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Properties

Label 690.2.a.g.1.1
Level $690$
Weight $2$
Character 690.1
Self dual yes
Analytic conductor $5.510$
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] = [690,2,Mod(1,690)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(690, 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("690.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 690.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: \(5.50967773947\)
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 \(\chi\) \(=\) 690.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -2.00000 q^{11} -1.00000 q^{12} -6.00000 q^{13} -2.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -4.00000 q^{17} +1.00000 q^{18} -1.00000 q^{20} +2.00000 q^{21} -2.00000 q^{22} +1.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} -6.00000 q^{26} -1.00000 q^{27} -2.00000 q^{28} +2.00000 q^{29} +1.00000 q^{30} +1.00000 q^{32} +2.00000 q^{33} -4.00000 q^{34} +2.00000 q^{35} +1.00000 q^{36} -8.00000 q^{37} +6.00000 q^{39} -1.00000 q^{40} -6.00000 q^{41} +2.00000 q^{42} -4.00000 q^{43} -2.00000 q^{44} -1.00000 q^{45} +1.00000 q^{46} -1.00000 q^{48} -3.00000 q^{49} +1.00000 q^{50} +4.00000 q^{51} -6.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} +2.00000 q^{55} -2.00000 q^{56} +2.00000 q^{58} +1.00000 q^{60} -8.00000 q^{61} -2.00000 q^{63} +1.00000 q^{64} +6.00000 q^{65} +2.00000 q^{66} -4.00000 q^{67} -4.00000 q^{68} -1.00000 q^{69} +2.00000 q^{70} +16.0000 q^{71} +1.00000 q^{72} +6.00000 q^{73} -8.00000 q^{74} -1.00000 q^{75} +4.00000 q^{77} +6.00000 q^{78} +14.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} +14.0000 q^{83} +2.00000 q^{84} +4.00000 q^{85} -4.00000 q^{86} -2.00000 q^{87} -2.00000 q^{88} -8.00000 q^{89} -1.00000 q^{90} +12.0000 q^{91} +1.00000 q^{92} -1.00000 q^{96} -6.00000 q^{97} -3.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −1.00000 −0.408248
\(7\) −2.00000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) −2.00000 −0.603023 −0.301511 − 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) −1.00000 −0.288675
\(13\) −6.00000 −1.66410 −0.832050 − 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) −2.00000 −0.534522
\(15\) 1.00000 0.258199
\(16\) 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 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −1.00000 −0.223607
\(21\) 2.00000 0.436436
\(22\) −2.00000 −0.426401
\(23\) 1.00000 0.208514
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) −6.00000 −1.17670
\(27\) −1.00000 −0.192450
\(28\) −2.00000 −0.377964
\(29\) 2.00000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 1.00000 0.182574
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000 0.176777
\(33\) 2.00000 0.348155
\(34\) −4.00000 −0.685994
\(35\) 2.00000 0.338062
\(36\) 1.00000 0.166667
\(37\) −8.00000 −1.31519 −0.657596 − 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 0 0
\(39\) 6.00000 0.960769
\(40\) −1.00000 −0.158114
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 2.00000 0.308607
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −2.00000 −0.301511
\(45\) −1.00000 −0.149071
\(46\) 1.00000 0.147442
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.00000 −0.144338
\(49\) −3.00000 −0.428571
\(50\) 1.00000 0.141421
\(51\) 4.00000 0.560112
\(52\) −6.00000 −0.832050
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) −1.00000 −0.136083
\(55\) 2.00000 0.269680
\(56\) −2.00000 −0.267261
\(57\) 0 0
\(58\) 2.00000 0.262613
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 1.00000 0.129099
\(61\) −8.00000 −1.02430 −0.512148 − 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 0 0
\(63\) −2.00000 −0.251976
\(64\) 1.00000 0.125000
\(65\) 6.00000 0.744208
\(66\) 2.00000 0.246183
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −4.00000 −0.485071
\(69\) −1.00000 −0.120386
\(70\) 2.00000 0.239046
\(71\) 16.0000 1.89885 0.949425 − 0.313993i \(-0.101667\pi\)
0.949425 + 0.313993i \(0.101667\pi\)
\(72\) 1.00000 0.117851
\(73\) 6.00000 0.702247 0.351123 − 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) −8.00000 −0.929981
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 4.00000 0.455842
\(78\) 6.00000 0.679366
\(79\) 14.0000 1.57512 0.787562 − 0.616236i \(-0.211343\pi\)
0.787562 + 0.616236i \(0.211343\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) 14.0000 1.53670 0.768350 − 0.640030i \(-0.221078\pi\)
0.768350 + 0.640030i \(0.221078\pi\)
\(84\) 2.00000 0.218218
\(85\) 4.00000 0.433861
\(86\) −4.00000 −0.431331
\(87\) −2.00000 −0.214423
\(88\) −2.00000 −0.213201
\(89\) −8.00000 −0.847998 −0.423999 − 0.905663i \(-0.639374\pi\)
