Properties

Label 8042.2.a.a.1.26
Level $8042$
Weight $2$
Character 8042.1
Self dual yes
Analytic conductor $64.216$
Analytic rank $1$
Dimension $67$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8042,2,Mod(1,8042)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8042, 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("8042.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8042 = 2 \cdot 4021 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8042.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: \(64.2156933055\)
Analytic rank: \(1\)
Dimension: \(67\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.26
Character \(\chi\) \(=\) 8042.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.08153 q^{3} +1.00000 q^{4} -2.98271 q^{5} -1.08153 q^{6} -1.13392 q^{7} +1.00000 q^{8} -1.83028 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.08153 q^{3} +1.00000 q^{4} -2.98271 q^{5} -1.08153 q^{6} -1.13392 q^{7} +1.00000 q^{8} -1.83028 q^{9} -2.98271 q^{10} -2.91322 q^{11} -1.08153 q^{12} -0.731076 q^{13} -1.13392 q^{14} +3.22590 q^{15} +1.00000 q^{16} +7.72696 q^{17} -1.83028 q^{18} -2.28749 q^{19} -2.98271 q^{20} +1.22637 q^{21} -2.91322 q^{22} +0.277997 q^{23} -1.08153 q^{24} +3.89656 q^{25} -0.731076 q^{26} +5.22412 q^{27} -1.13392 q^{28} +5.89169 q^{29} +3.22590 q^{30} +7.72320 q^{31} +1.00000 q^{32} +3.15075 q^{33} +7.72696 q^{34} +3.38214 q^{35} -1.83028 q^{36} +2.82143 q^{37} -2.28749 q^{38} +0.790684 q^{39} -2.98271 q^{40} -2.69161 q^{41} +1.22637 q^{42} +1.83122 q^{43} -2.91322 q^{44} +5.45921 q^{45} +0.277997 q^{46} -2.34973 q^{47} -1.08153 q^{48} -5.71423 q^{49} +3.89656 q^{50} -8.35697 q^{51} -0.731076 q^{52} +10.0741 q^{53} +5.22412 q^{54} +8.68931 q^{55} -1.13392 q^{56} +2.47400 q^{57} +5.89169 q^{58} -4.81439 q^{59} +3.22590 q^{60} -10.2163 q^{61} +7.72320 q^{62} +2.07539 q^{63} +1.00000 q^{64} +2.18059 q^{65} +3.15075 q^{66} +6.42078 q^{67} +7.72696 q^{68} -0.300664 q^{69} +3.38214 q^{70} -8.65038 q^{71} -1.83028 q^{72} -2.20145 q^{73} +2.82143 q^{74} -4.21427 q^{75} -2.28749 q^{76} +3.30335 q^{77} +0.790684 q^{78} +7.73841 q^{79} -2.98271 q^{80} -0.159210 q^{81} -2.69161 q^{82} +5.72485 q^{83} +1.22637 q^{84} -23.0473 q^{85} +1.83122 q^{86} -6.37207 q^{87} -2.91322 q^{88} -7.04382 q^{89} +5.45921 q^{90} +0.828979 q^{91} +0.277997 q^{92} -8.35291 q^{93} -2.34973 q^{94} +6.82291 q^{95} -1.08153 q^{96} -10.8995 q^{97} -5.71423 q^{98} +5.33203 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9} - 20 q^{10} - 13 q^{11} - 11 q^{12} - 51 q^{13} - 40 q^{14} - 31 q^{15} + 67 q^{16} - 34 q^{17} + 24 q^{18} - 33 q^{19} - 20 q^{20} - 39 q^{21} - 13 q^{22} - 43 q^{23} - 11 q^{24} - 9 q^{25} - 51 q^{26} - 29 q^{27} - 40 q^{28} - 63 q^{29} - 31 q^{30} - 43 q^{31} + 67 q^{32} - 49 q^{33} - 34 q^{34} - 20 q^{35} + 24 q^{36} - 77 q^{37} - 33 q^{38} - 40 q^{39} - 20 q^{40} - 50 q^{41} - 39 q^{42} - 56 q^{43} - 13 q^{44} - 48 q^{45} - 43 q^{46} - 48 q^{47} - 11 q^{48} + q^{49} - 9 q^{50} - 18 q^{51} - 51 q^{52} - 91 q^{53} - 29 q^{54} - 58 q^{55} - 40 q^{56} - 65 q^{57} - 63 q^{58} - 17 q^{59} - 31 q^{60} - 45 q^{61} - 43 q^{62} - 67 q^{63} + 67 q^{64} - 65 q^{65} - 49 q^{66} - 112 q^{67} - 34 q^{68} - 57 q^{69} - 20 q^{70} - 75 q^{71} + 24 q^{72} - 79 q^{73} - 77 q^{74} - 5 q^{75} - 33 q^{76} - 85 q^{77} - 40 q^{78} - 80 q^{79} - 20 q^{80} - 77 q^{81} - 50 q^{82} - 22 q^{83} - 39 q^{84} - 134 q^{85} - 56 q^{86} - 49 q^{87} - 13 q^{88} - 77 q^{89} - 48 q^{90} - 17 q^{91} - 43 q^{92} - 97 q^{93} - 48 q^{94} - 73 q^{95} - 11 q^{96} - 87 q^{97} + q^{98} - 44 q^{99}+O(q^{100}) \) 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
\(3\) −1.08153 −0.624424 −0.312212 0.950012i \(-0.601070\pi\)
−0.312212 + 0.950012i \(0.601070\pi\)
\(4\) 1.00000 0.500000
\(5\) −2.98271 −1.33391 −0.666954 0.745099i \(-0.732402\pi\)
−0.666954 + 0.745099i \(0.732402\pi\)
\(6\) −1.08153 −0.441534
\(7\) −1.13392 −0.428580 −0.214290 0.976770i \(-0.568744\pi\)
−0.214290 + 0.976770i \(0.568744\pi\)
\(8\) 1.00000 0.353553
\(9\) −1.83028 −0.610095
\(10\) −2.98271 −0.943216
\(11\) −2.91322 −0.878370 −0.439185 0.898397i \(-0.644733\pi\)
−0.439185 + 0.898397i \(0.644733\pi\)
\(12\) −1.08153 −0.312212
\(13\) −0.731076 −0.202764 −0.101382 0.994848i \(-0.532326\pi\)
−0.101382 + 0.994848i \(0.532326\pi\)
\(14\) −1.13392 −0.303052
\(15\) 3.22590 0.832925
\(16\) 1.00000 0.250000
\(17\) 7.72696 1.87406 0.937032 0.349244i \(-0.113562\pi\)
0.937032 + 0.349244i \(0.113562\pi\)
\(18\) −1.83028 −0.431402
\(19\) −2.28749 −0.524786 −0.262393 0.964961i \(-0.584512\pi\)
−0.262393 + 0.964961i \(0.584512\pi\)
\(20\) −2.98271 −0.666954
\(21\) 1.22637 0.267616
\(22\) −2.91322 −0.621102
\(23\) 0.277997 0.0579665 0.0289832 0.999580i \(-0.490773\pi\)
0.0289832 + 0.999580i \(0.490773\pi\)
\(24\) −1.08153 −0.220767
\(25\) 3.89656 0.779312
\(26\) −0.731076 −0.143376
\(27\) 5.22412 1.00538
\(28\) −1.13392 −0.214290
\(29\) 5.89169 1.09406 0.547030 0.837113i \(-0.315758\pi\)
0.547030 + 0.837113i \(0.315758\pi\)
\(30\) 3.22590 0.588967
\(31\) 7.72320 1.38713 0.693564 0.720395i \(-0.256039\pi\)
0.693564 + 0.720395i \(0.256039\pi\)
\(32\) 1.00000 0.176777
\(33\) 3.15075 0.548476
\(34\) 7.72696 1.32516
\(35\) 3.38214 0.571686
\(36\) −1.83028 −0.305047
\(37\) 2.82143 0.463839 0.231920 0.972735i \(-0.425499\pi\)
