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Properties

Label 2310.2.a.o.1.1
Level $2310$
Weight $2$
Character 2310.1
Self dual yes
Analytic conductor $18.445$
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] = [2310,2,Mod(1,2310)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2310, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("2310.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2310 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2310.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: \(18.4454428669\)
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 \(\chi\) \(=\) 2310.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} +1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} +1.00000 q^{20} -1.00000 q^{21} -1.00000 q^{22} +8.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} +6.00000 q^{29} -1.00000 q^{30} +8.00000 q^{31} +1.00000 q^{32} +1.00000 q^{33} +2.00000 q^{34} +1.00000 q^{35} +1.00000 q^{36} -2.00000 q^{37} -4.00000 q^{38} +2.00000 q^{39} +1.00000 q^{40} +2.00000 q^{41} -1.00000 q^{42} +4.00000 q^{43} -1.00000 q^{44} +1.00000 q^{45} +8.00000 q^{46} +8.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -2.00000 q^{51} -2.00000 q^{52} -10.0000 q^{53} -1.00000 q^{54} -1.00000 q^{55} +1.00000 q^{56} +4.00000 q^{57} +6.00000 q^{58} -12.0000 q^{59} -1.00000 q^{60} -2.00000 q^{61} +8.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} +1.00000 q^{66} -4.00000 q^{67} +2.00000 q^{68} -8.00000 q^{69} +1.00000 q^{70} +8.00000 q^{71} +1.00000 q^{72} +2.00000 q^{73} -2.00000 q^{74} -1.00000 q^{75} -4.00000 q^{76} -1.00000 q^{77} +2.00000 q^{78} +16.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} -4.00000 q^{83} -1.00000 q^{84} +2.00000 q^{85} +4.00000 q^{86} -6.00000 q^{87} -1.00000 q^{88} -14.0000 q^{89} +1.00000 q^{90} -2.00000 q^{91} +8.00000 q^{92} -8.00000 q^{93} +8.00000 q^{94} -4.00000 q^{95} -1.00000 q^{96} +10.0000 q^{97} +1.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −1.00000 −0.408248
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) −1.00000 −0.301511
\(12\) −1.00000 −0.288675
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 1.00000 0.267261
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 1.00000 0.235702
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 1.00000 0.223607
\(21\) −1.00000 −0.218218
\(22\) −1.00000 −0.213201
\(23\) 8.00000 1.66812 0.834058 − 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) −2.00000 −0.392232
\(27\) −1.00000 −0.192450
\(28\) 1.00000 0.188982
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) −1.00000 −0.182574
\(31\) 8.00000 1.43684 0.718421 − 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.00000 0.174078
\(34\) 2.00000 0.342997
\(35\) 1.00000 0.169031
\(36\) 1.00000 0.166667
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) −4.00000 −0.648886
\(39\) 2.00000 0.320256
\(40\) 1.00000 0.158114
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) −1.00000 −0.154303
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) −1.00000 −0.150756
\(45\) 1.00000 0.149071
\(46\) 8.00000 1.17954
\(47\) 8.00000 1.16692 0.583460 − 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) −2.00000 −0.280056
\(52\) −2.00000 −0.277350
\(53\) −10.0000 −1.37361 −0.686803 − 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) −1.00000 −0.136083
\(55\) −1.00000 −0.134840
\(56\) 1.00000 0.133631
\(57\) 4.00000 0.529813
\(58\) 6.00000 0.787839
\(59\) −12.0000 −1.56227 −0.781133 − 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) −1.00000 −0.129099
\(61\) −2.00000 −0.256074 −0.128037 − 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 8.00000 1.01600
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) 1.00000 0.123091
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 2.00000 0.242536
\(69\) −8.00000 −0.963087
\(70\) 1.00000 0.119523
\(71\) 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 \(0.462659\pi\)
\(74\) −2.00000 −0.232495
\(75\) −1.00000 −0.115470
\(76\) −4.00000 −0.458831
\(77\) −1.00000 −0.113961
\(78\) 2.00000 0.226455
\(79\) 16.0000 1.80014 0.900070 − 0.435745i \(-0.143515\pi\)
0.900070 + 0.435745i \(0.143515\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) −1.00000 −0.109109
\(85\) 2.00000 0.216930
\(86\) 4.00000 0.431331
\(87\) −6.00000 −0.643268
\(88\) −1.00000 −0.106600
\(89\) −14.0000 −1.48400 −0.741999 − 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 1.00000 0.105409
\(91\) −2.00000 −0.209657
\(92\) 8.00000 0.834058
\(93\) −8.00000 −0.829561
\(94\) 8.00000 0.825137
\(95\) −4.00000 −0.410391
\(96\) −1.00000 −0.102062
