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Properties

Label 930.2.a.l.1.1
Level $930$
Weight $2$
Character 930.1
Self dual yes
Analytic conductor $7.426$
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] = [930,2,Mod(1,930)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(930, 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("930.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.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: \(7.42608738798\)
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 \(\chi\) \(=\) 930.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -6.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} +1.00000 q^{15} +1.00000 q^{16} -4.00000 q^{17} +1.00000 q^{18} -1.00000 q^{20} -6.00000 q^{22} +2.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} -8.00000 q^{29} +1.00000 q^{30} +1.00000 q^{31} +1.00000 q^{32} +6.00000 q^{33} -4.00000 q^{34} +1.00000 q^{36} -6.00000 q^{37} +2.00000 q^{39} -1.00000 q^{40} -2.00000 q^{41} +4.00000 q^{43} -6.00000 q^{44} -1.00000 q^{45} +2.00000 q^{46} +4.00000 q^{47} -1.00000 q^{48} -7.00000 q^{49} +1.00000 q^{50} +4.00000 q^{51} -2.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} +6.00000 q^{55} -8.00000 q^{58} +1.00000 q^{60} +4.00000 q^{61} +1.00000 q^{62} +1.00000 q^{64} +2.00000 q^{65} +6.00000 q^{66} -4.00000 q^{67} -4.00000 q^{68} -2.00000 q^{69} -8.00000 q^{71} +1.00000 q^{72} -4.00000 q^{73} -6.00000 q^{74} -1.00000 q^{75} +2.00000 q^{78} -4.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} +4.00000 q^{85} +4.00000 q^{86} +8.00000 q^{87} -6.00000 q^{88} -2.00000 q^{89} -1.00000 q^{90} +2.00000 q^{92} -1.00000 q^{93} +4.00000 q^{94} -1.00000 q^{96} +14.0000 q^{97} -7.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

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
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) −6.00000 −1.80907 −0.904534 − 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) −1.00000 −0.288675
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 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
\(21\) 0 0
\(22\) −6.00000 −1.27920
\(23\) 2.00000 0.417029 0.208514 − 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) −2.00000 −0.392232
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −8.00000 −1.48556 −0.742781 − 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) 1.00000 0.182574
\(31\) 1.00000 0.179605
\(32\) 1.00000 0.176777
\(33\) 6.00000 1.04447
\(34\) −4.00000 −0.685994
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −6.00000 −0.986394 −0.493197 − 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) −1.00000 −0.158114
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) −6.00000 −0.904534
\(45\) −1.00000 −0.149071
\(46\) 2.00000 0.294884
\(47\) 4.00000 0.583460 0.291730 − 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) −1.00000 −0.144338
\(49\) −7.00000 −1.00000
\(50\) 1.00000 0.141421
\(51\) 4.00000 0.560112
\(52\) −2.00000 −0.277350
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) −1.00000 −0.136083
\(55\) 6.00000 0.809040
\(56\) 0 0
\(57\) 0 0
\(58\) −8.00000 −1.05045
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 1.00000 0.129099
\(61\) 4.00000 0.512148 0.256074 − 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) 1.00000 0.127000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) 6.00000 0.738549
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −4.00000 −0.485071
\(69\) −2.00000 −0.240772
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 1.00000 0.117851
\(73\) −4.00000 −0.468165 −0.234082 − 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) −6.00000 −0.697486
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 0 0
\(78\) 2.00000 0.226455
\(79\) −4.00000 −0.450035 −0.225018 − 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −2.00000 −0.220863
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 4.00000 0.433861
\(86\) 4.00000 0.431331
\(87\) 8.00000 0.857690
\(88\) −6.00000 −0.639602
\(89\) −2.00000 −0.212000 −0.106000 − 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) 2.00000 0.208514
\(93\) −1.00000 −0.103695
\(94\) 4.00000 0.412568
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 14.0000 1.42148 0.710742 − 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) −7.00000 −0.707107
\(99\) −6.00000 −0.603023
\(100\) 1.00000 0.100000