0.231920 + 0.972735i \(0.425499\pi\)
\(38\) −2.28749 −0.371079
\(39\) 0.790684 0.126611
\(40\) −2.98271 −0.471608
\(41\) −2.69161 −0.420359 −0.210180 0.977663i \(-0.567405\pi\)
−0.210180 + 0.977663i \(0.567405\pi\)
\(42\) 1.22637 0.189233
\(43\) 1.83122 0.279259 0.139629 0.990204i \(-0.455409\pi\)
0.139629 + 0.990204i \(0.455409\pi\)
\(44\) −2.91322 −0.439185
\(45\) 5.45921 0.813810
\(46\) 0.277997 0.0409885
\(47\) −2.34973 −0.342744 −0.171372 0.985206i \(-0.554820\pi\)
−0.171372 + 0.985206i \(0.554820\pi\)
\(48\) −1.08153 −0.156106
\(49\) −5.71423 −0.816319
\(50\) 3.89656 0.551057
\(51\) −8.35697 −1.17021
\(52\) −0.731076 −0.101382
\(53\) 10.0741 1.38378 0.691891 0.722002i \(-0.256778\pi\)
0.691891 + 0.722002i \(0.256778\pi\)
\(54\) 5.22412 0.710912
\(55\) 8.68931 1.17167
\(56\) −1.13392 −0.151526
\(57\) 2.47400 0.327689
\(58\) 5.89169 0.773617
\(59\) −4.81439 −0.626780 −0.313390 0.949625i \(-0.601465\pi\)
−0.313390 + 0.949625i \(0.601465\pi\)
\(60\) 3.22590 0.416462
\(61\) −10.2163 −1.30807 −0.654035 0.756465i \(-0.726925\pi\)
−0.654035 + 0.756465i \(0.726925\pi\)
\(62\) 7.72320 0.980848
\(63\) 2.07539 0.261474
\(64\) 1.00000 0.125000
\(65\) 2.18059 0.270469
\(66\) 3.15075 0.387831
\(67\) 6.42078 0.784423 0.392211 0.919875i \(-0.371710\pi\)
0.392211 + 0.919875i \(0.371710\pi\)
\(68\) 7.72696 0.937032
\(69\) −0.300664 −0.0361957
\(70\) 3.38214 0.404243
\(71\) −8.65038 −1.02661 −0.513305 0.858206i \(-0.671579\pi\)
−0.513305 + 0.858206i \(0.671579\pi\)
\(72\) −1.83028 −0.215701
\(73\) −2.20145 −0.257661 −0.128830 0.991667i \(-0.541122\pi\)
−0.128830 + 0.991667i \(0.541122\pi\)
\(74\) 2.82143 0.327984
\(75\) −4.21427 −0.486621
\(76\) −2.28749 −0.262393
\(77\) 3.30335 0.376452
\(78\) 0.790684 0.0895273
\(79\) 7.73841 0.870639 0.435320 0.900276i \(-0.356635\pi\)
0.435320 + 0.900276i \(0.356635\pi\)
\(80\) −2.98271 −0.333477
\(81\) −0.159210 −0.0176900
\(82\) −2.69161 −0.297239
\(83\) 5.72485 0.628384 0.314192 0.949360i \(-0.398266\pi\)
0.314192 + 0.949360i \(0.398266\pi\)
\(84\) 1.22637 0.133808
\(85\) −23.0473 −2.49983
\(86\) 1.83122 0.197466
\(87\) −6.37207 −0.683158
\(88\) −2.91322 −0.310551
\(89\) −7.04382 −0.746644 −0.373322 0.927702i \(-0.621781\pi\)
−0.373322 + 0.927702i \(0.621781\pi\)
\(90\) 5.45921 0.575451
\(91\) 0.828979 0.0869006
\(92\) 0.277997 0.0289832
\(93\) −8.35291 −0.866156
\(94\) −2.34973 −0.242356
\(95\) 6.82291 0.700016
\(96\) −1.08153 −0.110384
\(97\) −10.8995 −1.10667 −0.553337 0.832958i \(-0.686646\pi\)
−0.553337 + 0.832958i \(0.686646\pi\)
\(98\) −5.71423 −0.577225
\(99\) 5.33203 0.535889
\(100\) 3.89656 0.389656
\(101\) 7.85628 0.781729 0.390865 0.920448i \(-0.372176\pi\)
0.390865 + 0.920448i \(0.372176\pi\)
\(102\) −8.35697 −0.827464
\(103\) 8.57974 0.845387 0.422694 0.906273i \(-0.361085\pi\)
0.422694 + 0.906273i \(0.361085\pi\)
\(104\) −0.731076 −0.0716879
\(105\) −3.65790 −0.356975
\(106\) 10.0741 0.978482
\(107\) −17.5513 −1.69675 −0.848376 0.529394i \(-0.822419\pi\)
−0.848376 + 0.529394i \(0.822419\pi\)
\(108\) 5.22412 0.502691
\(109\) −2.48069 −0.237607 −0.118803 0.992918i \(-0.537906\pi\)
−0.118803 + 0.992918i \(0.537906\pi\)
\(110\) 8.68931 0.828493
\(111\) −3.05147 −0.289633
\(112\) −1.13392 −0.107145
\(113\) −11.6372 −1.09474 −0.547368 0.836892i \(-0.684370\pi\)
−0.547368 + 0.836892i \(0.684370\pi\)
\(114\) 2.47400 0.231711
\(115\) −0.829186 −0.0773220
\(116\) 5.89169 0.547030
\(117\) 1.33808 0.123705
\(118\) −4.81439 −0.443200
\(119\) −8.76173 −0.803186
\(120\) 3.22590 0.294483
\(121\) −2.51312 −0.228466
\(122\) −10.2163 −0.924945
\(123\) 2.91107 0.262482
\(124\) 7.72320 0.693564
\(125\) 3.29124 0.294377
\(126\) 2.07539 0.184890
\(127\) −8.57468 −0.760880 −0.380440 0.924806i \(-0.624227\pi\)
−0.380440 + 0.924806i \(0.624227\pi\)
\(128\) 1.00000 0.0883883
\(129\) −1.98053 −0.174376
\(130\) 2.18059 0.191250
\(131\) −12.4733 −1.08980 −0.544898 0.838502i \(-0.683432\pi\)
−0.544898 + 0.838502i \(0.683432\pi\)
\(132\) 3.15075 0.274238
\(133\) 2.59382 0.224913
\(134\) 6.42078 0.554670
\(135\) −15.5820 −1.34109
\(136\) 7.72696 0.662582
\(137\) −13.9245 −1.18965 −0.594826 0.803854i \(-0.702779\pi\)
−0.594826 + 0.803854i \(0.702779\pi\)
\(138\) −0.300664 −0.0255942
\(139\) 8.26207 0.700780 0.350390 0.936604i \(-0.386049\pi\)
0.350390 + 0.936604i \(0.386049\pi\)
\(140\) 3.38214 0.285843
\(141\) 2.54131 0.214017
\(142\) −8.65038 −0.725924
\(143\) 2.12979 0.178102
\(144\) −1.83028 −0.152524
\(145\) −17.5732 −1.45938
\(146\) −2.20145 −0.182194
\(147\) 6.18014 0.509729
\(148\) 2.82143 0.231920
\(149\) −10.0517 −0.823468 −0.411734 0.911304i \(-0.635077\pi\)
−0.411734 + 0.911304i \(0.635077\pi\)
\(150\) −4.21427 −0.344093
\(151\) 12.4850 1.01601 0.508006 0.861354i \(-0.330383\pi\)
0.508006 + 0.861354i \(0.330383\pi\)
\(152\) −2.28749 −0.185540
\(153\) −14.1425 −1.14336
\(154\) 3.30335 0.266192
\(155\) −23.0361 −1.85030
\(156\) 0.790684 0.0633053
\(157\) 5.48909 0.438077 0.219039 0.975716i \(-0.429708\pi\)
0.219039 + 0.975716i \(0.429708\pi\)
\(158\) 7.73841 0.615635
\(159\) −10.8955 −0.864067
\(160\) −2.98271 −0.235804
\(161\) −0.315226 −0.0248433
\(162\) −0.159210 −0.0125087
\(163\) −3.97886 −0.311648 −0.155824 0.987785i \(-0.549803\pi\)
−0.155824 + 0.987785i \(0.549803\pi\)
\(164\) −2.69161 −0.210180
\(165\) −9.39778 −0.731616
