Properties

Label 6075.2.a.bm.1.1
Level $6075$
Weight $2$
Character 6075.1
Self dual yes
Analytic conductor $48.509$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6075,2,Mod(1,6075)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6075, 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("6075.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6075 = 3^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6075.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: \(48.5091192279\)
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: no (minimal twist has level 243)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 6075.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73205 q^{2} +1.00000 q^{4} +1.00000 q^{7} +1.73205 q^{8} +O(q^{10})\) \(q-1.73205 q^{2} +1.00000 q^{4} +1.00000 q^{7} +1.73205 q^{8} -3.46410 q^{11} -5.00000 q^{13} -1.73205 q^{14} -5.00000 q^{16} -1.00000 q^{19} +6.00000 q^{22} +6.92820 q^{23} +8.66025 q^{26} +1.00000 q^{28} -3.46410 q^{29} +5.00000 q^{31} +5.19615 q^{32} +1.00000 q^{37} +1.73205 q^{38} +3.46410 q^{41} +1.00000 q^{43} -3.46410 q^{44} -12.0000 q^{46} +3.46410 q^{47} -6.00000 q^{49} -5.00000 q^{52} +10.3923 q^{53} +1.73205 q^{56} +6.00000 q^{58} +3.46410 q^{59} +2.00000 q^{61} -8.66025 q^{62} +1.00000 q^{64} -8.00000 q^{67} +10.3923 q^{71} -2.00000 q^{73} -1.73205 q^{74} -1.00000 q^{76} -3.46410 q^{77} -1.00000 q^{79} -6.00000 q^{82} -6.92820 q^{83} -1.73205 q^{86} -6.00000 q^{88} +10.3923 q^{89} -5.00000 q^{91} +6.92820 q^{92} -6.00000 q^{94} -17.0000 q^{97} +10.3923 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} + 2 q^{7} - 10 q^{13} - 10 q^{16} - 2 q^{19} + 12 q^{22} + 2 q^{28} + 10 q^{31} + 2 q^{37} + 2 q^{43} - 24 q^{46} - 12 q^{49} - 10 q^{52} + 12 q^{58} + 4 q^{61} + 2 q^{64} - 16 q^{67} - 4 q^{73} - 2 q^{76} - 2 q^{79} - 12 q^{82} - 12 q^{88} - 10 q^{91} - 12 q^{94} - 34 q^{97}+O(q^{100}) \) 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 \(-0.709785\pi\)
−0.612372 + 0.790569i \(0.709785\pi\)
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000 0.377964 0.188982 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) 1.73205 0.612372
\(9\) 0 0
\(10\) 0 0
\(11\) −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 \(-0.743877\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) −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 \(0.536593\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 6.00000 1.27920
\(23\) 6.92820 1.44463 0.722315 0.691564i \(-0.243078\pi\)
0.722315 + 0.691564i \(0.243078\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 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
\(35\) 0 0
\(36\) 0 0
\(37\) 1.00000 0.164399 0.0821995 0.996616i \(-0.473806\pi\)
0.0821995 + 0.996616i \(0.473806\pi\)
\(38\) 1.73205 0.280976
\(39\) 0 0
\(40\) 0 0
\(41\) 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 \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) −3.46410 −0.522233
\(45\) 0 0
\(46\) −12.0000 −1.76930
\(47\) 3.46410 0.505291 0.252646 0.967559i \(-0.418699\pi\)
0.252646 + 0.967559i \(0.418699\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) 0 0
\(52\) −5.00000 −0.693375
\(53\) 10.3923 1.42749 0.713746 0.700404i \(-0.246997\pi\)
0.713746 + 0.700404i \(0.246997\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 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
\(65\) 0 0
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 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 \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) −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 \(0.517916\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −6.00000 −0.662589
\(83\) −6.92820 −0.760469 −0.380235 0.924890i \(-0.624157\pi\)
−0.380235 + 0.924890i \(0.624157\pi\)
\(84\) 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
