Properties

Label 1911.2.a.s.1.4
Level $1911$
Weight $2$
Character 1911.1
Self dual yes
Analytic conductor $15.259$
Analytic rank $0$
Dimension $4$
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] = [1911,2,Mod(1,1911)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1911, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1911.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1911 = 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1911.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: \(15.2594118263\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.17428.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} + 4x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 273)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(2.36865\) of defining polynomial
Character \(\chi\) \(=\) 1911.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.61050 q^{2} -1.00000 q^{3} +4.81471 q^{4} +3.81471 q^{5} -2.61050 q^{6} +7.34780 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.61050 q^{2} -1.00000 q^{3} +4.81471 q^{4} +3.81471 q^{5} -2.61050 q^{6} +7.34780 q^{8} +1.00000 q^{9} +9.95830 q^{10} -4.73730 q^{11} -4.81471 q^{12} -1.00000 q^{13} -3.81471 q^{15} +9.55201 q^{16} +5.22100 q^{17} +2.61050 q^{18} -2.92259 q^{19} +18.3667 q^{20} -12.3667 q^{22} +3.33101 q^{23} -7.34780 q^{24} +9.55201 q^{25} -2.61050 q^{26} -1.00000 q^{27} -0.922589 q^{29} -9.95830 q^{30} +7.51941 q^{31} +10.2399 q^{32} +4.73730 q^{33} +13.6294 q^{34} +4.81471 q^{36} +0.154821 q^{37} -7.62942 q^{38} +1.00000 q^{39} +28.0297 q^{40} -6.36672 q^{41} -6.55201 q^{43} -22.8087 q^{44} +3.81471 q^{45} +8.69560 q^{46} -9.03571 q^{47} -9.55201 q^{48} +24.9355 q^{50} -5.22100 q^{51} -4.81471 q^{52} +8.55201 q^{53} -2.61050 q^{54} -18.0714 q^{55} +2.92259 q^{57} -2.40842 q^{58} -3.95830 q^{59} -18.3667 q^{60} -12.4420 q^{61} +19.6294 q^{62} +7.62729 q^{64} -3.81471 q^{65} +12.3667 q^{66} -10.6620 q^{67} +25.1376 q^{68} -3.33101 q^{69} -6.58248 q^{71} +7.34780 q^{72} +7.73517 q^{73} +0.404161 q^{74} -9.55201 q^{75} -14.0714 q^{76} +2.61050 q^{78} +13.3646 q^{79} +36.4381 q^{80} +1.00000 q^{81} -16.6203 q^{82} +1.40629 q^{83} +19.9166 q^{85} -17.1040 q^{86} +0.922589 q^{87} -34.8087 q^{88} +1.96953 q^{89} +9.95830 q^{90} +16.0378 q^{92} -7.51941 q^{93} -23.5877 q^{94} -11.1488 q^{95} -10.2399 q^{96} +2.11001 q^{97} -4.73730 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} - 4 q^{3} + 7 q^{4} + 3 q^{5} - q^{6} + 3 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} - 4 q^{3} + 7 q^{4} + 3 q^{5} - q^{6} + 3 q^{8} + 4 q^{9} + 4 q^{10} - 2 q^{11} - 7 q^{12} - 4 q^{13} - 3 q^{15} + 9 q^{16} + 2 q^{17} + q^{18} - 7 q^{19} + 32 q^{20} - 8 q^{22} + 3 q^{23} - 3 q^{24} + 9 q^{25} - q^{26} - 4 q^{27} + q^{29} - 4 q^{30} - 3 q^{31} + 7 q^{32} + 2 q^{33} + 30 q^{34} + 7 q^{36} + 10 q^{37} - 6 q^{38} + 4 q^{39} + 14 q^{40} + 16 q^{41} + 3 q^{43} - 12 q^{44} + 3 q^{45} - 18 q^{46} - 5 q^{47} - 9 q^{48} + 13 q^{50} - 2 q^{51} - 7 q^{52} + 5 q^{53} - q^{54} - 10 q^{55} + 7 q^{57} - 4 q^{58} + 20 q^{59} - 32 q^{60} - 12 q^{61} + 54 q^{62} + 5 q^{64} - 3 q^{65} + 8 q^{66} - 22 q^{67} + 10 q^{68} - 3 q^{69} + 3 q^{72} + 13 q^{73} - 6 q^{74} - 9 q^{75} + 6 q^{76} + q^{78} + 11 q^{79} + 42 q^{80} + 4 q^{81} - 10 q^{82} - q^{83} + 8 q^{85} - 10 q^{86} - q^{87} - 60 q^{88} + 5 q^{89} + 4 q^{90} + 34 q^{92} + 3 q^{93} - 34 q^{94} + 13 q^{95} - 7 q^{96} + 17 q^{97} - 2 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.61050 1.84590 0.922951 0.384917i \(-0.125770\pi\)
0.922951 + 0.384917i \(0.125770\pi\)
\(3\) −1.00000 −0.577350
\(4\) 4.81471 2.40735
\(5\) 3.81471 1.70599 0.852995 0.521919i \(-0.174784\pi\)
0.852995 + 0.521919i \(0.174784\pi\)
\(6\) −2.61050 −1.06573
\(7\) 0 0
\(8\) 7.34780 2.59784
\(9\) 1.00000 0.333333
\(10\) 9.95830 3.14909
\(11\) −4.73730 −1.42835 −0.714175 0.699968i \(-0.753198\pi\)
−0.714175 + 0.699968i \(0.753198\pi\)
\(12\) −4.81471 −1.38989
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) −3.81471 −0.984954
\(16\) 9.55201 2.38800
\(17\) 5.22100 1.26628 0.633139 0.774038i \(-0.281766\pi\)
0.633139 + 0.774038i \(0.281766\pi\)
\(18\) 2.61050 0.615301
\(19\) −2.92259 −0.670488 −0.335244 0.942131i \(-0.608819\pi\)
−0.335244 + 0.942131i \(0.608819\pi\)
\(20\) 18.3667 4.10692
\(21\) 0 0
\(22\) −12.3667 −2.63659
\(23\) 3.33101 0.694563 0.347282 0.937761i \(-0.387105\pi\)
0.347282 + 0.937761i \(0.387105\pi\)
\(24\) −7.34780 −1.49986
\(25\) 9.55201 1.91040
\(26\) −2.61050 −0.511961
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −0.922589 −0.171321 −0.0856603 0.996324i \(-0.527300\pi\)
−0.0856603 + 0.996324i \(0.527300\pi\)
\(30\) −9.95830 −1.81813
\(31\) 7.51941 1.35053 0.675263 0.737577i \(-0.264030\pi\)
0.675263 + 0.737577i \(0.264030\pi\)
\(32\) 10.2399 1.81018
\(33\) 4.73730 0.824658
\(34\) 13.6294 2.33743
\(35\) 0 0
\(36\) 4.81471 0.802452
\(37\) 0.154821 0.0254525 0.0127262 0.999919i \(-0.495949\pi\)
0.0127262 + 0.999919i \(0.495949\pi\)
\(38\) −7.62942 −1.23766
\(39\) 1.00000 0.160128