\(166\) 5.72485 0.444334
\(167\) 20.8740 1.61528 0.807640 0.589676i \(-0.200745\pi\)
0.807640 + 0.589676i \(0.200745\pi\)
\(168\) 1.22637 0.0946164
\(169\) −12.4655 −0.958887
\(170\) −23.0473 −1.76765
\(171\) 4.18675 0.320169
\(172\) 1.83122 0.139629
\(173\) −3.22557 −0.245236 −0.122618 0.992454i \(-0.539129\pi\)
−0.122618 + 0.992454i \(0.539129\pi\)
\(174\) −6.37207 −0.483065
\(175\) −4.41837 −0.333998
\(176\) −2.91322 −0.219593
\(177\) 5.20693 0.391376
\(178\) −7.04382 −0.527957
\(179\) −11.1498 −0.833375 −0.416687 0.909050i \(-0.636809\pi\)
−0.416687 + 0.909050i \(0.636809\pi\)
\(180\) 5.45921 0.406905
\(181\) 5.32402 0.395732 0.197866 0.980229i \(-0.436599\pi\)
0.197866 + 0.980229i \(0.436599\pi\)
\(182\) 0.828979 0.0614480
\(183\) 11.0493 0.816790
\(184\) 0.277997 0.0204942
\(185\) −8.41549 −0.618719
\(186\) −8.35291 −0.612465
\(187\) −22.5104 −1.64612
\(188\) −2.34973 −0.171372
\(189\) −5.92371 −0.430886
\(190\) 6.82291 0.494986
\(191\) −18.0496 −1.30603 −0.653013 0.757347i \(-0.726495\pi\)
−0.653013 + 0.757347i \(0.726495\pi\)
\(192\) −1.08153 −0.0780530
\(193\) −23.7304 −1.70815 −0.854075 0.520149i \(-0.825876\pi\)
−0.854075 + 0.520149i \(0.825876\pi\)
\(194\) −10.8995 −0.782536
\(195\) −2.35838 −0.168887
\(196\) −5.71423 −0.408160
\(197\) 5.26145 0.374863 0.187432 0.982278i \(-0.439984\pi\)
0.187432 + 0.982278i \(0.439984\pi\)
\(198\) 5.33203 0.378931
\(199\) −8.41324 −0.596399 −0.298199 0.954504i \(-0.596386\pi\)
−0.298199 + 0.954504i \(0.596386\pi\)
\(200\) 3.89656 0.275529
\(201\) −6.94429 −0.489812
\(202\) 7.85628 0.552766
\(203\) −6.68069 −0.468892
\(204\) −8.35697 −0.585105
\(205\) 8.02830 0.560721
\(206\) 8.57974 0.597779
\(207\) −0.508814 −0.0353650
\(208\) −0.731076 −0.0506910
\(209\) 6.66396 0.460956
\(210\) −3.65790 −0.252419
\(211\) 26.7402 1.84087 0.920437 0.390892i \(-0.127833\pi\)
0.920437 + 0.390892i \(0.127833\pi\)
\(212\) 10.0741 0.691891
\(213\) 9.35568 0.641041
\(214\) −17.5513 −1.19978
\(215\) −5.46201 −0.372506
\(216\) 5.22412 0.355456
\(217\) −8.75746 −0.594495
\(218\) −2.48069 −0.168013
\(219\) 2.38095 0.160890
\(220\) 8.68931 0.585833
\(221\) −5.64900 −0.379993
\(222\) −3.05147 −0.204801
\(223\) −20.4191 −1.36736 −0.683682 0.729780i \(-0.739623\pi\)
−0.683682 + 0.729780i \(0.739623\pi\)
\(224\) −1.13392 −0.0757629
\(225\) −7.13181 −0.475454
\(226\) −11.6372 −0.774095
\(227\) 4.05376 0.269058 0.134529 0.990910i \(-0.457048\pi\)
0.134529 + 0.990910i \(0.457048\pi\)
\(228\) 2.47400 0.163844
\(229\) −18.0760 −1.19450 −0.597249 0.802056i \(-0.703740\pi\)
−0.597249 + 0.802056i \(0.703740\pi\)
\(230\) −0.829186 −0.0546749
\(231\) −3.57269 −0.235066
\(232\) 5.89169 0.386809
\(233\) 26.7595 1.75307 0.876536 0.481337i \(-0.159849\pi\)
0.876536 + 0.481337i \(0.159849\pi\)
\(234\) 1.33808 0.0874728
\(235\) 7.00857 0.457189
\(236\) −4.81439 −0.313390
\(237\) −8.36936 −0.543648
\(238\) −8.76173 −0.567938
\(239\) 9.24933 0.598289 0.299145 0.954208i \(-0.403299\pi\)
0.299145 + 0.954208i \(0.403299\pi\)
\(240\) 3.22590 0.208231
\(241\) −2.15772 −0.138991 −0.0694954 0.997582i \(-0.522139\pi\)
−0.0694954 + 0.997582i \(0.522139\pi\)
\(242\) −2.51312 −0.161550
\(243\) −15.5002 −0.994336
\(244\) −10.2163 −0.654035
\(245\) 17.0439 1.08890
\(246\) 2.91107 0.185603
\(247\) 1.67233 0.106408
\(248\) 7.72320 0.490424
\(249\) −6.19162 −0.392378
\(250\) 3.29124 0.208156
\(251\) 22.0796 1.39365 0.696827 0.717239i \(-0.254595\pi\)
0.696827 + 0.717239i \(0.254595\pi\)
\(252\) 2.07539 0.130737
\(253\) −0.809869 −0.0509160
\(254\) −8.57468 −0.538023
\(255\) 24.9264 1.56095
\(256\) 1.00000 0.0625000
\(257\) 7.21757 0.450220 0.225110 0.974333i \(-0.427726\pi\)
0.225110 + 0.974333i \(0.427726\pi\)
\(258\) −1.98053 −0.123302
\(259\) −3.19926 −0.198792
\(260\) 2.18059 0.135234
\(261\) −10.7835 −0.667480
\(262\) −12.4733 −0.770602
\(263\) 15.3564 0.946918 0.473459 0.880816i \(-0.343005\pi\)
0.473459 + 0.880816i \(0.343005\pi\)
\(264\) 3.15075 0.193915
\(265\) −30.0481 −1.84584
\(266\) 2.59382 0.159037
\(267\) 7.61814 0.466222
\(268\) 6.42078 0.392211
\(269\) −22.8657 −1.39415 −0.697073 0.717000i \(-0.745515\pi\)
−0.697073 + 0.717000i \(0.745515\pi\)
\(270\) −15.5820 −0.948292
\(271\) −23.3920 −1.42096 −0.710482 0.703715i \(-0.751523\pi\)
−0.710482 + 0.703715i \(0.751523\pi\)
\(272\) 7.72696 0.468516
\(273\) −0.896569 −0.0542628
\(274\) −13.9245 −0.841211
\(275\) −11.3516 −0.684525
\(276\) −0.300664 −0.0180978
\(277\) 30.4657 1.83051 0.915253 0.402880i \(-0.131991\pi\)
0.915253 + 0.402880i \(0.131991\pi\)
\(278\) 8.26207 0.495526
\(279\) −14.1357 −0.846279
\(280\) 3.38214 0.202122
\(281\) −3.16439 −0.188772 −0.0943858 0.995536i \(-0.530089\pi\)
−0.0943858 + 0.995536i \(0.530089\pi\)
\(282\) 2.54131 0.151333
\(283\) 22.7438 1.35198 0.675990 0.736910i \(-0.263716\pi\)
0.675990 + 0.736910i \(0.263716\pi\)
\(284\) −8.65038 −0.513305
\(285\) −7.37921 −0.437107
\(286\) 2.12979 0.125937
\(287\) 3.05206 0.180158
\(288\) −1.83028 −0.107851
\(289\) 42.7060 2.51211
\(290\) −17.5732 −1.03194
\(291\) 11.7881 0.691034
\(292\) −2.20145 −0.128830
\(293\) 31.9816 1.86839 0.934193 0.356769i \(-0.116122\pi\)
0.934193 + 0.356769i \(0.116122\pi\)
\(294\) 6.18014 0.360433
\(295\) 14.3599 0.836067
\(296\) 2.82143 0.163992
\(297\) −15.2190 −0.883097
\(298\) −10.0517 −0.582280
\(299\) −0.203237 −0.0117535
