Properties

Label 8281.2.a.bg.1.3
Level $8281$
Weight $2$
Character 8281.1
Self dual yes
Analytic conductor $66.124$
Analytic rank $0$
Dimension $3$
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] = [8281,2,Mod(1,8281)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8281, 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("8281.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8281 = 7^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8281.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: \(66.1241179138\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.81361\) of defining polynomial
Character \(\chi\) \(=\) 8281.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.81361 q^{2} +3.10278 q^{3} +1.28917 q^{4} +2.81361 q^{5} +5.62721 q^{6} -1.28917 q^{8} +6.62721 q^{9} +O(q^{10})\) \(q+1.81361 q^{2} +3.10278 q^{3} +1.28917 q^{4} +2.81361 q^{5} +5.62721 q^{6} -1.28917 q^{8} +6.62721 q^{9} +5.10278 q^{10} -3.10278 q^{11} +4.00000 q^{12} +8.72999 q^{15} -4.91638 q^{16} +0.524438 q^{17} +12.0192 q^{18} +0.813607 q^{19} +3.62721 q^{20} -5.62721 q^{22} +7.33804 q^{23} -4.00000 q^{24} +2.91638 q^{25} +11.2544 q^{27} +8.28917 q^{29} +15.8328 q^{30} +1.39194 q^{31} -6.33804 q^{32} -9.62721 q^{33} +0.951124 q^{34} +8.54359 q^{36} +6.15165 q^{37} +1.47556 q^{38} -3.62721 q^{40} -4.20555 q^{41} +6.75971 q^{43} -4.00000 q^{44} +18.6464 q^{45} +13.3083 q^{46} -5.97028 q^{47} -15.2544 q^{48} +5.28917 q^{50} +1.62721 q^{51} -2.49472 q^{53} +20.4111 q^{54} -8.72999 q^{55} +2.52444 q^{57} +15.0333 q^{58} -4.47054 q^{59} +11.2544 q^{60} +2.00000 q^{61} +2.52444 q^{62} -1.66196 q^{64} -17.4600 q^{66} -10.0383 q^{67} +0.676089 q^{68} +22.7683 q^{69} +8.72999 q^{71} -8.54359 q^{72} -2.34307 q^{73} +11.1567 q^{74} +9.04888 q^{75} +1.04888 q^{76} -13.5436 q^{79} -13.8328 q^{80} +15.0383 q^{81} -7.62721 q^{82} +16.4791 q^{83} +1.47556 q^{85} +12.2594 q^{86} +25.7194 q^{87} +4.00000 q^{88} -10.6464 q^{89} +33.8172 q^{90} +9.45998 q^{92} +4.31889 q^{93} -10.8277 q^{94} +2.28917 q^{95} -19.6655 q^{96} -1.18639 q^{97} -20.5628 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 2 q^{3} + 3 q^{4} + 2 q^{5} + 4 q^{6} - 3 q^{8} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} + 2 q^{3} + 3 q^{4} + 2 q^{5} + 4 q^{6} - 3 q^{8} + 7 q^{9} + 8 q^{10} - 2 q^{11} + 12 q^{12} + 6 q^{15} - q^{16} - 4 q^{17} + 15 q^{18} - 4 q^{19} - 2 q^{20} - 4 q^{22} + 10 q^{23} - 12 q^{24} - 5 q^{25} + 8 q^{27} + 24 q^{29} + 20 q^{30} - 4 q^{31} - 7 q^{32} - 16 q^{33} + 14 q^{34} - q^{36} + 10 q^{38} + 2 q^{40} + 2 q^{41} + 10 q^{43} - 12 q^{44} + 22 q^{45} + 18 q^{46} - 8 q^{47} - 20 q^{48} + 15 q^{50} - 8 q^{51} + 8 q^{53} + 32 q^{54} - 6 q^{55} + 2 q^{57} - 12 q^{58} - 4 q^{59} + 8 q^{60} + 6 q^{61} + 2 q^{62} - 17 q^{64} - 12 q^{66} + 12 q^{67} - 22 q^{68} + 6 q^{69} + 6 q^{71} + q^{72} - 10 q^{73} + 30 q^{74} + 16 q^{75} - 8 q^{76} - 14 q^{79} - 14 q^{80} + 3 q^{81} - 10 q^{82} - 12 q^{83} + 10 q^{85} + 26 q^{86} + 26 q^{87} + 12 q^{88} + 2 q^{89} + 28 q^{90} - 12 q^{92} + 22 q^{93} + 10 q^{94} + 6 q^{95} - 4 q^{96} - 10 q^{97} - 14 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.81361 1.28241 0.641207 0.767368i \(-0.278434\pi\)
0.641207 + 0.767368i \(0.278434\pi\)
\(3\) 3.10278 1.79139 0.895694 0.444671i \(-0.146679\pi\)
0.895694 + 0.444671i \(0.146679\pi\)
\(4\) 1.28917 0.644584
\(5\) 2.81361 1.25828 0.629142 0.777291i \(-0.283407\pi\)
0.629142 + 0.777291i \(0.283407\pi\)
\(6\) 5.62721 2.29730
\(7\) 0 0
\(8\) −1.28917 −0.455790
\(9\) 6.62721 2.20907
\(10\) 5.10278 1.61364
\(11\) −3.10278 −0.935522 −0.467761 0.883855i \(-0.654939\pi\)
−0.467761 + 0.883855i \(0.654939\pi\)
\(12\) 4.00000 1.15470
\(13\) 0 0
\(14\) 0 0
\(15\) 8.72999 2.25407
\(16\) −4.91638 −1.22910
\(17\) 0.524438 0.127195 0.0635974 0.997976i \(-0.479743\pi\)
0.0635974 + 0.997976i \(0.479743\pi\)
\(18\) 12.0192 2.83294
\(19\) 0.813607 0.186654 0.0933271 0.995636i \(-0.470250\pi\)
0.0933271 + 0.995636i \(0.470250\pi\)
\(20\) 3.62721 0.811069
\(21\) 0 0
\(22\) −5.62721 −1.19973
\(23\) 7.33804 1.53009 0.765044 0.643978i \(-0.222717\pi\)
0.765044 + 0.643978i \(0.222717\pi\)
\(24\) −4.00000 −0.816497
\(25\) 2.91638 0.583276
\(26\) 0 0
\(27\) 11.2544 2.16592
\(28\) 0 0
\(29\) 8.28917 1.53926 0.769630 0.638490i \(-0.220441\pi\)
0.769630 + 0.638490i \(0.220441\pi\)
\(30\) 15.8328 2.89065
\(31\) 1.39194 0.250000 0.125000 0.992157i \(-0.460107\pi\)
0.125000 + 0.992157i \(0.460107\pi\)
\(32\) −6.33804 −1.12042
\(33\) −9.62721 −1.67588
\(34\) 0.951124 0.163116
\(35\) 0 0
\(36\) 8.54359 1.42393
\(37\) 6.15165 1.01133 0.505663 0.862731i \(-0.331248\pi\)
0.505663 + 0.862731i \(0.331248\pi\)
\(38\) 1.47556 0.239368
\(39\) 0 0
\(40\) −3.62721 −0.573513
\(41\) −4.20555 −0.656797 −0.328398 0.944539i \(-0.606509\pi\)
−0.328398 + 0.944539i \(0.606509\pi\)
\(42\) 0 0
\(43\) 6.75971 1.03085 0.515423 0.856936i \(-0.327635\pi\)
0.515423 + 0.856936i \(0.327635\pi\)
