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Properties

Label 2310.2.a.m.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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{11} -1.00000 q^{12} +2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} -1.00000 q^{20} +1.00000 q^{21} -1.00000 q^{22} -4.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} -1.00000 q^{28} -6.00000 q^{29} +1.00000 q^{30} +1.00000 q^{32} +1.00000 q^{33} +2.00000 q^{34} +1.00000 q^{35} +1.00000 q^{36} +2.00000 q^{37} +4.00000 q^{38} -2.00000 q^{39} -1.00000 q^{40} +6.00000 q^{41} +1.00000 q^{42} +12.0000 q^{43} -1.00000 q^{44} -1.00000 q^{45} -4.00000 q^{46} +8.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -2.00000 q^{51} +2.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} +1.00000 q^{55} -1.00000 q^{56} -4.00000 q^{57} -6.00000 q^{58} +8.00000 q^{59} +1.00000 q^{60} -14.0000 q^{61} -1.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} +1.00000 q^{66} +12.0000 q^{67} +2.00000 q^{68} +4.00000 q^{69} +1.00000 q^{70} +8.00000 q^{71} +1.00000 q^{72} +10.0000 q^{73} +2.00000 q^{74} -1.00000 q^{75} +4.00000 q^{76} +1.00000 q^{77} -2.00000 q^{78} +4.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} +16.0000 q^{83} +1.00000 q^{84} -2.00000 q^{85} +12.0000 q^{86} +6.00000 q^{87} -1.00000 q^{88} +10.0000 q^{89} -1.00000 q^{90} -2.00000 q^{91} -4.00000 q^{92} +8.00000 q^{94} -4.00000 q^{95} -1.00000 q^{96} +6.00000 q^{97} +1.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −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
\(23\) −4.00000 −0.834058 −0.417029 − 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) −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 \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 1.00000 0.182574
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 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 \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 4.00000 0.648886
\(39\) −2.00000 −0.320256
\(40\) −1.00000 −0.158114
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 1.00000 0.154303
\(43\) 12.0000 1.82998 0.914991 − 0.403473i \(-0.132197\pi\)
0.914991 + 0.403473i \(0.132197\pi\)
\(44\) −1.00000 −0.150756
\(45\) −1.00000 −0.149071
\(46\) −4.00000 −0.589768
\(47\) 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
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) −1.00000 −0.136083
\(55\) 1.00000 0.134840
\(56\) −1.00000 −0.133631
\(57\) −4.00000 −0.529813
\(58\) −6.00000 −0.787839
\(59\) 8.00000 1.04151 0.520756 − 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 1.00000 0.129099
\(61\) −14.0000 −1.79252 −0.896258 − 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) 1.00000 0.123091
\(67\) 12.0000 1.46603 0.733017 − 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) 2.00000 0.242536
\(69\) 4.00000 0.481543
\(70\) 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
\(73\) 10.0000 1.17041 0.585206 − 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 2.00000 0.232495
\(75\) −1.00000 −0.115470
\(76\) 4.00000 0.458831
\(77\) 1.00000 0.113961
\(78\) −2.00000 −0.226455
\(79\) 4.00000 0.450035 0.225018 − 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) 16.0000 1.75623 0.878114 − 0.478451i \(-0.158802\pi\)
0.878114 + 0.478451i \(0.158802\pi\)
\(84\) 1.00000 0.109109
\(85\) −2.00000 −0.216930
\(86\) 12.0000 1.29399
\(87\) 6.00000 0.643268
\(88\) −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
\(91\) −2.00000 −0.209657
\(92\) −4.00000 −0.417029
\(93\) 0 0
\(94\) 8.00000 0.825137
\(95\) −4.00000 −0.410391
\(96\) −1.00000 −0.102062
\(97\) 6.00000 0.609208 0.304604 − 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 1.00000 0.101015
\(99\) −1.00000 −0.100504
\(100\) 1.00000 0.100000
\(101\) −6.00000 −0.597022 −0.298511 − 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) −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
\(106\) −6.00000 −0.582772
\(107\) −12.0000 −1.16008 −0.580042 − 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 2.00000 0.191565 0.0957826 − 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 1.00000 0.0953463
\(111\) −2.00000 −0.189832
\(112\) −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
\(115\) 4.00000 0.373002
\(116\) −6.00000 −0.557086
\(117\) 2.00000 0.184900
\(118\) 8.00000 0.736460
\(119\) −2.00000 −0.183340
\(120\) 1.00000 0.0912871
\(121\) 1.00000 0.0909091
