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Properties

Label 1638.2.a.b.1.1
Level $1638$
Weight $2$
Character 1638.1
Self dual yes
Analytic conductor $13.079$
Analytic rank $1$
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] = [1638,2,Mod(1,1638)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1638, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1638.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1638.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: \(13.0794958511\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 1638.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +3.00000 q^{10} -3.00000 q^{11} +1.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +3.00000 q^{17} +5.00000 q^{19} -3.00000 q^{20} +3.00000 q^{22} +3.00000 q^{23} +4.00000 q^{25} -1.00000 q^{26} +1.00000 q^{28} +3.00000 q^{29} -10.0000 q^{31} -1.00000 q^{32} -3.00000 q^{34} -3.00000 q^{35} -7.00000 q^{37} -5.00000 q^{38} +3.00000 q^{40} -6.00000 q^{41} +5.00000 q^{43} -3.00000 q^{44} -3.00000 q^{46} +1.00000 q^{49} -4.00000 q^{50} +1.00000 q^{52} -6.00000 q^{53} +9.00000 q^{55} -1.00000 q^{56} -3.00000 q^{58} -6.00000 q^{59} -7.00000 q^{61} +10.0000 q^{62} +1.00000 q^{64} -3.00000 q^{65} -4.00000 q^{67} +3.00000 q^{68} +3.00000 q^{70} +6.00000 q^{71} -1.00000 q^{73} +7.00000 q^{74} +5.00000 q^{76} -3.00000 q^{77} -10.0000 q^{79} -3.00000 q^{80} +6.00000 q^{82} -6.00000 q^{83} -9.00000 q^{85} -5.00000 q^{86} +3.00000 q^{88} +6.00000 q^{89} +1.00000 q^{91} +3.00000 q^{92} -15.0000 q^{95} +2.00000 q^{97} -1.00000 q^{98} +O(q^{100})\)

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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −3.00000 −1.34164 −0.670820 − 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 3.00000 0.948683
\(11\) −3.00000 −0.904534 −0.452267 − 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.00000 0.727607 0.363803 − 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) 5.00000 1.14708 0.573539 − 0.819178i \(-0.305570\pi\)
0.573539 + 0.819178i \(0.305570\pi\)
\(20\) −3.00000 −0.670820
\(21\) 0 0
\(22\) 3.00000 0.639602
\(23\) 3.00000 0.625543 0.312772 − 0.949828i \(-0.398743\pi\)
0.312772 + 0.949828i \(0.398743\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) −1.00000 −0.196116
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) 3.00000 0.557086 0.278543 − 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) −10.0000 −1.79605 −0.898027 − 0.439941i \(-0.854999\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −3.00000 −0.514496
\(35\) −3.00000 −0.507093
\(36\) 0 0
\(37\) −7.00000 −1.15079 −0.575396 − 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) −5.00000 −0.811107
\(39\) 0 0
\(40\) 3.00000 0.474342
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 5.00000 0.762493 0.381246 − 0.924473i \(-0.375495\pi\)
0.381246 + 0.924473i \(0.375495\pi\)
\(44\) −3.00000 −0.452267
\(45\) 0 0
\(46\) −3.00000 −0.442326
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −4.00000 −0.565685
\(51\) 0 0
\(52\) 1.00000 0.138675
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 9.00000 1.21356
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −3.00000 −0.393919
\(59\) −6.00000 −0.781133 −0.390567 − 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) −7.00000 −0.896258 −0.448129 − 0.893969i \(-0.647910\pi\)
−0.448129 + 0.893969i \(0.647910\pi\)
\(62\) 10.0000 1.27000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −3.00000 −0.372104
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 3.00000 0.363803
\(69\) 0 0
\(70\) 3.00000 0.358569
\(71\) 6.00000 0.712069 0.356034 − 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 0 0
\(73\) −1.00000 −0.117041 −0.0585206 − 0.998286i \(-0.518638\pi\)
−0.0585206 + 0.998286i \(0.518638\pi\)
\(74\) 7.00000 0.813733
\(75\) 0 0
\(76\) 5.00000 0.573539
\(77\) −3.00000 −0.341882
\(78\) 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
\(82\) 6.00000 0.662589
\(83\) −6.00000 −0.658586 −0.329293 − 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) −9.00000 −0.976187
\(86\) −5.00000 −0.539164
\(87\) 0 0
\(88\) 3.00000 0.319801
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 3.00000 0.312772
