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Properties

Label 2310.2.a.g.1.1
Level $2310$
Weight $2$
Character 2310.1
Self dual yes
Analytic conductor $18.445$
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] = [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: \(1\)
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 \(\chi\) \(=\) 2310.1

$q$-expansion

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

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
\(13\) −6.00000 −1.66410 −0.832050 − 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) −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 \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) −1.00000 −0.223607
\(21\) 1.00000 0.218218
\(22\) 1.00000 0.213201
\(23\) −8.00000 −1.66812 −0.834058 − 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 6.00000 1.17670
\(27\) 1.00000 0.192450
\(28\) 1.00000 0.188982
\(29\) 2.00000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 1.00000 0.182574
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.00000 −0.174078
\(34\) −2.00000 −0.342997
\(35\) −1.00000 −0.169031
\(36\) 1.00000 0.166667
\(37\) −6.00000 −0.986394 −0.493197 − 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) −4.00000 −0.648886
\(39\) −6.00000 −0.960769
\(40\) 1.00000 0.158114
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) −1.00000 −0.154303
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −1.00000 −0.150756
\(45\) −1.00000 −0.149071
\(46\) 8.00000 1.17954
\(47\) −8.00000 −1.16692 −0.583460 − 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 2.00000 0.280056
\(52\) −6.00000 −0.832050
\(53\) 2.00000 0.274721 0.137361 − 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) −1.00000 −0.136083
\(55\) 1.00000 0.134840
\(56\) −1.00000 −0.133631
\(57\) 4.00000 0.529813
\(58\) −2.00000 −0.262613
\(59\) −12.0000 −1.56227 −0.781133 − 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) −1.00000 −0.129099
\(61\) −10.0000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) −4.00000 −0.508001
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 6.00000 0.744208
\(66\) 1.00000 0.123091
\(67\) 16.0000 1.95471 0.977356 − 0.211604i \(-0.0678686\pi\)
0.977356 + 0.211604i \(0.0678686\pi\)
\(68\) 2.00000 0.242536
\(69\) −8.00000 −0.963087
\(70\) 1.00000 0.119523
\(71\) 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 \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 6.00000 0.697486
\(75\) 1.00000 0.115470
\(76\) 4.00000 0.458831
\(77\) −1.00000 −0.113961
\(78\) 6.00000 0.679366
\(79\) −12.0000 −1.35011 −0.675053 − 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) −16.0000 −1.75623 −0.878114 − 0.478451i \(-0.841198\pi\)
−0.878114 + 0.478451i \(0.841198\pi\)
\(84\) 1.00000 0.109109
\(85\) −2.00000 −0.216930
\(86\) 4.00000 0.431331
\(87\) 2.00000 0.214423
\(88\) 1.00000 0.106600
\(89\) 10.0000 1.06000 0.529999 − 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 1.00000 0.105409
\(91\) −6.00000 −0.628971
\(92\) −8.00000 −0.834058
\(93\) 4.00000 0.414781
\(94\) 8.00000 0.825137
\(95\) −4.00000 −0.410391
\(96\) −1.00000 −0.102062
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) −1.00000 −0.101015
\(99\) −1.00000 −0.100504
\(100\) 1.00000 0.100000
\(101\) −10.0000 −0.995037 −0.497519 − 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) −2.00000 −0.198030
\(103\) 16.0000 1.57653 0.788263 − 0.615338i \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) 6.00000 0.588348
\(105\) −1.00000 −0.0975900
\(106\) −2.00000 −0.194257
\(107\) −12.0000 −1.16008 −0.580042 − 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 1.00000 0.0962250
\(109\) −6.00000 −0.574696 −0.287348 − 0.957826i \(-0.592774\pi\)
−0.287348 + 0.957826i \(0.592774\pi\)
\(110\) −1.00000 −0.0953463
\(111\) −6.00000 −0.569495
\(112\) 1.00000 0.0944911
\(113\) −6.00000 −0.564433 −0.282216 − 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) −4.00000 −0.374634
\(115\) 8.00000 0.746004
\(116\) 2.00000 0.185695
\(117\) −6.00000 −0.554700
\(118\) 12.0000 1.10469
\(119\) 2.00000 0.183340
\(120\) 1.00000 0.0912871
\(121\) 1.00000 0.0909091
\(122\) 10.0000 0.905357
\(123\) −6.00000 −0.541002
\(124\) 4.00000 0.359211
\(125\) −1.00000 −0.0894427
\(126\) −1.00000 −0.0890871
\(127\) −16.0000 −1.41977 −0.709885 − 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −4.00000 −0.352180
\(130\) −6.00000 −0.526235
\(131\) 4.00000 0.349482 0.174741 − 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) −1.00000 −0.0870388
