Properties

Label 1089.2.a.p.1.1
Level $1089$
Weight $2$
Character 1089.1
Self dual yes
Analytic conductor $8.696$
Analytic rank $0$
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] = [1089,2,Mod(1,1089)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1089, 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("1089.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1089.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: \(8.69570878012\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 363)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 1089.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.23607 q^{2} +3.00000 q^{4} -2.00000 q^{5} -4.47214 q^{7} -2.23607 q^{8} +O(q^{10})\) \(q-2.23607 q^{2} +3.00000 q^{4} -2.00000 q^{5} -4.47214 q^{7} -2.23607 q^{8} +4.47214 q^{10} +10.0000 q^{14} -1.00000 q^{16} -4.47214 q^{17} -4.47214 q^{19} -6.00000 q^{20} +4.00000 q^{23} -1.00000 q^{25} -13.4164 q^{28} -4.47214 q^{29} +6.70820 q^{32} +10.0000 q^{34} +8.94427 q^{35} +2.00000 q^{37} +10.0000 q^{38} +4.47214 q^{40} +4.47214 q^{41} +4.47214 q^{43} -8.94427 q^{46} -8.00000 q^{47} +13.0000 q^{49} +2.23607 q^{50} -6.00000 q^{53} +10.0000 q^{56} +10.0000 q^{58} +8.94427 q^{61} -13.0000 q^{64} -12.0000 q^{67} -13.4164 q^{68} -20.0000 q^{70} +8.00000 q^{71} -8.94427 q^{73} -4.47214 q^{74} -13.4164 q^{76} +13.4164 q^{79} +2.00000 q^{80} -10.0000 q^{82} +8.94427 q^{83} +8.94427 q^{85} -10.0000 q^{86} +14.0000 q^{89} +12.0000 q^{92} +17.8885 q^{94} +8.94427 q^{95} +2.00000 q^{97} -29.0689 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{4} - 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{4} - 4 q^{5} + 20 q^{14} - 2 q^{16} - 12 q^{20} + 8 q^{23} - 2 q^{25} + 20 q^{34} + 4 q^{37} + 20 q^{38} - 16 q^{47} + 26 q^{49} - 12 q^{53} + 20 q^{56} + 20 q^{58} - 26 q^{64} - 24 q^{67} - 40 q^{70} + 16 q^{71} + 4 q^{80} - 20 q^{82} - 20 q^{86} + 28 q^{89} + 24 q^{92} + 4 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.23607 −1.58114 −0.790569 0.612372i \(-0.790215\pi\)
−0.790569 + 0.612372i \(0.790215\pi\)
\(3\) 0 0
\(4\) 3.00000 1.50000
\(5\) −2.00000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 0 0
\(7\) −4.47214 −1.69031 −0.845154 0.534522i \(-0.820491\pi\)
−0.845154 + 0.534522i \(0.820491\pi\)
\(8\) −2.23607 −0.790569
\(9\) 0 0
\(10\) 4.47214 1.41421
\(11\) 0 0
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 10.0000 2.67261
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) −4.47214 −1.08465 −0.542326 0.840168i \(-0.682456\pi\)
−0.542326 + 0.840168i \(0.682456\pi\)
\(18\) 0 0
\(19\) −4.47214 −1.02598 −0.512989 0.858395i \(-0.671462\pi\)
−0.512989 + 0.858395i \(0.671462\pi\)
\(20\) −6.00000 −1.34164
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) −13.4164 −2.53546
\(29\) −4.47214 −0.830455 −0.415227 0.909718i \(-0.636298\pi\)
−0.415227 + 0.909718i \(0.636298\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 6.70820 1.18585
\(33\) 0 0
\(34\) 10.0000 1.71499
\(35\) 8.94427 1.51186
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 10.0000 1.62221
\(39\) 0 0
\(40\) 4.47214 0.707107
\(41\) 4.47214 0.698430 0.349215 0.937043i \(-0.386448\pi\)
0.349215 + 0.937043i \(0.386448\pi\)
\(42\) 0 0
\(43\) 4.47214 0.681994 0.340997 0.940064i \(-0.389235\pi\)
0.340997 + 0.940064i \(0.389235\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −8.94427 −1.31876
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) 13.0000 1.85714
\(50\) 2.23607 0.316228
\(51\) 0 0
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 10.0000 1.33631
\(57\) 0 0
\(58\) 10.0000 1.31306
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 8.94427 1.14520 0.572598 0.819836i \(-0.305935\pi\)
0.572598 + 0.819836i \(0.305935\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −13.0000 −1.62500
\(65\) 0 0
\(66\) 0 0
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) −13.4164 −1.62698
\(69\) 0 0
\(70\) −20.0000 −2.39046
