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Properties

Label 6042.2.a.m.1.1
Level $6042$
Weight $2$
Character 6042.1
Self dual yes
Analytic conductor $48.246$
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] = [6042,2,Mod(1,6042)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6042, 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("6042.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6042 = 2 \cdot 3 \cdot 19 \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6042.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: \(48.2456129013\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 6042.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} +1.00000 q^{12} +4.00000 q^{13} +4.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} +1.00000 q^{18} -1.00000 q^{19} +2.00000 q^{20} +4.00000 q^{21} +1.00000 q^{24} -1.00000 q^{25} +4.00000 q^{26} +1.00000 q^{27} +4.00000 q^{28} +6.00000 q^{29} +2.00000 q^{30} +4.00000 q^{31} +1.00000 q^{32} -6.00000 q^{34} +8.00000 q^{35} +1.00000 q^{36} -4.00000 q^{37} -1.00000 q^{38} +4.00000 q^{39} +2.00000 q^{40} +10.0000 q^{41} +4.00000 q^{42} -4.00000 q^{43} +2.00000 q^{45} -8.00000 q^{47} +1.00000 q^{48} +9.00000 q^{49} -1.00000 q^{50} -6.00000 q^{51} +4.00000 q^{52} +1.00000 q^{53} +1.00000 q^{54} +4.00000 q^{56} -1.00000 q^{57} +6.00000 q^{58} -14.0000 q^{59} +2.00000 q^{60} -4.00000 q^{61} +4.00000 q^{62} +4.00000 q^{63} +1.00000 q^{64} +8.00000 q^{65} +4.00000 q^{67} -6.00000 q^{68} +8.00000 q^{70} +1.00000 q^{72} -10.0000 q^{73} -4.00000 q^{74} -1.00000 q^{75} -1.00000 q^{76} +4.00000 q^{78} +12.0000 q^{79} +2.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} -6.00000 q^{83} +4.00000 q^{84} -12.0000 q^{85} -4.00000 q^{86} +6.00000 q^{87} +2.00000 q^{90} +16.0000 q^{91} +4.00000 q^{93} -8.00000 q^{94} -2.00000 q^{95} +1.00000 q^{96} +18.0000 q^{97} +9.00000 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 2.00000 0.894427 0.447214 − 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 1.00000 0.408248
\(7\) 4.00000 1.51186 0.755929 − 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 2.00000 0.632456
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 1.00000 0.288675
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 4.00000 1.06904
\(15\) 2.00000 0.516398
\(16\) 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
\(19\) −1.00000 −0.229416
\(20\) 2.00000 0.447214
\(21\) 4.00000 0.872872
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 1.00000 0.204124
\(25\) −1.00000 −0.200000
\(26\) 4.00000 0.784465
\(27\) 1.00000 0.192450
\(28\) 4.00000 0.755929
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 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
\(33\) 0 0
\(34\) −6.00000 −1.02899
\(35\) 8.00000 1.35225
\(36\) 1.00000 0.166667
\(37\) −4.00000 −0.657596 −0.328798 − 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) −1.00000 −0.162221
\(39\) 4.00000 0.640513
\(40\) 2.00000 0.316228
\(41\) 10.0000 1.56174 0.780869 − 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 4.00000 0.617213
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 2.00000 0.298142
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 − 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 1.00000 0.144338
\(49\) 9.00000 1.28571
\(50\) −1.00000 −0.141421
\(51\) −6.00000 −0.840168
\(52\) 4.00000 0.554700
\(53\) 1.00000 0.137361
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 4.00000 0.534522
\(57\) −1.00000 −0.132453
\(58\) 6.00000 0.787839
\(59\) −14.0000 −1.82264 −0.911322 − 0.411693i \(-0.864937\pi\)
−0.911322 + 0.411693i \(0.864937\pi\)
\(60\) 2.00000 0.258199
\(61\) −4.00000 −0.512148 −0.256074 − 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) 4.00000 0.508001
\(63\) 4.00000 0.503953
\(64\) 1.00000 0.125000
\(65\) 8.00000 0.992278
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 − 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) −6.00000 −0.727607
\(69\) 0 0
\(70\) 8.00000 0.956183
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 1.00000 0.117851
\(73\) −10.0000 −1.17041 −0.585206 − 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) −4.00000 −0.464991
\(75\) −1.00000 −0.115470
\(76\) −1.00000 −0.114708
\(77\) 0 0
\(78\) 4.00000 0.452911
\(79\) 12.0000 1.35011 0.675053 − 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) 2.00000 0.223607
\(81\) 1.00000 0.111111
\(82\) 10.0000 1.10432
\(83\) −6.00000 −0.658586 −0.329293 − 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 4.00000 0.436436
\(85\) −12.0000 −1.30158