\(93\) 0 0
\(94\) 0 0
\(95\) −15.0000 −1.53897
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 4.00000 0.400000
\(101\) −6.00000 −0.597022 −0.298511 − 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) −13.0000 −1.28093 −0.640464 − 0.767988i \(-0.721258\pi\)
−0.640464 + 0.767988i \(0.721258\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) −18.0000 −1.74013 −0.870063 − 0.492941i \(-0.835922\pi\)
−0.870063 + 0.492941i \(0.835922\pi\)
\(108\) 0 0
\(109\) −1.00000 −0.0957826 −0.0478913 − 0.998853i \(-0.515250\pi\)
−0.0478913 + 0.998853i \(0.515250\pi\)
\(110\) −9.00000 −0.858116
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) −9.00000 −0.839254
\(116\) 3.00000 0.278543
\(117\) 0 0
\(118\) 6.00000 0.552345
\(119\) 3.00000 0.275010
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 7.00000 0.633750
\(123\) 0 0
\(124\) −10.0000 −0.898027
\(125\) 3.00000 0.268328
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 − 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 3.00000 0.263117
\(131\) −15.0000 −1.31056 −0.655278 − 0.755388i \(-0.727449\pi\)
−0.655278 + 0.755388i \(0.727449\pi\)
\(132\) 0 0
\(133\) 5.00000 0.433555
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) −3.00000 −0.257248
\(137\) 9.00000 0.768922 0.384461 − 0.923141i \(-0.374387\pi\)
0.384461 + 0.923141i \(0.374387\pi\)
\(138\) 0 0
\(139\) 14.0000 1.18746 0.593732 − 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) −3.00000 −0.253546
\(141\) 0 0
\(142\) −6.00000 −0.503509
\(143\) −3.00000 −0.250873
\(144\) 0 0
\(145\) −9.00000 −0.747409
\(146\) 1.00000 0.0827606
\(147\) 0 0
\(148\) −7.00000 −0.575396
\(149\) 12.0000 0.983078 0.491539 − 0.870855i \(-0.336434\pi\)
0.491539 + 0.870855i \(0.336434\pi\)
\(150\) 0 0
\(151\) −13.0000 −1.05792 −0.528962 − 0.848645i \(-0.677419\pi\)
−0.528962 + 0.848645i \(0.677419\pi\)
\(152\) −5.00000 −0.405554
\(153\) 0 0
\(154\) 3.00000 0.241747
\(155\) 30.0000 2.40966
\(156\) 0 0
\(157\) 5.00000 0.399043 0.199522 − 0.979893i \(-0.436061\pi\)
0.199522 + 0.979893i \(0.436061\pi\)
\(158\) 10.0000 0.795557
\(159\) 0 0
\(160\) 3.00000 0.237171
\(161\) 3.00000 0.236433
\(162\) 0 0
\(163\) 2.00000 0.156652 0.0783260 − 0.996928i \(-0.475042\pi\)
0.0783260 + 0.996928i \(0.475042\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 6.00000 0.465690
\(167\) −21.0000 −1.62503 −0.812514 − 0.582941i \(-0.801902\pi\)
−0.812514 + 0.582941i \(0.801902\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 9.00000 0.690268
\(171\) 0 0
\(172\) 5.00000 0.381246
\(173\) −18.0000 −1.36851 −0.684257 − 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) 0 0
\(175\) 4.00000 0.302372
\(176\) −3.00000 −0.226134
\(177\) 0 0
\(178\) −6.00000 −0.449719
\(179\) −12.0000 −0.896922 −0.448461 − 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) −22.0000 −1.63525 −0.817624 − 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) −1.00000 −0.0741249
\(183\) 0 0
\(184\) −3.00000 −0.221163
\(185\) 21.0000 1.54395
\(186\) 0 0
\(187\) −9.00000 −0.658145
\(188\) 0 0
\(189\) 0 0
\(190\) 15.0000 1.08821
\(191\) 15.0000 1.08536 0.542681 − 0.839939i \(-0.317409\pi\)
0.542681 + 0.839939i \(0.317409\pi\)
\(192\) 0 0
\(193\) −22.0000 −1.58359 −0.791797 − 0.610784i \(-0.790854\pi\)
−0.791797 + 0.610784i \(0.790854\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −6.00000 −0.427482 −0.213741 − 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) 5.00000 0.354441 0.177220 − 0.984171i \(-0.443289\pi\)
0.177220 + 0.984171i \(0.443289\pi\)
\(200\) −4.00000 −0.282843
\(201\) 0 0
\(202\) 6.00000 0.422159
\(203\) 3.00000 0.210559
\(204\) 0 0
\(205\) 18.0000 1.25717
\(206\) 13.0000 0.905753
\(207\) 0 0
\(208\) 1.00000 0.0693375
\(209\) −15.0000 −1.03757
\(210\) 0 0
\(211\) 23.0000 1.58339 0.791693 − 0.610920i \(-0.209200\pi\)
0.791693 + 0.610920i \(0.209200\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) 18.0000 1.23045
\(215\) −15.0000 −1.02299
\(216\) 0 0
\(217\) −10.0000 −0.678844
\(218\) 1.00000 0.0677285
\(219\) 0 0
\(220\) 9.00000 0.606780
\(221\) 3.00000 0.201802
\(222\) 0 0
\(223\) 8.00000 0.535720 0.267860 − 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 0 0