\(44\) −4.00000 −0.603023
\(45\) 18.6464 2.77964
\(46\) 13.3083 1.96221
\(47\) −5.97028 −0.870855 −0.435427 0.900224i \(-0.643403\pi\)
−0.435427 + 0.900224i \(0.643403\pi\)
\(48\) −15.2544 −2.20179
\(49\) 0 0
\(50\) 5.28917 0.748001
\(51\) 1.62721 0.227855
\(52\) 0 0
\(53\) −2.49472 −0.342676 −0.171338 0.985212i \(-0.554809\pi\)
−0.171338 + 0.985212i \(0.554809\pi\)
\(54\) 20.4111 2.77760
\(55\) −8.72999 −1.17715
\(56\) 0 0
\(57\) 2.52444 0.334370
\(58\) 15.0333 1.97397
\(59\) −4.47054 −0.582015 −0.291007 0.956721i \(-0.593990\pi\)
−0.291007 + 0.956721i \(0.593990\pi\)
\(60\) 11.2544 1.45294
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 2.52444 0.320604
\(63\) 0 0
\(64\) −1.66196 −0.207744
\(65\) 0 0
\(66\) −17.4600 −2.14917
\(67\) −10.0383 −1.22638 −0.613188 0.789937i \(-0.710113\pi\)
−0.613188 + 0.789937i \(0.710113\pi\)
\(68\) 0.676089 0.0819878
\(69\) 22.7683 2.74098
\(70\) 0 0
\(71\) 8.72999 1.03606 0.518029 0.855363i \(-0.326666\pi\)
0.518029 + 0.855363i \(0.326666\pi\)
\(72\) −8.54359 −1.00687
\(73\) −2.34307 −0.274235 −0.137118 0.990555i \(-0.543784\pi\)
−0.137118 + 0.990555i \(0.543784\pi\)
\(74\) 11.1567 1.29694
\(75\) 9.04888 1.04487
\(76\) 1.04888 0.120314
\(77\) 0 0
\(78\) 0 0
\(79\) −13.5436 −1.52377 −0.761887 0.647710i \(-0.775727\pi\)
−0.761887 + 0.647710i \(0.775727\pi\)
\(80\) −13.8328 −1.54655
\(81\) 15.0383 1.67092
\(82\) −7.62721 −0.842285
\(83\) 16.4791 1.80882 0.904410 0.426665i \(-0.140312\pi\)
0.904410 + 0.426665i \(0.140312\pi\)
\(84\) 0 0
\(85\) 1.47556 0.160047
\(86\) 12.2594 1.32197
\(87\) 25.7194 2.75741
\(88\) 4.00000 0.426401
\(89\) −10.6464 −1.12851 −0.564256 0.825600i \(-0.690837\pi\)
−0.564256 + 0.825600i \(0.690837\pi\)
\(90\) 33.8172 3.56464
\(91\) 0 0
\(92\) 9.45998 0.986271
\(93\) 4.31889 0.447848
\(94\) −10.8277 −1.11680
\(95\) 2.28917 0.234864
\(96\) −19.6655 −2.00710
\(97\) −1.18639 −0.120460 −0.0602300 0.998185i \(-0.519183\pi\)
−0.0602300 + 0.998185i \(0.519183\pi\)
\(98\) 0 0
\(99\) −20.5628 −2.06663
\(100\) 3.75971 0.375971
\(101\) −13.1028 −1.30377 −0.651887 0.758316i \(-0.726023\pi\)
−0.651887 + 0.758316i \(0.726023\pi\)
\(102\) 2.95112 0.292205
\(103\) −4.41110 −0.434639 −0.217319 0.976101i \(-0.569731\pi\)
−0.217319 + 0.976101i \(0.569731\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −4.52444 −0.439452
\(107\) 0.578337 0.0559100 0.0279550 0.999609i \(-0.491100\pi\)
0.0279550 + 0.999609i \(0.491100\pi\)
\(108\) 14.5089 1.39611
\(109\) −5.57331 −0.533827 −0.266913 0.963721i \(-0.586004\pi\)
−0.266913 + 0.963721i \(0.586004\pi\)
\(110\) −15.8328 −1.50959
\(111\) 19.0872 1.81168
\(112\) 0 0
\(113\) 5.44584 0.512302 0.256151 0.966637i \(-0.417546\pi\)
0.256151 + 0.966637i \(0.417546\pi\)
\(114\) 4.57834 0.428801
\(115\) 20.6464 1.92528
\(116\) 10.6861 0.992183
\(117\) 0 0
\(118\) −8.10780 −0.746383
\(119\) 0 0
\(120\) −11.2544 −1.02738
\(121\) −1.37279 −0.124799
\(122\) 3.62721 0.328392
\(123\) −13.0489 −1.17658
\(124\) 1.79445 0.161146
\(125\) −5.86248 −0.524356
\(126\) 0 0
\(127\) −12.8816 −1.14306 −0.571530 0.820581i \(-0.693650\pi\)
−0.571530 + 0.820581i \(0.693650\pi\)
\(128\) 9.66196 0.854004
\(129\) 20.9739 1.84664
\(130\) 0 0
\(131\) −9.04888 −0.790604 −0.395302 0.918551i \(-0.629360\pi\)
−0.395302 + 0.918551i \(0.629360\pi\)
\(132\) −12.4111 −1.08025
\(133\) 0 0
\(134\) −18.2056 −1.57272
\(135\) 31.6655 2.72533
\(136\) −0.676089 −0.0579741
\(137\) 6.25945 0.534781 0.267390 0.963588i \(-0.413839\pi\)
0.267390 + 0.963588i \(0.413839\pi\)
\(138\) 41.2927 3.51507
\(139\) 11.5733 0.981636 0.490818 0.871262i \(-0.336698\pi\)
0.490818 + 0.871262i \(0.336698\pi\)
\(140\) 0 0
\(141\) −18.5244 −1.56004
\(142\) 15.8328 1.32866
\(143\) 0 0
\(144\) −32.5819 −2.71516
\(145\) 23.3225 1.93682
\(146\) −4.24940 −0.351683
\(147\) 0 0
\(148\) 7.93051 0.651884
\(149\) −8.52444 −0.698349 −0.349175 0.937058i \(-0.613538\pi\)
−0.349175 + 0.937058i \(0.613538\pi\)
\(150\) 16.4111 1.33996
\(151\) 11.9844 0.975278 0.487639 0.873045i \(-0.337858\pi\)
0.487639 + 0.873045i \(0.337858\pi\)
\(152\) −1.04888 −0.0850751
\(153\) 3.47556 0.280983
\(154\) 0 0
\(155\) 3.91638 0.314571
\(156\) 0 0
\(157\) 12.8277 1.02377 0.511883 0.859055i \(-0.328948\pi\)
0.511883 + 0.859055i \(0.328948\pi\)
\(158\) −24.5628 −1.95411
\(159\) −7.74055 −0.613866
\(160\) −17.8328 −1.40980
\(161\) 0 0
\(162\) 27.2736 2.14282
\(163\) 13.4600 1.05427 0.527133 0.849783i \(-0.323267\pi\)
0.527133 + 0.849783i \(0.323267\pi\)
\(164\) −5.42166 −0.423361
\(165\) −27.0872 −2.10873
\(166\) 29.8867 2.31965
\(167\) −2.02972 −0.157064 −0.0785322 0.996912i \(-0.525023\pi\)
−0.0785322 + 0.996912i \(0.525023\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 2.67609 0.205247
\(171\) 5.39194 0.412332
\(172\) 8.71440 0.664467
\(173\) −20.2978 −1.54321 −0.771605 0.636102i \(-0.780546\pi\)
−0.771605 + 0.636102i \(0.780546\pi\)
\(174\) 46.6449 3.53614
\(175\) 0 0
\(176\) 15.2544 1.14985
\(177\) −13.8711 −1.04261
