Properties

Label 322.2.a.c.1.1
Level $322$
Weight $2$
Character 322.1
Self dual yes
Analytic conductor $2.571$
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] = [322,2,Mod(1,322)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(322, 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("322.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 322 = 2 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 322.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: \(2.57118294509\)
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 \(\chi\) \(=\) 322.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -2.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -2.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -2.00000 q^{11} -2.00000 q^{12} -4.00000 q^{13} -1.00000 q^{14} +4.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} +1.00000 q^{18} -2.00000 q^{20} +2.00000 q^{21} -2.00000 q^{22} +1.00000 q^{23} -2.00000 q^{24} -1.00000 q^{25} -4.00000 q^{26} +4.00000 q^{27} -1.00000 q^{28} -2.00000 q^{29} +4.00000 q^{30} +4.00000 q^{31} +1.00000 q^{32} +4.00000 q^{33} -6.00000 q^{34} +2.00000 q^{35} +1.00000 q^{36} +8.00000 q^{39} -2.00000 q^{40} +6.00000 q^{41} +2.00000 q^{42} +6.00000 q^{43} -2.00000 q^{44} -2.00000 q^{45} +1.00000 q^{46} -2.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} +12.0000 q^{51} -4.00000 q^{52} -12.0000 q^{53} +4.00000 q^{54} +4.00000 q^{55} -1.00000 q^{56} -2.00000 q^{58} -10.0000 q^{59} +4.00000 q^{60} +2.00000 q^{61} +4.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +8.00000 q^{65} +4.00000 q^{66} -2.00000 q^{67} -6.00000 q^{68} -2.00000 q^{69} +2.00000 q^{70} +8.00000 q^{71} +1.00000 q^{72} +2.00000 q^{73} +2.00000 q^{75} +2.00000 q^{77} +8.00000 q^{78} +8.00000 q^{79} -2.00000 q^{80} -11.0000 q^{81} +6.00000 q^{82} -16.0000 q^{83} +2.00000 q^{84} +12.0000 q^{85} +6.00000 q^{86} +4.00000 q^{87} -2.00000 q^{88} +6.00000 q^{89} -2.00000 q^{90} +4.00000 q^{91} +1.00000 q^{92} -8.00000 q^{93} -2.00000 q^{96} -2.00000 q^{97} +1.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −2.00000 −1.15470 −0.577350 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 1.00000 0.500000
\(5\) −2.00000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) −2.00000 −0.816497
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −2.00000 −0.632456
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) −2.00000 −0.577350
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) −1.00000 −0.267261
\(15\) 4.00000 1.03280
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 1.00000 0.235702
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −2.00000 −0.447214
\(21\) 2.00000 0.436436
\(22\) −2.00000 −0.426401
\(23\) 1.00000 0.208514
\(24\) −2.00000 −0.408248
\(25\) −1.00000 −0.200000
\(26\) −4.00000 −0.784465
\(27\) 4.00000 0.769800
\(28\) −1.00000 −0.188982
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 4.00000 0.730297
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 1.00000 0.176777
\(33\) 4.00000 0.696311
\(34\) −6.00000 −1.02899
\(35\) 2.00000 0.338062
\(36\) 1.00000 0.166667
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 8.00000 1.28103
\(40\) −2.00000 −0.316228
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 2.00000 0.308607
\(43\) 6.00000 0.914991 0.457496 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) −2.00000 −0.301511
\(45\) −2.00000 −0.298142
\(46\) 1.00000 0.147442
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −2.00000 −0.288675
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 12.0000 1.68034
\(52\) −4.00000 −0.554700
\(53\) −12.0000 −1.64833 −0.824163 0.566352i \(-0.808354\pi\)
−0.824163 + 0.566352i \(0.808354\pi\)
\(54\) 4.00000 0.544331
\(55\) 4.00000 0.539360
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −2.00000 −0.262613
\(59\) −10.0000 −1.30189 −0.650945 0.759125i \(-0.725627\pi\)
−0.650945 + 0.759125i \(0.725627\pi\)
\(60\) 4.00000 0.516398
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 4.00000 0.508001
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 8.00000 0.992278
\(66\) 4.00000 0.492366
\(67\) −2.00000 −0.244339 −0.122169 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) −6.00000 −0.727607
\(69\) −2.00000 −0.240772
\(70\) 2.00000 0.239046
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 1.00000 0.117851
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 0 0
\(75\) 2.00000 0.230940
\(76\) 0 0
\(77\) 2.00000 0.227921
\(78\) 8.00000 0.905822
