Properties

Label 297.2.a.e.1.1
Level $297$
Weight $2$
Character 297.1
Self dual yes
Analytic conductor $2.372$
Analytic rank $1$
Dimension $2$
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] = [297,2,Mod(1,297)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(297, 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("297.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 297 = 3^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 297.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: \(2.37155694003\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 297.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.73205 q^{2} +5.46410 q^{4} +0.732051 q^{5} -3.73205 q^{7} -9.46410 q^{8} +O(q^{10})\) \(q-2.73205 q^{2} +5.46410 q^{4} +0.732051 q^{5} -3.73205 q^{7} -9.46410 q^{8} -2.00000 q^{10} -1.00000 q^{11} -0.267949 q^{13} +10.1962 q^{14} +14.9282 q^{16} +4.19615 q^{17} -5.19615 q^{19} +4.00000 q^{20} +2.73205 q^{22} -8.00000 q^{23} -4.46410 q^{25} +0.732051 q^{26} -20.3923 q^{28} +1.26795 q^{29} +0.535898 q^{31} -21.8564 q^{32} -11.4641 q^{34} -2.73205 q^{35} -6.46410 q^{37} +14.1962 q^{38} -6.92820 q^{40} -1.46410 q^{41} +3.46410 q^{43} -5.46410 q^{44} +21.8564 q^{46} -10.1962 q^{47} +6.92820 q^{49} +12.1962 q^{50} -1.46410 q^{52} -4.73205 q^{53} -0.732051 q^{55} +35.3205 q^{56} -3.46410 q^{58} -10.1962 q^{59} -4.26795 q^{61} -1.46410 q^{62} +29.8564 q^{64} -0.196152 q^{65} +5.92820 q^{67} +22.9282 q^{68} +7.46410 q^{70} +13.8564 q^{71} +3.19615 q^{73} +17.6603 q^{74} -28.3923 q^{76} +3.73205 q^{77} -5.73205 q^{79} +10.9282 q^{80} +4.00000 q^{82} +3.07180 q^{85} -9.46410 q^{86} +9.46410 q^{88} +0.339746 q^{89} +1.00000 q^{91} -43.7128 q^{92} +27.8564 q^{94} -3.80385 q^{95} -5.39230 q^{97} -18.9282 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 4 q^{4} - 2 q^{5} - 4 q^{7} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 4 q^{4} - 2 q^{5} - 4 q^{7} - 12 q^{8} - 4 q^{10} - 2 q^{11} - 4 q^{13} + 10 q^{14} + 16 q^{16} - 2 q^{17} + 8 q^{20} + 2 q^{22} - 16 q^{23} - 2 q^{25} - 2 q^{26} - 20 q^{28} + 6 q^{29} + 8 q^{31} - 16 q^{32} - 16 q^{34} - 2 q^{35} - 6 q^{37} + 18 q^{38} + 4 q^{41} - 4 q^{44} + 16 q^{46} - 10 q^{47} + 14 q^{50} + 4 q^{52} - 6 q^{53} + 2 q^{55} + 36 q^{56} - 10 q^{59} - 12 q^{61} + 4 q^{62} + 32 q^{64} + 10 q^{65} - 2 q^{67} + 32 q^{68} + 8 q^{70} - 4 q^{73} + 18 q^{74} - 36 q^{76} + 4 q^{77} - 8 q^{79} + 8 q^{80} + 8 q^{82} + 20 q^{85} - 12 q^{86} + 12 q^{88} + 18 q^{89} + 2 q^{91} - 32 q^{92} + 28 q^{94} - 18 q^{95} + 10 q^{97} - 24 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.73205 −1.93185 −0.965926 0.258819i \(-0.916667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(3\) 0 0
\(4\) 5.46410 2.73205
\(5\) 0.732051 0.327383 0.163692 0.986512i \(-0.447660\pi\)
0.163692 + 0.986512i \(0.447660\pi\)
\(6\) 0 0
\(7\) −3.73205 −1.41058 −0.705291 0.708918i \(-0.749184\pi\)
−0.705291 + 0.708918i \(0.749184\pi\)
\(8\) −9.46410 −3.34607
\(9\) 0 0
\(10\) −2.00000 −0.632456
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −0.267949 −0.0743157 −0.0371579 0.999309i \(-0.511830\pi\)
−0.0371579 + 0.999309i \(0.511830\pi\)
\(14\) 10.1962 2.72504
\(15\) 0 0
\(16\) 14.9282 3.73205
\(17\) 4.19615 1.01772 0.508858 0.860850i \(-0.330068\pi\)
0.508858 + 0.860850i \(0.330068\pi\)
\(18\) 0 0
\(19\) −5.19615 −1.19208 −0.596040 0.802955i \(-0.703260\pi\)
−0.596040 + 0.802955i \(0.703260\pi\)
\(20\) 4.00000 0.894427
\(21\) 0 0
\(22\) 2.73205 0.582475
\(23\) −8.00000 −1.66812 −0.834058 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 0 0
\(25\) −4.46410 −0.892820
\(26\) 0.732051 0.143567
\(27\) 0 0
\(28\) −20.3923 −3.85378
\(29\) 1.26795 0.235452 0.117726 0.993046i \(-0.462440\pi\)
0.117726 + 0.993046i \(0.462440\pi\)
\(30\) 0 0
\(31\) 0.535898 0.0962502 0.0481251 0.998841i \(-0.484675\pi\)
0.0481251 + 0.998841i \(0.484675\pi\)
\(32\) −21.8564 −3.86370
\(33\) 0 0
\(34\) −11.4641 −1.96608
\(35\) −2.73205 −0.461801
\(36\) 0 0
\(37\) −6.46410 −1.06269 −0.531346 0.847155i \(-0.678314\pi\)
−0.531346 + 0.847155i \(0.678314\pi\)
\(38\) 14.1962 2.30292
\(39\) 0 0
\(40\) −6.92820 −1.09545
\(41\) −1.46410 −0.228654 −0.114327 0.993443i \(-0.536471\pi\)
−0.114327 + 0.993443i \(0.536471\pi\)
\(42\) 0 0
\(43\) 3.46410 0.528271 0.264135 0.964486i \(-0.414913\pi\)
0.264135 + 0.964486i \(0.414913\pi\)
\(44\) −5.46410 −0.823744
\(45\) 0 0
\(46\) 21.8564 3.22255
\(47\) −10.1962 −1.48726 −0.743631 0.668590i \(-0.766898\pi\)
−0.743631 + 0.668590i \(0.766898\pi\)
\(48\) 0 0
\(49\) 6.92820 0.989743
\(50\) 12.1962 1.72480
\(51\) 0 0
\(52\) −1.46410 −0.203034