\(300\) −4.21427 −0.243311
\(301\) −2.07645 −0.119685
\(302\) 12.4850 0.718429
\(303\) −8.49684 −0.488131
\(304\) −2.28749 −0.131196
\(305\) 30.4724 1.74485
\(306\) −14.1425 −0.808475
\(307\) −23.5117 −1.34188 −0.670941 0.741510i \(-0.734110\pi\)
−0.670941 + 0.741510i \(0.734110\pi\)
\(308\) 3.30335 0.188226
\(309\) −9.27929 −0.527880
\(310\) −23.0361 −1.30836
\(311\) −17.2273 −0.976873 −0.488437 0.872599i \(-0.662433\pi\)
−0.488437 + 0.872599i \(0.662433\pi\)
\(312\) 0.790684 0.0447636
\(313\) −9.55032 −0.539816 −0.269908 0.962886i \(-0.586993\pi\)
−0.269908 + 0.962886i \(0.586993\pi\)
\(314\) 5.48909 0.309767
\(315\) −6.19028 −0.348783
\(316\) 7.73841 0.435320
\(317\) −5.00960 −0.281367 −0.140684 0.990055i \(-0.544930\pi\)
−0.140684 + 0.990055i \(0.544930\pi\)
\(318\) −10.8955 −0.610988
\(319\) −17.1638 −0.960990
\(320\) −2.98271 −0.166739
\(321\) 18.9824 1.05949
\(322\) −0.315226 −0.0175668
\(323\) −17.6753 −0.983482
\(324\) −0.159210 −0.00884500
\(325\) −2.84868 −0.158016
\(326\) −3.97886 −0.220369
\(327\) 2.68295 0.148367
\(328\) −2.69161 −0.148619
\(329\) 2.66440 0.146893
\(330\) −9.39778 −0.517331
\(331\) 12.8684 0.707312 0.353656 0.935375i \(-0.384938\pi\)
0.353656 + 0.935375i \(0.384938\pi\)
\(332\) 5.72485 0.314192
\(333\) −5.16401 −0.282986
\(334\) 20.8740 1.14218
\(335\) −19.1513 −1.04635
\(336\) 1.22637 0.0669039
\(337\) −7.29248 −0.397247 −0.198623 0.980076i \(-0.563647\pi\)
−0.198623 + 0.980076i \(0.563647\pi\)
\(338\) −12.4655 −0.678035
\(339\) 12.5860 0.683579
\(340\) −23.0473 −1.24991
\(341\) −22.4994 −1.21841
\(342\) 4.18675 0.226394
\(343\) 14.4169 0.778438
\(344\) 1.83122 0.0987330
\(345\) 0.896793 0.0482817
\(346\) −3.22557 −0.173408
\(347\) 14.7381 0.791185 0.395592 0.918426i \(-0.370539\pi\)
0.395592 + 0.918426i \(0.370539\pi\)
\(348\) −6.37207 −0.341579
\(349\) −25.4287 −1.36117 −0.680584 0.732670i \(-0.738274\pi\)
−0.680584 + 0.732670i \(0.738274\pi\)
\(350\) −4.41837 −0.236172
\(351\) −3.81923 −0.203855
\(352\) −2.91322 −0.155275
\(353\) −6.78268 −0.361005 −0.180503 0.983574i \(-0.557772\pi\)
−0.180503 + 0.983574i \(0.557772\pi\)
\(354\) 5.20693 0.276745
\(355\) 25.8016 1.36941
\(356\) −7.04382 −0.373322
\(357\) 9.47611 0.501529
\(358\) −11.1498 −0.589285
\(359\) −30.4679 −1.60803 −0.804016 0.594607i \(-0.797308\pi\)
−0.804016 + 0.594607i \(0.797308\pi\)
\(360\) 5.45921 0.287725
\(361\) −13.7674 −0.724600
\(362\) 5.32402 0.279824
\(363\) 2.71803 0.142659
\(364\) 0.828979 0.0434503
\(365\) 6.56630 0.343696
\(366\) 11.0493 0.577558
\(367\) 24.6983 1.28924 0.644620 0.764503i \(-0.277016\pi\)
0.644620 + 0.764503i \(0.277016\pi\)
\(368\) 0.277997 0.0144916
\(369\) 4.92641 0.256459
\(370\) −8.41549 −0.437501
\(371\) −11.4232 −0.593061
\(372\) −8.35291 −0.433078
\(373\) −10.4154 −0.539289 −0.269644 0.962960i \(-0.586906\pi\)
−0.269644 + 0.962960i \(0.586906\pi\)
\(374\) −22.5104 −1.16398
\(375\) −3.55958 −0.183816
\(376\) −2.34973 −0.121178
\(377\) −4.30728 −0.221836
\(378\) −5.92371 −0.304683
\(379\) −21.4886 −1.10379 −0.551897 0.833912i \(-0.686096\pi\)
−0.551897 + 0.833912i \(0.686096\pi\)
\(380\) 6.82291 0.350008
\(381\) 9.27381 0.475112
\(382\) −18.0496 −0.923500
\(383\) −18.9808 −0.969875 −0.484938 0.874549i \(-0.661158\pi\)
−0.484938 + 0.874549i \(0.661158\pi\)
\(384\) −1.08153 −0.0551918
\(385\) −9.85294 −0.502152
\(386\) −23.7304 −1.20785
\(387\) −3.35166 −0.170374
\(388\) −10.8995 −0.553337
\(389\) 1.81540 0.0920444 0.0460222 0.998940i \(-0.485346\pi\)
0.0460222 + 0.998940i \(0.485346\pi\)
\(390\) −2.35838 −0.119421
\(391\) 2.14808 0.108633
\(392\) −5.71423 −0.288612
\(393\) 13.4903 0.680495
\(394\) 5.26145 0.265068
\(395\) −23.0814 −1.16135
\(396\) 5.33203 0.267944
\(397\) 7.94724 0.398861 0.199430 0.979912i \(-0.436091\pi\)
0.199430 + 0.979912i \(0.436091\pi\)
\(398\) −8.41324 −0.421718
\(399\) −2.80530 −0.140441
\(400\) 3.89656 0.194828
\(401\) −25.5334 −1.27508 −0.637538 0.770419i \(-0.720047\pi\)
−0.637538 + 0.770419i \(0.720047\pi\)
\(402\) −6.94429 −0.346350
\(403\) −5.64625 −0.281260
\(404\) 7.85628 0.390865
\(405\) 0.474877 0.0235968
\(406\) −6.68069 −0.331557
\(407\) −8.21945 −0.407423
\(408\) −8.35697 −0.413732
\(409\) 9.59658 0.474520 0.237260 0.971446i \(-0.423751\pi\)
0.237260 + 0.971446i \(0.423751\pi\)
\(410\) 8.02830 0.396490
\(411\) 15.0598 0.742848
\(412\) 8.57974 0.422694
\(413\) 5.45911 0.268625
\(414\) −0.508814 −0.0250069
\(415\) −17.0756 −0.838207
\(416\) −0.731076 −0.0358439
\(417\) −8.93572 −0.437584
\(418\) 6.66396 0.325945
\(419\) −14.0049 −0.684184 −0.342092 0.939667i \(-0.611135\pi\)
−0.342092 + 0.939667i \(0.611135\pi\)
\(420\) −3.65790 −0.178487
\(421\) 13.7078 0.668079 0.334040 0.942559i \(-0.391588\pi\)
0.334040 + 0.942559i \(0.391588\pi\)
\(422\) 26.7402 1.30169
\(423\) 4.30067 0.209106
\(424\) 10.0741 0.489241
\(425\) 30.1086 1.46048
\(426\) 9.35568 0.453284
\(427\) 11.5845 0.560612
\(428\) −17.5513 −0.848376
\(429\) −2.30344 −0.111211
\(430\) −5.46201 −0.263402
\(431\) −32.6504 −1.57271 −0.786356 0.617773i \(-0.788035\pi\)
−0.786356 + 0.617773i \(0.788035\pi\)
\(432\) 5.22412 0.251345
\(433\) 13.3901 0.643489 0.321745 0.946827i \(-0.395731\pi\)
0.321745 + 0.946827i \(0.395731\pi\)
\(434\) −8.75746 −0.420372
\(435\) 19.0060 0.911270
\(436\) −2.48069 −0.118803
\(437\) −0.635916 −0.0304200
