Properties

Label 1134.2.a.g.1.1
Level $1134$
Weight $2$
Character 1134.1
Self dual yes
Analytic conductor $9.055$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1134,2,Mod(1,1134)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1134, 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("1134.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1134.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: \(9.05503558921\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1134.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{10} +2.00000 q^{11} +3.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -1.00000 q^{17} +2.00000 q^{19} +1.00000 q^{20} +2.00000 q^{22} +2.00000 q^{23} -4.00000 q^{25} +3.00000 q^{26} -1.00000 q^{28} +7.00000 q^{29} +6.00000 q^{31} +1.00000 q^{32} -1.00000 q^{34} -1.00000 q^{35} -7.00000 q^{37} +2.00000 q^{38} +1.00000 q^{40} +6.00000 q^{41} -4.00000 q^{43} +2.00000 q^{44} +2.00000 q^{46} +6.00000 q^{47} +1.00000 q^{49} -4.00000 q^{50} +3.00000 q^{52} -6.00000 q^{53} +2.00000 q^{55} -1.00000 q^{56} +7.00000 q^{58} +10.0000 q^{59} -9.00000 q^{61} +6.00000 q^{62} +1.00000 q^{64} +3.00000 q^{65} +10.0000 q^{67} -1.00000 q^{68} -1.00000 q^{70} -4.00000 q^{71} -11.0000 q^{73} -7.00000 q^{74} +2.00000 q^{76} -2.00000 q^{77} -6.00000 q^{79} +1.00000 q^{80} +6.00000 q^{82} -10.0000 q^{83} -1.00000 q^{85} -4.00000 q^{86} +2.00000 q^{88} +15.0000 q^{89} -3.00000 q^{91} +2.00000 q^{92} +6.00000 q^{94} +2.00000 q^{95} -2.00000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214 0.223607 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) 3.00000 0.832050 0.416025 0.909353i \(-0.363423\pi\)
0.416025 + 0.909353i \(0.363423\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −1.00000 −0.242536 −0.121268 0.992620i \(-0.538696\pi\)
−0.121268 + 0.992620i \(0.538696\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 2.00000 0.426401
\(23\) 2.00000 0.417029 0.208514 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 3.00000 0.588348
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) 7.00000 1.29987 0.649934 0.759991i \(-0.274797\pi\)
0.649934 + 0.759991i \(0.274797\pi\)
\(30\) 0 0
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −1.00000 −0.171499
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) −7.00000 −1.15079 −0.575396 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) 2.00000 0.324443
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) 2.00000 0.294884
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −4.00000 −0.565685
\(51\) 0 0
\(52\) 3.00000 0.416025
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 2.00000 0.269680
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 7.00000 0.919145
\(59\) 10.0000 1.30189 0.650945 0.759125i \(-0.274373\pi\)
0.650945 + 0.759125i \(0.274373\pi\)
\(60\) 0 0
\(61\) −9.00000 −1.15233 −0.576166 0.817333i \(-0.695452\pi\)
−0.576166 + 0.817333i \(0.695452\pi\)
\(62\) 6.00000 0.762001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 3.00000 0.372104
\(66\) 0 0
\(67\) 10.0000 1.22169 0.610847 0.791748i \(-0.290829\pi\)
0.610847 + 0.791748i \(0.290829\pi\)
\(68\) −1.00000 −0.121268
\(69\) 0 0
\(70\) −1.00000 −0.119523
\(71\) −4.00000 −0.474713 −0.237356 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 0 0
\(73\) −11.0000 −1.28745 −0.643726 0.765256i \(-0.722612\pi\)
−0.643726 + 0.765256i \(0.722612\pi\)
\(74\) −7.00000 −0.813733
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) −2.00000 −0.227921
\(78\) 0 0
\(79\) −6.00000 −0.675053 −0.337526 0.941316i \(-0.609590\pi\)
−0.337526 + 0.941316i \(0.609590\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) 6.00000 0.662589
\(83\) −10.0000 −1.09764 −0.548821 0.835940i \(-0.684923\pi\)
−0.548821 + 0.835940i \(0.684923\pi\)
\(84\) 0 0
\(85\) −1.00000 −0.108465
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) 2.00000 0.213201
\(89\) 15.0000 1.59000 0.794998 0.606612i \(-0.207472\pi\)
0.794998 + 0.606612i \(0.207472\pi\)
\(90\) 0 0
\(91\) −3.00000 −0.314485
\(92\) 2.00000 0.208514
\(93\) 0 0
\(94\) 6.00000 0.618853
\(95\) 2.00000 0.205196
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) −4.00000 −0.400000
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) 0 0