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) −8.94427 −1.04685 −0.523424 0.852072i \(-0.675346\pi\)
−0.523424 + 0.852072i \(0.675346\pi\)
\(74\) −4.47214 −0.519875
\(75\) 0 0
\(76\) −13.4164 −1.53897
\(77\) 0 0
\(78\) 0 0
\(79\) 13.4164 1.50946 0.754732 0.656033i \(-0.227767\pi\)
0.754732 + 0.656033i \(0.227767\pi\)
\(80\) 2.00000 0.223607
\(81\) 0 0
\(82\) −10.0000 −1.10432
\(83\) 8.94427 0.981761 0.490881 0.871227i \(-0.336675\pi\)
0.490881 + 0.871227i \(0.336675\pi\)
\(84\) 0 0
\(85\) 8.94427 0.970143
\(86\) −10.0000 −1.07833
\(87\) 0 0
\(88\) 0 0
\(89\) 14.0000 1.48400 0.741999 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 12.0000 1.25109
\(93\) 0 0
\(94\) 17.8885 1.84506
\(95\) 8.94427 0.917663
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) −29.0689 −2.93640
\(99\) 0 0
\(100\) −3.00000 −0.300000
\(101\) −4.47214 −0.444994 −0.222497 0.974933i \(-0.571421\pi\)
−0.222497 + 0.974933i \(0.571421\pi\)
\(102\) 0 0
\(103\) 16.0000 1.57653 0.788263 0.615338i \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 13.4164 1.30312
\(107\) −8.94427 −0.864675 −0.432338 0.901712i \(-0.642311\pi\)
−0.432338 + 0.901712i \(0.642311\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 4.47214 0.422577
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) −8.00000 −0.746004
\(116\) −13.4164 −1.24568
\(117\) 0 0
\(118\) 0 0
\(119\) 20.0000 1.83340
\(120\) 0 0
\(121\) 0 0
\(122\) −20.0000 −1.81071
\(123\) 0 0
\(124\) 0 0
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) 13.4164 1.19051 0.595257 0.803535i \(-0.297050\pi\)
0.595257 + 0.803535i \(0.297050\pi\)
\(128\) 15.6525 1.38350
\(129\) 0 0
\(130\) 0 0
\(131\) 17.8885 1.56293 0.781465 0.623949i \(-0.214473\pi\)
0.781465 + 0.623949i \(0.214473\pi\)
\(132\) 0 0
\(133\) 20.0000 1.73422
\(134\) 26.8328 2.31800
\(135\) 0 0
\(136\) 10.0000 0.857493
\(137\) −22.0000 −1.87959 −0.939793 0.341743i \(-0.888983\pi\)
−0.939793 + 0.341743i \(0.888983\pi\)
\(138\) 0 0
\(139\) 13.4164 1.13796 0.568982 0.822350i \(-0.307337\pi\)
0.568982 + 0.822350i \(0.307337\pi\)
\(140\) 26.8328 2.26779
\(141\) 0 0
\(142\) −17.8885 −1.50117
\(143\) 0 0
\(144\) 0 0
\(145\) 8.94427 0.742781
\(146\) 20.0000 1.65521
\(147\) 0 0
\(148\) 6.00000 0.493197
\(149\) 22.3607 1.83186 0.915929 0.401340i \(-0.131455\pi\)
0.915929 + 0.401340i \(0.131455\pi\)
\(150\) 0 0
\(151\) 13.4164 1.09181 0.545906 0.837846i \(-0.316186\pi\)
0.545906 + 0.837846i \(0.316186\pi\)
\(152\) 10.0000 0.811107
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) −30.0000 −2.38667
\(159\) 0 0
\(160\) −13.4164 −1.06066
\(161\) −17.8885 −1.40981
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 13.4164 1.04765
\(165\) 0 0
\(166\) −20.0000 −1.55230
\(167\) −8.94427 −0.692129 −0.346064 0.938211i \(-0.612482\pi\)
−0.346064 + 0.938211i \(0.612482\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) −20.0000 −1.53393
\(171\) 0 0
\(172\) 13.4164 1.02299
\(173\) 13.4164 1.02003 0.510015 0.860165i \(-0.329640\pi\)
0.510015 + 0.860165i \(0.329640\pi\)
\(174\) 0 0
\(175\) 4.47214 0.338062
\(176\) 0 0
\(177\) 0 0
\(178\) −31.3050 −2.34641
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −8.94427 −0.659380
\(185\) −4.00000 −0.294086
\(186\) 0 0
\(187\) 0 0
\(188\) −24.0000 −1.75038
\(189\) 0 0
\(190\) −20.0000 −1.45095
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −17.8885 −1.28765 −0.643823 0.765175i \(-0.722653\pi\)
−0.643823 + 0.765175i \(0.722653\pi\)
\(194\) −4.47214 −0.321081
\(195\) 0 0
\(196\) 39.0000 2.78571
\(197\) −22.3607 −1.59313 −0.796566 0.604551i \(-0.793352\pi\)
−0.796566 + 0.604551i \(0.793352\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 2.23607 0.158114
\(201\) 0 0
\(202\) 10.0000 0.703598
\(203\) 20.0000 1.40372
\(204\) 0 0
\(205\) −8.94427 −0.624695
\(206\) −35.7771 −2.49271
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −4.47214 −0.307875 −0.153937 0.988081i \(-0.549195\pi\)
