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Properties

Label 2310.2.a.u.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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} +1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} -1.00000 q^{20} +1.00000 q^{21} -1.00000 q^{22} +1.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} +1.00000 q^{27} +1.00000 q^{28} +6.00000 q^{29} -1.00000 q^{30} +8.00000 q^{31} +1.00000 q^{32} -1.00000 q^{33} +6.00000 q^{34} -1.00000 q^{35} +1.00000 q^{36} -10.0000 q^{37} -4.00000 q^{38} +2.00000 q^{39} -1.00000 q^{40} -6.00000 q^{41} +1.00000 q^{42} +8.00000 q^{43} -1.00000 q^{44} -1.00000 q^{45} +1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +6.00000 q^{51} +2.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} +1.00000 q^{55} +1.00000 q^{56} -4.00000 q^{57} +6.00000 q^{58} +12.0000 q^{59} -1.00000 q^{60} +2.00000 q^{61} +8.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} -1.00000 q^{66} -4.00000 q^{67} +6.00000 q^{68} -1.00000 q^{70} +12.0000 q^{71} +1.00000 q^{72} -10.0000 q^{73} -10.0000 q^{74} +1.00000 q^{75} -4.00000 q^{76} -1.00000 q^{77} +2.00000 q^{78} -4.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} -12.0000 q^{83} +1.00000 q^{84} -6.00000 q^{85} +8.00000 q^{86} +6.00000 q^{87} -1.00000 q^{88} +6.00000 q^{89} -1.00000 q^{90} +2.00000 q^{91} +8.00000 q^{93} +4.00000 q^{95} +1.00000 q^{96} -10.0000 q^{97} +1.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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 \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 1.00000 0.267261
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 − 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 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
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 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
\(34\) 6.00000 1.02899
\(35\) −1.00000 −0.169031
\(36\) 1.00000 0.166667
\(37\) −10.0000 −1.64399 −0.821995 − 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) −4.00000 −0.648886
\(39\) 2.00000 0.320256
\(40\) −1.00000 −0.158114
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 1.00000 0.154303
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) −1.00000 −0.150756
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 6.00000 0.840168
\(52\) 2.00000 0.277350
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 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 \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) −1.00000 −0.129099
\(61\) 2.00000 0.256074 0.128037 − 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 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 \(0.578571\pi\)
\(68\) 6.00000 0.727607
\(69\) 0 0
\(70\) −1.00000 −0.119523
\(71\) 12.0000 1.42414 0.712069 − 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 1.00000 0.117851
\(73\) −10.0000 −1.17041 −0.585206 − 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) −10.0000 −1.16248
\(75\) 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 \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) −12.0000 −1.31717 −0.658586 − 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 1.00000 0.109109
\(85\) −6.00000 −0.650791
\(86\) 8.00000 0.862662
\(87\) 6.00000 0.643268
\(88\) −1.00000 −0.106600
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) −1.00000 −0.105409
\(91\) 2.00000 0.209657
\(92\) 0 0
\(93\) 8.00000 0.829561
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) 1.00000 0.102062
\(97\) −10.0000 −1.01535 −0.507673 − 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 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 \(0.596490\pi\)
\(102\) 6.00000 0.594089
\(103\) −4.00000 −0.394132 −0.197066 − 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 2.00000 0.196116
\(105\) −1.00000 −0.0975900
\(106\) 6.00000 0.582772
\(107\) 12.0000 1.16008 0.580042 − 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 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
\(111\) −10.0000 −0.949158
\(112\) 1.00000 0.0944911
\(113\) 6.00000 0.564433 0.282216 − 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) −4.00000 −0.374634
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) 2.00000 0.184900
\(118\) 12.0000 1.10469
\(119\) 6.00000 0.550019