\(122\) −14.0000 −1.26750
\(123\) −6.00000 −0.541002
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) −1.00000 −0.0890871
\(127\) −8.00000 −0.709885 −0.354943 − 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 1.00000 0.0883883
\(129\) −12.0000 −1.05654
\(130\) −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
\(134\) 12.0000 1.03664
\(135\) 1.00000 0.0860663
\(136\) 2.00000 0.171499
\(137\) 2.00000 0.170872 0.0854358 − 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) 4.00000 0.340503
\(139\) −4.00000 −0.339276 −0.169638 − 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 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
\(146\) 10.0000 0.827606
\(147\) −1.00000 −0.0824786
\(148\) 2.00000 0.164399
\(149\) −6.00000 −0.491539 −0.245770 − 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −12.0000 −0.976546 −0.488273 − 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 4.00000 0.324443
\(153\) 2.00000 0.161690
\(154\) 1.00000 0.0805823
\(155\) 0 0
\(156\) −2.00000 −0.160128
\(157\) 2.00000 0.159617 0.0798087 − 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 4.00000 0.318223
\(159\) 6.00000 0.475831
\(160\) −1.00000 −0.0790569
\(161\) 4.00000 0.315244
\(162\) 1.00000 0.0785674
\(163\) 4.00000 0.313304 0.156652 − 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 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
\(169\) −9.00000 −0.692308
\(170\) −2.00000 −0.153393
\(171\) 4.00000 0.305888
\(172\) 12.0000 0.914991
\(173\) −6.00000 −0.456172 −0.228086 − 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 6.00000 0.454859
\(175\) −1.00000 −0.0755929
\(176\) −1.00000 −0.0753778
\(177\) −8.00000 −0.601317
\(178\) 10.0000 0.749532
\(179\) −20.0000 −1.49487 −0.747435 − 0.664335i \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 26.0000 1.93256 0.966282 − 0.257485i \(-0.0828937\pi\)
0.966282 + 0.257485i \(0.0828937\pi\)
\(182\) −2.00000 −0.148250
\(183\) 14.0000 1.03491
\(184\) −4.00000 −0.294884
\(185\) −2.00000 −0.147043
\(186\) 0 0
\(187\) −2.00000 −0.146254
\(188\) 8.00000 0.583460
\(189\) 1.00000 0.0727393
\(190\) −4.00000 −0.290191
\(191\) −24.0000 −1.73658 −0.868290 − 0.496058i \(-0.834780\pi\)
−0.868290 + 0.496058i \(0.834780\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 2.00000 0.143963 0.0719816 − 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 6.00000 0.430775
\(195\) 2.00000 0.143223
\(196\) 1.00000 0.0714286
\(197\) 6.00000 0.427482 0.213741 − 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) −1.00000 −0.0710669
\(199\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 1.00000 0.0707107
\(201\) −12.0000 −0.846415
\(202\) −6.00000 −0.422159
\(203\) 6.00000 0.421117
\(204\) −2.00000 −0.140028
\(205\) −6.00000 −0.419058
\(206\) 8.00000 0.557386
\(207\) −4.00000 −0.278019
\(208\) 2.00000 0.138675
\(209\) −4.00000 −0.276686
\(210\) −1.00000 −0.0690066
\(211\) 12.0000 0.826114 0.413057 − 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) −6.00000 −0.412082
\(213\) −8.00000 −0.548151
\(214\) −12.0000 −0.820303
\(215\) −12.0000 −0.818393
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 2.00000 0.135457
\(219\) −10.0000 −0.675737
\(220\) 1.00000 0.0674200
\(221\) 4.00000 0.269069
\(222\) −2.00000 −0.134231
\(223\) 24.0000 1.60716 0.803579 − 0.595198i \(-0.202926\pi\)
0.803579 + 0.595198i \(0.202926\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 1.00000 0.0666667
\(226\) −6.00000 −0.399114
\(227\) −8.00000 −0.530979 −0.265489 − 0.964114i \(-0.585534\pi\)
−0.265489 + 0.964114i \(0.585534\pi\)
\(228\) −4.00000 −0.264906
\(229\) 10.0000 0.660819 0.330409 − 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 4.00000 0.263752
\(231\) −1.00000 −0.0657952
\(232\) −6.00000 −0.393919
\(233\) 26.0000 1.70332 0.851658 − 0.524097i \(-0.175597\pi\)
0.851658 + 0.524097i \(0.175597\pi\)
\(234\) 2.00000 0.130744
\(235\) −8.00000 −0.521862
\(236\) 8.00000 0.520756
\(237\) −4.00000 −0.259828
\(238\) −2.00000 −0.129641
\(239\) 12.0000 0.776215 0.388108 − 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 1.00000 0.0645497
\(241\) −18.0000 −1.15948 −0.579741 − 0.814801i \(-0.696846\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(242\) 1.00000 0.0642824
\(243\) −1.00000 −0.0641500
\(244\) −14.0000 −0.896258
\(245\) −1.00000 −0.0638877
\(246\) −6.00000 −0.382546
\(247\) 8.00000 0.509028
\(248\) 0 0
\(249\) −16.0000 −1.01396
