Properties

Label 6039.2.a.f.1.10
Level $6039$
Weight $2$
Character 6039.1
Self dual yes
Analytic conductor $48.222$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6039,2,Mod(1,6039)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6039, 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("6039.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6039.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: \(48.2216577807\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 5x^{11} - 5x^{10} + 48x^{9} - 173x^{7} + 29x^{6} + 281x^{5} - 41x^{4} - 201x^{3} + 8x^{2} + 49x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2013)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Root \(-1.31792\) of defining polynomial
Character \(\chi\) \(=\) 6039.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.31792 q^{2} +3.37276 q^{4} +2.47951 q^{5} +3.26859 q^{7} +3.18195 q^{8} +O(q^{10})\) \(q+2.31792 q^{2} +3.37276 q^{4} +2.47951 q^{5} +3.26859 q^{7} +3.18195 q^{8} +5.74732 q^{10} -1.00000 q^{11} +6.05111 q^{13} +7.57633 q^{14} +0.629993 q^{16} +6.63735 q^{17} -4.68664 q^{19} +8.36281 q^{20} -2.31792 q^{22} +6.72397 q^{23} +1.14799 q^{25} +14.0260 q^{26} +11.0242 q^{28} +6.44005 q^{29} -6.17187 q^{31} -4.90363 q^{32} +15.3849 q^{34} +8.10451 q^{35} -6.59063 q^{37} -10.8633 q^{38} +7.88969 q^{40} -8.91287 q^{41} -8.98140 q^{43} -3.37276 q^{44} +15.5856 q^{46} -10.7447 q^{47} +3.68368 q^{49} +2.66095 q^{50} +20.4089 q^{52} -11.7082 q^{53} -2.47951 q^{55} +10.4005 q^{56} +14.9275 q^{58} -3.81524 q^{59} -1.00000 q^{61} -14.3059 q^{62} -12.6262 q^{64} +15.0038 q^{65} +3.27788 q^{67} +22.3862 q^{68} +18.7856 q^{70} +13.3606 q^{71} -6.29610 q^{73} -15.2766 q^{74} -15.8069 q^{76} -3.26859 q^{77} -5.89560 q^{79} +1.56208 q^{80} -20.6593 q^{82} +6.77257 q^{83} +16.4574 q^{85} -20.8182 q^{86} -3.18195 q^{88} +2.86664 q^{89} +19.7786 q^{91} +22.6783 q^{92} -24.9054 q^{94} -11.6206 q^{95} +8.49974 q^{97} +8.53848 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 7 q^{2} + 13 q^{4} + 7 q^{5} - 15 q^{7} + 18 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 7 q^{2} + 13 q^{4} + 7 q^{5} - 15 q^{7} + 18 q^{8} - 6 q^{10} - 12 q^{11} - 11 q^{13} - 3 q^{14} + 19 q^{16} + 33 q^{17} - 24 q^{19} + 11 q^{20} - 7 q^{22} + 9 q^{23} + 11 q^{25} + 16 q^{26} - 41 q^{28} + 16 q^{29} + q^{31} + 28 q^{32} + 32 q^{34} + 22 q^{35} - 6 q^{37} - 12 q^{38} + 26 q^{40} + 21 q^{41} - 39 q^{43} - 13 q^{44} + 18 q^{47} + 31 q^{49} + 44 q^{50} + 3 q^{52} + 14 q^{53} - 7 q^{55} - 16 q^{56} + 33 q^{58} + 23 q^{59} - 12 q^{61} + 25 q^{62} + 12 q^{64} + 29 q^{65} + 96 q^{68} + 44 q^{70} + 19 q^{71} - 42 q^{73} - 38 q^{74} + 11 q^{76} + 15 q^{77} - 11 q^{79} + 44 q^{80} - 14 q^{82} + 56 q^{83} + 16 q^{85} + 18 q^{86} - 18 q^{88} + 55 q^{89} + 11 q^{91} + 4 q^{92} - 5 q^{94} - 15 q^{95} - 7 q^{97} - 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.31792 1.63902 0.819509 0.573066i \(-0.194246\pi\)
0.819509 + 0.573066i \(0.194246\pi\)
\(3\) 0 0
\(4\) 3.37276 1.68638
\(5\) 2.47951 1.10887 0.554436 0.832226i \(-0.312934\pi\)
0.554436 + 0.832226i \(0.312934\pi\)
\(6\) 0 0
\(7\) 3.26859 1.23541 0.617705 0.786410i \(-0.288062\pi\)
0.617705 + 0.786410i \(0.288062\pi\)
\(8\) 3.18195 1.12499
\(9\) 0 0
\(10\) 5.74732 1.81746
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 6.05111 1.67827 0.839137 0.543920i \(-0.183060\pi\)
0.839137 + 0.543920i \(0.183060\pi\)
\(14\) 7.57633 2.02486
\(15\) 0 0
\(16\) 0.629993 0.157498
\(17\) 6.63735 1.60979 0.804897 0.593414i \(-0.202220\pi\)
0.804897 + 0.593414i \(0.202220\pi\)
\(18\) 0 0
\(19\) −4.68664 −1.07519 −0.537595 0.843203i \(-0.680667\pi\)
−0.537595 + 0.843203i \(0.680667\pi\)
\(20\) 8.36281 1.86998
\(21\) 0 0
\(22\) −2.31792 −0.494183
\(23\) 6.72397 1.40204 0.701022 0.713139i \(-0.252727\pi\)
0.701022 + 0.713139i \(0.252727\pi\)
\(24\) 0 0
\(25\) 1.14799 0.229598
\(26\) 14.0260 2.75072
\(27\) 0 0
\(28\) 11.0242 2.08337
\(29\) 6.44005 1.19589 0.597944 0.801538i \(-0.295985\pi\)
0.597944 + 0.801538i \(0.295985\pi\)
\(30\) 0 0
\(31\) −6.17187 −1.10850 −0.554250 0.832350i \(-0.686995\pi\)
−0.554250 + 0.832350i \(0.686995\pi\)
\(32\) −4.90363 −0.866847
\(33\) 0 0
\(34\) 15.3849 2.63848
\(35\) 8.10451 1.36991
\(36\) 0 0
\(37\) −6.59063 −1.08349 −0.541747 0.840542i \(-0.682237\pi\)
−0.541747 + 0.840542i \(0.682237\pi\)
\(38\) −10.8633 −1.76225
\(39\) 0 0
\(40\) 7.88969 1.24747
\(41\) −8.91287 −1.39196 −0.695978 0.718063i \(-0.745029\pi\)
−0.695978 + 0.718063i \(0.745029\pi\)
\(42\) 0 0
\(43\) −8.98140 −1.36965 −0.684825 0.728707i \(-0.740122\pi\)