\(438\) 2.38095 0.113766
\(439\) −24.4386 −1.16639 −0.583196 0.812332i \(-0.698198\pi\)
−0.583196 + 0.812332i \(0.698198\pi\)
\(440\) 8.68931 0.414246
\(441\) 10.4587 0.498032
\(442\) −5.64900 −0.268695
\(443\) 17.5279 0.832775 0.416388 0.909187i \(-0.363296\pi\)
0.416388 + 0.909187i \(0.363296\pi\)
\(444\) −3.05147 −0.144816
\(445\) 21.0097 0.995955
\(446\) −20.4191 −0.966873
\(447\) 10.8713 0.514194
\(448\) −1.13392 −0.0535725
\(449\) −20.4080 −0.963112 −0.481556 0.876415i \(-0.659928\pi\)
−0.481556 + 0.876415i \(0.659928\pi\)
\(450\) −7.13181 −0.336197
\(451\) 7.84127 0.369231
\(452\) −11.6372 −0.547368
\(453\) −13.5029 −0.634422
\(454\) 4.05376 0.190252
\(455\) −2.47260 −0.115917
\(456\) 2.47400 0.115855
\(457\) −2.60763 −0.121980 −0.0609898 0.998138i \(-0.519426\pi\)
−0.0609898 + 0.998138i \(0.519426\pi\)
\(458\) −18.0760 −0.844638
\(459\) 40.3666 1.88415
\(460\) −0.829186 −0.0386610
\(461\) 30.6105 1.42567 0.712837 0.701330i \(-0.247410\pi\)
0.712837 + 0.701330i \(0.247410\pi\)
\(462\) −3.57269 −0.166216
\(463\) −10.9539 −0.509073 −0.254536 0.967063i \(-0.581923\pi\)
−0.254536 + 0.967063i \(0.581923\pi\)
\(464\) 5.89169 0.273515
\(465\) 24.9143 1.15537
\(466\) 26.7595 1.23961
\(467\) 31.9401 1.47801 0.739005 0.673700i \(-0.235296\pi\)
0.739005 + 0.673700i \(0.235296\pi\)
\(468\) 1.33808 0.0618526
\(469\) −7.28062 −0.336188
\(470\) 7.00857 0.323281
\(471\) −5.93664 −0.273546
\(472\) −4.81439 −0.221600
\(473\) −5.33477 −0.245293
\(474\) −8.36936 −0.384417
\(475\) −8.91334 −0.408972
\(476\) −8.76173 −0.401593
\(477\) −18.4384 −0.844238
\(478\) 9.24933 0.423055
\(479\) 3.95132 0.180540 0.0902702 0.995917i \(-0.471227\pi\)
0.0902702 + 0.995917i \(0.471227\pi\)
\(480\) 3.22590 0.147242
\(481\) −2.06268 −0.0940499
\(482\) −2.15772 −0.0982814
\(483\) 0.340927 0.0155127
\(484\) −2.51312 −0.114233
\(485\) 32.5100 1.47620
\(486\) −15.5002 −0.703102
\(487\) 2.21635 0.100432 0.0502161 0.998738i \(-0.484009\pi\)
0.0502161 + 0.998738i \(0.484009\pi\)
\(488\) −10.2163 −0.462472
\(489\) 4.30327 0.194601
\(490\) 17.0439 0.769965
\(491\) −0.638437 −0.0288123 −0.0144061 0.999896i \(-0.504586\pi\)
−0.0144061 + 0.999896i \(0.504586\pi\)
\(492\) 2.91107 0.131241
\(493\) 45.5249 2.05034
\(494\) 1.67233 0.0752415
\(495\) −15.9039 −0.714827
\(496\) 7.72320 0.346782
\(497\) 9.80880 0.439985
\(498\) −6.19162 −0.277453
\(499\) 1.84868 0.0827583 0.0413791 0.999144i \(-0.486825\pi\)
0.0413791 + 0.999144i \(0.486825\pi\)
\(500\) 3.29124 0.147189
\(501\) −22.5760 −1.00862
\(502\) 22.0796 0.985462
\(503\) 14.9232 0.665392 0.332696 0.943034i \(-0.392042\pi\)
0.332696 + 0.943034i \(0.392042\pi\)
\(504\) 2.07539 0.0924451
\(505\) −23.4330 −1.04276
\(506\) −0.809869 −0.0360031
\(507\) 13.4819 0.598752
\(508\) −8.57468 −0.380440
\(509\) −39.3398 −1.74371 −0.871854 0.489766i \(-0.837082\pi\)
−0.871854 + 0.489766i \(0.837082\pi\)
\(510\) 24.9264 1.10376
\(511\) 2.49626 0.110428
\(512\) 1.00000 0.0441942
\(513\) −11.9501 −0.527610
\(514\) 7.21757 0.318354
\(515\) −25.5909 −1.12767
\(516\) −1.98053 −0.0871880
\(517\) 6.84529 0.301056
\(518\) −3.19926 −0.140567
\(519\) 3.48856 0.153131
\(520\) 2.18059 0.0956251
\(521\) −4.87042 −0.213377 −0.106688 0.994293i \(-0.534025\pi\)
−0.106688 + 0.994293i \(0.534025\pi\)
\(522\) −10.7835 −0.471980
\(523\) 17.3383 0.758152 0.379076 0.925366i \(-0.376242\pi\)
0.379076 + 0.925366i \(0.376242\pi\)
\(524\) −12.4733 −0.544898
\(525\) 4.77862 0.208556
\(526\) 15.3564 0.669572
\(527\) 59.6769 2.59957
\(528\) 3.15075 0.137119
\(529\) −22.9227 −0.996640
\(530\) −30.0481 −1.30521
\(531\) 8.81170 0.382395
\(532\) 2.59382 0.112456
\(533\) 1.96777 0.0852337
\(534\) 7.61814 0.329669
\(535\) 52.3506 2.26331
\(536\) 6.42078 0.277335
\(537\) 12.0589 0.520379
\(538\) −22.8657 −0.985810
\(539\) 16.6468 0.717031
\(540\) −15.5820 −0.670544
\(541\) −29.1340 −1.25257 −0.626283 0.779595i \(-0.715425\pi\)
−0.626283 + 0.779595i \(0.715425\pi\)
\(542\) −23.3920 −1.00477
\(543\) −5.75811 −0.247104
\(544\) 7.72696 0.331291
\(545\) 7.39917 0.316946
\(546\) −0.896569 −0.0383696
\(547\) 28.2825 1.20927 0.604637 0.796501i \(-0.293318\pi\)
0.604637 + 0.796501i \(0.293318\pi\)
\(548\) −13.9245 −0.594826
\(549\) 18.6988 0.798046
\(550\) −11.3516 −0.484032
\(551\) −13.4772 −0.574147
\(552\) −0.300664 −0.0127971
\(553\) −8.77471 −0.373139
\(554\) 30.4657 1.29436
\(555\) 9.10165 0.386343
\(556\) 8.26207 0.350390
\(557\) 40.8589 1.73125 0.865623 0.500696i \(-0.166923\pi\)
0.865623 + 0.500696i \(0.166923\pi\)
\(558\) −14.1357 −0.598410
\(559\) −1.33876 −0.0566237
\(560\) 3.38214 0.142922
\(561\) 24.3457 1.02788
\(562\) −3.16439 −0.133482
\(563\) −32.1670 −1.35568 −0.677838 0.735211i \(-0.737083\pi\)
−0.677838 + 0.735211i \(0.737083\pi\)
\(564\) 2.54131 0.107009
\(565\) 34.7104 1.46028
\(566\) 22.7438 0.955995
\(567\) 0.180531 0.00758157
\(568\) −8.65038 −0.362962
\(569\) −4.64164 −0.194588 −0.0972939 0.995256i \(-0.531019\pi\)
−0.0972939 + 0.995256i \(0.531019\pi\)
\(570\) −7.37921 −0.309081
\(571\) −32.1457 −1.34526 −0.672628 0.739981i \(-0.734835\pi\)
−0.672628 + 0.739981i \(0.734835\pi\)
\(572\) 2.12979 0.0890509
\(573\) 19.5213 0.815514
\(574\) 3.05206 0.127391
\(575\) 1.08323 0.0451740
\(576\) −1.83028 −0.0762618
\(577\) −27.9159 −1.16215 −0.581076 0.813849i \(-0.697368\pi\)
−0.581076 + 0.813849i \(0.697368\pi\)
