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Properties

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

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
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) −1.00000 −0.235702
\(19\) −8.00000 −1.83533 −0.917663 − 0.397360i \(-0.869927\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 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
\(26\) −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
\(37\) −6.00000 −0.986394 −0.493197 − 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 8.00000 1.29777
\(39\) 2.00000 0.320256
\(40\) −1.00000 −0.158114
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 1.00000 0.154303
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) −1.00000 −0.150756
\(45\) 1.00000 0.149071
\(46\) 8.00000 1.17954
\(47\) 12.0000 1.75038 0.875190 − 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) −2.00000 −0.280056
\(52\) 2.00000 0.277350
\(53\) −14.0000 −1.92305 −0.961524 − 0.274721i \(-0.911414\pi\)
−0.961524 + 0.274721i \(0.911414\pi\)
\(54\) −1.00000 −0.136083
\(55\) −1.00000 −0.134840
\(56\) 1.00000 0.133631
\(57\) −8.00000 −1.05963
\(58\) 6.00000 0.787839
\(59\) 4.00000 0.520756 0.260378 − 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 1.00000 0.129099
\(61\) −10.0000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) 1.00000 0.123091
\(67\) 8.00000 0.977356 0.488678 − 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) −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 \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 6.00000 0.697486
\(75\) 1.00000 0.115470
\(76\) −8.00000 −0.917663
\(77\) 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
\(82\) 2.00000 0.220863
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) −1.00000 −0.109109
\(85\) −2.00000 −0.216930
\(86\) −4.00000 −0.431331
\(87\) −6.00000 −0.643268
\(88\) 1.00000 0.106600
\(89\) −14.0000 −1.48400 −0.741999 − 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) −1.00000 −0.105409
\(91\) −2.00000 −0.209657
\(92\) −8.00000 −0.834058
\(93\) 0 0
\(94\) −12.0000 −1.23771
\(95\) −8.00000 −0.820783
\(96\) −1.00000 −0.102062
\(97\) 14.0000 1.42148 0.710742 − 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) −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
\(103\) 12.0000 1.18240 0.591198 − 0.806527i \(-0.298655\pi\)
0.591198 + 0.806527i \(0.298655\pi\)
\(104\) −2.00000 −0.196116
\(105\) −1.00000 −0.0975900
\(106\) 14.0000 1.35980
\(107\) 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
\(111\) −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 \(0.591070\pi\)
\(114\) 8.00000 0.749269
\(115\) −8.00000 −0.746004
\(116\) −6.00000 −0.557086
\(117\) 2.00000 0.184900
\(118\) −4.00000 −0.368230
\(119\) 2.00000 0.183340
\(120\) −1.00000 −0.0912871
\(121\) 1.00000 0.0909091
\(122\) 10.0000 0.905357
\(123\) −2.00000 −0.180334
\(124\) 0 0
\(125\) 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
\(129\) 4.00000 0.352180
\(130\) −2.00000 −0.175412
\(131\) −8.00000 −0.698963 −0.349482 − 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) −1.00000 −0.0870388
\(133\) 8.00000 0.693688
\(134\) −8.00000 −0.691095
\(135\) 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 \(0.472772\pi\)
\(138\) 8.00000 0.681005
\(139\) 16.0000 1.35710 0.678551 − 0.734553i \(-0.262608\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 12.0000 1.01058
\(142\) 0 0
\(143\) −2.00000 −0.167248
\(144\) 1.00000 0.0833333
\(145\) −6.00000 −0.498273
\(146\) 10.0000 0.827606
\(147\) 1.00000 0.0824786
\(148\) −6.00000 −0.493197
\(149\) −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 \(0.662373\pi\)
\(152\) 8.00000 0.648886
\(153\) −2.00000 −0.161690
\(154\) −1.00000 −0.0805823
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) 6.00000 0.478852 0.239426 − 0.970915i \(-0.423041\pi\)
0.239426 + 0.970915i \(0.423041\pi\)
\(158\) −4.00000 −0.318223
\(159\) −14.0000 −1.11027
\(160\) −1.00000 −0.0790569
\(161\) 8.00000 0.630488
\(162\) −1.00000 −0.0785674
\(163\) −8.00000 −0.626608 −0.313304 − 0.949653i \(-0.601436\pi\)
−0.313304 + 0.949653i \(0.601436\pi\)