−0.684825 + 0.728707i \(0.740122\pi\)
\(44\) −3.37276 −0.508463
\(45\) 0 0
\(46\) 15.5856 2.29798
\(47\) −10.7447 −1.56728 −0.783639 0.621217i \(-0.786639\pi\)
−0.783639 + 0.621217i \(0.786639\pi\)
\(48\) 0 0
\(49\) 3.68368 0.526240
\(50\) 2.66095 0.376315
\(51\) 0 0
\(52\) 20.4089 2.83021
\(53\) −11.7082 −1.60825 −0.804125 0.594460i \(-0.797366\pi\)
−0.804125 + 0.594460i \(0.797366\pi\)
\(54\) 0 0
\(55\) −2.47951 −0.334338
\(56\) 10.4005 1.38982
\(57\) 0 0
\(58\) 14.9275 1.96008
\(59\) −3.81524 −0.496702 −0.248351 0.968670i \(-0.579889\pi\)
−0.248351 + 0.968670i \(0.579889\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037
\(62\) −14.3059 −1.81685
\(63\) 0 0
\(64\) −12.6262 −1.57828
\(65\) 15.0038 1.86099
\(66\) 0 0
\(67\) 3.27788 0.400456 0.200228 0.979749i \(-0.435832\pi\)
0.200228 + 0.979749i \(0.435832\pi\)
\(68\) 22.3862 2.71473
\(69\) 0 0
\(70\) 18.7856 2.24531
\(71\) 13.3606 1.58561 0.792804 0.609476i \(-0.208620\pi\)
0.792804 + 0.609476i \(0.208620\pi\)
\(72\) 0 0
\(73\) −6.29610 −0.736903 −0.368451 0.929647i \(-0.620112\pi\)
−0.368451 + 0.929647i \(0.620112\pi\)
\(74\) −15.2766 −1.77587
\(75\) 0 0
\(76\) −15.8069 −1.81318
\(77\) −3.26859 −0.372490
\(78\) 0 0
\(79\) −5.89560 −0.663307 −0.331653 0.943401i \(-0.607606\pi\)
−0.331653 + 0.943401i \(0.607606\pi\)
\(80\) 1.56208 0.174645
\(81\) 0 0
\(82\) −20.6593 −2.28144
\(83\) 6.77257 0.743386 0.371693 0.928356i \(-0.378777\pi\)
0.371693 + 0.928356i \(0.378777\pi\)
\(84\) 0 0
\(85\) 16.4574 1.78506
\(86\) −20.8182 −2.24488
\(87\) 0 0
\(88\) −3.18195 −0.339197
\(89\) 2.86664 0.303863 0.151932 0.988391i \(-0.451451\pi\)
0.151932 + 0.988391i \(0.451451\pi\)
\(90\) 0 0
\(91\) 19.7786 2.07336
\(92\) 22.6783 2.36438
\(93\) 0 0
\(94\) −24.9054 −2.56880
\(95\) −11.6206 −1.19225
\(96\) 0 0
\(97\) 8.49974 0.863018 0.431509 0.902109i \(-0.357981\pi\)
0.431509 + 0.902109i \(0.357981\pi\)
\(98\) 8.53848 0.862517
\(99\) 0 0
\(100\) 3.87189 0.387189
\(101\) 4.21053 0.418964 0.209482 0.977813i \(-0.432822\pi\)
0.209482 + 0.977813i \(0.432822\pi\)
\(102\) 0 0
\(103\) −18.1476 −1.78813 −0.894067 0.447934i \(-0.852160\pi\)
−0.894067 + 0.447934i \(0.852160\pi\)
\(104\) 19.2543 1.88804
\(105\) 0 0
\(106\) −27.1388 −2.63595
\(107\) 6.78421 0.655854 0.327927 0.944703i \(-0.393650\pi\)
0.327927 + 0.944703i \(0.393650\pi\)
\(108\) 0 0
\(109\) −1.58488 −0.151804 −0.0759021 0.997115i \(-0.524184\pi\)
−0.0759021 + 0.997115i \(0.524184\pi\)
\(110\) −5.74732 −0.547985
\(111\) 0 0
\(112\) 2.05919 0.194575
\(113\) −0.642108 −0.0604045 −0.0302022 0.999544i \(-0.509615\pi\)
−0.0302022 + 0.999544i \(0.509615\pi\)
\(114\) 0 0
\(115\) 16.6722 1.55469
\(116\) 21.7207 2.01672
\(117\) 0 0
\(118\) −8.84342 −0.814103
\(119\) 21.6948 1.98876
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −2.31792 −0.209855
\(123\) 0 0
\(124\) −20.8162 −1.86935
\(125\) −9.55111 −0.854278
\(126\) 0 0
\(127\) 11.6746 1.03595 0.517975 0.855396i \(-0.326686\pi\)
0.517975 + 0.855396i \(0.326686\pi\)
\(128\) −19.4593 −1.71998
\(129\) 0 0
\(130\) 34.7776 3.05020
\(131\) 6.62541 0.578865 0.289432 0.957198i \(-0.406533\pi\)
0.289432 + 0.957198i \(0.406533\pi\)
\(132\) 0 0
\(133\) −15.3187 −1.32830
\(134\) 7.59786 0.656355
\(135\) 0 0
\(136\) 21.1197 1.81100
\(137\) 6.72646 0.574681 0.287340 0.957829i \(-0.407229\pi\)
0.287340 + 0.957829i \(0.407229\pi\)
\(138\) 0 0
\(139\) 17.5108 1.48525 0.742625 0.669707i \(-0.233580\pi\)
0.742625 + 0.669707i \(0.233580\pi\)
\(140\) 27.3346 2.31019
\(141\) 0 0
\(142\) 30.9688 2.59884
\(143\) −6.05111 −0.506019
\(144\) 0 0
\(145\) 15.9682 1.32609
\(146\) −14.5939 −1.20780
\(147\) 0 0
\(148\) −22.2286 −1.82718
\(149\) 21.6874 1.77670 0.888352 0.459163i \(-0.151851\pi\)
0.888352 + 0.459163i \(0.151851\pi\)
\(150\) 0 0
\(151\) −12.3753 −1.00709 −0.503544 0.863970i \(-0.667971\pi\)
−0.503544 + 0.863970i \(0.667971\pi\)
\(152\) −14.9127 −1.20958
\(153\) 0 0
\(154\) −7.57633 −0.610518
\(155\) −15.3032 −1.22919
\(156\) 0 0
\(157\) 6.60694 0.527292 0.263646 0.964620i \(-0.415075\pi\)
0.263646 + 0.964620i \(0.415075\pi\)
\(158\) −13.6655 −1.08717
\(159\) 0 0
\(160\) −12.1586 −0.961223
\(161\) 21.9779 1.73210
\(162\) 0 0
\(163\) −8.97660 −0.703102 −0.351551 0.936169i \(-0.614346\pi\)
−0.351551 + 0.936169i \(0.614346\pi\)
\(164\) −30.0610 −2.34737
\(165\) 0 0
\(166\) 15.6983 1.21842
\(167\) 6.34806 0.491228 0.245614 0.969368i \(-0.421010\pi\)
0.245614 + 0.969368i \(0.421010\pi\)
\(168\) 0 0
\(169\) 23.6159 1.81661
\(170\) 38.1470 2.92574
\(171\) 0 0
\(172\) −30.2921 −2.30975
\(173\) 10.8803 0.827215 0.413607 0.910455i \(-0.364269\pi\)