\(578\) 42.7060 1.77633
\(579\) 25.6652 1.06661
\(580\) −17.5732 −0.729688
\(581\) −6.49150 −0.269313
\(582\) 11.7881 0.488635
\(583\) −29.3481 −1.21547
\(584\) −2.20145 −0.0910968
\(585\) −3.99109 −0.165011
\(586\) 31.9816 1.32115
\(587\) −11.5842 −0.478133 −0.239066 0.971003i \(-0.576841\pi\)
−0.239066 + 0.971003i \(0.576841\pi\)
\(588\) 6.18014 0.254865
\(589\) −17.6667 −0.727945
\(590\) 14.3599 0.591189
\(591\) −5.69044 −0.234073
\(592\) 2.82143 0.115960
\(593\) 29.7714 1.22256 0.611282 0.791413i \(-0.290654\pi\)
0.611282 + 0.791413i \(0.290654\pi\)
\(594\) −15.2190 −0.624444
\(595\) 26.1337 1.07138
\(596\) −10.0517 −0.411734
\(597\) 9.09921 0.372406
\(598\) −0.203237 −0.00831099
\(599\) −21.5261 −0.879532 −0.439766 0.898112i \(-0.644939\pi\)
−0.439766 + 0.898112i \(0.644939\pi\)
\(600\) −4.21427 −0.172047
\(601\) 25.0311 1.02104 0.510520 0.859866i \(-0.329453\pi\)
0.510520 + 0.859866i \(0.329453\pi\)
\(602\) −2.07645 −0.0846299
\(603\) −11.7518 −0.478572
\(604\) 12.4850 0.508006
\(605\) 7.49592 0.304752
\(606\) −8.49684 −0.345160
\(607\) −9.77988 −0.396953 −0.198476 0.980106i \(-0.563599\pi\)
−0.198476 + 0.980106i \(0.563599\pi\)
\(608\) −2.28749 −0.0927699
\(609\) 7.22539 0.292788
\(610\) 30.4724 1.23379
\(611\) 1.71783 0.0694960
\(612\) −14.1425 −0.571678
\(613\) −12.6469 −0.510804 −0.255402 0.966835i \(-0.582208\pi\)
−0.255402 + 0.966835i \(0.582208\pi\)
\(614\) −23.5117 −0.948854
\(615\) −8.68288 −0.350128
\(616\) 3.30335 0.133096
\(617\) 12.4909 0.502865 0.251433 0.967875i \(-0.419098\pi\)
0.251433 + 0.967875i \(0.419098\pi\)
\(618\) −9.27929 −0.373268
\(619\) −2.13541 −0.0858295 −0.0429148 0.999079i \(-0.513664\pi\)
−0.0429148 + 0.999079i \(0.513664\pi\)
\(620\) −23.0361 −0.925151
\(621\) 1.45229 0.0582784
\(622\) −17.2273 −0.690754
\(623\) 7.98710 0.319997
\(624\) 0.790684 0.0316527
\(625\) −29.2996 −1.17198
\(626\) −9.55032 −0.381708
\(627\) −7.20730 −0.287832
\(628\) 5.48909 0.219039
\(629\) 21.8010 0.869265
\(630\) −6.19028 −0.246627
\(631\) −22.6430 −0.901404 −0.450702 0.892674i \(-0.648826\pi\)
−0.450702 + 0.892674i \(0.648826\pi\)
\(632\) 7.73841 0.307818
\(633\) −28.9205 −1.14949
\(634\) −5.00960 −0.198957
\(635\) 25.5758 1.01494
\(636\) −10.8955 −0.432033
\(637\) 4.17754 0.165520
\(638\) −17.1638 −0.679523
\(639\) 15.8326 0.626330
\(640\) −2.98271 −0.117902
\(641\) 17.2774 0.682416 0.341208 0.939988i \(-0.389164\pi\)
0.341208 + 0.939988i \(0.389164\pi\)
\(642\) 18.9824 0.749174
\(643\) −36.5920 −1.44305 −0.721523 0.692390i \(-0.756558\pi\)
−0.721523 + 0.692390i \(0.756558\pi\)
\(644\) −0.315226 −0.0124216
\(645\) 5.90735 0.232602
\(646\) −17.6753 −0.695426
\(647\) −9.12560 −0.358764 −0.179382 0.983779i \(-0.557410\pi\)
−0.179382 + 0.983779i \(0.557410\pi\)
\(648\) −0.159210 −0.00625436
\(649\) 14.0254 0.550545
\(650\) −2.84868 −0.111735
\(651\) 9.47150 0.371217
\(652\) −3.97886 −0.155824
\(653\) −39.8566 −1.55971 −0.779855 0.625960i \(-0.784707\pi\)
−0.779855 + 0.625960i \(0.784707\pi\)
\(654\) 2.68295 0.104912
\(655\) 37.2042 1.45369
\(656\) −2.69161 −0.105090
\(657\) 4.02929 0.157197
\(658\) 2.66440 0.103869
\(659\) 22.3262 0.869705 0.434852 0.900502i \(-0.356801\pi\)
0.434852 + 0.900502i \(0.356801\pi\)
\(660\) −9.39778 −0.365808
\(661\) −28.2513 −1.09885 −0.549423 0.835544i \(-0.685153\pi\)
−0.549423 + 0.835544i \(0.685153\pi\)
\(662\) 12.8684 0.500145
\(663\) 6.10958 0.237277
\(664\) 5.72485 0.222167
\(665\) −7.73661 −0.300013
\(666\) −5.16401 −0.200101
\(667\) 1.63788 0.0634188
\(668\) 20.8740 0.807640
\(669\) 22.0840 0.853815
\(670\) −19.1513 −0.739880
\(671\) 29.7625 1.14897
\(672\) 1.22637 0.0473082
\(673\) −43.1383 −1.66286 −0.831429 0.555631i \(-0.812477\pi\)
−0.831429 + 0.555631i \(0.812477\pi\)
\(674\) −7.29248 −0.280896
\(675\) 20.3561 0.783507
\(676\) −12.4655 −0.479443
\(677\) −13.9912 −0.537726 −0.268863 0.963178i \(-0.586648\pi\)
−0.268863 + 0.963178i \(0.586648\pi\)
\(678\) 12.5860 0.483364
\(679\) 12.3591 0.474298
\(680\) −23.0473 −0.883823
\(681\) −4.38428 −0.168006
\(682\) −22.4994 −0.861548
\(683\) 37.9890 1.45361 0.726805 0.686844i \(-0.241005\pi\)
0.726805 + 0.686844i \(0.241005\pi\)
\(684\) 4.18675 0.160084
\(685\) 41.5328 1.58689
\(686\) 14.4169 0.550439
\(687\) 19.5499 0.745874
\(688\) 1.83122 0.0698147
\(689\) −7.36492 −0.280581
\(690\) 0.896793 0.0341403
\(691\) −37.6898 −1.43379 −0.716894 0.697182i \(-0.754437\pi\)
−0.716894 + 0.697182i \(0.754437\pi\)
\(692\) −3.22557 −0.122618
\(693\) −6.04607 −0.229671
\(694\) 14.7381 0.559452
\(695\) −24.6434 −0.934776
\(696\) −6.37207 −0.241533
\(697\) −20.7980 −0.787780
\(698\) −25.4287 −0.962490
\(699\) −28.9413 −1.09466
\(700\) −4.41837 −0.166999
\(701\) 13.6542 0.515714 0.257857 0.966183i \(-0.416984\pi\)
0.257857 + 0.966183i \(0.416984\pi\)
\(702\) −3.81923 −0.144147
\(703\) −6.45397 −0.243416
\(704\) −2.91322 −0.109796
\(705\) −7.58001 −0.285480
\(706\) −6.78268 −0.255269
\(707\) −8.90836 −0.335034
\(708\) 5.20693 0.195688
\(709\) 3.21091 0.120588 0.0602942 0.998181i \(-0.480796\pi\)
0.0602942 + 0.998181i \(0.480796\pi\)
\(710\) 25.8016 0.968316
\(711\) −14.1635 −0.531172
\(712\) −7.04382 −0.263978
\(713\) 2.14703 0.0804069
\(714\) 9.47611 0.354634
\(715\) −6.35254 −0.237572
\(716\) −11.1498 −0.416687
\(717\) −10.0035 −0.373586
\(718\) −30.4679 −1.13705
