Properties

Label 8049.2.a.d.1.16
Level $8049$
Weight $2$
Character 8049.1
Self dual yes
Analytic conductor $64.272$
Analytic rank $0$
Dimension $129$
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] = [8049,2,Mod(1,8049)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8049, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8049.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8049 = 3 \cdot 2683 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8049.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: \(64.2715885869\)
Analytic rank: \(0\)
Dimension: \(129\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.16
Character \(\chi\) \(=\) 8049.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.30919 q^{2} +1.00000 q^{3} +3.33237 q^{4} -0.223778 q^{5} -2.30919 q^{6} +3.94364 q^{7} -3.07670 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.30919 q^{2} +1.00000 q^{3} +3.33237 q^{4} -0.223778 q^{5} -2.30919 q^{6} +3.94364 q^{7} -3.07670 q^{8} +1.00000 q^{9} +0.516746 q^{10} -4.65249 q^{11} +3.33237 q^{12} -0.868261 q^{13} -9.10664 q^{14} -0.223778 q^{15} +0.439960 q^{16} -2.19008 q^{17} -2.30919 q^{18} +8.29060 q^{19} -0.745711 q^{20} +3.94364 q^{21} +10.7435 q^{22} -2.92422 q^{23} -3.07670 q^{24} -4.94992 q^{25} +2.00498 q^{26} +1.00000 q^{27} +13.1417 q^{28} -1.95710 q^{29} +0.516746 q^{30} -8.87044 q^{31} +5.13746 q^{32} -4.65249 q^{33} +5.05731 q^{34} -0.882500 q^{35} +3.33237 q^{36} +2.47861 q^{37} -19.1446 q^{38} -0.868261 q^{39} +0.688498 q^{40} -4.80549 q^{41} -9.10664 q^{42} +3.31798 q^{43} -15.5038 q^{44} -0.223778 q^{45} +6.75259 q^{46} -0.547298 q^{47} +0.439960 q^{48} +8.55234 q^{49} +11.4303 q^{50} -2.19008 q^{51} -2.89337 q^{52} -0.651506 q^{53} -2.30919 q^{54} +1.04112 q^{55} -12.1334 q^{56} +8.29060 q^{57} +4.51932 q^{58} +9.59738 q^{59} -0.745711 q^{60} +12.5445 q^{61} +20.4836 q^{62} +3.94364 q^{63} -12.7433 q^{64} +0.194298 q^{65} +10.7435 q^{66} -4.05863 q^{67} -7.29815 q^{68} -2.92422 q^{69} +2.03786 q^{70} -4.25409 q^{71} -3.07670 q^{72} +2.99090 q^{73} -5.72359 q^{74} -4.94992 q^{75} +27.6274 q^{76} -18.3478 q^{77} +2.00498 q^{78} -9.51405 q^{79} -0.0984531 q^{80} +1.00000 q^{81} +11.0968 q^{82} +5.74416 q^{83} +13.1417 q^{84} +0.490091 q^{85} -7.66186 q^{86} -1.95710 q^{87} +14.3143 q^{88} +2.16881 q^{89} +0.516746 q^{90} -3.42411 q^{91} -9.74460 q^{92} -8.87044 q^{93} +1.26382 q^{94} -1.85525 q^{95} +5.13746 q^{96} +2.50946 q^{97} -19.7490 q^{98} -4.65249 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 129 q + 8 q^{2} + 129 q^{3} + 158 q^{4} + 11 q^{5} + 8 q^{6} + 40 q^{7} + 18 q^{8} + 129 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 129 q + 8 q^{2} + 129 q^{3} + 158 q^{4} + 11 q^{5} + 8 q^{6} + 40 q^{7} + 18 q^{8} + 129 q^{9} + 20 q^{10} + 48 q^{11} + 158 q^{12} + 77 q^{13} + 13 q^{14} + 11 q^{15} + 212 q^{16} + 9 q^{17} + 8 q^{18} + 68 q^{19} + 19 q^{20} + 40 q^{21} + 45 q^{22} + 64 q^{23} + 18 q^{24} + 188 q^{25} + 19 q^{26} + 129 q^{27} + 69 q^{28} + 23 q^{29} + 20 q^{30} + 133 q^{31} + 24 q^{32} + 48 q^{33} + 63 q^{34} + 26 q^{35} + 158 q^{36} + 147 q^{37} + 9 q^{38} + 77 q^{39} + 58 q^{40} + 21 q^{41} + 13 q^{42} + 76 q^{43} + 110 q^{44} + 11 q^{45} + 48 q^{46} + 85 q^{47} + 212 q^{48} + 213 q^{49} + 17 q^{50} + 9 q^{51} + 139 q^{52} + 30 q^{53} + 8 q^{54} + 103 q^{55} + 19 q^{56} + 68 q^{57} + 94 q^{58} + 64 q^{59} + 19 q^{60} + 110 q^{61} - 10 q^{62} + 40 q^{63} + 288 q^{64} - 8 q^{65} + 45 q^{66} + 118 q^{67} - 15 q^{68} + 64 q^{69} + 75 q^{70} + 154 q^{71} + 18 q^{72} + 137 q^{73} + 28 q^{74} + 188 q^{75} + 156 q^{76} + 17 q^{77} + 19 q^{78} + 157 q^{79} + 2 q^{80} + 129 q^{81} + 72 q^{82} + 39 q^{83} + 69 q^{84} + 127 q^{85} + 54 q^{86} + 23 q^{87} + 97 q^{88} + 31 q^{89} + 20 q^{90} + 137 q^{91} + 82 q^{92} + 133 q^{93} + 40 q^{94} + 68 q^{95} + 24 q^{96} + 170 q^{97} - 21 q^{98} + 48 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.30919 −1.63285 −0.816423 0.577454i \(-0.804046\pi\)
−0.816423 + 0.577454i \(0.804046\pi\)
\(3\) 1.00000 0.577350
\(4\) 3.33237 1.66619
\(5\) −0.223778 −0.100076 −0.0500382 0.998747i \(-0.515934\pi\)
−0.0500382 + 0.998747i \(0.515934\pi\)
\(6\) −2.30919 −0.942724
\(7\) 3.94364 1.49056 0.745279 0.666753i \(-0.232316\pi\)
0.745279 + 0.666753i \(0.232316\pi\)
\(8\) −3.07670 −1.08778
\(9\) 1.00000 0.333333
\(10\) 0.516746 0.163409
\(11\) −4.65249 −1.40278 −0.701389 0.712779i \(-0.747436\pi\)
−0.701389 + 0.712779i \(0.747436\pi\)
\(12\) 3.33237 0.961973
\(13\) −0.868261 −0.240812 −0.120406 0.992725i \(-0.538420\pi\)
−0.120406 + 0.992725i \(0.538420\pi\)
\(14\) −9.10664 −2.43385
\(15\) −0.223778 −0.0577792
\(16\) 0.439960 0.109990
\(17\) −2.19008 −0.531172 −0.265586 0.964087i \(-0.585565\pi\)
−0.265586 + 0.964087i \(0.585565\pi\)
\(18\) −2.30919 −0.544282
\(19\) 8.29060 1.90199 0.950997 0.309200i \(-0.100061\pi\)
0.950997 + 0.309200i \(0.100061\pi\)
\(20\) −0.745711 −0.166746
\(21\) 3.94364 0.860574
\(22\) 10.7435 2.29052
\(23\) −2.92422 −0.609743 −0.304871 0.952394i \(-0.598613\pi\)
−0.304871 + 0.952394i \(0.598613\pi\)
\(24\) −3.07670 −0.628030
\(25\) −4.94992 −0.989985
\(26\) 2.00498 0.393210
\(27\) 1.00000 0.192450
\(28\) 13.1417 2.48355
\(29\) −1.95710 −0.363424 −0.181712 0.983352i \(-0.558164\pi\)
−0.181712 + 0.983352i \(0.558164\pi\)
\(30\) 0.516746 0.0943445