\(120\) −1.00000 −0.0912871
\(121\) 1.00000 0.0909091
\(122\) 2.00000 0.181071
\(123\) −6.00000 −0.541002
\(124\) 8.00000 0.718421
\(125\) −1.00000 −0.0894427
\(126\) 1.00000 0.0890871
\(127\) 8.00000 0.709885 0.354943 − 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 1.00000 0.0883883
\(129\) 8.00000 0.704361
\(130\) −2.00000 −0.175412
\(131\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) −1.00000 −0.0870388
\(133\) −4.00000 −0.346844
\(134\) −4.00000 −0.345547
\(135\) −1.00000 −0.0860663
\(136\) 6.00000 0.514496
\(137\) 6.00000 0.512615 0.256307 − 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 − 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 0 0
\(142\) 12.0000 1.00702
\(143\) −2.00000 −0.167248
\(144\) 1.00000 0.0833333
\(145\) −6.00000 −0.498273
\(146\) −10.0000 −0.827606
\(147\) 1.00000 0.0824786
\(148\) −10.0000 −0.821995
\(149\) −18.0000 −1.47462 −0.737309 − 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 1.00000 0.0816497
\(151\) −4.00000 −0.325515 −0.162758 − 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) −4.00000 −0.324443
\(153\) 6.00000 0.485071
\(154\) −1.00000 −0.0805823
\(155\) −8.00000 −0.642575
\(156\) 2.00000 0.160128
\(157\) −10.0000 −0.798087 −0.399043 − 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) −4.00000 −0.318223
\(159\) 6.00000 0.475831
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −4.00000 −0.313304 −0.156652 − 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) −6.00000 −0.468521
\(165\) 1.00000 0.0778499
\(166\) −12.0000 −0.931381
\(167\) −12.0000 −0.928588 −0.464294 − 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 1.00000 0.0771517
\(169\) −9.00000 −0.692308
\(170\) −6.00000 −0.460179
\(171\) −4.00000 −0.305888
\(172\) 8.00000 0.609994
\(173\) 6.00000 0.456172 0.228086 − 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 6.00000 0.454859
\(175\) 1.00000 0.0755929
\(176\) −1.00000 −0.0753778
\(177\) 12.0000 0.901975
\(178\) 6.00000 0.449719
\(179\) −12.0000 −0.896922 −0.448461 − 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 2.00000 0.148659 0.0743294 − 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 2.00000 0.148250
\(183\) 2.00000 0.147844
\(184\) 0 0
\(185\) 10.0000 0.735215
\(186\) 8.00000 0.586588
\(187\) −6.00000 −0.438763
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 4.00000 0.290191
\(191\) −12.0000 −0.868290 −0.434145 − 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 1.00000 0.0721688
\(193\) 14.0000 1.00774 0.503871 − 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) −10.0000 −0.717958
\(195\) −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
\(199\) −16.0000 −1.13421 −0.567105 − 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 1.00000 0.0707107
\(201\) −4.00000 −0.282138
\(202\) −6.00000 −0.422159
\(203\) 6.00000 0.421117
\(204\) 6.00000 0.420084
\(205\) 6.00000 0.419058
\(206\) −4.00000 −0.278693
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) 4.00000 0.276686
\(210\) −1.00000 −0.0690066
\(211\) −4.00000 −0.275371 −0.137686 − 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 6.00000 0.412082
\(213\) 12.0000 0.822226
\(214\) 12.0000 0.820303
\(215\) −8.00000 −0.545595
\(216\) 1.00000 0.0680414
\(217\) 8.00000 0.543075
\(218\) 2.00000 0.135457
\(219\) −10.0000 −0.675737
\(220\) 1.00000 0.0674200
\(221\) 12.0000 0.807207
\(222\) −10.0000 −0.671156
\(223\) −4.00000 −0.267860 −0.133930 − 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) 1.00000 0.0668153
\(225\) 1.00000 0.0666667
\(226\) 6.00000 0.399114
\(227\) −12.0000 −0.796468 −0.398234 − 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) −4.00000 −0.264906
\(229\) −22.0000 −1.45380 −0.726900 − 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 0 0
\(231\) −1.00000 −0.0657952
\(232\) 6.00000 0.393919
\(233\) 18.0000 1.17922 0.589610 − 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) 12.0000 0.781133
\(237\) −4.00000 −0.259828
\(238\) 6.00000 0.388922
\(239\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 14.0000 0.901819 0.450910 − 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) 1.00000 0.0642824
\(243\) 1.00000 0.0641500
\(244\) 2.00000 0.128037
\(245\) −1.00000 −0.0638877
\(246\) −6.00000 −0.382546
\(247\) −8.00000 −0.509028
