Properties

Label 6039.2.a.e.1.12
Level $6039$
Weight $2$
Character 6039.1
Self dual yes
Analytic conductor $48.222$
Analytic rank $1$
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: \(1\)
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: \( x^{12} - x^{11} - 16 x^{10} + 13 x^{9} + 93 x^{8} - 59 x^{7} - 238 x^{6} + 108 x^{5} + 257 x^{4} - 71 x^{3} - 93 x^{2} + 13 x + 1 \) 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 1.12
Root \(-2.26879\) 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)
 
\(f(q)\) \(=\) \(q+2.26879 q^{2} +3.14741 q^{4} +2.83484 q^{5} -4.72992 q^{7} +2.60323 q^{8} +O(q^{10})\) \(q+2.26879 q^{2} +3.14741 q^{4} +2.83484 q^{5} -4.72992 q^{7} +2.60323 q^{8} +6.43165 q^{10} +1.00000 q^{11} -0.329272 q^{13} -10.7312 q^{14} -0.388631 q^{16} -6.43807 q^{17} -6.41366 q^{19} +8.92240 q^{20} +2.26879 q^{22} -0.000241797 q^{23} +3.03631 q^{25} -0.747050 q^{26} -14.8870 q^{28} -8.79711 q^{29} +5.66189 q^{31} -6.08819 q^{32} -14.6066 q^{34} -13.4086 q^{35} -0.385555 q^{37} -14.5513 q^{38} +7.37974 q^{40} -4.58536 q^{41} -2.75220 q^{43} +3.14741 q^{44} -0.000548586 q^{46} +4.95298 q^{47} +15.3722 q^{49} +6.88874 q^{50} -1.03636 q^{52} +2.98816 q^{53} +2.83484 q^{55} -12.3131 q^{56} -19.9588 q^{58} -11.8673 q^{59} -1.00000 q^{61} +12.8456 q^{62} -13.0356 q^{64} -0.933434 q^{65} -7.49341 q^{67} -20.2632 q^{68} -30.4212 q^{70} +3.18076 q^{71} +12.3598 q^{73} -0.874744 q^{74} -20.1864 q^{76} -4.72992 q^{77} +5.72569 q^{79} -1.10171 q^{80} -10.4032 q^{82} +10.3351 q^{83} -18.2509 q^{85} -6.24417 q^{86} +2.60323 q^{88} +3.95881 q^{89} +1.55743 q^{91} -0.000761033 q^{92} +11.2373 q^{94} -18.1817 q^{95} -11.5563 q^{97} +34.8763 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - q^{2} + 9 q^{4} + 3 q^{5} - 9 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - q^{2} + 9 q^{4} + 3 q^{5} - 9 q^{7} - 6 q^{8} - 8 q^{10} + 12 q^{11} - q^{13} + 3 q^{14} + 3 q^{16} - 9 q^{17} - 20 q^{19} + 9 q^{20} - q^{22} + 9 q^{23} + 3 q^{25} + 18 q^{26} - 31 q^{28} - 18 q^{29} - 21 q^{31} - 18 q^{32} - 12 q^{34} + 4 q^{35} - 18 q^{37} + 2 q^{38} - 26 q^{40} - 15 q^{41} - 33 q^{43} + 9 q^{44} - 28 q^{46} + 20 q^{47} + 15 q^{49} + 2 q^{50} - 27 q^{52} + 3 q^{55} + 8 q^{56} - 11 q^{58} + 21 q^{59} - 12 q^{61} + 9 q^{62} - 12 q^{64} - 17 q^{65} - 34 q^{67} + 16 q^{68} - 36 q^{70} + 5 q^{71} - 2 q^{73} - 6 q^{74} - 27 q^{76} - 9 q^{77} - 31 q^{79} + 60 q^{80} - 12 q^{82} + 32 q^{83} - 40 q^{85} - 18 q^{86} - 6 q^{88} - 27 q^{89} - 45 q^{91} + 78 q^{92} - 13 q^{94} - 37 q^{95} - 19 q^{97} - 4 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.26879 1.60428 0.802139 0.597138i \(-0.203696\pi\)
0.802139 + 0.597138i \(0.203696\pi\)
\(3\) 0 0
\(4\) 3.14741 1.57370
\(5\) 2.83484 1.26778 0.633889 0.773424i \(-0.281458\pi\)
0.633889 + 0.773424i \(0.281458\pi\)
\(6\) 0 0
\(7\) −4.72992 −1.78774 −0.893872 0.448323i \(-0.852022\pi\)
−0.893872 + 0.448323i \(0.852022\pi\)
\(8\) 2.60323 0.920382
\(9\) 0 0
\(10\) 6.43165 2.03387
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −0.329272 −0.0913237 −0.0456619 0.998957i \(-0.514540\pi\)
−0.0456619 + 0.998957i \(0.514540\pi\)
\(14\) −10.7312 −2.86804
\(15\) 0 0
\(16\) −0.388631 −0.0971577
\(17\) −6.43807 −1.56146 −0.780731 0.624868i \(-0.785153\pi\)
−0.780731 + 0.624868i \(0.785153\pi\)
\(18\) 0 0
\(19\) −6.41366 −1.47139 −0.735697 0.677310i \(-0.763145\pi\)
−0.735697 + 0.677310i \(0.763145\pi\)
\(20\) 8.92240 1.99511
\(21\) 0 0
\(22\) 2.26879 0.483708
\(23\) −0.000241797 0 −5.04181e−5 0 −2.52091e−5 1.00000i \(-0.500008\pi\)
−2.52091e−5 1.00000i \(0.500008\pi\)
\(24\) 0 0
\(25\) 3.03631 0.607261
\(26\) −0.747050 −0.146509
\(27\) 0 0
\(28\) −14.8870 −2.81338
\(29\) −8.79711 −1.63358 −0.816791 0.576933i \(-0.804249\pi\)
−0.816791 + 0.576933i \(0.804249\pi\)
\(30\) 0 0
\(31\) 5.66189 1.01691 0.508453 0.861090i \(-0.330218\pi\)
0.508453 + 0.861090i \(0.330218\pi\)
\(32\) −6.08819 −1.07625
\(33\) 0 0
\(34\) −14.6066 −2.50502
\(35\) −13.4086 −2.26646
\(36\) 0 0
\(37\) −0.385555 −0.0633849 −0.0316924 0.999498i \(-0.510090\pi\)
−0.0316924 + 0.999498i \(0.510090\pi\)
\(38\) −14.5513 −2.36052
\(39\) 0 0
\(40\) 7.37974 1.16684
\(41\) −4.58536 −0.716113 −0.358056 0.933700i \(-0.616560\pi\)
−0.358056 + 0.933700i \(0.616560\pi\)
\(42\) 0 0
\(43\) −2.75220 −0.419707 −0.209854 0.977733i \(-0.567299\pi\)
−0.209854 + 0.977733i \(0.567299\pi\)
\(44\) 3.14741 0.474490
\(45\) 0 0
\(46\) −0.000548586 0 −8.08846e−5 0
\(47\) 4.95298 0.722466 0.361233 0.932475i \(-0.382356\pi\)
0.361233 + 0.932475i \(0.382356\pi\)
\(48\) 0 0
\(49\) 15.3722 2.19603