\(53\) −4.73205 −0.649997 −0.324999 0.945715i \(-0.605364\pi\)
−0.324999 + 0.945715i \(0.605364\pi\)
\(54\) 0 0
\(55\) −0.732051 −0.0987097
\(56\) 35.3205 4.71990
\(57\) 0 0
\(58\) −3.46410 −0.454859
\(59\) −10.1962 −1.32743 −0.663713 0.747987i \(-0.731020\pi\)
−0.663713 + 0.747987i \(0.731020\pi\)
\(60\) 0 0
\(61\) −4.26795 −0.546455 −0.273227 0.961949i \(-0.588091\pi\)
−0.273227 + 0.961949i \(0.588091\pi\)
\(62\) −1.46410 −0.185941
\(63\) 0 0
\(64\) 29.8564 3.73205
\(65\) −0.196152 −0.0243297
\(66\) 0 0
\(67\) 5.92820 0.724245 0.362123 0.932130i \(-0.382052\pi\)
0.362123 + 0.932130i \(0.382052\pi\)
\(68\) 22.9282 2.78045
\(69\) 0 0
\(70\) 7.46410 0.892131
\(71\) 13.8564 1.64445 0.822226 0.569160i \(-0.192732\pi\)
0.822226 + 0.569160i \(0.192732\pi\)
\(72\) 0 0
\(73\) 3.19615 0.374081 0.187041 0.982352i \(-0.440110\pi\)
0.187041 + 0.982352i \(0.440110\pi\)
\(74\) 17.6603 2.05296
\(75\) 0 0
\(76\) −28.3923 −3.25682
\(77\) 3.73205 0.425307
\(78\) 0 0
\(79\) −5.73205 −0.644906 −0.322453 0.946585i \(-0.604507\pi\)
−0.322453 + 0.946585i \(0.604507\pi\)
\(80\) 10.9282 1.22181
\(81\) 0 0
\(82\) 4.00000 0.441726
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 3.07180 0.333183
\(86\) −9.46410 −1.02054
\(87\) 0 0
\(88\) 9.46410 1.00888
\(89\) 0.339746 0.0360130 0.0180065 0.999838i \(-0.494268\pi\)
0.0180065 + 0.999838i \(0.494268\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) −43.7128 −4.55738
\(93\) 0 0
\(94\) 27.8564 2.87317
\(95\) −3.80385 −0.390267
\(96\) 0 0
\(97\) −5.39230 −0.547506 −0.273753 0.961800i \(-0.588265\pi\)
−0.273753 + 0.961800i \(0.588265\pi\)
\(98\) −18.9282 −1.91204
\(99\) 0 0
\(100\) −24.3923 −2.43923
\(101\) 0.196152 0.0195179 0.00975895 0.999952i \(-0.496894\pi\)
0.00975895 + 0.999952i \(0.496894\pi\)
\(102\) 0 0
\(103\) 12.8564 1.26678 0.633390 0.773833i \(-0.281663\pi\)
0.633390 + 0.773833i \(0.281663\pi\)
\(104\) 2.53590 0.248665
\(105\) 0 0
\(106\) 12.9282 1.25570
\(107\) −5.66025 −0.547197 −0.273599 0.961844i \(-0.588214\pi\)
−0.273599 + 0.961844i \(0.588214\pi\)
\(108\) 0 0
\(109\) 11.8564 1.13564 0.567819 0.823154i \(-0.307787\pi\)
0.567819 + 0.823154i \(0.307787\pi\)
\(110\) 2.00000 0.190693
\(111\) 0 0
\(112\) −55.7128 −5.26437
\(113\) 11.6603 1.09690 0.548452 0.836182i \(-0.315217\pi\)
0.548452 + 0.836182i \(0.315217\pi\)
\(114\) 0 0
\(115\) −5.85641 −0.546113
\(116\) 6.92820 0.643268
\(117\) 0 0
\(118\) 27.8564 2.56439
\(119\) −15.6603 −1.43557
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 11.6603 1.05567
\(123\) 0 0
\(124\) 2.92820 0.262960
\(125\) −6.92820 −0.619677
\(126\) 0 0
\(127\) 5.46410 0.484861 0.242430 0.970169i \(-0.422055\pi\)
0.242430 + 0.970169i \(0.422055\pi\)
\(128\) −37.8564 −3.34607
\(129\) 0 0
\(130\) 0.535898 0.0470014
\(131\) 15.1244 1.32142 0.660711 0.750641i \(-0.270255\pi\)
0.660711 + 0.750641i \(0.270255\pi\)
\(132\) 0 0
\(133\) 19.3923 1.68153
\(134\) −16.1962 −1.39913
\(135\) 0 0
\(136\) −39.7128 −3.40535
\(137\) 8.39230 0.717003 0.358501 0.933529i \(-0.383288\pi\)
0.358501 + 0.933529i \(0.383288\pi\)
\(138\) 0 0
\(139\) −7.19615 −0.610370 −0.305185 0.952293i \(-0.598718\pi\)
−0.305185 + 0.952293i \(0.598718\pi\)
\(140\) −14.9282 −1.26166
\(141\) 0 0
\(142\) −37.8564 −3.17684
\(143\) 0.267949 0.0224070
\(144\) 0 0
\(145\) 0.928203 0.0770831
\(146\) −8.73205 −0.722670
\(147\) 0 0
\(148\) −35.3205 −2.90333
\(149\) −9.85641 −0.807468 −0.403734 0.914876i \(-0.632288\pi\)
−0.403734 + 0.914876i \(0.632288\pi\)
\(150\) 0 0
\(151\) −4.80385 −0.390932 −0.195466 0.980711i \(-0.562622\pi\)
−0.195466 + 0.980711i \(0.562622\pi\)
\(152\) 49.1769 3.98877
\(153\) 0 0
\(154\) −10.1962 −0.821629
\(155\) 0.392305 0.0315107
\(156\) 0 0
\(157\) 10.5359 0.840856 0.420428 0.907326i \(-0.361880\pi\)
0.420428 + 0.907326i \(0.361880\pi\)
\(158\) 15.6603 1.24586
\(159\) 0 0
\(160\) −16.0000 −1.26491
\(161\) 29.8564 2.35301
\(162\) 0 0
\(163\) 19.0000 1.48819 0.744097 0.668071i \(-0.232880\pi\)
0.744097 + 0.668071i \(0.232880\pi\)
\(164\) −8.00000 −0.624695
\(165\) 0 0
\(166\) 0 0
\(167\) −22.7321 −1.75906 −0.879529 0.475844i \(-0.842143\pi\)
−0.879529 + 0.475844i \(0.842143\pi\)
\(168\) 0 0
\(169\) −12.9282 −0.994477
\(170\) −8.39230 −0.643660
\(171\) 0 0
\(172\) 18.9282 1.44326
\(173\) 4.39230 0.333941 0.166970 0.985962i \(-0.446602\pi\)
0.166970 + 0.985962i \(0.446602\pi\)
\(174\) 0 0
\(175\) 16.6603 1.25940
\(176\) −14.9282 −1.12526
\(177\) 0 0
\(178\) −0.928203 −0.0695718
\(179\) 14.1962 1.06107 0.530535 0.847663i \(-0.321991\pi\)
0.530535 + 0.847663i \(0.321991\pi\)
\(180\) 0 0
\(181\) −16.3205 −1.21309 −0.606547 0.795048i \(-0.707446\pi\)
