 9.35560 0.306947 0.153474 0.988153i \(-0.450954\pi\)
0.153474 + 0.988153i \(0.450954\pi\)
\(930\) 33.7129 1.10549
\(931\) 2.74284 0.0898929
\(932\) −5.71207 −0.187105
\(933\) 7.71742 0.252657
\(934\) 5.16486 0.168999
\(935\) 66.8685 2.18683
\(936\) 0.115351 0.00377036
\(937\) −19.8079 −0.647097 −0.323549 0.946212i \(-0.604876\pi\)
−0.323549 + 0.946212i \(0.604876\pi\)
\(938\) 14.8846 0.485999
\(939\) 1.29804 0.0423598
\(940\) −18.5648 −0.605518
\(941\) −4.37434 −0.142599 −0.0712997 0.997455i \(-0.522715\pi\)
−0.0712997 + 0.997455i \(0.522715\pi\)
\(942\) −8.41386 −0.274139
\(943\) −24.3350 −0.792457
\(944\) 1.02706 0.0334280
\(945\) −13.0068 −0.423112
\(946\) 11.5044 0.374039
\(947\) 23.8564 0.775230 0.387615 0.921821i \(-0.373299\pi\)
0.387615 + 0.921821i \(0.373299\pi\)
\(948\) −11.0095 −0.357571
\(949\) 14.3936 0.467236
\(950\) 0.310536 0.0100751
\(951\) −44.2110 −1.43364
\(952\) −6.52006 −0.211316
\(953\) 56.3661 1.82588 0.912938 0.408098i \(-0.133808\pi\)
0.912938 + 0.408098i \(0.133808\pi\)
\(954\) −0.451436 −0.0146158
\(955\) 49.8902 1.61441
\(956\) −7.10847 −0.229904
\(957\) 86.4247 2.79371
\(958\) −33.7622 −1.09081
\(959\) −11.6811 −0.377201
\(960\) 3.66338 0.118235
\(961\) 53.6892 1.73191
\(962\) 0 0
\(963\) 0.762147 0.0245598
\(964\) −10.1248 −0.326096
\(965\) 35.2207 1.13379
\(966\) 4.96266 0.159671
\(967\) 46.6038 1.49868 0.749338 0.662188i \(-0.230372\pi\)
0.749338 + 0.662188i \(0.230372\pi\)
\(968\) 24.2785 0.780339
\(969\) −4.67462 −0.150170
\(970\) −16.9838 −0.545318
\(971\) 33.1861 1.06499 0.532496 0.846432i \(-0.321254\pi\)
0.532496 + 0.846432i \(0.321254\pi\)
\(972\) 0.711895 0.0228341
\(973\) 7.32990 0.234986
\(974\) 32.6970 1.04768
\(975\) 1.84746 0.0591662
\(976\) −1.46885 −0.0470169
\(977\) −20.7016 −0.662304 −0.331152 0.943577i \(-0.607437\pi\)
−0.331152 + 0.943577i \(0.607437\pi\)
\(978\) −10.3796 −0.331904
\(979\) 107.112 3.42333
\(980\) 11.5714 0.369634
\(981\) 0.301900 0.00963891
\(982\) −24.2572 −0.774079
\(983\) −16.2367 −0.517871 −0.258935 0.965895i \(-0.583372\pi\)
−0.258935 + 0.965895i \(0.583372\pi\)
\(984\) −18.2241 −0.580964
\(985\) −37.8432 −1.20578
\(986\) 44.7165 1.42406
\(987\) −18.8339 −0.599489
\(988\) −0.834575 −0.0265514
\(989\) −4.53060 −0.144065
\(990\) −0.851061 −0.0270485
\(991\) 2.25933 0.0717699 0.0358849 0.999356i \(-0.488575\pi\)
0.0358849 + 0.999356i \(0.488575\pi\)
\(992\) 9.20267 0.292185
\(993\) −8.03634 −0.255026
\(994\) −15.8312 −0.502136
\(995\) 33.3931 1.05863
\(996\) −6.85266 −0.217135
\(997\) −40.1141 −1.27043 −0.635213 0.772337i \(-0.719088\pi\)
−0.635213 + 0.772337i \(0.719088\pi\)
\(998\) −18.6640 −0.590798
\(999\) 0 0
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 2738.2.a.x.1.4 yes 18
37.36 even 2 2738.2.a.w.1.4 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2738.2.a.w.1.4 18 37.36 even 2
2738.2.a.x.1.4 yes 18 1.1 even 1 trivial