\(250\) −1.00000 −0.0632456
\(251\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 4.00000 0.251478
\(254\) −8.00000 −0.501965
\(255\) 2.00000 0.125245
\(256\) 1.00000 0.0625000
\(257\) −18.0000 −1.12281 −0.561405 − 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) −12.0000 −0.747087
\(259\) −2.00000 −0.124274
\(260\) −2.00000 −0.124035
\(261\) −6.00000 −0.371391
\(262\) 12.0000 0.741362
\(263\) 24.0000 1.47990 0.739952 − 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 1.00000 0.0615457
\(265\) 6.00000 0.368577
\(266\) −4.00000 −0.245256
\(267\) −10.0000 −0.611990
\(268\) 12.0000 0.733017
\(269\) −6.00000 −0.365826 −0.182913 − 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 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
\(274\) 2.00000 0.120824
\(275\) −1.00000 −0.0603023
\(276\) 4.00000 0.240772
\(277\) −18.0000 −1.08152 −0.540758 − 0.841178i \(-0.681862\pi\)
−0.540758 + 0.841178i \(0.681862\pi\)
\(278\) −4.00000 −0.239904
\(279\) 0 0
\(280\) 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
\(283\) 24.0000 1.42665 0.713326 − 0.700832i \(-0.247188\pi\)
0.713326 + 0.700832i \(0.247188\pi\)
\(284\) 8.00000 0.474713
\(285\) 4.00000 0.236940
\(286\) −2.00000 −0.118262
\(287\) −6.00000 −0.354169
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) 6.00000 0.352332
\(291\) −6.00000 −0.351726
\(292\) 10.0000 0.585206
\(293\) −22.0000 −1.28525 −0.642627 − 0.766179i \(-0.722155\pi\)
−0.642627 + 0.766179i \(0.722155\pi\)
\(294\) −1.00000 −0.0583212
\(295\) −8.00000 −0.465778
\(296\) 2.00000 0.116248
\(297\) 1.00000 0.0580259
\(298\) −6.00000 −0.347571
\(299\) −8.00000 −0.462652
\(300\) −1.00000 −0.0577350
\(301\) −12.0000 −0.691669
\(302\) −12.0000 −0.690522
\(303\) 6.00000 0.344691
\(304\) 4.00000 0.229416
\(305\) 14.0000 0.801638
\(306\) 2.00000 0.114332
\(307\) 8.00000 0.456584 0.228292 − 0.973593i \(-0.426686\pi\)
0.228292 + 0.973593i \(0.426686\pi\)
\(308\) 1.00000 0.0569803
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 − 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) −2.00000 −0.113228
\(313\) −18.0000 −1.01742 −0.508710 − 0.860938i \(-0.669877\pi\)
−0.508710 + 0.860938i \(0.669877\pi\)
\(314\) 2.00000 0.112867
\(315\) 1.00000 0.0563436
\(316\) 4.00000 0.225018
\(317\) −22.0000 −1.23564 −0.617822 − 0.786318i \(-0.711985\pi\)
−0.617822 + 0.786318i \(0.711985\pi\)
\(318\) 6.00000 0.336463
\(319\) 6.00000 0.335936
\(320\) −1.00000 −0.0559017
\(321\) 12.0000 0.669775
\(322\) 4.00000 0.222911
\(323\) 8.00000 0.445132
\(324\) 1.00000 0.0555556
\(325\) 2.00000 0.110940
\(326\) 4.00000 0.221540
\(327\) −2.00000 −0.110600
\(328\) 6.00000 0.331295
\(329\) −8.00000 −0.441054
\(330\) −1.00000 −0.0550482
\(331\) −4.00000 −0.219860 −0.109930 − 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 16.0000 0.878114
\(333\) 2.00000 0.109599
\(334\) −8.00000 −0.437741
\(335\) −12.0000 −0.655630
\(336\) 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
\(339\) 6.00000 0.325875
\(340\) −2.00000 −0.108465
\(341\) 0 0
\(342\) 4.00000 0.216295
\(343\) −1.00000 −0.0539949
\(344\) 12.0000 0.646997
\(345\) −4.00000 −0.215353
\(346\) −6.00000 −0.322562
\(347\) −20.0000 −1.07366 −0.536828 − 0.843692i \(-0.680378\pi\)
−0.536828 + 0.843692i \(0.680378\pi\)
\(348\) 6.00000 0.321634
\(349\) 2.00000 0.107058 0.0535288 − 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) −1.00000 −0.0534522
\(351\) −2.00000 −0.106752
\(352\) −1.00000 −0.0533002
\(353\) 30.0000 1.59674 0.798369 − 0.602168i \(-0.205696\pi\)
0.798369 + 0.602168i \(0.205696\pi\)
\(354\) −8.00000 −0.425195
\(355\) −8.00000 −0.424596
\(356\) 10.0000 0.529999
\(357\) 2.00000 0.105851
\(358\) −20.0000 −1.05703
\(359\) −4.00000 −0.211112 −0.105556 − 0.994413i \(-0.533662\pi\)
−0.105556 + 0.994413i \(0.533662\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −3.00000 −0.157895
\(362\) 26.0000 1.36653
\(363\) −1.00000 −0.0524864
\(364\) −2.00000 −0.104828
\(365\) −10.0000 −0.523424
\(366\) 14.0000 0.731792
\(367\) −8.00000 −0.417597 −0.208798 − 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) −4.00000 −0.208514
\(369\) 6.00000 0.312348
\(370\) −2.00000 −0.103975
\(371\) 6.00000 0.311504
\(372\) 0 0
\(373\) 6.00000 0.310668 0.155334 − 0.987862i \(-0.450355\pi\)
0.155334 + 0.987862i \(0.450355\pi\)
\(374\) −2.00000 −0.103418
\(375\) 1.00000 0.0516398
\(376\) 8.00000 0.412568
\(377\) −12.0000 −0.618031
\(378\) 1.00000 0.0514344