\(164\) −2.00000 −0.156174
\(165\) −1.00000 −0.0778499
\(166\) 4.00000 0.310460
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 1.00000 0.0771517
\(169\) −9.00000 −0.692308
\(170\) 2.00000 0.153393
\(171\) −8.00000 −0.611775
\(172\) 4.00000 0.304997
\(173\) −22.0000 −1.67263 −0.836315 − 0.548250i \(-0.815294\pi\)
−0.836315 + 0.548250i \(0.815294\pi\)
\(174\) 6.00000 0.454859
\(175\) −1.00000 −0.0755929
\(176\) −1.00000 −0.0753778
\(177\) 4.00000 0.300658
\(178\) 14.0000 1.04934
\(179\) −20.0000 −1.49487 −0.747435 − 0.664335i \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) 1.00000 0.0745356
\(181\) 22.0000 1.63525 0.817624 − 0.575753i \(-0.195291\pi\)
0.817624 + 0.575753i \(0.195291\pi\)
\(182\) 2.00000 0.148250
\(183\) −10.0000 −0.739221
\(184\) 8.00000 0.589768
\(185\) −6.00000 −0.441129
\(186\) 0 0
\(187\) 2.00000 0.146254
\(188\) 12.0000 0.875190
\(189\) −1.00000 −0.0727393
\(190\) 8.00000 0.580381
\(191\) 24.0000 1.73658 0.868290 − 0.496058i \(-0.165220\pi\)
0.868290 + 0.496058i \(0.165220\pi\)
\(192\) 1.00000 0.0721688
\(193\) 22.0000 1.58359 0.791797 − 0.610784i \(-0.209146\pi\)
0.791797 + 0.610784i \(0.209146\pi\)
\(194\) −14.0000 −1.00514
\(195\) 2.00000 0.143223
\(196\) 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
\(199\) 16.0000 1.13421 0.567105 − 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 8.00000 0.564276
\(202\) 10.0000 0.703598
\(203\) 6.00000 0.421117
\(204\) −2.00000 −0.140028
\(205\) −2.00000 −0.139686
\(206\) −12.0000 −0.836080
\(207\) −8.00000 −0.556038
\(208\) 2.00000 0.138675
\(209\) 8.00000 0.553372
\(210\) 1.00000 0.0690066
\(211\) −28.0000 −1.92760 −0.963800 − 0.266627i \(-0.914091\pi\)
−0.963800 + 0.266627i \(0.914091\pi\)
\(212\) −14.0000 −0.961524
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) 4.00000 0.272798
\(216\) −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
\(222\) 6.00000 0.402694
\(223\) −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 \(0.630377\pi\)
\(228\) −8.00000 −0.529813
\(229\) −26.0000 −1.71813 −0.859064 − 0.511868i \(-0.828954\pi\)
−0.859064 + 0.511868i \(0.828954\pi\)
\(230\) 8.00000 0.527504
\(231\) 1.00000 0.0657952
\(232\) 6.00000 0.393919
\(233\) −10.0000 −0.655122 −0.327561 − 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) −2.00000 −0.130744
\(235\) 12.0000 0.782794
\(236\) 4.00000 0.260378
\(237\) 4.00000 0.259828
\(238\) −2.00000 −0.129641
\(239\) −20.0000 −1.29369 −0.646846 − 0.762620i \(-0.723912\pi\)
−0.646846 + 0.762620i \(0.723912\pi\)
\(240\) 1.00000 0.0645497
\(241\) −10.0000 −0.644157 −0.322078 − 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 1.00000 0.0641500
\(244\) −10.0000 −0.640184
\(245\) 1.00000 0.0638877
\(246\) 2.00000 0.127515
\(247\) −16.0000 −1.01806
\(248\) 0 0
\(249\) −4.00000 −0.253490
\(250\) −1.00000 −0.0632456
\(251\) −4.00000 −0.252478 −0.126239 − 0.992000i \(-0.540291\pi\)
−0.126239 + 0.992000i \(0.540291\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 8.00000 0.502956
\(254\) −8.00000 −0.501965
\(255\) −2.00000 −0.125245
\(256\) 1.00000 0.0625000
\(257\) −10.0000 −0.623783 −0.311891 − 0.950118i \(-0.600963\pi\)
−0.311891 + 0.950118i \(0.600963\pi\)
\(258\) −4.00000 −0.249029
\(259\) 6.00000 0.372822
\(260\) 2.00000 0.124035
\(261\) −6.00000 −0.371391
\(262\) 8.00000 0.494242
\(263\) 16.0000 0.986602 0.493301 − 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) 1.00000 0.0615457
\(265\) −14.0000 −0.860013
\(266\) −8.00000 −0.490511
\(267\) −14.0000 −0.856786
\(268\) 8.00000 0.488678
\(269\) 14.0000 0.853595 0.426798 − 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) −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
\(276\) −8.00000 −0.481543
\(277\) 10.0000 0.600842 0.300421 − 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) −16.0000 −0.959616
\(279\) 0 0
\(280\) 1.00000 0.0597614
\(281\) −6.00000 −0.357930 −0.178965 − 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) −12.0000 −0.714590
\(283\) −4.00000 −0.237775 −0.118888 − 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 0 0
\(285\) −8.00000 −0.473879
\(286\) 2.00000 0.118262
\(287\) 2.00000 0.118056
\(288\) −1.00000 −0.0589256