−0.153937 + 0.988081i \(0.549195\pi\)
\(212\) −18.0000 −1.23625
\(213\) 0 0
\(214\) 20.0000 1.36717
\(215\) −8.94427 −0.609994
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) −30.0000 −2.00446
\(225\) 0 0
\(226\) 13.4164 0.892446
\(227\) −8.94427 −0.593652 −0.296826 0.954932i \(-0.595928\pi\)
−0.296826 + 0.954932i \(0.595928\pi\)
\(228\) 0 0
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 17.8885 1.17954
\(231\) 0 0
\(232\) 10.0000 0.656532
\(233\) 4.47214 0.292979 0.146490 0.989212i \(-0.453202\pi\)
0.146490 + 0.989212i \(0.453202\pi\)
\(234\) 0 0
\(235\) 16.0000 1.04372
\(236\) 0 0
\(237\) 0 0
\(238\) −44.7214 −2.89886
\(239\) −8.94427 −0.578557 −0.289278 0.957245i \(-0.593415\pi\)
−0.289278 + 0.957245i \(0.593415\pi\)
\(240\) 0 0
\(241\) 8.94427 0.576151 0.288076 0.957608i \(-0.406985\pi\)
0.288076 + 0.957608i \(0.406985\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 26.8328 1.71780
\(245\) −26.0000 −1.66108
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) −26.8328 −1.69706
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −30.0000 −1.88237
\(255\) 0 0
\(256\) −9.00000 −0.562500
\(257\) 22.0000 1.37232 0.686161 0.727450i \(-0.259294\pi\)
0.686161 + 0.727450i \(0.259294\pi\)
\(258\) 0 0
\(259\) −8.94427 −0.555770
\(260\) 0 0
\(261\) 0 0
\(262\) −40.0000 −2.47121
\(263\) −8.94427 −0.551527 −0.275764 0.961225i \(-0.588931\pi\)
−0.275764 + 0.961225i \(0.588931\pi\)
\(264\) 0 0
\(265\) 12.0000 0.737154
\(266\) −44.7214 −2.74204
\(267\) 0 0
\(268\) −36.0000 −2.19905
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 0 0
\(271\) −13.4164 −0.814989 −0.407494 0.913208i \(-0.633597\pi\)
−0.407494 + 0.913208i \(0.633597\pi\)
\(272\) 4.47214 0.271163
\(273\) 0 0
\(274\) 49.1935 2.97189
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(278\) −30.0000 −1.79928
\(279\) 0 0
\(280\) −20.0000 −1.19523
\(281\) −31.3050 −1.86750 −0.933748 0.357930i \(-0.883483\pi\)
−0.933748 + 0.357930i \(0.883483\pi\)
\(282\) 0 0
\(283\) −13.4164 −0.797523 −0.398761 0.917055i \(-0.630560\pi\)
−0.398761 + 0.917055i \(0.630560\pi\)
\(284\) 24.0000 1.42414
\(285\) 0 0
\(286\) 0 0
\(287\) −20.0000 −1.18056
\(288\) 0 0
\(289\) 3.00000 0.176471
\(290\) −20.0000 −1.17444
\(291\) 0 0
\(292\) −26.8328 −1.57027
\(293\) 22.3607 1.30632 0.653162 0.757218i \(-0.273442\pi\)
0.653162 + 0.757218i \(0.273442\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −4.47214 −0.259938
\(297\) 0 0
\(298\) −50.0000 −2.89642
\(299\) 0 0
\(300\) 0 0
\(301\) −20.0000 −1.15278
\(302\) −30.0000 −1.72631
\(303\) 0 0
\(304\) 4.47214 0.256495
\(305\) −17.8885 −1.02430
\(306\) 0 0
\(307\) −4.47214 −0.255238 −0.127619 0.991823i \(-0.540734\pi\)
−0.127619 + 0.991823i \(0.540734\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 0 0
\(313\) 14.0000 0.791327 0.395663 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) −4.47214 −0.252377
\(315\) 0 0
\(316\) 40.2492 2.26420
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 26.0000 1.45344
\(321\) 0 0
\(322\) 40.0000 2.22911
\(323\) 20.0000 1.11283
\(324\) 0 0
\(325\) 0 0
\(326\) −8.94427 −0.495377
\(327\) 0 0
\(328\) −10.0000 −0.552158
\(329\) 35.7771 1.97245
\(330\) 0 0
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) 26.8328 1.47264
\(333\) 0 0
\(334\) 20.0000 1.09435
\(335\) 24.0000 1.31126
\(336\) 0 0
\(337\) 8.94427 0.487226 0.243613 0.969873i \(-0.421667\pi\)
0.243613 + 0.969873i \(0.421667\pi\)
\(338\) 29.0689 1.58114
\(339\) 0 0
\(340\) 26.8328 1.45521
\(341\) 0 0
\(342\) 0 0
\(343\) −26.8328 −1.44884
\(344\) −10.0000 −0.539164
\(345\) 0 0
\(346\) −30.0000 −1.61281
\(347\) −8.94427 −0.480154 −0.240077 0.970754i \(-0.577173\pi\)
−0.240077 + 0.970754i \(0.577173\pi\)
\(348\) 0 0
\(349\) −26.8328 −1.43633 −0.718164 0.695874i \(-0.755017\pi\)
−0.718164 + 0.695874i \(0.755017\pi\)
\(350\) −10.0000 −0.534522
\(351\) 0 0
\(352\) 0 0
