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Properties

Label 7942.2.a.s.1.1
Level $7942$
Weight $2$
Character 7942.1
Self dual yes
Analytic conductor $63.417$
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] = [7942,2,Mod(1,7942)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7942, 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("7942.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7942 = 2 \cdot 11 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7942.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: \(63.4171892853\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 418)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 7942.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} -3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} -3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{11} +2.00000 q^{12} +4.00000 q^{13} -3.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} -4.00000 q^{17} +1.00000 q^{18} +1.00000 q^{20} -6.00000 q^{21} -1.00000 q^{22} +4.00000 q^{23} +2.00000 q^{24} -4.00000 q^{25} +4.00000 q^{26} -4.00000 q^{27} -3.00000 q^{28} +10.0000 q^{29} +2.00000 q^{30} +4.00000 q^{31} +1.00000 q^{32} -2.00000 q^{33} -4.00000 q^{34} -3.00000 q^{35} +1.00000 q^{36} +3.00000 q^{37} +8.00000 q^{39} +1.00000 q^{40} +6.00000 q^{41} -6.00000 q^{42} -4.00000 q^{43} -1.00000 q^{44} +1.00000 q^{45} +4.00000 q^{46} +6.00000 q^{47} +2.00000 q^{48} +2.00000 q^{49} -4.00000 q^{50} -8.00000 q^{51} +4.00000 q^{52} -1.00000 q^{53} -4.00000 q^{54} -1.00000 q^{55} -3.00000 q^{56} +10.0000 q^{58} +12.0000 q^{59} +2.00000 q^{60} +12.0000 q^{61} +4.00000 q^{62} -3.00000 q^{63} +1.00000 q^{64} +4.00000 q^{65} -2.00000 q^{66} +12.0000 q^{67} -4.00000 q^{68} +8.00000 q^{69} -3.00000 q^{70} +14.0000 q^{71} +1.00000 q^{72} +6.00000 q^{73} +3.00000 q^{74} -8.00000 q^{75} +3.00000 q^{77} +8.00000 q^{78} +17.0000 q^{79} +1.00000 q^{80} -11.0000 q^{81} +6.00000 q^{82} -13.0000 q^{83} -6.00000 q^{84} -4.00000 q^{85} -4.00000 q^{86} +20.0000 q^{87} -1.00000 q^{88} -2.00000 q^{89} +1.00000 q^{90} -12.0000 q^{91} +4.00000 q^{92} +8.00000 q^{93} +6.00000 q^{94} +2.00000 q^{96} +5.00000 q^{97} +2.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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
\(3\) 2.00000 1.15470 0.577350 − 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214 0.223607 − 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 2.00000 0.816497
\(7\) −3.00000 −1.13389 −0.566947 − 0.823754i \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) −1.00000 −0.301511
\(12\) 2.00000 0.577350
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) −3.00000 −0.801784
\(15\) 2.00000 0.516398
\(16\) 1.00000 0.250000
\(17\) −4.00000 −0.970143 −0.485071 − 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 1.00000 0.235702
\(19\) 0 0
\(20\) 1.00000 0.223607
\(21\) −6.00000 −1.30931
\(22\) −1.00000 −0.213201
\(23\) 4.00000 0.834058 0.417029 − 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 2.00000 0.408248
\(25\) −4.00000 −0.800000
\(26\) 4.00000 0.784465
\(27\) −4.00000 −0.769800
\(28\) −3.00000 −0.566947
\(29\) 10.0000 1.85695 0.928477 − 0.371391i \(-0.121119\pi\)
0.928477 + 0.371391i \(0.121119\pi\)
\(30\) 2.00000 0.365148
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 1.00000 0.176777
\(33\) −2.00000 −0.348155
\(34\) −4.00000 −0.685994
\(35\) −3.00000 −0.507093
\(36\) 1.00000 0.166667
\(37\) 3.00000 0.493197 0.246598 − 0.969118i \(-0.420687\pi\)
0.246598 + 0.969118i \(0.420687\pi\)
\(38\) 0 0
\(39\) 8.00000 1.28103
\(40\) 1.00000 0.158114
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) −6.00000 −0.925820
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −1.00000 −0.150756
\(45\) 1.00000 0.149071
\(46\) 4.00000 0.589768
\(47\) 6.00000 0.875190 0.437595 − 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 2.00000 0.288675
\(49\) 2.00000 0.285714
\(50\) −4.00000 −0.565685
\(51\) −8.00000 −1.12022
\(52\) 4.00000 0.554700
\(53\) −1.00000 −0.137361 −0.0686803 − 0.997639i \(-0.521879\pi\)
−0.0686803 + 0.997639i \(0.521879\pi\)
\(54\) −4.00000 −0.544331
\(55\) −1.00000 −0.134840
\(56\) −3.00000 −0.400892
\(57\) 0 0
\(58\) 10.0000 1.31306
\(59\) 12.0000 1.56227 0.781133 − 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 2.00000 0.258199
\(61\) 12.0000 1.53644 0.768221 − 0.640184i \(-0.221142\pi\)
0.768221 + 0.640184i \(0.221142\pi\)
\(62\) 4.00000 0.508001
\(63\) −3.00000 −0.377964
\(64\) 1.00000 0.125000
\(65\) 4.00000 0.496139
\(66\) −2.00000 −0.246183
\(67\) 12.0000 1.46603 0.733017 − 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) −4.00000 −0.485071
\(69\) 8.00000 0.963087
\(70\) −3.00000 −0.358569
\(71\) 14.0000 1.66149 0.830747 − 0.556650i \(-0.187914\pi\)
0.830747 + 0.556650i \(0.187914\pi\)
\(72\) 1.00000 0.117851
\(73\) 6.00000 0.702247 0.351123 − 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 3.00000 0.348743