\(40\) 28.0297 4.43189
\(41\) −6.36672 −0.994314 −0.497157 0.867661i \(-0.665623\pi\)
−0.497157 + 0.867661i \(0.665623\pi\)
\(42\) 0 0
\(43\) −6.55201 −0.999172 −0.499586 0.866264i \(-0.666514\pi\)
−0.499586 + 0.866264i \(0.666514\pi\)
\(44\) −22.8087 −3.43854
\(45\) 3.81471 0.568663
\(46\) 8.69560 1.28210
\(47\) −9.03571 −1.31799 −0.658997 0.752146i \(-0.729019\pi\)
−0.658997 + 0.752146i \(0.729019\pi\)
\(48\) −9.55201 −1.37871
\(49\) 0 0
\(50\) 24.9355 3.52641
\(51\) −5.22100 −0.731086
\(52\) −4.81471 −0.667680
\(53\) 8.55201 1.17471 0.587354 0.809330i \(-0.300170\pi\)
0.587354 + 0.809330i \(0.300170\pi\)
\(54\) −2.61050 −0.355244
\(55\) −18.0714 −2.43675
\(56\) 0 0
\(57\) 2.92259 0.387106
\(58\) −2.40842 −0.316241
\(59\) −3.95830 −0.515327 −0.257663 0.966235i \(-0.582953\pi\)
−0.257663 + 0.966235i \(0.582953\pi\)
\(60\) −18.3667 −2.37113
\(61\) −12.4420 −1.59303 −0.796517 0.604616i \(-0.793327\pi\)
−0.796517 + 0.604616i \(0.793327\pi\)
\(62\) 19.6294 2.49294
\(63\) 0 0
\(64\) 7.62729 0.953411
\(65\) −3.81471 −0.473156
\(66\) 12.3667 1.52224
\(67\) −10.6620 −1.30257 −0.651286 0.758832i \(-0.725770\pi\)
−0.651286 + 0.758832i \(0.725770\pi\)
\(68\) 25.1376 3.04838
\(69\) −3.33101 −0.401006
\(70\) 0 0
\(71\) −6.58248 −0.781196 −0.390598 0.920561i \(-0.627732\pi\)
−0.390598 + 0.920561i \(0.627732\pi\)
\(72\) 7.34780 0.865946
\(73\) 7.73517 0.905333 0.452667 0.891680i \(-0.350473\pi\)
0.452667 + 0.891680i \(0.350473\pi\)
\(74\) 0.404161 0.0469828
\(75\) −9.55201 −1.10297
\(76\) −14.0714 −1.61410
\(77\) 0 0
\(78\) 2.61050 0.295581
\(79\) 13.3646 1.50363 0.751817 0.659372i \(-0.229178\pi\)
0.751817 + 0.659372i \(0.229178\pi\)
\(80\) 36.4381 4.07391
\(81\) 1.00000 0.111111
\(82\) −16.6203 −1.83541
\(83\) 1.40629 0.154360 0.0771802 0.997017i \(-0.475408\pi\)
0.0771802 + 0.997017i \(0.475408\pi\)
\(84\) 0 0
\(85\) 19.9166 2.16026
\(86\) −17.1040 −1.84437
\(87\) 0.922589 0.0989120
\(88\) −34.8087 −3.71062
\(89\) 1.96953 0.208770 0.104385 0.994537i \(-0.466713\pi\)
0.104385 + 0.994537i \(0.466713\pi\)
\(90\) 9.95830 1.04970
\(91\) 0 0
\(92\) 16.0378 1.67206
\(93\) −7.51941 −0.779727
\(94\) −23.5877 −2.43289
\(95\) −11.1488 −1.14385
\(96\) −10.2399 −1.04511
\(97\) 2.11001 0.214239 0.107119 0.994246i \(-0.465837\pi\)
0.107119 + 0.994246i \(0.465837\pi\)
\(98\) 0 0
\(99\) −4.73730 −0.476116
\(100\) 45.9901 4.59901
\(101\) −0.850419 −0.0846198 −0.0423099 0.999105i \(-0.513472\pi\)
−0.0423099 + 0.999105i \(0.513472\pi\)
\(102\) −13.6294 −1.34951
\(103\) 1.47460 0.145296 0.0726482 0.997358i \(-0.476855\pi\)
0.0726482 + 0.997358i \(0.476855\pi\)
\(104\) −7.34780 −0.720511
\(105\) 0 0
\(106\) 22.3250 2.16840
\(107\) 2.62418 0.253689 0.126844 0.991923i \(-0.459515\pi\)
0.126844 + 0.991923i \(0.459515\pi\)
\(108\) −4.81471 −0.463296
\(109\) −17.9166 −1.71610 −0.858049 0.513567i \(-0.828324\pi\)
−0.858049 + 0.513567i \(0.828324\pi\)
\(110\) −47.1754 −4.49800
\(111\) −0.154821 −0.0146950
\(112\) 0 0
\(113\) 0.922589 0.0867899 0.0433950 0.999058i \(-0.486183\pi\)
0.0433950 + 0.999058i \(0.486183\pi\)
\(114\) 7.62942 0.714561
\(115\) 12.7068 1.18492
\(116\) −4.44200 −0.412429
\(117\) −1.00000 −0.0924500
\(118\) −10.3331 −0.951242
\(119\) 0 0
\(120\) −28.0297 −2.55875
\(121\) 11.4420 1.04018
\(122\) −32.4798 −2.94059
\(123\) 6.36672 0.574068
\(124\) 36.2038 3.25119
\(125\) 17.3646 1.55314
\(126\) 0 0
\(127\) 17.4746 1.55062 0.775310 0.631581i \(-0.217594\pi\)
0.775310 + 0.631581i \(0.217594\pi\)
\(128\) −0.568798 −0.0502751
\(129\) 6.55201 0.576872
\(130\) −9.95830 −0.873401
\(131\) −0.967402 −0.0845223 −0.0422611 0.999107i \(-0.513456\pi\)
−0.0422611 + 0.999107i \(0.513456\pi\)
\(132\) 22.8087 1.98524
\(133\) 0 0
\(134\) −27.8332 −2.40442
\(135\) −3.81471 −0.328318
\(136\) 38.3629 3.28959
\(137\) 3.29628 0.281620 0.140810 0.990037i \(-0.455029\pi\)
0.140810 + 0.990037i \(0.455029\pi\)
\(138\) −8.69560 −0.740218
\(139\) −0.370581 −0.0314323 −0.0157161 0.999876i \(-0.505003\pi\)
−0.0157161 + 0.999876i \(0.505003\pi\)
\(140\) 0 0
\(141\) 9.03571 0.760944
\(142\) −17.1836 −1.44201
\(143\) 4.73730 0.396153
\(144\) 9.55201 0.796001
\(145\) −3.51941 −0.292271
\(146\) 20.1927 1.67116
\(147\) 0 0
\(148\) 0.745420 0.0612732
\(149\) −15.7425 −1.28968 −0.644840 0.764318i \(-0.723076\pi\)
−0.644840 + 0.764318i \(0.723076\pi\)
\(150\) −24.9355 −2.03598
\(151\) 10.2914 0.837505 0.418753 0.908100i \(-0.362467\pi\)
0.418753 + 0.908100i \(0.362467\pi\)
\(152\) −21.4746 −1.74182
\(153\) 5.22100 0.422093
\(154\) 0 0
\(155\) 28.6844 2.30398
\(156\) 4.81471 0.385485
\(157\) 11.4137 0.910909 0.455455 0.890259i \(-0.349477\pi\)
0.455455 + 0.890259i \(0.349477\pi\)
\(158\) 34.8883 2.77556
\(159\) −8.55201 −0.678218
\(160\) 39.0623 3.08815
\(161\) 0 0
\(162\) 2.61050 0.205100
\(163\) −13.4746 −1.05541 −0.527706 0.849427i \(-0.676948\pi\)
−0.527706 + 0.849427i \(0.676948\pi\)
\(164\) −30.6539 −2.39367
\(165\) 18.0714 1.40686
\(166\) 3.67112 0.284934
\(167\) −19.1905 −1.48501 −0.742504 0.669842i \(-0.766362\pi\)