\(101\) 2.00000 0.199007 0.0995037 − 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 4.00000 0.396059
\(103\) 16.0000 1.57653 0.788263 − 0.615338i \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 12.0000 1.16008 0.580042 − 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −10.0000 −0.957826 −0.478913 − 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 6.00000 0.572078
\(111\) 6.00000 0.569495
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 − 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) −2.00000 −0.186501
\(116\) −8.00000 −0.742781
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) 0 0
\(120\) 1.00000 0.0912871
\(121\) 25.0000 2.27273
\(122\) 4.00000 0.362143
\(123\) 2.00000 0.180334
\(124\) 1.00000 0.0898027
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 14.0000 1.24230 0.621150 − 0.783692i \(-0.286666\pi\)
0.621150 + 0.783692i \(0.286666\pi\)
\(128\) 1.00000 0.0883883
\(129\) −4.00000 −0.352180
\(130\) 2.00000 0.175412
\(131\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 6.00000 0.522233
\(133\) 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 \(-0.388983\pi\)
0.341743 + 0.939793i \(0.388983\pi\)
\(138\) −2.00000 −0.170251
\(139\) −2.00000 −0.169638 −0.0848189 − 0.996396i \(-0.527031\pi\)
−0.0848189 + 0.996396i \(0.527031\pi\)
\(140\) 0 0
\(141\) −4.00000 −0.336861
\(142\) −8.00000 −0.671345
\(143\) 12.0000 1.00349
\(144\) 1.00000 0.0833333
\(145\) 8.00000 0.664364
\(146\) −4.00000 −0.331042
\(147\) 7.00000 0.577350
\(148\) −6.00000 −0.493197
\(149\) −14.0000 −1.14692 −0.573462 − 0.819232i \(-0.694400\pi\)
−0.573462 + 0.819232i \(0.694400\pi\)
\(150\) −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
\(154\) 0 0
\(155\) −1.00000 −0.0803219
\(156\) 2.00000 0.160128
\(157\) −2.00000 −0.159617 −0.0798087 − 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) −4.00000 −0.318223
\(159\) 6.00000 0.475831
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −4.00000 −0.313304 −0.156652 − 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) −2.00000 −0.156174
\(165\) −6.00000 −0.467099
\(166\) 0 0
\(167\) −10.0000 −0.773823 −0.386912 − 0.922117i \(-0.626458\pi\)
−0.386912 + 0.922117i \(0.626458\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 4.00000 0.306786
\(171\) 0 0
\(172\) 4.00000 0.304997
\(173\) 14.0000 1.06440 0.532200 − 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) 8.00000 0.606478
\(175\) 0 0
\(176\) −6.00000 −0.452267
\(177\) 0 0
\(178\) −2.00000 −0.149906
\(179\) −2.00000 −0.149487 −0.0747435 − 0.997203i \(-0.523814\pi\)
−0.0747435 + 0.997203i \(0.523814\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 8.00000 0.594635 0.297318 − 0.954779i \(-0.403908\pi\)
0.297318 + 0.954779i \(0.403908\pi\)
\(182\) 0 0
\(183\) −4.00000 −0.295689
\(184\) 2.00000 0.147442
\(185\) 6.00000 0.441129
\(186\) −1.00000 −0.0733236
\(187\) 24.0000 1.75505
\(188\) 4.00000 0.291730
\(189\) 0 0
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 − 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −18.0000 −1.29567 −0.647834 − 0.761781i \(-0.724325\pi\)
−0.647834 + 0.761781i \(0.724325\pi\)
\(194\) 14.0000 1.00514
\(195\) −2.00000 −0.143223
\(196\) −7.00000 −0.500000
\(197\) −18.0000 −1.28245 −0.641223 − 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) −6.00000 −0.426401
\(199\) −20.0000 −1.41776 −0.708881 − 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) 1.00000 0.0707107
\(201\) 4.00000 0.282138
\(202\) 2.00000 0.140720
\(203\) 0 0
\(204\) 4.00000 0.280056
\(205\) 2.00000 0.139686
\(206\) 16.0000 1.11477
\(207\) 2.00000 0.139010
\(208\) −2.00000 −0.138675
\(209\) 0 0
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 − 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) −6.00000 −0.412082
\(213\) 8.00000 0.548151
\(214\) 12.0000 0.820303
\(215\) −4.00000 −0.272798
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −10.0000 −0.677285
\(219\) 4.00000 0.270295
\(220\) 6.00000 0.404520
\(221\) 8.00000 0.538138
\(222\) 6.00000 0.402694
\(223\) 18.0000 1.20537 0.602685 − 0.797980i \(-0.294098\pi\)
0.602685 + 0.797980i \(0.294098\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 6.00000 0.399114
\(227\) 12.0000 0.796468 0.398234 − 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) 16.0000 1.05731 0.528655 − 0.848837i \(-0.322697\pi\)
0.528655 + 0.848837i \(0.322697\pi\)
\(230\) −2.00000 −0.131876
\(231\) 0 0
\(232\) −8.00000 −0.525226