\(75\) −8.00000 −0.923760
\(76\) 0 0
\(77\) 3.00000 0.341882
\(78\) 8.00000 0.905822
\(79\) 17.0000 1.91265 0.956325 − 0.292306i \(-0.0944227\pi\)
0.956325 + 0.292306i \(0.0944227\pi\)
\(80\) 1.00000 0.111803
\(81\) −11.0000 −1.22222
\(82\) 6.00000 0.662589
\(83\) −13.0000 −1.42694 −0.713468 − 0.700688i \(-0.752876\pi\)
−0.713468 + 0.700688i \(0.752876\pi\)
\(84\) −6.00000 −0.654654
\(85\) −4.00000 −0.433861
\(86\) −4.00000 −0.431331
\(87\) 20.0000 2.14423
\(88\) −1.00000 −0.106600
\(89\) −2.00000 −0.212000 −0.106000 − 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 1.00000 0.105409
\(91\) −12.0000 −1.25794
\(92\) 4.00000 0.417029
\(93\) 8.00000 0.829561
\(94\) 6.00000 0.618853
\(95\) 0 0
\(96\) 2.00000 0.204124
\(97\) 5.00000 0.507673 0.253837 − 0.967247i \(-0.418307\pi\)
0.253837 + 0.967247i \(0.418307\pi\)
\(98\) 2.00000 0.202031
\(99\) −1.00000 −0.100504
\(100\) −4.00000 −0.400000
\(101\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) −8.00000 −0.792118
\(103\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 4.00000 0.392232
\(105\) −6.00000 −0.585540
\(106\) −1.00000 −0.0971286
\(107\) −7.00000 −0.676716 −0.338358 − 0.941018i \(-0.609871\pi\)
−0.338358 + 0.941018i \(0.609871\pi\)
\(108\) −4.00000 −0.384900
\(109\) −4.00000 −0.383131 −0.191565 − 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) −1.00000 −0.0953463
\(111\) 6.00000 0.569495
\(112\) −3.00000 −0.283473
\(113\) −6.00000 −0.564433 −0.282216 − 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 4.00000 0.373002
\(116\) 10.0000 0.928477
\(117\) 4.00000 0.369800
\(118\) 12.0000 1.10469
\(119\) 12.0000 1.10004
\(120\) 2.00000 0.182574
\(121\) 1.00000 0.0909091
\(122\) 12.0000 1.08643
\(123\) 12.0000 1.08200
\(124\) 4.00000 0.359211
\(125\) −9.00000 −0.804984
\(126\) −3.00000 −0.267261
\(127\) −8.00000 −0.709885 −0.354943 − 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 1.00000 0.0883883
\(129\) −8.00000 −0.704361
\(130\) 4.00000 0.350823
\(131\) −16.0000 −1.39793 −0.698963 − 0.715158i \(-0.746355\pi\)
−0.698963 + 0.715158i \(0.746355\pi\)
\(132\) −2.00000 −0.174078
\(133\) 0 0
\(134\) 12.0000 1.03664
\(135\) −4.00000 −0.344265
\(136\) −4.00000 −0.342997
\(137\) 1.00000 0.0854358 0.0427179 − 0.999087i \(-0.486398\pi\)
0.0427179 + 0.999087i \(0.486398\pi\)
\(138\) 8.00000 0.681005
\(139\) −17.0000 −1.44192 −0.720961 − 0.692976i \(-0.756299\pi\)
−0.720961 + 0.692976i \(0.756299\pi\)
\(140\) −3.00000 −0.253546
\(141\) 12.0000 1.01058
\(142\) 14.0000 1.17485
\(143\) −4.00000 −0.334497
\(144\) 1.00000 0.0833333
\(145\) 10.0000 0.830455
\(146\) 6.00000 0.496564
\(147\) 4.00000 0.329914
\(148\) 3.00000 0.246598
\(149\) −6.00000 −0.491539 −0.245770 − 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) −8.00000 −0.653197
\(151\) 15.0000 1.22068 0.610341 − 0.792139i \(-0.291032\pi\)
0.610341 + 0.792139i \(0.291032\pi\)
\(152\) 0 0
\(153\) −4.00000 −0.323381
\(154\) 3.00000 0.241747
\(155\) 4.00000 0.321288
\(156\) 8.00000 0.640513
\(157\) 5.00000 0.399043 0.199522 − 0.979893i \(-0.436061\pi\)
0.199522 + 0.979893i \(0.436061\pi\)
\(158\) 17.0000 1.35245
\(159\) −2.00000 −0.158610
\(160\) 1.00000 0.0790569
\(161\) −12.0000 −0.945732
\(162\) −11.0000 −0.864242
\(163\) 4.00000 0.313304 0.156652 − 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 6.00000 0.468521
\(165\) −2.00000 −0.155700
\(166\) −13.0000 −1.00900
\(167\) 17.0000 1.31550 0.657750 − 0.753237i \(-0.271508\pi\)
0.657750 + 0.753237i \(0.271508\pi\)
\(168\) −6.00000 −0.462910
\(169\) 3.00000 0.230769
\(170\) −4.00000 −0.306786
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) −4.00000 −0.304114 −0.152057 − 0.988372i \(-0.548590\pi\)
−0.152057 + 0.988372i \(0.548590\pi\)
\(174\) 20.0000 1.51620
\(175\) 12.0000 0.907115
\(176\) −1.00000 −0.0753778
\(177\) 24.0000 1.80395
\(178\) −2.00000 −0.149906
\(179\) −20.0000 −1.49487 −0.747435 − 0.664335i \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) 1.00000 0.0745356
\(181\) −17.0000 −1.26360 −0.631800 − 0.775131i \(-0.717684\pi\)
−0.631800 + 0.775131i \(0.717684\pi\)
\(182\) −12.0000 −0.889499
\(183\) 24.0000 1.77413
\(184\) 4.00000 0.294884
\(185\) 3.00000 0.220564
\(186\) 8.00000 0.586588
\(187\) 4.00000 0.292509
\(188\) 6.00000 0.437595
\(189\) 12.0000 0.872872
\(190\) 0 0
\(191\) 18.0000 1.30243 0.651217 − 0.758891i \(-0.274259\pi\)
0.651217 + 0.758891i \(0.274259\pi\)
\(192\) 2.00000 0.144338
\(193\) −10.0000 −0.719816 −0.359908 − 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) 5.00000 0.358979
\(195\) 8.00000 0.572892
\(196\) 2.00000 0.142857
\(197\) −12.0000 −0.854965 −0.427482 − 0.904024i \(-0.640599\pi\)
−0.427482 + 0.904024i \(0.640599\pi\)
\(198\) −1.00000 −0.0710669
\(199\) −10.0000 −0.708881 −0.354441 − 0.935079i \(-0.615329\pi\)
−0.354441 + 0.935079i \(0.615329\pi\)
\(200\) −4.00000 −0.282843