−0.606547 + 0.795048i \(0.707446\pi\)
\(182\) −2.73205 −0.202513
\(183\) 0 0
\(184\) 75.7128 5.58162
\(185\) −4.73205 −0.347907
\(186\) 0 0
\(187\) −4.19615 −0.306853
\(188\) −55.7128 −4.06327
\(189\) 0 0
\(190\) 10.3923 0.753937
\(191\) 4.00000 0.289430 0.144715 0.989473i \(-0.453773\pi\)
0.144715 + 0.989473i \(0.453773\pi\)
\(192\) 0 0
\(193\) −2.80385 −0.201825 −0.100913 0.994895i \(-0.532176\pi\)
−0.100913 + 0.994895i \(0.532176\pi\)
\(194\) 14.7321 1.05770
\(195\) 0 0
\(196\) 37.8564 2.70403
\(197\) −9.66025 −0.688265 −0.344132 0.938921i \(-0.611827\pi\)
−0.344132 + 0.938921i \(0.611827\pi\)
\(198\) 0 0
\(199\) −15.0000 −1.06332 −0.531661 0.846957i \(-0.678432\pi\)
−0.531661 + 0.846957i \(0.678432\pi\)
\(200\) 42.2487 2.98744
\(201\) 0 0
\(202\) −0.535898 −0.0377057
\(203\) −4.73205 −0.332125
\(204\) 0 0
\(205\) −1.07180 −0.0748575
\(206\) −35.1244 −2.44723
\(207\) 0 0
\(208\) −4.00000 −0.277350
\(209\) 5.19615 0.359425
\(210\) 0 0
\(211\) −9.33975 −0.642975 −0.321487 0.946914i \(-0.604183\pi\)
−0.321487 + 0.946914i \(0.604183\pi\)
\(212\) −25.8564 −1.77583
\(213\) 0 0
\(214\) 15.4641 1.05710
\(215\) 2.53590 0.172947
\(216\) 0 0
\(217\) −2.00000 −0.135769
\(218\) −32.3923 −2.19388
\(219\) 0 0
\(220\) −4.00000 −0.269680
\(221\) −1.12436 −0.0756323
\(222\) 0 0
\(223\) 17.8564 1.19575 0.597877 0.801588i \(-0.296011\pi\)
0.597877 + 0.801588i \(0.296011\pi\)
\(224\) 81.5692 5.45007
\(225\) 0 0
\(226\) −31.8564 −2.11906
\(227\) 29.6603 1.96862 0.984310 0.176447i \(-0.0564605\pi\)
0.984310 + 0.176447i \(0.0564605\pi\)
\(228\) 0 0
\(229\) −16.3923 −1.08323 −0.541617 0.840625i \(-0.682188\pi\)
−0.541617 + 0.840625i \(0.682188\pi\)
\(230\) 16.0000 1.05501
\(231\) 0 0
\(232\) −12.0000 −0.787839
\(233\) 6.33975 0.415331 0.207665 0.978200i \(-0.433414\pi\)
0.207665 + 0.978200i \(0.433414\pi\)
\(234\) 0 0
\(235\) −7.46410 −0.486904
\(236\) −55.7128 −3.62660
\(237\) 0 0
\(238\) 42.7846 2.77331
\(239\) −16.3923 −1.06033 −0.530165 0.847894i \(-0.677870\pi\)
−0.530165 + 0.847894i \(0.677870\pi\)
\(240\) 0 0
\(241\) 15.1962 0.978870 0.489435 0.872040i \(-0.337203\pi\)
0.489435 + 0.872040i \(0.337203\pi\)
\(242\) −2.73205 −0.175623
\(243\) 0 0
\(244\) −23.3205 −1.49294
\(245\) 5.07180 0.324025
\(246\) 0 0
\(247\) 1.39230 0.0885902
\(248\) −5.07180 −0.322059
\(249\) 0 0
\(250\) 18.9282 1.19712
\(251\) −10.5885 −0.668337 −0.334169 0.942513i \(-0.608456\pi\)
−0.334169 + 0.942513i \(0.608456\pi\)
\(252\) 0 0
\(253\) 8.00000 0.502956
\(254\) −14.9282 −0.936679
\(255\) 0 0
\(256\) 43.7128 2.73205
\(257\) 12.7321 0.794204 0.397102 0.917775i \(-0.370016\pi\)
0.397102 + 0.917775i \(0.370016\pi\)
\(258\) 0 0
\(259\) 24.1244 1.49901
\(260\) −1.07180 −0.0664700
\(261\) 0 0
\(262\) −41.3205 −2.55279
\(263\) −4.87564 −0.300645 −0.150323 0.988637i \(-0.548031\pi\)
−0.150323 + 0.988637i \(0.548031\pi\)
\(264\) 0 0
\(265\) −3.46410 −0.212798
\(266\) −52.9808 −3.24846
\(267\) 0 0
\(268\) 32.3923 1.97867
\(269\) 26.9282 1.64184 0.820921 0.571042i \(-0.193461\pi\)
0.820921 + 0.571042i \(0.193461\pi\)
\(270\) 0 0
\(271\) −2.26795 −0.137768 −0.0688841 0.997625i \(-0.521944\pi\)
−0.0688841 + 0.997625i \(0.521944\pi\)
\(272\) 62.6410 3.79817
\(273\) 0 0
\(274\) −22.9282 −1.38514
\(275\) 4.46410 0.269195
\(276\) 0 0
\(277\) −9.85641 −0.592214 −0.296107 0.955155i \(-0.595689\pi\)
−0.296107 + 0.955155i \(0.595689\pi\)
\(278\) 19.6603 1.17914
\(279\) 0 0
\(280\) 25.8564 1.54522
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) −26.2487 −1.56032 −0.780162 0.625578i \(-0.784863\pi\)
−0.780162 + 0.625578i \(0.784863\pi\)
\(284\) 75.7128 4.49273
\(285\) 0 0
\(286\) −0.732051 −0.0432871
\(287\) 5.46410 0.322536
\(288\) 0 0
\(289\) 0.607695 0.0357468
\(290\) −2.53590 −0.148913
\(291\) 0 0
\(292\) 17.4641 1.02201
\(293\) 27.1244 1.58462 0.792311 0.610118i \(-0.208878\pi\)
0.792311 + 0.610118i \(0.208878\pi\)
\(294\) 0 0
\(295\) −7.46410 −0.434577
\(296\) 61.1769 3.55584
\(297\) 0 0
\(298\) 26.9282 1.55991
\(299\) 2.14359 0.123967
\(300\) 0 0
\(301\) −12.9282 −0.745169
\(302\) 13.1244 0.755222
\(303\) 0 0
\(304\) −77.5692 −4.44890
\(305\) −3.12436 −0.178900
\(306\) 0 0
\(307\) −27.3205 −1.55926 −0.779632 0.626238i \(-0.784594\pi\)
−0.779632 + 0.626238i \(0.784594\pi\)
\(308\) 20.3923 1.16196
\(309\) 0 0
\(310\) −1.07180 −0.0608740
\(311\) −30.5885 −1.73451 −0.867256 0.497862i \(-0.834119\pi\)
−0.867256 + 0.497862i \(0.834119\pi\)
\(312\) 0 0
\(313\) −1.92820 −0.108988 −0.0544942 0.998514i \(-0.517355\pi\)
−0.0544942 + 0.998514i \(0.517355\pi\)
\(314\) −28.7846 −1.62441
\(315\) 0 0
\(316\) −31.3205 −1.76192
