Properties

Label 8030.2.a.be.1.5
Level $8030$
Weight $2$
Character 8030.1
Self dual yes
Analytic conductor $64.120$
Analytic rank $0$
Dimension $15$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8030,2,Mod(1,8030)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8030, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8030.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8030 = 2 \cdot 5 \cdot 11 \cdot 73 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8030.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(64.1198728231\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 3 x^{14} - 24 x^{13} + 64 x^{12} + 237 x^{11} - 524 x^{10} - 1225 x^{9} + 2074 x^{8} + 3463 x^{7} - 4142 x^{6} - 5157 x^{5} + 3892 x^{4} + 3622 x^{3} - 1239 x^{2} + \cdots - 52 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-1.51118\) of defining polynomial
Character \(\chi\) \(=\) 8030.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.51118 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.51118 q^{6} -0.990203 q^{7} -1.00000 q^{8} -0.716341 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.51118 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.51118 q^{6} -0.990203 q^{7} -1.00000 q^{8} -0.716341 q^{9} -1.00000 q^{10} -1.00000 q^{11} -1.51118 q^{12} -4.51816 q^{13} +0.990203 q^{14} -1.51118 q^{15} +1.00000 q^{16} +4.85206 q^{17} +0.716341 q^{18} -6.04845 q^{19} +1.00000 q^{20} +1.49637 q^{21} +1.00000 q^{22} +6.91028 q^{23} +1.51118 q^{24} +1.00000 q^{25} +4.51816 q^{26} +5.61605 q^{27} -0.990203 q^{28} +7.68677 q^{29} +1.51118 q^{30} +5.17086 q^{31} -1.00000 q^{32} +1.51118 q^{33} -4.85206 q^{34} -0.990203 q^{35} -0.716341 q^{36} -5.91021 q^{37} +6.04845 q^{38} +6.82775 q^{39} -1.00000 q^{40} -6.51071 q^{41} -1.49637 q^{42} -1.00014 q^{43} -1.00000 q^{44} -0.716341 q^{45} -6.91028 q^{46} +1.74147 q^{47} -1.51118 q^{48} -6.01950 q^{49} -1.00000 q^{50} -7.33233 q^{51} -4.51816 q^{52} -7.27787 q^{53} -5.61605 q^{54} -1.00000 q^{55} +0.990203 q^{56} +9.14028 q^{57} -7.68677 q^{58} -6.92339 q^{59} -1.51118 q^{60} +7.09406 q^{61} -5.17086 q^{62} +0.709323 q^{63} +1.00000 q^{64} -4.51816 q^{65} -1.51118 q^{66} -8.62196 q^{67} +4.85206 q^{68} -10.4427 q^{69} +0.990203 q^{70} -10.7024 q^{71} +0.716341 q^{72} -1.00000 q^{73} +5.91021 q^{74} -1.51118 q^{75} -6.04845 q^{76} +0.990203 q^{77} -6.82775 q^{78} -1.71773 q^{79} +1.00000 q^{80} -6.33783 q^{81} +6.51071 q^{82} +3.56466 q^{83} +1.49637 q^{84} +4.85206 q^{85} +1.00014 q^{86} -11.6161 q^{87} +1.00000 q^{88} +6.63971 q^{89} +0.716341 q^{90} +4.47390 q^{91} +6.91028 q^{92} -7.81409 q^{93} -1.74147 q^{94} -6.04845 q^{95} +1.51118 q^{96} -2.00346 q^{97} +6.01950 q^{98} +0.716341 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - 15 q^{2} + 3 q^{3} + 15 q^{4} + 15 q^{5} - 3 q^{6} + 7 q^{7} - 15 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q - 15 q^{2} + 3 q^{3} + 15 q^{4} + 15 q^{5} - 3 q^{6} + 7 q^{7} - 15 q^{8} + 12 q^{9} - 15 q^{10} - 15 q^{11} + 3 q^{12} + 13 q^{13} - 7 q^{14} + 3 q^{15} + 15 q^{16} + 20 q^{17} - 12 q^{18} + 3 q^{19} + 15 q^{20} + 22 q^{21} + 15 q^{22} + 2 q^{23} - 3 q^{24} + 15 q^{25} - 13 q^{26} + 33 q^{27} + 7 q^{28} + 11 q^{29} - 3 q^{30} - 3 q^{31} - 15 q^{32} - 3 q^{33} - 20 q^{34} + 7 q^{35} + 12 q^{36} + 9 q^{37} - 3 q^{38} + 11 q^{39} - 15 q^{40} + 17 q^{41} - 22 q^{42} + 29 q^{43} - 15 q^{44} + 12 q^{45} - 2 q^{46} - 2 q^{47} + 3 q^{48} + 20 q^{49} - 15 q^{50} + 7 q^{51} + 13 q^{52} + 3 q^{53} - 33 q^{54} - 15 q^{55} - 7 q^{56} + 13 q^{57} - 11 q^{58} - 32 q^{59} + 3 q^{60} + 61 q^{61} + 3 q^{62} + 20 q^{63} + 15 q^{64} + 13 q^{65} + 3 q^{66} + 7 q^{67} + 20 q^{68} - 23 q^{69} - 7 q^{70} - 6 q^{71} - 12 q^{72} - 15 q^{73} - 9 q^{74} + 3 q^{75} + 3 q^{76} - 7 q^{77} - 11 q^{78} + 12 q^{79} + 15 q^{80} + 3 q^{81} - 17 q^{82} + 17 q^{83} + 22 q^{84} + 20 q^{85} - 29 q^{86} + 23 q^{87} + 15 q^{88} - 18 q^{89} - 12 q^{90} - 15 q^{91} + 2 q^{92} + 32 q^{93} + 2 q^{94} + 3 q^{95} - 3 q^{96} + 36 q^{97} - 20 q^{98} - 12 q^{99}+O(q^{100}) \) 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
\(3\) −1.51118 −0.872479 −0.436240 0.899831i \(-0.643690\pi\)
−0.436240 + 0.899831i \(0.643690\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 1.51118 0.616936
\(7\) −0.990203 −0.374262 −0.187131 0.982335i \(-0.559919\pi\)
−0.187131 + 0.982335i \(0.559919\pi\)
\(8\) −1.00000 −0.353553
\(9\) −0.716341 −0.238780
\(10\) −1.00000 −0.316228
\(11\) −1.00000 −0.301511
\(12\) −1.51118 −0.436240
\(13\) −4.51816 −1.25311 −0.626557 0.779376i \(-0.715536\pi\)
−0.626557 + 0.779376i \(0.715536\pi\)
\(14\) 0.990203 0.264643
\(15\) −1.51118 −0.390184
\(16\) 1.00000 0.250000
\(17\) 4.85206 1.17680 0.588399 0.808571i \(-0.299759\pi\)
0.588399 + 0.808571i \(0.299759\pi\)
\(18\) 0.716341 0.168843
\(19\) −6.04845 −1.38761 −0.693804 0.720164i \(-0.744067\pi\)
−0.693804 + 0.720164i \(0.744067\pi\)
\(20\) 1.00000 0.223607
\(21\) 1.49637 0.326535
\(22\) 1.00000 0.213201
\(23\) 6.91028 1.44089 0.720446 0.693511i \(-0.243937\pi\)
0.720446 + 0.693511i \(0.243937\pi\)
\(24\) 1.51118 0.308468
\(25\) 1.00000 0.200000
\(26\) 4.51816 0.886085
\(27\) 5.61605 1.08081
\(28\) −0.990203 −0.187131
\(29\) 7.68677 1.42740 0.713699 0.700453i \(-0.247019\pi\)
0.713699 + 0.700453i \(0.247019\pi\)
\(30\) 1.51118 0.275902