\(353\) 14.0000 0.745145 0.372572 0.928003i \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) 0 0
\(355\) −16.0000 −0.849192
\(356\) 42.0000 2.22600
\(357\) 0 0
\(358\) −8.94427 −0.472719
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 22.3607 1.17525
\(363\) 0 0
\(364\) 0 0
\(365\) 17.8885 0.936329
\(366\) 0 0
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) −4.00000 −0.208514
\(369\) 0 0
\(370\) 8.94427 0.464991
\(371\) 26.8328 1.39309
\(372\) 0 0
\(373\) −26.8328 −1.38935 −0.694675 0.719323i \(-0.744452\pi\)
−0.694675 + 0.719323i \(0.744452\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 17.8885 0.922531
\(377\) 0 0
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 26.8328 1.37649
\(381\) 0 0
\(382\) 0 0
\(383\) 36.0000 1.83951 0.919757 0.392488i \(-0.128386\pi\)
0.919757 + 0.392488i \(0.128386\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 40.0000 2.03595
\(387\) 0 0
\(388\) 6.00000 0.304604
\(389\) −10.0000 −0.507020 −0.253510 0.967333i \(-0.581585\pi\)
−0.253510 + 0.967333i \(0.581585\pi\)
\(390\) 0 0
\(391\) −17.8885 −0.904663
\(392\) −29.0689 −1.46820
\(393\) 0 0
\(394\) 50.0000 2.51896
\(395\) −26.8328 −1.35011
\(396\) 0 0
\(397\) 22.0000 1.10415 0.552074 0.833795i \(-0.313837\pi\)
0.552074 + 0.833795i \(0.313837\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −13.4164 −0.667491
\(405\) 0 0
\(406\) −44.7214 −2.21948
\(407\) 0 0
\(408\) 0 0
\(409\) −26.8328 −1.32680 −0.663399 0.748266i \(-0.730887\pi\)
−0.663399 + 0.748266i \(0.730887\pi\)
\(410\) 20.0000 0.987730
\(411\) 0 0
\(412\) 48.0000 2.36479
\(413\) 0 0
\(414\) 0 0
\(415\) −17.8885 −0.878114
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −4.00000 −0.195413 −0.0977064 0.995215i \(-0.531151\pi\)
−0.0977064 + 0.995215i \(0.531151\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) 10.0000 0.486792
\(423\) 0 0
\(424\) 13.4164 0.651558
\(425\) 4.47214 0.216930
\(426\) 0 0
\(427\) −40.0000 −1.93574
\(428\) −26.8328 −1.29701
\(429\) 0 0
\(430\) 20.0000 0.964486
\(431\) 8.94427 0.430830 0.215415 0.976523i \(-0.430890\pi\)
0.215415 + 0.976523i \(0.430890\pi\)
\(432\) 0 0
\(433\) −6.00000 −0.288342 −0.144171 0.989553i \(-0.546051\pi\)
−0.144171 + 0.989553i \(0.546051\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −17.8885 −0.855725
\(438\) 0 0
\(439\) −13.4164 −0.640330 −0.320165 0.947362i \(-0.603738\pi\)
−0.320165 + 0.947362i \(0.603738\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 24.0000 1.14027 0.570137 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) 0 0
\(445\) −28.0000 −1.32733
\(446\) 35.7771 1.69409
\(447\) 0 0
\(448\) 58.1378 2.74675
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −18.0000 −0.846649
\(453\) 0 0
\(454\) 20.0000 0.938647
\(455\) 0 0
\(456\) 0 0
\(457\) 26.8328 1.25519 0.627593 0.778542i \(-0.284040\pi\)
0.627593 + 0.778542i \(0.284040\pi\)
\(458\) −22.3607 −1.04485
\(459\) 0 0
\(460\) −24.0000 −1.11901
\(461\) −13.4164 −0.624864 −0.312432 0.949940i \(-0.601144\pi\)
−0.312432 + 0.949940i \(0.601144\pi\)
\(462\) 0 0
\(463\) 24.0000 1.11537 0.557687 0.830051i \(-0.311689\pi\)
0.557687 + 0.830051i \(0.311689\pi\)
\(464\) 4.47214 0.207614
\(465\) 0 0
\(466\) −10.0000 −0.463241
\(467\) 8.00000 0.370196 0.185098 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 0 0
\(469\) 53.6656 2.47805
\(470\) −35.7771 −1.65027
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 4.47214 0.205196
\(476\) 60.0000 2.75010
\(477\) 0 0
\(478\) 20.0000 0.914779
\(479\) 8.94427 0.408674 0.204337 0.978901i \(-0.434496\pi\)
0.204337 + 0.978901i \(0.434496\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −20.0000 −0.910975
\(483\) 0 0
\(484\) 0 0
\(485\) −4.00000 −0.181631
\(486\) 0 0
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) −20.0000 −0.905357
\(489\) 0 0
\(490\) 58.1378 2.62640
\(491\) 26.8328 1.21095 0.605474 0.795865i \(-0.292984\pi\)
0.605474 + 0.795865i \(0.292984\pi\)
\(492\) 0 0
\(493\) 20.0000 0.900755