\(233\) 6.00000 0.393073 0.196537 − 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) −2.00000 −0.130744
\(235\) −4.00000 −0.260931
\(236\) 0 0
\(237\) 4.00000 0.259828
\(238\) 0 0
\(239\) −24.0000 −1.55243 −0.776215 − 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 1.00000 0.0645497
\(241\) 18.0000 1.15948 0.579741 − 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) 25.0000 1.60706
\(243\) −1.00000 −0.0641500
\(244\) 4.00000 0.256074
\(245\) 7.00000 0.447214
\(246\) 2.00000 0.127515
\(247\) 0 0
\(248\) 1.00000 0.0635001
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) 18.0000 1.13615 0.568075 − 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) 0 0
\(253\) −12.0000 −0.754434
\(254\) 14.0000 0.878438
\(255\) −4.00000 −0.250490
\(256\) 1.00000 0.0625000
\(257\) −26.0000 −1.62184 −0.810918 − 0.585160i \(-0.801032\pi\)
−0.810918 + 0.585160i \(0.801032\pi\)
\(258\) −4.00000 −0.249029
\(259\) 0 0
\(260\) 2.00000 0.124035
\(261\) −8.00000 −0.495188
\(262\) 0 0
\(263\) 18.0000 1.10993 0.554964 − 0.831875i \(-0.312732\pi\)
0.554964 + 0.831875i \(0.312732\pi\)
\(264\) 6.00000 0.369274
\(265\) 6.00000 0.368577
\(266\) 0 0
\(267\) 2.00000 0.122398
\(268\) −4.00000 −0.244339
\(269\) −16.0000 −0.975537 −0.487769 − 0.872973i \(-0.662189\pi\)
−0.487769 + 0.872973i \(0.662189\pi\)
\(270\) 1.00000 0.0608581
\(271\) 16.0000 0.971931 0.485965 − 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) −4.00000 −0.242536
\(273\) 0 0
\(274\) 8.00000 0.483298
\(275\) −6.00000 −0.361814
\(276\) −2.00000 −0.120386
\(277\) 18.0000 1.08152 0.540758 − 0.841178i \(-0.318138\pi\)
0.540758 + 0.841178i \(0.318138\pi\)
\(278\) −2.00000 −0.119952
\(279\) 1.00000 0.0598684
\(280\) 0 0
\(281\) −26.0000 −1.55103 −0.775515 − 0.631329i \(-0.782510\pi\)
−0.775515 + 0.631329i \(0.782510\pi\)
\(282\) −4.00000 −0.238197
\(283\) −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
\(286\) 12.0000 0.709575
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −1.00000 −0.0588235
\(290\) 8.00000 0.469776
\(291\) −14.0000 −0.820695
\(292\) −4.00000 −0.234082
\(293\) −30.0000 −1.75262 −0.876309 − 0.481749i \(-0.840002\pi\)
−0.876309 + 0.481749i \(0.840002\pi\)
\(294\) 7.00000 0.408248
\(295\) 0 0
\(296\) −6.00000 −0.348743
\(297\) 6.00000 0.348155
\(298\) −14.0000 −0.810998
\(299\) −4.00000 −0.231326
\(300\) −1.00000 −0.0577350
\(301\) 0 0
\(302\) 12.0000 0.690522
\(303\) −2.00000 −0.114897
\(304\) 0 0
\(305\) −4.00000 −0.229039
\(306\) −4.00000 −0.228665
\(307\) 4.00000 0.228292 0.114146 − 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) 0 0
\(309\) −16.0000 −0.910208
\(310\) −1.00000 −0.0567962
\(311\) −24.0000 −1.36092 −0.680458 − 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 2.00000 0.113228
\(313\) −12.0000 −0.678280 −0.339140 − 0.940736i \(-0.610136\pi\)
−0.339140 + 0.940736i \(0.610136\pi\)
\(314\) −2.00000 −0.112867
\(315\) 0 0
\(316\) −4.00000 −0.225018
\(317\) −26.0000 −1.46031 −0.730153 − 0.683284i \(-0.760551\pi\)
−0.730153 + 0.683284i \(0.760551\pi\)
\(318\) 6.00000 0.336463
\(319\) 48.0000 2.68748
\(320\) −1.00000 −0.0559017
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) −2.00000 −0.110940
\(326\) −4.00000 −0.221540
\(327\) 10.0000 0.553001
\(328\) −2.00000 −0.110432
\(329\) 0 0
\(330\) −6.00000 −0.330289
\(331\) −18.0000 −0.989369 −0.494685 − 0.869072i \(-0.664716\pi\)
−0.494685 + 0.869072i \(0.664716\pi\)
\(332\) 0 0
\(333\) −6.00000 −0.328798
\(334\) −10.0000 −0.547176
\(335\) 4.00000 0.218543
\(336\) 0 0
\(337\) −8.00000 −0.435788 −0.217894 − 0.975972i \(-0.569919\pi\)
−0.217894 + 0.975972i \(0.569919\pi\)
\(338\) −9.00000 −0.489535
\(339\) −6.00000 −0.325875
\(340\) 4.00000 0.216930
\(341\) −6.00000 −0.324918
\(342\) 0 0
\(343\) 0 0
\(344\) 4.00000 0.215666
\(345\) 2.00000 0.107676
\(346\) 14.0000 0.752645
\(347\) 16.0000 0.858925 0.429463 − 0.903085i \(-0.358703\pi\)
0.429463 + 0.903085i \(0.358703\pi\)
\(348\) 8.00000 0.428845
\(349\) −30.0000 −1.60586 −0.802932 − 0.596071i \(-0.796728\pi\)
−0.802932 + 0.596071i \(0.796728\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) −6.00000 −0.319801
\(353\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) 8.00000 0.424596
\(356\) −2.00000 −0.106000
\(357\) 0 0
\(358\) −2.00000 −0.105703
\(359\) 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
\(362\) 8.00000 0.420471
\(363\) −25.0000 −1.31216
\(364\) 0 0
\(365\) 4.00000 0.209370
\(366\) −4.00000 −0.209083