0.413607 + 0.910455i \(0.364269\pi\)
\(174\) 0 0
\(175\) 3.75230 0.283648
\(176\) −0.629993 −0.0474875
\(177\) 0 0
\(178\) 6.64465 0.498038
\(179\) −21.7370 −1.62470 −0.812351 0.583168i \(-0.801813\pi\)
−0.812351 + 0.583168i \(0.801813\pi\)
\(180\) 0 0
\(181\) −2.85061 −0.211884 −0.105942 0.994372i \(-0.533786\pi\)
−0.105942 + 0.994372i \(0.533786\pi\)
\(182\) 45.8452 3.39827
\(183\) 0 0
\(184\) 21.3953 1.57729
\(185\) −16.3416 −1.20146
\(186\) 0 0
\(187\) −6.63735 −0.485371
\(188\) −36.2393 −2.64303
\(189\) 0 0
\(190\) −26.9356 −1.95412
\(191\) −16.8513 −1.21932 −0.609659 0.792664i \(-0.708693\pi\)
−0.609659 + 0.792664i \(0.708693\pi\)
\(192\) 0 0
\(193\) −0.0491126 −0.00353520 −0.00176760 0.999998i \(-0.500563\pi\)
−0.00176760 + 0.999998i \(0.500563\pi\)
\(194\) 19.7017 1.41450
\(195\) 0 0
\(196\) 12.4242 0.887440
\(197\) −6.18214 −0.440459 −0.220230 0.975448i \(-0.570681\pi\)
−0.220230 + 0.975448i \(0.570681\pi\)
\(198\) 0 0
\(199\) −7.69298 −0.545341 −0.272670 0.962108i \(-0.587907\pi\)
−0.272670 + 0.962108i \(0.587907\pi\)
\(200\) 3.65284 0.258295
\(201\) 0 0
\(202\) 9.75968 0.686689
\(203\) 21.0499 1.47741
\(204\) 0 0
\(205\) −22.0996 −1.54350
\(206\) −42.0647 −2.93078
\(207\) 0 0
\(208\) 3.81215 0.264325
\(209\) 4.68664 0.324182
\(210\) 0 0
\(211\) −25.7265 −1.77108 −0.885541 0.464561i \(-0.846212\pi\)
−0.885541 + 0.464561i \(0.846212\pi\)
\(212\) −39.4891 −2.71212
\(213\) 0 0
\(214\) 15.7253 1.07496
\(215\) −22.2695 −1.51877
\(216\) 0 0
\(217\) −20.1733 −1.36945
\(218\) −3.67363 −0.248810
\(219\) 0 0
\(220\) −8.36281 −0.563820
\(221\) 40.1633 2.70168
\(222\) 0 0
\(223\) 13.4211 0.898741 0.449371 0.893345i \(-0.351648\pi\)
0.449371 + 0.893345i \(0.351648\pi\)
\(224\) −16.0280 −1.07091
\(225\) 0 0
\(226\) −1.48836 −0.0990040
\(227\) 22.1736 1.47171 0.735856 0.677138i \(-0.236780\pi\)
0.735856 + 0.677138i \(0.236780\pi\)
\(228\) 0 0
\(229\) 8.20706 0.542338 0.271169 0.962532i \(-0.412590\pi\)
0.271169 + 0.962532i \(0.412590\pi\)
\(230\) 38.6448 2.54816
\(231\) 0 0
\(232\) 20.4919 1.34536
\(233\) 15.5362 1.01781 0.508906 0.860822i \(-0.330050\pi\)
0.508906 + 0.860822i \(0.330050\pi\)
\(234\) 0 0
\(235\) −26.6417 −1.73791
\(236\) −12.8679 −0.837628
\(237\) 0 0
\(238\) 50.2868 3.25961
\(239\) −7.51917 −0.486375 −0.243187 0.969979i \(-0.578193\pi\)
−0.243187 + 0.969979i \(0.578193\pi\)
\(240\) 0 0
\(241\) −15.5840 −1.00385 −0.501927 0.864910i \(-0.667376\pi\)
−0.501927 + 0.864910i \(0.667376\pi\)
\(242\) 2.31792 0.149002
\(243\) 0 0
\(244\) −3.37276 −0.215919
\(245\) 9.13373 0.583533
\(246\) 0 0
\(247\) −28.3594 −1.80446
\(248\) −19.6386 −1.24705
\(249\) 0 0
\(250\) −22.1387 −1.40018
\(251\) −29.0362 −1.83275 −0.916373 0.400326i \(-0.868897\pi\)
−0.916373 + 0.400326i \(0.868897\pi\)
\(252\) 0 0
\(253\) −6.72397 −0.422732
\(254\) 27.0607 1.69794
\(255\) 0 0
\(256\) −19.8527 −1.24080
\(257\) −7.42017 −0.462858 −0.231429 0.972852i \(-0.574340\pi\)
−0.231429 + 0.972852i \(0.574340\pi\)
\(258\) 0 0
\(259\) −21.5421 −1.33856
\(260\) 50.6042 3.13834
\(261\) 0 0
\(262\) 15.3572 0.948770
\(263\) −5.43921 −0.335396 −0.167698 0.985838i \(-0.553633\pi\)
−0.167698 + 0.985838i \(0.553633\pi\)
\(264\) 0 0
\(265\) −29.0307 −1.78334
\(266\) −35.5076 −2.17711
\(267\) 0 0
\(268\) 11.0555 0.675322
\(269\) 10.2198 0.623113 0.311557 0.950228i \(-0.399150\pi\)
0.311557 + 0.950228i \(0.399150\pi\)
\(270\) 0 0
\(271\) 7.36832 0.447593 0.223797 0.974636i \(-0.428155\pi\)
0.223797 + 0.974636i \(0.428155\pi\)
\(272\) 4.18148 0.253540
\(273\) 0 0
\(274\) 15.5914 0.941912
\(275\) −1.14799 −0.0692263
\(276\) 0 0
\(277\) −19.2638 −1.15745 −0.578725 0.815523i \(-0.696450\pi\)
−0.578725 + 0.815523i \(0.696450\pi\)
\(278\) 40.5888 2.43435
\(279\) 0 0
\(280\) 25.7882 1.54114
\(281\) −1.00514 −0.0599619 −0.0299809 0.999550i \(-0.509545\pi\)
−0.0299809 + 0.999550i \(0.509545\pi\)
\(282\) 0 0
\(283\) 14.9883 0.890965 0.445482 0.895291i \(-0.353032\pi\)
0.445482 + 0.895291i \(0.353032\pi\)
\(284\) 45.0620 2.67394
\(285\) 0 0
\(286\) −14.0260 −0.829374
\(287\) −29.1325 −1.71964
\(288\) 0 0
\(289\) 27.0544 1.59144
\(290\) 37.0130 2.17348
\(291\) 0 0
\(292\) −21.2352 −1.24270
\(293\) 32.8105 1.91681 0.958404 0.285416i \(-0.0921318\pi\)
0.958404 + 0.285416i \(0.0921318\pi\)
\(294\) 0 0
\(295\) −9.45993 −0.550779
\(296\) −20.9711 −1.21892
\(297\) 0 0
\(298\) 50.2698 2.91205
\(299\) 40.6875 2.35302
\(300\) 0 0
\(301\) −29.3565 −1.69208
\(302\) −28.6850 −1.65063
\(303\) 0 0
\(304\) −2.95255 −0.169340
\(305\) −2.47951 −0.141977
\(306\) 0 0
\(307\) 7.14961 0.408050 0.204025 0.978966i \(-0.434598\pi\)
