Properties

Label 1502.2.a.g.1.7
Level $1502$
Weight $2$
Character 1502.1
Self dual yes
Analytic conductor $11.994$
Analytic rank $0$
Dimension $16$
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] = [1502,2,Mod(1,1502)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1502, 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("1502.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1502 = 2 \cdot 751 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1502.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: \(11.9935303836\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 3 x^{15} - 25 x^{14} + 59 x^{13} + 273 x^{12} - 443 x^{11} - 1620 x^{10} + 1595 x^{9} + 5490 x^{8} - 2787 x^{7} - 10540 x^{6} + 1919 x^{5} + 10822 x^{4} + 132 x^{3} + \cdots + 864 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(0.861451\) of defining polynomial
Character \(\chi\) \(=\) 1502.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +0.138549 q^{3} +1.00000 q^{4} +1.13854 q^{5} +0.138549 q^{6} -2.28657 q^{7} +1.00000 q^{8} -2.98080 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +0.138549 q^{3} +1.00000 q^{4} +1.13854 q^{5} +0.138549 q^{6} -2.28657 q^{7} +1.00000 q^{8} -2.98080 q^{9} +1.13854 q^{10} +4.66208 q^{11} +0.138549 q^{12} +1.82268 q^{13} -2.28657 q^{14} +0.157743 q^{15} +1.00000 q^{16} +5.36722 q^{17} -2.98080 q^{18} -1.97980 q^{19} +1.13854 q^{20} -0.316803 q^{21} +4.66208 q^{22} +4.84322 q^{23} +0.138549 q^{24} -3.70374 q^{25} +1.82268 q^{26} -0.828635 q^{27} -2.28657 q^{28} +3.73505 q^{29} +0.157743 q^{30} +2.93872 q^{31} +1.00000 q^{32} +0.645926 q^{33} +5.36722 q^{34} -2.60335 q^{35} -2.98080 q^{36} +4.76643 q^{37} -1.97980 q^{38} +0.252531 q^{39} +1.13854 q^{40} +3.64130 q^{41} -0.316803 q^{42} +6.16489 q^{43} +4.66208 q^{44} -3.39375 q^{45} +4.84322 q^{46} +0.388521 q^{47} +0.138549 q^{48} -1.77158 q^{49} -3.70374 q^{50} +0.743623 q^{51} +1.82268 q^{52} -0.525829 q^{53} -0.828635 q^{54} +5.30794 q^{55} -2.28657 q^{56} -0.274299 q^{57} +3.73505 q^{58} +14.4444 q^{59} +0.157743 q^{60} -10.2226 q^{61} +2.93872 q^{62} +6.81583 q^{63} +1.00000 q^{64} +2.07519 q^{65} +0.645926 q^{66} -10.4758 q^{67} +5.36722 q^{68} +0.671023 q^{69} -2.60335 q^{70} -0.842117 q^{71} -2.98080 q^{72} -7.06349 q^{73} +4.76643 q^{74} -0.513149 q^{75} -1.97980 q^{76} -10.6602 q^{77} +0.252531 q^{78} -3.58053 q^{79} +1.13854 q^{80} +8.82761 q^{81} +3.64130 q^{82} +5.10442 q^{83} -0.316803 q^{84} +6.11077 q^{85} +6.16489 q^{86} +0.517488 q^{87} +4.66208 q^{88} -0.209053 q^{89} -3.39375 q^{90} -4.16770 q^{91} +4.84322 q^{92} +0.407157 q^{93} +0.388521 q^{94} -2.25407 q^{95} +0.138549 q^{96} -10.5510 q^{97} -1.77158 q^{98} -13.8967 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{2} + 13 q^{3} + 16 q^{4} + 4 q^{5} + 13 q^{6} + 7 q^{7} + 16 q^{8} + 21 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{2} + 13 q^{3} + 16 q^{4} + 4 q^{5} + 13 q^{6} + 7 q^{7} + 16 q^{8} + 21 q^{9} + 4 q^{10} + 4 q^{11} + 13 q^{12} + 17 q^{13} + 7 q^{14} + 8 q^{15} + 16 q^{16} - q^{17} + 21 q^{18} + 23 q^{19} + 4 q^{20} + 9 q^{21} + 4 q^{22} + 15 q^{23} + 13 q^{24} + 24 q^{25} + 17 q^{26} + 31 q^{27} + 7 q^{28} + 4 q^{29} + 8 q^{30} + 42 q^{31} + 16 q^{32} + 3 q^{33} - q^{34} - 13 q^{35} + 21 q^{36} + 31 q^{37} + 23 q^{38} - 2 q^{39} + 4 q^{40} - 9 q^{41} + 9 q^{42} + 13 q^{43} + 4 q^{44} - 2 q^{45} + 15 q^{46} + 18 q^{47} + 13 q^{48} - 9 q^{49} + 24 q^{50} - 2 q^{51} + 17 q^{52} - 14 q^{53} + 31 q^{54} - 2 q^{55} + 7 q^{56} - 18 q^{57} + 4 q^{58} + 4 q^{59} + 8 q^{60} + q^{61} + 42 q^{62} + 17 q^{63} + 16 q^{64} - 32 q^{65} + 3 q^{66} + 5 q^{67} - q^{68} + 6 q^{69} - 13 q^{70} + 9 q^{71} + 21 q^{72} + 28 q^{73} + 31 q^{74} + 16 q^{75} + 23 q^{76} - 30 q^{77} - 2 q^{78} + 10 q^{79} + 4 q^{80} + 12 q^{81} - 9 q^{82} + 3 q^{83} + 9 q^{84} - 7 q^{85} + 13 q^{86} - 22 q^{87} + 4 q^{88} - 17 q^{89} - 2 q^{90} + 12 q^{91} + 15 q^{92} - q^{93} + 18 q^{94} - 4 q^{95} + 13 q^{96} - 17 q^{97} - 9 q^{98} - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0.138549 0.0799913 0.0399957 0.999200i \(-0.487266\pi\)
0.0399957 + 0.999200i \(0.487266\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.13854 0.509169 0.254584 0.967051i \(-0.418061\pi\)
0.254584 + 0.967051i \(0.418061\pi\)
\(6\) 0.138549 0.0565624
\(7\) −2.28657 −0.864244 −0.432122 0.901815i \(-0.642235\pi\)
−0.432122 + 0.901815i \(0.642235\pi\)
\(8\) 1.00000 0.353553
\(9\) −2.98080 −0.993601
\(10\) 1.13854 0.360037
\(11\) 4.66208 1.40567 0.702835 0.711353i \(-0.251917\pi\)
0.702835 + 0.711353i \(0.251917\pi\)
\(12\) 0.138549 0.0399957
\(13\) 1.82268 0.505521 0.252760 0.967529i \(-0.418662\pi\)
0.252760 + 0.967529i \(0.418662\pi\)
\(14\) −2.28657 −0.611113
\(15\) 0.157743 0.0407291
\(16\) 1.00000 0.250000
\(17\) 5.36722 1.30174 0.650871 0.759189i \(-0.274404\pi\)
0.650871 + 0.759189i \(0.274404\pi\)
\(18\) −2.98080 −0.702582
\(19\) −1.97980 −0.454197 −0.227098 0.973872i \(-0.572924\pi\)
−0.227098 + 0.973872i \(0.572924\pi\)
\(20\) 1.13854 0.254584
\(21\) −0.316803 −0.0691320
\(22\) 4.66208 0.993958
\(23\) 4.84322 1.00988 0.504940 0.863154i \(-0.331514\pi\)
0.504940 + 0.863154i \(0.331514\pi\)
\(24\) 0.138549 0.0282812
\(25\) −3.70374 −0.740747
\(26\) 1.82268 0.357457
\(27\) −0.828635 −0.159471
\(28\) −2.28657 −0.432122
\(29\) 3.73505 0.693581 0.346791 0.937943i \(-0.387271\pi\)
0.346791 + 0.937943i \(0.387271\pi\)