−0.423999 + 0.905663i \(0.639374\pi\)
\(90\) −1.00000 −0.105409
\(91\) 12.0000 1.25794
\(92\) 1.00000 0.104257
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −6.00000 −0.609208 −0.304604 − 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) −3.00000 −0.303046
\(99\) −2.00000 −0.201008
\(100\) 1.00000 0.100000
\(101\) 6.00000 0.597022 0.298511 − 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 4.00000 0.396059
\(103\) −2.00000 −0.197066 −0.0985329 − 0.995134i \(-0.531415\pi\)
−0.0985329 + 0.995134i \(0.531415\pi\)
\(104\) −6.00000 −0.588348
\(105\) −2.00000 −0.195180
\(106\) 6.00000 0.582772
\(107\) −18.0000 −1.74013 −0.870063 − 0.492941i \(-0.835922\pi\)
−0.870063 + 0.492941i \(0.835922\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −4.00000 −0.383131 −0.191565 − 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 2.00000 0.190693
\(111\) 8.00000 0.759326
\(112\) −2.00000 −0.188982
\(113\) 12.0000 1.12887 0.564433 − 0.825479i \(-0.309095\pi\)
0.564433 + 0.825479i \(0.309095\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) 2.00000 0.185695
\(117\) −6.00000 −0.554700
\(118\) 0 0
\(119\) 8.00000 0.733359
\(120\) 1.00000 0.0912871
\(121\) −7.00000 −0.636364
\(122\) −8.00000 −0.724286
\(123\) 6.00000 0.541002
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) −2.00000 −0.178174
\(127\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 1.00000 0.0883883
\(129\) 4.00000 0.352180
\(130\) 6.00000 0.526235
\(131\) 12.0000 1.04844 0.524222 − 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 2.00000 0.174078
\(133\) 0 0
\(134\) −4.00000 −0.345547
\(135\) 1.00000 0.0860663
\(136\) −4.00000 −0.342997
\(137\) −8.00000 −0.683486 −0.341743 − 0.939793i \(-0.611017\pi\)
−0.341743 + 0.939793i \(0.611017\pi\)
\(138\) −1.00000 −0.0851257
\(139\) −4.00000 −0.339276 −0.169638 − 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 2.00000 0.169031
\(141\) 0 0
\(142\) 16.0000 1.34269
\(143\) 12.0000 1.00349
\(144\) 1.00000 0.0833333
\(145\) −2.00000 −0.166091
\(146\) 6.00000 0.496564
\(147\) 3.00000 0.247436
\(148\) −8.00000 −0.657596
\(149\) 10.0000 0.819232 0.409616 − 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 12.0000 0.976546 0.488273 − 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) 0 0
\(153\) −4.00000 −0.323381
\(154\) 4.00000 0.322329
\(155\) 0 0
\(156\) 6.00000 0.480384
\(157\) −16.0000 −1.27694 −0.638470 − 0.769647i \(-0.720432\pi\)
−0.638470 + 0.769647i \(0.720432\pi\)
\(158\) 14.0000 1.11378
\(159\) −6.00000 −0.475831
\(160\) −1.00000 −0.0790569
\(161\) −2.00000 −0.157622
\(162\) 1.00000 0.0785674
\(163\) −4.00000 −0.313304 −0.156652 − 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) −6.00000 −0.468521
\(165\) −2.00000 −0.155700
\(166\) 14.0000 1.08661
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 2.00000 0.154303
\(169\) 23.0000 1.76923
\(170\) 4.00000 0.306786
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) 18.0000 1.36851 0.684257 − 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) −2.00000 −0.151620
\(175\) −2.00000 −0.151186
\(176\) −2.00000 −0.150756
\(177\) 0 0
\(178\) −8.00000 −0.599625
\(179\) −20.0000 −1.49487 −0.747435 − 0.664335i \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −12.0000 −0.891953 −0.445976 − 0.895045i \(-0.647144\pi\)
−0.445976 + 0.895045i \(0.647144\pi\)
\(182\) 12.0000 0.889499
\(183\) 8.00000 0.591377
\(184\) 1.00000 0.0737210
\(185\) 8.00000 0.588172
\(186\) 0 0
\(187\) 8.00000 0.585018
\(188\) 0 0
\(189\) 2.00000 0.145479
\(190\) 0 0
\(191\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −2.00000 −0.143963 −0.0719816 − 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) −6.00000 −0.430775
\(195\) −6.00000 −0.429669
\(196\) −3.00000 −0.214286
\(197\) 6.00000 0.427482 0.213741 − 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) −2.00000 −0.142134
\(199\) −2.00000 −0.141776 −0.0708881 − 0.997484i \(-0.522583\pi\)
−0.0708881 + 0.997484i \(0.522583\pi\)
\(200\) 1.00000 0.0707107
\(201\) 4.00000 0.282138
\(202\) 6.00000 0.422159
\(203\) −4.00000 −0.280745
\(204\) 4.00000 0.280056
\(205\) 6.00000 0.419058
\(206\) −2.00000 −0.139347
\(207\) 1.00000 0.0695048
\(208\) −6.00000 −0.416025
\(209\) 0 0
\(210\) −2.00000 −0.138013
\(211\) −20.0000 −1.37686 −0.688428 − 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 6.00000 0.412082
\(213\) −16.0000 −1.09630
\(214\) −18.0000 −1.23045
\(215\) 4.00000 0.272798
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −4.00000 −0.270914
\(219\) −6.00000 −0.405442
\(220\) 2.00000 0.134840
\(221\) 24.0000 1.61441
\(222\) 8.00000 0.536925