\(379\) 20.0000 1.02733 0.513665 − 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) −4.00000 −0.205196
\(381\) 8.00000 0.409852
\(382\) −24.0000 −1.22795
\(383\) 32.0000 1.63512 0.817562 − 0.575841i \(-0.195325\pi\)
0.817562 + 0.575841i \(0.195325\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −1.00000 −0.0509647
\(386\) 2.00000 0.101797
\(387\) 12.0000 0.609994
\(388\) 6.00000 0.304604
\(389\) −10.0000 −0.507020 −0.253510 − 0.967333i \(-0.581585\pi\)
−0.253510 + 0.967333i \(0.581585\pi\)
\(390\) 2.00000 0.101274
\(391\) −8.00000 −0.404577
\(392\) 1.00000 0.0505076
\(393\) −12.0000 −0.605320
\(394\) 6.00000 0.302276
\(395\) −4.00000 −0.201262
\(396\) −1.00000 −0.0502519
\(397\) −14.0000 −0.702640 −0.351320 − 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) 0 0
\(399\) 4.00000 0.200250
\(400\) 1.00000 0.0500000
\(401\) 2.00000 0.0998752 0.0499376 − 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) −12.0000 −0.598506
\(403\) 0 0
\(404\) −6.00000 −0.298511
\(405\) −1.00000 −0.0496904
\(406\) 6.00000 0.297775
\(407\) −2.00000 −0.0991363
\(408\) −2.00000 −0.0990148
\(409\) −10.0000 −0.494468 −0.247234 − 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) −6.00000 −0.296319
\(411\) −2.00000 −0.0986527
\(412\) 8.00000 0.394132
\(413\) −8.00000 −0.393654
\(414\) −4.00000 −0.196589
\(415\) −16.0000 −0.785409
\(416\) 2.00000 0.0980581
\(417\) 4.00000 0.195881
\(418\) −4.00000 −0.195646
\(419\) −40.0000 −1.95413 −0.977064 − 0.212946i \(-0.931694\pi\)
−0.977064 + 0.212946i \(0.931694\pi\)
\(420\) −1.00000 −0.0487950
\(421\) 22.0000 1.07221 0.536107 − 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 12.0000 0.584151
\(423\) 8.00000 0.388973
\(424\) −6.00000 −0.291386
\(425\) 2.00000 0.0970143
\(426\) −8.00000 −0.387601
\(427\) 14.0000 0.677507
\(428\) −12.0000 −0.580042
\(429\) 2.00000 0.0965609
\(430\) −12.0000 −0.578691
\(431\) 36.0000 1.73406 0.867029 − 0.498257i \(-0.166026\pi\)
0.867029 + 0.498257i \(0.166026\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −2.00000 −0.0961139 −0.0480569 − 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) −6.00000 −0.287678
\(436\) 2.00000 0.0957826
\(437\) −16.0000 −0.765384
\(438\) −10.0000 −0.477818
\(439\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 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
\(445\) −10.0000 −0.474045
\(446\) 24.0000 1.13643
\(447\) 6.00000 0.283790
\(448\) −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
\(453\) 12.0000 0.563809
\(454\) −8.00000 −0.375459
\(455\) 2.00000 0.0937614
\(456\) −4.00000 −0.187317
\(457\) 10.0000 0.467780 0.233890 − 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) 10.0000 0.467269
\(459\) −2.00000 −0.0933520
\(460\) 4.00000 0.186501
\(461\) 18.0000 0.838344 0.419172 − 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) −1.00000 −0.0465242
\(463\) 36.0000 1.67306 0.836531 − 0.547920i \(-0.184580\pi\)
0.836531 + 0.547920i \(0.184580\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 26.0000 1.20443
\(467\) −20.0000 −0.925490 −0.462745 − 0.886492i \(-0.653135\pi\)
−0.462745 + 0.886492i \(0.653135\pi\)
\(468\) 2.00000 0.0924500
\(469\) −12.0000 −0.554109
\(470\) −8.00000 −0.369012
\(471\) −2.00000 −0.0921551
\(472\) 8.00000 0.368230
\(473\) −12.0000 −0.551761
\(474\) −4.00000 −0.183726
\(475\) 4.00000 0.183533
\(476\) −2.00000 −0.0916698
\(477\) −6.00000 −0.274721
\(478\) 12.0000 0.548867
\(479\) −24.0000 −1.09659 −0.548294 − 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 1.00000 0.0456435
\(481\) 4.00000 0.182384
\(482\) −18.0000 −0.819878
\(483\) −4.00000 −0.182006
\(484\) 1.00000 0.0454545
\(485\) −6.00000 −0.272446
\(486\) −1.00000 −0.0453609
\(487\) 4.00000 0.181257 0.0906287 − 0.995885i \(-0.471112\pi\)
0.0906287 + 0.995885i \(0.471112\pi\)
\(488\) −14.0000 −0.633750
\(489\) −4.00000 −0.180886
\(490\) −1.00000 −0.0451754
\(491\) 12.0000 0.541552 0.270776 − 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) −6.00000 −0.270501
\(493\) −12.0000 −0.540453
\(494\) 8.00000 0.359937
\(495\) 1.00000 0.0449467
\(496\) 0 0
\(497\) −8.00000 −0.358849
\(498\) −16.0000 −0.716977
\(499\) −36.0000 −1.61158 −0.805791 − 0.592200i \(-0.798259\pi\)
−0.805791 + 0.592200i \(0.798259\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 8.00000 0.357414
\(502\) 0 0
\(503\) 16.0000 0.713405 0.356702 − 0.934218i \(-0.383901\pi\)