\(227\) −18.0000 −1.19470 −0.597351 − 0.801980i \(-0.703780\pi\)
−0.597351 + 0.801980i \(0.703780\pi\)
\(228\) 0 0
\(229\) −22.0000 −1.45380 −0.726900 − 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 9.00000 0.593442
\(231\) 0 0
\(232\) −3.00000 −0.196960
\(233\) 24.0000 1.57229 0.786146 − 0.618041i \(-0.212073\pi\)
0.786146 + 0.618041i \(0.212073\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −6.00000 −0.390567
\(237\) 0 0
\(238\) −3.00000 −0.194461
\(239\) 12.0000 0.776215 0.388108 − 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) −22.0000 −1.41714 −0.708572 − 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) 2.00000 0.128565
\(243\) 0 0
\(244\) −7.00000 −0.448129
\(245\) −3.00000 −0.191663
\(246\) 0 0
\(247\) 5.00000 0.318142
\(248\) 10.0000 0.635001
\(249\) 0 0
\(250\) −3.00000 −0.189737
\(251\) 9.00000 0.568075 0.284037 − 0.958813i \(-0.408326\pi\)
0.284037 + 0.958813i \(0.408326\pi\)
\(252\) 0 0
\(253\) −9.00000 −0.565825
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −6.00000 −0.374270 −0.187135 − 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 0 0
\(259\) −7.00000 −0.434959
\(260\) −3.00000 −0.186052
\(261\) 0 0
\(262\) 15.0000 0.926703
\(263\) 24.0000 1.47990 0.739952 − 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 0 0
\(265\) 18.0000 1.10573
\(266\) −5.00000 −0.306570
\(267\) 0 0
\(268\) −4.00000 −0.244339
\(269\) 18.0000 1.09748 0.548740 − 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) 2.00000 0.121491 0.0607457 − 0.998153i \(-0.480652\pi\)
0.0607457 + 0.998153i \(0.480652\pi\)
\(272\) 3.00000 0.181902
\(273\) 0 0
\(274\) −9.00000 −0.543710
\(275\) −12.0000 −0.723627
\(276\) 0 0
\(277\) −28.0000 −1.68236 −0.841178 − 0.540758i \(-0.818138\pi\)
−0.841178 + 0.540758i \(0.818138\pi\)
\(278\) −14.0000 −0.839664
\(279\) 0 0
\(280\) 3.00000 0.179284
\(281\) 30.0000 1.78965 0.894825 − 0.446417i \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) 0 0
\(283\) −16.0000 −0.951101 −0.475551 − 0.879688i \(-0.657751\pi\)
−0.475551 + 0.879688i \(0.657751\pi\)
\(284\) 6.00000 0.356034
\(285\) 0 0
\(286\) 3.00000 0.177394
\(287\) −6.00000 −0.354169
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 9.00000 0.528498
\(291\) 0 0
\(292\) −1.00000 −0.0585206
\(293\) −30.0000 −1.75262 −0.876309 − 0.481749i \(-0.840002\pi\)
−0.876309 + 0.481749i \(0.840002\pi\)
\(294\) 0 0
\(295\) 18.0000 1.04800
\(296\) 7.00000 0.406867
\(297\) 0 0
\(298\) −12.0000 −0.695141
\(299\) 3.00000 0.173494
\(300\) 0 0
\(301\) 5.00000 0.288195
\(302\) 13.0000 0.748066
\(303\) 0 0
\(304\) 5.00000 0.286770
\(305\) 21.0000 1.20246
\(306\) 0 0
\(307\) −4.00000 −0.228292 −0.114146 − 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(308\) −3.00000 −0.170941
\(309\) 0 0
\(310\) −30.0000 −1.70389
\(311\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 14.0000 0.791327 0.395663 − 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) −5.00000 −0.282166
\(315\) 0 0
\(316\) −10.0000 −0.562544
\(317\) −18.0000 −1.01098 −0.505490 − 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 0 0
\(319\) −9.00000 −0.503903
\(320\) −3.00000 −0.167705
\(321\) 0 0
\(322\) −3.00000 −0.167183
\(323\) 15.0000 0.834622
\(324\) 0 0
\(325\) 4.00000 0.221880
\(326\) −2.00000 −0.110770
\(327\) 0 0
\(328\) 6.00000 0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) 14.0000 0.769510 0.384755 − 0.923019i \(-0.374286\pi\)
0.384755 + 0.923019i \(0.374286\pi\)
\(332\) −6.00000 −0.329293
\(333\) 0 0
\(334\) 21.0000 1.14907
\(335\) 12.0000 0.655630
\(336\) 0 0
\(337\) −25.0000 −1.36184 −0.680918 − 0.732359i \(-0.738419\pi\)
−0.680918 + 0.732359i \(0.738419\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 0 0
\(340\) −9.00000 −0.488094
\(341\) 30.0000 1.62459
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −5.00000 −0.269582
\(345\) 0 0
\(346\) 18.0000 0.967686
\(347\) 12.0000 0.644194 0.322097 − 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 0 0
\(349\) −16.0000 −0.856460 −0.428230 − 0.903670i \(-0.640863\pi\)
−0.428230 + 0.903670i \(0.640863\pi\)
\(350\) −4.00000 −0.213809
\(351\) 0 0
\(352\) 3.00000 0.159901
\(353\) 30.0000 1.59674 0.798369 − 0.602168i \(-0.205696\pi\)