\(201\) 24.0000 1.69283
\(202\) 0 0
\(203\) −30.0000 −2.10559
\(204\) −8.00000 −0.560112
\(205\) 6.00000 0.419058
\(206\) 0 0
\(207\) 4.00000 0.278019
\(208\) 4.00000 0.277350
\(209\) 0 0
\(210\) −6.00000 −0.414039
\(211\) −23.0000 −1.58339 −0.791693 − 0.610920i \(-0.790800\pi\)
−0.791693 + 0.610920i \(0.790800\pi\)
\(212\) −1.00000 −0.0686803
\(213\) 28.0000 1.91853
\(214\) −7.00000 −0.478510
\(215\) −4.00000 −0.272798
\(216\) −4.00000 −0.272166
\(217\) −12.0000 −0.814613
\(218\) −4.00000 −0.270914
\(219\) 12.0000 0.810885
\(220\) −1.00000 −0.0674200
\(221\) −16.0000 −1.07628
\(222\) 6.00000 0.402694
\(223\) 16.0000 1.07144 0.535720 − 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) −3.00000 −0.200446
\(225\) −4.00000 −0.266667
\(226\) −6.00000 −0.399114
\(227\) −3.00000 −0.199117 −0.0995585 − 0.995032i \(-0.531743\pi\)
−0.0995585 + 0.995032i \(0.531743\pi\)
\(228\) 0 0
\(229\) −13.0000 −0.859064 −0.429532 − 0.903052i \(-0.641321\pi\)
−0.429532 + 0.903052i \(0.641321\pi\)
\(230\) 4.00000 0.263752
\(231\) 6.00000 0.394771
\(232\) 10.0000 0.656532
\(233\) −16.0000 −1.04819 −0.524097 − 0.851658i \(-0.675597\pi\)
−0.524097 + 0.851658i \(0.675597\pi\)
\(234\) 4.00000 0.261488
\(235\) 6.00000 0.391397
\(236\) 12.0000 0.781133
\(237\) 34.0000 2.20854
\(238\) 12.0000 0.777844
\(239\) −5.00000 −0.323423 −0.161712 − 0.986838i \(-0.551701\pi\)
−0.161712 + 0.986838i \(0.551701\pi\)
\(240\) 2.00000 0.129099
\(241\) −10.0000 −0.644157 −0.322078 − 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 1.00000 0.0642824
\(243\) −10.0000 −0.641500
\(244\) 12.0000 0.768221
\(245\) 2.00000 0.127775
\(246\) 12.0000 0.765092
\(247\) 0 0
\(248\) 4.00000 0.254000
\(249\) −26.0000 −1.64768
\(250\) −9.00000 −0.569210
\(251\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) −3.00000 −0.188982
\(253\) −4.00000 −0.251478
\(254\) −8.00000 −0.501965
\(255\) −8.00000 −0.500979
\(256\) 1.00000 0.0625000
\(257\) −1.00000 −0.0623783 −0.0311891 − 0.999514i \(-0.509929\pi\)
−0.0311891 + 0.999514i \(0.509929\pi\)
\(258\) −8.00000 −0.498058
\(259\) −9.00000 −0.559233
\(260\) 4.00000 0.248069
\(261\) 10.0000 0.618984
\(262\) −16.0000 −0.988483
\(263\) −23.0000 −1.41824 −0.709120 − 0.705087i \(-0.750908\pi\)
−0.709120 + 0.705087i \(0.750908\pi\)
\(264\) −2.00000 −0.123091
\(265\) −1.00000 −0.0614295
\(266\) 0 0
\(267\) −4.00000 −0.244796
\(268\) 12.0000 0.733017
\(269\) 15.0000 0.914566 0.457283 − 0.889321i \(-0.348823\pi\)
0.457283 + 0.889321i \(0.348823\pi\)
\(270\) −4.00000 −0.243432
\(271\) −21.0000 −1.27566 −0.637830 − 0.770178i \(-0.720168\pi\)
−0.637830 + 0.770178i \(0.720168\pi\)
\(272\) −4.00000 −0.242536
\(273\) −24.0000 −1.45255
\(274\) 1.00000 0.0604122
\(275\) 4.00000 0.241209
\(276\) 8.00000 0.481543
\(277\) −4.00000 −0.240337 −0.120168 − 0.992754i \(-0.538343\pi\)
−0.120168 + 0.992754i \(0.538343\pi\)
\(278\) −17.0000 −1.01959
\(279\) 4.00000 0.239474
\(280\) −3.00000 −0.179284
\(281\) 4.00000 0.238620 0.119310 − 0.992857i \(-0.461932\pi\)
0.119310 + 0.992857i \(0.461932\pi\)
\(282\) 12.0000 0.714590
\(283\) 13.0000 0.772770 0.386385 − 0.922338i \(-0.373724\pi\)
0.386385 + 0.922338i \(0.373724\pi\)
\(284\) 14.0000 0.830747
\(285\) 0 0
\(286\) −4.00000 −0.236525
\(287\) −18.0000 −1.06251
\(288\) 1.00000 0.0589256
\(289\) −1.00000 −0.0588235
\(290\) 10.0000 0.587220
\(291\) 10.0000 0.586210
\(292\) 6.00000 0.351123
\(293\) 30.0000 1.75262 0.876309 − 0.481749i \(-0.159998\pi\)
0.876309 + 0.481749i \(0.159998\pi\)
\(294\) 4.00000 0.233285
\(295\) 12.0000 0.698667
\(296\) 3.00000 0.174371
\(297\) 4.00000 0.232104
\(298\) −6.00000 −0.347571
\(299\) 16.0000 0.925304
\(300\) −8.00000 −0.461880
\(301\) 12.0000 0.691669
\(302\) 15.0000 0.863153
\(303\) 0 0
\(304\) 0 0
\(305\) 12.0000 0.687118
\(306\) −4.00000 −0.228665
\(307\) 29.0000 1.65512 0.827559 − 0.561379i \(-0.189729\pi\)
0.827559 + 0.561379i \(0.189729\pi\)
\(308\) 3.00000 0.170941
\(309\) 0 0
\(310\) 4.00000 0.227185
\(311\) −18.0000 −1.02069 −0.510343 − 0.859971i \(-0.670482\pi\)
−0.510343 + 0.859971i \(0.670482\pi\)
\(312\) 8.00000 0.452911
\(313\) −9.00000 −0.508710 −0.254355 − 0.967111i \(-0.581863\pi\)
−0.254355 + 0.967111i \(0.581863\pi\)
\(314\) 5.00000 0.282166
\(315\) −3.00000 −0.169031
\(316\) 17.0000 0.956325
\(317\) −18.0000 −1.01098 −0.505490 − 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) −2.00000 −0.112154
\(319\) −10.0000 −0.559893
\(320\) 1.00000 0.0559017
\(321\) −14.0000 −0.781404
\(322\) −12.0000 −0.668734
\(323\) 0 0
\(324\) −11.0000 −0.611111
\(325\) −16.0000 −0.887520
\(326\) 4.00000 0.221540
\(327\) −8.00000 −0.442401
\(328\) 6.00000 0.331295
\(329\) −18.0000 −0.992372
\(330\) −2.00000 −0.110096
\(331\) 20.0000 1.09930 0.549650 − 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) −13.0000 −0.713468
\(333\) 3.00000 0.164399
\(334\) 17.0000 0.930199
\(335\) 12.0000 0.655630
\(336\) −6.00000 −0.327327