−0.742504 + 0.669842i \(0.766362\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 51.9923 3.98763
\(171\) −2.92259 −0.223496
\(172\) −31.5460 −2.40536
\(173\) 19.5124 1.48350 0.741752 0.670675i \(-0.233995\pi\)
0.741752 + 0.670675i \(0.233995\pi\)
\(174\) 2.40842 0.182582
\(175\) 0 0
\(176\) −45.2507 −3.41090
\(177\) 3.95830 0.297524
\(178\) 5.14146 0.385369
\(179\) −16.5856 −1.23967 −0.619833 0.784734i \(-0.712799\pi\)
−0.619833 + 0.784734i \(0.712799\pi\)
\(180\) 18.3667 1.36897
\(181\) 2.81684 0.209374 0.104687 0.994505i \(-0.466616\pi\)
0.104687 + 0.994505i \(0.466616\pi\)
\(182\) 0 0
\(183\) 12.4420 0.919739
\(184\) 24.4756 1.80436
\(185\) 0.590599 0.0434217
\(186\) −19.6294 −1.43930
\(187\) −24.7334 −1.80869
\(188\) −43.5043 −3.17288
\(189\) 0 0
\(190\) −29.1040 −2.11143
\(191\) 15.1601 1.09694 0.548472 0.836169i \(-0.315210\pi\)
0.548472 + 0.836169i \(0.315210\pi\)
\(192\) −7.62729 −0.550452
\(193\) 0.0651962 0.00469293 0.00234646 0.999997i \(-0.499253\pi\)
0.00234646 + 0.999997i \(0.499253\pi\)
\(194\) 5.50818 0.395464
\(195\) 3.81471 0.273177
\(196\) 0 0
\(197\) −17.1415 −1.22128 −0.610639 0.791909i \(-0.709087\pi\)
−0.610639 + 0.791909i \(0.709087\pi\)
\(198\) −12.3667 −0.878864
\(199\) −6.44200 −0.456661 −0.228331 0.973584i \(-0.573327\pi\)
−0.228331 + 0.973584i \(0.573327\pi\)
\(200\) 70.1862 4.96292
\(201\) 10.6620 0.752041
\(202\) −2.22002 −0.156200
\(203\) 0 0
\(204\) −25.1376 −1.75998
\(205\) −24.2872 −1.69629
\(206\) 3.84944 0.268203
\(207\) 3.33101 0.231521
\(208\) −9.55201 −0.662313
\(209\) 13.8452 0.957691
\(210\) 0 0
\(211\) −16.0266 −1.10332 −0.551659 0.834070i \(-0.686005\pi\)
−0.551659 + 0.834070i \(0.686005\pi\)
\(212\) 41.1754 2.82794
\(213\) 6.58248 0.451024
\(214\) 6.85042 0.468285
\(215\) −24.9940 −1.70458
\(216\) −7.34780 −0.499954
\(217\) 0 0
\(218\) −46.7713 −3.16775
\(219\) −7.73517 −0.522694
\(220\) −87.0086 −5.86612
\(221\) −5.22100 −0.351202
\(222\) −0.404161 −0.0271255
\(223\) −20.0266 −1.34108 −0.670540 0.741873i \(-0.733938\pi\)
−0.670540 + 0.741873i \(0.733938\pi\)
\(224\) 0 0
\(225\) 9.55201 0.636801
\(226\) 2.40842 0.160206
\(227\) −19.2171 −1.27549 −0.637743 0.770249i \(-0.720132\pi\)
−0.637743 + 0.770249i \(0.720132\pi\)
\(228\) 14.0714 0.931902
\(229\) −5.25884 −0.347514 −0.173757 0.984789i \(-0.555591\pi\)
−0.173757 + 0.984789i \(0.555591\pi\)
\(230\) 33.1712 2.18724
\(231\) 0 0
\(232\) −6.77900 −0.445063
\(233\) −26.6234 −1.74416 −0.872079 0.489365i \(-0.837229\pi\)
−0.872079 + 0.489365i \(0.837229\pi\)
\(234\) −2.61050 −0.170654
\(235\) −34.4686 −2.24848
\(236\) −19.0581 −1.24057
\(237\) −13.3646 −0.868123
\(238\) 0 0
\(239\) 4.29104 0.277564 0.138782 0.990323i \(-0.455681\pi\)
0.138782 + 0.990323i \(0.455681\pi\)
\(240\) −36.4381 −2.35207
\(241\) −7.52367 −0.484642 −0.242321 0.970196i \(-0.577909\pi\)
−0.242321 + 0.970196i \(0.577909\pi\)
\(242\) 29.8693 1.92007
\(243\) −1.00000 −0.0641500
\(244\) −59.9046 −3.83500
\(245\) 0 0
\(246\) 16.6203 1.05967
\(247\) 2.92259 0.185960
\(248\) 55.2511 3.50845
\(249\) −1.40629 −0.0891200
\(250\) 45.3303 2.86694
\(251\) −11.3198 −0.714498 −0.357249 0.934009i \(-0.616285\pi\)
−0.357249 + 0.934009i \(0.616285\pi\)
\(252\) 0 0
\(253\) −15.7800 −0.992079
\(254\) 45.6174 2.86229
\(255\) −19.9166 −1.24723
\(256\) −16.7394 −1.04621
\(257\) 24.8504 1.55013 0.775063 0.631884i \(-0.217718\pi\)
0.775063 + 0.631884i \(0.217718\pi\)
\(258\) 17.1040 1.06485
\(259\) 0 0
\(260\) −18.3667 −1.13906
\(261\) −0.922589 −0.0571068
\(262\) −2.52540 −0.156020
\(263\) 17.1762 1.05913 0.529565 0.848270i \(-0.322355\pi\)
0.529565 + 0.848270i \(0.322355\pi\)
\(264\) 34.8087 2.14233
\(265\) 32.6234 2.00404
\(266\) 0 0
\(267\) −1.96953 −0.120533
\(268\) −51.3345 −3.13575
\(269\) 7.74640 0.472306 0.236153 0.971716i \(-0.424113\pi\)
0.236153 + 0.971716i \(0.424113\pi\)
\(270\) −9.95830 −0.606043
\(271\) 29.7008 1.80420 0.902099 0.431530i \(-0.142026\pi\)
0.902099 + 0.431530i \(0.142026\pi\)
\(272\) 49.8710 3.02388
\(273\) 0 0
\(274\) 8.60494 0.519844
\(275\) −45.2507 −2.72872
\(276\) −16.0378 −0.965364
\(277\) 25.1488 1.51105 0.755523 0.655122i \(-0.227383\pi\)
0.755523 + 0.655122i \(0.227383\pi\)
\(278\) −0.967402 −0.0580209
\(279\) 7.51941 0.450175
\(280\) 0 0
\(281\) 8.40030 0.501120 0.250560 0.968101i \(-0.419385\pi\)
0.250560 + 0.968101i \(0.419385\pi\)
\(282\) 23.5877 1.40463
\(283\) 21.3912 1.27157 0.635787 0.771864i \(-0.280676\pi\)
0.635787 + 0.771864i \(0.280676\pi\)
\(284\) −31.6927 −1.88062
\(285\) 11.1488 0.660400
\(286\) 12.3667 0.731259
\(287\) 0 0
\(288\) 10.2399 0.603393
\(289\) 10.2588 0.603461
\(290\) −9.18742 −0.539504
\(291\) −2.11001 −0.123691
\(292\) 37.2426 2.17946
\(293\) 9.59895 0.560777 0.280388 0.959887i \(-0.409537\pi\)
0.280388 + 0.959887i \(0.409537\pi\)
\(294\) 0 0
\(295\) −15.0998 −0.879142
\(296\) 1.13760 0.0661215
\(297\) 4.73730 0.274886
\(298\) −41.0959 −2.38062
\(299\) −3.33101 −0.192637
\(300\) −45.9901 −2.65524
\(301\) 0 0
\(302\) 26.8658 1.54595
\(303\) 0.850419 0.0488553