\(97\) 10.0000 1.01535 0.507673 − 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 1.00000 0.101015
\(99\) −1.00000 −0.100504
\(100\) 1.00000 0.100000
\(101\) 6.00000 0.597022 0.298511 − 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) −2.00000 −0.198030
\(103\) 8.00000 0.788263 0.394132 − 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) −2.00000 −0.196116
\(105\) −1.00000 −0.0975900
\(106\) −10.0000 −0.971286
\(107\) 12.0000 1.16008 0.580042 − 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 14.0000 1.34096 0.670478 − 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) −1.00000 −0.0953463
\(111\) 2.00000 0.189832
\(112\) 1.00000 0.0944911
\(113\) −14.0000 −1.31701 −0.658505 − 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 4.00000 0.374634
\(115\) 8.00000 0.746004
\(116\) 6.00000 0.557086
\(117\) −2.00000 −0.184900
\(118\) −12.0000 −1.10469
\(119\) 2.00000 0.183340
\(120\) −1.00000 −0.0912871
\(121\) 1.00000 0.0909091
\(122\) −2.00000 −0.181071
\(123\) −2.00000 −0.180334
\(124\) 8.00000 0.718421
\(125\) 1.00000 0.0894427
\(126\) 1.00000 0.0890871
\(127\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 1.00000 0.0883883
\(129\) −4.00000 −0.352180
\(130\) −2.00000 −0.175412
\(131\) 12.0000 1.04844 0.524222 − 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 1.00000 0.0870388
\(133\) −4.00000 −0.346844
\(134\) −4.00000 −0.345547
\(135\) −1.00000 −0.0860663
\(136\) 2.00000 0.171499
\(137\) 10.0000 0.854358 0.427179 − 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) −8.00000 −0.681005
\(139\) −12.0000 −1.01783 −0.508913 − 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 1.00000 0.0845154
\(141\) −8.00000 −0.673722
\(142\) 8.00000 0.671345
\(143\) 2.00000 0.167248
\(144\) 1.00000 0.0833333
\(145\) 6.00000 0.498273
\(146\) 2.00000 0.165521
\(147\) −1.00000 −0.0824786
\(148\) −2.00000 −0.164399
\(149\) 14.0000 1.14692 0.573462 − 0.819232i \(-0.305600\pi\)
0.573462 + 0.819232i \(0.305600\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 24.0000 1.95309 0.976546 − 0.215308i \(-0.0690756\pi\)
0.976546 + 0.215308i \(0.0690756\pi\)
\(152\) −4.00000 −0.324443
\(153\) 2.00000 0.161690
\(154\) −1.00000 −0.0805823
\(155\) 8.00000 0.642575
\(156\) 2.00000 0.160128
\(157\) −18.0000 −1.43656 −0.718278 − 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) 16.0000 1.27289
\(159\) 10.0000 0.793052
\(160\) 1.00000 0.0790569
\(161\) 8.00000 0.630488
\(162\) 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
\(165\) 1.00000 0.0778499
\(166\) −4.00000 −0.310460
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) −1.00000 −0.0771517
\(169\) −9.00000 −0.692308
\(170\) 2.00000 0.153393
\(171\) −4.00000 −0.305888
\(172\) 4.00000 0.304997
\(173\) −18.0000 −1.36851 −0.684257 − 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) −6.00000 −0.454859
\(175\) 1.00000 0.0755929
\(176\) −1.00000 −0.0753778
\(177\) 12.0000 0.901975
\(178\) −14.0000 −1.04934
\(179\) −4.00000 −0.298974 −0.149487 − 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 1.00000 0.0745356
\(181\) 22.0000 1.63525 0.817624 − 0.575753i \(-0.195291\pi\)
0.817624 + 0.575753i \(0.195291\pi\)
\(182\) −2.00000 −0.148250
\(183\) 2.00000 0.147844
\(184\) 8.00000 0.589768
\(185\) −2.00000 −0.147043
\(186\) −8.00000 −0.586588
\(187\) −2.00000 −0.146254
\(188\) 8.00000 0.583460
\(189\) −1.00000 −0.0727393
\(190\) −4.00000 −0.290191
\(191\) −16.0000 −1.15772 −0.578860 − 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −14.0000 −1.00774 −0.503871 − 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 10.0000 0.717958
\(195\) 2.00000 0.143223
\(196\) 1.00000 0.0714286
\(197\) −26.0000 −1.85242 −0.926212 − 0.377004i \(-0.876954\pi\)
−0.926212 + 0.377004i \(0.876954\pi\)
\(198\) −1.00000 −0.0710669
\(199\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 1.00000 0.0707107
\(201\) 4.00000 0.282138
\(202\) 6.00000 0.422159
\(203\) 6.00000 0.421117
\(204\) −2.00000 −0.140028
\(205\) 2.00000 0.139686
\(206\) 8.00000 0.557386
\(207\) 8.00000 0.556038
\(208\) −2.00000 −0.138675
\(209\) 4.00000 0.276686
\(210\) −1.00000 −0.0690066
\(211\) −12.0000 −0.826114 −0.413057 − 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) −10.0000 −0.686803
\(213\) −8.00000 −0.548151
\(214\) 12.0000 0.820303
\(215\) 4.00000 0.272798
\(216\) −1.00000 −0.0680414
\(217\) 8.00000 0.543075
\(218\) 14.0000 0.948200
\(219\) −2.00000 −0.135147
\(220\) −1.00000 −0.0674200
\(221\) −4.00000 −0.269069
\(222\) 2.00000 0.134231
\(223\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 1.00000 0.0668153
\(225\) 1.00000 0.0666667
\(226\) −14.0000 −0.931266