\(337\) 28.0000 1.52526 0.762629 − 0.646837i \(-0.223908\pi\)
0.762629 + 0.646837i \(0.223908\pi\)
\(338\) 3.00000 0.163178
\(339\) −12.0000 −0.651751
\(340\) −4.00000 −0.216930
\(341\) −4.00000 −0.216612
\(342\) 0 0
\(343\) 15.0000 0.809924
\(344\) −4.00000 −0.215666
\(345\) 8.00000 0.430706
\(346\) −4.00000 −0.215041
\(347\) 13.0000 0.697877 0.348938 − 0.937146i \(-0.386542\pi\)
0.348938 + 0.937146i \(0.386542\pi\)
\(348\) 20.0000 1.07211
\(349\) −26.0000 −1.39175 −0.695874 − 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) 12.0000 0.641427
\(351\) −16.0000 −0.854017
\(352\) −1.00000 −0.0533002
\(353\) 19.0000 1.01127 0.505634 − 0.862748i \(-0.331259\pi\)
0.505634 + 0.862748i \(0.331259\pi\)
\(354\) 24.0000 1.27559
\(355\) 14.0000 0.743043
\(356\) −2.00000 −0.106000
\(357\) 24.0000 1.27021
\(358\) −20.0000 −1.05703
\(359\) 7.00000 0.369446 0.184723 − 0.982791i \(-0.440861\pi\)
0.184723 + 0.982791i \(0.440861\pi\)
\(360\) 1.00000 0.0527046
\(361\) 0 0
\(362\) −17.0000 −0.893500
\(363\) 2.00000 0.104973
\(364\) −12.0000 −0.628971
\(365\) 6.00000 0.314054
\(366\) 24.0000 1.25450
\(367\) 16.0000 0.835193 0.417597 − 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) 4.00000 0.208514
\(369\) 6.00000 0.312348
\(370\) 3.00000 0.155963
\(371\) 3.00000 0.155752
\(372\) 8.00000 0.414781
\(373\) −28.0000 −1.44979 −0.724893 − 0.688862i \(-0.758111\pi\)
−0.724893 + 0.688862i \(0.758111\pi\)
\(374\) 4.00000 0.206835
\(375\) −18.0000 −0.929516
\(376\) 6.00000 0.309426
\(377\) 40.0000 2.06010
\(378\) 12.0000 0.617213
\(379\) 4.00000 0.205466 0.102733 − 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) 0 0
\(381\) −16.0000 −0.819705
\(382\) 18.0000 0.920960
\(383\) 8.00000 0.408781 0.204390 − 0.978889i \(-0.434479\pi\)
0.204390 + 0.978889i \(0.434479\pi\)
\(384\) 2.00000 0.102062
\(385\) 3.00000 0.152894
\(386\) −10.0000 −0.508987
\(387\) −4.00000 −0.203331
\(388\) 5.00000 0.253837
\(389\) 21.0000 1.06474 0.532371 − 0.846511i \(-0.321301\pi\)
0.532371 + 0.846511i \(0.321301\pi\)
\(390\) 8.00000 0.405096
\(391\) −16.0000 −0.809155
\(392\) 2.00000 0.101015
\(393\) −32.0000 −1.61419
\(394\) −12.0000 −0.604551
\(395\) 17.0000 0.855363
\(396\) −1.00000 −0.0502519
\(397\) 25.0000 1.25471 0.627357 − 0.778732i \(-0.284137\pi\)
0.627357 + 0.778732i \(0.284137\pi\)
\(398\) −10.0000 −0.501255
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) 10.0000 0.499376 0.249688 − 0.968326i \(-0.419672\pi\)
0.249688 + 0.968326i \(0.419672\pi\)
\(402\) 24.0000 1.19701
\(403\) 16.0000 0.797017
\(404\) 0 0
\(405\) −11.0000 −0.546594
\(406\) −30.0000 −1.48888
\(407\) −3.00000 −0.148704
\(408\) −8.00000 −0.396059
\(409\) −20.0000 −0.988936 −0.494468 − 0.869196i \(-0.664637\pi\)
−0.494468 + 0.869196i \(0.664637\pi\)
\(410\) 6.00000 0.296319
\(411\) 2.00000 0.0986527
\(412\) 0 0
\(413\) −36.0000 −1.77144
\(414\) 4.00000 0.196589
\(415\) −13.0000 −0.638145
\(416\) 4.00000 0.196116
\(417\) −34.0000 −1.66499
\(418\) 0 0
\(419\) −38.0000 −1.85642 −0.928211 − 0.372055i \(-0.878653\pi\)
−0.928211 + 0.372055i \(0.878653\pi\)
\(420\) −6.00000 −0.292770
\(421\) −3.00000 −0.146211 −0.0731055 − 0.997324i \(-0.523291\pi\)
−0.0731055 + 0.997324i \(0.523291\pi\)
\(422\) −23.0000 −1.11962
\(423\) 6.00000 0.291730
\(424\) −1.00000 −0.0485643
\(425\) 16.0000 0.776114
\(426\) 28.0000 1.35660
\(427\) −36.0000 −1.74216
\(428\) −7.00000 −0.338358
\(429\) −8.00000 −0.386244
\(430\) −4.00000 −0.192897
\(431\) 13.0000 0.626188 0.313094 − 0.949722i \(-0.398635\pi\)
0.313094 + 0.949722i \(0.398635\pi\)
\(432\) −4.00000 −0.192450
\(433\) −18.0000 −0.865025 −0.432512 − 0.901628i \(-0.642373\pi\)
−0.432512 + 0.901628i \(0.642373\pi\)
\(434\) −12.0000 −0.576018
\(435\) 20.0000 0.958927
\(436\) −4.00000 −0.191565
\(437\) 0 0
\(438\) 12.0000 0.573382
\(439\) 1.00000 0.0477274 0.0238637 − 0.999715i \(-0.492403\pi\)
0.0238637 + 0.999715i \(0.492403\pi\)
\(440\) −1.00000 −0.0476731
\(441\) 2.00000 0.0952381
\(442\) −16.0000 −0.761042
\(443\) 24.0000 1.14027 0.570137 − 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) 6.00000 0.284747
\(445\) −2.00000 −0.0948091
\(446\) 16.0000 0.757622
\(447\) −12.0000 −0.567581
\(448\) −3.00000 −0.141737
\(449\) −33.0000 −1.55737 −0.778683 − 0.627417i \(-0.784112\pi\)
−0.778683 + 0.627417i \(0.784112\pi\)
\(450\) −4.00000 −0.188562
\(451\) −6.00000 −0.282529
\(452\) −6.00000 −0.282216
\(453\) 30.0000 1.40952
\(454\) −3.00000 −0.140797
\(455\) −12.0000 −0.562569
\(456\) 0 0
\(457\) 38.0000 1.77757 0.888783 − 0.458329i \(-0.151552\pi\)
0.888783 + 0.458329i \(0.151552\pi\)
\(458\) −13.0000 −0.607450
\(459\) 16.0000 0.746816
\(460\) 4.00000 0.186501
\(461\) −2.00000 −0.0931493 −0.0465746 − 0.998915i \(-0.514831\pi\)
−0.0465746 + 0.998915i \(0.514831\pi\)
\(462\) 6.00000 0.279145
\(463\) 4.00000 0.185896 0.0929479 − 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) 10.0000 0.464238