\(31\) 5.17086 0.928714 0.464357 0.885648i \(-0.346285\pi\)
0.464357 + 0.885648i \(0.346285\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.51118 0.263062
\(34\) −4.85206 −0.832122
\(35\) −0.990203 −0.167375
\(36\) −0.716341 −0.119390
\(37\) −5.91021 −0.971633 −0.485816 0.874061i \(-0.661478\pi\)
−0.485816 + 0.874061i \(0.661478\pi\)
\(38\) 6.04845 0.981188
\(39\) 6.82775 1.09331
\(40\) −1.00000 −0.158114
\(41\) −6.51071 −1.01680 −0.508401 0.861121i \(-0.669763\pi\)
−0.508401 + 0.861121i \(0.669763\pi\)
\(42\) −1.49637 −0.230895
\(43\) −1.00014 −0.152520 −0.0762602 0.997088i \(-0.524298\pi\)
−0.0762602 + 0.997088i \(0.524298\pi\)
\(44\) −1.00000 −0.150756
\(45\) −0.716341 −0.106786
\(46\) −6.91028 −1.01886
\(47\) 1.74147 0.254019 0.127010 0.991901i \(-0.459462\pi\)
0.127010 + 0.991901i \(0.459462\pi\)
\(48\) −1.51118 −0.218120
\(49\) −6.01950 −0.859928
\(50\) −1.00000 −0.141421
\(51\) −7.33233 −1.02673
\(52\) −4.51816 −0.626557
\(53\) −7.27787 −0.999692 −0.499846 0.866114i \(-0.666610\pi\)
−0.499846 + 0.866114i \(0.666610\pi\)
\(54\) −5.61605 −0.764248
\(55\) −1.00000 −0.134840
\(56\) 0.990203 0.132321
\(57\) 9.14028 1.21066
\(58\) −7.68677 −1.00932
\(59\) −6.92339 −0.901349 −0.450675 0.892688i \(-0.648817\pi\)
−0.450675 + 0.892688i \(0.648817\pi\)
\(60\) −1.51118 −0.195092
\(61\) 7.09406 0.908301 0.454150 0.890925i \(-0.349943\pi\)
0.454150 + 0.890925i \(0.349943\pi\)
\(62\) −5.17086 −0.656700
\(63\) 0.709323 0.0893663
\(64\) 1.00000 0.125000
\(65\) −4.51816 −0.560409
\(66\) −1.51118 −0.186013
\(67\) −8.62196 −1.05334 −0.526670 0.850070i \(-0.676560\pi\)
−0.526670 + 0.850070i \(0.676560\pi\)
\(68\) 4.85206 0.588399
\(69\) −10.4427 −1.25715
\(70\) 0.990203 0.118352
\(71\) −10.7024 −1.27014 −0.635068 0.772456i \(-0.719028\pi\)
−0.635068 + 0.772456i \(0.719028\pi\)
\(72\) 0.716341 0.0844216
\(73\) −1.00000 −0.117041
\(74\) 5.91021 0.687048
\(75\) −1.51118 −0.174496
\(76\) −6.04845 −0.693804
\(77\) 0.990203 0.112844
\(78\) −6.82775 −0.773090
\(79\) −1.71773 −0.193260 −0.0966299 0.995320i \(-0.530806\pi\)
−0.0966299 + 0.995320i \(0.530806\pi\)
\(80\) 1.00000 0.111803
\(81\) −6.33783 −0.704204
\(82\) 6.51071 0.718987
\(83\) 3.56466 0.391272 0.195636 0.980677i \(-0.437323\pi\)
0.195636 + 0.980677i \(0.437323\pi\)
\(84\) 1.49637 0.163268
\(85\) 4.85206 0.526280
\(86\) 1.00014 0.107848
\(87\) −11.6161 −1.24537
\(88\) 1.00000 0.106600
\(89\) 6.63971 0.703808 0.351904 0.936036i \(-0.385534\pi\)
0.351904 + 0.936036i \(0.385534\pi\)
\(90\) 0.716341 0.0755090
\(91\) 4.47390 0.468992
\(92\) 6.91028 0.720446
\(93\) −7.81409 −0.810284
\(94\) −1.74147 −0.179619
\(95\) −6.04845 −0.620557
\(96\) 1.51118 0.154234
\(97\) −2.00346 −0.203421 −0.101710 0.994814i \(-0.532432\pi\)
−0.101710 + 0.994814i \(0.532432\pi\)
\(98\) 6.01950 0.608061
\(99\) 0.716341 0.0719950
\(100\) 1.00000 0.100000
\(101\) 0.947272 0.0942571 0.0471285 0.998889i \(-0.484993\pi\)
0.0471285 + 0.998889i \(0.484993\pi\)
\(102\) 7.33233 0.726009
\(103\) −14.1558 −1.39482 −0.697408 0.716674i \(-0.745664\pi\)
−0.697408 + 0.716674i \(0.745664\pi\)
\(104\) 4.51816 0.443042
\(105\) 1.49637 0.146031
\(106\) 7.27787 0.706889
\(107\) 6.31073 0.610081 0.305040 0.952339i \(-0.401330\pi\)
0.305040 + 0.952339i \(0.401330\pi\)
\(108\) 5.61605 0.540405
\(109\) −17.9388 −1.71823 −0.859115 0.511783i \(-0.828985\pi\)
−0.859115 + 0.511783i \(0.828985\pi\)
\(110\) 1.00000 0.0953463
\(111\) 8.93138 0.847729
\(112\) −0.990203 −0.0935654
\(113\) 13.8348 1.30147 0.650734 0.759306i \(-0.274461\pi\)
0.650734 + 0.759306i \(0.274461\pi\)
\(114\) −9.14028 −0.856066
\(115\) 6.91028 0.644387
\(116\) 7.68677 0.713699
\(117\) 3.23655 0.299219
\(118\) 6.92339 0.637350
\(119\) −4.80453 −0.440430
\(120\) 1.51118 0.137951
\(121\) 1.00000 0.0909091
\(122\) −7.09406 −0.642266
\(123\) 9.83884 0.887138
\(124\) 5.17086 0.464357
\(125\) 1.00000 0.0894427
\(126\) −0.709323 −0.0631915
\(127\) 6.19048 0.549316 0.274658 0.961542i \(-0.411435\pi\)
0.274658 + 0.961542i \(0.411435\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 1.51139 0.133071
\(130\) 4.51816 0.396269
\(131\) −1.73521 −0.151606 −0.0758032 0.997123i \(-0.524152\pi\)
−0.0758032 + 0.997123i \(0.524152\pi\)
\(132\) 1.51118 0.131531
\(133\) 5.98919 0.519329
\(134\) 8.62196 0.744824
\(135\) 5.61605 0.483353
\(136\) −4.85206 −0.416061
\(137\) 5.75063 0.491309 0.245655 0.969357i \(-0.420997\pi\)
0.245655 + 0.969357i \(0.420997\pi\)
\(138\) 10.4427 0.888938
\(139\) −22.3773 −1.89802 −0.949009 0.315249i \(-0.897912\pi\)
−0.949009 + 0.315249i \(0.897912\pi\)
\(140\) −0.990203 −0.0836874
\(141\) −2.63167 −0.221626
\(142\) 10.7024 0.898122
\(143\) 4.51816 0.377828
\(144\) −0.716341 −0.0596951
\(145\) 7.68677 0.638351
\(146\) 1.00000 0.0827606
\(147\) 9.09653 0.750269
\(148\) −5.91021 −0.485816
\(149\) 5.81913 0.476722 0.238361 0.971177i \(-0.423390\pi\)
0.238361 + 0.971177i \(0.423390\pi\)
\(150\) 1.51118 0.123387
\(151\) 16.6560 1.35545 0.677725 0.735316i \(-0.262966\pi\)
0.677725 + 0.735316i \(0.262966\pi\)
\(152\) 6.04845 0.490594
\(153\) −3.47573 −0.280996
\(154\) −0.990203 −0.0797928
\(155\) 5.17086 0.415334
\(156\) 6.82775 0.546657
\(157\) 2.55559 0.203959 0.101979 0.994787i \(-0.467482\pi\)
0.101979 + 0.994787i \(0.467482\pi\)
\(158\) 1.71773 0.136655
\(159\) 10.9982 0.872210
\(160\) −1.00000 −0.0790569
\(161\) −6.84258 −0.539271