\(31\) −8.87044 −1.59318 −0.796589 0.604521i \(-0.793365\pi\)
−0.796589 + 0.604521i \(0.793365\pi\)
\(32\) 5.13746 0.908183
\(33\) −4.65249 −0.809894
\(34\) 5.05731 0.867322
\(35\) −0.882500 −0.149170
\(36\) 3.33237 0.555395
\(37\) 2.47861 0.407481 0.203740 0.979025i \(-0.434690\pi\)
0.203740 + 0.979025i \(0.434690\pi\)
\(38\) −19.1446 −3.10566
\(39\) −0.868261 −0.139033
\(40\) 0.688498 0.108861
\(41\) −4.80549 −0.750492 −0.375246 0.926925i \(-0.622442\pi\)
−0.375246 + 0.926925i \(0.622442\pi\)
\(42\) −9.10664 −1.40518
\(43\) 3.31798 0.505987 0.252994 0.967468i \(-0.418585\pi\)
0.252994 + 0.967468i \(0.418585\pi\)
\(44\) −15.5038 −2.33729
\(45\) −0.223778 −0.0333588
\(46\) 6.75259 0.995616
\(47\) −0.547298 −0.0798317 −0.0399158 0.999203i \(-0.512709\pi\)
−0.0399158 + 0.999203i \(0.512709\pi\)
\(48\) 0.439960 0.0635027
\(49\) 8.55234 1.22176
\(50\) 11.4303 1.61649
\(51\) −2.19008 −0.306672
\(52\) −2.89337 −0.401238
\(53\) −0.651506 −0.0894912 −0.0447456 0.998998i \(-0.514248\pi\)
−0.0447456 + 0.998998i \(0.514248\pi\)
\(54\) −2.30919 −0.314241
\(55\) 1.04112 0.140385
\(56\) −12.1334 −1.62140
\(57\) 8.29060 1.09812
\(58\) 4.51932 0.593415
\(59\) 9.59738 1.24947 0.624736 0.780836i \(-0.285207\pi\)
0.624736 + 0.780836i \(0.285207\pi\)
\(60\) −0.745711 −0.0962708
\(61\) 12.5445 1.60616 0.803078 0.595873i \(-0.203194\pi\)
0.803078 + 0.595873i \(0.203194\pi\)
\(62\) 20.4836 2.60142
\(63\) 3.94364 0.496853
\(64\) −12.7433 −1.59291
\(65\) 0.194298 0.0240996
\(66\) 10.7435 1.32243
\(67\) −4.05863 −0.495841 −0.247920 0.968780i \(-0.579747\pi\)
−0.247920 + 0.968780i \(0.579747\pi\)
\(68\) −7.29815 −0.885031
\(69\) −2.92422 −0.352035
\(70\) 2.03786 0.243571
\(71\) −4.25409 −0.504868 −0.252434 0.967614i \(-0.581231\pi\)
−0.252434 + 0.967614i \(0.581231\pi\)
\(72\) −3.07670 −0.362593
\(73\) 2.99090 0.350058 0.175029 0.984563i \(-0.443998\pi\)
0.175029 + 0.984563i \(0.443998\pi\)
\(74\) −5.72359 −0.665353
\(75\) −4.94992 −0.571568
\(76\) 27.6274 3.16908
\(77\) −18.3478 −2.09092
\(78\) 2.00498 0.227020
\(79\) −9.51405 −1.07041 −0.535207 0.844721i \(-0.679766\pi\)
−0.535207 + 0.844721i \(0.679766\pi\)
\(80\) −0.0984531 −0.0110074
\(81\) 1.00000 0.111111
\(82\) 11.0968 1.22544
\(83\) 5.74416 0.630504 0.315252 0.949008i \(-0.397911\pi\)
0.315252 + 0.949008i \(0.397911\pi\)
\(84\) 13.1417 1.43388
\(85\) 0.490091 0.0531578
\(86\) −7.66186 −0.826199
\(87\) −1.95710 −0.209823
\(88\) 14.3143 1.52591
\(89\) 2.16881 0.229893 0.114947 0.993372i \(-0.463330\pi\)
0.114947 + 0.993372i \(0.463330\pi\)
\(90\) 0.516746 0.0544698
\(91\) −3.42411 −0.358945
\(92\) −9.74460 −1.01594
\(93\) −8.87044 −0.919822
\(94\) 1.26382 0.130353
\(95\) −1.85525 −0.190345
\(96\) 5.13746 0.524340
\(97\) 2.50946 0.254798 0.127399 0.991852i \(-0.459337\pi\)
0.127399 + 0.991852i \(0.459337\pi\)
\(98\) −19.7490 −1.99495
\(99\) −4.65249 −0.467592
\(100\) −16.4950 −1.64950
\(101\) −8.39236 −0.835071 −0.417536 0.908661i \(-0.637106\pi\)
−0.417536 + 0.908661i \(0.637106\pi\)
\(102\) 5.05731 0.500749
\(103\) 3.24018 0.319264 0.159632 0.987177i \(-0.448969\pi\)
0.159632 + 0.987177i \(0.448969\pi\)
\(104\) 2.67138 0.261951
\(105\) −0.882500 −0.0861232
\(106\) 1.50445 0.146125
\(107\) 19.7559 1.90988 0.954938 0.296804i \(-0.0959208\pi\)
0.954938 + 0.296804i \(0.0959208\pi\)
\(108\) 3.33237 0.320658
\(109\) 16.2859 1.55990 0.779951 0.625840i \(-0.215244\pi\)
0.779951 + 0.625840i \(0.215244\pi\)
\(110\) −2.40415 −0.229227
\(111\) 2.47861 0.235259
\(112\) 1.73504 0.163946
\(113\) −6.70705 −0.630946 −0.315473 0.948935i \(-0.602163\pi\)
−0.315473 + 0.948935i \(0.602163\pi\)
\(114\) −19.1446 −1.79306
\(115\) 0.654376 0.0610209
\(116\) −6.52178 −0.605532
\(117\) −0.868261 −0.0802708
\(118\) −22.1622 −2.04020
\(119\) −8.63689 −0.791742
\(120\) 0.688498 0.0628510
\(121\) 10.6456 0.967784
\(122\) −28.9676 −2.62261
\(123\) −4.80549 −0.433297
\(124\) −29.5596 −2.65453
\(125\) 2.22657 0.199151
\(126\) −9.10664 −0.811284
\(127\) 12.7044 1.12733 0.563666 0.826003i \(-0.309390\pi\)
0.563666 + 0.826003i \(0.309390\pi\)
\(128\) 19.1518 1.69280
\(129\) 3.31798 0.292132
\(130\) −0.448671 −0.0393510
\(131\) 13.7546 1.20175 0.600874 0.799343i \(-0.294819\pi\)
0.600874 + 0.799343i \(0.294819\pi\)
\(132\) −15.5038 −1.34943
\(133\) 32.6952 2.83503
\(134\) 9.37217 0.809632
\(135\) −0.223778 −0.0192597
\(136\) 6.73822 0.577798
\(137\) 8.65141 0.739140 0.369570 0.929203i \(-0.379505\pi\)
0.369570 + 0.929203i \(0.379505\pi\)
\(138\) 6.75259 0.574819
\(139\) 6.89645 0.584949 0.292474 0.956273i \(-0.405521\pi\)
0.292474 + 0.956273i \(0.405521\pi\)
\(140\) −2.94082 −0.248544
\(141\) −0.547298 −0.0460908
\(142\) 9.82351 0.824371
\(143\) 4.03957 0.337806
\(144\) 0.439960 0.0366633
\(145\) 0.437955 0.0363702
\(146\) −6.90656 −0.571591
\(147\) 8.55234 0.705385
\(148\) 8.25965 0.678939
\(149\) 7.88506 0.645969 0.322985 0.946404i \(-0.395314\pi\)
0.322985 + 0.946404i \(0.395314\pi\)
\(150\) 11.4303 0.933282
\(151\) 7.16906 0.583410 0.291705 0.956508i \(-0.405777\pi\)
0.291705 + 0.956508i \(0.405777\pi\)
\(152\) −25.5077 −2.06895
\(153\) −2.19008 −0.177057
\(154\) 42.3685 3.41415
\(155\) 1.98501 0.159440
\(156\) −2.89337 −0.231655
\(157\) −0.499226 −0.0398426 −0.0199213 0.999802i \(-0.506342\pi\)
−0.0199213 + 0.999802i \(0.506342\pi\)
\(158\) 21.9698 1.74782
\(159\) −0.651506 −0.0516678
\(160\) −1.14965 −0.0908877