\(223\) −16.0000 −1.07144 −0.535720 − 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) −2.00000 −0.133631
\(225\) 1.00000 0.0666667
\(226\) 12.0000 0.798228
\(227\) −18.0000 −1.19470 −0.597351 − 0.801980i \(-0.703780\pi\)
−0.597351 + 0.801980i \(0.703780\pi\)
\(228\) 0 0
\(229\) −20.0000 −1.32164 −0.660819 − 0.750546i \(-0.729791\pi\)
−0.660819 + 0.750546i \(0.729791\pi\)
\(230\) −1.00000 −0.0659380
\(231\) −4.00000 −0.263181
\(232\) 2.00000 0.131306
\(233\) −26.0000 −1.70332 −0.851658 − 0.524097i \(-0.824403\pi\)
−0.851658 + 0.524097i \(0.824403\pi\)
\(234\) −6.00000 −0.392232
\(235\) 0 0
\(236\) 0 0
\(237\) −14.0000 −0.909398
\(238\) 8.00000 0.518563
\(239\) 16.0000 1.03495 0.517477 − 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) 1.00000 0.0645497
\(241\) −14.0000 −0.901819 −0.450910 − 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) −7.00000 −0.449977
\(243\) −1.00000 −0.0641500
\(244\) −8.00000 −0.512148
\(245\) 3.00000 0.191663
\(246\) 6.00000 0.382546
\(247\) 0 0
\(248\) 0 0
\(249\) −14.0000 −0.887214
\(250\) −1.00000 −0.0632456
\(251\) 6.00000 0.378717 0.189358 − 0.981908i \(-0.439359\pi\)
0.189358 + 0.981908i \(0.439359\pi\)
\(252\) −2.00000 −0.125988
\(253\) −2.00000 −0.125739
\(254\) 0 0
\(255\) −4.00000 −0.250490
\(256\) 1.00000 0.0625000
\(257\) 14.0000 0.873296 0.436648 − 0.899632i \(-0.356166\pi\)
0.436648 + 0.899632i \(0.356166\pi\)
\(258\) 4.00000 0.249029
\(259\) 16.0000 0.994192
\(260\) 6.00000 0.372104
\(261\) 2.00000 0.123797
\(262\) 12.0000 0.741362
\(263\) 4.00000 0.246651 0.123325 − 0.992366i \(-0.460644\pi\)
0.123325 + 0.992366i \(0.460644\pi\)
\(264\) 2.00000 0.123091
\(265\) −6.00000 −0.368577
\(266\) 0 0
\(267\) 8.00000 0.489592
\(268\) −4.00000 −0.244339
\(269\) −18.0000 −1.09748 −0.548740 − 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 1.00000 0.0608581
\(271\) 20.0000 1.21491 0.607457 − 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) −4.00000 −0.242536
\(273\) −12.0000 −0.726273
\(274\) −8.00000 −0.483298
\(275\) −2.00000 −0.120605
\(276\) −1.00000 −0.0601929
\(277\) −10.0000 −0.600842 −0.300421 − 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) −4.00000 −0.239904
\(279\) 0 0
\(280\) 2.00000 0.119523
\(281\) 8.00000 0.477240 0.238620 − 0.971113i \(-0.423305\pi\)
0.238620 + 0.971113i \(0.423305\pi\)
\(282\) 0 0
\(283\) −16.0000 −0.951101 −0.475551 − 0.879688i \(-0.657751\pi\)
−0.475551 + 0.879688i \(0.657751\pi\)
\(284\) 16.0000 0.949425
\(285\) 0 0
\(286\) 12.0000 0.709575
\(287\) 12.0000 0.708338
\(288\) 1.00000 0.0589256
\(289\) −1.00000 −0.0588235
\(290\) −2.00000 −0.117444
\(291\) 6.00000 0.351726
\(292\) 6.00000 0.351123
\(293\) −10.0000 −0.584206 −0.292103 − 0.956387i \(-0.594355\pi\)
−0.292103 + 0.956387i \(0.594355\pi\)
\(294\) 3.00000 0.174964
\(295\) 0 0
\(296\) −8.00000 −0.464991
\(297\) 2.00000 0.116052
\(298\) 10.0000 0.579284
\(299\) −6.00000 −0.346989
\(300\) −1.00000 −0.0577350
\(301\) 8.00000 0.461112
\(302\) 12.0000 0.690522
\(303\) −6.00000 −0.344691
\(304\) 0 0
\(305\) 8.00000 0.458079
\(306\) −4.00000 −0.228665
\(307\) 28.0000 1.59804 0.799022 − 0.601302i \(-0.205351\pi\)
0.799022 + 0.601302i \(0.205351\pi\)
\(308\) 4.00000 0.227921
\(309\) 2.00000 0.113776
\(310\) 0 0
\(311\) −32.0000 −1.81455 −0.907277 − 0.420534i \(-0.861843\pi\)
−0.907277 + 0.420534i \(0.861843\pi\)
\(312\) 6.00000 0.339683
\(313\) −34.0000 −1.92179 −0.960897 − 0.276907i \(-0.910691\pi\)
−0.960897 + 0.276907i \(0.910691\pi\)
\(314\) −16.0000 −0.902932
\(315\) 2.00000 0.112687
\(316\) 14.0000 0.787562
\(317\) −22.0000 −1.23564 −0.617822 − 0.786318i \(-0.711985\pi\)
−0.617822 + 0.786318i \(0.711985\pi\)
\(318\) −6.00000 −0.336463
\(319\) −4.00000 −0.223957
\(320\) −1.00000 −0.0559017
\(321\) 18.0000 1.00466
\(322\) −2.00000 −0.111456
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) −6.00000 −0.332820
\(326\) −4.00000 −0.221540
\(327\) 4.00000 0.221201
\(328\) −6.00000 −0.331295
\(329\) 0 0
\(330\) −2.00000 −0.110096
\(331\) 20.0000 1.09930 0.549650 − 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 14.0000 0.768350
\(333\) −8.00000 −0.438397
\(334\) 0 0
\(335\) 4.00000 0.218543
\(336\) 2.00000 0.109109
\(337\) −34.0000 −1.85210 −0.926049 − 0.377403i \(-0.876817\pi\)
−0.926049 + 0.377403i \(0.876817\pi\)
\(338\) 23.0000 1.25104
\(339\) −12.0000 −0.651751
\(340\) 4.00000 0.216930
\(341\) 0 0
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) −4.00000 −0.215666
\(345\) 1.00000 0.0538382
\(346\) 18.0000 0.967686
\(347\) −8.00000 −0.429463 −0.214731 − 0.976673i \(-0.568888\pi\)
−0.214731 + 0.976673i \(0.568888\pi\)
\(348\) −2.00000 −0.107211