\(289\) −13.0000 −0.764706
\(290\) 6.00000 0.352332
\(291\) 14.0000 0.820695
\(292\) −10.0000 −0.585206
\(293\) 26.0000 1.51894 0.759468 − 0.650545i \(-0.225459\pi\)
0.759468 + 0.650545i \(0.225459\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 4.00000 0.232889
\(296\) 6.00000 0.348743
\(297\) −1.00000 −0.0580259
\(298\) 6.00000 0.347571
\(299\) −16.0000 −0.925304
\(300\) 1.00000 0.0577350
\(301\) −4.00000 −0.230556
\(302\) 12.0000 0.690522
\(303\) −10.0000 −0.574485
\(304\) −8.00000 −0.458831
\(305\) −10.0000 −0.572598
\(306\) 2.00000 0.114332
\(307\) 20.0000 1.14146 0.570730 − 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 1.00000 0.0569803
\(309\) 12.0000 0.682656
\(310\) 0 0
\(311\) −16.0000 −0.907277 −0.453638 − 0.891186i \(-0.649874\pi\)
−0.453638 + 0.891186i \(0.649874\pi\)
\(312\) −2.00000 −0.113228
\(313\) −18.0000 −1.01742 −0.508710 − 0.860938i \(-0.669877\pi\)
−0.508710 + 0.860938i \(0.669877\pi\)
\(314\) −6.00000 −0.338600
\(315\) −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 \(0.711985\pi\)
\(318\) 14.0000 0.785081
\(319\) 6.00000 0.335936
\(320\) 1.00000 0.0559017
\(321\) 12.0000 0.669775
\(322\) −8.00000 −0.445823
\(323\) 16.0000 0.890264
\(324\) 1.00000 0.0555556
\(325\) 2.00000 0.110940
\(326\) 8.00000 0.443079
\(327\) 2.00000 0.110600
\(328\) 2.00000 0.110432
\(329\) −12.0000 −0.661581
\(330\) 1.00000 0.0550482
\(331\) 28.0000 1.53902 0.769510 − 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) −4.00000 −0.219529
\(333\) −6.00000 −0.328798
\(334\) 0 0
\(335\) 8.00000 0.437087
\(336\) −1.00000 −0.0545545
\(337\) −10.0000 −0.544735 −0.272367 − 0.962193i \(-0.587807\pi\)
−0.272367 + 0.962193i \(0.587807\pi\)
\(338\) 9.00000 0.489535
\(339\) −6.00000 −0.325875
\(340\) −2.00000 −0.108465
\(341\) 0 0
\(342\) 8.00000 0.432590
\(343\) −1.00000 −0.0539949
\(344\) −4.00000 −0.215666
\(345\) −8.00000 −0.430706
\(346\) 22.0000 1.18273
\(347\) 28.0000 1.50312 0.751559 − 0.659665i \(-0.229302\pi\)
0.751559 + 0.659665i \(0.229302\pi\)
\(348\) −6.00000 −0.321634
\(349\) −2.00000 −0.107058 −0.0535288 − 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 1.00000 0.0534522
\(351\) 2.00000 0.106752
\(352\) 1.00000 0.0533002
\(353\) 22.0000 1.17094 0.585471 − 0.810693i \(-0.300910\pi\)
0.585471 + 0.810693i \(0.300910\pi\)
\(354\) −4.00000 −0.212598
\(355\) 0 0
\(356\) −14.0000 −0.741999
\(357\) 2.00000 0.105851
\(358\) 20.0000 1.05703
\(359\) −20.0000 −1.05556 −0.527780 − 0.849381i \(-0.676975\pi\)
−0.527780 + 0.849381i \(0.676975\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 45.0000 2.36842
\(362\) −22.0000 −1.15629
\(363\) 1.00000 0.0524864
\(364\) −2.00000 −0.104828
\(365\) −10.0000 −0.523424
\(366\) 10.0000 0.522708
\(367\) 4.00000 0.208798 0.104399 − 0.994535i \(-0.466708\pi\)
0.104399 + 0.994535i \(0.466708\pi\)
\(368\) −8.00000 −0.417029
\(369\) −2.00000 −0.104116
\(370\) 6.00000 0.311925
\(371\) 14.0000 0.726844
\(372\) 0 0
\(373\) 10.0000 0.517780 0.258890 − 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) −2.00000 −0.103418
\(375\) 1.00000 0.0516398
\(376\) −12.0000 −0.618853
\(377\) −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 \(0.467241\pi\)
\(380\) −8.00000 −0.410391
\(381\) 8.00000 0.409852
\(382\) −24.0000 −1.22795
\(383\) −12.0000 −0.613171 −0.306586 − 0.951843i \(-0.599187\pi\)
−0.306586 + 0.951843i \(0.599187\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 1.00000 0.0509647
\(386\) −22.0000 −1.11977
\(387\) 4.00000 0.203331
\(388\) 14.0000 0.710742
\(389\) −18.0000 −0.912636 −0.456318 − 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) −2.00000 −0.101274
\(391\) 16.0000 0.809155
\(392\) −1.00000 −0.0505076
\(393\) −8.00000 −0.403547
\(394\) 6.00000 0.302276
\(395\) 4.00000 0.201262
\(396\) −1.00000 −0.0502519
\(397\) 22.0000 1.10415 0.552074 − 0.833795i \(-0.313837\pi\)
0.552074 + 0.833795i \(0.313837\pi\)
\(398\) −16.0000 −0.802008
\(399\) 8.00000 0.400501
\(400\) 1.00000 0.0500000
\(401\) −14.0000 −0.699127 −0.349563 − 0.936913i \(-0.613670\pi\)
−0.349563 + 0.936913i \(0.613670\pi\)
\(402\) −8.00000 −0.399004
\(403\) 0 0
\(404\) −10.0000 −0.497519
\(405\) 1.00000 0.0496904