\(133\) 4.00000 0.346844
\(134\) −16.0000 −1.38219
\(135\) −1.00000 −0.0860663
\(136\) −2.00000 −0.171499
\(137\) −14.0000 −1.19610 −0.598050 − 0.801459i \(-0.704058\pi\)
−0.598050 + 0.801459i \(0.704058\pi\)
\(138\) 8.00000 0.681005
\(139\) 20.0000 1.69638 0.848189 − 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) −1.00000 −0.0845154
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) 6.00000 0.501745
\(144\) 1.00000 0.0833333
\(145\) −2.00000 −0.166091
\(146\) −10.0000 −0.827606
\(147\) 1.00000 0.0824786
\(148\) −6.00000 −0.493197
\(149\) 18.0000 1.47462 0.737309 − 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −20.0000 −1.62758 −0.813788 − 0.581161i \(-0.802599\pi\)
−0.813788 + 0.581161i \(0.802599\pi\)
\(152\) −4.00000 −0.324443
\(153\) 2.00000 0.161690
\(154\) 1.00000 0.0805823
\(155\) −4.00000 −0.321288
\(156\) −6.00000 −0.480384
\(157\) −14.0000 −1.11732 −0.558661 − 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) 12.0000 0.954669
\(159\) 2.00000 0.158610
\(160\) 1.00000 0.0790569
\(161\) −8.00000 −0.630488
\(162\) −1.00000 −0.0785674
\(163\) 8.00000 0.626608 0.313304 − 0.949653i \(-0.398564\pi\)
0.313304 + 0.949653i \(0.398564\pi\)
\(164\) −6.00000 −0.468521
\(165\) 1.00000 0.0778499
\(166\) 16.0000 1.24184
\(167\) −8.00000 −0.619059 −0.309529 − 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) −1.00000 −0.0771517
\(169\) 23.0000 1.76923
\(170\) 2.00000 0.153393
\(171\) 4.00000 0.305888
\(172\) −4.00000 −0.304997
\(173\) −6.00000 −0.456172 −0.228086 − 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) −2.00000 −0.151620
\(175\) 1.00000 0.0755929
\(176\) −1.00000 −0.0753778
\(177\) −12.0000 −0.901975
\(178\) −10.0000 −0.749532
\(179\) 4.00000 0.298974 0.149487 − 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 26.0000 1.93256 0.966282 − 0.257485i \(-0.0828937\pi\)
0.966282 + 0.257485i \(0.0828937\pi\)
\(182\) 6.00000 0.444750
\(183\) −10.0000 −0.739221
\(184\) 8.00000 0.589768
\(185\) 6.00000 0.441129
\(186\) −4.00000 −0.293294
\(187\) −2.00000 −0.146254
\(188\) −8.00000 −0.583460
\(189\) 1.00000 0.0727393
\(190\) 4.00000 0.290191
\(191\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 1.00000 0.0721688
\(193\) −10.0000 −0.719816 −0.359908 − 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) 2.00000 0.143592
\(195\) 6.00000 0.429669
\(196\) 1.00000 0.0714286
\(197\) −6.00000 −0.427482 −0.213741 − 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 1.00000 0.0710669
\(199\) −4.00000 −0.283552 −0.141776 − 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 16.0000 1.12855
\(202\) 10.0000 0.703598
\(203\) 2.00000 0.140372
\(204\) 2.00000 0.140028
\(205\) 6.00000 0.419058
\(206\) −16.0000 −1.11477
\(207\) −8.00000 −0.556038
\(208\) −6.00000 −0.416025
\(209\) −4.00000 −0.276686
\(210\) 1.00000 0.0690066
\(211\) −20.0000 −1.37686 −0.688428 − 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 2.00000 0.137361
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) 4.00000 0.272798
\(216\) −1.00000 −0.0680414
\(217\) 4.00000 0.271538
\(218\) 6.00000 0.406371
\(219\) 10.0000 0.675737
\(220\) 1.00000 0.0674200
\(221\) −12.0000 −0.807207
\(222\) 6.00000 0.402694
\(223\) −8.00000 −0.535720 −0.267860 − 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 1.00000 0.0666667
\(226\) 6.00000 0.399114
\(227\) −16.0000 −1.06196 −0.530979 − 0.847385i \(-0.678176\pi\)
−0.530979 + 0.847385i \(0.678176\pi\)
\(228\) 4.00000 0.264906
\(229\) −22.0000 −1.45380 −0.726900 − 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) −8.00000 −0.527504
\(231\) −1.00000 −0.0657952
\(232\) −2.00000 −0.131306
\(233\) 6.00000 0.393073 0.196537 − 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 6.00000 0.392232
\(235\) 8.00000 0.521862
\(236\) −12.0000 −0.781133
\(237\) −12.0000 −0.779484
\(238\) −2.00000 −0.129641
\(239\) 20.0000 1.29369 0.646846 − 0.762620i \(-0.276088\pi\)
0.646846 + 0.762620i \(0.276088\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −14.0000 −0.901819 −0.450910 − 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 1.00000 0.0641500
\(244\) −10.0000 −0.640184
\(245\) −1.00000 −0.0638877
\(246\) 6.00000 0.382546
\(247\) −24.0000 −1.52708
\(248\) −4.00000 −0.254000
\(249\) −16.0000 −1.01396
\(250\) 1.00000 0.0632456
\(251\) −20.0000 −1.26239 −0.631194 − 0.775625i \(-0.717435\pi\)
−0.631194 + 0.775625i \(0.717435\pi\)
\(252\) 1.00000 0.0629941
\(253\) 8.00000 0.502956
\(254\) 16.0000 1.00393
\(255\) −2.00000 −0.125245
\(256\) 1.00000 0.0625000