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −35.7771 −1.60482
\(498\) 0 0
\(499\) −20.0000 −0.895323 −0.447661 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) 36.0000 1.60997
\(501\) 0 0
\(502\) −26.8328 −1.19761
\(503\) 26.8328 1.19642 0.598208 0.801341i \(-0.295880\pi\)
0.598208 + 0.801341i \(0.295880\pi\)
\(504\) 0 0
\(505\) 8.94427 0.398015
\(506\) 0 0
\(507\) 0 0
\(508\) 40.2492 1.78577
\(509\) −30.0000 −1.32973 −0.664863 0.746965i \(-0.731510\pi\)
−0.664863 + 0.746965i \(0.731510\pi\)
\(510\) 0 0
\(511\) 40.0000 1.76950
\(512\) −11.1803 −0.494106
\(513\) 0 0
\(514\) −49.1935 −2.16983
\(515\) −32.0000 −1.41009
\(516\) 0 0
\(517\) 0 0
\(518\) 20.0000 0.878750
\(519\) 0 0
\(520\) 0 0
\(521\) −30.0000 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) 0 0
\(523\) 4.47214 0.195553 0.0977764 0.995208i \(-0.468827\pi\)
0.0977764 + 0.995208i \(0.468827\pi\)
\(524\) 53.6656 2.34439
\(525\) 0 0
\(526\) 20.0000 0.872041
\(527\) 0 0
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) −26.8328 −1.16554
\(531\) 0 0
\(532\) 60.0000 2.60133
\(533\) 0 0
\(534\) 0 0
\(535\) 17.8885 0.773389
\(536\) 26.8328 1.15900
\(537\) 0 0
\(538\) −22.3607 −0.964037
\(539\) 0 0
\(540\) 0 0
\(541\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(542\) 30.0000 1.28861
\(543\) 0 0
\(544\) −30.0000 −1.28624
\(545\) 0 0
\(546\) 0 0
\(547\) 40.2492 1.72093 0.860466 0.509507i \(-0.170172\pi\)
0.860466 + 0.509507i \(0.170172\pi\)
\(548\) −66.0000 −2.81938
\(549\) 0 0
\(550\) 0 0
\(551\) 20.0000 0.852029
\(552\) 0 0
\(553\) −60.0000 −2.55146
\(554\) 0 0
\(555\) 0 0
\(556\) 40.2492 1.70695
\(557\) 40.2492 1.70541 0.852707 0.522389i \(-0.174959\pi\)
0.852707 + 0.522389i \(0.174959\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −8.94427 −0.377964
\(561\) 0 0
\(562\) 70.0000 2.95277
\(563\) −17.8885 −0.753912 −0.376956 0.926231i \(-0.623029\pi\)
−0.376956 + 0.926231i \(0.623029\pi\)
\(564\) 0 0
\(565\) 12.0000 0.504844
\(566\) 30.0000 1.26099
\(567\) 0 0
\(568\) −17.8885 −0.750587
\(569\) −4.47214 −0.187482 −0.0937408 0.995597i \(-0.529883\pi\)
−0.0937408 + 0.995597i \(0.529883\pi\)
\(570\) 0 0
\(571\) −13.4164 −0.561459 −0.280730 0.959787i \(-0.590576\pi\)
−0.280730 + 0.959787i \(0.590576\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 44.7214 1.86663
\(575\) −4.00000 −0.166812
\(576\) 0 0
\(577\) −2.00000 −0.0832611 −0.0416305 0.999133i \(-0.513255\pi\)
−0.0416305 + 0.999133i \(0.513255\pi\)
\(578\) −6.70820 −0.279024
\(579\) 0 0
\(580\) 26.8328 1.11417
\(581\) −40.0000 −1.65948
\(582\) 0 0
\(583\) 0 0
\(584\) 20.0000 0.827606
\(585\) 0 0
\(586\) −50.0000 −2.06548
\(587\) 8.00000 0.330195 0.165098 0.986277i \(-0.447206\pi\)
0.165098 + 0.986277i \(0.447206\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −2.00000 −0.0821995
\(593\) 22.3607 0.918243 0.459122 0.888373i \(-0.348164\pi\)
0.459122 + 0.888373i \(0.348164\pi\)
\(594\) 0 0
\(595\) −40.0000 −1.63984
\(596\) 67.0820 2.74779
\(597\) 0 0
\(598\) 0 0
\(599\) 36.0000 1.47092 0.735460 0.677568i \(-0.236966\pi\)
0.735460 + 0.677568i \(0.236966\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) 44.7214 1.82271
\(603\) 0 0
\(604\) 40.2492 1.63772
\(605\) 0 0
\(606\) 0 0
\(607\) −4.47214 −0.181518 −0.0907592 0.995873i \(-0.528929\pi\)
−0.0907592 + 0.995873i \(0.528929\pi\)
\(608\) −30.0000 −1.21666
\(609\) 0 0
\(610\) 40.0000 1.61955
\(611\) 0 0
\(612\) 0 0
\(613\) 44.7214 1.80628 0.903139 0.429348i \(-0.141256\pi\)
0.903139 + 0.429348i \(0.141256\pi\)
\(614\) 10.0000 0.403567
\(615\) 0 0
\(616\) 0 0
\(617\) 2.00000 0.0805170 0.0402585 0.999189i \(-0.487182\pi\)
0.0402585 + 0.999189i \(0.487182\pi\)
\(618\) 0 0
\(619\) 44.0000 1.76851 0.884255 0.467005i \(-0.154667\pi\)
0.884255 + 0.467005i \(0.154667\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −26.8328 −1.07590
\(623\) −62.6099 −2.50841
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) −31.3050 −1.25120