\(50\) 6.88874 0.974216
\(51\) 0 0
\(52\) −1.03636 −0.143717
\(53\) 2.98816 0.410455 0.205228 0.978714i \(-0.434207\pi\)
0.205228 + 0.978714i \(0.434207\pi\)
\(54\) 0 0
\(55\) 2.83484 0.382249
\(56\) −12.3131 −1.64541
\(57\) 0 0
\(58\) −19.9588 −2.62072
\(59\) −11.8673 −1.54499 −0.772494 0.635022i \(-0.780991\pi\)
−0.772494 + 0.635022i \(0.780991\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037
\(62\) 12.8456 1.63140
\(63\) 0 0
\(64\) −13.0356 −1.62944
\(65\) −0.933434 −0.115778
\(66\) 0 0
\(67\) −7.49341 −0.915465 −0.457733 0.889090i \(-0.651338\pi\)
−0.457733 + 0.889090i \(0.651338\pi\)
\(68\) −20.2632 −2.45728
\(69\) 0 0
\(70\) −30.4212 −3.63603
\(71\) 3.18076 0.377487 0.188743 0.982026i \(-0.439559\pi\)
0.188743 + 0.982026i \(0.439559\pi\)
\(72\) 0 0
\(73\) 12.3598 1.44661 0.723303 0.690530i \(-0.242623\pi\)
0.723303 + 0.690530i \(0.242623\pi\)
\(74\) −0.874744 −0.101687
\(75\) 0 0
\(76\) −20.1864 −2.31554
\(77\) −4.72992 −0.539025
\(78\) 0 0
\(79\) 5.72569 0.644190 0.322095 0.946707i \(-0.395613\pi\)
0.322095 + 0.946707i \(0.395613\pi\)
\(80\) −1.10171 −0.123174
\(81\) 0 0
\(82\) −10.4032 −1.14884
\(83\) 10.3351 1.13443 0.567214 0.823570i \(-0.308021\pi\)
0.567214 + 0.823570i \(0.308021\pi\)
\(84\) 0 0
\(85\) −18.2509 −1.97959
\(86\) −6.24417 −0.673326
\(87\) 0 0
\(88\) 2.60323 0.277506
\(89\) 3.95881 0.419634 0.209817 0.977741i \(-0.432713\pi\)
0.209817 + 0.977741i \(0.432713\pi\)
\(90\) 0 0
\(91\) 1.55743 0.163263
\(92\) −0.000761033 0 −7.93432e−5 0
\(93\) 0 0
\(94\) 11.2373 1.15904
\(95\) −18.1817 −1.86540
\(96\) 0 0
\(97\) −11.5563 −1.17337 −0.586684 0.809816i \(-0.699567\pi\)
−0.586684 + 0.809816i \(0.699567\pi\)
\(98\) 34.8763 3.52304
\(99\) 0 0
\(100\) 9.55650 0.955650
\(101\) −16.8491 −1.67655 −0.838276 0.545247i \(-0.816436\pi\)
−0.838276 + 0.545247i \(0.816436\pi\)
\(102\) 0 0
\(103\) 2.38045 0.234553 0.117276 0.993099i \(-0.462584\pi\)
0.117276 + 0.993099i \(0.462584\pi\)
\(104\) −0.857173 −0.0840527
\(105\) 0 0
\(106\) 6.77951 0.658484
\(107\) 12.2216 1.18150 0.590751 0.806854i \(-0.298831\pi\)
0.590751 + 0.806854i \(0.298831\pi\)
\(108\) 0 0
\(109\) 6.27529 0.601064 0.300532 0.953772i \(-0.402836\pi\)
0.300532 + 0.953772i \(0.402836\pi\)
\(110\) 6.43165 0.613234
\(111\) 0 0
\(112\) 1.83820 0.173693
\(113\) −12.6571 −1.19068 −0.595342 0.803472i \(-0.702984\pi\)
−0.595342 + 0.803472i \(0.702984\pi\)
\(114\) 0 0
\(115\) −0.000685455 0 −6.39190e−5 0
\(116\) −27.6881 −2.57078
\(117\) 0 0
\(118\) −26.9244 −2.47859
\(119\) 30.4516 2.79149
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −2.26879 −0.205407
\(123\) 0 0
\(124\) 17.8203 1.60031
\(125\) −5.56675 −0.497905
\(126\) 0 0
\(127\) 18.2420 1.61871 0.809356 0.587318i \(-0.199816\pi\)
0.809356 + 0.587318i \(0.199816\pi\)
\(128\) −17.3986 −1.53783
\(129\) 0 0
\(130\) −2.11777 −0.185740
\(131\) 9.31461 0.813821 0.406911 0.913468i \(-0.366606\pi\)
0.406911 + 0.913468i \(0.366606\pi\)
\(132\) 0 0
\(133\) 30.3361 2.63048
\(134\) −17.0010 −1.46866
\(135\) 0 0
\(136\) −16.7598 −1.43714
\(137\) 11.2029 0.957129 0.478565 0.878052i \(-0.341157\pi\)
0.478565 + 0.878052i \(0.341157\pi\)
\(138\) 0 0
\(139\) 17.8177 1.51127 0.755637 0.654990i \(-0.227327\pi\)
0.755637 + 0.654990i \(0.227327\pi\)
\(140\) −42.2023 −3.56674
\(141\) 0 0
\(142\) 7.21648 0.605594
\(143\) −0.329272 −0.0275351
\(144\) 0 0
\(145\) −24.9384 −2.07102
\(146\) 28.0418 2.32076
\(147\) 0 0
\(148\) −1.21350 −0.0997491
\(149\) 8.25059 0.675915 0.337958 0.941161i \(-0.390264\pi\)
0.337958 + 0.941161i \(0.390264\pi\)
\(150\) 0 0
\(151\) −9.72341 −0.791280 −0.395640 0.918406i \(-0.629477\pi\)
−0.395640 + 0.918406i \(0.629477\pi\)
\(152\) −16.6962 −1.35424
\(153\) 0 0
\(154\) −10.7312 −0.864745
\(155\) 16.0505 1.28921
\(156\) 0 0
\(157\) −10.6011 −0.846059 −0.423030 0.906116i \(-0.639033\pi\)
−0.423030 + 0.906116i \(0.639033\pi\)
\(158\) 12.9904 1.03346
\(159\) 0 0
\(160\) −17.2590 −1.36445
\(161\) 0.00114368 9.01346e−5 0
\(162\) 0 0
\(163\) −10.7755 −0.843999 −0.422000 0.906596i \(-0.638672\pi\)
−0.422000 + 0.906596i \(0.638672\pi\)
\(164\) −14.4320 −1.12695
\(165\) 0 0
\(166\) 23.4483 1.81994
\(167\) −0.523617 −0.0405187 −0.0202594 0.999795i \(-0.506449\pi\)
−0.0202594 + 0.999795i \(0.506449\pi\)
\(168\) 0 0
\(169\) −12.8916 −0.991660
\(170\) −41.4074 −3.17581
\(171\) 0 0
\(172\) −8.66231 −0.660495
\(173\) −6.06345 −0.460995 −0.230498 0.973073i \(-0.574035\pi\)
−0.230498 + 0.973073i \(0.574035\pi\)
\(174\) 0 0
\(175\) −14.3615 −1.08563
\(176\) −0.388631 −0.0292942
\(177\) 0 0
\(178\) 8.98172 0.673208