\(162\) 6.33783 0.497947
\(163\) −8.37109 −0.655675 −0.327837 0.944734i \(-0.606320\pi\)
−0.327837 + 0.944734i \(0.606320\pi\)
\(164\) −6.51071 −0.508401
\(165\) 1.51118 0.117645
\(166\) −3.56466 −0.276671
\(167\) 23.7180 1.83535 0.917677 0.397326i \(-0.130062\pi\)
0.917677 + 0.397326i \(0.130062\pi\)
\(168\) −1.49637 −0.115448
\(169\) 7.41380 0.570293
\(170\) −4.85206 −0.372136
\(171\) 4.33275 0.331334
\(172\) −1.00014 −0.0762602
\(173\) 20.5031 1.55882 0.779411 0.626513i \(-0.215519\pi\)
0.779411 + 0.626513i \(0.215519\pi\)
\(174\) 11.6161 0.880612
\(175\) −0.990203 −0.0748523
\(176\) −1.00000 −0.0753778
\(177\) 10.4625 0.786408
\(178\) −6.63971 −0.497667
\(179\) −6.97635 −0.521437 −0.260719 0.965415i \(-0.583959\pi\)
−0.260719 + 0.965415i \(0.583959\pi\)
\(180\) −0.716341 −0.0533929
\(181\) 13.2575 0.985422 0.492711 0.870193i \(-0.336006\pi\)
0.492711 + 0.870193i \(0.336006\pi\)
\(182\) −4.47390 −0.331627
\(183\) −10.7204 −0.792473
\(184\) −6.91028 −0.509432
\(185\) −5.91021 −0.434527
\(186\) 7.81409 0.572957
\(187\) −4.85206 −0.354818
\(188\) 1.74147 0.127010
\(189\) −5.56103 −0.404506
\(190\) 6.04845 0.438800
\(191\) −11.9475 −0.864490 −0.432245 0.901756i \(-0.642279\pi\)
−0.432245 + 0.901756i \(0.642279\pi\)
\(192\) −1.51118 −0.109060
\(193\) −14.4429 −1.03962 −0.519811 0.854281i \(-0.673998\pi\)
−0.519811 + 0.854281i \(0.673998\pi\)
\(194\) 2.00346 0.143840
\(195\) 6.82775 0.488945
\(196\) −6.01950 −0.429964
\(197\) 7.88836 0.562022 0.281011 0.959704i \(-0.409330\pi\)
0.281011 + 0.959704i \(0.409330\pi\)
\(198\) −0.716341 −0.0509081
\(199\) −9.33569 −0.661790 −0.330895 0.943668i \(-0.607351\pi\)
−0.330895 + 0.943668i \(0.607351\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 13.0293 0.919017
\(202\) −0.947272 −0.0666498
\(203\) −7.61146 −0.534220
\(204\) −7.33233 −0.513366
\(205\) −6.51071 −0.454727
\(206\) 14.1558 0.986284
\(207\) −4.95012 −0.344057
\(208\) −4.51816 −0.313278
\(209\) 6.04845 0.418380
\(210\) −1.49637 −0.103260
\(211\) 18.7964 1.29400 0.646998 0.762491i \(-0.276024\pi\)
0.646998 + 0.762491i \(0.276024\pi\)
\(212\) −7.27787 −0.499846
\(213\) 16.1732 1.10817
\(214\) −6.31073 −0.431392
\(215\) −1.00014 −0.0682092
\(216\) −5.61605 −0.382124
\(217\) −5.12020 −0.347582
\(218\) 17.9388 1.21497
\(219\) 1.51118 0.102116
\(220\) −1.00000 −0.0674200
\(221\) −21.9224 −1.47466
\(222\) −8.93138 −0.599435
\(223\) −12.8435 −0.860061 −0.430031 0.902814i \(-0.641497\pi\)
−0.430031 + 0.902814i \(0.641497\pi\)
\(224\) 0.990203 0.0661607
\(225\) −0.716341 −0.0477561
\(226\) −13.8348 −0.920276
\(227\) 22.0558 1.46389 0.731946 0.681362i \(-0.238612\pi\)
0.731946 + 0.681362i \(0.238612\pi\)
\(228\) 9.14028 0.605330
\(229\) 3.31830 0.219279 0.109640 0.993971i \(-0.465030\pi\)
0.109640 + 0.993971i \(0.465030\pi\)
\(230\) −6.91028 −0.455650
\(231\) −1.49637 −0.0984541
\(232\) −7.68677 −0.504661
\(233\) 16.4539 1.07793 0.538964 0.842329i \(-0.318816\pi\)
0.538964 + 0.842329i \(0.318816\pi\)
\(234\) −3.23655 −0.211580
\(235\) 1.74147 0.113601
\(236\) −6.92339 −0.450675
\(237\) 2.59580 0.168615
\(238\) 4.80453 0.311431
\(239\) −6.03920 −0.390643 −0.195322 0.980739i \(-0.562575\pi\)
−0.195322 + 0.980739i \(0.562575\pi\)
\(240\) −1.51118 −0.0975461
\(241\) 3.84885 0.247926 0.123963 0.992287i \(-0.460440\pi\)
0.123963 + 0.992287i \(0.460440\pi\)
\(242\) −1.00000 −0.0642824
\(243\) −7.27057 −0.466407
\(244\) 7.09406 0.454150
\(245\) −6.01950 −0.384572
\(246\) −9.83884 −0.627301
\(247\) 27.3279 1.73883
\(248\) −5.17086 −0.328350
\(249\) −5.38683 −0.341377
\(250\) −1.00000 −0.0632456
\(251\) −20.9190 −1.32040 −0.660198 0.751092i \(-0.729527\pi\)
−0.660198 + 0.751092i \(0.729527\pi\)
\(252\) 0.709323 0.0446832
\(253\) −6.91028 −0.434445
\(254\) −6.19048 −0.388425
\(255\) −7.33233 −0.459168
\(256\) 1.00000 0.0625000
\(257\) −17.4654 −1.08946 −0.544732 0.838610i \(-0.683369\pi\)
−0.544732 + 0.838610i \(0.683369\pi\)
\(258\) −1.51139 −0.0940953
\(259\) 5.85231 0.363645
\(260\) −4.51816 −0.280205
\(261\) −5.50635 −0.340834
\(262\) 1.73521 0.107202
\(263\) 26.9377 1.66105 0.830526 0.556980i \(-0.188040\pi\)
0.830526 + 0.556980i \(0.188040\pi\)
\(264\) −1.51118 −0.0930066
\(265\) −7.27787 −0.447076
\(266\) −5.98919 −0.367221
\(267\) −10.0338 −0.614057
\(268\) −8.62196 −0.526670
\(269\) −10.2138 −0.622748 −0.311374 0.950287i \(-0.600789\pi\)
−0.311374 + 0.950287i \(0.600789\pi\)
\(270\) −5.61605 −0.341782
\(271\) −20.8175 −1.26457 −0.632286 0.774735i \(-0.717883\pi\)
−0.632286 + 0.774735i \(0.717883\pi\)
\(272\) 4.85206 0.294199
\(273\) −6.76086 −0.409186
\(274\) −5.75063 −0.347408
\(275\) −1.00000 −0.0603023
\(276\) −10.4427 −0.628574
\(277\) 11.5058 0.691319 0.345659 0.938360i \(-0.387655\pi\)
0.345659 + 0.938360i \(0.387655\pi\)
\(278\) 22.3773 1.34210
\(279\) −3.70410 −0.221759
\(280\) 0.990203 0.0591759
\(281\) 27.8850 1.66348 0.831739 0.555167i \(-0.187346\pi\)
0.831739 + 0.555167i \(0.187346\pi\)
\(282\) 2.63167 0.156714
\(283\) 5.13300 0.305125 0.152562 0.988294i \(-0.451247\pi\)
0.152562 + 0.988294i \(0.451247\pi\)
\(284\) −10.7024 −0.635068
\(285\) 9.14028 0.541423
\(286\) −4.51816 −0.267165
\(287\) 6.44692 0.380550
\(288\) 0.716341 0.0422108
\(289\) 6.54250 0.384853
\(290\) −7.68677 −0.451383
\(291\) 3.02759 0.177480
\(292\) −1.00000 −0.0585206
\(293\) 2.28640 0.133573 0.0667864 0.997767i \(-0.478725\pi\)