\(86\) −4.00000 −0.431331
\(87\) 6.00000 0.643268
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 2.00000 0.210819
\(91\) 16.0000 1.67726
\(92\) 0 0
\(93\) 4.00000 0.414781
\(94\) −8.00000 −0.825137
\(95\) −2.00000 −0.205196
\(96\) 1.00000 0.102062
\(97\) 18.0000 1.82762 0.913812 − 0.406138i \(-0.133125\pi\)
0.913812 + 0.406138i \(0.133125\pi\)
\(98\) 9.00000 0.909137
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) 14.0000 1.39305 0.696526 − 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) −6.00000 −0.594089
\(103\) 12.0000 1.18240 0.591198 − 0.806527i \(-0.298655\pi\)
0.591198 + 0.806527i \(0.298655\pi\)
\(104\) 4.00000 0.392232
\(105\) 8.00000 0.780720
\(106\) 1.00000 0.0971286
\(107\) −10.0000 −0.966736 −0.483368 − 0.875417i \(-0.660587\pi\)
−0.483368 + 0.875417i \(0.660587\pi\)
\(108\) 1.00000 0.0962250
\(109\) −6.00000 −0.574696 −0.287348 − 0.957826i \(-0.592774\pi\)
−0.287348 + 0.957826i \(0.592774\pi\)
\(110\) 0 0
\(111\) −4.00000 −0.379663
\(112\) 4.00000 0.377964
\(113\) 12.0000 1.12887 0.564433 − 0.825479i \(-0.309095\pi\)
0.564433 + 0.825479i \(0.309095\pi\)
\(114\) −1.00000 −0.0936586
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) 4.00000 0.369800
\(118\) −14.0000 −1.28880
\(119\) −24.0000 −2.20008
\(120\) 2.00000 0.182574
\(121\) −11.0000 −1.00000
\(122\) −4.00000 −0.362143
\(123\) 10.0000 0.901670
\(124\) 4.00000 0.359211
\(125\) −12.0000 −1.07331
\(126\) 4.00000 0.356348
\(127\) −4.00000 −0.354943 −0.177471 − 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) 1.00000 0.0883883
\(129\) −4.00000 −0.352180
\(130\) 8.00000 0.701646
\(131\) −12.0000 −1.04844 −0.524222 − 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) −4.00000 −0.346844
\(134\) 4.00000 0.345547
\(135\) 2.00000 0.172133
\(136\) −6.00000 −0.514496
\(137\) −12.0000 −1.02523 −0.512615 − 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 0 0
\(139\) 16.0000 1.35710 0.678551 − 0.734553i \(-0.262608\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) 8.00000 0.676123
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 12.0000 0.996546
\(146\) −10.0000 −0.827606
\(147\) 9.00000 0.742307
\(148\) −4.00000 −0.328798
\(149\) −22.0000 −1.80231 −0.901155 − 0.433497i \(-0.857280\pi\)
−0.901155 + 0.433497i \(0.857280\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) −1.00000 −0.0811107
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) 8.00000 0.642575
\(156\) 4.00000 0.320256
\(157\) 4.00000 0.319235 0.159617 − 0.987179i \(-0.448974\pi\)
0.159617 + 0.987179i \(0.448974\pi\)
\(158\) 12.0000 0.954669
\(159\) 1.00000 0.0793052
\(160\) 2.00000 0.158114
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 12.0000 0.939913 0.469956 − 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) 10.0000 0.780869
\(165\) 0 0
\(166\) −6.00000 −0.465690
\(167\) −8.00000 −0.619059 −0.309529 − 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 4.00000 0.308607
\(169\) 3.00000 0.230769
\(170\) −12.0000 −0.920358
\(171\) −1.00000 −0.0764719
\(172\) −4.00000 −0.304997
\(173\) −14.0000 −1.06440 −0.532200 − 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) 6.00000 0.454859
\(175\) −4.00000 −0.302372
\(176\) 0 0
\(177\) −14.0000 −1.05230
\(178\) 0 0
\(179\) −24.0000 −1.79384 −0.896922 − 0.442189i \(-0.854202\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) 2.00000 0.149071
\(181\) −22.0000 −1.63525 −0.817624 − 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) 16.0000 1.18600
\(183\) −4.00000 −0.295689
\(184\) 0 0
\(185\) −8.00000 −0.588172
\(186\) 4.00000 0.293294
\(187\) 0 0
\(188\) −8.00000 −0.583460
\(189\) 4.00000 0.290957
\(190\) −2.00000 −0.145095
\(191\) 16.0000 1.15772 0.578860 − 0.815427i \(-0.303498\pi\)
0.578860 + 0.815427i \(0.303498\pi\)
\(192\) 1.00000 0.0721688
\(193\) −18.0000 −1.29567 −0.647834 − 0.761781i \(-0.724325\pi\)
−0.647834 + 0.761781i \(0.724325\pi\)
\(194\) 18.0000 1.29232
\(195\) 8.00000 0.572892
\(196\) 9.00000 0.642857
\(197\) 6.00000 0.427482 0.213741 − 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 − 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 4.00000 0.282138
\(202\) 14.0000 0.985037
\(203\) 24.0000 1.68447
\(204\) −6.00000 −0.420084
\(205\) 20.0000 1.39686
\(206\) 12.0000 0.836080
\(207\) 0 0
\(208\) 4.00000 0.277350
\(209\) 0 0
\(210\) 8.00000 0.552052
\(211\) −8.00000 −0.550743 −0.275371 − 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) 1.00000 0.0686803
\(213\) 0 0
\(214\) −10.0000 −0.683586
\(215\) −8.00000 −0.545595
\(216\) 1.00000 0.0680414
\(217\) 16.0000 1.08615