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) −2.00000 −0.223607
\(81\) −11.0000 −1.22222
\(82\) 6.00000 0.662589
\(83\) −16.0000 −1.75623 −0.878114 0.478451i \(-0.841198\pi\)
−0.878114 + 0.478451i \(0.841198\pi\)
\(84\) 2.00000 0.218218
\(85\) 12.0000 1.30158
\(86\) 6.00000 0.646997
\(87\) 4.00000 0.428845
\(88\) −2.00000 −0.213201
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) −2.00000 −0.210819
\(91\) 4.00000 0.419314
\(92\) 1.00000 0.104257
\(93\) −8.00000 −0.829561
\(94\) 0 0
\(95\) 0 0
\(96\) −2.00000 −0.204124
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 1.00000 0.101015
\(99\) −2.00000 −0.201008
\(100\) −1.00000 −0.100000
\(101\) −8.00000 −0.796030 −0.398015 0.917379i \(-0.630301\pi\)
−0.398015 + 0.917379i \(0.630301\pi\)
\(102\) 12.0000 1.18818
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) −4.00000 −0.392232
\(105\) −4.00000 −0.390360
\(106\) −12.0000 −1.16554
\(107\) −18.0000 −1.74013 −0.870063 0.492941i \(-0.835922\pi\)
−0.870063 + 0.492941i \(0.835922\pi\)
\(108\) 4.00000 0.384900
\(109\) −20.0000 −1.91565 −0.957826 0.287348i \(-0.907226\pi\)
−0.957826 + 0.287348i \(0.907226\pi\)
\(110\) 4.00000 0.381385
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) −2.00000 −0.186501
\(116\) −2.00000 −0.185695
\(117\) −4.00000 −0.369800
\(118\) −10.0000 −0.920575
\(119\) 6.00000 0.550019
\(120\) 4.00000 0.365148
\(121\) −7.00000 −0.636364
\(122\) 2.00000 0.181071
\(123\) −12.0000 −1.08200
\(124\) 4.00000 0.359211
\(125\) 12.0000 1.07331
\(126\) −1.00000 −0.0890871
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 1.00000 0.0883883
\(129\) −12.0000 −1.05654
\(130\) 8.00000 0.701646
\(131\) 22.0000 1.92215 0.961074 0.276289i \(-0.0891049\pi\)
0.961074 + 0.276289i \(0.0891049\pi\)
\(132\) 4.00000 0.348155
\(133\) 0 0
\(134\) −2.00000 −0.172774
\(135\) −8.00000 −0.688530
\(136\) −6.00000 −0.514496
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) −2.00000 −0.170251
\(139\) −14.0000 −1.18746 −0.593732 0.804663i \(-0.702346\pi\)
−0.593732 + 0.804663i \(0.702346\pi\)
\(140\) 2.00000 0.169031
\(141\) 0 0
\(142\) 8.00000 0.671345
\(143\) 8.00000 0.668994
\(144\) 1.00000 0.0833333
\(145\) 4.00000 0.332182
\(146\) 2.00000 0.165521
\(147\) −2.00000 −0.164957
\(148\) 0 0
\(149\) −12.0000 −0.983078 −0.491539 0.870855i \(-0.663566\pi\)
−0.491539 + 0.870855i \(0.663566\pi\)
\(150\) 2.00000 0.163299
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) −6.00000 −0.485071
\(154\) 2.00000 0.161165
\(155\) −8.00000 −0.642575
\(156\) 8.00000 0.640513
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) 8.00000 0.636446
\(159\) 24.0000 1.90332
\(160\) −2.00000 −0.158114
\(161\) −1.00000 −0.0788110
\(162\) −11.0000 −0.864242
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 6.00000 0.468521
\(165\) −8.00000 −0.622799
\(166\) −16.0000 −1.24184
\(167\) −16.0000 −1.23812 −0.619059 0.785345i \(-0.712486\pi\)
−0.619059 + 0.785345i \(0.712486\pi\)
\(168\) 2.00000 0.154303
\(169\) 3.00000 0.230769
\(170\) 12.0000 0.920358
\(171\) 0 0
\(172\) 6.00000 0.457496
\(173\) −24.0000 −1.82469 −0.912343 0.409426i \(-0.865729\pi\)
−0.912343 + 0.409426i \(0.865729\pi\)
\(174\) 4.00000 0.303239
\(175\) 1.00000 0.0755929
\(176\) −2.00000 −0.150756
\(177\) 20.0000 1.50329
\(178\) 6.00000 0.449719
\(179\) 16.0000 1.19590 0.597948 0.801535i \(-0.295983\pi\)
0.597948 + 0.801535i \(0.295983\pi\)
\(180\) −2.00000 −0.149071
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 4.00000 0.296500
\(183\) −4.00000 −0.295689
\(184\) 1.00000 0.0737210
\(185\) 0 0
\(186\) −8.00000 −0.586588
\(187\) 12.0000 0.877527
\(188\) 0 0
\(189\) −4.00000 −0.290957
\(190\) 0 0
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) −2.00000 −0.144338
\(193\) −18.0000 −1.29567 −0.647834 0.761781i \(-0.724325\pi\)
−0.647834 + 0.761781i \(0.724325\pi\)
\(194\) −2.00000 −0.143592
\(195\) −16.0000 −1.14578
\(196\) 1.00000 0.0714286
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) −2.00000 −0.142134
\(199\) 24.0000 1.70131 0.850657 0.525720i \(-0.176204\pi\)
0.850657 + 0.525720i \(0.176204\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 4.00000 0.282138
\(202\) −8.00000 −0.562878
\(203\) 2.00000 0.140372
\(204\) 12.0000 0.840168
\(205\) −12.0000 −0.838116
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) −4.00000 −0.277350
\(209\) 0 0
\(210\) −4.00000 −0.276026
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) −12.0000 −0.824163
\(213\) −16.0000 −1.09630
\(214\) −18.0000 −1.23045
