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Properties

Label 1323.2.a.g.1.1
Level $1323$
Weight $2$
Character 1323.1
Self dual yes
Analytic conductor $10.564$
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] = [1323,2,Mod(1,1323)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1323, 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("1323.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1323.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: \(10.5642081874\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 189)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1323.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{4} +4.00000 q^{5} +3.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{4} +4.00000 q^{5} +3.00000 q^{8} -4.00000 q^{10} -2.00000 q^{11} +1.00000 q^{13} -1.00000 q^{16} +6.00000 q^{17} +4.00000 q^{19} -4.00000 q^{20} +2.00000 q^{22} -6.00000 q^{23} +11.0000 q^{25} -1.00000 q^{26} -2.00000 q^{29} +3.00000 q^{31} -5.00000 q^{32} -6.00000 q^{34} +3.00000 q^{37} -4.00000 q^{38} +12.0000 q^{40} +2.00000 q^{41} -1.00000 q^{43} +2.00000 q^{44} +6.00000 q^{46} -6.00000 q^{47} -11.0000 q^{50} -1.00000 q^{52} +6.00000 q^{53} -8.00000 q^{55} +2.00000 q^{58} -6.00000 q^{59} -5.00000 q^{61} -3.00000 q^{62} +7.00000 q^{64} +4.00000 q^{65} +7.00000 q^{67} -6.00000 q^{68} -6.00000 q^{73} -3.00000 q^{74} -4.00000 q^{76} +11.0000 q^{79} -4.00000 q^{80} -2.00000 q^{82} -6.00000 q^{83} +24.0000 q^{85} +1.00000 q^{86} -6.00000 q^{88} +4.00000 q^{89} +6.00000 q^{92} +6.00000 q^{94} +16.0000 q^{95} +9.00000 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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 −0.353553 − 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 4.00000 1.78885 0.894427 − 0.447214i \(-0.147584\pi\)
0.894427 + 0.447214i \(0.147584\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 3.00000 1.06066
\(9\) 0 0
\(10\) −4.00000 −1.26491
\(11\) −2.00000 −0.603023 −0.301511 − 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350 0.138675 − 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 6.00000 1.45521 0.727607 − 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) −4.00000 −0.894427
\(21\) 0 0
\(22\) 2.00000 0.426401
\(23\) −6.00000 −1.25109 −0.625543 − 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) 11.0000 2.20000
\(26\) −1.00000 −0.196116
\(27\) 0 0
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 − 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 3.00000 0.538816 0.269408 − 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) −5.00000 −0.883883
\(33\) 0 0
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 0 0
\(37\) 3.00000 0.493197 0.246598 − 0.969118i \(-0.420687\pi\)
0.246598 + 0.969118i \(0.420687\pi\)
\(38\) −4.00000 −0.648886
\(39\) 0 0
\(40\) 12.0000 1.89737
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 − 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) −6.00000 −0.875190 −0.437595 − 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −11.0000 −1.55563
\(51\) 0 0
\(52\) −1.00000 −0.138675
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) −8.00000 −1.07872
\(56\) 0 0
\(57\) 0 0
\(58\) 2.00000 0.262613
\(59\) −6.00000 −0.781133 −0.390567 − 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) −5.00000 −0.640184 −0.320092 − 0.947386i \(-0.603714\pi\)
−0.320092 + 0.947386i \(0.603714\pi\)
\(62\) −3.00000 −0.381000
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) 7.00000 0.855186 0.427593 − 0.903971i \(-0.359362\pi\)
0.427593 + 0.903971i \(0.359362\pi\)
\(68\) −6.00000 −0.727607
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −6.00000 −0.702247 −0.351123 − 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) −3.00000 −0.348743
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) 0 0
\(79\) 11.0000 1.23760 0.618798 − 0.785550i \(-0.287620\pi\)
0.618798 + 0.785550i \(0.287620\pi\)
\(80\) −4.00000 −0.447214
\(81\) 0 0
\(82\) −2.00000 −0.220863
\(83\) −6.00000 −0.658586 −0.329293 − 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) 24.0000 2.60317
\(86\) 1.00000 0.107833
\(87\) 0 0
\(88\) −6.00000 −0.639602
\(89\) 4.00000 0.423999 0.212000 − 0.977270i \(-0.432002\pi\)
