Properties

Label 1323.2.a.f.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: yes
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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{4} +3.00000 q^{5} +3.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{4} +3.00000 q^{5} +3.00000 q^{8} -3.00000 q^{10} +5.00000 q^{11} +6.00000 q^{13} -1.00000 q^{16} -6.00000 q^{17} +3.00000 q^{19} -3.00000 q^{20} -5.00000 q^{22} +1.00000 q^{23} +4.00000 q^{25} -6.00000 q^{26} -2.00000 q^{29} -3.00000 q^{31} -5.00000 q^{32} +6.00000 q^{34} +3.00000 q^{37} -3.00000 q^{38} +9.00000 q^{40} -9.00000 q^{41} +6.00000 q^{43} -5.00000 q^{44} -1.00000 q^{46} +6.00000 q^{47} -4.00000 q^{50} -6.00000 q^{52} -8.00000 q^{53} +15.0000 q^{55} +2.00000 q^{58} +6.00000 q^{59} +12.0000 q^{61} +3.00000 q^{62} +7.00000 q^{64} +18.0000 q^{65} -14.0000 q^{67} +6.00000 q^{68} -7.00000 q^{71} +6.00000 q^{73} -3.00000 q^{74} -3.00000 q^{76} -10.0000 q^{79} -3.00000 q^{80} +9.00000 q^{82} +6.00000 q^{83} -18.0000 q^{85} -6.00000 q^{86} +15.0000 q^{88} +3.00000 q^{89} -1.00000 q^{92} -6.00000 q^{94} +9.00000 q^{95} +12.0000 q^{97} +O(q^{100})\)

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 −0.353553 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 3.00000 1.34164 0.670820 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 3.00000 1.06066
\(9\) 0 0
\(10\) −3.00000 −0.948683
\(11\) 5.00000 1.50756 0.753778 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) 0 0
\(13\) 6.00000 1.66410 0.832050 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) 3.00000 0.688247 0.344124 0.938924i \(-0.388176\pi\)
0.344124 + 0.938924i \(0.388176\pi\)
\(20\) −3.00000 −0.670820
\(21\) 0 0
\(22\) −5.00000 −1.06600
\(23\) 1.00000 0.208514 0.104257 0.994550i \(-0.466753\pi\)
0.104257 + 0.994550i \(0.466753\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) −6.00000 −1.17670
\(27\) 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 \(-0.586828\pi\)
−0.269408 + 0.963026i \(0.586828\pi\)
\(32\) −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 \(0.420687\pi\)
\(38\) −3.00000 −0.486664
\(39\) 0 0
\(40\) 9.00000 1.42302
\(41\) −9.00000 −1.40556 −0.702782 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) 0 0
\(43\) 6.00000 0.914991 0.457496 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) −5.00000 −0.753778
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −4.00000 −0.565685
\(51\) 0 0
\(52\) −6.00000 −0.832050
\(53\) −8.00000 −1.09888 −0.549442 0.835532i \(-0.685160\pi\)
−0.549442 + 0.835532i \(0.685160\pi\)
\(54\) 0 0
\(55\) 15.0000 2.02260
\(56\) 0 0
\(57\) 0 0
\(58\) 2.00000 0.262613
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) 12.0000 1.53644 0.768221 0.640184i \(-0.221142\pi\)
0.768221 + 0.640184i \(0.221142\pi\)
\(62\) 3.00000 0.381000
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) 18.0000 2.23263
\(66\) 0 0
\(67\) −14.0000 −1.71037 −0.855186 0.518321i \(-0.826557\pi\)
−0.855186 + 0.518321i \(0.826557\pi\)
\(68\) 6.00000 0.727607
\(69\) 0 0
\(70\) 0 0
\(71\) −7.00000 −0.830747 −0.415374 0.909651i \(-0.636349\pi\)
−0.415374 + 0.909651i \(0.636349\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) −3.00000 −0.348743
\(75\) 0 0
\(76\) −3.00000 −0.344124
\(77\) 0 0
\(78\) 0 0
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) −3.00000 −0.335410
\(81\) 0 0
\(82\) 9.00000 0.993884
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) −18.0000 −1.95237
\(86\) −6.00000 −0.646997
\(87\) 0 0
\(88\) 15.0000 1.59901
\(89\) 3.00000 0.317999 0.159000 0.987279i \(-0.449173\pi\)
0.159000 + 0.987279i \(0.449173\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −1.00000 −0.104257
\(93\) 0 0
\(94\) −6.00000 −0.618853
\(95\) 9.00000 0.923381
\(96\) 0 0
\(97\) 12.0000 1.21842 0.609208 0.793011i \(-0.291488\pi\)
0.609208 + 0.793011i \(0.291488\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −4.00000 −0.400000