\(215\) −12.0000 −0.818393
\(216\) 4.00000 0.272166
\(217\) −4.00000 −0.271538
\(218\) −20.0000 −1.35457
\(219\) −4.00000 −0.270295
\(220\) 4.00000 0.269680
\(221\) 24.0000 1.61441
\(222\) 0 0
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −1.00000 −0.0666667
\(226\) −6.00000 −0.399114
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 0 0
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) −2.00000 −0.131876
\(231\) −4.00000 −0.263181
\(232\) −2.00000 −0.131306
\(233\) 10.0000 0.655122 0.327561 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) −4.00000 −0.261488
\(235\) 0 0
\(236\) −10.0000 −0.650945
\(237\) −16.0000 −1.03931
\(238\) 6.00000 0.388922
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 4.00000 0.258199
\(241\) 14.0000 0.901819 0.450910 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) −7.00000 −0.449977
\(243\) 10.0000 0.641500
\(244\) 2.00000 0.128037
\(245\) −2.00000 −0.127775
\(246\) −12.0000 −0.765092
\(247\) 0 0
\(248\) 4.00000 0.254000
\(249\) 32.0000 2.02792
\(250\) 12.0000 0.758947
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) −1.00000 −0.0629941
\(253\) −2.00000 −0.125739
\(254\) 16.0000 1.00393
\(255\) −24.0000 −1.50294
\(256\) 1.00000 0.0625000
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) −12.0000 −0.747087
\(259\) 0 0
\(260\) 8.00000 0.496139
\(261\) −2.00000 −0.123797
\(262\) 22.0000 1.35916
\(263\) −12.0000 −0.739952 −0.369976 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) 4.00000 0.246183
\(265\) 24.0000 1.47431
\(266\) 0 0
\(267\) −12.0000 −0.734388
\(268\) −2.00000 −0.122169
\(269\) 24.0000 1.46331 0.731653 0.681677i \(-0.238749\pi\)
0.731653 + 0.681677i \(0.238749\pi\)
\(270\) −8.00000 −0.486864
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) −6.00000 −0.363803
\(273\) −8.00000 −0.484182
\(274\) −2.00000 −0.120824
\(275\) 2.00000 0.120605
\(276\) −2.00000 −0.120386
\(277\) 26.0000 1.56219 0.781094 0.624413i \(-0.214662\pi\)
0.781094 + 0.624413i \(0.214662\pi\)
\(278\) −14.0000 −0.839664
\(279\) 4.00000 0.239474
\(280\) 2.00000 0.119523
\(281\) −22.0000 −1.31241 −0.656205 0.754583i \(-0.727839\pi\)
−0.656205 + 0.754583i \(0.727839\pi\)
\(282\) 0 0
\(283\) −12.0000 −0.713326 −0.356663 0.934233i \(-0.616086\pi\)
−0.356663 + 0.934233i \(0.616086\pi\)
\(284\) 8.00000 0.474713
\(285\) 0 0
\(286\) 8.00000 0.473050
\(287\) −6.00000 −0.354169
\(288\) 1.00000 0.0589256
\(289\) 19.0000 1.11765
\(290\) 4.00000 0.234888
\(291\) 4.00000 0.234484
\(292\) 2.00000 0.117041
\(293\) 14.0000 0.817889 0.408944 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) −2.00000 −0.116642
\(295\) 20.0000 1.16445
\(296\) 0 0
\(297\) −8.00000 −0.464207
\(298\) −12.0000 −0.695141
\(299\) −4.00000 −0.231326
\(300\) 2.00000 0.115470
\(301\) −6.00000 −0.345834
\(302\) 0 0
\(303\) 16.0000 0.919176
\(304\) 0 0
\(305\) −4.00000 −0.229039
\(306\) −6.00000 −0.342997
\(307\) 10.0000 0.570730 0.285365 0.958419i \(-0.407885\pi\)
0.285365 + 0.958419i \(0.407885\pi\)
\(308\) 2.00000 0.113961
\(309\) 0 0
\(310\) −8.00000 −0.454369
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 8.00000 0.452911
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 18.0000 1.01580
\(315\) 2.00000 0.112687
\(316\) 8.00000 0.450035
\(317\) 2.00000 0.112331 0.0561656 0.998421i \(-0.482113\pi\)
0.0561656 + 0.998421i \(0.482113\pi\)
\(318\) 24.0000 1.34585
\(319\) 4.00000 0.223957
\(320\) −2.00000 −0.111803
\(321\) 36.0000 2.00932
\(322\) −1.00000 −0.0557278
\(323\) 0 0
\(324\) −11.0000 −0.611111
\(325\) 4.00000 0.221880
\(326\) −4.00000 −0.221540
\(327\) 40.0000 2.21201
\(328\) 6.00000 0.331295
\(329\) 0 0
\(330\) −8.00000 −0.440386
\(331\) 16.0000 0.879440 0.439720 0.898135i \(-0.355078\pi\)
0.439720 + 0.898135i \(0.355078\pi\)
\(332\) −16.0000 −0.878114
\(333\) 0 0
\(334\) −16.0000 −0.875481
\(335\) 4.00000 0.218543
\(336\) 2.00000 0.109109
\(337\) 26.0000 1.41631 0.708155 0.706057i \(-0.249528\pi\)
0.708155 + 0.706057i \(0.249528\pi\)
\(338\) 3.00000 0.163178
\(339\) 12.0000 0.651751
\(340\) 12.0000 0.650791
\(341\) −8.00000 −0.433224
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 6.00000 0.323498
\(345\) 4.00000 0.215353
\(346\) −24.0000 −1.29025
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 4.00000 0.214423
\(349\) 24.0000 1.28469 0.642345 0.766415i \(-0.277962\pi\)
0.642345 + 0.766415i \(0.277962\pi\)
\(350\) 1.00000 0.0534522
\(351\) −16.0000 −0.854017
\(352\) −2.00000 −0.106600
\(353\) 10.0000 0.532246 0.266123 0.963939i \(-0.414257\pi\)