\(349\) −6.00000 −0.321173 −0.160586 − 0.987022i \(-0.551338\pi\)
−0.160586 + 0.987022i \(0.551338\pi\)
\(350\) −2.00000 −0.106904
\(351\) 6.00000 0.320256
\(352\) −2.00000 −0.106600
\(353\) 26.0000 1.38384 0.691920 − 0.721974i \(-0.256765\pi\)
0.691920 + 0.721974i \(0.256765\pi\)
\(354\) 0 0
\(355\) −16.0000 −0.849192
\(356\) −8.00000 −0.423999
\(357\) −8.00000 −0.423405
\(358\) −20.0000 −1.05703
\(359\) 24.0000 1.26667 0.633336 − 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −19.0000 −1.00000
\(362\) −12.0000 −0.630706
\(363\) 7.00000 0.367405
\(364\) 12.0000 0.628971
\(365\) −6.00000 −0.314054
\(366\) 8.00000 0.418167
\(367\) 10.0000 0.521996 0.260998 − 0.965339i \(-0.415948\pi\)
0.260998 + 0.965339i \(0.415948\pi\)
\(368\) 1.00000 0.0521286
\(369\) −6.00000 −0.312348
\(370\) 8.00000 0.415900
\(371\) −12.0000 −0.623009
\(372\) 0 0
\(373\) 32.0000 1.65690 0.828449 − 0.560065i \(-0.189224\pi\)
0.828449 + 0.560065i \(0.189224\pi\)
\(374\) 8.00000 0.413670
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) 2.00000 0.102869
\(379\) 16.0000 0.821865 0.410932 − 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −12.0000 −0.613171 −0.306586 − 0.951843i \(-0.599187\pi\)
−0.306586 + 0.951843i \(0.599187\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −4.00000 −0.203859
\(386\) −2.00000 −0.101797
\(387\) −4.00000 −0.203331
\(388\) −6.00000 −0.304604
\(389\) 10.0000 0.507020 0.253510 − 0.967333i \(-0.418415\pi\)
0.253510 + 0.967333i \(0.418415\pi\)
\(390\) −6.00000 −0.303822
\(391\) −4.00000 −0.202289
\(392\) −3.00000 −0.151523
\(393\) −12.0000 −0.605320
\(394\) 6.00000 0.302276
\(395\) −14.0000 −0.704416
\(396\) −2.00000 −0.100504
\(397\) −22.0000 −1.10415 −0.552074 − 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) −2.00000 −0.100251
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −24.0000 −1.19850 −0.599251 − 0.800561i \(-0.704535\pi\)
−0.599251 + 0.800561i \(0.704535\pi\)
\(402\) 4.00000 0.199502
\(403\) 0 0
\(404\) 6.00000 0.298511
\(405\) −1.00000 −0.0496904
\(406\) −4.00000 −0.198517
\(407\) 16.0000 0.793091
\(408\) 4.00000 0.198030
\(409\) 38.0000 1.87898 0.939490 − 0.342578i \(-0.111300\pi\)
0.939490 + 0.342578i \(0.111300\pi\)
\(410\) 6.00000 0.296319
\(411\) 8.00000 0.394611
\(412\) −2.00000 −0.0985329
\(413\) 0 0
\(414\) 1.00000 0.0491473
\(415\) −14.0000 −0.687233
\(416\) −6.00000 −0.294174
\(417\) 4.00000 0.195881
\(418\) 0 0
\(419\) −14.0000 −0.683945 −0.341972 − 0.939710i \(-0.611095\pi\)
−0.341972 + 0.939710i \(0.611095\pi\)
\(420\) −2.00000 −0.0975900
\(421\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(422\) −20.0000 −0.973585
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) −4.00000 −0.194029
\(426\) −16.0000 −0.775203
\(427\) 16.0000 0.774294
\(428\) −18.0000 −0.870063
\(429\) −12.0000 −0.579365
\(430\) 4.00000 0.192897
\(431\) −36.0000 −1.73406 −0.867029 − 0.498257i \(-0.833974\pi\)
−0.867029 + 0.498257i \(0.833974\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 30.0000 1.44171 0.720854 − 0.693087i \(-0.243750\pi\)
0.720854 + 0.693087i \(0.243750\pi\)
\(434\) 0 0
\(435\) 2.00000 0.0958927
\(436\) −4.00000 −0.191565
\(437\) 0 0
\(438\) −6.00000 −0.286691
\(439\) 32.0000 1.52728 0.763638 − 0.645644i \(-0.223411\pi\)
0.763638 + 0.645644i \(0.223411\pi\)
\(440\) 2.00000 0.0953463
\(441\) −3.00000 −0.142857
\(442\) 24.0000 1.14156
\(443\) 4.00000 0.190046 0.0950229 − 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) 8.00000 0.379663
\(445\) 8.00000 0.379236
\(446\) −16.0000 −0.757622
\(447\) −10.0000 −0.472984
\(448\) −2.00000 −0.0944911
\(449\) −22.0000 −1.03824 −0.519122 − 0.854700i \(-0.673741\pi\)
−0.519122 + 0.854700i \(0.673741\pi\)
\(450\) 1.00000 0.0471405
\(451\) 12.0000 0.565058
\(452\) 12.0000 0.564433
\(453\) −12.0000 −0.563809
\(454\) −18.0000 −0.844782
\(455\) −12.0000 −0.562569
\(456\) 0 0
\(457\) 18.0000 0.842004 0.421002 − 0.907060i \(-0.361678\pi\)
0.421002 + 0.907060i \(0.361678\pi\)
\(458\) −20.0000 −0.934539
\(459\) 4.00000 0.186704
\(460\) −1.00000 −0.0466252
\(461\) −10.0000 −0.465746 −0.232873 − 0.972507i \(-0.574813\pi\)
−0.232873 + 0.972507i \(0.574813\pi\)
\(462\) −4.00000 −0.186097
\(463\) −36.0000 −1.67306 −0.836531 − 0.547920i \(-0.815420\pi\)
−0.836531 + 0.547920i \(0.815420\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) −26.0000 −1.20443
\(467\) −30.0000 −1.38823 −0.694117 − 0.719862i \(-0.744205\pi\)
−0.694117 + 0.719862i \(0.744205\pi\)
\(468\) −6.00000 −0.277350
\(469\) 8.00000 0.369406
\(470\) 0 0
\(471\) 16.0000 0.737241
\(472\) 0 0
\(473\) 8.00000 0.367840
\(474\) −14.0000 −0.643041
\(475\) 0 0
\(476\) 8.00000 0.366679
\(477\) 6.00000 0.274721