\(30\) 0.157743 0.0287998
\(31\) 2.93872 0.527810 0.263905 0.964549i \(-0.414989\pi\)
0.263905 + 0.964549i \(0.414989\pi\)
\(32\) 1.00000 0.176777
\(33\) 0.645926 0.112441
\(34\) 5.36722 0.920470
\(35\) −2.60335 −0.440046
\(36\) −2.98080 −0.496801
\(37\) 4.76643 0.783596 0.391798 0.920051i \(-0.371853\pi\)
0.391798 + 0.920051i \(0.371853\pi\)
\(38\) −1.97980 −0.321166
\(39\) 0.252531 0.0404373
\(40\) 1.13854 0.180018
\(41\) 3.64130 0.568676 0.284338 0.958724i \(-0.408226\pi\)
0.284338 + 0.958724i \(0.408226\pi\)
\(42\) −0.316803 −0.0488837
\(43\) 6.16489 0.940136 0.470068 0.882630i \(-0.344229\pi\)
0.470068 + 0.882630i \(0.344229\pi\)
\(44\) 4.66208 0.702835
\(45\) −3.39375 −0.505911
\(46\) 4.84322 0.714094
\(47\) 0.388521 0.0566716 0.0283358 0.999598i \(-0.490979\pi\)
0.0283358 + 0.999598i \(0.490979\pi\)
\(48\) 0.138549 0.0199978
\(49\) −1.77158 −0.253082
\(50\) −3.70374 −0.523787
\(51\) 0.743623 0.104128
\(52\) 1.82268 0.252760
\(53\) −0.525829 −0.0722282 −0.0361141 0.999348i \(-0.511498\pi\)
−0.0361141 + 0.999348i \(0.511498\pi\)
\(54\) −0.828635 −0.112763
\(55\) 5.30794 0.715723
\(56\) −2.28657 −0.305556
\(57\) −0.274299 −0.0363318
\(58\) 3.73505 0.490436
\(59\) 14.4444 1.88050 0.940248 0.340490i \(-0.110593\pi\)
0.940248 + 0.340490i \(0.110593\pi\)
\(60\) 0.157743 0.0203645
\(61\) −10.2226 −1.30888 −0.654438 0.756116i \(-0.727095\pi\)
−0.654438 + 0.756116i \(0.727095\pi\)
\(62\) 2.93872 0.373218
\(63\) 6.81583 0.858714
\(64\) 1.00000 0.125000
\(65\) 2.07519 0.257395
\(66\) 0.645926 0.0795080
\(67\) −10.4758 −1.27982 −0.639910 0.768450i \(-0.721028\pi\)
−0.639910 + 0.768450i \(0.721028\pi\)
\(68\) 5.36722 0.650871
\(69\) 0.671023 0.0807817
\(70\) −2.60335 −0.311160
\(71\) −0.842117 −0.0999409 −0.0499704 0.998751i \(-0.515913\pi\)
−0.0499704 + 0.998751i \(0.515913\pi\)
\(72\) −2.98080 −0.351291
\(73\) −7.06349 −0.826718 −0.413359 0.910568i \(-0.635645\pi\)
−0.413359 + 0.910568i \(0.635645\pi\)
\(74\) 4.76643 0.554086
\(75\) −0.513149 −0.0592533
\(76\) −1.97980 −0.227098
\(77\) −10.6602 −1.21484
\(78\) 0.252531 0.0285935
\(79\) −3.58053 −0.402841 −0.201420 0.979505i \(-0.564556\pi\)
−0.201420 + 0.979505i \(0.564556\pi\)
\(80\) 1.13854 0.127292
\(81\) 8.82761 0.980845
\(82\) 3.64130 0.402115
\(83\) 5.10442 0.560283 0.280142 0.959959i \(-0.409619\pi\)
0.280142 + 0.959959i \(0.409619\pi\)
\(84\) −0.316803 −0.0345660
\(85\) 6.11077 0.662806
\(86\) 6.16489 0.664777
\(87\) 0.517488 0.0554805
\(88\) 4.66208 0.496979
\(89\) −0.209053 −0.0221596 −0.0110798 0.999939i \(-0.503527\pi\)
−0.0110798 + 0.999939i \(0.503527\pi\)
\(90\) −3.39375 −0.357733
\(91\) −4.16770 −0.436894
\(92\) 4.84322 0.504940
\(93\) 0.407157 0.0422202
\(94\) 0.388521 0.0400729
\(95\) −2.25407 −0.231263
\(96\) 0.138549 0.0141406
\(97\) −10.5510 −1.07129 −0.535646 0.844443i \(-0.679932\pi\)
−0.535646 + 0.844443i \(0.679932\pi\)
\(98\) −1.77158 −0.178956
\(99\) −13.8967 −1.39667
\(100\) −3.70374 −0.370374
\(101\) 8.90464 0.886045 0.443023 0.896510i \(-0.353906\pi\)
0.443023 + 0.896510i \(0.353906\pi\)
\(102\) 0.743623 0.0736296
\(103\) −10.6656 −1.05091 −0.525455 0.850822i \(-0.676105\pi\)
−0.525455 + 0.850822i \(0.676105\pi\)
\(104\) 1.82268 0.178729
\(105\) −0.360691 −0.0351999
\(106\) −0.525829 −0.0510730
\(107\) −10.7423 −1.03849 −0.519246 0.854625i \(-0.673787\pi\)
−0.519246 + 0.854625i \(0.673787\pi\)
\(108\) −0.828635 −0.0797354
\(109\) −11.6993 −1.12059 −0.560297 0.828292i \(-0.689313\pi\)
−0.560297 + 0.828292i \(0.689313\pi\)
\(110\) 5.30794 0.506092
\(111\) 0.660384 0.0626809
\(112\) −2.28657 −0.216061
\(113\) 15.2792 1.43735 0.718675 0.695346i \(-0.244749\pi\)
0.718675 + 0.695346i \(0.244749\pi\)
\(114\) −0.274299 −0.0256905
\(115\) 5.51418 0.514200
\(116\) 3.73505 0.346791
\(117\) −5.43306 −0.502286
\(118\) 14.4444 1.32971
\(119\) −12.2725 −1.12502
\(120\) 0.157743 0.0143999
\(121\) 10.7350 0.975906
\(122\) −10.2226 −0.925515
\(123\) 0.504499 0.0454891
\(124\) 2.93872 0.263905
\(125\) −9.90952 −0.886334
\(126\) 6.81583 0.607203
\(127\) −6.13297 −0.544213 −0.272107 0.962267i \(-0.587720\pi\)
−0.272107 + 0.962267i \(0.587720\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0.854139 0.0752027
\(130\) 2.07519 0.182006
\(131\) −7.17094 −0.626528 −0.313264 0.949666i \(-0.601422\pi\)
−0.313264 + 0.949666i \(0.601422\pi\)
\(132\) 0.645926 0.0562207
\(133\) 4.52696 0.392537
\(134\) −10.4758 −0.904969
\(135\) −0.943430 −0.0811975
\(136\) 5.36722 0.460235
\(137\) −2.24576 −0.191868 −0.0959340 0.995388i \(-0.530584\pi\)
−0.0959340 + 0.995388i \(0.530584\pi\)
\(138\) 0.671023 0.0571213
\(139\) 12.4554 1.05645 0.528226 0.849104i \(-0.322857\pi\)
0.528226 + 0.849104i \(0.322857\pi\)
\(140\) −2.60335 −0.220023
\(141\) 0.0538292 0.00453323
\(142\) −0.842117 −0.0706689
\(143\) 8.49748 0.710595
\(144\) −2.98080 −0.248400
\(145\) 4.25249 0.353150
\(146\) −7.06349 −0.584578
\(147\) −0.245450 −0.0202444
\(148\) 4.76643 0.391798
\(149\) −15.6328 −1.28069 −0.640343 0.768089i \(-0.721208\pi\)
−0.640343 + 0.768089i \(0.721208\pi\)
\(150\) −0.513149 −0.0418984
\(151\) 13.1386 1.06920 0.534602 0.845104i \(-0.320462\pi\)
0.534602 + 0.845104i \(0.320462\pi\)
\(152\) −1.97980 −0.160583
\(153\) −15.9986 −1.29341
\(154\) −10.6602 −0.859022
\(155\) 3.34584 0.268744
\(156\) 0.252531 0.0202186
\(157\) −17.4965 −1.39638 −0.698188 0.715914i \(-0.746010\pi\)
−0.698188 + 0.715914i \(0.746010\pi\)
\(158\) −3.58053 −0.284851