0.356702 + 0.934218i \(0.383901\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 6.00000 0.266996
\(506\) 4.00000 0.177822
\(507\) 9.00000 0.399704
\(508\) −8.00000 −0.354943
\(509\) 10.0000 0.443242 0.221621 − 0.975133i \(-0.428865\pi\)
0.221621 + 0.975133i \(0.428865\pi\)
\(510\) 2.00000 0.0885615
\(511\) −10.0000 −0.442374
\(512\) 1.00000 0.0441942
\(513\) −4.00000 −0.176604
\(514\) −18.0000 −0.793946
\(515\) −8.00000 −0.352522
\(516\) −12.0000 −0.528271
\(517\) −8.00000 −0.351840
\(518\) −2.00000 −0.0878750
\(519\) 6.00000 0.263371
\(520\) −2.00000 −0.0877058
\(521\) −30.0000 −1.31432 −0.657162 − 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) −6.00000 −0.262613
\(523\) −16.0000 −0.699631 −0.349816 − 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) 12.0000 0.524222
\(525\) 1.00000 0.0436436
\(526\) 24.0000 1.04645
\(527\) 0 0
\(528\) 1.00000 0.0435194
\(529\) −7.00000 −0.304348
\(530\) 6.00000 0.260623
\(531\) 8.00000 0.347170
\(532\) −4.00000 −0.173422
\(533\) 12.0000 0.519778
\(534\) −10.0000 −0.432742
\(535\) 12.0000 0.518805
\(536\) 12.0000 0.518321
\(537\) 20.0000 0.863064
\(538\) −6.00000 −0.258678
\(539\) −1.00000 −0.0430730
\(540\) 1.00000 0.0430331
\(541\) −46.0000 −1.97769 −0.988847 − 0.148933i \(-0.952416\pi\)
−0.988847 + 0.148933i \(0.952416\pi\)
\(542\) −8.00000 −0.343629
\(543\) −26.0000 −1.11577
\(544\) 2.00000 0.0857493
\(545\) −2.00000 −0.0856706
\(546\) 2.00000 0.0855921
\(547\) 28.0000 1.19719 0.598597 − 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) 2.00000 0.0854358
\(549\) −14.0000 −0.597505
\(550\) −1.00000 −0.0426401
\(551\) −24.0000 −1.02243
\(552\) 4.00000 0.170251
\(553\) −4.00000 −0.170097
\(554\) −18.0000 −0.764747
\(555\) 2.00000 0.0848953
\(556\) −4.00000 −0.169638
\(557\) −18.0000 −0.762684 −0.381342 − 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 0 0
\(559\) 24.0000 1.01509
\(560\) 1.00000 0.0422577
\(561\) 2.00000 0.0844401
\(562\) −22.0000 −0.928014
\(563\) −16.0000 −0.674320 −0.337160 − 0.941447i \(-0.609466\pi\)
−0.337160 + 0.941447i \(0.609466\pi\)
\(564\) −8.00000 −0.336861
\(565\) 6.00000 0.252422
\(566\) 24.0000 1.00880
\(567\) −1.00000 −0.0419961
\(568\) 8.00000 0.335673
\(569\) 2.00000 0.0838444 0.0419222 − 0.999121i \(-0.486652\pi\)
0.0419222 + 0.999121i \(0.486652\pi\)
\(570\) 4.00000 0.167542
\(571\) 20.0000 0.836974 0.418487 − 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) −2.00000 −0.0836242
\(573\) 24.0000 1.00261
\(574\) −6.00000 −0.250435
\(575\) −4.00000 −0.166812
\(576\) 1.00000 0.0416667
\(577\) −2.00000 −0.0832611 −0.0416305 − 0.999133i \(-0.513255\pi\)
−0.0416305 + 0.999133i \(0.513255\pi\)
\(578\) −13.0000 −0.540729
\(579\) −2.00000 −0.0831172
\(580\) 6.00000 0.249136
\(581\) −16.0000 −0.663792
\(582\) −6.00000 −0.248708
\(583\) 6.00000 0.248495
\(584\) 10.0000 0.413803
\(585\) −2.00000 −0.0826898
\(586\) −22.0000 −0.908812
\(587\) 12.0000 0.495293 0.247647 − 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) −1.00000 −0.0412393
\(589\) 0 0
\(590\) −8.00000 −0.329355
\(591\) −6.00000 −0.246807
\(592\) 2.00000 0.0821995
\(593\) 26.0000 1.06769 0.533846 − 0.845582i \(-0.320746\pi\)
0.533846 + 0.845582i \(0.320746\pi\)
\(594\) 1.00000 0.0410305
\(595\) 2.00000 0.0819920
\(596\) −6.00000 −0.245770
\(597\) 0 0
\(598\) −8.00000 −0.327144
\(599\) 24.0000 0.980613 0.490307 − 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −26.0000 −1.06056 −0.530281 − 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) −12.0000 −0.489083
\(603\) 12.0000 0.488678
\(604\) −12.0000 −0.488273
\(605\) −1.00000 −0.0406558
\(606\) 6.00000 0.243733
\(607\) 8.00000 0.324710 0.162355 − 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) 4.00000 0.162221
\(609\) −6.00000 −0.243132
\(610\) 14.0000 0.566843
\(611\) 16.0000 0.647291
\(612\) 2.00000 0.0808452
\(613\) −2.00000 −0.0807792 −0.0403896 − 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) 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 \(0.538538\pi\)
\(618\) −8.00000 −0.321807
\(619\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) −24.0000 −0.962312
\(623\) −10.0000 −0.400642
\(624\) −2.00000 −0.0800641
\(625\) 1.00000 0.0400000
\(626\) −18.0000 −0.719425
\(627\) 4.00000 0.159745
\(628\) 2.00000 0.0798087
\(629\) 4.00000 0.159490
\(630\) 1.00000 0.0398410
\(631\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 4.00000 0.159111
\(633\) −12.0000 −0.476957
\(634\) −22.0000 −0.873732
