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Properties

Label 8670.2.a.i.1.1
Level $8670$
Weight $2$
Character 8670.1
Self dual yes
Analytic conductor $69.230$
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] = [8670,2,Mod(1,8670)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8670, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("8670.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8670 = 2 \cdot 3 \cdot 5 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8670.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: \(69.2302985525\)
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 \(\chi\) \(=\) 8670.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +3.00000 q^{11} +1.00000 q^{12} -1.00000 q^{13} -3.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} +4.00000 q^{19} -1.00000 q^{20} +3.00000 q^{21} -3.00000 q^{22} +1.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{26} +1.00000 q^{27} +3.00000 q^{28} +1.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} +3.00000 q^{33} -3.00000 q^{35} +1.00000 q^{36} +6.00000 q^{37} -4.00000 q^{38} -1.00000 q^{39} +1.00000 q^{40} -2.00000 q^{41} -3.00000 q^{42} +10.0000 q^{43} +3.00000 q^{44} -1.00000 q^{45} -1.00000 q^{46} -9.00000 q^{47} +1.00000 q^{48} +2.00000 q^{49} -1.00000 q^{50} -1.00000 q^{52} -1.00000 q^{53} -1.00000 q^{54} -3.00000 q^{55} -3.00000 q^{56} +4.00000 q^{57} +9.00000 q^{59} -1.00000 q^{60} +10.0000 q^{61} +4.00000 q^{62} +3.00000 q^{63} +1.00000 q^{64} +1.00000 q^{65} -3.00000 q^{66} +8.00000 q^{67} +1.00000 q^{69} +3.00000 q^{70} -2.00000 q^{71} -1.00000 q^{72} -6.00000 q^{74} +1.00000 q^{75} +4.00000 q^{76} +9.00000 q^{77} +1.00000 q^{78} -16.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} -2.00000 q^{83} +3.00000 q^{84} -10.0000 q^{86} -3.00000 q^{88} +11.0000 q^{89} +1.00000 q^{90} -3.00000 q^{91} +1.00000 q^{92} -4.00000 q^{93} +9.00000 q^{94} -4.00000 q^{95} -1.00000 q^{96} +4.00000 q^{97} -2.00000 q^{98} +3.00000 q^{99} +O(q^{100})\)

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
\(7\) 3.00000 1.13389 0.566947 − 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 3.00000 0.904534 0.452267 − 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 1.00000 0.288675
\(13\) −1.00000 −0.277350 −0.138675 − 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) −3.00000 −0.801784
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 0 0
\(18\) −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
\(21\) 3.00000 0.654654
\(22\) −3.00000 −0.639602
\(23\) 1.00000 0.208514 0.104257 − 0.994550i \(-0.466753\pi\)
0.104257 + 0.994550i \(0.466753\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 1.00000 0.196116
\(27\) 1.00000 0.192450
\(28\) 3.00000 0.566947
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 1.00000 0.182574
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −1.00000 −0.176777
\(33\) 3.00000 0.522233
\(34\) 0 0
\(35\) −3.00000 −0.507093
\(36\) 1.00000 0.166667
\(37\) 6.00000 0.986394 0.493197 − 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) −4.00000 −0.648886
\(39\) −1.00000 −0.160128
\(40\) 1.00000 0.158114
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) −3.00000 −0.462910
\(43\) 10.0000 1.52499 0.762493 − 0.646997i \(-0.223975\pi\)
0.762493 + 0.646997i \(0.223975\pi\)
\(44\) 3.00000 0.452267
\(45\) −1.00000 −0.149071
\(46\) −1.00000 −0.147442
\(47\) −9.00000 −1.31278 −0.656392 − 0.754420i \(-0.727918\pi\)
−0.656392 + 0.754420i \(0.727918\pi\)
\(48\) 1.00000 0.144338
\(49\) 2.00000 0.285714
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −1.00000 −0.138675
\(53\) −1.00000 −0.137361 −0.0686803 − 0.997639i \(-0.521879\pi\)
−0.0686803 + 0.997639i \(0.521879\pi\)
\(54\) −1.00000 −0.136083
\(55\) −3.00000 −0.404520
\(56\) −3.00000 −0.400892
\(57\) 4.00000 0.529813
\(58\) 0 0
\(59\) 9.00000 1.17170 0.585850 − 0.810419i \(-0.300761\pi\)
0.585850 + 0.810419i \(0.300761\pi\)
\(60\) −1.00000 −0.129099
\(61\) 10.0000 1.28037 0.640184 − 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 4.00000 0.508001
\(63\) 3.00000 0.377964
\(64\) 1.00000 0.125000
\(65\) 1.00000 0.124035
\(66\) −3.00000 −0.369274
\(67\) 8.00000 0.977356 0.488678 − 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 3.00000 0.358569
\(71\) −2.00000 −0.237356 −0.118678 − 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) −1.00000 −0.117851
\(73\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) −6.00000 −0.697486
\(75\) 1.00000 0.115470
\(76\) 4.00000 0.458831
\(77\) 9.00000 1.02565
\(78\) 1.00000 0.113228
\(79\) −16.0000 −1.80014 −0.900070 − 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) −2.00000 −0.219529 −0.109764 − 0.993958i \(-0.535010\pi\)
−0.109764 + 0.993958i \(0.535010\pi\)
\(84\) 3.00000 0.327327