0.798369 + 0.602168i \(0.205696\pi\)
\(354\) 0 0
\(355\) −18.0000 −0.955341
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) 30.0000 1.58334 0.791670 − 0.610949i \(-0.209212\pi\)
0.791670 + 0.610949i \(0.209212\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) 22.0000 1.15629
\(363\) 0 0
\(364\) 1.00000 0.0524142
\(365\) 3.00000 0.157027
\(366\) 0 0
\(367\) 32.0000 1.67039 0.835193 − 0.549957i \(-0.185356\pi\)
0.835193 + 0.549957i \(0.185356\pi\)
\(368\) 3.00000 0.156386
\(369\) 0 0
\(370\) −21.0000 −1.09174
\(371\) −6.00000 −0.311504
\(372\) 0 0
\(373\) 14.0000 0.724893 0.362446 − 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) 9.00000 0.465379
\(375\) 0 0
\(376\) 0 0
\(377\) 3.00000 0.154508
\(378\) 0 0
\(379\) 2.00000 0.102733 0.0513665 − 0.998680i \(-0.483642\pi\)
0.0513665 + 0.998680i \(0.483642\pi\)
\(380\) −15.0000 −0.769484
\(381\) 0 0
\(382\) −15.0000 −0.767467
\(383\) −9.00000 −0.459879 −0.229939 − 0.973205i \(-0.573853\pi\)
−0.229939 + 0.973205i \(0.573853\pi\)
\(384\) 0 0
\(385\) 9.00000 0.458682
\(386\) 22.0000 1.11977
\(387\) 0 0
\(388\) 2.00000 0.101535
\(389\) −6.00000 −0.304212 −0.152106 − 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 9.00000 0.455150
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) 6.00000 0.302276
\(395\) 30.0000 1.50946
\(396\) 0 0
\(397\) 8.00000 0.401508 0.200754 − 0.979642i \(-0.435661\pi\)
0.200754 + 0.979642i \(0.435661\pi\)
\(398\) −5.00000 −0.250627
\(399\) 0 0
\(400\) 4.00000 0.200000
\(401\) −18.0000 −0.898877 −0.449439 − 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) −10.0000 −0.498135
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) −3.00000 −0.148888
\(407\) 21.0000 1.04093
\(408\) 0 0
\(409\) 5.00000 0.247234 0.123617 − 0.992330i \(-0.460551\pi\)
0.123617 + 0.992330i \(0.460551\pi\)
\(410\) −18.0000 −0.888957
\(411\) 0 0
\(412\) −13.0000 −0.640464
\(413\) −6.00000 −0.295241
\(414\) 0 0
\(415\) 18.0000 0.883585
\(416\) −1.00000 −0.0490290
\(417\) 0 0
\(418\) 15.0000 0.733674
\(419\) 3.00000 0.146560 0.0732798 − 0.997311i \(-0.476653\pi\)
0.0732798 + 0.997311i \(0.476653\pi\)
\(420\) 0 0
\(421\) 26.0000 1.26716 0.633581 − 0.773676i \(-0.281584\pi\)
0.633581 + 0.773676i \(0.281584\pi\)
\(422\) −23.0000 −1.11962
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) 12.0000 0.582086
\(426\) 0 0
\(427\) −7.00000 −0.338754
\(428\) −18.0000 −0.870063
\(429\) 0 0
\(430\) 15.0000 0.723364
\(431\) 18.0000 0.867029 0.433515 − 0.901146i \(-0.357273\pi\)
0.433515 + 0.901146i \(0.357273\pi\)
\(432\) 0 0
\(433\) 14.0000 0.672797 0.336399 − 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 10.0000 0.480015
\(435\) 0 0
\(436\) −1.00000 −0.0478913
\(437\) 15.0000 0.717547
\(438\) 0 0
\(439\) 35.0000 1.67046 0.835229 − 0.549902i \(-0.185335\pi\)
0.835229 + 0.549902i \(0.185335\pi\)
\(440\) −9.00000 −0.429058
\(441\) 0 0
\(442\) −3.00000 −0.142695
\(443\) 6.00000 0.285069 0.142534 − 0.989790i \(-0.454475\pi\)
0.142534 + 0.989790i \(0.454475\pi\)
\(444\) 0 0
\(445\) −18.0000 −0.853282
\(446\) −8.00000 −0.378811
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) 21.0000 0.991051 0.495526 − 0.868593i \(-0.334975\pi\)
0.495526 + 0.868593i \(0.334975\pi\)
\(450\) 0 0
\(451\) 18.0000 0.847587
\(452\) 0 0
\(453\) 0 0
\(454\) 18.0000 0.844782
\(455\) −3.00000 −0.140642
\(456\) 0 0
\(457\) 20.0000 0.935561 0.467780 − 0.883845i \(-0.345054\pi\)
0.467780 + 0.883845i \(0.345054\pi\)
\(458\) 22.0000 1.02799
\(459\) 0 0
\(460\) −9.00000 −0.419627
\(461\) −33.0000 −1.53696 −0.768482 − 0.639872i \(-0.778987\pi\)
−0.768482 + 0.639872i \(0.778987\pi\)
\(462\) 0 0
\(463\) −25.0000 −1.16185 −0.580924 − 0.813958i \(-0.697309\pi\)
−0.580924 + 0.813958i \(0.697309\pi\)
\(464\) 3.00000 0.139272
\(465\) 0 0
\(466\) −24.0000 −1.11178
\(467\) −3.00000 −0.138823 −0.0694117 − 0.997588i \(-0.522112\pi\)
−0.0694117 + 0.997588i \(0.522112\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) 0 0
\(472\) 6.00000 0.276172
\(473\) −15.0000 −0.689701
\(474\) 0 0
\(475\) 20.0000 0.917663
\(476\) 3.00000 0.137505
\(477\) 0 0
\(478\) −12.0000 −0.548867
\(479\) −15.0000 −0.685367 −0.342684 − 0.939451i \(-0.611336\pi\)
−0.342684 + 0.939451i \(0.611336\pi\)