\(317\) −10.5885 −0.594707 −0.297354 0.954767i \(-0.596104\pi\)
−0.297354 + 0.954767i \(0.596104\pi\)
\(318\) 0 0
\(319\) −1.26795 −0.0709915
\(320\) 21.8564 1.22181
\(321\) 0 0
\(322\) −81.5692 −4.54567
\(323\) −21.8038 −1.21320
\(324\) 0 0
\(325\) 1.19615 0.0663506
\(326\) −51.9090 −2.87497
\(327\) 0 0
\(328\) 13.8564 0.765092
\(329\) 38.0526 2.09791
\(330\) 0 0
\(331\) −27.3923 −1.50562 −0.752809 0.658239i \(-0.771301\pi\)
−0.752809 + 0.658239i \(0.771301\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 62.1051 3.39824
\(335\) 4.33975 0.237106
\(336\) 0 0
\(337\) −29.9808 −1.63316 −0.816578 0.577235i \(-0.804132\pi\)
−0.816578 + 0.577235i \(0.804132\pi\)
\(338\) 35.3205 1.92118
\(339\) 0 0
\(340\) 16.7846 0.910273
\(341\) −0.535898 −0.0290205
\(342\) 0 0
\(343\) 0.267949 0.0144679
\(344\) −32.7846 −1.76763
\(345\) 0 0
\(346\) −12.0000 −0.645124
\(347\) −10.3397 −0.555067 −0.277533 0.960716i \(-0.589517\pi\)
−0.277533 + 0.960716i \(0.589517\pi\)
\(348\) 0 0
\(349\) −5.87564 −0.314516 −0.157258 0.987558i \(-0.550265\pi\)
−0.157258 + 0.987558i \(0.550265\pi\)
\(350\) −45.5167 −2.43297
\(351\) 0 0
\(352\) 21.8564 1.16495
\(353\) −21.1244 −1.12434 −0.562168 0.827023i \(-0.690033\pi\)
−0.562168 + 0.827023i \(0.690033\pi\)
\(354\) 0 0
\(355\) 10.1436 0.538366
\(356\) 1.85641 0.0983893
\(357\) 0 0
\(358\) −38.7846 −2.04983
\(359\) −6.53590 −0.344952 −0.172476 0.985014i \(-0.555177\pi\)
−0.172476 + 0.985014i \(0.555177\pi\)
\(360\) 0 0
\(361\) 8.00000 0.421053
\(362\) 44.5885 2.34352
\(363\) 0 0
\(364\) 5.46410 0.286397
\(365\) 2.33975 0.122468
\(366\) 0 0
\(367\) 13.9282 0.727046 0.363523 0.931585i \(-0.381574\pi\)
0.363523 + 0.931585i \(0.381574\pi\)
\(368\) −119.426 −6.22549
\(369\) 0 0
\(370\) 12.9282 0.672105
\(371\) 17.6603 0.916875
\(372\) 0 0
\(373\) −20.1244 −1.04200 −0.521000 0.853557i \(-0.674441\pi\)
−0.521000 + 0.853557i \(0.674441\pi\)
\(374\) 11.4641 0.592795
\(375\) 0 0
\(376\) 96.4974 4.97647
\(377\) −0.339746 −0.0174978
\(378\) 0 0
\(379\) 14.8564 0.763122 0.381561 0.924344i \(-0.375387\pi\)
0.381561 + 0.924344i \(0.375387\pi\)
\(380\) −20.7846 −1.06623
\(381\) 0 0
\(382\) −10.9282 −0.559136
\(383\) −24.7321 −1.26375 −0.631874 0.775071i \(-0.717714\pi\)
−0.631874 + 0.775071i \(0.717714\pi\)
\(384\) 0 0
\(385\) 2.73205 0.139238
\(386\) 7.66025 0.389897
\(387\) 0 0
\(388\) −29.4641 −1.49581
\(389\) −21.4641 −1.08827 −0.544137 0.838997i \(-0.683143\pi\)
−0.544137 + 0.838997i \(0.683143\pi\)
\(390\) 0 0
\(391\) −33.5692 −1.69767
\(392\) −65.5692 −3.31175
\(393\) 0 0
\(394\) 26.3923 1.32963
\(395\) −4.19615 −0.211131
\(396\) 0 0
\(397\) 20.3923 1.02346 0.511730 0.859146i \(-0.329005\pi\)
0.511730 + 0.859146i \(0.329005\pi\)
\(398\) 40.9808 2.05418
\(399\) 0 0
\(400\) −66.6410 −3.33205
\(401\) 6.53590 0.326387 0.163194 0.986594i \(-0.447820\pi\)
0.163194 + 0.986594i \(0.447820\pi\)
\(402\) 0 0
\(403\) −0.143594 −0.00715290
\(404\) 1.07180 0.0533239
\(405\) 0 0
\(406\) 12.9282 0.641616
\(407\) 6.46410 0.320414
\(408\) 0 0
\(409\) 32.9090 1.62724 0.813622 0.581394i \(-0.197493\pi\)
0.813622 + 0.581394i \(0.197493\pi\)
\(410\) 2.92820 0.144614
\(411\) 0 0
\(412\) 70.2487 3.46091
\(413\) 38.0526 1.87244
\(414\) 0 0
\(415\) 0 0
\(416\) 5.85641 0.287134
\(417\) 0 0
\(418\) −14.1962 −0.694357
\(419\) −23.7128 −1.15845 −0.579223 0.815169i \(-0.696644\pi\)
−0.579223 + 0.815169i \(0.696644\pi\)
\(420\) 0 0
\(421\) 20.8564 1.01648 0.508240 0.861216i \(-0.330296\pi\)
0.508240 + 0.861216i \(0.330296\pi\)
\(422\) 25.5167 1.24213
\(423\) 0 0
\(424\) 44.7846 2.17493
\(425\) −18.7321 −0.908638
\(426\) 0 0
\(427\) 15.9282 0.770820
\(428\) −30.9282 −1.49497
\(429\) 0 0
\(430\) −6.92820 −0.334108
\(431\) −14.5359 −0.700170 −0.350085 0.936718i \(-0.613847\pi\)
−0.350085 + 0.936718i \(0.613847\pi\)
\(432\) 0 0
\(433\) −3.60770 −0.173375 −0.0866874 0.996236i \(-0.527628\pi\)
−0.0866874 + 0.996236i \(0.527628\pi\)
\(434\) 5.46410 0.262285
\(435\) 0 0
\(436\) 64.7846 3.10262
\(437\) 41.5692 1.98853
\(438\) 0 0
\(439\) 30.5359 1.45740 0.728699 0.684834i \(-0.240125\pi\)
0.728699 + 0.684834i \(0.240125\pi\)
\(440\) 6.92820 0.330289
\(441\) 0 0
\(442\) 3.07180 0.146110
\(443\) 3.26795 0.155265 0.0776325 0.996982i \(-0.475264\pi\)
0.0776325 + 0.996982i \(0.475264\pi\)
\(444\) 0 0
\(445\) 0.248711 0.0117900
\(446\) −48.7846 −2.31002
\(447\) 0 0
\(448\) −111.426 −5.26437
\(449\) 10.2487 0.483667 0.241833 0.970318i \(-0.422251\pi\)
0.241833 + 0.970318i \(0.422251\pi\)
\(450\) 0 0
\(451\) 1.46410 0.0689419
\(452\) 63.7128 2.99680
\(453\) 0 0
\(454\) −81.0333 −3.80308
\(455\) 0.732051 0.0343191
\(456\) 0 0