0.266123 + 0.963939i \(0.414257\pi\)
\(354\) 20.0000 1.06299
\(355\) −16.0000 −0.849192
\(356\) 6.00000 0.317999
\(357\) −12.0000 −0.635107
\(358\) 16.0000 0.845626
\(359\) −4.00000 −0.211112 −0.105556 0.994413i \(-0.533662\pi\)
−0.105556 + 0.994413i \(0.533662\pi\)
\(360\) −2.00000 −0.105409
\(361\) −19.0000 −1.00000
\(362\) −2.00000 −0.105118
\(363\) 14.0000 0.734809
\(364\) 4.00000 0.209657
\(365\) −4.00000 −0.209370
\(366\) −4.00000 −0.209083
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 1.00000 0.0521286
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) 12.0000 0.623009
\(372\) −8.00000 −0.414781
\(373\) 4.00000 0.207112 0.103556 0.994624i \(-0.466978\pi\)
0.103556 + 0.994624i \(0.466978\pi\)
\(374\) 12.0000 0.620505
\(375\) −24.0000 −1.23935
\(376\) 0 0
\(377\) 8.00000 0.412021
\(378\) −4.00000 −0.205738
\(379\) 2.00000 0.102733 0.0513665 0.998680i \(-0.483642\pi\)
0.0513665 + 0.998680i \(0.483642\pi\)
\(380\) 0 0
\(381\) −32.0000 −1.63941
\(382\) −16.0000 −0.818631
\(383\) −24.0000 −1.22634 −0.613171 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) −2.00000 −0.102062
\(385\) −4.00000 −0.203859
\(386\) −18.0000 −0.916176
\(387\) 6.00000 0.304997
\(388\) −2.00000 −0.101535
\(389\) −28.0000 −1.41966 −0.709828 0.704375i \(-0.751227\pi\)
−0.709828 + 0.704375i \(0.751227\pi\)
\(390\) −16.0000 −0.810191
\(391\) −6.00000 −0.303433
\(392\) 1.00000 0.0505076
\(393\) −44.0000 −2.21951
\(394\) 18.0000 0.906827
\(395\) −16.0000 −0.805047
\(396\) −2.00000 −0.100504
\(397\) −20.0000 −1.00377 −0.501886 0.864934i \(-0.667360\pi\)
−0.501886 + 0.864934i \(0.667360\pi\)
\(398\) 24.0000 1.20301
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 6.00000 0.299626 0.149813 0.988714i \(-0.452133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(402\) 4.00000 0.199502
\(403\) −16.0000 −0.797017
\(404\) −8.00000 −0.398015
\(405\) 22.0000 1.09319
\(406\) 2.00000 0.0992583
\(407\) 0 0
\(408\) 12.0000 0.594089
\(409\) −34.0000 −1.68119 −0.840596 0.541663i \(-0.817795\pi\)
−0.840596 + 0.541663i \(0.817795\pi\)
\(410\) −12.0000 −0.592638
\(411\) 4.00000 0.197305
\(412\) 0 0
\(413\) 10.0000 0.492068
\(414\) 1.00000 0.0491473
\(415\) 32.0000 1.57082
\(416\) −4.00000 −0.196116
\(417\) 28.0000 1.37117
\(418\) 0 0
\(419\) 4.00000 0.195413 0.0977064 0.995215i \(-0.468849\pi\)
0.0977064 + 0.995215i \(0.468849\pi\)
\(420\) −4.00000 −0.195180
\(421\) −20.0000 −0.974740 −0.487370 0.873195i \(-0.662044\pi\)
−0.487370 + 0.873195i \(0.662044\pi\)
\(422\) 4.00000 0.194717
\(423\) 0 0
\(424\) −12.0000 −0.582772
\(425\) 6.00000 0.291043
\(426\) −16.0000 −0.775203
\(427\) −2.00000 −0.0967868
\(428\) −18.0000 −0.870063
\(429\) −16.0000 −0.772487
\(430\) −12.0000 −0.578691
\(431\) 28.0000 1.34871 0.674356 0.738406i \(-0.264421\pi\)
0.674356 + 0.738406i \(0.264421\pi\)
\(432\) 4.00000 0.192450
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) −4.00000 −0.192006
\(435\) −8.00000 −0.383571
\(436\) −20.0000 −0.957826
\(437\) 0 0
\(438\) −4.00000 −0.191127
\(439\) 20.0000 0.954548 0.477274 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 4.00000 0.190693
\(441\) 1.00000 0.0476190
\(442\) 24.0000 1.14156
\(443\) −16.0000 −0.760183 −0.380091 0.924949i \(-0.624107\pi\)
−0.380091 + 0.924949i \(0.624107\pi\)
\(444\) 0 0
\(445\) −12.0000 −0.568855
\(446\) −16.0000 −0.757622
\(447\) 24.0000 1.13516
\(448\) −1.00000 −0.0472456
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −12.0000 −0.565058
\(452\) −6.00000 −0.282216
\(453\) 0 0
\(454\) −12.0000 −0.563188
\(455\) −8.00000 −0.375046
\(456\) 0 0
\(457\) −18.0000 −0.842004 −0.421002 0.907060i \(-0.638322\pi\)
−0.421002 + 0.907060i \(0.638322\pi\)
\(458\) 6.00000 0.280362
\(459\) −24.0000 −1.12022
\(460\) −2.00000 −0.0932505
\(461\) 8.00000 0.372597 0.186299 0.982493i \(-0.440351\pi\)
0.186299 + 0.982493i \(0.440351\pi\)
\(462\) −4.00000 −0.186097
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 16.0000 0.741982
\(466\) 10.0000 0.463241
\(467\) −40.0000 −1.85098 −0.925490 0.378773i \(-0.876346\pi\)
−0.925490 + 0.378773i \(0.876346\pi\)
\(468\) −4.00000 −0.184900
\(469\) 2.00000 0.0923514
\(470\) 0 0
\(471\) −36.0000 −1.65879
\(472\) −10.0000 −0.460287
\(473\) −12.0000 −0.551761
\(474\) −16.0000 −0.734904
\(475\) 0 0
\(476\) 6.00000 0.275010
\(477\) −12.0000 −0.549442
\(478\) −16.0000 −0.731823
\(479\) −8.00000 −0.365529 −0.182765 0.983157i \(-0.558505\pi\)
−0.182765 + 0.983157i \(0.558505\pi\)
\(480\) 4.00000 0.182574
\(481\) 0 0
