Properties

Label 243.2.a.c.1.2
Level $243$
Weight $2$
Character 243.1
Self dual yes
Analytic conductor $1.940$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [243,2,Mod(1,243)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(243, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("243.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 243 = 3^{5} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 243.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: \(1.94036476912\)
Analytic rank: \(0\)
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 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 243.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.73205 q^{2} +1.00000 q^{4} +3.46410 q^{5} -1.00000 q^{7} -1.73205 q^{8} +O(q^{10})\) \(q+1.73205 q^{2} +1.00000 q^{4} +3.46410 q^{5} -1.00000 q^{7} -1.73205 q^{8} +6.00000 q^{10} -3.46410 q^{11} +5.00000 q^{13} -1.73205 q^{14} -5.00000 q^{16} -1.00000 q^{19} +3.46410 q^{20} -6.00000 q^{22} -6.92820 q^{23} +7.00000 q^{25} +8.66025 q^{26} -1.00000 q^{28} -3.46410 q^{29} +5.00000 q^{31} -5.19615 q^{32} -3.46410 q^{35} -1.00000 q^{37} -1.73205 q^{38} -6.00000 q^{40} +3.46410 q^{41} -1.00000 q^{43} -3.46410 q^{44} -12.0000 q^{46} -3.46410 q^{47} -6.00000 q^{49} +12.1244 q^{50} +5.00000 q^{52} -10.3923 q^{53} -12.0000 q^{55} +1.73205 q^{56} -6.00000 q^{58} +3.46410 q^{59} +2.00000 q^{61} +8.66025 q^{62} +1.00000 q^{64} +17.3205 q^{65} +8.00000 q^{67} -6.00000 q^{70} +10.3923 q^{71} +2.00000 q^{73} -1.73205 q^{74} -1.00000 q^{76} +3.46410 q^{77} -1.00000 q^{79} -17.3205 q^{80} +6.00000 q^{82} +6.92820 q^{83} -1.73205 q^{86} +6.00000 q^{88} +10.3923 q^{89} -5.00000 q^{91} -6.92820 q^{92} -6.00000 q^{94} -3.46410 q^{95} +17.0000 q^{97} -10.3923 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} - 2 q^{7} + 12 q^{10} + 10 q^{13} - 10 q^{16} - 2 q^{19} - 12 q^{22} + 14 q^{25} - 2 q^{28} + 10 q^{31} - 2 q^{37} - 12 q^{40} - 2 q^{43} - 24 q^{46} - 12 q^{49} + 10 q^{52} - 24 q^{55} - 12 q^{58} + 4 q^{61} + 2 q^{64} + 16 q^{67} - 12 q^{70} + 4 q^{73} - 2 q^{76} - 2 q^{79} + 12 q^{82} + 12 q^{88} - 10 q^{91} - 12 q^{94} + 34 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.73205 1.22474 0.612372 0.790569i \(-0.290215\pi\)
0.612372 + 0.790569i \(0.290215\pi\)
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 3.46410 1.54919 0.774597 0.632456i \(-0.217953\pi\)
0.774597 + 0.632456i \(0.217953\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964 −0.188982 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) −1.73205 −0.612372
\(9\) 0 0
\(10\) 6.00000 1.89737
\(11\) −3.46410 −1.04447 −0.522233 0.852803i \(-0.674901\pi\)
−0.522233 + 0.852803i \(0.674901\pi\)
\(12\) 0 0
\(13\) 5.00000 1.38675 0.693375 0.720577i \(-0.256123\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) −1.73205 −0.462910
\(15\) 0 0
\(16\) −5.00000 −1.25000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 3.46410 0.774597
\(21\) 0 0
\(22\) −6.00000 −1.27920
\(23\) −6.92820 −1.44463 −0.722315 0.691564i \(-0.756922\pi\)
−0.722315 + 0.691564i \(0.756922\pi\)
\(24\) 0 0
\(25\) 7.00000 1.40000
\(26\) 8.66025 1.69842
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) −3.46410 −0.643268 −0.321634 0.946864i \(-0.604232\pi\)
−0.321634 + 0.946864i \(0.604232\pi\)
\(30\) 0 0
\(31\) 5.00000 0.898027 0.449013 0.893525i \(-0.351776\pi\)
0.449013 + 0.893525i \(0.351776\pi\)
\(32\) −5.19615 −0.918559
\(33\) 0 0
\(34\) 0 0
\(35\) −3.46410 −0.585540
\(36\) 0 0
\(37\) −1.00000 −0.164399 −0.0821995 0.996616i \(-0.526194\pi\)
−0.0821995 + 0.996616i \(0.526194\pi\)
\(38\) −1.73205 −0.280976
\(39\) 0 0
\(40\) −6.00000 −0.948683
\(41\) 3.46410 0.541002 0.270501 0.962720i \(-0.412811\pi\)
0.270501 + 0.962720i \(0.412811\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) −3.46410 −0.522233
\(45\) 0 0
\(46\) −12.0000 −1.76930
\(47\) −3.46410 −0.505291 −0.252646 0.967559i \(-0.581301\pi\)
−0.252646 + 0.967559i \(0.581301\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 12.1244 1.71464
\(51\) 0 0
\(52\) 5.00000 0.693375
\(53\) −10.3923 −1.42749 −0.713746 0.700404i \(-0.753003\pi\)
−0.713746 + 0.700404i \(0.753003\pi\)
\(54\) 0 0
\(55\) −12.0000 −1.61808
\(56\) 1.73205 0.231455
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) 3.46410 0.450988 0.225494 0.974245i \(-0.427600\pi\)
0.225494 + 0.974245i \(0.427600\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 8.66025 1.09985