\(227\) −4.00000 −0.265489 −0.132745 − 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(228\) 4.00000 0.264906
\(229\) 6.00000 0.396491 0.198246 − 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 8.00000 0.527504
\(231\) 1.00000 0.0657952
\(232\) 6.00000 0.393919
\(233\) 10.0000 0.655122 0.327561 − 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) −2.00000 −0.130744
\(235\) 8.00000 0.521862
\(236\) −12.0000 −0.781133
\(237\) −16.0000 −1.03931
\(238\) 2.00000 0.129641
\(239\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 2.00000 0.128831 0.0644157 − 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 1.00000 0.0642824
\(243\) −1.00000 −0.0641500
\(244\) −2.00000 −0.128037
\(245\) 1.00000 0.0638877
\(246\) −2.00000 −0.127515
\(247\) 8.00000 0.509028
\(248\) 8.00000 0.508001
\(249\) 4.00000 0.253490
\(250\) 1.00000 0.0632456
\(251\) −12.0000 −0.757433 −0.378717 − 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 1.00000 0.0629941
\(253\) −8.00000 −0.502956
\(254\) 0 0
\(255\) −2.00000 −0.125245
\(256\) 1.00000 0.0625000
\(257\) −14.0000 −0.873296 −0.436648 − 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) −4.00000 −0.249029
\(259\) −2.00000 −0.124274
\(260\) −2.00000 −0.124035
\(261\) 6.00000 0.371391
\(262\) 12.0000 0.741362
\(263\) −8.00000 −0.493301 −0.246651 − 0.969104i \(-0.579330\pi\)
−0.246651 + 0.969104i \(0.579330\pi\)
\(264\) 1.00000 0.0615457
\(265\) −10.0000 −0.614295
\(266\) −4.00000 −0.245256
\(267\) 14.0000 0.856786
\(268\) −4.00000 −0.244339
\(269\) −2.00000 −0.121942 −0.0609711 − 0.998140i \(-0.519420\pi\)
−0.0609711 + 0.998140i \(0.519420\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −8.00000 −0.485965 −0.242983 − 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 2.00000 0.121268
\(273\) 2.00000 0.121046
\(274\) 10.0000 0.604122
\(275\) −1.00000 −0.0603023
\(276\) −8.00000 −0.481543
\(277\) −2.00000 −0.120168 −0.0600842 − 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) −12.0000 −0.719712
\(279\) 8.00000 0.478947
\(280\) 1.00000 0.0597614
\(281\) −22.0000 −1.31241 −0.656205 − 0.754583i \(-0.727839\pi\)
−0.656205 + 0.754583i \(0.727839\pi\)
\(282\) −8.00000 −0.476393
\(283\) 4.00000 0.237775 0.118888 − 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 8.00000 0.474713
\(285\) 4.00000 0.236940
\(286\) 2.00000 0.118262
\(287\) 2.00000 0.118056
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) 6.00000 0.352332
\(291\) −10.0000 −0.586210
\(292\) 2.00000 0.117041
\(293\) −26.0000 −1.51894 −0.759468 − 0.650545i \(-0.774541\pi\)
−0.759468 + 0.650545i \(0.774541\pi\)
\(294\) −1.00000 −0.0583212
\(295\) −12.0000 −0.698667
\(296\) −2.00000 −0.116248
\(297\) 1.00000 0.0580259
\(298\) 14.0000 0.810998
\(299\) −16.0000 −0.925304
\(300\) −1.00000 −0.0577350
\(301\) 4.00000 0.230556
\(302\) 24.0000 1.38104
\(303\) −6.00000 −0.344691
\(304\) −4.00000 −0.229416
\(305\) −2.00000 −0.114520
\(306\) 2.00000 0.114332
\(307\) −20.0000 −1.14146 −0.570730 − 0.821138i \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) −1.00000 −0.0569803
\(309\) −8.00000 −0.455104
\(310\) 8.00000 0.454369
\(311\) −24.0000 −1.36092 −0.680458 − 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 2.00000 0.113228
\(313\) −30.0000 −1.69570 −0.847850 − 0.530236i \(-0.822103\pi\)
−0.847850 + 0.530236i \(0.822103\pi\)
\(314\) −18.0000 −1.01580
\(315\) 1.00000 0.0563436
\(316\) 16.0000 0.900070
\(317\) 30.0000 1.68497 0.842484 − 0.538721i \(-0.181092\pi\)
0.842484 + 0.538721i \(0.181092\pi\)
\(318\) 10.0000 0.560772
\(319\) −6.00000 −0.335936
\(320\) 1.00000 0.0559017
\(321\) −12.0000 −0.669775
\(322\) 8.00000 0.445823
\(323\) −8.00000 −0.445132
\(324\) 1.00000 0.0555556
\(325\) −2.00000 −0.110940
\(326\) −4.00000 −0.221540
\(327\) −14.0000 −0.774202
\(328\) 2.00000 0.110432
\(329\) 8.00000 0.441054
\(330\) 1.00000 0.0550482
\(331\) 28.0000 1.53902 0.769510 − 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) −4.00000 −0.219529
\(333\) −2.00000 −0.109599
\(334\) 0 0
\(335\) −4.00000 −0.218543
\(336\) −1.00000 −0.0545545
\(337\) 2.00000 0.108947 0.0544735 − 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) −9.00000 −0.489535
\(339\) 14.0000 0.760376
\(340\) 2.00000 0.108465
\(341\) −8.00000 −0.433224
\(342\) −4.00000 −0.216295
\(343\) 1.00000 0.0539949
\(344\) 4.00000 0.215666
\(345\) −8.00000 −0.430706
\(346\) −18.0000 −0.967686
\(347\) 12.0000 0.644194 0.322097 − 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) −6.00000 −0.321634
\(349\) −2.00000 −0.107058 −0.0535288 − 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 1.00000 0.0534522
\(351\) 2.00000 0.106752
\(352\) −1.00000 −0.0533002