\(482\) 14.0000 0.637683
\(483\) 2.00000 0.0910032
\(484\) −7.00000 −0.318182
\(485\) 4.00000 0.181631
\(486\) 10.0000 0.453609
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 2.00000 0.0905357
\(489\) 8.00000 0.361773
\(490\) −2.00000 −0.0903508
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) −12.0000 −0.541002
\(493\) 12.0000 0.540453
\(494\) 0 0
\(495\) 4.00000 0.179787
\(496\) 4.00000 0.179605
\(497\) −8.00000 −0.358849
\(498\) 32.0000 1.43395
\(499\) −24.0000 −1.07439 −0.537194 0.843459i \(-0.680516\pi\)
−0.537194 + 0.843459i \(0.680516\pi\)
\(500\) 12.0000 0.536656
\(501\) 32.0000 1.42965
\(502\) −12.0000 −0.535586
\(503\) −32.0000 −1.42681 −0.713405 0.700752i \(-0.752848\pi\)
−0.713405 + 0.700752i \(0.752848\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 16.0000 0.711991
\(506\) −2.00000 −0.0889108
\(507\) −6.00000 −0.266469
\(508\) 16.0000 0.709885
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) −24.0000 −1.06274
\(511\) −2.00000 −0.0884748
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −6.00000 −0.264649
\(515\) 0 0
\(516\) −12.0000 −0.528271
\(517\) 0 0
\(518\) 0 0
\(519\) 48.0000 2.10697
\(520\) 8.00000 0.350823
\(521\) 10.0000 0.438108 0.219054 0.975713i \(-0.429703\pi\)
0.219054 + 0.975713i \(0.429703\pi\)
\(522\) −2.00000 −0.0875376
\(523\) 16.0000 0.699631 0.349816 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) 22.0000 0.961074
\(525\) −2.00000 −0.0872872
\(526\) −12.0000 −0.523225
\(527\) −24.0000 −1.04546
\(528\) 4.00000 0.174078
\(529\) 1.00000 0.0434783
\(530\) 24.0000 1.04249
\(531\) −10.0000 −0.433963
\(532\) 0 0
\(533\) −24.0000 −1.03956
\(534\) −12.0000 −0.519291
\(535\) 36.0000 1.55642
\(536\) −2.00000 −0.0863868
\(537\) −32.0000 −1.38090
\(538\) 24.0000 1.03471
\(539\) −2.00000 −0.0861461
\(540\) −8.00000 −0.344265
\(541\) −6.00000 −0.257960 −0.128980 0.991647i \(-0.541170\pi\)
−0.128980 + 0.991647i \(0.541170\pi\)
\(542\) 20.0000 0.859074
\(543\) 4.00000 0.171656
\(544\) −6.00000 −0.257248
\(545\) 40.0000 1.71341
\(546\) −8.00000 −0.342368
\(547\) −40.0000 −1.71028 −0.855138 0.518400i \(-0.826528\pi\)
−0.855138 + 0.518400i \(0.826528\pi\)
\(548\) −2.00000 −0.0854358
\(549\) 2.00000 0.0853579
\(550\) 2.00000 0.0852803
\(551\) 0 0
\(552\) −2.00000 −0.0851257
\(553\) −8.00000 −0.340195
\(554\) 26.0000 1.10463
\(555\) 0 0
\(556\) −14.0000 −0.593732
\(557\) 32.0000 1.35588 0.677942 0.735116i \(-0.262872\pi\)
0.677942 + 0.735116i \(0.262872\pi\)
\(558\) 4.00000 0.169334
\(559\) −24.0000 −1.01509
\(560\) 2.00000 0.0845154
\(561\) −24.0000 −1.01328
\(562\) −22.0000 −0.928014
\(563\) 24.0000 1.01148 0.505740 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) 0 0
\(565\) 12.0000 0.504844
\(566\) −12.0000 −0.504398
\(567\) 11.0000 0.461957
\(568\) 8.00000 0.335673
\(569\) −42.0000 −1.76073 −0.880366 0.474295i \(-0.842703\pi\)
−0.880366 + 0.474295i \(0.842703\pi\)
\(570\) 0 0
\(571\) −38.0000 −1.59025 −0.795125 0.606445i \(-0.792595\pi\)
−0.795125 + 0.606445i \(0.792595\pi\)
\(572\) 8.00000 0.334497
\(573\) 32.0000 1.33682
\(574\) −6.00000 −0.250435
\(575\) −1.00000 −0.0417029
\(576\) 1.00000 0.0416667
\(577\) −38.0000 −1.58196 −0.790980 0.611842i \(-0.790429\pi\)
−0.790980 + 0.611842i \(0.790429\pi\)
\(578\) 19.0000 0.790296
\(579\) 36.0000 1.49611
\(580\) 4.00000 0.166091
\(581\) 16.0000 0.663792
\(582\) 4.00000 0.165805
\(583\) 24.0000 0.993978
\(584\) 2.00000 0.0827606
\(585\) 8.00000 0.330759
\(586\) 14.0000 0.578335
\(587\) 18.0000 0.742940 0.371470 0.928445i \(-0.378854\pi\)
0.371470 + 0.928445i \(0.378854\pi\)
\(588\) −2.00000 −0.0824786
\(589\) 0 0
\(590\) 20.0000 0.823387
\(591\) −36.0000 −1.48084
\(592\) 0 0
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) −8.00000 −0.328244
\(595\) −12.0000 −0.491952
\(596\) −12.0000 −0.491539
\(597\) −48.0000 −1.96451
\(598\) −4.00000 −0.163572
\(599\) 40.0000 1.63436 0.817178 0.576386i \(-0.195537\pi\)
0.817178 + 0.576386i \(0.195537\pi\)
\(600\) 2.00000 0.0816497
\(601\) −46.0000 −1.87638 −0.938190 0.346122i \(-0.887498\pi\)
−0.938190 + 0.346122i \(0.887498\pi\)
\(602\) −6.00000 −0.244542
\(603\) −2.00000 −0.0814463
\(604\) 0 0
\(605\) 14.0000 0.569181
\(606\) 16.0000 0.649956
\(607\) −40.0000 −1.62355 −0.811775 0.583970i \(-0.801498\pi\)
−0.811775 + 0.583970i \(0.801498\pi\)
\(608\) 0 0
\(609\) −4.00000 −0.162088
\(610\) −4.00000 −0.161955
\(611\) 0 0
\(612\) −6.00000 −0.242536
\(613\) 44.0000 1.77714 0.888572 0.458738i \(-0.151698\pi\)
0.888572 + 0.458738i \(0.151698\pi\)