\(465\) 8.00000 0.370991
\(466\) −16.0000 −0.741186
\(467\) 14.0000 0.647843 0.323921 − 0.946084i \(-0.394999\pi\)
0.323921 + 0.946084i \(0.394999\pi\)
\(468\) 4.00000 0.184900
\(469\) −36.0000 −1.66233
\(470\) 6.00000 0.276759
\(471\) 10.0000 0.460776
\(472\) 12.0000 0.552345
\(473\) 4.00000 0.183920
\(474\) 34.0000 1.56167
\(475\) 0 0
\(476\) 12.0000 0.550019
\(477\) −1.00000 −0.0457869
\(478\) −5.00000 −0.228695
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 2.00000 0.0912871
\(481\) 12.0000 0.547153
\(482\) −10.0000 −0.455488
\(483\) −24.0000 −1.09204
\(484\) 1.00000 0.0454545
\(485\) 5.00000 0.227038
\(486\) −10.0000 −0.453609
\(487\) 18.0000 0.815658 0.407829 − 0.913058i \(-0.366286\pi\)
0.407829 + 0.913058i \(0.366286\pi\)
\(488\) 12.0000 0.543214
\(489\) 8.00000 0.361773
\(490\) 2.00000 0.0903508
\(491\) 1.00000 0.0451294 0.0225647 − 0.999745i \(-0.492817\pi\)
0.0225647 + 0.999745i \(0.492817\pi\)
\(492\) 12.0000 0.541002
\(493\) −40.0000 −1.80151
\(494\) 0 0
\(495\) −1.00000 −0.0449467
\(496\) 4.00000 0.179605
\(497\) −42.0000 −1.88396
\(498\) −26.0000 −1.16509
\(499\) −40.0000 −1.79065 −0.895323 − 0.445418i \(-0.853055\pi\)
−0.895323 + 0.445418i \(0.853055\pi\)
\(500\) −9.00000 −0.402492
\(501\) 34.0000 1.51901
\(502\) 0 0
\(503\) 8.00000 0.356702 0.178351 − 0.983967i \(-0.442924\pi\)
0.178351 + 0.983967i \(0.442924\pi\)
\(504\) −3.00000 −0.133631
\(505\) 0 0
\(506\) −4.00000 −0.177822
\(507\) 6.00000 0.266469
\(508\) −8.00000 −0.354943
\(509\) −13.0000 −0.576215 −0.288107 − 0.957598i \(-0.593026\pi\)
−0.288107 + 0.957598i \(0.593026\pi\)
\(510\) −8.00000 −0.354246
\(511\) −18.0000 −0.796273
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −1.00000 −0.0441081
\(515\) 0 0
\(516\) −8.00000 −0.352180
\(517\) −6.00000 −0.263880
\(518\) −9.00000 −0.395437
\(519\) −8.00000 −0.351161
\(520\) 4.00000 0.175412
\(521\) 25.0000 1.09527 0.547635 − 0.836717i \(-0.315528\pi\)
0.547635 + 0.836717i \(0.315528\pi\)
\(522\) 10.0000 0.437688
\(523\) 9.00000 0.393543 0.196771 − 0.980449i \(-0.436954\pi\)
0.196771 + 0.980449i \(0.436954\pi\)
\(524\) −16.0000 −0.698963
\(525\) 24.0000 1.04745
\(526\) −23.0000 −1.00285
\(527\) −16.0000 −0.696971
\(528\) −2.00000 −0.0870388
\(529\) −7.00000 −0.304348
\(530\) −1.00000 −0.0434372
\(531\) 12.0000 0.520756
\(532\) 0 0
\(533\) 24.0000 1.03956
\(534\) −4.00000 −0.173097
\(535\) −7.00000 −0.302636
\(536\) 12.0000 0.518321
\(537\) −40.0000 −1.72613
\(538\) 15.0000 0.646696
\(539\) −2.00000 −0.0861461
\(540\) −4.00000 −0.172133
\(541\) −22.0000 −0.945854 −0.472927 − 0.881102i \(-0.656803\pi\)
−0.472927 + 0.881102i \(0.656803\pi\)
\(542\) −21.0000 −0.902027
\(543\) −34.0000 −1.45908
\(544\) −4.00000 −0.171499
\(545\) −4.00000 −0.171341
\(546\) −24.0000 −1.02711
\(547\) 35.0000 1.49649 0.748246 − 0.663421i \(-0.230896\pi\)
0.748246 + 0.663421i \(0.230896\pi\)
\(548\) 1.00000 0.0427179
\(549\) 12.0000 0.512148
\(550\) 4.00000 0.170561
\(551\) 0 0
\(552\) 8.00000 0.340503
\(553\) −51.0000 −2.16874
\(554\) −4.00000 −0.169944
\(555\) 6.00000 0.254686
\(556\) −17.0000 −0.720961
\(557\) −12.0000 −0.508456 −0.254228 − 0.967144i \(-0.581821\pi\)
−0.254228 + 0.967144i \(0.581821\pi\)
\(558\) 4.00000 0.169334
\(559\) −16.0000 −0.676728
\(560\) −3.00000 −0.126773
\(561\) 8.00000 0.337760
\(562\) 4.00000 0.168730
\(563\) −33.0000 −1.39078 −0.695392 − 0.718631i \(-0.744769\pi\)
−0.695392 + 0.718631i \(0.744769\pi\)
\(564\) 12.0000 0.505291
\(565\) −6.00000 −0.252422
\(566\) 13.0000 0.546431
\(567\) 33.0000 1.38587
\(568\) 14.0000 0.587427
\(569\) −36.0000 −1.50920 −0.754599 − 0.656186i \(-0.772169\pi\)
−0.754599 + 0.656186i \(0.772169\pi\)
\(570\) 0 0
\(571\) 19.0000 0.795125 0.397563 − 0.917575i \(-0.369856\pi\)
0.397563 + 0.917575i \(0.369856\pi\)
\(572\) −4.00000 −0.167248
\(573\) 36.0000 1.50392
\(574\) −18.0000 −0.751305
\(575\) −16.0000 −0.667246
\(576\) 1.00000 0.0416667
\(577\) 17.0000 0.707719 0.353860 − 0.935299i \(-0.384869\pi\)
0.353860 + 0.935299i \(0.384869\pi\)
\(578\) −1.00000 −0.0415945
\(579\) −20.0000 −0.831172
\(580\) 10.0000 0.415227
\(581\) 39.0000 1.61799
\(582\) 10.0000 0.414513
\(583\) 1.00000 0.0414158
\(584\) 6.00000 0.248282
\(585\) 4.00000 0.165380
\(586\) 30.0000 1.23929
\(587\) 24.0000 0.990586 0.495293 − 0.868726i \(-0.335061\pi\)
0.495293 + 0.868726i \(0.335061\pi\)
\(588\) 4.00000 0.164957
\(589\) 0 0
\(590\) 12.0000 0.494032
\(591\) −24.0000 −0.987228
\(592\) 3.00000 0.123299
\(593\) −12.0000 −0.492781 −0.246390 − 0.969171i \(-0.579245\pi\)
−0.246390 + 0.969171i \(0.579245\pi\)
\(594\) 4.00000 0.164122
\(595\) 12.0000 0.491952
\(596\) −6.00000 −0.245770
\(597\) −20.0000 −0.818546
\(598\) 16.0000 0.654289
\(599\) 32.0000 1.30748 0.653742 − 0.756717i \(-0.273198\pi\)
0.653742 + 0.756717i \(0.273198\pi\)
\(600\) −8.00000 −0.326599
\(601\) −10.0000 −0.407909 −0.203954 − 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 12.0000 0.489083