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) −3.00000 −0.295599 −0.147799 0.989017i \(-0.547219\pi\)
−0.147799 + 0.989017i \(0.547219\pi\)
\(104\) 18.0000 1.76505
\(105\) 0 0
\(106\) 8.00000 0.777029
\(107\) −16.0000 −1.54678 −0.773389 0.633932i \(-0.781440\pi\)
−0.773389 + 0.633932i \(0.781440\pi\)
\(108\) 0 0
\(109\) 9.00000 0.862044 0.431022 0.902342i \(-0.358153\pi\)
0.431022 + 0.902342i \(0.358153\pi\)
\(110\) −15.0000 −1.43019
\(111\) 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
\(115\) 3.00000 0.279751
\(116\) 2.00000 0.185695
\(117\) 0 0
\(118\) −6.00000 −0.552345
\(119\) 0 0
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) −12.0000 −1.08643
\(123\) 0 0
\(124\) 3.00000 0.269408
\(125\) −3.00000 −0.268328
\(126\) 0 0
\(127\) −2.00000 −0.177471 −0.0887357 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) 3.00000 0.265165
\(129\) 0 0
\(130\) −18.0000 −1.57870
\(131\) −18.0000 −1.57267 −0.786334 0.617802i \(-0.788023\pi\)
−0.786334 + 0.617802i \(0.788023\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 14.0000 1.20942
\(135\) 0 0
\(136\) −18.0000 −1.54349
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) 0 0
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 7.00000 0.587427
\(143\) 30.0000 2.50873
\(144\) 0 0
\(145\) −6.00000 −0.498273
\(146\) −6.00000 −0.496564
\(147\) 0 0
\(148\) −3.00000 −0.246598
\(149\) −4.00000 −0.327693 −0.163846 0.986486i \(-0.552390\pi\)
−0.163846 + 0.986486i \(0.552390\pi\)
\(150\) 0 0
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) 9.00000 0.729996
\(153\) 0 0
\(154\) 0 0
\(155\) −9.00000 −0.722897
\(156\) 0 0
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) 10.0000 0.795557
\(159\) 0 0
\(160\) −15.0000 −1.18585
\(161\) 0 0
\(162\) 0 0
\(163\) −2.00000 −0.156652 −0.0783260 0.996928i \(-0.524958\pi\)
−0.0783260 + 0.996928i \(0.524958\pi\)
\(164\) 9.00000 0.702782
\(165\) 0 0
\(166\) −6.00000 −0.465690
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 18.0000 1.38054
\(171\) 0 0
\(172\) −6.00000 −0.457496
\(173\) 9.00000 0.684257 0.342129 0.939653i \(-0.388852\pi\)
0.342129 + 0.939653i \(0.388852\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −5.00000 −0.376889
\(177\) 0 0
\(178\) −3.00000 −0.224860
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) −12.0000 −0.891953 −0.445976 0.895045i \(-0.647144\pi\)
−0.445976 + 0.895045i \(0.647144\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 3.00000 0.221163
\(185\) 9.00000 0.661693
\(186\) 0 0
\(187\) −30.0000 −2.19382
\(188\) −6.00000 −0.437595
\(189\) 0 0
\(190\) −9.00000 −0.652929
\(191\) −17.0000 −1.23008 −0.615038 0.788497i \(-0.710860\pi\)
−0.615038 + 0.788497i \(0.710860\pi\)
\(192\) 0 0
\(193\) −18.0000 −1.29567 −0.647834 0.761781i \(-0.724325\pi\)
−0.647834 + 0.761781i \(0.724325\pi\)
\(194\) −12.0000 −0.861550
\(195\) 0 0
\(196\) 0 0
\(197\) 8.00000 0.569976 0.284988 0.958531i \(-0.408010\pi\)
0.284988 + 0.958531i \(0.408010\pi\)
\(198\) 0 0
\(199\) 9.00000 0.637993 0.318997 0.947756i \(-0.396654\pi\)
0.318997 + 0.947756i \(0.396654\pi\)
\(200\) 12.0000 0.848528
\(201\) 0 0
\(202\) −6.00000 −0.422159
\(203\) 0 0
\(204\) 0 0
\(205\) −27.0000 −1.88576
\(206\) 3.00000 0.209020
\(207\) 0 0
\(208\) −6.00000 −0.416025
\(209\) 15.0000 1.03757
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 8.00000 0.549442
\(213\) 0 0
\(214\) 16.0000 1.09374
\(215\) 18.0000 1.22759
\(216\) 0 0
\(217\) 0 0
\(218\) −9.00000 −0.609557
\(219\) 0 0
\(220\) −15.0000 −1.01130
\(221\) −36.0000 −2.42162
\(222\) 0 0
\(223\) −9.00000 −0.602685 −0.301342 0.953516i \(-0.597435\pi\)
−0.301342 + 0.953516i \(0.597435\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −16.0000 −1.06430
\(227\) −24.0000 −1.59294 −0.796468 0.604681i \(-0.793301\pi\)
−0.796468 + 0.604681i \(0.793301\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) −3.00000 −0.197814
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) −4.00000 −0.262049 −0.131024 0.991379i \(-0.541827\pi\)