\(85\) 0 0
\(86\) −10.0000 −1.07833
\(87\) 0 0
\(88\) −3.00000 −0.319801
\(89\) 11.0000 1.16600 0.582999 − 0.812473i \(-0.301879\pi\)
0.582999 + 0.812473i \(0.301879\pi\)
\(90\) 1.00000 0.105409
\(91\) −3.00000 −0.314485
\(92\) 1.00000 0.104257
\(93\) −4.00000 −0.414781
\(94\) 9.00000 0.928279
\(95\) −4.00000 −0.410391
\(96\) −1.00000 −0.102062
\(97\) 4.00000 0.406138 0.203069 − 0.979164i \(-0.434908\pi\)
0.203069 + 0.979164i \(0.434908\pi\)
\(98\) −2.00000 −0.202031
\(99\) 3.00000 0.301511
\(100\) 1.00000 0.100000
\(101\) 10.0000 0.995037 0.497519 − 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 − 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 1.00000 0.0980581
\(105\) −3.00000 −0.292770
\(106\) 1.00000 0.0971286
\(107\) −8.00000 −0.773389 −0.386695 − 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) 1.00000 0.0962250
\(109\) 4.00000 0.383131 0.191565 − 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) 3.00000 0.286039
\(111\) 6.00000 0.569495
\(112\) 3.00000 0.283473
\(113\) −4.00000 −0.376288 −0.188144 − 0.982141i \(-0.560247\pi\)
−0.188144 + 0.982141i \(0.560247\pi\)
\(114\) −4.00000 −0.374634
\(115\) −1.00000 −0.0932505
\(116\) 0 0
\(117\) −1.00000 −0.0924500
\(118\) −9.00000 −0.828517
\(119\) 0 0
\(120\) 1.00000 0.0912871
\(121\) −2.00000 −0.181818
\(122\) −10.0000 −0.905357
\(123\) −2.00000 −0.180334
\(124\) −4.00000 −0.359211
\(125\) −1.00000 −0.0894427
\(126\) −3.00000 −0.267261
\(127\) 12.0000 1.06483 0.532414 − 0.846484i \(-0.321285\pi\)
0.532414 + 0.846484i \(0.321285\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 10.0000 0.880451
\(130\) −1.00000 −0.0877058
\(131\) 15.0000 1.31056 0.655278 − 0.755388i \(-0.272551\pi\)
0.655278 + 0.755388i \(0.272551\pi\)
\(132\) 3.00000 0.261116
\(133\) 12.0000 1.04053
\(134\) −8.00000 −0.691095
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −16.0000 −1.36697 −0.683486 − 0.729964i \(-0.739537\pi\)
−0.683486 + 0.729964i \(0.739537\pi\)
\(138\) −1.00000 −0.0851257
\(139\) 20.0000 1.69638 0.848189 − 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) −3.00000 −0.253546
\(141\) −9.00000 −0.757937
\(142\) 2.00000 0.167836
\(143\) −3.00000 −0.250873
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 0 0
\(147\) 2.00000 0.164957
\(148\) 6.00000 0.493197
\(149\) −20.0000 −1.63846 −0.819232 − 0.573462i \(-0.805600\pi\)
−0.819232 + 0.573462i \(0.805600\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 12.0000 0.976546 0.488273 − 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) −4.00000 −0.324443
\(153\) 0 0
\(154\) −9.00000 −0.725241
\(155\) 4.00000 0.321288
\(156\) −1.00000 −0.0800641
\(157\) −9.00000 −0.718278 −0.359139 − 0.933284i \(-0.616930\pi\)
−0.359139 + 0.933284i \(0.616930\pi\)
\(158\) 16.0000 1.27289
\(159\) −1.00000 −0.0793052
\(160\) 1.00000 0.0790569
\(161\) 3.00000 0.236433
\(162\) −1.00000 −0.0785674
\(163\) −6.00000 −0.469956 −0.234978 − 0.972001i \(-0.575502\pi\)
−0.234978 + 0.972001i \(0.575502\pi\)
\(164\) −2.00000 −0.156174
\(165\) −3.00000 −0.233550
\(166\) 2.00000 0.155230
\(167\) 15.0000 1.16073 0.580367 − 0.814355i \(-0.302909\pi\)
0.580367 + 0.814355i \(0.302909\pi\)
\(168\) −3.00000 −0.231455
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) 10.0000 0.762493
\(173\) −15.0000 −1.14043 −0.570214 − 0.821496i \(-0.693140\pi\)
−0.570214 + 0.821496i \(0.693140\pi\)
\(174\) 0 0
\(175\) 3.00000 0.226779
\(176\) 3.00000 0.226134
\(177\) 9.00000 0.676481
\(178\) −11.0000 −0.824485
\(179\) 21.0000 1.56961 0.784807 − 0.619740i \(-0.212762\pi\)
0.784807 + 0.619740i \(0.212762\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −10.0000 −0.743294 −0.371647 − 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 3.00000 0.222375
\(183\) 10.0000 0.739221
\(184\) −1.00000 −0.0737210
\(185\) −6.00000 −0.441129
\(186\) 4.00000 0.293294
\(187\) 0 0
\(188\) −9.00000 −0.656392
\(189\) 3.00000 0.218218
\(190\) 4.00000 0.290191
\(191\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 1.00000 0.0721688
\(193\) 12.0000 0.863779 0.431889 − 0.901927i \(-0.357847\pi\)
0.431889 + 0.901927i \(0.357847\pi\)
\(194\) −4.00000 −0.287183
\(195\) 1.00000 0.0716115
\(196\) 2.00000 0.142857
\(197\) 3.00000 0.213741 0.106871 − 0.994273i \(-0.465917\pi\)
0.106871 + 0.994273i \(0.465917\pi\)
\(198\) −3.00000 −0.213201
\(199\) −20.0000 −1.41776 −0.708881 − 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 8.00000 0.564276
\(202\) −10.0000 −0.703598
\(203\) 0 0
\(204\) 0 0
\(205\) 2.00000 0.139686
\(206\) 8.00000 0.557386
\(207\) 1.00000 0.0695048
\(208\) −1.00000 −0.0693375
\(209\) 12.0000 0.830057
\(210\) 3.00000 0.207020
\(211\) 27.0000 1.85876 0.929378 − 0.369129i \(-0.120344\pi\)
0.929378 + 0.369129i \(0.120344\pi\)
\(212\) −1.00000 −0.0686803
\(213\) −2.00000 −0.137038