\(603\) 12.0000 0.488678
\(604\) 15.0000 0.610341
\(605\) 1.00000 0.0406558
\(606\) 0 0
\(607\) −40.0000 −1.62355 −0.811775 − 0.583970i \(-0.801498\pi\)
−0.811775 + 0.583970i \(0.801498\pi\)
\(608\) 0 0
\(609\) −60.0000 −2.43132
\(610\) 12.0000 0.485866
\(611\) 24.0000 0.970936
\(612\) −4.00000 −0.161690
\(613\) −16.0000 −0.646234 −0.323117 − 0.946359i \(-0.604731\pi\)
−0.323117 + 0.946359i \(0.604731\pi\)
\(614\) 29.0000 1.17034
\(615\) 12.0000 0.483887
\(616\) 3.00000 0.120873
\(617\) 33.0000 1.32853 0.664265 − 0.747497i \(-0.268745\pi\)
0.664265 + 0.747497i \(0.268745\pi\)
\(618\) 0 0
\(619\) −32.0000 −1.28619 −0.643094 − 0.765787i \(-0.722350\pi\)
−0.643094 + 0.765787i \(0.722350\pi\)
\(620\) 4.00000 0.160644
\(621\) −16.0000 −0.642058
\(622\) −18.0000 −0.721734
\(623\) 6.00000 0.240385
\(624\) 8.00000 0.320256
\(625\) 11.0000 0.440000
\(626\) −9.00000 −0.359712
\(627\) 0 0
\(628\) 5.00000 0.199522
\(629\) −12.0000 −0.478471
\(630\) −3.00000 −0.119523
\(631\) 40.0000 1.59237 0.796187 − 0.605050i \(-0.206847\pi\)
0.796187 + 0.605050i \(0.206847\pi\)
\(632\) 17.0000 0.676224
\(633\) −46.0000 −1.82834
\(634\) −18.0000 −0.714871
\(635\) −8.00000 −0.317470
\(636\) −2.00000 −0.0793052
\(637\) 8.00000 0.316972
\(638\) −10.0000 −0.395904
\(639\) 14.0000 0.553831
\(640\) 1.00000 0.0395285
\(641\) −41.0000 −1.61940 −0.809701 − 0.586842i \(-0.800371\pi\)
−0.809701 + 0.586842i \(0.800371\pi\)
\(642\) −14.0000 −0.552536
\(643\) −26.0000 −1.02534 −0.512670 − 0.858586i \(-0.671344\pi\)
−0.512670 + 0.858586i \(0.671344\pi\)
\(644\) −12.0000 −0.472866
\(645\) −8.00000 −0.315000
\(646\) 0 0
\(647\) 42.0000 1.65119 0.825595 − 0.564263i \(-0.190840\pi\)
0.825595 + 0.564263i \(0.190840\pi\)
\(648\) −11.0000 −0.432121
\(649\) −12.0000 −0.471041
\(650\) −16.0000 −0.627572
\(651\) −24.0000 −0.940634
\(652\) 4.00000 0.156652
\(653\) 10.0000 0.391330 0.195665 − 0.980671i \(-0.437313\pi\)
0.195665 + 0.980671i \(0.437313\pi\)
\(654\) −8.00000 −0.312825
\(655\) −16.0000 −0.625172
\(656\) 6.00000 0.234261
\(657\) 6.00000 0.234082
\(658\) −18.0000 −0.701713
\(659\) −31.0000 −1.20759 −0.603794 − 0.797140i \(-0.706345\pi\)
−0.603794 + 0.797140i \(0.706345\pi\)
\(660\) −2.00000 −0.0778499
\(661\) 5.00000 0.194477 0.0972387 − 0.995261i \(-0.468999\pi\)
0.0972387 + 0.995261i \(0.468999\pi\)
\(662\) 20.0000 0.777322
\(663\) −32.0000 −1.24278
\(664\) −13.0000 −0.504498
\(665\) 0 0
\(666\) 3.00000 0.116248
\(667\) 40.0000 1.54881
\(668\) 17.0000 0.657750
\(669\) 32.0000 1.23719
\(670\) 12.0000 0.463600
\(671\) −12.0000 −0.463255
\(672\) −6.00000 −0.231455
\(673\) −42.0000 −1.61898 −0.809491 − 0.587133i \(-0.800257\pi\)
−0.809491 + 0.587133i \(0.800257\pi\)
\(674\) 28.0000 1.07852
\(675\) 16.0000 0.615840
\(676\) 3.00000 0.115385
\(677\) −38.0000 −1.46046 −0.730229 − 0.683202i \(-0.760587\pi\)
−0.730229 + 0.683202i \(0.760587\pi\)
\(678\) −12.0000 −0.460857
\(679\) −15.0000 −0.575647
\(680\) −4.00000 −0.153393
\(681\) −6.00000 −0.229920
\(682\) −4.00000 −0.153168
\(683\) −2.00000 −0.0765279 −0.0382639 − 0.999268i \(-0.512183\pi\)
−0.0382639 + 0.999268i \(0.512183\pi\)
\(684\) 0 0
\(685\) 1.00000 0.0382080
\(686\) 15.0000 0.572703
\(687\) −26.0000 −0.991962
\(688\) −4.00000 −0.152499
\(689\) −4.00000 −0.152388
\(690\) 8.00000 0.304555
\(691\) 16.0000 0.608669 0.304334 − 0.952565i \(-0.401566\pi\)
0.304334 + 0.952565i \(0.401566\pi\)
\(692\) −4.00000 −0.152057
\(693\) 3.00000 0.113961
\(694\) 13.0000 0.493473
\(695\) −17.0000 −0.644847
\(696\) 20.0000 0.758098
\(697\) −24.0000 −0.909065
\(698\) −26.0000 −0.984115
\(699\) −32.0000 −1.21035
\(700\) 12.0000 0.453557
\(701\) 12.0000 0.453234 0.226617 − 0.973984i \(-0.427233\pi\)
0.226617 + 0.973984i \(0.427233\pi\)
\(702\) −16.0000 −0.603881
\(703\) 0 0
\(704\) −1.00000 −0.0376889
\(705\) 12.0000 0.451946
\(706\) 19.0000 0.715074
\(707\) 0 0
\(708\) 24.0000 0.901975
\(709\) 19.0000 0.713560 0.356780 − 0.934188i \(-0.383875\pi\)
0.356780 + 0.934188i \(0.383875\pi\)
\(710\) 14.0000 0.525411
\(711\) 17.0000 0.637550
\(712\) −2.00000 −0.0749532
\(713\) 16.0000 0.599205
\(714\) 24.0000 0.898177
\(715\) −4.00000 −0.149592
\(716\) −20.0000 −0.747435
\(717\) −10.0000 −0.373457
\(718\) 7.00000 0.261238
\(719\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 1.00000 0.0372678
\(721\) 0 0
\(722\) 0 0
\(723\) −20.0000 −0.743808
\(724\) −17.0000 −0.631800
\(725\) −40.0000 −1.48556
\(726\) 2.00000 0.0742270
\(727\) 14.0000 0.519231 0.259616 − 0.965712i \(-0.416404\pi\)
0.259616 + 0.965712i \(0.416404\pi\)
\(728\) −12.0000 −0.444750
\(729\) 13.0000 0.481481
\(730\) 6.00000 0.222070
\(731\) 16.0000 0.591781
\(732\) 24.0000 0.887066
\(733\) −32.0000 −1.18195 −0.590973 − 0.806691i \(-0.701256\pi\)
−0.590973 + 0.806691i \(0.701256\pi\)
\(734\) 16.0000 0.590571
\(735\) 4.00000 0.147542
\(736\) 4.00000 0.147442
\(737\) −12.0000 −0.442026