−0.131024 + 0.991379i \(0.541827\pi\)
\(234\) 0 0
\(235\) 18.0000 1.17419
\(236\) −6.00000 −0.390567
\(237\) 0 0
\(238\) 0 0
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 0 0
\(241\) 18.0000 1.15948 0.579741 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) −14.0000 −0.899954
\(243\) 0 0
\(244\) −12.0000 −0.768221
\(245\) 0 0
\(246\) 0 0
\(247\) 18.0000 1.14531
\(248\) −9.00000 −0.571501
\(249\) 0 0
\(250\) 3.00000 0.189737
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) 5.00000 0.314347
\(254\) 2.00000 0.125491
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) 3.00000 0.187135 0.0935674 0.995613i \(-0.470173\pi\)
0.0935674 + 0.995613i \(0.470173\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −18.0000 −1.11631
\(261\) 0 0
\(262\) 18.0000 1.11204
\(263\) −23.0000 −1.41824 −0.709120 0.705087i \(-0.750908\pi\)
−0.709120 + 0.705087i \(0.750908\pi\)
\(264\) 0 0
\(265\) −24.0000 −1.47431
\(266\) 0 0
\(267\) 0 0
\(268\) 14.0000 0.855186
\(269\) −21.0000 −1.28039 −0.640196 0.768211i \(-0.721147\pi\)
−0.640196 + 0.768211i \(0.721147\pi\)
\(270\) 0 0
\(271\) −24.0000 −1.45790 −0.728948 0.684569i \(-0.759990\pi\)
−0.728948 + 0.684569i \(0.759990\pi\)
\(272\) 6.00000 0.363803
\(273\) 0 0
\(274\) −10.0000 −0.604122
\(275\) 20.0000 1.20605
\(276\) 0 0
\(277\) −1.00000 −0.0600842 −0.0300421 0.999549i \(-0.509564\pi\)
−0.0300421 + 0.999549i \(0.509564\pi\)
\(278\) −12.0000 −0.719712
\(279\) 0 0
\(280\) 0 0
\(281\) 26.0000 1.55103 0.775515 0.631329i \(-0.217490\pi\)
0.775515 + 0.631329i \(0.217490\pi\)
\(282\) 0 0
\(283\) 12.0000 0.713326 0.356663 0.934233i \(-0.383914\pi\)
0.356663 + 0.934233i \(0.383914\pi\)
\(284\) 7.00000 0.415374
\(285\) 0 0
\(286\) −30.0000 −1.77394
\(287\) 0 0
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 6.00000 0.352332
\(291\) 0 0
\(292\) −6.00000 −0.351123
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 0 0
\(295\) 18.0000 1.04800
\(296\) 9.00000 0.523114
\(297\) 0 0
\(298\) 4.00000 0.231714
\(299\) 6.00000 0.346989
\(300\) 0 0
\(301\) 0 0
\(302\) −12.0000 −0.690522
\(303\) 0 0
\(304\) −3.00000 −0.172062
\(305\) 36.0000 2.06135
\(306\) 0 0
\(307\) 3.00000 0.171219 0.0856095 0.996329i \(-0.472716\pi\)
0.0856095 + 0.996329i \(0.472716\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 9.00000 0.511166
\(311\) −18.0000 −1.02069 −0.510343 0.859971i \(-0.670482\pi\)
−0.510343 + 0.859971i \(0.670482\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) −18.0000 −1.01580
\(315\) 0 0
\(316\) 10.0000 0.562544
\(317\) −2.00000 −0.112331 −0.0561656 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) 0 0
\(319\) −10.0000 −0.559893
\(320\) 21.0000 1.17394
\(321\) 0 0
\(322\) 0 0
\(323\) −18.0000 −1.00155
\(324\) 0 0
\(325\) 24.0000 1.33128
\(326\) 2.00000 0.110770
\(327\) 0 0
\(328\) −27.0000 −1.49083
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) −6.00000 −0.329293
\(333\) 0 0
\(334\) 0 0
\(335\) −42.0000 −2.29471
\(336\) 0 0
\(337\) 3.00000 0.163420 0.0817102 0.996656i \(-0.473962\pi\)
0.0817102 + 0.996656i \(0.473962\pi\)
\(338\) −23.0000 −1.25104
\(339\) 0 0
\(340\) 18.0000 0.976187
\(341\) −15.0000 −0.812296
\(342\) 0 0
\(343\) 0 0
\(344\) 18.0000 0.970495
\(345\) 0 0
\(346\) −9.00000 −0.483843
\(347\) 5.00000 0.268414 0.134207 0.990953i \(-0.457151\pi\)
0.134207 + 0.990953i \(0.457151\pi\)
\(348\) 0 0
\(349\) 12.0000 0.642345 0.321173 0.947021i \(-0.395923\pi\)
0.321173 + 0.947021i \(0.395923\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −25.0000 −1.33250
\(353\) 21.0000 1.11772 0.558859 0.829263i \(-0.311239\pi\)
0.558859 + 0.829263i \(0.311239\pi\)
\(354\) 0 0
\(355\) −21.0000 −1.11456
\(356\) −3.00000 −0.159000
\(357\) 0 0
\(358\) 4.00000 0.211407
\(359\) −8.00000 −0.422224 −0.211112 0.977462i \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) 0 0
\(361\) −10.0000 −0.526316
\(362\) 12.0000 0.630706
\(363\) 0 0
\(364\) 0 0
\(365\) 18.0000 0.942163
\(366\) 0 0
\(367\) 9.00000 0.469796 0.234898 0.972020i \(-0.424524\pi\)