0.0667864 + 0.997767i \(0.478725\pi\)
\(294\) −9.09653 −0.530521
\(295\) −6.92339 −0.403096
\(296\) 5.91021 0.343524
\(297\) −5.61605 −0.325876
\(298\) −5.81913 −0.337093
\(299\) −31.2218 −1.80560
\(300\) −1.51118 −0.0872479
\(301\) 0.990345 0.0570825
\(302\) −16.6560 −0.958447
\(303\) −1.43150 −0.0822373
\(304\) −6.04845 −0.346902
\(305\) 7.09406 0.406205
\(306\) 3.47573 0.198694
\(307\) −21.6710 −1.23683 −0.618415 0.785851i \(-0.712225\pi\)
−0.618415 + 0.785851i \(0.712225\pi\)
\(308\) 0.990203 0.0564221
\(309\) 21.3920 1.21695
\(310\) −5.17086 −0.293685
\(311\) 12.5364 0.710871 0.355436 0.934701i \(-0.384333\pi\)
0.355436 + 0.934701i \(0.384333\pi\)
\(312\) −6.82775 −0.386545
\(313\) 25.4308 1.43743 0.718716 0.695303i \(-0.244730\pi\)
0.718716 + 0.695303i \(0.244730\pi\)
\(314\) −2.55559 −0.144221
\(315\) 0.709323 0.0399658
\(316\) −1.71773 −0.0966299
\(317\) −5.99644 −0.336794 −0.168397 0.985719i \(-0.553859\pi\)
−0.168397 + 0.985719i \(0.553859\pi\)
\(318\) −10.9982 −0.616746
\(319\) −7.68677 −0.430376
\(320\) 1.00000 0.0559017
\(321\) −9.53663 −0.532283
\(322\) 6.84258 0.381322
\(323\) −29.3474 −1.63293
\(324\) −6.33783 −0.352102
\(325\) −4.51816 −0.250623
\(326\) 8.37109 0.463632
\(327\) 27.1088 1.49912
\(328\) 6.51071 0.359494
\(329\) −1.72441 −0.0950696
\(330\) −1.51118 −0.0831876
\(331\) −21.5793 −1.18611 −0.593053 0.805163i \(-0.702078\pi\)
−0.593053 + 0.805163i \(0.702078\pi\)
\(332\) 3.56466 0.195636
\(333\) 4.23373 0.232007
\(334\) −23.7180 −1.29779
\(335\) −8.62196 −0.471068
\(336\) 1.49637 0.0816338
\(337\) 29.7494 1.62055 0.810275 0.586049i \(-0.199318\pi\)
0.810275 + 0.586049i \(0.199318\pi\)
\(338\) −7.41380 −0.403258
\(339\) −20.9068 −1.13550
\(340\) 4.85206 0.263140
\(341\) −5.17086 −0.280018
\(342\) −4.33275 −0.234288
\(343\) 12.8919 0.696100
\(344\) 1.00014 0.0539241
\(345\) −10.4427 −0.562214
\(346\) −20.5031 −1.10225
\(347\) 8.44737 0.453479 0.226739 0.973955i \(-0.427193\pi\)
0.226739 + 0.973955i \(0.427193\pi\)
\(348\) −11.6161 −0.622687
\(349\) 9.34190 0.500061 0.250030 0.968238i \(-0.419559\pi\)
0.250030 + 0.968238i \(0.419559\pi\)
\(350\) 0.990203 0.0529286
\(351\) −25.3742 −1.35438
\(352\) 1.00000 0.0533002
\(353\) 33.4444 1.78007 0.890033 0.455895i \(-0.150681\pi\)
0.890033 + 0.455895i \(0.150681\pi\)
\(354\) −10.4625 −0.556075
\(355\) −10.7024 −0.568022
\(356\) 6.63971 0.351904
\(357\) 7.26049 0.384266
\(358\) 6.97635 0.368712
\(359\) 16.6322 0.877813 0.438906 0.898533i \(-0.355366\pi\)
0.438906 + 0.898533i \(0.355366\pi\)
\(360\) 0.716341 0.0377545
\(361\) 17.5837 0.925458
\(362\) −13.2575 −0.696799
\(363\) −1.51118 −0.0793163
\(364\) 4.47390 0.234496
\(365\) −1.00000 −0.0523424
\(366\) 10.7204 0.560363
\(367\) 20.0788 1.04810 0.524052 0.851686i \(-0.324420\pi\)
0.524052 + 0.851686i \(0.324420\pi\)
\(368\) 6.91028 0.360223
\(369\) 4.66389 0.242792
\(370\) 5.91021 0.307257
\(371\) 7.20657 0.374146
\(372\) −7.81409 −0.405142
\(373\) 5.90606 0.305804 0.152902 0.988241i \(-0.451138\pi\)
0.152902 + 0.988241i \(0.451138\pi\)
\(374\) 4.85206 0.250894
\(375\) −1.51118 −0.0780369
\(376\) −1.74147 −0.0898093
\(377\) −34.7301 −1.78869
\(378\) 5.56103 0.286029
\(379\) 7.88648 0.405101 0.202551 0.979272i \(-0.435077\pi\)
0.202551 + 0.979272i \(0.435077\pi\)
\(380\) −6.04845 −0.310279
\(381\) −9.35491 −0.479267
\(382\) 11.9475 0.611287
\(383\) 17.7655 0.907776 0.453888 0.891059i \(-0.350037\pi\)
0.453888 + 0.891059i \(0.350037\pi\)
\(384\) 1.51118 0.0771170
\(385\) 0.990203 0.0504654
\(386\) 14.4429 0.735124
\(387\) 0.716444 0.0364189
\(388\) −2.00346 −0.101710
\(389\) 2.64850 0.134284 0.0671421 0.997743i \(-0.478612\pi\)
0.0671421 + 0.997743i \(0.478612\pi\)
\(390\) −6.82775 −0.345737
\(391\) 33.5291 1.69564
\(392\) 6.01950 0.304031
\(393\) 2.62222 0.132273
\(394\) −7.88836 −0.397410
\(395\) −1.71773 −0.0864284
\(396\) 0.716341 0.0359975
\(397\) 23.6146 1.18518 0.592592 0.805503i \(-0.298105\pi\)
0.592592 + 0.805503i \(0.298105\pi\)
\(398\) 9.33569 0.467956
\(399\) −9.05073 −0.453103
\(400\) 1.00000 0.0500000
\(401\) −26.8925 −1.34295 −0.671474 0.741029i \(-0.734338\pi\)
−0.671474 + 0.741029i \(0.734338\pi\)
\(402\) −13.0293 −0.649843
\(403\) −23.3628 −1.16378
\(404\) 0.947272 0.0471285
\(405\) −6.33783 −0.314929
\(406\) 7.61146 0.377750
\(407\) 5.91021 0.292958
\(408\) 7.33233 0.363004
\(409\) 9.94490 0.491744 0.245872 0.969302i \(-0.420926\pi\)
0.245872 + 0.969302i \(0.420926\pi\)
\(410\) 6.51071 0.321541
\(411\) −8.69022 −0.428657
\(412\) −14.1558 −0.697408
\(413\) 6.85557 0.337340
\(414\) 4.95012 0.243285
\(415\) 3.56466 0.174982
\(416\) 4.51816 0.221521
\(417\) 33.8161 1.65598
\(418\) −6.04845 −0.295839
\(419\) −7.37525 −0.360305 −0.180152 0.983639i \(-0.557659\pi\)
−0.180152 + 0.983639i \(0.557659\pi\)
\(420\) 1.49637 0.0730155
\(421\) 15.5669 0.758685 0.379343 0.925256i \(-0.376150\pi\)
0.379343 + 0.925256i \(0.376150\pi\)
\(422\) −18.7964 −0.914994
\(423\) −1.24748 −0.0606548
\(424\) 7.27787 0.353445
\(425\) 4.85206 0.235360
\(426\) −16.1732 −0.783592
\(427\) −7.02456 −0.339942
\(428\) 6.31073 0.305040
\(429\) −6.82775 −0.329647
\(430\) 1.00014 0.0482312
\(431\) −3.21997 −0.155101 −0.0775503 0.996988i \(-0.524710\pi\)
−0.0775503 + 0.996988i \(0.524710\pi\)
\(432\) 5.61605 0.270202
\(433\) 5.14929 0.247459 0.123729 0.992316i \(-0.460514\pi\)