\(95\) 0 0
\(96\) 0 0
\(97\) −17.0000 −1.72609 −0.863044 0.505128i \(-0.831445\pi\)
−0.863044 + 0.505128i \(0.831445\pi\)
\(98\) 10.3923 1.04978
\(99\) 0 0
\(100\) 0 0
\(101\) −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 \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) −8.66025 −0.849208
\(105\) 0 0
\(106\) −18.0000 −1.74831
\(107\) −10.3923 −1.00466 −0.502331 0.864675i \(-0.667524\pi\)
−0.502331 + 0.864675i \(0.667524\pi\)
\(108\) 0 0
\(109\) 17.0000 1.62830 0.814152 0.580651i \(-0.197202\pi\)
0.814152 + 0.580651i \(0.197202\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −5.00000 −0.472456
\(113\) 17.3205 1.62938 0.814688 0.579899i \(-0.196908\pi\)
0.814688 + 0.579899i \(0.196908\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −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
\(125\) 0 0
\(126\) 0 0
\(127\) −17.0000 −1.50851 −0.754253 0.656584i \(-0.772001\pi\)
−0.754253 + 0.656584i \(0.772001\pi\)
\(128\) −12.1244 −1.07165
\(129\) 0 0
\(130\) 0 0
\(131\) 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 \(-0.595639\pi\)
−0.295958 + 0.955201i \(0.595639\pi\)
\(138\) 0 0
\(139\) −13.0000 −1.10265 −0.551323 0.834292i \(-0.685877\pi\)
−0.551323 + 0.834292i \(0.685877\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −18.0000 −1.51053
\(143\) 17.3205 1.44841
\(144\) 0 0
\(145\) 0 0
\(146\) 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
\(155\) 0 0
\(156\) 0 0
\(157\) 13.0000 1.03751 0.518756 0.854922i \(-0.326395\pi\)
0.518756 + 0.854922i \(0.326395\pi\)
\(158\) 1.73205 0.137795
\(159\) 0 0
\(160\) 0 0
\(161\) 6.92820 0.546019
\(162\) 0 0
\(163\) 1.00000 0.0783260 0.0391630 0.999233i \(-0.487531\pi\)
0.0391630 + 0.999233i \(0.487531\pi\)
\(164\) 3.46410 0.270501
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) −24.2487 −1.87642 −0.938211 0.346064i \(-0.887518\pi\)
−0.938211 + 0.346064i \(0.887518\pi\)
\(168\) 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 \(-0.323414\pi\)
0.526742 + 0.850026i \(0.323414\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 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
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 3.46410 0.252646
\(189\) 0 0
\(190\) 0 0
\(191\) 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 \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) 29.4449 2.11402
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) 10.3923 0.740421 0.370211 0.928948i \(-0.379286\pi\)
0.370211 + 0.928948i \(0.379286\pi\)
\(198\) 0 0
\(199\) −19.0000 −1.34687 −0.673437 0.739244i \(-0.735183\pi\)
−0.673437 + 0.739244i \(0.735183\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 24.0000 1.68863
\(203\) −3.46410 −0.243132
\(204\) 0 0
\(205\) 0 0
\(206\) 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
\(215\) 0 0
\(216\) 0 0
\(217\) 5.00000 0.339422
\(218\) −29.4449 −1.99426
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 19.0000 1.27233 0.636167 0.771551i \(-0.280519\pi\)
0.636167 + 0.771551i \(0.280519\pi\)
\(224\) 5.19615 0.347183
\(225\) 0 0
\(226\) −30.0000 −1.99557
\(227\) 13.8564 0.919682 0.459841 0.888001i \(-0.347906\pi\)
0.459841 + 0.888001i \(0.347906\pi\)
\(228\) 0 0
\(229\) 5.00000 0.330409 0.165205 0.986259i \(-0.447172\pi\)
0.165205 + 0.986259i \(0.447172\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 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
\(245\) 0 0
\(246\) 0 0
\(247\) 5.00000 0.318142
\(248\) 8.66025 0.549927
\(249\) 0 0
\(250\) 0 0
\(251\) 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 \(-0.534458\pi\)
−0.108042 + 0.994146i \(0.534458\pi\)
\(258\) 0 0
\(259\) 1.00000 0.0621370
\(260\) 0 0
\(261\) 0 0
\(262\) −6.00000 −0.370681
\(263\) 13.8564 0.854423 0.427211 0.904152i \(-0.359496\pi\)