\(406\) −6.00000 −0.297775
\(407\) 6.00000 0.297409
\(408\) 2.00000 0.0990148
\(409\) −10.0000 −0.494468 −0.247234 − 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 2.00000 0.0987730
\(411\) 2.00000 0.0986527
\(412\) 12.0000 0.591198
\(413\) −4.00000 −0.196827
\(414\) 8.00000 0.393179
\(415\) −4.00000 −0.196352
\(416\) −2.00000 −0.0980581
\(417\) 16.0000 0.783523
\(418\) −8.00000 −0.391293
\(419\) −12.0000 −0.586238 −0.293119 − 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) −1.00000 −0.0487950
\(421\) 22.0000 1.07221 0.536107 − 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 28.0000 1.36302
\(423\) 12.0000 0.583460
\(424\) 14.0000 0.679900
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) 10.0000 0.483934
\(428\) 12.0000 0.580042
\(429\) −2.00000 −0.0965609
\(430\) −4.00000 −0.192897
\(431\) 36.0000 1.73406 0.867029 − 0.498257i \(-0.166026\pi\)
0.867029 + 0.498257i \(0.166026\pi\)
\(432\) 1.00000 0.0481125
\(433\) −26.0000 −1.24948 −0.624740 − 0.780833i \(-0.714795\pi\)
−0.624740 + 0.780833i \(0.714795\pi\)
\(434\) 0 0
\(435\) −6.00000 −0.287678
\(436\) 2.00000 0.0957826
\(437\) 64.0000 3.06154
\(438\) 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
\(443\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) −6.00000 −0.284747
\(445\) −14.0000 −0.663664
\(446\) 4.00000 0.189405
\(447\) −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
\(451\) 2.00000 0.0941763
\(452\) −6.00000 −0.282216
\(453\) −12.0000 −0.563809
\(454\) 12.0000 0.563188
\(455\) −2.00000 −0.0937614
\(456\) 8.00000 0.374634
\(457\) 6.00000 0.280668 0.140334 − 0.990104i \(-0.455182\pi\)
0.140334 + 0.990104i \(0.455182\pi\)
\(458\) 26.0000 1.21490
\(459\) −2.00000 −0.0933520
\(460\) −8.00000 −0.373002
\(461\) −10.0000 −0.465746 −0.232873 − 0.972507i \(-0.574813\pi\)
−0.232873 + 0.972507i \(0.574813\pi\)
\(462\) −1.00000 −0.0465242
\(463\) −16.0000 −0.743583 −0.371792 − 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 10.0000 0.463241
\(467\) 28.0000 1.29569 0.647843 − 0.761774i \(-0.275671\pi\)
0.647843 + 0.761774i \(0.275671\pi\)
\(468\) 2.00000 0.0924500
\(469\) −8.00000 −0.369406
\(470\) −12.0000 −0.553519
\(471\) 6.00000 0.276465
\(472\) −4.00000 −0.184115
\(473\) −4.00000 −0.183920
\(474\) −4.00000 −0.183726
\(475\) −8.00000 −0.367065
\(476\) 2.00000 0.0916698
\(477\) −14.0000 −0.641016
\(478\) 20.0000 0.914779
\(479\) −8.00000 −0.365529 −0.182765 − 0.983157i \(-0.558505\pi\)
−0.182765 + 0.983157i \(0.558505\pi\)
\(480\) −1.00000 −0.0456435
\(481\) −12.0000 −0.547153
\(482\) 10.0000 0.455488
\(483\) 8.00000 0.364013
\(484\) 1.00000 0.0454545
\(485\) 14.0000 0.635707
\(486\) −1.00000 −0.0453609
\(487\) −16.0000 −0.725029 −0.362515 − 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) 10.0000 0.452679
\(489\) −8.00000 −0.361773
\(490\) −1.00000 −0.0451754
\(491\) −28.0000 −1.26362 −0.631811 − 0.775122i \(-0.717688\pi\)
−0.631811 + 0.775122i \(0.717688\pi\)
\(492\) −2.00000 −0.0901670
\(493\) 12.0000 0.540453
\(494\) 16.0000 0.719874
\(495\) −1.00000 −0.0449467
\(496\) 0 0
\(497\) 0 0
\(498\) 4.00000 0.179244
\(499\) 4.00000 0.179065 0.0895323 − 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) 4.00000 0.178529
\(503\) 24.0000 1.07011 0.535054 − 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 1.00000 0.0445435
\(505\) −10.0000 −0.444994
\(506\) −8.00000 −0.355643
\(507\) −9.00000 −0.399704
\(508\) 8.00000 0.354943
\(509\) −34.0000 −1.50702 −0.753512 − 0.657434i \(-0.771642\pi\)
−0.753512 + 0.657434i \(0.771642\pi\)
\(510\) 2.00000 0.0885615
\(511\) 10.0000 0.442374
\(512\) −1.00000 −0.0441942
\(513\) −8.00000 −0.353209
\(514\) 10.0000 0.441081
\(515\) 12.0000 0.528783
\(516\) 4.00000 0.176090
\(517\) −12.0000 −0.527759
\(518\) −6.00000 −0.263625
\(519\) −22.0000 −0.965693
\(520\) −2.00000 −0.0877058
\(521\) 18.0000 0.788594 0.394297 − 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 6.00000 0.262613
\(523\) −36.0000 −1.57417 −0.787085 − 0.616844i \(-0.788411\pi\)
−0.787085 + 0.616844i \(0.788411\pi\)
\(524\) −8.00000 −0.349482
\(525\) −1.00000 −0.0436436
\(526\) −16.0000 −0.697633
\(527\) 0 0
\(528\) −1.00000 −0.0435194