\(159\) −0.0728531 −0.00577762
\(160\) 1.13854 0.0900092
\(161\) −11.0744 −0.872783
\(162\) 8.82761 0.693562
\(163\) −5.75782 −0.450987 −0.225494 0.974245i \(-0.572399\pi\)
−0.225494 + 0.974245i \(0.572399\pi\)
\(164\) 3.64130 0.284338
\(165\) 0.735410 0.0572516
\(166\) 5.10442 0.396180
\(167\) 16.6673 1.28975 0.644876 0.764287i \(-0.276909\pi\)
0.644876 + 0.764287i \(0.276909\pi\)
\(168\) −0.316803 −0.0244419
\(169\) −9.67783 −0.744449
\(170\) 6.11077 0.468675
\(171\) 5.90139 0.451291
\(172\) 6.16489 0.470068
\(173\) −5.18347 −0.394092 −0.197046 0.980394i \(-0.563135\pi\)
−0.197046 + 0.980394i \(0.563135\pi\)
\(174\) 0.517488 0.0392306
\(175\) 8.46887 0.640186
\(176\) 4.66208 0.351417
\(177\) 2.00125 0.150423
\(178\) −0.209053 −0.0156692
\(179\) −1.73198 −0.129455 −0.0647273 0.997903i \(-0.520618\pi\)
−0.0647273 + 0.997903i \(0.520618\pi\)
\(180\) −3.39375 −0.252955
\(181\) 17.2557 1.28261 0.641303 0.767288i \(-0.278394\pi\)
0.641303 + 0.767288i \(0.278394\pi\)
\(182\) −4.16770 −0.308930
\(183\) −1.41634 −0.104699
\(184\) 4.84322 0.357047
\(185\) 5.42675 0.398983
\(186\) 0.407157 0.0298542
\(187\) 25.0224 1.82982
\(188\) 0.388521 0.0283358
\(189\) 1.89473 0.137822
\(190\) −2.25407 −0.163528
\(191\) 23.1592 1.67574 0.837872 0.545867i \(-0.183800\pi\)
0.837872 + 0.545867i \(0.183800\pi\)
\(192\) 0.138549 0.00999891
\(193\) 1.10413 0.0794771 0.0397386 0.999210i \(-0.487347\pi\)
0.0397386 + 0.999210i \(0.487347\pi\)
\(194\) −10.5510 −0.757518
\(195\) 0.287515 0.0205894
\(196\) −1.77158 −0.126541
\(197\) −4.14299 −0.295176 −0.147588 0.989049i \(-0.547151\pi\)
−0.147588 + 0.989049i \(0.547151\pi\)
\(198\) −13.8967 −0.987598
\(199\) 9.85628 0.698693 0.349347 0.936994i \(-0.386404\pi\)
0.349347 + 0.936994i \(0.386404\pi\)
\(200\) −3.70374 −0.261894
\(201\) −1.45141 −0.102374
\(202\) 8.90464 0.626528
\(203\) −8.54047 −0.599424
\(204\) 0.743623 0.0520640
\(205\) 4.14575 0.289552
\(206\) −10.6656 −0.743105
\(207\) −14.4367 −1.00342
\(208\) 1.82268 0.126380
\(209\) −9.22997 −0.638451
\(210\) −0.360691 −0.0248901
\(211\) −18.3543 −1.26356 −0.631780 0.775148i \(-0.717675\pi\)
−0.631780 + 0.775148i \(0.717675\pi\)
\(212\) −0.525829 −0.0361141
\(213\) −0.116674 −0.00799440
\(214\) −10.7423 −0.734325
\(215\) 7.01894 0.478688
\(216\) −0.828635 −0.0563814
\(217\) −6.71961 −0.456157
\(218\) −11.6993 −0.792379
\(219\) −0.978639 −0.0661303
\(220\) 5.30794 0.357861
\(221\) 9.78273 0.658058
\(222\) 0.660384 0.0443221
\(223\) −1.09531 −0.0733471 −0.0366735 0.999327i \(-0.511676\pi\)
−0.0366735 + 0.999327i \(0.511676\pi\)
\(224\) −2.28657 −0.152778
\(225\) 11.0401 0.736007
\(226\) 15.2792 1.01636
\(227\) −2.25099 −0.149404 −0.0747018 0.997206i \(-0.523800\pi\)
−0.0747018 + 0.997206i \(0.523800\pi\)
\(228\) −0.274299 −0.0181659
\(229\) −13.2224 −0.873764 −0.436882 0.899519i \(-0.643917\pi\)
−0.436882 + 0.899519i \(0.643917\pi\)
\(230\) 5.51418 0.363594
\(231\) −1.47696 −0.0971767
\(232\) 3.73505 0.245218
\(233\) −4.72089 −0.309276 −0.154638 0.987971i \(-0.549421\pi\)
−0.154638 + 0.987971i \(0.549421\pi\)
\(234\) −5.43306 −0.355170
\(235\) 0.442345 0.0288554
\(236\) 14.4444 0.940248
\(237\) −0.496078 −0.0322237
\(238\) −12.2725 −0.795511
\(239\) 5.84687 0.378203 0.189101 0.981958i \(-0.439443\pi\)
0.189101 + 0.981958i \(0.439443\pi\)
\(240\) 0.157743 0.0101823
\(241\) −0.115891 −0.00746520 −0.00373260 0.999993i \(-0.501188\pi\)
−0.00373260 + 0.999993i \(0.501188\pi\)
\(242\) 10.7350 0.690070
\(243\) 3.70896 0.237930
\(244\) −10.2226 −0.654438
\(245\) −2.01700 −0.128862
\(246\) 0.504499 0.0321657
\(247\) −3.60854 −0.229606
\(248\) 2.93872 0.186609
\(249\) 0.707213 0.0448178
\(250\) −9.90952 −0.626733
\(251\) −7.68083 −0.484809 −0.242405 0.970175i \(-0.577936\pi\)
−0.242405 + 0.970175i \(0.577936\pi\)
\(252\) 6.81583 0.429357
\(253\) 22.5795 1.41956
\(254\) −6.13297 −0.384817
\(255\) 0.846641 0.0530187
\(256\) 1.00000 0.0625000
\(257\) −22.7498 −1.41909 −0.709546 0.704660i \(-0.751100\pi\)
−0.709546 + 0.704660i \(0.751100\pi\)
\(258\) 0.854139 0.0531764
\(259\) −10.8988 −0.677218
\(260\) 2.07519 0.128698
\(261\) −11.1335 −0.689143
\(262\) −7.17094 −0.443022
\(263\) 28.3260 1.74666 0.873329 0.487130i \(-0.161956\pi\)
0.873329 + 0.487130i \(0.161956\pi\)
\(264\) 0.645926 0.0397540
\(265\) −0.598675 −0.0367763
\(266\) 4.52696 0.277566
\(267\) −0.0289641 −0.00177257
\(268\) −10.4758 −0.639910
\(269\) −25.1503 −1.53344 −0.766721 0.641981i \(-0.778113\pi\)
−0.766721 + 0.641981i \(0.778113\pi\)
\(270\) −0.943430 −0.0574153
\(271\) −21.3951 −1.29966 −0.649829 0.760080i \(-0.725160\pi\)
−0.649829 + 0.760080i \(0.725160\pi\)
\(272\) 5.36722 0.325435
\(273\) −0.577431 −0.0349477
\(274\) −2.24576 −0.135671
\(275\) −17.2671 −1.04125
\(276\) 0.671023 0.0403908
\(277\) −9.82730 −0.590465 −0.295233 0.955425i \(-0.595397\pi\)
−0.295233 + 0.955425i \(0.595397\pi\)
\(278\) 12.4554 0.747025
\(279\) −8.75976 −0.524433
\(280\) −2.60335 −0.155580
\(281\) −17.7452 −1.05859 −0.529295 0.848438i \(-0.677544\pi\)
−0.529295 + 0.848438i \(0.677544\pi\)
\(282\) 0.0538292 0.00320548
\(283\) −27.8139 −1.65336 −0.826682 0.562669i \(-0.809774\pi\)
−0.826682 + 0.562669i \(0.809774\pi\)
\(284\) −0.842117 −0.0499704
\(285\) −0.312299 −0.0184990
\(286\) 8.49748 0.502467
\(287\) −8.32611 −0.491475
\(288\) −2.98080 −0.175646
\(289\) 11.8070 0.694530
\(290\) 4.25249 0.249715
\(291\) −1.46183 −0.0856940
\(292\) −7.06349 −0.413359