\(248\) 8.00000 0.508001
\(249\) −12.0000 −0.760469
\(250\) −1.00000 −0.0632456
\(251\) 12.0000 0.757433 0.378717 − 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 1.00000 0.0629941
\(253\) 0 0
\(254\) 8.00000 0.501965
\(255\) −6.00000 −0.375735
\(256\) 1.00000 0.0625000
\(257\) −18.0000 −1.12281 −0.561405 − 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 8.00000 0.498058
\(259\) −10.0000 −0.621370
\(260\) −2.00000 −0.124035
\(261\) 6.00000 0.371391
\(262\) 0 0
\(263\) 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
\(267\) 6.00000 0.367194
\(268\) −4.00000 −0.244339
\(269\) −30.0000 −1.82913 −0.914566 − 0.404436i \(-0.867468\pi\)
−0.914566 + 0.404436i \(0.867468\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −16.0000 −0.971931 −0.485965 − 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 6.00000 0.363803
\(273\) 2.00000 0.121046
\(274\) 6.00000 0.362473
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 − 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) −4.00000 −0.239904
\(279\) 8.00000 0.478947
\(280\) −1.00000 −0.0597614
\(281\) −6.00000 −0.357930 −0.178965 − 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 − 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 12.0000 0.712069
\(285\) 4.00000 0.236940
\(286\) −2.00000 −0.118262
\(287\) −6.00000 −0.354169
\(288\) 1.00000 0.0589256
\(289\) 19.0000 1.11765
\(290\) −6.00000 −0.352332
\(291\) −10.0000 −0.586210
\(292\) −10.0000 −0.585206
\(293\) −18.0000 −1.05157 −0.525786 − 0.850617i \(-0.676229\pi\)
−0.525786 + 0.850617i \(0.676229\pi\)
\(294\) 1.00000 0.0583212
\(295\) −12.0000 −0.698667
\(296\) −10.0000 −0.581238
\(297\) −1.00000 −0.0580259
\(298\) −18.0000 −1.04271
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) 8.00000 0.461112
\(302\) −4.00000 −0.230174
\(303\) −6.00000 −0.344691
\(304\) −4.00000 −0.229416
\(305\) −2.00000 −0.114520
\(306\) 6.00000 0.342997
\(307\) −28.0000 −1.59804 −0.799022 − 0.601302i \(-0.794649\pi\)
−0.799022 + 0.601302i \(0.794649\pi\)
\(308\) −1.00000 −0.0569803
\(309\) −4.00000 −0.227552
\(310\) −8.00000 −0.454369
\(311\) 24.0000 1.36092 0.680458 − 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 2.00000 0.113228
\(313\) −10.0000 −0.565233 −0.282617 − 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) −10.0000 −0.564333
\(315\) −1.00000 −0.0563436
\(316\) −4.00000 −0.225018
\(317\) −18.0000 −1.01098 −0.505490 − 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 6.00000 0.336463
\(319\) −6.00000 −0.335936
\(320\) −1.00000 −0.0559017
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) −24.0000 −1.33540
\(324\) 1.00000 0.0555556
\(325\) 2.00000 0.110940
\(326\) −4.00000 −0.221540
\(327\) 2.00000 0.110600
\(328\) −6.00000 −0.331295
\(329\) 0 0
\(330\) 1.00000 0.0550482
\(331\) 20.0000 1.09930 0.549650 − 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) −12.0000 −0.658586
\(333\) −10.0000 −0.547997
\(334\) −12.0000 −0.656611
\(335\) 4.00000 0.218543
\(336\) 1.00000 0.0545545
\(337\) 14.0000 0.762629 0.381314 − 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) −9.00000 −0.489535
\(339\) 6.00000 0.325875
\(340\) −6.00000 −0.325396
\(341\) −8.00000 −0.433224
\(342\) −4.00000 −0.216295
\(343\) 1.00000 0.0539949
\(344\) 8.00000 0.431331
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) −12.0000 −0.644194 −0.322097 − 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 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 \(0.205696\pi\)
\(354\) 12.0000 0.637793
\(355\) −12.0000 −0.636894
\(356\) 6.00000 0.317999
\(357\) 6.00000 0.317554
\(358\) −12.0000 −0.634220
\(359\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −3.00000 −0.157895
\(362\) 2.00000 0.105118
\(363\) 1.00000 0.0524864
\(364\) 2.00000 0.104828
\(365\) 10.0000 0.523424
\(366\) 2.00000 0.104542
\(367\) −4.00000 −0.208798 −0.104399 − 0.994535i \(-0.533292\pi\)
−0.104399 + 0.994535i \(0.533292\pi\)
\(368\) 0 0
\(369\) −6.00000 −0.312348
\(370\) 10.0000 0.519875
\(371\) 6.00000 0.311504
\(372\) 8.00000 0.414781
\(373\) 14.0000 0.724893 0.362446 − 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) −6.00000 −0.310253
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) 1.00000 0.0514344
\(379\) −4.00000 −0.205466 −0.102733 − 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 4.00000 0.205196