\(218\) −6.00000 −0.406371
\(219\) −10.0000 −0.675737
\(220\) 0 0
\(221\) −24.0000 −1.61441
\(222\) −4.00000 −0.268462
\(223\) 2.00000 0.133930 0.0669650 − 0.997755i \(-0.478668\pi\)
0.0669650 + 0.997755i \(0.478668\pi\)
\(224\) 4.00000 0.267261
\(225\) −1.00000 −0.0666667
\(226\) 12.0000 0.798228
\(227\) 2.00000 0.132745 0.0663723 − 0.997795i \(-0.478857\pi\)
0.0663723 + 0.997795i \(0.478857\pi\)
\(228\) −1.00000 −0.0662266
\(229\) −22.0000 −1.45380 −0.726900 − 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) −4.00000 −0.262049 −0.131024 − 0.991379i \(-0.541827\pi\)
−0.131024 + 0.991379i \(0.541827\pi\)
\(234\) 4.00000 0.261488
\(235\) −16.0000 −1.04372
\(236\) −14.0000 −0.911322
\(237\) 12.0000 0.779484
\(238\) −24.0000 −1.55569
\(239\) 24.0000 1.55243 0.776215 − 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 2.00000 0.129099
\(241\) 10.0000 0.644157 0.322078 − 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) −11.0000 −0.707107
\(243\) 1.00000 0.0641500
\(244\) −4.00000 −0.256074
\(245\) 18.0000 1.14998
\(246\) 10.0000 0.637577
\(247\) −4.00000 −0.254514
\(248\) 4.00000 0.254000
\(249\) −6.00000 −0.380235
\(250\) −12.0000 −0.758947
\(251\) 6.00000 0.378717 0.189358 − 0.981908i \(-0.439359\pi\)
0.189358 + 0.981908i \(0.439359\pi\)
\(252\) 4.00000 0.251976
\(253\) 0 0
\(254\) −4.00000 −0.250982
\(255\) −12.0000 −0.751469
\(256\) 1.00000 0.0625000
\(257\) −18.0000 −1.12281 −0.561405 − 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) −4.00000 −0.249029
\(259\) −16.0000 −0.994192
\(260\) 8.00000 0.496139
\(261\) 6.00000 0.371391
\(262\) −12.0000 −0.741362
\(263\) −12.0000 −0.739952 −0.369976 − 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) 0 0
\(265\) 2.00000 0.122859
\(266\) −4.00000 −0.245256
\(267\) 0 0
\(268\) 4.00000 0.244339
\(269\) 6.00000 0.365826 0.182913 − 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 2.00000 0.121716
\(271\) −20.0000 −1.21491 −0.607457 − 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) −6.00000 −0.363803
\(273\) 16.0000 0.968364
\(274\) −12.0000 −0.724947
\(275\) 0 0
\(276\) 0 0
\(277\) 28.0000 1.68236 0.841178 − 0.540758i \(-0.181862\pi\)
0.841178 + 0.540758i \(0.181862\pi\)
\(278\) 16.0000 0.959616
\(279\) 4.00000 0.239474
\(280\) 8.00000 0.478091
\(281\) 28.0000 1.67034 0.835170 − 0.549992i \(-0.185369\pi\)
0.835170 + 0.549992i \(0.185369\pi\)
\(282\) −8.00000 −0.476393
\(283\) 16.0000 0.951101 0.475551 − 0.879688i \(-0.342249\pi\)
0.475551 + 0.879688i \(0.342249\pi\)
\(284\) 0 0
\(285\) −2.00000 −0.118470
\(286\) 0 0
\(287\) 40.0000 2.36113
\(288\) 1.00000 0.0589256
\(289\) 19.0000 1.11765
\(290\) 12.0000 0.704664
\(291\) 18.0000 1.05518
\(292\) −10.0000 −0.585206
\(293\) 22.0000 1.28525 0.642627 − 0.766179i \(-0.277845\pi\)
0.642627 + 0.766179i \(0.277845\pi\)
\(294\) 9.00000 0.524891
\(295\) −28.0000 −1.63022
\(296\) −4.00000 −0.232495
\(297\) 0 0
\(298\) −22.0000 −1.27443
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) −16.0000 −0.922225
\(302\) 0 0
\(303\) 14.0000 0.804279
\(304\) −1.00000 −0.0573539
\(305\) −8.00000 −0.458079
\(306\) −6.00000 −0.342997
\(307\) 8.00000 0.456584 0.228292 − 0.973593i \(-0.426686\pi\)
0.228292 + 0.973593i \(0.426686\pi\)
\(308\) 0 0
\(309\) 12.0000 0.682656
\(310\) 8.00000 0.454369
\(311\) 8.00000 0.453638 0.226819 − 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) 4.00000 0.226455
\(313\) 22.0000 1.24351 0.621757 − 0.783210i \(-0.286419\pi\)
0.621757 + 0.783210i \(0.286419\pi\)
\(314\) 4.00000 0.225733
\(315\) 8.00000 0.450749
\(316\) 12.0000 0.675053
\(317\) −10.0000 −0.561656 −0.280828 − 0.959758i \(-0.590609\pi\)
−0.280828 + 0.959758i \(0.590609\pi\)
\(318\) 1.00000 0.0560772
\(319\) 0 0
\(320\) 2.00000 0.111803
\(321\) −10.0000 −0.558146
\(322\) 0 0
\(323\) 6.00000 0.333849
\(324\) 1.00000 0.0555556
\(325\) −4.00000 −0.221880
\(326\) 12.0000 0.664619
\(327\) −6.00000 −0.331801
\(328\) 10.0000 0.552158
\(329\) −32.0000 −1.76422
\(330\) 0 0
\(331\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) −6.00000 −0.329293
\(333\) −4.00000 −0.219199
\(334\) −8.00000 −0.437741
\(335\) 8.00000 0.437087
\(336\) 4.00000 0.218218
\(337\) 18.0000 0.980522 0.490261 − 0.871576i \(-0.336901\pi\)
0.490261 + 0.871576i \(0.336901\pi\)
\(338\) 3.00000 0.163178
\(339\) 12.0000 0.651751
\(340\) −12.0000 −0.650791
\(341\) 0 0
\(342\) −1.00000 −0.0540738
\(343\) 8.00000 0.431959
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) −14.0000 −0.752645
\(347\) 12.0000 0.644194 0.322097 − 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 6.00000 0.321634