0.212000 + 0.977270i \(0.432002\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 6.00000 0.625543
\(93\) 0 0
\(94\) 6.00000 0.618853
\(95\) 16.0000 1.64157
\(96\) 0 0
\(97\) 9.00000 0.913812 0.456906 − 0.889515i \(-0.348958\pi\)
0.456906 + 0.889515i \(0.348958\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −11.0000 −1.10000
\(101\) 8.00000 0.796030 0.398015 − 0.917379i \(-0.369699\pi\)
0.398015 + 0.917379i \(0.369699\pi\)
\(102\) 0 0
\(103\) 17.0000 1.67506 0.837530 − 0.546392i \(-0.183999\pi\)
0.837530 + 0.546392i \(0.183999\pi\)
\(104\) 3.00000 0.294174
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) −2.00000 −0.193347 −0.0966736 − 0.995316i \(-0.530820\pi\)
−0.0966736 + 0.995316i \(0.530820\pi\)
\(108\) 0 0
\(109\) 9.00000 0.862044 0.431022 − 0.902342i \(-0.358153\pi\)
0.431022 + 0.902342i \(0.358153\pi\)
\(110\) 8.00000 0.762770
\(111\) 0 0
\(112\) 0 0
\(113\) 16.0000 1.50515 0.752577 − 0.658505i \(-0.228811\pi\)
0.752577 + 0.658505i \(0.228811\pi\)
\(114\) 0 0
\(115\) −24.0000 −2.23801
\(116\) 2.00000 0.185695
\(117\) 0 0
\(118\) 6.00000 0.552345
\(119\) 0 0
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 5.00000 0.452679
\(123\) 0 0
\(124\) −3.00000 −0.269408
\(125\) 24.0000 2.14663
\(126\) 0 0
\(127\) −9.00000 −0.798621 −0.399310 − 0.916816i \(-0.630750\pi\)
−0.399310 + 0.916816i \(0.630750\pi\)
\(128\) 3.00000 0.265165
\(129\) 0 0
\(130\) −4.00000 −0.350823
\(131\) 4.00000 0.349482 0.174741 − 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −7.00000 −0.604708
\(135\) 0 0
\(136\) 18.0000 1.54349
\(137\) −18.0000 −1.53784 −0.768922 − 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 0 0
\(139\) 9.00000 0.763370 0.381685 − 0.924292i \(-0.375344\pi\)
0.381685 + 0.924292i \(0.375344\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −2.00000 −0.167248
\(144\) 0 0
\(145\) −8.00000 −0.664364
\(146\) 6.00000 0.496564
\(147\) 0 0
\(148\) −3.00000 −0.246598
\(149\) 24.0000 1.96616 0.983078 − 0.183186i \(-0.0586410\pi\)
0.983078 + 0.183186i \(0.0586410\pi\)
\(150\) 0 0
\(151\) 5.00000 0.406894 0.203447 − 0.979086i \(-0.434786\pi\)
0.203447 + 0.979086i \(0.434786\pi\)
\(152\) 12.0000 0.973329
\(153\) 0 0
\(154\) 0 0
\(155\) 12.0000 0.963863
\(156\) 0 0
\(157\) 10.0000 0.798087 0.399043 − 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) −11.0000 −0.875113
\(159\) 0 0
\(160\) −20.0000 −1.58114
\(161\) 0 0
\(162\) 0 0
\(163\) 19.0000 1.48819 0.744097 − 0.668071i \(-0.232880\pi\)
0.744097 + 0.668071i \(0.232880\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) 6.00000 0.465690
\(167\) −14.0000 −1.08335 −0.541676 − 0.840587i \(-0.682210\pi\)
−0.541676 + 0.840587i \(0.682210\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) −24.0000 −1.84072
\(171\) 0 0
\(172\) 1.00000 0.0762493
\(173\) −16.0000 −1.21646 −0.608229 − 0.793762i \(-0.708120\pi\)
−0.608229 + 0.793762i \(0.708120\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) 0 0
\(178\) −4.00000 −0.299813
\(179\) −4.00000 −0.298974 −0.149487 − 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 − 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −18.0000 −1.32698
\(185\) 12.0000 0.882258
\(186\) 0 0
\(187\) −12.0000 −0.877527
\(188\) 6.00000 0.437595
\(189\) 0 0
\(190\) −16.0000 −1.16076
\(191\) 4.00000 0.289430 0.144715 − 0.989473i \(-0.453773\pi\)
0.144715 + 0.989473i \(0.453773\pi\)
\(192\) 0 0
\(193\) −25.0000 −1.79954 −0.899770 − 0.436365i \(-0.856266\pi\)
−0.899770 + 0.436365i \(0.856266\pi\)
\(194\) −9.00000 −0.646162
\(195\) 0 0
\(196\) 0 0
\(197\) −20.0000 −1.42494 −0.712470 − 0.701702i \(-0.752424\pi\)
−0.712470 + 0.701702i \(0.752424\pi\)
\(198\) 0 0
\(199\) −9.00000 −0.637993 −0.318997 − 0.947756i \(-0.603346\pi\)
−0.318997 + 0.947756i \(0.603346\pi\)
\(200\) 33.0000 2.33345
\(201\) 0 0
\(202\) −8.00000 −0.562878
\(203\) 0 0
\(204\) 0 0
\(205\) 8.00000 0.558744
\(206\) −17.0000 −1.18445
\(207\) 0 0
\(208\) −1.00000 −0.0693375
\(209\) −8.00000 −0.553372
\(210\) 0 0
\(211\) −5.00000 −0.344214 −0.172107 − 0.985078i \(-0.555058\pi\)
−0.172107 + 0.985078i \(0.555058\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) 2.00000 0.136717
\(215\) −4.00000 −0.272798
\(216\) 0 0
\(217\) 0 0
\(218\) −9.00000 −0.609557
\(219\) 0 0
\(220\) 8.00000 0.539360
\(221\) 6.00000 0.403604