0.204025 + 0.978966i \(0.434598\pi\)
\(308\) −11.0242 −0.628160
\(309\) 0 0
\(310\) −35.4717 −2.01466
\(311\) 18.8314 1.06783 0.533916 0.845538i \(-0.320720\pi\)
0.533916 + 0.845538i \(0.320720\pi\)
\(312\) 0 0
\(313\) 23.7753 1.34386 0.671928 0.740616i \(-0.265466\pi\)
0.671928 + 0.740616i \(0.265466\pi\)
\(314\) 15.3144 0.864240
\(315\) 0 0
\(316\) −19.8845 −1.11859
\(317\) 15.5940 0.875848 0.437924 0.899012i \(-0.355714\pi\)
0.437924 + 0.899012i \(0.355714\pi\)
\(318\) 0 0
\(319\) −6.44005 −0.360573
\(320\) −31.3069 −1.75011
\(321\) 0 0
\(322\) 50.9431 2.83895
\(323\) −31.1069 −1.73083
\(324\) 0 0
\(325\) 6.94660 0.385328
\(326\) −20.8071 −1.15240
\(327\) 0 0
\(328\) −28.3603 −1.56594
\(329\) −35.1200 −1.93623
\(330\) 0 0
\(331\) −18.3454 −1.00835 −0.504177 0.863600i \(-0.668204\pi\)
−0.504177 + 0.863600i \(0.668204\pi\)
\(332\) 22.8422 1.25363
\(333\) 0 0
\(334\) 14.7143 0.805131
\(335\) 8.12754 0.444055
\(336\) 0 0
\(337\) −0.343178 −0.0186941 −0.00934704 0.999956i \(-0.502975\pi\)
−0.00934704 + 0.999956i \(0.502975\pi\)
\(338\) 54.7397 2.97745
\(339\) 0 0
\(340\) 55.5069 3.01028
\(341\) 6.17187 0.334226
\(342\) 0 0
\(343\) −10.8397 −0.585288
\(344\) −28.5784 −1.54084
\(345\) 0 0
\(346\) 25.2197 1.35582
\(347\) 2.46579 0.132370 0.0661852 0.997807i \(-0.478917\pi\)
0.0661852 + 0.997807i \(0.478917\pi\)
\(348\) 0 0
\(349\) 26.2419 1.40470 0.702348 0.711834i \(-0.252135\pi\)
0.702348 + 0.711834i \(0.252135\pi\)
\(350\) 8.69755 0.464904
\(351\) 0 0
\(352\) 4.90363 0.261364
\(353\) 8.93973 0.475814 0.237907 0.971288i \(-0.423539\pi\)
0.237907 + 0.971288i \(0.423539\pi\)
\(354\) 0 0
\(355\) 33.1277 1.75824
\(356\) 9.66849 0.512429
\(357\) 0 0
\(358\) −50.3848 −2.66292
\(359\) −24.1204 −1.27303 −0.636513 0.771266i \(-0.719624\pi\)
−0.636513 + 0.771266i \(0.719624\pi\)
\(360\) 0 0
\(361\) 2.96461 0.156032
\(362\) −6.60750 −0.347282
\(363\) 0 0
\(364\) 66.7084 3.49647
\(365\) −15.6113 −0.817131
\(366\) 0 0
\(367\) 4.09954 0.213994 0.106997 0.994259i \(-0.465876\pi\)
0.106997 + 0.994259i \(0.465876\pi\)
\(368\) 4.23605 0.220819
\(369\) 0 0
\(370\) −37.8785 −1.96921
\(371\) −38.2694 −1.98685
\(372\) 0 0
\(373\) 4.36673 0.226101 0.113050 0.993589i \(-0.463938\pi\)
0.113050 + 0.993589i \(0.463938\pi\)
\(374\) −15.3849 −0.795532
\(375\) 0 0
\(376\) −34.1891 −1.76317
\(377\) 38.9694 2.00703
\(378\) 0 0
\(379\) 38.3572 1.97028 0.985139 0.171761i \(-0.0549457\pi\)
0.985139 + 0.171761i \(0.0549457\pi\)
\(380\) −39.1935 −2.01058
\(381\) 0 0
\(382\) −39.0600 −1.99848
\(383\) −3.77120 −0.192699 −0.0963497 0.995348i \(-0.530717\pi\)
−0.0963497 + 0.995348i \(0.530717\pi\)
\(384\) 0 0
\(385\) −8.10451 −0.413044
\(386\) −0.113839 −0.00579426
\(387\) 0 0
\(388\) 28.6676 1.45538
\(389\) −33.2927 −1.68800 −0.844002 0.536339i \(-0.819807\pi\)
−0.844002 + 0.536339i \(0.819807\pi\)
\(390\) 0 0
\(391\) 44.6294 2.25700
\(392\) 11.7213 0.592014
\(393\) 0 0
\(394\) −14.3297 −0.721921
\(395\) −14.6182 −0.735523
\(396\) 0 0
\(397\) −19.5260 −0.979980 −0.489990 0.871728i \(-0.663000\pi\)
−0.489990 + 0.871728i \(0.663000\pi\)
\(398\) −17.8317 −0.893823
\(399\) 0 0
\(400\) 0.723225 0.0361612
\(401\) 38.0053 1.89789 0.948946 0.315438i \(-0.102152\pi\)
0.948946 + 0.315438i \(0.102152\pi\)
\(402\) 0 0
\(403\) −37.3466 −1.86037
\(404\) 14.2011 0.706532
\(405\) 0 0
\(406\) 48.7920 2.42150
\(407\) 6.59063 0.326686
\(408\) 0 0
\(409\) −2.79051 −0.137982 −0.0689909 0.997617i \(-0.521978\pi\)
−0.0689909 + 0.997617i \(0.521978\pi\)
\(410\) −51.2251 −2.52983
\(411\) 0 0
\(412\) −61.2074 −3.01547
\(413\) −12.4704 −0.613630
\(414\) 0 0
\(415\) 16.7927 0.824320
\(416\) −29.6724 −1.45481
\(417\) 0 0
\(418\) 10.8633 0.531340
\(419\) 1.30588 0.0637964 0.0318982 0.999491i \(-0.489845\pi\)
0.0318982 + 0.999491i \(0.489845\pi\)
\(420\) 0 0
\(421\) −18.6901 −0.910897 −0.455449 0.890262i \(-0.650521\pi\)
−0.455449 + 0.890262i \(0.650521\pi\)
\(422\) −59.6319 −2.90284
\(423\) 0 0
\(424\) −37.2550 −1.80927
\(425\) 7.61961 0.369605
\(426\) 0 0
\(427\) −3.26859 −0.158178
\(428\) 22.8815 1.10602
\(429\) 0 0
\(430\) −51.6190 −2.48929
\(431\) 4.71349 0.227041 0.113520 0.993536i \(-0.463787\pi\)
0.113520 + 0.993536i \(0.463787\pi\)
\(432\) 0 0
\(433\) −27.0077 −1.29791 −0.648953 0.760829i \(-0.724793\pi\)
−0.648953 + 0.760829i \(0.724793\pi\)
\(434\) −46.7602 −2.24456
\(435\) 0 0
\(436\) −5.34543 −0.255999
\(437\) −31.5128 −1.50746
\(438\) 0 0
\(439\) 28.0507 1.33879 0.669394 0.742907i \(-0.266554\pi\)
0.669394 + 0.742907i \(0.266554\pi\)
\(440\) −7.88969 −0.376126
\(441\) 0 0
\(442\) 93.0954 4.42810