\(367\) −10.0000 −0.521996 −0.260998 − 0.965339i \(-0.584052\pi\)
−0.260998 + 0.965339i \(0.584052\pi\)
\(368\) 2.00000 0.104257
\(369\) −2.00000 −0.104116
\(370\) 6.00000 0.311925
\(371\) 0 0
\(372\) −1.00000 −0.0518476
\(373\) 18.0000 0.932005 0.466002 − 0.884783i \(-0.345694\pi\)
0.466002 + 0.884783i \(0.345694\pi\)
\(374\) 24.0000 1.24101
\(375\) 1.00000 0.0516398
\(376\) 4.00000 0.206284
\(377\) 16.0000 0.824042
\(378\) 0 0
\(379\) −24.0000 −1.23280 −0.616399 − 0.787434i \(-0.711409\pi\)
−0.616399 + 0.787434i \(0.711409\pi\)
\(380\) 0 0
\(381\) −14.0000 −0.717242
\(382\) −8.00000 −0.409316
\(383\) 6.00000 0.306586 0.153293 − 0.988181i \(-0.451012\pi\)
0.153293 + 0.988181i \(0.451012\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −18.0000 −0.916176
\(387\) 4.00000 0.203331
\(388\) 14.0000 0.710742
\(389\) −20.0000 −1.01404 −0.507020 − 0.861934i \(-0.669253\pi\)
−0.507020 + 0.861934i \(0.669253\pi\)
\(390\) −2.00000 −0.101274
\(391\) −8.00000 −0.404577
\(392\) −7.00000 −0.353553
\(393\) 0 0
\(394\) −18.0000 −0.906827
\(395\) 4.00000 0.201262
\(396\) −6.00000 −0.301511
\(397\) 38.0000 1.90717 0.953583 − 0.301131i \(-0.0973643\pi\)
0.953583 + 0.301131i \(0.0973643\pi\)
\(398\) −20.0000 −1.00251
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −26.0000 −1.29838 −0.649189 − 0.760627i \(-0.724892\pi\)
−0.649189 + 0.760627i \(0.724892\pi\)
\(402\) 4.00000 0.199502
\(403\) −2.00000 −0.0996271
\(404\) 2.00000 0.0995037
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 36.0000 1.78445
\(408\) 4.00000 0.198030
\(409\) 14.0000 0.692255 0.346128 − 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 2.00000 0.0987730
\(411\) −8.00000 −0.394611
\(412\) 16.0000 0.788263
\(413\) 0 0
\(414\) 2.00000 0.0982946
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) 2.00000 0.0979404
\(418\) 0 0
\(419\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 26.0000 1.26716 0.633581 − 0.773676i \(-0.281584\pi\)
0.633581 + 0.773676i \(0.281584\pi\)
\(422\) −4.00000 −0.194717
\(423\) 4.00000 0.194487
\(424\) −6.00000 −0.291386
\(425\) −4.00000 −0.194029
\(426\) 8.00000 0.387601
\(427\) 0 0
\(428\) 12.0000 0.580042
\(429\) −12.0000 −0.579365
\(430\) −4.00000 −0.192897
\(431\) 40.0000 1.92673 0.963366 − 0.268190i \(-0.0864254\pi\)
0.963366 + 0.268190i \(0.0864254\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −28.0000 −1.34559 −0.672797 − 0.739827i \(-0.734907\pi\)
−0.672797 + 0.739827i \(0.734907\pi\)
\(434\) 0 0
\(435\) −8.00000 −0.383571
\(436\) −10.0000 −0.478913
\(437\) 0 0
\(438\) 4.00000 0.191127
\(439\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 6.00000 0.286039
\(441\) −7.00000 −0.333333
\(442\) 8.00000 0.380521
\(443\) 12.0000 0.570137 0.285069 − 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 6.00000 0.284747
\(445\) 2.00000 0.0948091
\(446\) 18.0000 0.852325
\(447\) 14.0000 0.662177
\(448\) 0 0
\(449\) 14.0000 0.660701 0.330350 − 0.943858i \(-0.392833\pi\)
0.330350 + 0.943858i \(0.392833\pi\)
\(450\) 1.00000 0.0471405
\(451\) 12.0000 0.565058
\(452\) 6.00000 0.282216
\(453\) −12.0000 −0.563809
\(454\) 12.0000 0.563188
\(455\) 0 0
\(456\) 0 0
\(457\) −8.00000 −0.374224 −0.187112 − 0.982339i \(-0.559913\pi\)
−0.187112 + 0.982339i \(0.559913\pi\)
\(458\) 16.0000 0.747631
\(459\) 4.00000 0.186704
\(460\) −2.00000 −0.0932505
\(461\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) −14.0000 −0.650635 −0.325318 − 0.945605i \(-0.605471\pi\)
−0.325318 + 0.945605i \(0.605471\pi\)
\(464\) −8.00000 −0.371391
\(465\) 1.00000 0.0463739
\(466\) 6.00000 0.277945
\(467\) −36.0000 −1.66588 −0.832941 − 0.553362i \(-0.813345\pi\)
−0.832941 + 0.553362i \(0.813345\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 0 0
\(470\) −4.00000 −0.184506
\(471\) 2.00000 0.0921551
\(472\) 0 0
\(473\) −24.0000 −1.10352
\(474\) 4.00000 0.183726
\(475\) 0 0
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) −24.0000 −1.09773
\(479\) 24.0000 1.09659 0.548294 − 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 1.00000 0.0456435
\(481\) 12.0000 0.547153
\(482\) 18.0000 0.819878
\(483\) 0 0
\(484\) 25.0000 1.13636
\(485\) −14.0000 −0.635707
\(486\) −1.00000 −0.0453609
\(487\) −34.0000 −1.54069 −0.770344 − 0.637629i \(-0.779915\pi\)
−0.770344 + 0.637629i \(0.779915\pi\)
\(488\) 4.00000 0.181071
\(489\) 4.00000 0.180886
\(490\) 7.00000 0.316228
\(491\) −34.0000 −1.53440 −0.767199 − 0.641409i \(-0.778350\pi\)
−0.767199 + 0.641409i \(0.778350\pi\)