\(738\) 6.00000 0.220863
\(739\) −4.00000 −0.147142 −0.0735712 − 0.997290i \(-0.523440\pi\)
−0.0735712 + 0.997290i \(0.523440\pi\)
\(740\) 3.00000 0.110282
\(741\) 0 0
\(742\) 3.00000 0.110133
\(743\) 9.00000 0.330178 0.165089 − 0.986279i \(-0.447209\pi\)
0.165089 + 0.986279i \(0.447209\pi\)
\(744\) 8.00000 0.293294
\(745\) −6.00000 −0.219823
\(746\) −28.0000 −1.02515
\(747\) −13.0000 −0.475645
\(748\) 4.00000 0.146254
\(749\) 21.0000 0.767323
\(750\) −18.0000 −0.657267
\(751\) 10.0000 0.364905 0.182453 − 0.983215i \(-0.441596\pi\)
0.182453 + 0.983215i \(0.441596\pi\)
\(752\) 6.00000 0.218797
\(753\) 0 0
\(754\) 40.0000 1.45671
\(755\) 15.0000 0.545906
\(756\) 12.0000 0.436436
\(757\) −22.0000 −0.799604 −0.399802 − 0.916602i \(-0.630921\pi\)
−0.399802 + 0.916602i \(0.630921\pi\)
\(758\) 4.00000 0.145287
\(759\) −8.00000 −0.290382
\(760\) 0 0
\(761\) 22.0000 0.797499 0.398750 − 0.917060i \(-0.369444\pi\)
0.398750 + 0.917060i \(0.369444\pi\)
\(762\) −16.0000 −0.579619
\(763\) 12.0000 0.434429
\(764\) 18.0000 0.651217
\(765\) −4.00000 −0.144620
\(766\) 8.00000 0.289052
\(767\) 48.0000 1.73318
\(768\) 2.00000 0.0721688
\(769\) −34.0000 −1.22607 −0.613036 − 0.790055i \(-0.710052\pi\)
−0.613036 + 0.790055i \(0.710052\pi\)
\(770\) 3.00000 0.108112
\(771\) −2.00000 −0.0720282
\(772\) −10.0000 −0.359908
\(773\) −26.0000 −0.935155 −0.467578 − 0.883952i \(-0.654873\pi\)
−0.467578 + 0.883952i \(0.654873\pi\)
\(774\) −4.00000 −0.143777
\(775\) −16.0000 −0.574737
\(776\) 5.00000 0.179490
\(777\) −18.0000 −0.645746
\(778\) 21.0000 0.752886
\(779\) 0 0
\(780\) 8.00000 0.286446
\(781\) −14.0000 −0.500959
\(782\) −16.0000 −0.572159
\(783\) −40.0000 −1.42948
\(784\) 2.00000 0.0714286
\(785\) 5.00000 0.178458
\(786\) −32.0000 −1.14140
\(787\) 7.00000 0.249523 0.124762 − 0.992187i \(-0.460183\pi\)
0.124762 + 0.992187i \(0.460183\pi\)
\(788\) −12.0000 −0.427482
\(789\) −46.0000 −1.63764
\(790\) 17.0000 0.604833
\(791\) 18.0000 0.640006
\(792\) −1.00000 −0.0355335
\(793\) 48.0000 1.70453
\(794\) 25.0000 0.887217
\(795\) −2.00000 −0.0709327
\(796\) −10.0000 −0.354441
\(797\) 37.0000 1.31061 0.655304 − 0.755366i \(-0.272541\pi\)
0.655304 + 0.755366i \(0.272541\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) −4.00000 −0.141421
\(801\) −2.00000 −0.0706665
\(802\) 10.0000 0.353112
\(803\) −6.00000 −0.211735
\(804\) 24.0000 0.846415
\(805\) −12.0000 −0.422944
\(806\) 16.0000 0.563576
\(807\) 30.0000 1.05605
\(808\) 0 0
\(809\) −28.0000 −0.984428 −0.492214 − 0.870474i \(-0.663812\pi\)
−0.492214 + 0.870474i \(0.663812\pi\)
\(810\) −11.0000 −0.386501
\(811\) 35.0000 1.22902 0.614508 − 0.788911i \(-0.289355\pi\)
0.614508 + 0.788911i \(0.289355\pi\)
\(812\) −30.0000 −1.05279
\(813\) −42.0000 −1.47300
\(814\) −3.00000 −0.105150
\(815\) 4.00000 0.140114
\(816\) −8.00000 −0.280056
\(817\) 0 0
\(818\) −20.0000 −0.699284
\(819\) −12.0000 −0.419314
\(820\) 6.00000 0.209529
\(821\) −52.0000 −1.81481 −0.907406 − 0.420255i \(-0.861941\pi\)
−0.907406 + 0.420255i \(0.861941\pi\)
\(822\) 2.00000 0.0697580
\(823\) 2.00000 0.0697156 0.0348578 − 0.999392i \(-0.488902\pi\)
0.0348578 + 0.999392i \(0.488902\pi\)
\(824\) 0 0
\(825\) 8.00000 0.278524
\(826\) −36.0000 −1.25260
\(827\) −44.0000 −1.53003 −0.765015 − 0.644013i \(-0.777268\pi\)
−0.765015 + 0.644013i \(0.777268\pi\)
\(828\) 4.00000 0.139010
\(829\) 46.0000 1.59765 0.798823 − 0.601566i \(-0.205456\pi\)
0.798823 + 0.601566i \(0.205456\pi\)
\(830\) −13.0000 −0.451237
\(831\) −8.00000 −0.277517
\(832\) 4.00000 0.138675
\(833\) −8.00000 −0.277184
\(834\) −34.0000 −1.17732
\(835\) 17.0000 0.588309
\(836\) 0 0
\(837\) −16.0000 −0.553041
\(838\) −38.0000 −1.31269
\(839\) −56.0000 −1.93333 −0.966667 − 0.256036i \(-0.917584\pi\)
−0.966667 + 0.256036i \(0.917584\pi\)
\(840\) −6.00000 −0.207020
\(841\) 71.0000 2.44828
\(842\) −3.00000 −0.103387
\(843\) 8.00000 0.275535
\(844\) −23.0000 −0.791693
\(845\) 3.00000 0.103203
\(846\) 6.00000 0.206284
\(847\) −3.00000 −0.103081
\(848\) −1.00000 −0.0343401
\(849\) 26.0000 0.892318
\(850\) 16.0000 0.548795
\(851\) 12.0000 0.411355
\(852\) 28.0000 0.959264
\(853\) 42.0000 1.43805 0.719026 − 0.694983i \(-0.244588\pi\)
0.719026 + 0.694983i \(0.244588\pi\)
\(854\) −36.0000 −1.23189
\(855\) 0 0
\(856\) −7.00000 −0.239255
\(857\) 48.0000 1.63965 0.819824 − 0.572615i \(-0.194071\pi\)
0.819824 + 0.572615i \(0.194071\pi\)
\(858\) −8.00000 −0.273115
\(859\) −6.00000 −0.204717 −0.102359 − 0.994748i \(-0.532639\pi\)
−0.102359 + 0.994748i \(0.532639\pi\)
\(860\) −4.00000 −0.136399
\(861\) −36.0000 −1.22688
\(862\) 13.0000 0.442782
\(863\) −6.00000 −0.204242 −0.102121 − 0.994772i \(-0.532563\pi\)
−0.102121 + 0.994772i \(0.532563\pi\)
\(864\) −4.00000 −0.136083
\(865\) −4.00000 −0.136004
\(866\) −18.0000 −0.611665
\(867\) −2.00000 −0.0679236
\(868\) −12.0000 −0.407307
\(869\) −17.0000 −0.576686
\(870\) 20.0000 0.678064