\(222\) 0 0
\(223\) 16.0000 1.07144 0.535720 − 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −16.0000 −1.06430
\(227\) −18.0000 −1.19470 −0.597351 − 0.801980i \(-0.703780\pi\)
−0.597351 + 0.801980i \(0.703780\pi\)
\(228\) 0 0
\(229\) −7.00000 −0.462573 −0.231287 − 0.972886i \(-0.574293\pi\)
−0.231287 + 0.972886i \(0.574293\pi\)
\(230\) 24.0000 1.58251
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) 24.0000 1.57229 0.786146 − 0.618041i \(-0.212073\pi\)
0.786146 + 0.618041i \(0.212073\pi\)
\(234\) 0 0
\(235\) −24.0000 −1.56559
\(236\) 6.00000 0.390567
\(237\) 0 0
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 − 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) 17.0000 1.09507 0.547533 − 0.836784i \(-0.315567\pi\)
0.547533 + 0.836784i \(0.315567\pi\)
\(242\) 7.00000 0.449977
\(243\) 0 0
\(244\) 5.00000 0.320092
\(245\) 0 0
\(246\) 0 0
\(247\) 4.00000 0.254514
\(248\) 9.00000 0.571501
\(249\) 0 0
\(250\) −24.0000 −1.51789
\(251\) −26.0000 −1.64111 −0.820553 − 0.571571i \(-0.806334\pi\)
−0.820553 + 0.571571i \(0.806334\pi\)
\(252\) 0 0
\(253\) 12.0000 0.754434
\(254\) 9.00000 0.564710
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) −10.0000 −0.623783 −0.311891 − 0.950118i \(-0.600963\pi\)
−0.311891 + 0.950118i \(0.600963\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −4.00000 −0.248069
\(261\) 0 0
\(262\) −4.00000 −0.247121
\(263\) −2.00000 −0.123325 −0.0616626 − 0.998097i \(-0.519640\pi\)
−0.0616626 + 0.998097i \(0.519640\pi\)
\(264\) 0 0
\(265\) 24.0000 1.47431
\(266\) 0 0
\(267\) 0 0
\(268\) −7.00000 −0.427593
\(269\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) −11.0000 −0.668202 −0.334101 − 0.942537i \(-0.608433\pi\)
−0.334101 + 0.942537i \(0.608433\pi\)
\(272\) −6.00000 −0.363803
\(273\) 0 0
\(274\) 18.0000 1.08742
\(275\) −22.0000 −1.32665
\(276\) 0 0
\(277\) 13.0000 0.781094 0.390547 − 0.920583i \(-0.372286\pi\)
0.390547 + 0.920583i \(0.372286\pi\)
\(278\) −9.00000 −0.539784
\(279\) 0 0
\(280\) 0 0
\(281\) −16.0000 −0.954480 −0.477240 − 0.878773i \(-0.658363\pi\)
−0.477240 + 0.878773i \(0.658363\pi\)
\(282\) 0 0
\(283\) −5.00000 −0.297219 −0.148610 − 0.988896i \(-0.547480\pi\)
−0.148610 + 0.988896i \(0.547480\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 2.00000 0.118262
\(287\) 0 0
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 8.00000 0.469776
\(291\) 0 0
\(292\) 6.00000 0.351123
\(293\) −22.0000 −1.28525 −0.642627 − 0.766179i \(-0.722155\pi\)
−0.642627 + 0.766179i \(0.722155\pi\)
\(294\) 0 0
\(295\) −24.0000 −1.39733
\(296\) 9.00000 0.523114
\(297\) 0 0
\(298\) −24.0000 −1.39028
\(299\) −6.00000 −0.346989
\(300\) 0 0
\(301\) 0 0
\(302\) −5.00000 −0.287718
\(303\) 0 0
\(304\) −4.00000 −0.229416
\(305\) −20.0000 −1.14520
\(306\) 0 0
\(307\) 25.0000 1.42683 0.713413 − 0.700744i \(-0.247149\pi\)
0.713413 + 0.700744i \(0.247149\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −12.0000 −0.681554
\(311\) −24.0000 −1.36092 −0.680458 − 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 0 0
\(313\) −22.0000 −1.24351 −0.621757 − 0.783210i \(-0.713581\pi\)
−0.621757 + 0.783210i \(0.713581\pi\)
\(314\) −10.0000 −0.564333
\(315\) 0 0
\(316\) −11.0000 −0.618798
\(317\) −30.0000 −1.68497 −0.842484 − 0.538721i \(-0.818908\pi\)
−0.842484 + 0.538721i \(0.818908\pi\)
\(318\) 0 0
\(319\) 4.00000 0.223957
\(320\) 28.0000 1.56525
\(321\) 0 0
\(322\) 0 0
\(323\) 24.0000 1.33540
\(324\) 0 0
\(325\) 11.0000 0.610170
\(326\) −19.0000 −1.05231
\(327\) 0 0
\(328\) 6.00000 0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) −28.0000 −1.53902 −0.769510 − 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) 6.00000 0.329293
\(333\) 0 0
\(334\) 14.0000 0.766046
\(335\) 28.0000 1.52980
\(336\) 0 0
\(337\) 10.0000 0.544735 0.272367 − 0.962193i \(-0.412193\pi\)
0.272367 + 0.962193i \(0.412193\pi\)
\(338\) 12.0000 0.652714
\(339\) 0 0
\(340\) −24.0000 −1.30158
\(341\) −6.00000 −0.324918
\(342\) 0 0
\(343\) 0 0
\(344\) −3.00000 −0.161749
\(345\) 0 0
\(346\) 16.0000 0.860165
\(347\) −16.0000 −0.858925 −0.429463 − 0.903085i \(-0.641297\pi\)
−0.429463 + 0.903085i \(0.641297\pi\)
\(348\) 0 0
\(349\) −5.00000 −0.267644 −0.133822 − 0.991005i \(-0.542725\pi\)