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) 3.00000 0.294174
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) −11.0000 −1.05361 −0.526804 0.849987i \(-0.676610\pi\)
−0.526804 + 0.849987i \(0.676610\pi\)
\(110\) 2.00000 0.190693
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) 1.00000 0.0940721 0.0470360 0.998893i \(-0.485022\pi\)
0.0470360 + 0.998893i \(0.485022\pi\)
\(114\) 0 0
\(115\) 2.00000 0.186501
\(116\) 7.00000 0.649934
\(117\) 0 0
\(118\) 10.0000 0.920575
\(119\) 1.00000 0.0916698
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) −9.00000 −0.814822
\(123\) 0 0
\(124\) 6.00000 0.538816
\(125\) −9.00000 −0.804984
\(126\) 0 0
\(127\) 12.0000 1.06483 0.532414 0.846484i \(-0.321285\pi\)
0.532414 + 0.846484i \(0.321285\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 3.00000 0.263117
\(131\) 16.0000 1.39793 0.698963 0.715158i \(-0.253645\pi\)
0.698963 + 0.715158i \(0.253645\pi\)
\(132\) 0 0
\(133\) −2.00000 −0.173422
\(134\) 10.0000 0.863868
\(135\) 0 0
\(136\) −1.00000 −0.0857493
\(137\) −11.0000 −0.939793 −0.469897 0.882721i \(-0.655709\pi\)
−0.469897 + 0.882721i \(0.655709\pi\)
\(138\) 0 0
\(139\) −2.00000 −0.169638 −0.0848189 0.996396i \(-0.527031\pi\)
−0.0848189 + 0.996396i \(0.527031\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 0 0
\(142\) −4.00000 −0.335673
\(143\) 6.00000 0.501745
\(144\) 0 0
\(145\) 7.00000 0.581318
\(146\) −11.0000 −0.910366
\(147\) 0 0
\(148\) −7.00000 −0.575396
\(149\) −21.0000 −1.72039 −0.860194 0.509968i \(-0.829657\pi\)
−0.860194 + 0.509968i \(0.829657\pi\)
\(150\) 0 0
\(151\) 10.0000 0.813788 0.406894 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) 2.00000 0.162221
\(153\) 0 0
\(154\) −2.00000 −0.161165
\(155\) 6.00000 0.481932
\(156\) 0 0
\(157\) −13.0000 −1.03751 −0.518756 0.854922i \(-0.673605\pi\)
−0.518756 + 0.854922i \(0.673605\pi\)
\(158\) −6.00000 −0.477334
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) −2.00000 −0.157622
\(162\) 0 0
\(163\) −16.0000 −1.25322 −0.626608 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) −10.0000 −0.776151
\(167\) 22.0000 1.70241 0.851206 0.524832i \(-0.175872\pi\)
0.851206 + 0.524832i \(0.175872\pi\)
\(168\) 0 0
\(169\) −4.00000 −0.307692
\(170\) −1.00000 −0.0766965
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) 13.0000 0.988372 0.494186 0.869356i \(-0.335466\pi\)
0.494186 + 0.869356i \(0.335466\pi\)
\(174\) 0 0
\(175\) 4.00000 0.302372
\(176\) 2.00000 0.150756
\(177\) 0 0
\(178\) 15.0000 1.12430
\(179\) −18.0000 −1.34538 −0.672692 0.739923i \(-0.734862\pi\)
−0.672692 + 0.739923i \(0.734862\pi\)
\(180\) 0 0
\(181\) 18.0000 1.33793 0.668965 0.743294i \(-0.266738\pi\)
0.668965 + 0.743294i \(0.266738\pi\)
\(182\) −3.00000 −0.222375
\(183\) 0 0
\(184\) 2.00000 0.147442
\(185\) −7.00000 −0.514650
\(186\) 0 0
\(187\) −2.00000 −0.146254
\(188\) 6.00000 0.437595
\(189\) 0 0
\(190\) 2.00000 0.145095
\(191\) −18.0000 −1.30243 −0.651217 0.758891i \(-0.725741\pi\)
−0.651217 + 0.758891i \(0.725741\pi\)
\(192\) 0 0
\(193\) 11.0000 0.791797 0.395899 0.918294i \(-0.370433\pi\)
0.395899 + 0.918294i \(0.370433\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −1.00000 −0.0712470 −0.0356235 0.999365i \(-0.511342\pi\)
−0.0356235 + 0.999365i \(0.511342\pi\)
\(198\) 0 0
\(199\) −10.0000 −0.708881 −0.354441 0.935079i \(-0.615329\pi\)
−0.354441 + 0.935079i \(0.615329\pi\)
\(200\) −4.00000 −0.282843
\(201\) 0 0
\(202\) −2.00000 −0.140720
\(203\) −7.00000 −0.491304
\(204\) 0 0
\(205\) 6.00000 0.419058
\(206\) −16.0000 −1.11477
\(207\) 0 0
\(208\) 3.00000 0.208013
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) −22.0000 −1.51454 −0.757271 0.653101i \(-0.773468\pi\)
−0.757271 + 0.653101i \(0.773468\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) 0 0
\(215\) −4.00000 −0.272798
\(216\) 0 0
\(217\) −6.00000 −0.407307
\(218\) −11.0000 −0.745014
\(219\) 0 0
\(220\) 2.00000 0.134840
\(221\) −3.00000 −0.201802
\(222\) 0 0
\(223\) 20.0000 1.33930 0.669650 0.742677i \(-0.266444\pi\)
0.669650 + 0.742677i \(0.266444\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 1.00000 0.0665190
\(227\) −24.0000 −1.59294 −0.796468 0.604681i \(-0.793301\pi\)