\(178\) −19.3083 −1.44722
\(179\) −11.0036 −0.822445 −0.411223 0.911535i \(-0.634898\pi\)
−0.411223 + 0.911535i \(0.634898\pi\)
\(180\) 24.0383 1.79171
\(181\) −0.691675 −0.0514118 −0.0257059 0.999670i \(-0.508183\pi\)
−0.0257059 + 0.999670i \(0.508183\pi\)
\(182\) 0 0
\(183\) 6.20555 0.458727
\(184\) −9.45998 −0.697399
\(185\) 17.3083 1.27253
\(186\) 7.83276 0.574326
\(187\) −1.62721 −0.118994
\(188\) −7.69670 −0.561339
\(189\) 0 0
\(190\) 4.15165 0.301192
\(191\) 7.83276 0.566759 0.283379 0.959008i \(-0.408544\pi\)
0.283379 + 0.959008i \(0.408544\pi\)
\(192\) −5.15667 −0.372151
\(193\) 12.2056 0.878575 0.439287 0.898347i \(-0.355231\pi\)
0.439287 + 0.898347i \(0.355231\pi\)
\(194\) −2.15165 −0.154480
\(195\) 0 0
\(196\) 0 0
\(197\) 18.8222 1.34103 0.670513 0.741898i \(-0.266074\pi\)
0.670513 + 0.741898i \(0.266074\pi\)
\(198\) −37.2927 −2.65028
\(199\) −21.6116 −1.53201 −0.766004 0.642836i \(-0.777758\pi\)
−0.766004 + 0.642836i \(0.777758\pi\)
\(200\) −3.75971 −0.265851
\(201\) −31.1466 −2.19691
\(202\) −23.7633 −1.67198
\(203\) 0 0
\(204\) 2.09775 0.146872
\(205\) −11.8328 −0.826436
\(206\) −8.00000 −0.557386
\(207\) 48.6308 3.38007
\(208\) 0 0
\(209\) −2.52444 −0.174619
\(210\) 0 0
\(211\) −17.3764 −1.19624 −0.598119 0.801407i \(-0.704085\pi\)
−0.598119 + 0.801407i \(0.704085\pi\)
\(212\) −3.21611 −0.220884
\(213\) 27.0872 1.85598
\(214\) 1.04888 0.0716997
\(215\) 19.0192 1.29710
\(216\) −14.5089 −0.987202
\(217\) 0 0
\(218\) −10.1078 −0.684586
\(219\) −7.27001 −0.491262
\(220\) −11.2544 −0.758773
\(221\) 0 0
\(222\) 34.6167 2.32332
\(223\) −10.5486 −0.706388 −0.353194 0.935550i \(-0.614904\pi\)
−0.353194 + 0.935550i \(0.614904\pi\)
\(224\) 0 0
\(225\) 19.3275 1.28850
\(226\) 9.87662 0.656983
\(227\) −6.95112 −0.461362 −0.230681 0.973029i \(-0.574095\pi\)
−0.230681 + 0.973029i \(0.574095\pi\)
\(228\) 3.25443 0.215530
\(229\) −21.0872 −1.39348 −0.696740 0.717323i \(-0.745367\pi\)
−0.696740 + 0.717323i \(0.745367\pi\)
\(230\) 37.4444 2.46901
\(231\) 0 0
\(232\) −10.6861 −0.701579
\(233\) 6.08362 0.398551 0.199276 0.979943i \(-0.436141\pi\)
0.199276 + 0.979943i \(0.436141\pi\)
\(234\) 0 0
\(235\) −16.7980 −1.09578
\(236\) −5.76328 −0.375157
\(237\) −42.0227 −2.72967
\(238\) 0 0
\(239\) 14.2056 0.918881 0.459440 0.888209i \(-0.348050\pi\)
0.459440 + 0.888209i \(0.348050\pi\)
\(240\) −42.9200 −2.77047
\(241\) 8.44082 0.543721 0.271860 0.962337i \(-0.412361\pi\)
0.271860 + 0.962337i \(0.412361\pi\)
\(242\) −2.48970 −0.160044
\(243\) 12.8972 0.827357
\(244\) 2.57834 0.165061
\(245\) 0 0
\(246\) −23.6655 −1.50886
\(247\) 0 0
\(248\) −1.79445 −0.113948
\(249\) 51.1310 3.24030
\(250\) −10.6322 −0.672442
\(251\) 23.9844 1.51388 0.756941 0.653483i \(-0.226693\pi\)
0.756941 + 0.653483i \(0.226693\pi\)
\(252\) 0 0
\(253\) −22.7683 −1.43143
\(254\) −23.3622 −1.46588
\(255\) 4.57834 0.286707
\(256\) 20.8469 1.30293
\(257\) −15.6116 −0.973827 −0.486913 0.873450i \(-0.661877\pi\)
−0.486913 + 0.873450i \(0.661877\pi\)
\(258\) 38.0383 2.36816
\(259\) 0 0
\(260\) 0 0
\(261\) 54.9341 3.40033
\(262\) −16.4111 −1.01388
\(263\) −15.1708 −0.935472 −0.467736 0.883868i \(-0.654930\pi\)
−0.467736 + 0.883868i \(0.654930\pi\)
\(264\) 12.4111 0.763850
\(265\) −7.01916 −0.431183
\(266\) 0 0
\(267\) −33.0333 −2.02160
\(268\) −12.9411 −0.790502
\(269\) −23.2927 −1.42018 −0.710092 0.704109i \(-0.751347\pi\)
−0.710092 + 0.704109i \(0.751347\pi\)
\(270\) 57.4288 3.49501
\(271\) 12.7456 0.774238 0.387119 0.922030i \(-0.373470\pi\)
0.387119 + 0.922030i \(0.373470\pi\)
\(272\) −2.57834 −0.156335
\(273\) 0 0
\(274\) 11.3522 0.685810
\(275\) −9.04888 −0.545668
\(276\) 29.3522 1.76679
\(277\) 8.12193 0.488000 0.244000 0.969775i \(-0.421540\pi\)
0.244000 + 0.969775i \(0.421540\pi\)
\(278\) 20.9894 1.25886
\(279\) 9.22471 0.552269
\(280\) 0 0
\(281\) −19.0333 −1.13543 −0.567715 0.823225i \(-0.692173\pi\)
−0.567715 + 0.823225i \(0.692173\pi\)
\(282\) −33.5960 −2.00062
\(283\) −11.1466 −0.662598 −0.331299 0.943526i \(-0.607487\pi\)
−0.331299 + 0.943526i \(0.607487\pi\)
\(284\) 11.2544 0.667827
\(285\) 7.10278 0.420732
\(286\) 0 0
\(287\) 0 0
\(288\) −42.0036 −2.47508
\(289\) −16.7250 −0.983821
\(290\) 42.2978 2.48381
\(291\) −3.68111 −0.215791
\(292\) −3.02061 −0.176768
\(293\) −14.1758 −0.828161 −0.414080 0.910240i \(-0.635897\pi\)
−0.414080 + 0.910240i \(0.635897\pi\)
\(294\) 0 0
\(295\) −12.5783 −0.732339
\(296\) −7.93051 −0.460952
\(297\) −34.9200 −2.02626
\(298\) −15.4600 −0.895572
\(299\) 0 0
\(300\) 11.6655 0.673509
\(301\) 0 0
\(302\) 21.7350 1.25071
\(303\) −40.6550 −2.33557
\(304\) −4.00000 −0.229416
\(305\) 5.62721 0.322213
\(306\) 6.30330 0.360336
\(307\) 13.5592 0.773863 0.386932 0.922108i \(-0.373535\pi\)
0.386932 + 0.922108i \(0.373535\pi\)
\(308\) 0 0
\(309\) −13.6867 −0.778606
\(310\) 7.10278 0.403411
\(311\) −0.426686 −0.0241952 −0.0120976 0.999927i \(-0.503851\pi\)
−0.0120976 + 0.999927i \(0.503851\pi\)
\(312\) 0 0