\(214\) 8.00000 0.546869
\(215\) −10.0000 −0.681994
\(216\) −1.00000 −0.0680414
\(217\) −12.0000 −0.814613
\(218\) −4.00000 −0.270914
\(219\) 0 0
\(220\) −3.00000 −0.202260
\(221\) 0 0
\(222\) −6.00000 −0.402694
\(223\) 17.0000 1.13840 0.569202 − 0.822198i \(-0.307252\pi\)
0.569202 + 0.822198i \(0.307252\pi\)
\(224\) −3.00000 −0.200446
\(225\) 1.00000 0.0666667
\(226\) 4.00000 0.266076
\(227\) 6.00000 0.398234 0.199117 − 0.979976i \(-0.436193\pi\)
0.199117 + 0.979976i \(0.436193\pi\)
\(228\) 4.00000 0.264906
\(229\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) 1.00000 0.0659380
\(231\) 9.00000 0.592157
\(232\) 0 0
\(233\) 4.00000 0.262049 0.131024 − 0.991379i \(-0.458173\pi\)
0.131024 + 0.991379i \(0.458173\pi\)
\(234\) 1.00000 0.0653720
\(235\) 9.00000 0.587095
\(236\) 9.00000 0.585850
\(237\) −16.0000 −1.03931
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 − 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −15.0000 −0.966235 −0.483117 − 0.875556i \(-0.660496\pi\)
−0.483117 + 0.875556i \(0.660496\pi\)
\(242\) 2.00000 0.128565
\(243\) 1.00000 0.0641500
\(244\) 10.0000 0.640184
\(245\) −2.00000 −0.127775
\(246\) 2.00000 0.127515
\(247\) −4.00000 −0.254514
\(248\) 4.00000 0.254000
\(249\) −2.00000 −0.126745
\(250\) 1.00000 0.0632456
\(251\) −15.0000 −0.946792 −0.473396 − 0.880850i \(-0.656972\pi\)
−0.473396 + 0.880850i \(0.656972\pi\)
\(252\) 3.00000 0.188982
\(253\) 3.00000 0.188608
\(254\) −12.0000 −0.752947
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −14.0000 −0.873296 −0.436648 − 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) −10.0000 −0.622573
\(259\) 18.0000 1.11847
\(260\) 1.00000 0.0620174
\(261\) 0 0
\(262\) −15.0000 −0.926703
\(263\) −25.0000 −1.54157 −0.770783 − 0.637098i \(-0.780135\pi\)
−0.770783 + 0.637098i \(0.780135\pi\)
\(264\) −3.00000 −0.184637
\(265\) 1.00000 0.0614295
\(266\) −12.0000 −0.735767
\(267\) 11.0000 0.673189
\(268\) 8.00000 0.488678
\(269\) −2.00000 −0.121942 −0.0609711 − 0.998140i \(-0.519420\pi\)
−0.0609711 + 0.998140i \(0.519420\pi\)
\(270\) 1.00000 0.0608581
\(271\) 26.0000 1.57939 0.789694 − 0.613501i \(-0.210239\pi\)
0.789694 + 0.613501i \(0.210239\pi\)
\(272\) 0 0
\(273\) −3.00000 −0.181568
\(274\) 16.0000 0.966595
\(275\) 3.00000 0.180907
\(276\) 1.00000 0.0601929
\(277\) −1.00000 −0.0600842 −0.0300421 − 0.999549i \(-0.509564\pi\)
−0.0300421 + 0.999549i \(0.509564\pi\)
\(278\) −20.0000 −1.19952
\(279\) −4.00000 −0.239474
\(280\) 3.00000 0.179284
\(281\) −21.0000 −1.25275 −0.626377 − 0.779520i \(-0.715463\pi\)
−0.626377 + 0.779520i \(0.715463\pi\)
\(282\) 9.00000 0.535942
\(283\) −10.0000 −0.594438 −0.297219 − 0.954809i \(-0.596059\pi\)
−0.297219 + 0.954809i \(0.596059\pi\)
\(284\) −2.00000 −0.118678
\(285\) −4.00000 −0.236940
\(286\) 3.00000 0.177394
\(287\) −6.00000 −0.354169
\(288\) −1.00000 −0.0589256
\(289\) 0 0
\(290\) 0 0
\(291\) 4.00000 0.234484
\(292\) 0 0
\(293\) 9.00000 0.525786 0.262893 − 0.964825i \(-0.415323\pi\)
0.262893 + 0.964825i \(0.415323\pi\)
\(294\) −2.00000 −0.116642
\(295\) −9.00000 −0.524000
\(296\) −6.00000 −0.348743
\(297\) 3.00000 0.174078
\(298\) 20.0000 1.15857
\(299\) −1.00000 −0.0578315
\(300\) 1.00000 0.0577350
\(301\) 30.0000 1.72917
\(302\) −12.0000 −0.690522
\(303\) 10.0000 0.574485
\(304\) 4.00000 0.229416
\(305\) −10.0000 −0.572598
\(306\) 0 0
\(307\) 2.00000 0.114146 0.0570730 − 0.998370i \(-0.481823\pi\)
0.0570730 + 0.998370i \(0.481823\pi\)
\(308\) 9.00000 0.512823
\(309\) −8.00000 −0.455104
\(310\) −4.00000 −0.227185
\(311\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 1.00000 0.0566139
\(313\) 28.0000 1.58265 0.791327 − 0.611393i \(-0.209391\pi\)
0.791327 + 0.611393i \(0.209391\pi\)
\(314\) 9.00000 0.507899
\(315\) −3.00000 −0.169031
\(316\) −16.0000 −0.900070
\(317\) −2.00000 −0.112331 −0.0561656 − 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) 1.00000 0.0560772
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) −8.00000 −0.446516
\(322\) −3.00000 −0.167183
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) −1.00000 −0.0554700
\(326\) 6.00000 0.332309
\(327\) 4.00000 0.221201
\(328\) 2.00000 0.110432
\(329\) −27.0000 −1.48856
\(330\) 3.00000 0.165145
\(331\) 13.0000 0.714545 0.357272 − 0.934000i \(-0.383707\pi\)
0.357272 + 0.934000i \(0.383707\pi\)
\(332\) −2.00000 −0.109764
\(333\) 6.00000 0.328798
\(334\) −15.0000 −0.820763
\(335\) −8.00000 −0.437087
\(336\) 3.00000 0.163663
\(337\) 12.0000 0.653682 0.326841 − 0.945079i \(-0.394016\pi\)
0.326841 + 0.945079i \(0.394016\pi\)
\(338\) 12.0000 0.652714
\(339\) −4.00000 −0.217250
\(340\) 0 0
\(341\) −12.0000 −0.649836
\(342\) −4.00000 −0.216295
\(343\) −15.0000 −0.809924
\(344\) −10.0000 −0.539164
\(345\) −1.00000 −0.0538382
\(346\) 15.0000 0.806405