0.123729 + 0.992316i \(0.460514\pi\)
\(434\) 5.12020 0.245778
\(435\) −11.6161 −0.556948
\(436\) −17.9388 −0.859115
\(437\) −41.7964 −1.99940
\(438\) −1.51118 −0.0722069
\(439\) −11.6466 −0.555863 −0.277931 0.960601i \(-0.589649\pi\)
−0.277931 + 0.960601i \(0.589649\pi\)
\(440\) 1.00000 0.0476731
\(441\) 4.31201 0.205334
\(442\) 21.9224 1.04274
\(443\) −19.0393 −0.904583 −0.452291 0.891870i \(-0.649393\pi\)
−0.452291 + 0.891870i \(0.649393\pi\)
\(444\) 8.93138 0.423865
\(445\) 6.63971 0.314752
\(446\) 12.8435 0.608155
\(447\) −8.79374 −0.415930
\(448\) −0.990203 −0.0467827
\(449\) 10.8077 0.510049 0.255025 0.966935i \(-0.417916\pi\)
0.255025 + 0.966935i \(0.417916\pi\)
\(450\) 0.716341 0.0337686
\(451\) 6.51071 0.306577
\(452\) 13.8348 0.650734
\(453\) −25.1702 −1.18260
\(454\) −22.0558 −1.03513
\(455\) 4.47390 0.209740
\(456\) −9.14028 −0.428033
\(457\) 16.7154 0.781914 0.390957 0.920409i \(-0.372144\pi\)
0.390957 + 0.920409i \(0.372144\pi\)
\(458\) −3.31830 −0.155054
\(459\) 27.2494 1.27189
\(460\) 6.91028 0.322193
\(461\) 9.61907 0.448005 0.224002 0.974589i \(-0.428088\pi\)
0.224002 + 0.974589i \(0.428088\pi\)
\(462\) 1.49637 0.0696176
\(463\) 28.1461 1.30806 0.654029 0.756469i \(-0.273077\pi\)
0.654029 + 0.756469i \(0.273077\pi\)
\(464\) 7.68677 0.356849
\(465\) −7.81409 −0.362370
\(466\) −16.4539 −0.762210
\(467\) 22.0985 1.02260 0.511298 0.859404i \(-0.329165\pi\)
0.511298 + 0.859404i \(0.329165\pi\)
\(468\) 3.23655 0.149609
\(469\) 8.53749 0.394225
\(470\) −1.74147 −0.0803279
\(471\) −3.86196 −0.177950
\(472\) 6.92339 0.318675
\(473\) 1.00014 0.0459866
\(474\) −2.59580 −0.119229
\(475\) −6.04845 −0.277522
\(476\) −4.80453 −0.220215
\(477\) 5.21344 0.238707
\(478\) 6.03920 0.276227
\(479\) 27.6449 1.26313 0.631564 0.775324i \(-0.282413\pi\)
0.631564 + 0.775324i \(0.282413\pi\)
\(480\) 1.51118 0.0689755
\(481\) 26.7033 1.21757
\(482\) −3.84885 −0.175310
\(483\) 10.3404 0.470502
\(484\) 1.00000 0.0454545
\(485\) −2.00346 −0.0909726
\(486\) 7.27057 0.329800
\(487\) 5.60480 0.253978 0.126989 0.991904i \(-0.459469\pi\)
0.126989 + 0.991904i \(0.459469\pi\)
\(488\) −7.09406 −0.321133
\(489\) 12.6502 0.572062
\(490\) 6.01950 0.271933
\(491\) −26.4531 −1.19381 −0.596905 0.802312i \(-0.703603\pi\)
−0.596905 + 0.802312i \(0.703603\pi\)
\(492\) 9.83884 0.443569
\(493\) 37.2967 1.67976
\(494\) −27.3279 −1.22954
\(495\) 0.716341 0.0321971
\(496\) 5.17086 0.232179
\(497\) 10.5975 0.475363
\(498\) 5.38683 0.241390
\(499\) −8.25403 −0.369501 −0.184750 0.982785i \(-0.559148\pi\)
−0.184750 + 0.982785i \(0.559148\pi\)
\(500\) 1.00000 0.0447214
\(501\) −35.8421 −1.60131
\(502\) 20.9190 0.933660
\(503\) −28.3431 −1.26375 −0.631877 0.775069i \(-0.717715\pi\)
−0.631877 + 0.775069i \(0.717715\pi\)
\(504\) −0.709323 −0.0315958
\(505\) 0.947272 0.0421531
\(506\) 6.91028 0.307199
\(507\) −11.2036 −0.497568
\(508\) 6.19048 0.274658
\(509\) 32.7171 1.45016 0.725080 0.688665i \(-0.241803\pi\)
0.725080 + 0.688665i \(0.241803\pi\)
\(510\) 7.33233 0.324681
\(511\) 0.990203 0.0438040
\(512\) −1.00000 −0.0441942
\(513\) −33.9684 −1.49974
\(514\) 17.4654 0.770367
\(515\) −14.1558 −0.623781
\(516\) 1.51139 0.0665354
\(517\) −1.74147 −0.0765897
\(518\) −5.85231 −0.257136
\(519\) −30.9838 −1.36004
\(520\) 4.51816 0.198135
\(521\) −8.12552 −0.355986 −0.177993 0.984032i \(-0.556960\pi\)
−0.177993 + 0.984032i \(0.556960\pi\)
\(522\) 5.50635 0.241006
\(523\) 23.5251 1.02868 0.514340 0.857586i \(-0.328037\pi\)
0.514340 + 0.857586i \(0.328037\pi\)
\(524\) −1.73521 −0.0758032
\(525\) 1.49637 0.0653071
\(526\) −26.9377 −1.17454
\(527\) 25.0893 1.09291
\(528\) 1.51118 0.0657656
\(529\) 24.7519 1.07617
\(530\) 7.27787 0.316130
\(531\) 4.95951 0.215224
\(532\) 5.98919 0.259664
\(533\) 29.4164 1.27417
\(534\) 10.0338 0.434204
\(535\) 6.31073 0.272836
\(536\) 8.62196 0.372412
\(537\) 10.5425 0.454943
\(538\) 10.2138 0.440349
\(539\) 6.01950 0.259278
\(540\) 5.61605 0.241676
\(541\) 16.4866 0.708813 0.354406 0.935091i \(-0.384683\pi\)
0.354406 + 0.935091i \(0.384683\pi\)
\(542\) 20.8175 0.894187
\(543\) −20.0344 −0.859760
\(544\) −4.85206 −0.208030
\(545\) −17.9388 −0.768416
\(546\) 6.76086 0.289338
\(547\) −26.1362 −1.11750 −0.558752 0.829335i \(-0.688720\pi\)
−0.558752 + 0.829335i \(0.688720\pi\)
\(548\) 5.75063 0.245655
\(549\) −5.08176 −0.216884
\(550\) 1.00000 0.0426401
\(551\) −46.4930 −1.98067
\(552\) 10.4427 0.444469
\(553\) 1.70090 0.0723297
\(554\) −11.5058 −0.488836
\(555\) 8.93138 0.379116
\(556\) −22.3773 −0.949009
\(557\) 10.2298 0.433449 0.216724 0.976233i \(-0.430463\pi\)
0.216724 + 0.976233i \(0.430463\pi\)
\(558\) 3.70410 0.156807
\(559\) 4.51881 0.191125
\(560\) −0.990203 −0.0418437
\(561\) 7.33233 0.309571
\(562\) −27.8850 −1.17626
\(563\) −25.4980 −1.07461 −0.537307 0.843387i \(-0.680558\pi\)
−0.537307 + 0.843387i \(0.680558\pi\)
\(564\) −2.63167 −0.110813
\(565\) 13.8348 0.582034
\(566\) −5.13300 −0.215756
\(567\) 6.27574 0.263556
\(568\) 10.7024 0.449061
\(569\) 10.0550 0.421528 0.210764 0.977537i \(-0.432405\pi\)
0.210764 + 0.977537i \(0.432405\pi\)
\(570\) −9.14028 −0.382844
\(571\) −11.7141 −0.490218 −0.245109 0.969496i \(-0.578824\pi\)
−0.245109 + 0.969496i \(0.578824\pi\)
\(572\) 4.51816 0.188914
\(573\) 18.0548 0.754250
\(574\) −6.44692 −0.269089
\(575\) 6.91028 0.288179
\(576\) −0.716341 −0.0298475