\(353\) −30.0000 −1.59674 −0.798369 − 0.602168i \(-0.794304\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) 12.0000 0.637793
\(355\) 8.00000 0.424596
\(356\) −14.0000 −0.741999
\(357\) −2.00000 −0.105851
\(358\) −4.00000 −0.211407
\(359\) 8.00000 0.422224 0.211112 − 0.977462i \(-0.432292\pi\)
0.211112 + 0.977462i \(0.432292\pi\)
\(360\) 1.00000 0.0527046
\(361\) −3.00000 −0.157895
\(362\) 22.0000 1.15629
\(363\) −1.00000 −0.0524864
\(364\) −2.00000 −0.104828
\(365\) 2.00000 0.104685
\(366\) 2.00000 0.104542
\(367\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 8.00000 0.417029
\(369\) 2.00000 0.104116
\(370\) −2.00000 −0.103975
\(371\) −10.0000 −0.519174
\(372\) −8.00000 −0.414781
\(373\) 30.0000 1.55334 0.776671 − 0.629907i \(-0.216907\pi\)
0.776671 + 0.629907i \(0.216907\pi\)
\(374\) −2.00000 −0.103418
\(375\) −1.00000 −0.0516398
\(376\) 8.00000 0.412568
\(377\) −12.0000 −0.618031
\(378\) −1.00000 −0.0514344
\(379\) 28.0000 1.43826 0.719132 − 0.694874i \(-0.244540\pi\)
0.719132 + 0.694874i \(0.244540\pi\)
\(380\) −4.00000 −0.205196
\(381\) 0 0
\(382\) −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
\(385\) −1.00000 −0.0509647
\(386\) −14.0000 −0.712581
\(387\) 4.00000 0.203331
\(388\) 10.0000 0.507673
\(389\) −2.00000 −0.101404 −0.0507020 − 0.998714i \(-0.516146\pi\)
−0.0507020 + 0.998714i \(0.516146\pi\)
\(390\) 2.00000 0.101274
\(391\) 16.0000 0.809155
\(392\) 1.00000 0.0505076
\(393\) −12.0000 −0.605320
\(394\) −26.0000 −1.30986
\(395\) 16.0000 0.805047
\(396\) −1.00000 −0.0502519
\(397\) 30.0000 1.50566 0.752828 − 0.658217i \(-0.228689\pi\)
0.752828 + 0.658217i \(0.228689\pi\)
\(398\) 0 0
\(399\) 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
\(404\) 6.00000 0.298511
\(405\) 1.00000 0.0496904
\(406\) 6.00000 0.297775
\(407\) 2.00000 0.0991363
\(408\) −2.00000 −0.0990148
\(409\) 26.0000 1.28562 0.642809 − 0.766027i \(-0.277769\pi\)
0.642809 + 0.766027i \(0.277769\pi\)
\(410\) 2.00000 0.0987730
\(411\) −10.0000 −0.493264
\(412\) 8.00000 0.394132
\(413\) −12.0000 −0.590481
\(414\) 8.00000 0.393179
\(415\) −4.00000 −0.196352
\(416\) −2.00000 −0.0980581
\(417\) 12.0000 0.587643
\(418\) 4.00000 0.195646
\(419\) −4.00000 −0.195413 −0.0977064 − 0.995215i \(-0.531151\pi\)
−0.0977064 + 0.995215i \(0.531151\pi\)
\(420\) −1.00000 −0.0487950
\(421\) −26.0000 −1.26716 −0.633581 − 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) −12.0000 −0.584151
\(423\) 8.00000 0.388973
\(424\) −10.0000 −0.485643
\(425\) 2.00000 0.0970143
\(426\) −8.00000 −0.387601
\(427\) −2.00000 −0.0967868
\(428\) 12.0000 0.580042
\(429\) −2.00000 −0.0965609
\(430\) 4.00000 0.192897
\(431\) 16.0000 0.770693 0.385346 − 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −38.0000 −1.82616 −0.913082 − 0.407777i \(-0.866304\pi\)
−0.913082 + 0.407777i \(0.866304\pi\)
\(434\) 8.00000 0.384012
\(435\) −6.00000 −0.287678
\(436\) 14.0000 0.670478
\(437\) −32.0000 −1.53077
\(438\) −2.00000 −0.0955637
\(439\) 16.0000 0.763638 0.381819 − 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) −1.00000 −0.0476731
\(441\) 1.00000 0.0476190
\(442\) −4.00000 −0.190261
\(443\) −36.0000 −1.71041 −0.855206 − 0.518289i \(-0.826569\pi\)
−0.855206 + 0.518289i \(0.826569\pi\)
\(444\) 2.00000 0.0949158
\(445\) −14.0000 −0.663664
\(446\) 0 0
\(447\) −14.0000 −0.662177
\(448\) 1.00000 0.0472456
\(449\) 18.0000 0.849473 0.424736 − 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 1.00000 0.0471405
\(451\) −2.00000 −0.0941763
\(452\) −14.0000 −0.658505
\(453\) −24.0000 −1.12762
\(454\) −4.00000 −0.187729
\(455\) −2.00000 −0.0937614
\(456\) 4.00000 0.187317
\(457\) −38.0000 −1.77757 −0.888783 − 0.458329i \(-0.848448\pi\)
−0.888783 + 0.458329i \(0.848448\pi\)
\(458\) 6.00000 0.280362
\(459\) −2.00000 −0.0933520
\(460\) 8.00000 0.373002
\(461\) −34.0000 −1.58354 −0.791769 − 0.610821i \(-0.790840\pi\)
−0.791769 + 0.610821i \(0.790840\pi\)
\(462\) 1.00000 0.0465242
\(463\) −32.0000 −1.48717 −0.743583 − 0.668644i \(-0.766875\pi\)
−0.743583 + 0.668644i \(0.766875\pi\)
\(464\) 6.00000 0.278543
\(465\) −8.00000 −0.370991
\(466\) 10.0000 0.463241
\(467\) 12.0000 0.555294 0.277647 − 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) −2.00000 −0.0924500
\(469\) −4.00000 −0.184703
\(470\) 8.00000 0.369012
\(471\) 18.0000 0.829396
\(472\) −12.0000 −0.552345
\(473\) −4.00000 −0.183920
\(474\) −16.0000 −0.734904
\(475\) −4.00000 −0.183533
\(476\) 2.00000 0.0916698
\(477\) −10.0000 −0.457869
\(478\) 0 0
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 4.00000 0.182384
\(482\) 2.00000 0.0910975
\(483\) −8.00000 −0.364013
\(484\) 1.00000 0.0454545
\(485\) 10.0000 0.454077