\(492\) 2.00000 0.0901670
\(493\) 32.0000 1.44121
\(494\) 0 0
\(495\) 6.00000 0.269680
\(496\) 1.00000 0.0449013
\(497\) 0 0
\(498\) 0 0
\(499\) −38.0000 −1.70111 −0.850557 − 0.525883i \(-0.823735\pi\)
−0.850557 + 0.525883i \(0.823735\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 10.0000 0.446767
\(502\) 18.0000 0.803379
\(503\) −16.0000 −0.713405 −0.356702 − 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) 0 0
\(505\) −2.00000 −0.0889988
\(506\) −12.0000 −0.533465
\(507\) 9.00000 0.399704
\(508\) 14.0000 0.621150
\(509\) −36.0000 −1.59567 −0.797836 − 0.602875i \(-0.794022\pi\)
−0.797836 + 0.602875i \(0.794022\pi\)
\(510\) −4.00000 −0.177123
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −26.0000 −1.14681
\(515\) −16.0000 −0.705044
\(516\) −4.00000 −0.176090
\(517\) −24.0000 −1.05552
\(518\) 0 0
\(519\) −14.0000 −0.614532
\(520\) 2.00000 0.0877058
\(521\) 6.00000 0.262865 0.131432 − 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) −8.00000 −0.350150
\(523\) 20.0000 0.874539 0.437269 − 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 18.0000 0.784837
\(527\) −4.00000 −0.174243
\(528\) 6.00000 0.261116
\(529\) −19.0000 −0.826087
\(530\) 6.00000 0.260623
\(531\) 0 0
\(532\) 0 0
\(533\) 4.00000 0.173259
\(534\) 2.00000 0.0865485
\(535\) −12.0000 −0.518805
\(536\) −4.00000 −0.172774
\(537\) 2.00000 0.0863064
\(538\) −16.0000 −0.689809
\(539\) 42.0000 1.80907
\(540\) 1.00000 0.0430331
\(541\) −2.00000 −0.0859867 −0.0429934 − 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) 16.0000 0.687259
\(543\) −8.00000 −0.343313
\(544\) −4.00000 −0.171499
\(545\) 10.0000 0.428353
\(546\) 0 0
\(547\) −44.0000 −1.88130 −0.940652 − 0.339372i \(-0.889785\pi\)
−0.940652 + 0.339372i \(0.889785\pi\)
\(548\) 8.00000 0.341743
\(549\) 4.00000 0.170716
\(550\) −6.00000 −0.255841
\(551\) 0 0
\(552\) −2.00000 −0.0851257
\(553\) 0 0
\(554\) 18.0000 0.764747
\(555\) −6.00000 −0.254686
\(556\) −2.00000 −0.0848189
\(557\) 38.0000 1.61011 0.805056 − 0.593199i \(-0.202135\pi\)
0.805056 + 0.593199i \(0.202135\pi\)
\(558\) 1.00000 0.0423334
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) −24.0000 −1.01328
\(562\) −26.0000 −1.09674
\(563\) 44.0000 1.85438 0.927189 − 0.374593i \(-0.122217\pi\)
0.927189 + 0.374593i \(0.122217\pi\)
\(564\) −4.00000 −0.168430
\(565\) −6.00000 −0.252422
\(566\) −12.0000 −0.504398
\(567\) 0 0
\(568\) −8.00000 −0.335673
\(569\) −2.00000 −0.0838444 −0.0419222 − 0.999121i \(-0.513348\pi\)
−0.0419222 + 0.999121i \(0.513348\pi\)
\(570\) 0 0
\(571\) −38.0000 −1.59025 −0.795125 − 0.606445i \(-0.792595\pi\)
−0.795125 + 0.606445i \(0.792595\pi\)
\(572\) 12.0000 0.501745
\(573\) 8.00000 0.334205
\(574\) 0 0
\(575\) 2.00000 0.0834058
\(576\) 1.00000 0.0416667
\(577\) −38.0000 −1.58196 −0.790980 − 0.611842i \(-0.790429\pi\)
−0.790980 + 0.611842i \(0.790429\pi\)
\(578\) −1.00000 −0.0415945
\(579\) 18.0000 0.748054
\(580\) 8.00000 0.332182
\(581\) 0 0
\(582\) −14.0000 −0.580319
\(583\) 36.0000 1.49097
\(584\) −4.00000 −0.165521
\(585\) 2.00000 0.0826898
\(586\) −30.0000 −1.23929
\(587\) 32.0000 1.32078 0.660391 − 0.750922i \(-0.270391\pi\)
0.660391 + 0.750922i \(0.270391\pi\)
\(588\) 7.00000 0.288675
\(589\) 0 0
\(590\) 0 0
\(591\) 18.0000 0.740421
\(592\) −6.00000 −0.246598
\(593\) 34.0000 1.39621 0.698106 − 0.715994i \(-0.254026\pi\)
0.698106 + 0.715994i \(0.254026\pi\)
\(594\) 6.00000 0.246183
\(595\) 0 0
\(596\) −14.0000 −0.573462
\(597\) 20.0000 0.818546
\(598\) −4.00000 −0.163572
\(599\) 48.0000 1.96123 0.980613 − 0.195952i \(-0.0627798\pi\)
0.980613 + 0.195952i \(0.0627798\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 30.0000 1.22373 0.611863 − 0.790964i \(-0.290420\pi\)
0.611863 + 0.790964i \(0.290420\pi\)
\(602\) 0 0
\(603\) −4.00000 −0.162893
\(604\) 12.0000 0.488273
\(605\) −25.0000 −1.01639
\(606\) −2.00000 −0.0812444
\(607\) −28.0000 −1.13648 −0.568242 − 0.822861i \(-0.692376\pi\)
−0.568242 + 0.822861i \(0.692376\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) −4.00000 −0.161955
\(611\) −8.00000 −0.323645
\(612\) −4.00000 −0.161690
\(613\) −22.0000 −0.888572 −0.444286 − 0.895885i \(-0.646543\pi\)
−0.444286 + 0.895885i \(0.646543\pi\)
\(614\) 4.00000 0.161427
\(615\) −2.00000 −0.0806478
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 − 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) −16.0000 −0.643614
\(619\) −26.0000 −1.04503 −0.522514 − 0.852631i \(-0.675006\pi\)