\(443\) −12.6047 −0.598866 −0.299433 0.954117i \(-0.596798\pi\)
−0.299433 + 0.954117i \(0.596798\pi\)
\(444\) 0 0
\(445\) 7.10788 0.336946
\(446\) 31.1090 1.47305
\(447\) 0 0
\(448\) −41.2699 −1.94982
\(449\) −9.95289 −0.469706 −0.234853 0.972031i \(-0.575461\pi\)
−0.234853 + 0.972031i \(0.575461\pi\)
\(450\) 0 0
\(451\) 8.91287 0.419691
\(452\) −2.16568 −0.101865
\(453\) 0 0
\(454\) 51.3966 2.41216
\(455\) 49.0413 2.29909
\(456\) 0 0
\(457\) 31.4066 1.46914 0.734569 0.678534i \(-0.237384\pi\)
0.734569 + 0.678534i \(0.237384\pi\)
\(458\) 19.0233 0.888902
\(459\) 0 0
\(460\) 56.2313 2.62180
\(461\) −29.2165 −1.36075 −0.680373 0.732866i \(-0.738182\pi\)
−0.680373 + 0.732866i \(0.738182\pi\)
\(462\) 0 0
\(463\) −26.3649 −1.22528 −0.612640 0.790362i \(-0.709892\pi\)
−0.612640 + 0.790362i \(0.709892\pi\)
\(464\) 4.05718 0.188350
\(465\) 0 0
\(466\) 36.0117 1.66821
\(467\) 14.1452 0.654563 0.327281 0.944927i \(-0.393868\pi\)
0.327281 + 0.944927i \(0.393868\pi\)
\(468\) 0 0
\(469\) 10.7140 0.494728
\(470\) −61.7533 −2.84847
\(471\) 0 0
\(472\) −12.1399 −0.558784
\(473\) 8.98140 0.412965
\(474\) 0 0
\(475\) −5.38021 −0.246861
\(476\) 73.1713 3.35380
\(477\) 0 0
\(478\) −17.4289 −0.797177
\(479\) −3.16963 −0.144824 −0.0724121 0.997375i \(-0.523070\pi\)
−0.0724121 + 0.997375i \(0.523070\pi\)
\(480\) 0 0
\(481\) −39.8806 −1.81840
\(482\) −36.1225 −1.64534
\(483\) 0 0
\(484\) 3.37276 0.153307
\(485\) 21.0752 0.956976
\(486\) 0 0
\(487\) −1.73207 −0.0784876 −0.0392438 0.999230i \(-0.512495\pi\)
−0.0392438 + 0.999230i \(0.512495\pi\)
\(488\) −3.18195 −0.144040
\(489\) 0 0
\(490\) 21.1713 0.956421
\(491\) 33.0415 1.49114 0.745572 0.666425i \(-0.232176\pi\)
0.745572 + 0.666425i \(0.232176\pi\)
\(492\) 0 0
\(493\) 42.7449 1.92513
\(494\) −65.7348 −2.95755
\(495\) 0 0
\(496\) −3.88823 −0.174587
\(497\) 43.6702 1.95888
\(498\) 0 0
\(499\) −14.4982 −0.649027 −0.324514 0.945881i \(-0.605201\pi\)
−0.324514 + 0.945881i \(0.605201\pi\)
\(500\) −32.2136 −1.44064
\(501\) 0 0
\(502\) −67.3035 −3.00390
\(503\) 38.2279 1.70450 0.852250 0.523136i \(-0.175238\pi\)
0.852250 + 0.523136i \(0.175238\pi\)
\(504\) 0 0
\(505\) 10.4401 0.464577
\(506\) −15.5856 −0.692866
\(507\) 0 0
\(508\) 39.3755 1.74701
\(509\) 0.825848 0.0366051 0.0183025 0.999832i \(-0.494174\pi\)
0.0183025 + 0.999832i \(0.494174\pi\)
\(510\) 0 0
\(511\) −20.5794 −0.910378
\(512\) −7.09846 −0.313711
\(513\) 0 0
\(514\) −17.1994 −0.758632
\(515\) −44.9972 −1.98281
\(516\) 0 0
\(517\) 10.7447 0.472552
\(518\) −49.9329 −2.19392
\(519\) 0 0
\(520\) 47.7414 2.09360
\(521\) −12.2959 −0.538695 −0.269348 0.963043i \(-0.586808\pi\)
−0.269348 + 0.963043i \(0.586808\pi\)
\(522\) 0 0
\(523\) −23.6237 −1.03299 −0.516496 0.856290i \(-0.672764\pi\)
−0.516496 + 0.856290i \(0.672764\pi\)
\(524\) 22.3459 0.976186
\(525\) 0 0
\(526\) −12.6077 −0.549721
\(527\) −40.9649 −1.78446
\(528\) 0 0
\(529\) 22.2118 0.965730
\(530\) −67.2910 −2.92293
\(531\) 0 0
\(532\) −51.6663 −2.24002
\(533\) −53.9327 −2.33608
\(534\) 0 0
\(535\) 16.8215 0.727259
\(536\) 10.4300 0.450509
\(537\) 0 0
\(538\) 23.6887 1.02129
\(539\) −3.68368 −0.158667
\(540\) 0 0
\(541\) 10.5431 0.453283 0.226642 0.973978i \(-0.427225\pi\)
0.226642 + 0.973978i \(0.427225\pi\)
\(542\) 17.0792 0.733614
\(543\) 0 0
\(544\) −32.5471 −1.39545
\(545\) −3.92974 −0.168331
\(546\) 0 0
\(547\) −9.08152 −0.388298 −0.194149 0.980972i \(-0.562194\pi\)
−0.194149 + 0.980972i \(0.562194\pi\)
\(548\) 22.6868 0.969130
\(549\) 0 0
\(550\) −2.66095 −0.113463
\(551\) −30.1822 −1.28580
\(552\) 0 0
\(553\) −19.2703 −0.819457
\(554\) −44.6520 −1.89708
\(555\) 0 0
\(556\) 59.0599 2.50470
\(557\) −5.77966 −0.244892 −0.122446 0.992475i \(-0.539074\pi\)
−0.122446 + 0.992475i \(0.539074\pi\)
\(558\) 0 0
\(559\) −54.3474 −2.29865
\(560\) 5.10578 0.215759
\(561\) 0 0
\(562\) −2.32985 −0.0982786
\(563\) 15.7767 0.664909 0.332455 0.943119i \(-0.392123\pi\)
0.332455 + 0.943119i \(0.392123\pi\)
\(564\) 0 0
\(565\) −1.59212 −0.0669808
\(566\) 34.7418 1.46031
\(567\) 0 0
\(568\) 42.5127 1.78379
\(569\) 21.8817 0.917329 0.458665 0.888609i \(-0.348328\pi\)
0.458665 + 0.888609i \(0.348328\pi\)
\(570\) 0 0
\(571\) 0.534112 0.0223519 0.0111759 0.999938i \(-0.496443\pi\)
0.0111759 + 0.999938i \(0.496443\pi\)
\(572\) −20.4089 −0.853340
\(573\) 0 0
\(574\) −67.5269 −2.81852
\(575\) 7.71904 0.321906
\(576\) 0 0
\(577\) −8.32355 −0.346514 −0.173257 0.984877i \(-0.555429\pi\)
−0.173257 + 0.984877i \(0.555429\pi\)
\(578\) 62.7101 2.60839
\(579\) 0 0
\(580\) 53.8569 2.23629
\(581\) 22.1367 0.918387
\(582\) 0 0