\(313\) 18.1517 1.02599 0.512996 0.858391i \(-0.328536\pi\)
0.512996 + 0.858391i \(0.328536\pi\)
\(314\) 23.2645 1.31289
\(315\) 0 0
\(316\) −17.4600 −0.982200
\(317\) −9.42166 −0.529173 −0.264587 0.964362i \(-0.585236\pi\)
−0.264587 + 0.964362i \(0.585236\pi\)
\(318\) −14.0383 −0.787230
\(319\) −25.7194 −1.44001
\(320\) −4.67609 −0.261401
\(321\) 1.79445 0.100156
\(322\) 0 0
\(323\) 0.426686 0.0237415
\(324\) 19.3869 1.07705
\(325\) 0 0
\(326\) 24.4111 1.35201
\(327\) −17.2927 −0.956291
\(328\) 5.42166 0.299361
\(329\) 0 0
\(330\) −49.1255 −2.70427
\(331\) 17.4005 0.956420 0.478210 0.878246i \(-0.341286\pi\)
0.478210 + 0.878246i \(0.341286\pi\)
\(332\) 21.2444 1.16594
\(333\) 40.7683 2.23409
\(334\) −3.68111 −0.201421
\(335\) −28.2439 −1.54313
\(336\) 0 0
\(337\) 22.0524 1.20127 0.600637 0.799522i \(-0.294914\pi\)
0.600637 + 0.799522i \(0.294914\pi\)
\(338\) 0 0
\(339\) 16.8972 0.917731
\(340\) 1.90225 0.103164
\(341\) −4.31889 −0.233881
\(342\) 9.77886 0.528780
\(343\) 0 0
\(344\) −8.71440 −0.469849
\(345\) 64.0610 3.44893
\(346\) −36.8122 −1.97903
\(347\) 25.3522 1.36098 0.680488 0.732759i \(-0.261768\pi\)
0.680488 + 0.732759i \(0.261768\pi\)
\(348\) 33.1567 1.77738
\(349\) −5.70529 −0.305397 −0.152699 0.988273i \(-0.548796\pi\)
−0.152699 + 0.988273i \(0.548796\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 19.6655 1.04818
\(353\) −28.6761 −1.52627 −0.763137 0.646237i \(-0.776342\pi\)
−0.763137 + 0.646237i \(0.776342\pi\)
\(354\) −25.1567 −1.33706
\(355\) 24.5628 1.30366
\(356\) −13.7250 −0.727422
\(357\) 0 0
\(358\) −19.9561 −1.05472
\(359\) −11.0433 −0.582845 −0.291423 0.956594i \(-0.594129\pi\)
−0.291423 + 0.956594i \(0.594129\pi\)
\(360\) −24.0383 −1.26693
\(361\) −18.3380 −0.965160
\(362\) −1.25443 −0.0659312
\(363\) −4.25945 −0.223563
\(364\) 0 0
\(365\) −6.59247 −0.345066
\(366\) 11.2544 0.588278
\(367\) −27.3466 −1.42748 −0.713741 0.700409i \(-0.753001\pi\)
−0.713741 + 0.700409i \(0.753001\pi\)
\(368\) −36.0766 −1.88062
\(369\) −27.8711 −1.45091
\(370\) 31.3905 1.63191
\(371\) 0 0
\(372\) 5.56777 0.288676
\(373\) 16.1461 0.836014 0.418007 0.908444i \(-0.362729\pi\)
0.418007 + 0.908444i \(0.362729\pi\)
\(374\) −2.95112 −0.152599
\(375\) −18.1900 −0.939326
\(376\) 7.69670 0.396927
\(377\) 0 0
\(378\) 0 0
\(379\) −26.1305 −1.34223 −0.671117 0.741351i \(-0.734185\pi\)
−0.671117 + 0.741351i \(0.734185\pi\)
\(380\) 2.95112 0.151389
\(381\) −39.9688 −2.04767
\(382\) 14.2056 0.726819
\(383\) −21.0489 −1.07555 −0.537774 0.843089i \(-0.680735\pi\)
−0.537774 + 0.843089i \(0.680735\pi\)
\(384\) 29.9789 1.52985
\(385\) 0 0
\(386\) 22.1361 1.12670
\(387\) 44.7980 2.27721
\(388\) −1.52946 −0.0776466
\(389\) −21.6061 −1.09547 −0.547736 0.836651i \(-0.684510\pi\)
−0.547736 + 0.836651i \(0.684510\pi\)
\(390\) 0 0
\(391\) 3.84835 0.194619
\(392\) 0 0
\(393\) −28.0766 −1.41628
\(394\) 34.1361 1.71975
\(395\) −38.1063 −1.91734
\(396\) −26.5089 −1.33212
\(397\) −27.6952 −1.38998 −0.694992 0.719017i \(-0.744592\pi\)
−0.694992 + 0.719017i \(0.744592\pi\)
\(398\) −39.1950 −1.96467
\(399\) 0 0
\(400\) −14.3380 −0.716902
\(401\) 2.57834 0.128756 0.0643780 0.997926i \(-0.479494\pi\)
0.0643780 + 0.997926i \(0.479494\pi\)
\(402\) −56.4877 −2.81735
\(403\) 0 0
\(404\) −16.8917 −0.840393
\(405\) 42.3119 2.10250
\(406\) 0 0
\(407\) −19.0872 −0.946117
\(408\) −2.09775 −0.103854
\(409\) 15.1169 0.747483 0.373742 0.927533i \(-0.378075\pi\)
0.373742 + 0.927533i \(0.378075\pi\)
\(410\) −21.4600 −1.05983
\(411\) 19.4217 0.958000
\(412\) −5.68665 −0.280161
\(413\) 0 0
\(414\) 88.1971 4.33465
\(415\) 46.3658 2.27601
\(416\) 0 0
\(417\) 35.9094 1.75849
\(418\) −4.57834 −0.223934
\(419\) 9.99446 0.488261 0.244131 0.969742i \(-0.421497\pi\)
0.244131 + 0.969742i \(0.421497\pi\)
\(420\) 0 0
\(421\) 25.9250 1.26351 0.631753 0.775170i \(-0.282336\pi\)
0.631753 + 0.775170i \(0.282336\pi\)
\(422\) −31.5139 −1.53407
\(423\) −39.5663 −1.92378
\(424\) 3.21611 0.156188
\(425\) 1.52946 0.0741898
\(426\) 49.1255 2.38014
\(427\) 0 0
\(428\) 0.745574 0.0360387
\(429\) 0 0
\(430\) 34.4933 1.66341
\(431\) −30.6761 −1.47762 −0.738808 0.673916i \(-0.764611\pi\)
−0.738808 + 0.673916i \(0.764611\pi\)
\(432\) −55.3311 −2.66212
\(433\) 3.51941 0.169132 0.0845661 0.996418i \(-0.473050\pi\)
0.0845661 + 0.996418i \(0.473050\pi\)
\(434\) 0 0
\(435\) 72.3643 3.46960
\(436\) −7.18494 −0.344096
\(437\) 5.97028 0.285597
\(438\) −13.1849 −0.630001
\(439\) −32.3517 −1.54406 −0.772030 0.635586i \(-0.780759\pi\)
−0.772030 + 0.635586i \(0.780759\pi\)
\(440\) 11.2544 0.536534
\(441\) 0 0
\(442\) 0 0
\(443\) 15.4458 0.733854 0.366927 0.930250i \(-0.380410\pi\)
0.366927 + 0.930250i \(0.380410\pi\)
\(444\) 24.6066 1.16778
\(445\) −29.9547 −1.41999
\(446\) −19.1310 −0.905881
\(447\) −26.4494 −1.25101
\(448\) 0 0
\(449\) 14.4705 0.682907 0.341453 0.939899i \(-0.389081\pi\)
0.341453 + 0.939899i \(0.389081\pi\)
\(450\) 35.0524 1.65239
\(451\) 13.0489 0.614448
\(452\) 7.02061 0.330222