\(304\) −27.9166 −1.60113
\(305\) −47.4626 −2.71770
\(306\) 13.6294 0.779142
\(307\) 15.1488 0.864589 0.432295 0.901732i \(-0.357704\pi\)
0.432295 + 0.901732i \(0.357704\pi\)
\(308\) 0 0
\(309\) −1.47460 −0.0838869
\(310\) 74.8805 4.25293
\(311\) 4.37058 0.247833 0.123916 0.992293i \(-0.460455\pi\)
0.123916 + 0.992293i \(0.460455\pi\)
\(312\) 7.34780 0.415987
\(313\) 1.49280 0.0843783 0.0421891 0.999110i \(-0.486567\pi\)
0.0421891 + 0.999110i \(0.486567\pi\)
\(314\) 29.7954 1.68145
\(315\) 0 0
\(316\) 64.3466 3.61978
\(317\) 23.2129 1.30377 0.651883 0.758320i \(-0.273979\pi\)
0.651883 + 0.758320i \(0.273979\pi\)
\(318\) −22.3250 −1.25192
\(319\) 4.37058 0.244706
\(320\) 29.0959 1.62651
\(321\) −2.62418 −0.146467
\(322\) 0 0
\(323\) −15.2588 −0.849024
\(324\) 4.81471 0.267484
\(325\) −9.55201 −0.529850
\(326\) −35.1754 −1.94819
\(327\) 17.9166 0.990790
\(328\) −46.7814 −2.58307
\(329\) 0 0
\(330\) 47.1754 2.59692
\(331\) 1.10402 0.0606822 0.0303411 0.999540i \(-0.490341\pi\)
0.0303411 + 0.999540i \(0.490341\pi\)
\(332\) 6.77088 0.371600
\(333\) 0.154821 0.00848416
\(334\) −50.0969 −2.74118
\(335\) −40.6725 −2.22218
\(336\) 0 0
\(337\) −4.24237 −0.231096 −0.115548 0.993302i \(-0.536862\pi\)
−0.115548 + 0.993302i \(0.536862\pi\)
\(338\) 2.61050 0.141992
\(339\) −0.922589 −0.0501082
\(340\) 95.8926 5.20051
\(341\) −35.6217 −1.92902
\(342\) −7.62942 −0.412552
\(343\) 0 0
\(344\) −48.1428 −2.59569
\(345\) −12.7068 −0.684113
\(346\) 50.9372 2.73840
\(347\) 14.0336 0.753362 0.376681 0.926343i \(-0.377065\pi\)
0.376681 + 0.926343i \(0.377065\pi\)
\(348\) 4.44200 0.238116
\(349\) −4.10575 −0.219776 −0.109888 0.993944i \(-0.535049\pi\)
−0.109888 + 0.993944i \(0.535049\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) −48.5096 −2.58557
\(353\) 16.0753 0.855601 0.427800 0.903873i \(-0.359289\pi\)
0.427800 + 0.903873i \(0.359289\pi\)
\(354\) 10.3331 0.549200
\(355\) −25.1102 −1.33271
\(356\) 9.48272 0.502583
\(357\) 0 0
\(358\) −43.2967 −2.28830
\(359\) −18.1510 −0.957971 −0.478985 0.877823i \(-0.658995\pi\)
−0.478985 + 0.877823i \(0.658995\pi\)
\(360\) 28.0297 1.47730
\(361\) −10.4585 −0.550446
\(362\) 7.35336 0.386484
\(363\) −11.4420 −0.600549
\(364\) 0 0
\(365\) 29.5074 1.54449
\(366\) 32.4798 1.69775
\(367\) 23.3955 1.22123 0.610616 0.791927i \(-0.290922\pi\)
0.610616 + 0.791927i \(0.290922\pi\)
\(368\) 31.8178 1.65862
\(369\) −6.36672 −0.331438
\(370\) 1.54176 0.0801522
\(371\) 0 0
\(372\) −36.2038 −1.87708
\(373\) −4.75164 −0.246031 −0.123015 0.992405i \(-0.539256\pi\)
−0.123015 + 0.992405i \(0.539256\pi\)
\(374\) −64.5666 −3.33866
\(375\) −17.3646 −0.896704
\(376\) −66.3926 −3.42394
\(377\) 0.922589 0.0475158
\(378\) 0 0
\(379\) −6.81258 −0.349939 −0.174969 0.984574i \(-0.555983\pi\)
−0.174969 + 0.984574i \(0.555983\pi\)
\(380\) −53.6784 −2.75364
\(381\) −17.4746 −0.895251
\(382\) 39.5753 2.02485
\(383\) −2.25746 −0.115351 −0.0576754 0.998335i \(-0.518369\pi\)
−0.0576754 + 0.998335i \(0.518369\pi\)
\(384\) 0.568798 0.0290264
\(385\) 0 0
\(386\) 0.170195 0.00866268
\(387\) −6.55201 −0.333057
\(388\) 10.1591 0.515749
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 9.95830 0.504258
\(391\) 17.3912 0.879511
\(392\) 0 0
\(393\) 0.967402 0.0487990
\(394\) −44.7478 −2.25436
\(395\) 50.9820 2.56518
\(396\) −22.8087 −1.14618
\(397\) −24.9897 −1.25420 −0.627100 0.778939i \(-0.715758\pi\)
−0.627100 + 0.778939i \(0.715758\pi\)
\(398\) −16.8168 −0.842952
\(399\) 0 0
\(400\) 91.2409 4.56204
\(401\) 27.6529 1.38092 0.690460 0.723370i \(-0.257408\pi\)
0.690460 + 0.723370i \(0.257408\pi\)
\(402\) 27.8332 1.38819
\(403\) −7.51941 −0.374568
\(404\) −4.09452 −0.203710
\(405\) 3.81471 0.189554
\(406\) 0 0
\(407\) −0.733435 −0.0363550
\(408\) −38.3629 −1.89924
\(409\) −13.1488 −0.650168 −0.325084 0.945685i \(-0.605393\pi\)
−0.325084 + 0.945685i \(0.605393\pi\)
\(410\) −63.4017 −3.13119
\(411\) −3.29628 −0.162594
\(412\) 7.09976 0.349780
\(413\) 0 0
\(414\) 8.69560 0.427365
\(415\) 5.36459 0.263337
\(416\) −10.2399 −0.502053
\(417\) 0.370581 0.0181474
\(418\) 36.1428 1.76780
\(419\) 5.69462 0.278200 0.139100 0.990278i \(-0.455579\pi\)
0.139100 + 0.990278i \(0.455579\pi\)
\(420\) 0 0
\(421\) −23.0206 −1.12196 −0.560978 0.827831i \(-0.689575\pi\)
−0.560978 + 0.827831i \(0.689575\pi\)
\(422\) −41.8375 −2.03662
\(423\) −9.03571 −0.439331
\(424\) 62.8384 3.05170
\(425\) 49.8710 2.41910
\(426\) 17.1836 0.832546
\(427\) 0 0
\(428\) 12.6347 0.610719
\(429\) −4.73730 −0.228719
\(430\) −65.2469 −3.14648
\(431\) 20.8045 1.00212 0.501058 0.865414i \(-0.332944\pi\)
0.501058 + 0.865414i \(0.332944\pi\)
\(432\) −9.55201 −0.459571
\(433\) −31.4808 −1.51287 −0.756436 0.654068i \(-0.773061\pi\)
−0.756436 + 0.654068i \(0.773061\pi\)
\(434\) 0 0
\(435\) 3.51941 0.168743
\(436\) −86.2632 −4.13126
\(437\) −9.73517 −0.465696
\(438\) −20.1927 −0.964843
\(439\) −22.3811 −1.06819 −0.534095 0.845425i \(-0.679347\pi\)
−0.534095 + 0.845425i \(0.679347\pi\)
\(440\) −132.785 −6.33028
\(441\) 0 0
\(442\) −13.6294 −0.648285