\(614\) 10.0000 0.403567
\(615\) 24.0000 0.967773
\(616\) 2.00000 0.0805823
\(617\) 18.0000 0.724653 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 0 0
\(619\) −28.0000 −1.12542 −0.562708 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) −8.00000 −0.321288
\(621\) 4.00000 0.160514
\(622\) 24.0000 0.962312
\(623\) −6.00000 −0.240385
\(624\) 8.00000 0.320256
\(625\) −19.0000 −0.760000
\(626\) −10.0000 −0.399680
\(627\) 0 0
\(628\) 18.0000 0.718278
\(629\) 0 0
\(630\) 2.00000 0.0796819
\(631\) −20.0000 −0.796187 −0.398094 0.917345i \(-0.630328\pi\)
−0.398094 + 0.917345i \(0.630328\pi\)
\(632\) 8.00000 0.318223
\(633\) −8.00000 −0.317971
\(634\) 2.00000 0.0794301
\(635\) −32.0000 −1.26988
\(636\) 24.0000 0.951662
\(637\) −4.00000 −0.158486
\(638\) 4.00000 0.158362
\(639\) 8.00000 0.316475
\(640\) −2.00000 −0.0790569
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) 36.0000 1.42081
\(643\) 24.0000 0.946468 0.473234 0.880937i \(-0.343087\pi\)
0.473234 + 0.880937i \(0.343087\pi\)
\(644\) −1.00000 −0.0394055
\(645\) 24.0000 0.944999
\(646\) 0 0
\(647\) 12.0000 0.471769 0.235884 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) −11.0000 −0.432121
\(649\) 20.0000 0.785069
\(650\) 4.00000 0.156893
\(651\) 8.00000 0.313545
\(652\) −4.00000 −0.156652
\(653\) 26.0000 1.01746 0.508729 0.860927i \(-0.330115\pi\)
0.508729 + 0.860927i \(0.330115\pi\)
\(654\) 40.0000 1.56412
\(655\) −44.0000 −1.71922
\(656\) 6.00000 0.234261
\(657\) 2.00000 0.0780274
\(658\) 0 0
\(659\) −6.00000 −0.233727 −0.116863 0.993148i \(-0.537284\pi\)
−0.116863 + 0.993148i \(0.537284\pi\)
\(660\) −8.00000 −0.311400
\(661\) −30.0000 −1.16686 −0.583432 0.812162i \(-0.698291\pi\)
−0.583432 + 0.812162i \(0.698291\pi\)
\(662\) 16.0000 0.621858
\(663\) −48.0000 −1.86417
\(664\) −16.0000 −0.620920
\(665\) 0 0
\(666\) 0 0
\(667\) −2.00000 −0.0774403
\(668\) −16.0000 −0.619059
\(669\) 32.0000 1.23719
\(670\) 4.00000 0.154533
\(671\) −4.00000 −0.154418
\(672\) 2.00000 0.0771517
\(673\) −50.0000 −1.92736 −0.963679 0.267063i \(-0.913947\pi\)
−0.963679 + 0.267063i \(0.913947\pi\)
\(674\) 26.0000 1.00148
\(675\) −4.00000 −0.153960
\(676\) 3.00000 0.115385
\(677\) −22.0000 −0.845529 −0.422764 0.906240i \(-0.638940\pi\)
−0.422764 + 0.906240i \(0.638940\pi\)
\(678\) 12.0000 0.460857
\(679\) 2.00000 0.0767530
\(680\) 12.0000 0.460179
\(681\) 24.0000 0.919682
\(682\) −8.00000 −0.306336
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 0 0
\(685\) 4.00000 0.152832
\(686\) −1.00000 −0.0381802
\(687\) −12.0000 −0.457829
\(688\) 6.00000 0.228748
\(689\) 48.0000 1.82865
\(690\) 4.00000 0.152277
\(691\) 46.0000 1.74992 0.874961 0.484193i \(-0.160887\pi\)
0.874961 + 0.484193i \(0.160887\pi\)
\(692\) −24.0000 −0.912343
\(693\) 2.00000 0.0759737
\(694\) 12.0000 0.455514
\(695\) 28.0000 1.06210
\(696\) 4.00000 0.151620
\(697\) −36.0000 −1.36360
\(698\) 24.0000 0.908413
\(699\) −20.0000 −0.756469
\(700\) 1.00000 0.0377964
\(701\) 20.0000 0.755390 0.377695 0.925930i \(-0.376717\pi\)
0.377695 + 0.925930i \(0.376717\pi\)
\(702\) −16.0000 −0.603881
\(703\) 0 0
\(704\) −2.00000 −0.0753778
\(705\) 0 0
\(706\) 10.0000 0.376355
\(707\) 8.00000 0.300871
\(708\) 20.0000 0.751646
\(709\) −24.0000 −0.901339 −0.450669 0.892691i \(-0.648815\pi\)
−0.450669 + 0.892691i \(0.648815\pi\)
\(710\) −16.0000 −0.600469
\(711\) 8.00000 0.300023
\(712\) 6.00000 0.224860
\(713\) 4.00000 0.149801
\(714\) −12.0000 −0.449089
\(715\) −16.0000 −0.598366
\(716\) 16.0000 0.597948
\(717\) 32.0000 1.19506
\(718\) −4.00000 −0.149279
\(719\) −20.0000 −0.745874 −0.372937 0.927857i \(-0.621649\pi\)
−0.372937 + 0.927857i \(0.621649\pi\)
\(720\) −2.00000 −0.0745356
\(721\) 0 0
\(722\) −19.0000 −0.707107
\(723\) −28.0000 −1.04133
\(724\) −2.00000 −0.0743294
\(725\) 2.00000 0.0742781
\(726\) 14.0000 0.519589
\(727\) −8.00000 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 4.00000 0.148250
\(729\) 13.0000 0.481481
\(730\) −4.00000 −0.148047
\(731\) −36.0000 −1.33151
\(732\) −4.00000 −0.147844
\(733\) 38.0000 1.40356 0.701781 0.712393i \(-0.252388\pi\)
0.701781 + 0.712393i \(0.252388\pi\)
\(734\) 0 0
\(735\) 4.00000 0.147542
\(736\) 1.00000 0.0368605
\(737\) 4.00000 0.147342
\(738\) 6.00000 0.220863
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 12.0000 0.440534
\(743\) 48.0000 1.76095 0.880475 0.474093i \(-0.157224\pi\)
0.880475 + 0.474093i \(0.157224\pi\)
\(744\) −8.00000 −0.293294
\(745\) 24.0000 0.879292
\(746\) 4.00000 0.146450
\(747\) −16.0000 −0.585409