\(583\) 11.7082 0.484906
\(584\) −20.0339 −0.829008
\(585\) 0 0
\(586\) 76.0521 3.14168
\(587\) 19.5104 0.805281 0.402641 0.915358i \(-0.368092\pi\)
0.402641 + 0.915358i \(0.368092\pi\)
\(588\) 0 0
\(589\) 28.9253 1.19185
\(590\) −21.9274 −0.902736
\(591\) 0 0
\(592\) −4.15205 −0.170648
\(593\) 20.9055 0.858486 0.429243 0.903189i \(-0.358780\pi\)
0.429243 + 0.903189i \(0.358780\pi\)
\(594\) 0 0
\(595\) 53.7925 2.20528
\(596\) 73.1465 2.99620
\(597\) 0 0
\(598\) 94.3103 3.85664
\(599\) −17.3363 −0.708341 −0.354170 0.935181i \(-0.615237\pi\)
−0.354170 + 0.935181i \(0.615237\pi\)
\(600\) 0 0
\(601\) 10.8920 0.444293 0.222147 0.975013i \(-0.428694\pi\)
0.222147 + 0.975013i \(0.428694\pi\)
\(602\) −68.0461 −2.77335
\(603\) 0 0
\(604\) −41.7389 −1.69833
\(605\) 2.47951 0.100807
\(606\) 0 0
\(607\) 34.3377 1.39372 0.696862 0.717205i \(-0.254579\pi\)
0.696862 + 0.717205i \(0.254579\pi\)
\(608\) 22.9816 0.932025
\(609\) 0 0
\(610\) −5.74732 −0.232702
\(611\) −65.0174 −2.63032
\(612\) 0 0
\(613\) −7.44572 −0.300730 −0.150365 0.988631i \(-0.548045\pi\)
−0.150365 + 0.988631i \(0.548045\pi\)
\(614\) 16.5722 0.668801
\(615\) 0 0
\(616\) −10.4005 −0.419048
\(617\) 23.0634 0.928497 0.464249 0.885705i \(-0.346324\pi\)
0.464249 + 0.885705i \(0.346324\pi\)
\(618\) 0 0
\(619\) −16.8519 −0.677334 −0.338667 0.940906i \(-0.609976\pi\)
−0.338667 + 0.940906i \(0.609976\pi\)
\(620\) −51.6142 −2.07287
\(621\) 0 0
\(622\) 43.6497 1.75020
\(623\) 9.36987 0.375396
\(624\) 0 0
\(625\) −29.4221 −1.17688
\(626\) 55.1092 2.20261
\(627\) 0 0
\(628\) 22.2836 0.889214
\(629\) −43.7444 −1.74420
\(630\) 0 0
\(631\) 30.8979 1.23002 0.615012 0.788518i \(-0.289151\pi\)
0.615012 + 0.788518i \(0.289151\pi\)
\(632\) −18.7595 −0.746213
\(633\) 0 0
\(634\) 36.1458 1.43553
\(635\) 28.9473 1.14874
\(636\) 0 0
\(637\) 22.2903 0.883175
\(638\) −14.9275 −0.590986
\(639\) 0 0
\(640\) −48.2496 −1.90723
\(641\) −7.28363 −0.287686 −0.143843 0.989601i \(-0.545946\pi\)
−0.143843 + 0.989601i \(0.545946\pi\)
\(642\) 0 0
\(643\) −37.6899 −1.48634 −0.743172 0.669101i \(-0.766679\pi\)
−0.743172 + 0.669101i \(0.766679\pi\)
\(644\) 74.1262 2.92098
\(645\) 0 0
\(646\) −72.1033 −2.83687
\(647\) −49.6098 −1.95036 −0.975181 0.221409i \(-0.928934\pi\)
−0.975181 + 0.221409i \(0.928934\pi\)
\(648\) 0 0
\(649\) 3.81524 0.149761
\(650\) 16.1017 0.631560
\(651\) 0 0
\(652\) −30.2759 −1.18570
\(653\) 23.6399 0.925101 0.462551 0.886593i \(-0.346934\pi\)
0.462551 + 0.886593i \(0.346934\pi\)
\(654\) 0 0
\(655\) 16.4278 0.641887
\(656\) −5.61504 −0.219231
\(657\) 0 0
\(658\) −81.4055 −3.17352
\(659\) −28.7430 −1.11967 −0.559834 0.828605i \(-0.689135\pi\)
−0.559834 + 0.828605i \(0.689135\pi\)
\(660\) 0 0
\(661\) 5.89740 0.229382 0.114691 0.993401i \(-0.463412\pi\)
0.114691 + 0.993401i \(0.463412\pi\)
\(662\) −42.5232 −1.65271
\(663\) 0 0
\(664\) 21.5500 0.836301
\(665\) −37.9829 −1.47292
\(666\) 0 0
\(667\) 43.3027 1.67669
\(668\) 21.4105 0.828397
\(669\) 0 0
\(670\) 18.8390 0.727814
\(671\) 1.00000 0.0386046
\(672\) 0 0
\(673\) 23.9587 0.923539 0.461769 0.887000i \(-0.347215\pi\)
0.461769 + 0.887000i \(0.347215\pi\)
\(674\) −0.795459 −0.0306400
\(675\) 0 0
\(676\) 79.6507 3.06349
\(677\) −10.7860 −0.414539 −0.207270 0.978284i \(-0.566458\pi\)
−0.207270 + 0.978284i \(0.566458\pi\)
\(678\) 0 0
\(679\) 27.7822 1.06618
\(680\) 52.3667 2.00817
\(681\) 0 0
\(682\) 14.3059 0.547802
\(683\) −28.2231 −1.07993 −0.539964 0.841688i \(-0.681562\pi\)
−0.539964 + 0.841688i \(0.681562\pi\)
\(684\) 0 0
\(685\) 16.6784 0.637247
\(686\) −25.1256 −0.959298
\(687\) 0 0
\(688\) −5.65822 −0.215718
\(689\) −70.8478 −2.69909
\(690\) 0 0
\(691\) −9.63955 −0.366706 −0.183353 0.983047i \(-0.558695\pi\)
−0.183353 + 0.983047i \(0.558695\pi\)
\(692\) 36.6967 1.39500
\(693\) 0 0
\(694\) 5.71550 0.216958
\(695\) 43.4184 1.64695
\(696\) 0 0
\(697\) −59.1578 −2.24076
\(698\) 60.8266 2.30232
\(699\) 0 0
\(700\) 12.6556 0.478338
\(701\) 5.33675 0.201566 0.100783 0.994908i \(-0.467865\pi\)
0.100783 + 0.994908i \(0.467865\pi\)
\(702\) 0 0
\(703\) 30.8879 1.16496
\(704\) 12.6262 0.475868
\(705\) 0 0
\(706\) 20.7216 0.779867
\(707\) 13.7625 0.517592
\(708\) 0 0
\(709\) 41.2039 1.54744 0.773722 0.633525i \(-0.218393\pi\)
0.773722 + 0.633525i \(0.218393\pi\)
\(710\) 76.7875 2.88178
\(711\) 0 0
\(712\) 9.12151 0.341843
\(713\) −41.4995 −1.55417
\(714\) 0 0
\(715\) −15.0038 −0.561110
\(716\) −73.3138 −2.73987
\(717\) 0 0
\(718\) −55.9092 −2.08651
\(719\) −13.1824 −0.491620 −0.245810 0.969318i \(-0.579054\pi\)
−0.245810 + 0.969318i \(0.579054\pi\)
\(720\) 0 0
\(721\) −59.3170 −2.20908