−0.133822 + 0.991005i \(0.542725\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 10.0000 0.533002
\(353\) 28.0000 1.49029 0.745145 − 0.666903i \(-0.232380\pi\)
0.745145 + 0.666903i \(0.232380\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −4.00000 −0.212000
\(357\) 0 0
\(358\) 4.00000 0.211407
\(359\) −22.0000 −1.16112 −0.580558 − 0.814219i \(-0.697165\pi\)
−0.580558 + 0.814219i \(0.697165\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 2.00000 0.105118
\(363\) 0 0
\(364\) 0 0
\(365\) −24.0000 −1.25622
\(366\) 0 0
\(367\) 12.0000 0.626395 0.313197 − 0.949688i \(-0.398600\pi\)
0.313197 + 0.949688i \(0.398600\pi\)
\(368\) 6.00000 0.312772
\(369\) 0 0
\(370\) −12.0000 −0.623850
\(371\) 0 0
\(372\) 0 0
\(373\) 26.0000 1.34623 0.673114 − 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(374\) 12.0000 0.620505
\(375\) 0 0
\(376\) −18.0000 −0.928279
\(377\) −2.00000 −0.103005
\(378\) 0 0
\(379\) −9.00000 −0.462299 −0.231149 − 0.972918i \(-0.574249\pi\)
−0.231149 + 0.972918i \(0.574249\pi\)
\(380\) −16.0000 −0.820783
\(381\) 0 0
\(382\) −4.00000 −0.204658
\(383\) 18.0000 0.919757 0.459879 − 0.887982i \(-0.347893\pi\)
0.459879 + 0.887982i \(0.347893\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 25.0000 1.27247
\(387\) 0 0
\(388\) −9.00000 −0.456906
\(389\) 12.0000 0.608424 0.304212 − 0.952604i \(-0.401607\pi\)
0.304212 + 0.952604i \(0.401607\pi\)
\(390\) 0 0
\(391\) −36.0000 −1.82060
\(392\) 0 0
\(393\) 0 0
\(394\) 20.0000 1.00759
\(395\) 44.0000 2.21388
\(396\) 0 0
\(397\) −33.0000 −1.65622 −0.828111 − 0.560564i \(-0.810584\pi\)
−0.828111 + 0.560564i \(0.810584\pi\)
\(398\) 9.00000 0.451129
\(399\) 0 0
\(400\) −11.0000 −0.550000
\(401\) −30.0000 −1.49813 −0.749064 − 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 0 0
\(403\) 3.00000 0.149441
\(404\) −8.00000 −0.398015
\(405\) 0 0
\(406\) 0 0
\(407\) −6.00000 −0.297409
\(408\) 0 0
\(409\) 5.00000 0.247234 0.123617 − 0.992330i \(-0.460551\pi\)
0.123617 + 0.992330i \(0.460551\pi\)
\(410\) −8.00000 −0.395092
\(411\) 0 0
\(412\) −17.0000 −0.837530
\(413\) 0 0
\(414\) 0 0
\(415\) −24.0000 −1.17811
\(416\) −5.00000 −0.245145
\(417\) 0 0
\(418\) 8.00000 0.391293
\(419\) 12.0000 0.586238 0.293119 − 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 − 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 5.00000 0.243396
\(423\) 0 0
\(424\) 18.0000 0.874157
\(425\) 66.0000 3.20147
\(426\) 0 0
\(427\) 0 0
\(428\) 2.00000 0.0966736
\(429\) 0 0
\(430\) 4.00000 0.192897
\(431\) −18.0000 −0.867029 −0.433515 − 0.901146i \(-0.642727\pi\)
−0.433515 + 0.901146i \(0.642727\pi\)
\(432\) 0 0
\(433\) −17.0000 −0.816968 −0.408484 − 0.912766i \(-0.633942\pi\)
−0.408484 + 0.912766i \(0.633942\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −9.00000 −0.431022
\(437\) −24.0000 −1.14808
\(438\) 0 0
\(439\) 24.0000 1.14546 0.572729 − 0.819745i \(-0.305885\pi\)
0.572729 + 0.819745i \(0.305885\pi\)
\(440\) −24.0000 −1.14416
\(441\) 0 0
\(442\) −6.00000 −0.285391
\(443\) 24.0000 1.14027 0.570137 − 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) 0 0
\(445\) 16.0000 0.758473
\(446\) −16.0000 −0.757622
\(447\) 0 0
\(448\) 0 0
\(449\) 36.0000 1.69895 0.849473 − 0.527633i \(-0.176920\pi\)
0.849473 + 0.527633i \(0.176920\pi\)
\(450\) 0 0
\(451\) −4.00000 −0.188353
\(452\) −16.0000 −0.752577
\(453\) 0 0
\(454\) 18.0000 0.844782
\(455\) 0 0
\(456\) 0 0
\(457\) 13.0000 0.608114 0.304057 − 0.952654i \(-0.401659\pi\)
0.304057 + 0.952654i \(0.401659\pi\)
\(458\) 7.00000 0.327089
\(459\) 0 0
\(460\) 24.0000 1.11901
\(461\) 2.00000 0.0931493 0.0465746 − 0.998915i \(-0.485169\pi\)
0.0465746 + 0.998915i \(0.485169\pi\)
\(462\) 0 0
\(463\) −32.0000 −1.48717 −0.743583 − 0.668644i \(-0.766875\pi\)
−0.743583 + 0.668644i \(0.766875\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) −24.0000 −1.11178
\(467\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 24.0000 1.10704
\(471\) 0 0
\(472\) −18.0000 −0.828517
\(473\) 2.00000 0.0919601
\(474\) 0 0
\(475\) 44.0000 2.01886
\(476\) 0 0
\(477\) 0 0
\(478\) −12.0000 −0.548867
\(479\) −16.0000 −0.731059 −0.365529 − 0.930800i \(-0.619112\pi\)
−0.365529 + 0.930800i \(0.619112\pi\)
\(480\) 0 0
\(481\) 3.00000 0.136788
\(482\) −17.0000 −0.774329
\(483\) 0 0
\(484\) 7.00000 0.318182