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 17.3205 2.14834
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) −6.00000 −0.717137
\(71\) 10.3923 1.23334 0.616670 0.787222i \(-0.288481\pi\)
0.616670 + 0.787222i \(0.288481\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) −1.73205 −0.201347
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) 3.46410 0.394771
\(78\) 0 0
\(79\) −1.00000 −0.112509 −0.0562544 0.998416i \(-0.517916\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) −17.3205 −1.93649
\(81\) 0 0
\(82\) 6.00000 0.662589
\(83\) 6.92820 0.760469 0.380235 0.924890i \(-0.375843\pi\)
0.380235 + 0.924890i \(0.375843\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1.73205 −0.186772
\(87\) 0 0
\(88\) 6.00000 0.639602
\(89\) 10.3923 1.10158 0.550791 0.834643i \(-0.314326\pi\)
0.550791 + 0.834643i \(0.314326\pi\)
\(90\) 0 0
\(91\) −5.00000 −0.524142
\(92\) −6.92820 −0.722315
\(93\) 0 0
\(94\) −6.00000 −0.618853
\(95\) −3.46410 −0.355409
\(96\) 0 0
\(97\) 17.0000 1.72609 0.863044 0.505128i \(-0.168555\pi\)
0.863044 + 0.505128i \(0.168555\pi\)
\(98\) −10.3923 −1.04978
\(99\) 0 0
\(100\) 7.00000 0.700000
\(101\) −13.8564 −1.37876 −0.689382 0.724398i \(-0.742118\pi\)
−0.689382 + 0.724398i \(0.742118\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) −8.66025 −0.849208
\(105\) 0 0
\(106\) −18.0000 −1.74831
\(107\) 10.3923 1.00466 0.502331 0.864675i \(-0.332476\pi\)
0.502331 + 0.864675i \(0.332476\pi\)
\(108\) 0 0
\(109\) 17.0000 1.62830 0.814152 0.580651i \(-0.197202\pi\)
0.814152 + 0.580651i \(0.197202\pi\)
\(110\) −20.7846 −1.98173
\(111\) 0 0
\(112\) 5.00000 0.472456
\(113\) −17.3205 −1.62938 −0.814688 0.579899i \(-0.803092\pi\)
−0.814688 + 0.579899i \(0.803092\pi\)
\(114\) 0 0
\(115\) −24.0000 −2.23801
\(116\) −3.46410 −0.321634
\(117\) 0 0
\(118\) 6.00000 0.552345
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 3.46410 0.313625
\(123\) 0 0
\(124\) 5.00000 0.449013
\(125\) 6.92820 0.619677
\(126\) 0 0
\(127\) 17.0000 1.50851 0.754253 0.656584i \(-0.227999\pi\)
0.754253 + 0.656584i \(0.227999\pi\)
\(128\) 12.1244 1.07165
\(129\) 0 0
\(130\) 30.0000 2.63117
\(131\) 3.46410 0.302660 0.151330 0.988483i \(-0.451644\pi\)
0.151330 + 0.988483i \(0.451644\pi\)
\(132\) 0 0
\(133\) 1.00000 0.0867110
\(134\) 13.8564 1.19701
\(135\) 0 0
\(136\) 0 0
\(137\) 6.92820 0.591916 0.295958 0.955201i \(-0.404361\pi\)
0.295958 + 0.955201i \(0.404361\pi\)
\(138\) 0 0
\(139\) −13.0000 −1.10265 −0.551323 0.834292i \(-0.685877\pi\)
−0.551323 + 0.834292i \(0.685877\pi\)
\(140\) −3.46410 −0.292770
\(141\) 0 0
\(142\) 18.0000 1.51053
\(143\) −17.3205 −1.44841
\(144\) 0 0
\(145\) −12.0000 −0.996546
\(146\) 3.46410 0.286691
\(147\) 0 0
\(148\) −1.00000 −0.0821995
\(149\) −6.92820 −0.567581 −0.283790 0.958886i \(-0.591592\pi\)
−0.283790 + 0.958886i \(0.591592\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 1.73205 0.140488
\(153\) 0 0
\(154\) 6.00000 0.483494
\(155\) 17.3205 1.39122
\(156\) 0 0
\(157\) −13.0000 −1.03751 −0.518756 0.854922i \(-0.673605\pi\)
−0.518756 + 0.854922i \(0.673605\pi\)
\(158\) −1.73205 −0.137795
\(159\) 0 0
\(160\) −18.0000 −1.42302
\(161\) 6.92820 0.546019
\(162\) 0 0
\(163\) −1.00000 −0.0783260 −0.0391630 0.999233i \(-0.512469\pi\)
−0.0391630 + 0.999233i \(0.512469\pi\)
\(164\) 3.46410 0.270501
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) 24.2487 1.87642 0.938211 0.346064i \(-0.112482\pi\)
0.938211 + 0.346064i \(0.112482\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) −1.00000 −0.0762493
\(173\) −13.8564 −1.05348 −0.526742 0.850026i \(-0.676586\pi\)
−0.526742 + 0.850026i \(0.676586\pi\)
\(174\) 0 0
\(175\) −7.00000 −0.529150
\(176\) 17.3205 1.30558
\(177\) 0 0
\(178\) 18.0000 1.34916
\(179\) −20.7846 −1.55351 −0.776757 0.629800i \(-0.783137\pi\)
−0.776757 + 0.629800i \(0.783137\pi\)
\(180\) 0 0
\(181\) 17.0000 1.26360 0.631800 0.775131i \(-0.282316\pi\)
0.631800 + 0.775131i \(0.282316\pi\)
\(182\) −8.66025 −0.641941
\(183\) 0 0
\(184\) 12.0000 0.884652
\(185\) −3.46410 −0.254686
\(186\) 0 0
\(187\) 0 0
\(188\) −3.46410 −0.252646
\(189\) 0 0
\(190\) −6.00000 −0.435286
\(191\) 6.92820 0.501307 0.250654 0.968077i \(-0.419354\pi\)
0.250654 + 0.968077i \(0.419354\pi\)
\(192\) 0 0
\(193\) −10.0000 −0.719816 −0.359908 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) 29.4449 2.11402
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) −10.3923 −0.740421 −0.370211 0.928948i \(-0.620714\pi\)