0.234898 + 0.972020i \(0.424524\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 0 0
\(370\) −9.00000 −0.467888
\(371\) 0 0
\(372\) 0 0
\(373\) −23.0000 −1.19089 −0.595447 0.803394i \(-0.703025\pi\)
−0.595447 + 0.803394i \(0.703025\pi\)
\(374\) 30.0000 1.55126
\(375\) 0 0
\(376\) 18.0000 0.928279
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) −30.0000 −1.54100 −0.770498 0.637442i \(-0.779993\pi\)
−0.770498 + 0.637442i \(0.779993\pi\)
\(380\) −9.00000 −0.461690
\(381\) 0 0
\(382\) 17.0000 0.869796
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 18.0000 0.916176
\(387\) 0 0
\(388\) −12.0000 −0.609208
\(389\) −16.0000 −0.811232 −0.405616 0.914044i \(-0.632943\pi\)
−0.405616 + 0.914044i \(0.632943\pi\)
\(390\) 0 0
\(391\) −6.00000 −0.303433
\(392\) 0 0
\(393\) 0 0
\(394\) −8.00000 −0.403034
\(395\) −30.0000 −1.50946
\(396\) 0 0
\(397\) 12.0000 0.602263 0.301131 0.953583i \(-0.402636\pi\)
0.301131 + 0.953583i \(0.402636\pi\)
\(398\) −9.00000 −0.451129
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) −2.00000 −0.0998752 −0.0499376 0.998752i \(-0.515902\pi\)
−0.0499376 + 0.998752i \(0.515902\pi\)
\(402\) 0 0
\(403\) −18.0000 −0.896644
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) 0 0
\(407\) 15.0000 0.743522
\(408\) 0 0
\(409\) 30.0000 1.48340 0.741702 0.670729i \(-0.234019\pi\)
0.741702 + 0.670729i \(0.234019\pi\)
\(410\) 27.0000 1.33343
\(411\) 0 0
\(412\) 3.00000 0.147799
\(413\) 0 0
\(414\) 0 0
\(415\) 18.0000 0.883585
\(416\) −30.0000 −1.47087
\(417\) 0 0
\(418\) −15.0000 −0.733674
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) −17.0000 −0.828529 −0.414265 0.910156i \(-0.635961\pi\)
−0.414265 + 0.910156i \(0.635961\pi\)
\(422\) 12.0000 0.584151
\(423\) 0 0
\(424\) −24.0000 −1.16554
\(425\) −24.0000 −1.16417
\(426\) 0 0
\(427\) 0 0
\(428\) 16.0000 0.773389
\(429\) 0 0
\(430\) −18.0000 −0.868037
\(431\) 31.0000 1.49322 0.746609 0.665263i \(-0.231681\pi\)
0.746609 + 0.665263i \(0.231681\pi\)
\(432\) 0 0
\(433\) −18.0000 −0.865025 −0.432512 0.901628i \(-0.642373\pi\)
−0.432512 + 0.901628i \(0.642373\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −9.00000 −0.431022
\(437\) 3.00000 0.143509
\(438\) 0 0
\(439\) −24.0000 −1.14546 −0.572729 0.819745i \(-0.694115\pi\)
−0.572729 + 0.819745i \(0.694115\pi\)
\(440\) 45.0000 2.14529
\(441\) 0 0
\(442\) 36.0000 1.71235
\(443\) −25.0000 −1.18779 −0.593893 0.804544i \(-0.702410\pi\)
−0.593893 + 0.804544i \(0.702410\pi\)
\(444\) 0 0
\(445\) 9.00000 0.426641
\(446\) 9.00000 0.426162
\(447\) 0 0
\(448\) 0 0
\(449\) −20.0000 −0.943858 −0.471929 0.881636i \(-0.656442\pi\)
−0.471929 + 0.881636i \(0.656442\pi\)
\(450\) 0 0
\(451\) −45.0000 −2.11897
\(452\) −16.0000 −0.752577
\(453\) 0 0
\(454\) 24.0000 1.12638
\(455\) 0 0
\(456\) 0 0
\(457\) −15.0000 −0.701670 −0.350835 0.936437i \(-0.614102\pi\)
−0.350835 + 0.936437i \(0.614102\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) −3.00000 −0.139876
\(461\) 33.0000 1.53696 0.768482 0.639872i \(-0.221013\pi\)
0.768482 + 0.639872i \(0.221013\pi\)
\(462\) 0 0
\(463\) −4.00000 −0.185896 −0.0929479 0.995671i \(-0.529629\pi\)
−0.0929479 + 0.995671i \(0.529629\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) 4.00000 0.185296
\(467\) −42.0000 −1.94353 −0.971764 0.235954i \(-0.924178\pi\)
−0.971764 + 0.235954i \(0.924178\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −18.0000 −0.830278
\(471\) 0 0
\(472\) 18.0000 0.828517
\(473\) 30.0000 1.37940
\(474\) 0 0
\(475\) 12.0000 0.550598
\(476\) 0 0
\(477\) 0 0
\(478\) 16.0000 0.731823
\(479\) −12.0000 −0.548294 −0.274147 0.961688i \(-0.588395\pi\)
−0.274147 + 0.961688i \(0.588395\pi\)
\(480\) 0 0
\(481\) 18.0000 0.820729
\(482\) −18.0000 −0.819878
\(483\) 0 0
\(484\) −14.0000 −0.636364
\(485\) 36.0000 1.63468
\(486\) 0 0
\(487\) 30.0000 1.35943 0.679715 0.733476i \(-0.262104\pi\)
0.679715 + 0.733476i \(0.262104\pi\)
\(488\) 36.0000 1.62964
\(489\) 0 0
\(490\) 0 0