\(457\) −11.0718 −0.517917 −0.258958 0.965888i \(-0.583379\pi\)
−0.258958 + 0.965888i \(0.583379\pi\)
\(458\) 44.7846 2.09265
\(459\) 0 0
\(460\) −32.0000 −1.49201
\(461\) 29.6603 1.38142 0.690708 0.723134i \(-0.257299\pi\)
0.690708 + 0.723134i \(0.257299\pi\)
\(462\) 0 0
\(463\) −25.7846 −1.19831 −0.599156 0.800632i \(-0.704497\pi\)
−0.599156 + 0.800632i \(0.704497\pi\)
\(464\) 18.9282 0.878720
\(465\) 0 0
\(466\) −17.3205 −0.802357
\(467\) 18.9282 0.875893 0.437946 0.899001i \(-0.355706\pi\)
0.437946 + 0.899001i \(0.355706\pi\)
\(468\) 0 0
\(469\) −22.1244 −1.02161
\(470\) 20.3923 0.940627
\(471\) 0 0
\(472\) 96.4974 4.44165
\(473\) −3.46410 −0.159280
\(474\) 0 0
\(475\) 23.1962 1.06431
\(476\) −85.5692 −3.92206
\(477\) 0 0
\(478\) 44.7846 2.04840
\(479\) 11.5167 0.526210 0.263105 0.964767i \(-0.415253\pi\)
0.263105 + 0.964767i \(0.415253\pi\)
\(480\) 0 0
\(481\) 1.73205 0.0789747
\(482\) −41.5167 −1.89103
\(483\) 0 0
\(484\) 5.46410 0.248368
\(485\) −3.94744 −0.179244
\(486\) 0 0
\(487\) 18.8564 0.854465 0.427233 0.904142i \(-0.359489\pi\)
0.427233 + 0.904142i \(0.359489\pi\)
\(488\) 40.3923 1.82847
\(489\) 0 0
\(490\) −13.8564 −0.625969
\(491\) 30.0526 1.35625 0.678126 0.734945i \(-0.262792\pi\)
0.678126 + 0.734945i \(0.262792\pi\)
\(492\) 0 0
\(493\) 5.32051 0.239624
\(494\) −3.80385 −0.171143
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) −51.7128 −2.31964
\(498\) 0 0
\(499\) −19.4641 −0.871333 −0.435666 0.900108i \(-0.643487\pi\)
−0.435666 + 0.900108i \(0.643487\pi\)
\(500\) −37.8564 −1.69299
\(501\) 0 0
\(502\) 28.9282 1.29113
\(503\) −22.9282 −1.02232 −0.511159 0.859486i \(-0.670784\pi\)
−0.511159 + 0.859486i \(0.670784\pi\)
\(504\) 0 0
\(505\) 0.143594 0.00638983
\(506\) −21.8564 −0.971636
\(507\) 0 0
\(508\) 29.8564 1.32466
\(509\) −16.0526 −0.711517 −0.355759 0.934578i \(-0.615777\pi\)
−0.355759 + 0.934578i \(0.615777\pi\)
\(510\) 0 0
\(511\) −11.9282 −0.527673
\(512\) −43.7128 −1.93185
\(513\) 0 0
\(514\) −34.7846 −1.53428
\(515\) 9.41154 0.414722
\(516\) 0 0
\(517\) 10.1962 0.448426
\(518\) −65.9090 −2.89587
\(519\) 0 0
\(520\) 1.85641 0.0814088
\(521\) −20.7846 −0.910590 −0.455295 0.890341i \(-0.650466\pi\)
−0.455295 + 0.890341i \(0.650466\pi\)
\(522\) 0 0
\(523\) 39.9808 1.74824 0.874118 0.485713i \(-0.161440\pi\)
0.874118 + 0.485713i \(0.161440\pi\)
\(524\) 82.6410 3.61019
\(525\) 0 0
\(526\) 13.3205 0.580802
\(527\) 2.24871 0.0979554
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) 9.46410 0.411094
\(531\) 0 0
\(532\) 105.962 4.59401
\(533\) 0.392305 0.0169926
\(534\) 0 0
\(535\) −4.14359 −0.179143
\(536\) −56.1051 −2.42337
\(537\) 0 0
\(538\) −73.5692 −3.17179
\(539\) −6.92820 −0.298419
\(540\) 0 0
\(541\) −23.5885 −1.01415 −0.507073 0.861903i \(-0.669273\pi\)
−0.507073 + 0.861903i \(0.669273\pi\)
\(542\) 6.19615 0.266148
\(543\) 0 0
\(544\) −91.7128 −3.93215
\(545\) 8.67949 0.371789
\(546\) 0 0
\(547\) −16.8038 −0.718481 −0.359240 0.933245i \(-0.616964\pi\)
−0.359240 + 0.933245i \(0.616964\pi\)
\(548\) 45.8564 1.95889
\(549\) 0 0
\(550\) −12.1962 −0.520046
\(551\) −6.58846 −0.280678
\(552\) 0 0
\(553\) 21.3923 0.909693
\(554\) 26.9282 1.14407
\(555\) 0 0
\(556\) −39.3205 −1.66756
\(557\) −29.2679 −1.24012 −0.620061 0.784553i \(-0.712892\pi\)
−0.620061 + 0.784553i \(0.712892\pi\)
\(558\) 0 0
\(559\) −0.928203 −0.0392588
\(560\) −40.7846 −1.72346
\(561\) 0 0
\(562\) 0 0
\(563\) 35.7128 1.50512 0.752558 0.658526i \(-0.228820\pi\)
0.752558 + 0.658526i \(0.228820\pi\)
\(564\) 0 0
\(565\) 8.53590 0.359108
\(566\) 71.7128 3.01431
\(567\) 0 0
\(568\) −131.138 −5.50245
\(569\) −35.3205 −1.48071 −0.740356 0.672215i \(-0.765343\pi\)
−0.740356 + 0.672215i \(0.765343\pi\)
\(570\) 0 0
\(571\) 32.1244 1.34436 0.672181 0.740387i \(-0.265358\pi\)
0.672181 + 0.740387i \(0.265358\pi\)
\(572\) 1.46410 0.0612172
\(573\) 0 0
\(574\) −14.9282 −0.623091
\(575\) 35.7128 1.48933
\(576\) 0 0
\(577\) −3.00000 −0.124892 −0.0624458 0.998048i \(-0.519890\pi\)
−0.0624458 + 0.998048i \(0.519890\pi\)
\(578\) −1.66025 −0.0690575
\(579\) 0 0
\(580\) 5.07180 0.210595
\(581\) 0 0
\(582\) 0 0
\(583\) 4.73205 0.195982
\(584\) −30.2487 −1.25170
\(585\) 0 0
\(586\) −74.1051 −3.06125
\(587\) 1.41154 0.0582606 0.0291303 0.999576i \(-0.490726\pi\)
0.0291303 + 0.999576i \(0.490726\pi\)
\(588\) 0 0
\(589\) −2.78461 −0.114738
\(590\) 20.3923 0.839538
\(591\) 0 0
\(592\) −96.4974 −3.96602
\(593\) −47.7128 −1.95933 −0.979665 0.200639i \(-0.935698\pi\)
−0.979665 + 0.200639i \(0.935698\pi\)
\(594\) 0 0
\(595\) −11.4641 −0.469982
\(596\) −53.8564 −2.20604
\(597\) 0 0
\(598\) −5.85641 −0.239486
\(599\) −46.2487 −1.88967 −0.944836 0.327545i \(-0.893779\pi\)