−0.796468 + 0.604681i \(0.793301\pi\)
\(228\) 0 0
\(229\) −29.0000 −1.91637 −0.958187 0.286143i \(-0.907627\pi\)
−0.958187 + 0.286143i \(0.907627\pi\)
\(230\) 2.00000 0.131876
\(231\) 0 0
\(232\) 7.00000 0.459573
\(233\) −19.0000 −1.24473 −0.622366 0.782727i \(-0.713828\pi\)
−0.622366 + 0.782727i \(0.713828\pi\)
\(234\) 0 0
\(235\) 6.00000 0.391397
\(236\) 10.0000 0.650945
\(237\) 0 0
\(238\) 1.00000 0.0648204
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) −7.00000 −0.450910 −0.225455 0.974254i \(-0.572387\pi\)
−0.225455 + 0.974254i \(0.572387\pi\)
\(242\) −7.00000 −0.449977
\(243\) 0 0
\(244\) −9.00000 −0.576166
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) 6.00000 0.381771
\(248\) 6.00000 0.381000
\(249\) 0 0
\(250\) −9.00000 −0.569210
\(251\) −6.00000 −0.378717 −0.189358 0.981908i \(-0.560641\pi\)
−0.189358 + 0.981908i \(0.560641\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) 12.0000 0.752947
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −9.00000 −0.561405 −0.280702 0.959795i \(-0.590567\pi\)
−0.280702 + 0.959795i \(0.590567\pi\)
\(258\) 0 0
\(259\) 7.00000 0.434959
\(260\) 3.00000 0.186052
\(261\) 0 0
\(262\) 16.0000 0.988483
\(263\) −18.0000 −1.10993 −0.554964 0.831875i \(-0.687268\pi\)
−0.554964 + 0.831875i \(0.687268\pi\)
\(264\) 0 0
\(265\) −6.00000 −0.368577
\(266\) −2.00000 −0.122628
\(267\) 0 0
\(268\) 10.0000 0.610847
\(269\) 1.00000 0.0609711 0.0304855 0.999535i \(-0.490295\pi\)
0.0304855 + 0.999535i \(0.490295\pi\)
\(270\) 0 0
\(271\) 30.0000 1.82237 0.911185 0.411997i \(-0.135169\pi\)
0.911185 + 0.411997i \(0.135169\pi\)
\(272\) −1.00000 −0.0606339
\(273\) 0 0
\(274\) −11.0000 −0.664534
\(275\) −8.00000 −0.482418
\(276\) 0 0
\(277\) 14.0000 0.841178 0.420589 0.907251i \(-0.361823\pi\)
0.420589 + 0.907251i \(0.361823\pi\)
\(278\) −2.00000 −0.119952
\(279\) 0 0
\(280\) −1.00000 −0.0597614
\(281\) 29.0000 1.72999 0.864997 0.501776i \(-0.167320\pi\)
0.864997 + 0.501776i \(0.167320\pi\)
\(282\) 0 0
\(283\) 16.0000 0.951101 0.475551 0.879688i \(-0.342249\pi\)
0.475551 + 0.879688i \(0.342249\pi\)
\(284\) −4.00000 −0.237356
\(285\) 0 0
\(286\) 6.00000 0.354787
\(287\) −6.00000 −0.354169
\(288\) 0 0
\(289\) −16.0000 −0.941176
\(290\) 7.00000 0.411054
\(291\) 0 0
\(292\) −11.0000 −0.643726
\(293\) −15.0000 −0.876309 −0.438155 0.898900i \(-0.644368\pi\)
−0.438155 + 0.898900i \(0.644368\pi\)
\(294\) 0 0
\(295\) 10.0000 0.582223
\(296\) −7.00000 −0.406867
\(297\) 0 0
\(298\) −21.0000 −1.21650
\(299\) 6.00000 0.346989
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) 10.0000 0.575435
\(303\) 0 0
\(304\) 2.00000 0.114708
\(305\) −9.00000 −0.515339
\(306\) 0 0
\(307\) 16.0000 0.913168 0.456584 0.889680i \(-0.349073\pi\)
0.456584 + 0.889680i \(0.349073\pi\)
\(308\) −2.00000 −0.113961
\(309\) 0 0
\(310\) 6.00000 0.340777
\(311\) −14.0000 −0.793867 −0.396934 0.917847i \(-0.629926\pi\)
−0.396934 + 0.917847i \(0.629926\pi\)
\(312\) 0 0
\(313\) 1.00000 0.0565233 0.0282617 0.999601i \(-0.491003\pi\)
0.0282617 + 0.999601i \(0.491003\pi\)
\(314\) −13.0000 −0.733632
\(315\) 0 0
\(316\) −6.00000 −0.337526
\(317\) 3.00000 0.168497 0.0842484 0.996445i \(-0.473151\pi\)
0.0842484 + 0.996445i \(0.473151\pi\)
\(318\) 0 0
\(319\) 14.0000 0.783850
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) −2.00000 −0.111456
\(323\) −2.00000 −0.111283
\(324\) 0 0
\(325\) −12.0000 −0.665640
\(326\) −16.0000 −0.886158
\(327\) 0 0
\(328\) 6.00000 0.331295
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) −10.0000 −0.548821
\(333\) 0 0
\(334\) 22.0000 1.20379
\(335\) 10.0000 0.546358
\(336\) 0 0
\(337\) −18.0000 −0.980522 −0.490261 0.871576i \(-0.663099\pi\)
−0.490261 + 0.871576i \(0.663099\pi\)
\(338\) −4.00000 −0.217571
\(339\) 0 0
\(340\) −1.00000 −0.0542326
\(341\) 12.0000 0.649836
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) 13.0000 0.698884
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 0 0
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 4.00000 0.213809
\(351\) 0 0
\(352\) 2.00000 0.106600
\(353\) −30.0000 −1.59674 −0.798369 0.602168i \(-0.794304\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) 0 0
\(355\) −4.00000 −0.212298
\(356\) 15.0000 0.794998