\(748\) 12.0000 0.438763
\(749\) 18.0000 0.657706
\(750\) −24.0000 −0.876356
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) 24.0000 0.874609
\(754\) 8.00000 0.291343
\(755\) 0 0
\(756\) −4.00000 −0.145479
\(757\) 44.0000 1.59921 0.799604 0.600528i \(-0.205043\pi\)
0.799604 + 0.600528i \(0.205043\pi\)
\(758\) 2.00000 0.0726433
\(759\) 4.00000 0.145191
\(760\) 0 0
\(761\) −18.0000 −0.652499 −0.326250 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) −32.0000 −1.15924
\(763\) 20.0000 0.724049
\(764\) −16.0000 −0.578860
\(765\) 12.0000 0.433861
\(766\) −24.0000 −0.867155
\(767\) 40.0000 1.44432
\(768\) −2.00000 −0.0721688
\(769\) 26.0000 0.937584 0.468792 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) −4.00000 −0.144150
\(771\) 12.0000 0.432169
\(772\) −18.0000 −0.647834
\(773\) 10.0000 0.359675 0.179838 0.983696i \(-0.442443\pi\)
0.179838 + 0.983696i \(0.442443\pi\)
\(774\) 6.00000 0.215666
\(775\) −4.00000 −0.143684
\(776\) −2.00000 −0.0717958
\(777\) 0 0
\(778\) −28.0000 −1.00385
\(779\) 0 0
\(780\) −16.0000 −0.572892
\(781\) −16.0000 −0.572525
\(782\) −6.00000 −0.214560
\(783\) −8.00000 −0.285897
\(784\) 1.00000 0.0357143
\(785\) −36.0000 −1.28490
\(786\) −44.0000 −1.56943
\(787\) 20.0000 0.712923 0.356462 0.934310i \(-0.383983\pi\)
0.356462 + 0.934310i \(0.383983\pi\)
\(788\) 18.0000 0.641223
\(789\) 24.0000 0.854423
\(790\) −16.0000 −0.569254
\(791\) 6.00000 0.213335
\(792\) −2.00000 −0.0710669
\(793\) −8.00000 −0.284088
\(794\) −20.0000 −0.709773
\(795\) −48.0000 −1.70238
\(796\) 24.0000 0.850657
\(797\) 14.0000 0.495905 0.247953 0.968772i \(-0.420242\pi\)
0.247953 + 0.968772i \(0.420242\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) 6.00000 0.212000
\(802\) 6.00000 0.211867
\(803\) −4.00000 −0.141157
\(804\) 4.00000 0.141069
\(805\) 2.00000 0.0704907
\(806\) −16.0000 −0.563576
\(807\) −48.0000 −1.68968
\(808\) −8.00000 −0.281439
\(809\) −42.0000 −1.47664 −0.738321 0.674450i \(-0.764381\pi\)
−0.738321 + 0.674450i \(0.764381\pi\)
\(810\) 22.0000 0.773001
\(811\) −2.00000 −0.0702295 −0.0351147 0.999383i \(-0.511180\pi\)
−0.0351147 + 0.999383i \(0.511180\pi\)
\(812\) 2.00000 0.0701862
\(813\) −40.0000 −1.40286
\(814\) 0 0
\(815\) 8.00000 0.280228
\(816\) 12.0000 0.420084
\(817\) 0 0
\(818\) −34.0000 −1.18878
\(819\) 4.00000 0.139771
\(820\) −12.0000 −0.419058
\(821\) 54.0000 1.88461 0.942306 0.334751i \(-0.108652\pi\)
0.942306 + 0.334751i \(0.108652\pi\)
\(822\) 4.00000 0.139516
\(823\) 8.00000 0.278862 0.139431 0.990232i \(-0.455473\pi\)
0.139431 + 0.990232i \(0.455473\pi\)
\(824\) 0 0
\(825\) −4.00000 −0.139262
\(826\) 10.0000 0.347945
\(827\) 14.0000 0.486828 0.243414 0.969923i \(-0.421733\pi\)
0.243414 + 0.969923i \(0.421733\pi\)
\(828\) 1.00000 0.0347524
\(829\) 12.0000 0.416777 0.208389 0.978046i \(-0.433178\pi\)
0.208389 + 0.978046i \(0.433178\pi\)
\(830\) 32.0000 1.11074
\(831\) −52.0000 −1.80386
\(832\) −4.00000 −0.138675
\(833\) −6.00000 −0.207888
\(834\) 28.0000 0.969561
\(835\) 32.0000 1.10741
\(836\) 0 0
\(837\) 16.0000 0.553041
\(838\) 4.00000 0.138178
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) −4.00000 −0.138013
\(841\) −25.0000 −0.862069
\(842\) −20.0000 −0.689246
\(843\) 44.0000 1.51544
\(844\) 4.00000 0.137686
\(845\) −6.00000 −0.206406
\(846\) 0 0
\(847\) 7.00000 0.240523
\(848\) −12.0000 −0.412082
\(849\) 24.0000 0.823678
\(850\) 6.00000 0.205798
\(851\) 0 0
\(852\) −16.0000 −0.548151
\(853\) −44.0000 −1.50653 −0.753266 0.657716i \(-0.771523\pi\)
−0.753266 + 0.657716i \(0.771523\pi\)
\(854\) −2.00000 −0.0684386
\(855\) 0 0
\(856\) −18.0000 −0.615227
\(857\) 38.0000 1.29806 0.649028 0.760765i \(-0.275176\pi\)
0.649028 + 0.760765i \(0.275176\pi\)
\(858\) −16.0000 −0.546231
\(859\) −42.0000 −1.43302 −0.716511 0.697576i \(-0.754262\pi\)
−0.716511 + 0.697576i \(0.754262\pi\)
\(860\) −12.0000 −0.409197
\(861\) 12.0000 0.408959
\(862\) 28.0000 0.953684
\(863\) −8.00000 −0.272323 −0.136162 0.990687i \(-0.543477\pi\)
−0.136162 + 0.990687i \(0.543477\pi\)
\(864\) 4.00000 0.136083
\(865\) 48.0000 1.63205
\(866\) −14.0000 −0.475739
\(867\) −38.0000 −1.29055
\(868\) −4.00000 −0.135769
\(869\) −16.0000 −0.542763
\(870\) −8.00000 −0.271225
\(871\) 8.00000 0.271070
\(872\) −20.0000 −0.677285
\(873\) −2.00000 −0.0676897
\(874\) 0 0
\(875\) −12.0000 −0.405674
\(876\) −4.00000 −0.135147
\(877\) −2.00000 −0.0675352 −0.0337676 0.999430i \(-0.510751\pi\)
−0.0337676 + 0.999430i \(0.510751\pi\)
\(878\) 20.0000 0.674967