\(381\) 8.00000 0.409852
\(382\) −12.0000 −0.613973
\(383\) 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
\(386\) 14.0000 0.712581
\(387\) 8.00000 0.406663
\(388\) −10.0000 −0.507673
\(389\) −30.0000 −1.52106 −0.760530 − 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) −2.00000 −0.101274
\(391\) 0 0
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) 6.00000 0.302276
\(395\) 4.00000 0.201262
\(396\) −1.00000 −0.0502519
\(397\) −34.0000 −1.70641 −0.853206 − 0.521575i \(-0.825345\pi\)
−0.853206 + 0.521575i \(0.825345\pi\)
\(398\) −16.0000 −0.802008
\(399\) −4.00000 −0.200250
\(400\) 1.00000 0.0500000
\(401\) −30.0000 −1.49813 −0.749064 − 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) −4.00000 −0.199502
\(403\) 16.0000 0.797017
\(404\) −6.00000 −0.298511
\(405\) −1.00000 −0.0496904
\(406\) 6.00000 0.297775
\(407\) 10.0000 0.495682
\(408\) 6.00000 0.297044
\(409\) 14.0000 0.692255 0.346128 − 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 6.00000 0.296319
\(411\) 6.00000 0.295958
\(412\) −4.00000 −0.197066
\(413\) 12.0000 0.590481
\(414\) 0 0
\(415\) 12.0000 0.589057
\(416\) 2.00000 0.0980581
\(417\) −4.00000 −0.195881
\(418\) 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
\(421\) −10.0000 −0.487370 −0.243685 − 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) −4.00000 −0.194717
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) 6.00000 0.291043
\(426\) 12.0000 0.581402
\(427\) 2.00000 0.0967868
\(428\) 12.0000 0.580042
\(429\) −2.00000 −0.0965609
\(430\) −8.00000 −0.385794
\(431\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 1.00000 0.0481125
\(433\) −34.0000 −1.63394 −0.816968 − 0.576683i \(-0.804347\pi\)
−0.816968 + 0.576683i \(0.804347\pi\)
\(434\) 8.00000 0.384012
\(435\) −6.00000 −0.287678
\(436\) 2.00000 0.0957826
\(437\) 0 0
\(438\) −10.0000 −0.477818
\(439\) 8.00000 0.381819 0.190910 − 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 1.00000 0.0476731
\(441\) 1.00000 0.0476190
\(442\) 12.0000 0.570782
\(443\) 24.0000 1.14027 0.570137 − 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) −10.0000 −0.474579
\(445\) −6.00000 −0.284427
\(446\) −4.00000 −0.189405
\(447\) −18.0000 −0.851371
\(448\) 1.00000 0.0472456
\(449\) −6.00000 −0.283158 −0.141579 − 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 1.00000 0.0471405
\(451\) 6.00000 0.282529
\(452\) 6.00000 0.282216
\(453\) −4.00000 −0.187936
\(454\) −12.0000 −0.563188
\(455\) −2.00000 −0.0937614
\(456\) −4.00000 −0.187317
\(457\) −10.0000 −0.467780 −0.233890 − 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) −22.0000 −1.02799
\(459\) 6.00000 0.280056
\(460\) 0 0
\(461\) −6.00000 −0.279448 −0.139724 − 0.990190i \(-0.544622\pi\)
−0.139724 + 0.990190i \(0.544622\pi\)
\(462\) −1.00000 −0.0465242
\(463\) 32.0000 1.48717 0.743583 − 0.668644i \(-0.233125\pi\)
0.743583 + 0.668644i \(0.233125\pi\)
\(464\) 6.00000 0.278543
\(465\) −8.00000 −0.370991
\(466\) 18.0000 0.833834
\(467\) 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
\(470\) 0 0
\(471\) −10.0000 −0.460776
\(472\) 12.0000 0.552345
\(473\) −8.00000 −0.367840
\(474\) −4.00000 −0.183726
\(475\) −4.00000 −0.183533
\(476\) 6.00000 0.275010
\(477\) 6.00000 0.274721
\(478\) 0 0
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) −1.00000 −0.0456435
\(481\) −20.0000 −0.911922
\(482\) 14.0000 0.637683
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 10.0000 0.454077
\(486\) 1.00000 0.0453609
\(487\) −40.0000 −1.81257 −0.906287 − 0.422664i \(-0.861095\pi\)
−0.906287 + 0.422664i \(0.861095\pi\)
\(488\) 2.00000 0.0905357
\(489\) −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
\(493\) 36.0000 1.62136
\(494\) −8.00000 −0.359937
\(495\) 1.00000 0.0449467
\(496\) 8.00000 0.359211
\(497\) 12.0000 0.538274
\(498\) −12.0000 −0.537733
\(499\) −4.00000 −0.179065 −0.0895323 − 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −12.0000 −0.536120
\(502\) 12.0000 0.535586
\(503\) 12.0000 0.535054 0.267527 − 0.963550i \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) 1.00000 0.0445435
\(505\) 6.00000 0.266996
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) 8.00000 0.354943
\(509\) −30.0000 −1.32973 −0.664863 − 0.746965i \(-0.731510\pi\)
−0.664863 + 0.746965i \(0.731510\pi\)