\(161\) −11.5321 −0.908856
\(162\) −2.30919 −0.181427
\(163\) 5.37514 0.421013 0.210507 0.977592i \(-0.432489\pi\)
0.210507 + 0.977592i \(0.432489\pi\)
\(164\) −16.0137 −1.25046
\(165\) 1.04112 0.0810513
\(166\) −13.2644 −1.02952
\(167\) 11.7716 0.910910 0.455455 0.890259i \(-0.349477\pi\)
0.455455 + 0.890259i \(0.349477\pi\)
\(168\) −12.1334 −0.936114
\(169\) −12.2461 −0.942009
\(170\) −1.13171 −0.0867985
\(171\) 8.29060 0.633998
\(172\) 11.0567 0.843069
\(173\) 24.9254 1.89504 0.947522 0.319691i \(-0.103579\pi\)
0.947522 + 0.319691i \(0.103579\pi\)
\(174\) 4.51932 0.342609
\(175\) −19.5207 −1.47563
\(176\) −2.04691 −0.154291
\(177\) 9.59738 0.721383
\(178\) −5.00820 −0.375381
\(179\) −6.84478 −0.511603 −0.255801 0.966729i \(-0.582339\pi\)
−0.255801 + 0.966729i \(0.582339\pi\)
\(180\) −0.745711 −0.0555820
\(181\) 21.2839 1.58202 0.791009 0.611804i \(-0.209556\pi\)
0.791009 + 0.611804i \(0.209556\pi\)
\(182\) 7.90694 0.586101
\(183\) 12.5445 0.927315
\(184\) 8.99697 0.663265
\(185\) −0.554657 −0.0407792
\(186\) 20.4836 1.50193
\(187\) 10.1893 0.745116
\(188\) −1.82380 −0.133014
\(189\) 3.94364 0.286858
\(190\) 4.28413 0.310804
\(191\) 22.6969 1.64229 0.821147 0.570717i \(-0.193335\pi\)
0.821147 + 0.570717i \(0.193335\pi\)
\(192\) −12.7433 −0.919668
\(193\) −1.86094 −0.133953 −0.0669765 0.997755i \(-0.521335\pi\)
−0.0669765 + 0.997755i \(0.521335\pi\)
\(194\) −5.79484 −0.416045
\(195\) 0.194298 0.0139139
\(196\) 28.4996 2.03568
\(197\) 8.82171 0.628521 0.314261 0.949337i \(-0.398243\pi\)
0.314261 + 0.949337i \(0.398243\pi\)
\(198\) 10.7435 0.763506
\(199\) −5.84192 −0.414123 −0.207061 0.978328i \(-0.566390\pi\)
−0.207061 + 0.978328i \(0.566390\pi\)
\(200\) 15.2295 1.07688
\(201\) −4.05863 −0.286274
\(202\) 19.3796 1.36354
\(203\) −7.71810 −0.541704
\(204\) −7.29815 −0.510973
\(205\) 1.07536 0.0751065
\(206\) −7.48220 −0.521309
\(207\) −2.92422 −0.203248
\(208\) −0.382000 −0.0264869
\(209\) −38.5719 −2.66807
\(210\) 2.03786 0.140626
\(211\) 15.1432 1.04250 0.521250 0.853404i \(-0.325466\pi\)
0.521250 + 0.853404i \(0.325466\pi\)
\(212\) −2.17106 −0.149109
\(213\) −4.25409 −0.291485
\(214\) −45.6202 −3.11853
\(215\) −0.742490 −0.0506374
\(216\) −3.07670 −0.209343
\(217\) −34.9819 −2.37472
\(218\) −37.6072 −2.54708
\(219\) 2.99090 0.202106
\(220\) 3.46941 0.233907
\(221\) 1.90156 0.127913
\(222\) −5.72359 −0.384142
\(223\) 0.556387 0.0372584 0.0186292 0.999826i \(-0.494070\pi\)
0.0186292 + 0.999826i \(0.494070\pi\)
\(224\) 20.2603 1.35370
\(225\) −4.94992 −0.329995
\(226\) 15.4879 1.03024
\(227\) −18.4884 −1.22712 −0.613560 0.789648i \(-0.710263\pi\)
−0.613560 + 0.789648i \(0.710263\pi\)
\(228\) 27.6274 1.82967
\(229\) −8.71380 −0.575824 −0.287912 0.957657i \(-0.592961\pi\)
−0.287912 + 0.957657i \(0.592961\pi\)
\(230\) −1.51108 −0.0996377
\(231\) −18.3478 −1.20719
\(232\) 6.02141 0.395325
\(233\) 22.0312 1.44331 0.721656 0.692252i \(-0.243381\pi\)
0.721656 + 0.692252i \(0.243381\pi\)
\(234\) 2.00498 0.131070
\(235\) 0.122473 0.00798927
\(236\) 31.9820 2.08185
\(237\) −9.51405 −0.618004
\(238\) 19.9442 1.29279
\(239\) 3.06543 0.198286 0.0991430 0.995073i \(-0.468390\pi\)
0.0991430 + 0.995073i \(0.468390\pi\)
\(240\) −0.0984531 −0.00635512
\(241\) −17.4603 −1.12471 −0.562357 0.826894i \(-0.690105\pi\)
−0.562357 + 0.826894i \(0.690105\pi\)
\(242\) −24.5828 −1.58024
\(243\) 1.00000 0.0641500
\(244\) 41.8029 2.67616
\(245\) −1.91382 −0.122270
\(246\) 11.0968 0.707507
\(247\) −7.19841 −0.458024
\(248\) 27.2917 1.73303
\(249\) 5.74416 0.364022
\(250\) −5.14158 −0.325182
\(251\) 7.68374 0.484993 0.242497 0.970152i \(-0.422034\pi\)
0.242497 + 0.970152i \(0.422034\pi\)
\(252\) 13.1417 0.827849
\(253\) 13.6049 0.855333
\(254\) −29.3369 −1.84076
\(255\) 0.490091 0.0306907
\(256\) −18.7387 −1.17117
\(257\) 21.0624 1.31384 0.656919 0.753961i \(-0.271859\pi\)
0.656919 + 0.753961i \(0.271859\pi\)
\(258\) −7.66186 −0.477006
\(259\) 9.77475 0.607374
\(260\) 0.647472 0.0401545
\(261\) −1.95710 −0.121141
\(262\) −31.7621 −1.96227
\(263\) −14.6359 −0.902491 −0.451245 0.892400i \(-0.649020\pi\)
−0.451245 + 0.892400i \(0.649020\pi\)
\(264\) 14.3143 0.880986
\(265\) 0.145793 0.00895596
\(266\) −75.4995 −4.62917
\(267\) 2.16881 0.132729
\(268\) −13.5249 −0.826163
\(269\) 9.26375 0.564821 0.282410 0.959294i \(-0.408866\pi\)
0.282410 + 0.959294i \(0.408866\pi\)
\(270\) 0.516746 0.0314482
\(271\) 12.8171 0.778584 0.389292 0.921114i \(-0.372720\pi\)
0.389292 + 0.921114i \(0.372720\pi\)
\(272\) −0.963546 −0.0584235
\(273\) −3.42411 −0.207237
\(274\) −19.9778 −1.20690
\(275\) 23.0294 1.38873
\(276\) −9.74460 −0.586556
\(277\) 18.6361 1.11974 0.559868 0.828582i \(-0.310852\pi\)
0.559868 + 0.828582i \(0.310852\pi\)
\(278\) −15.9252 −0.955131
\(279\) −8.87044 −0.531059
\(280\) 2.71519 0.162264
\(281\) 6.31691 0.376835 0.188418 0.982089i \(-0.439664\pi\)
0.188418 + 0.982089i \(0.439664\pi\)
\(282\) 1.26382 0.0752592
\(283\) −23.8199 −1.41595 −0.707973 0.706239i \(-0.750390\pi\)
−0.707973 + 0.706239i \(0.750390\pi\)
\(284\) −14.1762 −0.841203
\(285\) −1.85525 −0.109896
\(286\) −9.32815 −0.551585
\(287\) −18.9512 −1.11865
\(288\) 5.13746 0.302728
\(289\) −12.2036 −0.717856
\(290\) −1.01132 −0.0593869
\(291\) 2.50946 0.147107
\(292\) 9.96678 0.583262
\(293\) 11.6479 0.680479 0.340239 0.940339i \(-0.389492\pi\)