0.427211 + 0.904152i \(0.359496\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 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
\(275\) 0 0
\(276\) 0 0
\(277\) −17.0000 −1.02143 −0.510716 0.859750i \(-0.670619\pi\)
−0.510716 + 0.859750i \(0.670619\pi\)
\(278\) 22.5167 1.35046
\(279\) 0 0
\(280\) 0 0
\(281\) −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 \(-0.373724\pi\)
0.386385 + 0.922338i \(0.373724\pi\)
\(284\) 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
\(290\) 0 0
\(291\) 0 0
\(292\) −2.00000 −0.117041
\(293\) −13.8564 −0.809500 −0.404750 0.914427i \(-0.632641\pi\)
−0.404750 + 0.914427i \(0.632641\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 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
\(305\) 0 0
\(306\) 0 0
\(307\) −20.0000 −1.14146 −0.570730 0.821138i \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) −3.46410 −0.197386
\(309\) 0 0
\(310\) 0 0
\(311\) 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 \(-0.491003\pi\)
0.0282617 + 0.999601i \(0.491003\pi\)
\(314\) −22.5167 −1.27069
\(315\) 0 0
\(316\) −1.00000 −0.0562544
\(317\) −27.7128 −1.55651 −0.778253 0.627950i \(-0.783894\pi\)
−0.778253 + 0.627950i \(0.783894\pi\)
\(318\) 0 0
\(319\) 12.0000 0.671871
\(320\) 0 0
\(321\) 0 0
\(322\) −12.0000 −0.668734
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) −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
\(335\) 0 0
\(336\) 0 0
\(337\) −5.00000 −0.272367 −0.136184 0.990684i \(-0.543484\pi\)
−0.136184 + 0.990684i \(0.543484\pi\)
\(338\) −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 \(-0.725596\pi\)
−0.650870 + 0.759190i \(0.725596\pi\)
\(348\) 0 0
\(349\) −1.00000 −0.0535288 −0.0267644 0.999642i \(-0.508520\pi\)
−0.0267644 + 0.999642i \(0.508520\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −18.0000 −0.959403
\(353\) −17.3205 −0.921878 −0.460939 0.887432i \(-0.652487\pi\)
−0.460939 + 0.887432i \(0.652487\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 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
\(365\) 0 0
\(366\) 0 0
\(367\) 16.0000 0.835193 0.417597 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) −34.6410 −1.80579
\(369\) 0 0
\(370\) 0 0
\(371\) 10.3923 0.539542
\(372\) 0 0
\(373\) −23.0000 −1.19089 −0.595447 0.803394i \(-0.703025\pi\)
−0.595447 + 0.803394i \(0.703025\pi\)
\(374\) 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 \(0.662267\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −12.0000 −0.613973
\(383\) 17.3205 0.885037 0.442518 0.896759i \(-0.354085\pi\)
0.442518 + 0.896759i \(0.354085\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −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
\(395\) 0 0
\(396\) 0 0
\(397\) 1.00000 0.0501886 0.0250943 0.999685i \(-0.492011\pi\)
0.0250943 + 0.999685i \(0.492011\pi\)
\(398\) 32.9090 1.64958
\(399\) 0 0
\(400\) 0 0
\(401\) 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 \(0.460551\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −8.00000 −0.394132
\(413\) 3.46410 0.170457
\(414\) 0 0
\(415\) 0 0
\(416\) −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
\(430\) 0 0
\(431\) −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 \(-0.492351\pi\)
0.0240285 + 0.999711i \(0.492351\pi\)
\(434\) −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 \(0.341625\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 13.8564 0.658338 0.329169 0.944271i \(-0.393231\pi\)
0.329169 + 0.944271i \(0.393231\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −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
\(455\) 0 0
\(456\) 0 0
\(457\) −17.0000 −0.795226 −0.397613 0.917553i \(-0.630161\pi\)
−0.397613 + 0.917553i \(0.630161\pi\)
\(458\) −8.66025 −0.404667
\(459\) 0 0
\(460\) 0 0
\(461\) 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 \(-0.243983\pi\)
0.720346 + 0.693615i \(0.243983\pi\)