\(722\) 6.87173 0.255739
\(723\) 0 0
\(724\) −9.61444 −0.357318
\(725\) 7.39310 0.274573
\(726\) 0 0
\(727\) −9.00135 −0.333842 −0.166921 0.985970i \(-0.553382\pi\)
−0.166921 + 0.985970i \(0.553382\pi\)
\(728\) 62.9345 2.33251
\(729\) 0 0
\(730\) −36.1857 −1.33929
\(731\) −59.6127 −2.20486
\(732\) 0 0
\(733\) −29.6369 −1.09467 −0.547333 0.836915i \(-0.684357\pi\)
−0.547333 + 0.836915i \(0.684357\pi\)
\(734\) 9.50240 0.350740
\(735\) 0 0
\(736\) −32.9719 −1.21536
\(737\) −3.27788 −0.120742
\(738\) 0 0
\(739\) −23.8776 −0.878350 −0.439175 0.898401i \(-0.644729\pi\)
−0.439175 + 0.898401i \(0.644729\pi\)
\(740\) −55.1162 −2.02611
\(741\) 0 0
\(742\) −88.7055 −3.25648
\(743\) 46.5931 1.70933 0.854667 0.519177i \(-0.173762\pi\)
0.854667 + 0.519177i \(0.173762\pi\)
\(744\) 0 0
\(745\) 53.7743 1.97014
\(746\) 10.1217 0.370583
\(747\) 0 0
\(748\) −22.3862 −0.818520
\(749\) 22.1748 0.810250
\(750\) 0 0
\(751\) −11.2932 −0.412095 −0.206047 0.978542i \(-0.566060\pi\)
−0.206047 + 0.978542i \(0.566060\pi\)
\(752\) −6.76909 −0.246843
\(753\) 0 0
\(754\) 90.3280 3.28955
\(755\) −30.6847 −1.11673
\(756\) 0 0
\(757\) 24.6228 0.894931 0.447466 0.894301i \(-0.352327\pi\)
0.447466 + 0.894301i \(0.352327\pi\)
\(758\) 88.9090 3.22932
\(759\) 0 0
\(760\) −36.9762 −1.34127
\(761\) −37.1943 −1.34829 −0.674146 0.738598i \(-0.735488\pi\)
−0.674146 + 0.738598i \(0.735488\pi\)
\(762\) 0 0
\(763\) −5.18033 −0.187540
\(764\) −56.8354 −2.05623
\(765\) 0 0
\(766\) −8.74135 −0.315838
\(767\) −23.0864 −0.833602
\(768\) 0 0
\(769\) −29.5613 −1.06601 −0.533004 0.846113i \(-0.678937\pi\)
−0.533004 + 0.846113i \(0.678937\pi\)
\(770\) −18.7856 −0.676987
\(771\) 0 0
\(772\) −0.165645 −0.00596170
\(773\) −9.44618 −0.339755 −0.169878 0.985465i \(-0.554337\pi\)
−0.169878 + 0.985465i \(0.554337\pi\)
\(774\) 0 0
\(775\) −7.08524 −0.254509
\(776\) 27.0458 0.970886
\(777\) 0 0
\(778\) −77.1698 −2.76667
\(779\) 41.7714 1.49662
\(780\) 0 0
\(781\) −13.3606 −0.478079
\(782\) 103.447 3.69927
\(783\) 0 0
\(784\) 2.32069 0.0828818
\(785\) 16.3820 0.584699
\(786\) 0 0
\(787\) 19.3687 0.690421 0.345211 0.938525i \(-0.387807\pi\)
0.345211 + 0.938525i \(0.387807\pi\)
\(788\) −20.8509 −0.742782
\(789\) 0 0
\(790\) −33.8839 −1.20554
\(791\) −2.09879 −0.0746243
\(792\) 0 0
\(793\) −6.05111 −0.214881
\(794\) −45.2596 −1.60620
\(795\) 0 0
\(796\) −25.9466 −0.919652
\(797\) −6.30009 −0.223161 −0.111580 0.993755i \(-0.535591\pi\)
−0.111580 + 0.993755i \(0.535591\pi\)
\(798\) 0 0
\(799\) −71.3164 −2.52299
\(800\) −5.62931 −0.199026
\(801\) 0 0
\(802\) 88.0932 3.11068
\(803\) 6.29610 0.222185
\(804\) 0 0
\(805\) 54.4945 1.92068
\(806\) −86.5666 −3.04918
\(807\) 0 0
\(808\) 13.3977 0.471330
\(809\) −5.41537 −0.190394 −0.0951971 0.995458i \(-0.530348\pi\)
−0.0951971 + 0.995458i \(0.530348\pi\)
\(810\) 0 0
\(811\) −29.7092 −1.04323 −0.521616 0.853180i \(-0.674671\pi\)
−0.521616 + 0.853180i \(0.674671\pi\)
\(812\) 70.9962 2.49148
\(813\) 0 0
\(814\) 15.2766 0.535444
\(815\) −22.2576 −0.779650
\(816\) 0 0
\(817\) 42.0926 1.47263
\(818\) −6.46818 −0.226155
\(819\) 0 0
\(820\) −74.5366 −2.60293
\(821\) 22.5460 0.786862 0.393431 0.919354i \(-0.371288\pi\)
0.393431 + 0.919354i \(0.371288\pi\)
\(822\) 0 0
\(823\) 26.8510 0.935967 0.467984 0.883737i \(-0.344981\pi\)
0.467984 + 0.883737i \(0.344981\pi\)
\(824\) −57.7447 −2.01163
\(825\) 0 0
\(826\) −28.9055 −1.00575
\(827\) 24.0270 0.835502 0.417751 0.908562i \(-0.362818\pi\)
0.417751 + 0.908562i \(0.362818\pi\)
\(828\) 0 0
\(829\) 38.6281 1.34161 0.670805 0.741634i \(-0.265949\pi\)
0.670805 + 0.741634i \(0.265949\pi\)
\(830\) 38.9241 1.35108
\(831\) 0 0
\(832\) −76.4025 −2.64878
\(833\) 24.4499 0.847138
\(834\) 0 0
\(835\) 15.7401 0.544709
\(836\) 15.8069 0.546694
\(837\) 0 0
\(838\) 3.02693 0.104563
\(839\) −49.6307 −1.71344 −0.856720 0.515781i \(-0.827502\pi\)
−0.856720 + 0.515781i \(0.827502\pi\)
\(840\) 0 0
\(841\) 12.4742 0.430146
\(842\) −43.3221 −1.49298
\(843\) 0 0
\(844\) −86.7692 −2.98672
\(845\) 58.5559 2.01438
\(846\) 0 0
\(847\) 3.26859 0.112310
\(848\) −7.37610 −0.253296
\(849\) 0 0
\(850\) 17.6616 0.605790
\(851\) −44.3152 −1.51911
\(852\) 0 0
\(853\) 1.49047 0.0510326 0.0255163 0.999674i \(-0.491877\pi\)
0.0255163 + 0.999674i \(0.491877\pi\)
\(854\) −7.57633 −0.259257
\(855\) 0 0
\(856\) 21.5870 0.737830
\(857\) −5.29697 −0.180941 −0.0904705 0.995899i \(-0.528837\pi\)
−0.0904705 + 0.995899i \(0.528837\pi\)
\(858\) 0 0
\(859\) −46.5876 −1.58955 −0.794774 0.606905i \(-0.792411\pi\)
−0.794774 + 0.606905i \(0.792411\pi\)
\(860\) −75.1097 −2.56122
\(861\) 0 0