\(443\) 8.52465 0.405018 0.202509 0.979280i \(-0.435090\pi\)
0.202509 + 0.979280i \(0.435090\pi\)
\(444\) −0.745420 −0.0353761
\(445\) 7.51319 0.356159
\(446\) −52.2795 −2.47550
\(447\) 15.7425 0.744597
\(448\) 0 0
\(449\) 18.7142 0.883178 0.441589 0.897218i \(-0.354415\pi\)
0.441589 + 0.897218i \(0.354415\pi\)
\(450\) 24.9355 1.17547
\(451\) 30.1610 1.42023
\(452\) 4.44200 0.208934
\(453\) −10.2914 −0.483534
\(454\) −50.1663 −2.35442
\(455\) 0 0
\(456\) 21.4746 1.00564
\(457\) −14.1366 −0.661283 −0.330641 0.943756i \(-0.607265\pi\)
−0.330641 + 0.943756i \(0.607265\pi\)
\(458\) −13.7282 −0.641476
\(459\) −5.22100 −0.243695
\(460\) 61.1797 2.85252
\(461\) 2.67636 0.124651 0.0623253 0.998056i \(-0.480148\pi\)
0.0623253 + 0.998056i \(0.480148\pi\)
\(462\) 0 0
\(463\) −2.53162 −0.117655 −0.0588273 0.998268i \(-0.518736\pi\)
−0.0588273 + 0.998268i \(0.518736\pi\)
\(464\) −8.81258 −0.409114
\(465\) −28.6844 −1.33021
\(466\) −69.5005 −3.21955
\(467\) −2.00426 −0.0927460 −0.0463730 0.998924i \(-0.514766\pi\)
−0.0463730 + 0.998924i \(0.514766\pi\)
\(468\) −4.81471 −0.222560
\(469\) 0 0
\(470\) −89.9803 −4.15048
\(471\) −11.4137 −0.525914
\(472\) −29.0848 −1.33874
\(473\) 31.0388 1.42717
\(474\) −34.8883 −1.60247
\(475\) −27.9166 −1.28090
\(476\) 0 0
\(477\) 8.55201 0.391570
\(478\) 11.2018 0.512357
\(479\) 14.6609 0.669872 0.334936 0.942241i \(-0.391285\pi\)
0.334936 + 0.942241i \(0.391285\pi\)
\(480\) −39.0623 −1.78294
\(481\) −0.154821 −0.00705925
\(482\) −19.6405 −0.894602
\(483\) 0 0
\(484\) 55.0899 2.50409
\(485\) 8.04907 0.365489
\(486\) −2.61050 −0.118415
\(487\) 38.2143 1.73165 0.865827 0.500344i \(-0.166793\pi\)
0.865827 + 0.500344i \(0.166793\pi\)
\(488\) −91.4213 −4.13845
\(489\) 13.4746 0.609342
\(490\) 0 0
\(491\) 21.3758 0.964677 0.482339 0.875985i \(-0.339787\pi\)
0.482339 + 0.875985i \(0.339787\pi\)
\(492\) 30.6539 1.38198
\(493\) −4.81684 −0.216939
\(494\) 7.62942 0.343264
\(495\) −18.0714 −0.812250
\(496\) 71.8255 3.22506
\(497\) 0 0
\(498\) −3.67112 −0.164507
\(499\) 27.0920 1.21281 0.606403 0.795158i \(-0.292612\pi\)
0.606403 + 0.795158i \(0.292612\pi\)
\(500\) 83.6054 3.73895
\(501\) 19.1905 0.857370
\(502\) −29.5503 −1.31889
\(503\) −24.8778 −1.10925 −0.554623 0.832102i \(-0.687137\pi\)
−0.554623 + 0.832102i \(0.687137\pi\)
\(504\) 0 0
\(505\) −3.24410 −0.144361
\(506\) −41.1936 −1.83128
\(507\) −1.00000 −0.0444116
\(508\) 84.1351 3.73289
\(509\) −2.70297 −0.119807 −0.0599034 0.998204i \(-0.519079\pi\)
−0.0599034 + 0.998204i \(0.519079\pi\)
\(510\) −51.9923 −2.30226
\(511\) 0 0
\(512\) −42.5607 −1.88093
\(513\) 2.92259 0.129035
\(514\) 64.8720 2.86138
\(515\) 5.62516 0.247874
\(516\) 31.5460 1.38874
\(517\) 42.8049 1.88256
\(518\) 0 0
\(519\) −19.5124 −0.856501
\(520\) −28.0297 −1.22918
\(521\) 33.4472 1.46535 0.732675 0.680579i \(-0.238272\pi\)
0.732675 + 0.680579i \(0.238272\pi\)
\(522\) −2.40842 −0.105414
\(523\) −19.3198 −0.844795 −0.422397 0.906411i \(-0.638811\pi\)
−0.422397 + 0.906411i \(0.638811\pi\)
\(524\) −4.65776 −0.203475
\(525\) 0 0
\(526\) 44.8384 1.95505
\(527\) 39.2588 1.71014
\(528\) 45.2507 1.96928
\(529\) −11.9044 −0.517582
\(530\) 85.1634 3.69926
\(531\) −3.95830 −0.171776
\(532\) 0 0
\(533\) 6.36672 0.275773
\(534\) −5.14146 −0.222493
\(535\) 10.0105 0.432791
\(536\) −78.3424 −3.38387
\(537\) 16.5856 0.715721
\(538\) 20.2220 0.871832
\(539\) 0 0
\(540\) −18.3667 −0.790378
\(541\) −24.9554 −1.07292 −0.536459 0.843927i \(-0.680238\pi\)
−0.536459 + 0.843927i \(0.680238\pi\)
\(542\) 77.5340 3.33037
\(543\) −2.81684 −0.120882
\(544\) 53.4626 2.29219
\(545\) −68.3466 −2.92765
\(546\) 0 0
\(547\) 3.80037 0.162492 0.0812460 0.996694i \(-0.474110\pi\)
0.0812460 + 0.996694i \(0.474110\pi\)
\(548\) 15.8706 0.677960
\(549\) −12.4420 −0.531012
\(550\) −118.127 −5.03695
\(551\) 2.69635 0.114868
\(552\) −24.4756 −1.04175
\(553\) 0 0
\(554\) 65.6510 2.78924
\(555\) −0.590599 −0.0250695
\(556\) −1.78424 −0.0756686
\(557\) 43.8792 1.85922 0.929610 0.368546i \(-0.120144\pi\)
0.929610 + 0.368546i \(0.120144\pi\)
\(558\) 19.6294 0.830980
\(559\) 6.55201 0.277120
\(560\) 0 0
\(561\) 24.7334 1.04425
\(562\) 21.9290 0.925018
\(563\) −12.3768 −0.521620 −0.260810 0.965390i \(-0.583990\pi\)
−0.260810 + 0.965390i \(0.583990\pi\)
\(564\) 43.5043 1.83186
\(565\) 3.51941 0.148063
\(566\) 55.8417 2.34720
\(567\) 0 0
\(568\) −48.3667 −2.02942
\(569\) −26.6234 −1.11611 −0.558056 0.829803i \(-0.688453\pi\)
−0.558056 + 0.829803i \(0.688453\pi\)
\(570\) 29.1040 1.21903
\(571\) 10.9983 0.460263 0.230132 0.973160i \(-0.426084\pi\)
0.230132 + 0.973160i \(0.426084\pi\)
\(572\) 22.8087 0.953680
\(573\) −15.1601 −0.633321
\(574\) 0 0
\(575\) 31.8178 1.32689
\(576\) 7.62729 0.317804
\(577\) −30.4238 −1.26656 −0.633280 0.773923i \(-0.718292\pi\)
−0.633280 + 0.773923i \(0.718292\pi\)
\(578\) 26.7807 1.11393
\(579\) −0.0651962 −0.00270946
\(580\) −16.9449 −0.703600
\(581\) 0 0
\(582\) −5.50818 −0.228321
\(583\) −40.5134 −1.67789
\(584\) 56.8365 2.35191
\(585\) −3.81471 −0.157719