\(635\) 8.00000 0.317470
\(636\) 6.00000 0.237915
\(637\) 2.00000 0.0792429
\(638\) 6.00000 0.237542
\(639\) 8.00000 0.316475
\(640\) −1.00000 −0.0395285
\(641\) 2.00000 0.0789953 0.0394976 − 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) 12.0000 0.473602
\(643\) 44.0000 1.73519 0.867595 − 0.497271i \(-0.165665\pi\)
0.867595 + 0.497271i \(0.165665\pi\)
\(644\) 4.00000 0.157622
\(645\) 12.0000 0.472500
\(646\) 8.00000 0.314756
\(647\) −24.0000 −0.943537 −0.471769 − 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 1.00000 0.0392837
\(649\) −8.00000 −0.314027
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) 4.00000 0.156652
\(653\) −22.0000 −0.860927 −0.430463 − 0.902608i \(-0.641650\pi\)
−0.430463 + 0.902608i \(0.641650\pi\)
\(654\) −2.00000 −0.0782062
\(655\) −12.0000 −0.468879
\(656\) 6.00000 0.234261
\(657\) 10.0000 0.390137
\(658\) −8.00000 −0.311872
\(659\) −44.0000 −1.71400 −0.856998 − 0.515319i \(-0.827673\pi\)
−0.856998 + 0.515319i \(0.827673\pi\)
\(660\) −1.00000 −0.0389249
\(661\) 2.00000 0.0777910 0.0388955 − 0.999243i \(-0.487616\pi\)
0.0388955 + 0.999243i \(0.487616\pi\)
\(662\) −4.00000 −0.155464
\(663\) −4.00000 −0.155347
\(664\) 16.0000 0.620920
\(665\) 4.00000 0.155113
\(666\) 2.00000 0.0774984
\(667\) 24.0000 0.929284
\(668\) −8.00000 −0.309529
\(669\) −24.0000 −0.927894
\(670\) −12.0000 −0.463600
\(671\) 14.0000 0.540464
\(672\) 1.00000 0.0385758
\(673\) 50.0000 1.92736 0.963679 − 0.267063i \(-0.0860531\pi\)
0.963679 + 0.267063i \(0.0860531\pi\)
\(674\) 2.00000 0.0770371
\(675\) −1.00000 −0.0384900
\(676\) −9.00000 −0.346154
\(677\) 18.0000 0.691796 0.345898 − 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) 6.00000 0.230429
\(679\) −6.00000 −0.230259
\(680\) −2.00000 −0.0766965
\(681\) 8.00000 0.306561
\(682\) 0 0
\(683\) −28.0000 −1.07139 −0.535695 − 0.844411i \(-0.679950\pi\)
−0.535695 + 0.844411i \(0.679950\pi\)
\(684\) 4.00000 0.152944
\(685\) −2.00000 −0.0764161
\(686\) −1.00000 −0.0381802
\(687\) −10.0000 −0.381524
\(688\) 12.0000 0.457496
\(689\) −12.0000 −0.457164
\(690\) −4.00000 −0.152277
\(691\) 32.0000 1.21734 0.608669 − 0.793424i \(-0.291704\pi\)
0.608669 + 0.793424i \(0.291704\pi\)
\(692\) −6.00000 −0.228086
\(693\) 1.00000 0.0379869
\(694\) −20.0000 −0.759190
\(695\) 4.00000 0.151729
\(696\) 6.00000 0.227429
\(697\) 12.0000 0.454532
\(698\) 2.00000 0.0757011
\(699\) −26.0000 −0.983410
\(700\) −1.00000 −0.0377964
\(701\) −30.0000 −1.13308 −0.566542 − 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) −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
\(708\) −8.00000 −0.300658
\(709\) −10.0000 −0.375558 −0.187779 − 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) −8.00000 −0.300235
\(711\) 4.00000 0.150012
\(712\) 10.0000 0.374766
\(713\) 0 0
\(714\) 2.00000 0.0748481
\(715\) 2.00000 0.0747958
\(716\) −20.0000 −0.747435
\(717\) −12.0000 −0.448148
\(718\) −4.00000 −0.149279
\(719\) −16.0000 −0.596699 −0.298350 − 0.954457i \(-0.596436\pi\)
−0.298350 + 0.954457i \(0.596436\pi\)
\(720\) −1.00000 −0.0372678
\(721\) −8.00000 −0.297936
\(722\) −3.00000 −0.111648
\(723\) 18.0000 0.669427
\(724\) 26.0000 0.966282
\(725\) −6.00000 −0.222834
\(726\) −1.00000 −0.0371135
\(727\) 16.0000 0.593407 0.296704 − 0.954970i \(-0.404113\pi\)
0.296704 + 0.954970i \(0.404113\pi\)
\(728\) −2.00000 −0.0741249
\(729\) 1.00000 0.0370370
\(730\) −10.0000 −0.370117
\(731\) 24.0000 0.887672
\(732\) 14.0000 0.517455
\(733\) 10.0000 0.369358 0.184679 − 0.982799i \(-0.440875\pi\)
0.184679 + 0.982799i \(0.440875\pi\)
\(734\) −8.00000 −0.295285
\(735\) 1.00000 0.0368856
\(736\) −4.00000 −0.147442
\(737\) −12.0000 −0.442026
\(738\) 6.00000 0.220863
\(739\) 36.0000 1.32428 0.662141 − 0.749380i \(-0.269648\pi\)
0.662141 + 0.749380i \(0.269648\pi\)
\(740\) −2.00000 −0.0735215
\(741\) −8.00000 −0.293887
\(742\) 6.00000 0.220267
\(743\) 16.0000 0.586983 0.293492 − 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) 0 0
\(745\) 6.00000 0.219823
\(746\) 6.00000 0.219676
\(747\) 16.0000 0.585409
\(748\) −2.00000 −0.0731272
\(749\) 12.0000 0.438470
\(750\) 1.00000 0.0365148
\(751\) 40.0000 1.45962 0.729810 − 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) 8.00000 0.291730
\(753\) 0 0
\(754\) −12.0000 −0.437014
\(755\) 12.0000 0.436725
\(756\) 1.00000 0.0363696
\(757\) 10.0000 0.363456 0.181728 − 0.983349i \(-0.441831\pi\)
0.181728 + 0.983349i \(0.441831\pi\)
\(758\) 20.0000 0.726433