−0.522514 + 0.852631i \(0.675006\pi\)
\(620\) −1.00000 −0.0401610
\(621\) −2.00000 −0.0802572
\(622\) −24.0000 −0.962312
\(623\) 0 0
\(624\) 2.00000 0.0800641
\(625\) 1.00000 0.0400000
\(626\) −12.0000 −0.479616
\(627\) 0 0
\(628\) −2.00000 −0.0798087
\(629\) 24.0000 0.956943
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 − 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) −4.00000 −0.159111
\(633\) 4.00000 0.158986
\(634\) −26.0000 −1.03259
\(635\) −14.0000 −0.555573
\(636\) 6.00000 0.237915
\(637\) 14.0000 0.554700
\(638\) 48.0000 1.90034
\(639\) −8.00000 −0.316475
\(640\) −1.00000 −0.0395285
\(641\) −22.0000 −0.868948 −0.434474 − 0.900684i \(-0.643066\pi\)
−0.434474 + 0.900684i \(0.643066\pi\)
\(642\) −12.0000 −0.473602
\(643\) −44.0000 −1.73519 −0.867595 − 0.497271i \(-0.834335\pi\)
−0.867595 + 0.497271i \(0.834335\pi\)
\(644\) 0 0
\(645\) 4.00000 0.157500
\(646\) 0 0
\(647\) −18.0000 −0.707653 −0.353827 − 0.935311i \(-0.615120\pi\)
−0.353827 + 0.935311i \(0.615120\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) −2.00000 −0.0784465
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) 34.0000 1.33052 0.665261 − 0.746611i \(-0.268320\pi\)
0.665261 + 0.746611i \(0.268320\pi\)
\(654\) 10.0000 0.391031
\(655\) 0 0
\(656\) −2.00000 −0.0780869
\(657\) −4.00000 −0.156055
\(658\) 0 0
\(659\) −16.0000 −0.623272 −0.311636 − 0.950202i \(-0.600877\pi\)
−0.311636 + 0.950202i \(0.600877\pi\)
\(660\) −6.00000 −0.233550
\(661\) −26.0000 −1.01128 −0.505641 − 0.862744i \(-0.668744\pi\)
−0.505641 + 0.862744i \(0.668744\pi\)
\(662\) −18.0000 −0.699590
\(663\) −8.00000 −0.310694
\(664\) 0 0
\(665\) 0 0
\(666\) −6.00000 −0.232495
\(667\) −16.0000 −0.619522
\(668\) −10.0000 −0.386912
\(669\) −18.0000 −0.695920
\(670\) 4.00000 0.154533
\(671\) −24.0000 −0.926510
\(672\) 0 0
\(673\) 32.0000 1.23351 0.616755 − 0.787155i \(-0.288447\pi\)
0.616755 + 0.787155i \(0.288447\pi\)
\(674\) −8.00000 −0.308148
\(675\) −1.00000 −0.0384900
\(676\) −9.00000 −0.346154
\(677\) 6.00000 0.230599 0.115299 − 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) −6.00000 −0.230429
\(679\) 0 0
\(680\) 4.00000 0.153393
\(681\) −12.0000 −0.459841
\(682\) −6.00000 −0.229752
\(683\) −4.00000 −0.153056 −0.0765279 − 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) 0 0
\(685\) −8.00000 −0.305664
\(686\) 0 0
\(687\) −16.0000 −0.610438
\(688\) 4.00000 0.152499
\(689\) 12.0000 0.457164
\(690\) 2.00000 0.0761387
\(691\) 44.0000 1.67384 0.836919 − 0.547326i \(-0.184354\pi\)
0.836919 + 0.547326i \(0.184354\pi\)
\(692\) 14.0000 0.532200
\(693\) 0 0
\(694\) 16.0000 0.607352
\(695\) 2.00000 0.0758643
\(696\) 8.00000 0.303239
\(697\) 8.00000 0.303022
\(698\) −30.0000 −1.13552
\(699\) −6.00000 −0.226941
\(700\) 0 0
\(701\) 30.0000 1.13308 0.566542 − 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 2.00000 0.0754851
\(703\) 0 0
\(704\) −6.00000 −0.226134
\(705\) 4.00000 0.150649
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −4.00000 −0.150223 −0.0751116 − 0.997175i \(-0.523931\pi\)
−0.0751116 + 0.997175i \(0.523931\pi\)
\(710\) 8.00000 0.300235
\(711\) −4.00000 −0.150012
\(712\) −2.00000 −0.0749532
\(713\) 2.00000 0.0749006
\(714\) 0 0
\(715\) −12.0000 −0.448775
\(716\) −2.00000 −0.0747435
\(717\) 24.0000 0.896296
\(718\) 24.0000 0.895672
\(719\) 20.0000 0.745874 0.372937 − 0.927857i \(-0.378351\pi\)
0.372937 + 0.927857i \(0.378351\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 0 0
\(722\) −19.0000 −0.707107
\(723\) −18.0000 −0.669427
\(724\) 8.00000 0.297318
\(725\) −8.00000 −0.297113
\(726\) −25.0000 −0.927837
\(727\) −12.0000 −0.445055 −0.222528 − 0.974926i \(-0.571431\pi\)
−0.222528 + 0.974926i \(0.571431\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 4.00000 0.148047
\(731\) −16.0000 −0.591781
\(732\) −4.00000 −0.147844
\(733\) 6.00000 0.221615 0.110808 − 0.993842i \(-0.464656\pi\)
0.110808 + 0.993842i \(0.464656\pi\)
\(734\) −10.0000 −0.369107
\(735\) −7.00000 −0.258199
\(736\) 2.00000 0.0737210
\(737\) 24.0000 0.884051
\(738\) −2.00000 −0.0736210
\(739\) −2.00000 −0.0735712 −0.0367856 − 0.999323i \(-0.511712\pi\)
−0.0367856 + 0.999323i \(0.511712\pi\)
\(740\) 6.00000 0.220564
\(741\) 0 0
\(742\) 0 0
\(743\) −22.0000 −0.807102 −0.403551 − 0.914957i \(-0.632224\pi\)
−0.403551 + 0.914957i \(0.632224\pi\)
\(744\) −1.00000 −0.0366618
\(745\) 14.0000 0.512920
\(746\) 18.0000 0.659027
\(747\) 0 0