\(347\) 18.0000 0.966291 0.483145 − 0.875540i \(-0.339494\pi\)
0.483145 + 0.875540i \(0.339494\pi\)
\(348\) 0 0
\(349\) −16.0000 −0.856460 −0.428230 − 0.903670i \(-0.640863\pi\)
−0.428230 + 0.903670i \(0.640863\pi\)
\(350\) −3.00000 −0.160357
\(351\) −1.00000 −0.0533761
\(352\) −3.00000 −0.159901
\(353\) −32.0000 −1.70319 −0.851594 − 0.524202i \(-0.824364\pi\)
−0.851594 + 0.524202i \(0.824364\pi\)
\(354\) −9.00000 −0.478345
\(355\) 2.00000 0.106149
\(356\) 11.0000 0.582999
\(357\) 0 0
\(358\) −21.0000 −1.10988
\(359\) 26.0000 1.37223 0.686114 − 0.727494i \(-0.259315\pi\)
0.686114 + 0.727494i \(0.259315\pi\)
\(360\) 1.00000 0.0527046
\(361\) −3.00000 −0.157895
\(362\) 10.0000 0.525588
\(363\) −2.00000 −0.104973
\(364\) −3.00000 −0.157243
\(365\) 0 0
\(366\) −10.0000 −0.522708
\(367\) 17.0000 0.887393 0.443696 − 0.896177i \(-0.353667\pi\)
0.443696 + 0.896177i \(0.353667\pi\)
\(368\) 1.00000 0.0521286
\(369\) −2.00000 −0.104116
\(370\) 6.00000 0.311925
\(371\) −3.00000 −0.155752
\(372\) −4.00000 −0.207390
\(373\) 31.0000 1.60512 0.802560 − 0.596572i \(-0.203471\pi\)
0.802560 + 0.596572i \(0.203471\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 9.00000 0.464140
\(377\) 0 0
\(378\) −3.00000 −0.154303
\(379\) −4.00000 −0.205466 −0.102733 − 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) −4.00000 −0.205196
\(381\) 12.0000 0.614779
\(382\) 0 0
\(383\) −16.0000 −0.817562 −0.408781 − 0.912633i \(-0.634046\pi\)
−0.408781 + 0.912633i \(0.634046\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −9.00000 −0.458682
\(386\) −12.0000 −0.610784
\(387\) 10.0000 0.508329
\(388\) 4.00000 0.203069
\(389\) 16.0000 0.811232 0.405616 − 0.914044i \(-0.367057\pi\)
0.405616 + 0.914044i \(0.367057\pi\)
\(390\) −1.00000 −0.0506370
\(391\) 0 0
\(392\) −2.00000 −0.101015
\(393\) 15.0000 0.756650
\(394\) −3.00000 −0.151138
\(395\) 16.0000 0.805047
\(396\) 3.00000 0.150756
\(397\) 15.0000 0.752828 0.376414 − 0.926451i \(-0.377157\pi\)
0.376414 + 0.926451i \(0.377157\pi\)
\(398\) 20.0000 1.00251
\(399\) 12.0000 0.600751
\(400\) 1.00000 0.0500000
\(401\) 33.0000 1.64794 0.823971 − 0.566632i \(-0.191754\pi\)
0.823971 + 0.566632i \(0.191754\pi\)
\(402\) −8.00000 −0.399004
\(403\) 4.00000 0.199254
\(404\) 10.0000 0.497519
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 18.0000 0.892227
\(408\) 0 0
\(409\) 5.00000 0.247234 0.123617 − 0.992330i \(-0.460551\pi\)
0.123617 + 0.992330i \(0.460551\pi\)
\(410\) −2.00000 −0.0987730
\(411\) −16.0000 −0.789222
\(412\) −8.00000 −0.394132
\(413\) 27.0000 1.32858
\(414\) −1.00000 −0.0491473
\(415\) 2.00000 0.0981761
\(416\) 1.00000 0.0490290
\(417\) 20.0000 0.979404
\(418\) −12.0000 −0.586939
\(419\) 17.0000 0.830504 0.415252 − 0.909706i \(-0.363693\pi\)
0.415252 + 0.909706i \(0.363693\pi\)
\(420\) −3.00000 −0.146385
\(421\) 8.00000 0.389896 0.194948 − 0.980814i \(-0.437546\pi\)
0.194948 + 0.980814i \(0.437546\pi\)
\(422\) −27.0000 −1.31434
\(423\) −9.00000 −0.437595
\(424\) 1.00000 0.0485643
\(425\) 0 0
\(426\) 2.00000 0.0969003
\(427\) 30.0000 1.45180
\(428\) −8.00000 −0.386695
\(429\) −3.00000 −0.144841
\(430\) 10.0000 0.482243
\(431\) −32.0000 −1.54139 −0.770693 − 0.637207i \(-0.780090\pi\)
−0.770693 + 0.637207i \(0.780090\pi\)
\(432\) 1.00000 0.0481125
\(433\) −24.0000 −1.15337 −0.576683 − 0.816968i \(-0.695653\pi\)
−0.576683 + 0.816968i \(0.695653\pi\)
\(434\) 12.0000 0.576018
\(435\) 0 0
\(436\) 4.00000 0.191565
\(437\) 4.00000 0.191346
\(438\) 0 0
\(439\) −14.0000 −0.668184 −0.334092 − 0.942541i \(-0.608430\pi\)
−0.334092 + 0.942541i \(0.608430\pi\)
\(440\) 3.00000 0.143019
\(441\) 2.00000 0.0952381
\(442\) 0 0
\(443\) −26.0000 −1.23530 −0.617649 − 0.786454i \(-0.711915\pi\)
−0.617649 + 0.786454i \(0.711915\pi\)
\(444\) 6.00000 0.284747
\(445\) −11.0000 −0.521450
\(446\) −17.0000 −0.804973
\(447\) −20.0000 −0.945968
\(448\) 3.00000 0.141737
\(449\) 10.0000 0.471929 0.235965 − 0.971762i \(-0.424175\pi\)
0.235965 + 0.971762i \(0.424175\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −6.00000 −0.282529
\(452\) −4.00000 −0.188144
\(453\) 12.0000 0.563809
\(454\) −6.00000 −0.281594
\(455\) 3.00000 0.140642
\(456\) −4.00000 −0.187317
\(457\) 32.0000 1.49690 0.748448 − 0.663193i \(-0.230799\pi\)
0.748448 + 0.663193i \(0.230799\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) −1.00000 −0.0466252
\(461\) −14.0000 −0.652045 −0.326023 − 0.945362i \(-0.605709\pi\)
−0.326023 + 0.945362i \(0.605709\pi\)
\(462\) −9.00000 −0.418718
\(463\) 33.0000 1.53364 0.766820 − 0.641862i \(-0.221838\pi\)
0.766820 + 0.641862i \(0.221838\pi\)
\(464\) 0 0
\(465\) 4.00000 0.185496
\(466\) −4.00000 −0.185296
\(467\) 18.0000 0.832941 0.416470 − 0.909149i \(-0.363267\pi\)
0.416470 + 0.909149i \(0.363267\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 24.0000 1.10822