\(510\) −6.00000 −0.265684
\(511\) −10.0000 −0.442374
\(512\) 1.00000 0.0441942
\(513\) −4.00000 −0.176604
\(514\) −18.0000 −0.793946
\(515\) 4.00000 0.176261
\(516\) 8.00000 0.352180
\(517\) 0 0
\(518\) −10.0000 −0.439375
\(519\) 6.00000 0.263371
\(520\) −2.00000 −0.0877058
\(521\) −42.0000 −1.84005 −0.920027 − 0.391856i \(-0.871833\pi\)
−0.920027 + 0.391856i \(0.871833\pi\)
\(522\) 6.00000 0.262613
\(523\) 20.0000 0.874539 0.437269 − 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 0 0
\(525\) 1.00000 0.0436436
\(526\) 24.0000 1.04645
\(527\) 48.0000 2.09091
\(528\) −1.00000 −0.0435194
\(529\) −23.0000 −1.00000
\(530\) −6.00000 −0.260623
\(531\) 12.0000 0.520756
\(532\) −4.00000 −0.173422
\(533\) −12.0000 −0.519778
\(534\) 6.00000 0.259645
\(535\) −12.0000 −0.518805
\(536\) −4.00000 −0.172774
\(537\) −12.0000 −0.517838
\(538\) −30.0000 −1.29339
\(539\) −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 \(0.486311\pi\)
\(542\) −16.0000 −0.687259
\(543\) 2.00000 0.0858282
\(544\) 6.00000 0.257248
\(545\) −2.00000 −0.0856706
\(546\) 2.00000 0.0855921
\(547\) 8.00000 0.342055 0.171028 − 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) 6.00000 0.256307
\(549\) 2.00000 0.0853579
\(550\) −1.00000 −0.0426401
\(551\) −24.0000 −1.02243
\(552\) 0 0
\(553\) −4.00000 −0.170097
\(554\) −10.0000 −0.424859
\(555\) 10.0000 0.424476
\(556\) −4.00000 −0.169638
\(557\) 30.0000 1.27114 0.635570 − 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) 8.00000 0.338667
\(559\) 16.0000 0.676728
\(560\) −1.00000 −0.0422577
\(561\) −6.00000 −0.253320
\(562\) −6.00000 −0.253095
\(563\) 12.0000 0.505740 0.252870 − 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 0 0
\(565\) −6.00000 −0.252422
\(566\) −4.00000 −0.168133
\(567\) 1.00000 0.0419961
\(568\) 12.0000 0.503509
\(569\) 42.0000 1.76073 0.880366 − 0.474295i \(-0.157297\pi\)
0.880366 + 0.474295i \(0.157297\pi\)
\(570\) 4.00000 0.167542
\(571\) −4.00000 −0.167395 −0.0836974 − 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) −2.00000 −0.0836242
\(573\) −12.0000 −0.501307
\(574\) −6.00000 −0.250435
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 38.0000 1.58196 0.790980 − 0.611842i \(-0.209571\pi\)
0.790980 + 0.611842i \(0.209571\pi\)
\(578\) 19.0000 0.790296
\(579\) 14.0000 0.581820
\(580\) −6.00000 −0.249136
\(581\) −12.0000 −0.497844
\(582\) −10.0000 −0.414513
\(583\) −6.00000 −0.248495
\(584\) −10.0000 −0.413803
\(585\) −2.00000 −0.0826898
\(586\) −18.0000 −0.743573
\(587\) 12.0000 0.495293 0.247647 − 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 1.00000 0.0412393
\(589\) −32.0000 −1.31854
\(590\) −12.0000 −0.494032
\(591\) 6.00000 0.246807
\(592\) −10.0000 −0.410997
\(593\) 6.00000 0.246390 0.123195 − 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) −1.00000 −0.0410305
\(595\) −6.00000 −0.245976
\(596\) −18.0000 −0.737309
\(597\) −16.0000 −0.654836
\(598\) 0 0
\(599\) 12.0000 0.490307 0.245153 − 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) 1.00000 0.0408248
\(601\) 14.0000 0.571072 0.285536 − 0.958368i \(-0.407828\pi\)
0.285536 + 0.958368i \(0.407828\pi\)
\(602\) 8.00000 0.326056
\(603\) −4.00000 −0.162893
\(604\) −4.00000 −0.162758
\(605\) −1.00000 −0.0406558
\(606\) −6.00000 −0.243733
\(607\) −40.0000 −1.62355 −0.811775 − 0.583970i \(-0.801498\pi\)
−0.811775 + 0.583970i \(0.801498\pi\)
\(608\) −4.00000 −0.162221
\(609\) 6.00000 0.243132
\(610\) −2.00000 −0.0809776
\(611\) 0 0
\(612\) 6.00000 0.242536
\(613\) 14.0000 0.565455 0.282727 − 0.959200i \(-0.408761\pi\)
0.282727 + 0.959200i \(0.408761\pi\)
\(614\) −28.0000 −1.12999
\(615\) 6.00000 0.241943
\(616\) −1.00000 −0.0402911
\(617\) −42.0000 −1.69086 −0.845428 − 0.534089i \(-0.820655\pi\)
−0.845428 + 0.534089i \(0.820655\pi\)
\(618\) −4.00000 −0.160904
\(619\) 8.00000 0.321547 0.160774 − 0.986991i \(-0.448601\pi\)
0.160774 + 0.986991i \(0.448601\pi\)
\(620\) −8.00000 −0.321288
\(621\) 0 0
\(622\) 24.0000 0.962312
\(623\) 6.00000 0.240385
\(624\) 2.00000 0.0800641
\(625\) 1.00000 0.0400000
\(626\) −10.0000 −0.399680
\(627\) 4.00000 0.159745
\(628\) −10.0000 −0.399043
\(629\) −60.0000 −2.39236
\(630\) −1.00000 −0.0398410
\(631\) 8.00000 0.318475 0.159237 − 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) −4.00000 −0.159111