\(486\) −1.00000 −0.0453609
\(487\) 24.0000 1.08754 0.543772 − 0.839233i \(-0.316996\pi\)
0.543772 + 0.839233i \(0.316996\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 4.00000 0.180886
\(490\) 1.00000 0.0451754
\(491\) 20.0000 0.902587 0.451294 − 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) −2.00000 −0.0901670
\(493\) 12.0000 0.540453
\(494\) 8.00000 0.359937
\(495\) −1.00000 −0.0449467
\(496\) 8.00000 0.359211
\(497\) 8.00000 0.358849
\(498\) 4.00000 0.179244
\(499\) 4.00000 0.179065 0.0895323 − 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) −12.0000 −0.535586
\(503\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 1.00000 0.0445435
\(505\) 6.00000 0.266996
\(506\) −8.00000 −0.355643
\(507\) 9.00000 0.399704
\(508\) 0 0
\(509\) −18.0000 −0.797836 −0.398918 − 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) −2.00000 −0.0885615
\(511\) 2.00000 0.0884748
\(512\) 1.00000 0.0441942
\(513\) 4.00000 0.176604
\(514\) −14.0000 −0.617514
\(515\) 8.00000 0.352522
\(516\) −4.00000 −0.176090
\(517\) −8.00000 −0.351840
\(518\) −2.00000 −0.0878750
\(519\) 18.0000 0.790112
\(520\) −2.00000 −0.0877058
\(521\) −14.0000 −0.613351 −0.306676 − 0.951814i \(-0.599217\pi\)
−0.306676 + 0.951814i \(0.599217\pi\)
\(522\) 6.00000 0.262613
\(523\) −12.0000 −0.524723 −0.262362 − 0.964970i \(-0.584501\pi\)
−0.262362 + 0.964970i \(0.584501\pi\)
\(524\) 12.0000 0.524222
\(525\) −1.00000 −0.0436436
\(526\) −8.00000 −0.348817
\(527\) 16.0000 0.696971
\(528\) 1.00000 0.0435194
\(529\) 41.0000 1.78261
\(530\) −10.0000 −0.434372
\(531\) −12.0000 −0.520756
\(532\) −4.00000 −0.173422
\(533\) −4.00000 −0.173259
\(534\) 14.0000 0.605839
\(535\) 12.0000 0.518805
\(536\) −4.00000 −0.172774
\(537\) 4.00000 0.172613
\(538\) −2.00000 −0.0862261
\(539\) −1.00000 −0.0430730
\(540\) −1.00000 −0.0430331
\(541\) −2.00000 −0.0859867 −0.0429934 − 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) −8.00000 −0.343629
\(543\) −22.0000 −0.944110
\(544\) 2.00000 0.0857493
\(545\) 14.0000 0.599694
\(546\) 2.00000 0.0855921
\(547\) −20.0000 −0.855138 −0.427569 − 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) 10.0000 0.427179
\(549\) −2.00000 −0.0853579
\(550\) −1.00000 −0.0426401
\(551\) −24.0000 −1.02243
\(552\) −8.00000 −0.340503
\(553\) 16.0000 0.680389
\(554\) −2.00000 −0.0849719
\(555\) 2.00000 0.0848953
\(556\) −12.0000 −0.508913
\(557\) −2.00000 −0.0847427 −0.0423714 − 0.999102i \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) 8.00000 0.338667
\(559\) −8.00000 −0.338364
\(560\) 1.00000 0.0422577
\(561\) 2.00000 0.0844401
\(562\) −22.0000 −0.928014
\(563\) 12.0000 0.505740 0.252870 − 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) −8.00000 −0.336861
\(565\) −14.0000 −0.588984
\(566\) 4.00000 0.168133
\(567\) 1.00000 0.0419961
\(568\) 8.00000 0.335673
\(569\) −38.0000 −1.59304 −0.796521 − 0.604610i \(-0.793329\pi\)
−0.796521 + 0.604610i \(0.793329\pi\)
\(570\) 4.00000 0.167542
\(571\) 12.0000 0.502184 0.251092 − 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) 2.00000 0.0836242
\(573\) 16.0000 0.668410
\(574\) 2.00000 0.0834784
\(575\) 8.00000 0.333623
\(576\) 1.00000 0.0416667
\(577\) −6.00000 −0.249783 −0.124892 − 0.992170i \(-0.539858\pi\)
−0.124892 + 0.992170i \(0.539858\pi\)
\(578\) −13.0000 −0.540729
\(579\) 14.0000 0.581820
\(580\) 6.00000 0.249136
\(581\) −4.00000 −0.165948
\(582\) −10.0000 −0.414513
\(583\) 10.0000 0.414158
\(584\) 2.00000 0.0827606
\(585\) −2.00000 −0.0826898
\(586\) −26.0000 −1.07405
\(587\) −12.0000 −0.495293 −0.247647 − 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) −1.00000 −0.0412393
\(589\) −32.0000 −1.31854
\(590\) −12.0000 −0.494032
\(591\) 26.0000 1.06950
\(592\) −2.00000 −0.0821995
\(593\) −14.0000 −0.574911 −0.287456 − 0.957794i \(-0.592809\pi\)
−0.287456 + 0.957794i \(0.592809\pi\)
\(594\) 1.00000 0.0410305
\(595\) 2.00000 0.0819920
\(596\) 14.0000 0.573462
\(597\) 0 0
\(598\) −16.0000 −0.654289
\(599\) −24.0000 −0.980613 −0.490307 − 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 26.0000 1.06056 0.530281 − 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 4.00000 0.163028
\(603\) −4.00000 −0.162893
\(604\) 24.0000 0.976546
\(605\) 1.00000 0.0406558
\(606\) −6.00000 −0.243733
\(607\) 32.0000 1.29884 0.649420 − 0.760430i \(-0.275012\pi\)
0.649420 + 0.760430i \(0.275012\pi\)
\(608\) −4.00000 −0.162221
\(609\) −6.00000 −0.243132
\(610\) −2.00000 −0.0809776
\(611\) −16.0000 −0.647291
\(612\) 2.00000 0.0808452
\(613\) −18.0000 −0.727013 −0.363507 − 0.931592i \(-0.618421\pi\)
−0.363507 + 0.931592i \(0.618421\pi\)
\(614\) −20.0000 −0.807134
\(615\) −2.00000 −0.0806478