\(349\) −12.0000 −0.642345 −0.321173 − 0.947021i \(-0.604077\pi\)
−0.321173 + 0.947021i \(0.604077\pi\)
\(350\) −4.00000 −0.213809
\(351\) 4.00000 0.213504
\(352\) 0 0
\(353\) 24.0000 1.27739 0.638696 − 0.769460i \(-0.279474\pi\)
0.638696 + 0.769460i \(0.279474\pi\)
\(354\) −14.0000 −0.744092
\(355\) 0 0
\(356\) 0 0
\(357\) −24.0000 −1.27021
\(358\) −24.0000 −1.26844
\(359\) −24.0000 −1.26667 −0.633336 − 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 2.00000 0.105409
\(361\) 1.00000 0.0526316
\(362\) −22.0000 −1.15629
\(363\) −11.0000 −0.577350
\(364\) 16.0000 0.838628
\(365\) −20.0000 −1.04685
\(366\) −4.00000 −0.209083
\(367\) 20.0000 1.04399 0.521996 − 0.852948i \(-0.325188\pi\)
0.521996 + 0.852948i \(0.325188\pi\)
\(368\) 0 0
\(369\) 10.0000 0.520579
\(370\) −8.00000 −0.415900
\(371\) 4.00000 0.207670
\(372\) 4.00000 0.207390
\(373\) −6.00000 −0.310668 −0.155334 − 0.987862i \(-0.549645\pi\)
−0.155334 + 0.987862i \(0.549645\pi\)
\(374\) 0 0
\(375\) −12.0000 −0.619677
\(376\) −8.00000 −0.412568
\(377\) 24.0000 1.23606
\(378\) 4.00000 0.205738
\(379\) −36.0000 −1.84920 −0.924598 − 0.380945i \(-0.875599\pi\)
−0.924598 + 0.380945i \(0.875599\pi\)
\(380\) −2.00000 −0.102598
\(381\) −4.00000 −0.204926
\(382\) 16.0000 0.818631
\(383\) −24.0000 −1.22634 −0.613171 − 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −18.0000 −0.916176
\(387\) −4.00000 −0.203331
\(388\) 18.0000 0.913812
\(389\) −6.00000 −0.304212 −0.152106 − 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 8.00000 0.405096
\(391\) 0 0
\(392\) 9.00000 0.454569
\(393\) −12.0000 −0.605320
\(394\) 6.00000 0.302276
\(395\) 24.0000 1.20757
\(396\) 0 0
\(397\) 12.0000 0.602263 0.301131 − 0.953583i \(-0.402636\pi\)
0.301131 + 0.953583i \(0.402636\pi\)
\(398\) −8.00000 −0.401004
\(399\) −4.00000 −0.200250
\(400\) −1.00000 −0.0500000
\(401\) 18.0000 0.898877 0.449439 − 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 4.00000 0.199502
\(403\) 16.0000 0.797017
\(404\) 14.0000 0.696526
\(405\) 2.00000 0.0993808
\(406\) 24.0000 1.19110
\(407\) 0 0
\(408\) −6.00000 −0.297044
\(409\) 18.0000 0.890043 0.445021 − 0.895520i \(-0.353196\pi\)
0.445021 + 0.895520i \(0.353196\pi\)
\(410\) 20.0000 0.987730
\(411\) −12.0000 −0.591916
\(412\) 12.0000 0.591198
\(413\) −56.0000 −2.75558
\(414\) 0 0
\(415\) −12.0000 −0.589057
\(416\) 4.00000 0.196116
\(417\) 16.0000 0.783523
\(418\) 0 0
\(419\) 30.0000 1.46560 0.732798 − 0.680446i \(-0.238214\pi\)
0.732798 + 0.680446i \(0.238214\pi\)
\(420\) 8.00000 0.390360
\(421\) 30.0000 1.46211 0.731055 − 0.682318i \(-0.239028\pi\)
0.731055 + 0.682318i \(0.239028\pi\)
\(422\) −8.00000 −0.389434
\(423\) −8.00000 −0.388973
\(424\) 1.00000 0.0485643
\(425\) 6.00000 0.291043
\(426\) 0 0
\(427\) −16.0000 −0.774294
\(428\) −10.0000 −0.483368
\(429\) 0 0
\(430\) −8.00000 −0.385794
\(431\) −32.0000 −1.54139 −0.770693 − 0.637207i \(-0.780090\pi\)
−0.770693 + 0.637207i \(0.780090\pi\)
\(432\) 1.00000 0.0481125
\(433\) −10.0000 −0.480569 −0.240285 − 0.970702i \(-0.577241\pi\)
−0.240285 + 0.970702i \(0.577241\pi\)
\(434\) 16.0000 0.768025
\(435\) 12.0000 0.575356
\(436\) −6.00000 −0.287348
\(437\) 0 0
\(438\) −10.0000 −0.477818
\(439\) 26.0000 1.24091 0.620456 − 0.784241i \(-0.286947\pi\)
0.620456 + 0.784241i \(0.286947\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) −24.0000 −1.14156
\(443\) 34.0000 1.61539 0.807694 − 0.589601i \(-0.200715\pi\)
0.807694 + 0.589601i \(0.200715\pi\)
\(444\) −4.00000 −0.189832
\(445\) 0 0
\(446\) 2.00000 0.0947027
\(447\) −22.0000 −1.04056
\(448\) 4.00000 0.188982
\(449\) −40.0000 −1.88772 −0.943858 − 0.330350i \(-0.892833\pi\)
−0.943858 + 0.330350i \(0.892833\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 0 0
\(452\) 12.0000 0.564433
\(453\) 0 0
\(454\) 2.00000 0.0938647
\(455\) 32.0000 1.50018
\(456\) −1.00000 −0.0468293
\(457\) −22.0000 −1.02912 −0.514558 − 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) −22.0000 −1.02799
\(459\) −6.00000 −0.280056
\(460\) 0 0
\(461\) −14.0000 −0.652045 −0.326023 − 0.945362i \(-0.605709\pi\)
−0.326023 + 0.945362i \(0.605709\pi\)
\(462\) 0 0
\(463\) −22.0000 −1.02243 −0.511213 − 0.859454i \(-0.670804\pi\)
−0.511213 + 0.859454i \(0.670804\pi\)
\(464\) 6.00000 0.278543
\(465\) 8.00000 0.370991
\(466\) −4.00000 −0.185296
\(467\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 4.00000 0.184900
\(469\) 16.0000 0.738811
\(470\) −16.0000 −0.738025
\(471\) 4.00000 0.184310
\(472\) −14.0000 −0.644402
\(473\) 0 0
\(474\) 12.0000 0.551178
\(475\) 1.00000 0.0458831