\(485\) 36.0000 1.63468
\(486\) 0 0
\(487\) 16.0000 0.725029 0.362515 − 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(488\) −15.0000 −0.679018
\(489\) 0 0
\(490\) 0 0
\(491\) −34.0000 −1.53440 −0.767199 − 0.641409i \(-0.778350\pi\)
−0.767199 + 0.641409i \(0.778350\pi\)
\(492\) 0 0
\(493\) −12.0000 −0.540453
\(494\) −4.00000 −0.179969
\(495\) 0 0
\(496\) −3.00000 −0.134704
\(497\) 0 0
\(498\) 0 0
\(499\) −17.0000 −0.761025 −0.380512 − 0.924776i \(-0.624252\pi\)
−0.380512 + 0.924776i \(0.624252\pi\)
\(500\) −24.0000 −1.07331
\(501\) 0 0
\(502\) 26.0000 1.16044
\(503\) −6.00000 −0.267527 −0.133763 − 0.991013i \(-0.542706\pi\)
−0.133763 + 0.991013i \(0.542706\pi\)
\(504\) 0 0
\(505\) 32.0000 1.42398
\(506\) −12.0000 −0.533465
\(507\) 0 0
\(508\) 9.00000 0.399310
\(509\) −22.0000 −0.975133 −0.487566 − 0.873086i \(-0.662115\pi\)
−0.487566 + 0.873086i \(0.662115\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 11.0000 0.486136
\(513\) 0 0
\(514\) 10.0000 0.441081
\(515\) 68.0000 2.99644
\(516\) 0 0
\(517\) 12.0000 0.527759
\(518\) 0 0
\(519\) 0 0
\(520\) 12.0000 0.526235
\(521\) −18.0000 −0.788594 −0.394297 − 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 0 0
\(523\) −17.0000 −0.743358 −0.371679 − 0.928361i \(-0.621218\pi\)
−0.371679 + 0.928361i \(0.621218\pi\)
\(524\) −4.00000 −0.174741
\(525\) 0 0
\(526\) 2.00000 0.0872041
\(527\) 18.0000 0.784092
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) −24.0000 −1.04249
\(531\) 0 0
\(532\) 0 0
\(533\) 2.00000 0.0866296
\(534\) 0 0
\(535\) −8.00000 −0.345870
\(536\) 21.0000 0.907062
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 − 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 11.0000 0.472490
\(543\) 0 0
\(544\) −30.0000 −1.28624
\(545\) 36.0000 1.54207
\(546\) 0 0
\(547\) 43.0000 1.83855 0.919274 − 0.393619i \(-0.128777\pi\)
0.919274 + 0.393619i \(0.128777\pi\)
\(548\) 18.0000 0.768922
\(549\) 0 0
\(550\) 22.0000 0.938083
\(551\) −8.00000 −0.340811
\(552\) 0 0
\(553\) 0 0
\(554\) −13.0000 −0.552317
\(555\) 0 0
\(556\) −9.00000 −0.381685
\(557\) 28.0000 1.18640 0.593199 − 0.805056i \(-0.297865\pi\)
0.593199 + 0.805056i \(0.297865\pi\)
\(558\) 0 0
\(559\) −1.00000 −0.0422955
\(560\) 0 0
\(561\) 0 0
\(562\) 16.0000 0.674919
\(563\) 10.0000 0.421450 0.210725 − 0.977545i \(-0.432418\pi\)
0.210725 + 0.977545i \(0.432418\pi\)
\(564\) 0 0
\(565\) 64.0000 2.69250
\(566\) 5.00000 0.210166
\(567\) 0 0
\(568\) 0 0
\(569\) −32.0000 −1.34151 −0.670755 − 0.741679i \(-0.734030\pi\)
−0.670755 + 0.741679i \(0.734030\pi\)
\(570\) 0 0
\(571\) −16.0000 −0.669579 −0.334790 − 0.942293i \(-0.608665\pi\)
−0.334790 + 0.942293i \(0.608665\pi\)
\(572\) 2.00000 0.0836242
\(573\) 0 0
\(574\) 0 0
\(575\) −66.0000 −2.75239
\(576\) 0 0
\(577\) −17.0000 −0.707719 −0.353860 − 0.935299i \(-0.615131\pi\)
−0.353860 + 0.935299i \(0.615131\pi\)
\(578\) −19.0000 −0.790296
\(579\) 0 0
\(580\) 8.00000 0.332182
\(581\) 0 0
\(582\) 0 0
\(583\) −12.0000 −0.496989
\(584\) −18.0000 −0.744845
\(585\) 0 0
\(586\) 22.0000 0.908812
\(587\) 28.0000 1.15568 0.577842 − 0.816149i \(-0.303895\pi\)
0.577842 + 0.816149i \(0.303895\pi\)
\(588\) 0 0
\(589\) 12.0000 0.494451
\(590\) 24.0000 0.988064
\(591\) 0 0
\(592\) −3.00000 −0.123299
\(593\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −24.0000 −0.983078
\(597\) 0 0
\(598\) 6.00000 0.245358
\(599\) −12.0000 −0.490307 −0.245153 − 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) 0 0
\(601\) −33.0000 −1.34610 −0.673049 − 0.739598i \(-0.735016\pi\)
−0.673049 + 0.739598i \(0.735016\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −5.00000 −0.203447
\(605\) −28.0000 −1.13836
\(606\) 0 0
\(607\) −16.0000 −0.649420 −0.324710 − 0.945814i \(-0.605267\pi\)
−0.324710 + 0.945814i \(0.605267\pi\)
\(608\) −20.0000 −0.811107
\(609\) 0 0
\(610\) 20.0000 0.809776
\(611\) −6.00000 −0.242734
\(612\) 0 0
\(613\) 7.00000 0.282727 0.141364 − 0.989958i \(-0.454851\pi\)
0.141364 + 0.989958i \(0.454851\pi\)
\(614\) −25.0000 −1.00892
\(615\) 0 0
\(616\) 0 0
\(617\) 18.0000 0.724653 0.362326 − 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 0 0
\(619\) −41.0000 −1.64793 −0.823965 − 0.566641i \(-0.808243\pi\)
−0.823965 + 0.566641i \(0.808243\pi\)
\(620\) −12.0000 −0.481932
\(621\) 0 0