\(529\) 41.0000 1.78261
\(530\) 14.0000 0.608121
\(531\) 4.00000 0.173585
\(532\) 8.00000 0.346844
\(533\) −4.00000 −0.173259
\(534\) 14.0000 0.605839
\(535\) 12.0000 0.518805
\(536\) −8.00000 −0.345547
\(537\) −20.0000 −0.863064
\(538\) −14.0000 −0.603583
\(539\) −1.00000 −0.0430730
\(540\) 1.00000 0.0430331
\(541\) −22.0000 −0.945854 −0.472927 − 0.881102i \(-0.656803\pi\)
−0.472927 + 0.881102i \(0.656803\pi\)
\(542\) 8.00000 0.343629
\(543\) 22.0000 0.944110
\(544\) 2.00000 0.0857493
\(545\) 2.00000 0.0856706
\(546\) 2.00000 0.0855921
\(547\) 20.0000 0.855138 0.427569 − 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) 2.00000 0.0854358
\(549\) −10.0000 −0.426790
\(550\) 1.00000 0.0426401
\(551\) 48.0000 2.04487
\(552\) 8.00000 0.340503
\(553\) −4.00000 −0.170097
\(554\) −10.0000 −0.424859
\(555\) −6.00000 −0.254686
\(556\) 16.0000 0.678551
\(557\) 18.0000 0.762684 0.381342 − 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) −1.00000 −0.0422577
\(561\) 2.00000 0.0844401
\(562\) 6.00000 0.253095
\(563\) 12.0000 0.505740 0.252870 − 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 12.0000 0.505291
\(565\) −6.00000 −0.252422
\(566\) 4.00000 0.168133
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 26.0000 1.08998 0.544988 − 0.838444i \(-0.316534\pi\)
0.544988 + 0.838444i \(0.316534\pi\)
\(570\) 8.00000 0.335083
\(571\) 28.0000 1.17176 0.585882 − 0.810397i \(-0.300748\pi\)
0.585882 + 0.810397i \(0.300748\pi\)
\(572\) −2.00000 −0.0836242
\(573\) 24.0000 1.00261
\(574\) −2.00000 −0.0834784
\(575\) −8.00000 −0.333623
\(576\) 1.00000 0.0416667
\(577\) 30.0000 1.24892 0.624458 − 0.781058i \(-0.285320\pi\)
0.624458 + 0.781058i \(0.285320\pi\)
\(578\) 13.0000 0.540729
\(579\) 22.0000 0.914289
\(580\) −6.00000 −0.249136
\(581\) 4.00000 0.165948
\(582\) −14.0000 −0.580319
\(583\) 14.0000 0.579821
\(584\) 10.0000 0.413803
\(585\) 2.00000 0.0826898
\(586\) −26.0000 −1.07405
\(587\) 20.0000 0.825488 0.412744 − 0.910847i \(-0.364570\pi\)
0.412744 + 0.910847i \(0.364570\pi\)
\(588\) 1.00000 0.0412393
\(589\) 0 0
\(590\) −4.00000 −0.164677
\(591\) −6.00000 −0.246807
\(592\) −6.00000 −0.246598
\(593\) 46.0000 1.88899 0.944497 − 0.328521i \(-0.106550\pi\)
0.944497 + 0.328521i \(0.106550\pi\)
\(594\) 1.00000 0.0410305
\(595\) 2.00000 0.0819920
\(596\) −6.00000 −0.245770
\(597\) 16.0000 0.654836
\(598\) 16.0000 0.654289
\(599\) −16.0000 −0.653742 −0.326871 − 0.945069i \(-0.605994\pi\)
−0.326871 + 0.945069i \(0.605994\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −10.0000 −0.407909 −0.203954 − 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 4.00000 0.163028
\(603\) 8.00000 0.325785
\(604\) −12.0000 −0.488273
\(605\) 1.00000 0.0406558
\(606\) 10.0000 0.406222
\(607\) −32.0000 −1.29884 −0.649420 − 0.760430i \(-0.724988\pi\)
−0.649420 + 0.760430i \(0.724988\pi\)
\(608\) 8.00000 0.324443
\(609\) 6.00000 0.243132
\(610\) 10.0000 0.404888
\(611\) 24.0000 0.970936
\(612\) −2.00000 −0.0808452
\(613\) −38.0000 −1.53481 −0.767403 − 0.641165i \(-0.778451\pi\)
−0.767403 + 0.641165i \(0.778451\pi\)
\(614\) −20.0000 −0.807134
\(615\) −2.00000 −0.0806478
\(616\) −1.00000 −0.0402911
\(617\) 26.0000 1.04672 0.523360 − 0.852111i \(-0.324678\pi\)
0.523360 + 0.852111i \(0.324678\pi\)
\(618\) −12.0000 −0.482711
\(619\) −20.0000 −0.803868 −0.401934 − 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) −8.00000 −0.321029
\(622\) 16.0000 0.641542
\(623\) 14.0000 0.560898
\(624\) 2.00000 0.0800641
\(625\) 1.00000 0.0400000
\(626\) 18.0000 0.719425
\(627\) 8.00000 0.319489
\(628\) 6.00000 0.239426
\(629\) 12.0000 0.478471
\(630\) 1.00000 0.0398410
\(631\) 40.0000 1.59237 0.796187 − 0.605050i \(-0.206847\pi\)
0.796187 + 0.605050i \(0.206847\pi\)
\(632\) −4.00000 −0.159111
\(633\) −28.0000 −1.11290
\(634\) 22.0000 0.873732
\(635\) 8.00000 0.317470
\(636\) −14.0000 −0.555136
\(637\) 2.00000 0.0792429
\(638\) −6.00000 −0.237542
\(639\) 0 0
\(640\) −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
\(643\) 4.00000 0.157745 0.0788723 − 0.996885i \(-0.474868\pi\)
0.0788723 + 0.996885i \(0.474868\pi\)
\(644\) 8.00000 0.315244
\(645\) 4.00000 0.157500
\(646\) −16.0000 −0.629512