\(464\) 17.3205 0.804084
\(465\) 0 0
\(466\) 0 0
\(467\) 10.3923 0.480899 0.240449 0.970662i \(-0.422705\pi\)
0.240449 + 0.970662i \(0.422705\pi\)
\(468\) 0 0
\(469\) −8.00000 −0.369406
\(470\) 0 0
\(471\) 0 0
\(472\) 6.00000 0.276172
\(473\) −3.46410 −0.159280
\(474\) 0 0
\(475\) 0 0
\(476\) 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
\(485\) 0 0
\(486\) 0 0
\(487\) 19.0000 0.860972 0.430486 0.902597i \(-0.358342\pi\)
0.430486 + 0.902597i \(0.358342\pi\)
\(488\) 3.46410 0.156813
\(489\) 0 0
\(490\) 0 0
\(491\) −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 \(0.715605\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −36.0000 −1.60676
\(503\) −41.5692 −1.85348 −0.926740 0.375703i \(-0.877401\pi\)
−0.926740 + 0.375703i \(0.877401\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 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
\(515\) 0 0
\(516\) 0 0
\(517\) −12.0000 −0.527759
\(518\) −1.73205 −0.0761019
\(519\) 0 0
\(520\) 0 0
\(521\) −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 \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 3.46410 0.151330
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) 0 0
\(528\) 0 0
\(529\) 25.0000 1.08696
\(530\) 0 0
\(531\) 0 0
\(532\) −1.00000 −0.0433555
\(533\) −17.3205 −0.750234
\(534\) 0 0
\(535\) 0 0
\(536\) −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
\(545\) 0 0
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) −6.92820 −0.295958
\(549\) 0 0
\(550\) 0 0
\(551\) 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 \(-0.570661\pi\)
−0.220168 + 0.975462i \(0.570661\pi\)
\(558\) 0 0
\(559\) −5.00000 −0.211477
\(560\) 0 0
\(561\) 0 0
\(562\) 24.0000 1.01238
\(563\) −34.6410 −1.45994 −0.729972 0.683477i \(-0.760467\pi\)
−0.729972 + 0.683477i \(0.760467\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −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
\(575\) 0 0
\(576\) 0 0
\(577\) 34.0000 1.41544 0.707719 0.706494i \(-0.249724\pi\)
0.707719 + 0.706494i \(0.249724\pi\)
\(578\) 29.4449 1.22474
\(579\) 0 0
\(580\) 0 0
\(581\) −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 \(-0.545668\pi\)
−0.142979 + 0.989726i \(0.545668\pi\)
\(588\) 0 0
\(589\) −5.00000 −0.206021
\(590\) 0 0
\(591\) 0 0
\(592\) −5.00000 −0.205499
\(593\) −20.7846 −0.853522 −0.426761 0.904365i \(-0.640345\pi\)
−0.426761 + 0.904365i \(0.640345\pi\)
\(594\) 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
\(605\) 0 0
\(606\) 0 0
\(607\) 13.0000 0.527654 0.263827 0.964570i \(-0.415015\pi\)
0.263827 + 0.964570i \(0.415015\pi\)
\(608\) −5.19615 −0.210732
\(609\) 0 0
\(610\) 0 0
\(611\) −17.3205 −0.700713
\(612\) 0 0
\(613\) −2.00000 −0.0807792 −0.0403896 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) 34.6410 1.39800
\(615\) 0 0
\(616\) −6.00000 −0.241747
\(617\) 6.92820 0.278919 0.139459 0.990228i \(-0.455464\pi\)
0.139459 + 0.990228i \(0.455464\pi\)
\(618\) 0 0
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −24.0000 −0.962312
\(623\) 10.3923 0.416359
\(624\) 0 0
\(625\) 0 0
\(626\) −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
\(635\) 0 0
\(636\) 0 0
\(637\) 30.0000 1.18864
\(638\) −20.7846 −0.822871
\(639\) 0 0
\(640\) 0 0
\(641\) −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 \(-0.834335\pi\)
−0.867595 + 0.497271i \(0.834335\pi\)
\(644\) 6.92820 0.273009
\(645\) 0 0
\(646\) 0 0
\(647\) 20.7846 0.817127 0.408564 0.912730i \(-0.366030\pi\)
0.408564 + 0.912730i \(0.366030\pi\)
\(648\) 0 0
\(649\) −12.0000 −0.471041
\(650\) 0 0
\(651\) 0 0
\(652\) 1.00000 0.0391630
\(653\) 17.3205 0.677804 0.338902 0.940822i \(-0.389945\pi\)
0.338902 + 0.940822i \(0.389945\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −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
\(665\) 0 0
\(666\) 0 0
\(667\) −24.0000 −0.929284