0.340239 + 0.940339i \(0.389492\pi\)
\(294\) −19.7490 −1.15178
\(295\) −2.14768 −0.125043
\(296\) −7.62595 −0.443249
\(297\) −4.65249 −0.269965
\(298\) −18.2081 −1.05477
\(299\) 2.53899 0.146834
\(300\) −16.4950 −0.952339
\(301\) 13.0849 0.754203
\(302\) −16.5547 −0.952618
\(303\) −8.39236 −0.482129
\(304\) 3.64753 0.209200
\(305\) −2.80718 −0.160738
\(306\) 5.05731 0.289107
\(307\) −6.97570 −0.398124 −0.199062 0.979987i \(-0.563790\pi\)
−0.199062 + 0.979987i \(0.563790\pi\)
\(308\) −61.1415 −3.48386
\(309\) 3.24018 0.184327
\(310\) −4.58376 −0.260340
\(311\) −2.58733 −0.146714 −0.0733571 0.997306i \(-0.523371\pi\)
−0.0733571 + 0.997306i \(0.523371\pi\)
\(312\) 2.67138 0.151237
\(313\) −0.196515 −0.0111077 −0.00555384 0.999985i \(-0.501768\pi\)
−0.00555384 + 0.999985i \(0.501768\pi\)
\(314\) 1.15281 0.0650568
\(315\) −0.882500 −0.0497232
\(316\) −31.7043 −1.78351
\(317\) −22.0936 −1.24090 −0.620450 0.784246i \(-0.713050\pi\)
−0.620450 + 0.784246i \(0.713050\pi\)
\(318\) 1.50445 0.0843655
\(319\) 9.10537 0.509803
\(320\) 2.85167 0.159413
\(321\) 19.7559 1.10267
\(322\) 26.6298 1.48402
\(323\) −18.1571 −1.01029
\(324\) 3.33237 0.185132
\(325\) 4.29783 0.238401
\(326\) −12.4122 −0.687450
\(327\) 16.2859 0.900610
\(328\) 14.7851 0.816369
\(329\) −2.15835 −0.118994
\(330\) −2.40415 −0.132344
\(331\) −0.551627 −0.0303201 −0.0151601 0.999885i \(-0.504826\pi\)
−0.0151601 + 0.999885i \(0.504826\pi\)
\(332\) 19.1417 1.05054
\(333\) 2.47861 0.135827
\(334\) −27.1828 −1.48738
\(335\) 0.908232 0.0496220
\(336\) 1.73504 0.0946544
\(337\) 30.6111 1.66749 0.833747 0.552146i \(-0.186191\pi\)
0.833747 + 0.552146i \(0.186191\pi\)
\(338\) 28.2787 1.53816
\(339\) −6.70705 −0.364277
\(340\) 1.63316 0.0885708
\(341\) 41.2696 2.23487
\(342\) −19.1446 −1.03522
\(343\) 6.12186 0.330549
\(344\) −10.2084 −0.550402
\(345\) 0.654376 0.0352304
\(346\) −57.5576 −3.09431
\(347\) −0.682669 −0.0366476 −0.0183238 0.999832i \(-0.505833\pi\)
−0.0183238 + 0.999832i \(0.505833\pi\)
\(348\) −6.52178 −0.349604
\(349\) −9.13117 −0.488780 −0.244390 0.969677i \(-0.578588\pi\)
−0.244390 + 0.969677i \(0.578588\pi\)
\(350\) 45.0772 2.40948
\(351\) −0.868261 −0.0463444
\(352\) −23.9019 −1.27398
\(353\) −29.5873 −1.57477 −0.787387 0.616459i \(-0.788566\pi\)
−0.787387 + 0.616459i \(0.788566\pi\)
\(354\) −22.1622 −1.17791
\(355\) 0.951970 0.0505253
\(356\) 7.22728 0.383045
\(357\) −8.63689 −0.457113
\(358\) 15.8059 0.835368
\(359\) 15.2420 0.804443 0.402221 0.915542i \(-0.368238\pi\)
0.402221 + 0.915542i \(0.368238\pi\)
\(360\) 0.688498 0.0362870
\(361\) 49.7341 2.61758
\(362\) −49.1486 −2.58319
\(363\) 10.6456 0.558750
\(364\) −11.4104 −0.598069
\(365\) −0.669296 −0.0350325
\(366\) −28.9676 −1.51416
\(367\) −4.86360 −0.253878 −0.126939 0.991911i \(-0.540515\pi\)
−0.126939 + 0.991911i \(0.540515\pi\)
\(368\) −1.28654 −0.0670655
\(369\) −4.80549 −0.250164
\(370\) 1.28081 0.0665862
\(371\) −2.56931 −0.133392
\(372\) −29.5596 −1.53259
\(373\) 9.87652 0.511387 0.255693 0.966758i \(-0.417696\pi\)
0.255693 + 0.966758i \(0.417696\pi\)
\(374\) −23.5291 −1.21666
\(375\) 2.22657 0.114980
\(376\) 1.68388 0.0868392
\(377\) 1.69927 0.0875170
\(378\) −9.10664 −0.468395
\(379\) −37.6086 −1.93182 −0.965912 0.258869i \(-0.916650\pi\)
−0.965912 + 0.258869i \(0.916650\pi\)
\(380\) −6.18239 −0.317150
\(381\) 12.7044 0.650866
\(382\) −52.4116 −2.68161
\(383\) −3.79669 −0.194002 −0.0970008 0.995284i \(-0.530925\pi\)
−0.0970008 + 0.995284i \(0.530925\pi\)
\(384\) 19.1518 0.977337
\(385\) 4.10582 0.209252
\(386\) 4.29726 0.218725
\(387\) 3.31798 0.168662
\(388\) 8.36247 0.424540
\(389\) −32.3551 −1.64047 −0.820234 0.572027i \(-0.806157\pi\)
−0.820234 + 0.572027i \(0.806157\pi\)
\(390\) −0.448671 −0.0227193
\(391\) 6.40428 0.323878
\(392\) −26.3130 −1.32901
\(393\) 13.7546 0.693830
\(394\) −20.3710 −1.02628
\(395\) 2.12903 0.107123
\(396\) −15.5038 −0.779096
\(397\) 15.9887 0.802448 0.401224 0.915980i \(-0.368585\pi\)
0.401224 + 0.915980i \(0.368585\pi\)
\(398\) 13.4901 0.676199
\(399\) 32.6952 1.63681
\(400\) −2.17777 −0.108888
\(401\) −20.3510 −1.01628 −0.508140 0.861275i \(-0.669667\pi\)
−0.508140 + 0.861275i \(0.669667\pi\)
\(402\) 9.37217 0.467441
\(403\) 7.70186 0.383657
\(404\) −27.9665 −1.39138
\(405\) −0.223778 −0.0111196
\(406\) 17.8226 0.884520
\(407\) −11.5317 −0.571605
\(408\) 6.73822 0.333592
\(409\) 25.0573 1.23900 0.619502 0.784995i \(-0.287334\pi\)
0.619502 + 0.784995i \(0.287334\pi\)
\(410\) −2.48322 −0.122637
\(411\) 8.65141 0.426742
\(412\) 10.7975 0.531954
\(413\) 37.8486 1.86241
\(414\) 6.75259 0.331872
\(415\) −1.28542 −0.0630986
\(416\) −4.46066 −0.218702
\(417\) 6.89645 0.337720
\(418\) 89.0700 4.35655
\(419\) −17.7321 −0.866272 −0.433136 0.901329i \(-0.642593\pi\)
−0.433136 + 0.901329i \(0.642593\pi\)
\(420\) −2.94082 −0.143497
\(421\) −21.8440 −1.06461 −0.532305 0.846552i \(-0.678674\pi\)
−0.532305 + 0.846552i \(0.678674\pi\)
\(422\) −34.9685 −1.70224
\(423\) −0.547298 −0.0266106
\(424\) 2.00449 0.0973467
\(425\) 10.8407 0.525852
\(426\) 9.82351 0.475951
\(427\) 49.4710 2.39407
\(428\) 65.8341 3.18221
\(429\) 4.03957 0.195032
\(430\) 1.71455 0.0826831
\(431\) 6.86119 0.330492 0.165246 0.986252i \(-0.447158\pi\)
0.165246 + 0.986252i \(0.447158\pi\)
\(432\) 0.439960 0.0211676
\(433\) 28.7591 1.38207 0.691037 0.722819i \(-0.257154\pi\)
0.691037 + 0.722819i \(0.257154\pi\)