\(577\) 8.15872 0.339652 0.169826 0.985474i \(-0.445679\pi\)
0.169826 + 0.985474i \(0.445679\pi\)
\(578\) −6.54250 −0.272132
\(579\) 21.8258 0.907048
\(580\) 7.68677 0.319176
\(581\) −3.52974 −0.146438
\(582\) −3.02759 −0.125498
\(583\) 7.27787 0.301419
\(584\) 1.00000 0.0413803
\(585\) 3.23655 0.133815
\(586\) −2.28640 −0.0944502
\(587\) −9.99356 −0.412478 −0.206239 0.978502i \(-0.566122\pi\)
−0.206239 + 0.978502i \(0.566122\pi\)
\(588\) 9.09653 0.375135
\(589\) −31.2757 −1.28869
\(590\) 6.92339 0.285032
\(591\) −11.9207 −0.490353
\(592\) −5.91021 −0.242908
\(593\) −36.9339 −1.51669 −0.758346 0.651852i \(-0.773992\pi\)
−0.758346 + 0.651852i \(0.773992\pi\)
\(594\) 5.61605 0.230429
\(595\) −4.80453 −0.196966
\(596\) 5.81913 0.238361
\(597\) 14.1079 0.577398
\(598\) 31.2218 1.27675
\(599\) −19.5459 −0.798623 −0.399311 0.916815i \(-0.630751\pi\)
−0.399311 + 0.916815i \(0.630751\pi\)
\(600\) 1.51118 0.0616936
\(601\) 30.4933 1.24385 0.621924 0.783077i \(-0.286351\pi\)
0.621924 + 0.783077i \(0.286351\pi\)
\(602\) −0.990345 −0.0403635
\(603\) 6.17627 0.251517
\(604\) 16.6560 0.677725
\(605\) 1.00000 0.0406558
\(606\) 1.43150 0.0581506
\(607\) 43.3623 1.76002 0.880010 0.474954i \(-0.157535\pi\)
0.880010 + 0.474954i \(0.157535\pi\)
\(608\) 6.04845 0.245297
\(609\) 11.5023 0.466096
\(610\) −7.09406 −0.287230
\(611\) −7.86824 −0.318315
\(612\) −3.47573 −0.140498
\(613\) 14.4218 0.582490 0.291245 0.956649i \(-0.405931\pi\)
0.291245 + 0.956649i \(0.405931\pi\)
\(614\) 21.6710 0.874571
\(615\) 9.83884 0.396740
\(616\) −0.990203 −0.0398964
\(617\) 8.07656 0.325150 0.162575 0.986696i \(-0.448020\pi\)
0.162575 + 0.986696i \(0.448020\pi\)
\(618\) −21.3920 −0.860512
\(619\) −20.9320 −0.841328 −0.420664 0.907216i \(-0.638203\pi\)
−0.420664 + 0.907216i \(0.638203\pi\)
\(620\) 5.17086 0.207667
\(621\) 38.8085 1.55733
\(622\) −12.5364 −0.502662
\(623\) −6.57466 −0.263408
\(624\) 6.82775 0.273329
\(625\) 1.00000 0.0400000
\(626\) −25.4308 −1.01642
\(627\) −9.14028 −0.365028
\(628\) 2.55559 0.101979
\(629\) −28.6767 −1.14342
\(630\) −0.709323 −0.0282601
\(631\) −24.0015 −0.955485 −0.477742 0.878500i \(-0.658545\pi\)
−0.477742 + 0.878500i \(0.658545\pi\)
\(632\) 1.71773 0.0683277
\(633\) −28.4047 −1.12898
\(634\) 5.99644 0.238149
\(635\) 6.19048 0.245662
\(636\) 10.9982 0.436105
\(637\) 27.1971 1.07759
\(638\) 7.68677 0.304322
\(639\) 7.66654 0.303283
\(640\) −1.00000 −0.0395285
\(641\) −28.0727 −1.10881 −0.554403 0.832248i \(-0.687053\pi\)
−0.554403 + 0.832248i \(0.687053\pi\)
\(642\) 9.53663 0.376381
\(643\) 13.5364 0.533823 0.266912 0.963721i \(-0.413997\pi\)
0.266912 + 0.963721i \(0.413997\pi\)
\(644\) −6.84258 −0.269635
\(645\) 1.51139 0.0595111
\(646\) 29.3474 1.15466
\(647\) 32.7173 1.28625 0.643125 0.765761i \(-0.277638\pi\)
0.643125 + 0.765761i \(0.277638\pi\)
\(648\) 6.33783 0.248974
\(649\) 6.92339 0.271767
\(650\) 4.51816 0.177217
\(651\) 7.73754 0.303258
\(652\) −8.37109 −0.327837
\(653\) 1.52324 0.0596089 0.0298045 0.999556i \(-0.490512\pi\)
0.0298045 + 0.999556i \(0.490512\pi\)
\(654\) −27.1088 −1.06004
\(655\) −1.73521 −0.0678005
\(656\) −6.51071 −0.254200
\(657\) 0.716341 0.0279471
\(658\) 1.72441 0.0672244
\(659\) −47.6697 −1.85695 −0.928474 0.371398i \(-0.878878\pi\)
−0.928474 + 0.371398i \(0.878878\pi\)
\(660\) 1.51118 0.0588225
\(661\) 18.0440 0.701831 0.350915 0.936407i \(-0.385870\pi\)
0.350915 + 0.936407i \(0.385870\pi\)
\(662\) 21.5793 0.838704
\(663\) 33.1287 1.28661
\(664\) −3.56466 −0.138336
\(665\) 5.98919 0.232251
\(666\) −4.23373 −0.164054
\(667\) 53.1177 2.05673
\(668\) 23.7180 0.917677
\(669\) 19.4087 0.750385
\(670\) 8.62196 0.333095
\(671\) −7.09406 −0.273863
\(672\) −1.49637 −0.0577238
\(673\) −25.7292 −0.991789 −0.495895 0.868383i \(-0.665160\pi\)
−0.495895 + 0.868383i \(0.665160\pi\)
\(674\) −29.7494 −1.14590
\(675\) 5.61605 0.216162
\(676\) 7.41380 0.285146
\(677\) 17.9743 0.690808 0.345404 0.938454i \(-0.387742\pi\)
0.345404 + 0.938454i \(0.387742\pi\)
\(678\) 20.9068 0.802922
\(679\) 1.98384 0.0761326
\(680\) −4.85206 −0.186068
\(681\) −33.3302 −1.27722
\(682\) 5.17086 0.198003
\(683\) 14.6123 0.559126 0.279563 0.960127i \(-0.409810\pi\)
0.279563 + 0.960127i \(0.409810\pi\)
\(684\) 4.33275 0.165667
\(685\) 5.75063 0.219720
\(686\) −12.8919 −0.492217
\(687\) −5.01454 −0.191317
\(688\) −1.00014 −0.0381301
\(689\) 32.8826 1.25273
\(690\) 10.4427 0.397545
\(691\) 35.2528 1.34108 0.670540 0.741873i \(-0.266062\pi\)
0.670540 + 0.741873i \(0.266062\pi\)
\(692\) 20.5031 0.779411
\(693\) −0.709323 −0.0269450
\(694\) −8.44737 −0.320658
\(695\) −22.3773 −0.848819
\(696\) 11.6161 0.440306
\(697\) −31.5903 −1.19657
\(698\) −9.34190 −0.353596
\(699\) −24.8647 −0.940470
\(700\) −0.990203 −0.0374262
\(701\) −3.51044 −0.132587 −0.0662937 0.997800i \(-0.521117\pi\)
−0.0662937 + 0.997800i \(0.521117\pi\)
\(702\) 25.3742 0.957689
\(703\) 35.7476 1.34825
\(704\) −1.00000 −0.0376889
\(705\) −2.63167 −0.0991143
\(706\) −33.4444 −1.25870
\(707\) −0.937992 −0.0352768
\(708\) 10.4625 0.393204
\(709\) 6.83301 0.256619 0.128310 0.991734i \(-0.459045\pi\)
0.128310 + 0.991734i \(0.459045\pi\)
\(710\) 10.7024 0.401652
\(711\) 1.23048 0.0461466
\(712\) −6.63971 −0.248834
\(713\) 35.7321 1.33818
\(714\) −7.26049 −0.271717
\(715\) 4.51816 0.168970
\(716\) −6.97635 −0.260719
\(717\) 9.12631 0.340828
\(718\) −16.6322 −0.620707