\(470\) −9.00000 −0.415139
\(471\) −9.00000 −0.414698
\(472\) −9.00000 −0.414259
\(473\) 30.0000 1.37940
\(474\) 16.0000 0.734904
\(475\) 4.00000 0.183533
\(476\) 0 0
\(477\) −1.00000 −0.0457869
\(478\) 12.0000 0.548867
\(479\) 28.0000 1.27935 0.639676 − 0.768644i \(-0.279068\pi\)
0.639676 + 0.768644i \(0.279068\pi\)
\(480\) 1.00000 0.0456435
\(481\) −6.00000 −0.273576
\(482\) 15.0000 0.683231
\(483\) 3.00000 0.136505
\(484\) −2.00000 −0.0909091
\(485\) −4.00000 −0.181631
\(486\) −1.00000 −0.0453609
\(487\) −32.0000 −1.45006 −0.725029 − 0.688718i \(-0.758174\pi\)
−0.725029 + 0.688718i \(0.758174\pi\)
\(488\) −10.0000 −0.452679
\(489\) −6.00000 −0.271329
\(490\) 2.00000 0.0903508
\(491\) −37.0000 −1.66979 −0.834893 − 0.550412i \(-0.814471\pi\)
−0.834893 + 0.550412i \(0.814471\pi\)
\(492\) −2.00000 −0.0901670
\(493\) 0 0
\(494\) 4.00000 0.179969
\(495\) −3.00000 −0.134840
\(496\) −4.00000 −0.179605
\(497\) −6.00000 −0.269137
\(498\) 2.00000 0.0896221
\(499\) 41.0000 1.83541 0.917706 − 0.397260i \(-0.130039\pi\)
0.917706 + 0.397260i \(0.130039\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 15.0000 0.670151
\(502\) 15.0000 0.669483
\(503\) 23.0000 1.02552 0.512760 − 0.858532i \(-0.328623\pi\)
0.512760 + 0.858532i \(0.328623\pi\)
\(504\) −3.00000 −0.133631
\(505\) −10.0000 −0.444994
\(506\) −3.00000 −0.133366
\(507\) −12.0000 −0.532939
\(508\) 12.0000 0.532414
\(509\) 18.0000 0.797836 0.398918 − 0.916987i \(-0.369386\pi\)
0.398918 + 0.916987i \(0.369386\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 4.00000 0.176604
\(514\) 14.0000 0.617514
\(515\) 8.00000 0.352522
\(516\) 10.0000 0.440225
\(517\) −27.0000 −1.18746
\(518\) −18.0000 −0.790875
\(519\) −15.0000 −0.658427
\(520\) −1.00000 −0.0438529
\(521\) −6.00000 −0.262865 −0.131432 − 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 0 0
\(523\) −26.0000 −1.13690 −0.568450 − 0.822718i \(-0.692457\pi\)
−0.568450 + 0.822718i \(0.692457\pi\)
\(524\) 15.0000 0.655278
\(525\) 3.00000 0.130931
\(526\) 25.0000 1.09005
\(527\) 0 0
\(528\) 3.00000 0.130558
\(529\) −22.0000 −0.956522
\(530\) −1.00000 −0.0434372
\(531\) 9.00000 0.390567
\(532\) 12.0000 0.520266
\(533\) 2.00000 0.0866296
\(534\) −11.0000 −0.476017
\(535\) 8.00000 0.345870
\(536\) −8.00000 −0.345547
\(537\) 21.0000 0.906217
\(538\) 2.00000 0.0862261
\(539\) 6.00000 0.258438
\(540\) −1.00000 −0.0430331
\(541\) −32.0000 −1.37579 −0.687894 − 0.725811i \(-0.741464\pi\)
−0.687894 + 0.725811i \(0.741464\pi\)
\(542\) −26.0000 −1.11680
\(543\) −10.0000 −0.429141
\(544\) 0 0
\(545\) −4.00000 −0.171341
\(546\) 3.00000 0.128388
\(547\) −2.00000 −0.0855138 −0.0427569 − 0.999086i \(-0.513614\pi\)
−0.0427569 + 0.999086i \(0.513614\pi\)
\(548\) −16.0000 −0.683486
\(549\) 10.0000 0.426790
\(550\) −3.00000 −0.127920
\(551\) 0 0
\(552\) −1.00000 −0.0425628
\(553\) −48.0000 −2.04117
\(554\) 1.00000 0.0424859
\(555\) −6.00000 −0.254686
\(556\) 20.0000 0.848189
\(557\) −35.0000 −1.48300 −0.741499 − 0.670954i \(-0.765885\pi\)
−0.741499 + 0.670954i \(0.765885\pi\)
\(558\) 4.00000 0.169334
\(559\) −10.0000 −0.422955
\(560\) −3.00000 −0.126773
\(561\) 0 0
\(562\) 21.0000 0.885832
\(563\) −12.0000 −0.505740 −0.252870 − 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) −9.00000 −0.378968
\(565\) 4.00000 0.168281
\(566\) 10.0000 0.420331
\(567\) 3.00000 0.125988
\(568\) 2.00000 0.0839181
\(569\) 46.0000 1.92842 0.964210 − 0.265139i \(-0.0854179\pi\)
0.964210 + 0.265139i \(0.0854179\pi\)
\(570\) 4.00000 0.167542
\(571\) 43.0000 1.79949 0.899747 − 0.436412i \(-0.143751\pi\)
0.899747 + 0.436412i \(0.143751\pi\)
\(572\) −3.00000 −0.125436
\(573\) 0 0
\(574\) 6.00000 0.250435
\(575\) 1.00000 0.0417029
\(576\) 1.00000 0.0416667
\(577\) 2.00000 0.0832611 0.0416305 − 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 0 0
\(579\) 12.0000 0.498703
\(580\) 0 0
\(581\) −6.00000 −0.248922
\(582\) −4.00000 −0.165805
\(583\) −3.00000 −0.124247
\(584\) 0 0
\(585\) 1.00000 0.0413449
\(586\) −9.00000 −0.371787
\(587\) 18.0000 0.742940 0.371470 − 0.928445i \(-0.378854\pi\)
0.371470 + 0.928445i \(0.378854\pi\)
\(588\) 2.00000 0.0824786
\(589\) −16.0000 −0.659269
\(590\) 9.00000 0.370524
\(591\) 3.00000 0.123404
\(592\) 6.00000 0.246598
\(593\) −36.0000 −1.47834 −0.739171 − 0.673517i \(-0.764783\pi\)
−0.739171 + 0.673517i \(0.764783\pi\)
\(594\) −3.00000 −0.123091
\(595\) 0 0
\(596\) −20.0000 −0.819232
\(597\) −20.0000 −0.818546
\(598\) 1.00000 0.0408930
\(599\) 10.0000 0.408589 0.204294 − 0.978909i \(-0.434510\pi\)
0.204294 + 0.978909i \(0.434510\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 19.0000 0.775026 0.387513 − 0.921864i \(-0.373334\pi\)
0.387513 + 0.921864i \(0.373334\pi\)
\(602\) −30.0000 −1.22271
\(603\) 8.00000 0.325785