\(179\) 22.6875 1.69574 0.847871 0.530202i \(-0.177884\pi\)
0.847871 + 0.530202i \(0.177884\pi\)
\(180\) 0 0
\(181\) −22.9241 −1.70394 −0.851969 0.523593i \(-0.824591\pi\)
−0.851969 + 0.523593i \(0.824591\pi\)
\(182\) 3.53349 0.261920
\(183\) 0 0
\(184\) −0.000629453 0 −4.64039e−5 0
\(185\) −1.09299 −0.0803579
\(186\) 0 0
\(187\) −6.43807 −0.470798
\(188\) 15.5891 1.13695
\(189\) 0 0
\(190\) −41.2504 −2.99262
\(191\) −22.9356 −1.65956 −0.829780 0.558091i \(-0.811534\pi\)
−0.829780 + 0.558091i \(0.811534\pi\)
\(192\) 0 0
\(193\) −18.8045 −1.35358 −0.676789 0.736177i \(-0.736629\pi\)
−0.676789 + 0.736177i \(0.736629\pi\)
\(194\) −26.2189 −1.88241
\(195\) 0 0
\(196\) 48.3826 3.45590
\(197\) −8.55113 −0.609243 −0.304621 0.952474i \(-0.598530\pi\)
−0.304621 + 0.952474i \(0.598530\pi\)
\(198\) 0 0
\(199\) 10.5926 0.750890 0.375445 0.926845i \(-0.377490\pi\)
0.375445 + 0.926845i \(0.377490\pi\)
\(200\) 7.90421 0.558912
\(201\) 0 0
\(202\) −38.2271 −2.68965
\(203\) 41.6097 2.92043
\(204\) 0 0
\(205\) −12.9988 −0.907872
\(206\) 5.40074 0.376287
\(207\) 0 0
\(208\) 0.127965 0.00887281
\(209\) −6.41366 −0.443642
\(210\) 0 0
\(211\) −18.4180 −1.26794 −0.633972 0.773356i \(-0.718577\pi\)
−0.633972 + 0.773356i \(0.718577\pi\)
\(212\) 9.40496 0.645936
\(213\) 0 0
\(214\) 27.7281 1.89546
\(215\) −7.80205 −0.532095
\(216\) 0 0
\(217\) −26.7803 −1.81797
\(218\) 14.2373 0.964273
\(219\) 0 0
\(220\) 8.92240 0.601548
\(221\) 2.11988 0.142598
\(222\) 0 0
\(223\) −3.52442 −0.236013 −0.118006 0.993013i \(-0.537650\pi\)
−0.118006 + 0.993013i \(0.537650\pi\)
\(224\) 28.7967 1.92406
\(225\) 0 0
\(226\) −28.7164 −1.91019
\(227\) 3.96905 0.263435 0.131718 0.991287i \(-0.457951\pi\)
0.131718 + 0.991287i \(0.457951\pi\)
\(228\) 0 0
\(229\) 22.0640 1.45803 0.729013 0.684499i \(-0.239979\pi\)
0.729013 + 0.684499i \(0.239979\pi\)
\(230\) −0.00155515 −0.000102544 0
\(231\) 0 0
\(232\) −22.9009 −1.50352
\(233\) 25.1375 1.64681 0.823405 0.567454i \(-0.192072\pi\)
0.823405 + 0.567454i \(0.192072\pi\)
\(234\) 0 0
\(235\) 14.0409 0.915927
\(236\) −37.3512 −2.43136
\(237\) 0 0
\(238\) 69.0883 4.47833
\(239\) −14.8916 −0.963259 −0.481629 0.876375i \(-0.659955\pi\)
−0.481629 + 0.876375i \(0.659955\pi\)
\(240\) 0 0
\(241\) −19.7364 −1.27134 −0.635668 0.771963i \(-0.719275\pi\)
−0.635668 + 0.771963i \(0.719275\pi\)
\(242\) 2.26879 0.145843
\(243\) 0 0
\(244\) −3.14741 −0.201492
\(245\) 43.5777 2.78407
\(246\) 0 0
\(247\) 2.11184 0.134373
\(248\) 14.7392 0.935941
\(249\) 0 0
\(250\) −12.6298 −0.798778
\(251\) 22.4411 1.41647 0.708235 0.705977i \(-0.249492\pi\)
0.708235 + 0.705977i \(0.249492\pi\)
\(252\) 0 0
\(253\) −0.000241797 0 −1.52016e−5 0
\(254\) 41.3872 2.59686
\(255\) 0 0
\(256\) −13.4026 −0.837663
\(257\) −7.08903 −0.442201 −0.221101 0.975251i \(-0.570965\pi\)
−0.221101 + 0.975251i \(0.570965\pi\)
\(258\) 0 0
\(259\) 1.82365 0.113316
\(260\) −2.93790 −0.182201
\(261\) 0 0
\(262\) 21.1329 1.30559
\(263\) −6.00950 −0.370562 −0.185281 0.982686i \(-0.559319\pi\)
−0.185281 + 0.982686i \(0.559319\pi\)
\(264\) 0 0
\(265\) 8.47095 0.520366
\(266\) 68.8263 4.22001
\(267\) 0 0
\(268\) −23.5848 −1.44067
\(269\) 20.6863 1.26126 0.630632 0.776082i \(-0.282796\pi\)
0.630632 + 0.776082i \(0.282796\pi\)
\(270\) 0 0
\(271\) −10.2338 −0.621661 −0.310831 0.950465i \(-0.600607\pi\)
−0.310831 + 0.950465i \(0.600607\pi\)
\(272\) 2.50203 0.151708
\(273\) 0 0
\(274\) 25.4171 1.53550
\(275\) 3.03631 0.183096
\(276\) 0 0
\(277\) 8.41552 0.505640 0.252820 0.967513i \(-0.418642\pi\)
0.252820 + 0.967513i \(0.418642\pi\)
\(278\) 40.4245 2.42450
\(279\) 0 0
\(280\) −34.9056 −2.08601
\(281\) −24.4089 −1.45611 −0.728057 0.685516i \(-0.759577\pi\)
−0.728057 + 0.685516i \(0.759577\pi\)
\(282\) 0 0
\(283\) −20.5811 −1.22342 −0.611709 0.791083i \(-0.709518\pi\)
−0.611709 + 0.791083i \(0.709518\pi\)
\(284\) 10.0112 0.594053
\(285\) 0 0
\(286\) −0.747050 −0.0441740
\(287\) 21.6884 1.28023
\(288\) 0 0
\(289\) 24.4488 1.43816
\(290\) −56.5800 −3.32249
\(291\) 0 0
\(292\) 38.9014 2.27653
\(293\) 2.38810 0.139515 0.0697573 0.997564i \(-0.477778\pi\)
0.0697573 + 0.997564i \(0.477778\pi\)
\(294\) 0 0
\(295\) −33.6418 −1.95870
\(296\) −1.00369 −0.0583383
\(297\) 0 0
\(298\) 18.7189 1.08436
\(299\) 7.96170e−5 0 4.60437e−6 0
\(300\) 0 0
\(301\) 13.0177 0.750329
\(302\) −22.0604 −1.26943
\(303\) 0 0
\(304\) 2.49255 0.142957
\(305\) −2.83484 −0.162322
\(306\) 0 0
\(307\) 24.0250 1.37118 0.685588 0.727990i \(-0.259545\pi\)
0.685588 + 0.727990i \(0.259545\pi\)
\(308\) −14.8870 −0.848266
\(309\) 0 0
\(310\) 36.4153 2.06825
\(311\) −16.3407 −0.926595 −0.463297 0.886203i \(-0.653334\pi\)