\(719\) −22.0697 −0.823060 −0.411530 0.911396i \(-0.635005\pi\)
−0.411530 + 0.911396i \(0.635005\pi\)
\(720\) −0.716341 −0.0266965
\(721\) 14.0172 0.522026
\(722\) −17.5837 −0.654398
\(723\) −5.81629 −0.216310
\(724\) 13.2575 0.492711
\(725\) 7.68677 0.285479
\(726\) 1.51118 0.0560851
\(727\) 6.99505 0.259432 0.129716 0.991551i \(-0.458593\pi\)
0.129716 + 0.991551i \(0.458593\pi\)
\(728\) −4.47390 −0.165814
\(729\) 30.0006 1.11113
\(730\) 1.00000 0.0370117
\(731\) −4.85276 −0.179486
\(732\) −10.7204 −0.396237
\(733\) −6.06166 −0.223893 −0.111946 0.993714i \(-0.535708\pi\)
−0.111946 + 0.993714i \(0.535708\pi\)
\(734\) −20.0788 −0.741122
\(735\) 9.09653 0.335531
\(736\) −6.91028 −0.254716
\(737\) 8.62196 0.317594
\(738\) −4.66389 −0.171680
\(739\) 22.4896 0.827293 0.413646 0.910438i \(-0.364255\pi\)
0.413646 + 0.910438i \(0.364255\pi\)
\(740\) −5.91021 −0.217264
\(741\) −41.2973 −1.51709
\(742\) −7.20657 −0.264561
\(743\) −16.4312 −0.602804 −0.301402 0.953497i \(-0.597455\pi\)
−0.301402 + 0.953497i \(0.597455\pi\)
\(744\) 7.81409 0.286479
\(745\) 5.81913 0.213196
\(746\) −5.90606 −0.216236
\(747\) −2.55351 −0.0934281
\(748\) −4.85206 −0.177409
\(749\) −6.24890 −0.228330
\(750\) 1.51118 0.0551804
\(751\) 40.0323 1.46080 0.730399 0.683020i \(-0.239334\pi\)
0.730399 + 0.683020i \(0.239334\pi\)
\(752\) 1.74147 0.0635048
\(753\) 31.6123 1.15202
\(754\) 34.7301 1.26479
\(755\) 16.6560 0.606175
\(756\) −5.56103 −0.202253
\(757\) −31.1912 −1.13367 −0.566833 0.823833i \(-0.691831\pi\)
−0.566833 + 0.823833i \(0.691831\pi\)
\(758\) −7.88648 −0.286450
\(759\) 10.4427 0.379045
\(760\) 6.04845 0.219400
\(761\) −25.7187 −0.932304 −0.466152 0.884705i \(-0.654360\pi\)
−0.466152 + 0.884705i \(0.654360\pi\)
\(762\) 9.35491 0.338893
\(763\) 17.7631 0.643067
\(764\) −11.9475 −0.432245
\(765\) −3.47573 −0.125665
\(766\) −17.7655 −0.641894
\(767\) 31.2810 1.12949
\(768\) −1.51118 −0.0545299
\(769\) 17.2208 0.620997 0.310499 0.950574i \(-0.399504\pi\)
0.310499 + 0.950574i \(0.399504\pi\)
\(770\) −0.990203 −0.0356844
\(771\) 26.3934 0.950534
\(772\) −14.4429 −0.519811
\(773\) −16.7962 −0.604118 −0.302059 0.953289i \(-0.597674\pi\)
−0.302059 + 0.953289i \(0.597674\pi\)
\(774\) −0.716444 −0.0257520
\(775\) 5.17086 0.185743
\(776\) 2.00346 0.0719201
\(777\) −8.84388 −0.317272
\(778\) −2.64850 −0.0949532
\(779\) 39.3797 1.41092
\(780\) 6.82775 0.244473
\(781\) 10.7024 0.382960
\(782\) −33.5291 −1.19900
\(783\) 43.1693 1.54274
\(784\) −6.01950 −0.214982
\(785\) 2.55559 0.0912131
\(786\) −2.62222 −0.0935314
\(787\) −2.09610 −0.0747178 −0.0373589 0.999302i \(-0.511894\pi\)
−0.0373589 + 0.999302i \(0.511894\pi\)
\(788\) 7.88836 0.281011
\(789\) −40.7077 −1.44923
\(790\) 1.71773 0.0611141
\(791\) −13.6992 −0.487089
\(792\) −0.716341 −0.0254541
\(793\) −32.0521 −1.13820
\(794\) −23.6146 −0.838052
\(795\) 10.9982 0.390064
\(796\) −9.33569 −0.330895
\(797\) −5.58804 −0.197938 −0.0989692 0.995090i \(-0.531555\pi\)
−0.0989692 + 0.995090i \(0.531555\pi\)
\(798\) 9.05073 0.320392
\(799\) 8.44971 0.298929
\(800\) −1.00000 −0.0353553
\(801\) −4.75630 −0.168055
\(802\) 26.8925 0.949607
\(803\) 1.00000 0.0352892
\(804\) 13.0293 0.459509
\(805\) −6.84258 −0.241169
\(806\) 23.3628 0.822920
\(807\) 15.4349 0.543334
\(808\) −0.947272 −0.0333249
\(809\) 11.8188 0.415527 0.207763 0.978179i \(-0.433382\pi\)
0.207763 + 0.978179i \(0.433382\pi\)
\(810\) 6.33783 0.222689
\(811\) 17.8463 0.626667 0.313334 0.949643i \(-0.398554\pi\)
0.313334 + 0.949643i \(0.398554\pi\)
\(812\) −7.61146 −0.267110
\(813\) 31.4589 1.10331
\(814\) −5.91021 −0.207153
\(815\) −8.37109 −0.293227
\(816\) −7.33233 −0.256683
\(817\) 6.04931 0.211639
\(818\) −9.94490 −0.347715
\(819\) −3.20484 −0.111986
\(820\) −6.51071 −0.227364
\(821\) 36.0498 1.25815 0.629073 0.777346i \(-0.283435\pi\)
0.629073 + 0.777346i \(0.283435\pi\)
\(822\) 8.69022 0.303106
\(823\) −4.30228 −0.149968 −0.0749841 0.997185i \(-0.523891\pi\)
−0.0749841 + 0.997185i \(0.523891\pi\)
\(824\) 14.1558 0.493142
\(825\) 1.51118 0.0526125
\(826\) −6.85557 −0.238536
\(827\) −37.4901 −1.30366 −0.651830 0.758365i \(-0.725998\pi\)
−0.651830 + 0.758365i \(0.725998\pi\)
\(828\) −4.95012 −0.172028
\(829\) 19.3863 0.673313 0.336657 0.941628i \(-0.390704\pi\)
0.336657 + 0.941628i \(0.390704\pi\)
\(830\) −3.56466 −0.123731
\(831\) −17.3874 −0.603161
\(832\) −4.51816 −0.156639
\(833\) −29.2070 −1.01196
\(834\) −33.8161 −1.17096
\(835\) 23.7180 0.820796
\(836\) 6.04845 0.209190
\(837\) 29.0398 1.00376
\(838\) 7.37525 0.254774
\(839\) −22.1036 −0.763100 −0.381550 0.924348i \(-0.624610\pi\)
−0.381550 + 0.924348i \(0.624610\pi\)
\(840\) −1.49637 −0.0516298
\(841\) 30.0864 1.03746
\(842\) −15.5669 −0.536471
\(843\) −42.1392 −1.45135
\(844\) 18.7964 0.646998
\(845\) 7.41380 0.255043
\(846\) 1.24748 0.0428894
\(847\) −0.990203 −0.0340238
\(848\) −7.27787 −0.249923
\(849\) −7.75687 −0.266215
\(850\) −4.85206 −0.166424
\(851\) −40.8412 −1.40002
\(852\) 16.1732 0.554083
\(853\) −25.3138 −0.866730 −0.433365 0.901219i \(-0.642674\pi\)
−0.433365 + 0.901219i \(0.642674\pi\)
\(854\) 7.02456 0.240375
\(855\) 4.33275 0.148177
\(856\) −6.31073 −0.215696
\(857\) −50.4853 −1.72455 −0.862273 0.506444i \(-0.830960\pi\)
−0.862273 + 0.506444i \(0.830960\pi\)
\(858\) 6.82775 0.233096
\(859\) 14.5480 0.496372 0.248186 0.968712i \(-0.420166\pi\)