\(480\) 0 0
\(481\) −7.00000 −0.319173
\(482\) 22.0000 1.00207
\(483\) 0 0
\(484\) −2.00000 −0.0909091
\(485\) −6.00000 −0.272446
\(486\) 0 0
\(487\) 32.0000 1.45006 0.725029 − 0.688718i \(-0.241826\pi\)
0.725029 + 0.688718i \(0.241826\pi\)
\(488\) 7.00000 0.316875
\(489\) 0 0
\(490\) 3.00000 0.135526
\(491\) 36.0000 1.62466 0.812329 − 0.583200i \(-0.198200\pi\)
0.812329 + 0.583200i \(0.198200\pi\)
\(492\) 0 0
\(493\) 9.00000 0.405340
\(494\) −5.00000 −0.224961
\(495\) 0 0
\(496\) −10.0000 −0.449013
\(497\) 6.00000 0.269137
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 − 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 3.00000 0.134164
\(501\) 0 0
\(502\) −9.00000 −0.401690
\(503\) −12.0000 −0.535054 −0.267527 − 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) 0 0
\(505\) 18.0000 0.800989
\(506\) 9.00000 0.400099
\(507\) 0 0
\(508\) 8.00000 0.354943
\(509\) −33.0000 −1.46270 −0.731350 − 0.682003i \(-0.761109\pi\)
−0.731350 + 0.682003i \(0.761109\pi\)
\(510\) 0 0
\(511\) −1.00000 −0.0442374
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 6.00000 0.264649
\(515\) 39.0000 1.71855
\(516\) 0 0
\(517\) 0 0
\(518\) 7.00000 0.307562
\(519\) 0 0
\(520\) 3.00000 0.131559
\(521\) −3.00000 −0.131432 −0.0657162 − 0.997838i \(-0.520933\pi\)
−0.0657162 + 0.997838i \(0.520933\pi\)
\(522\) 0 0
\(523\) −34.0000 −1.48672 −0.743358 − 0.668894i \(-0.766768\pi\)
−0.743358 + 0.668894i \(0.766768\pi\)
\(524\) −15.0000 −0.655278
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) −30.0000 −1.30682
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) −18.0000 −0.781870
\(531\) 0 0
\(532\) 5.00000 0.216777
\(533\) −6.00000 −0.259889
\(534\) 0 0
\(535\) 54.0000 2.33462
\(536\) 4.00000 0.172774
\(537\) 0 0
\(538\) −18.0000 −0.776035
\(539\) −3.00000 −0.129219
\(540\) 0 0
\(541\) −1.00000 −0.0429934 −0.0214967 − 0.999769i \(-0.506843\pi\)
−0.0214967 + 0.999769i \(0.506843\pi\)
\(542\) −2.00000 −0.0859074
\(543\) 0 0
\(544\) −3.00000 −0.128624
\(545\) 3.00000 0.128506
\(546\) 0 0
\(547\) 20.0000 0.855138 0.427569 − 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) 9.00000 0.384461
\(549\) 0 0
\(550\) 12.0000 0.511682
\(551\) 15.0000 0.639021
\(552\) 0 0
\(553\) −10.0000 −0.425243
\(554\) 28.0000 1.18961
\(555\) 0 0
\(556\) 14.0000 0.593732
\(557\) 6.00000 0.254228 0.127114 − 0.991888i \(-0.459429\pi\)
0.127114 + 0.991888i \(0.459429\pi\)
\(558\) 0 0
\(559\) 5.00000 0.211477
\(560\) −3.00000 −0.126773
\(561\) 0 0
\(562\) −30.0000 −1.26547
\(563\) −39.0000 −1.64365 −0.821827 − 0.569737i \(-0.807045\pi\)
−0.821827 + 0.569737i \(0.807045\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 16.0000 0.672530
\(567\) 0 0
\(568\) −6.00000 −0.251754
\(569\) −42.0000 −1.76073 −0.880366 − 0.474295i \(-0.842703\pi\)
−0.880366 + 0.474295i \(0.842703\pi\)
\(570\) 0 0
\(571\) 44.0000 1.84134 0.920671 − 0.390339i \(-0.127642\pi\)
0.920671 + 0.390339i \(0.127642\pi\)
\(572\) −3.00000 −0.125436
\(573\) 0 0
\(574\) 6.00000 0.250435
\(575\) 12.0000 0.500435
\(576\) 0 0
\(577\) 2.00000 0.0832611 0.0416305 − 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 8.00000 0.332756
\(579\) 0 0
\(580\) −9.00000 −0.373705
\(581\) −6.00000 −0.248922
\(582\) 0 0
\(583\) 18.0000 0.745484
\(584\) 1.00000 0.0413803
\(585\) 0 0
\(586\) 30.0000 1.23929
\(587\) 6.00000 0.247647 0.123823 − 0.992304i \(-0.460484\pi\)
0.123823 + 0.992304i \(0.460484\pi\)
\(588\) 0 0
\(589\) −50.0000 −2.06021
\(590\) −18.0000 −0.741048
\(591\) 0 0
\(592\) −7.00000 −0.287698
\(593\) 12.0000 0.492781 0.246390 − 0.969171i \(-0.420755\pi\)
0.246390 + 0.969171i \(0.420755\pi\)
\(594\) 0 0
\(595\) −9.00000 −0.368964
\(596\) 12.0000 0.491539
\(597\) 0 0
\(598\) −3.00000 −0.122679
\(599\) 9.00000 0.367730 0.183865 − 0.982952i \(-0.441139\pi\)
0.183865 + 0.982952i \(0.441139\pi\)
\(600\) 0 0
\(601\) 26.0000 1.06056 0.530281 − 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) −5.00000 −0.203785
\(603\) 0 0
\(604\) −13.0000 −0.528962
\(605\) 6.00000 0.243935
\(606\) 0 0
\(607\) −1.00000 −0.0405887 −0.0202944 − 0.999794i \(-0.506460\pi\)
−0.0202944 + 0.999794i \(0.506460\pi\)
\(608\) −5.00000 −0.202777
\(609\) 0 0