\(647\) −12.0000 −0.471769 −0.235884 − 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −4.00000 −0.157014
\(650\) −2.00000 −0.0784465
\(651\) 0 0
\(652\) −8.00000 −0.313304
\(653\) 26.0000 1.01746 0.508729 − 0.860927i \(-0.330115\pi\)
0.508729 + 0.860927i \(0.330115\pi\)
\(654\) −2.00000 −0.0782062
\(655\) −8.00000 −0.312586
\(656\) −2.00000 −0.0780869
\(657\) −10.0000 −0.390137
\(658\) 12.0000 0.467809
\(659\) −4.00000 −0.155818 −0.0779089 − 0.996960i \(-0.524824\pi\)
−0.0779089 + 0.996960i \(0.524824\pi\)
\(660\) −1.00000 −0.0389249
\(661\) −10.0000 −0.388955 −0.194477 − 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) −28.0000 −1.08825
\(663\) −4.00000 −0.155347
\(664\) 4.00000 0.155230
\(665\) 8.00000 0.310227
\(666\) 6.00000 0.232495
\(667\) 48.0000 1.85857
\(668\) 0 0
\(669\) −4.00000 −0.154649
\(670\) −8.00000 −0.309067
\(671\) 10.0000 0.386046
\(672\) 1.00000 0.0385758
\(673\) 38.0000 1.46479 0.732396 − 0.680879i \(-0.238402\pi\)
0.732396 + 0.680879i \(0.238402\pi\)
\(674\) 10.0000 0.385186
\(675\) 1.00000 0.0384900
\(676\) −9.00000 −0.346154
\(677\) −38.0000 −1.46046 −0.730229 − 0.683202i \(-0.760587\pi\)
−0.730229 + 0.683202i \(0.760587\pi\)
\(678\) 6.00000 0.230429
\(679\) −14.0000 −0.537271
\(680\) 2.00000 0.0766965
\(681\) −12.0000 −0.459841
\(682\) 0 0
\(683\) 48.0000 1.83667 0.918334 − 0.395805i \(-0.129534\pi\)
0.918334 + 0.395805i \(0.129534\pi\)
\(684\) −8.00000 −0.305888
\(685\) 2.00000 0.0764161
\(686\) 1.00000 0.0381802
\(687\) −26.0000 −0.991962
\(688\) 4.00000 0.152499
\(689\) −28.0000 −1.06672
\(690\) 8.00000 0.304555
\(691\) −20.0000 −0.760836 −0.380418 − 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) −22.0000 −0.836315
\(693\) 1.00000 0.0379869
\(694\) −28.0000 −1.06287
\(695\) 16.0000 0.606915
\(696\) 6.00000 0.227429
\(697\) 4.00000 0.151511
\(698\) 2.00000 0.0757011
\(699\) −10.0000 −0.378235
\(700\) −1.00000 −0.0377964
\(701\) 2.00000 0.0755390 0.0377695 − 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) −2.00000 −0.0754851
\(703\) 48.0000 1.81035
\(704\) −1.00000 −0.0376889
\(705\) 12.0000 0.451946
\(706\) −22.0000 −0.827981
\(707\) 10.0000 0.376089
\(708\) 4.00000 0.150329
\(709\) 46.0000 1.72757 0.863783 − 0.503864i \(-0.168089\pi\)
0.863783 + 0.503864i \(0.168089\pi\)
\(710\) 0 0
\(711\) 4.00000 0.150012
\(712\) 14.0000 0.524672
\(713\) 0 0
\(714\) −2.00000 −0.0748481
\(715\) −2.00000 −0.0747958
\(716\) −20.0000 −0.747435
\(717\) −20.0000 −0.746914
\(718\) 20.0000 0.746393
\(719\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 1.00000 0.0372678
\(721\) −12.0000 −0.446903
\(722\) −45.0000 −1.67473
\(723\) −10.0000 −0.371904
\(724\) 22.0000 0.817624
\(725\) −6.00000 −0.222834
\(726\) −1.00000 −0.0371135
\(727\) −20.0000 −0.741759 −0.370879 − 0.928681i \(-0.620944\pi\)
−0.370879 + 0.928681i \(0.620944\pi\)
\(728\) 2.00000 0.0741249
\(729\) 1.00000 0.0370370
\(730\) 10.0000 0.370117
\(731\) −8.00000 −0.295891
\(732\) −10.0000 −0.369611
\(733\) 26.0000 0.960332 0.480166 − 0.877178i \(-0.340576\pi\)
0.480166 + 0.877178i \(0.340576\pi\)
\(734\) −4.00000 −0.147643
\(735\) 1.00000 0.0368856
\(736\) 8.00000 0.294884
\(737\) −8.00000 −0.294684
\(738\) 2.00000 0.0736210
\(739\) −12.0000 −0.441427 −0.220714 − 0.975339i \(-0.570839\pi\)
−0.220714 + 0.975339i \(0.570839\pi\)
\(740\) −6.00000 −0.220564
\(741\) −16.0000 −0.587775
\(742\) −14.0000 −0.513956
\(743\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) −6.00000 −0.219823
\(746\) −10.0000 −0.366126
\(747\) −4.00000 −0.146352
\(748\) 2.00000 0.0731272
\(749\) −12.0000 −0.438470
\(750\) −1.00000 −0.0365148
\(751\) −40.0000 −1.45962 −0.729810 − 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 12.0000 0.437595
\(753\) −4.00000 −0.145768
\(754\) 12.0000 0.437014
\(755\) −12.0000 −0.436725
\(756\) −1.00000 −0.0363696
\(757\) −38.0000 −1.38113 −0.690567 − 0.723269i \(-0.742639\pi\)
−0.690567 + 0.723269i \(0.742639\pi\)
\(758\) −4.00000 −0.145287
\(759\) 8.00000 0.290382
\(760\) 8.00000 0.290191
\(761\) 38.0000 1.37750 0.688749 − 0.724999i \(-0.258160\pi\)
0.688749 + 0.724999i \(0.258160\pi\)
\(762\) −8.00000 −0.289809
\(763\) −2.00000 −0.0724049