\(293\) 0.342825 0.0200281 0.0100140 0.999950i \(-0.496812\pi\)
0.0100140 + 0.999950i \(0.496812\pi\)
\(294\) −0.245450 −0.0143149
\(295\) 16.4454 0.957490
\(296\) 4.76643 0.277043
\(297\) −3.86316 −0.224163
\(298\) −15.6328 −0.905582
\(299\) 8.82765 0.510516
\(300\) −0.513149 −0.0296267
\(301\) −14.0965 −0.812507
\(302\) 13.1386 0.756041
\(303\) 1.23373 0.0708759
\(304\) −1.97980 −0.113549
\(305\) −11.6389 −0.666439
\(306\) −15.9986 −0.914580
\(307\) 19.5763 1.11728 0.558640 0.829410i \(-0.311323\pi\)
0.558640 + 0.829410i \(0.311323\pi\)
\(308\) −10.6602 −0.607421
\(309\) −1.47770 −0.0840636
\(310\) 3.34584 0.190031
\(311\) −23.1403 −1.31217 −0.656084 0.754688i \(-0.727788\pi\)
−0.656084 + 0.754688i \(0.727788\pi\)
\(312\) 0.252531 0.0142967
\(313\) 7.89543 0.446276 0.223138 0.974787i \(-0.428370\pi\)
0.223138 + 0.974787i \(0.428370\pi\)
\(314\) −17.4965 −0.987387
\(315\) 7.76007 0.437230
\(316\) −3.58053 −0.201420
\(317\) −9.95610 −0.559190 −0.279595 0.960118i \(-0.590200\pi\)
−0.279595 + 0.960118i \(0.590200\pi\)
\(318\) −0.0728531 −0.00408540
\(319\) 17.4131 0.974946
\(320\) 1.13854 0.0636461
\(321\) −1.48833 −0.0830704
\(322\) −11.0744 −0.617151
\(323\) −10.6260 −0.591247
\(324\) 8.82761 0.490423
\(325\) −6.75073 −0.374463
\(326\) −5.75782 −0.318896
\(327\) −1.62093 −0.0896377
\(328\) 3.64130 0.201057
\(329\) −0.888382 −0.0489781
\(330\) 0.735410 0.0404830
\(331\) −14.4038 −0.791704 −0.395852 0.918314i \(-0.629551\pi\)
−0.395852 + 0.918314i \(0.629551\pi\)
\(332\) 5.10442 0.280142
\(333\) −14.2078 −0.778582
\(334\) 16.6673 0.911992
\(335\) −11.9270 −0.651644
\(336\) −0.316803 −0.0172830
\(337\) −1.71653 −0.0935056 −0.0467528 0.998906i \(-0.514887\pi\)
−0.0467528 + 0.998906i \(0.514887\pi\)
\(338\) −9.67783 −0.526405
\(339\) 2.11692 0.114976
\(340\) 6.11077 0.331403
\(341\) 13.7006 0.741927
\(342\) 5.90139 0.319111
\(343\) 20.0569 1.08297
\(344\) 6.16489 0.332388
\(345\) 0.763984 0.0411315
\(346\) −5.18347 −0.278665
\(347\) 10.7239 0.575689 0.287845 0.957677i \(-0.407061\pi\)
0.287845 + 0.957677i \(0.407061\pi\)
\(348\) 0.517488 0.0277402
\(349\) 13.0689 0.699563 0.349782 0.936831i \(-0.386256\pi\)
0.349782 + 0.936831i \(0.386256\pi\)
\(350\) 8.46887 0.452680
\(351\) −1.51034 −0.0806158
\(352\) 4.66208 0.248490
\(353\) 10.7805 0.573788 0.286894 0.957962i \(-0.407377\pi\)
0.286894 + 0.957962i \(0.407377\pi\)
\(354\) 2.00125 0.106365
\(355\) −0.958780 −0.0508868
\(356\) −0.209053 −0.0110798
\(357\) −1.70035 −0.0899920
\(358\) −1.73198 −0.0915382
\(359\) 2.01409 0.106299 0.0531497 0.998587i \(-0.483074\pi\)
0.0531497 + 0.998587i \(0.483074\pi\)
\(360\) −3.39375 −0.178866
\(361\) −15.0804 −0.793705
\(362\) 17.2557 0.906939
\(363\) 1.48732 0.0780640
\(364\) −4.16770 −0.218447
\(365\) −8.04203 −0.420939
\(366\) −1.41634 −0.0740332
\(367\) −8.65410 −0.451740 −0.225870 0.974157i \(-0.572523\pi\)
−0.225870 + 0.974157i \(0.572523\pi\)
\(368\) 4.84322 0.252470
\(369\) −10.8540 −0.565037
\(370\) 5.42675 0.282123
\(371\) 1.20235 0.0624227
\(372\) 0.407157 0.0211101
\(373\) 1.86032 0.0963239 0.0481619 0.998840i \(-0.484664\pi\)
0.0481619 + 0.998840i \(0.484664\pi\)
\(374\) 25.0224 1.29388
\(375\) −1.37295 −0.0708990
\(376\) 0.388521 0.0200364
\(377\) 6.80781 0.350620
\(378\) 1.89473 0.0974546
\(379\) −31.4273 −1.61431 −0.807157 0.590337i \(-0.798995\pi\)
−0.807157 + 0.590337i \(0.798995\pi\)
\(380\) −2.25407 −0.115631
\(381\) −0.849717 −0.0435323
\(382\) 23.1592 1.18493
\(383\) −4.84782 −0.247712 −0.123856 0.992300i \(-0.539526\pi\)
−0.123856 + 0.992300i \(0.539526\pi\)
\(384\) 0.138549 0.00707030
\(385\) −12.1370 −0.618559
\(386\) 1.10413 0.0561988
\(387\) −18.3763 −0.934121
\(388\) −10.5510 −0.535646
\(389\) −15.2510 −0.773256 −0.386628 0.922236i \(-0.626360\pi\)
−0.386628 + 0.922236i \(0.626360\pi\)
\(390\) 0.287515 0.0145589
\(391\) 25.9946 1.31460
\(392\) −1.77158 −0.0894781
\(393\) −0.993527 −0.0501168
\(394\) −4.14299 −0.208721
\(395\) −4.07656 −0.205114
\(396\) −13.8967 −0.698337
\(397\) 17.2077 0.863630 0.431815 0.901962i \(-0.357873\pi\)
0.431815 + 0.901962i \(0.357873\pi\)
\(398\) 9.85628 0.494051
\(399\) 0.627206 0.0313996
\(400\) −3.70374 −0.185187
\(401\) 36.1817 1.80683 0.903413 0.428771i \(-0.141053\pi\)
0.903413 + 0.428771i \(0.141053\pi\)
\(402\) −1.45141 −0.0723897
\(403\) 5.35636 0.266819
\(404\) 8.90464 0.443023
\(405\) 10.0505 0.499416
\(406\) −8.54047 −0.423856
\(407\) 22.2215 1.10148
\(408\) 0.743623 0.0368148
\(409\) −16.9164 −0.836463 −0.418232 0.908340i \(-0.637350\pi\)
−0.418232 + 0.908340i \(0.637350\pi\)
\(410\) 4.14575 0.204744
\(411\) −0.311147 −0.0153478
\(412\) −10.6656 −0.525455
\(413\) −33.0281 −1.62521
\(414\) −14.4367 −0.709524
\(415\) 5.81157 0.285279
\(416\) 1.82268 0.0893643
\(417\) 1.72568 0.0845070
\(418\) −9.22997 −0.451453
\(419\) 25.6805 1.25458 0.627288 0.778787i \(-0.284165\pi\)
0.627288 + 0.778787i \(0.284165\pi\)
\(420\) −0.360691 −0.0175999
\(421\) 15.1695 0.739315 0.369657 0.929168i \(-0.379475\pi\)
0.369657 + 0.929168i \(0.379475\pi\)
\(422\) −18.3543 −0.893472
\(423\) −1.15810 −0.0563090
\(424\) −0.525829 −0.0255365
\(425\) −19.8788 −0.964261
\(426\) −0.116674 −0.00565289
\(427\) 23.3749 1.13119
\(428\) −10.7423 −0.519246
\(429\) 1.17732 0.0568415
\(430\) 7.01894 0.338484
\(431\) −22.0541 −1.06231 −0.531153 0.847276i \(-0.678241\pi\)
−0.531153 + 0.847276i \(0.678241\pi\)
\(432\) −0.828635 −0.0398677