−0.463297 + 0.886203i \(0.653334\pi\)
\(312\) 0 0
\(313\) −25.9694 −1.46787 −0.733937 0.679217i \(-0.762319\pi\)
−0.733937 + 0.679217i \(0.762319\pi\)
\(314\) −24.0517 −1.35731
\(315\) 0 0
\(316\) 18.0211 1.01377
\(317\) 0.168883 0.00948539 0.00474270 0.999989i \(-0.498490\pi\)
0.00474270 + 0.999989i \(0.498490\pi\)
\(318\) 0 0
\(319\) −8.79711 −0.492544
\(320\) −36.9537 −2.06577
\(321\) 0 0
\(322\) 0.00259477 0.000144601 0
\(323\) 41.2916 2.29753
\(324\) 0 0
\(325\) −0.999772 −0.0554574
\(326\) −24.4473 −1.35401
\(327\) 0 0
\(328\) −11.9368 −0.659097
\(329\) −23.4272 −1.29158
\(330\) 0 0
\(331\) 15.0242 0.825804 0.412902 0.910776i \(-0.364515\pi\)
0.412902 + 0.910776i \(0.364515\pi\)
\(332\) 32.5289 1.78526
\(333\) 0 0
\(334\) −1.18798 −0.0650033
\(335\) −21.2426 −1.16061
\(336\) 0 0
\(337\) 11.6051 0.632172 0.316086 0.948731i \(-0.397631\pi\)
0.316086 + 0.948731i \(0.397631\pi\)
\(338\) −29.2483 −1.59090
\(339\) 0 0
\(340\) −57.4430 −3.11529
\(341\) 5.66189 0.306609
\(342\) 0 0
\(343\) −39.5998 −2.13819
\(344\) −7.16463 −0.386291
\(345\) 0 0
\(346\) −13.7567 −0.739564
\(347\) −35.9376 −1.92923 −0.964615 0.263661i \(-0.915070\pi\)
−0.964615 + 0.263661i \(0.915070\pi\)
\(348\) 0 0
\(349\) −3.25292 −0.174125 −0.0870624 0.996203i \(-0.527748\pi\)
−0.0870624 + 0.996203i \(0.527748\pi\)
\(350\) −32.5832 −1.74165
\(351\) 0 0
\(352\) −6.08819 −0.324501
\(353\) 23.5713 1.25457 0.627286 0.778789i \(-0.284166\pi\)
0.627286 + 0.778789i \(0.284166\pi\)
\(354\) 0 0
\(355\) 9.01694 0.478570
\(356\) 12.4600 0.660379
\(357\) 0 0
\(358\) 51.4732 2.72044
\(359\) 6.43776 0.339772 0.169886 0.985464i \(-0.445660\pi\)
0.169886 + 0.985464i \(0.445660\pi\)
\(360\) 0 0
\(361\) 22.1350 1.16500
\(362\) −52.0101 −2.73359
\(363\) 0 0
\(364\) 4.90188 0.256928
\(365\) 35.0381 1.83398
\(366\) 0 0
\(367\) −10.8180 −0.564693 −0.282346 0.959313i \(-0.591113\pi\)
−0.282346 + 0.959313i \(0.591113\pi\)
\(368\) 9.39697e−5 0 4.89851e−6 0
\(369\) 0 0
\(370\) −2.47976 −0.128916
\(371\) −14.1338 −0.733789
\(372\) 0 0
\(373\) −23.6020 −1.22207 −0.611033 0.791605i \(-0.709246\pi\)
−0.611033 + 0.791605i \(0.709246\pi\)
\(374\) −14.6066 −0.755291
\(375\) 0 0
\(376\) 12.8938 0.664945
\(377\) 2.89665 0.149185
\(378\) 0 0
\(379\) 26.0391 1.33754 0.668769 0.743470i \(-0.266821\pi\)
0.668769 + 0.743470i \(0.266821\pi\)
\(380\) −57.2252 −2.93559
\(381\) 0 0
\(382\) −52.0360 −2.66239
\(383\) 17.5297 0.895726 0.447863 0.894102i \(-0.352185\pi\)
0.447863 + 0.894102i \(0.352185\pi\)
\(384\) 0 0
\(385\) −13.4086 −0.683364
\(386\) −42.6635 −2.17151
\(387\) 0 0
\(388\) −36.3725 −1.84653
\(389\) −21.2431 −1.07707 −0.538535 0.842603i \(-0.681022\pi\)
−0.538535 + 0.842603i \(0.681022\pi\)
\(390\) 0 0
\(391\) 0.00155670 7.87259e−5 0
\(392\) 40.0174 2.02118
\(393\) 0 0
\(394\) −19.4007 −0.977394
\(395\) 16.2314 0.816690
\(396\) 0 0
\(397\) −16.0596 −0.806006 −0.403003 0.915199i \(-0.632034\pi\)
−0.403003 + 0.915199i \(0.632034\pi\)
\(398\) 24.0324 1.20464
\(399\) 0 0
\(400\) −1.18000 −0.0590001
\(401\) −12.2725 −0.612861 −0.306431 0.951893i \(-0.599135\pi\)
−0.306431 + 0.951893i \(0.599135\pi\)
\(402\) 0 0
\(403\) −1.86430 −0.0928676
\(404\) −53.0311 −2.63840
\(405\) 0 0
\(406\) 94.4037 4.68517
\(407\) −0.385555 −0.0191113
\(408\) 0 0
\(409\) −8.71520 −0.430939 −0.215470 0.976511i \(-0.569128\pi\)
−0.215470 + 0.976511i \(0.569128\pi\)
\(410\) −29.4914 −1.45648
\(411\) 0 0
\(412\) 7.49225 0.369117
\(413\) 56.1313 2.76204
\(414\) 0 0
\(415\) 29.2984 1.43820
\(416\) 2.00467 0.0982871
\(417\) 0 0
\(418\) −14.5513 −0.711725
\(419\) 27.6500 1.35079 0.675396 0.737455i \(-0.263973\pi\)
0.675396 + 0.737455i \(0.263973\pi\)
\(420\) 0 0
\(421\) −21.5245 −1.04904 −0.524520 0.851398i \(-0.675755\pi\)
−0.524520 + 0.851398i \(0.675755\pi\)
\(422\) −41.7865 −2.03413
\(423\) 0 0
\(424\) 7.77887 0.377776
\(425\) −19.5480 −0.948215
\(426\) 0 0
\(427\) 4.72992 0.228897
\(428\) 38.4662 1.85934
\(429\) 0 0
\(430\) −17.7012 −0.853629
\(431\) −35.6418 −1.71681 −0.858403 0.512977i \(-0.828543\pi\)
−0.858403 + 0.512977i \(0.828543\pi\)
\(432\) 0 0
\(433\) −25.8670 −1.24309 −0.621543 0.783380i \(-0.713494\pi\)
−0.621543 + 0.783380i \(0.713494\pi\)
\(434\) −60.7589 −2.91652
\(435\) 0 0
\(436\) 19.7509 0.945897
\(437\) 0.00155080 7.41849e−5 0
\(438\) 0 0
\(439\) −6.60137 −0.315066 −0.157533 0.987514i \(-0.550354\pi\)
−0.157533 + 0.987514i \(0.550354\pi\)
\(440\) 7.37974 0.351815
\(441\) 0 0
\(442\) 4.80956 0.228767
\(443\) −20.7297 −0.984900 −0.492450 0.870341i \(-0.663898\pi\)
−0.492450 + 0.870341i \(0.663898\pi\)