\(622\) 24.0000 0.962312
\(623\) 0 0
\(624\) 0 0
\(625\) 41.0000 1.64000
\(626\) 22.0000 0.879297
\(627\) 0 0
\(628\) −10.0000 −0.399043
\(629\) 18.0000 0.717707
\(630\) 0 0
\(631\) 5.00000 0.199047 0.0995234 − 0.995035i \(-0.468268\pi\)
0.0995234 + 0.995035i \(0.468268\pi\)
\(632\) 33.0000 1.31267
\(633\) 0 0
\(634\) 30.0000 1.19145
\(635\) −36.0000 −1.42862
\(636\) 0 0
\(637\) 0 0
\(638\) −4.00000 −0.158362
\(639\) 0 0
\(640\) 12.0000 0.474342
\(641\) −6.00000 −0.236986 −0.118493 − 0.992955i \(-0.537806\pi\)
−0.118493 + 0.992955i \(0.537806\pi\)
\(642\) 0 0
\(643\) −25.0000 −0.985904 −0.492952 − 0.870057i \(-0.664082\pi\)
−0.492952 + 0.870057i \(0.664082\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −24.0000 −0.944267
\(647\) −28.0000 −1.10079 −0.550397 − 0.834903i \(-0.685524\pi\)
−0.550397 + 0.834903i \(0.685524\pi\)
\(648\) 0 0
\(649\) 12.0000 0.471041
\(650\) −11.0000 −0.431455
\(651\) 0 0
\(652\) −19.0000 −0.744097
\(653\) −6.00000 −0.234798 −0.117399 − 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) 0 0
\(655\) 16.0000 0.625172
\(656\) −2.00000 −0.0780869
\(657\) 0 0
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 − 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) −14.0000 −0.544537 −0.272268 − 0.962221i \(-0.587774\pi\)
−0.272268 + 0.962221i \(0.587774\pi\)
\(662\) 28.0000 1.08825
\(663\) 0 0
\(664\) −18.0000 −0.698535
\(665\) 0 0
\(666\) 0 0
\(667\) 12.0000 0.464642
\(668\) 14.0000 0.541676
\(669\) 0 0
\(670\) −28.0000 −1.08173
\(671\) 10.0000 0.386046
\(672\) 0 0
\(673\) 22.0000 0.848038 0.424019 − 0.905653i \(-0.360619\pi\)
0.424019 + 0.905653i \(0.360619\pi\)
\(674\) −10.0000 −0.385186
\(675\) 0 0
\(676\) 12.0000 0.461538
\(677\) −12.0000 −0.461197 −0.230599 − 0.973049i \(-0.574068\pi\)
−0.230599 + 0.973049i \(0.574068\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 72.0000 2.76107
\(681\) 0 0
\(682\) 6.00000 0.229752
\(683\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) −72.0000 −2.75098
\(686\) 0 0
\(687\) 0 0
\(688\) 1.00000 0.0381246
\(689\) 6.00000 0.228582
\(690\) 0 0
\(691\) 41.0000 1.55971 0.779857 − 0.625958i \(-0.215292\pi\)
0.779857 + 0.625958i \(0.215292\pi\)
\(692\) 16.0000 0.608229
\(693\) 0 0
\(694\) 16.0000 0.607352
\(695\) 36.0000 1.36556
\(696\) 0 0
\(697\) 12.0000 0.454532
\(698\) 5.00000 0.189253
\(699\) 0 0
\(700\) 0 0
\(701\) −24.0000 −0.906467 −0.453234 − 0.891392i \(-0.649730\pi\)
−0.453234 + 0.891392i \(0.649730\pi\)
\(702\) 0 0
\(703\) 12.0000 0.452589
\(704\) −14.0000 −0.527645
\(705\) 0 0
\(706\) −28.0000 −1.05379
\(707\) 0 0
\(708\) 0 0
\(709\) 21.0000 0.788672 0.394336 − 0.918966i \(-0.370975\pi\)
0.394336 + 0.918966i \(0.370975\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 12.0000 0.449719
\(713\) −18.0000 −0.674105
\(714\) 0 0
\(715\) −8.00000 −0.299183
\(716\) 4.00000 0.149487
\(717\) 0 0
\(718\) 22.0000 0.821033
\(719\) 48.0000 1.79010 0.895049 − 0.445968i \(-0.147140\pi\)
0.895049 + 0.445968i \(0.147140\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 3.00000 0.111648
\(723\) 0 0
\(724\) 2.00000 0.0743294
\(725\) −22.0000 −0.817059
\(726\) 0 0
\(727\) 41.0000 1.52061 0.760303 − 0.649569i \(-0.225051\pi\)
0.760303 + 0.649569i \(0.225051\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 24.0000 0.888280
\(731\) −6.00000 −0.221918
\(732\) 0 0
\(733\) 21.0000 0.775653 0.387826 − 0.921732i \(-0.373226\pi\)
0.387826 + 0.921732i \(0.373226\pi\)
\(734\) −12.0000 −0.442928
\(735\) 0 0
\(736\) 30.0000 1.10581
\(737\) −14.0000 −0.515697
\(738\) 0 0
\(739\) −9.00000 −0.331070 −0.165535 − 0.986204i \(-0.552935\pi\)
−0.165535 + 0.986204i \(0.552935\pi\)
\(740\) −12.0000 −0.441129
\(741\) 0 0
\(742\) 0 0
\(743\) 42.0000 1.54083 0.770415 − 0.637542i \(-0.220049\pi\)
0.770415 + 0.637542i \(0.220049\pi\)
\(744\) 0 0
\(745\) 96.0000 3.51717
\(746\) −26.0000 −0.951928
\(747\) 0 0
\(748\) 12.0000 0.438763
\(749\) 0 0
\(750\) 0 0
\(751\) −8.00000 −0.291924 −0.145962 − 0.989290i \(-0.546628\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) 6.00000 0.218797
\(753\) 0 0
\(754\) 2.00000 0.0728357
\(755\) 20.0000 0.727875
\(756\) 0 0
\(757\) 17.0000 0.617876 0.308938 − 0.951082i \(-0.400027\pi\)
0.308938 + 0.951082i \(0.400027\pi\)
\(758\) 9.00000 0.326895