\(357\) 0 0
\(358\) −18.0000 −0.951330
\(359\) 16.0000 0.844448 0.422224 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 18.0000 0.946059
\(363\) 0 0
\(364\) −3.00000 −0.157243
\(365\) −11.0000 −0.575766
\(366\) 0 0
\(367\) −10.0000 −0.521996 −0.260998 0.965339i \(-0.584052\pi\)
−0.260998 + 0.965339i \(0.584052\pi\)
\(368\) 2.00000 0.104257
\(369\) 0 0
\(370\) −7.00000 −0.363913
\(371\) 6.00000 0.311504
\(372\) 0 0
\(373\) 10.0000 0.517780 0.258890 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) −2.00000 −0.103418
\(375\) 0 0
\(376\) 6.00000 0.309426
\(377\) 21.0000 1.08156
\(378\) 0 0
\(379\) 10.0000 0.513665 0.256833 0.966456i \(-0.417321\pi\)
0.256833 + 0.966456i \(0.417321\pi\)
\(380\) 2.00000 0.102598
\(381\) 0 0
\(382\) −18.0000 −0.920960
\(383\) 32.0000 1.63512 0.817562 0.575841i \(-0.195325\pi\)
0.817562 + 0.575841i \(0.195325\pi\)
\(384\) 0 0
\(385\) −2.00000 −0.101929
\(386\) 11.0000 0.559885
\(387\) 0 0
\(388\) −2.00000 −0.101535
\(389\) 34.0000 1.72387 0.861934 0.507020i \(-0.169253\pi\)
0.861934 + 0.507020i \(0.169253\pi\)
\(390\) 0 0
\(391\) −2.00000 −0.101144
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) −1.00000 −0.0503793
\(395\) −6.00000 −0.301893
\(396\) 0 0
\(397\) 39.0000 1.95735 0.978677 0.205406i \(-0.0658513\pi\)
0.978677 + 0.205406i \(0.0658513\pi\)
\(398\) −10.0000 −0.501255
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) −3.00000 −0.149813 −0.0749064 0.997191i \(-0.523866\pi\)
−0.0749064 + 0.997191i \(0.523866\pi\)
\(402\) 0 0
\(403\) 18.0000 0.896644
\(404\) −2.00000 −0.0995037
\(405\) 0 0
\(406\) −7.00000 −0.347404
\(407\) −14.0000 −0.693954
\(408\) 0 0
\(409\) −7.00000 −0.346128 −0.173064 0.984911i \(-0.555367\pi\)
−0.173064 + 0.984911i \(0.555367\pi\)
\(410\) 6.00000 0.296319
\(411\) 0 0
\(412\) −16.0000 −0.788263
\(413\) −10.0000 −0.492068
\(414\) 0 0
\(415\) −10.0000 −0.490881
\(416\) 3.00000 0.147087
\(417\) 0 0
\(418\) 4.00000 0.195646
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) 9.00000 0.438633 0.219317 0.975654i \(-0.429617\pi\)
0.219317 + 0.975654i \(0.429617\pi\)
\(422\) −22.0000 −1.07094
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) 4.00000 0.194029
\(426\) 0 0
\(427\) 9.00000 0.435541
\(428\) 0 0
\(429\) 0 0
\(430\) −4.00000 −0.192897
\(431\) 12.0000 0.578020 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) 0 0
\(433\) 5.00000 0.240285 0.120142 0.992757i \(-0.461665\pi\)
0.120142 + 0.992757i \(0.461665\pi\)
\(434\) −6.00000 −0.288009
\(435\) 0 0
\(436\) −11.0000 −0.526804
\(437\) 4.00000 0.191346
\(438\) 0 0
\(439\) −12.0000 −0.572729 −0.286364 0.958121i \(-0.592447\pi\)
−0.286364 + 0.958121i \(0.592447\pi\)
\(440\) 2.00000 0.0953463
\(441\) 0 0
\(442\) −3.00000 −0.142695
\(443\) −22.0000 −1.04525 −0.522626 0.852562i \(-0.675047\pi\)
−0.522626 + 0.852562i \(0.675047\pi\)
\(444\) 0 0
\(445\) 15.0000 0.711068
\(446\) 20.0000 0.947027
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) 34.0000 1.60456 0.802280 0.596948i \(-0.203620\pi\)
0.802280 + 0.596948i \(0.203620\pi\)
\(450\) 0 0
\(451\) 12.0000 0.565058
\(452\) 1.00000 0.0470360
\(453\) 0 0
\(454\) −24.0000 −1.12638
\(455\) −3.00000 −0.140642
\(456\) 0 0
\(457\) −41.0000 −1.91790 −0.958950 0.283577i \(-0.908479\pi\)
−0.958950 + 0.283577i \(0.908479\pi\)
\(458\) −29.0000 −1.35508
\(459\) 0 0
\(460\) 2.00000 0.0932505
\(461\) 22.0000 1.02464 0.512321 0.858794i \(-0.328786\pi\)
0.512321 + 0.858794i \(0.328786\pi\)
\(462\) 0 0
\(463\) 34.0000 1.58011 0.790057 0.613033i \(-0.210051\pi\)
0.790057 + 0.613033i \(0.210051\pi\)
\(464\) 7.00000 0.324967
\(465\) 0 0
\(466\) −19.0000 −0.880158
\(467\) 34.0000 1.57333 0.786666 0.617379i \(-0.211805\pi\)
0.786666 + 0.617379i \(0.211805\pi\)
\(468\) 0 0
\(469\) −10.0000 −0.461757
\(470\) 6.00000 0.276759
\(471\) 0 0
\(472\) 10.0000 0.460287
\(473\) −8.00000 −0.367840
\(474\) 0 0
\(475\) −8.00000 −0.367065
\(476\) 1.00000 0.0458349
\(477\) 0 0
\(478\) −6.00000 −0.274434
\(479\) −22.0000 −1.00521 −0.502603 0.864517i \(-0.667624\pi\)
−0.502603 + 0.864517i \(0.667624\pi\)
\(480\) 0 0
\(481\) −21.0000 −0.957518
\(482\) −7.00000 −0.318841
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) −2.00000 −0.0908153
\(486\) 0 0