\(719\) 27.2625 1.01672 0.508359 0.861145i \(-0.330252\pi\)
0.508359 + 0.861145i \(0.330252\pi\)
\(720\) 5.45921 0.203453
\(721\) −9.72871 −0.362316
\(722\) −13.7674 −0.512370
\(723\) 2.33365 0.0867893
\(724\) 5.32402 0.197866
\(725\) 22.9574 0.852615
\(726\) 2.71803 0.100875
\(727\) 6.03486 0.223821 0.111910 0.993718i \(-0.464303\pi\)
0.111910 + 0.993718i \(0.464303\pi\)
\(728\) 0.828979 0.0307240
\(729\) 17.2416 0.638577
\(730\) 6.56630 0.243030
\(731\) 14.1498 0.523349
\(732\) 11.0493 0.408395
\(733\) −46.7308 −1.72604 −0.863020 0.505170i \(-0.831430\pi\)
−0.863020 + 0.505170i \(0.831430\pi\)
\(734\) 24.6983 0.911630
\(735\) −18.4336 −0.679932
\(736\) 0.277997 0.0102471
\(737\) −18.7052 −0.689013
\(738\) 4.92641 0.181344
\(739\) 1.82661 0.0671928 0.0335964 0.999435i \(-0.489304\pi\)
0.0335964 + 0.999435i \(0.489304\pi\)
\(740\) −8.41549 −0.309360
\(741\) −1.80868 −0.0664435
\(742\) −11.4232 −0.419358
\(743\) −18.5274 −0.679703 −0.339852 0.940479i \(-0.610377\pi\)
−0.339852 + 0.940479i \(0.610377\pi\)
\(744\) −8.35291 −0.306232
\(745\) 29.9814 1.09843
\(746\) −10.4154 −0.381335
\(747\) −10.4781 −0.383374
\(748\) −22.5104 −0.823061
\(749\) 19.9017 0.727194
\(750\) −3.55958 −0.129978
\(751\) −34.9022 −1.27360 −0.636799 0.771030i \(-0.719742\pi\)
−0.636799 + 0.771030i \(0.719742\pi\)
\(752\) −2.34973 −0.0856859
\(753\) −23.8799 −0.870231
\(754\) −4.30728 −0.156862
\(755\) −37.2390 −1.35527
\(756\) −5.92371 −0.215443
\(757\) −51.3967 −1.86805 −0.934023 0.357214i \(-0.883727\pi\)
−0.934023 + 0.357214i \(0.883727\pi\)
\(758\) −21.4886 −0.780500
\(759\) 0.875901 0.0317932
\(760\) 6.82291 0.247493
\(761\) 38.7851 1.40596 0.702980 0.711210i \(-0.251852\pi\)
0.702980 + 0.711210i \(0.251852\pi\)
\(762\) 9.27381 0.335955
\(763\) 2.81289 0.101834
\(764\) −18.0496 −0.653013
\(765\) 42.1831 1.52513
\(766\) −18.9808 −0.685806
\(767\) 3.51968 0.127088
\(768\) −1.08153 −0.0390265
\(769\) −16.1838 −0.583604 −0.291802 0.956479i \(-0.594255\pi\)
−0.291802 + 0.956479i \(0.594255\pi\)
\(770\) −9.85294 −0.355075
\(771\) −7.80605 −0.281128
\(772\) −23.7304 −0.854075
\(773\) −24.5627 −0.883459 −0.441729 0.897148i \(-0.645635\pi\)
−0.441729 + 0.897148i \(0.645635\pi\)
\(774\) −3.35166 −0.120473
\(775\) 30.0939 1.08101
\(776\) −10.8995 −0.391268
\(777\) 3.46011 0.124131
\(778\) 1.81540 0.0650852
\(779\) 6.15703 0.220598
\(780\) −2.35838 −0.0844436
\(781\) 25.2005 0.901745
\(782\) 2.14808 0.0768150
\(783\) 30.7789 1.09995
\(784\) −5.71423 −0.204080
\(785\) −16.3724 −0.584355
\(786\) 13.4903 0.481182
\(787\) 37.8219 1.34821 0.674103 0.738637i \(-0.264530\pi\)
0.674103 + 0.738637i \(0.264530\pi\)
\(788\) 5.26145 0.187432
\(789\) −16.6085 −0.591278
\(790\) −23.0814 −0.821201
\(791\) 13.1956 0.469182
\(792\) 5.33203 0.189465
\(793\) 7.46893 0.265229
\(794\) 7.94724 0.282037
\(795\) 32.4980 1.15259
\(796\) −8.41324 −0.298199
\(797\) −2.13168 −0.0755081 −0.0377540 0.999287i \(-0.512020\pi\)
−0.0377540 + 0.999287i \(0.512020\pi\)
\(798\) −2.80530 −0.0993067
\(799\) −18.1563 −0.642323
\(800\) 3.89656 0.137764
\(801\) 12.8922 0.455523
\(802\) −25.5334 −0.901615
\(803\) 6.41333 0.226322
\(804\) −6.94429 −0.244906
\(805\) 0.940227 0.0331386
\(806\) −5.64625 −0.198881
\(807\) 24.7300 0.870538
\(808\) 7.85628 0.276383
\(809\) −15.6387 −0.549826 −0.274913 0.961469i \(-0.588649\pi\)
−0.274913 + 0.961469i \(0.588649\pi\)
\(810\) 0.474877 0.0166855
\(811\) 37.2490 1.30799 0.653995 0.756499i \(-0.273092\pi\)
0.653995 + 0.756499i \(0.273092\pi\)
\(812\) −6.68069 −0.234446
\(813\) 25.2993 0.887284
\(814\) −8.21945 −0.288091
\(815\) 11.8678 0.415711
\(816\) −8.35697 −0.292553
\(817\) −4.18890 −0.146551
\(818\) 9.59658 0.335536
\(819\) −1.51727 −0.0530176
\(820\) 8.02830 0.280360
\(821\) −36.3634 −1.26909 −0.634546 0.772885i \(-0.718813\pi\)
−0.634546 + 0.772885i \(0.718813\pi\)
\(822\) 15.0598 0.525273
\(823\) 13.0658 0.455447 0.227723 0.973726i \(-0.426872\pi\)
0.227723 + 0.973726i \(0.426872\pi\)
\(824\) 8.57974 0.298890
\(825\) 12.2771 0.427434
\(826\) 5.45911 0.189947
\(827\) −18.3079 −0.636628 −0.318314 0.947985i \(-0.603117\pi\)
−0.318314 + 0.947985i \(0.603117\pi\)
\(828\) −0.508814 −0.0176825
\(829\) −51.4908 −1.78835 −0.894174 0.447719i \(-0.852236\pi\)
−0.894174 + 0.447719i \(0.852236\pi\)
\(830\) −17.0756 −0.592702
\(831\) −32.9497 −1.14301
\(832\) −0.731076 −0.0253455
\(833\) −44.1537 −1.52983
\(834\) −8.93572 −0.309419
\(835\) −62.2611 −2.15464
\(836\) 6.66396 0.230478
\(837\) 40.3469 1.39459
\(838\) −14.0049 −0.483791
\(839\) 11.6875 0.403497 0.201748 0.979437i \(-0.435338\pi\)
0.201748 + 0.979437i \(0.435338\pi\)
\(840\) −3.65790 −0.126210
\(841\) 5.71207 0.196968
\(842\) 13.7078 0.472403
\(843\) 3.42239 0.117874
\(844\) 26.7402 0.920437
\(845\) 37.1811 1.27907
\(846\) 4.30067 0.147860
\(847\) 2.84967 0.0979158
\(848\) 10.0741 0.345946
\(849\) −24.5982 −0.844209
\(850\) 30.1086 1.03272
\(851\) 0.784349 0.0268871
\(852\) 9.35568 0.320520
\(853\) 2.33987 0.0801156 0.0400578 0.999197i \(-0.487246\pi\)
0.0400578 + 0.999197i \(0.487246\pi\)
\(854\) 11.5845 0.396413
\(855\) −12.4879 −0.427076
\(856\) −17.5513 −0.599892
\(857\) 11.8398 0.404441 0.202221 0.979340i \(-0.435184\pi\)
0.202221 + 0.979340i \(0.435184\pi\)
\(858\) −2.30344 −0.0786381
\(859\) 42.6766 1.45611 0.728053 0.685521i \(-0.240426\pi\)