\(453\) 37.1849 1.74710
\(454\) −12.6066 −0.591657
\(455\) 0 0
\(456\) −3.25443 −0.152402
\(457\) 34.6705 1.62182 0.810910 0.585171i \(-0.198973\pi\)
0.810910 + 0.585171i \(0.198973\pi\)
\(458\) −38.2439 −1.78702
\(459\) 5.90225 0.275493
\(460\) 26.6167 1.24101
\(461\) 12.5400 0.584047 0.292024 0.956411i \(-0.405671\pi\)
0.292024 + 0.956411i \(0.405671\pi\)
\(462\) 0 0
\(463\) −12.1517 −0.564735 −0.282368 0.959306i \(-0.591120\pi\)
−0.282368 + 0.959306i \(0.591120\pi\)
\(464\) −40.7527 −1.89190
\(465\) 12.1517 0.563519
\(466\) 11.0333 0.511107
\(467\) 37.0333 1.71370 0.856848 0.515569i \(-0.172419\pi\)
0.856848 + 0.515569i \(0.172419\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −30.4650 −1.40525
\(471\) 39.8016 1.83396
\(472\) 5.76328 0.265276
\(473\) −20.9739 −0.964379
\(474\) −76.2127 −3.50056
\(475\) 2.37279 0.108871
\(476\) 0 0
\(477\) −16.5330 −0.756996
\(478\) 25.7633 1.17838
\(479\) 12.0086 0.548687 0.274343 0.961632i \(-0.411540\pi\)
0.274343 + 0.961632i \(0.411540\pi\)
\(480\) −55.3311 −2.52551
\(481\) 0 0
\(482\) 15.3083 0.697275
\(483\) 0 0
\(484\) −1.76975 −0.0804434
\(485\) −3.33804 −0.151573
\(486\) 23.3905 1.06101
\(487\) −11.1184 −0.503821 −0.251911 0.967751i \(-0.581059\pi\)
−0.251911 + 0.967751i \(0.581059\pi\)
\(488\) −2.57834 −0.116716
\(489\) 41.7633 1.88860
\(490\) 0 0
\(491\) 0.0594386 0.00268243 0.00134121 0.999999i \(-0.499573\pi\)
0.00134121 + 0.999999i \(0.499573\pi\)
\(492\) −16.8222 −0.758403
\(493\) 4.34715 0.195786
\(494\) 0 0
\(495\) −57.8555 −2.60041
\(496\) −6.84333 −0.307274
\(497\) 0 0
\(498\) 92.7316 4.15540
\(499\) −10.2978 −0.460991 −0.230496 0.973073i \(-0.574035\pi\)
−0.230496 + 0.973073i \(0.574035\pi\)
\(500\) −7.55773 −0.337992
\(501\) −6.29776 −0.281363
\(502\) 43.4983 1.94142
\(503\) 9.32391 0.415733 0.207866 0.978157i \(-0.433348\pi\)
0.207866 + 0.978157i \(0.433348\pi\)
\(504\) 0 0
\(505\) −36.8661 −1.64052
\(506\) −41.2927 −1.83569
\(507\) 0 0
\(508\) −16.6066 −0.736799
\(509\) 39.6952 1.75946 0.879730 0.475473i \(-0.157723\pi\)
0.879730 + 0.475473i \(0.157723\pi\)
\(510\) 8.30330 0.367676
\(511\) 0 0
\(512\) 18.4842 0.816892
\(513\) 9.15667 0.404277
\(514\) −28.3133 −1.24885
\(515\) −12.4111 −0.546898
\(516\) 27.0388 1.19032
\(517\) 18.5244 0.814704
\(518\) 0 0
\(519\) −62.9794 −2.76449
\(520\) 0 0
\(521\) −22.3627 −0.979729 −0.489865 0.871798i \(-0.662954\pi\)
−0.489865 + 0.871798i \(0.662954\pi\)
\(522\) 99.6288 4.36063
\(523\) 20.6550 0.903178 0.451589 0.892226i \(-0.350857\pi\)
0.451589 + 0.892226i \(0.350857\pi\)
\(524\) −11.6655 −0.509611
\(525\) 0 0
\(526\) −27.5139 −1.19966
\(527\) 0.729988 0.0317988
\(528\) 47.3311 2.05982
\(529\) 30.8469 1.34117
\(530\) −12.7300 −0.552955
\(531\) −29.6272 −1.28571
\(532\) 0 0
\(533\) 0 0
\(534\) −59.9094 −2.59253
\(535\) 1.62721 0.0703506
\(536\) 12.9411 0.558969
\(537\) −34.1416 −1.47332
\(538\) −42.2439 −1.82126
\(539\) 0 0
\(540\) 40.8222 1.75671
\(541\) −5.62167 −0.241695 −0.120847 0.992671i \(-0.538561\pi\)
−0.120847 + 0.992671i \(0.538561\pi\)
\(542\) 23.1155 0.992894
\(543\) −2.14611 −0.0920985
\(544\) −3.32391 −0.142512
\(545\) −15.6811 −0.671705
\(546\) 0 0
\(547\) −10.3970 −0.444542 −0.222271 0.974985i \(-0.571347\pi\)
−0.222271 + 0.974985i \(0.571347\pi\)
\(548\) 8.06949 0.344711
\(549\) 13.2544 0.565685
\(550\) −16.4111 −0.699772
\(551\) 6.74412 0.287309
\(552\) −29.3522 −1.24931
\(553\) 0 0
\(554\) 14.7300 0.625817
\(555\) 53.7038 2.27960
\(556\) 14.9200 0.632747
\(557\) −14.6550 −0.620951 −0.310475 0.950581i \(-0.600488\pi\)
−0.310475 + 0.950581i \(0.600488\pi\)
\(558\) 16.7300 0.708237
\(559\) 0 0
\(560\) 0 0
\(561\) −5.04888 −0.213164
\(562\) −34.5189 −1.45609
\(563\) 24.7456 1.04290 0.521451 0.853281i \(-0.325391\pi\)
0.521451 + 0.853281i \(0.325391\pi\)
\(564\) −23.8811 −1.00558
\(565\) 15.3225 0.644621
\(566\) −20.2156 −0.849725
\(567\) 0 0
\(568\) −11.2544 −0.472225
\(569\) −20.5330 −0.860789 −0.430395 0.902641i \(-0.641626\pi\)
−0.430395 + 0.902641i \(0.641626\pi\)
\(570\) 12.8816 0.539552
\(571\) −41.8953 −1.75326 −0.876631 0.481163i \(-0.840214\pi\)
−0.876631 + 0.481163i \(0.840214\pi\)
\(572\) 0 0
\(573\) 24.3033 1.01529
\(574\) 0 0
\(575\) 21.4005 0.892464
\(576\) −11.0141 −0.458922
\(577\) −20.1744 −0.839870 −0.419935 0.907554i \(-0.637947\pi\)
−0.419935 + 0.907554i \(0.637947\pi\)
\(578\) −30.3325 −1.26167
\(579\) 37.8711 1.57387
\(580\) 30.0666 1.24845
\(581\) 0 0
\(582\) −6.67609 −0.276733
\(583\) 7.74055 0.320581
\(584\) 3.02061 0.124994
\(585\) 0 0
\(586\) −25.7094 −1.06204
\(587\) −18.7441 −0.773653 −0.386826 0.922153i \(-0.626429\pi\)
−0.386826 + 0.922153i \(0.626429\pi\)
\(588\) 0 0
\(589\) 1.13249 0.0466636
\(590\) −22.8122 −0.939162
\(591\) 58.4011 2.40230
\(592\) −30.2439 −1.24302
\(593\) −2.98084 −0.122409 −0.0612043 0.998125i \(-0.519494\pi\)
−0.0612043 + 0.998125i \(0.519494\pi\)
\(594\) −63.3311 −2.59850
\(595\) 0 0
\(596\) −10.9894 −0.450145