\(487\) −10.0000 −0.453143 −0.226572 0.973995i \(-0.572752\pi\)
−0.226572 + 0.973995i \(0.572752\pi\)
\(488\) −9.00000 −0.407411
\(489\) 0 0
\(490\) 1.00000 0.0451754
\(491\) 36.0000 1.62466 0.812329 0.583200i \(-0.198200\pi\)
0.812329 + 0.583200i \(0.198200\pi\)
\(492\) 0 0
\(493\) −7.00000 −0.315264
\(494\) 6.00000 0.269953
\(495\) 0 0
\(496\) 6.00000 0.269408
\(497\) 4.00000 0.179425
\(498\) 0 0
\(499\) 22.0000 0.984855 0.492428 0.870353i \(-0.336110\pi\)
0.492428 + 0.870353i \(0.336110\pi\)
\(500\) −9.00000 −0.402492
\(501\) 0 0
\(502\) −6.00000 −0.267793
\(503\) −6.00000 −0.267527 −0.133763 0.991013i \(-0.542706\pi\)
−0.133763 + 0.991013i \(0.542706\pi\)
\(504\) 0 0
\(505\) −2.00000 −0.0889988
\(506\) 4.00000 0.177822
\(507\) 0 0
\(508\) 12.0000 0.532414
\(509\) −18.0000 −0.797836 −0.398918 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) 0 0
\(511\) 11.0000 0.486611
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −9.00000 −0.396973
\(515\) −16.0000 −0.705044
\(516\) 0 0
\(517\) 12.0000 0.527759
\(518\) 7.00000 0.307562
\(519\) 0 0
\(520\) 3.00000 0.131559
\(521\) 42.0000 1.84005 0.920027 0.391856i \(-0.128167\pi\)
0.920027 + 0.391856i \(0.128167\pi\)
\(522\) 0 0
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) 16.0000 0.698963
\(525\) 0 0
\(526\) −18.0000 −0.784837
\(527\) −6.00000 −0.261364
\(528\) 0 0
\(529\) −19.0000 −0.826087
\(530\) −6.00000 −0.260623
\(531\) 0 0
\(532\) −2.00000 −0.0867110
\(533\) 18.0000 0.779667
\(534\) 0 0
\(535\) 0 0
\(536\) 10.0000 0.431934
\(537\) 0 0
\(538\) 1.00000 0.0431131
\(539\) 2.00000 0.0861461
\(540\) 0 0
\(541\) −3.00000 −0.128980 −0.0644900 0.997918i \(-0.520542\pi\)
−0.0644900 + 0.997918i \(0.520542\pi\)
\(542\) 30.0000 1.28861
\(543\) 0 0
\(544\) −1.00000 −0.0428746
\(545\) −11.0000 −0.471188
\(546\) 0 0
\(547\) −6.00000 −0.256541 −0.128271 0.991739i \(-0.540943\pi\)
−0.128271 + 0.991739i \(0.540943\pi\)
\(548\) −11.0000 −0.469897
\(549\) 0 0
\(550\) −8.00000 −0.341121
\(551\) 14.0000 0.596420
\(552\) 0 0
\(553\) 6.00000 0.255146
\(554\) 14.0000 0.594803
\(555\) 0 0
\(556\) −2.00000 −0.0848189
\(557\) −17.0000 −0.720313 −0.360157 0.932892i \(-0.617277\pi\)
−0.360157 + 0.932892i \(0.617277\pi\)
\(558\) 0 0
\(559\) −12.0000 −0.507546
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) 29.0000 1.22329
\(563\) 8.00000 0.337160 0.168580 0.985688i \(-0.446082\pi\)
0.168580 + 0.985688i \(0.446082\pi\)
\(564\) 0 0
\(565\) 1.00000 0.0420703
\(566\) 16.0000 0.672530
\(567\) 0 0
\(568\) −4.00000 −0.167836
\(569\) −27.0000 −1.13190 −0.565949 0.824440i \(-0.691490\pi\)
−0.565949 + 0.824440i \(0.691490\pi\)
\(570\) 0 0
\(571\) 18.0000 0.753277 0.376638 0.926360i \(-0.377080\pi\)
0.376638 + 0.926360i \(0.377080\pi\)
\(572\) 6.00000 0.250873
\(573\) 0 0
\(574\) −6.00000 −0.250435
\(575\) −8.00000 −0.333623
\(576\) 0 0
\(577\) −35.0000 −1.45707 −0.728535 0.685009i \(-0.759798\pi\)
−0.728535 + 0.685009i \(0.759798\pi\)
\(578\) −16.0000 −0.665512
\(579\) 0 0
\(580\) 7.00000 0.290659
\(581\) 10.0000 0.414870
\(582\) 0 0
\(583\) −12.0000 −0.496989
\(584\) −11.0000 −0.455183
\(585\) 0 0
\(586\) −15.0000 −0.619644
\(587\) 2.00000 0.0825488 0.0412744 0.999148i \(-0.486858\pi\)
0.0412744 + 0.999148i \(0.486858\pi\)
\(588\) 0 0
\(589\) 12.0000 0.494451
\(590\) 10.0000 0.411693
\(591\) 0 0
\(592\) −7.00000 −0.287698
\(593\) 39.0000 1.60154 0.800769 0.598973i \(-0.204424\pi\)
0.800769 + 0.598973i \(0.204424\pi\)
\(594\) 0 0
\(595\) 1.00000 0.0409960
\(596\) −21.0000 −0.860194
\(597\) 0 0
\(598\) 6.00000 0.245358
\(599\) −6.00000 −0.245153 −0.122577 0.992459i \(-0.539116\pi\)
−0.122577 + 0.992459i \(0.539116\pi\)
\(600\) 0 0
\(601\) −19.0000 −0.775026 −0.387513 0.921864i \(-0.626666\pi\)
−0.387513 + 0.921864i \(0.626666\pi\)
\(602\) 4.00000 0.163028
\(603\) 0 0
\(604\) 10.0000 0.406894
\(605\) −7.00000 −0.284590
\(606\) 0 0
\(607\) −6.00000 −0.243532 −0.121766 0.992559i \(-0.538856\pi\)
−0.121766 + 0.992559i \(0.538856\pi\)
\(608\) 2.00000 0.0811107
\(609\) 0 0
\(610\) −9.00000 −0.364399
\(611\) 18.0000 0.728202
\(612\) 0 0
\(613\) −18.0000 −0.727013 −0.363507 0.931592i \(-0.618421\pi\)
−0.363507 + 0.931592i \(0.618421\pi\)
\(614\) 16.0000 0.645707
\(615\) 0 0