\(616\) −1.00000 −0.0402911
\(617\) −22.0000 −0.885687 −0.442843 − 0.896599i \(-0.646030\pi\)
−0.442843 + 0.896599i \(0.646030\pi\)
\(618\) −8.00000 −0.321807
\(619\) 4.00000 0.160774 0.0803868 − 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 8.00000 0.321288
\(621\) −8.00000 −0.321029
\(622\) −24.0000 −0.962312
\(623\) −14.0000 −0.560898
\(624\) 2.00000 0.0800641
\(625\) 1.00000 0.0400000
\(626\) −30.0000 −1.19904
\(627\) −4.00000 −0.159745
\(628\) −18.0000 −0.718278
\(629\) −4.00000 −0.159490
\(630\) 1.00000 0.0398410
\(631\) −40.0000 −1.59237 −0.796187 − 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) 16.0000 0.636446
\(633\) 12.0000 0.476957
\(634\) 30.0000 1.19145
\(635\) 0 0
\(636\) 10.0000 0.396526
\(637\) −2.00000 −0.0792429
\(638\) −6.00000 −0.237542
\(639\) 8.00000 0.316475
\(640\) 1.00000 0.0395285
\(641\) −30.0000 −1.18493 −0.592464 − 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) −12.0000 −0.473602
\(643\) 12.0000 0.473234 0.236617 − 0.971603i \(-0.423961\pi\)
0.236617 + 0.971603i \(0.423961\pi\)
\(644\) 8.00000 0.315244
\(645\) −4.00000 −0.157500
\(646\) −8.00000 −0.314756
\(647\) 48.0000 1.88707 0.943537 − 0.331266i \(-0.107476\pi\)
0.943537 + 0.331266i \(0.107476\pi\)
\(648\) 1.00000 0.0392837
\(649\) 12.0000 0.471041
\(650\) −2.00000 −0.0784465
\(651\) −8.00000 −0.313545
\(652\) −4.00000 −0.156652
\(653\) 14.0000 0.547862 0.273931 − 0.961749i \(-0.411676\pi\)
0.273931 + 0.961749i \(0.411676\pi\)
\(654\) −14.0000 −0.547443
\(655\) 12.0000 0.468879
\(656\) 2.00000 0.0780869
\(657\) 2.00000 0.0780274
\(658\) 8.00000 0.311872
\(659\) 12.0000 0.467454 0.233727 − 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 1.00000 0.0389249
\(661\) 22.0000 0.855701 0.427850 − 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) 28.0000 1.08825
\(663\) 4.00000 0.155347
\(664\) −4.00000 −0.155230
\(665\) −4.00000 −0.155113
\(666\) −2.00000 −0.0774984
\(667\) 48.0000 1.85857
\(668\) 0 0
\(669\) 0 0
\(670\) −4.00000 −0.154533
\(671\) 2.00000 0.0772091
\(672\) −1.00000 −0.0385758
\(673\) 34.0000 1.31060 0.655302 − 0.755367i \(-0.272541\pi\)
0.655302 + 0.755367i \(0.272541\pi\)
\(674\) 2.00000 0.0770371
\(675\) −1.00000 −0.0384900
\(676\) −9.00000 −0.346154
\(677\) −26.0000 −0.999261 −0.499631 − 0.866239i \(-0.666531\pi\)
−0.499631 + 0.866239i \(0.666531\pi\)
\(678\) 14.0000 0.537667
\(679\) 10.0000 0.383765
\(680\) 2.00000 0.0766965
\(681\) 4.00000 0.153280
\(682\) −8.00000 −0.306336
\(683\) −36.0000 −1.37750 −0.688751 − 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) −4.00000 −0.152944
\(685\) 10.0000 0.382080
\(686\) 1.00000 0.0381802
\(687\) −6.00000 −0.228914
\(688\) 4.00000 0.152499
\(689\) 20.0000 0.761939
\(690\) −8.00000 −0.304555
\(691\) 28.0000 1.06517 0.532585 − 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) −18.0000 −0.684257
\(693\) −1.00000 −0.0379869
\(694\) 12.0000 0.455514
\(695\) −12.0000 −0.455186
\(696\) −6.00000 −0.227429
\(697\) 4.00000 0.151511
\(698\) −2.00000 −0.0757011
\(699\) −10.0000 −0.378235
\(700\) 1.00000 0.0377964
\(701\) 38.0000 1.43524 0.717620 − 0.696435i \(-0.245231\pi\)
0.717620 + 0.696435i \(0.245231\pi\)
\(702\) 2.00000 0.0754851
\(703\) 8.00000 0.301726
\(704\) −1.00000 −0.0376889
\(705\) −8.00000 −0.301297
\(706\) −30.0000 −1.12906
\(707\) 6.00000 0.225653
\(708\) 12.0000 0.450988
\(709\) 6.00000 0.225335 0.112667 − 0.993633i \(-0.464061\pi\)
0.112667 + 0.993633i \(0.464061\pi\)
\(710\) 8.00000 0.300235
\(711\) 16.0000 0.600047
\(712\) −14.0000 −0.524672
\(713\) 64.0000 2.39682
\(714\) −2.00000 −0.0748481
\(715\) 2.00000 0.0747958
\(716\) −4.00000 −0.149487
\(717\) 0 0
\(718\) 8.00000 0.298557
\(719\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 1.00000 0.0372678
\(721\) 8.00000 0.297936
\(722\) −3.00000 −0.111648
\(723\) −2.00000 −0.0743808
\(724\) 22.0000 0.817624
\(725\) 6.00000 0.222834
\(726\) −1.00000 −0.0371135
\(727\) −24.0000 −0.890111 −0.445055 − 0.895503i \(-0.646816\pi\)
−0.445055 + 0.895503i \(0.646816\pi\)
\(728\) −2.00000 −0.0741249
\(729\) 1.00000 0.0370370
\(730\) 2.00000 0.0740233
\(731\) 8.00000 0.295891
\(732\) 2.00000 0.0739221
\(733\) −2.00000 −0.0738717 −0.0369358 − 0.999318i \(-0.511760\pi\)
−0.0369358 + 0.999318i \(0.511760\pi\)
\(734\) 0 0
\(735\) −1.00000 −0.0368856
\(736\) 8.00000 0.294884
\(737\) 4.00000 0.147342
\(738\) 2.00000 0.0736210
\(739\) 4.00000 0.147142 0.0735712 − 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) −2.00000 −0.0735215
\(741\) −8.00000 −0.293887
\(742\) −10.0000 −0.367112
\(743\) −8.00000 −0.293492 −0.146746 − 0.989174i \(-0.546880\pi\)
−0.146746 + 0.989174i \(0.546880\pi\)