\(871\) 48.0000 1.62642
\(872\) −4.00000 −0.135457
\(873\) 5.00000 0.169224
\(874\) 0 0
\(875\) 27.0000 0.912767
\(876\) 12.0000 0.405442
\(877\) −12.0000 −0.405211 −0.202606 − 0.979260i \(-0.564941\pi\)
−0.202606 + 0.979260i \(0.564941\pi\)
\(878\) 1.00000 0.0337484
\(879\) 60.0000 2.02375
\(880\) −1.00000 −0.0337100
\(881\) −6.00000 −0.202145 −0.101073 − 0.994879i \(-0.532227\pi\)
−0.101073 + 0.994879i \(0.532227\pi\)
\(882\) 2.00000 0.0673435
\(883\) −26.0000 −0.874970 −0.437485 − 0.899226i \(-0.644131\pi\)
−0.437485 + 0.899226i \(0.644131\pi\)
\(884\) −16.0000 −0.538138
\(885\) 24.0000 0.806751
\(886\) 24.0000 0.806296
\(887\) 29.0000 0.973725 0.486862 − 0.873479i \(-0.338141\pi\)
0.486862 + 0.873479i \(0.338141\pi\)
\(888\) 6.00000 0.201347
\(889\) 24.0000 0.804934
\(890\) −2.00000 −0.0670402
\(891\) 11.0000 0.368514
\(892\) 16.0000 0.535720
\(893\) 0 0
\(894\) −12.0000 −0.401340
\(895\) −20.0000 −0.668526
\(896\) −3.00000 −0.100223
\(897\) 32.0000 1.06845
\(898\) −33.0000 −1.10122
\(899\) 40.0000 1.33407
\(900\) −4.00000 −0.133333
\(901\) 4.00000 0.133259
\(902\) −6.00000 −0.199778
\(903\) 24.0000 0.798670
\(904\) −6.00000 −0.199557
\(905\) −17.0000 −0.565099
\(906\) 30.0000 0.996683
\(907\) −2.00000 −0.0664089 −0.0332045 − 0.999449i \(-0.510571\pi\)
−0.0332045 + 0.999449i \(0.510571\pi\)
\(908\) −3.00000 −0.0995585
\(909\) 0 0
\(910\) −12.0000 −0.397796
\(911\) 30.0000 0.993944 0.496972 − 0.867766i \(-0.334445\pi\)
0.496972 + 0.867766i \(0.334445\pi\)
\(912\) 0 0
\(913\) 13.0000 0.430237
\(914\) 38.0000 1.25693
\(915\) 24.0000 0.793416
\(916\) −13.0000 −0.429532
\(917\) 48.0000 1.58510
\(918\) 16.0000 0.528079
\(919\) 45.0000 1.48441 0.742207 − 0.670171i \(-0.233779\pi\)
0.742207 + 0.670171i \(0.233779\pi\)
\(920\) 4.00000 0.131876
\(921\) 58.0000 1.91116
\(922\) −2.00000 −0.0658665
\(923\) 56.0000 1.84326
\(924\) 6.00000 0.197386
\(925\) −12.0000 −0.394558
\(926\) 4.00000 0.131448
\(927\) 0 0
\(928\) 10.0000 0.328266
\(929\) 14.0000 0.459325 0.229663 − 0.973270i \(-0.426238\pi\)
0.229663 + 0.973270i \(0.426238\pi\)
\(930\) 8.00000 0.262330
\(931\) 0 0
\(932\) −16.0000 −0.524097
\(933\) −36.0000 −1.17859
\(934\) 14.0000 0.458094
\(935\) 4.00000 0.130814
\(936\) 4.00000 0.130744
\(937\) −38.0000 −1.24141 −0.620703 − 0.784046i \(-0.713153\pi\)
−0.620703 + 0.784046i \(0.713153\pi\)
\(938\) −36.0000 −1.17544
\(939\) −18.0000 −0.587408
\(940\) 6.00000 0.195698
\(941\) −24.0000 −0.782378 −0.391189 − 0.920310i \(-0.627936\pi\)
−0.391189 + 0.920310i \(0.627936\pi\)
\(942\) 10.0000 0.325818
\(943\) 24.0000 0.781548
\(944\) 12.0000 0.390567
\(945\) 12.0000 0.390360
\(946\) 4.00000 0.130051
\(947\) −30.0000 −0.974869 −0.487435 − 0.873160i \(-0.662067\pi\)
−0.487435 + 0.873160i \(0.662067\pi\)
\(948\) 34.0000 1.10427
\(949\) 24.0000 0.779073
\(950\) 0 0
\(951\) −36.0000 −1.16738
\(952\) 12.0000 0.388922
\(953\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) −1.00000 −0.0323762
\(955\) 18.0000 0.582466
\(956\) −5.00000 −0.161712
\(957\) −20.0000 −0.646508
\(958\) 0 0
\(959\) −3.00000 −0.0968751
\(960\) 2.00000 0.0645497
\(961\) −15.0000 −0.483871
\(962\) 12.0000 0.386896
\(963\) −7.00000 −0.225572
\(964\) −10.0000 −0.322078
\(965\) −10.0000 −0.321911
\(966\) −24.0000 −0.772187
\(967\) −37.0000 −1.18984 −0.594920 − 0.803785i \(-0.702816\pi\)
−0.594920 + 0.803785i \(0.702816\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) 5.00000 0.160540
\(971\) −50.0000 −1.60458 −0.802288 − 0.596937i \(-0.796384\pi\)
−0.802288 + 0.596937i \(0.796384\pi\)
\(972\) −10.0000 −0.320750
\(973\) 51.0000 1.63498
\(974\) 18.0000 0.576757
\(975\) −32.0000 −1.02482
\(976\) 12.0000 0.384111
\(977\) 9.00000 0.287936 0.143968 − 0.989582i \(-0.454014\pi\)
0.143968 + 0.989582i \(0.454014\pi\)
\(978\) 8.00000 0.255812
\(979\) 2.00000 0.0639203
\(980\) 2.00000 0.0638877
\(981\) −4.00000 −0.127710
\(982\) 1.00000 0.0319113
\(983\) 38.0000 1.21201 0.606006 − 0.795460i \(-0.292771\pi\)
0.606006 + 0.795460i \(0.292771\pi\)
\(984\) 12.0000 0.382546
\(985\) −12.0000 −0.382352
\(986\) −40.0000 −1.27386
\(987\) −36.0000 −1.14589
\(988\) 0 0
\(989\) −16.0000 −0.508770
\(990\) −1.00000 −0.0317821
\(991\) 38.0000 1.20711 0.603555 − 0.797321i \(-0.293750\pi\)
0.603555 + 0.797321i \(0.293750\pi\)
\(992\) 4.00000 0.127000
\(993\) 40.0000 1.26936
\(994\) −42.0000 −1.33216
\(995\) −10.0000 −0.317021
\(996\) −26.0000 −0.823842
\(997\) 44.0000 1.39349 0.696747 − 0.717317i \(-0.254630\pi\)
0.696747 + 0.717317i \(0.254630\pi\)
\(998\) −40.0000 −1.26618
\(999\) −12.0000 −0.379663
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 7942.2.a.s.1.1 1
19.7 even 3 418.2.e.a.353.1 yes 2
19.11 even 3 418.2.e.a.45.1 ✓ 2
19.18 odd 2 7942.2.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.e.a.45.1 ✓ 2 19.11 even 3
418.2.e.a.353.1 yes 2 19.7 even 3
7942.2.a.c.1.1 1 19.18 odd 2
7942.2.a.s.1.1 1 1.1 even 1 trivial