−0.370211 + 0.928948i \(0.620714\pi\)
\(198\) 0 0
\(199\) −19.0000 −1.34687 −0.673437 0.739244i \(-0.735183\pi\)
−0.673437 + 0.739244i \(0.735183\pi\)
\(200\) −12.1244 −0.857321
\(201\) 0 0
\(202\) −24.0000 −1.68863
\(203\) 3.46410 0.243132
\(204\) 0 0
\(205\) 12.0000 0.838116
\(206\) 13.8564 0.965422
\(207\) 0 0
\(208\) −25.0000 −1.73344
\(209\) 3.46410 0.239617
\(210\) 0 0
\(211\) 5.00000 0.344214 0.172107 0.985078i \(-0.444942\pi\)
0.172107 + 0.985078i \(0.444942\pi\)
\(212\) −10.3923 −0.713746
\(213\) 0 0
\(214\) 18.0000 1.23045
\(215\) −3.46410 −0.236250
\(216\) 0 0
\(217\) −5.00000 −0.339422
\(218\) 29.4449 1.99426
\(219\) 0 0
\(220\) −12.0000 −0.809040
\(221\) 0 0
\(222\) 0 0
\(223\) −19.0000 −1.27233 −0.636167 0.771551i \(-0.719481\pi\)
−0.636167 + 0.771551i \(0.719481\pi\)
\(224\) 5.19615 0.347183
\(225\) 0 0
\(226\) −30.0000 −1.99557
\(227\) −13.8564 −0.919682 −0.459841 0.888001i \(-0.652094\pi\)
−0.459841 + 0.888001i \(0.652094\pi\)
\(228\) 0 0
\(229\) 5.00000 0.330409 0.165205 0.986259i \(-0.447172\pi\)
0.165205 + 0.986259i \(0.447172\pi\)
\(230\) −41.5692 −2.74099
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) −12.0000 −0.782794
\(236\) 3.46410 0.225494
\(237\) 0 0
\(238\) 0 0
\(239\) −6.92820 −0.448148 −0.224074 0.974572i \(-0.571936\pi\)
−0.224074 + 0.974572i \(0.571936\pi\)
\(240\) 0 0
\(241\) −19.0000 −1.22390 −0.611949 0.790897i \(-0.709614\pi\)
−0.611949 + 0.790897i \(0.709614\pi\)
\(242\) 1.73205 0.111340
\(243\) 0 0
\(244\) 2.00000 0.128037
\(245\) −20.7846 −1.32788
\(246\) 0 0
\(247\) −5.00000 −0.318142
\(248\) −8.66025 −0.549927
\(249\) 0 0
\(250\) 12.0000 0.758947
\(251\) 20.7846 1.31191 0.655956 0.754799i \(-0.272265\pi\)
0.655956 + 0.754799i \(0.272265\pi\)
\(252\) 0 0
\(253\) 24.0000 1.50887
\(254\) 29.4449 1.84754
\(255\) 0 0
\(256\) 19.0000 1.18750
\(257\) 3.46410 0.216085 0.108042 0.994146i \(-0.465542\pi\)
0.108042 + 0.994146i \(0.465542\pi\)
\(258\) 0 0
\(259\) 1.00000 0.0621370
\(260\) 17.3205 1.07417
\(261\) 0 0
\(262\) 6.00000 0.370681
\(263\) −13.8564 −0.854423 −0.427211 0.904152i \(-0.640504\pi\)
−0.427211 + 0.904152i \(0.640504\pi\)
\(264\) 0 0
\(265\) −36.0000 −2.21146
\(266\) 1.73205 0.106199
\(267\) 0 0
\(268\) 8.00000 0.488678
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 12.0000 0.724947
\(275\) −24.2487 −1.46225
\(276\) 0 0
\(277\) 17.0000 1.02143 0.510716 0.859750i \(-0.329381\pi\)
0.510716 + 0.859750i \(0.329381\pi\)
\(278\) −22.5167 −1.35046
\(279\) 0 0
\(280\) 6.00000 0.358569
\(281\) −13.8564 −0.826604 −0.413302 0.910594i \(-0.635625\pi\)
−0.413302 + 0.910594i \(0.635625\pi\)
\(282\) 0 0
\(283\) −13.0000 −0.772770 −0.386385 0.922338i \(-0.626276\pi\)
−0.386385 + 0.922338i \(0.626276\pi\)
\(284\) 10.3923 0.616670
\(285\) 0 0
\(286\) −30.0000 −1.77394
\(287\) −3.46410 −0.204479
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) −20.7846 −1.22051
\(291\) 0 0
\(292\) 2.00000 0.117041
\(293\) 13.8564 0.809500 0.404750 0.914427i \(-0.367359\pi\)
0.404750 + 0.914427i \(0.367359\pi\)
\(294\) 0 0
\(295\) 12.0000 0.698667
\(296\) 1.73205 0.100673
\(297\) 0 0
\(298\) −12.0000 −0.695141
\(299\) −34.6410 −2.00334
\(300\) 0 0
\(301\) 1.00000 0.0576390
\(302\) −27.7128 −1.59469
\(303\) 0 0
\(304\) 5.00000 0.286770
\(305\) 6.92820 0.396708
\(306\) 0 0
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 3.46410 0.197386
\(309\) 0 0
\(310\) 30.0000 1.70389
\(311\) 13.8564 0.785725 0.392862 0.919597i \(-0.371485\pi\)
0.392862 + 0.919597i \(0.371485\pi\)
\(312\) 0 0
\(313\) −1.00000 −0.0565233 −0.0282617 0.999601i \(-0.508997\pi\)
−0.0282617 + 0.999601i \(0.508997\pi\)
\(314\) −22.5167 −1.27069
\(315\) 0 0
\(316\) −1.00000 −0.0562544
\(317\) 27.7128 1.55651 0.778253 0.627950i \(-0.216106\pi\)
0.778253 + 0.627950i \(0.216106\pi\)
\(318\) 0 0
\(319\) 12.0000 0.671871
\(320\) 3.46410 0.193649
\(321\) 0 0
\(322\) 12.0000 0.668734
\(323\) 0 0
\(324\) 0 0
\(325\) 35.0000 1.94145
\(326\) −1.73205 −0.0959294
\(327\) 0 0
\(328\) −6.00000 −0.331295
\(329\) 3.46410 0.190982
\(330\) 0 0
\(331\) −19.0000 −1.04433 −0.522167 0.852843i \(-0.674876\pi\)
−0.522167 + 0.852843i \(0.674876\pi\)
\(332\) 6.92820 0.380235
\(333\) 0 0
\(334\) 42.0000 2.29814
\(335\) 27.7128 1.51411
\(336\) 0 0
\(337\) 5.00000 0.272367 0.136184 0.990684i \(-0.456516\pi\)
0.136184 + 0.990684i \(0.456516\pi\)
\(338\) 20.7846 1.13053
\(339\) 0 0
\(340\) 0 0
\(341\) −17.3205 −0.937958