\(434\) 80.7799 3.87756
\(435\) 0.437955 0.0209983
\(436\) 54.2705 2.59909
\(437\) −24.2436 −1.15973
\(438\) −6.90656 −0.330008
\(439\) −10.2911 −0.491165 −0.245583 0.969376i \(-0.578979\pi\)
−0.245583 + 0.969376i \(0.578979\pi\)
\(440\) −3.20323 −0.152708
\(441\) 8.55234 0.407254
\(442\) −4.39107 −0.208862
\(443\) 6.34405 0.301415 0.150707 0.988578i \(-0.451845\pi\)
0.150707 + 0.988578i \(0.451845\pi\)
\(444\) 8.25965 0.391986
\(445\) −0.485332 −0.0230069
\(446\) −1.28480 −0.0608373
\(447\) 7.88506 0.372951
\(448\) −50.2550 −2.37433
\(449\) −34.0688 −1.60780 −0.803902 0.594762i \(-0.797246\pi\)
−0.803902 + 0.594762i \(0.797246\pi\)
\(450\) 11.4303 0.538831
\(451\) 22.3575 1.05277
\(452\) −22.3504 −1.05127
\(453\) 7.16906 0.336832
\(454\) 42.6933 2.00370
\(455\) 0.766241 0.0359219
\(456\) −25.5077 −1.19451
\(457\) 28.8395 1.34905 0.674527 0.738250i \(-0.264347\pi\)
0.674527 + 0.738250i \(0.264347\pi\)
\(458\) 20.1218 0.940232
\(459\) −2.19008 −0.102224
\(460\) 2.18062 0.101672
\(461\) 33.0622 1.53986 0.769931 0.638127i \(-0.220291\pi\)
0.769931 + 0.638127i \(0.220291\pi\)
\(462\) 42.3685 1.97116
\(463\) −25.8383 −1.20081 −0.600403 0.799698i \(-0.704993\pi\)
−0.600403 + 0.799698i \(0.704993\pi\)
\(464\) −0.861044 −0.0399730
\(465\) 1.98501 0.0920525
\(466\) −50.8743 −2.35671
\(467\) 16.7010 0.772831 0.386415 0.922325i \(-0.373713\pi\)
0.386415 + 0.922325i \(0.373713\pi\)
\(468\) −2.89337 −0.133746
\(469\) −16.0058 −0.739079
\(470\) −0.282814 −0.0130452
\(471\) −0.499226 −0.0230031
\(472\) −29.5283 −1.35915
\(473\) −15.4369 −0.709787
\(474\) 21.9698 1.00910
\(475\) −41.0378 −1.88295
\(476\) −28.7813 −1.31919
\(477\) −0.651506 −0.0298304
\(478\) −7.07866 −0.323770
\(479\) 32.7043 1.49430 0.747149 0.664657i \(-0.231422\pi\)
0.747149 + 0.664657i \(0.231422\pi\)
\(480\) −1.14965 −0.0524740
\(481\) −2.15208 −0.0981264
\(482\) 40.3191 1.83649
\(483\) −11.5321 −0.524729
\(484\) 35.4752 1.61251
\(485\) −0.561562 −0.0254992
\(486\) −2.30919 −0.104747
\(487\) 15.1562 0.686794 0.343397 0.939190i \(-0.388422\pi\)
0.343397 + 0.939190i \(0.388422\pi\)
\(488\) −38.5957 −1.74714
\(489\) 5.37514 0.243072
\(490\) 4.41938 0.199647
\(491\) −14.3531 −0.647745 −0.323872 0.946101i \(-0.604985\pi\)
−0.323872 + 0.946101i \(0.604985\pi\)
\(492\) −16.0137 −0.721953
\(493\) 4.28620 0.193041
\(494\) 16.6225 0.747882
\(495\) 1.04112 0.0467950
\(496\) −3.90264 −0.175234
\(497\) −16.7766 −0.752534
\(498\) −13.2644 −0.594391
\(499\) −28.7162 −1.28551 −0.642756 0.766071i \(-0.722209\pi\)
−0.642756 + 0.766071i \(0.722209\pi\)
\(500\) 7.41976 0.331822
\(501\) 11.7716 0.525914
\(502\) −17.7432 −0.791919
\(503\) −37.3063 −1.66341 −0.831703 0.555221i \(-0.812634\pi\)
−0.831703 + 0.555221i \(0.812634\pi\)
\(504\) −12.1334 −0.540466
\(505\) 1.87802 0.0835709
\(506\) −31.4163 −1.39663
\(507\) −12.2461 −0.543869
\(508\) 42.3357 1.87835
\(509\) −2.32115 −0.102883 −0.0514417 0.998676i \(-0.516382\pi\)
−0.0514417 + 0.998676i \(0.516382\pi\)
\(510\) −1.13171 −0.0501131
\(511\) 11.7950 0.521781
\(512\) 4.96752 0.219536
\(513\) 8.29060 0.366039
\(514\) −48.6372 −2.14530
\(515\) −0.725080 −0.0319508
\(516\) 11.0567 0.486746
\(517\) 2.54630 0.111986
\(518\) −22.5718 −0.991748
\(519\) 24.9254 1.09410
\(520\) −0.597796 −0.0262151
\(521\) −21.3185 −0.933980 −0.466990 0.884262i \(-0.654662\pi\)
−0.466990 + 0.884262i \(0.654662\pi\)
\(522\) 4.51932 0.197805
\(523\) −33.0074 −1.44331 −0.721656 0.692251i \(-0.756619\pi\)
−0.721656 + 0.692251i \(0.756619\pi\)
\(524\) 45.8356 2.00234
\(525\) −19.5207 −0.851955
\(526\) 33.7972 1.47363
\(527\) 19.4270 0.846252
\(528\) −2.04691 −0.0890801
\(529\) −14.4489 −0.628214
\(530\) −0.336663 −0.0146237
\(531\) 9.59738 0.416491
\(532\) 108.953 4.72369
\(533\) 4.17242 0.180728
\(534\) −5.00820 −0.216726
\(535\) −4.42093 −0.191134
\(536\) 12.4872 0.539365
\(537\) −6.84478 −0.295374
\(538\) −21.3918 −0.922265
\(539\) −39.7896 −1.71386
\(540\) −0.745711 −0.0320903
\(541\) −40.7528 −1.75210 −0.876050 0.482219i \(-0.839831\pi\)
−0.876050 + 0.482219i \(0.839831\pi\)
\(542\) −29.5972 −1.27131
\(543\) 21.2839 0.913379
\(544\) −11.2514 −0.482401
\(545\) −3.64441 −0.156109
\(546\) 7.90694 0.338386
\(547\) −4.75585 −0.203345 −0.101673 0.994818i \(-0.532419\pi\)
−0.101673 + 0.994818i \(0.532419\pi\)
\(548\) 28.8297 1.23154
\(549\) 12.5445 0.535386
\(550\) −53.1794 −2.26758
\(551\) −16.2255 −0.691230
\(552\) 8.99697 0.382936
\(553\) −37.5200 −1.59551
\(554\) −43.0344 −1.82836
\(555\) −0.554657 −0.0235439
\(556\) 22.9815 0.974634
\(557\) −2.97886 −0.126219 −0.0631093 0.998007i \(-0.520102\pi\)
−0.0631093 + 0.998007i \(0.520102\pi\)
\(558\) 20.4836 0.867138
\(559\) −2.88087 −0.121848
\(560\) −0.388264 −0.0164072
\(561\) 10.1893 0.430193
\(562\) −14.5870 −0.615314
\(563\) −15.1944 −0.640367 −0.320183 0.947356i \(-0.603745\pi\)
−0.320183 + 0.947356i \(0.603745\pi\)
\(564\) −1.82380 −0.0767959
\(565\) 1.50089 0.0631428
\(566\) 55.0047 2.31202
\(567\) 3.94364 0.165618
\(568\) 13.0886 0.549184
\(569\) 5.16884 0.216689 0.108345 0.994113i \(-0.465445\pi\)
0.108345 + 0.994113i \(0.465445\pi\)
\(570\) 4.28413 0.179443
\(571\) −6.79866 −0.284515 −0.142257 0.989830i \(-0.545436\pi\)
−0.142257 + 0.989830i \(0.545436\pi\)
\(572\) 13.4614 0.562848
\(573\) 22.6969 0.948178
\(574\) 43.7619 1.82659
\(575\) 14.4747 0.603636