0.728053 + 0.685521i \(0.240426\pi\)
\(860\) −5.46201 −0.186253
\(861\) −3.30091 −0.112495
\(862\) −32.6504 −1.11208
\(863\) −3.29402 −0.112130 −0.0560649 0.998427i \(-0.517855\pi\)
−0.0560649 + 0.998427i \(0.517855\pi\)
\(864\) 5.22412 0.177728
\(865\) 9.62094 0.327122
\(866\) 13.3901 0.455016
\(867\) −46.1879 −1.56862
\(868\) −8.75746 −0.297248
\(869\) −22.5437 −0.764744
\(870\) 19.0060 0.644365
\(871\) −4.69407 −0.159053
\(872\) −2.48069 −0.0840067
\(873\) 19.9491 0.675176
\(874\) −0.635916 −0.0215102
\(875\) −3.73198 −0.126164
\(876\) 2.38095 0.0804448
\(877\) −52.5111 −1.77317 −0.886587 0.462562i \(-0.846930\pi\)
−0.886587 + 0.462562i \(0.846930\pi\)
\(878\) −24.4386 −0.824763
\(879\) −34.5892 −1.16666
\(880\) 8.68931 0.292916
\(881\) 29.9596 1.00936 0.504682 0.863305i \(-0.331610\pi\)
0.504682 + 0.863305i \(0.331610\pi\)
\(882\) 10.4587 0.352162
\(883\) −26.0284 −0.875926 −0.437963 0.898993i \(-0.644300\pi\)
−0.437963 + 0.898993i \(0.644300\pi\)
\(884\) −5.64900 −0.189996
\(885\) −15.5308 −0.522060
\(886\) 17.5279 0.588861
\(887\) −49.5212 −1.66276 −0.831379 0.555706i \(-0.812448\pi\)
−0.831379 + 0.555706i \(0.812448\pi\)
\(888\) −3.05147 −0.102401
\(889\) 9.72297 0.326098
\(890\) 21.0097 0.704246
\(891\) 0.463814 0.0155384
\(892\) −20.4191 −0.683682
\(893\) 5.37498 0.179867
\(894\) 10.8713 0.363590
\(895\) 33.2566 1.11165
\(896\) −1.13392 −0.0378815
\(897\) 0.219808 0.00733917
\(898\) −20.4080 −0.681023
\(899\) 45.5028 1.51760
\(900\) −7.13181 −0.237727
\(901\) 77.8421 2.59330
\(902\) 7.84127 0.261086
\(903\) 2.24576 0.0747341
\(904\) −11.6372 −0.387048
\(905\) −15.8800 −0.527870
\(906\) −13.5029 −0.448604
\(907\) −32.4478 −1.07741 −0.538706 0.842494i \(-0.681086\pi\)
−0.538706 + 0.842494i \(0.681086\pi\)
\(908\) 4.05376 0.134529
\(909\) −14.3792 −0.476929
\(910\) −2.47260 −0.0819660
\(911\) 52.1417 1.72753 0.863765 0.503894i \(-0.168100\pi\)
0.863765 + 0.503894i \(0.168100\pi\)
\(912\) 2.47400 0.0819222
\(913\) −16.6778 −0.551954
\(914\) −2.60763 −0.0862526
\(915\) −32.9570 −1.08952
\(916\) −18.0760 −0.597249
\(917\) 14.1437 0.467065
\(918\) 40.3666 1.33229
\(919\) −26.4609 −0.872865 −0.436433 0.899737i \(-0.643758\pi\)
−0.436433 + 0.899737i \(0.643758\pi\)
\(920\) −0.829186 −0.0273374
\(921\) 25.4287 0.837904
\(922\) 30.6105 1.00810
\(923\) 6.32408 0.208160
\(924\) −3.57269 −0.117533
\(925\) 10.9939 0.361476
\(926\) −10.9539 −0.359969
\(927\) −15.7034 −0.515766
\(928\) 5.89169 0.193404
\(929\) 51.5944 1.69276 0.846379 0.532581i \(-0.178778\pi\)
0.846379 + 0.532581i \(0.178778\pi\)
\(930\) 24.9143 0.816972
\(931\) 13.0712 0.428393
\(932\) 26.7595 0.876536
\(933\) 18.6320 0.609983
\(934\) 31.9401 1.04511
\(935\) 67.1419 2.19578
\(936\) 1.33808 0.0437364
\(937\) −31.7036 −1.03571 −0.517855 0.855468i \(-0.673269\pi\)
−0.517855 + 0.855468i \(0.673269\pi\)
\(938\) −7.28062 −0.237721
\(939\) 10.3290 0.337074
\(940\) 7.00857 0.228594
\(941\) −8.22125 −0.268005 −0.134003 0.990981i \(-0.542783\pi\)
−0.134003 + 0.990981i \(0.542783\pi\)
\(942\) −5.93664 −0.193426
\(943\) −0.748261 −0.0243667
\(944\) −4.81439 −0.156695
\(945\) 17.6687 0.574763
\(946\) −5.33477 −0.173448
\(947\) −19.8767 −0.645906 −0.322953 0.946415i \(-0.604676\pi\)
−0.322953 + 0.946415i \(0.604676\pi\)
\(948\) −8.36936 −0.271824
\(949\) 1.60943 0.0522443
\(950\) −8.91334 −0.289187
\(951\) 5.41806 0.175693
\(952\) −8.76173 −0.283969
\(953\) −7.75630 −0.251251 −0.125626 0.992078i \(-0.540094\pi\)
−0.125626 + 0.992078i \(0.540094\pi\)
\(954\) −18.4384 −0.596966
\(955\) 53.8369 1.74212
\(956\) 9.24933 0.299145
\(957\) 18.5633 0.600065
\(958\) 3.95132 0.127661
\(959\) 15.7892 0.509861
\(960\) 3.22590 0.104116
\(961\) 28.6479 0.924125
\(962\) −2.06268 −0.0665033
\(963\) 32.1239 1.03518
\(964\) −2.15772 −0.0694954
\(965\) 70.7809 2.27852
\(966\) 0.340927 0.0109692
\(967\) −8.03035 −0.258239 −0.129119 0.991629i \(-0.541215\pi\)
−0.129119 + 0.991629i \(0.541215\pi\)
\(968\) −2.51312 −0.0807748
\(969\) 19.1165 0.614110
\(970\) 32.5100 1.04383
\(971\) 30.6975 0.985130 0.492565 0.870276i \(-0.336059\pi\)
0.492565 + 0.870276i \(0.336059\pi\)
\(972\) −15.5002 −0.497168
\(973\) −9.36850 −0.300340
\(974\) 2.21635 0.0710163
\(975\) 3.08095 0.0986693
\(976\) −10.2163 −0.327017
\(977\) 3.99588 0.127840 0.0639198 0.997955i \(-0.479640\pi\)
0.0639198 + 0.997955i \(0.479640\pi\)
\(978\) 4.30327 0.137604
\(979\) 20.5202 0.655830
\(980\) 17.0439 0.544448
\(981\) 4.54036 0.144963
\(982\) −0.638437 −0.0203734
\(983\) −14.6984 −0.468806 −0.234403 0.972140i \(-0.575314\pi\)
−0.234403 + 0.972140i \(0.575314\pi\)
\(984\) 2.91107 0.0928016
\(985\) −15.6934 −0.500033
\(986\) 45.5249 1.44981
\(987\) −2.88164 −0.0917235
\(988\) 1.67233 0.0532038
\(989\) 0.509075 0.0161877
\(990\) −15.9039 −0.505459
\(991\) 20.5451 0.652636 0.326318 0.945260i \(-0.394192\pi\)
0.326318 + 0.945260i \(0.394192\pi\)
\(992\) 7.72320 0.245212
\(993\) −13.9176 −0.441663
\(994\) 9.80880 0.311116
\(995\) 25.0943 0.795541
\(996\) −6.19162 −0.196189
\(997\) 61.2662 1.94032 0.970160 0.242464i \(-0.0779557\pi\)
0.970160 + 0.242464i \(0.0779557\pi\)
\(998\) 1.84868 0.0585189
\(999\) 14.7395 0.466336
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 8042.2.a.a.1.26 67
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8042.2.a.a.1.26 67 1.1 even 1 trivial