\(744\) −8.00000 −0.293294
\(745\) 14.0000 0.512920
\(746\) 30.0000 1.09838
\(747\) −4.00000 −0.146352
\(748\) −2.00000 −0.0731272
\(749\) 12.0000 0.438470
\(750\) −1.00000 −0.0365148
\(751\) 32.0000 1.16770 0.583848 − 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) 8.00000 0.291730
\(753\) 12.0000 0.437304
\(754\) −12.0000 −0.437014
\(755\) 24.0000 0.873449
\(756\) −1.00000 −0.0363696
\(757\) −34.0000 −1.23575 −0.617876 − 0.786276i \(-0.712006\pi\)
−0.617876 + 0.786276i \(0.712006\pi\)
\(758\) 28.0000 1.01701
\(759\) 8.00000 0.290382
\(760\) −4.00000 −0.145095
\(761\) −30.0000 −1.08750 −0.543750 − 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) 0 0
\(763\) 14.0000 0.506834
\(764\) −16.0000 −0.578860
\(765\) 2.00000 0.0723102
\(766\) −24.0000 −0.867155
\(767\) 24.0000 0.866590
\(768\) −1.00000 −0.0360844
\(769\) 34.0000 1.22607 0.613036 − 0.790055i \(-0.289948\pi\)
0.613036 + 0.790055i \(0.289948\pi\)
\(770\) −1.00000 −0.0360375
\(771\) 14.0000 0.504198
\(772\) −14.0000 −0.503871
\(773\) 38.0000 1.36677 0.683383 − 0.730061i \(-0.260508\pi\)
0.683383 + 0.730061i \(0.260508\pi\)
\(774\) 4.00000 0.143777
\(775\) 8.00000 0.287368
\(776\) 10.0000 0.358979
\(777\) 2.00000 0.0717496
\(778\) −2.00000 −0.0717035
\(779\) −8.00000 −0.286630
\(780\) 2.00000 0.0716115
\(781\) −8.00000 −0.286263
\(782\) 16.0000 0.572159
\(783\) −6.00000 −0.214423
\(784\) 1.00000 0.0357143
\(785\) −18.0000 −0.642448
\(786\) −12.0000 −0.428026
\(787\) 12.0000 0.427754 0.213877 − 0.976861i \(-0.431391\pi\)
0.213877 + 0.976861i \(0.431391\pi\)
\(788\) −26.0000 −0.926212
\(789\) 8.00000 0.284808
\(790\) 16.0000 0.569254
\(791\) −14.0000 −0.497783
\(792\) −1.00000 −0.0355335
\(793\) 4.00000 0.142044
\(794\) 30.0000 1.06466
\(795\) 10.0000 0.354663
\(796\) 0 0
\(797\) −2.00000 −0.0708436 −0.0354218 − 0.999372i \(-0.511277\pi\)
−0.0354218 + 0.999372i \(0.511277\pi\)
\(798\) 4.00000 0.141598
\(799\) 16.0000 0.566039
\(800\) 1.00000 0.0353553
\(801\) −14.0000 −0.494666
\(802\) 18.0000 0.635602
\(803\) −2.00000 −0.0705785
\(804\) 4.00000 0.141069
\(805\) 8.00000 0.281963
\(806\) −16.0000 −0.563576
\(807\) 2.00000 0.0704033
\(808\) 6.00000 0.211079
\(809\) −6.00000 −0.210949 −0.105474 − 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 1.00000 0.0351364
\(811\) −28.0000 −0.983213 −0.491606 − 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) 6.00000 0.210559
\(813\) 8.00000 0.280572
\(814\) 2.00000 0.0701000
\(815\) −4.00000 −0.140114
\(816\) −2.00000 −0.0700140
\(817\) −16.0000 −0.559769
\(818\) 26.0000 0.909069
\(819\) −2.00000 −0.0698857
\(820\) 2.00000 0.0698430
\(821\) −18.0000 −0.628204 −0.314102 − 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) −10.0000 −0.348790
\(823\) 40.0000 1.39431 0.697156 − 0.716919i \(-0.254448\pi\)
0.697156 + 0.716919i \(0.254448\pi\)
\(824\) 8.00000 0.278693
\(825\) 1.00000 0.0348155
\(826\) −12.0000 −0.417533
\(827\) −20.0000 −0.695468 −0.347734 − 0.937593i \(-0.613049\pi\)
−0.347734 + 0.937593i \(0.613049\pi\)
\(828\) 8.00000 0.278019
\(829\) −2.00000 −0.0694629 −0.0347314 − 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) −4.00000 −0.138842
\(831\) 2.00000 0.0693792
\(832\) −2.00000 −0.0693375
\(833\) 2.00000 0.0692959
\(834\) 12.0000 0.415526
\(835\) 0 0
\(836\) 4.00000 0.138343
\(837\) −8.00000 −0.276520
\(838\) −4.00000 −0.138178
\(839\) −24.0000 −0.828572 −0.414286 − 0.910147i \(-0.635969\pi\)
−0.414286 + 0.910147i \(0.635969\pi\)
\(840\) −1.00000 −0.0345033
\(841\) 7.00000 0.241379
\(842\) −26.0000 −0.896019
\(843\) 22.0000 0.757720
\(844\) −12.0000 −0.413057
\(845\) −9.00000 −0.309609
\(846\) 8.00000 0.275046
\(847\) 1.00000 0.0343604
\(848\) −10.0000 −0.343401
\(849\) −4.00000 −0.137280
\(850\) 2.00000 0.0685994
\(851\) −16.0000 −0.548473
\(852\) −8.00000 −0.274075
\(853\) −26.0000 −0.890223 −0.445112 − 0.895475i \(-0.646836\pi\)
−0.445112 + 0.895475i \(0.646836\pi\)
\(854\) −2.00000 −0.0684386
\(855\) −4.00000 −0.136797
\(856\) 12.0000 0.410152
\(857\) 42.0000 1.43469 0.717346 − 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) −2.00000 −0.0682789
\(859\) 20.0000 0.682391 0.341196 − 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 4.00000 0.136399
\(861\) −2.00000 −0.0681598
\(862\) 16.0000 0.544962
\(863\) 16.0000 0.544646 0.272323 − 0.962206i \(-0.412208\pi\)
0.272323 + 0.962206i \(0.412208\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −18.0000 −0.612018
\(866\) −38.0000 −1.29129
\(867\) 13.0000 0.441503
\(868\) 8.00000 0.271538
\(869\) −16.0000 −0.542763
\(870\) −6.00000 −0.203419
\(871\) 8.00000 0.271070
\(872\) 14.0000 0.474100
\(873\) 10.0000 0.338449