\(433\) 16.9027 0.812293 0.406147 0.913808i \(-0.366872\pi\)
0.406147 + 0.913808i \(0.366872\pi\)
\(434\) −6.71961 −0.322552
\(435\) 0.589178 0.0282489
\(436\) −11.6993 −0.560297
\(437\) −9.58860 −0.458685
\(438\) −0.978639 −0.0467612
\(439\) 39.1863 1.87026 0.935131 0.354302i \(-0.115281\pi\)
0.935131 + 0.354302i \(0.115281\pi\)
\(440\) 5.30794 0.253046
\(441\) 5.28072 0.251463
\(442\) 9.78273 0.465317
\(443\) −26.0277 −1.23661 −0.618307 0.785936i \(-0.712181\pi\)
−0.618307 + 0.785936i \(0.712181\pi\)
\(444\) 0.660384 0.0313404
\(445\) −0.238014 −0.0112830
\(446\) −1.09531 −0.0518642
\(447\) −2.16591 −0.102444
\(448\) −2.28657 −0.108031
\(449\) −24.2305 −1.14351 −0.571754 0.820425i \(-0.693737\pi\)
−0.571754 + 0.820425i \(0.693737\pi\)
\(450\) 11.0401 0.520436
\(451\) 16.9760 0.799370
\(452\) 15.2792 0.718675
\(453\) 1.82034 0.0855270
\(454\) −2.25099 −0.105644
\(455\) −4.74507 −0.222452
\(456\) −0.274299 −0.0128452
\(457\) −29.9564 −1.40130 −0.700650 0.713505i \(-0.747107\pi\)
−0.700650 + 0.713505i \(0.747107\pi\)
\(458\) −13.2224 −0.617844
\(459\) −4.44746 −0.207590
\(460\) 5.51418 0.257100
\(461\) 7.05161 0.328426 0.164213 0.986425i \(-0.447492\pi\)
0.164213 + 0.986425i \(0.447492\pi\)
\(462\) −1.47696 −0.0687143
\(463\) 33.0534 1.53612 0.768060 0.640378i \(-0.221222\pi\)
0.768060 + 0.640378i \(0.221222\pi\)
\(464\) 3.73505 0.173395
\(465\) 0.463563 0.0214972
\(466\) −4.72089 −0.218691
\(467\) 32.7686 1.51635 0.758175 0.652051i \(-0.226091\pi\)
0.758175 + 0.652051i \(0.226091\pi\)
\(468\) −5.43306 −0.251143
\(469\) 23.9536 1.10608
\(470\) 0.442345 0.0204038
\(471\) −2.42413 −0.111698
\(472\) 14.4444 0.664856
\(473\) 28.7412 1.32152
\(474\) −0.496078 −0.0227856
\(475\) 7.33265 0.336445
\(476\) −12.2725 −0.562511
\(477\) 1.56739 0.0717660
\(478\) 5.84687 0.267430
\(479\) −17.9402 −0.819710 −0.409855 0.912151i \(-0.634421\pi\)
−0.409855 + 0.912151i \(0.634421\pi\)
\(480\) 0.157743 0.00719995
\(481\) 8.68768 0.396124
\(482\) −0.115891 −0.00527869
\(483\) −1.53434 −0.0698151
\(484\) 10.7350 0.487953
\(485\) −12.0127 −0.545468
\(486\) 3.70896 0.168242
\(487\) 34.1072 1.54555 0.772773 0.634682i \(-0.218869\pi\)
0.772773 + 0.634682i \(0.218869\pi\)
\(488\) −10.2226 −0.462758
\(489\) −0.797740 −0.0360750
\(490\) −2.01700 −0.0911189
\(491\) 5.20616 0.234951 0.117475 0.993076i \(-0.462520\pi\)
0.117475 + 0.993076i \(0.462520\pi\)
\(492\) 0.504499 0.0227446
\(493\) 20.0468 0.902864
\(494\) −3.60854 −0.162356
\(495\) −15.8219 −0.711143
\(496\) 2.93872 0.131953
\(497\) 1.92556 0.0863733
\(498\) 0.707213 0.0316910
\(499\) −15.5744 −0.697206 −0.348603 0.937271i \(-0.613344\pi\)
−0.348603 + 0.937271i \(0.613344\pi\)
\(500\) −9.90952 −0.443167
\(501\) 2.30923 0.103169
\(502\) −7.68083 −0.342812
\(503\) 28.1844 1.25668 0.628341 0.777938i \(-0.283734\pi\)
0.628341 + 0.777938i \(0.283734\pi\)
\(504\) 6.81583 0.303601
\(505\) 10.1383 0.451146
\(506\) 22.5795 1.00378
\(507\) −1.34085 −0.0595494
\(508\) −6.13297 −0.272107
\(509\) −26.7550 −1.18589 −0.592947 0.805241i \(-0.702036\pi\)
−0.592947 + 0.805241i \(0.702036\pi\)
\(510\) 0.846641 0.0374899
\(511\) 16.1512 0.714486
\(512\) 1.00000 0.0441942
\(513\) 1.64053 0.0724311
\(514\) −22.7498 −1.00345
\(515\) −12.1431 −0.535090
\(516\) 0.854139 0.0376014
\(517\) 1.81131 0.0796615
\(518\) −10.8988 −0.478866
\(519\) −0.718164 −0.0315239
\(520\) 2.07519 0.0910030
\(521\) −38.9717 −1.70738 −0.853690 0.520782i \(-0.825640\pi\)
−0.853690 + 0.520782i \(0.825640\pi\)
\(522\) −11.1335 −0.487298
\(523\) 22.3230 0.976116 0.488058 0.872811i \(-0.337706\pi\)
0.488058 + 0.872811i \(0.337706\pi\)
\(524\) −7.17094 −0.313264
\(525\) 1.17335 0.0512094
\(526\) 28.3260 1.23507
\(527\) 15.7728 0.687072
\(528\) 0.645926 0.0281103
\(529\) 0.456764 0.0198593
\(530\) −0.598675 −0.0260048
\(531\) −43.0558 −1.86846
\(532\) 4.52696 0.196269
\(533\) 6.63694 0.287478
\(534\) −0.0289641 −0.00125340
\(535\) −12.2304 −0.528768
\(536\) −10.4758 −0.452484
\(537\) −0.239965 −0.0103552
\(538\) −25.1503 −1.08431
\(539\) −8.25922 −0.355750
\(540\) −0.943430 −0.0405988
\(541\) 35.1038 1.50923 0.754616 0.656167i \(-0.227823\pi\)
0.754616 + 0.656167i \(0.227823\pi\)
\(542\) −21.3951 −0.918998
\(543\) 2.39076 0.102597
\(544\) 5.36722 0.230118
\(545\) −13.3201 −0.570571
\(546\) −0.577431 −0.0247117
\(547\) −9.37171 −0.400705 −0.200353 0.979724i \(-0.564209\pi\)
−0.200353 + 0.979724i \(0.564209\pi\)
\(548\) −2.24576 −0.0959340
\(549\) 30.4717 1.30050
\(550\) −17.2671 −0.736272
\(551\) −7.39465 −0.315023
\(552\) 0.671023 0.0285606
\(553\) 8.18714 0.348153
\(554\) −9.82730 −0.417522
\(555\) 0.751871 0.0319151
\(556\) 12.4554 0.528226
\(557\) −22.4550 −0.951450 −0.475725 0.879594i \(-0.657814\pi\)
−0.475725 + 0.879594i \(0.657814\pi\)
\(558\) −8.75976 −0.370830
\(559\) 11.2366 0.475259
\(560\) −2.60335 −0.110012
\(561\) 3.46683 0.146370
\(562\) −17.7452 −0.748536
\(563\) 17.6918 0.745622 0.372811 0.927907i \(-0.378394\pi\)
0.372811 + 0.927907i \(0.378394\pi\)
\(564\) 0.0538292 0.00226662
\(565\) 17.3960 0.731854
\(566\) −27.8139 −1.16911
\(567\) −20.1850 −0.847690
\(568\) −0.842117 −0.0353344
\(569\) 1.79786 0.0753701 0.0376851 0.999290i \(-0.488002\pi\)
0.0376851 + 0.999290i \(0.488002\pi\)
\(570\) −0.312299 −0.0130808
\(571\) −20.4821 −0.857150 −0.428575 0.903506i \(-0.640984\pi\)
−0.428575 + 0.903506i \(0.640984\pi\)
\(572\) 8.49748 0.355298
\(573\) 3.20869 0.134045