\(604\) 12.0000 0.488273
\(605\) 2.00000 0.0813116
\(606\) −10.0000 −0.406222
\(607\) 29.0000 1.17707 0.588537 − 0.808470i \(-0.299704\pi\)
0.588537 + 0.808470i \(0.299704\pi\)
\(608\) −4.00000 −0.162221
\(609\) 0 0
\(610\) 10.0000 0.404888
\(611\) 9.00000 0.364101
\(612\) 0 0
\(613\) −11.0000 −0.444286 −0.222143 − 0.975014i \(-0.571305\pi\)
−0.222143 + 0.975014i \(0.571305\pi\)
\(614\) −2.00000 −0.0807134
\(615\) 2.00000 0.0806478
\(616\) −9.00000 −0.362620
\(617\) 12.0000 0.483102 0.241551 − 0.970388i \(-0.422344\pi\)
0.241551 + 0.970388i \(0.422344\pi\)
\(618\) 8.00000 0.321807
\(619\) 25.0000 1.00483 0.502417 − 0.864625i \(-0.332444\pi\)
0.502417 + 0.864625i \(0.332444\pi\)
\(620\) 4.00000 0.160644
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) 33.0000 1.32212
\(624\) −1.00000 −0.0400320
\(625\) 1.00000 0.0400000
\(626\) −28.0000 −1.11911
\(627\) 12.0000 0.479234
\(628\) −9.00000 −0.359139
\(629\) 0 0
\(630\) 3.00000 0.119523
\(631\) −4.00000 −0.159237 −0.0796187 − 0.996825i \(-0.525370\pi\)
−0.0796187 + 0.996825i \(0.525370\pi\)
\(632\) 16.0000 0.636446
\(633\) 27.0000 1.07315
\(634\) 2.00000 0.0794301
\(635\) −12.0000 −0.476205
\(636\) −1.00000 −0.0396526
\(637\) −2.00000 −0.0792429
\(638\) 0 0
\(639\) −2.00000 −0.0791188
\(640\) 1.00000 0.0395285
\(641\) −9.00000 −0.355479 −0.177739 − 0.984078i \(-0.556878\pi\)
−0.177739 + 0.984078i \(0.556878\pi\)
\(642\) 8.00000 0.315735
\(643\) −32.0000 −1.26196 −0.630978 − 0.775800i \(-0.717346\pi\)
−0.630978 + 0.775800i \(0.717346\pi\)
\(644\) 3.00000 0.118217
\(645\) −10.0000 −0.393750
\(646\) 0 0
\(647\) 25.0000 0.982851 0.491426 − 0.870919i \(-0.336476\pi\)
0.491426 + 0.870919i \(0.336476\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 27.0000 1.05984
\(650\) 1.00000 0.0392232
\(651\) −12.0000 −0.470317
\(652\) −6.00000 −0.234978
\(653\) 15.0000 0.586995 0.293498 − 0.955960i \(-0.405181\pi\)
0.293498 + 0.955960i \(0.405181\pi\)
\(654\) −4.00000 −0.156412
\(655\) −15.0000 −0.586098
\(656\) −2.00000 −0.0780869
\(657\) 0 0
\(658\) 27.0000 1.05257
\(659\) 4.00000 0.155818 0.0779089 − 0.996960i \(-0.475176\pi\)
0.0779089 + 0.996960i \(0.475176\pi\)
\(660\) −3.00000 −0.116775
\(661\) 8.00000 0.311164 0.155582 − 0.987823i \(-0.450275\pi\)
0.155582 + 0.987823i \(0.450275\pi\)
\(662\) −13.0000 −0.505259
\(663\) 0 0
\(664\) 2.00000 0.0776151
\(665\) −12.0000 −0.465340
\(666\) −6.00000 −0.232495
\(667\) 0 0
\(668\) 15.0000 0.580367
\(669\) 17.0000 0.657258
\(670\) 8.00000 0.309067
\(671\) 30.0000 1.15814
\(672\) −3.00000 −0.115728
\(673\) −34.0000 −1.31060 −0.655302 − 0.755367i \(-0.727459\pi\)
−0.655302 + 0.755367i \(0.727459\pi\)
\(674\) −12.0000 −0.462223
\(675\) 1.00000 0.0384900
\(676\) −12.0000 −0.461538
\(677\) 15.0000 0.576497 0.288248 − 0.957556i \(-0.406927\pi\)
0.288248 + 0.957556i \(0.406927\pi\)
\(678\) 4.00000 0.153619
\(679\) 12.0000 0.460518
\(680\) 0 0
\(681\) 6.00000 0.229920
\(682\) 12.0000 0.459504
\(683\) 36.0000 1.37750 0.688751 − 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 4.00000 0.152944
\(685\) 16.0000 0.611329
\(686\) 15.0000 0.572703
\(687\) 0 0
\(688\) 10.0000 0.381246
\(689\) 1.00000 0.0380970
\(690\) 1.00000 0.0380693
\(691\) 15.0000 0.570627 0.285313 − 0.958434i \(-0.407902\pi\)
0.285313 + 0.958434i \(0.407902\pi\)
\(692\) −15.0000 −0.570214
\(693\) 9.00000 0.341882
\(694\) −18.0000 −0.683271
\(695\) −20.0000 −0.758643
\(696\) 0 0
\(697\) 0 0
\(698\) 16.0000 0.605609
\(699\) 4.00000 0.151294
\(700\) 3.00000 0.113389
\(701\) −50.0000 −1.88847 −0.944237 − 0.329267i \(-0.893198\pi\)
−0.944237 + 0.329267i \(0.893198\pi\)
\(702\) 1.00000 0.0377426
\(703\) 24.0000 0.905177
\(704\) 3.00000 0.113067
\(705\) 9.00000 0.338960
\(706\) 32.0000 1.20434
\(707\) 30.0000 1.12827
\(708\) 9.00000 0.338241
\(709\) −8.00000 −0.300446 −0.150223 − 0.988652i \(-0.547999\pi\)
−0.150223 + 0.988652i \(0.547999\pi\)
\(710\) −2.00000 −0.0750587
\(711\) −16.0000 −0.600047
\(712\) −11.0000 −0.412242
\(713\) −4.00000 −0.149801
\(714\) 0 0
\(715\) 3.00000 0.112194
\(716\) 21.0000 0.784807
\(717\) −12.0000 −0.448148
\(718\) −26.0000 −0.970311
\(719\) 30.0000 1.11881 0.559406 − 0.828894i \(-0.311029\pi\)
0.559406 + 0.828894i \(0.311029\pi\)
\(720\) −1.00000 −0.0372678
\(721\) −24.0000 −0.893807
\(722\) 3.00000 0.111648
\(723\) −15.0000 −0.557856
\(724\) −10.0000 −0.371647
\(725\) 0 0
\(726\) 2.00000 0.0742270
\(727\) −13.0000 −0.482143 −0.241072 − 0.970507i \(-0.577499\pi\)
−0.241072 + 0.970507i \(0.577499\pi\)
\(728\) 3.00000 0.111187
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 10.0000 0.369611
\(733\) 2.00000 0.0738717 0.0369358 − 0.999318i \(-0.488240\pi\)
0.0369358 + 0.999318i \(0.488240\pi\)
\(734\) −17.0000 −0.627481
\(735\) −2.00000 −0.0737711
\(736\) −1.00000 −0.0368605