−0.944836 + 0.327545i \(0.893779\pi\)
\(600\) 0 0
\(601\) 3.85641 0.157306 0.0786531 0.996902i \(-0.474938\pi\)
0.0786531 + 0.996902i \(0.474938\pi\)
\(602\) 35.3205 1.43956
\(603\) 0 0
\(604\) −26.2487 −1.06804
\(605\) 0.732051 0.0297621
\(606\) 0 0
\(607\) −37.5885 −1.52567 −0.762834 0.646594i \(-0.776193\pi\)
−0.762834 + 0.646594i \(0.776193\pi\)
\(608\) 113.569 4.60584
\(609\) 0 0
\(610\) 8.53590 0.345608
\(611\) 2.73205 0.110527
\(612\) 0 0
\(613\) 22.9090 0.925284 0.462642 0.886545i \(-0.346901\pi\)
0.462642 + 0.886545i \(0.346901\pi\)
\(614\) 74.6410 3.01227
\(615\) 0 0
\(616\) −35.3205 −1.42310
\(617\) −37.8564 −1.52404 −0.762021 0.647553i \(-0.775793\pi\)
−0.762021 + 0.647553i \(0.775793\pi\)
\(618\) 0 0
\(619\) 20.4641 0.822522 0.411261 0.911518i \(-0.365089\pi\)
0.411261 + 0.911518i \(0.365089\pi\)
\(620\) 2.14359 0.0860888
\(621\) 0 0
\(622\) 83.5692 3.35082
\(623\) −1.26795 −0.0507993
\(624\) 0 0
\(625\) 17.2487 0.689948
\(626\) 5.26795 0.210550
\(627\) 0 0
\(628\) 57.5692 2.29726
\(629\) −27.1244 −1.08152
\(630\) 0 0
\(631\) −3.39230 −0.135046 −0.0675228 0.997718i \(-0.521510\pi\)
−0.0675228 + 0.997718i \(0.521510\pi\)
\(632\) 54.2487 2.15790
\(633\) 0 0
\(634\) 28.9282 1.14889
\(635\) 4.00000 0.158735
\(636\) 0 0
\(637\) −1.85641 −0.0735535
\(638\) 3.46410 0.137145
\(639\) 0 0
\(640\) −27.7128 −1.09545
\(641\) −2.87564 −0.113581 −0.0567906 0.998386i \(-0.518087\pi\)
−0.0567906 + 0.998386i \(0.518087\pi\)
\(642\) 0 0
\(643\) 30.3923 1.19856 0.599278 0.800541i \(-0.295455\pi\)
0.599278 + 0.800541i \(0.295455\pi\)
\(644\) 163.138 6.42856
\(645\) 0 0
\(646\) 59.5692 2.34372
\(647\) −26.2487 −1.03194 −0.515972 0.856606i \(-0.672569\pi\)
−0.515972 + 0.856606i \(0.672569\pi\)
\(648\) 0 0
\(649\) 10.1962 0.400234
\(650\) −3.26795 −0.128180
\(651\) 0 0
\(652\) 103.818 4.06582
\(653\) 29.4641 1.15302 0.576510 0.817090i \(-0.304414\pi\)
0.576510 + 0.817090i \(0.304414\pi\)
\(654\) 0 0
\(655\) 11.0718 0.432611
\(656\) −21.8564 −0.853349
\(657\) 0 0
\(658\) −103.962 −4.05284
\(659\) 37.6603 1.46704 0.733518 0.679670i \(-0.237877\pi\)
0.733518 + 0.679670i \(0.237877\pi\)
\(660\) 0 0
\(661\) 2.60770 0.101428 0.0507138 0.998713i \(-0.483850\pi\)
0.0507138 + 0.998713i \(0.483850\pi\)
\(662\) 74.8372 2.90863
\(663\) 0 0
\(664\) 0 0
\(665\) 14.1962 0.550503
\(666\) 0 0
\(667\) −10.1436 −0.392762
\(668\) −124.210 −4.80584
\(669\) 0 0
\(670\) −11.8564 −0.458053
\(671\) 4.26795 0.164762
\(672\) 0 0
\(673\) −9.44486 −0.364073 −0.182036 0.983292i \(-0.558269\pi\)
−0.182036 + 0.983292i \(0.558269\pi\)
\(674\) 81.9090 3.15502
\(675\) 0 0
\(676\) −70.6410 −2.71696
\(677\) −30.2487 −1.16255 −0.581276 0.813706i \(-0.697446\pi\)
−0.581276 + 0.813706i \(0.697446\pi\)
\(678\) 0 0
\(679\) 20.1244 0.772302
\(680\) −29.0718 −1.11485
\(681\) 0 0
\(682\) 1.46410 0.0560633
\(683\) 22.5885 0.864323 0.432162 0.901796i \(-0.357751\pi\)
0.432162 + 0.901796i \(0.357751\pi\)
\(684\) 0 0
\(685\) 6.14359 0.234735
\(686\) −0.732051 −0.0279498
\(687\) 0 0
\(688\) 51.7128 1.97153
\(689\) 1.26795 0.0483050
\(690\) 0 0
\(691\) −7.21539 −0.274486 −0.137243 0.990537i \(-0.543824\pi\)
−0.137243 + 0.990537i \(0.543824\pi\)
\(692\) 24.0000 0.912343
\(693\) 0 0
\(694\) 28.2487 1.07231
\(695\) −5.26795 −0.199825
\(696\) 0 0
\(697\) −6.14359 −0.232705
\(698\) 16.0526 0.607598
\(699\) 0 0
\(700\) 91.0333 3.44074
\(701\) −9.94744 −0.375710 −0.187855 0.982197i \(-0.560153\pi\)
−0.187855 + 0.982197i \(0.560153\pi\)
\(702\) 0 0
\(703\) 33.5885 1.26681
\(704\) −29.8564 −1.12526
\(705\) 0 0
\(706\) 57.7128 2.17205
\(707\) −0.732051 −0.0275316
\(708\) 0 0
\(709\) −35.3923 −1.32919 −0.664593 0.747206i \(-0.731395\pi\)
−0.664593 + 0.747206i \(0.731395\pi\)
\(710\) −27.7128 −1.04004
\(711\) 0 0
\(712\) −3.21539 −0.120502
\(713\) −4.28719 −0.160556
\(714\) 0 0
\(715\) 0.196152 0.00733568
\(716\) 77.5692 2.89890
\(717\) 0 0
\(718\) 17.8564 0.666395
\(719\) −22.9808 −0.857038 −0.428519 0.903533i \(-0.640964\pi\)
−0.428519 + 0.903533i \(0.640964\pi\)
\(720\) 0 0
\(721\) −47.9808 −1.78690
\(722\) −21.8564 −0.813411
\(723\) 0 0
\(724\) −89.1769 −3.31423
\(725\) −5.66025 −0.210217
\(726\) 0 0
\(727\) −29.3205 −1.08744 −0.543719 0.839268i \(-0.682984\pi\)
−0.543719 + 0.839268i \(0.682984\pi\)
\(728\) −9.46410 −0.350763
\(729\) 0 0
\(730\) −6.39230 −0.236590
\(731\) 14.5359 0.537630
\(732\) 0 0
\(733\) 24.9282 0.920744 0.460372 0.887726i \(-0.347716\pi\)
0.460372 + 0.887726i \(0.347716\pi\)
\(734\) −38.0526 −1.40455
\(735\) 0 0
\(736\) 174.851 6.44510
\(737\) −5.92820 −0.218368
\(738\) 0 0