\(476\) −24.0000 −1.10004
\(477\) 1.00000 0.0457869
\(478\) 24.0000 1.09773
\(479\) −28.0000 −1.27935 −0.639676 − 0.768644i \(-0.720932\pi\)
−0.639676 + 0.768644i \(0.720932\pi\)
\(480\) 2.00000 0.0912871
\(481\) −16.0000 −0.729537
\(482\) 10.0000 0.455488
\(483\) 0 0
\(484\) −11.0000 −0.500000
\(485\) 36.0000 1.63468
\(486\) 1.00000 0.0453609
\(487\) 14.0000 0.634401 0.317200 − 0.948359i \(-0.397257\pi\)
0.317200 + 0.948359i \(0.397257\pi\)
\(488\) −4.00000 −0.181071
\(489\) 12.0000 0.542659
\(490\) 18.0000 0.813157
\(491\) 30.0000 1.35388 0.676941 − 0.736038i \(-0.263305\pi\)
0.676941 + 0.736038i \(0.263305\pi\)
\(492\) 10.0000 0.450835
\(493\) −36.0000 −1.62136
\(494\) −4.00000 −0.179969
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) 0 0
\(498\) −6.00000 −0.268866
\(499\) 12.0000 0.537194 0.268597 − 0.963253i \(-0.413440\pi\)
0.268597 + 0.963253i \(0.413440\pi\)
\(500\) −12.0000 −0.536656
\(501\) −8.00000 −0.357414
\(502\) 6.00000 0.267793
\(503\) −36.0000 −1.60516 −0.802580 − 0.596544i \(-0.796540\pi\)
−0.802580 + 0.596544i \(0.796540\pi\)
\(504\) 4.00000 0.178174
\(505\) 28.0000 1.24598
\(506\) 0 0
\(507\) 3.00000 0.133235
\(508\) −4.00000 −0.177471
\(509\) 30.0000 1.32973 0.664863 − 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) −12.0000 −0.531369
\(511\) −40.0000 −1.76950
\(512\) 1.00000 0.0441942
\(513\) −1.00000 −0.0441511
\(514\) −18.0000 −0.793946
\(515\) 24.0000 1.05757
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) −16.0000 −0.703000
\(519\) −14.0000 −0.614532
\(520\) 8.00000 0.350823
\(521\) −32.0000 −1.40195 −0.700973 − 0.713188i \(-0.747251\pi\)
−0.700973 + 0.713188i \(0.747251\pi\)
\(522\) 6.00000 0.262613
\(523\) 40.0000 1.74908 0.874539 − 0.484955i \(-0.161164\pi\)
0.874539 + 0.484955i \(0.161164\pi\)
\(524\) −12.0000 −0.524222
\(525\) −4.00000 −0.174574
\(526\) −12.0000 −0.523225
\(527\) −24.0000 −1.04546
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 2.00000 0.0868744
\(531\) −14.0000 −0.607548
\(532\) −4.00000 −0.173422
\(533\) 40.0000 1.73259
\(534\) 0 0
\(535\) −20.0000 −0.864675
\(536\) 4.00000 0.172774
\(537\) −24.0000 −1.03568
\(538\) 6.00000 0.258678
\(539\) 0 0
\(540\) 2.00000 0.0860663
\(541\) −2.00000 −0.0859867 −0.0429934 − 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) −20.0000 −0.859074
\(543\) −22.0000 −0.944110
\(544\) −6.00000 −0.257248
\(545\) −12.0000 −0.514024
\(546\) 16.0000 0.684737
\(547\) 24.0000 1.02617 0.513083 − 0.858339i \(-0.328503\pi\)
0.513083 + 0.858339i \(0.328503\pi\)
\(548\) −12.0000 −0.512615
\(549\) −4.00000 −0.170716
\(550\) 0 0
\(551\) −6.00000 −0.255609
\(552\) 0 0
\(553\) 48.0000 2.04117
\(554\) 28.0000 1.18961
\(555\) −8.00000 −0.339581
\(556\) 16.0000 0.678551
\(557\) 18.0000 0.762684 0.381342 − 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) 4.00000 0.169334
\(559\) −16.0000 −0.676728
\(560\) 8.00000 0.338062
\(561\) 0 0
\(562\) 28.0000 1.18111
\(563\) −28.0000 −1.18006 −0.590030 − 0.807382i \(-0.700884\pi\)
−0.590030 + 0.807382i \(0.700884\pi\)
\(564\) −8.00000 −0.336861
\(565\) 24.0000 1.00969
\(566\) 16.0000 0.672530
\(567\) 4.00000 0.167984
\(568\) 0 0
\(569\) 30.0000 1.25767 0.628833 − 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) −2.00000 −0.0837708
\(571\) −16.0000 −0.669579 −0.334790 − 0.942293i \(-0.608665\pi\)
−0.334790 + 0.942293i \(0.608665\pi\)
\(572\) 0 0
\(573\) 16.0000 0.668410
\(574\) 40.0000 1.66957
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −2.00000 −0.0832611 −0.0416305 − 0.999133i \(-0.513255\pi\)
−0.0416305 + 0.999133i \(0.513255\pi\)
\(578\) 19.0000 0.790296
\(579\) −18.0000 −0.748054
\(580\) 12.0000 0.498273
\(581\) −24.0000 −0.995688
\(582\) 18.0000 0.746124
\(583\) 0 0
\(584\) −10.0000 −0.413803
\(585\) 8.00000 0.330759
\(586\) 22.0000 0.908812
\(587\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 9.00000 0.371154
\(589\) −4.00000 −0.164817
\(590\) −28.0000 −1.15274
\(591\) 6.00000 0.246807
\(592\) −4.00000 −0.164399
\(593\) −42.0000 −1.72473 −0.862367 − 0.506284i \(-0.831019\pi\)
−0.862367 + 0.506284i \(0.831019\pi\)
\(594\) 0 0
\(595\) −48.0000 −1.96781
\(596\) −22.0000 −0.901155
\(597\) −8.00000 −0.327418
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 − 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −2.00000 −0.0815817 −0.0407909 − 0.999168i \(-0.512988\pi\)
−0.0407909 + 0.999168i \(0.512988\pi\)
\(602\) −16.0000 −0.652111
\(603\) 4.00000 0.162893
\(604\) 0 0
\(605\) −22.0000 −0.894427
\(606\) 14.0000 0.568711
\(607\) −6.00000 −0.243532 −0.121766 − 0.992559i \(-0.538856\pi\)