\(874\) −32.0000 −1.08242
\(875\) 1.00000 0.0338062
\(876\) −2.00000 −0.0675737
\(877\) 38.0000 1.28317 0.641584 − 0.767052i \(-0.278277\pi\)
0.641584 + 0.767052i \(0.278277\pi\)
\(878\) 16.0000 0.539974
\(879\) 26.0000 0.876958
\(880\) −1.00000 −0.0337100
\(881\) −38.0000 −1.28025 −0.640126 − 0.768270i \(-0.721118\pi\)
−0.640126 + 0.768270i \(0.721118\pi\)
\(882\) 1.00000 0.0336718
\(883\) −20.0000 −0.673054 −0.336527 − 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) −4.00000 −0.134535
\(885\) 12.0000 0.403376
\(886\) −36.0000 −1.20944
\(887\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 2.00000 0.0671156
\(889\) 0 0
\(890\) −14.0000 −0.469281
\(891\) −1.00000 −0.0335013
\(892\) 0 0
\(893\) −32.0000 −1.07084
\(894\) −14.0000 −0.468230
\(895\) −4.00000 −0.133705
\(896\) 1.00000 0.0334077
\(897\) 16.0000 0.534224
\(898\) 18.0000 0.600668
\(899\) 48.0000 1.60089
\(900\) 1.00000 0.0333333
\(901\) −20.0000 −0.666297
\(902\) −2.00000 −0.0665927
\(903\) −4.00000 −0.133112
\(904\) −14.0000 −0.465633
\(905\) 22.0000 0.731305
\(906\) −24.0000 −0.797347
\(907\) 4.00000 0.132818 0.0664089 − 0.997792i \(-0.478846\pi\)
0.0664089 + 0.997792i \(0.478846\pi\)
\(908\) −4.00000 −0.132745
\(909\) 6.00000 0.199007
\(910\) −2.00000 −0.0662994
\(911\) −16.0000 −0.530104 −0.265052 − 0.964234i \(-0.585389\pi\)
−0.265052 + 0.964234i \(0.585389\pi\)
\(912\) 4.00000 0.132453
\(913\) 4.00000 0.132381
\(914\) −38.0000 −1.25693
\(915\) 2.00000 0.0661180
\(916\) 6.00000 0.198246
\(917\) 12.0000 0.396275
\(918\) −2.00000 −0.0660098
\(919\) −24.0000 −0.791687 −0.395843 − 0.918318i \(-0.629548\pi\)
−0.395843 + 0.918318i \(0.629548\pi\)
\(920\) 8.00000 0.263752
\(921\) 20.0000 0.659022
\(922\) −34.0000 −1.11973
\(923\) −16.0000 −0.526646
\(924\) 1.00000 0.0328976
\(925\) −2.00000 −0.0657596
\(926\) −32.0000 −1.05159
\(927\) 8.00000 0.262754
\(928\) 6.00000 0.196960
\(929\) −6.00000 −0.196854 −0.0984268 − 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) −8.00000 −0.262330
\(931\) −4.00000 −0.131095
\(932\) 10.0000 0.327561
\(933\) 24.0000 0.785725
\(934\) 12.0000 0.392652
\(935\) −2.00000 −0.0654070
\(936\) −2.00000 −0.0653720
\(937\) 2.00000 0.0653372 0.0326686 − 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) −4.00000 −0.130605
\(939\) 30.0000 0.979013
\(940\) 8.00000 0.260931
\(941\) 46.0000 1.49956 0.749779 − 0.661689i \(-0.230160\pi\)
0.749779 + 0.661689i \(0.230160\pi\)
\(942\) 18.0000 0.586472
\(943\) 16.0000 0.521032
\(944\) −12.0000 −0.390567
\(945\) −1.00000 −0.0325300
\(946\) −4.00000 −0.130051
\(947\) 52.0000 1.68977 0.844886 − 0.534946i \(-0.179668\pi\)
0.844886 + 0.534946i \(0.179668\pi\)
\(948\) −16.0000 −0.519656
\(949\) −4.00000 −0.129845
\(950\) −4.00000 −0.129777
\(951\) −30.0000 −0.972817
\(952\) 2.00000 0.0648204
\(953\) −6.00000 −0.194359 −0.0971795 − 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) −10.0000 −0.323762
\(955\) −16.0000 −0.517748
\(956\) 0 0
\(957\) 6.00000 0.193952
\(958\) 0 0
\(959\) 10.0000 0.322917
\(960\) −1.00000 −0.0322749
\(961\) 33.0000 1.06452
\(962\) 4.00000 0.128965
\(963\) 12.0000 0.386695
\(964\) 2.00000 0.0644157
\(965\) −14.0000 −0.450676
\(966\) −8.00000 −0.257396
\(967\) −40.0000 −1.28631 −0.643157 − 0.765735i \(-0.722376\pi\)
−0.643157 + 0.765735i \(0.722376\pi\)
\(968\) 1.00000 0.0321412
\(969\) 8.00000 0.256997
\(970\) 10.0000 0.321081
\(971\) −12.0000 −0.385098 −0.192549 − 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −12.0000 −0.384702
\(974\) 24.0000 0.769010
\(975\) 2.00000 0.0640513
\(976\) −2.00000 −0.0640184
\(977\) −46.0000 −1.47167 −0.735835 − 0.677161i \(-0.763210\pi\)
−0.735835 + 0.677161i \(0.763210\pi\)
\(978\) 4.00000 0.127906
\(979\) 14.0000 0.447442
\(980\) 1.00000 0.0319438
\(981\) 14.0000 0.446986
\(982\) 20.0000 0.638226
\(983\) −48.0000 −1.53096 −0.765481 − 0.643458i \(-0.777499\pi\)
−0.765481 + 0.643458i \(0.777499\pi\)
\(984\) −2.00000 −0.0637577
\(985\) −26.0000 −0.828429
\(986\) 12.0000 0.382158
\(987\) −8.00000 −0.254643
\(988\) 8.00000 0.254514
\(989\) 32.0000 1.01754
\(990\) −1.00000 −0.0317821
\(991\) 32.0000 1.01651 0.508257 − 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) 8.00000 0.254000
\(993\) −28.0000 −0.888553
\(994\) 8.00000 0.253745
\(995\) 0 0
\(996\) 4.00000 0.126745
\(997\) −58.0000 −1.83688 −0.918439 − 0.395562i \(-0.870550\pi\)
−0.918439 + 0.395562i \(0.870550\pi\)
\(998\) 4.00000 0.126618
\(999\) 2.00000 0.0632772
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 2310.2.a.o.1.1 ✓ 1
3.2 odd 2 6930.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2310.2.a.o.1.1 ✓ 1 1.1 even 1 trivial
6930.2.a.d.1.1 1 3.2 odd 2