\(879\) −28.0000 −0.944417
\(880\) 4.00000 0.134840
\(881\) 18.0000 0.606435 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) 1.00000 0.0336718
\(883\) 12.0000 0.403832 0.201916 0.979403i \(-0.435283\pi\)
0.201916 + 0.979403i \(0.435283\pi\)
\(884\) 24.0000 0.807207
\(885\) −40.0000 −1.34459
\(886\) −16.0000 −0.537531
\(887\) −8.00000 −0.268614 −0.134307 0.990940i \(-0.542881\pi\)
−0.134307 + 0.990940i \(0.542881\pi\)
\(888\) 0 0
\(889\) −16.0000 −0.536623
\(890\) −12.0000 −0.402241
\(891\) 22.0000 0.737028
\(892\) −16.0000 −0.535720
\(893\) 0 0
\(894\) 24.0000 0.802680
\(895\) −32.0000 −1.06964
\(896\) −1.00000 −0.0334077
\(897\) 8.00000 0.267112
\(898\) 30.0000 1.00111
\(899\) −8.00000 −0.266815
\(900\) −1.00000 −0.0333333
\(901\) 72.0000 2.39867
\(902\) −12.0000 −0.399556
\(903\) 12.0000 0.399335
\(904\) −6.00000 −0.199557
\(905\) 4.00000 0.132964
\(906\) 0 0
\(907\) −46.0000 −1.52740 −0.763702 0.645568i \(-0.776621\pi\)
−0.763702 + 0.645568i \(0.776621\pi\)
\(908\) −12.0000 −0.398234
\(909\) −8.00000 −0.265343
\(910\) −8.00000 −0.265197
\(911\) 36.0000 1.19273 0.596367 0.802712i \(-0.296610\pi\)
0.596367 + 0.802712i \(0.296610\pi\)
\(912\) 0 0
\(913\) 32.0000 1.05905
\(914\) −18.0000 −0.595387
\(915\) 8.00000 0.264472
\(916\) 6.00000 0.198246
\(917\) −22.0000 −0.726504
\(918\) −24.0000 −0.792118
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) −2.00000 −0.0659380
\(921\) −20.0000 −0.659022
\(922\) 8.00000 0.263466
\(923\) −32.0000 −1.05329
\(924\) −4.00000 −0.131590
\(925\) 0 0
\(926\) 16.0000 0.525793
\(927\) 0 0
\(928\) −2.00000 −0.0656532
\(929\) −34.0000 −1.11550 −0.557752 0.830008i \(-0.688336\pi\)
−0.557752 + 0.830008i \(0.688336\pi\)
\(930\) 16.0000 0.524661
\(931\) 0 0
\(932\) 10.0000 0.327561
\(933\) −48.0000 −1.57145
\(934\) −40.0000 −1.30884
\(935\) −24.0000 −0.784884
\(936\) −4.00000 −0.130744
\(937\) 30.0000 0.980057 0.490029 0.871706i \(-0.336986\pi\)
0.490029 + 0.871706i \(0.336986\pi\)
\(938\) 2.00000 0.0653023
\(939\) 20.0000 0.652675
\(940\) 0 0
\(941\) −10.0000 −0.325991 −0.162995 0.986627i \(-0.552116\pi\)
−0.162995 + 0.986627i \(0.552116\pi\)
\(942\) −36.0000 −1.17294
\(943\) 6.00000 0.195387
\(944\) −10.0000 −0.325472
\(945\) 8.00000 0.260240
\(946\) −12.0000 −0.390154
\(947\) 60.0000 1.94974 0.974869 0.222779i \(-0.0715128\pi\)
0.974869 + 0.222779i \(0.0715128\pi\)
\(948\) −16.0000 −0.519656
\(949\) −8.00000 −0.259691
\(950\) 0 0
\(951\) −4.00000 −0.129709
\(952\) 6.00000 0.194461
\(953\) −26.0000 −0.842223 −0.421111 0.907009i \(-0.638360\pi\)
−0.421111 + 0.907009i \(0.638360\pi\)
\(954\) −12.0000 −0.388514
\(955\) 32.0000 1.03550
\(956\) −16.0000 −0.517477
\(957\) −8.00000 −0.258603
\(958\) −8.00000 −0.258468
\(959\) 2.00000 0.0645834
\(960\) 4.00000 0.129099
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) −18.0000 −0.580042
\(964\) 14.0000 0.450910
\(965\) 36.0000 1.15888
\(966\) 2.00000 0.0643489
\(967\) 24.0000 0.771788 0.385894 0.922543i \(-0.373893\pi\)
0.385894 + 0.922543i \(0.373893\pi\)
\(968\) −7.00000 −0.224989
\(969\) 0 0
\(970\) 4.00000 0.128432
\(971\) 36.0000 1.15529 0.577647 0.816286i \(-0.303971\pi\)
0.577647 + 0.816286i \(0.303971\pi\)
\(972\) 10.0000 0.320750
\(973\) 14.0000 0.448819
\(974\) 8.00000 0.256337
\(975\) −8.00000 −0.256205
\(976\) 2.00000 0.0640184
\(977\) 30.0000 0.959785 0.479893 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(978\) 8.00000 0.255812
\(979\) −12.0000 −0.383522
\(980\) −2.00000 −0.0638877
\(981\) −20.0000 −0.638551
\(982\) 0 0
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) −12.0000 −0.382546
\(985\) −36.0000 −1.14706
\(986\) 12.0000 0.382158
\(987\) 0 0
\(988\) 0 0
\(989\) 6.00000 0.190789
\(990\) 4.00000 0.127128
\(991\) 40.0000 1.27064 0.635321 0.772248i \(-0.280868\pi\)
0.635321 + 0.772248i \(0.280868\pi\)
\(992\) 4.00000 0.127000
\(993\) −32.0000 −1.01549
\(994\) −8.00000 −0.253745
\(995\) −48.0000 −1.52170
\(996\) 32.0000 1.01396
\(997\) −36.0000 −1.14013 −0.570066 0.821599i \(-0.693082\pi\)
−0.570066 + 0.821599i \(0.693082\pi\)
\(998\) −24.0000 −0.759707
\(999\) 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 322.2.a.c.1.1 1
3.2 odd 2 2898.2.a.g.1.1 1
4.3 odd 2 2576.2.a.m.1.1 1
5.4 even 2 8050.2.a.m.1.1 1
7.6 odd 2 2254.2.a.g.1.1 1
23.22 odd 2 7406.2.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
322.2.a.c.1.1 1 1.1 even 1 trivial
2254.2.a.g.1.1 1 7.6 odd 2
2576.2.a.m.1.1 1 4.3 odd 2
2898.2.a.g.1.1 1 3.2 odd 2
7406.2.a.f.1.1 1 23.22 odd 2
8050.2.a.m.1.1 1 5.4 even 2