−0.121766 + 0.992559i \(0.538856\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 24.0000 0.972529
\(610\) −8.00000 −0.323911
\(611\) −32.0000 −1.29458
\(612\) −6.00000 −0.242536
\(613\) 8.00000 0.323117 0.161558 − 0.986863i \(-0.448348\pi\)
0.161558 + 0.986863i \(0.448348\pi\)
\(614\) 8.00000 0.322854
\(615\) 20.0000 0.806478
\(616\) 0 0
\(617\) −12.0000 −0.483102 −0.241551 − 0.970388i \(-0.577656\pi\)
−0.241551 + 0.970388i \(0.577656\pi\)
\(618\) 12.0000 0.482711
\(619\) −36.0000 −1.44696 −0.723481 − 0.690344i \(-0.757459\pi\)
−0.723481 + 0.690344i \(0.757459\pi\)
\(620\) 8.00000 0.321288
\(621\) 0 0
\(622\) 8.00000 0.320771
\(623\) 0 0
\(624\) 4.00000 0.160128
\(625\) −19.0000 −0.760000
\(626\) 22.0000 0.879297
\(627\) 0 0
\(628\) 4.00000 0.159617
\(629\) 24.0000 0.956943
\(630\) 8.00000 0.318728
\(631\) 10.0000 0.398094 0.199047 − 0.979990i \(-0.436215\pi\)
0.199047 + 0.979990i \(0.436215\pi\)
\(632\) 12.0000 0.477334
\(633\) −8.00000 −0.317971
\(634\) −10.0000 −0.397151
\(635\) −8.00000 −0.317470
\(636\) 1.00000 0.0396526
\(637\) 36.0000 1.42637
\(638\) 0 0
\(639\) 0 0
\(640\) 2.00000 0.0790569
\(641\) 42.0000 1.65890 0.829450 − 0.558581i \(-0.188654\pi\)
0.829450 + 0.558581i \(0.188654\pi\)
\(642\) −10.0000 −0.394669
\(643\) 44.0000 1.73519 0.867595 − 0.497271i \(-0.165665\pi\)
0.867595 + 0.497271i \(0.165665\pi\)
\(644\) 0 0
\(645\) −8.00000 −0.315000
\(646\) 6.00000 0.236067
\(647\) −32.0000 −1.25805 −0.629025 − 0.777385i \(-0.716546\pi\)
−0.629025 + 0.777385i \(0.716546\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) −4.00000 −0.156893
\(651\) 16.0000 0.627089
\(652\) 12.0000 0.469956
\(653\) 50.0000 1.95665 0.978326 − 0.207072i \(-0.0663936\pi\)
0.978326 + 0.207072i \(0.0663936\pi\)
\(654\) −6.00000 −0.234619
\(655\) −24.0000 −0.937758
\(656\) 10.0000 0.390434
\(657\) −10.0000 −0.390137
\(658\) −32.0000 −1.24749
\(659\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 40.0000 1.55582 0.777910 − 0.628376i \(-0.216280\pi\)
0.777910 + 0.628376i \(0.216280\pi\)
\(662\) 0 0
\(663\) −24.0000 −0.932083
\(664\) −6.00000 −0.232845
\(665\) −8.00000 −0.310227
\(666\) −4.00000 −0.154997
\(667\) 0 0
\(668\) −8.00000 −0.309529
\(669\) 2.00000 0.0773245
\(670\) 8.00000 0.309067
\(671\) 0 0
\(672\) 4.00000 0.154303
\(673\) 6.00000 0.231283 0.115642 − 0.993291i \(-0.463108\pi\)
0.115642 + 0.993291i \(0.463108\pi\)
\(674\) 18.0000 0.693334
\(675\) −1.00000 −0.0384900
\(676\) 3.00000 0.115385
\(677\) 26.0000 0.999261 0.499631 − 0.866239i \(-0.333469\pi\)
0.499631 + 0.866239i \(0.333469\pi\)
\(678\) 12.0000 0.460857
\(679\) 72.0000 2.76311
\(680\) −12.0000 −0.460179
\(681\) 2.00000 0.0766402
\(682\) 0 0
\(683\) −26.0000 −0.994862 −0.497431 − 0.867503i \(-0.665723\pi\)
−0.497431 + 0.867503i \(0.665723\pi\)
\(684\) −1.00000 −0.0382360
\(685\) −24.0000 −0.916993
\(686\) 8.00000 0.305441
\(687\) −22.0000 −0.839352
\(688\) −4.00000 −0.152499
\(689\) 4.00000 0.152388
\(690\) 0 0
\(691\) 36.0000 1.36950 0.684752 − 0.728776i \(-0.259910\pi\)
0.684752 + 0.728776i \(0.259910\pi\)
\(692\) −14.0000 −0.532200
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) 32.0000 1.21383
\(696\) 6.00000 0.227429
\(697\) −60.0000 −2.27266
\(698\) −12.0000 −0.454207
\(699\) −4.00000 −0.151294
\(700\) −4.00000 −0.151186
\(701\) 30.0000 1.13308 0.566542 − 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 4.00000 0.150970
\(703\) 4.00000 0.150863
\(704\) 0 0
\(705\) −16.0000 −0.602595
\(706\) 24.0000 0.903252
\(707\) 56.0000 2.10610
\(708\) −14.0000 −0.526152
\(709\) −36.0000 −1.35201 −0.676004 − 0.736898i \(-0.736290\pi\)
−0.676004 + 0.736898i \(0.736290\pi\)
\(710\) 0 0
\(711\) 12.0000 0.450035
\(712\) 0 0
\(713\) 0 0
\(714\) −24.0000 −0.898177
\(715\) 0 0
\(716\) −24.0000 −0.896922
\(717\) 24.0000 0.896296
\(718\) −24.0000 −0.895672
\(719\) 4.00000 0.149175 0.0745874 − 0.997214i \(-0.476236\pi\)
0.0745874 + 0.997214i \(0.476236\pi\)
\(720\) 2.00000 0.0745356
\(721\) 48.0000 1.78761
\(722\) 1.00000 0.0372161
\(723\) 10.0000 0.371904
\(724\) −22.0000 −0.817624
\(725\) −6.00000 −0.222834
\(726\) −11.0000 −0.408248
\(727\) 40.0000 1.48352 0.741759 − 0.670667i \(-0.233992\pi\)
0.741759 + 0.670667i \(0.233992\pi\)
\(728\) 16.0000 0.592999
\(729\) 1.00000 0.0370370
\(730\) −20.0000 −0.740233
\(731\) 24.0000 0.887672
\(732\) −4.00000 −0.147844
\(733\) −30.0000 −1.10808 −0.554038 − 0.832492i \(-0.686914\pi\)
−0.554038 + 0.832492i \(0.686914\pi\)
\(734\) 20.0000 0.738213
\(735\) 18.0000 0.663940
\(736\) 0 0
\(737\) 0 0
\(738\) 10.0000 0.368105