\(491\) 1.00000 0.0451294 0.0225647 0.999745i \(-0.492817\pi\)
0.0225647 + 0.999745i \(0.492817\pi\)
\(492\) 0 0
\(493\) 12.0000 0.540453
\(494\) −18.0000 −0.809858
\(495\) 0 0
\(496\) 3.00000 0.134704
\(497\) 0 0
\(498\) 0 0
\(499\) 18.0000 0.805791 0.402895 0.915246i \(-0.368004\pi\)
0.402895 + 0.915246i \(0.368004\pi\)
\(500\) 3.00000 0.134164
\(501\) 0 0
\(502\) −12.0000 −0.535586
\(503\) 6.00000 0.267527 0.133763 0.991013i \(-0.457294\pi\)
0.133763 + 0.991013i \(0.457294\pi\)
\(504\) 0 0
\(505\) 18.0000 0.800989
\(506\) −5.00000 −0.222277
\(507\) 0 0
\(508\) 2.00000 0.0887357
\(509\) −6.00000 −0.265945 −0.132973 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 11.0000 0.486136
\(513\) 0 0
\(514\) −3.00000 −0.132324
\(515\) −9.00000 −0.396587
\(516\) 0 0
\(517\) 30.0000 1.31940
\(518\) 0 0
\(519\) 0 0
\(520\) 54.0000 2.36806
\(521\) −3.00000 −0.131432 −0.0657162 0.997838i \(-0.520933\pi\)
−0.0657162 + 0.997838i \(0.520933\pi\)
\(522\) 0 0
\(523\) 3.00000 0.131181 0.0655904 0.997847i \(-0.479107\pi\)
0.0655904 + 0.997847i \(0.479107\pi\)
\(524\) 18.0000 0.786334
\(525\) 0 0
\(526\) 23.0000 1.00285
\(527\) 18.0000 0.784092
\(528\) 0 0
\(529\) −22.0000 −0.956522
\(530\) 24.0000 1.04249
\(531\) 0 0
\(532\) 0 0
\(533\) −54.0000 −2.33900
\(534\) 0 0
\(535\) −48.0000 −2.07522
\(536\) −42.0000 −1.81412
\(537\) 0 0
\(538\) 21.0000 0.905374
\(539\) 0 0
\(540\) 0 0
\(541\) 11.0000 0.472927 0.236463 0.971640i \(-0.424012\pi\)
0.236463 + 0.971640i \(0.424012\pi\)
\(542\) 24.0000 1.03089
\(543\) 0 0
\(544\) 30.0000 1.28624
\(545\) 27.0000 1.15655
\(546\) 0 0
\(547\) 8.00000 0.342055 0.171028 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) −10.0000 −0.427179
\(549\) 0 0
\(550\) −20.0000 −0.852803
\(551\) −6.00000 −0.255609
\(552\) 0 0
\(553\) 0 0
\(554\) 1.00000 0.0424859
\(555\) 0 0
\(556\) −12.0000 −0.508913
\(557\) −28.0000 −1.18640 −0.593199 0.805056i \(-0.702135\pi\)
−0.593199 + 0.805056i \(0.702135\pi\)
\(558\) 0 0
\(559\) 36.0000 1.52264
\(560\) 0 0
\(561\) 0 0
\(562\) −26.0000 −1.09674
\(563\) 18.0000 0.758610 0.379305 0.925272i \(-0.376163\pi\)
0.379305 + 0.925272i \(0.376163\pi\)
\(564\) 0 0
\(565\) 48.0000 2.01938
\(566\) −12.0000 −0.504398
\(567\) 0 0
\(568\) −21.0000 −0.881140
\(569\) −4.00000 −0.167689 −0.0838444 0.996479i \(-0.526720\pi\)
−0.0838444 + 0.996479i \(0.526720\pi\)
\(570\) 0 0
\(571\) 40.0000 1.67395 0.836974 0.547243i \(-0.184323\pi\)
0.836974 + 0.547243i \(0.184323\pi\)
\(572\) −30.0000 −1.25436
\(573\) 0 0
\(574\) 0 0
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) 24.0000 0.999133 0.499567 0.866276i \(-0.333493\pi\)
0.499567 + 0.866276i \(0.333493\pi\)
\(578\) −19.0000 −0.790296
\(579\) 0 0
\(580\) 6.00000 0.249136
\(581\) 0 0
\(582\) 0 0
\(583\) −40.0000 −1.65663
\(584\) 18.0000 0.744845
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) −9.00000 −0.370839
\(590\) −18.0000 −0.741048
\(591\) 0 0
\(592\) −3.00000 −0.123299
\(593\) 21.0000 0.862367 0.431183 0.902264i \(-0.358096\pi\)
0.431183 + 0.902264i \(0.358096\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 4.00000 0.163846
\(597\) 0 0
\(598\) −6.00000 −0.245358
\(599\) −5.00000 −0.204294 −0.102147 0.994769i \(-0.532571\pi\)
−0.102147 + 0.994769i \(0.532571\pi\)
\(600\) 0 0
\(601\) −30.0000 −1.22373 −0.611863 0.790964i \(-0.709580\pi\)
−0.611863 + 0.790964i \(0.709580\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −12.0000 −0.488273
\(605\) 42.0000 1.70754
\(606\) 0 0
\(607\) −12.0000 −0.487065 −0.243532 0.969893i \(-0.578306\pi\)
−0.243532 + 0.969893i \(0.578306\pi\)
\(608\) −15.0000 −0.608330
\(609\) 0 0
\(610\) −36.0000 −1.45760
\(611\) 36.0000 1.45640
\(612\) 0 0
\(613\) 7.00000 0.282727 0.141364 0.989958i \(-0.454851\pi\)
0.141364 + 0.989958i \(0.454851\pi\)
\(614\) −3.00000 −0.121070
\(615\) 0 0
\(616\) 0 0
\(617\) −10.0000 −0.402585 −0.201292 0.979531i \(-0.564514\pi\)
−0.201292 + 0.979531i \(0.564514\pi\)
\(618\) 0 0
\(619\) −15.0000 −0.602901 −0.301450 0.953482i \(-0.597471\pi\)