\(478\) 16.0000 0.731823
\(479\) 8.00000 0.365529 0.182765 − 0.983157i \(-0.441495\pi\)
0.182765 + 0.983157i \(0.441495\pi\)
\(480\) 1.00000 0.0456435
\(481\) 48.0000 2.18861
\(482\) −14.0000 −0.637683
\(483\) 2.00000 0.0910032
\(484\) −7.00000 −0.318182
\(485\) 6.00000 0.272446
\(486\) −1.00000 −0.0453609
\(487\) 12.0000 0.543772 0.271886 − 0.962329i \(-0.412353\pi\)
0.271886 + 0.962329i \(0.412353\pi\)
\(488\) −8.00000 −0.362143
\(489\) 4.00000 0.180886
\(490\) 3.00000 0.135526
\(491\) −12.0000 −0.541552 −0.270776 − 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 6.00000 0.270501
\(493\) −8.00000 −0.360302
\(494\) 0 0
\(495\) 2.00000 0.0898933
\(496\) 0 0
\(497\) −32.0000 −1.43540
\(498\) −14.0000 −0.627355
\(499\) −28.0000 −1.25345 −0.626726 − 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) 6.00000 0.267793
\(503\) 4.00000 0.178351 0.0891756 − 0.996016i \(-0.471577\pi\)
0.0891756 + 0.996016i \(0.471577\pi\)
\(504\) −2.00000 −0.0890871
\(505\) −6.00000 −0.266996
\(506\) −2.00000 −0.0889108
\(507\) −23.0000 −1.02147
\(508\) 0 0
\(509\) −14.0000 −0.620539 −0.310270 − 0.950649i \(-0.600419\pi\)
−0.310270 + 0.950649i \(0.600419\pi\)
\(510\) −4.00000 −0.177123
\(511\) −12.0000 −0.530849
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 14.0000 0.617514
\(515\) 2.00000 0.0881305
\(516\) 4.00000 0.176090
\(517\) 0 0
\(518\) 16.0000 0.703000
\(519\) −18.0000 −0.790112
\(520\) 6.00000 0.263117
\(521\) −4.00000 −0.175243 −0.0876216 − 0.996154i \(-0.527927\pi\)
−0.0876216 + 0.996154i \(0.527927\pi\)
\(522\) 2.00000 0.0875376
\(523\) 4.00000 0.174908 0.0874539 − 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) 12.0000 0.524222
\(525\) 2.00000 0.0872872
\(526\) 4.00000 0.174408
\(527\) 0 0
\(528\) 2.00000 0.0870388
\(529\) 1.00000 0.0434783
\(530\) −6.00000 −0.260623
\(531\) 0 0
\(532\) 0 0
\(533\) 36.0000 1.55933
\(534\) 8.00000 0.346194
\(535\) 18.0000 0.778208
\(536\) −4.00000 −0.172774
\(537\) 20.0000 0.863064
\(538\) −18.0000 −0.776035
\(539\) 6.00000 0.258438
\(540\) 1.00000 0.0430331
\(541\) 26.0000 1.11783 0.558914 − 0.829226i \(-0.311218\pi\)
0.558914 + 0.829226i \(0.311218\pi\)
\(542\) 20.0000 0.859074
\(543\) 12.0000 0.514969
\(544\) −4.00000 −0.171499
\(545\) 4.00000 0.171341
\(546\) −12.0000 −0.513553
\(547\) 12.0000 0.513083 0.256541 − 0.966533i \(-0.417417\pi\)
0.256541 + 0.966533i \(0.417417\pi\)
\(548\) −8.00000 −0.341743
\(549\) −8.00000 −0.341432
\(550\) −2.00000 −0.0852803
\(551\) 0 0
\(552\) −1.00000 −0.0425628
\(553\) −28.0000 −1.19068
\(554\) −10.0000 −0.424859
\(555\) −8.00000 −0.339581
\(556\) −4.00000 −0.169638
\(557\) −22.0000 −0.932170 −0.466085 − 0.884740i \(-0.654336\pi\)
−0.466085 + 0.884740i \(0.654336\pi\)
\(558\) 0 0
\(559\) 24.0000 1.01509
\(560\) 2.00000 0.0845154
\(561\) −8.00000 −0.337760
\(562\) 8.00000 0.337460
\(563\) −10.0000 −0.421450 −0.210725 − 0.977545i \(-0.567582\pi\)
−0.210725 + 0.977545i \(0.567582\pi\)
\(564\) 0 0
\(565\) −12.0000 −0.504844
\(566\) −16.0000 −0.672530
\(567\) −2.00000 −0.0839921
\(568\) 16.0000 0.671345
\(569\) 24.0000 1.00613 0.503066 − 0.864248i \(-0.332205\pi\)
0.503066 + 0.864248i \(0.332205\pi\)
\(570\) 0 0
\(571\) 12.0000 0.502184 0.251092 − 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) 12.0000 0.501745
\(573\) 0 0
\(574\) 12.0000 0.500870
\(575\) 1.00000 0.0417029
\(576\) 1.00000 0.0416667
\(577\) 2.00000 0.0832611 0.0416305 − 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) −1.00000 −0.0415945
\(579\) 2.00000 0.0831172
\(580\) −2.00000 −0.0830455
\(581\) −28.0000 −1.16164
\(582\) 6.00000 0.248708
\(583\) −12.0000 −0.496989
\(584\) 6.00000 0.248282
\(585\) 6.00000 0.248069
\(586\) −10.0000 −0.413096
\(587\) 8.00000 0.330195 0.165098 − 0.986277i \(-0.447206\pi\)
0.165098 + 0.986277i \(0.447206\pi\)
\(588\) 3.00000 0.123718
\(589\) 0 0
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) −8.00000 −0.328798
\(593\) 30.0000 1.23195 0.615976 − 0.787765i \(-0.288762\pi\)
0.615976 + 0.787765i \(0.288762\pi\)
\(594\) 2.00000 0.0820610
\(595\) −8.00000 −0.327968
\(596\) 10.0000 0.409616
\(597\) 2.00000 0.0818546
\(598\) −6.00000 −0.245358
\(599\) −24.0000 −0.980613 −0.490307 − 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −22.0000 −0.897399 −0.448699 − 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 8.00000 0.326056
\(603\) −4.00000 −0.162893
\(604\) 12.0000 0.488273
\(605\) 7.00000 0.284590
\(606\) −6.00000 −0.243733
\(607\) 40.0000 1.62355 0.811775 − 0.583970i \(-0.198502\pi\)
0.811775 + 0.583970i \(0.198502\pi\)
\(608\) 0 0
\(609\) 4.00000 0.162088
\(610\) 8.00000 0.323911
\(611\) 0 0
\(612\) −4.00000 −0.161690