\(627\) 0 0
\(628\) 6.00000 0.239426
\(629\) −8.94427 −0.356631
\(630\) 0 0
\(631\) 32.0000 1.27390 0.636950 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) −30.0000 −1.19334
\(633\) 0 0
\(634\) 40.2492 1.59850
\(635\) −26.8328 −1.06483
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) −31.3050 −1.23744
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) 0 0
\(643\) 36.0000 1.41970 0.709851 0.704352i \(-0.248762\pi\)
0.709851 + 0.704352i \(0.248762\pi\)
\(644\) −53.6656 −2.11472
\(645\) 0 0
\(646\) −44.7214 −1.75954
\(647\) 12.0000 0.471769 0.235884 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 12.0000 0.469956
\(653\) −46.0000 −1.80012 −0.900060 0.435767i \(-0.856477\pi\)
−0.900060 + 0.435767i \(0.856477\pi\)
\(654\) 0 0
\(655\) −35.7771 −1.39793
\(656\) −4.47214 −0.174608
\(657\) 0 0
\(658\) −80.0000 −3.11872
\(659\) −17.8885 −0.696839 −0.348419 0.937339i \(-0.613281\pi\)
−0.348419 + 0.937339i \(0.613281\pi\)
\(660\) 0 0
\(661\) −22.0000 −0.855701 −0.427850 0.903850i \(-0.640729\pi\)
−0.427850 + 0.903850i \(0.640729\pi\)
\(662\) 44.7214 1.73814
\(663\) 0 0
\(664\) −20.0000 −0.776151
\(665\) −40.0000 −1.55113
\(666\) 0 0
\(667\) −17.8885 −0.692647
\(668\) −26.8328 −1.03819
\(669\) 0 0
\(670\) −53.6656 −2.07328
\(671\) 0 0
\(672\) 0 0
\(673\) −17.8885 −0.689553 −0.344776 0.938685i \(-0.612045\pi\)
−0.344776 + 0.938685i \(0.612045\pi\)
\(674\) −20.0000 −0.770371
\(675\) 0 0
\(676\) −39.0000 −1.50000
\(677\) 31.3050 1.20315 0.601574 0.798817i \(-0.294541\pi\)
0.601574 + 0.798817i \(0.294541\pi\)
\(678\) 0 0
\(679\) −8.94427 −0.343250
\(680\) −20.0000 −0.766965
\(681\) 0 0
\(682\) 0 0
\(683\) −44.0000 −1.68361 −0.841807 0.539779i \(-0.818508\pi\)
−0.841807 + 0.539779i \(0.818508\pi\)
\(684\) 0 0
\(685\) 44.0000 1.68115
\(686\) 60.0000 2.29081
\(687\) 0 0
\(688\) −4.47214 −0.170499
\(689\) 0 0
\(690\) 0 0
\(691\) 12.0000 0.456502 0.228251 0.973602i \(-0.426699\pi\)
0.228251 + 0.973602i \(0.426699\pi\)
\(692\) 40.2492 1.53005
\(693\) 0 0
\(694\) 20.0000 0.759190
\(695\) −26.8328 −1.01783
\(696\) 0 0
\(697\) −20.0000 −0.757554
\(698\) 60.0000 2.27103
\(699\) 0 0
\(700\) 13.4164 0.507093
\(701\) 22.3607 0.844551 0.422276 0.906467i \(-0.361231\pi\)
0.422276 + 0.906467i \(0.361231\pi\)
\(702\) 0 0
\(703\) −8.94427 −0.337340
\(704\) 0 0
\(705\) 0 0
\(706\) −31.3050 −1.17818
\(707\) 20.0000 0.752177
\(708\) 0 0
\(709\) −6.00000 −0.225335 −0.112667 0.993633i \(-0.535939\pi\)
−0.112667 + 0.993633i \(0.535939\pi\)
\(710\) 35.7771 1.34269
\(711\) 0 0
\(712\) −31.3050 −1.17320
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) 0 0
\(718\) 0 0
\(719\) 24.0000 0.895049 0.447524 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 0 0
\(721\) −71.5542 −2.66482
\(722\) −2.23607 −0.0832178
\(723\) 0 0
\(724\) −30.0000 −1.11494
\(725\) 4.47214 0.166091
\(726\) 0 0
\(727\) −8.00000 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −40.0000 −1.48047
\(731\) −20.0000 −0.739727
\(732\) 0 0
\(733\) −17.8885 −0.660728 −0.330364 0.943854i \(-0.607172\pi\)
−0.330364 + 0.943854i \(0.607172\pi\)
\(734\) −17.8885 −0.660278
\(735\) 0 0
\(736\) 26.8328 0.989071
\(737\) 0 0
\(738\) 0 0
\(739\) 4.47214 0.164510 0.0822551 0.996611i \(-0.473788\pi\)
0.0822551 + 0.996611i \(0.473788\pi\)
\(740\) −12.0000 −0.441129
\(741\) 0 0
\(742\) −60.0000 −2.20267
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) −44.7214 −1.63846
\(746\) 60.0000 2.19676
\(747\) 0 0
\(748\) 0 0
\(749\) 40.0000 1.46157
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) 8.00000 0.291730
\(753\) 0 0
\(754\) 0 0
\(755\) −26.8328 −0.976546
\(756\) 0 0
\(757\) 42.0000 1.52652 0.763258 0.646094i \(-0.223599\pi\)
0.763258 + 0.646094i \(0.223599\pi\)
\(758\) 44.7214 1.62435
\(759\) 0 0
\(760\) −20.0000 −0.725476
\(761\) 31.3050 1.13480 0.567402 0.823441i \(-0.307949\pi\)
0.567402 + 0.823441i \(0.307949\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) −80.4984 −2.90853