\(342\) 0 0
\(343\) 13.0000 0.701934
\(344\) 1.73205 0.0933859
\(345\) 0 0
\(346\) −24.0000 −1.29025
\(347\) 24.2487 1.30174 0.650870 0.759190i \(-0.274404\pi\)
0.650870 + 0.759190i \(0.274404\pi\)
\(348\) 0 0
\(349\) −1.00000 −0.0535288 −0.0267644 0.999642i \(-0.508520\pi\)
−0.0267644 + 0.999642i \(0.508520\pi\)
\(350\) −12.1244 −0.648074
\(351\) 0 0
\(352\) 18.0000 0.959403
\(353\) 17.3205 0.921878 0.460939 0.887432i \(-0.347513\pi\)
0.460939 + 0.887432i \(0.347513\pi\)
\(354\) 0 0
\(355\) 36.0000 1.91068
\(356\) 10.3923 0.550791
\(357\) 0 0
\(358\) −36.0000 −1.90266
\(359\) 31.1769 1.64545 0.822727 0.568436i \(-0.192451\pi\)
0.822727 + 0.568436i \(0.192451\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 29.4449 1.54759
\(363\) 0 0
\(364\) −5.00000 −0.262071
\(365\) 6.92820 0.362639
\(366\) 0 0
\(367\) −16.0000 −0.835193 −0.417597 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) 34.6410 1.80579
\(369\) 0 0
\(370\) −6.00000 −0.311925
\(371\) 10.3923 0.539542
\(372\) 0 0
\(373\) 23.0000 1.19089 0.595447 0.803394i \(-0.296975\pi\)
0.595447 + 0.803394i \(0.296975\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 6.00000 0.309426
\(377\) −17.3205 −0.892052
\(378\) 0 0
\(379\) −19.0000 −0.975964 −0.487982 0.872854i \(-0.662267\pi\)
−0.487982 + 0.872854i \(0.662267\pi\)
\(380\) −3.46410 −0.177705
\(381\) 0 0
\(382\) 12.0000 0.613973
\(383\) −17.3205 −0.885037 −0.442518 0.896759i \(-0.645915\pi\)
−0.442518 + 0.896759i \(0.645915\pi\)
\(384\) 0 0
\(385\) 12.0000 0.611577
\(386\) −17.3205 −0.881591
\(387\) 0 0
\(388\) 17.0000 0.863044
\(389\) 6.92820 0.351274 0.175637 0.984455i \(-0.443802\pi\)
0.175637 + 0.984455i \(0.443802\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 10.3923 0.524891
\(393\) 0 0
\(394\) −18.0000 −0.906827
\(395\) −3.46410 −0.174298
\(396\) 0 0
\(397\) −1.00000 −0.0501886 −0.0250943 0.999685i \(-0.507989\pi\)
−0.0250943 + 0.999685i \(0.507989\pi\)
\(398\) −32.9090 −1.64958
\(399\) 0 0
\(400\) −35.0000 −1.75000
\(401\) 3.46410 0.172989 0.0864945 0.996252i \(-0.472434\pi\)
0.0864945 + 0.996252i \(0.472434\pi\)
\(402\) 0 0
\(403\) 25.0000 1.24534
\(404\) −13.8564 −0.689382
\(405\) 0 0
\(406\) 6.00000 0.297775
\(407\) 3.46410 0.171709
\(408\) 0 0
\(409\) 5.00000 0.247234 0.123617 0.992330i \(-0.460551\pi\)
0.123617 + 0.992330i \(0.460551\pi\)
\(410\) 20.7846 1.02648
\(411\) 0 0
\(412\) 8.00000 0.394132
\(413\) −3.46410 −0.170457
\(414\) 0 0
\(415\) 24.0000 1.17811
\(416\) −25.9808 −1.27381
\(417\) 0 0
\(418\) 6.00000 0.293470
\(419\) −38.1051 −1.86156 −0.930778 0.365584i \(-0.880869\pi\)
−0.930778 + 0.365584i \(0.880869\pi\)
\(420\) 0 0
\(421\) −19.0000 −0.926003 −0.463002 0.886357i \(-0.653228\pi\)
−0.463002 + 0.886357i \(0.653228\pi\)
\(422\) 8.66025 0.421575
\(423\) 0 0
\(424\) 18.0000 0.874157
\(425\) 0 0
\(426\) 0 0
\(427\) −2.00000 −0.0967868
\(428\) 10.3923 0.502331
\(429\) 0 0
\(430\) −6.00000 −0.289346
\(431\) −20.7846 −1.00116 −0.500580 0.865690i \(-0.666880\pi\)
−0.500580 + 0.865690i \(0.666880\pi\)
\(432\) 0 0
\(433\) −1.00000 −0.0480569 −0.0240285 0.999711i \(-0.507649\pi\)
−0.0240285 + 0.999711i \(0.507649\pi\)
\(434\) −8.66025 −0.415705
\(435\) 0 0
\(436\) 17.0000 0.814152
\(437\) 6.92820 0.331421
\(438\) 0 0
\(439\) 20.0000 0.954548 0.477274 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 20.7846 0.990867
\(441\) 0 0
\(442\) 0 0
\(443\) −13.8564 −0.658338 −0.329169 0.944271i \(-0.606769\pi\)
−0.329169 + 0.944271i \(0.606769\pi\)
\(444\) 0 0
\(445\) 36.0000 1.70656
\(446\) −32.9090 −1.55828
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) −10.3923 −0.490443 −0.245222 0.969467i \(-0.578861\pi\)
−0.245222 + 0.969467i \(0.578861\pi\)
\(450\) 0 0
\(451\) −12.0000 −0.565058
\(452\) −17.3205 −0.814688
\(453\) 0 0
\(454\) −24.0000 −1.12638
\(455\) −17.3205 −0.811998
\(456\) 0 0
\(457\) 17.0000 0.795226 0.397613 0.917553i \(-0.369839\pi\)
0.397613 + 0.917553i \(0.369839\pi\)
\(458\) 8.66025 0.404667
\(459\) 0 0
\(460\) −24.0000 −1.11901
\(461\) 27.7128 1.29071 0.645357 0.763881i \(-0.276709\pi\)
0.645357 + 0.763881i \(0.276709\pi\)
\(462\) 0 0
\(463\) −31.0000 −1.44069 −0.720346 0.693615i \(-0.756017\pi\)
−0.720346 + 0.693615i \(0.756017\pi\)
\(464\) 17.3205 0.804084
\(465\) 0 0
\(466\) 0 0
\(467\) −10.3923 −0.480899 −0.240449 0.970662i \(-0.577295\pi\)
−0.240449 + 0.970662i \(0.577295\pi\)
\(468\) 0 0
\(469\) −8.00000 −0.369406
\(470\) −20.7846 −0.958723
\(471\) 0 0