\(748\) 24.0000 0.877527
\(749\) 0 0
\(750\) 1.00000 0.0365148
\(751\) −8.00000 −0.291924 −0.145962 − 0.989290i \(-0.546628\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) 4.00000 0.145865
\(753\) −18.0000 −0.655956
\(754\) 16.0000 0.582686
\(755\) −12.0000 −0.436725
\(756\) 0 0
\(757\) 2.00000 0.0726912 0.0363456 − 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) −24.0000 −0.871719
\(759\) 12.0000 0.435572
\(760\) 0 0
\(761\) −18.0000 −0.652499 −0.326250 − 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) −14.0000 −0.507166
\(763\) 0 0
\(764\) −8.00000 −0.289430
\(765\) 4.00000 0.144620
\(766\) 6.00000 0.216789
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 30.0000 1.08183 0.540914 − 0.841078i \(-0.318079\pi\)
0.540914 + 0.841078i \(0.318079\pi\)
\(770\) 0 0
\(771\) 26.0000 0.936367
\(772\) −18.0000 −0.647834
\(773\) 26.0000 0.935155 0.467578 − 0.883952i \(-0.345127\pi\)
0.467578 + 0.883952i \(0.345127\pi\)
\(774\) 4.00000 0.143777
\(775\) 1.00000 0.0359211
\(776\) 14.0000 0.502571
\(777\) 0 0
\(778\) −20.0000 −0.717035
\(779\) 0 0
\(780\) −2.00000 −0.0716115
\(781\) 48.0000 1.71758
\(782\) −8.00000 −0.286079
\(783\) 8.00000 0.285897
\(784\) −7.00000 −0.250000
\(785\) 2.00000 0.0713831
\(786\) 0 0
\(787\) −8.00000 −0.285169 −0.142585 − 0.989783i \(-0.545541\pi\)
−0.142585 + 0.989783i \(0.545541\pi\)
\(788\) −18.0000 −0.641223
\(789\) −18.0000 −0.640817
\(790\) 4.00000 0.142314
\(791\) 0 0
\(792\) −6.00000 −0.213201
\(793\) −8.00000 −0.284088
\(794\) 38.0000 1.34857
\(795\) −6.00000 −0.212798
\(796\) −20.0000 −0.708881
\(797\) 2.00000 0.0708436 0.0354218 − 0.999372i \(-0.488723\pi\)
0.0354218 + 0.999372i \(0.488723\pi\)
\(798\) 0 0
\(799\) −16.0000 −0.566039
\(800\) 1.00000 0.0353553
\(801\) −2.00000 −0.0706665
\(802\) −26.0000 −0.918092
\(803\) 24.0000 0.846942
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) −2.00000 −0.0704470
\(807\) 16.0000 0.563227
\(808\) 2.00000 0.0703598
\(809\) −46.0000 −1.61727 −0.808637 − 0.588308i \(-0.799794\pi\)
−0.808637 + 0.588308i \(0.799794\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 24.0000 0.842754 0.421377 − 0.906886i \(-0.361547\pi\)
0.421377 + 0.906886i \(0.361547\pi\)
\(812\) 0 0
\(813\) −16.0000 −0.561144
\(814\) 36.0000 1.26180
\(815\) 4.00000 0.140114
\(816\) 4.00000 0.140028
\(817\) 0 0
\(818\) 14.0000 0.489499
\(819\) 0 0
\(820\) 2.00000 0.0698430
\(821\) 16.0000 0.558404 0.279202 − 0.960232i \(-0.409930\pi\)
0.279202 + 0.960232i \(0.409930\pi\)
\(822\) −8.00000 −0.279032
\(823\) 6.00000 0.209147 0.104573 − 0.994517i \(-0.466652\pi\)
0.104573 + 0.994517i \(0.466652\pi\)
\(824\) 16.0000 0.557386
\(825\) 6.00000 0.208893
\(826\) 0 0
\(827\) 32.0000 1.11275 0.556375 − 0.830932i \(-0.312192\pi\)
0.556375 + 0.830932i \(0.312192\pi\)
\(828\) 2.00000 0.0695048
\(829\) −32.0000 −1.11141 −0.555703 − 0.831381i \(-0.687551\pi\)
−0.555703 + 0.831381i \(0.687551\pi\)
\(830\) 0 0
\(831\) −18.0000 −0.624413
\(832\) −2.00000 −0.0693375
\(833\) 28.0000 0.970143
\(834\) 2.00000 0.0692543
\(835\) 10.0000 0.346064
\(836\) 0 0
\(837\) −1.00000 −0.0345651
\(838\) 0 0
\(839\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) 26.0000 0.896019
\(843\) 26.0000 0.895488
\(844\) −4.00000 −0.137686
\(845\) 9.00000 0.309609
\(846\) 4.00000 0.137523
\(847\) 0 0
\(848\) −6.00000 −0.206041
\(849\) 12.0000 0.411839
\(850\) −4.00000 −0.137199
\(851\) −12.0000 −0.411355
\(852\) 8.00000 0.274075
\(853\) 10.0000 0.342393 0.171197 − 0.985237i \(-0.445237\pi\)
0.171197 + 0.985237i \(0.445237\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) −18.0000 −0.614868 −0.307434 − 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) −12.0000 −0.409673
\(859\) −22.0000 −0.750630 −0.375315 − 0.926897i \(-0.622466\pi\)
−0.375315 + 0.926897i \(0.622466\pi\)
\(860\) −4.00000 −0.136399
\(861\) 0 0
\(862\) 40.0000 1.36241
\(863\) 22.0000 0.748889 0.374444 − 0.927249i \(-0.377833\pi\)
0.374444 + 0.927249i \(0.377833\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −14.0000 −0.476014
\(866\) −28.0000 −0.951479
\(867\) 1.00000 0.0339618
\(868\) 0 0
\(869\) 24.0000 0.814144
\(870\) −8.00000 −0.271225
\(871\) 8.00000 0.271070
\(872\) −10.0000 −0.338643
\(873\) 14.0000 0.473828
\(874\) 0 0
\(875\) 0 0
\(876\) 4.00000 0.135147
\(877\) −50.0000 −1.68838 −0.844190 − 0.536044i \(-0.819918\pi\)
−0.844190 + 0.536044i \(0.819918\pi\)
\(878\) 0 0
\(879\) 30.0000 1.01187
\(880\) 6.00000 0.202260