\(586\) 25.0581 1.03514
\(587\) −3.84207 −0.158579 −0.0792895 0.996852i \(-0.525265\pi\)
−0.0792895 + 0.996852i \(0.525265\pi\)
\(588\) 0 0
\(589\) −21.9761 −0.905511
\(590\) −39.4179 −1.62281
\(591\) 17.1415 0.705105
\(592\) 1.47886 0.0607806
\(593\) 23.8904 0.981061 0.490530 0.871424i \(-0.336803\pi\)
0.490530 + 0.871424i \(0.336803\pi\)
\(594\) 12.3667 0.507413
\(595\) 0 0
\(596\) −75.7958 −3.10471
\(597\) 6.44200 0.263653
\(598\) −8.69560 −0.355589
\(599\) −15.4206 −0.630070 −0.315035 0.949080i \(-0.602016\pi\)
−0.315035 + 0.949080i \(0.602016\pi\)
\(600\) −70.1862 −2.86534
\(601\) 1.49280 0.0608928 0.0304464 0.999536i \(-0.490307\pi\)
0.0304464 + 0.999536i \(0.490307\pi\)
\(602\) 0 0
\(603\) −10.6620 −0.434191
\(604\) 49.5503 2.01617
\(605\) 43.6479 1.77454
\(606\) 2.22002 0.0901821
\(607\) 4.44626 0.180468 0.0902340 0.995921i \(-0.471239\pi\)
0.0902340 + 0.995921i \(0.471239\pi\)
\(608\) −29.9271 −1.21370
\(609\) 0 0
\(610\) −123.901 −5.01661
\(611\) 9.03571 0.365546
\(612\) 25.1376 1.01613
\(613\) 26.6640 1.07695 0.538474 0.842642i \(-0.319001\pi\)
0.538474 + 0.842642i \(0.319001\pi\)
\(614\) 39.5460 1.59595
\(615\) 24.2872 0.979354
\(616\) 0 0
\(617\) 27.3495 1.10105 0.550525 0.834819i \(-0.314428\pi\)
0.550525 + 0.834819i \(0.314428\pi\)
\(618\) −3.84944 −0.154847
\(619\) 9.24836 0.371723 0.185861 0.982576i \(-0.440492\pi\)
0.185861 + 0.982576i \(0.440492\pi\)
\(620\) 138.107 5.54651
\(621\) −3.33101 −0.133669
\(622\) 11.4094 0.457475
\(623\) 0 0
\(624\) 9.55201 0.382386
\(625\) 18.4808 0.739233
\(626\) 3.89697 0.155754
\(627\) −13.8452 −0.552923
\(628\) 54.9535 2.19288
\(629\) 0.808323 0.0322299
\(630\) 0 0
\(631\) 24.5134 0.975864 0.487932 0.872882i \(-0.337751\pi\)
0.487932 + 0.872882i \(0.337751\pi\)
\(632\) 98.2003 3.90620
\(633\) 16.0266 0.637000
\(634\) 60.5972 2.40662
\(635\) 66.6605 2.64534
\(636\) −41.1754 −1.63271
\(637\) 0 0
\(638\) 11.4094 0.451703
\(639\) −6.58248 −0.260399
\(640\) −2.16980 −0.0857689
\(641\) 34.3972 1.35861 0.679304 0.733857i \(-0.262282\pi\)
0.679304 + 0.733857i \(0.262282\pi\)
\(642\) −6.85042 −0.270364
\(643\) 7.69036 0.303278 0.151639 0.988436i \(-0.451545\pi\)
0.151639 + 0.988436i \(0.451545\pi\)
\(644\) 0 0
\(645\) 24.9940 0.984138
\(646\) −39.8332 −1.56722
\(647\) −41.9123 −1.64774 −0.823872 0.566776i \(-0.808191\pi\)
−0.823872 + 0.566776i \(0.808191\pi\)
\(648\) 7.34780 0.288649
\(649\) 18.7516 0.736066
\(650\) −24.9355 −0.978051
\(651\) 0 0
\(652\) −64.8763 −2.54075
\(653\) 34.2262 1.33938 0.669688 0.742642i \(-0.266428\pi\)
0.669688 + 0.742642i \(0.266428\pi\)
\(654\) 46.7713 1.82890
\(655\) −3.69036 −0.144194
\(656\) −60.8149 −2.37442
\(657\) 7.73517 0.301778
\(658\) 0 0
\(659\) 45.1685 1.75951 0.879757 0.475424i \(-0.157705\pi\)
0.879757 + 0.475424i \(0.157705\pi\)
\(660\) 87.0086 3.38681
\(661\) −11.9839 −0.466119 −0.233059 0.972463i \(-0.574874\pi\)
−0.233059 + 0.972463i \(0.574874\pi\)
\(662\) 2.88203 0.112013
\(663\) 5.22100 0.202767
\(664\) 10.3331 0.401004
\(665\) 0 0
\(666\) 0.404161 0.0156609
\(667\) −3.07315 −0.118993
\(668\) −92.3968 −3.57494
\(669\) 20.0266 0.774273
\(670\) −106.176 −4.10192
\(671\) 58.9415 2.27541
\(672\) 0 0
\(673\) −29.5117 −1.13759 −0.568796 0.822479i \(-0.692591\pi\)
−0.568796 + 0.822479i \(0.692591\pi\)
\(674\) −11.0747 −0.426581
\(675\) −9.55201 −0.367657
\(676\) 4.81471 0.185181
\(677\) 31.0662 1.19397 0.596985 0.802252i \(-0.296365\pi\)
0.596985 + 0.802252i \(0.296365\pi\)
\(678\) −2.40842 −0.0924948
\(679\) 0 0
\(680\) 146.343 5.61200
\(681\) 19.2171 0.736402
\(682\) −92.9904 −3.56079
\(683\) 5.79433 0.221714 0.110857 0.993836i \(-0.464640\pi\)
0.110857 + 0.993836i \(0.464640\pi\)
\(684\) −14.0714 −0.538034
\(685\) 12.5744 0.480441
\(686\) 0 0
\(687\) 5.25884 0.200637
\(688\) −62.5848 −2.38602
\(689\) −8.55201 −0.325806
\(690\) −33.1712 −1.26281
\(691\) −18.8617 −0.717531 −0.358766 0.933428i \(-0.616802\pi\)
−0.358766 + 0.933428i \(0.616802\pi\)
\(692\) 93.9467 3.57132
\(693\) 0 0
\(694\) 36.6347 1.39063
\(695\) −1.41366 −0.0536232
\(696\) 6.77900 0.256957
\(697\) −33.2406 −1.25908
\(698\) −10.7181 −0.405685
\(699\) 26.6234 1.00699
\(700\) 0 0
\(701\) −32.3180 −1.22064 −0.610318 0.792157i \(-0.708958\pi\)
−0.610318 + 0.792157i \(0.708958\pi\)
\(702\) 2.61050 0.0985270
\(703\) −0.452479 −0.0170656
\(704\) −36.1328 −1.36180
\(705\) 34.4686 1.29816
\(706\) 41.9645 1.57936
\(707\) 0 0
\(708\) 19.0581 0.716246
\(709\) 4.72296 0.177374 0.0886872 0.996060i \(-0.471733\pi\)
0.0886872 + 0.996060i \(0.471733\pi\)
\(710\) −65.5503 −2.46006
\(711\) 13.3646 0.501211
\(712\) 14.4717 0.542350
\(713\) 25.0472 0.938026
\(714\) 0 0
\(715\) 18.0714 0.675833
\(716\) −79.8548 −2.98431
\(717\) −4.29104 −0.160252
\(718\) −47.3831 −1.76832
\(719\) 40.8902 1.52495 0.762474 0.647019i \(-0.223985\pi\)
0.762474 + 0.647019i \(0.223985\pi\)
\(720\) 36.4381 1.35797
\(721\) 0 0
\(722\) −27.3018 −1.01607
\(723\) 7.52367 0.279808
\(724\) 13.5623 0.504037
\(725\) −8.81258 −0.327291
\(726\) −29.8693 −1.10856