\(257\) 30.0000 1.87135 0.935674 − 0.352865i \(-0.114792\pi\)
0.935674 + 0.352865i \(0.114792\pi\)
\(258\) 4.00000 0.249029
\(259\) −6.00000 −0.372822
\(260\) 6.00000 0.372104
\(261\) 2.00000 0.123797
\(262\) −4.00000 −0.247121
\(263\) −8.00000 −0.493301 −0.246651 − 0.969104i \(-0.579330\pi\)
−0.246651 + 0.969104i \(0.579330\pi\)
\(264\) 1.00000 0.0615457
\(265\) −2.00000 −0.122859
\(266\) −4.00000 −0.245256
\(267\) 10.0000 0.611990
\(268\) 16.0000 0.977356
\(269\) 2.00000 0.121942 0.0609711 − 0.998140i \(-0.480580\pi\)
0.0609711 + 0.998140i \(0.480580\pi\)
\(270\) 1.00000 0.0608581
\(271\) 32.0000 1.94386 0.971931 − 0.235267i \(-0.0755965\pi\)
0.971931 + 0.235267i \(0.0755965\pi\)
\(272\) 2.00000 0.121268
\(273\) −6.00000 −0.363137
\(274\) 14.0000 0.845771
\(275\) −1.00000 −0.0603023
\(276\) −8.00000 −0.481543
\(277\) 10.0000 0.600842 0.300421 − 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) −20.0000 −1.19952
\(279\) 4.00000 0.239474
\(280\) 1.00000 0.0597614
\(281\) 10.0000 0.596550 0.298275 − 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 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
\(285\) −4.00000 −0.236940
\(286\) −6.00000 −0.354787
\(287\) −6.00000 −0.354169
\(288\) −1.00000 −0.0589256
\(289\) −13.0000 −0.764706
\(290\) 2.00000 0.117444
\(291\) −2.00000 −0.117242
\(292\) 10.0000 0.585206
\(293\) −6.00000 −0.350524 −0.175262 − 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 12.0000 0.698667
\(296\) 6.00000 0.348743
\(297\) −1.00000 −0.0580259
\(298\) −18.0000 −1.04271
\(299\) 48.0000 2.77591
\(300\) 1.00000 0.0577350
\(301\) −4.00000 −0.230556
\(302\) 20.0000 1.15087
\(303\) −10.0000 −0.574485
\(304\) 4.00000 0.229416
\(305\) 10.0000 0.572598
\(306\) −2.00000 −0.114332
\(307\) −8.00000 −0.456584 −0.228292 − 0.973593i \(-0.573314\pi\)
−0.228292 + 0.973593i \(0.573314\pi\)
\(308\) −1.00000 −0.0569803
\(309\) 16.0000 0.910208
\(310\) 4.00000 0.227185
\(311\) 12.0000 0.680458 0.340229 − 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 6.00000 0.339683
\(313\) 6.00000 0.339140 0.169570 − 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) 14.0000 0.790066
\(315\) −1.00000 −0.0563436
\(316\) −12.0000 −0.675053
\(317\) −6.00000 −0.336994 −0.168497 − 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) −2.00000 −0.112154
\(319\) −2.00000 −0.111979
\(320\) −1.00000 −0.0559017
\(321\) −12.0000 −0.669775
\(322\) 8.00000 0.445823
\(323\) 8.00000 0.445132
\(324\) 1.00000 0.0555556
\(325\) −6.00000 −0.332820
\(326\) −8.00000 −0.443079
\(327\) −6.00000 −0.331801
\(328\) 6.00000 0.331295
\(329\) −8.00000 −0.441054
\(330\) −1.00000 −0.0550482
\(331\) 4.00000 0.219860 0.109930 − 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) −16.0000 −0.878114
\(333\) −6.00000 −0.328798
\(334\) 8.00000 0.437741
\(335\) −16.0000 −0.874173
\(336\) 1.00000 0.0545545
\(337\) −34.0000 −1.85210 −0.926049 − 0.377403i \(-0.876817\pi\)
−0.926049 + 0.377403i \(0.876817\pi\)
\(338\) −23.0000 −1.25104
\(339\) −6.00000 −0.325875
\(340\) −2.00000 −0.108465
\(341\) −4.00000 −0.216612
\(342\) −4.00000 −0.216295
\(343\) 1.00000 0.0539949
\(344\) 4.00000 0.215666
\(345\) 8.00000 0.430706
\(346\) 6.00000 0.322562
\(347\) −28.0000 −1.50312 −0.751559 − 0.659665i \(-0.770698\pi\)
−0.751559 + 0.659665i \(0.770698\pi\)
\(348\) 2.00000 0.107211
\(349\) −26.0000 −1.39175 −0.695874 − 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) −1.00000 −0.0534522
\(351\) −6.00000 −0.320256
\(352\) 1.00000 0.0533002
\(353\) −2.00000 −0.106449 −0.0532246 − 0.998583i \(-0.516950\pi\)
−0.0532246 + 0.998583i \(0.516950\pi\)
\(354\) 12.0000 0.637793
\(355\) 0 0
\(356\) 10.0000 0.529999
\(357\) 2.00000 0.105851
\(358\) −4.00000 −0.211407
\(359\) 12.0000 0.633336 0.316668 − 0.948536i \(-0.397436\pi\)
0.316668 + 0.948536i \(0.397436\pi\)
\(360\) 1.00000 0.0527046
\(361\) −3.00000 −0.157895
\(362\) −26.0000 −1.36653
\(363\) 1.00000 0.0524864
\(364\) −6.00000 −0.314485
\(365\) −10.0000 −0.523424
\(366\) 10.0000 0.522708
\(367\) −16.0000 −0.835193 −0.417597 − 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) −8.00000 −0.417029
\(369\) −6.00000 −0.312348
\(370\) −6.00000 −0.311925
\(371\) 2.00000 0.103835
\(372\) 4.00000 0.207390
\(373\) −6.00000 −0.310668 −0.155334 − 0.987862i \(-0.549645\pi\)
−0.155334 + 0.987862i \(0.549645\pi\)
\(374\) 2.00000 0.103418
\(375\) −1.00000 −0.0516398
\(376\) 8.00000 0.412568
\(377\) −12.0000 −0.618031
\(378\) −1.00000 −0.0514344
\(379\) 4.00000 0.205466 0.102733 − 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) −4.00000 −0.205196