\(616\) −2.00000 −0.0805823
\(617\) −7.00000 −0.281809 −0.140905 0.990023i \(-0.545001\pi\)
−0.140905 + 0.990023i \(0.545001\pi\)
\(618\) 0 0
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 6.00000 0.240966
\(621\) 0 0
\(622\) −14.0000 −0.561349
\(623\) −15.0000 −0.600962
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 1.00000 0.0399680
\(627\) 0 0
\(628\) −13.0000 −0.518756
\(629\) 7.00000 0.279108
\(630\) 0 0
\(631\) 38.0000 1.51276 0.756378 0.654135i \(-0.226967\pi\)
0.756378 + 0.654135i \(0.226967\pi\)
\(632\) −6.00000 −0.238667
\(633\) 0 0
\(634\) 3.00000 0.119145
\(635\) 12.0000 0.476205
\(636\) 0 0
\(637\) 3.00000 0.118864
\(638\) 14.0000 0.554265
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) 17.0000 0.671460 0.335730 0.941958i \(-0.391017\pi\)
0.335730 + 0.941958i \(0.391017\pi\)
\(642\) 0 0
\(643\) 26.0000 1.02534 0.512670 0.858586i \(-0.328656\pi\)
0.512670 + 0.858586i \(0.328656\pi\)
\(644\) −2.00000 −0.0788110
\(645\) 0 0
\(646\) −2.00000 −0.0786889
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) 0 0
\(649\) 20.0000 0.785069
\(650\) −12.0000 −0.470679
\(651\) 0 0
\(652\) −16.0000 −0.626608
\(653\) 18.0000 0.704394 0.352197 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) 0 0
\(655\) 16.0000 0.625172
\(656\) 6.00000 0.234261
\(657\) 0 0
\(658\) −6.00000 −0.233904
\(659\) 32.0000 1.24654 0.623272 0.782006i \(-0.285803\pi\)
0.623272 + 0.782006i \(0.285803\pi\)
\(660\) 0 0
\(661\) 7.00000 0.272268 0.136134 0.990690i \(-0.456532\pi\)
0.136134 + 0.990690i \(0.456532\pi\)
\(662\) 20.0000 0.777322
\(663\) 0 0
\(664\) −10.0000 −0.388075
\(665\) −2.00000 −0.0775567
\(666\) 0 0
\(667\) 14.0000 0.542082
\(668\) 22.0000 0.851206
\(669\) 0 0
\(670\) 10.0000 0.386334
\(671\) −18.0000 −0.694882
\(672\) 0 0
\(673\) 23.0000 0.886585 0.443292 0.896377i \(-0.353810\pi\)
0.443292 + 0.896377i \(0.353810\pi\)
\(674\) −18.0000 −0.693334
\(675\) 0 0
\(676\) −4.00000 −0.153846
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) 0 0
\(679\) 2.00000 0.0767530
\(680\) −1.00000 −0.0383482
\(681\) 0 0
\(682\) 12.0000 0.459504
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 0 0
\(685\) −11.0000 −0.420288
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) −4.00000 −0.152499
\(689\) −18.0000 −0.685745
\(690\) 0 0
\(691\) −4.00000 −0.152167 −0.0760836 0.997101i \(-0.524242\pi\)
−0.0760836 + 0.997101i \(0.524242\pi\)
\(692\) 13.0000 0.494186
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) −2.00000 −0.0758643
\(696\) 0 0
\(697\) −6.00000 −0.227266
\(698\) 10.0000 0.378506
\(699\) 0 0
\(700\) 4.00000 0.151186
\(701\) −5.00000 −0.188847 −0.0944237 0.995532i \(-0.530101\pi\)
−0.0944237 + 0.995532i \(0.530101\pi\)
\(702\) 0 0
\(703\) −14.0000 −0.528020
\(704\) 2.00000 0.0753778
\(705\) 0 0
\(706\) −30.0000 −1.12906
\(707\) 2.00000 0.0752177
\(708\) 0 0
\(709\) 5.00000 0.187779 0.0938895 0.995583i \(-0.470070\pi\)
0.0938895 + 0.995583i \(0.470070\pi\)
\(710\) −4.00000 −0.150117
\(711\) 0 0
\(712\) 15.0000 0.562149
\(713\) 12.0000 0.449404
\(714\) 0 0
\(715\) 6.00000 0.224387
\(716\) −18.0000 −0.672692
\(717\) 0 0
\(718\) 16.0000 0.597115
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) −15.0000 −0.558242
\(723\) 0 0
\(724\) 18.0000 0.668965
\(725\) −28.0000 −1.03989
\(726\) 0 0
\(727\) −8.00000 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) −3.00000 −0.111187
\(729\) 0 0
\(730\) −11.0000 −0.407128
\(731\) 4.00000 0.147945
\(732\) 0 0
\(733\) 42.0000 1.55131 0.775653 0.631160i \(-0.217421\pi\)
0.775653 + 0.631160i \(0.217421\pi\)
\(734\) −10.0000 −0.369107
\(735\) 0 0
\(736\) 2.00000 0.0737210
\(737\) 20.0000 0.736709
\(738\) 0 0
\(739\) −6.00000 −0.220714 −0.110357 0.993892i \(-0.535199\pi\)
−0.110357 + 0.993892i \(0.535199\pi\)
\(740\) −7.00000 −0.257325
\(741\) 0 0
\(742\) 6.00000 0.220267
\(743\) 24.0000 0.880475 0.440237 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) 0 0
\(745\) −21.0000 −0.769380
\(746\) 10.0000 0.366126
\(747\) 0 0
\(748\) −2.00000 −0.0731272
\(749\) 0 0
\(750\) 0 0
\(751\) 12.0000 0.437886 0.218943 0.975738i \(-0.429739\pi\)
0.218943 + 0.975738i \(0.429739\pi\)
\(752\) 6.00000 0.218797
\(753\) 0 0
\(754\) 21.0000 0.764775