\(737\) 24.0000 0.884051
\(738\) 2.00000 0.0736210
\(739\) −1.00000 −0.0367856 −0.0183928 − 0.999831i \(-0.505855\pi\)
−0.0183928 + 0.999831i \(0.505855\pi\)
\(740\) −6.00000 −0.220564
\(741\) −4.00000 −0.146944
\(742\) 3.00000 0.110133
\(743\) −11.0000 −0.403551 −0.201775 − 0.979432i \(-0.564671\pi\)
−0.201775 + 0.979432i \(0.564671\pi\)
\(744\) 4.00000 0.146647
\(745\) 20.0000 0.732743
\(746\) −31.0000 −1.13499
\(747\) −2.00000 −0.0731762
\(748\) 0 0
\(749\) −24.0000 −0.876941
\(750\) 1.00000 0.0365148
\(751\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) −9.00000 −0.328196
\(753\) −15.0000 −0.546630
\(754\) 0 0
\(755\) −12.0000 −0.436725
\(756\) 3.00000 0.109109
\(757\) −11.0000 −0.399802 −0.199901 − 0.979816i \(-0.564062\pi\)
−0.199901 + 0.979816i \(0.564062\pi\)
\(758\) 4.00000 0.145287
\(759\) 3.00000 0.108893
\(760\) 4.00000 0.145095
\(761\) 27.0000 0.978749 0.489375 − 0.872074i \(-0.337225\pi\)
0.489375 + 0.872074i \(0.337225\pi\)
\(762\) −12.0000 −0.434714
\(763\) 12.0000 0.434429
\(764\) 0 0
\(765\) 0 0
\(766\) 16.0000 0.578103
\(767\) −9.00000 −0.324971
\(768\) 1.00000 0.0360844
\(769\) −13.0000 −0.468792 −0.234396 − 0.972141i \(-0.575311\pi\)
−0.234396 + 0.972141i \(0.575311\pi\)
\(770\) 9.00000 0.324337
\(771\) −14.0000 −0.504198
\(772\) 12.0000 0.431889
\(773\) −30.0000 −1.07903 −0.539513 − 0.841978i \(-0.681391\pi\)
−0.539513 + 0.841978i \(0.681391\pi\)
\(774\) −10.0000 −0.359443
\(775\) −4.00000 −0.143684
\(776\) −4.00000 −0.143592
\(777\) 18.0000 0.645746
\(778\) −16.0000 −0.573628
\(779\) −8.00000 −0.286630
\(780\) 1.00000 0.0358057
\(781\) −6.00000 −0.214697
\(782\) 0 0
\(783\) 0 0
\(784\) 2.00000 0.0714286
\(785\) 9.00000 0.321224
\(786\) −15.0000 −0.535032
\(787\) −24.0000 −0.855508 −0.427754 − 0.903895i \(-0.640695\pi\)
−0.427754 + 0.903895i \(0.640695\pi\)
\(788\) 3.00000 0.106871
\(789\) −25.0000 −0.890024
\(790\) −16.0000 −0.569254
\(791\) −12.0000 −0.426671
\(792\) −3.00000 −0.106600
\(793\) −10.0000 −0.355110
\(794\) −15.0000 −0.532330
\(795\) 1.00000 0.0354663
\(796\) −20.0000 −0.708881
\(797\) −49.0000 −1.73567 −0.867835 − 0.496853i \(-0.834489\pi\)
−0.867835 + 0.496853i \(0.834489\pi\)
\(798\) −12.0000 −0.424795
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) 11.0000 0.388666
\(802\) −33.0000 −1.16527
\(803\) 0 0
\(804\) 8.00000 0.282138
\(805\) −3.00000 −0.105736
\(806\) −4.00000 −0.140894
\(807\) −2.00000 −0.0704033
\(808\) −10.0000 −0.351799
\(809\) 30.0000 1.05474 0.527372 − 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 1.00000 0.0351364
\(811\) 7.00000 0.245803 0.122902 − 0.992419i \(-0.460780\pi\)
0.122902 + 0.992419i \(0.460780\pi\)
\(812\) 0 0
\(813\) 26.0000 0.911860
\(814\) −18.0000 −0.630900
\(815\) 6.00000 0.210171
\(816\) 0 0
\(817\) 40.0000 1.39942
\(818\) −5.00000 −0.174821
\(819\) −3.00000 −0.104828
\(820\) 2.00000 0.0698430
\(821\) 12.0000 0.418803 0.209401 − 0.977830i \(-0.432848\pi\)
0.209401 + 0.977830i \(0.432848\pi\)
\(822\) 16.0000 0.558064
\(823\) 32.0000 1.11545 0.557725 − 0.830026i \(-0.311674\pi\)
0.557725 + 0.830026i \(0.311674\pi\)
\(824\) 8.00000 0.278693
\(825\) 3.00000 0.104447
\(826\) −27.0000 −0.939450
\(827\) −2.00000 −0.0695468 −0.0347734 − 0.999395i \(-0.511071\pi\)
−0.0347734 + 0.999395i \(0.511071\pi\)
\(828\) 1.00000 0.0347524
\(829\) 16.0000 0.555703 0.277851 − 0.960624i \(-0.410378\pi\)
0.277851 + 0.960624i \(0.410378\pi\)
\(830\) −2.00000 −0.0694210
\(831\) −1.00000 −0.0346896
\(832\) −1.00000 −0.0346688
\(833\) 0 0
\(834\) −20.0000 −0.692543
\(835\) −15.0000 −0.519096
\(836\) 12.0000 0.415029
\(837\) −4.00000 −0.138260
\(838\) −17.0000 −0.587255
\(839\) −30.0000 −1.03572 −0.517858 − 0.855467i \(-0.673270\pi\)
−0.517858 + 0.855467i \(0.673270\pi\)
\(840\) 3.00000 0.103510
\(841\) −29.0000 −1.00000
\(842\) −8.00000 −0.275698
\(843\) −21.0000 −0.723278
\(844\) 27.0000 0.929378
\(845\) 12.0000 0.412813
\(846\) 9.00000 0.309426
\(847\) −6.00000 −0.206162
\(848\) −1.00000 −0.0343401
\(849\) −10.0000 −0.343199
\(850\) 0 0
\(851\) 6.00000 0.205677
\(852\) −2.00000 −0.0685189
\(853\) 3.00000 0.102718 0.0513590 − 0.998680i \(-0.483645\pi\)
0.0513590 + 0.998680i \(0.483645\pi\)
\(854\) −30.0000 −1.02658
\(855\) −4.00000 −0.136797
\(856\) 8.00000 0.273434
\(857\) −8.00000 −0.273275 −0.136637 − 0.990621i \(-0.543630\pi\)
−0.136637 + 0.990621i \(0.543630\pi\)
\(858\) 3.00000 0.102418
\(859\) 44.0000 1.50126 0.750630 − 0.660722i \(-0.229750\pi\)
0.750630 + 0.660722i \(0.229750\pi\)
\(860\) −10.0000 −0.340997
\(861\) −6.00000 −0.204479
\(862\) 32.0000 1.08992
\(863\) −37.0000 −1.25949 −0.629747 − 0.776800i \(-0.716842\pi\)
−0.629747 + 0.776800i \(0.716842\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 15.0000 0.510015
\(866\) 24.0000 0.815553
\(867\) 0 0