\(381\) −16.0000 −0.819705
\(382\) 0 0
\(383\) 24.0000 1.22634 0.613171 − 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 1.00000 0.0509647
\(386\) 10.0000 0.508987
\(387\) −4.00000 −0.203331
\(388\) −2.00000 −0.101535
\(389\) −18.0000 −0.912636 −0.456318 − 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) −6.00000 −0.303822
\(391\) −16.0000 −0.809155
\(392\) −1.00000 −0.0505076
\(393\) 4.00000 0.201773
\(394\) 6.00000 0.302276
\(395\) 12.0000 0.603786
\(396\) −1.00000 −0.0502519
\(397\) 18.0000 0.903394 0.451697 − 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) 4.00000 0.200502
\(399\) 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 \(0.351624\pi\)
\(402\) −16.0000 −0.798007
\(403\) −24.0000 −1.19553
\(404\) −10.0000 −0.497519
\(405\) −1.00000 −0.0496904
\(406\) −2.00000 −0.0992583
\(407\) 6.00000 0.297409
\(408\) −2.00000 −0.0990148
\(409\) 10.0000 0.494468 0.247234 − 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) −6.00000 −0.296319
\(411\) −14.0000 −0.690569
\(412\) 16.0000 0.788263
\(413\) −12.0000 −0.590481
\(414\) 8.00000 0.393179
\(415\) 16.0000 0.785409
\(416\) 6.00000 0.294174
\(417\) 20.0000 0.979404
\(418\) 4.00000 0.195646
\(419\) 12.0000 0.586238 0.293119 − 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) −1.00000 −0.0487950
\(421\) 22.0000 1.07221 0.536107 − 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 20.0000 0.973585
\(423\) −8.00000 −0.388973
\(424\) −2.00000 −0.0971286
\(425\) 2.00000 0.0970143
\(426\) 0 0
\(427\) −10.0000 −0.483934
\(428\) −12.0000 −0.580042
\(429\) 6.00000 0.289683
\(430\) −4.00000 −0.192897
\(431\) −12.0000 −0.578020 −0.289010 − 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 1.00000 0.0481125
\(433\) 6.00000 0.288342 0.144171 − 0.989553i \(-0.453949\pi\)
0.144171 + 0.989553i \(0.453949\pi\)
\(434\) −4.00000 −0.192006
\(435\) −2.00000 −0.0958927
\(436\) −6.00000 −0.287348
\(437\) −32.0000 −1.53077
\(438\) −10.0000 −0.477818
\(439\) 24.0000 1.14546 0.572729 − 0.819745i \(-0.305885\pi\)
0.572729 + 0.819745i \(0.305885\pi\)
\(440\) −1.00000 −0.0476731
\(441\) 1.00000 0.0476190
\(442\) 12.0000 0.570782
\(443\) −40.0000 −1.90046 −0.950229 − 0.311553i \(-0.899151\pi\)
−0.950229 + 0.311553i \(0.899151\pi\)
\(444\) −6.00000 −0.284747
\(445\) −10.0000 −0.474045
\(446\) 8.00000 0.378811
\(447\) 18.0000 0.851371
\(448\) 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
\(451\) 6.00000 0.282529
\(452\) −6.00000 −0.282216
\(453\) −20.0000 −0.939682
\(454\) 16.0000 0.750917
\(455\) 6.00000 0.281284
\(456\) −4.00000 −0.187317
\(457\) 22.0000 1.02912 0.514558 − 0.857455i \(-0.327956\pi\)
0.514558 + 0.857455i \(0.327956\pi\)
\(458\) 22.0000 1.02799
\(459\) 2.00000 0.0933520
\(460\) 8.00000 0.373002
\(461\) −42.0000 −1.95614 −0.978068 − 0.208288i \(-0.933211\pi\)
−0.978068 + 0.208288i \(0.933211\pi\)
\(462\) 1.00000 0.0465242
\(463\) 24.0000 1.11537 0.557687 − 0.830051i \(-0.311689\pi\)
0.557687 + 0.830051i \(0.311689\pi\)
\(464\) 2.00000 0.0928477
\(465\) −4.00000 −0.185496
\(466\) −6.00000 −0.277945
\(467\) 12.0000 0.555294 0.277647 − 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) −6.00000 −0.277350
\(469\) 16.0000 0.738811
\(470\) −8.00000 −0.369012
\(471\) −14.0000 −0.645086
\(472\) 12.0000 0.552345
\(473\) 4.00000 0.183920
\(474\) 12.0000 0.551178
\(475\) 4.00000 0.183533
\(476\) 2.00000 0.0916698
\(477\) 2.00000 0.0915737
\(478\) −20.0000 −0.914779
\(479\) 40.0000 1.82765 0.913823 − 0.406112i \(-0.133116\pi\)
0.913823 + 0.406112i \(0.133116\pi\)
\(480\) 1.00000 0.0456435
\(481\) 36.0000 1.64146
\(482\) 14.0000 0.637683
\(483\) −8.00000 −0.364013
\(484\) 1.00000 0.0454545
\(485\) 2.00000 0.0908153
\(486\) −1.00000 −0.0453609
\(487\) 16.0000 0.725029 0.362515 − 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(488\) 10.0000 0.452679
\(489\) 8.00000 0.361773
\(490\) 1.00000 0.0451754
\(491\) 4.00000 0.180517 0.0902587 − 0.995918i \(-0.471231\pi\)
0.0902587 + 0.995918i \(0.471231\pi\)
\(492\) −6.00000 −0.270501
\(493\) 4.00000 0.180151
\(494\) 24.0000 1.07981
\(495\) 1.00000 0.0449467
\(496\) 4.00000 0.179605
\(497\) 0 0
\(498\) 16.0000 0.716977
\(499\) 28.0000 1.25345 0.626726 − 0.779240i \(-0.284395\pi\)
0.626726 + 0.779240i \(0.284395\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −8.00000 −0.357414
\(502\) 20.0000 0.892644
\(503\) 32.0000 1.42681 0.713405 − 0.700752i \(-0.247152\pi\)
0.713405 + 0.700752i \(0.247152\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 10.0000 0.444994
\(506\) −8.00000 −0.355643