\(576\) −12.7433 −0.530971
\(577\) 8.60525 0.358241 0.179121 0.983827i \(-0.442675\pi\)
0.179121 + 0.983827i \(0.442675\pi\)
\(578\) 28.1804 1.17215
\(579\) −1.86094 −0.0773379
\(580\) 1.45943 0.0605995
\(581\) 22.6529 0.939802
\(582\) −5.79484 −0.240204
\(583\) 3.03112 0.125536
\(584\) −9.20210 −0.380786
\(585\) 0.194298 0.00803321
\(586\) −26.8973 −1.11112
\(587\) 31.2044 1.28794 0.643972 0.765049i \(-0.277285\pi\)
0.643972 + 0.765049i \(0.277285\pi\)
\(588\) 28.4996 1.17530
\(589\) −73.5413 −3.03022
\(590\) 4.95940 0.204175
\(591\) 8.82171 0.362877
\(592\) 1.09049 0.0448188
\(593\) −9.34742 −0.383852 −0.191926 0.981409i \(-0.561473\pi\)
−0.191926 + 0.981409i \(0.561473\pi\)
\(594\) 10.7435 0.440811
\(595\) 1.93274 0.0792348
\(596\) 26.2760 1.07631
\(597\) −5.84192 −0.239094
\(598\) −5.86302 −0.239757
\(599\) 2.83717 0.115924 0.0579619 0.998319i \(-0.481540\pi\)
0.0579619 + 0.998319i \(0.481540\pi\)
\(600\) 15.2295 0.621740
\(601\) 42.2604 1.72384 0.861919 0.507046i \(-0.169262\pi\)
0.861919 + 0.507046i \(0.169262\pi\)
\(602\) −30.2156 −1.23150
\(603\) −4.05863 −0.165280
\(604\) 23.8900 0.972069
\(605\) −2.38225 −0.0968523
\(606\) 19.3796 0.787242
\(607\) −11.6474 −0.472753 −0.236377 0.971662i \(-0.575960\pi\)
−0.236377 + 0.971662i \(0.575960\pi\)
\(608\) 42.5926 1.72736
\(609\) −7.71810 −0.312753
\(610\) 6.48231 0.262461
\(611\) 0.475198 0.0192245
\(612\) −7.29815 −0.295010
\(613\) −1.57469 −0.0636011 −0.0318005 0.999494i \(-0.510124\pi\)
−0.0318005 + 0.999494i \(0.510124\pi\)
\(614\) 16.1082 0.650076
\(615\) 1.07536 0.0433628
\(616\) 56.4506 2.27446
\(617\) −9.63509 −0.387894 −0.193947 0.981012i \(-0.562129\pi\)
−0.193947 + 0.981012i \(0.562129\pi\)
\(618\) −7.48220 −0.300978
\(619\) 13.0019 0.522589 0.261294 0.965259i \(-0.415851\pi\)
0.261294 + 0.965259i \(0.415851\pi\)
\(620\) 6.61478 0.265656
\(621\) −2.92422 −0.117345
\(622\) 5.97465 0.239562
\(623\) 8.55302 0.342670
\(624\) −0.382000 −0.0152922
\(625\) 24.2514 0.970054
\(626\) 0.453791 0.0181371
\(627\) −38.5719 −1.54041
\(628\) −1.66361 −0.0663851
\(629\) −5.42835 −0.216442
\(630\) 2.03786 0.0811904
\(631\) 16.0331 0.638266 0.319133 0.947710i \(-0.396608\pi\)
0.319133 + 0.947710i \(0.396608\pi\)
\(632\) 29.2719 1.16437
\(633\) 15.1432 0.601887
\(634\) 51.0184 2.02620
\(635\) −2.84296 −0.112819
\(636\) −2.17106 −0.0860882
\(637\) −7.42566 −0.294215
\(638\) −21.0261 −0.832430
\(639\) −4.25409 −0.168289
\(640\) −4.28575 −0.169409
\(641\) −20.9742 −0.828429 −0.414215 0.910179i \(-0.635944\pi\)
−0.414215 + 0.910179i \(0.635944\pi\)
\(642\) −45.6202 −1.80049
\(643\) 22.3851 0.882782 0.441391 0.897315i \(-0.354485\pi\)
0.441391 + 0.897315i \(0.354485\pi\)
\(644\) −38.4292 −1.51432
\(645\) −0.742490 −0.0292355
\(646\) 41.9282 1.64964
\(647\) 36.7408 1.44443 0.722216 0.691668i \(-0.243124\pi\)
0.722216 + 0.691668i \(0.243124\pi\)
\(648\) −3.07670 −0.120864
\(649\) −44.6516 −1.75273
\(650\) −9.92451 −0.389271
\(651\) −34.9819 −1.37105
\(652\) 17.9120 0.701486
\(653\) −18.7159 −0.732409 −0.366205 0.930534i \(-0.619343\pi\)
−0.366205 + 0.930534i \(0.619343\pi\)
\(654\) −37.6072 −1.47056
\(655\) −3.07798 −0.120267
\(656\) −2.11422 −0.0825465
\(657\) 2.99090 0.116686
\(658\) 4.98405 0.194298
\(659\) −8.26000 −0.321764 −0.160882 0.986974i \(-0.551434\pi\)
−0.160882 + 0.986974i \(0.551434\pi\)
\(660\) 3.46941 0.135047
\(661\) −7.46824 −0.290481 −0.145240 0.989396i \(-0.546396\pi\)
−0.145240 + 0.989396i \(0.546396\pi\)
\(662\) 1.27381 0.0495081
\(663\) 1.90156 0.0738505
\(664\) −17.6731 −0.685849
\(665\) −7.31645 −0.283720
\(666\) −5.72359 −0.221784
\(667\) 5.72299 0.221595
\(668\) 39.2272 1.51775
\(669\) 0.556387 0.0215112
\(670\) −2.09728 −0.0810251
\(671\) −58.3630 −2.25308
\(672\) 20.2603 0.781558
\(673\) −10.6084 −0.408924 −0.204462 0.978874i \(-0.565545\pi\)
−0.204462 + 0.978874i \(0.565545\pi\)
\(674\) −70.6870 −2.72276
\(675\) −4.94992 −0.190523
\(676\) −40.8086 −1.56956
\(677\) 30.8256 1.18472 0.592362 0.805672i \(-0.298195\pi\)
0.592362 + 0.805672i \(0.298195\pi\)
\(678\) 15.4879 0.594808
\(679\) 9.89644 0.379790
\(680\) −1.50786 −0.0578239
\(681\) −18.4884 −0.708478
\(682\) −95.2995 −3.64921
\(683\) 16.6597 0.637465 0.318732 0.947845i \(-0.396743\pi\)
0.318732 + 0.947845i \(0.396743\pi\)
\(684\) 27.6274 1.05636
\(685\) −1.93599 −0.0739704
\(686\) −14.1366 −0.539736
\(687\) −8.71380 −0.332452
\(688\) 1.45978 0.0556535
\(689\) 0.565678 0.0215506
\(690\) −1.51108 −0.0575258
\(691\) −27.2864 −1.03802 −0.519011 0.854767i \(-0.673700\pi\)
−0.519011 + 0.854767i \(0.673700\pi\)
\(692\) 83.0607 3.15750
\(693\) −18.3478 −0.696973
\(694\) 1.57641 0.0598398
\(695\) −1.54327 −0.0585396
\(696\) 6.02141 0.228241
\(697\) 10.5244 0.398640
\(698\) 21.0856 0.798103
\(699\) 22.0312 0.833297
\(700\) −65.0504 −2.45867
\(701\) 50.5627 1.90973 0.954863 0.297046i \(-0.0960014\pi\)
0.954863 + 0.297046i \(0.0960014\pi\)
\(702\) 2.00498 0.0756732
\(703\) 20.5492 0.775026
\(704\) 59.2880 2.23450
\(705\) 0.122473 0.00461261
\(706\) 68.3228 2.57136
\(707\) −33.0965 −1.24472
\(708\) 31.9820 1.20196
\(709\) 2.32108 0.0871699 0.0435850 0.999050i \(-0.486122\pi\)
0.0435850 + 0.999050i \(0.486122\pi\)
\(710\) −2.19828 −0.0825001
\(711\) −9.51405 −0.356805
\(712\) −6.67279 −0.250073
\(713\) 25.9391 0.971429
\(714\) 19.9442 0.746395
\(715\) −0.903966 −0.0338064