−0.301450 + 0.953482i \(0.597471\pi\)
\(620\) 9.00000 0.361449
\(621\) 0 0
\(622\) 18.0000 0.721734
\(623\) 0 0
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 6.00000 0.239808
\(627\) 0 0
\(628\) −18.0000 −0.718278
\(629\) −18.0000 −0.717707
\(630\) 0 0
\(631\) −2.00000 −0.0796187 −0.0398094 0.999207i \(-0.512675\pi\)
−0.0398094 + 0.999207i \(0.512675\pi\)
\(632\) −30.0000 −1.19334
\(633\) 0 0
\(634\) 2.00000 0.0794301
\(635\) −6.00000 −0.238103
\(636\) 0 0
\(637\) 0 0
\(638\) 10.0000 0.395904
\(639\) 0 0
\(640\) 9.00000 0.355756
\(641\) 8.00000 0.315981 0.157991 0.987441i \(-0.449498\pi\)
0.157991 + 0.987441i \(0.449498\pi\)
\(642\) 0 0
\(643\) 39.0000 1.53801 0.769005 0.639243i \(-0.220752\pi\)
0.769005 + 0.639243i \(0.220752\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 18.0000 0.708201
\(647\) 42.0000 1.65119 0.825595 0.564263i \(-0.190840\pi\)
0.825595 + 0.564263i \(0.190840\pi\)
\(648\) 0 0
\(649\) 30.0000 1.17760
\(650\) −24.0000 −0.941357
\(651\) 0 0
\(652\) 2.00000 0.0783260
\(653\) 22.0000 0.860927 0.430463 0.902608i \(-0.358350\pi\)
0.430463 + 0.902608i \(0.358350\pi\)
\(654\) 0 0
\(655\) −54.0000 −2.10995
\(656\) 9.00000 0.351391
\(657\) 0 0
\(658\) 0 0
\(659\) 37.0000 1.44132 0.720658 0.693291i \(-0.243840\pi\)
0.720658 + 0.693291i \(0.243840\pi\)
\(660\) 0 0
\(661\) −42.0000 −1.63361 −0.816805 0.576913i \(-0.804257\pi\)
−0.816805 + 0.576913i \(0.804257\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 18.0000 0.698535
\(665\) 0 0
\(666\) 0 0
\(667\) −2.00000 −0.0774403
\(668\) 0 0
\(669\) 0 0
\(670\) 42.0000 1.62260
\(671\) 60.0000 2.31627
\(672\) 0 0
\(673\) −6.00000 −0.231283 −0.115642 0.993291i \(-0.536892\pi\)
−0.115642 + 0.993291i \(0.536892\pi\)
\(674\) −3.00000 −0.115556
\(675\) 0 0
\(676\) −23.0000 −0.884615
\(677\) −51.0000 −1.96009 −0.980045 0.198778i \(-0.936303\pi\)
−0.980045 + 0.198778i \(0.936303\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −54.0000 −2.07081
\(681\) 0 0
\(682\) 15.0000 0.574380
\(683\) 7.00000 0.267848 0.133924 0.990992i \(-0.457242\pi\)
0.133924 + 0.990992i \(0.457242\pi\)
\(684\) 0 0
\(685\) 30.0000 1.14624
\(686\) 0 0
\(687\) 0 0
\(688\) −6.00000 −0.228748
\(689\) −48.0000 −1.82865
\(690\) 0 0
\(691\) 36.0000 1.36950 0.684752 0.728776i \(-0.259910\pi\)
0.684752 + 0.728776i \(0.259910\pi\)
\(692\) −9.00000 −0.342129
\(693\) 0 0
\(694\) −5.00000 −0.189797
\(695\) 36.0000 1.36556
\(696\) 0 0
\(697\) 54.0000 2.04540
\(698\) −12.0000 −0.454207
\(699\) 0 0
\(700\) 0 0
\(701\) −38.0000 −1.43524 −0.717620 0.696435i \(-0.754769\pi\)
−0.717620 + 0.696435i \(0.754769\pi\)
\(702\) 0 0
\(703\) 9.00000 0.339441
\(704\) 35.0000 1.31911
\(705\) 0 0
\(706\) −21.0000 −0.790345
\(707\) 0 0
\(708\) 0 0
\(709\) 21.0000 0.788672 0.394336 0.918966i \(-0.370975\pi\)
0.394336 + 0.918966i \(0.370975\pi\)
\(710\) 21.0000 0.788116
\(711\) 0 0
\(712\) 9.00000 0.337289
\(713\) −3.00000 −0.112351
\(714\) 0 0
\(715\) 90.0000 3.36581
\(716\) 4.00000 0.149487
\(717\) 0 0
\(718\) 8.00000 0.298557
\(719\) −6.00000 −0.223762 −0.111881 0.993722i \(-0.535688\pi\)
−0.111881 + 0.993722i \(0.535688\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 10.0000 0.372161
\(723\) 0 0
\(724\) 12.0000 0.445976
\(725\) −8.00000 −0.297113
\(726\) 0 0
\(727\) 36.0000 1.33517 0.667583 0.744535i \(-0.267329\pi\)
0.667583 + 0.744535i \(0.267329\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −18.0000 −0.666210
\(731\) −36.0000 −1.33151
\(732\) 0 0
\(733\) 42.0000 1.55131 0.775653 0.631160i \(-0.217421\pi\)
0.775653 + 0.631160i \(0.217421\pi\)
\(734\) −9.00000 −0.332196
\(735\) 0 0
\(736\) −5.00000 −0.184302
\(737\) −70.0000 −2.57848
\(738\) 0 0
\(739\) −16.0000 −0.588570 −0.294285 0.955718i \(-0.595081\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) −9.00000 −0.330847
\(741\) 0 0
\(742\) 0 0
\(743\) 49.0000 1.79764 0.898818 0.438322i \(-0.144427\pi\)
0.898818 + 0.438322i \(0.144427\pi\)
\(744\) 0 0