\(472\) −6.00000 −0.276172
\(473\) 3.46410 0.159280
\(474\) 0 0
\(475\) −7.00000 −0.321182
\(476\) 0 0
\(477\) 0 0
\(478\) −12.0000 −0.548867
\(479\) −13.8564 −0.633115 −0.316558 0.948573i \(-0.602527\pi\)
−0.316558 + 0.948573i \(0.602527\pi\)
\(480\) 0 0
\(481\) −5.00000 −0.227980
\(482\) −32.9090 −1.49896
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 58.8897 2.67404
\(486\) 0 0
\(487\) −19.0000 −0.860972 −0.430486 0.902597i \(-0.641658\pi\)
−0.430486 + 0.902597i \(0.641658\pi\)
\(488\) −3.46410 −0.156813
\(489\) 0 0
\(490\) −36.0000 −1.62631
\(491\) −38.1051 −1.71966 −0.859830 0.510581i \(-0.829431\pi\)
−0.859830 + 0.510581i \(0.829431\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) −8.66025 −0.389643
\(495\) 0 0
\(496\) −25.0000 −1.12253
\(497\) −10.3923 −0.466159
\(498\) 0 0
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) 6.92820 0.309839
\(501\) 0 0
\(502\) 36.0000 1.60676
\(503\) 41.5692 1.85348 0.926740 0.375703i \(-0.122599\pi\)
0.926740 + 0.375703i \(0.122599\pi\)
\(504\) 0 0
\(505\) −48.0000 −2.13597
\(506\) 41.5692 1.84798
\(507\) 0 0
\(508\) 17.0000 0.754253
\(509\) −27.7128 −1.22835 −0.614174 0.789170i \(-0.710511\pi\)
−0.614174 + 0.789170i \(0.710511\pi\)
\(510\) 0 0
\(511\) −2.00000 −0.0884748
\(512\) 8.66025 0.382733
\(513\) 0 0
\(514\) 6.00000 0.264649
\(515\) 27.7128 1.22117
\(516\) 0 0
\(517\) 12.0000 0.527759
\(518\) 1.73205 0.0761019
\(519\) 0 0
\(520\) −30.0000 −1.31559
\(521\) −20.7846 −0.910590 −0.455295 0.890341i \(-0.650466\pi\)
−0.455295 + 0.890341i \(0.650466\pi\)
\(522\) 0 0
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 3.46410 0.151330
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) 0 0
\(528\) 0 0
\(529\) 25.0000 1.08696
\(530\) −62.3538 −2.70848
\(531\) 0 0
\(532\) 1.00000 0.0433555
\(533\) 17.3205 0.750234
\(534\) 0 0
\(535\) 36.0000 1.55642
\(536\) −13.8564 −0.598506
\(537\) 0 0
\(538\) 0 0
\(539\) 20.7846 0.895257
\(540\) 0 0
\(541\) 17.0000 0.730887 0.365444 0.930834i \(-0.380917\pi\)
0.365444 + 0.930834i \(0.380917\pi\)
\(542\) −27.7128 −1.19037
\(543\) 0 0
\(544\) 0 0
\(545\) 58.8897 2.52256
\(546\) 0 0
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) 6.92820 0.295958
\(549\) 0 0
\(550\) −42.0000 −1.79089
\(551\) 3.46410 0.147576
\(552\) 0 0
\(553\) 1.00000 0.0425243
\(554\) 29.4449 1.25099
\(555\) 0 0
\(556\) −13.0000 −0.551323
\(557\) 10.3923 0.440336 0.220168 0.975462i \(-0.429339\pi\)
0.220168 + 0.975462i \(0.429339\pi\)
\(558\) 0 0
\(559\) −5.00000 −0.211477
\(560\) 17.3205 0.731925
\(561\) 0 0
\(562\) −24.0000 −1.01238
\(563\) 34.6410 1.45994 0.729972 0.683477i \(-0.239533\pi\)
0.729972 + 0.683477i \(0.239533\pi\)
\(564\) 0 0
\(565\) −60.0000 −2.52422
\(566\) −22.5167 −0.946446
\(567\) 0 0
\(568\) −18.0000 −0.755263
\(569\) −24.2487 −1.01656 −0.508279 0.861192i \(-0.669718\pi\)
−0.508279 + 0.861192i \(0.669718\pi\)
\(570\) 0 0
\(571\) 41.0000 1.71580 0.857898 0.513820i \(-0.171770\pi\)
0.857898 + 0.513820i \(0.171770\pi\)
\(572\) −17.3205 −0.724207
\(573\) 0 0
\(574\) −6.00000 −0.250435
\(575\) −48.4974 −2.02248
\(576\) 0 0
\(577\) −34.0000 −1.41544 −0.707719 0.706494i \(-0.750276\pi\)
−0.707719 + 0.706494i \(0.750276\pi\)
\(578\) −29.4449 −1.22474
\(579\) 0 0
\(580\) −12.0000 −0.498273
\(581\) −6.92820 −0.287430
\(582\) 0 0
\(583\) 36.0000 1.49097
\(584\) −3.46410 −0.143346
\(585\) 0 0
\(586\) 24.0000 0.991431
\(587\) 6.92820 0.285958 0.142979 0.989726i \(-0.454332\pi\)
0.142979 + 0.989726i \(0.454332\pi\)
\(588\) 0 0
\(589\) −5.00000 −0.206021
\(590\) 20.7846 0.855689
\(591\) 0 0
\(592\) 5.00000 0.205499
\(593\) 20.7846 0.853522 0.426761 0.904365i \(-0.359655\pi\)
0.426761 + 0.904365i \(0.359655\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −6.92820 −0.283790
\(597\) 0 0
\(598\) −60.0000 −2.45358
\(599\) 24.2487 0.990775 0.495388 0.868672i \(-0.335026\pi\)
0.495388 + 0.868672i \(0.335026\pi\)
\(600\) 0 0
\(601\) 35.0000 1.42768 0.713840 0.700309i \(-0.246954\pi\)
0.713840 + 0.700309i \(0.246954\pi\)
\(602\) 1.73205 0.0705931
\(603\) 0 0
\(604\) −16.0000 −0.651031
\(605\) 3.46410 0.140836
\(606\) 0 0
\(607\) −13.0000 −0.527654 −0.263827 0.964570i \(-0.584985\pi\)
−0.263827 + 0.964570i \(0.584985\pi\)
\(608\) 5.19615 0.210732
\(609\) 0 0
\(610\) 12.0000 0.485866
\(611\) −17.3205 −0.700713
\(612\) 0 0
\(613\) 2.00000 0.0807792 0.0403896 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) 34.6410 1.39800
\(615\) 0 0