\(613\) −4.00000 −0.161558 −0.0807792 − 0.996732i \(-0.525741\pi\)
−0.0807792 + 0.996732i \(0.525741\pi\)
\(614\) 28.0000 1.12999
\(615\) −6.00000 −0.241943
\(616\) 4.00000 0.161165
\(617\) 28.0000 1.12724 0.563619 − 0.826035i \(-0.309409\pi\)
0.563619 + 0.826035i \(0.309409\pi\)
\(618\) 2.00000 0.0804518
\(619\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) −32.0000 −1.28308
\(623\) 16.0000 0.641026
\(624\) 6.00000 0.240192
\(625\) 1.00000 0.0400000
\(626\) −34.0000 −1.35891
\(627\) 0 0
\(628\) −16.0000 −0.638470
\(629\) 32.0000 1.27592
\(630\) 2.00000 0.0796819
\(631\) −22.0000 −0.875806 −0.437903 − 0.899022i \(-0.644279\pi\)
−0.437903 + 0.899022i \(0.644279\pi\)
\(632\) 14.0000 0.556890
\(633\) 20.0000 0.794929
\(634\) −22.0000 −0.873732
\(635\) 0 0
\(636\) −6.00000 −0.237915
\(637\) 18.0000 0.713186
\(638\) −4.00000 −0.158362
\(639\) 16.0000 0.632950
\(640\) −1.00000 −0.0395285
\(641\) 16.0000 0.631962 0.315981 − 0.948766i \(-0.397666\pi\)
0.315981 + 0.948766i \(0.397666\pi\)
\(642\) 18.0000 0.710403
\(643\) −28.0000 −1.10421 −0.552106 − 0.833774i \(-0.686176\pi\)
−0.552106 + 0.833774i \(0.686176\pi\)
\(644\) −2.00000 −0.0788110
\(645\) −4.00000 −0.157500
\(646\) 0 0
\(647\) −24.0000 −0.943537 −0.471769 − 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) −6.00000 −0.235339
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) 22.0000 0.860927 0.430463 − 0.902608i \(-0.358350\pi\)
0.430463 + 0.902608i \(0.358350\pi\)
\(654\) 4.00000 0.156412
\(655\) −12.0000 −0.468879
\(656\) −6.00000 −0.234261
\(657\) 6.00000 0.234082
\(658\) 0 0
\(659\) 6.00000 0.233727 0.116863 − 0.993148i \(-0.462716\pi\)
0.116863 + 0.993148i \(0.462716\pi\)
\(660\) −2.00000 −0.0778499
\(661\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(662\) 20.0000 0.777322
\(663\) −24.0000 −0.932083
\(664\) 14.0000 0.543305
\(665\) 0 0
\(666\) −8.00000 −0.309994
\(667\) 2.00000 0.0774403
\(668\) 0 0
\(669\) 16.0000 0.618596
\(670\) 4.00000 0.154533
\(671\) 16.0000 0.617673
\(672\) 2.00000 0.0771517
\(673\) 42.0000 1.61898 0.809491 − 0.587133i \(-0.199743\pi\)
0.809491 + 0.587133i \(0.199743\pi\)
\(674\) −34.0000 −1.30963
\(675\) −1.00000 −0.0384900
\(676\) 23.0000 0.884615
\(677\) 2.00000 0.0768662 0.0384331 − 0.999261i \(-0.487763\pi\)
0.0384331 + 0.999261i \(0.487763\pi\)
\(678\) −12.0000 −0.460857
\(679\) 12.0000 0.460518
\(680\) 4.00000 0.153393
\(681\) 18.0000 0.689761
\(682\) 0 0
\(683\) 48.0000 1.83667 0.918334 − 0.395805i \(-0.129534\pi\)
0.918334 + 0.395805i \(0.129534\pi\)
\(684\) 0 0
\(685\) 8.00000 0.305664
\(686\) 20.0000 0.763604
\(687\) 20.0000 0.763048
\(688\) −4.00000 −0.152499
\(689\) −36.0000 −1.37149
\(690\) 1.00000 0.0380693
\(691\) 28.0000 1.06517 0.532585 − 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) 18.0000 0.684257
\(693\) 4.00000 0.151947
\(694\) −8.00000 −0.303676
\(695\) 4.00000 0.151729
\(696\) −2.00000 −0.0758098
\(697\) 24.0000 0.909065
\(698\) −6.00000 −0.227103
\(699\) 26.0000 0.983410
\(700\) −2.00000 −0.0755929
\(701\) 30.0000 1.13308 0.566542 − 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 6.00000 0.226455
\(703\) 0 0
\(704\) −2.00000 −0.0753778
\(705\) 0 0
\(706\) 26.0000 0.978523
\(707\) −12.0000 −0.451306
\(708\) 0 0
\(709\) 16.0000 0.600893 0.300446 − 0.953799i \(-0.402864\pi\)
0.300446 + 0.953799i \(0.402864\pi\)
\(710\) −16.0000 −0.600469
\(711\) 14.0000 0.525041
\(712\) −8.00000 −0.299813
\(713\) 0 0
\(714\) −8.00000 −0.299392
\(715\) −12.0000 −0.448775
\(716\) −20.0000 −0.747435
\(717\) −16.0000 −0.597531
\(718\) 24.0000 0.895672
\(719\) 24.0000 0.895049 0.447524 − 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 4.00000 0.148968
\(722\) −19.0000 −0.707107
\(723\) 14.0000 0.520666
\(724\) −12.0000 −0.445976
\(725\) 2.00000 0.0742781
\(726\) 7.00000 0.259794
\(727\) 38.0000 1.40934 0.704671 − 0.709534i \(-0.251095\pi\)
0.704671 + 0.709534i \(0.251095\pi\)
\(728\) 12.0000 0.444750
\(729\) 1.00000 0.0370370
\(730\) −6.00000 −0.222070
\(731\) 16.0000 0.591781
\(732\) 8.00000 0.295689
\(733\) 36.0000 1.32969 0.664845 − 0.746981i \(-0.268498\pi\)
0.664845 + 0.746981i \(0.268498\pi\)
\(734\) 10.0000 0.369107
\(735\) −3.00000 −0.110657
\(736\) 1.00000 0.0368605
\(737\) 8.00000 0.294684
\(738\) −6.00000 −0.220863
\(739\) 12.0000 0.441427 0.220714 − 0.975339i \(-0.429161\pi\)
0.220714 + 0.975339i \(0.429161\pi\)
\(740\) 8.00000 0.294086
\(741\) 0 0
\(742\) −12.0000 −0.440534
\(743\) 4.00000 0.146746 0.0733729 − 0.997305i \(-0.476624\pi\)
0.0733729 + 0.997305i \(0.476624\pi\)
\(744\) 0 0
\(745\) −10.0000 −0.366372
\(746\) 32.0000 1.17160
\(747\) 14.0000 0.512233
\(748\) 8.00000 0.292509