\(755\) 10.0000 0.363937
\(756\) 0 0
\(757\) 42.0000 1.52652 0.763258 0.646094i \(-0.223599\pi\)
0.763258 + 0.646094i \(0.223599\pi\)
\(758\) 10.0000 0.363216
\(759\) 0 0
\(760\) 2.00000 0.0725476
\(761\) 11.0000 0.398750 0.199375 0.979923i \(-0.436109\pi\)
0.199375 + 0.979923i \(0.436109\pi\)
\(762\) 0 0
\(763\) 11.0000 0.398227
\(764\) −18.0000 −0.651217
\(765\) 0 0
\(766\) 32.0000 1.15621
\(767\) 30.0000 1.08324
\(768\) 0 0
\(769\) −23.0000 −0.829401 −0.414701 0.909958i \(-0.636114\pi\)
−0.414701 + 0.909958i \(0.636114\pi\)
\(770\) −2.00000 −0.0720750
\(771\) 0 0
\(772\) 11.0000 0.395899
\(773\) −35.0000 −1.25886 −0.629431 0.777056i \(-0.716712\pi\)
−0.629431 + 0.777056i \(0.716712\pi\)
\(774\) 0 0
\(775\) −24.0000 −0.862105
\(776\) −2.00000 −0.0717958
\(777\) 0 0
\(778\) 34.0000 1.21896
\(779\) 12.0000 0.429945
\(780\) 0 0
\(781\) −8.00000 −0.286263
\(782\) −2.00000 −0.0715199
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) −13.0000 −0.463990
\(786\) 0 0
\(787\) −24.0000 −0.855508 −0.427754 0.903895i \(-0.640695\pi\)
−0.427754 + 0.903895i \(0.640695\pi\)
\(788\) −1.00000 −0.0356235
\(789\) 0 0
\(790\) −6.00000 −0.213470
\(791\) −1.00000 −0.0355559
\(792\) 0 0
\(793\) −27.0000 −0.958798
\(794\) 39.0000 1.38406
\(795\) 0 0
\(796\) −10.0000 −0.354441
\(797\) −3.00000 −0.106265 −0.0531327 0.998587i \(-0.516921\pi\)
−0.0531327 + 0.998587i \(0.516921\pi\)
\(798\) 0 0
\(799\) −6.00000 −0.212265
\(800\) −4.00000 −0.141421
\(801\) 0 0
\(802\) −3.00000 −0.105934
\(803\) −22.0000 −0.776363
\(804\) 0 0
\(805\) −2.00000 −0.0704907
\(806\) 18.0000 0.634023
\(807\) 0 0
\(808\) −2.00000 −0.0703598
\(809\) 1.00000 0.0351581 0.0175791 0.999845i \(-0.494404\pi\)
0.0175791 + 0.999845i \(0.494404\pi\)
\(810\) 0 0
\(811\) 34.0000 1.19390 0.596951 0.802278i \(-0.296379\pi\)
0.596951 + 0.802278i \(0.296379\pi\)
\(812\) −7.00000 −0.245652
\(813\) 0 0
\(814\) −14.0000 −0.490700
\(815\) −16.0000 −0.560456
\(816\) 0 0
\(817\) −8.00000 −0.279885
\(818\) −7.00000 −0.244749
\(819\) 0 0
\(820\) 6.00000 0.209529
\(821\) 11.0000 0.383903 0.191951 0.981404i \(-0.438518\pi\)
0.191951 + 0.981404i \(0.438518\pi\)
\(822\) 0 0
\(823\) 16.0000 0.557725 0.278862 0.960331i \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) −16.0000 −0.557386
\(825\) 0 0
\(826\) −10.0000 −0.347945
\(827\) −48.0000 −1.66912 −0.834562 0.550914i \(-0.814279\pi\)
−0.834562 + 0.550914i \(0.814279\pi\)
\(828\) 0 0
\(829\) −2.00000 −0.0694629 −0.0347314 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) −10.0000 −0.347105
\(831\) 0 0
\(832\) 3.00000 0.104006
\(833\) −1.00000 −0.0346479
\(834\) 0 0
\(835\) 22.0000 0.761341
\(836\) 4.00000 0.138343
\(837\) 0 0
\(838\) 12.0000 0.414533
\(839\) 32.0000 1.10476 0.552381 0.833592i \(-0.313719\pi\)
0.552381 + 0.833592i \(0.313719\pi\)
\(840\) 0 0
\(841\) 20.0000 0.689655
\(842\) 9.00000 0.310160
\(843\) 0 0
\(844\) −22.0000 −0.757271
\(845\) −4.00000 −0.137604
\(846\) 0 0
\(847\) 7.00000 0.240523
\(848\) −6.00000 −0.206041
\(849\) 0 0
\(850\) 4.00000 0.137199
\(851\) −14.0000 −0.479914
\(852\) 0 0
\(853\) −10.0000 −0.342393 −0.171197 0.985237i \(-0.554763\pi\)
−0.171197 + 0.985237i \(0.554763\pi\)
\(854\) 9.00000 0.307974
\(855\) 0 0
\(856\) 0 0
\(857\) 3.00000 0.102478 0.0512390 0.998686i \(-0.483683\pi\)
0.0512390 + 0.998686i \(0.483683\pi\)
\(858\) 0 0
\(859\) −52.0000 −1.77422 −0.887109 0.461561i \(-0.847290\pi\)
−0.887109 + 0.461561i \(0.847290\pi\)
\(860\) −4.00000 −0.136399
\(861\) 0 0
\(862\) 12.0000 0.408722
\(863\) −14.0000 −0.476566 −0.238283 0.971196i \(-0.576585\pi\)
−0.238283 + 0.971196i \(0.576585\pi\)
\(864\) 0 0
\(865\) 13.0000 0.442013
\(866\) 5.00000 0.169907
\(867\) 0 0
\(868\) −6.00000 −0.203653
\(869\) −12.0000 −0.407072
\(870\) 0 0
\(871\) 30.0000 1.01651
\(872\) −11.0000 −0.372507
\(873\) 0 0
\(874\) 4.00000 0.135302
\(875\) 9.00000 0.304256
\(876\) 0 0
\(877\) −23.0000 −0.776655 −0.388327 0.921521i \(-0.626947\pi\)
−0.388327 + 0.921521i \(0.626947\pi\)
\(878\) −12.0000 −0.404980
\(879\) 0 0
\(880\) 2.00000 0.0674200
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 0 0
\(883\) −10.0000 −0.336527 −0.168263 0.985742i \(-0.553816\pi\)
−0.168263 + 0.985742i \(0.553816\pi\)