\(616\) −6.00000 −0.241747
\(617\) −6.92820 −0.278919 −0.139459 0.990228i \(-0.544536\pi\)
−0.139459 + 0.990228i \(0.544536\pi\)
\(618\) 0 0
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 17.3205 0.695608
\(621\) 0 0
\(622\) 24.0000 0.962312
\(623\) −10.3923 −0.416359
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) −1.73205 −0.0692267
\(627\) 0 0
\(628\) −13.0000 −0.518756
\(629\) 0 0
\(630\) 0 0
\(631\) 17.0000 0.676759 0.338380 0.941010i \(-0.390121\pi\)
0.338380 + 0.941010i \(0.390121\pi\)
\(632\) 1.73205 0.0688973
\(633\) 0 0
\(634\) 48.0000 1.90632
\(635\) 58.8897 2.33697
\(636\) 0 0
\(637\) −30.0000 −1.18864
\(638\) 20.7846 0.822871
\(639\) 0 0
\(640\) 42.0000 1.66020
\(641\) −45.0333 −1.77871 −0.889355 0.457218i \(-0.848846\pi\)
−0.889355 + 0.457218i \(0.848846\pi\)
\(642\) 0 0
\(643\) 44.0000 1.73519 0.867595 0.497271i \(-0.165665\pi\)
0.867595 + 0.497271i \(0.165665\pi\)
\(644\) 6.92820 0.273009
\(645\) 0 0
\(646\) 0 0
\(647\) −20.7846 −0.817127 −0.408564 0.912730i \(-0.633970\pi\)
−0.408564 + 0.912730i \(0.633970\pi\)
\(648\) 0 0
\(649\) −12.0000 −0.471041
\(650\) 60.6218 2.37778
\(651\) 0 0
\(652\) −1.00000 −0.0391630
\(653\) −17.3205 −0.677804 −0.338902 0.940822i \(-0.610055\pi\)
−0.338902 + 0.940822i \(0.610055\pi\)
\(654\) 0 0
\(655\) 12.0000 0.468879
\(656\) −17.3205 −0.676252
\(657\) 0 0
\(658\) 6.00000 0.233904
\(659\) 27.7128 1.07954 0.539769 0.841813i \(-0.318512\pi\)
0.539769 + 0.841813i \(0.318512\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) −32.9090 −1.27904
\(663\) 0 0
\(664\) −12.0000 −0.465690
\(665\) 3.46410 0.134332
\(666\) 0 0
\(667\) 24.0000 0.929284
\(668\) 24.2487 0.938211
\(669\) 0 0
\(670\) 48.0000 1.85440
\(671\) −6.92820 −0.267460
\(672\) 0 0
\(673\) −37.0000 −1.42625 −0.713123 0.701039i \(-0.752720\pi\)
−0.713123 + 0.701039i \(0.752720\pi\)
\(674\) 8.66025 0.333581
\(675\) 0 0
\(676\) 12.0000 0.461538
\(677\) 17.3205 0.665681 0.332841 0.942983i \(-0.391993\pi\)
0.332841 + 0.942983i \(0.391993\pi\)
\(678\) 0 0
\(679\) −17.0000 −0.652400
\(680\) 0 0
\(681\) 0 0
\(682\) −30.0000 −1.14876
\(683\) 41.5692 1.59060 0.795301 0.606215i \(-0.207313\pi\)
0.795301 + 0.606215i \(0.207313\pi\)
\(684\) 0 0
\(685\) 24.0000 0.916993
\(686\) 22.5167 0.859690
\(687\) 0 0
\(688\) 5.00000 0.190623
\(689\) −51.9615 −1.97958
\(690\) 0 0
\(691\) 17.0000 0.646710 0.323355 0.946278i \(-0.395189\pi\)
0.323355 + 0.946278i \(0.395189\pi\)
\(692\) −13.8564 −0.526742
\(693\) 0 0
\(694\) 42.0000 1.59430
\(695\) −45.0333 −1.70821
\(696\) 0 0
\(697\) 0 0
\(698\) −1.73205 −0.0655591
\(699\) 0 0
\(700\) −7.00000 −0.264575
\(701\) −41.5692 −1.57005 −0.785024 0.619466i \(-0.787349\pi\)
−0.785024 + 0.619466i \(0.787349\pi\)
\(702\) 0 0
\(703\) 1.00000 0.0377157
\(704\) −3.46410 −0.130558
\(705\) 0 0
\(706\) 30.0000 1.12906
\(707\) 13.8564 0.521124
\(708\) 0 0
\(709\) −19.0000 −0.713560 −0.356780 0.934188i \(-0.616125\pi\)
−0.356780 + 0.934188i \(0.616125\pi\)
\(710\) 62.3538 2.34010
\(711\) 0 0
\(712\) −18.0000 −0.674579
\(713\) −34.6410 −1.29732
\(714\) 0 0
\(715\) −60.0000 −2.24387
\(716\) −20.7846 −0.776757
\(717\) 0 0
\(718\) 54.0000 2.01526
\(719\) 10.3923 0.387568 0.193784 0.981044i \(-0.437924\pi\)
0.193784 + 0.981044i \(0.437924\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) −31.1769 −1.16028
\(723\) 0 0
\(724\) 17.0000 0.631800
\(725\) −24.2487 −0.900575
\(726\) 0 0
\(727\) −16.0000 −0.593407 −0.296704 0.954970i \(-0.595887\pi\)
−0.296704 + 0.954970i \(0.595887\pi\)
\(728\) 8.66025 0.320970
\(729\) 0 0
\(730\) 12.0000 0.444140
\(731\) 0 0
\(732\) 0 0
\(733\) 41.0000 1.51437 0.757185 0.653201i \(-0.226574\pi\)
0.757185 + 0.653201i \(0.226574\pi\)
\(734\) −27.7128 −1.02290
\(735\) 0 0
\(736\) 36.0000 1.32698
\(737\) −27.7128 −1.02081
\(738\) 0 0
\(739\) −19.0000 −0.698926 −0.349463 0.936950i \(-0.613636\pi\)
−0.349463 + 0.936950i \(0.613636\pi\)
\(740\) −3.46410 −0.127343
\(741\) 0 0
\(742\) 18.0000 0.660801
\(743\) −6.92820 −0.254171 −0.127086 0.991892i \(-0.540562\pi\)
−0.127086 + 0.991892i \(0.540562\pi\)
\(744\) 0 0
\(745\) −24.0000 −0.879292
\(746\) 39.8372 1.45854
\(747\) 0 0
\(748\) 0 0
\(749\) −10.3923 −0.379727
\(750\) 0 0
\(751\) 5.00000 0.182453 0.0912263 0.995830i \(-0.470921\pi\)
0.0912263 + 0.995830i \(0.470921\pi\)
\(752\) 17.3205 0.631614
\(753\) 0 0
\(754\) −30.0000 −1.09254
\(755\) −55.4256 −2.01715
\(756\) 0 0
\(757\) 38.0000 1.38113 0.690567 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) −32.9090 −1.19531