\(633\) −4.00000 −0.158986
\(634\) −18.0000 −0.714871
\(635\) −8.00000 −0.317470
\(636\) 6.00000 0.237915
\(637\) 2.00000 0.0792429
\(638\) −6.00000 −0.237542
\(639\) 12.0000 0.474713
\(640\) −1.00000 −0.0395285
\(641\) 42.0000 1.65890 0.829450 − 0.558581i \(-0.188654\pi\)
0.829450 + 0.558581i \(0.188654\pi\)
\(642\) 12.0000 0.473602
\(643\) −28.0000 −1.10421 −0.552106 − 0.833774i \(-0.686176\pi\)
−0.552106 + 0.833774i \(0.686176\pi\)
\(644\) 0 0
\(645\) −8.00000 −0.315000
\(646\) −24.0000 −0.944267
\(647\) 24.0000 0.943537 0.471769 − 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) 1.00000 0.0392837
\(649\) −12.0000 −0.471041
\(650\) 2.00000 0.0784465
\(651\) 8.00000 0.313545
\(652\) −4.00000 −0.156652
\(653\) 30.0000 1.17399 0.586995 − 0.809590i \(-0.300311\pi\)
0.586995 + 0.809590i \(0.300311\pi\)
\(654\) 2.00000 0.0782062
\(655\) 0 0
\(656\) −6.00000 −0.234261
\(657\) −10.0000 −0.390137
\(658\) 0 0
\(659\) −36.0000 −1.40236 −0.701180 − 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 1.00000 0.0389249
\(661\) 2.00000 0.0777910 0.0388955 − 0.999243i \(-0.487616\pi\)
0.0388955 + 0.999243i \(0.487616\pi\)
\(662\) 20.0000 0.777322
\(663\) 12.0000 0.466041
\(664\) −12.0000 −0.465690
\(665\) 4.00000 0.155113
\(666\) −10.0000 −0.387492
\(667\) 0 0
\(668\) −12.0000 −0.464294
\(669\) −4.00000 −0.154649
\(670\) 4.00000 0.154533
\(671\) −2.00000 −0.0772091
\(672\) 1.00000 0.0385758
\(673\) −34.0000 −1.31060 −0.655302 − 0.755367i \(-0.727459\pi\)
−0.655302 + 0.755367i \(0.727459\pi\)
\(674\) 14.0000 0.539260
\(675\) 1.00000 0.0384900
\(676\) −9.00000 −0.346154
\(677\) 6.00000 0.230599 0.115299 − 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 6.00000 0.230429
\(679\) −10.0000 −0.383765
\(680\) −6.00000 −0.230089
\(681\) −12.0000 −0.459841
\(682\) −8.00000 −0.306336
\(683\) 48.0000 1.83667 0.918334 − 0.395805i \(-0.129534\pi\)
0.918334 + 0.395805i \(0.129534\pi\)
\(684\) −4.00000 −0.152944
\(685\) −6.00000 −0.229248
\(686\) 1.00000 0.0381802
\(687\) −22.0000 −0.839352
\(688\) 8.00000 0.304997
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) 8.00000 0.304334 0.152167 − 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) 6.00000 0.228086
\(693\) −1.00000 −0.0379869
\(694\) −12.0000 −0.455514
\(695\) 4.00000 0.151729
\(696\) 6.00000 0.227429
\(697\) −36.0000 −1.36360
\(698\) 2.00000 0.0757011
\(699\) 18.0000 0.680823
\(700\) 1.00000 0.0377964
\(701\) −18.0000 −0.679851 −0.339925 − 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 2.00000 0.0754851
\(703\) 40.0000 1.50863
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) 30.0000 1.12906
\(707\) −6.00000 −0.225653
\(708\) 12.0000 0.450988
\(709\) −10.0000 −0.375558 −0.187779 − 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) −12.0000 −0.450352
\(711\) −4.00000 −0.150012
\(712\) 6.00000 0.224860
\(713\) 0 0
\(714\) 6.00000 0.224544
\(715\) 2.00000 0.0747958
\(716\) −12.0000 −0.448461
\(717\) 0 0
\(718\) 0 0
\(719\) −24.0000 −0.895049 −0.447524 − 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) −1.00000 −0.0372678
\(721\) −4.00000 −0.148968
\(722\) −3.00000 −0.111648
\(723\) 14.0000 0.520666
\(724\) 2.00000 0.0743294
\(725\) 6.00000 0.222834
\(726\) 1.00000 0.0371135
\(727\) −4.00000 −0.148352 −0.0741759 − 0.997245i \(-0.523633\pi\)
−0.0741759 + 0.997245i \(0.523633\pi\)
\(728\) 2.00000 0.0741249
\(729\) 1.00000 0.0370370
\(730\) 10.0000 0.370117
\(731\) 48.0000 1.77534
\(732\) 2.00000 0.0739221
\(733\) −22.0000 −0.812589 −0.406294 − 0.913742i \(-0.633179\pi\)
−0.406294 + 0.913742i \(0.633179\pi\)
\(734\) −4.00000 −0.147643
\(735\) −1.00000 −0.0368856
\(736\) 0 0
\(737\) 4.00000 0.147342
\(738\) −6.00000 −0.220863
\(739\) −4.00000 −0.147142 −0.0735712 − 0.997290i \(-0.523440\pi\)
−0.0735712 + 0.997290i \(0.523440\pi\)
\(740\) 10.0000 0.367607
\(741\) −8.00000 −0.293887
\(742\) 6.00000 0.220267
\(743\) −48.0000 −1.76095 −0.880475 − 0.474093i \(-0.842776\pi\)
−0.880475 + 0.474093i \(0.842776\pi\)
\(744\) 8.00000 0.293294
\(745\) 18.0000 0.659469
\(746\) 14.0000 0.512576
\(747\) −12.0000 −0.439057
\(748\) −6.00000 −0.219382
\(749\) 12.0000 0.438470
\(750\) −1.00000 −0.0365148
\(751\) −40.0000 −1.45962 −0.729810 − 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 0 0
\(753\) 12.0000 0.437304
\(754\) 12.0000 0.437014