\(716\) −22.8093 −0.852425
\(717\) 3.06543 0.114480
\(718\) −35.1968 −1.31353
\(719\) −49.1036 −1.83126 −0.915628 0.402026i \(-0.868306\pi\)
−0.915628 + 0.402026i \(0.868306\pi\)
\(720\) −0.0984531 −0.00366913
\(721\) 12.7781 0.475882
\(722\) −114.846 −4.27411
\(723\) −17.4603 −0.649354
\(724\) 70.9258 2.63594
\(725\) 9.68749 0.359784
\(726\) −24.5828 −0.912353
\(727\) 28.4757 1.05610 0.528052 0.849212i \(-0.322922\pi\)
0.528052 + 0.849212i \(0.322922\pi\)
\(728\) 10.5350 0.390453
\(729\) 1.00000 0.0370370
\(730\) 1.54553 0.0572027
\(731\) −7.26664 −0.268766
\(732\) 41.8029 1.54508
\(733\) −40.6325 −1.50080 −0.750399 0.660985i \(-0.770138\pi\)
−0.750399 + 0.660985i \(0.770138\pi\)
\(734\) 11.2310 0.414543
\(735\) −1.91382 −0.0705924
\(736\) −15.0231 −0.553758
\(737\) 18.8827 0.695554
\(738\) 11.0968 0.408479
\(739\) 39.9846 1.47086 0.735428 0.677603i \(-0.236981\pi\)
0.735428 + 0.677603i \(0.236981\pi\)
\(740\) −1.84833 −0.0679458
\(741\) −7.19841 −0.264440
\(742\) 5.93303 0.217808
\(743\) −6.98238 −0.256159 −0.128079 0.991764i \(-0.540881\pi\)
−0.128079 + 0.991764i \(0.540881\pi\)
\(744\) 27.2917 1.00056
\(745\) −1.76450 −0.0646463
\(746\) −22.8068 −0.835016
\(747\) 5.74416 0.210168
\(748\) 33.9546 1.24150
\(749\) 77.9103 2.84678
\(750\) −5.14158 −0.187744
\(751\) 18.4358 0.672732 0.336366 0.941731i \(-0.390802\pi\)
0.336366 + 0.941731i \(0.390802\pi\)
\(752\) −0.240789 −0.00878068
\(753\) 7.68374 0.280011
\(754\) −3.92395 −0.142902
\(755\) −1.60428 −0.0583856
\(756\) 13.1417 0.477959
\(757\) −33.6474 −1.22293 −0.611467 0.791270i \(-0.709420\pi\)
−0.611467 + 0.791270i \(0.709420\pi\)
\(758\) 86.8455 3.15437
\(759\) 13.6049 0.493827
\(760\) 5.70806 0.207053
\(761\) −3.71822 −0.134785 −0.0673927 0.997727i \(-0.521468\pi\)
−0.0673927 + 0.997727i \(0.521468\pi\)
\(762\) −29.3369 −1.06276
\(763\) 64.2257 2.32512
\(764\) 75.6347 2.73637
\(765\) 0.490091 0.0177193
\(766\) 8.76728 0.316775
\(767\) −8.33303 −0.300888
\(768\) −18.7387 −0.676173
\(769\) 14.1764 0.511214 0.255607 0.966781i \(-0.417725\pi\)
0.255607 + 0.966781i \(0.417725\pi\)
\(770\) −9.48112 −0.341676
\(771\) 21.0624 0.758545
\(772\) −6.20133 −0.223191
\(773\) 43.3850 1.56045 0.780225 0.625499i \(-0.215104\pi\)
0.780225 + 0.625499i \(0.215104\pi\)
\(774\) −7.66186 −0.275400
\(775\) 43.9080 1.57722
\(776\) −7.72088 −0.277163
\(777\) 9.77475 0.350667
\(778\) 74.7142 2.67863
\(779\) −39.8404 −1.42743
\(780\) 0.647472 0.0231832
\(781\) 19.7921 0.708217
\(782\) −14.7887 −0.528843
\(783\) −1.95710 −0.0699410
\(784\) 3.76268 0.134381
\(785\) 0.111716 0.00398730
\(786\) −31.7621 −1.13292
\(787\) 29.6983 1.05863 0.529315 0.848425i \(-0.322449\pi\)
0.529315 + 0.848425i \(0.322449\pi\)
\(788\) 29.3972 1.04723
\(789\) −14.6359 −0.521053
\(790\) −4.91634 −0.174916
\(791\) −26.4502 −0.940461
\(792\) 14.3143 0.508637
\(793\) −10.8919 −0.386782
\(794\) −36.9209 −1.31027
\(795\) 0.145793 0.00517073
\(796\) −19.4675 −0.690006
\(797\) −46.5653 −1.64943 −0.824713 0.565551i \(-0.808663\pi\)
−0.824713 + 0.565551i \(0.808663\pi\)
\(798\) −75.4995 −2.67265
\(799\) 1.19863 0.0424043
\(800\) −25.4300 −0.899087
\(801\) 2.16881 0.0766312
\(802\) 46.9943 1.65943
\(803\) −13.9151 −0.491053
\(804\) −13.5249 −0.476986
\(805\) 2.58063 0.0909551
\(806\) −17.7851 −0.626453
\(807\) 9.26375 0.326099
\(808\) 25.8208 0.908373
\(809\) 34.9838 1.22997 0.614983 0.788540i \(-0.289163\pi\)
0.614983 + 0.788540i \(0.289163\pi\)
\(810\) 0.516746 0.0181566
\(811\) 25.1624 0.883573 0.441787 0.897120i \(-0.354345\pi\)
0.441787 + 0.897120i \(0.354345\pi\)
\(812\) −25.7196 −0.902580
\(813\) 12.8171 0.449516
\(814\) 26.6289 0.933343
\(815\) −1.20284 −0.0421335
\(816\) −0.963546 −0.0337308
\(817\) 27.5080 0.962385
\(818\) −57.8622 −2.02310
\(819\) −3.42411 −0.119648
\(820\) 3.58351 0.125141
\(821\) 49.2886 1.72018 0.860092 0.510139i \(-0.170406\pi\)
0.860092 + 0.510139i \(0.170406\pi\)
\(822\) −19.9778 −0.696805
\(823\) −10.3308 −0.360109 −0.180055 0.983657i \(-0.557627\pi\)
−0.180055 + 0.983657i \(0.557627\pi\)
\(824\) −9.96907 −0.347289
\(825\) 23.0294 0.801782
\(826\) −87.3998 −3.04103
\(827\) −7.70542 −0.267944 −0.133972 0.990985i \(-0.542773\pi\)
−0.133972 + 0.990985i \(0.542773\pi\)
\(828\) −9.74460 −0.338648
\(829\) 46.4413 1.61297 0.806486 0.591253i \(-0.201366\pi\)
0.806486 + 0.591253i \(0.201366\pi\)
\(830\) 2.96827 0.103030
\(831\) 18.6361 0.646480
\(832\) 11.0645 0.383593
\(833\) −18.7303 −0.648966
\(834\) −15.9252 −0.551445
\(835\) −2.63421 −0.0911606
\(836\) −128.536 −4.44551
\(837\) −8.87044 −0.306607
\(838\) 40.9469 1.41449
\(839\) −17.0035 −0.587026 −0.293513 0.955955i \(-0.594824\pi\)
−0.293513 + 0.955955i \(0.594824\pi\)
\(840\) 2.71519 0.0936830
\(841\) −25.1698 −0.867923
\(842\) 50.4420 1.73835
\(843\) 6.31691 0.217566
\(844\) 50.4627 1.73700
\(845\) 2.74041 0.0942729
\(846\) 1.26382 0.0434509
\(847\) 41.9825 1.44254
\(848\) −0.286636 −0.00984313
\(849\) −23.8199 −0.817497
\(850\) −25.0333 −0.858635
\(851\) −7.24801 −0.248458
\(852\) −14.1762 −0.485669
\(853\) 12.2993 0.421121 0.210561 0.977581i \(-0.432471\pi\)
0.210561 + 0.977581i \(0.432471\pi\)
\(854\) −114.238 −3.90915
\(855\) −1.85525 −0.0634483
\(856\) −60.7831 −2.07752
\(857\) −11.7667 −0.401941 −0.200971 0.979597i \(-0.564410\pi\)
−0.200971 + 0.979597i \(0.564410\pi\)
\(858\) −9.32815 −0.318458