\(610\) −21.0000 −0.850265
\(611\) 0 0
\(612\) 0 0
\(613\) −49.0000 −1.97909 −0.989546 − 0.144220i \(-0.953933\pi\)
−0.989546 + 0.144220i \(0.953933\pi\)
\(614\) 4.00000 0.161427
\(615\) 0 0
\(616\) 3.00000 0.120873
\(617\) −33.0000 −1.32853 −0.664265 − 0.747497i \(-0.731255\pi\)
−0.664265 + 0.747497i \(0.731255\pi\)
\(618\) 0 0
\(619\) 17.0000 0.683288 0.341644 − 0.939829i \(-0.389016\pi\)
0.341644 + 0.939829i \(0.389016\pi\)
\(620\) 30.0000 1.20483
\(621\) 0 0
\(622\) 0 0
\(623\) 6.00000 0.240385
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) −14.0000 −0.559553
\(627\) 0 0
\(628\) 5.00000 0.199522
\(629\) −21.0000 −0.837325
\(630\) 0 0
\(631\) −13.0000 −0.517522 −0.258761 − 0.965941i \(-0.583314\pi\)
−0.258761 + 0.965941i \(0.583314\pi\)
\(632\) 10.0000 0.397779
\(633\) 0 0
\(634\) 18.0000 0.714871
\(635\) −24.0000 −0.952411
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) 9.00000 0.356313
\(639\) 0 0
\(640\) 3.00000 0.118585
\(641\) −42.0000 −1.65890 −0.829450 − 0.558581i \(-0.811346\pi\)
−0.829450 + 0.558581i \(0.811346\pi\)
\(642\) 0 0
\(643\) −25.0000 −0.985904 −0.492952 − 0.870057i \(-0.664082\pi\)
−0.492952 + 0.870057i \(0.664082\pi\)
\(644\) 3.00000 0.118217
\(645\) 0 0
\(646\) −15.0000 −0.590167
\(647\) 12.0000 0.471769 0.235884 − 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) 0 0
\(649\) 18.0000 0.706562
\(650\) −4.00000 −0.156893
\(651\) 0 0
\(652\) 2.00000 0.0783260
\(653\) −15.0000 −0.586995 −0.293498 − 0.955960i \(-0.594819\pi\)
−0.293498 + 0.955960i \(0.594819\pi\)
\(654\) 0 0
\(655\) 45.0000 1.75830
\(656\) −6.00000 −0.234261
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −4.00000 −0.155582 −0.0777910 − 0.996970i \(-0.524787\pi\)
−0.0777910 + 0.996970i \(0.524787\pi\)
\(662\) −14.0000 −0.544125
\(663\) 0 0
\(664\) 6.00000 0.232845
\(665\) −15.0000 −0.581675
\(666\) 0 0
\(667\) 9.00000 0.348481
\(668\) −21.0000 −0.812514
\(669\) 0 0
\(670\) −12.0000 −0.463600
\(671\) 21.0000 0.810696
\(672\) 0 0
\(673\) −13.0000 −0.501113 −0.250557 − 0.968102i \(-0.580614\pi\)
−0.250557 + 0.968102i \(0.580614\pi\)
\(674\) 25.0000 0.962964
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) −42.0000 −1.61419 −0.807096 − 0.590421i \(-0.798962\pi\)
−0.807096 + 0.590421i \(0.798962\pi\)
\(678\) 0 0
\(679\) 2.00000 0.0767530
\(680\) 9.00000 0.345134
\(681\) 0 0
\(682\) −30.0000 −1.14876
\(683\) −21.0000 −0.803543 −0.401771 − 0.915740i \(-0.631605\pi\)
−0.401771 + 0.915740i \(0.631605\pi\)
\(684\) 0 0
\(685\) −27.0000 −1.03162
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) 5.00000 0.190623
\(689\) −6.00000 −0.228582
\(690\) 0 0
\(691\) 32.0000 1.21734 0.608669 − 0.793424i \(-0.291704\pi\)
0.608669 + 0.793424i \(0.291704\pi\)
\(692\) −18.0000 −0.684257
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) −42.0000 −1.59315
\(696\) 0 0
\(697\) −18.0000 −0.681799
\(698\) 16.0000 0.605609
\(699\) 0 0
\(700\) 4.00000 0.151186
\(701\) 30.0000 1.13308 0.566542 − 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 0 0
\(703\) −35.0000 −1.32005
\(704\) −3.00000 −0.113067
\(705\) 0 0
\(706\) −30.0000 −1.12906
\(707\) −6.00000 −0.225653
\(708\) 0 0
\(709\) 26.0000 0.976450 0.488225 − 0.872718i \(-0.337644\pi\)
0.488225 + 0.872718i \(0.337644\pi\)
\(710\) 18.0000 0.675528
\(711\) 0 0
\(712\) −6.00000 −0.224860
\(713\) −30.0000 −1.12351
\(714\) 0 0
\(715\) 9.00000 0.336581
\(716\) −12.0000 −0.448461
\(717\) 0 0
\(718\) −30.0000 −1.11959
\(719\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) −13.0000 −0.484145
\(722\) −6.00000 −0.223297
\(723\) 0 0
\(724\) −22.0000 −0.817624
\(725\) 12.0000 0.445669
\(726\) 0 0
\(727\) −25.0000 −0.927199 −0.463599 − 0.886045i \(-0.653442\pi\)
−0.463599 + 0.886045i \(0.653442\pi\)
\(728\) −1.00000 −0.0370625
\(729\) 0 0
\(730\) −3.00000 −0.111035
\(731\) 15.0000 0.554795
\(732\) 0 0
\(733\) −28.0000 −1.03420 −0.517102 − 0.855924i \(-0.672989\pi\)
−0.517102 + 0.855924i \(0.672989\pi\)
\(734\) −32.0000 −1.18114
\(735\) 0 0
\(736\) −3.00000 −0.110581
\(737\) 12.0000 0.442026
\(738\) 0 0
\(739\) −34.0000 −1.25071 −0.625355 − 0.780340i \(-0.715046\pi\)
−0.625355 + 0.780340i \(0.715046\pi\)
\(740\) 21.0000 0.771975
\(741\) 0 0