\(745\) −12.0000 −0.439646
\(746\) 23.0000 0.842090
\(747\) 0 0
\(748\) 30.0000 1.09691
\(749\) 0 0
\(750\) 0 0
\(751\) 6.00000 0.218943 0.109472 0.993990i \(-0.465084\pi\)
0.109472 + 0.993990i \(0.465084\pi\)
\(752\) −6.00000 −0.218797
\(753\) 0 0
\(754\) 12.0000 0.437014
\(755\) 36.0000 1.31017
\(756\) 0 0
\(757\) −18.0000 −0.654221 −0.327111 0.944986i \(-0.606075\pi\)
−0.327111 + 0.944986i \(0.606075\pi\)
\(758\) 30.0000 1.08965
\(759\) 0 0
\(760\) 27.0000 0.979393
\(761\) −6.00000 −0.217500 −0.108750 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 17.0000 0.615038
\(765\) 0 0
\(766\) −24.0000 −0.867155
\(767\) 36.0000 1.29988
\(768\) 0 0
\(769\) −18.0000 −0.649097 −0.324548 0.945869i \(-0.605212\pi\)
−0.324548 + 0.945869i \(0.605212\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 18.0000 0.647834
\(773\) 9.00000 0.323708 0.161854 0.986815i \(-0.448253\pi\)
0.161854 + 0.986815i \(0.448253\pi\)
\(774\) 0 0
\(775\) −12.0000 −0.431053
\(776\) 36.0000 1.29232
\(777\) 0 0
\(778\) 16.0000 0.573628
\(779\) −27.0000 −0.967375
\(780\) 0 0
\(781\) −35.0000 −1.25240
\(782\) 6.00000 0.214560
\(783\) 0 0
\(784\) 0 0
\(785\) 54.0000 1.92734
\(786\) 0 0
\(787\) 12.0000 0.427754 0.213877 0.976861i \(-0.431391\pi\)
0.213877 + 0.976861i \(0.431391\pi\)
\(788\) −8.00000 −0.284988
\(789\) 0 0
\(790\) 30.0000 1.06735
\(791\) 0 0
\(792\) 0 0
\(793\) 72.0000 2.55679
\(794\) −12.0000 −0.425864
\(795\) 0 0
\(796\) −9.00000 −0.318997
\(797\) 21.0000 0.743858 0.371929 0.928261i \(-0.378696\pi\)
0.371929 + 0.928261i \(0.378696\pi\)
\(798\) 0 0
\(799\) −36.0000 −1.27359
\(800\) −20.0000 −0.707107
\(801\) 0 0
\(802\) 2.00000 0.0706225
\(803\) 30.0000 1.05868
\(804\) 0 0
\(805\) 0 0
\(806\) 18.0000 0.634023
\(807\) 0 0
\(808\) 18.0000 0.633238
\(809\) 2.00000 0.0703163 0.0351581 0.999382i \(-0.488807\pi\)
0.0351581 + 0.999382i \(0.488807\pi\)
\(810\) 0 0
\(811\) −39.0000 −1.36948 −0.684738 0.728790i \(-0.740083\pi\)
−0.684738 + 0.728790i \(0.740083\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −15.0000 −0.525750
\(815\) −6.00000 −0.210171
\(816\) 0 0
\(817\) 18.0000 0.629740
\(818\) −30.0000 −1.04893
\(819\) 0 0
\(820\) 27.0000 0.942881
\(821\) −22.0000 −0.767805 −0.383903 0.923374i \(-0.625420\pi\)
−0.383903 + 0.923374i \(0.625420\pi\)
\(822\) 0 0
\(823\) −4.00000 −0.139431 −0.0697156 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) −9.00000 −0.313530
\(825\) 0 0
\(826\) 0 0
\(827\) 19.0000 0.660695 0.330347 0.943859i \(-0.392834\pi\)
0.330347 + 0.943859i \(0.392834\pi\)
\(828\) 0 0
\(829\) −18.0000 −0.625166 −0.312583 0.949890i \(-0.601194\pi\)
−0.312583 + 0.949890i \(0.601194\pi\)
\(830\) −18.0000 −0.624789
\(831\) 0 0
\(832\) 42.0000 1.45609
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) −15.0000 −0.518786
\(837\) 0 0
\(838\) 12.0000 0.414533
\(839\) −30.0000 −1.03572 −0.517858 0.855467i \(-0.673270\pi\)
−0.517858 + 0.855467i \(0.673270\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 17.0000 0.585859
\(843\) 0 0
\(844\) 12.0000 0.413057
\(845\) 69.0000 2.37367
\(846\) 0 0
\(847\) 0 0
\(848\) 8.00000 0.274721
\(849\) 0 0
\(850\) 24.0000 0.823193
\(851\) 3.00000 0.102839
\(852\) 0 0
\(853\) −12.0000 −0.410872 −0.205436 0.978671i \(-0.565861\pi\)
−0.205436 + 0.978671i \(0.565861\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −48.0000 −1.64061
\(857\) −9.00000 −0.307434 −0.153717 0.988115i \(-0.549124\pi\)
−0.153717 + 0.988115i \(0.549124\pi\)
\(858\) 0 0
\(859\) −45.0000 −1.53538 −0.767690 0.640821i \(-0.778594\pi\)
−0.767690 + 0.640821i \(0.778594\pi\)
\(860\) −18.0000 −0.613795
\(861\) 0 0
\(862\) −31.0000 −1.05586
\(863\) −8.00000 −0.272323 −0.136162 0.990687i \(-0.543477\pi\)
−0.136162 + 0.990687i \(0.543477\pi\)
\(864\) 0 0
\(865\) 27.0000 0.918028
\(866\) 18.0000 0.611665
\(867\) 0 0
\(868\) 0 0
\(869\) −50.0000 −1.69613
\(870\) 0 0
\(871\) −84.0000 −2.84623
\(872\) 27.0000 0.914335
\(873\) 0 0
\(874\) −3.00000 −0.101477
\(875\) 0 0
\(876\) 0 0