\(739\) −34.2487 −1.25986 −0.629930 0.776652i \(-0.716916\pi\)
−0.629930 + 0.776652i \(0.716916\pi\)
\(740\) −25.8564 −0.950500
\(741\) 0 0
\(742\) −48.2487 −1.77127
\(743\) −28.5885 −1.04881 −0.524404 0.851469i \(-0.675712\pi\)
−0.524404 + 0.851469i \(0.675712\pi\)
\(744\) 0 0
\(745\) −7.21539 −0.264351
\(746\) 54.9808 2.01299
\(747\) 0 0
\(748\) −22.9282 −0.838338
\(749\) 21.1244 0.771867
\(750\) 0 0
\(751\) −22.3205 −0.814487 −0.407243 0.913320i \(-0.633510\pi\)
−0.407243 + 0.913320i \(0.633510\pi\)
\(752\) −152.210 −5.55054
\(753\) 0 0
\(754\) 0.928203 0.0338032
\(755\) −3.51666 −0.127984
\(756\) 0 0
\(757\) 28.3205 1.02933 0.514663 0.857392i \(-0.327917\pi\)
0.514663 + 0.857392i \(0.327917\pi\)
\(758\) −40.5885 −1.47424
\(759\) 0 0
\(760\) 36.0000 1.30586
\(761\) 0.679492 0.0246316 0.0123158 0.999924i \(-0.496080\pi\)
0.0123158 + 0.999924i \(0.496080\pi\)
\(762\) 0 0
\(763\) −44.2487 −1.60191
\(764\) 21.8564 0.790737
\(765\) 0 0
\(766\) 67.5692 2.44138
\(767\) 2.73205 0.0986486
\(768\) 0 0
\(769\) −25.5885 −0.922743 −0.461372 0.887207i \(-0.652643\pi\)
−0.461372 + 0.887207i \(0.652643\pi\)
\(770\) −7.46410 −0.268988
\(771\) 0 0
\(772\) −15.3205 −0.551397
\(773\) −21.1244 −0.759790 −0.379895 0.925030i \(-0.624040\pi\)
−0.379895 + 0.925030i \(0.624040\pi\)
\(774\) 0 0
\(775\) −2.39230 −0.0859341
\(776\) 51.0333 1.83199
\(777\) 0 0
\(778\) 58.6410 2.10238
\(779\) 7.60770 0.272574
\(780\) 0 0
\(781\) −13.8564 −0.495821
\(782\) 91.7128 3.27964
\(783\) 0 0
\(784\) 103.426 3.69377
\(785\) 7.71281 0.275282
\(786\) 0 0
\(787\) 23.0526 0.821735 0.410867 0.911695i \(-0.365226\pi\)
0.410867 + 0.911695i \(0.365226\pi\)
\(788\) −52.7846 −1.88037
\(789\) 0 0
\(790\) 11.4641 0.407874
\(791\) −43.5167 −1.54727
\(792\) 0 0
\(793\) 1.14359 0.0406102
\(794\) −55.7128 −1.97717
\(795\) 0 0
\(796\) −81.9615 −2.90505
\(797\) −8.00000 −0.283375 −0.141687 0.989911i \(-0.545253\pi\)
−0.141687 + 0.989911i \(0.545253\pi\)
\(798\) 0 0
\(799\) −42.7846 −1.51361
\(800\) 97.5692 3.44959
\(801\) 0 0
\(802\) −17.8564 −0.630532
\(803\) −3.19615 −0.112790
\(804\) 0 0
\(805\) 21.8564 0.770337
\(806\) 0.392305 0.0138183
\(807\) 0 0
\(808\) −1.85641 −0.0653082
\(809\) −23.4115 −0.823106 −0.411553 0.911386i \(-0.635013\pi\)
−0.411553 + 0.911386i \(0.635013\pi\)
\(810\) 0 0
\(811\) 20.3923 0.716071 0.358035 0.933708i \(-0.383447\pi\)
0.358035 + 0.933708i \(0.383447\pi\)
\(812\) −25.8564 −0.907382
\(813\) 0 0
\(814\) −17.6603 −0.618992
\(815\) 13.9090 0.487210
\(816\) 0 0
\(817\) −18.0000 −0.629740
\(818\) −89.9090 −3.14359
\(819\) 0 0
\(820\) −5.85641 −0.204515
\(821\) 12.3923 0.432494 0.216247 0.976339i \(-0.430618\pi\)
0.216247 + 0.976339i \(0.430618\pi\)
\(822\) 0 0
\(823\) 48.7128 1.69802 0.849011 0.528375i \(-0.177199\pi\)
0.849011 + 0.528375i \(0.177199\pi\)
\(824\) −121.674 −4.23873
\(825\) 0 0
\(826\) −103.962 −3.61728
\(827\) 44.9808 1.56413 0.782067 0.623194i \(-0.214165\pi\)
0.782067 + 0.623194i \(0.214165\pi\)
\(828\) 0 0
\(829\) −0.607695 −0.0211061 −0.0105531 0.999944i \(-0.503359\pi\)
−0.0105531 + 0.999944i \(0.503359\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −8.00000 −0.277350
\(833\) 29.0718 1.00728
\(834\) 0 0
\(835\) −16.6410 −0.575886
\(836\) 28.3923 0.981968
\(837\) 0 0
\(838\) 64.7846 2.23795
\(839\) 35.0333 1.20948 0.604742 0.796421i \(-0.293276\pi\)
0.604742 + 0.796421i \(0.293276\pi\)
\(840\) 0 0
\(841\) −27.3923 −0.944562
\(842\) −56.9808 −1.96369
\(843\) 0 0
\(844\) −51.0333 −1.75664
\(845\) −9.46410 −0.325575
\(846\) 0 0
\(847\) −3.73205 −0.128235
\(848\) −70.6410 −2.42582
\(849\) 0 0
\(850\) 51.1769 1.75535
\(851\) 51.7128 1.77269
\(852\) 0 0
\(853\) 12.2679 0.420047 0.210023 0.977696i \(-0.432646\pi\)
0.210023 + 0.977696i \(0.432646\pi\)
\(854\) −43.5167 −1.48911
\(855\) 0 0
\(856\) 53.5692 1.83096
\(857\) −23.8038 −0.813124 −0.406562 0.913623i \(-0.633272\pi\)
−0.406562 + 0.913623i \(0.633272\pi\)
\(858\) 0 0
\(859\) 28.8564 0.984568 0.492284 0.870435i \(-0.336162\pi\)
0.492284 + 0.870435i \(0.336162\pi\)
\(860\) 13.8564 0.472500
\(861\) 0 0
\(862\) 39.7128 1.35262
\(863\) 17.9090 0.609628 0.304814 0.952412i \(-0.401406\pi\)
0.304814 + 0.952412i \(0.401406\pi\)
\(864\) 0 0
\(865\) 3.21539 0.109327
\(866\) 9.85641 0.334934
\(867\) 0 0
\(868\) −10.9282 −0.370927
\(869\) 5.73205 0.194447
\(870\) 0 0
\(871\) −1.58846 −0.0538228
\(872\) −112.210 −3.79992
\(873\) 0 0
\(874\) −113.569 −3.84154
\(875\) 25.8564 0.874106
\(876\) 0 0
\(877\) −11.1962 −0.378067 −0.189034 0.981971i \(-0.560536\pi\)
−0.189034 + 0.981971i \(0.560536\pi\)
\(878\) −83.4256 −2.81548
\(879\) 0 0
\(880\) −10.9282 −0.368390