\(742\) 6.00000 0.220267
\(743\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) −36.0000 −1.31894
\(746\) −14.0000 −0.512576
\(747\) 0 0
\(748\) −9.00000 −0.329073
\(749\) −18.0000 −0.657706
\(750\) 0 0
\(751\) 44.0000 1.60558 0.802791 − 0.596260i \(-0.203347\pi\)
0.802791 + 0.596260i \(0.203347\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) −3.00000 −0.109254
\(755\) 39.0000 1.41936
\(756\) 0 0
\(757\) 8.00000 0.290765 0.145382 − 0.989376i \(-0.453559\pi\)
0.145382 + 0.989376i \(0.453559\pi\)
\(758\) −2.00000 −0.0726433
\(759\) 0 0
\(760\) 15.0000 0.544107
\(761\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) −1.00000 −0.0362024
\(764\) 15.0000 0.542681
\(765\) 0 0
\(766\) 9.00000 0.325183
\(767\) −6.00000 −0.216647
\(768\) 0 0
\(769\) 53.0000 1.91123 0.955614 − 0.294620i \(-0.0951931\pi\)
0.955614 + 0.294620i \(0.0951931\pi\)
\(770\) −9.00000 −0.324337
\(771\) 0 0
\(772\) −22.0000 −0.791797
\(773\) 39.0000 1.40273 0.701366 − 0.712801i \(-0.252574\pi\)
0.701366 + 0.712801i \(0.252574\pi\)
\(774\) 0 0
\(775\) −40.0000 −1.43684
\(776\) −2.00000 −0.0717958
\(777\) 0 0
\(778\) 6.00000 0.215110
\(779\) −30.0000 −1.07486
\(780\) 0 0
\(781\) −18.0000 −0.644091
\(782\) −9.00000 −0.321839
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) −15.0000 −0.535373
\(786\) 0 0
\(787\) 17.0000 0.605985 0.302992 − 0.952993i \(-0.402014\pi\)
0.302992 + 0.952993i \(0.402014\pi\)
\(788\) −6.00000 −0.213741
\(789\) 0 0
\(790\) −30.0000 −1.06735
\(791\) 0 0
\(792\) 0 0
\(793\) −7.00000 −0.248577
\(794\) −8.00000 −0.283909
\(795\) 0 0
\(796\) 5.00000 0.177220
\(797\) −48.0000 −1.70025 −0.850124 − 0.526583i \(-0.823473\pi\)
−0.850124 + 0.526583i \(0.823473\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −4.00000 −0.141421
\(801\) 0 0
\(802\) 18.0000 0.635602
\(803\) 3.00000 0.105868
\(804\) 0 0
\(805\) −9.00000 −0.317208
\(806\) 10.0000 0.352235
\(807\) 0 0
\(808\) 6.00000 0.211079
\(809\) 18.0000 0.632846 0.316423 − 0.948618i \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) 0 0
\(811\) −7.00000 −0.245803 −0.122902 − 0.992419i \(-0.539220\pi\)
−0.122902 + 0.992419i \(0.539220\pi\)
\(812\) 3.00000 0.105279
\(813\) 0 0
\(814\) −21.0000 −0.736050
\(815\) −6.00000 −0.210171
\(816\) 0 0
\(817\) 25.0000 0.874639
\(818\) −5.00000 −0.174821
\(819\) 0 0
\(820\) 18.0000 0.628587
\(821\) −36.0000 −1.25641 −0.628204 − 0.778048i \(-0.716210\pi\)
−0.628204 + 0.778048i \(0.716210\pi\)
\(822\) 0 0
\(823\) −34.0000 −1.18517 −0.592583 − 0.805510i \(-0.701892\pi\)
−0.592583 + 0.805510i \(0.701892\pi\)
\(824\) 13.0000 0.452876
\(825\) 0 0
\(826\) 6.00000 0.208767
\(827\) 21.0000 0.730242 0.365121 − 0.930960i \(-0.381028\pi\)
0.365121 + 0.930960i \(0.381028\pi\)
\(828\) 0 0
\(829\) −55.0000 −1.91023 −0.955114 − 0.296237i \(-0.904268\pi\)
−0.955114 + 0.296237i \(0.904268\pi\)
\(830\) −18.0000 −0.624789
\(831\) 0 0
\(832\) 1.00000 0.0346688
\(833\) 3.00000 0.103944
\(834\) 0 0
\(835\) 63.0000 2.18020
\(836\) −15.0000 −0.518786
\(837\) 0 0
\(838\) −3.00000 −0.103633
\(839\) −48.0000 −1.65714 −0.828572 − 0.559883i \(-0.810846\pi\)
−0.828572 + 0.559883i \(0.810846\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) −26.0000 −0.896019
\(843\) 0 0
\(844\) 23.0000 0.791693
\(845\) −3.00000 −0.103203
\(846\) 0 0
\(847\) −2.00000 −0.0687208
\(848\) −6.00000 −0.206041
\(849\) 0 0
\(850\) −12.0000 −0.411597
\(851\) −21.0000 −0.719871
\(852\) 0 0
\(853\) 8.00000 0.273915 0.136957 − 0.990577i \(-0.456268\pi\)
0.136957 + 0.990577i \(0.456268\pi\)
\(854\) 7.00000 0.239535
\(855\) 0 0
\(856\) 18.0000 0.615227
\(857\) 54.0000 1.84460 0.922302 − 0.386469i \(-0.126305\pi\)
0.922302 + 0.386469i \(0.126305\pi\)
\(858\) 0 0
\(859\) 14.0000 0.477674 0.238837 − 0.971060i \(-0.423234\pi\)
0.238837 + 0.971060i \(0.423234\pi\)
\(860\) −15.0000 −0.511496
\(861\) 0 0
\(862\) −18.0000 −0.613082
\(863\) −12.0000 −0.408485 −0.204242 − 0.978920i \(-0.565473\pi\)
−0.204242 + 0.978920i \(0.565473\pi\)
\(864\) 0 0
\(865\) 54.0000 1.83606
\(866\) −14.0000 −0.475739
\(867\) 0 0
\(868\) −10.0000 −0.339422
\(869\) 30.0000 1.01768
\(870\) 0 0
\(871\) −4.00000 −0.135535
\(872\) 1.00000 0.0338643
\(873\) 0 0
\(874\) −15.0000 −0.507383