\(868\) −12.0000 −0.407307
\(869\) −48.0000 −1.62829
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) −4.00000 −0.135457
\(873\) 4.00000 0.135379
\(874\) −4.00000 −0.135302
\(875\) −3.00000 −0.101419
\(876\) 0 0
\(877\) 17.0000 0.574049 0.287025 − 0.957923i \(-0.407334\pi\)
0.287025 + 0.957923i \(0.407334\pi\)
\(878\) 14.0000 0.472477
\(879\) 9.00000 0.303562
\(880\) −3.00000 −0.101130
\(881\) −31.0000 −1.04442 −0.522208 − 0.852818i \(-0.674892\pi\)
−0.522208 + 0.852818i \(0.674892\pi\)
\(882\) −2.00000 −0.0673435
\(883\) −54.0000 −1.81724 −0.908622 − 0.417619i \(-0.862865\pi\)
−0.908622 + 0.417619i \(0.862865\pi\)
\(884\) 0 0
\(885\) −9.00000 −0.302532
\(886\) 26.0000 0.873487
\(887\) 48.0000 1.61168 0.805841 − 0.592132i \(-0.201714\pi\)
0.805841 + 0.592132i \(0.201714\pi\)
\(888\) −6.00000 −0.201347
\(889\) 36.0000 1.20740
\(890\) 11.0000 0.368721
\(891\) 3.00000 0.100504
\(892\) 17.0000 0.569202
\(893\) −36.0000 −1.20469
\(894\) 20.0000 0.668900
\(895\) −21.0000 −0.701953
\(896\) −3.00000 −0.100223
\(897\) −1.00000 −0.0333890
\(898\) −10.0000 −0.333704
\(899\) 0 0
\(900\) 1.00000 0.0333333
\(901\) 0 0
\(902\) 6.00000 0.199778
\(903\) 30.0000 0.998337
\(904\) 4.00000 0.133038
\(905\) 10.0000 0.332411
\(906\) −12.0000 −0.398673
\(907\) −44.0000 −1.46100 −0.730498 − 0.682915i \(-0.760712\pi\)
−0.730498 + 0.682915i \(0.760712\pi\)
\(908\) 6.00000 0.199117
\(909\) 10.0000 0.331679
\(910\) −3.00000 −0.0994490
\(911\) 8.00000 0.265052 0.132526 − 0.991180i \(-0.457691\pi\)
0.132526 + 0.991180i \(0.457691\pi\)
\(912\) 4.00000 0.132453
\(913\) −6.00000 −0.198571
\(914\) −32.0000 −1.05847
\(915\) −10.0000 −0.330590
\(916\) 0 0
\(917\) 45.0000 1.48603
\(918\) 0 0
\(919\) 16.0000 0.527791 0.263896 − 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 1.00000 0.0329690
\(921\) 2.00000 0.0659022
\(922\) 14.0000 0.461065
\(923\) 2.00000 0.0658308
\(924\) 9.00000 0.296078
\(925\) 6.00000 0.197279
\(926\) −33.0000 −1.08445
\(927\) −8.00000 −0.262754
\(928\) 0 0
\(929\) 34.0000 1.11550 0.557752 − 0.830008i \(-0.311664\pi\)
0.557752 + 0.830008i \(0.311664\pi\)
\(930\) −4.00000 −0.131165
\(931\) 8.00000 0.262189
\(932\) 4.00000 0.131024
\(933\) 0 0
\(934\) −18.0000 −0.588978
\(935\) 0 0
\(936\) 1.00000 0.0326860
\(937\) 40.0000 1.30674 0.653372 − 0.757037i \(-0.273354\pi\)
0.653372 + 0.757037i \(0.273354\pi\)
\(938\) −24.0000 −0.783628
\(939\) 28.0000 0.913745
\(940\) 9.00000 0.293548
\(941\) 2.00000 0.0651981 0.0325991 − 0.999469i \(-0.489622\pi\)
0.0325991 + 0.999469i \(0.489622\pi\)
\(942\) 9.00000 0.293236
\(943\) −2.00000 −0.0651290
\(944\) 9.00000 0.292925
\(945\) −3.00000 −0.0975900
\(946\) −30.0000 −0.975384
\(947\) −18.0000 −0.584921 −0.292461 − 0.956278i \(-0.594474\pi\)
−0.292461 + 0.956278i \(0.594474\pi\)
\(948\) −16.0000 −0.519656
\(949\) 0 0
\(950\) −4.00000 −0.129777
\(951\) −2.00000 −0.0648544
\(952\) 0 0
\(953\) 4.00000 0.129573 0.0647864 − 0.997899i \(-0.479363\pi\)
0.0647864 + 0.997899i \(0.479363\pi\)
\(954\) 1.00000 0.0323762
\(955\) 0 0
\(956\) −12.0000 −0.388108
\(957\) 0 0
\(958\) −28.0000 −0.904639
\(959\) −48.0000 −1.55000
\(960\) −1.00000 −0.0322749
\(961\) −15.0000 −0.483871
\(962\) 6.00000 0.193448
\(963\) −8.00000 −0.257796
\(964\) −15.0000 −0.483117
\(965\) −12.0000 −0.386294
\(966\) −3.00000 −0.0965234
\(967\) 48.0000 1.54358 0.771788 − 0.635880i \(-0.219363\pi\)
0.771788 + 0.635880i \(0.219363\pi\)
\(968\) 2.00000 0.0642824
\(969\) 0 0
\(970\) 4.00000 0.128432
\(971\) −12.0000 −0.385098 −0.192549 − 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) 1.00000 0.0320750
\(973\) 60.0000 1.92351
\(974\) 32.0000 1.02535
\(975\) −1.00000 −0.0320256
\(976\) 10.0000 0.320092
\(977\) −58.0000 −1.85558 −0.927792 − 0.373097i \(-0.878296\pi\)
−0.927792 + 0.373097i \(0.878296\pi\)
\(978\) 6.00000 0.191859
\(979\) 33.0000 1.05468
\(980\) −2.00000 −0.0638877
\(981\) 4.00000 0.127710
\(982\) 37.0000 1.18072
\(983\) 56.0000 1.78612 0.893061 − 0.449935i \(-0.148553\pi\)
0.893061 + 0.449935i \(0.148553\pi\)
\(984\) 2.00000 0.0637577
\(985\) −3.00000 −0.0955879
\(986\) 0 0
\(987\) −27.0000 −0.859419
\(988\) −4.00000 −0.127257
\(989\) 10.0000 0.317982
\(990\) 3.00000 0.0953463
\(991\) −10.0000 −0.317660 −0.158830 − 0.987306i \(-0.550772\pi\)
−0.158830 + 0.987306i \(0.550772\pi\)
\(992\) 4.00000 0.127000
\(993\) 13.0000 0.412543
\(994\) 6.00000 0.190308
\(995\) 20.0000 0.634043
\(996\) −2.00000 −0.0633724
\(997\) 41.0000 1.29848 0.649242 − 0.760582i \(-0.275086\pi\)
0.649242 + 0.760582i \(0.275086\pi\)
\(998\) −41.0000 −1.29783
\(999\) 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 8670.2.a.i.1.1 yes 1
17.16 even 2 8670.2.a.e.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8670.2.a.e.1.1 ✓ 1 17.16 even 2
8670.2.a.i.1.1 yes 1 1.1 even 1 trivial