\(859\) 42.4042 1.44681 0.723406 0.690423i \(-0.242576\pi\)
0.723406 + 0.690423i \(0.242576\pi\)
\(860\) −2.47425 −0.0843713
\(861\) −18.9512 −0.645854
\(862\) −15.8438 −0.539642
\(863\) −43.8008 −1.49100 −0.745499 0.666507i \(-0.767789\pi\)
−0.745499 + 0.666507i \(0.767789\pi\)
\(864\) 5.13746 0.174780
\(865\) −5.57775 −0.189649
\(866\) −66.4103 −2.25672
\(867\) −12.2036 −0.414455
\(868\) −116.573 −3.95673
\(869\) 44.2640 1.50155
\(870\) −1.01132 −0.0342870
\(871\) 3.52395 0.119405
\(872\) −50.1068 −1.69683
\(873\) 2.50946 0.0849325
\(874\) 55.9831 1.89366
\(875\) 8.78081 0.296845
\(876\) 9.96678 0.336746
\(877\) 7.34368 0.247979 0.123989 0.992284i \(-0.460431\pi\)
0.123989 + 0.992284i \(0.460431\pi\)
\(878\) 23.7640 0.801997
\(879\) 11.6479 0.392875
\(880\) 0.458052 0.0154409
\(881\) 11.3578 0.382654 0.191327 0.981526i \(-0.438721\pi\)
0.191327 + 0.981526i \(0.438721\pi\)
\(882\) −19.7490 −0.664983
\(883\) 38.4711 1.29466 0.647328 0.762211i \(-0.275886\pi\)
0.647328 + 0.762211i \(0.275886\pi\)
\(884\) 6.33671 0.213126
\(885\) −2.14768 −0.0721934
\(886\) −14.6496 −0.492164
\(887\) 6.88399 0.231142 0.115571 0.993299i \(-0.463130\pi\)
0.115571 + 0.993299i \(0.463130\pi\)
\(888\) −7.62595 −0.255910
\(889\) 50.1016 1.68035
\(890\) 1.12072 0.0375668
\(891\) −4.65249 −0.155864
\(892\) 1.85409 0.0620795
\(893\) −4.53743 −0.151839
\(894\) −18.2081 −0.608971
\(895\) 1.53171 0.0511994
\(896\) 75.5280 2.52321
\(897\) 2.53899 0.0847744
\(898\) 78.6713 2.62530
\(899\) 17.3603 0.578999
\(900\) −16.4950 −0.549833
\(901\) 1.42685 0.0475352
\(902\) −51.6277 −1.71902
\(903\) 13.0849 0.435439
\(904\) 20.6356 0.686330
\(905\) −4.76286 −0.158323
\(906\) −16.5547 −0.549995
\(907\) 28.8725 0.958696 0.479348 0.877625i \(-0.340873\pi\)
0.479348 + 0.877625i \(0.340873\pi\)
\(908\) −61.6103 −2.04461
\(909\) −8.39236 −0.278357
\(910\) −1.76940 −0.0586549
\(911\) 29.6630 0.982778 0.491389 0.870940i \(-0.336489\pi\)
0.491389 + 0.870940i \(0.336489\pi\)
\(912\) 3.64753 0.120782
\(913\) −26.7246 −0.884457
\(914\) −66.5960 −2.20280
\(915\) −2.80718 −0.0928024
\(916\) −29.0376 −0.959430
\(917\) 54.2434 1.79128
\(918\) 5.05731 0.166916
\(919\) 4.45092 0.146822 0.0734112 0.997302i \(-0.476611\pi\)
0.0734112 + 0.997302i \(0.476611\pi\)
\(920\) −2.01332 −0.0663772
\(921\) −6.97570 −0.229857
\(922\) −76.3471 −2.51436
\(923\) 3.69366 0.121578
\(924\) −61.1415 −2.01141
\(925\) −12.2689 −0.403400
\(926\) 59.6655 1.96073
\(927\) 3.24018 0.106421
\(928\) −10.0545 −0.330055
\(929\) −8.06216 −0.264511 −0.132255 0.991216i \(-0.542222\pi\)
−0.132255 + 0.991216i \(0.542222\pi\)
\(930\) −4.58376 −0.150308
\(931\) 70.9040 2.32378
\(932\) 73.4162 2.40483
\(933\) −2.58733 −0.0847055
\(934\) −38.5659 −1.26191
\(935\) −2.28014 −0.0745685
\(936\) 2.67138 0.0873169
\(937\) −23.0363 −0.752562 −0.376281 0.926506i \(-0.622797\pi\)
−0.376281 + 0.926506i \(0.622797\pi\)
\(938\) 36.9605 1.20680
\(939\) −0.196515 −0.00641302
\(940\) 0.408126 0.0133116
\(941\) 45.6609 1.48850 0.744252 0.667899i \(-0.232806\pi\)
0.744252 + 0.667899i \(0.232806\pi\)
\(942\) 1.15281 0.0375606
\(943\) 14.0523 0.457607
\(944\) 4.22246 0.137429
\(945\) −0.882500 −0.0287077
\(946\) 35.6467 1.15897
\(947\) −33.1849 −1.07836 −0.539182 0.842189i \(-0.681266\pi\)
−0.539182 + 0.842189i \(0.681266\pi\)
\(948\) −31.7043 −1.02971
\(949\) −2.59688 −0.0842983
\(950\) 94.7643 3.07456
\(951\) −22.0936 −0.716434
\(952\) 26.5732 0.861241
\(953\) −4.18617 −0.135603 −0.0678016 0.997699i \(-0.521598\pi\)
−0.0678016 + 0.997699i \(0.521598\pi\)
\(954\) 1.50445 0.0487085
\(955\) −5.07907 −0.164355
\(956\) 10.2151 0.330381
\(957\) 9.10537 0.294335
\(958\) −75.5205 −2.43996
\(959\) 34.1181 1.10173
\(960\) 2.85167 0.0920371
\(961\) 47.6847 1.53822
\(962\) 4.96957 0.160225
\(963\) 19.7559 0.636626
\(964\) −58.1841 −1.87398
\(965\) 0.416436 0.0134055
\(966\) 26.6298 0.856801
\(967\) 30.8911 0.993391 0.496695 0.867925i \(-0.334547\pi\)
0.496695 + 0.867925i \(0.334547\pi\)
\(968\) −32.7534 −1.05273
\(969\) −18.1571 −0.583289
\(970\) 1.29676 0.0416363
\(971\) −2.64645 −0.0849285 −0.0424643 0.999098i \(-0.513521\pi\)
−0.0424643 + 0.999098i \(0.513521\pi\)
\(972\) 3.33237 0.106886
\(973\) 27.1971 0.871900
\(974\) −34.9987 −1.12143
\(975\) 4.29783 0.137641
\(976\) 5.51907 0.176661
\(977\) 36.9313 1.18154 0.590769 0.806841i \(-0.298824\pi\)
0.590769 + 0.806841i \(0.298824\pi\)
\(978\) −12.4122 −0.396899
\(979\) −10.0904 −0.322489
\(980\) −6.37757 −0.203724
\(981\) 16.2859 0.519968
\(982\) 33.1440 1.05767
\(983\) −25.8869 −0.825663 −0.412831 0.910808i \(-0.635460\pi\)
−0.412831 + 0.910808i \(0.635460\pi\)
\(984\) 14.7851 0.471331
\(985\) −1.97410 −0.0629001
\(986\) −9.89766 −0.315206
\(987\) −2.15835 −0.0687011
\(988\) −23.9878 −0.763153
\(989\) −9.70251 −0.308522
\(990\) −2.40415 −0.0764090
\(991\) −58.6070 −1.86171 −0.930856 0.365386i \(-0.880937\pi\)
−0.930856 + 0.365386i \(0.880937\pi\)
\(992\) −45.5715 −1.44690
\(993\) −0.551627 −0.0175053
\(994\) 38.7405 1.22877
\(995\) 1.30729 0.0414439
\(996\) 19.1417 0.606528
\(997\) −12.2278 −0.387258 −0.193629 0.981075i \(-0.562026\pi\)
−0.193629 + 0.981075i \(0.562026\pi\)
\(998\) 66.3111 2.09904
\(999\) 2.47861 0.0784197
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 8049.2.a.d.1.16 129
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8049.2.a.d.1.16 129 1.1 even 1 trivial