\(875\) 3.00000 0.101419
\(876\) 0 0
\(877\) 2.00000 0.0675352 0.0337676 − 0.999430i \(-0.489249\pi\)
0.0337676 + 0.999430i \(0.489249\pi\)
\(878\) −35.0000 −1.18119
\(879\) 0 0
\(880\) 9.00000 0.303390
\(881\) 3.00000 0.101073 0.0505363 − 0.998722i \(-0.483907\pi\)
0.0505363 + 0.998722i \(0.483907\pi\)
\(882\) 0 0
\(883\) 17.0000 0.572096 0.286048 − 0.958215i \(-0.407658\pi\)
0.286048 + 0.958215i \(0.407658\pi\)
\(884\) 3.00000 0.100901
\(885\) 0 0
\(886\) −6.00000 −0.201574
\(887\) −42.0000 −1.41022 −0.705111 − 0.709097i \(-0.749103\pi\)
−0.705111 + 0.709097i \(0.749103\pi\)
\(888\) 0 0
\(889\) 8.00000 0.268311
\(890\) 18.0000 0.603361
\(891\) 0 0
\(892\) 8.00000 0.267860
\(893\) 0 0
\(894\) 0 0
\(895\) 36.0000 1.20335
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −21.0000 −0.700779
\(899\) −30.0000 −1.00056
\(900\) 0 0
\(901\) −18.0000 −0.599667
\(902\) −18.0000 −0.599334
\(903\) 0 0
\(904\) 0 0
\(905\) 66.0000 2.19391
\(906\) 0 0
\(907\) 44.0000 1.46100 0.730498 − 0.682915i \(-0.239288\pi\)
0.730498 + 0.682915i \(0.239288\pi\)
\(908\) −18.0000 −0.597351
\(909\) 0 0
\(910\) 3.00000 0.0994490
\(911\) −27.0000 −0.894550 −0.447275 − 0.894397i \(-0.647605\pi\)
−0.447275 + 0.894397i \(0.647605\pi\)
\(912\) 0 0
\(913\) 18.0000 0.595713
\(914\) −20.0000 −0.661541
\(915\) 0 0
\(916\) −22.0000 −0.726900
\(917\) −15.0000 −0.495344
\(918\) 0 0
\(919\) −34.0000 −1.12156 −0.560778 − 0.827966i \(-0.689498\pi\)
−0.560778 + 0.827966i \(0.689498\pi\)
\(920\) 9.00000 0.296721
\(921\) 0 0
\(922\) 33.0000 1.08680
\(923\) 6.00000 0.197492
\(924\) 0 0
\(925\) −28.0000 −0.920634
\(926\) 25.0000 0.821551
\(927\) 0 0
\(928\) −3.00000 −0.0984798
\(929\) 18.0000 0.590561 0.295280 − 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) 0 0
\(931\) 5.00000 0.163868
\(932\) 24.0000 0.786146
\(933\) 0 0
\(934\) 3.00000 0.0981630
\(935\) 27.0000 0.882994
\(936\) 0 0
\(937\) 2.00000 0.0653372 0.0326686 − 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) 4.00000 0.130605
\(939\) 0 0
\(940\) 0 0
\(941\) 42.0000 1.36916 0.684580 − 0.728937i \(-0.259985\pi\)
0.684580 + 0.728937i \(0.259985\pi\)
\(942\) 0 0
\(943\) −18.0000 −0.586161
\(944\) −6.00000 −0.195283
\(945\) 0 0
\(946\) 15.0000 0.487692
\(947\) −27.0000 −0.877382 −0.438691 − 0.898638i \(-0.644558\pi\)
−0.438691 + 0.898638i \(0.644558\pi\)
\(948\) 0 0
\(949\) −1.00000 −0.0324614
\(950\) −20.0000 −0.648886
\(951\) 0 0
\(952\) −3.00000 −0.0972306
\(953\) −36.0000 −1.16615 −0.583077 − 0.812417i \(-0.698151\pi\)
−0.583077 + 0.812417i \(0.698151\pi\)
\(954\) 0 0
\(955\) −45.0000 −1.45617
\(956\) 12.0000 0.388108
\(957\) 0 0
\(958\) 15.0000 0.484628
\(959\) 9.00000 0.290625
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) 7.00000 0.225689
\(963\) 0 0
\(964\) −22.0000 −0.708572
\(965\) 66.0000 2.12462
\(966\) 0 0
\(967\) 41.0000 1.31847 0.659236 − 0.751936i \(-0.270880\pi\)
0.659236 + 0.751936i \(0.270880\pi\)
\(968\) 2.00000 0.0642824
\(969\) 0 0
\(970\) 6.00000 0.192648
\(971\) 48.0000 1.54039 0.770197 − 0.637806i \(-0.220158\pi\)
0.770197 + 0.637806i \(0.220158\pi\)
\(972\) 0 0
\(973\) 14.0000 0.448819
\(974\) −32.0000 −1.02535
\(975\) 0 0
\(976\) −7.00000 −0.224065
\(977\) −57.0000 −1.82359 −0.911796 − 0.410644i \(-0.865304\pi\)
−0.911796 + 0.410644i \(0.865304\pi\)
\(978\) 0 0
\(979\) −18.0000 −0.575282
\(980\) −3.00000 −0.0958315
\(981\) 0 0
\(982\) −36.0000 −1.14881
\(983\) 21.0000 0.669796 0.334898 − 0.942254i \(-0.391298\pi\)
0.334898 + 0.942254i \(0.391298\pi\)
\(984\) 0 0
\(985\) 18.0000 0.573528
\(986\) −9.00000 −0.286618
\(987\) 0 0
\(988\) 5.00000 0.159071
\(989\) 15.0000 0.476972
\(990\) 0 0
\(991\) −34.0000 −1.08005 −0.540023 − 0.841650i \(-0.681584\pi\)
−0.540023 + 0.841650i \(0.681584\pi\)
\(992\) 10.0000 0.317500
\(993\) 0 0
\(994\) −6.00000 −0.190308
\(995\) −15.0000 −0.475532
\(996\) 0 0
\(997\) −58.0000 −1.83688 −0.918439 − 0.395562i \(-0.870550\pi\)
−0.918439 + 0.395562i \(0.870550\pi\)
\(998\) 4.00000 0.126618
\(999\) 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 1638.2.a.b.1.1 ✓ 1
3.2 odd 2 1638.2.a.t.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1638.2.a.b.1.1 ✓ 1 1.1 even 1 trivial
1638.2.a.t.1.1 yes 1 3.2 odd 2