\(507\) 23.0000 1.02147
\(508\) −16.0000 −0.709885
\(509\) −30.0000 −1.32973 −0.664863 − 0.746965i \(-0.731510\pi\)
−0.664863 + 0.746965i \(0.731510\pi\)
\(510\) 2.00000 0.0885615
\(511\) 10.0000 0.442374
\(512\) −1.00000 −0.0441942
\(513\) 4.00000 0.176604
\(514\) −30.0000 −1.32324
\(515\) −16.0000 −0.705044
\(516\) −4.00000 −0.176090
\(517\) 8.00000 0.351840
\(518\) 6.00000 0.263625
\(519\) −6.00000 −0.263371
\(520\) −6.00000 −0.263117
\(521\) 26.0000 1.13908 0.569540 − 0.821963i \(-0.307121\pi\)
0.569540 + 0.821963i \(0.307121\pi\)
\(522\) −2.00000 −0.0875376
\(523\) 16.0000 0.699631 0.349816 − 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) 4.00000 0.174741
\(525\) 1.00000 0.0436436
\(526\) 8.00000 0.348817
\(527\) 8.00000 0.348485
\(528\) −1.00000 −0.0435194
\(529\) 41.0000 1.78261
\(530\) 2.00000 0.0868744
\(531\) −12.0000 −0.520756
\(532\) 4.00000 0.173422
\(533\) 36.0000 1.55933
\(534\) −10.0000 −0.432742
\(535\) 12.0000 0.518805
\(536\) −16.0000 −0.691095
\(537\) 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 \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) −32.0000 −1.37452
\(543\) 26.0000 1.11577
\(544\) −2.00000 −0.0857493
\(545\) 6.00000 0.257012
\(546\) 6.00000 0.256776
\(547\) 4.00000 0.171028 0.0855138 − 0.996337i \(-0.472747\pi\)
0.0855138 + 0.996337i \(0.472747\pi\)
\(548\) −14.0000 −0.598050
\(549\) −10.0000 −0.426790
\(550\) 1.00000 0.0426401
\(551\) 8.00000 0.340811
\(552\) 8.00000 0.340503
\(553\) −12.0000 −0.510292
\(554\) −10.0000 −0.424859
\(555\) 6.00000 0.254686
\(556\) 20.0000 0.848189
\(557\) 18.0000 0.762684 0.381342 − 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) −4.00000 −0.169334
\(559\) 24.0000 1.01509
\(560\) −1.00000 −0.0422577
\(561\) −2.00000 −0.0844401
\(562\) −10.0000 −0.421825
\(563\) 24.0000 1.01148 0.505740 − 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) −8.00000 −0.336861
\(565\) 6.00000 0.252422
\(566\) −16.0000 −0.672530
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −6.00000 −0.251533 −0.125767 − 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 4.00000 0.167542
\(571\) 28.0000 1.17176 0.585882 − 0.810397i \(-0.300748\pi\)
0.585882 + 0.810397i \(0.300748\pi\)
\(572\) 6.00000 0.250873
\(573\) 0 0
\(574\) 6.00000 0.250435
\(575\) −8.00000 −0.333623
\(576\) 1.00000 0.0416667
\(577\) 22.0000 0.915872 0.457936 − 0.888985i \(-0.348589\pi\)
0.457936 + 0.888985i \(0.348589\pi\)
\(578\) 13.0000 0.540729
\(579\) −10.0000 −0.415586
\(580\) −2.00000 −0.0830455
\(581\) −16.0000 −0.663792
\(582\) 2.00000 0.0829027
\(583\) −2.00000 −0.0828315
\(584\) −10.0000 −0.413803
\(585\) 6.00000 0.248069
\(586\) 6.00000 0.247858
\(587\) 12.0000 0.495293 0.247647 − 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 1.00000 0.0412393
\(589\) 16.0000 0.659269
\(590\) −12.0000 −0.494032
\(591\) −6.00000 −0.246807
\(592\) −6.00000 −0.246598
\(593\) 18.0000 0.739171 0.369586 − 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) 1.00000 0.0410305
\(595\) −2.00000 −0.0819920
\(596\) 18.0000 0.737309
\(597\) −4.00000 −0.163709
\(598\) −48.0000 −1.96287
\(599\) 32.0000 1.30748 0.653742 − 0.756717i \(-0.273198\pi\)
0.653742 + 0.756717i \(0.273198\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −38.0000 −1.55005 −0.775026 − 0.631929i \(-0.782263\pi\)
−0.775026 + 0.631929i \(0.782263\pi\)
\(602\) 4.00000 0.163028
\(603\) 16.0000 0.651570
\(604\) −20.0000 −0.813788
\(605\) −1.00000 −0.0406558
\(606\) 10.0000 0.406222
\(607\) −16.0000 −0.649420 −0.324710 − 0.945814i \(-0.605267\pi\)
−0.324710 + 0.945814i \(0.605267\pi\)
\(608\) −4.00000 −0.162221
\(609\) 2.00000 0.0810441
\(610\) −10.0000 −0.404888
\(611\) 48.0000 1.94187
\(612\) 2.00000 0.0808452
\(613\) −6.00000 −0.242338 −0.121169 − 0.992632i \(-0.538664\pi\)
−0.121169 + 0.992632i \(0.538664\pi\)
\(614\) 8.00000 0.322854
\(615\) 6.00000 0.241943
\(616\) 1.00000 0.0402911
\(617\) −6.00000 −0.241551 −0.120775 − 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) −16.0000 −0.643614
\(619\) 20.0000 0.803868 0.401934 − 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) −4.00000 −0.160644
\(621\) −8.00000 −0.321029
\(622\) −12.0000 −0.481156
\(623\) 10.0000 0.400642
\(624\) −6.00000 −0.240192
\(625\) 1.00000 0.0400000
\(626\) −6.00000 −0.239808
\(627\) −4.00000 −0.159745
\(628\) −14.0000 −0.558661
\(629\) −12.0000 −0.478471
\(630\) 1.00000 0.0398410
\(631\) −32.0000 −1.27390 −0.636950 − 0.770905i \(-0.719804\pi\)
−0.636950 + 0.770905i \(0.719804\pi\)
\(632\) 12.0000 0.477334