\(574\) −8.32611 −0.347525
\(575\) −17.9380 −0.748066
\(576\) −2.98080 −0.124200
\(577\) −2.55098 −0.106199 −0.0530993 0.998589i \(-0.516910\pi\)
−0.0530993 + 0.998589i \(0.516910\pi\)
\(578\) 11.8070 0.491107
\(579\) 0.152976 0.00635748
\(580\) 4.25249 0.176575
\(581\) −11.6716 −0.484221
\(582\) −1.46183 −0.0605948
\(583\) −2.45145 −0.101529
\(584\) −7.06349 −0.292289
\(585\) −6.18573 −0.255748
\(586\) 0.342825 0.0141620
\(587\) 30.7292 1.26833 0.634166 0.773197i \(-0.281344\pi\)
0.634166 + 0.773197i \(0.281344\pi\)
\(588\) −0.245450 −0.0101222
\(589\) −5.81808 −0.239730
\(590\) 16.4454 0.677047
\(591\) −0.574007 −0.0236115
\(592\) 4.76643 0.195899
\(593\) 8.38904 0.344496 0.172248 0.985054i \(-0.444897\pi\)
0.172248 + 0.985054i \(0.444897\pi\)
\(594\) −3.86316 −0.158507
\(595\) −13.9727 −0.572826
\(596\) −15.6328 −0.640343
\(597\) 1.36558 0.0558894
\(598\) 8.82765 0.360989
\(599\) 32.1056 1.31180 0.655899 0.754849i \(-0.272290\pi\)
0.655899 + 0.754849i \(0.272290\pi\)
\(600\) −0.513149 −0.0209492
\(601\) −10.6968 −0.436333 −0.218166 0.975912i \(-0.570008\pi\)
−0.218166 + 0.975912i \(0.570008\pi\)
\(602\) −14.0965 −0.574529
\(603\) 31.2262 1.27163
\(604\) 13.1386 0.534602
\(605\) 12.2221 0.496901
\(606\) 1.23373 0.0501168
\(607\) −9.33240 −0.378790 −0.189395 0.981901i \(-0.560653\pi\)
−0.189395 + 0.981901i \(0.560653\pi\)
\(608\) −1.97980 −0.0802914
\(609\) −1.18327 −0.0479487
\(610\) −11.6389 −0.471243
\(611\) 0.708150 0.0286487
\(612\) −15.9986 −0.646706
\(613\) 26.3233 1.06319 0.531595 0.846999i \(-0.321593\pi\)
0.531595 + 0.846999i \(0.321593\pi\)
\(614\) 19.5763 0.790036
\(615\) 0.574390 0.0231616
\(616\) −10.6602 −0.429511
\(617\) 23.1241 0.930940 0.465470 0.885064i \(-0.345885\pi\)
0.465470 + 0.885064i \(0.345885\pi\)
\(618\) −1.47770 −0.0594420
\(619\) −36.0721 −1.44986 −0.724930 0.688822i \(-0.758128\pi\)
−0.724930 + 0.688822i \(0.758128\pi\)
\(620\) 3.34584 0.134372
\(621\) −4.01326 −0.161047
\(622\) −23.1403 −0.927843
\(623\) 0.478015 0.0191513
\(624\) 0.252531 0.0101093
\(625\) 7.23634 0.289454
\(626\) 7.89543 0.315565
\(627\) −1.27880 −0.0510705
\(628\) −17.4965 −0.698188
\(629\) 25.5825 1.02004
\(630\) 7.76007 0.309169
\(631\) −24.5142 −0.975896 −0.487948 0.872873i \(-0.662254\pi\)
−0.487948 + 0.872873i \(0.662254\pi\)
\(632\) −3.58053 −0.142426
\(633\) −2.54297 −0.101074
\(634\) −9.95610 −0.395407
\(635\) −6.98261 −0.277096
\(636\) −0.0728531 −0.00288881
\(637\) −3.22902 −0.127938
\(638\) 17.4131 0.689391
\(639\) 2.51018 0.0993014
\(640\) 1.13854 0.0450046
\(641\) 3.57085 0.141040 0.0705201 0.997510i \(-0.477534\pi\)
0.0705201 + 0.997510i \(0.477534\pi\)
\(642\) −1.48833 −0.0587396
\(643\) −5.61347 −0.221374 −0.110687 0.993855i \(-0.535305\pi\)
−0.110687 + 0.993855i \(0.535305\pi\)
\(644\) −11.0744 −0.436392
\(645\) 0.972468 0.0382909
\(646\) −10.6260 −0.418075
\(647\) 44.9967 1.76900 0.884501 0.466538i \(-0.154499\pi\)
0.884501 + 0.466538i \(0.154499\pi\)
\(648\) 8.82761 0.346781
\(649\) 67.3407 2.64336
\(650\) −6.75073 −0.264786
\(651\) −0.930995 −0.0364886
\(652\) −5.75782 −0.225494
\(653\) 3.02430 0.118350 0.0591749 0.998248i \(-0.481153\pi\)
0.0591749 + 0.998248i \(0.481153\pi\)
\(654\) −1.62093 −0.0633835
\(655\) −8.16437 −0.319009
\(656\) 3.64130 0.142169
\(657\) 21.0549 0.821429
\(658\) −0.888382 −0.0346327
\(659\) 14.2081 0.553467 0.276734 0.960947i \(-0.410748\pi\)
0.276734 + 0.960947i \(0.410748\pi\)
\(660\) 0.735410 0.0286258
\(661\) 0.681590 0.0265108 0.0132554 0.999912i \(-0.495781\pi\)
0.0132554 + 0.999912i \(0.495781\pi\)
\(662\) −14.4038 −0.559819
\(663\) 1.35539 0.0526389
\(664\) 5.10442 0.198090
\(665\) 5.15410 0.199868
\(666\) −14.2078 −0.550541
\(667\) 18.0897 0.700435
\(668\) 16.6673 0.644876
\(669\) −0.151754 −0.00586713
\(670\) −11.9270 −0.460782
\(671\) −47.6588 −1.83985
\(672\) −0.316803 −0.0122209
\(673\) −7.02794 −0.270907 −0.135454 0.990784i \(-0.543249\pi\)
−0.135454 + 0.990784i \(0.543249\pi\)
\(674\) −1.71653 −0.0661184
\(675\) 3.06904 0.118128
\(676\) −9.67783 −0.372224
\(677\) −9.53803 −0.366576 −0.183288 0.983059i \(-0.558674\pi\)
−0.183288 + 0.983059i \(0.558674\pi\)
\(678\) 2.11692 0.0813000
\(679\) 24.1256 0.925857
\(680\) 6.11077 0.234337
\(681\) −0.311873 −0.0119510
\(682\) 13.7006 0.524621
\(683\) 20.8725 0.798663 0.399332 0.916807i \(-0.369242\pi\)
0.399332 + 0.916807i \(0.369242\pi\)
\(684\) 5.90139 0.225645
\(685\) −2.55687 −0.0976932
\(686\) 20.0569 0.765775
\(687\) −1.83196 −0.0698935
\(688\) 6.16489 0.235034
\(689\) −0.958419 −0.0365128
\(690\) 0.763984 0.0290844
\(691\) 19.9094 0.757391 0.378695 0.925521i \(-0.376373\pi\)
0.378695 + 0.925521i \(0.376373\pi\)
\(692\) −5.18347 −0.197046
\(693\) 31.7759 1.20707
\(694\) 10.7239 0.407074
\(695\) 14.1809 0.537913
\(696\) 0.517488 0.0196153
\(697\) 19.5437 0.740269
\(698\) 13.0689 0.494666
\(699\) −0.654075 −0.0247394
\(700\) 8.46887 0.320093
\(701\) 27.8722 1.05272 0.526359 0.850262i \(-0.323557\pi\)
0.526359 + 0.850262i \(0.323557\pi\)
\(702\) −1.51034 −0.0570040
\(703\) −9.43657 −0.355907
\(704\) 4.66208 0.175709
\(705\) 0.0612864 0.00230818
\(706\) 10.7805 0.405729
\(707\) −20.3611 −0.765759
\(708\) 2.00125 0.0752117
\(709\) −47.6371 −1.78905 −0.894524 0.447021i \(-0.852485\pi\)
−0.894524 + 0.447021i \(0.852485\pi\)
\(710\) −0.958780 −0.0359824
\(711\) 10.6728 0.400263
\(712\) −0.209053 −0.00783459
\(713\) 14.2329 0.533025