\(764\) 24.0000 0.868290
\(765\) −2.00000 −0.0723102
\(766\) 12.0000 0.433578
\(767\) 8.00000 0.288863
\(768\) 1.00000 0.0360844
\(769\) 14.0000 0.504853 0.252426 − 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) −1.00000 −0.0360375
\(771\) −10.0000 −0.360141
\(772\) 22.0000 0.791797
\(773\) 14.0000 0.503545 0.251773 − 0.967786i \(-0.418987\pi\)
0.251773 + 0.967786i \(0.418987\pi\)
\(774\) −4.00000 −0.143777
\(775\) 0 0
\(776\) −14.0000 −0.502571
\(777\) 6.00000 0.215249
\(778\) 18.0000 0.645331
\(779\) 16.0000 0.573259
\(780\) 2.00000 0.0716115
\(781\) 0 0
\(782\) −16.0000 −0.572159
\(783\) −6.00000 −0.214423
\(784\) 1.00000 0.0357143
\(785\) 6.00000 0.214149
\(786\) 8.00000 0.285351
\(787\) 12.0000 0.427754 0.213877 − 0.976861i \(-0.431391\pi\)
0.213877 + 0.976861i \(0.431391\pi\)
\(788\) −6.00000 −0.213741
\(789\) 16.0000 0.569615
\(790\) −4.00000 −0.142314
\(791\) 6.00000 0.213335
\(792\) 1.00000 0.0355335
\(793\) −20.0000 −0.710221
\(794\) −22.0000 −0.780751
\(795\) −14.0000 −0.496529
\(796\) 16.0000 0.567105
\(797\) 22.0000 0.779280 0.389640 − 0.920967i \(-0.372599\pi\)
0.389640 + 0.920967i \(0.372599\pi\)
\(798\) −8.00000 −0.283197
\(799\) −24.0000 −0.849059
\(800\) −1.00000 −0.0353553
\(801\) −14.0000 −0.494666
\(802\) 14.0000 0.494357
\(803\) 10.0000 0.352892
\(804\) 8.00000 0.282138
\(805\) 8.00000 0.281963
\(806\) 0 0
\(807\) 14.0000 0.492823
\(808\) 10.0000 0.351799
\(809\) −6.00000 −0.210949 −0.105474 − 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −40.0000 −1.40459 −0.702295 − 0.711886i \(-0.747841\pi\)
−0.702295 + 0.711886i \(0.747841\pi\)
\(812\) 6.00000 0.210559
\(813\) −8.00000 −0.280572
\(814\) −6.00000 −0.210300
\(815\) −8.00000 −0.280228
\(816\) −2.00000 −0.0700140
\(817\) −32.0000 −1.11954
\(818\) 10.0000 0.349642
\(819\) −2.00000 −0.0698857
\(820\) −2.00000 −0.0698430
\(821\) −6.00000 −0.209401 −0.104701 − 0.994504i \(-0.533388\pi\)
−0.104701 + 0.994504i \(0.533388\pi\)
\(822\) −2.00000 −0.0697580
\(823\) −16.0000 −0.557725 −0.278862 − 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) −12.0000 −0.418040
\(825\) −1.00000 −0.0348155
\(826\) 4.00000 0.139178
\(827\) 20.0000 0.695468 0.347734 − 0.937593i \(-0.386951\pi\)
0.347734 + 0.937593i \(0.386951\pi\)
\(828\) −8.00000 −0.278019
\(829\) 30.0000 1.04194 0.520972 − 0.853574i \(-0.325570\pi\)
0.520972 + 0.853574i \(0.325570\pi\)
\(830\) 4.00000 0.138842
\(831\) 10.0000 0.346896
\(832\) 2.00000 0.0693375
\(833\) −2.00000 −0.0692959
\(834\) −16.0000 −0.554035
\(835\) 0 0
\(836\) 8.00000 0.276686
\(837\) 0 0
\(838\) 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
\(842\) −22.0000 −0.758170
\(843\) −6.00000 −0.206651
\(844\) −28.0000 −0.963800
\(845\) −9.00000 −0.309609
\(846\) −12.0000 −0.412568
\(847\) −1.00000 −0.0343604
\(848\) −14.0000 −0.480762
\(849\) −4.00000 −0.137280
\(850\) 2.00000 0.0685994
\(851\) 48.0000 1.64542
\(852\) 0 0
\(853\) 34.0000 1.16414 0.582069 − 0.813139i \(-0.302243\pi\)
0.582069 + 0.813139i \(0.302243\pi\)
\(854\) −10.0000 −0.342193
\(855\) −8.00000 −0.273594
\(856\) −12.0000 −0.410152
\(857\) −42.0000 −1.43469 −0.717346 − 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) 2.00000 0.0682789
\(859\) −20.0000 −0.682391 −0.341196 − 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) 4.00000 0.136399
\(861\) 2.00000 0.0681598
\(862\) −36.0000 −1.22616
\(863\) −24.0000 −0.816970 −0.408485 − 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −22.0000 −0.748022
\(866\) 26.0000 0.883516
\(867\) −13.0000 −0.441503
\(868\) 0 0
\(869\) −4.00000 −0.135691
\(870\) 6.00000 0.203419
\(871\) 16.0000 0.542139
\(872\) −2.00000 −0.0677285
\(873\) 14.0000 0.473828
\(874\) −64.0000 −2.16483
\(875\) −1.00000 −0.0338062
\(876\) −10.0000 −0.337869
\(877\) 2.00000 0.0675352 0.0337676 − 0.999430i \(-0.489249\pi\)
0.0337676 + 0.999430i \(0.489249\pi\)
\(878\) 0 0
\(879\) 26.0000 0.876958
\(880\) −1.00000 −0.0337100
\(881\) 18.0000 0.606435 0.303218 − 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) −1.00000 −0.0336718
\(883\) 16.0000 0.538443 0.269221 − 0.963078i \(-0.413234\pi\)
0.269221 + 0.963078i \(0.413234\pi\)
\(884\) −4.00000 −0.134535