\(759\) 0 0
\(760\) 6.00000 0.217643
\(761\) −27.7128 −1.00459 −0.502294 0.864697i \(-0.667511\pi\)
−0.502294 + 0.864697i \(0.667511\pi\)
\(762\) 0 0
\(763\) −17.0000 −0.615441
\(764\) 6.92820 0.250654
\(765\) 0 0
\(766\) −30.0000 −1.08394
\(767\) 17.3205 0.625407
\(768\) 0 0
\(769\) −13.0000 −0.468792 −0.234396 0.972141i \(-0.575311\pi\)
−0.234396 + 0.972141i \(0.575311\pi\)
\(770\) 20.7846 0.749025
\(771\) 0 0
\(772\) −10.0000 −0.359908
\(773\) 20.7846 0.747570 0.373785 0.927515i \(-0.378060\pi\)
0.373785 + 0.927515i \(0.378060\pi\)
\(774\) 0 0
\(775\) 35.0000 1.25724
\(776\) −29.4449 −1.05701
\(777\) 0 0
\(778\) 12.0000 0.430221
\(779\) −3.46410 −0.124114
\(780\) 0 0
\(781\) −36.0000 −1.28818
\(782\) 0 0
\(783\) 0 0
\(784\) 30.0000 1.07143
\(785\) −45.0333 −1.60731
\(786\) 0 0
\(787\) −28.0000 −0.998092 −0.499046 0.866575i \(-0.666316\pi\)
−0.499046 + 0.866575i \(0.666316\pi\)
\(788\) −10.3923 −0.370211
\(789\) 0 0
\(790\) −6.00000 −0.213470
\(791\) 17.3205 0.615846
\(792\) 0 0
\(793\) 10.0000 0.355110
\(794\) −1.73205 −0.0614682
\(795\) 0 0
\(796\) −19.0000 −0.673437
\(797\) 13.8564 0.490819 0.245410 0.969419i \(-0.421078\pi\)
0.245410 + 0.969419i \(0.421078\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −36.3731 −1.28598
\(801\) 0 0
\(802\) 6.00000 0.211867
\(803\) −6.92820 −0.244491
\(804\) 0 0
\(805\) 24.0000 0.845889
\(806\) 43.3013 1.52522
\(807\) 0 0
\(808\) 24.0000 0.844317
\(809\) −31.1769 −1.09612 −0.548061 0.836438i \(-0.684634\pi\)
−0.548061 + 0.836438i \(0.684634\pi\)
\(810\) 0 0
\(811\) 35.0000 1.22902 0.614508 0.788911i \(-0.289355\pi\)
0.614508 + 0.788911i \(0.289355\pi\)
\(812\) 3.46410 0.121566
\(813\) 0 0
\(814\) 6.00000 0.210300
\(815\) −3.46410 −0.121342
\(816\) 0 0
\(817\) 1.00000 0.0349856
\(818\) 8.66025 0.302799
\(819\) 0 0
\(820\) 12.0000 0.419058
\(821\) 38.1051 1.32988 0.664939 0.746898i \(-0.268458\pi\)
0.664939 + 0.746898i \(0.268458\pi\)
\(822\) 0 0
\(823\) 23.0000 0.801730 0.400865 0.916137i \(-0.368710\pi\)
0.400865 + 0.916137i \(0.368710\pi\)
\(824\) −13.8564 −0.482711
\(825\) 0 0
\(826\) −6.00000 −0.208767
\(827\) −10.3923 −0.361376 −0.180688 0.983540i \(-0.557832\pi\)
−0.180688 + 0.983540i \(0.557832\pi\)
\(828\) 0 0
\(829\) −19.0000 −0.659897 −0.329949 0.943999i \(-0.607031\pi\)
−0.329949 + 0.943999i \(0.607031\pi\)
\(830\) 41.5692 1.44289
\(831\) 0 0
\(832\) 5.00000 0.173344
\(833\) 0 0
\(834\) 0 0
\(835\) 84.0000 2.90694
\(836\) 3.46410 0.119808
\(837\) 0 0
\(838\) −66.0000 −2.27993
\(839\) −3.46410 −0.119594 −0.0597970 0.998211i \(-0.519045\pi\)
−0.0597970 + 0.998211i \(0.519045\pi\)
\(840\) 0 0
\(841\) −17.0000 −0.586207
\(842\) −32.9090 −1.13412
\(843\) 0 0
\(844\) 5.00000 0.172107
\(845\) 41.5692 1.43002
\(846\) 0 0
\(847\) −1.00000 −0.0343604
\(848\) 51.9615 1.78437
\(849\) 0 0
\(850\) 0 0
\(851\) 6.92820 0.237496
\(852\) 0 0
\(853\) 2.00000 0.0684787 0.0342393 0.999414i \(-0.489099\pi\)
0.0342393 + 0.999414i \(0.489099\pi\)
\(854\) −3.46410 −0.118539
\(855\) 0 0
\(856\) −18.0000 −0.615227
\(857\) 38.1051 1.30165 0.650823 0.759229i \(-0.274424\pi\)
0.650823 + 0.759229i \(0.274424\pi\)
\(858\) 0 0
\(859\) −49.0000 −1.67186 −0.835929 0.548837i \(-0.815071\pi\)
−0.835929 + 0.548837i \(0.815071\pi\)
\(860\) −3.46410 −0.118125
\(861\) 0 0
\(862\) −36.0000 −1.22616
\(863\) 10.3923 0.353758 0.176879 0.984233i \(-0.443400\pi\)
0.176879 + 0.984233i \(0.443400\pi\)
\(864\) 0 0
\(865\) −48.0000 −1.63205
\(866\) −1.73205 −0.0588575
\(867\) 0 0
\(868\) −5.00000 −0.169711
\(869\) 3.46410 0.117512
\(870\) 0 0
\(871\) 40.0000 1.35535
\(872\) −29.4449 −0.997129
\(873\) 0 0
\(874\) 12.0000 0.405906
\(875\) −6.92820 −0.234216
\(876\) 0 0
\(877\) 23.0000 0.776655 0.388327 0.921521i \(-0.373053\pi\)
0.388327 + 0.921521i \(0.373053\pi\)
\(878\) 34.6410 1.16908
\(879\) 0 0
\(880\) 60.0000 2.02260
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) −19.0000 −0.639401 −0.319700 0.947519i \(-0.603582\pi\)
−0.319700 + 0.947519i \(0.603582\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −24.0000 −0.806296
\(887\) 13.8564 0.465253 0.232626 0.972566i \(-0.425268\pi\)
0.232626 + 0.972566i \(0.425268\pi\)
\(888\) 0 0
\(889\) −17.0000 −0.570162
\(890\) 62.3538 2.09011
\(891\) 0 0
\(892\) −19.0000 −0.636167
\(893\) 3.46410 0.115922
\(894\) 0 0
\(895\) −72.0000 −2.40669
\(896\) −12.1244 −0.405046
\(897\) 0 0
\(898\) −18.0000 −0.600668