\(714\) −1.70035 −0.0636340
\(715\) 9.67469 0.361813
\(716\) −1.73198 −0.0647273
\(717\) 0.810078 0.0302529
\(718\) 2.01409 0.0751650
\(719\) −23.7039 −0.884008 −0.442004 0.897013i \(-0.645732\pi\)
−0.442004 + 0.897013i \(0.645732\pi\)
\(720\) −3.39375 −0.126478
\(721\) 24.3876 0.908242
\(722\) −15.0804 −0.561234
\(723\) −0.0160566 −0.000597151 0
\(724\) 17.2557 0.641303
\(725\) −13.8336 −0.513769
\(726\) 1.48732 0.0551996
\(727\) 0.112244 0.00416292 0.00208146 0.999998i \(-0.499337\pi\)
0.00208146 + 0.999998i \(0.499337\pi\)
\(728\) −4.16770 −0.154465
\(729\) −25.9689 −0.961813
\(730\) −8.04203 −0.297649
\(731\) 33.0883 1.22381
\(732\) −1.41634 −0.0523494
\(733\) 12.7854 0.472238 0.236119 0.971724i \(-0.424124\pi\)
0.236119 + 0.971724i \(0.424124\pi\)
\(734\) −8.65410 −0.319429
\(735\) −0.279454 −0.0103078
\(736\) 4.84322 0.178523
\(737\) −48.8389 −1.79900
\(738\) −10.8540 −0.399542
\(739\) −43.9324 −1.61608 −0.808039 0.589128i \(-0.799471\pi\)
−0.808039 + 0.589128i \(0.799471\pi\)
\(740\) 5.42675 0.199491
\(741\) −0.499960 −0.0183665
\(742\) 1.20235 0.0441395
\(743\) −17.7507 −0.651210 −0.325605 0.945506i \(-0.605568\pi\)
−0.325605 + 0.945506i \(0.605568\pi\)
\(744\) 0.407157 0.0149271
\(745\) −17.7985 −0.652086
\(746\) 1.86032 0.0681113
\(747\) −15.2153 −0.556698
\(748\) 25.0224 0.914909
\(749\) 24.5630 0.897511
\(750\) −1.37295 −0.0501332
\(751\) −1.00000 −0.0364905
\(752\) 0.388521 0.0141679
\(753\) −1.06417 −0.0387805
\(754\) 6.80781 0.247926
\(755\) 14.9588 0.544405
\(756\) 1.89473 0.0689108
\(757\) 25.6276 0.931450 0.465725 0.884930i \(-0.345794\pi\)
0.465725 + 0.884930i \(0.345794\pi\)
\(758\) −31.4273 −1.14149
\(759\) 3.12836 0.113552
\(760\) −2.25407 −0.0817638
\(761\) 3.12192 0.113170 0.0565848 0.998398i \(-0.481979\pi\)
0.0565848 + 0.998398i \(0.481979\pi\)
\(762\) −0.849717 −0.0307820
\(763\) 26.7514 0.968466
\(764\) 23.1592 0.837872
\(765\) −18.2150 −0.658565
\(766\) −4.84782 −0.175159
\(767\) 26.3275 0.950630
\(768\) 0.138549 0.00499946
\(769\) 39.8944 1.43863 0.719314 0.694685i \(-0.244456\pi\)
0.719314 + 0.694685i \(0.244456\pi\)
\(770\) −12.1370 −0.437387
\(771\) −3.15196 −0.113515
\(772\) 1.10413 0.0397386
\(773\) −34.2216 −1.23087 −0.615433 0.788190i \(-0.711019\pi\)
−0.615433 + 0.788190i \(0.711019\pi\)
\(774\) −18.3763 −0.660523
\(775\) −10.8843 −0.390974
\(776\) −10.5510 −0.378759
\(777\) −1.51002 −0.0541716
\(778\) −15.2510 −0.546774
\(779\) −7.20905 −0.258291
\(780\) 0.287515 0.0102947
\(781\) −3.92601 −0.140484
\(782\) 25.9946 0.929565
\(783\) −3.09499 −0.110606
\(784\) −1.77158 −0.0632706
\(785\) −19.9204 −0.710991
\(786\) −0.993527 −0.0354379
\(787\) −2.56575 −0.0914591 −0.0457296 0.998954i \(-0.514561\pi\)
−0.0457296 + 0.998954i \(0.514561\pi\)
\(788\) −4.14299 −0.147588
\(789\) 3.92454 0.139718
\(790\) −4.07656 −0.145037
\(791\) −34.9371 −1.24222
\(792\) −13.8967 −0.493799
\(793\) −18.6326 −0.661664
\(794\) 17.2077 0.610679
\(795\) −0.0829458 −0.00294179
\(796\) 9.85628 0.349347
\(797\) −4.15598 −0.147212 −0.0736062 0.997287i \(-0.523451\pi\)
−0.0736062 + 0.997287i \(0.523451\pi\)
\(798\) 0.627206 0.0222028
\(799\) 2.08528 0.0737717
\(800\) −3.70374 −0.130947
\(801\) 0.623146 0.0220178
\(802\) 36.1817 1.27762
\(803\) −32.9305 −1.16209
\(804\) −1.45141 −0.0511872
\(805\) −12.6086 −0.444394
\(806\) 5.35636 0.188670
\(807\) −3.48455 −0.122662
\(808\) 8.90464 0.313264
\(809\) 21.2844 0.748319 0.374159 0.927364i \(-0.377931\pi\)
0.374159 + 0.927364i \(0.377931\pi\)
\(810\) 10.0505 0.353140
\(811\) −19.8330 −0.696432 −0.348216 0.937414i \(-0.613212\pi\)
−0.348216 + 0.937414i \(0.613212\pi\)
\(812\) −8.54047 −0.299712
\(813\) −2.96427 −0.103961
\(814\) 22.2215 0.778862
\(815\) −6.55548 −0.229628
\(816\) 0.743623 0.0260320
\(817\) −12.2052 −0.427007
\(818\) −16.9164 −0.591469
\(819\) 12.4231 0.434098
\(820\) 4.14575 0.144776
\(821\) −17.1919 −0.600001 −0.300001 0.953939i \(-0.596987\pi\)
−0.300001 + 0.953939i \(0.596987\pi\)
\(822\) −0.311147 −0.0108525
\(823\) −26.3722 −0.919277 −0.459639 0.888106i \(-0.652021\pi\)
−0.459639 + 0.888106i \(0.652021\pi\)
\(824\) −10.6656 −0.371553
\(825\) −2.39234 −0.0832906
\(826\) −33.0281 −1.14920
\(827\) 18.8219 0.654500 0.327250 0.944938i \(-0.393878\pi\)
0.327250 + 0.944938i \(0.393878\pi\)
\(828\) −14.4367 −0.501710
\(829\) −5.71767 −0.198583 −0.0992915 0.995058i \(-0.531658\pi\)
−0.0992915 + 0.995058i \(0.531658\pi\)
\(830\) 5.81157 0.201723
\(831\) −1.36156 −0.0472321
\(832\) 1.82268 0.0631901
\(833\) −9.50843 −0.329448
\(834\) 1.72568 0.0597555
\(835\) 18.9763 0.656701
\(836\) −9.22997 −0.319225
\(837\) −2.43513 −0.0841703
\(838\) 25.6805 0.887119
\(839\) 52.0932 1.79846 0.899228 0.437481i \(-0.144129\pi\)
0.899228 + 0.437481i \(0.144129\pi\)
\(840\) −0.360691 −0.0124450
\(841\) −15.0494 −0.518945
\(842\) 15.1695 0.522774
\(843\) −2.45858 −0.0846780
\(844\) −18.3543 −0.631780
\(845\) −11.0186 −0.379050
\(846\) −1.15810 −0.0398165
\(847\) −24.5463 −0.843421
\(848\) −0.525829 −0.0180570
\(849\) −3.85359 −0.132255
\(850\) −19.8788 −0.681836
\(851\) 23.0849 0.791339
\(852\) −0.116674 −0.00399720
\(853\) 37.6986 1.29077 0.645387 0.763855i \(-0.276696\pi\)
0.645387 + 0.763855i \(0.276696\pi\)
\(854\) 23.3749 0.799871
\(855\) 6.71895 0.229783
\(856\) −10.7423 −0.367163
\(857\) 2.71656 0.0927959 0.0463980 0.998923i \(-0.485226\pi\)
0.0463980 + 0.998923i \(0.485226\pi\)
\(858\) 1.17732 0.0401930