\(759\) 0 0
\(760\) 48.0000 1.74114
\(761\) 48.0000 1.74000 0.869999 − 0.493053i \(-0.164119\pi\)
0.869999 + 0.493053i \(0.164119\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −4.00000 −0.144715
\(765\) 0 0
\(766\) −18.0000 −0.650366
\(767\) −6.00000 −0.216647
\(768\) 0 0
\(769\) −10.0000 −0.360609 −0.180305 − 0.983611i \(-0.557708\pi\)
−0.180305 + 0.983611i \(0.557708\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 25.0000 0.899770
\(773\) −16.0000 −0.575480 −0.287740 − 0.957709i \(-0.592904\pi\)
−0.287740 + 0.957709i \(0.592904\pi\)
\(774\) 0 0
\(775\) 33.0000 1.18539
\(776\) 27.0000 0.969244
\(777\) 0 0
\(778\) −12.0000 −0.430221
\(779\) 8.00000 0.286630
\(780\) 0 0
\(781\) 0 0
\(782\) 36.0000 1.28736
\(783\) 0 0
\(784\) 0 0
\(785\) 40.0000 1.42766
\(786\) 0 0
\(787\) −47.0000 −1.67537 −0.837685 − 0.546154i \(-0.816091\pi\)
−0.837685 + 0.546154i \(0.816091\pi\)
\(788\) 20.0000 0.712470
\(789\) 0 0
\(790\) −44.0000 −1.56545
\(791\) 0 0
\(792\) 0 0
\(793\) −5.00000 −0.177555
\(794\) 33.0000 1.17113
\(795\) 0 0
\(796\) 9.00000 0.318997
\(797\) −14.0000 −0.495905 −0.247953 − 0.968772i \(-0.579758\pi\)
−0.247953 + 0.968772i \(0.579758\pi\)
\(798\) 0 0
\(799\) −36.0000 −1.27359
\(800\) −55.0000 −1.94454
\(801\) 0 0
\(802\) 30.0000 1.05934
\(803\) 12.0000 0.423471
\(804\) 0 0
\(805\) 0 0
\(806\) −3.00000 −0.105670
\(807\) 0 0
\(808\) 24.0000 0.844317
\(809\) 30.0000 1.05474 0.527372 − 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) 32.0000 1.12367 0.561836 − 0.827249i \(-0.310095\pi\)
0.561836 + 0.827249i \(0.310095\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 6.00000 0.210300
\(815\) 76.0000 2.66216
\(816\) 0 0
\(817\) −4.00000 −0.139942
\(818\) −5.00000 −0.174821
\(819\) 0 0
\(820\) −8.00000 −0.279372
\(821\) −22.0000 −0.767805 −0.383903 − 0.923374i \(-0.625420\pi\)
−0.383903 + 0.923374i \(0.625420\pi\)
\(822\) 0 0
\(823\) 45.0000 1.56860 0.784301 − 0.620381i \(-0.213022\pi\)
0.784301 + 0.620381i \(0.213022\pi\)
\(824\) 51.0000 1.77667
\(825\) 0 0
\(826\) 0 0
\(827\) 12.0000 0.417281 0.208640 − 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 0 0
\(829\) −10.0000 −0.347314 −0.173657 − 0.984806i \(-0.555558\pi\)
−0.173657 + 0.984806i \(0.555558\pi\)
\(830\) 24.0000 0.833052
\(831\) 0 0
\(832\) 7.00000 0.242681
\(833\) 0 0
\(834\) 0 0
\(835\) −56.0000 −1.93796
\(836\) 8.00000 0.276686
\(837\) 0 0
\(838\) −12.0000 −0.414533
\(839\) −26.0000 −0.897620 −0.448810 − 0.893627i \(-0.648152\pi\)
−0.448810 + 0.893627i \(0.648152\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 10.0000 0.344623
\(843\) 0 0
\(844\) 5.00000 0.172107
\(845\) −48.0000 −1.65125
\(846\) 0 0
\(847\) 0 0
\(848\) −6.00000 −0.206041
\(849\) 0 0
\(850\) −66.0000 −2.26378
\(851\) −18.0000 −0.617032
\(852\) 0 0
\(853\) 26.0000 0.890223 0.445112 − 0.895475i \(-0.353164\pi\)
0.445112 + 0.895475i \(0.353164\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −6.00000 −0.205076
\(857\) 16.0000 0.546550 0.273275 − 0.961936i \(-0.411893\pi\)
0.273275 + 0.961936i \(0.411893\pi\)
\(858\) 0 0
\(859\) −25.0000 −0.852989 −0.426494 − 0.904490i \(-0.640252\pi\)
−0.426494 + 0.904490i \(0.640252\pi\)
\(860\) 4.00000 0.136399
\(861\) 0 0
\(862\) 18.0000 0.613082
\(863\) −36.0000 −1.22545 −0.612727 − 0.790295i \(-0.709928\pi\)
−0.612727 + 0.790295i \(0.709928\pi\)
\(864\) 0 0
\(865\) −64.0000 −2.17607
\(866\) 17.0000 0.577684
\(867\) 0 0
\(868\) 0 0
\(869\) −22.0000 −0.746299
\(870\) 0 0
\(871\) 7.00000 0.237186
\(872\) 27.0000 0.914335
\(873\) 0 0
\(874\) 24.0000 0.811812
\(875\) 0 0
\(876\) 0 0
\(877\) 7.00000 0.236373 0.118187 − 0.992991i \(-0.462292\pi\)
0.118187 + 0.992991i \(0.462292\pi\)
\(878\) −24.0000 −0.809961
\(879\) 0 0
\(880\) 8.00000 0.269680
\(881\) −6.00000 −0.202145 −0.101073 − 0.994879i \(-0.532227\pi\)
−0.101073 + 0.994879i \(0.532227\pi\)
\(882\) 0 0
\(883\) 8.00000 0.269221 0.134611 − 0.990899i \(-0.457022\pi\)
0.134611 + 0.990899i \(0.457022\pi\)
\(884\) −6.00000 −0.201802
\(885\) 0 0
\(886\) −24.0000 −0.806296
\(887\) −22.0000 −0.738688 −0.369344 − 0.929293i \(-0.620418\pi\)
−0.369344 + 0.929293i \(0.620418\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −16.0000 −0.536321
\(891\) 0 0
\(892\) −16.0000 −0.535720
\(893\) −24.0000 −0.803129