\(885\) 4.00000 0.134459
\(886\) 0 0
\(887\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 6.00000 0.201347
\(889\) −8.00000 −0.268311
\(890\) 14.0000 0.469281
\(891\) −1.00000 −0.0335013
\(892\) −4.00000 −0.133930
\(893\) −96.0000 −3.21252
\(894\) 6.00000 0.200670
\(895\) −20.0000 −0.668526
\(896\) 1.00000 0.0334077
\(897\) −16.0000 −0.534224
\(898\) −18.0000 −0.600668
\(899\) 0 0
\(900\) 1.00000 0.0333333
\(901\) 28.0000 0.932815
\(902\) −2.00000 −0.0665927
\(903\) −4.00000 −0.133112
\(904\) 6.00000 0.199557
\(905\) 22.0000 0.731305
\(906\) 12.0000 0.398673
\(907\) 40.0000 1.32818 0.664089 − 0.747653i \(-0.268820\pi\)
0.664089 + 0.747653i \(0.268820\pi\)
\(908\) −12.0000 −0.398234
\(909\) −10.0000 −0.331679
\(910\) 2.00000 0.0662994
\(911\) −16.0000 −0.530104 −0.265052 − 0.964234i \(-0.585389\pi\)
−0.265052 + 0.964234i \(0.585389\pi\)
\(912\) −8.00000 −0.264906
\(913\) 4.00000 0.132381
\(914\) −6.00000 −0.198462
\(915\) −10.0000 −0.330590
\(916\) −26.0000 −0.859064
\(917\) 8.00000 0.264183
\(918\) 2.00000 0.0660098
\(919\) 4.00000 0.131948 0.0659739 − 0.997821i \(-0.478985\pi\)
0.0659739 + 0.997821i \(0.478985\pi\)
\(920\) 8.00000 0.263752
\(921\) 20.0000 0.659022
\(922\) 10.0000 0.329332
\(923\) 0 0
\(924\) 1.00000 0.0328976
\(925\) −6.00000 −0.197279
\(926\) 16.0000 0.525793
\(927\) 12.0000 0.394132
\(928\) 6.00000 0.196960
\(929\) 26.0000 0.853032 0.426516 − 0.904480i \(-0.359741\pi\)
0.426516 + 0.904480i \(0.359741\pi\)
\(930\) 0 0
\(931\) −8.00000 −0.262189
\(932\) −10.0000 −0.327561
\(933\) −16.0000 −0.523816
\(934\) −28.0000 −0.916188
\(935\) 2.00000 0.0654070
\(936\) −2.00000 −0.0653720
\(937\) −26.0000 −0.849383 −0.424691 − 0.905338i \(-0.639617\pi\)
−0.424691 + 0.905338i \(0.639617\pi\)
\(938\) 8.00000 0.261209
\(939\) −18.0000 −0.587408
\(940\) 12.0000 0.391397
\(941\) 30.0000 0.977972 0.488986 − 0.872292i \(-0.337367\pi\)
0.488986 + 0.872292i \(0.337367\pi\)
\(942\) −6.00000 −0.195491
\(943\) 16.0000 0.521032
\(944\) 4.00000 0.130189
\(945\) −1.00000 −0.0325300
\(946\) 4.00000 0.130051
\(947\) 32.0000 1.03986 0.519930 − 0.854209i \(-0.325958\pi\)
0.519930 + 0.854209i \(0.325958\pi\)
\(948\) 4.00000 0.129914
\(949\) −20.0000 −0.649227
\(950\) 8.00000 0.259554
\(951\) −22.0000 −0.713399
\(952\) −2.00000 −0.0648204
\(953\) −26.0000 −0.842223 −0.421111 − 0.907009i \(-0.638360\pi\)
−0.421111 + 0.907009i \(0.638360\pi\)
\(954\) 14.0000 0.453267
\(955\) 24.0000 0.776622
\(956\) −20.0000 −0.646846
\(957\) 6.00000 0.193952
\(958\) 8.00000 0.258468
\(959\) −2.00000 −0.0645834
\(960\) 1.00000 0.0322749
\(961\) −31.0000 −1.00000
\(962\) 12.0000 0.386896
\(963\) 12.0000 0.386695
\(964\) −10.0000 −0.322078
\(965\) 22.0000 0.708205
\(966\) −8.00000 −0.257396
\(967\) −32.0000 −1.02905 −0.514525 − 0.857475i \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 16.0000 0.513994
\(970\) −14.0000 −0.449513
\(971\) −60.0000 −1.92549 −0.962746 − 0.270408i \(-0.912841\pi\)
−0.962746 + 0.270408i \(0.912841\pi\)
\(972\) 1.00000 0.0320750
\(973\) −16.0000 −0.512936
\(974\) 16.0000 0.512673
\(975\) 2.00000 0.0640513
\(976\) −10.0000 −0.320092
\(977\) 42.0000 1.34370 0.671850 − 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) 8.00000 0.255812
\(979\) 14.0000 0.447442
\(980\) 1.00000 0.0319438
\(981\) 2.00000 0.0638551
\(982\) 28.0000 0.893516
\(983\) −36.0000 −1.14822 −0.574111 − 0.818778i \(-0.694652\pi\)
−0.574111 + 0.818778i \(0.694652\pi\)
\(984\) 2.00000 0.0637577
\(985\) −6.00000 −0.191176
\(986\) −12.0000 −0.382158
\(987\) −12.0000 −0.381964
\(988\) −16.0000 −0.509028
\(989\) −32.0000 −1.01754
\(990\) 1.00000 0.0317821
\(991\) 56.0000 1.77890 0.889449 − 0.457034i \(-0.151088\pi\)
0.889449 + 0.457034i \(0.151088\pi\)
\(992\) 0 0
\(993\) 28.0000 0.888553
\(994\) 0 0
\(995\) 16.0000 0.507234
\(996\) −4.00000 −0.126745
\(997\) −62.0000 −1.96356 −0.981780 − 0.190022i \(-0.939144\pi\)
−0.981780 + 0.190022i \(0.939144\pi\)
\(998\) −4.00000 −0.126618
\(999\) −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.k.1.1 ✓ 1
3.2 odd 2 6930.2.a.w.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2310.2.a.k.1.1 ✓ 1 1.1 even 1 trivial
6930.2.a.w.1.1 1 3.2 odd 2