\(668\) −24.2487 −0.938211
\(669\) 0 0
\(670\) 0 0
\(671\) −6.92820 −0.267460
\(672\) 0 0
\(673\) 37.0000 1.42625 0.713123 0.701039i \(-0.247280\pi\)
0.713123 + 0.701039i \(0.247280\pi\)
\(674\) 8.66025 0.333581
\(675\) 0 0
\(676\) 12.0000 0.461538
\(677\) −17.3205 −0.665681 −0.332841 0.942983i \(-0.608007\pi\)
−0.332841 + 0.942983i \(0.608007\pi\)
\(678\) 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 \(-0.792687\pi\)
−0.795301 + 0.606215i \(0.792687\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 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
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 1.73205 0.0655591
\(699\) 0 0
\(700\) 0 0
\(701\) −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 \(0.616125\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 18.0000 0.674579
\(713\) 34.6410 1.29732
\(714\) 0 0
\(715\) 0 0
\(716\) −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
\(725\) 0 0
\(726\) 0 0
\(727\) 16.0000 0.593407 0.296704 0.954970i \(-0.404113\pi\)
0.296704 + 0.954970i \(0.404113\pi\)
\(728\) −8.66025 −0.320970
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −41.0000 −1.51437 −0.757185 0.653201i \(-0.773426\pi\)
−0.757185 + 0.653201i \(0.773426\pi\)
\(734\) −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 \(0.613636\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −18.0000 −0.660801
\(743\) 6.92820 0.254171 0.127086 0.991892i \(-0.459438\pi\)
0.127086 + 0.991892i \(0.459438\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 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
\(755\) 0 0
\(756\) 0 0
\(757\) −38.0000 −1.38113 −0.690567 0.723269i \(-0.742639\pi\)
−0.690567 + 0.723269i \(0.742639\pi\)
\(758\) 32.9090 1.19531
\(759\) 0 0
\(760\) 0 0
\(761\) −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 \(0.575311\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 10.0000 0.359908
\(773\) −20.7846 −0.747570 −0.373785 0.927515i \(-0.621940\pi\)
−0.373785 + 0.927515i \(0.621940\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −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
\(785\) 0 0
\(786\) 0 0
\(787\) 28.0000 0.998092 0.499046 0.866575i \(-0.333684\pi\)
0.499046 + 0.866575i \(0.333684\pi\)
\(788\) 10.3923 0.370211
\(789\) 0 0
\(790\) 0 0
\(791\) 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 \(-0.578922\pi\)
−0.245410 + 0.969419i \(0.578922\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) −6.00000 −0.211867
\(803\) 6.92820 0.244491
\(804\) 0 0
\(805\) 0 0
\(806\) 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
\(815\) 0 0
\(816\) 0 0
\(817\) −1.00000 −0.0349856
\(818\) −8.66025 −0.302799
\(819\) 0 0
\(820\) 0 0
\(821\) 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 \(-0.631290\pi\)
−0.400865 + 0.916137i \(0.631290\pi\)
\(824\) −13.8564 −0.482711
\(825\) 0 0
\(826\) −6.00000 −0.208767
\(827\) 10.3923 0.361376 0.180688 0.983540i \(-0.442168\pi\)
0.180688 + 0.983540i \(0.442168\pi\)
\(828\) 0 0
\(829\) −19.0000 −0.659897 −0.329949 0.943999i \(-0.607031\pi\)
−0.329949 + 0.943999i \(0.607031\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −5.00000 −0.173344
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 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
\(845\) 0 0
\(846\) 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 \(-0.510901\pi\)
−0.0342393 + 0.999414i \(0.510901\pi\)
\(854\) −3.46410 −0.118539
\(855\) 0 0
\(856\) −18.0000 −0.615227
\(857\) −38.1051 −1.30165 −0.650823 0.759229i \(-0.725576\pi\)
−0.650823 + 0.759229i \(0.725576\pi\)
\(858\) 0 0
\(859\) −49.0000 −1.67186 −0.835929 0.548837i \(-0.815071\pi\)
−0.835929 + 0.548837i \(0.815071\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 36.0000 1.22616
\(863\) −10.3923 −0.353758 −0.176879 0.984233i \(-0.556600\pi\)