\(894\) 0 0
\(895\) −16.0000 −0.534821
\(896\) 0 0
\(897\) 0 0
\(898\) −36.0000 −1.20134
\(899\) −6.00000 −0.200111
\(900\) 0 0
\(901\) 36.0000 1.19933
\(902\) 4.00000 0.133185
\(903\) 0 0
\(904\) 48.0000 1.59646
\(905\) −8.00000 −0.265929
\(906\) 0 0
\(907\) −13.0000 −0.431658 −0.215829 − 0.976431i \(-0.569245\pi\)
−0.215829 + 0.976431i \(0.569245\pi\)
\(908\) 18.0000 0.597351
\(909\) 0 0
\(910\) 0 0
\(911\) 18.0000 0.596367 0.298183 − 0.954509i \(-0.403619\pi\)
0.298183 + 0.954509i \(0.403619\pi\)
\(912\) 0 0
\(913\) 12.0000 0.397142
\(914\) −13.0000 −0.430002
\(915\) 0 0
\(916\) 7.00000 0.231287
\(917\) 0 0
\(918\) 0 0
\(919\) 29.0000 0.956622 0.478311 − 0.878191i \(-0.341249\pi\)
0.478311 + 0.878191i \(0.341249\pi\)
\(920\) −72.0000 −2.37377
\(921\) 0 0
\(922\) −2.00000 −0.0658665
\(923\) 0 0
\(924\) 0 0
\(925\) 33.0000 1.08503
\(926\) 32.0000 1.05159
\(927\) 0 0
\(928\) 10.0000 0.328266
\(929\) −4.00000 −0.131236 −0.0656179 − 0.997845i \(-0.520902\pi\)
−0.0656179 + 0.997845i \(0.520902\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −24.0000 −0.786146
\(933\) 0 0
\(934\) 0 0
\(935\) −48.0000 −1.56977
\(936\) 0 0
\(937\) −21.0000 −0.686040 −0.343020 − 0.939328i \(-0.611450\pi\)
−0.343020 + 0.939328i \(0.611450\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 24.0000 0.782794
\(941\) −46.0000 −1.49956 −0.749779 − 0.661689i \(-0.769840\pi\)
−0.749779 + 0.661689i \(0.769840\pi\)
\(942\) 0 0
\(943\) −12.0000 −0.390774
\(944\) 6.00000 0.195283
\(945\) 0 0
\(946\) −2.00000 −0.0650256
\(947\) 26.0000 0.844886 0.422443 − 0.906389i \(-0.361173\pi\)
0.422443 + 0.906389i \(0.361173\pi\)
\(948\) 0 0
\(949\) −6.00000 −0.194768
\(950\) −44.0000 −1.42755
\(951\) 0 0
\(952\) 0 0
\(953\) 14.0000 0.453504 0.226752 − 0.973952i \(-0.427189\pi\)
0.226752 + 0.973952i \(0.427189\pi\)
\(954\) 0 0
\(955\) 16.0000 0.517748
\(956\) −12.0000 −0.388108
\(957\) 0 0
\(958\) 16.0000 0.516937
\(959\) 0 0
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) −3.00000 −0.0967239
\(963\) 0 0
\(964\) −17.0000 −0.547533
\(965\) −100.000 −3.21911
\(966\) 0 0
\(967\) −17.0000 −0.546683 −0.273342 − 0.961917i \(-0.588129\pi\)
−0.273342 + 0.961917i \(0.588129\pi\)
\(968\) −21.0000 −0.674966
\(969\) 0 0
\(970\) −36.0000 −1.15589
\(971\) 12.0000 0.385098 0.192549 − 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −16.0000 −0.512673
\(975\) 0 0
\(976\) 5.00000 0.160046
\(977\) −6.00000 −0.191957 −0.0959785 − 0.995383i \(-0.530598\pi\)
−0.0959785 + 0.995383i \(0.530598\pi\)
\(978\) 0 0
\(979\) −8.00000 −0.255681
\(980\) 0 0
\(981\) 0 0
\(982\) 34.0000 1.08498
\(983\) 36.0000 1.14822 0.574111 − 0.818778i \(-0.305348\pi\)
0.574111 + 0.818778i \(0.305348\pi\)
\(984\) 0 0
\(985\) −80.0000 −2.54901
\(986\) 12.0000 0.382158
\(987\) 0 0
\(988\) −4.00000 −0.127257
\(989\) 6.00000 0.190789
\(990\) 0 0
\(991\) −55.0000 −1.74713 −0.873566 − 0.486705i \(-0.838199\pi\)
−0.873566 + 0.486705i \(0.838199\pi\)
\(992\) −15.0000 −0.476250
\(993\) 0 0
\(994\) 0 0
\(995\) −36.0000 −1.14128
\(996\) 0 0
\(997\) −29.0000 −0.918439 −0.459220 − 0.888323i \(-0.651871\pi\)
−0.459220 + 0.888323i \(0.651871\pi\)
\(998\) 17.0000 0.538126
\(999\) 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 1323.2.a.g.1.1 1
3.2 odd 2 1323.2.a.m.1.1 1
7.2 even 3 189.2.e.c.109.1 yes 2
7.4 even 3 189.2.e.c.163.1 yes 2
7.6 odd 2 1323.2.a.d.1.1 1
21.2 odd 6 189.2.e.a.109.1 ✓ 2
21.11 odd 6 189.2.e.a.163.1 yes 2
21.20 even 2 1323.2.a.p.1.1 1
63.2 odd 6 567.2.g.b.109.1 2
63.4 even 3 567.2.g.e.541.1 2
63.11 odd 6 567.2.h.e.352.1 2
63.16 even 3 567.2.g.e.109.1 2
63.23 odd 6 567.2.h.e.298.1 2
63.25 even 3 567.2.h.b.352.1 2
63.32 odd 6 567.2.g.b.541.1 2
63.58 even 3 567.2.h.b.298.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.e.a.109.1 ✓ 2 21.2 odd 6
189.2.e.a.163.1 yes 2 21.11 odd 6
189.2.e.c.109.1 yes 2 7.2 even 3
189.2.e.c.163.1 yes 2 7.4 even 3
567.2.g.b.109.1 2 63.2 odd 6
567.2.g.b.541.1 2 63.32 odd 6
567.2.g.e.109.1 2 63.16 even 3
567.2.g.e.541.1 2 63.4 even 3
567.2.h.b.298.1 2 63.58 even 3
567.2.h.b.352.1 2 63.25 even 3
567.2.h.e.298.1 2 63.23 odd 6
567.2.h.e.352.1 2 63.11 odd 6
1323.2.a.d.1.1 1 7.6 odd 2
1323.2.a.g.1.1 1 1.1 even 1 trivial
1323.2.a.m.1.1 1 3.2 odd 2
1323.2.a.p.1.1 1 21.20 even 2