\(739\) −40.0000 −1.47142 −0.735712 − 0.677295i \(-0.763152\pi\)
−0.735712 + 0.677295i \(0.763152\pi\)
\(740\) −8.00000 −0.294086
\(741\) −4.00000 −0.146944
\(742\) 4.00000 0.146845
\(743\) −36.0000 −1.32071 −0.660356 − 0.750953i \(-0.729595\pi\)
−0.660356 + 0.750953i \(0.729595\pi\)
\(744\) 4.00000 0.146647
\(745\) −44.0000 −1.61204
\(746\) −6.00000 −0.219676
\(747\) −6.00000 −0.219529
\(748\) 0 0
\(749\) −40.0000 −1.46157
\(750\) −12.0000 −0.438178
\(751\) −42.0000 −1.53260 −0.766301 − 0.642482i \(-0.777905\pi\)
−0.766301 + 0.642482i \(0.777905\pi\)
\(752\) −8.00000 −0.291730
\(753\) 6.00000 0.218652
\(754\) 24.0000 0.874028
\(755\) 0 0
\(756\) 4.00000 0.145479
\(757\) 22.0000 0.799604 0.399802 − 0.916602i \(-0.369079\pi\)
0.399802 + 0.916602i \(0.369079\pi\)
\(758\) −36.0000 −1.30758
\(759\) 0 0
\(760\) −2.00000 −0.0725476
\(761\) 12.0000 0.435000 0.217500 − 0.976060i \(-0.430210\pi\)
0.217500 + 0.976060i \(0.430210\pi\)
\(762\) −4.00000 −0.144905
\(763\) −24.0000 −0.868858
\(764\) 16.0000 0.578860
\(765\) −12.0000 −0.433861
\(766\) −24.0000 −0.867155
\(767\) −56.0000 −2.02204
\(768\) 1.00000 0.0360844
\(769\) 22.0000 0.793340 0.396670 − 0.917961i \(-0.370166\pi\)
0.396670 + 0.917961i \(0.370166\pi\)
\(770\) 0 0
\(771\) −18.0000 −0.648254
\(772\) −18.0000 −0.647834
\(773\) −46.0000 −1.65451 −0.827253 − 0.561830i \(-0.810097\pi\)
−0.827253 + 0.561830i \(0.810097\pi\)
\(774\) −4.00000 −0.143777
\(775\) −4.00000 −0.143684
\(776\) 18.0000 0.646162
\(777\) −16.0000 −0.573997
\(778\) −6.00000 −0.215110
\(779\) −10.0000 −0.358287
\(780\) 8.00000 0.286446
\(781\) 0 0
\(782\) 0 0
\(783\) 6.00000 0.214423
\(784\) 9.00000 0.321429
\(785\) 8.00000 0.285532
\(786\) −12.0000 −0.428026
\(787\) 4.00000 0.142585 0.0712923 − 0.997455i \(-0.477288\pi\)
0.0712923 + 0.997455i \(0.477288\pi\)
\(788\) 6.00000 0.213741
\(789\) −12.0000 −0.427211
\(790\) 24.0000 0.853882
\(791\) 48.0000 1.70668
\(792\) 0 0
\(793\) −16.0000 −0.568177
\(794\) 12.0000 0.425864
\(795\) 2.00000 0.0709327
\(796\) −8.00000 −0.283552
\(797\) 18.0000 0.637593 0.318796 − 0.947823i \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(798\) −4.00000 −0.141598
\(799\) 48.0000 1.69812
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) 18.0000 0.635602
\(803\) 0 0
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) 16.0000 0.563576
\(807\) 6.00000 0.211210
\(808\) 14.0000 0.492518
\(809\) 48.0000 1.68759 0.843795 − 0.536666i \(-0.180316\pi\)
0.843795 + 0.536666i \(0.180316\pi\)
\(810\) 2.00000 0.0702728
\(811\) −20.0000 −0.702295 −0.351147 − 0.936320i \(-0.614208\pi\)
−0.351147 + 0.936320i \(0.614208\pi\)
\(812\) 24.0000 0.842235
\(813\) −20.0000 −0.701431
\(814\) 0 0
\(815\) 24.0000 0.840683
\(816\) −6.00000 −0.210042
\(817\) 4.00000 0.139942
\(818\) 18.0000 0.629355
\(819\) 16.0000 0.559085
\(820\) 20.0000 0.698430
\(821\) −26.0000 −0.907406 −0.453703 − 0.891153i \(-0.649897\pi\)
−0.453703 + 0.891153i \(0.649897\pi\)
\(822\) −12.0000 −0.418548
\(823\) 52.0000 1.81261 0.906303 − 0.422628i \(-0.138892\pi\)
0.906303 + 0.422628i \(0.138892\pi\)
\(824\) 12.0000 0.418040
\(825\) 0 0
\(826\) −56.0000 −1.94849
\(827\) −16.0000 −0.556375 −0.278187 − 0.960527i \(-0.589734\pi\)
−0.278187 + 0.960527i \(0.589734\pi\)
\(828\) 0 0
\(829\) 6.00000 0.208389 0.104194 − 0.994557i \(-0.466774\pi\)
0.104194 + 0.994557i \(0.466774\pi\)
\(830\) −12.0000 −0.416526
\(831\) 28.0000 0.971309
\(832\) 4.00000 0.138675
\(833\) −54.0000 −1.87099
\(834\) 16.0000 0.554035
\(835\) −16.0000 −0.553703
\(836\) 0 0
\(837\) 4.00000 0.138260
\(838\) 30.0000 1.03633
\(839\) −44.0000 −1.51905 −0.759524 − 0.650479i \(-0.774568\pi\)
−0.759524 + 0.650479i \(0.774568\pi\)
\(840\) 8.00000 0.276026
\(841\) 7.00000 0.241379
\(842\) 30.0000 1.03387
\(843\) 28.0000 0.964371
\(844\) −8.00000 −0.275371
\(845\) 6.00000 0.206406
\(846\) −8.00000 −0.275046
\(847\) −44.0000 −1.51186
\(848\) 1.00000 0.0343401
\(849\) 16.0000 0.549119
\(850\) 6.00000 0.205798
\(851\) 0 0
\(852\) 0 0
\(853\) −36.0000 −1.23262 −0.616308 − 0.787505i \(-0.711372\pi\)
−0.616308 + 0.787505i \(0.711372\pi\)
\(854\) −16.0000 −0.547509
\(855\) −2.00000 −0.0683986
\(856\) −10.0000 −0.341793
\(857\) −36.0000 −1.22974 −0.614868 − 0.788630i \(-0.710791\pi\)
−0.614868 + 0.788630i \(0.710791\pi\)
\(858\) 0 0
\(859\) 4.00000 0.136478 0.0682391 − 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) −8.00000 −0.272798
\(861\) 40.0000 1.36320
\(862\) −32.0000 −1.08992
\(863\) −36.0000 −1.22545 −0.612727 − 0.790295i \(-0.709928\pi\)
−0.612727 + 0.790295i \(0.709928\pi\)
\(864\) 1.00000 0.0340207