\(727\) −18.7292 −0.694627 −0.347313 0.937749i \(-0.612906\pi\)
−0.347313 + 0.937749i \(0.612906\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 77.0291 2.85098
\(731\) −34.2080 −1.26523
\(732\) 59.9046 2.21414
\(733\) 14.1100 0.521165 0.260583 0.965452i \(-0.416085\pi\)
0.260583 + 0.965452i \(0.416085\pi\)
\(734\) 61.0738 2.25428
\(735\) 0 0
\(736\) 34.1093 1.25728
\(737\) 50.5092 1.86053
\(738\) −16.6203 −0.611802
\(739\) −49.9209 −1.83637 −0.918184 0.396154i \(-0.870345\pi\)
−0.918184 + 0.396154i \(0.870345\pi\)
\(740\) 2.84356 0.104531
\(741\) −2.92259 −0.107364
\(742\) 0 0
\(743\) 2.15868 0.0791945 0.0395972 0.999216i \(-0.487393\pi\)
0.0395972 + 0.999216i \(0.487393\pi\)
\(744\) −55.2511 −2.02560
\(745\) −60.0532 −2.20018
\(746\) −12.4042 −0.454149
\(747\) 1.40629 0.0514535
\(748\) −119.084 −4.35415
\(749\) 0 0
\(750\) −45.3303 −1.65523
\(751\) 16.8372 0.614399 0.307199 0.951645i \(-0.400608\pi\)
0.307199 + 0.951645i \(0.400608\pi\)
\(752\) −86.3092 −3.14737
\(753\) 11.3198 0.412516
\(754\) 2.40842 0.0877095
\(755\) 39.2588 1.42878
\(756\) 0 0
\(757\) −17.6028 −0.639785 −0.319893 0.947454i \(-0.603647\pi\)
−0.319893 + 0.947454i \(0.603647\pi\)
\(758\) −17.7842 −0.645953
\(759\) 15.7800 0.572777
\(760\) −81.9193 −2.97153
\(761\) −19.2179 −0.696648 −0.348324 0.937374i \(-0.613249\pi\)
−0.348324 + 0.937374i \(0.613249\pi\)
\(762\) −45.6174 −1.65255
\(763\) 0 0
\(764\) 72.9913 2.64073
\(765\) 19.9166 0.720086
\(766\) −5.89310 −0.212926
\(767\) 3.95830 0.142926
\(768\) 16.7394 0.604032
\(769\) 14.0406 0.506315 0.253158 0.967425i \(-0.418531\pi\)
0.253158 + 0.967425i \(0.418531\pi\)
\(770\) 0 0
\(771\) −24.8504 −0.894966
\(772\) 0.313901 0.0112975
\(773\) −22.4339 −0.806891 −0.403445 0.915004i \(-0.632187\pi\)
−0.403445 + 0.915004i \(0.632187\pi\)
\(774\) −17.1040 −0.614791
\(775\) 71.8255 2.58005
\(776\) 15.5039 0.556558
\(777\) 0 0
\(778\) 15.6630 0.561546
\(779\) 18.6073 0.666676
\(780\) 18.3667 0.657634
\(781\) 31.1832 1.11582
\(782\) 45.3997 1.62349
\(783\) 0.922589 0.0329707
\(784\) 0 0
\(785\) 43.5398 1.55400
\(786\) 2.52540 0.0900781
\(787\) 0.926847 0.0330385 0.0165193 0.999864i \(-0.494742\pi\)
0.0165193 + 0.999864i \(0.494742\pi\)
\(788\) −82.5311 −2.94005
\(789\) −17.1762 −0.611488
\(790\) 133.089 4.73508
\(791\) 0 0
\(792\) −34.8087 −1.23687
\(793\) 12.4420 0.441828
\(794\) −65.2357 −2.31513
\(795\) −32.6234 −1.15703
\(796\) −31.0164 −1.09935
\(797\) 26.0154 0.921512 0.460756 0.887527i \(-0.347578\pi\)
0.460756 + 0.887527i \(0.347578\pi\)
\(798\) 0 0
\(799\) −47.1754 −1.66895
\(800\) 97.8118 3.45817
\(801\) 1.96953 0.0695900
\(802\) 72.1879 2.54904
\(803\) −36.6438 −1.29313
\(804\) 51.3345 1.81043
\(805\) 0 0
\(806\) −19.6294 −0.691417
\(807\) −7.74640 −0.272686
\(808\) −6.24870 −0.219829
\(809\) −44.8539 −1.57698 −0.788490 0.615048i \(-0.789137\pi\)
−0.788490 + 0.615048i \(0.789137\pi\)
\(810\) 9.95830 0.349899
\(811\) −27.2511 −0.956916 −0.478458 0.878110i \(-0.658804\pi\)
−0.478458 + 0.878110i \(0.658804\pi\)
\(812\) 0 0
\(813\) −29.7008 −1.04165
\(814\) −1.91463 −0.0671078
\(815\) −51.4017 −1.80052
\(816\) −49.8710 −1.74584
\(817\) 19.1488 0.669933
\(818\) −34.3250 −1.20015
\(819\) 0 0
\(820\) −116.936 −4.08357
\(821\) 20.4003 0.711975 0.355988 0.934491i \(-0.384145\pi\)
0.355988 + 0.934491i \(0.384145\pi\)
\(822\) −8.60494 −0.300132
\(823\) 54.3509 1.89455 0.947276 0.320418i \(-0.103824\pi\)
0.947276 + 0.320418i \(0.103824\pi\)
\(824\) 10.8350 0.377457
\(825\) 45.2507 1.57543
\(826\) 0 0
\(827\) 3.48272 0.121106 0.0605530 0.998165i \(-0.480714\pi\)
0.0605530 + 0.998165i \(0.480714\pi\)
\(828\) 16.0378 0.557353
\(829\) 29.8417 1.03645 0.518223 0.855246i \(-0.326594\pi\)
0.518223 + 0.855246i \(0.326594\pi\)
\(830\) 14.0043 0.486095
\(831\) −25.1488 −0.872403
\(832\) −7.62729 −0.264429
\(833\) 0 0
\(834\) 0.967402 0.0334984
\(835\) −73.2063 −2.53341
\(836\) 66.6605 2.30550
\(837\) −7.51941 −0.259909
\(838\) 14.8658 0.513530
\(839\) 43.0033 1.48464 0.742320 0.670045i \(-0.233725\pi\)
0.742320 + 0.670045i \(0.233725\pi\)
\(840\) 0 0
\(841\) −28.1488 −0.970649
\(842\) −60.0953 −2.07102
\(843\) −8.40030 −0.289322
\(844\) −77.1634 −2.65608
\(845\) 3.81471 0.131230
\(846\) −23.5877 −0.810962
\(847\) 0 0
\(848\) 81.6889 2.80521
\(849\) −21.3912 −0.734144
\(850\) 130.188 4.46542
\(851\) 0.515711 0.0176784
\(852\) 31.6927 1.08577
\(853\) −34.1023 −1.16764 −0.583820 0.811883i \(-0.698443\pi\)
−0.583820 + 0.811883i \(0.698443\pi\)
\(854\) 0 0
\(855\) −11.1488 −0.381282
\(856\) 19.2819 0.659043
\(857\) 45.7344 1.56226 0.781129 0.624370i \(-0.214644\pi\)
0.781129 + 0.624370i \(0.214644\pi\)
\(858\) −12.3667 −0.422193
\(859\) −15.9861 −0.545437 −0.272719 0.962094i \(-0.587923\pi\)
−0.272719 + 0.962094i \(0.587923\pi\)
\(860\) −120.339 −4.10352
\(861\) 0 0
\(862\) 54.3100 1.84981
\(863\) −24.5096 −0.834315 −0.417157 0.908834i \(-0.636974\pi\)
−0.417157 + 0.908834i \(0.636974\pi\)
\(864\) −10.2399 −0.348369
\(865\) 74.4343 2.53084
\(866\) −82.1807 −2.79261
\(867\) −10.2588 −0.348408