\(877\) 14.0000 0.472746 0.236373 0.971662i \(-0.424041\pi\)
0.236373 + 0.971662i \(0.424041\pi\)
\(878\) 24.0000 0.809961
\(879\) 0 0
\(880\) −15.0000 −0.505650
\(881\) −15.0000 −0.505363 −0.252681 0.967550i \(-0.581312\pi\)
−0.252681 + 0.967550i \(0.581312\pi\)
\(882\) 0 0
\(883\) −48.0000 −1.61533 −0.807664 0.589643i \(-0.799269\pi\)
−0.807664 + 0.589643i \(0.799269\pi\)
\(884\) 36.0000 1.21081
\(885\) 0 0
\(886\) 25.0000 0.839891
\(887\) −6.00000 −0.201460 −0.100730 0.994914i \(-0.532118\pi\)
−0.100730 + 0.994914i \(0.532118\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −9.00000 −0.301681
\(891\) 0 0
\(892\) 9.00000 0.301342
\(893\) 18.0000 0.602347
\(894\) 0 0
\(895\) −12.0000 −0.401116
\(896\) 0 0
\(897\) 0 0
\(898\) 20.0000 0.667409
\(899\) 6.00000 0.200111
\(900\) 0 0
\(901\) 48.0000 1.59911
\(902\) 45.0000 1.49834
\(903\) 0 0
\(904\) 48.0000 1.59646
\(905\) −36.0000 −1.19668
\(906\) 0 0
\(907\) −48.0000 −1.59381 −0.796907 0.604102i \(-0.793532\pi\)
−0.796907 + 0.604102i \(0.793532\pi\)
\(908\) 24.0000 0.796468
\(909\) 0 0
\(910\) 0 0
\(911\) −52.0000 −1.72284 −0.861418 0.507896i \(-0.830423\pi\)
−0.861418 + 0.507896i \(0.830423\pi\)
\(912\) 0 0
\(913\) 30.0000 0.992855
\(914\) 15.0000 0.496156
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −48.0000 −1.58337 −0.791687 0.610927i \(-0.790797\pi\)
−0.791687 + 0.610927i \(0.790797\pi\)
\(920\) 9.00000 0.296721
\(921\) 0 0
\(922\) −33.0000 −1.08680
\(923\) −42.0000 −1.38245
\(924\) 0 0
\(925\) 12.0000 0.394558
\(926\) 4.00000 0.131448
\(927\) 0 0
\(928\) 10.0000 0.328266
\(929\) 18.0000 0.590561 0.295280 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 4.00000 0.131024
\(933\) 0 0
\(934\) 42.0000 1.37428
\(935\) −90.0000 −2.94331
\(936\) 0 0
\(937\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −18.0000 −0.587095
\(941\) −45.0000 −1.46696 −0.733479 0.679712i \(-0.762105\pi\)
−0.733479 + 0.679712i \(0.762105\pi\)
\(942\) 0 0
\(943\) −9.00000 −0.293080
\(944\) −6.00000 −0.195283
\(945\) 0 0
\(946\) −30.0000 −0.975384
\(947\) 47.0000 1.52729 0.763647 0.645634i \(-0.223407\pi\)
0.763647 + 0.645634i \(0.223407\pi\)
\(948\) 0 0
\(949\) 36.0000 1.16861
\(950\) −12.0000 −0.389331
\(951\) 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
\(955\) −51.0000 −1.65032
\(956\) 16.0000 0.517477
\(957\) 0 0
\(958\) 12.0000 0.387702
\(959\) 0 0
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) −18.0000 −0.580343
\(963\) 0 0
\(964\) −18.0000 −0.579741
\(965\) −54.0000 −1.73832
\(966\) 0 0
\(967\) 32.0000 1.02905 0.514525 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) 42.0000 1.34993
\(969\) 0 0
\(970\) −36.0000 −1.15589
\(971\) −54.0000 −1.73294 −0.866471 0.499227i \(-0.833617\pi\)
−0.866471 + 0.499227i \(0.833617\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −30.0000 −0.961262
\(975\) 0 0
\(976\) −12.0000 −0.384111
\(977\) 8.00000 0.255943 0.127971 0.991778i \(-0.459153\pi\)
0.127971 + 0.991778i \(0.459153\pi\)
\(978\) 0 0
\(979\) 15.0000 0.479402
\(980\) 0 0
\(981\) 0 0
\(982\) −1.00000 −0.0319113
\(983\) 6.00000 0.191370 0.0956851 0.995412i \(-0.469496\pi\)
0.0956851 + 0.995412i \(0.469496\pi\)
\(984\) 0 0
\(985\) 24.0000 0.764704
\(986\) −12.0000 −0.382158
\(987\) 0 0
\(988\) −18.0000 −0.572656
\(989\) 6.00000 0.190789
\(990\) 0 0
\(991\) −6.00000 −0.190596 −0.0952981 0.995449i \(-0.530380\pi\)
−0.0952981 + 0.995449i \(0.530380\pi\)
\(992\) 15.0000 0.476250
\(993\) 0 0
\(994\) 0 0
\(995\) 27.0000 0.855958
\(996\) 0 0
\(997\) 36.0000 1.14013 0.570066 0.821599i \(-0.306918\pi\)
0.570066 + 0.821599i \(0.306918\pi\)
\(998\) −18.0000 −0.569780
\(999\) 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.f.1.1 yes 1
3.2 odd 2 1323.2.a.n.1.1 yes 1
7.6 odd 2 1323.2.a.e.1.1 1
21.20 even 2 1323.2.a.o.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1323.2.a.e.1.1 1 7.6 odd 2
1323.2.a.f.1.1 yes 1 1.1 even 1 trivial
1323.2.a.n.1.1 yes 1 3.2 odd 2
1323.2.a.o.1.1 yes 1 21.20 even 2