\(759\) −4.00000 −0.145191
\(760\) −4.00000 −0.145095
\(761\) 22.0000 0.797499 0.398750 − 0.917060i \(-0.369444\pi\)
0.398750 + 0.917060i \(0.369444\pi\)
\(762\) 8.00000 0.289809
\(763\) −2.00000 −0.0724049
\(764\) −24.0000 −0.868290
\(765\) −2.00000 −0.0723102
\(766\) 32.0000 1.15621
\(767\) 16.0000 0.577727
\(768\) −1.00000 −0.0360844
\(769\) −10.0000 −0.360609 −0.180305 − 0.983611i \(-0.557708\pi\)
−0.180305 + 0.983611i \(0.557708\pi\)
\(770\) −1.00000 −0.0360375
\(771\) 18.0000 0.648254
\(772\) 2.00000 0.0719816
\(773\) −38.0000 −1.36677 −0.683383 − 0.730061i \(-0.739492\pi\)
−0.683383 + 0.730061i \(0.739492\pi\)
\(774\) 12.0000 0.431331
\(775\) 0 0
\(776\) 6.00000 0.215387
\(777\) 2.00000 0.0717496
\(778\) −10.0000 −0.358517
\(779\) 24.0000 0.859889
\(780\) 2.00000 0.0716115
\(781\) −8.00000 −0.286263
\(782\) −8.00000 −0.286079
\(783\) 6.00000 0.214423
\(784\) 1.00000 0.0357143
\(785\) −2.00000 −0.0713831
\(786\) −12.0000 −0.428026
\(787\) −48.0000 −1.71102 −0.855508 − 0.517790i \(-0.826755\pi\)
−0.855508 + 0.517790i \(0.826755\pi\)
\(788\) 6.00000 0.213741
\(789\) −24.0000 −0.854423
\(790\) −4.00000 −0.142314
\(791\) 6.00000 0.213335
\(792\) −1.00000 −0.0355335
\(793\) −28.0000 −0.994309
\(794\) −14.0000 −0.496841
\(795\) −6.00000 −0.212798
\(796\) 0 0
\(797\) 18.0000 0.637593 0.318796 − 0.947823i \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(798\) 4.00000 0.141598
\(799\) 16.0000 0.566039
\(800\) 1.00000 0.0353553
\(801\) 10.0000 0.353333
\(802\) 2.00000 0.0706225
\(803\) −10.0000 −0.352892
\(804\) −12.0000 −0.423207
\(805\) −4.00000 −0.140981
\(806\) 0 0
\(807\) 6.00000 0.211210
\(808\) −6.00000 −0.211079
\(809\) 26.0000 0.914111 0.457056 − 0.889438i \(-0.348904\pi\)
0.457056 + 0.889438i \(0.348904\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −44.0000 −1.54505 −0.772524 − 0.634985i \(-0.781006\pi\)
−0.772524 + 0.634985i \(0.781006\pi\)
\(812\) 6.00000 0.210559
\(813\) 8.00000 0.280572
\(814\) −2.00000 −0.0701000
\(815\) −4.00000 −0.140114
\(816\) −2.00000 −0.0700140
\(817\) 48.0000 1.67931
\(818\) −10.0000 −0.349642
\(819\) −2.00000 −0.0698857
\(820\) −6.00000 −0.209529
\(821\) −14.0000 −0.488603 −0.244302 − 0.969699i \(-0.578559\pi\)
−0.244302 + 0.969699i \(0.578559\pi\)
\(822\) −2.00000 −0.0697580
\(823\) −4.00000 −0.139431 −0.0697156 − 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) 8.00000 0.278693
\(825\) 1.00000 0.0348155
\(826\) −8.00000 −0.278356
\(827\) −36.0000 −1.25184 −0.625921 − 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) −4.00000 −0.139010
\(829\) 10.0000 0.347314 0.173657 − 0.984806i \(-0.444442\pi\)
0.173657 + 0.984806i \(0.444442\pi\)
\(830\) −16.0000 −0.555368
\(831\) 18.0000 0.624413
\(832\) 2.00000 0.0693375
\(833\) 2.00000 0.0692959
\(834\) 4.00000 0.138509
\(835\) 8.00000 0.276851
\(836\) −4.00000 −0.138343
\(837\) 0 0
\(838\) −40.0000 −1.38178
\(839\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) −1.00000 −0.0345033
\(841\) 7.00000 0.241379
\(842\) 22.0000 0.758170
\(843\) 22.0000 0.757720
\(844\) 12.0000 0.413057
\(845\) 9.00000 0.309609
\(846\) 8.00000 0.275046
\(847\) −1.00000 −0.0343604
\(848\) −6.00000 −0.206041
\(849\) −24.0000 −0.823678
\(850\) 2.00000 0.0685994
\(851\) −8.00000 −0.274236
\(852\) −8.00000 −0.274075
\(853\) −6.00000 −0.205436 −0.102718 − 0.994711i \(-0.532754\pi\)
−0.102718 + 0.994711i \(0.532754\pi\)
\(854\) 14.0000 0.479070
\(855\) −4.00000 −0.136797
\(856\) −12.0000 −0.410152
\(857\) −38.0000 −1.29806 −0.649028 − 0.760765i \(-0.724824\pi\)
−0.649028 + 0.760765i \(0.724824\pi\)
\(858\) 2.00000 0.0682789
\(859\) 40.0000 1.36478 0.682391 − 0.730987i \(-0.260940\pi\)
0.682391 + 0.730987i \(0.260940\pi\)
\(860\) −12.0000 −0.409197
\(861\) 6.00000 0.204479
\(862\) 36.0000 1.22616
\(863\) 12.0000 0.408485 0.204242 − 0.978920i \(-0.434527\pi\)
0.204242 + 0.978920i \(0.434527\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 6.00000 0.204006
\(866\) −2.00000 −0.0679628
\(867\) 13.0000 0.441503
\(868\) 0 0
\(869\) −4.00000 −0.135691
\(870\) −6.00000 −0.203419
\(871\) 24.0000 0.813209
\(872\) 2.00000 0.0677285
\(873\) 6.00000 0.203069
\(874\) −16.0000 −0.541208
\(875\) 1.00000 0.0338062
\(876\) −10.0000 −0.337869
\(877\) 22.0000 0.742887 0.371444 − 0.928456i \(-0.378863\pi\)
0.371444 + 0.928456i \(0.378863\pi\)
\(878\) 0 0
\(879\) 22.0000 0.742042
\(880\) 1.00000 0.0337100