\(597\) −67.0560 −2.74442
\(598\) 0 0
\(599\) 7.47411 0.305384 0.152692 0.988274i \(-0.451206\pi\)
0.152692 + 0.988274i \(0.451206\pi\)
\(600\) −11.6655 −0.476243
\(601\) −21.4700 −0.875780 −0.437890 0.899028i \(-0.644274\pi\)
−0.437890 + 0.899028i \(0.644274\pi\)
\(602\) 0 0
\(603\) −66.5260 −2.70915
\(604\) 15.4499 0.628649
\(605\) −3.86248 −0.157032
\(606\) −73.7321 −2.99516
\(607\) 22.9044 0.929660 0.464830 0.885400i \(-0.346116\pi\)
0.464830 + 0.885400i \(0.346116\pi\)
\(608\) −5.15667 −0.209131
\(609\) 0 0
\(610\) 10.2056 0.413211
\(611\) 0 0
\(612\) 4.48059 0.181117
\(613\) 20.1461 0.813694 0.406847 0.913496i \(-0.366628\pi\)
0.406847 + 0.913496i \(0.366628\pi\)
\(614\) 24.5910 0.992413
\(615\) −36.7144 −1.48047
\(616\) 0 0
\(617\) 13.7844 0.554939 0.277470 0.960734i \(-0.410504\pi\)
0.277470 + 0.960734i \(0.410504\pi\)
\(618\) −24.8222 −0.998495
\(619\) −19.6655 −0.790424 −0.395212 0.918590i \(-0.629329\pi\)
−0.395212 + 0.918590i \(0.629329\pi\)
\(620\) 5.04888 0.202768
\(621\) 82.5855 3.31404
\(622\) −0.773841 −0.0310282
\(623\) 0 0
\(624\) 0 0
\(625\) −31.0766 −1.24307
\(626\) 32.9200 1.31575
\(627\) −7.83276 −0.312810
\(628\) 16.5371 0.659903
\(629\) 3.22616 0.128635
\(630\) 0 0
\(631\) 16.1672 0.643608 0.321804 0.946806i \(-0.395711\pi\)
0.321804 + 0.946806i \(0.395711\pi\)
\(632\) 17.4600 0.694521
\(633\) −53.9149 −2.14293
\(634\) −17.0872 −0.678619
\(635\) −36.2439 −1.43829
\(636\) −9.97887 −0.395688
\(637\) 0 0
\(638\) −46.6449 −1.84669
\(639\) 57.8555 2.28873
\(640\) 27.1849 1.07458
\(641\) −29.0036 −1.14557 −0.572786 0.819705i \(-0.694137\pi\)
−0.572786 + 0.819705i \(0.694137\pi\)
\(642\) 3.25443 0.128442
\(643\) −39.2233 −1.54681 −0.773407 0.633910i \(-0.781449\pi\)
−0.773407 + 0.633910i \(0.781449\pi\)
\(644\) 0 0
\(645\) 59.0122 2.32360
\(646\) 0.773841 0.0304464
\(647\) 11.9844 0.471156 0.235578 0.971855i \(-0.424302\pi\)
0.235578 + 0.971855i \(0.424302\pi\)
\(648\) −19.3869 −0.761590
\(649\) 13.8711 0.544487
\(650\) 0 0
\(651\) 0 0
\(652\) 17.3522 0.679564
\(653\) 45.3311 1.77394 0.886971 0.461826i \(-0.152806\pi\)
0.886971 + 0.461826i \(0.152806\pi\)
\(654\) −31.3622 −1.22636
\(655\) −25.4600 −0.994804
\(656\) 20.6761 0.807266
\(657\) −15.5280 −0.605805
\(658\) 0 0
\(659\) 6.12193 0.238477 0.119238 0.992866i \(-0.461955\pi\)
0.119238 + 0.992866i \(0.461955\pi\)
\(660\) −34.9200 −1.35926
\(661\) −27.5280 −1.07072 −0.535358 0.844625i \(-0.679823\pi\)
−0.535358 + 0.844625i \(0.679823\pi\)
\(662\) 31.5577 1.22653
\(663\) 0 0
\(664\) −21.2444 −0.824442
\(665\) 0 0
\(666\) 73.9377 2.86503
\(667\) 60.8263 2.35520
\(668\) −2.61665 −0.101241
\(669\) −32.7300 −1.26541
\(670\) −51.2233 −1.97893
\(671\) −6.20555 −0.239563
\(672\) 0 0
\(673\) 27.9547 1.07757 0.538787 0.842442i \(-0.318883\pi\)
0.538787 + 0.842442i \(0.318883\pi\)
\(674\) 39.9945 1.54053
\(675\) 32.8222 1.26333
\(676\) 0 0
\(677\) 12.6605 0.486583 0.243291 0.969953i \(-0.421773\pi\)
0.243291 + 0.969953i \(0.421773\pi\)
\(678\) 30.6449 1.17691
\(679\) 0 0
\(680\) −1.90225 −0.0729479
\(681\) −21.5678 −0.826479
\(682\) −7.83276 −0.299932
\(683\) 28.3033 1.08300 0.541498 0.840702i \(-0.317857\pi\)
0.541498 + 0.840702i \(0.317857\pi\)
\(684\) 6.95112 0.265783
\(685\) 17.6116 0.672906
\(686\) 0 0
\(687\) −65.4288 −2.49626
\(688\) −33.2333 −1.26701
\(689\) 0 0
\(690\) 116.182 4.42295
\(691\) 12.2353 0.465452 0.232726 0.972542i \(-0.425236\pi\)
0.232726 + 0.972542i \(0.425236\pi\)
\(692\) −26.1672 −0.994729
\(693\) 0 0
\(694\) 45.9789 1.74533
\(695\) 32.5628 1.23518
\(696\) −33.1567 −1.25680
\(697\) −2.20555 −0.0835412
\(698\) −10.3472 −0.391646
\(699\) 18.8761 0.713960
\(700\) 0 0
\(701\) 51.0419 1.92783 0.963913 0.266219i \(-0.0857743\pi\)
0.963913 + 0.266219i \(0.0857743\pi\)
\(702\) 0 0
\(703\) 5.00502 0.188768
\(704\) 5.15667 0.194349
\(705\) −52.1205 −1.96297
\(706\) −52.0071 −1.95731
\(707\) 0 0
\(708\) −17.8822 −0.672053
\(709\) −42.5910 −1.59954 −0.799770 0.600307i \(-0.795045\pi\)
−0.799770 + 0.600307i \(0.795045\pi\)
\(710\) 44.5472 1.67183
\(711\) −89.7563 −3.36612
\(712\) 13.7250 0.514365
\(713\) 10.2141 0.382523
\(714\) 0 0
\(715\) 0 0
\(716\) −14.1855 −0.530135
\(717\) 44.0766 1.64607
\(718\) −20.0283 −0.747448
\(719\) 42.4933 1.58473 0.792366 0.610046i \(-0.208849\pi\)
0.792366 + 0.610046i \(0.208849\pi\)
\(720\) −91.6727 −3.41644
\(721\) 0 0
\(722\) −33.2580 −1.23773
\(723\) 26.1900 0.974015
\(724\) −0.891685 −0.0331392
\(725\) 24.1744 0.897814
\(726\) −7.72496 −0.286700
\(727\) 3.75614 0.139307 0.0696537 0.997571i \(-0.477811\pi\)
0.0696537 + 0.997571i \(0.477811\pi\)
\(728\) 0 0
\(729\) −5.09775 −0.188806
\(730\) −11.9561 −0.442517
\(731\) 3.54505 0.131118
\(732\) 8.00000 0.295689
\(733\) 45.7819 1.69099 0.845497 0.533980i \(-0.179304\pi\)
0.845497 + 0.533980i \(0.179304\pi\)
\(734\) −49.5960 −1.83062
\(735\) 0 0
\(736\) −46.5089 −1.71434
\(737\) 31.1466 1.14730
\(738\) −50.5472 −1.86067
\(739\) 14.0539 0.516981 0.258491 0.966014i \(-0.416775\pi\)