\(767\) 0 0
\(768\) 0 0
\(769\) 35.7771 1.29015 0.645077 0.764117i \(-0.276825\pi\)
0.645077 + 0.764117i \(0.276825\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −53.6656 −1.93147
\(773\) −14.0000 −0.503545 −0.251773 0.967786i \(-0.581013\pi\)
−0.251773 + 0.967786i \(0.581013\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −4.47214 −0.160540
\(777\) 0 0
\(778\) 22.3607 0.801669
\(779\) −20.0000 −0.716574
\(780\) 0 0
\(781\) 0 0
\(782\) 40.0000 1.43040
\(783\) 0 0
\(784\) −13.0000 −0.464286
\(785\) −4.00000 −0.142766
\(786\) 0 0
\(787\) 40.2492 1.43473 0.717365 0.696698i \(-0.245348\pi\)
0.717365 + 0.696698i \(0.245348\pi\)
\(788\) −67.0820 −2.38970
\(789\) 0 0
\(790\) 60.0000 2.13470
\(791\) 26.8328 0.954065
\(792\) 0 0
\(793\) 0 0
\(794\) −49.1935 −1.74581
\(795\) 0 0
\(796\) 0 0
\(797\) −42.0000 −1.48772 −0.743858 0.668338i \(-0.767006\pi\)
−0.743858 + 0.668338i \(0.767006\pi\)
\(798\) 0 0
\(799\) 35.7771 1.26570
\(800\) −6.70820 −0.237171
\(801\) 0 0
\(802\) 67.0820 2.36875
\(803\) 0 0
\(804\) 0 0
\(805\) 35.7771 1.26098
\(806\) 0 0
\(807\) 0 0
\(808\) 10.0000 0.351799
\(809\) 13.4164 0.471696 0.235848 0.971790i \(-0.424213\pi\)
0.235848 + 0.971790i \(0.424213\pi\)
\(810\) 0 0
\(811\) −4.47214 −0.157038 −0.0785190 0.996913i \(-0.525019\pi\)
−0.0785190 + 0.996913i \(0.525019\pi\)
\(812\) 60.0000 2.10559
\(813\) 0 0
\(814\) 0 0
\(815\) −8.00000 −0.280228
\(816\) 0 0
\(817\) −20.0000 −0.699711
\(818\) 60.0000 2.09785
\(819\) 0 0
\(820\) −26.8328 −0.937043
\(821\) −22.3607 −0.780393 −0.390197 0.920732i \(-0.627593\pi\)
−0.390197 + 0.920732i \(0.627593\pi\)
\(822\) 0 0
\(823\) 16.0000 0.557725 0.278862 0.960331i \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) −35.7771 −1.24635
\(825\) 0 0
\(826\) 0 0
\(827\) −44.7214 −1.55511 −0.777557 0.628812i \(-0.783541\pi\)
−0.777557 + 0.628812i \(0.783541\pi\)
\(828\) 0 0
\(829\) 14.0000 0.486240 0.243120 0.969996i \(-0.421829\pi\)
0.243120 + 0.969996i \(0.421829\pi\)
\(830\) 40.0000 1.38842
\(831\) 0 0
\(832\) 0 0
\(833\) −58.1378 −2.01435
\(834\) 0 0
\(835\) 17.8885 0.619059
\(836\) 0 0
\(837\) 0 0
\(838\) 8.94427 0.308975
\(839\) 20.0000 0.690477 0.345238 0.938515i \(-0.387798\pi\)
0.345238 + 0.938515i \(0.387798\pi\)
\(840\) 0 0
\(841\) −9.00000 −0.310345
\(842\) −22.3607 −0.770600
\(843\) 0 0
\(844\) −13.4164 −0.461812
\(845\) 26.0000 0.894427
\(846\) 0 0
\(847\) 0 0
\(848\) 6.00000 0.206041
\(849\) 0 0
\(850\) −10.0000 −0.342997
\(851\) 8.00000 0.274236
\(852\) 0 0
\(853\) 8.94427 0.306246 0.153123 0.988207i \(-0.451067\pi\)
0.153123 + 0.988207i \(0.451067\pi\)
\(854\) 89.4427 3.06067
\(855\) 0 0
\(856\) 20.0000 0.683586
\(857\) −40.2492 −1.37489 −0.687444 0.726238i \(-0.741267\pi\)
−0.687444 + 0.726238i \(0.741267\pi\)
\(858\) 0 0
\(859\) 4.00000 0.136478 0.0682391 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) −26.8328 −0.914991
\(861\) 0 0
\(862\) −20.0000 −0.681203
\(863\) −4.00000 −0.136162 −0.0680808 0.997680i \(-0.521688\pi\)
−0.0680808 + 0.997680i \(0.521688\pi\)
\(864\) 0 0
\(865\) −26.8328 −0.912343
\(866\) 13.4164 0.455908
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 40.0000 1.35302
\(875\) −53.6656 −1.81423
\(876\) 0 0
\(877\) −8.94427 −0.302027 −0.151013 0.988532i \(-0.548254\pi\)
−0.151013 + 0.988532i \(0.548254\pi\)
\(878\) 30.0000 1.01245
\(879\) 0 0
\(880\) 0 0
\(881\) 18.0000 0.606435 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) 0 0
\(883\) 44.0000 1.48072 0.740359 0.672212i \(-0.234656\pi\)
0.740359 + 0.672212i \(0.234656\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −53.6656 −1.80293
\(887\) 53.6656 1.80192 0.900958 0.433907i \(-0.142865\pi\)
0.900958 + 0.433907i \(0.142865\pi\)
\(888\) 0 0
\(889\) −60.0000 −2.01234
\(890\) 62.6099 2.09869
\(891\) 0 0
\(892\) −48.0000 −1.60716
\(893\) 35.7771 1.19723
\(894\) 0 0
\(895\) −8.00000 −0.267411
\(896\) −70.0000 −2.33854
\(897\) 0 0