\(859\) 11.3921 0.388692 0.194346 0.980933i \(-0.437742\pi\)
0.194346 + 0.980933i \(0.437742\pi\)
\(860\) 7.01894 0.239344
\(861\) −1.15357 −0.0393137
\(862\) −22.0541 −0.751164
\(863\) −17.5668 −0.597981 −0.298991 0.954256i \(-0.596650\pi\)
−0.298991 + 0.954256i \(0.596650\pi\)
\(864\) −0.828635 −0.0281907
\(865\) −5.90156 −0.200659
\(866\) 16.9027 0.574378
\(867\) 1.63585 0.0555564
\(868\) −6.71961 −0.228078
\(869\) −16.6927 −0.566261
\(870\) 0.589178 0.0199750
\(871\) −19.0940 −0.646976
\(872\) −11.6993 −0.396190
\(873\) 31.4505 1.06444
\(874\) −9.58860 −0.324339
\(875\) 22.6588 0.766009
\(876\) −0.978639 −0.0330651
\(877\) −2.40801 −0.0813126 −0.0406563 0.999173i \(-0.512945\pi\)
−0.0406563 + 0.999173i \(0.512945\pi\)
\(878\) 39.1863 1.32248
\(879\) 0.0474981 0.00160207
\(880\) 5.30794 0.178931
\(881\) −41.5528 −1.39995 −0.699975 0.714168i \(-0.746805\pi\)
−0.699975 + 0.714168i \(0.746805\pi\)
\(882\) 5.28072 0.177811
\(883\) 49.8958 1.67913 0.839563 0.543262i \(-0.182811\pi\)
0.839563 + 0.543262i \(0.182811\pi\)
\(884\) 9.78273 0.329029
\(885\) 2.27850 0.0765909
\(886\) −26.0277 −0.874419
\(887\) −26.4940 −0.889583 −0.444792 0.895634i \(-0.646722\pi\)
−0.444792 + 0.895634i \(0.646722\pi\)
\(888\) 0.660384 0.0221610
\(889\) 14.0235 0.470333
\(890\) −0.238014 −0.00797825
\(891\) 41.1550 1.37874
\(892\) −1.09531 −0.0366735
\(893\) −0.769193 −0.0257401
\(894\) −2.16591 −0.0724387
\(895\) −1.97193 −0.0659142
\(896\) −2.28657 −0.0763891
\(897\) 1.22306 0.0408368
\(898\) −24.2305 −0.808582
\(899\) 10.9763 0.366079
\(900\) 11.0401 0.368004
\(901\) −2.82224 −0.0940224
\(902\) 16.9760 0.565240
\(903\) −1.95305 −0.0649935
\(904\) 15.2792 0.508180
\(905\) 19.6462 0.653063
\(906\) 1.82034 0.0604767
\(907\) 46.7312 1.55168 0.775841 0.630928i \(-0.217326\pi\)
0.775841 + 0.630928i \(0.217326\pi\)
\(908\) −2.25099 −0.0747018
\(909\) −26.5430 −0.880376
\(910\) −4.74507 −0.157298
\(911\) 3.07178 0.101772 0.0508862 0.998704i \(-0.483795\pi\)
0.0508862 + 0.998704i \(0.483795\pi\)
\(912\) −0.274299 −0.00908295
\(913\) 23.7972 0.787573
\(914\) −29.9564 −0.990869
\(915\) −1.61255 −0.0533093
\(916\) −13.2224 −0.436882
\(917\) 16.3969 0.541473
\(918\) −4.44746 −0.146788
\(919\) −10.7399 −0.354278 −0.177139 0.984186i \(-0.556684\pi\)
−0.177139 + 0.984186i \(0.556684\pi\)
\(920\) 5.51418 0.181797
\(921\) 2.71228 0.0893727
\(922\) 7.05161 0.232232
\(923\) −1.53491 −0.0505222
\(924\) −1.47696 −0.0485884
\(925\) −17.6536 −0.580447
\(926\) 33.0534 1.08620
\(927\) 31.7920 1.04419
\(928\) 3.73505 0.122609
\(929\) 23.3730 0.766844 0.383422 0.923573i \(-0.374745\pi\)
0.383422 + 0.923573i \(0.374745\pi\)
\(930\) 0.463563 0.0152008
\(931\) 3.50736 0.114949
\(932\) −4.72089 −0.154638
\(933\) −3.20607 −0.104962
\(934\) 32.7686 1.07222
\(935\) 28.4889 0.931686
\(936\) −5.43306 −0.177585
\(937\) 21.9987 0.718666 0.359333 0.933209i \(-0.383004\pi\)
0.359333 + 0.933209i \(0.383004\pi\)
\(938\) 23.9536 0.782114
\(939\) 1.09390 0.0356982
\(940\) 0.442345 0.0144277
\(941\) 28.3308 0.923559 0.461780 0.886995i \(-0.347211\pi\)
0.461780 + 0.886995i \(0.347211\pi\)
\(942\) −2.42413 −0.0789824
\(943\) 17.6356 0.574295
\(944\) 14.4444 0.470124
\(945\) 2.15722 0.0701745
\(946\) 28.7412 0.934456
\(947\) −21.5386 −0.699910 −0.349955 0.936767i \(-0.613803\pi\)
−0.349955 + 0.936767i \(0.613803\pi\)
\(948\) −0.496078 −0.0161119
\(949\) −12.8745 −0.417924
\(950\) 7.33265 0.237903
\(951\) −1.37941 −0.0447304
\(952\) −12.2725 −0.397755
\(953\) −60.3153 −1.95380 −0.976902 0.213688i \(-0.931453\pi\)
−0.976902 + 0.213688i \(0.931453\pi\)
\(954\) 1.56739 0.0507462
\(955\) 26.3676 0.853236
\(956\) 5.84687 0.189101
\(957\) 2.41257 0.0779872
\(958\) −17.9402 −0.579622
\(959\) 5.13509 0.165821
\(960\) 0.157743 0.00509113
\(961\) −22.3639 −0.721416
\(962\) 8.68768 0.280102
\(963\) 32.0205 1.03185
\(964\) −0.115891 −0.00373260
\(965\) 1.25709 0.0404673
\(966\) −1.53434 −0.0493667
\(967\) 4.86306 0.156386 0.0781928 0.996938i \(-0.475085\pi\)
0.0781928 + 0.996938i \(0.475085\pi\)
\(968\) 10.7350 0.345035
\(969\) −1.47222 −0.0472946
\(970\) −12.0127 −0.385704
\(971\) 38.1127 1.22310 0.611548 0.791208i \(-0.290547\pi\)
0.611548 + 0.791208i \(0.290547\pi\)
\(972\) 3.70896 0.118965
\(973\) −28.4802 −0.913033
\(974\) 34.1072 1.09287
\(975\) −0.935307 −0.0299538
\(976\) −10.2226 −0.327219
\(977\) −60.7118 −1.94234 −0.971171 0.238384i \(-0.923382\pi\)
−0.971171 + 0.238384i \(0.923382\pi\)
\(978\) −0.797740 −0.0255089
\(979\) −0.974620 −0.0311490
\(980\) −2.01700 −0.0644308
\(981\) 34.8734 1.11342
\(982\) 5.20616 0.166135
\(983\) 43.8145 1.39747 0.698733 0.715383i \(-0.253748\pi\)
0.698733 + 0.715383i \(0.253748\pi\)
\(984\) 0.504499 0.0160828
\(985\) −4.71694 −0.150294
\(986\) 20.0468 0.638421
\(987\) −0.123084 −0.00391782
\(988\) −3.60854 −0.114803
\(989\) 29.8579 0.949426
\(990\) −15.8219 −0.502854
\(991\) −3.06703 −0.0974274 −0.0487137 0.998813i \(-0.515512\pi\)
−0.0487137 + 0.998813i \(0.515512\pi\)
\(992\) 2.93872 0.0933045
\(993\) −1.99563 −0.0633295
\(994\) 1.92556 0.0610751
\(995\) 11.2217 0.355753
\(996\) 0.707213 0.0224089
\(997\) 15.7466 0.498700 0.249350 0.968413i \(-0.419783\pi\)
0.249350 + 0.968413i \(0.419783\pi\)
\(998\) −15.5744 −0.492999
\(999\) −3.94963 −0.124961
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 1502.2.a.g.1.7 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1502.2.a.g.1.7 16 1.1 even 1 trivial