\(862\) 10.9255 0.372124
\(863\) 56.2138 1.91354 0.956770 0.290847i \(-0.0939369\pi\)
0.956770 + 0.290847i \(0.0939369\pi\)
\(864\) 0 0
\(865\) 26.9779 0.917276
\(866\) −62.6017 −2.12729
\(867\) 0 0
\(868\) −68.0397 −2.30942
\(869\) 5.89560 0.199995
\(870\) 0 0
\(871\) 19.8348 0.672076
\(872\) −5.04302 −0.170778
\(873\) 0 0
\(874\) −73.0443 −2.47076
\(875\) −31.2187 −1.05538
\(876\) 0 0
\(877\) 14.3453 0.484408 0.242204 0.970225i \(-0.422130\pi\)
0.242204 + 0.970225i \(0.422130\pi\)
\(878\) 65.0194 2.19430
\(879\) 0 0
\(880\) −1.56208 −0.0526576
\(881\) −23.0396 −0.776225 −0.388112 0.921612i \(-0.626873\pi\)
−0.388112 + 0.921612i \(0.626873\pi\)
\(882\) 0 0
\(883\) −19.9199 −0.670358 −0.335179 0.942155i \(-0.608797\pi\)
−0.335179 + 0.942155i \(0.608797\pi\)
\(884\) 135.461 4.55605
\(885\) 0 0
\(886\) −29.2167 −0.981553
\(887\) 7.91120 0.265632 0.132816 0.991141i \(-0.457598\pi\)
0.132816 + 0.991141i \(0.457598\pi\)
\(888\) 0 0
\(889\) 38.1594 1.27982
\(890\) 16.4755 0.552260
\(891\) 0 0
\(892\) 45.2661 1.51562
\(893\) 50.3566 1.68512
\(894\) 0 0
\(895\) −53.8973 −1.80159
\(896\) −63.6045 −2.12488
\(897\) 0 0
\(898\) −23.0700 −0.769857
\(899\) −39.7471 −1.32564
\(900\) 0 0
\(901\) −77.7117 −2.58895
\(902\) 20.6593 0.687880
\(903\) 0 0
\(904\) −2.04316 −0.0679544
\(905\) −7.06814 −0.234953
\(906\) 0 0
\(907\) −14.9038 −0.494872 −0.247436 0.968904i \(-0.579588\pi\)
−0.247436 + 0.968904i \(0.579588\pi\)
\(908\) 74.7862 2.48187
\(909\) 0 0
\(910\) 113.674 3.76825
\(911\) 52.3222 1.73351 0.866756 0.498732i \(-0.166201\pi\)
0.866756 + 0.498732i \(0.166201\pi\)
\(912\) 0 0
\(913\) −6.77257 −0.224139
\(914\) 72.7980 2.40794
\(915\) 0 0
\(916\) 27.6805 0.914588
\(917\) 21.6557 0.715136
\(918\) 0 0
\(919\) 21.3325 0.703693 0.351846 0.936058i \(-0.385554\pi\)
0.351846 + 0.936058i \(0.385554\pi\)
\(920\) 53.0501 1.74901
\(921\) 0 0
\(922\) −67.7215 −2.23029
\(923\) 80.8462 2.66109
\(924\) 0 0
\(925\) −7.56598 −0.248768
\(926\) −61.1117 −2.00826
\(927\) 0 0
\(928\) −31.5796 −1.03665
\(929\) −22.5813 −0.740867 −0.370433 0.928859i \(-0.620791\pi\)
−0.370433 + 0.928859i \(0.620791\pi\)
\(930\) 0 0
\(931\) −17.2641 −0.565807
\(932\) 52.3999 1.71642
\(933\) 0 0
\(934\) 32.7875 1.07284
\(935\) −16.4574 −0.538215
\(936\) 0 0
\(937\) −34.3641 −1.12263 −0.561314 0.827603i \(-0.689704\pi\)
−0.561314 + 0.827603i \(0.689704\pi\)
\(938\) 24.8343 0.810868
\(939\) 0 0
\(940\) −89.8559 −2.93078
\(941\) 6.54597 0.213393 0.106696 0.994292i \(-0.465973\pi\)
0.106696 + 0.994292i \(0.465973\pi\)
\(942\) 0 0
\(943\) −59.9299 −1.95158
\(944\) −2.40357 −0.0782296
\(945\) 0 0
\(946\) 20.8182 0.676858
\(947\) 54.5844 1.77375 0.886877 0.462005i \(-0.152870\pi\)
0.886877 + 0.462005i \(0.152870\pi\)
\(948\) 0 0
\(949\) −38.0984 −1.23673
\(950\) −12.4709 −0.404610
\(951\) 0 0
\(952\) 69.0317 2.23733
\(953\) −22.4941 −0.728654 −0.364327 0.931271i \(-0.618701\pi\)
−0.364327 + 0.931271i \(0.618701\pi\)
\(954\) 0 0
\(955\) −41.7830 −1.35207
\(956\) −25.3604 −0.820213
\(957\) 0 0
\(958\) −7.34696 −0.237369
\(959\) 21.9860 0.709967
\(960\) 0 0
\(961\) 7.09198 0.228774
\(962\) −92.4402 −2.98039
\(963\) 0 0
\(964\) −52.5612 −1.69288
\(965\) −0.121775 −0.00392009
\(966\) 0 0
\(967\) 12.2301 0.393294 0.196647 0.980474i \(-0.436995\pi\)
0.196647 + 0.980474i \(0.436995\pi\)
\(968\) 3.18195 0.102272
\(969\) 0 0
\(970\) 48.8507 1.56850
\(971\) −43.7846 −1.40511 −0.702557 0.711627i \(-0.747959\pi\)
−0.702557 + 0.711627i \(0.747959\pi\)
\(972\) 0 0
\(973\) 57.2358 1.83489
\(974\) −4.01480 −0.128643
\(975\) 0 0
\(976\) −0.629993 −0.0201656
\(977\) −31.3439 −1.00278 −0.501390 0.865221i \(-0.667178\pi\)
−0.501390 + 0.865221i \(0.667178\pi\)
\(978\) 0 0
\(979\) −2.86664 −0.0916183
\(980\) 30.8059 0.984058
\(981\) 0 0
\(982\) 76.5877 2.44401
\(983\) 44.4268 1.41699 0.708497 0.705713i \(-0.249373\pi\)
0.708497 + 0.705713i \(0.249373\pi\)
\(984\) 0 0
\(985\) −15.3287 −0.488413
\(986\) 99.0792 3.15533
\(987\) 0 0
\(988\) −95.6493 −3.04301
\(989\) −60.3907 −1.92031
\(990\) 0 0
\(991\) 8.16593 0.259399 0.129700 0.991553i \(-0.458599\pi\)
0.129700 + 0.991553i \(0.458599\pi\)
\(992\) 30.2646 0.960901
\(993\) 0 0
\(994\) 101.224 3.21064
\(995\) −19.0748 −0.604713
\(996\) 0 0
\(997\) −11.9667 −0.378989 −0.189494 0.981882i \(-0.560685\pi\)
−0.189494 + 0.981882i \(0.560685\pi\)
\(998\) −33.6056 −1.06377
\(999\) 0 0
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 6039.2.a.f.1.10 12
3.2 odd 2 2013.2.a.c.1.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2013.2.a.c.1.3 12 3.2 odd 2
6039.2.a.f.1.10 12 1.1 even 1 trivial