\(444\) 0 0
\(445\) 11.2226 0.532002
\(446\) −7.99617 −0.378630
\(447\) 0 0
\(448\) 61.6572 2.91303
\(449\) −11.8139 −0.557534 −0.278767 0.960359i \(-0.589926\pi\)
−0.278767 + 0.960359i \(0.589926\pi\)
\(450\) 0 0
\(451\) −4.58536 −0.215916
\(452\) −39.8372 −1.87379
\(453\) 0 0
\(454\) 9.00495 0.422623
\(455\) 4.41507 0.206982
\(456\) 0 0
\(457\) 28.7970 1.34707 0.673535 0.739156i \(-0.264775\pi\)
0.673535 + 0.739156i \(0.264775\pi\)
\(458\) 50.0585 2.33908
\(459\) 0 0
\(460\) −0.00215741 −0.000100590 0
\(461\) 0.622970 0.0290146 0.0145073 0.999895i \(-0.495382\pi\)
0.0145073 + 0.999895i \(0.495382\pi\)
\(462\) 0 0
\(463\) 20.1613 0.936974 0.468487 0.883470i \(-0.344799\pi\)
0.468487 + 0.883470i \(0.344799\pi\)
\(464\) 3.41883 0.158715
\(465\) 0 0
\(466\) 57.0316 2.64194
\(467\) 10.3857 0.480594 0.240297 0.970699i \(-0.422755\pi\)
0.240297 + 0.970699i \(0.422755\pi\)
\(468\) 0 0
\(469\) 35.4433 1.63662
\(470\) 31.8559 1.46940
\(471\) 0 0
\(472\) −30.8933 −1.42198
\(473\) −2.75220 −0.126546
\(474\) 0 0
\(475\) −19.4738 −0.893521
\(476\) 95.8436 4.39299
\(477\) 0 0
\(478\) −33.7860 −1.54533
\(479\) −24.6590 −1.12670 −0.563348 0.826220i \(-0.690487\pi\)
−0.563348 + 0.826220i \(0.690487\pi\)
\(480\) 0 0
\(481\) 0.126953 0.00578854
\(482\) −44.7778 −2.03958
\(483\) 0 0
\(484\) 3.14741 0.143064
\(485\) −32.7603 −1.48757
\(486\) 0 0
\(487\) 26.7342 1.21144 0.605720 0.795678i \(-0.292885\pi\)
0.605720 + 0.795678i \(0.292885\pi\)
\(488\) −2.60323 −0.117843
\(489\) 0 0
\(490\) 98.8686 4.46643
\(491\) −26.1263 −1.17906 −0.589532 0.807745i \(-0.700688\pi\)
−0.589532 + 0.807745i \(0.700688\pi\)
\(492\) 0 0
\(493\) 56.6364 2.55078
\(494\) 4.79132 0.215572
\(495\) 0 0
\(496\) −2.20039 −0.0988002
\(497\) −15.0448 −0.674850
\(498\) 0 0
\(499\) 26.3139 1.17797 0.588987 0.808143i \(-0.299527\pi\)
0.588987 + 0.808143i \(0.299527\pi\)
\(500\) −17.5208 −0.783556
\(501\) 0 0
\(502\) 50.9142 2.27241
\(503\) −1.89384 −0.0844421 −0.0422211 0.999108i \(-0.513443\pi\)
−0.0422211 + 0.999108i \(0.513443\pi\)
\(504\) 0 0
\(505\) −47.7646 −2.12549
\(506\) −0.000548586 0 −2.43876e−5 0
\(507\) 0 0
\(508\) 57.4149 2.54738
\(509\) −4.01793 −0.178092 −0.0890458 0.996028i \(-0.528382\pi\)
−0.0890458 + 0.996028i \(0.528382\pi\)
\(510\) 0 0
\(511\) −58.4610 −2.58616
\(512\) 4.38945 0.193988
\(513\) 0 0
\(514\) −16.0835 −0.709414
\(515\) 6.74819 0.297361
\(516\) 0 0
\(517\) 4.95298 0.217832
\(518\) 4.13747 0.181790
\(519\) 0 0
\(520\) −2.42995 −0.106560
\(521\) 31.3527 1.37359 0.686794 0.726852i \(-0.259018\pi\)
0.686794 + 0.726852i \(0.259018\pi\)
\(522\) 0 0
\(523\) 34.6005 1.51297 0.756487 0.654009i \(-0.226914\pi\)
0.756487 + 0.654009i \(0.226914\pi\)
\(524\) 29.3169 1.28071
\(525\) 0 0
\(526\) −13.6343 −0.594484
\(527\) −36.4516 −1.58786
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 19.2188 0.834812
\(531\) 0 0
\(532\) 95.4802 4.13959
\(533\) 1.50983 0.0653981
\(534\) 0 0
\(535\) 34.6461 1.49788
\(536\) −19.5071 −0.842578
\(537\) 0 0
\(538\) 46.9328 2.02342
\(539\) 15.3722 0.662127
\(540\) 0 0
\(541\) 15.6021 0.670786 0.335393 0.942078i \(-0.391131\pi\)
0.335393 + 0.942078i \(0.391131\pi\)
\(542\) −23.2184 −0.997317
\(543\) 0 0
\(544\) 39.1962 1.68052
\(545\) 17.7894 0.762015
\(546\) 0 0
\(547\) −29.9091 −1.27882 −0.639411 0.768865i \(-0.720822\pi\)
−0.639411 + 0.768865i \(0.720822\pi\)
\(548\) 35.2601 1.50624
\(549\) 0 0
\(550\) 6.88874 0.293737
\(551\) 56.4217 2.40365
\(552\) 0 0
\(553\) −27.0821 −1.15165
\(554\) 19.0931 0.811186
\(555\) 0 0
\(556\) 56.0795 2.37830
\(557\) −28.1835 −1.19417 −0.597087 0.802177i \(-0.703675\pi\)
−0.597087 + 0.802177i \(0.703675\pi\)
\(558\) 0 0
\(559\) 0.906225 0.0383292
\(560\) 5.21099 0.220204
\(561\) 0 0
\(562\) −55.3787 −2.33601
\(563\) −9.46021 −0.398700 −0.199350 0.979928i \(-0.563883\pi\)
−0.199350 + 0.979928i \(0.563883\pi\)
\(564\) 0 0
\(565\) −35.8810 −1.50952
\(566\) −46.6942 −1.96270
\(567\) 0 0
\(568\) 8.28026 0.347432
\(569\) 11.2370 0.471082 0.235541 0.971864i \(-0.424314\pi\)
0.235541 + 0.971864i \(0.424314\pi\)
\(570\) 0 0
\(571\) 5.43482 0.227440 0.113720 0.993513i \(-0.463723\pi\)
0.113720 + 0.993513i \(0.463723\pi\)
\(572\) −1.03636 −0.0433322
\(573\) 0 0
\(574\) 49.2065 2.05384
\(575\) −0.000734169 0 −3.06170e−5 0
\(576\) 0 0
\(577\) 16.5185 0.687676 0.343838 0.939029i \(-0.388273\pi\)
0.343838 + 0.939029i \(0.388273\pi\)
\(578\) 55.4691 2.30721
\(579\) 0 0
\(580\) −78.4913 −3.25918
\(581\) −48.8844 −2.02807
\(582\) 0 0
\(583\) 2.98816 0.123757
\(584\) 32.1755 1.33143
\(585\) 0 0
\(586\) 5.41811 0.223820