\(755\) 4.00000 0.145575
\(756\) 1.00000 0.0363696
\(757\) 38.0000 1.38113 0.690567 − 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) −4.00000 −0.145287
\(759\) 0 0
\(760\) 4.00000 0.145095
\(761\) 42.0000 1.52250 0.761249 − 0.648459i \(-0.224586\pi\)
0.761249 + 0.648459i \(0.224586\pi\)
\(762\) 8.00000 0.289809
\(763\) 2.00000 0.0724049
\(764\) −12.0000 −0.434145
\(765\) −6.00000 −0.216930
\(766\) 24.0000 0.867155
\(767\) 24.0000 0.866590
\(768\) 1.00000 0.0360844
\(769\) −34.0000 −1.22607 −0.613036 − 0.790055i \(-0.710052\pi\)
−0.613036 + 0.790055i \(0.710052\pi\)
\(770\) 1.00000 0.0360375
\(771\) −18.0000 −0.648254
\(772\) 14.0000 0.503871
\(773\) −6.00000 −0.215805 −0.107903 − 0.994161i \(-0.534413\pi\)
−0.107903 + 0.994161i \(0.534413\pi\)
\(774\) 8.00000 0.287554
\(775\) 8.00000 0.287368
\(776\) −10.0000 −0.358979
\(777\) −10.0000 −0.358748
\(778\) −30.0000 −1.07555
\(779\) 24.0000 0.859889
\(780\) −2.00000 −0.0716115
\(781\) −12.0000 −0.429394
\(782\) 0 0
\(783\) 6.00000 0.214423
\(784\) 1.00000 0.0357143
\(785\) 10.0000 0.356915
\(786\) 0 0
\(787\) −52.0000 −1.85360 −0.926800 − 0.375555i \(-0.877452\pi\)
−0.926800 + 0.375555i \(0.877452\pi\)
\(788\) 6.00000 0.213741
\(789\) 24.0000 0.854423
\(790\) 4.00000 0.142314
\(791\) 6.00000 0.213335
\(792\) −1.00000 −0.0355335
\(793\) 4.00000 0.142044
\(794\) −34.0000 −1.20661
\(795\) −6.00000 −0.212798
\(796\) −16.0000 −0.567105
\(797\) −54.0000 −1.91278 −0.956389 − 0.292096i \(-0.905647\pi\)
−0.956389 + 0.292096i \(0.905647\pi\)
\(798\) −4.00000 −0.141598
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) 6.00000 0.212000
\(802\) −30.0000 −1.05934
\(803\) 10.0000 0.352892
\(804\) −4.00000 −0.141069
\(805\) 0 0
\(806\) 16.0000 0.563576
\(807\) −30.0000 −1.05605
\(808\) −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
\(811\) 20.0000 0.702295 0.351147 − 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 6.00000 0.210559
\(813\) −16.0000 −0.561144
\(814\) 10.0000 0.350500
\(815\) 4.00000 0.140114
\(816\) 6.00000 0.210042
\(817\) −32.0000 −1.11954
\(818\) 14.0000 0.489499
\(819\) 2.00000 0.0698857
\(820\) 6.00000 0.209529
\(821\) 54.0000 1.88461 0.942306 − 0.334751i \(-0.108652\pi\)
0.942306 + 0.334751i \(0.108652\pi\)
\(822\) 6.00000 0.209274
\(823\) 32.0000 1.11545 0.557725 − 0.830026i \(-0.311674\pi\)
0.557725 + 0.830026i \(0.311674\pi\)
\(824\) −4.00000 −0.139347
\(825\) −1.00000 −0.0348155
\(826\) 12.0000 0.417533
\(827\) 12.0000 0.417281 0.208640 − 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 0 0
\(829\) −46.0000 −1.59765 −0.798823 − 0.601566i \(-0.794544\pi\)
−0.798823 + 0.601566i \(0.794544\pi\)
\(830\) 12.0000 0.416526
\(831\) −10.0000 −0.346896
\(832\) 2.00000 0.0693375
\(833\) 6.00000 0.207888
\(834\) −4.00000 −0.138509
\(835\) 12.0000 0.415277
\(836\) 4.00000 0.138343
\(837\) 8.00000 0.276520
\(838\) 12.0000 0.414533
\(839\) −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
\(842\) −10.0000 −0.344623
\(843\) −6.00000 −0.206651
\(844\) −4.00000 −0.137686
\(845\) 9.00000 0.309609
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) 6.00000 0.206041
\(849\) −4.00000 −0.137280
\(850\) 6.00000 0.205798
\(851\) 0 0
\(852\) 12.0000 0.411113
\(853\) 2.00000 0.0684787 0.0342393 − 0.999414i \(-0.489099\pi\)
0.0342393 + 0.999414i \(0.489099\pi\)
\(854\) 2.00000 0.0684386
\(855\) 4.00000 0.136797
\(856\) 12.0000 0.410152
\(857\) 54.0000 1.84460 0.922302 − 0.386469i \(-0.126305\pi\)
0.922302 + 0.386469i \(0.126305\pi\)
\(858\) −2.00000 −0.0682789
\(859\) 8.00000 0.272956 0.136478 − 0.990643i \(-0.456422\pi\)
0.136478 + 0.990643i \(0.456422\pi\)
\(860\) −8.00000 −0.272798
\(861\) −6.00000 −0.204479
\(862\) 0 0
\(863\) −24.0000 −0.816970 −0.408485 − 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) 1.00000 0.0340207
\(865\) −6.00000 −0.204006
\(866\) −34.0000 −1.15537
\(867\) 19.0000 0.645274
\(868\) 8.00000 0.271538
\(869\) 4.00000 0.135691
\(870\) −6.00000 −0.203419
\(871\) −8.00000 −0.271070
\(872\) 2.00000 0.0677285
\(873\) −10.0000 −0.338449
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) −10.0000 −0.337869
\(877\) −34.0000 −1.14810 −0.574049 − 0.818821i \(-0.694628\pi\)
−0.574049 + 0.818821i \(0.694628\pi\)
\(878\) 8.00000 0.269987
\(879\) −18.0000 −0.607125
\(880\) 1.00000 0.0337100