\(865\) −28.0000 −0.952029
\(866\) −10.0000 −0.339814
\(867\) 19.0000 0.645274
\(868\) 16.0000 0.543075
\(869\) 0 0
\(870\) 12.0000 0.406838
\(871\) 16.0000 0.542139
\(872\) −6.00000 −0.203186
\(873\) 18.0000 0.609208
\(874\) 0 0
\(875\) −48.0000 −1.62270
\(876\) −10.0000 −0.337869
\(877\) −44.0000 −1.48577 −0.742887 − 0.669417i \(-0.766544\pi\)
−0.742887 + 0.669417i \(0.766544\pi\)
\(878\) 26.0000 0.877457
\(879\) 22.0000 0.742042
\(880\) 0 0
\(881\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 9.00000 0.303046
\(883\) −8.00000 −0.269221 −0.134611 − 0.990899i \(-0.542978\pi\)
−0.134611 + 0.990899i \(0.542978\pi\)
\(884\) −24.0000 −0.807207
\(885\) −28.0000 −0.941210
\(886\) 34.0000 1.14225
\(887\) −48.0000 −1.61168 −0.805841 − 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) −4.00000 −0.134231
\(889\) −16.0000 −0.536623
\(890\) 0 0
\(891\) 0 0
\(892\) 2.00000 0.0669650
\(893\) 8.00000 0.267710
\(894\) −22.0000 −0.735790
\(895\) −48.0000 −1.60446
\(896\) 4.00000 0.133631
\(897\) 0 0
\(898\) −40.0000 −1.33482
\(899\) 24.0000 0.800445
\(900\) −1.00000 −0.0333333
\(901\) −6.00000 −0.199889
\(902\) 0 0
\(903\) −16.0000 −0.532447
\(904\) 12.0000 0.399114
\(905\) −44.0000 −1.46261
\(906\) 0 0
\(907\) −44.0000 −1.46100 −0.730498 − 0.682915i \(-0.760712\pi\)
−0.730498 + 0.682915i \(0.760712\pi\)
\(908\) 2.00000 0.0663723
\(909\) 14.0000 0.464351
\(910\) 32.0000 1.06079
\(911\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) −1.00000 −0.0331133
\(913\) 0 0
\(914\) −22.0000 −0.727695
\(915\) −8.00000 −0.264472
\(916\) −22.0000 −0.726900
\(917\) −48.0000 −1.58510
\(918\) −6.00000 −0.198030
\(919\) −10.0000 −0.329870 −0.164935 − 0.986304i \(-0.552741\pi\)
−0.164935 + 0.986304i \(0.552741\pi\)
\(920\) 0 0
\(921\) 8.00000 0.263609
\(922\) −14.0000 −0.461065
\(923\) 0 0
\(924\) 0 0
\(925\) 4.00000 0.131519
\(926\) −22.0000 −0.722965
\(927\) 12.0000 0.394132
\(928\) 6.00000 0.196960
\(929\) −30.0000 −0.984268 −0.492134 − 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 8.00000 0.262330
\(931\) −9.00000 −0.294963
\(932\) −4.00000 −0.131024
\(933\) 8.00000 0.261908
\(934\) 0 0
\(935\) 0 0
\(936\) 4.00000 0.130744
\(937\) 26.0000 0.849383 0.424691 − 0.905338i \(-0.360383\pi\)
0.424691 + 0.905338i \(0.360383\pi\)
\(938\) 16.0000 0.522419
\(939\) 22.0000 0.717943
\(940\) −16.0000 −0.521862
\(941\) −22.0000 −0.717180 −0.358590 − 0.933495i \(-0.616742\pi\)
−0.358590 + 0.933495i \(0.616742\pi\)
\(942\) 4.00000 0.130327
\(943\) 0 0
\(944\) −14.0000 −0.455661
\(945\) 8.00000 0.260240
\(946\) 0 0
\(947\) 32.0000 1.03986 0.519930 − 0.854209i \(-0.325958\pi\)
0.519930 + 0.854209i \(0.325958\pi\)
\(948\) 12.0000 0.389742
\(949\) −40.0000 −1.29845
\(950\) 1.00000 0.0324443
\(951\) −10.0000 −0.324272
\(952\) −24.0000 −0.777844
\(953\) 12.0000 0.388718 0.194359 − 0.980930i \(-0.437737\pi\)
0.194359 + 0.980930i \(0.437737\pi\)
\(954\) 1.00000 0.0323762
\(955\) 32.0000 1.03550
\(956\) 24.0000 0.776215
\(957\) 0 0
\(958\) −28.0000 −0.904639
\(959\) −48.0000 −1.55000
\(960\) 2.00000 0.0645497
\(961\) −15.0000 −0.483871
\(962\) −16.0000 −0.515861
\(963\) −10.0000 −0.322245
\(964\) 10.0000 0.322078
\(965\) −36.0000 −1.15888
\(966\) 0 0
\(967\) 44.0000 1.41494 0.707472 − 0.706741i \(-0.249835\pi\)
0.707472 + 0.706741i \(0.249835\pi\)
\(968\) −11.0000 −0.353553
\(969\) 6.00000 0.192748
\(970\) 36.0000 1.15589
\(971\) 18.0000 0.577647 0.288824 − 0.957382i \(-0.406736\pi\)
0.288824 + 0.957382i \(0.406736\pi\)
\(972\) 1.00000 0.0320750
\(973\) 64.0000 2.05175
\(974\) 14.0000 0.448589
\(975\) −4.00000 −0.128103
\(976\) −4.00000 −0.128037
\(977\) −14.0000 −0.447900 −0.223950 − 0.974601i \(-0.571895\pi\)
−0.223950 + 0.974601i \(0.571895\pi\)
\(978\) 12.0000 0.383718
\(979\) 0 0
\(980\) 18.0000 0.574989
\(981\) −6.00000 −0.191565
\(982\) 30.0000 0.957338
\(983\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 10.0000 0.318788
\(985\) 12.0000 0.382352
\(986\) −36.0000 −1.14647
\(987\) −32.0000 −1.01857
\(988\) −4.00000 −0.127257
\(989\) 0 0
\(990\) 0 0
\(991\) −46.0000 −1.46124 −0.730619 − 0.682785i \(-0.760768\pi\)
−0.730619 + 0.682785i \(0.760768\pi\)
\(992\) 4.00000 0.127000
\(993\) 0 0
\(994\) 0 0
\(995\) −16.0000 −0.507234
\(996\) −6.00000 −0.190117
\(997\) −42.0000 −1.33015 −0.665077 − 0.746775i \(-0.731601\pi\)
−0.665077 + 0.746775i \(0.731601\pi\)
\(998\) 12.0000 0.379853
\(999\) −4.00000 −0.126554
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 6042.2.a.m.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6042.2.a.m.1.1 ✓ 1 1.1 even 1 trivial