0.258491 + 0.966014i \(0.416775\pi\)
\(740\) 22.3133 0.820255
\(741\) 0 0
\(742\) 0 0
\(743\) −4.74557 −0.174098 −0.0870491 0.996204i \(-0.527744\pi\)
−0.0870491 + 0.996204i \(0.527744\pi\)
\(744\) −5.56777 −0.204125
\(745\) −23.9844 −0.878721
\(746\) 29.2827 1.07212
\(747\) 109.211 3.99581
\(748\) −2.09775 −0.0767014
\(749\) 0 0
\(750\) −32.9894 −1.20460
\(751\) 36.1008 1.31734 0.658669 0.752433i \(-0.271120\pi\)
0.658669 + 0.752433i \(0.271120\pi\)
\(752\) 29.3522 1.07036
\(753\) 74.4182 2.71195
\(754\) 0 0
\(755\) 33.7194 1.22718
\(756\) 0 0
\(757\) −1.03474 −0.0376084 −0.0188042 0.999823i \(-0.505986\pi\)
−0.0188042 + 0.999823i \(0.505986\pi\)
\(758\) −47.3905 −1.72130
\(759\) −70.6449 −2.56425
\(760\) −2.95112 −0.107049
\(761\) 29.8414 1.08175 0.540874 0.841104i \(-0.318094\pi\)
0.540874 + 0.841104i \(0.318094\pi\)
\(762\) −72.4877 −2.62595
\(763\) 0 0
\(764\) 10.0978 0.365324
\(765\) 9.77886 0.353556
\(766\) −38.1744 −1.37930
\(767\) 0 0
\(768\) 64.6832 2.33406
\(769\) 23.6358 0.852329 0.426164 0.904646i \(-0.359864\pi\)
0.426164 + 0.904646i \(0.359864\pi\)
\(770\) 0 0
\(771\) −48.4394 −1.74450
\(772\) 15.7350 0.566315
\(773\) 13.0278 0.468576 0.234288 0.972167i \(-0.424724\pi\)
0.234288 + 0.972167i \(0.424724\pi\)
\(774\) 81.2460 2.92033
\(775\) 4.05944 0.145819
\(776\) 1.52946 0.0549045
\(777\) 0 0
\(778\) −39.1849 −1.40485
\(779\) −3.42166 −0.122594
\(780\) 0 0
\(781\) −27.0872 −0.969256
\(782\) 6.97939 0.249583
\(783\) 93.2898 3.33391
\(784\) 0 0
\(785\) 36.0922 1.28819
\(786\) −50.9200 −1.81625
\(787\) 46.2141 1.64736 0.823678 0.567058i \(-0.191918\pi\)
0.823678 + 0.567058i \(0.191918\pi\)
\(788\) 24.2650 0.864404
\(789\) −47.0716 −1.67579
\(790\) −69.1099 −2.45882
\(791\) 0 0
\(792\) 26.5089 0.941951
\(793\) 0 0
\(794\) −50.2283 −1.78253
\(795\) −21.7789 −0.772417
\(796\) −27.8610 −0.987508
\(797\) −53.1155 −1.88145 −0.940723 0.339176i \(-0.889852\pi\)
−0.940723 + 0.339176i \(0.889852\pi\)
\(798\) 0 0
\(799\) −3.13104 −0.110768
\(800\) −18.4842 −0.653514
\(801\) −70.5558 −2.49297
\(802\) 4.67609 0.165118
\(803\) 7.27001 0.256553
\(804\) −40.1533 −1.41610
\(805\) 0 0
\(806\) 0 0
\(807\) −72.2721 −2.54410
\(808\) 16.8917 0.594247
\(809\) −54.4635 −1.91484 −0.957418 0.288705i \(-0.906775\pi\)
−0.957418 + 0.288705i \(0.906775\pi\)
\(810\) 76.7371 2.69627
\(811\) 38.0978 1.33779 0.668897 0.743356i \(-0.266767\pi\)
0.668897 + 0.743356i \(0.266767\pi\)
\(812\) 0 0
\(813\) 39.5466 1.38696
\(814\) −34.6167 −1.21331
\(815\) 37.8711 1.32657
\(816\) −8.00000 −0.280056
\(817\) 5.49974 0.192412
\(818\) 27.4161 0.958582
\(819\) 0 0
\(820\) −15.2544 −0.532708
\(821\) 2.30330 0.0803858 0.0401929 0.999192i \(-0.487203\pi\)
0.0401929 + 0.999192i \(0.487203\pi\)
\(822\) 35.2233 1.22855
\(823\) 23.6172 0.823243 0.411621 0.911355i \(-0.364963\pi\)
0.411621 + 0.911355i \(0.364963\pi\)
\(824\) 5.68665 0.198104
\(825\) −28.0766 −0.977503
\(826\) 0 0
\(827\) 48.1643 1.67484 0.837419 0.546562i \(-0.184064\pi\)
0.837419 + 0.546562i \(0.184064\pi\)
\(828\) 62.6933 2.17874
\(829\) −13.0716 −0.453996 −0.226998 0.973895i \(-0.572891\pi\)
−0.226998 + 0.973895i \(0.572891\pi\)
\(830\) 84.0893 2.91878
\(831\) 25.2005 0.874197
\(832\) 0 0
\(833\) 0 0
\(834\) 65.1255 2.25511
\(835\) −5.71083 −0.197631
\(836\) −3.25443 −0.112557
\(837\) 15.6655 0.541480
\(838\) 18.1260 0.626153
\(839\) −17.6756 −0.610229 −0.305114 0.952316i \(-0.598695\pi\)
−0.305114 + 0.952316i \(0.598695\pi\)
\(840\) 0 0
\(841\) 39.7103 1.36932
\(842\) 47.0177 1.62034
\(843\) −59.0560 −2.03400
\(844\) −22.4011 −0.771076
\(845\) 0 0
\(846\) −71.7577 −2.46708
\(847\) 0 0
\(848\) 12.2650 0.421181
\(849\) −34.5855 −1.18697
\(850\) 2.77384 0.0951420
\(851\) 45.1411 1.54742
\(852\) 34.9200 1.19634
\(853\) −5.48970 −0.187964 −0.0939818 0.995574i \(-0.529960\pi\)
−0.0939818 + 0.995574i \(0.529960\pi\)
\(854\) 0 0
\(855\) 15.1708 0.518831
\(856\) −0.745574 −0.0254832
\(857\) 11.0489 0.377422 0.188711 0.982033i \(-0.439569\pi\)
0.188711 + 0.982033i \(0.439569\pi\)
\(858\) 0 0
\(859\) −45.2616 −1.54430 −0.772152 0.635437i \(-0.780820\pi\)
−0.772152 + 0.635437i \(0.780820\pi\)
\(860\) 24.5189 0.836087
\(861\) 0 0
\(862\) −55.6344 −1.89491
\(863\) 5.90225 0.200915 0.100457 0.994941i \(-0.467969\pi\)
0.100457 + 0.994941i \(0.467969\pi\)
\(864\) −71.3311 −2.42673
\(865\) −57.1099 −1.94180
\(866\) 6.38283 0.216898
\(867\) −51.8938 −1.76241
\(868\) 0 0
\(869\) 42.0227 1.42552
\(870\) 131.240 4.44947
\(871\) 0 0
\(872\) 7.18494 0.243313
\(873\) −7.86248 −0.266105
\(874\) 10.8277 0.366254
\(875\) 0 0
\(876\) −9.37227 −0.316660
\(877\) 4.90727 0.165707 0.0828534 0.996562i \(-0.473597\pi\)
0.0828534 + 0.996562i \(0.473597\pi\)
\(878\) −58.6732 −1.98012
\(879\) −43.9844 −1.48356
\(880\) 42.9200 1.44683
\(881\) −44.2822 −1.49190 −0.745952 0.665999i \(-0.768005\pi\)
−0.745952 + 0.665999i \(0.768005\pi\)
\(882\) 0 0
\(883\) −58.8605 −1.98081 −0.990407 0.138181i \(-0.955874\pi\)