\(881\) −18.0000 −0.606435 −0.303218 − 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 1.00000 0.0336718
\(883\) 20.0000 0.673054 0.336527 − 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) 12.0000 0.403604
\(885\) −12.0000 −0.403376
\(886\) 24.0000 0.806296
\(887\) 36.0000 1.20876 0.604381 − 0.796696i \(-0.293421\pi\)
0.604381 + 0.796696i \(0.293421\pi\)
\(888\) −10.0000 −0.335578
\(889\) 8.00000 0.268311
\(890\) −6.00000 −0.201120
\(891\) −1.00000 −0.0335013
\(892\) −4.00000 −0.133930
\(893\) 0 0
\(894\) −18.0000 −0.602010
\(895\) 12.0000 0.401116
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) −6.00000 −0.200223
\(899\) 48.0000 1.60089
\(900\) 1.00000 0.0333333
\(901\) 36.0000 1.19933
\(902\) 6.00000 0.199778
\(903\) 8.00000 0.266223
\(904\) 6.00000 0.199557
\(905\) −2.00000 −0.0664822
\(906\) −4.00000 −0.132891
\(907\) −52.0000 −1.72663 −0.863316 − 0.504664i \(-0.831616\pi\)
−0.863316 + 0.504664i \(0.831616\pi\)
\(908\) −12.0000 −0.398234
\(909\) −6.00000 −0.199007
\(910\) −2.00000 −0.0662994
\(911\) −36.0000 −1.19273 −0.596367 − 0.802712i \(-0.703390\pi\)
−0.596367 + 0.802712i \(0.703390\pi\)
\(912\) −4.00000 −0.132453
\(913\) 12.0000 0.397142
\(914\) −10.0000 −0.330771
\(915\) −2.00000 −0.0661180
\(916\) −22.0000 −0.726900
\(917\) 0 0
\(918\) 6.00000 0.198030
\(919\) 20.0000 0.659739 0.329870 − 0.944027i \(-0.392995\pi\)
0.329870 + 0.944027i \(0.392995\pi\)
\(920\) 0 0
\(921\) −28.0000 −0.922631
\(922\) −6.00000 −0.197599
\(923\) 24.0000 0.789970
\(924\) −1.00000 −0.0328976
\(925\) −10.0000 −0.328798
\(926\) 32.0000 1.05159
\(927\) −4.00000 −0.131377
\(928\) 6.00000 0.196960
\(929\) 6.00000 0.196854 0.0984268 − 0.995144i \(-0.468619\pi\)
0.0984268 + 0.995144i \(0.468619\pi\)
\(930\) −8.00000 −0.262330
\(931\) −4.00000 −0.131095
\(932\) 18.0000 0.589610
\(933\) 24.0000 0.785725
\(934\) 12.0000 0.392652
\(935\) 6.00000 0.196221
\(936\) 2.00000 0.0653720
\(937\) 38.0000 1.24141 0.620703 − 0.784046i \(-0.286847\pi\)
0.620703 + 0.784046i \(0.286847\pi\)
\(938\) −4.00000 −0.130605
\(939\) −10.0000 −0.326338
\(940\) 0 0
\(941\) −30.0000 −0.977972 −0.488986 − 0.872292i \(-0.662633\pi\)
−0.488986 + 0.872292i \(0.662633\pi\)
\(942\) −10.0000 −0.325818
\(943\) 0 0
\(944\) 12.0000 0.390567
\(945\) −1.00000 −0.0325300
\(946\) −8.00000 −0.260102
\(947\) 48.0000 1.55979 0.779895 − 0.625910i \(-0.215272\pi\)
0.779895 + 0.625910i \(0.215272\pi\)
\(948\) −4.00000 −0.129914
\(949\) −20.0000 −0.649227
\(950\) −4.00000 −0.129777
\(951\) −18.0000 −0.583690
\(952\) 6.00000 0.194461
\(953\) 18.0000 0.583077 0.291539 − 0.956559i \(-0.405833\pi\)
0.291539 + 0.956559i \(0.405833\pi\)
\(954\) 6.00000 0.194257
\(955\) 12.0000 0.388311
\(956\) 0 0
\(957\) −6.00000 −0.193952
\(958\) 0 0
\(959\) 6.00000 0.193750
\(960\) −1.00000 −0.0322749
\(961\) 33.0000 1.06452
\(962\) −20.0000 −0.644826
\(963\) 12.0000 0.386695
\(964\) 14.0000 0.450910
\(965\) −14.0000 −0.450676
\(966\) 0 0
\(967\) 32.0000 1.02905 0.514525 − 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) 1.00000 0.0321412
\(969\) −24.0000 −0.770991
\(970\) 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
\(973\) −4.00000 −0.128234
\(974\) −40.0000 −1.28168
\(975\) 2.00000 0.0640513
\(976\) 2.00000 0.0640184
\(977\) 54.0000 1.72761 0.863807 − 0.503824i \(-0.168074\pi\)
0.863807 + 0.503824i \(0.168074\pi\)
\(978\) −4.00000 −0.127906
\(979\) −6.00000 −0.191761
\(980\) −1.00000 −0.0319438
\(981\) 2.00000 0.0638551
\(982\) 12.0000 0.382935
\(983\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) −6.00000 −0.191273
\(985\) −6.00000 −0.191176
\(986\) 36.0000 1.14647
\(987\) 0 0
\(988\) −8.00000 −0.254514
\(989\) 0 0
\(990\) 1.00000 0.0317821
\(991\) 56.0000 1.77890 0.889449 − 0.457034i \(-0.151088\pi\)
0.889449 + 0.457034i \(0.151088\pi\)
\(992\) 8.00000 0.254000
\(993\) 20.0000 0.634681
\(994\) 12.0000 0.380617
\(995\) 16.0000 0.507234
\(996\) −12.0000 −0.380235
\(997\) 50.0000 1.58352 0.791758 − 0.610835i \(-0.209166\pi\)
0.791758 + 0.610835i \(0.209166\pi\)
\(998\) −4.00000 −0.126618
\(999\) −10.0000 −0.316386
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.u.1.1 ✓ 1
3.2 odd 2 6930.2.a.q.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2310.2.a.u.1.1 ✓ 1 1.1 even 1 trivial
6930.2.a.q.1.1 1 3.2 odd 2