\(633\) −20.0000 −0.794929
\(634\) 6.00000 0.238290
\(635\) 16.0000 0.634941
\(636\) 2.00000 0.0793052
\(637\) −6.00000 −0.237729
\(638\) 2.00000 0.0791808
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) 18.0000 0.710957 0.355479 − 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 12.0000 0.473602
\(643\) −36.0000 −1.41970 −0.709851 − 0.704352i \(-0.751238\pi\)
−0.709851 + 0.704352i \(0.751238\pi\)
\(644\) −8.00000 −0.315244
\(645\) 4.00000 0.157500
\(646\) −8.00000 −0.314756
\(647\) −8.00000 −0.314512 −0.157256 − 0.987558i \(-0.550265\pi\)
−0.157256 + 0.987558i \(0.550265\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 12.0000 0.471041
\(650\) 6.00000 0.235339
\(651\) 4.00000 0.156772
\(652\) 8.00000 0.313304
\(653\) −30.0000 −1.17399 −0.586995 − 0.809590i \(-0.699689\pi\)
−0.586995 + 0.809590i \(0.699689\pi\)
\(654\) 6.00000 0.234619
\(655\) −4.00000 −0.156293
\(656\) −6.00000 −0.234261
\(657\) 10.0000 0.390137
\(658\) 8.00000 0.311872
\(659\) 36.0000 1.40236 0.701180 − 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 1.00000 0.0389249
\(661\) 10.0000 0.388955 0.194477 − 0.980907i \(-0.437699\pi\)
0.194477 + 0.980907i \(0.437699\pi\)
\(662\) −4.00000 −0.155464
\(663\) −12.0000 −0.466041
\(664\) 16.0000 0.620920
\(665\) −4.00000 −0.155113
\(666\) 6.00000 0.232495
\(667\) −16.0000 −0.619522
\(668\) −8.00000 −0.309529
\(669\) −8.00000 −0.309298
\(670\) 16.0000 0.618134
\(671\) 10.0000 0.386046
\(672\) −1.00000 −0.0385758
\(673\) 14.0000 0.539660 0.269830 − 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) 34.0000 1.30963
\(675\) 1.00000 0.0384900
\(676\) 23.0000 0.884615
\(677\) 42.0000 1.61419 0.807096 − 0.590421i \(-0.201038\pi\)
0.807096 + 0.590421i \(0.201038\pi\)
\(678\) 6.00000 0.230429
\(679\) −2.00000 −0.0767530
\(680\) 2.00000 0.0766965
\(681\) −16.0000 −0.613121
\(682\) 4.00000 0.153168
\(683\) −8.00000 −0.306111 −0.153056 − 0.988218i \(-0.548911\pi\)
−0.153056 + 0.988218i \(0.548911\pi\)
\(684\) 4.00000 0.152944
\(685\) 14.0000 0.534913
\(686\) −1.00000 −0.0381802
\(687\) −22.0000 −0.839352
\(688\) −4.00000 −0.152499
\(689\) −12.0000 −0.457164
\(690\) −8.00000 −0.304555
\(691\) 44.0000 1.67384 0.836919 − 0.547326i \(-0.184354\pi\)
0.836919 + 0.547326i \(0.184354\pi\)
\(692\) −6.00000 −0.228086
\(693\) −1.00000 −0.0379869
\(694\) 28.0000 1.06287
\(695\) −20.0000 −0.758643
\(696\) −2.00000 −0.0758098
\(697\) −12.0000 −0.454532
\(698\) 26.0000 0.984115
\(699\) 6.00000 0.226941
\(700\) 1.00000 0.0377964
\(701\) −22.0000 −0.830929 −0.415464 − 0.909610i \(-0.636381\pi\)
−0.415464 + 0.909610i \(0.636381\pi\)
\(702\) 6.00000 0.226455
\(703\) −24.0000 −0.905177
\(704\) −1.00000 −0.0376889
\(705\) 8.00000 0.301297
\(706\) 2.00000 0.0752710
\(707\) −10.0000 −0.376089
\(708\) −12.0000 −0.450988
\(709\) 6.00000 0.225335 0.112667 − 0.993633i \(-0.464061\pi\)
0.112667 + 0.993633i \(0.464061\pi\)
\(710\) 0 0
\(711\) −12.0000 −0.450035
\(712\) −10.0000 −0.374766
\(713\) −32.0000 −1.19841
\(714\) −2.00000 −0.0748481
\(715\) −6.00000 −0.224387
\(716\) 4.00000 0.149487
\(717\) 20.0000 0.746914
\(718\) −12.0000 −0.447836
\(719\) 28.0000 1.04422 0.522112 − 0.852877i \(-0.325144\pi\)
0.522112 + 0.852877i \(0.325144\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 16.0000 0.595871
\(722\) 3.00000 0.111648
\(723\) −14.0000 −0.520666
\(724\) 26.0000 0.966282
\(725\) 2.00000 0.0742781
\(726\) −1.00000 −0.0371135
\(727\) 40.0000 1.48352 0.741759 − 0.670667i \(-0.233992\pi\)
0.741759 + 0.670667i \(0.233992\pi\)
\(728\) 6.00000 0.222375
\(729\) 1.00000 0.0370370
\(730\) 10.0000 0.370117
\(731\) −8.00000 −0.295891
\(732\) −10.0000 −0.369611
\(733\) −14.0000 −0.517102 −0.258551 − 0.965998i \(-0.583245\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) 16.0000 0.590571
\(735\) −1.00000 −0.0368856
\(736\) 8.00000 0.294884
\(737\) −16.0000 −0.589368
\(738\) 6.00000 0.220863
\(739\) −28.0000 −1.03000 −0.514998 − 0.857191i \(-0.672207\pi\)
−0.514998 + 0.857191i \(0.672207\pi\)
\(740\) 6.00000 0.220564
\(741\) −24.0000 −0.881662
\(742\) −2.00000 −0.0734223
\(743\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) −4.00000 −0.146647
\(745\) −18.0000 −0.659469
\(746\) 6.00000 0.219676
\(747\) −16.0000 −0.585409
\(748\) −2.00000 −0.0731272
\(749\) −12.0000 −0.438470
\(750\) 1.00000 0.0365148
\(751\) −48.0000 −1.75154 −0.875772 − 0.482724i \(-0.839647\pi\)
−0.875772 + 0.482724i \(0.839647\pi\)
\(752\) −8.00000 −0.291730
\(753\) −20.0000 −0.728841
\(754\) 12.0000 0.437014
\(755\) 20.0000 0.727875
\(756\) 1.00000 0.0363696