\(587\) 26.9141 1.11086 0.555432 0.831562i \(-0.312553\pi\)
0.555432 + 0.831562i \(0.312553\pi\)
\(588\) 0 0
\(589\) −36.3134 −1.49627
\(590\) −76.3262 −3.14230
\(591\) 0 0
\(592\) 0.149839 0.00615833
\(593\) 19.3983 0.796593 0.398296 0.917257i \(-0.369602\pi\)
0.398296 + 0.917257i \(0.369602\pi\)
\(594\) 0 0
\(595\) 86.3253 3.53899
\(596\) 25.9680 1.06369
\(597\) 0 0
\(598\) 0.000180634 0 7.38668e−6 0
\(599\) 21.6311 0.883825 0.441912 0.897058i \(-0.354300\pi\)
0.441912 + 0.897058i \(0.354300\pi\)
\(600\) 0 0
\(601\) 40.4181 1.64869 0.824345 0.566088i \(-0.191544\pi\)
0.824345 + 0.566088i \(0.191544\pi\)
\(602\) 29.5345 1.20374
\(603\) 0 0
\(604\) −30.6036 −1.24524
\(605\) 2.83484 0.115253
\(606\) 0 0
\(607\) −4.54357 −0.184418 −0.0922089 0.995740i \(-0.529393\pi\)
−0.0922089 + 0.995740i \(0.529393\pi\)
\(608\) 39.0476 1.58359
\(609\) 0 0
\(610\) −6.43165 −0.260410
\(611\) −1.63088 −0.0659783
\(612\) 0 0
\(613\) −31.1325 −1.25743 −0.628715 0.777636i \(-0.716419\pi\)
−0.628715 + 0.777636i \(0.716419\pi\)
\(614\) 54.5076 2.19975
\(615\) 0 0
\(616\) −12.3131 −0.496109
\(617\) 8.28328 0.333472 0.166736 0.986002i \(-0.446677\pi\)
0.166736 + 0.986002i \(0.446677\pi\)
\(618\) 0 0
\(619\) −28.5704 −1.14834 −0.574170 0.818736i \(-0.694675\pi\)
−0.574170 + 0.818736i \(0.694675\pi\)
\(620\) 50.5176 2.02884
\(621\) 0 0
\(622\) −37.0736 −1.48651
\(623\) −18.7249 −0.750197
\(624\) 0 0
\(625\) −30.9624 −1.23849
\(626\) −58.9190 −2.35488
\(627\) 0 0
\(628\) −33.3660 −1.33145
\(629\) 2.48223 0.0989730
\(630\) 0 0
\(631\) −16.8212 −0.669641 −0.334820 0.942282i \(-0.608676\pi\)
−0.334820 + 0.942282i \(0.608676\pi\)
\(632\) 14.9053 0.592901
\(633\) 0 0
\(634\) 0.383159 0.0152172
\(635\) 51.7130 2.05217
\(636\) 0 0
\(637\) −5.06164 −0.200549
\(638\) −19.9588 −0.790177
\(639\) 0 0
\(640\) −49.3221 −1.94963
\(641\) 13.3915 0.528931 0.264466 0.964395i \(-0.414804\pi\)
0.264466 + 0.964395i \(0.414804\pi\)
\(642\) 0 0
\(643\) −25.2370 −0.995249 −0.497625 0.867393i \(-0.665794\pi\)
−0.497625 + 0.867393i \(0.665794\pi\)
\(644\) 0.00359963 0.000141845 0
\(645\) 0 0
\(646\) 93.6820 3.68587
\(647\) −17.6479 −0.693812 −0.346906 0.937900i \(-0.612768\pi\)
−0.346906 + 0.937900i \(0.612768\pi\)
\(648\) 0 0
\(649\) −11.8673 −0.465832
\(650\) −2.26827 −0.0889690
\(651\) 0 0
\(652\) −33.9148 −1.32821
\(653\) −45.1007 −1.76493 −0.882464 0.470380i \(-0.844117\pi\)
−0.882464 + 0.470380i \(0.844117\pi\)
\(654\) 0 0
\(655\) 26.4054 1.03174
\(656\) 1.78201 0.0695759
\(657\) 0 0
\(658\) −53.1515 −2.07206
\(659\) 10.7525 0.418857 0.209428 0.977824i \(-0.432840\pi\)
0.209428 + 0.977824i \(0.432840\pi\)
\(660\) 0 0
\(661\) −19.4223 −0.755438 −0.377719 0.925920i \(-0.623292\pi\)
−0.377719 + 0.925920i \(0.623292\pi\)
\(662\) 34.0867 1.32482
\(663\) 0 0
\(664\) 26.9048 1.04411
\(665\) 85.9980 3.33486
\(666\) 0 0
\(667\) 0.00212711 8.23621e−5 0
\(668\) −1.64804 −0.0637645
\(669\) 0 0
\(670\) −48.1950 −1.86194
\(671\) −1.00000 −0.0386046
\(672\) 0 0
\(673\) 4.38436 0.169005 0.0845024 0.996423i \(-0.473070\pi\)
0.0845024 + 0.996423i \(0.473070\pi\)
\(674\) 26.3296 1.01418
\(675\) 0 0
\(676\) −40.5751 −1.56058
\(677\) −7.27907 −0.279757 −0.139879 0.990169i \(-0.544671\pi\)
−0.139879 + 0.990169i \(0.544671\pi\)
\(678\) 0 0
\(679\) 54.6606 2.09768
\(680\) −47.5113 −1.82198
\(681\) 0 0
\(682\) 12.8456 0.491885
\(683\) −3.20425 −0.122607 −0.0613035 0.998119i \(-0.519526\pi\)
−0.0613035 + 0.998119i \(0.519526\pi\)
\(684\) 0 0
\(685\) 31.7584 1.21343
\(686\) −89.8437 −3.43025
\(687\) 0 0
\(688\) 1.06959 0.0407778
\(689\) −0.983918 −0.0374843
\(690\) 0 0
\(691\) −41.5148 −1.57930 −0.789649 0.613559i \(-0.789737\pi\)
−0.789649 + 0.613559i \(0.789737\pi\)
\(692\) −19.0842 −0.725471
\(693\) 0 0
\(694\) −81.5348 −3.09502
\(695\) 50.5102 1.91596
\(696\) 0 0
\(697\) 29.5209 1.11818
\(698\) −7.38020 −0.279345
\(699\) 0 0
\(700\) −45.2015 −1.70846
\(701\) 6.73449 0.254358 0.127179 0.991880i \(-0.459408\pi\)
0.127179 + 0.991880i \(0.459408\pi\)
\(702\) 0 0
\(703\) 2.47282 0.0932641
\(704\) −13.0356 −0.491296
\(705\) 0 0
\(706\) 53.4782 2.01268
\(707\) 79.6951 2.99724
\(708\) 0 0
\(709\) 19.8524 0.745574 0.372787 0.927917i \(-0.378402\pi\)
0.372787 + 0.927917i \(0.378402\pi\)
\(710\) 20.4576 0.767758
\(711\) 0 0
\(712\) 10.3057 0.386223
\(713\) −0.00136903 −5.12704e−5 0
\(714\) 0 0
\(715\) −0.933434 −0.0349084
\(716\) 71.4068 2.66860
\(717\) 0 0
\(718\) 14.6059 0.545088
\(719\) 40.2186 1.49990 0.749950 0.661494i \(-0.230077\pi\)
0.749950 + 0.661494i \(0.230077\pi\)
\(720\) 0 0
\(721\) −11.2593 −0.419320
\(722\) 50.2198 1.86899
\(723\) 0 0