\(881\) 42.0000 1.41502 0.707508 − 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) 1.00000 0.0336718
\(883\) 20.0000 0.673054 0.336527 − 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) 4.00000 0.134535
\(885\) 8.00000 0.268917
\(886\) −36.0000 −1.20944
\(887\) 24.0000 0.805841 0.402921 − 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) −2.00000 −0.0671156
\(889\) 8.00000 0.268311
\(890\) −10.0000 −0.335201
\(891\) −1.00000 −0.0335013
\(892\) 24.0000 0.803579
\(893\) 32.0000 1.07084
\(894\) 6.00000 0.200670
\(895\) 20.0000 0.668526
\(896\) −1.00000 −0.0334077
\(897\) 8.00000 0.267112
\(898\) 18.0000 0.600668
\(899\) 0 0
\(900\) 1.00000 0.0333333
\(901\) −12.0000 −0.399778
\(902\) −6.00000 −0.199778
\(903\) 12.0000 0.399335
\(904\) −6.00000 −0.199557
\(905\) −26.0000 −0.864269
\(906\) 12.0000 0.398673
\(907\) −4.00000 −0.132818 −0.0664089 − 0.997792i \(-0.521154\pi\)
−0.0664089 + 0.997792i \(0.521154\pi\)
\(908\) −8.00000 −0.265489
\(909\) −6.00000 −0.199007
\(910\) 2.00000 0.0662994
\(911\) 24.0000 0.795155 0.397578 − 0.917568i \(-0.369851\pi\)
0.397578 + 0.917568i \(0.369851\pi\)
\(912\) −4.00000 −0.132453
\(913\) −16.0000 −0.529523
\(914\) 10.0000 0.330771
\(915\) −14.0000 −0.462826
\(916\) 10.0000 0.330409
\(917\) −12.0000 −0.396275
\(918\) −2.00000 −0.0660098
\(919\) −36.0000 −1.18753 −0.593765 − 0.804638i \(-0.702359\pi\)
−0.593765 + 0.804638i \(0.702359\pi\)
\(920\) 4.00000 0.131876
\(921\) −8.00000 −0.263609
\(922\) 18.0000 0.592798
\(923\) 16.0000 0.526646
\(924\) −1.00000 −0.0328976
\(925\) 2.00000 0.0657596
\(926\) 36.0000 1.18303
\(927\) 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 \(0.531381\pi\)
\(930\) 0 0
\(931\) 4.00000 0.131095
\(932\) 26.0000 0.851658
\(933\) 24.0000 0.785725
\(934\) −20.0000 −0.654420
\(935\) 2.00000 0.0654070
\(936\) 2.00000 0.0653720
\(937\) 50.0000 1.63343 0.816714 − 0.577042i \(-0.195793\pi\)
0.816714 + 0.577042i \(0.195793\pi\)
\(938\) −12.0000 −0.391814
\(939\) 18.0000 0.587408
\(940\) −8.00000 −0.260931
\(941\) −14.0000 −0.456387 −0.228193 − 0.973616i \(-0.573282\pi\)
−0.228193 + 0.973616i \(0.573282\pi\)
\(942\) −2.00000 −0.0651635
\(943\) −24.0000 −0.781548
\(944\) 8.00000 0.260378
\(945\) −1.00000 −0.0325300
\(946\) −12.0000 −0.390154
\(947\) −28.0000 −0.909878 −0.454939 − 0.890523i \(-0.650339\pi\)
−0.454939 + 0.890523i \(0.650339\pi\)
\(948\) −4.00000 −0.129914
\(949\) 20.0000 0.649227
\(950\) 4.00000 0.129777
\(951\) 22.0000 0.713399
\(952\) −2.00000 −0.0648204
\(953\) 42.0000 1.36051 0.680257 − 0.732974i \(-0.261868\pi\)
0.680257 + 0.732974i \(0.261868\pi\)
\(954\) −6.00000 −0.194257
\(955\) 24.0000 0.776622
\(956\) 12.0000 0.388108
\(957\) −6.00000 −0.193952
\(958\) −24.0000 −0.775405
\(959\) −2.00000 −0.0645834
\(960\) 1.00000 0.0322749
\(961\) −31.0000 −1.00000
\(962\) 4.00000 0.128965
\(963\) −12.0000 −0.386695
\(964\) −18.0000 −0.579741
\(965\) −2.00000 −0.0643823
\(966\) −4.00000 −0.128698
\(967\) 48.0000 1.54358 0.771788 − 0.635880i \(-0.219363\pi\)
0.771788 + 0.635880i \(0.219363\pi\)
\(968\) 1.00000 0.0321412
\(969\) −8.00000 −0.256997
\(970\) −6.00000 −0.192648
\(971\) −40.0000 −1.28366 −0.641831 − 0.766846i \(-0.721825\pi\)
−0.641831 + 0.766846i \(0.721825\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 4.00000 0.128234
\(974\) 4.00000 0.128168
\(975\) −2.00000 −0.0640513
\(976\) −14.0000 −0.448129
\(977\) 10.0000 0.319928 0.159964 − 0.987123i \(-0.448862\pi\)
0.159964 + 0.987123i \(0.448862\pi\)
\(978\) −4.00000 −0.127906
\(979\) −10.0000 −0.319601
\(980\) −1.00000 −0.0319438
\(981\) 2.00000 0.0638551
\(982\) 12.0000 0.382935
\(983\) −8.00000 −0.255160 −0.127580 − 0.991828i \(-0.540721\pi\)
−0.127580 + 0.991828i \(0.540721\pi\)
\(984\) −6.00000 −0.191273
\(985\) −6.00000 −0.191176
\(986\) −12.0000 −0.382158
\(987\) 8.00000 0.254643
\(988\) 8.00000 0.254514
\(989\) −48.0000 −1.52631
\(990\) 1.00000 0.0317821
\(991\) 24.0000 0.762385 0.381193 − 0.924496i \(-0.375513\pi\)
0.381193 + 0.924496i \(0.375513\pi\)
\(992\) 0 0
\(993\) 4.00000 0.126936
\(994\) −8.00000 −0.253745
\(995\) 0 0
\(996\) −16.0000 −0.506979
\(997\) −14.0000 −0.443384 −0.221692 − 0.975117i \(-0.571158\pi\)
−0.221692 + 0.975117i \(0.571158\pi\)
\(998\) −36.0000 −1.13956
\(999\) −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.m.1.1 ✓ 1
3.2 odd 2 6930.2.a.m.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2310.2.a.m.1.1 ✓ 1 1.1 even 1 trivial
6930.2.a.m.1.1 1 3.2 odd 2