\(757\) −22.0000 −0.799604 −0.399802 − 0.916602i \(-0.630921\pi\)
−0.399802 + 0.916602i \(0.630921\pi\)
\(758\) −4.00000 −0.145287
\(759\) 8.00000 0.290382
\(760\) 4.00000 0.145095
\(761\) 10.0000 0.362500 0.181250 − 0.983437i \(-0.441986\pi\)
0.181250 + 0.983437i \(0.441986\pi\)
\(762\) 16.0000 0.579619
\(763\) −6.00000 −0.217215
\(764\) 0 0
\(765\) −2.00000 −0.0723102
\(766\) −24.0000 −0.867155
\(767\) 72.0000 2.59977
\(768\) 1.00000 0.0360844
\(769\) −14.0000 −0.504853 −0.252426 − 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) −1.00000 −0.0360375
\(771\) 30.0000 1.08042
\(772\) −10.0000 −0.359908
\(773\) 42.0000 1.51064 0.755318 − 0.655359i \(-0.227483\pi\)
0.755318 + 0.655359i \(0.227483\pi\)
\(774\) 4.00000 0.143777
\(775\) 4.00000 0.143684
\(776\) 2.00000 0.0717958
\(777\) −6.00000 −0.215249
\(778\) 18.0000 0.645331
\(779\) −24.0000 −0.859889
\(780\) 6.00000 0.214834
\(781\) 0 0
\(782\) 16.0000 0.572159
\(783\) 2.00000 0.0714742
\(784\) 1.00000 0.0357143
\(785\) 14.0000 0.499681
\(786\) −4.00000 −0.142675
\(787\) −8.00000 −0.285169 −0.142585 − 0.989783i \(-0.545541\pi\)
−0.142585 + 0.989783i \(0.545541\pi\)
\(788\) −6.00000 −0.213741
\(789\) −8.00000 −0.284808
\(790\) −12.0000 −0.426941
\(791\) −6.00000 −0.213335
\(792\) 1.00000 0.0355335
\(793\) 60.0000 2.13066
\(794\) −18.0000 −0.638796
\(795\) −2.00000 −0.0709327
\(796\) −4.00000 −0.141776
\(797\) 50.0000 1.77109 0.885545 − 0.464553i \(-0.153785\pi\)
0.885545 + 0.464553i \(0.153785\pi\)
\(798\) −4.00000 −0.141598
\(799\) −16.0000 −0.566039
\(800\) −1.00000 −0.0353553
\(801\) 10.0000 0.353333
\(802\) −18.0000 −0.635602
\(803\) −10.0000 −0.352892
\(804\) 16.0000 0.564276
\(805\) 8.00000 0.281963
\(806\) 24.0000 0.845364
\(807\) 2.00000 0.0704033
\(808\) 10.0000 0.351799
\(809\) −14.0000 −0.492214 −0.246107 − 0.969243i \(-0.579151\pi\)
−0.246107 + 0.969243i \(0.579151\pi\)
\(810\) 1.00000 0.0351364
\(811\) 28.0000 0.983213 0.491606 − 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) 2.00000 0.0701862
\(813\) 32.0000 1.12229
\(814\) −6.00000 −0.210300
\(815\) −8.00000 −0.280228
\(816\) 2.00000 0.0700140
\(817\) −16.0000 −0.559769
\(818\) −10.0000 −0.349642
\(819\) −6.00000 −0.209657
\(820\) 6.00000 0.209529
\(821\) −30.0000 −1.04701 −0.523504 − 0.852023i \(-0.675375\pi\)
−0.523504 + 0.852023i \(0.675375\pi\)
\(822\) 14.0000 0.488306
\(823\) −32.0000 −1.11545 −0.557725 − 0.830026i \(-0.688326\pi\)
−0.557725 + 0.830026i \(0.688326\pi\)
\(824\) −16.0000 −0.557386
\(825\) −1.00000 −0.0348155
\(826\) 12.0000 0.417533
\(827\) −12.0000 −0.417281 −0.208640 − 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) −8.00000 −0.278019
\(829\) −14.0000 −0.486240 −0.243120 − 0.969996i \(-0.578171\pi\)
−0.243120 + 0.969996i \(0.578171\pi\)
\(830\) −16.0000 −0.555368
\(831\) 10.0000 0.346896
\(832\) −6.00000 −0.208013
\(833\) 2.00000 0.0692959
\(834\) −20.0000 −0.692543
\(835\) 8.00000 0.276851
\(836\) −4.00000 −0.138343
\(837\) 4.00000 0.138260
\(838\) −12.0000 −0.414533
\(839\) 20.0000 0.690477 0.345238 − 0.938515i \(-0.387798\pi\)
0.345238 + 0.938515i \(0.387798\pi\)
\(840\) 1.00000 0.0345033
\(841\) −25.0000 −0.862069
\(842\) −22.0000 −0.758170
\(843\) 10.0000 0.344418
\(844\) −20.0000 −0.688428
\(845\) −23.0000 −0.791224
\(846\) 8.00000 0.275046
\(847\) 1.00000 0.0343604
\(848\) 2.00000 0.0686803
\(849\) 16.0000 0.549119
\(850\) −2.00000 −0.0685994
\(851\) 48.0000 1.64542
\(852\) 0 0
\(853\) 42.0000 1.43805 0.719026 − 0.694983i \(-0.244588\pi\)
0.719026 + 0.694983i \(0.244588\pi\)
\(854\) 10.0000 0.342193
\(855\) −4.00000 −0.136797
\(856\) 12.0000 0.410152
\(857\) 18.0000 0.614868 0.307434 − 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) −6.00000 −0.204837
\(859\) −52.0000 −1.77422 −0.887109 − 0.461561i \(-0.847290\pi\)
−0.887109 + 0.461561i \(0.847290\pi\)
\(860\) 4.00000 0.136399
\(861\) −6.00000 −0.204479
\(862\) 12.0000 0.408722
\(863\) 8.00000 0.272323 0.136162 − 0.990687i \(-0.456523\pi\)
0.136162 + 0.990687i \(0.456523\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 6.00000 0.204006
\(866\) −6.00000 −0.203888
\(867\) −13.0000 −0.441503
\(868\) 4.00000 0.135769
\(869\) 12.0000 0.407072
\(870\) 2.00000 0.0678064
\(871\) −96.0000 −3.25284
\(872\) 6.00000 0.203186
\(873\) −2.00000 −0.0676897
\(874\) 32.0000 1.08242
\(875\) −1.00000 −0.0338062
\(876\) 10.0000 0.337869
\(877\) −14.0000 −0.472746 −0.236373 − 0.971662i \(-0.575959\pi\)
−0.236373 + 0.971662i \(0.575959\pi\)
\(878\) −24.0000 −0.809961
\(879\) −6.00000 −0.202375