−0.176879 + 0.984233i \(0.556600\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −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
\(875\) 0 0
\(876\) 0 0
\(877\) −23.0000 −0.776655 −0.388327 0.921521i \(-0.626947\pi\)
−0.388327 + 0.921521i \(0.626947\pi\)
\(878\) −34.6410 −1.16908
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 19.0000 0.639401 0.319700 0.947519i \(-0.396418\pi\)
0.319700 + 0.947519i \(0.396418\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −24.0000 −0.806296
\(887\) −13.8564 −0.465253 −0.232626 0.972566i \(-0.574732\pi\)
−0.232626 + 0.972566i \(0.574732\pi\)
\(888\) 0 0
\(889\) −17.0000 −0.570162
\(890\) 0 0
\(891\) 0 0
\(892\) 19.0000 0.636167
\(893\) −3.46410 −0.115922
\(894\) 0 0
\(895\) 0 0
\(896\) −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
\(905\) 0 0
\(906\) 0 0
\(907\) 37.0000 1.22856 0.614282 0.789086i \(-0.289446\pi\)
0.614282 + 0.789086i \(0.289446\pi\)
\(908\) 13.8564 0.459841
\(909\) 0 0
\(910\) 0 0
\(911\) 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 \(0.585007\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −48.0000 −1.58080
\(923\) −51.9615 −1.71033
\(924\) 0 0
\(925\) 0 0
\(926\) −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 \(-0.293425\pi\)
0.604369 + 0.796705i \(0.293425\pi\)
\(938\) 13.8564 0.452428
\(939\) 0 0
\(940\) 0 0
\(941\) 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 \(-0.427717\pi\)
0.225136 + 0.974327i \(0.427717\pi\)
\(948\) 0 0
\(949\) 10.0000 0.324614
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −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
\(965\) 0 0
\(966\) 0 0
\(967\) −8.00000 −0.257263 −0.128631 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) 1.73205 0.0556702
\(969\) 0 0
\(970\) 0 0
\(971\) 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 \(-0.517648\pi\)
−0.0554132 + 0.998464i \(0.517648\pi\)
\(978\) 0 0
\(979\) −36.0000 −1.15056
\(980\) 0 0
\(981\) 0 0
\(982\) 66.0000 2.10614
\(983\) −6.92820 −0.220975 −0.110488 0.993877i \(-0.535241\pi\)
−0.110488 + 0.993877i \(0.535241\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 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
\(995\) 0 0
\(996\) 0 0
\(997\) −2.00000 −0.0633406 −0.0316703 0.999498i \(-0.510083\pi\)
−0.0316703 + 0.999498i \(0.510083\pi\)
\(998\) 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 6075.2.a.bm.1.1 2
3.2 odd 2 inner 6075.2.a.bm.1.2 2
5.4 even 2 243.2.a.c.1.2 yes 2
15.14 odd 2 243.2.a.c.1.1 2
20.19 odd 2 3888.2.a.ba.1.2 2
45.4 even 6 243.2.c.d.163.1 4
45.14 odd 6 243.2.c.d.163.2 4
45.29 odd 6 243.2.c.d.82.2 4
45.34 even 6 243.2.c.d.82.1 4
60.59 even 2 3888.2.a.ba.1.1 2
135.4 even 18 729.2.e.n.406.2 12
135.14 odd 18 729.2.e.n.163.2 12
135.29 odd 18 729.2.e.n.568.2 12
135.34 even 18 729.2.e.n.325.2 12
135.49 even 18 729.2.e.n.649.2 12
135.59 odd 18 729.2.e.n.649.1 12
135.74 odd 18 729.2.e.n.325.1 12
135.79 even 18 729.2.e.n.568.1 12
135.94 even 18 729.2.e.n.163.1 12
135.104 odd 18 729.2.e.n.406.1 12
135.119 odd 18 729.2.e.n.82.1 12
135.124 even 18 729.2.e.n.82.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
243.2.a.c.1.1 2 15.14 odd 2
243.2.a.c.1.2 yes 2 5.4 even 2
243.2.c.d.82.1 4 45.34 even 6
243.2.c.d.82.2 4 45.29 odd 6
243.2.c.d.163.1 4 45.4 even 6
243.2.c.d.163.2 4 45.14 odd 6
729.2.e.n.82.1 12 135.119 odd 18
729.2.e.n.82.2 12 135.124 even 18
729.2.e.n.163.1 12 135.94 even 18
729.2.e.n.163.2 12 135.14 odd 18
729.2.e.n.325.1 12 135.74 odd 18
729.2.e.n.325.2 12 135.34 even 18
729.2.e.n.406.1 12 135.104 odd 18
729.2.e.n.406.2 12 135.4 even 18
729.2.e.n.568.1 12 135.79 even 18
729.2.e.n.568.2 12 135.29 odd 18
729.2.e.n.649.1 12 135.59 odd 18
729.2.e.n.649.2 12 135.49 even 18
3888.2.a.ba.1.1 2 60.59 even 2
3888.2.a.ba.1.2 2 20.19 odd 2
6075.2.a.bm.1.1 2 1.1 even 1 trivial
6075.2.a.bm.1.2 2 3.2 odd 2 inner