\(749\) 36.0000 1.31541
\(750\) 1.00000 0.0365148
\(751\) −50.0000 −1.82453 −0.912263 − 0.409605i \(-0.865667\pi\)
−0.912263 + 0.409605i \(0.865667\pi\)
\(752\) 0 0
\(753\) −6.00000 −0.218652
\(754\) −12.0000 −0.437014
\(755\) −12.0000 −0.436725
\(756\) 2.00000 0.0727393
\(757\) 40.0000 1.45382 0.726912 − 0.686730i \(-0.240955\pi\)
0.726912 + 0.686730i \(0.240955\pi\)
\(758\) 16.0000 0.581146
\(759\) 2.00000 0.0725954
\(760\) 0 0
\(761\) −10.0000 −0.362500 −0.181250 − 0.983437i \(-0.558014\pi\)
−0.181250 + 0.983437i \(0.558014\pi\)
\(762\) 0 0
\(763\) 8.00000 0.289619
\(764\) 0 0
\(765\) 4.00000 0.144620
\(766\) −12.0000 −0.433578
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 6.00000 0.216366 0.108183 − 0.994131i \(-0.465497\pi\)
0.108183 + 0.994131i \(0.465497\pi\)
\(770\) −4.00000 −0.144150
\(771\) −14.0000 −0.504198
\(772\) −2.00000 −0.0719816
\(773\) −18.0000 −0.647415 −0.323708 − 0.946157i \(-0.604929\pi\)
−0.323708 + 0.946157i \(0.604929\pi\)
\(774\) −4.00000 −0.143777
\(775\) 0 0
\(776\) −6.00000 −0.215387
\(777\) −16.0000 −0.573997
\(778\) 10.0000 0.358517
\(779\) 0 0
\(780\) −6.00000 −0.214834
\(781\) −32.0000 −1.14505
\(782\) −4.00000 −0.143040
\(783\) −2.00000 −0.0714742
\(784\) −3.00000 −0.107143
\(785\) 16.0000 0.571064
\(786\) −12.0000 −0.428026
\(787\) −4.00000 −0.142585 −0.0712923 − 0.997455i \(-0.522712\pi\)
−0.0712923 + 0.997455i \(0.522712\pi\)
\(788\) 6.00000 0.213741
\(789\) −4.00000 −0.142404
\(790\) −14.0000 −0.498098
\(791\) −24.0000 −0.853342
\(792\) −2.00000 −0.0710669
\(793\) 48.0000 1.70453
\(794\) −22.0000 −0.780751
\(795\) 6.00000 0.212798
\(796\) −2.00000 −0.0708881
\(797\) −46.0000 −1.62940 −0.814702 − 0.579880i \(-0.803099\pi\)
−0.814702 + 0.579880i \(0.803099\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) −8.00000 −0.282666
\(802\) −24.0000 −0.847469
\(803\) −12.0000 −0.423471
\(804\) 4.00000 0.141069
\(805\) 2.00000 0.0704907
\(806\) 0 0
\(807\) 18.0000 0.633630
\(808\) 6.00000 0.211079
\(809\) −18.0000 −0.632846 −0.316423 − 0.948618i \(-0.602482\pi\)
−0.316423 + 0.948618i \(0.602482\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −44.0000 −1.54505 −0.772524 − 0.634985i \(-0.781006\pi\)
−0.772524 + 0.634985i \(0.781006\pi\)
\(812\) −4.00000 −0.140372
\(813\) −20.0000 −0.701431
\(814\) 16.0000 0.560800
\(815\) 4.00000 0.140114
\(816\) 4.00000 0.140028
\(817\) 0 0
\(818\) 38.0000 1.32864
\(819\) 12.0000 0.419314
\(820\) 6.00000 0.209529
\(821\) −10.0000 −0.349002 −0.174501 − 0.984657i \(-0.555831\pi\)
−0.174501 + 0.984657i \(0.555831\pi\)
\(822\) 8.00000 0.279032
\(823\) −4.00000 −0.139431 −0.0697156 − 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) −2.00000 −0.0696733
\(825\) 2.00000 0.0696311
\(826\) 0 0
\(827\) −22.0000 −0.765015 −0.382507 − 0.923952i \(-0.624939\pi\)
−0.382507 + 0.923952i \(0.624939\pi\)
\(828\) 1.00000 0.0347524
\(829\) −38.0000 −1.31979 −0.659897 − 0.751356i \(-0.729400\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(830\) −14.0000 −0.485947
\(831\) 10.0000 0.346896
\(832\) −6.00000 −0.208013
\(833\) 12.0000 0.415775
\(834\) 4.00000 0.138509
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) −14.0000 −0.483622
\(839\) −20.0000 −0.690477 −0.345238 − 0.938515i \(-0.612202\pi\)
−0.345238 + 0.938515i \(0.612202\pi\)
\(840\) −2.00000 −0.0690066
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) −8.00000 −0.275535
\(844\) −20.0000 −0.688428
\(845\) −23.0000 −0.791224
\(846\) 0 0
\(847\) 14.0000 0.481046
\(848\) 6.00000 0.206041
\(849\) 16.0000 0.549119
\(850\) −4.00000 −0.137199
\(851\) −8.00000 −0.274236
\(852\) −16.0000 −0.548151
\(853\) 26.0000 0.890223 0.445112 − 0.895475i \(-0.353164\pi\)
0.445112 + 0.895475i \(0.353164\pi\)
\(854\) 16.0000 0.547509
\(855\) 0 0
\(856\) −18.0000 −0.615227
\(857\) 30.0000 1.02478 0.512390 − 0.858753i \(-0.328760\pi\)
0.512390 + 0.858753i \(0.328760\pi\)
\(858\) −12.0000 −0.409673
\(859\) −12.0000 −0.409435 −0.204717 − 0.978821i \(-0.565628\pi\)
−0.204717 + 0.978821i \(0.565628\pi\)
\(860\) 4.00000 0.136399
\(861\) −12.0000 −0.408959
\(862\) −36.0000 −1.22616
\(863\) 24.0000 0.816970 0.408485 − 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −18.0000 −0.612018
\(866\) 30.0000 1.01944
\(867\) 1.00000 0.0339618
\(868\) 0 0
\(869\) −28.0000 −0.949835
\(870\) 2.00000 0.0678064
\(871\) 24.0000 0.813209
\(872\) −4.00000 −0.135457
\(873\) −6.00000 −0.203069
\(874\) 0 0
\(875\) 2.00000 0.0676123
\(876\) −6.00000 −0.202721
\(877\) −2.00000 −0.0675352 −0.0337676 − 0.999430i \(-0.510751\pi\)
−0.0337676 + 0.999430i \(0.510751\pi\)
\(878\) 32.0000 1.07995
\(879\) 10.0000 0.337292