\(868\) 0 0
\(869\) −63.3120 −2.14771
\(870\) 9.18742 0.311483
\(871\) 10.6620 0.361269
\(872\) −131.648 −4.45815
\(873\) 2.11001 0.0714130
\(874\) −25.4137 −0.859630
\(875\) 0 0
\(876\) −37.2426 −1.25831
\(877\) 53.4913 1.80627 0.903136 0.429354i \(-0.141259\pi\)
0.903136 + 0.429354i \(0.141259\pi\)
\(878\) −58.4258 −1.97177
\(879\) −9.59895 −0.323765
\(880\) −172.618 −5.81896
\(881\) −26.1745 −0.881840 −0.440920 0.897546i \(-0.645348\pi\)
−0.440920 + 0.897546i \(0.645348\pi\)
\(882\) 0 0
\(883\) 23.1840 0.780202 0.390101 0.920772i \(-0.372440\pi\)
0.390101 + 0.920772i \(0.372440\pi\)
\(884\) −25.1376 −0.845469
\(885\) 15.0998 0.507573
\(886\) 22.2536 0.747624
\(887\) 13.7008 0.460029 0.230015 0.973187i \(-0.426123\pi\)
0.230015 + 0.973187i \(0.426123\pi\)
\(888\) −1.13760 −0.0381752
\(889\) 0 0
\(890\) 19.6132 0.657435
\(891\) −4.73730 −0.158705
\(892\) −96.4223 −3.22846
\(893\) 26.4077 0.883699
\(894\) 41.0959 1.37445
\(895\) −63.2692 −2.11486
\(896\) 0 0
\(897\) 3.33101 0.111219
\(898\) 48.8534 1.63026
\(899\) −6.93733 −0.231373
\(900\) 45.9901 1.53300
\(901\) 44.6500 1.48751
\(902\) 78.7354 2.62160
\(903\) 0 0
\(904\) 6.77900 0.225466
\(905\) 10.7454 0.357190
\(906\) −26.8658 −0.892556
\(907\) 25.5621 0.848777 0.424388 0.905480i \(-0.360489\pi\)
0.424388 + 0.905480i \(0.360489\pi\)
\(908\) −92.5249 −3.07055
\(909\) −0.850419 −0.0282066
\(910\) 0 0
\(911\) 2.07643 0.0687951 0.0343976 0.999408i \(-0.489049\pi\)
0.0343976 + 0.999408i \(0.489049\pi\)
\(912\) 27.9166 0.924411
\(913\) −6.66202 −0.220481
\(914\) −36.9036 −1.22066
\(915\) 47.4626 1.56907
\(916\) −25.3198 −0.836589
\(917\) 0 0
\(918\) −13.6294 −0.449838
\(919\) 17.8514 0.588863 0.294432 0.955673i \(-0.404870\pi\)
0.294432 + 0.955673i \(0.404870\pi\)
\(920\) 93.3672 3.07823
\(921\) −15.1488 −0.499171
\(922\) 6.98664 0.230093
\(923\) 6.58248 0.216665
\(924\) 0 0
\(925\) 1.47886 0.0486245
\(926\) −6.60881 −0.217179
\(927\) 1.47460 0.0484321
\(928\) −9.44724 −0.310121
\(929\) 37.0553 1.21575 0.607873 0.794034i \(-0.292023\pi\)
0.607873 + 0.794034i \(0.292023\pi\)
\(930\) −74.8805 −2.45543
\(931\) 0 0
\(932\) −128.184 −4.19881
\(933\) −4.37058 −0.143086
\(934\) −5.23212 −0.171200
\(935\) −94.3509 −3.08560
\(936\) −7.34780 −0.240170
\(937\) −39.7540 −1.29871 −0.649354 0.760486i \(-0.724961\pi\)
−0.649354 + 0.760486i \(0.724961\pi\)
\(938\) 0 0
\(939\) −1.49280 −0.0487158
\(940\) −165.956 −5.41290
\(941\) 47.3530 1.54366 0.771832 0.635827i \(-0.219341\pi\)
0.771832 + 0.635827i \(0.219341\pi\)
\(942\) −29.7954 −0.970785
\(943\) −21.2076 −0.690614
\(944\) −37.8097 −1.23060
\(945\) 0 0
\(946\) 81.0268 2.63441
\(947\) 8.80446 0.286106 0.143053 0.989715i \(-0.454308\pi\)
0.143053 + 0.989715i \(0.454308\pi\)
\(948\) −64.3466 −2.08988
\(949\) −7.73517 −0.251094
\(950\) −72.8763 −2.36442
\(951\) −23.2129 −0.752729
\(952\) 0 0
\(953\) 7.72895 0.250365 0.125183 0.992134i \(-0.460048\pi\)
0.125183 + 0.992134i \(0.460048\pi\)
\(954\) 22.3250 0.722799
\(955\) 57.8312 1.87137
\(956\) 20.6601 0.668196
\(957\) −4.37058 −0.141281
\(958\) 38.2722 1.23652
\(959\) 0 0
\(960\) −29.0959 −0.939066
\(961\) 25.5415 0.823920
\(962\) −0.404161 −0.0130307
\(963\) 2.62418 0.0845630
\(964\) −36.2243 −1.16671
\(965\) 0.248705 0.00800608
\(966\) 0 0
\(967\) 14.3054 0.460030 0.230015 0.973187i \(-0.426122\pi\)
0.230015 + 0.973187i \(0.426122\pi\)
\(968\) 84.0735 2.70222
\(969\) 15.2588 0.490184
\(970\) 21.0121 0.674658
\(971\) 15.1692 0.486803 0.243402 0.969926i \(-0.421737\pi\)
0.243402 + 0.969926i \(0.421737\pi\)
\(972\) −4.81471 −0.154432
\(973\) 0 0
\(974\) 99.7583 3.19646
\(975\) 9.55201 0.305909
\(976\) −118.846 −3.80417
\(977\) −8.25746 −0.264180 −0.132090 0.991238i \(-0.542169\pi\)
−0.132090 + 0.991238i \(0.542169\pi\)
\(978\) 35.1754 1.12479
\(979\) −9.33026 −0.298196
\(980\) 0 0
\(981\) −17.9166 −0.572033
\(982\) 55.8016 1.78070
\(983\) 25.8525 0.824568 0.412284 0.911055i \(-0.364731\pi\)
0.412284 + 0.911055i \(0.364731\pi\)
\(984\) 46.7814 1.49134
\(985\) −65.3897 −2.08349
\(986\) −12.5744 −0.400449
\(987\) 0 0
\(988\) 14.0714 0.447671
\(989\) −21.8248 −0.693988
\(990\) −47.1754 −1.49933
\(991\) −56.2100 −1.78557 −0.892785 0.450484i \(-0.851252\pi\)
−0.892785 + 0.450484i \(0.851252\pi\)
\(992\) 76.9981 2.44469
\(993\) −1.10402 −0.0350349
\(994\) 0 0
\(995\) −24.5744 −0.779059
\(996\) −6.77088 −0.214544
\(997\) −37.4789 −1.18697 −0.593484 0.804846i \(-0.702248\pi\)
−0.593484 + 0.804846i \(0.702248\pi\)
\(998\) 70.7237 2.23872
\(999\) −0.154821 −0.00489833
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 1911.2.a.s.1.4 4
3.2 odd 2 5733.2.a.bf.1.1 4
7.6 odd 2 273.2.a.e.1.4 4
21.20 even 2 819.2.a.k.1.1 4
28.27 even 2 4368.2.a.br.1.1 4
35.34 odd 2 6825.2.a.bg.1.1 4
91.90 odd 2 3549.2.a.w.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
273.2.a.e.1.4 4 7.6 odd 2
819.2.a.k.1.1 4 21.20 even 2
1911.2.a.s.1.4 4 1.1 even 1 trivial
3549.2.a.w.1.1 4 91.90 odd 2
4368.2.a.br.1.1 4 28.27 even 2
5733.2.a.bf.1.1 4 3.2 odd 2
6825.2.a.bg.1.1 4 35.34 odd 2