\(884\) −3.00000 −0.100901
\(885\) 0 0
\(886\) −22.0000 −0.739104
\(887\) 42.0000 1.41022 0.705111 0.709097i \(-0.250897\pi\)
0.705111 + 0.709097i \(0.250897\pi\)
\(888\) 0 0
\(889\) −12.0000 −0.402467
\(890\) 15.0000 0.502801
\(891\) 0 0
\(892\) 20.0000 0.669650
\(893\) 12.0000 0.401565
\(894\) 0 0
\(895\) −18.0000 −0.601674
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 34.0000 1.13459
\(899\) 42.0000 1.40078
\(900\) 0 0
\(901\) 6.00000 0.199889
\(902\) 12.0000 0.399556
\(903\) 0 0
\(904\) 1.00000 0.0332595
\(905\) 18.0000 0.598340
\(906\) 0 0
\(907\) −12.0000 −0.398453 −0.199227 0.979953i \(-0.563843\pi\)
−0.199227 + 0.979953i \(0.563843\pi\)
\(908\) −24.0000 −0.796468
\(909\) 0 0
\(910\) −3.00000 −0.0994490
\(911\) 12.0000 0.397578 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) 0 0
\(913\) −20.0000 −0.661903
\(914\) −41.0000 −1.35616
\(915\) 0 0
\(916\) −29.0000 −0.958187
\(917\) −16.0000 −0.528367
\(918\) 0 0
\(919\) 10.0000 0.329870 0.164935 0.986304i \(-0.447259\pi\)
0.164935 + 0.986304i \(0.447259\pi\)
\(920\) 2.00000 0.0659380
\(921\) 0 0
\(922\) 22.0000 0.724531
\(923\) −12.0000 −0.394985
\(924\) 0 0
\(925\) 28.0000 0.920634
\(926\) 34.0000 1.11731
\(927\) 0 0
\(928\) 7.00000 0.229786
\(929\) −33.0000 −1.08269 −0.541347 0.840799i \(-0.682086\pi\)
−0.541347 + 0.840799i \(0.682086\pi\)
\(930\) 0 0
\(931\) 2.00000 0.0655474
\(932\) −19.0000 −0.622366
\(933\) 0 0
\(934\) 34.0000 1.11251
\(935\) −2.00000 −0.0654070
\(936\) 0 0
\(937\) −43.0000 −1.40475 −0.702374 0.711808i \(-0.747877\pi\)
−0.702374 + 0.711808i \(0.747877\pi\)
\(938\) −10.0000 −0.326512
\(939\) 0 0
\(940\) 6.00000 0.195698
\(941\) 25.0000 0.814977 0.407488 0.913210i \(-0.366405\pi\)
0.407488 + 0.913210i \(0.366405\pi\)
\(942\) 0 0
\(943\) 12.0000 0.390774
\(944\) 10.0000 0.325472
\(945\) 0 0
\(946\) −8.00000 −0.260102
\(947\) 4.00000 0.129983 0.0649913 0.997886i \(-0.479298\pi\)
0.0649913 + 0.997886i \(0.479298\pi\)
\(948\) 0 0
\(949\) −33.0000 −1.07123
\(950\) −8.00000 −0.259554
\(951\) 0 0
\(952\) 1.00000 0.0324102
\(953\) −19.0000 −0.615470 −0.307735 0.951472i \(-0.599571\pi\)
−0.307735 + 0.951472i \(0.599571\pi\)
\(954\) 0 0
\(955\) −18.0000 −0.582466
\(956\) −6.00000 −0.194054
\(957\) 0 0
\(958\) −22.0000 −0.710788
\(959\) 11.0000 0.355209
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) −21.0000 −0.677067
\(963\) 0 0
\(964\) −7.00000 −0.225455
\(965\) 11.0000 0.354103
\(966\) 0 0
\(967\) −22.0000 −0.707472 −0.353736 0.935345i \(-0.615089\pi\)
−0.353736 + 0.935345i \(0.615089\pi\)
\(968\) −7.00000 −0.224989
\(969\) 0 0
\(970\) −2.00000 −0.0642161
\(971\) −42.0000 −1.34784 −0.673922 0.738802i \(-0.735392\pi\)
−0.673922 + 0.738802i \(0.735392\pi\)
\(972\) 0 0
\(973\) 2.00000 0.0641171
\(974\) −10.0000 −0.320421
\(975\) 0 0
\(976\) −9.00000 −0.288083
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) 0 0
\(979\) 30.0000 0.958804
\(980\) 1.00000 0.0319438
\(981\) 0 0
\(982\) 36.0000 1.14881
\(983\) −60.0000 −1.91370 −0.956851 0.290578i \(-0.906153\pi\)
−0.956851 + 0.290578i \(0.906153\pi\)
\(984\) 0 0
\(985\) −1.00000 −0.0318626
\(986\) −7.00000 −0.222925
\(987\) 0 0
\(988\) 6.00000 0.190885
\(989\) −8.00000 −0.254385
\(990\) 0 0
\(991\) 62.0000 1.96949 0.984747 0.173990i \(-0.0556660\pi\)
0.984747 + 0.173990i \(0.0556660\pi\)
\(992\) 6.00000 0.190500
\(993\) 0 0
\(994\) 4.00000 0.126872
\(995\) −10.0000 −0.317021
\(996\) 0 0
\(997\) 23.0000 0.728417 0.364209 0.931317i \(-0.381339\pi\)
0.364209 + 0.931317i \(0.381339\pi\)
\(998\) 22.0000 0.696398
\(999\) 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 1134.2.a.g.1.1 yes 1
3.2 odd 2 1134.2.a.b.1.1 1
4.3 odd 2 9072.2.a.p.1.1 1
7.6 odd 2 7938.2.a.w.1.1 1
9.2 odd 6 1134.2.f.m.757.1 2
9.4 even 3 1134.2.f.d.379.1 2
9.5 odd 6 1134.2.f.m.379.1 2
9.7 even 3 1134.2.f.d.757.1 2
12.11 even 2 9072.2.a.k.1.1 1
21.20 even 2 7938.2.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1134.2.a.b.1.1 1 3.2 odd 2
1134.2.a.g.1.1 yes 1 1.1 even 1 trivial
1134.2.f.d.379.1 2 9.4 even 3
1134.2.f.d.757.1 2 9.7 even 3
1134.2.f.m.379.1 2 9.5 odd 6
1134.2.f.m.757.1 2 9.2 odd 6
7938.2.a.j.1.1 1 21.20 even 2
7938.2.a.w.1.1 1 7.6 odd 2
9072.2.a.k.1.1 1 12.11 even 2
9072.2.a.p.1.1 1 4.3 odd 2