−0.990407 + 0.138181i \(0.955874\pi\)
\(884\) 0 0
\(885\) −39.0278 −1.31190
\(886\) 28.0127 0.941104
\(887\) 10.1289 0.340096 0.170048 0.985436i \(-0.445608\pi\)
0.170048 + 0.985436i \(0.445608\pi\)
\(888\) −24.6066 −0.825744
\(889\) 0 0
\(890\) −54.3260 −1.82101
\(891\) −46.6605 −1.56319
\(892\) −13.5989 −0.455326
\(893\) −4.85746 −0.162549
\(894\) −47.9688 −1.60432
\(895\) −30.9597 −1.03487
\(896\) 0 0
\(897\) 0 0
\(898\) 26.2439 0.875769
\(899\) 11.5381 0.384816
\(900\) 24.9164 0.830546
\(901\) −1.30833 −0.0435866
\(902\) 23.6655 0.787976
\(903\) 0 0
\(904\) −7.02061 −0.233502
\(905\) −1.94610 −0.0646906
\(906\) 67.4389 2.24051
\(907\) 37.9547 1.26026 0.630132 0.776488i \(-0.283001\pi\)
0.630132 + 0.776488i \(0.283001\pi\)
\(908\) −8.96117 −0.297387
\(909\) −86.8349 −2.88013
\(910\) 0 0
\(911\) 5.57477 0.184700 0.0923501 0.995727i \(-0.470562\pi\)
0.0923501 + 0.995727i \(0.470562\pi\)
\(912\) −12.4111 −0.410973
\(913\) −51.1310 −1.69219
\(914\) 62.8787 2.07984
\(915\) 17.4600 0.577209
\(916\) −27.1849 −0.898216
\(917\) 0 0
\(918\) 10.7044 0.353296
\(919\) 15.7844 0.520679 0.260340 0.965517i \(-0.416165\pi\)
0.260340 + 0.965517i \(0.416165\pi\)
\(920\) −26.6167 −0.877525
\(921\) 42.0711 1.38629
\(922\) 22.7427 0.748990
\(923\) 0 0
\(924\) 0 0
\(925\) 17.9406 0.589882
\(926\) −22.0383 −0.724224
\(927\) −29.2333 −0.960148
\(928\) −52.5371 −1.72462
\(929\) −45.2630 −1.48503 −0.742516 0.669829i \(-0.766368\pi\)
−0.742516 + 0.669829i \(0.766368\pi\)
\(930\) 22.0383 0.722665
\(931\) 0 0
\(932\) 7.84281 0.256900
\(933\) −1.32391 −0.0433429
\(934\) 67.1638 2.19767
\(935\) −4.57834 −0.149728
\(936\) 0 0
\(937\) −53.6188 −1.75165 −0.875824 0.482630i \(-0.839682\pi\)
−0.875824 + 0.482630i \(0.839682\pi\)
\(938\) 0 0
\(939\) 56.3205 1.83795
\(940\) −21.6555 −0.706324
\(941\) 20.7753 0.677255 0.338628 0.940920i \(-0.390037\pi\)
0.338628 + 0.940920i \(0.390037\pi\)
\(942\) 72.1844 2.35190
\(943\) −30.8605 −1.00496
\(944\) 21.9789 0.715351
\(945\) 0 0
\(946\) −38.0383 −1.23673
\(947\) 10.8605 0.352919 0.176460 0.984308i \(-0.443535\pi\)
0.176460 + 0.984308i \(0.443535\pi\)
\(948\) −54.1744 −1.75950
\(949\) 0 0
\(950\) 4.30330 0.139618
\(951\) −29.2333 −0.947955
\(952\) 0 0
\(953\) −25.7180 −0.833087 −0.416543 0.909116i \(-0.636759\pi\)
−0.416543 + 0.909116i \(0.636759\pi\)
\(954\) −29.9844 −0.970781
\(955\) 22.0383 0.713143
\(956\) 18.3133 0.592296
\(957\) −79.8016 −2.57962
\(958\) 21.7789 0.703643
\(959\) 0 0
\(960\) −14.5089 −0.468271
\(961\) −29.0625 −0.937500
\(962\) 0 0
\(963\) 3.83276 0.123509
\(964\) 10.8816 0.350474
\(965\) 34.3416 1.10550
\(966\) 0 0
\(967\) −33.5038 −1.07741 −0.538705 0.842494i \(-0.681086\pi\)
−0.538705 + 0.842494i \(0.681086\pi\)
\(968\) 1.76975 0.0568820
\(969\) 1.32391 0.0425302
\(970\) −6.05390 −0.194379
\(971\) −2.03831 −0.0654126 −0.0327063 0.999465i \(-0.510413\pi\)
−0.0327063 + 0.999465i \(0.510413\pi\)
\(972\) 16.6267 0.533302
\(973\) 0 0
\(974\) −20.1643 −0.646107
\(975\) 0 0
\(976\) −9.83276 −0.314739
\(977\) −15.1411 −0.484406 −0.242203 0.970226i \(-0.577870\pi\)
−0.242203 + 0.970226i \(0.577870\pi\)
\(978\) 75.7422 2.42197
\(979\) 33.0333 1.05575
\(980\) 0 0
\(981\) −36.9355 −1.17926
\(982\) 0.107798 0.00343998
\(983\) 49.3124 1.57282 0.786411 0.617704i \(-0.211937\pi\)
0.786411 + 0.617704i \(0.211937\pi\)
\(984\) 16.8222 0.536272
\(985\) 52.9583 1.68739
\(986\) 7.88403 0.251079
\(987\) 0 0
\(988\) 0 0
\(989\) 49.6030 1.57728
\(990\) −104.927 −3.33480
\(991\) 5.43171 0.172544 0.0862720 0.996272i \(-0.472505\pi\)
0.0862720 + 0.996272i \(0.472505\pi\)
\(992\) −8.82220 −0.280105
\(993\) 53.9900 1.71332
\(994\) 0 0
\(995\) −60.8066 −1.92770
\(996\) 65.9165 2.08865
\(997\) 53.6061 1.69772 0.848861 0.528616i \(-0.177289\pi\)
0.848861 + 0.528616i \(0.177289\pi\)
\(998\) −18.6761 −0.591181
\(999\) 69.2333 2.19044
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 8281.2.a.bg.1.3 3
7.6 odd 2 1183.2.a.i.1.3 3
13.12 even 2 637.2.a.j.1.1 3
39.38 odd 2 5733.2.a.x.1.3 3
91.12 odd 6 637.2.e.j.508.3 6
91.25 even 6 637.2.e.i.79.3 6
91.34 even 4 1183.2.c.f.337.5 6
91.38 odd 6 637.2.e.j.79.3 6
91.51 even 6 637.2.e.i.508.3 6
91.83 even 4 1183.2.c.f.337.2 6
91.90 odd 2 91.2.a.d.1.1 3
273.272 even 2 819.2.a.i.1.3 3
364.363 even 2 1456.2.a.t.1.3 3
455.454 odd 2 2275.2.a.m.1.3 3
728.181 odd 2 5824.2.a.by.1.3 3
728.363 even 2 5824.2.a.bs.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.a.d.1.1 3 91.90 odd 2
637.2.a.j.1.1 3 13.12 even 2
637.2.e.i.79.3 6 91.25 even 6
637.2.e.i.508.3 6 91.51 even 6
637.2.e.j.79.3 6 91.38 odd 6
637.2.e.j.508.3 6 91.12 odd 6
819.2.a.i.1.3 3 273.272 even 2
1183.2.a.i.1.3 3 7.6 odd 2
1183.2.c.f.337.2 6 91.83 even 4
1183.2.c.f.337.5 6 91.34 even 4
1456.2.a.t.1.3 3 364.363 even 2
2275.2.a.m.1.3 3 455.454 odd 2
5733.2.a.x.1.3 3 39.38 odd 2
5824.2.a.bs.1.1 3 728.363 even 2
5824.2.a.by.1.3 3 728.181 odd 2
8281.2.a.bg.1.3 3 1.1 even 1 trivial