0.248186 + 0.968712i \(0.420166\pi\)
\(860\) −1.00014 −0.0341046
\(861\) −9.74245 −0.332022
\(862\) 3.21997 0.109673
\(863\) 35.0414 1.19282 0.596412 0.802678i \(-0.296592\pi\)
0.596412 + 0.802678i \(0.296592\pi\)
\(864\) −5.61605 −0.191062
\(865\) 20.5031 0.697126
\(866\) −5.14929 −0.174980
\(867\) −9.88688 −0.335776
\(868\) −5.12020 −0.173791
\(869\) 1.71773 0.0582700
\(870\) 11.6161 0.393822
\(871\) 38.9554 1.31995
\(872\) 17.9388 0.607486
\(873\) 1.43516 0.0485729
\(874\) 41.7964 1.41379
\(875\) −0.990203 −0.0334750
\(876\) 1.51118 0.0510580
\(877\) 2.46213 0.0831402 0.0415701 0.999136i \(-0.486764\pi\)
0.0415701 + 0.999136i \(0.486764\pi\)
\(878\) 11.6466 0.393054
\(879\) −3.45515 −0.116539
\(880\) −1.00000 −0.0337100
\(881\) 46.7674 1.57563 0.787817 0.615909i \(-0.211211\pi\)
0.787817 + 0.615909i \(0.211211\pi\)
\(882\) −4.31201 −0.145193
\(883\) −12.8721 −0.433181 −0.216591 0.976262i \(-0.569494\pi\)
−0.216591 + 0.976262i \(0.569494\pi\)
\(884\) −21.9224 −0.737330
\(885\) 10.4625 0.351692
\(886\) 19.0393 0.639637
\(887\) −2.68379 −0.0901127 −0.0450563 0.998984i \(-0.514347\pi\)
−0.0450563 + 0.998984i \(0.514347\pi\)
\(888\) −8.93138 −0.299718
\(889\) −6.12983 −0.205588
\(890\) −6.63971 −0.222564
\(891\) 6.33783 0.212325
\(892\) −12.8435 −0.430031
\(893\) −10.5332 −0.352479
\(894\) 8.79374 0.294107
\(895\) −6.97635 −0.233194
\(896\) 0.990203 0.0330804
\(897\) 47.1816 1.57535
\(898\) −10.8077 −0.360659
\(899\) 39.7472 1.32564
\(900\) −0.716341 −0.0238780
\(901\) −35.3127 −1.17644
\(902\) −6.51071 −0.216783
\(903\) −1.49659 −0.0498033
\(904\) −13.8348 −0.460138
\(905\) 13.2575 0.440694
\(906\) 25.1702 0.836225
\(907\) 37.4729 1.24427 0.622134 0.782910i \(-0.286266\pi\)
0.622134 + 0.782910i \(0.286266\pi\)
\(908\) 22.0558 0.731946
\(909\) −0.678570 −0.0225067
\(910\) −4.47390 −0.148308
\(911\) −22.1590 −0.734160 −0.367080 0.930189i \(-0.619642\pi\)
−0.367080 + 0.930189i \(0.619642\pi\)
\(912\) 9.14028 0.302665
\(913\) −3.56466 −0.117973
\(914\) −16.7154 −0.552897
\(915\) −10.7204 −0.354405
\(916\) 3.31830 0.109640
\(917\) 1.71821 0.0567405
\(918\) −27.2494 −0.899365
\(919\) 52.9631 1.74709 0.873546 0.486741i \(-0.161815\pi\)
0.873546 + 0.486741i \(0.161815\pi\)
\(920\) −6.91028 −0.227825
\(921\) 32.7488 1.07911
\(922\) −9.61907 −0.316787
\(923\) 48.3550 1.59162
\(924\) −1.49637 −0.0492271
\(925\) −5.91021 −0.194327
\(926\) −28.1461 −0.924937
\(927\) 10.1404 0.333055
\(928\) −7.68677 −0.252331
\(929\) 6.89444 0.226199 0.113100 0.993584i \(-0.463922\pi\)
0.113100 + 0.993584i \(0.463922\pi\)
\(930\) 7.81409 0.256234
\(931\) 36.4086 1.19324
\(932\) 16.4539 0.538964
\(933\) −18.9447 −0.620220
\(934\) −22.0985 −0.723084
\(935\) −4.85206 −0.158679
\(936\) −3.23655 −0.105790
\(937\) 33.0471 1.07960 0.539800 0.841793i \(-0.318500\pi\)
0.539800 + 0.841793i \(0.318500\pi\)
\(938\) −8.53749 −0.278759
\(939\) −38.4304 −1.25413
\(940\) 1.74147 0.0568004
\(941\) −3.06232 −0.0998287 −0.0499144 0.998754i \(-0.515895\pi\)
−0.0499144 + 0.998754i \(0.515895\pi\)
\(942\) 3.86196 0.125829
\(943\) −44.9908 −1.46510
\(944\) −6.92339 −0.225337
\(945\) −5.56103 −0.180900
\(946\) −1.00014 −0.0325175
\(947\) 7.60881 0.247253 0.123627 0.992329i \(-0.460548\pi\)
0.123627 + 0.992329i \(0.460548\pi\)
\(948\) 2.59580 0.0843076
\(949\) 4.51816 0.146666
\(950\) 6.04845 0.196238
\(951\) 9.06169 0.293845
\(952\) 4.80453 0.155716
\(953\) −2.91733 −0.0945017 −0.0472508 0.998883i \(-0.515046\pi\)
−0.0472508 + 0.998883i \(0.515046\pi\)
\(954\) −5.21344 −0.168791
\(955\) −11.9475 −0.386612
\(956\) −6.03920 −0.195322
\(957\) 11.6161 0.375494
\(958\) −27.6449 −0.893166
\(959\) −5.69429 −0.183878
\(960\) −1.51118 −0.0487731
\(961\) −4.26217 −0.137490
\(962\) −26.7033 −0.860949
\(963\) −4.52063 −0.145675
\(964\) 3.84885 0.123963
\(965\) −14.4429 −0.464933
\(966\) −10.3404 −0.332695
\(967\) 29.7568 0.956915 0.478457 0.878111i \(-0.341196\pi\)
0.478457 + 0.878111i \(0.341196\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 44.3492 1.42470
\(970\) 2.00346 0.0643273
\(971\) 53.3495 1.71207 0.856033 0.516921i \(-0.172922\pi\)
0.856033 + 0.516921i \(0.172922\pi\)
\(972\) −7.27057 −0.233204
\(973\) 22.1581 0.710355
\(974\) −5.60480 −0.179589
\(975\) 6.82775 0.218663
\(976\) 7.09406 0.227075
\(977\) 49.1738 1.57321 0.786605 0.617457i \(-0.211837\pi\)
0.786605 + 0.617457i \(0.211837\pi\)
\(978\) −12.6502 −0.404509
\(979\) −6.63971 −0.212206
\(980\) −6.01950 −0.192286
\(981\) 12.8503 0.410279
\(982\) 26.4531 0.844152
\(983\) −17.0572 −0.544042 −0.272021 0.962291i \(-0.587692\pi\)
−0.272021 + 0.962291i \(0.587692\pi\)
\(984\) −9.83884 −0.313651
\(985\) 7.88836 0.251344
\(986\) −37.2967 −1.18777
\(987\) 2.60589 0.0829462
\(988\) 27.3279 0.869415
\(989\) −6.91127 −0.219766
\(990\) −0.716341 −0.0227668
\(991\) −35.9102 −1.14073 −0.570363 0.821393i \(-0.693198\pi\)
−0.570363 + 0.821393i \(0.693198\pi\)
\(992\) −5.17086 −0.164175
\(993\) 32.6102 1.03485
\(994\) −10.5975 −0.336132
\(995\) −9.33569 −0.295961
\(996\) −5.38683 −0.170688
\(997\) 27.2512 0.863055 0.431527 0.902100i \(-0.357975\pi\)
0.431527 + 0.902100i \(0.357975\pi\)
\(998\) 8.25403 0.261277
\(999\) −33.1921 −1.05015
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 8030.2.a.be.1.5 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8030.2.a.be.1.5 15 1.1 even 1 trivial