\(899\) −17.3205 −0.577671
\(900\) 0 0
\(901\) 0 0
\(902\) −20.7846 −0.692052
\(903\) 0 0
\(904\) 30.0000 0.997785
\(905\) 58.8897 1.95756
\(906\) 0 0
\(907\) −37.0000 −1.22856 −0.614282 0.789086i \(-0.710554\pi\)
−0.614282 + 0.789086i \(0.710554\pi\)
\(908\) −13.8564 −0.459841
\(909\) 0 0
\(910\) −30.0000 −0.994490
\(911\) 27.7128 0.918166 0.459083 0.888393i \(-0.348178\pi\)
0.459083 + 0.888393i \(0.348178\pi\)
\(912\) 0 0
\(913\) −24.0000 −0.794284
\(914\) 29.4449 0.973950
\(915\) 0 0
\(916\) 5.00000 0.165205
\(917\) −3.46410 −0.114395
\(918\) 0 0
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 41.5692 1.37050
\(921\) 0 0
\(922\) 48.0000 1.58080
\(923\) 51.9615 1.71033
\(924\) 0 0
\(925\) −7.00000 −0.230159
\(926\) −53.6936 −1.76448
\(927\) 0 0
\(928\) 18.0000 0.590879
\(929\) 6.92820 0.227307 0.113653 0.993520i \(-0.463745\pi\)
0.113653 + 0.993520i \(0.463745\pi\)
\(930\) 0 0
\(931\) 6.00000 0.196642
\(932\) 0 0
\(933\) 0 0
\(934\) −18.0000 −0.588978
\(935\) 0 0
\(936\) 0 0
\(937\) −37.0000 −1.20874 −0.604369 0.796705i \(-0.706575\pi\)
−0.604369 + 0.796705i \(0.706575\pi\)
\(938\) −13.8564 −0.452428
\(939\) 0 0
\(940\) −12.0000 −0.391397
\(941\) 3.46410 0.112926 0.0564632 0.998405i \(-0.482018\pi\)
0.0564632 + 0.998405i \(0.482018\pi\)
\(942\) 0 0
\(943\) −24.0000 −0.781548
\(944\) −17.3205 −0.563735
\(945\) 0 0
\(946\) 6.00000 0.195077
\(947\) −13.8564 −0.450273 −0.225136 0.974327i \(-0.572283\pi\)
−0.225136 + 0.974327i \(0.572283\pi\)
\(948\) 0 0
\(949\) 10.0000 0.324614
\(950\) −12.1244 −0.393366
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) 24.0000 0.776622
\(956\) −6.92820 −0.224074
\(957\) 0 0
\(958\) −24.0000 −0.775405
\(959\) −6.92820 −0.223723
\(960\) 0 0
\(961\) −6.00000 −0.193548
\(962\) −8.66025 −0.279218
\(963\) 0 0
\(964\) −19.0000 −0.611949
\(965\) −34.6410 −1.11513
\(966\) 0 0
\(967\) 8.00000 0.257263 0.128631 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) −1.73205 −0.0556702
\(969\) 0 0
\(970\) 102.000 3.27502
\(971\) 51.9615 1.66752 0.833762 0.552124i \(-0.186182\pi\)
0.833762 + 0.552124i \(0.186182\pi\)
\(972\) 0 0
\(973\) 13.0000 0.416761
\(974\) −32.9090 −1.05447
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) 3.46410 0.110826 0.0554132 0.998464i \(-0.482352\pi\)
0.0554132 + 0.998464i \(0.482352\pi\)
\(978\) 0 0
\(979\) −36.0000 −1.15056
\(980\) −20.7846 −0.663940
\(981\) 0 0
\(982\) −66.0000 −2.10614
\(983\) 6.92820 0.220975 0.110488 0.993877i \(-0.464759\pi\)
0.110488 + 0.993877i \(0.464759\pi\)
\(984\) 0 0
\(985\) −36.0000 −1.14706
\(986\) 0 0
\(987\) 0 0
\(988\) −5.00000 −0.159071
\(989\) 6.92820 0.220304
\(990\) 0 0
\(991\) 20.0000 0.635321 0.317660 0.948205i \(-0.397103\pi\)
0.317660 + 0.948205i \(0.397103\pi\)
\(992\) −25.9808 −0.824890
\(993\) 0 0
\(994\) −18.0000 −0.570925
\(995\) −65.8179 −2.08657
\(996\) 0 0
\(997\) 2.00000 0.0633406 0.0316703 0.999498i \(-0.489917\pi\)
0.0316703 + 0.999498i \(0.489917\pi\)
\(998\) −48.4974 −1.53516
\(999\) 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 243.2.a.c.1.2 yes 2
3.2 odd 2 inner 243.2.a.c.1.1 2
4.3 odd 2 3888.2.a.ba.1.2 2
5.4 even 2 6075.2.a.bm.1.1 2
9.2 odd 6 243.2.c.d.82.2 4
9.4 even 3 243.2.c.d.163.1 4
9.5 odd 6 243.2.c.d.163.2 4
9.7 even 3 243.2.c.d.82.1 4
12.11 even 2 3888.2.a.ba.1.1 2
15.14 odd 2 6075.2.a.bm.1.2 2
27.2 odd 18 729.2.e.n.568.2 12
27.4 even 9 729.2.e.n.406.2 12
27.5 odd 18 729.2.e.n.649.1 12
27.7 even 9 729.2.e.n.325.2 12
27.11 odd 18 729.2.e.n.82.1 12
27.13 even 9 729.2.e.n.163.1 12
27.14 odd 18 729.2.e.n.163.2 12
27.16 even 9 729.2.e.n.82.2 12
27.20 odd 18 729.2.e.n.325.1 12
27.22 even 9 729.2.e.n.649.2 12
27.23 odd 18 729.2.e.n.406.1 12
27.25 even 9 729.2.e.n.568.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
243.2.a.c.1.1 2 3.2 odd 2 inner
243.2.a.c.1.2 yes 2 1.1 even 1 trivial
243.2.c.d.82.1 4 9.7 even 3
243.2.c.d.82.2 4 9.2 odd 6
243.2.c.d.163.1 4 9.4 even 3
243.2.c.d.163.2 4 9.5 odd 6
729.2.e.n.82.1 12 27.11 odd 18
729.2.e.n.82.2 12 27.16 even 9
729.2.e.n.163.1 12 27.13 even 9
729.2.e.n.163.2 12 27.14 odd 18
729.2.e.n.325.1 12 27.20 odd 18
729.2.e.n.325.2 12 27.7 even 9
729.2.e.n.406.1 12 27.23 odd 18
729.2.e.n.406.2 12 27.4 even 9
729.2.e.n.568.1 12 27.25 even 9
729.2.e.n.568.2 12 27.2 odd 18
729.2.e.n.649.1 12 27.5 odd 18
729.2.e.n.649.2 12 27.22 even 9
3888.2.a.ba.1.1 2 12.11 even 2
3888.2.a.ba.1.2 2 4.3 odd 2
6075.2.a.bm.1.1 2 5.4 even 2
6075.2.a.bm.1.2 2 15.14 odd 2