\(724\) −72.1516 −2.68149
\(725\) −26.7107 −0.992012
\(726\) 0 0
\(727\) −8.55895 −0.317434 −0.158717 0.987324i \(-0.550736\pi\)
−0.158717 + 0.987324i \(0.550736\pi\)
\(728\) 4.05436 0.150265
\(729\) 0 0
\(730\) 79.4940 2.94221
\(731\) 17.7189 0.655356
\(732\) 0 0
\(733\) 20.8567 0.770360 0.385180 0.922842i \(-0.374139\pi\)
0.385180 + 0.922842i \(0.374139\pi\)
\(734\) −24.5437 −0.905924
\(735\) 0 0
\(736\) 0.00147210 5.42625e−5 0
\(737\) −7.49341 −0.276023
\(738\) 0 0
\(739\) 2.12261 0.0780816 0.0390408 0.999238i \(-0.487570\pi\)
0.0390408 + 0.999238i \(0.487570\pi\)
\(740\) −3.44008 −0.126460
\(741\) 0 0
\(742\) −32.0666 −1.17720
\(743\) −24.8996 −0.913476 −0.456738 0.889601i \(-0.650982\pi\)
−0.456738 + 0.889601i \(0.650982\pi\)
\(744\) 0 0
\(745\) 23.3891 0.856910
\(746\) −53.5480 −1.96053
\(747\) 0 0
\(748\) −20.2632 −0.740898
\(749\) −57.8070 −2.11222
\(750\) 0 0
\(751\) −17.6679 −0.644709 −0.322355 0.946619i \(-0.604474\pi\)
−0.322355 + 0.946619i \(0.604474\pi\)
\(752\) −1.92488 −0.0701932
\(753\) 0 0
\(754\) 6.57188 0.239334
\(755\) −27.5643 −1.00317
\(756\) 0 0
\(757\) 45.7774 1.66381 0.831903 0.554921i \(-0.187252\pi\)
0.831903 + 0.554921i \(0.187252\pi\)
\(758\) 59.0772 2.14578
\(759\) 0 0
\(760\) −47.3312 −1.71688
\(761\) −37.3960 −1.35561 −0.677803 0.735244i \(-0.737068\pi\)
−0.677803 + 0.735244i \(0.737068\pi\)
\(762\) 0 0
\(763\) −29.6816 −1.07455
\(764\) −72.1876 −2.61166
\(765\) 0 0
\(766\) 39.7712 1.43699
\(767\) 3.90757 0.141094
\(768\) 0 0
\(769\) −0.617028 −0.0222506 −0.0111253 0.999938i \(-0.503541\pi\)
−0.0111253 + 0.999938i \(0.503541\pi\)
\(770\) −30.4212 −1.09631
\(771\) 0 0
\(772\) −59.1855 −2.13013
\(773\) 3.60037 0.129496 0.0647482 0.997902i \(-0.479376\pi\)
0.0647482 + 0.997902i \(0.479376\pi\)
\(774\) 0 0
\(775\) 17.1912 0.617527
\(776\) −30.0838 −1.07995
\(777\) 0 0
\(778\) −48.1963 −1.72792
\(779\) 29.4089 1.05368
\(780\) 0 0
\(781\) 3.18076 0.113817
\(782\) 0.00353184 0.000126298 0
\(783\) 0 0
\(784\) −5.97411 −0.213361
\(785\) −30.0524 −1.07262
\(786\) 0 0
\(787\) −0.622093 −0.0221752 −0.0110876 0.999939i \(-0.503529\pi\)
−0.0110876 + 0.999939i \(0.503529\pi\)
\(788\) −26.9139 −0.958768
\(789\) 0 0
\(790\) 36.8256 1.31020
\(791\) 59.8674 2.12864
\(792\) 0 0
\(793\) 0.329272 0.0116928
\(794\) −36.4358 −1.29306
\(795\) 0 0
\(796\) 33.3393 1.18168
\(797\) 37.7027 1.33550 0.667749 0.744387i \(-0.267258\pi\)
0.667749 + 0.744387i \(0.267258\pi\)
\(798\) 0 0
\(799\) −31.8876 −1.12810
\(800\) −18.4856 −0.653565
\(801\) 0 0
\(802\) −27.8438 −0.983199
\(803\) 12.3598 0.436168
\(804\) 0 0
\(805\) 0.00324215 0.000114271 0
\(806\) −4.22971 −0.148985
\(807\) 0 0
\(808\) −43.8622 −1.54307
\(809\) −4.49791 −0.158138 −0.0790690 0.996869i \(-0.525195\pi\)
−0.0790690 + 0.996869i \(0.525195\pi\)
\(810\) 0 0
\(811\) −13.6392 −0.478939 −0.239469 0.970904i \(-0.576973\pi\)
−0.239469 + 0.970904i \(0.576973\pi\)
\(812\) 130.963 4.59589
\(813\) 0 0
\(814\) −0.874744 −0.0306597
\(815\) −30.5467 −1.07000
\(816\) 0 0
\(817\) 17.6517 0.617555
\(818\) −19.7730 −0.691346
\(819\) 0 0
\(820\) −40.9124 −1.42872
\(821\) 30.9469 1.08006 0.540028 0.841647i \(-0.318414\pi\)
0.540028 + 0.841647i \(0.318414\pi\)
\(822\) 0 0
\(823\) −19.6941 −0.686492 −0.343246 0.939246i \(-0.611526\pi\)
−0.343246 + 0.939246i \(0.611526\pi\)
\(824\) 6.19686 0.215878
\(825\) 0 0
\(826\) 127.350 4.43108
\(827\) −15.5540 −0.540866 −0.270433 0.962739i \(-0.587167\pi\)
−0.270433 + 0.962739i \(0.587167\pi\)
\(828\) 0 0
\(829\) −31.4496 −1.09229 −0.546145 0.837691i \(-0.683905\pi\)
−0.546145 + 0.837691i \(0.683905\pi\)
\(830\) 66.4720 2.30728
\(831\) 0 0
\(832\) 4.29225 0.148807
\(833\) −98.9672 −3.42901
\(834\) 0 0
\(835\) −1.48437 −0.0513687
\(836\) −20.1864 −0.698162
\(837\) 0 0
\(838\) 62.7322 2.16705
\(839\) −35.4721 −1.22463 −0.612316 0.790613i \(-0.709762\pi\)
−0.612316 + 0.790613i \(0.709762\pi\)
\(840\) 0 0
\(841\) 48.3892 1.66859
\(842\) −48.8345 −1.68295
\(843\) 0 0
\(844\) −57.9688 −1.99537
\(845\) −36.5455 −1.25720
\(846\) 0 0
\(847\) −4.72992 −0.162522
\(848\) −1.16129 −0.0398789
\(849\) 0 0
\(850\) −44.3502 −1.52120
\(851\) 9.32259e−5 0 3.19574e−6 0
\(852\) 0 0
\(853\) −49.8898 −1.70819 −0.854097 0.520113i \(-0.825890\pi\)
−0.854097 + 0.520113i \(0.825890\pi\)
\(854\) 10.7312 0.367214
\(855\) 0 0
\(856\) 31.8156 1.08743
\(857\) −25.0717 −0.856434 −0.428217 0.903676i \(-0.640858\pi\)
−0.428217 + 0.903676i \(0.640858\pi\)
\(858\) 0 0
\(859\) 11.1566 0.380657 0.190328 0.981720i \(-0.439045\pi\)
0.190328 + 0.981720i \(0.439045\pi\)
\(860\) −24.5563 −0.837361
\(861\) 0 0
\(862\) −80.8638 −2.75423