\(880\) 2.00000 0.0674200
\(881\) −56.0000 −1.88669 −0.943344 − 0.331816i \(-0.892339\pi\)
−0.943344 + 0.331816i \(0.892339\pi\)
\(882\) −3.00000 −0.101015
\(883\) −36.0000 −1.21150 −0.605748 − 0.795656i \(-0.707126\pi\)
−0.605748 + 0.795656i \(0.707126\pi\)
\(884\) 24.0000 0.807207
\(885\) 0 0
\(886\) 4.00000 0.134383
\(887\) 48.0000 1.61168 0.805841 − 0.592132i \(-0.201714\pi\)
0.805841 + 0.592132i \(0.201714\pi\)
\(888\) 8.00000 0.268462
\(889\) 0 0
\(890\) 8.00000 0.268161
\(891\) −2.00000 −0.0670025
\(892\) −16.0000 −0.535720
\(893\) 0 0
\(894\) −10.0000 −0.334450
\(895\) 20.0000 0.668526
\(896\) −2.00000 −0.0668153
\(897\) 6.00000 0.200334
\(898\) −22.0000 −0.734150
\(899\) 0 0
\(900\) 1.00000 0.0333333
\(901\) −24.0000 −0.799556
\(902\) 12.0000 0.399556
\(903\) −8.00000 −0.266223
\(904\) 12.0000 0.399114
\(905\) 12.0000 0.398893
\(906\) −12.0000 −0.398673
\(907\) −24.0000 −0.796907 −0.398453 − 0.917189i \(-0.630453\pi\)
−0.398453 + 0.917189i \(0.630453\pi\)
\(908\) −18.0000 −0.597351
\(909\) 6.00000 0.199007
\(910\) −12.0000 −0.397796
\(911\) 36.0000 1.19273 0.596367 − 0.802712i \(-0.296610\pi\)
0.596367 + 0.802712i \(0.296610\pi\)
\(912\) 0 0
\(913\) −28.0000 −0.926665
\(914\) 18.0000 0.595387
\(915\) −8.00000 −0.264472
\(916\) −20.0000 −0.660819
\(917\) −24.0000 −0.792550
\(918\) 4.00000 0.132020
\(919\) −2.00000 −0.0659739 −0.0329870 − 0.999456i \(-0.510502\pi\)
−0.0329870 + 0.999456i \(0.510502\pi\)
\(920\) −1.00000 −0.0329690
\(921\) −28.0000 −0.922631
\(922\) −10.0000 −0.329332
\(923\) −96.0000 −3.15988
\(924\) −4.00000 −0.131590
\(925\) −8.00000 −0.263038
\(926\) −36.0000 −1.18303
\(927\) −2.00000 −0.0656886
\(928\) 2.00000 0.0656532
\(929\) 18.0000 0.590561 0.295280 − 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −26.0000 −0.851658
\(933\) 32.0000 1.04763
\(934\) −30.0000 −0.981630
\(935\) −8.00000 −0.261628
\(936\) −6.00000 −0.196116
\(937\) 42.0000 1.37208 0.686040 − 0.727564i \(-0.259347\pi\)
0.686040 + 0.727564i \(0.259347\pi\)
\(938\) 8.00000 0.261209
\(939\) 34.0000 1.10955
\(940\) 0 0
\(941\) −22.0000 −0.717180 −0.358590 − 0.933495i \(-0.616742\pi\)
−0.358590 + 0.933495i \(0.616742\pi\)
\(942\) 16.0000 0.521308
\(943\) −6.00000 −0.195387
\(944\) 0 0
\(945\) −2.00000 −0.0650600
\(946\) 8.00000 0.260102
\(947\) 8.00000 0.259965 0.129983 − 0.991516i \(-0.458508\pi\)
0.129983 + 0.991516i \(0.458508\pi\)
\(948\) −14.0000 −0.454699
\(949\) −36.0000 −1.16861
\(950\) 0 0
\(951\) 22.0000 0.713399
\(952\) 8.00000 0.259281
\(953\) 44.0000 1.42530 0.712650 − 0.701520i \(-0.247495\pi\)
0.712650 + 0.701520i \(0.247495\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) 16.0000 0.517477
\(957\) 4.00000 0.129302
\(958\) 8.00000 0.258468
\(959\) 16.0000 0.516667
\(960\) 1.00000 0.0322749
\(961\) −31.0000 −1.00000
\(962\) 48.0000 1.54758
\(963\) −18.0000 −0.580042
\(964\) −14.0000 −0.450910
\(965\) 2.00000 0.0643823
\(966\) 2.00000 0.0643489
\(967\) 52.0000 1.67221 0.836104 − 0.548572i \(-0.184828\pi\)
0.836104 + 0.548572i \(0.184828\pi\)
\(968\) −7.00000 −0.224989
\(969\) 0 0
\(970\) 6.00000 0.192648
\(971\) −46.0000 −1.47621 −0.738105 − 0.674686i \(-0.764279\pi\)
−0.738105 + 0.674686i \(0.764279\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 8.00000 0.256468
\(974\) 12.0000 0.384505
\(975\) 6.00000 0.192154
\(976\) −8.00000 −0.256074
\(977\) 24.0000 0.767828 0.383914 − 0.923369i \(-0.374576\pi\)
0.383914 + 0.923369i \(0.374576\pi\)
\(978\) 4.00000 0.127906
\(979\) 16.0000 0.511362
\(980\) 3.00000 0.0958315
\(981\) −4.00000 −0.127710
\(982\) −12.0000 −0.382935
\(983\) −12.0000 −0.382741 −0.191370 − 0.981518i \(-0.561293\pi\)
−0.191370 + 0.981518i \(0.561293\pi\)
\(984\) 6.00000 0.191273
\(985\) −6.00000 −0.191176
\(986\) −8.00000 −0.254772
\(987\) 0 0
\(988\) 0 0
\(989\) −4.00000 −0.127193
\(990\) 2.00000 0.0635642
\(991\) −32.0000 −1.01651 −0.508257 − 0.861206i \(-0.669710\pi\)
−0.508257 + 0.861206i \(0.669710\pi\)
\(992\) 0 0
\(993\) −20.0000 −0.634681
\(994\) −32.0000 −1.01498
\(995\) 2.00000 0.0634043
\(996\) −14.0000 −0.443607
\(997\) −2.00000 −0.0633406 −0.0316703 − 0.999498i \(-0.510083\pi\)
−0.0316703 + 0.999498i \(0.510083\pi\)
\(998\) −28.0000 −0.886325
\(999\) 8.00000 0.253109
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 690.2.a.g.1.1 ✓ 1
3.2 odd 2 2070.2.a.e.1.1 1
4.3 odd 2 5520.2.a.z.1.1 1
5.2 odd 4 3450.2.d.d.2899.2 2
5.3 odd 4 3450.2.d.d.2899.1 2
5.4 even 2 3450.2.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.a.g.1.1 ✓ 1 1.1 even 1 trivial
2070.2.a.e.1.1 1 3.2 odd 2
3450.2.a.l.1.1 1 5.4 even 2
3450.2.d.d.2899.1 2 5.3 odd 4
3450.2.d.d.2899.2 2 5.2 odd 4
5520.2.a.z.1.1 1 4.3 odd 2