\(880\) 1.00000 0.0337100
\(881\) −14.0000 −0.471672 −0.235836 − 0.971793i \(-0.575783\pi\)
−0.235836 + 0.971793i \(0.575783\pi\)
\(882\) −1.00000 −0.0336718
\(883\) −48.0000 −1.61533 −0.807664 − 0.589643i \(-0.799269\pi\)
−0.807664 + 0.589643i \(0.799269\pi\)
\(884\) −12.0000 −0.403604
\(885\) 12.0000 0.403376
\(886\) 40.0000 1.34383
\(887\) −48.0000 −1.61168 −0.805841 − 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) 6.00000 0.201347
\(889\) −16.0000 −0.536623
\(890\) 10.0000 0.335201
\(891\) −1.00000 −0.0335013
\(892\) −8.00000 −0.267860
\(893\) −32.0000 −1.07084
\(894\) −18.0000 −0.602010
\(895\) −4.00000 −0.133705
\(896\) −1.00000 −0.0334077
\(897\) 48.0000 1.60267
\(898\) −18.0000 −0.600668
\(899\) 8.00000 0.266815
\(900\) 1.00000 0.0333333
\(901\) 4.00000 0.133259
\(902\) −6.00000 −0.199778
\(903\) −4.00000 −0.133112
\(904\) 6.00000 0.199557
\(905\) −26.0000 −0.864269
\(906\) 20.0000 0.664455
\(907\) −32.0000 −1.06254 −0.531271 − 0.847202i \(-0.678286\pi\)
−0.531271 + 0.847202i \(0.678286\pi\)
\(908\) −16.0000 −0.530979
\(909\) −10.0000 −0.331679
\(910\) −6.00000 −0.198898
\(911\) 40.0000 1.32526 0.662630 − 0.748947i \(-0.269440\pi\)
0.662630 + 0.748947i \(0.269440\pi\)
\(912\) 4.00000 0.132453
\(913\) 16.0000 0.529523
\(914\) −22.0000 −0.727695
\(915\) 10.0000 0.330590
\(916\) −22.0000 −0.726900
\(917\) 4.00000 0.132092
\(918\) −2.00000 −0.0660098
\(919\) 28.0000 0.923635 0.461817 − 0.886975i \(-0.347198\pi\)
0.461817 + 0.886975i \(0.347198\pi\)
\(920\) −8.00000 −0.263752
\(921\) −8.00000 −0.263609
\(922\) 42.0000 1.38320
\(923\) 0 0
\(924\) −1.00000 −0.0328976
\(925\) −6.00000 −0.197279
\(926\) −24.0000 −0.788689
\(927\) 16.0000 0.525509
\(928\) −2.00000 −0.0656532
\(929\) 18.0000 0.590561 0.295280 − 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) 4.00000 0.131165
\(931\) 4.00000 0.131095
\(932\) 6.00000 0.196537
\(933\) 12.0000 0.392862
\(934\) −12.0000 −0.392652
\(935\) 2.00000 0.0654070
\(936\) 6.00000 0.196116
\(937\) −38.0000 −1.24141 −0.620703 − 0.784046i \(-0.713153\pi\)
−0.620703 + 0.784046i \(0.713153\pi\)
\(938\) −16.0000 −0.522419
\(939\) 6.00000 0.195803
\(940\) 8.00000 0.260931
\(941\) −42.0000 −1.36916 −0.684580 − 0.728937i \(-0.740015\pi\)
−0.684580 + 0.728937i \(0.740015\pi\)
\(942\) 14.0000 0.456145
\(943\) 48.0000 1.56310
\(944\) −12.0000 −0.390567
\(945\) −1.00000 −0.0325300
\(946\) −4.00000 −0.130051
\(947\) 16.0000 0.519930 0.259965 − 0.965618i \(-0.416289\pi\)
0.259965 + 0.965618i \(0.416289\pi\)
\(948\) −12.0000 −0.389742
\(949\) −60.0000 −1.94768
\(950\) −4.00000 −0.129777
\(951\) −6.00000 −0.194563
\(952\) −2.00000 −0.0648204
\(953\) −50.0000 −1.61966 −0.809829 − 0.586665i \(-0.800440\pi\)
−0.809829 + 0.586665i \(0.800440\pi\)
\(954\) −2.00000 −0.0647524
\(955\) 0 0
\(956\) 20.0000 0.646846
\(957\) −2.00000 −0.0646508
\(958\) −40.0000 −1.29234
\(959\) −14.0000 −0.452084
\(960\) −1.00000 −0.0322749
\(961\) −15.0000 −0.483871
\(962\) −36.0000 −1.16069
\(963\) −12.0000 −0.386695
\(964\) −14.0000 −0.450910
\(965\) 10.0000 0.321911
\(966\) 8.00000 0.257396
\(967\) 32.0000 1.02905 0.514525 − 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 8.00000 0.256997
\(970\) −2.00000 −0.0642161
\(971\) −4.00000 −0.128366 −0.0641831 − 0.997938i \(-0.520444\pi\)
−0.0641831 + 0.997938i \(0.520444\pi\)
\(972\) 1.00000 0.0320750
\(973\) 20.0000 0.641171
\(974\) −16.0000 −0.512673
\(975\) −6.00000 −0.192154
\(976\) −10.0000 −0.320092
\(977\) −38.0000 −1.21573 −0.607864 − 0.794041i \(-0.707973\pi\)
−0.607864 + 0.794041i \(0.707973\pi\)
\(978\) −8.00000 −0.255812
\(979\) −10.0000 −0.319601
\(980\) −1.00000 −0.0319438
\(981\) −6.00000 −0.191565
\(982\) −4.00000 −0.127645
\(983\) 56.0000 1.78612 0.893061 − 0.449935i \(-0.148553\pi\)
0.893061 + 0.449935i \(0.148553\pi\)
\(984\) 6.00000 0.191273
\(985\) 6.00000 0.191176
\(986\) −4.00000 −0.127386
\(987\) −8.00000 −0.254643
\(988\) −24.0000 −0.763542
\(989\) 32.0000 1.01754
\(990\) −1.00000 −0.0317821
\(991\) −8.00000 −0.254128 −0.127064 − 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) −4.00000 −0.127000
\(993\) 4.00000 0.126936
\(994\) 0 0
\(995\) 4.00000 0.126809
\(996\) −16.0000 −0.506979
\(997\) 2.00000 0.0633406 0.0316703 − 0.999498i \(-0.489917\pi\)
0.0316703 + 0.999498i \(0.489917\pi\)
\(998\) −28.0000 −0.886325
\(999\) −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 2310.2.a.g.1.1 ✓ 1
3.2 odd 2 6930.2.a.bi.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2310.2.a.g.1.1 ✓ 1 1.1 even 1 trivial
6930.2.a.bi.1.1 1 3.2 odd 2