\(863\) 32.0664 1.09155 0.545777 0.837930i \(-0.316235\pi\)
0.545777 + 0.837930i \(0.316235\pi\)
\(864\) 0 0
\(865\) −17.1889 −0.584440
\(866\) −58.6867 −1.99425
\(867\) 0 0
\(868\) −84.2886 −2.86094
\(869\) 5.72569 0.194231
\(870\) 0 0
\(871\) 2.46737 0.0836037
\(872\) 16.3360 0.553208
\(873\) 0 0
\(874\) 0.00351844 0.000119013 0
\(875\) 26.3303 0.890127
\(876\) 0 0
\(877\) −20.0650 −0.677547 −0.338774 0.940868i \(-0.610012\pi\)
−0.338774 + 0.940868i \(0.610012\pi\)
\(878\) −14.9771 −0.505454
\(879\) 0 0
\(880\) −1.10171 −0.0371385
\(881\) −7.46246 −0.251417 −0.125708 0.992067i \(-0.540120\pi\)
−0.125708 + 0.992067i \(0.540120\pi\)
\(882\) 0 0
\(883\) 24.8751 0.837114 0.418557 0.908190i \(-0.362536\pi\)
0.418557 + 0.908190i \(0.362536\pi\)
\(884\) 6.67213 0.224408
\(885\) 0 0
\(886\) −47.0314 −1.58005
\(887\) 15.9928 0.536986 0.268493 0.963282i \(-0.413474\pi\)
0.268493 + 0.963282i \(0.413474\pi\)
\(888\) 0 0
\(889\) −86.2831 −2.89384
\(890\) 25.4617 0.853479
\(891\) 0 0
\(892\) −11.0928 −0.371414
\(893\) −31.7667 −1.06303
\(894\) 0 0
\(895\) 64.3154 2.14983
\(896\) 82.2940 2.74925
\(897\) 0 0
\(898\) −26.8033 −0.894439
\(899\) −49.8083 −1.66120
\(900\) 0 0
\(901\) −19.2380 −0.640910
\(902\) −10.4032 −0.346389
\(903\) 0 0
\(904\) −32.9495 −1.09588
\(905\) −64.9862 −2.16021
\(906\) 0 0
\(907\) −32.8839 −1.09189 −0.545946 0.837820i \(-0.683830\pi\)
−0.545946 + 0.837820i \(0.683830\pi\)
\(908\) 12.4922 0.414569
\(909\) 0 0
\(910\) 10.0169 0.332056
\(911\) 32.1720 1.06590 0.532952 0.846145i \(-0.321082\pi\)
0.532952 + 0.846145i \(0.321082\pi\)
\(912\) 0 0
\(913\) 10.3351 0.342043
\(914\) 65.3345 2.16107
\(915\) 0 0
\(916\) 69.4443 2.29450
\(917\) −44.0574 −1.45490
\(918\) 0 0
\(919\) 9.19518 0.303321 0.151660 0.988433i \(-0.451538\pi\)
0.151660 + 0.988433i \(0.451538\pi\)
\(920\) −0.00178440 −5.88298e−5 0
\(921\) 0 0
\(922\) 1.41339 0.0465474
\(923\) −1.04734 −0.0344735
\(924\) 0 0
\(925\) −1.17066 −0.0384912
\(926\) 45.7417 1.50317
\(927\) 0 0
\(928\) 53.5585 1.75814
\(929\) 28.3672 0.930699 0.465349 0.885127i \(-0.345929\pi\)
0.465349 + 0.885127i \(0.345929\pi\)
\(930\) 0 0
\(931\) −98.5920 −3.23122
\(932\) 79.1179 2.59159
\(933\) 0 0
\(934\) 23.5630 0.771006
\(935\) −18.2509 −0.596868
\(936\) 0 0
\(937\) 6.49399 0.212149 0.106075 0.994358i \(-0.466172\pi\)
0.106075 + 0.994358i \(0.466172\pi\)
\(938\) 80.4133 2.62559
\(939\) 0 0
\(940\) 44.1925 1.44140
\(941\) 46.4185 1.51320 0.756600 0.653878i \(-0.226859\pi\)
0.756600 + 0.653878i \(0.226859\pi\)
\(942\) 0 0
\(943\) 0.00110873 3.61051e−5 0
\(944\) 4.61199 0.150108
\(945\) 0 0
\(946\) −6.24417 −0.203016
\(947\) 37.2359 1.21000 0.605001 0.796224i \(-0.293173\pi\)
0.605001 + 0.796224i \(0.293173\pi\)
\(948\) 0 0
\(949\) −4.06975 −0.132110
\(950\) −44.1821 −1.43346
\(951\) 0 0
\(952\) 79.2726 2.56924
\(953\) −0.668787 −0.0216641 −0.0108321 0.999941i \(-0.503448\pi\)
−0.0108321 + 0.999941i \(0.503448\pi\)
\(954\) 0 0
\(955\) −65.0186 −2.10395
\(956\) −46.8700 −1.51588
\(957\) 0 0
\(958\) −55.9460 −1.80753
\(959\) −52.9889 −1.71110
\(960\) 0 0
\(961\) 1.05700 0.0340967
\(962\) 0.288029 0.00928642
\(963\) 0 0
\(964\) −62.1187 −2.00071
\(965\) −53.3077 −1.71604
\(966\) 0 0
\(967\) 0.461767 0.0148494 0.00742471 0.999972i \(-0.497637\pi\)
0.00742471 + 0.999972i \(0.497637\pi\)
\(968\) 2.60323 0.0836711
\(969\) 0 0
\(970\) −74.3263 −2.38647
\(971\) −18.7299 −0.601071 −0.300536 0.953771i \(-0.597165\pi\)
−0.300536 + 0.953771i \(0.597165\pi\)
\(972\) 0 0
\(973\) −84.2762 −2.70177
\(974\) 60.6542 1.94349
\(975\) 0 0
\(976\) 0.388631 0.0124398
\(977\) −59.5054 −1.90375 −0.951874 0.306490i \(-0.900845\pi\)
−0.951874 + 0.306490i \(0.900845\pi\)
\(978\) 0 0
\(979\) 3.95881 0.126524
\(980\) 137.157 4.38131
\(981\) 0 0
\(982\) −59.2751 −1.89154
\(983\) −13.0408 −0.415936 −0.207968 0.978136i \(-0.566685\pi\)
−0.207968 + 0.978136i \(0.566685\pi\)
\(984\) 0 0
\(985\) −24.2411 −0.772384
\(986\) 128.496 4.09215
\(987\) 0 0
\(988\) 6.64683 0.211464
\(989\) 0.000665474 0 2.11608e−5 0
\(990\) 0 0
\(991\) 32.1630 1.02169 0.510846 0.859672i \(-0.329332\pi\)
0.510846 + 0.859672i \(0.329332\pi\)
\(992\) −34.4706 −1.09444
\(993\) 0 0
\(994\) −34.1334 −1.08265
\(995\) 30.0283 0.951962
\(996\) 0 0
\(997\) 6.44042 0.203970 0.101985 0.994786i \(-0.467481\pi\)
0.101985 + 0.994786i \(0.467481\pi\)
\(998\) 59.7008 1.88980
\(999\) 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.e.1.12 12
3.2 odd 2 2013.2.a.d.1.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2013.2.a.d.1.1 12 3.2 odd 2
6039.2.a.e.1.12 12 1.1 even 1 trivial