\(881\) 51.3205 1.72903 0.864516 0.502605i \(-0.167625\pi\)
0.864516 + 0.502605i \(0.167625\pi\)
\(882\) 0 0
\(883\) −5.67949 −0.191130 −0.0955651 0.995423i \(-0.530466\pi\)
−0.0955651 + 0.995423i \(0.530466\pi\)
\(884\) −6.14359 −0.206631
\(885\) 0 0
\(886\) −8.92820 −0.299949
\(887\) −6.73205 −0.226040 −0.113020 0.993593i \(-0.536052\pi\)
−0.113020 + 0.993593i \(0.536052\pi\)
\(888\) 0 0
\(889\) −20.3923 −0.683936
\(890\) −0.679492 −0.0227766
\(891\) 0 0
\(892\) 97.5692 3.26686
\(893\) 52.9808 1.77293
\(894\) 0 0
\(895\) 10.3923 0.347376
\(896\) 141.282 4.71990
\(897\) 0 0
\(898\) −28.0000 −0.934372
\(899\) 0.679492 0.0226623
\(900\) 0 0
\(901\) −19.8564 −0.661513
\(902\) −4.00000 −0.133185
\(903\) 0 0
\(904\) −110.354 −3.67031
\(905\) −11.9474 −0.397146
\(906\) 0 0
\(907\) 12.4641 0.413864 0.206932 0.978355i \(-0.433652\pi\)
0.206932 + 0.978355i \(0.433652\pi\)
\(908\) 162.067 5.37837
\(909\) 0 0
\(910\) −2.00000 −0.0662994
\(911\) 9.46410 0.313560 0.156780 0.987634i \(-0.449889\pi\)
0.156780 + 0.987634i \(0.449889\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 30.2487 1.00054
\(915\) 0 0
\(916\) −89.5692 −2.95945
\(917\) −56.4449 −1.86397
\(918\) 0 0
\(919\) −43.3205 −1.42901 −0.714506 0.699629i \(-0.753348\pi\)
−0.714506 + 0.699629i \(0.753348\pi\)
\(920\) 55.4256 1.82733
\(921\) 0 0
\(922\) −81.0333 −2.66869
\(923\) −3.71281 −0.122209
\(924\) 0 0
\(925\) 28.8564 0.948793
\(926\) 70.4449 2.31496
\(927\) 0 0
\(928\) −27.7128 −0.909718
\(929\) −43.2679 −1.41958 −0.709788 0.704416i \(-0.751209\pi\)
−0.709788 + 0.704416i \(0.751209\pi\)
\(930\) 0 0
\(931\) −36.0000 −1.17985
\(932\) 34.6410 1.13470
\(933\) 0 0
\(934\) −51.7128 −1.69209
\(935\) −3.07180 −0.100458
\(936\) 0 0
\(937\) −45.1962 −1.47649 −0.738247 0.674531i \(-0.764346\pi\)
−0.738247 + 0.674531i \(0.764346\pi\)
\(938\) 60.4449 1.97359
\(939\) 0 0
\(940\) −40.7846 −1.33025
\(941\) −26.5359 −0.865046 −0.432523 0.901623i \(-0.642376\pi\)
−0.432523 + 0.901623i \(0.642376\pi\)
\(942\) 0 0
\(943\) 11.7128 0.381422
\(944\) −152.210 −4.95402
\(945\) 0 0
\(946\) 9.46410 0.307704
\(947\) 23.6603 0.768855 0.384427 0.923155i \(-0.374399\pi\)
0.384427 + 0.923155i \(0.374399\pi\)
\(948\) 0 0
\(949\) −0.856406 −0.0278001
\(950\) −63.3731 −2.05609
\(951\) 0 0
\(952\) 148.210 4.80352
\(953\) 6.14359 0.199011 0.0995053 0.995037i \(-0.468274\pi\)
0.0995053 + 0.995037i \(0.468274\pi\)
\(954\) 0 0
\(955\) 2.92820 0.0947544
\(956\) −89.5692 −2.89688
\(957\) 0 0
\(958\) −31.4641 −1.01656
\(959\) −31.3205 −1.01139
\(960\) 0 0
\(961\) −30.7128 −0.990736
\(962\) −4.73205 −0.152567
\(963\) 0 0
\(964\) 83.0333 2.67432
\(965\) −2.05256 −0.0660742
\(966\) 0 0
\(967\) 30.4115 0.977969 0.488985 0.872292i \(-0.337367\pi\)
0.488985 + 0.872292i \(0.337367\pi\)
\(968\) −9.46410 −0.304188
\(969\) 0 0
\(970\) 10.7846 0.346273
\(971\) −2.58846 −0.0830675 −0.0415338 0.999137i \(-0.513224\pi\)
−0.0415338 + 0.999137i \(0.513224\pi\)
\(972\) 0 0
\(973\) 26.8564 0.860977
\(974\) −51.5167 −1.65070
\(975\) 0 0
\(976\) −63.7128 −2.03940
\(977\) 35.6603 1.14087 0.570436 0.821342i \(-0.306774\pi\)
0.570436 + 0.821342i \(0.306774\pi\)
\(978\) 0 0
\(979\) −0.339746 −0.0108583
\(980\) 27.7128 0.885253
\(981\) 0 0
\(982\) −82.1051 −2.62008
\(983\) 16.0526 0.511997 0.255999 0.966677i \(-0.417596\pi\)
0.255999 + 0.966677i \(0.417596\pi\)
\(984\) 0 0
\(985\) −7.07180 −0.225326
\(986\) −14.5359 −0.462917
\(987\) 0 0
\(988\) 7.60770 0.242033
\(989\) −27.7128 −0.881216
\(990\) 0 0
\(991\) −26.4641 −0.840660 −0.420330 0.907371i \(-0.638086\pi\)
−0.420330 + 0.907371i \(0.638086\pi\)
\(992\) −11.7128 −0.371882
\(993\) 0 0
\(994\) 141.282 4.48119
\(995\) −10.9808 −0.348114
\(996\) 0 0
\(997\) 0.143594 0.00454765 0.00227383 0.999997i \(-0.499276\pi\)
0.00227383 + 0.999997i \(0.499276\pi\)
\(998\) 53.1769 1.68329
\(999\) 0 0
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 297.2.a.e.1.1 2
3.2 odd 2 297.2.a.f.1.2 yes 2
4.3 odd 2 4752.2.a.w.1.2 2
5.4 even 2 7425.2.a.bl.1.2 2
9.2 odd 6 891.2.e.m.595.1 4
9.4 even 3 891.2.e.p.298.2 4
9.5 odd 6 891.2.e.m.298.1 4
9.7 even 3 891.2.e.p.595.2 4
11.10 odd 2 3267.2.a.q.1.2 2
12.11 even 2 4752.2.a.bf.1.1 2
15.14 odd 2 7425.2.a.z.1.1 2
33.32 even 2 3267.2.a.l.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
297.2.a.e.1.1 2 1.1 even 1 trivial
297.2.a.f.1.2 yes 2 3.2 odd 2
891.2.e.m.298.1 4 9.5 odd 6
891.2.e.m.595.1 4 9.2 odd 6
891.2.e.p.298.2 4 9.4 even 3
891.2.e.p.595.2 4 9.7 even 3
3267.2.a.l.1.1 2 33.32 even 2
3267.2.a.q.1.2 2 11.10 odd 2
4752.2.a.w.1.2 2 4.3 odd 2
4752.2.a.bf.1.1 2 12.11 even 2
7425.2.a.z.1.1 2 15.14 odd 2
7425.2.a.bl.1.2 2 5.4 even 2