\(881\) 38.0000 1.28025 0.640126 − 0.768270i \(-0.278882\pi\)
0.640126 + 0.768270i \(0.278882\pi\)
\(882\) −7.00000 −0.235702
\(883\) 20.0000 0.673054 0.336527 − 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) 8.00000 0.269069
\(885\) 0 0
\(886\) 12.0000 0.403148
\(887\) −4.00000 −0.134307 −0.0671534 − 0.997743i \(-0.521392\pi\)
−0.0671534 + 0.997743i \(0.521392\pi\)
\(888\) 6.00000 0.201347
\(889\) 0 0
\(890\) 2.00000 0.0670402
\(891\) −6.00000 −0.201008
\(892\) 18.0000 0.602685
\(893\) 0 0
\(894\) 14.0000 0.468230
\(895\) 2.00000 0.0668526
\(896\) 0 0
\(897\) 4.00000 0.133556
\(898\) 14.0000 0.467186
\(899\) −8.00000 −0.266815
\(900\) 1.00000 0.0333333
\(901\) 24.0000 0.799556
\(902\) 12.0000 0.399556
\(903\) 0 0
\(904\) 6.00000 0.199557
\(905\) −8.00000 −0.265929
\(906\) −12.0000 −0.398673
\(907\) 44.0000 1.46100 0.730498 − 0.682915i \(-0.239288\pi\)
0.730498 + 0.682915i \(0.239288\pi\)
\(908\) 12.0000 0.398234
\(909\) 2.00000 0.0663358
\(910\) 0 0
\(911\) −16.0000 −0.530104 −0.265052 − 0.964234i \(-0.585389\pi\)
−0.265052 + 0.964234i \(0.585389\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −8.00000 −0.264616
\(915\) 4.00000 0.132236
\(916\) 16.0000 0.528655
\(917\) 0 0
\(918\) 4.00000 0.132020
\(919\) 16.0000 0.527791 0.263896 − 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) −2.00000 −0.0659380
\(921\) −4.00000 −0.131804
\(922\) 0 0
\(923\) 16.0000 0.526646
\(924\) 0 0
\(925\) −6.00000 −0.197279
\(926\) −14.0000 −0.460069
\(927\) 16.0000 0.525509
\(928\) −8.00000 −0.262613
\(929\) 46.0000 1.50921 0.754606 − 0.656179i \(-0.227828\pi\)
0.754606 + 0.656179i \(0.227828\pi\)
\(930\) 1.00000 0.0327913
\(931\) 0 0
\(932\) 6.00000 0.196537
\(933\) 24.0000 0.785725
\(934\) −36.0000 −1.17796
\(935\) −24.0000 −0.784884
\(936\) −2.00000 −0.0653720
\(937\) 14.0000 0.457360 0.228680 − 0.973502i \(-0.426559\pi\)
0.228680 + 0.973502i \(0.426559\pi\)
\(938\) 0 0
\(939\) 12.0000 0.391605
\(940\) −4.00000 −0.130466
\(941\) 52.0000 1.69515 0.847576 − 0.530674i \(-0.178061\pi\)
0.847576 + 0.530674i \(0.178061\pi\)
\(942\) 2.00000 0.0651635
\(943\) −4.00000 −0.130258
\(944\) 0 0
\(945\) 0 0
\(946\) −24.0000 −0.780307
\(947\) −12.0000 −0.389948 −0.194974 − 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) 4.00000 0.129914
\(949\) 8.00000 0.259691
\(950\) 0 0
\(951\) 26.0000 0.843108
\(952\) 0 0
\(953\) 8.00000 0.259145 0.129573 − 0.991570i \(-0.458639\pi\)
0.129573 + 0.991570i \(0.458639\pi\)
\(954\) −6.00000 −0.194257
\(955\) 8.00000 0.258874
\(956\) −24.0000 −0.776215
\(957\) −48.0000 −1.55162
\(958\) 24.0000 0.775405
\(959\) 0 0
\(960\) 1.00000 0.0322749
\(961\) 1.00000 0.0322581
\(962\) 12.0000 0.386896
\(963\) 12.0000 0.386695
\(964\) 18.0000 0.579741
\(965\) 18.0000 0.579441
\(966\) 0 0
\(967\) 6.00000 0.192947 0.0964735 − 0.995336i \(-0.469244\pi\)
0.0964735 + 0.995336i \(0.469244\pi\)
\(968\) 25.0000 0.803530
\(969\) 0 0
\(970\) −14.0000 −0.449513
\(971\) 36.0000 1.15529 0.577647 − 0.816286i \(-0.303971\pi\)
0.577647 + 0.816286i \(0.303971\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −34.0000 −1.08943
\(975\) 2.00000 0.0640513
\(976\) 4.00000 0.128037
\(977\) 26.0000 0.831814 0.415907 − 0.909407i \(-0.363464\pi\)
0.415907 + 0.909407i \(0.363464\pi\)
\(978\) 4.00000 0.127906
\(979\) 12.0000 0.383522
\(980\) 7.00000 0.223607
\(981\) −10.0000 −0.319275
\(982\) −34.0000 −1.08498
\(983\) 10.0000 0.318950 0.159475 − 0.987202i \(-0.449020\pi\)
0.159475 + 0.987202i \(0.449020\pi\)
\(984\) 2.00000 0.0637577
\(985\) 18.0000 0.573528
\(986\) 32.0000 1.01909
\(987\) 0 0
\(988\) 0 0
\(989\) 8.00000 0.254385
\(990\) 6.00000 0.190693
\(991\) 20.0000 0.635321 0.317660 − 0.948205i \(-0.397103\pi\)
0.317660 + 0.948205i \(0.397103\pi\)
\(992\) 1.00000 0.0317500
\(993\) 18.0000 0.571213
\(994\) 0 0
\(995\) 20.0000 0.634043
\(996\) 0 0
\(997\) 18.0000 0.570066 0.285033 − 0.958518i \(-0.407995\pi\)
0.285033 + 0.958518i \(0.407995\pi\)
\(998\) −38.0000 −1.20287
\(999\) 6.00000 0.189832
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 930.2.a.l.1.1 ✓ 1
3.2 odd 2 2790.2.a.j.1.1 1
4.3 odd 2 7440.2.a.r.1.1 1
5.2 odd 4 4650.2.d.a.3349.2 2
5.3 odd 4 4650.2.d.a.3349.1 2
5.4 even 2 4650.2.a.r.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.l.1.1 ✓ 1 1.1 even 1 trivial
2790.2.a.j.1.1 1 3.2 odd 2
4650.2.a.r.1.1 1 5.4 even 2
4650.2.d.a.3349.1 2 5.3 odd 4
4650.2.d.a.3349.2 2 5.2 odd 4
7440.2.a.r.1.1 1 4.3 odd 2