\(898\) −13.4164 −0.447711
\(899\) 0 0
\(900\) 0 0
\(901\) 26.8328 0.893931
\(902\) 0 0
\(903\) 0 0
\(904\) 13.4164 0.446223
\(905\) 20.0000 0.664822
\(906\) 0 0
\(907\) −12.0000 −0.398453 −0.199227 0.979953i \(-0.563843\pi\)
−0.199227 + 0.979953i \(0.563843\pi\)
\(908\) −26.8328 −0.890478
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −60.0000 −1.98462
\(915\) 0 0
\(916\) 30.0000 0.991228
\(917\) −80.0000 −2.64183
\(918\) 0 0
\(919\) 22.3607 0.737611 0.368805 0.929507i \(-0.379767\pi\)
0.368805 + 0.929507i \(0.379767\pi\)
\(920\) 17.8885 0.589768
\(921\) 0 0
\(922\) 30.0000 0.987997
\(923\) 0 0
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) −53.6656 −1.76356
\(927\) 0 0
\(928\) −30.0000 −0.984798
\(929\) −30.0000 −0.984268 −0.492134 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 0 0
\(931\) −58.1378 −1.90539
\(932\) 13.4164 0.439469
\(933\) 0 0
\(934\) −17.8885 −0.585331
\(935\) 0 0
\(936\) 0 0
\(937\) 53.6656 1.75318 0.876590 0.481238i \(-0.159813\pi\)
0.876590 + 0.481238i \(0.159813\pi\)
\(938\) −120.000 −3.91814
\(939\) 0 0
\(940\) 48.0000 1.56559
\(941\) −22.3607 −0.728937 −0.364469 0.931216i \(-0.618749\pi\)
−0.364469 + 0.931216i \(0.618749\pi\)
\(942\) 0 0
\(943\) 17.8885 0.582531
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −52.0000 −1.68977 −0.844886 0.534946i \(-0.820332\pi\)
−0.844886 + 0.534946i \(0.820332\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −10.0000 −0.324443
\(951\) 0 0
\(952\) −44.7214 −1.44943
\(953\) −22.3607 −0.724333 −0.362167 0.932113i \(-0.617963\pi\)
−0.362167 + 0.932113i \(0.617963\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −26.8328 −0.867835
\(957\) 0 0
\(958\) −20.0000 −0.646171
\(959\) 98.3870 3.17708
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 26.8328 0.864227
\(965\) 35.7771 1.15171
\(966\) 0 0
\(967\) 13.4164 0.431443 0.215721 0.976455i \(-0.430790\pi\)
0.215721 + 0.976455i \(0.430790\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 8.94427 0.287183
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) −60.0000 −1.92351
\(974\) −17.8885 −0.573186
\(975\) 0 0
\(976\) −8.94427 −0.286299
\(977\) 42.0000 1.34370 0.671850 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −78.0000 −2.49162
\(981\) 0 0
\(982\) −60.0000 −1.91468
\(983\) −36.0000 −1.14822 −0.574111 0.818778i \(-0.694652\pi\)
−0.574111 + 0.818778i \(0.694652\pi\)
\(984\) 0 0
\(985\) 44.7214 1.42494
\(986\) −44.7214 −1.42422
\(987\) 0 0
\(988\) 0 0
\(989\) 17.8885 0.568823
\(990\) 0 0
\(991\) 40.0000 1.27064 0.635321 0.772248i \(-0.280868\pi\)
0.635321 + 0.772248i \(0.280868\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 80.0000 2.53745
\(995\) 0 0
\(996\) 0 0
\(997\) −26.8328 −0.849804 −0.424902 0.905239i \(-0.639691\pi\)
−0.424902 + 0.905239i \(0.639691\pi\)
\(998\) 44.7214 1.41563
\(999\) 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 1089.2.a.p.1.1 2
3.2 odd 2 363.2.a.g.1.2 yes 2
11.10 odd 2 inner 1089.2.a.p.1.2 2
12.11 even 2 5808.2.a.bx.1.2 2
15.14 odd 2 9075.2.a.bi.1.1 2
33.2 even 10 363.2.e.l.202.1 4
33.5 odd 10 363.2.e.a.124.1 4
33.8 even 10 363.2.e.a.130.1 4
33.14 odd 10 363.2.e.l.130.1 4
33.17 even 10 363.2.e.l.124.1 4
33.20 odd 10 363.2.e.a.202.1 4
33.26 odd 10 363.2.e.l.148.1 4
33.29 even 10 363.2.e.a.148.1 4
33.32 even 2 363.2.a.g.1.1 2
132.131 odd 2 5808.2.a.bx.1.1 2
165.164 even 2 9075.2.a.bi.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
363.2.a.g.1.1 2 33.32 even 2
363.2.a.g.1.2 yes 2 3.2 odd 2
363.2.e.a.124.1 4 33.5 odd 10
363.2.e.a.130.1 4 33.8 even 10
363.2.e.a.148.1 4 33.29 even 10
363.2.e.a.202.1 4 33.20 odd 10
363.2.e.l.124.1 4 33.17 even 10
363.2.e.l.130.1 4 33.14 odd 10
363.2.e.l.148.1 4 33.26 odd 10
363.2.e.l.202.1 4 33.2 even 10
1089.2.a.p.1.1 2 1.1 even 1 trivial
1089.2.a.p.1.2 2 11.10 odd 2 inner
5808.2.a.bx.1.1 2 132.131 odd 2
5808.2.a.bx.1.2 2 12.11 even 2
9075.2.a.bi.1.1 2 15.14 odd 2
9075.2.a.bi.1.2 2 165.164 even 2