Properties

Label 6046.2.a.e.1.1
Level $6046$
Weight $2$
Character 6046.1
Self dual yes
Analytic conductor $48.278$
Analytic rank $1$
Dimension $56$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6046,2,Mod(1,6046)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6046, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6046.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6046 = 2 \cdot 3023 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6046.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: \(48.2775530621\)
Analytic rank: \(1\)
Dimension: \(56\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 6046.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -3.35147 q^{3} +1.00000 q^{4} +0.434868 q^{5} -3.35147 q^{6} +3.30614 q^{7} +1.00000 q^{8} +8.23237 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -3.35147 q^{3} +1.00000 q^{4} +0.434868 q^{5} -3.35147 q^{6} +3.30614 q^{7} +1.00000 q^{8} +8.23237 q^{9} +0.434868 q^{10} -0.217260 q^{11} -3.35147 q^{12} -3.38686 q^{13} +3.30614 q^{14} -1.45745 q^{15} +1.00000 q^{16} +5.65181 q^{17} +8.23237 q^{18} -2.20866 q^{19} +0.434868 q^{20} -11.0804 q^{21} -0.217260 q^{22} +0.600212 q^{23} -3.35147 q^{24} -4.81089 q^{25} -3.38686 q^{26} -17.5362 q^{27} +3.30614 q^{28} -4.63290 q^{29} -1.45745 q^{30} -5.72971 q^{31} +1.00000 q^{32} +0.728143 q^{33} +5.65181 q^{34} +1.43774 q^{35} +8.23237 q^{36} -9.14910 q^{37} -2.20866 q^{38} +11.3510 q^{39} +0.434868 q^{40} +0.00426203 q^{41} -11.0804 q^{42} -1.55279 q^{43} -0.217260 q^{44} +3.58000 q^{45} +0.600212 q^{46} +2.85449 q^{47} -3.35147 q^{48} +3.93056 q^{49} -4.81089 q^{50} -18.9419 q^{51} -3.38686 q^{52} -10.1788 q^{53} -17.5362 q^{54} -0.0944797 q^{55} +3.30614 q^{56} +7.40225 q^{57} -4.63290 q^{58} -7.46902 q^{59} -1.45745 q^{60} +8.79215 q^{61} -5.72971 q^{62} +27.2174 q^{63} +1.00000 q^{64} -1.47284 q^{65} +0.728143 q^{66} +2.66054 q^{67} +5.65181 q^{68} -2.01160 q^{69} +1.43774 q^{70} -2.21118 q^{71} +8.23237 q^{72} -8.40293 q^{73} -9.14910 q^{74} +16.1236 q^{75} -2.20866 q^{76} -0.718294 q^{77} +11.3510 q^{78} -4.85441 q^{79} +0.434868 q^{80} +34.0748 q^{81} +0.00426203 q^{82} -9.12566 q^{83} -11.0804 q^{84} +2.45779 q^{85} -1.55279 q^{86} +15.5270 q^{87} -0.217260 q^{88} +13.1009 q^{89} +3.58000 q^{90} -11.1974 q^{91} +0.600212 q^{92} +19.2030 q^{93} +2.85449 q^{94} -0.960474 q^{95} -3.35147 q^{96} +17.7175 q^{97} +3.93056 q^{98} -1.78857 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56 q + 56 q^{2} - 18 q^{3} + 56 q^{4} - 17 q^{5} - 18 q^{6} - 35 q^{7} + 56 q^{8} + 34 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 56 q + 56 q^{2} - 18 q^{3} + 56 q^{4} - 17 q^{5} - 18 q^{6} - 35 q^{7} + 56 q^{8} + 34 q^{9} - 17 q^{10} - 53 q^{11} - 18 q^{12} - 21 q^{13} - 35 q^{14} - 36 q^{15} + 56 q^{16} - 22 q^{17} + 34 q^{18} - 31 q^{19} - 17 q^{20} - 23 q^{21} - 53 q^{22} - 59 q^{23} - 18 q^{24} + 41 q^{25} - 21 q^{26} - 63 q^{27} - 35 q^{28} - 88 q^{29} - 36 q^{30} - 44 q^{31} + 56 q^{32} + 4 q^{33} - 22 q^{34} - 51 q^{35} + 34 q^{36} - 60 q^{37} - 31 q^{38} - 62 q^{39} - 17 q^{40} - 39 q^{41} - 23 q^{42} - 66 q^{43} - 53 q^{44} - 34 q^{45} - 59 q^{46} - 51 q^{47} - 18 q^{48} + 41 q^{49} + 41 q^{50} - 48 q^{51} - 21 q^{52} - 75 q^{53} - 63 q^{54} - 41 q^{55} - 35 q^{56} - 12 q^{57} - 88 q^{58} - 77 q^{59} - 36 q^{60} - 43 q^{61} - 44 q^{62} - 88 q^{63} + 56 q^{64} - 54 q^{65} + 4 q^{66} - 62 q^{67} - 22 q^{68} - 48 q^{69} - 51 q^{70} - 122 q^{71} + 34 q^{72} - 7 q^{73} - 60 q^{74} - 63 q^{75} - 31 q^{76} - 39 q^{77} - 62 q^{78} - 91 q^{79} - 17 q^{80} + 8 q^{81} - 39 q^{82} - 51 q^{83} - 23 q^{84} - 72 q^{85} - 66 q^{86} - 19 q^{87} - 53 q^{88} - 62 q^{89} - 34 q^{90} - 48 q^{91} - 59 q^{92} - 41 q^{93} - 51 q^{94} - 120 q^{95} - 18 q^{96} + 6 q^{97} + 41 q^{98} - 128 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −3.35147 −1.93497 −0.967487 0.252921i \(-0.918609\pi\)
−0.967487 + 0.252921i \(0.918609\pi\)
\(4\) 1.00000 0.500000
\(5\) 0.434868 0.194479 0.0972395 0.995261i \(-0.468999\pi\)
0.0972395 + 0.995261i \(0.468999\pi\)
\(6\) −3.35147 −1.36823
\(7\) 3.30614 1.24960 0.624802 0.780783i \(-0.285180\pi\)
0.624802 + 0.780783i \(0.285180\pi\)
\(8\) 1.00000 0.353553
\(9\) 8.23237 2.74412
\(10\) 0.434868 0.137517
\(11\) −0.217260 −0.0655065 −0.0327533 0.999463i \(-0.510428\pi\)
−0.0327533 + 0.999463i \(0.510428\pi\)
\(12\) −3.35147 −0.967487
\(13\) −3.38686 −0.939346 −0.469673 0.882840i \(-0.655628\pi\)
−0.469673 + 0.882840i \(0.655628\pi\)
\(14\) 3.30614 0.883603
\(15\) −1.45745 −0.376312
\(16\) 1.00000 0.250000
\(17\) 5.65181 1.37077 0.685383 0.728183i \(-0.259635\pi\)
0.685383 + 0.728183i \(0.259635\pi\)
\(18\) 8.23237 1.94039
\(19\) −2.20866 −0.506700 −0.253350 0.967375i \(-0.581532\pi\)
−0.253350 + 0.967375i \(0.581532\pi\)
\(20\) 0.434868 0.0972395
\(21\) −11.0804 −2.41795
\(22\) −0.217260 −0.0463201
\(23\) 0.600212 0.125153 0.0625765 0.998040i \(-0.480068\pi\)
0.0625765 + 0.998040i \(0.480068\pi\)
\(24\) −3.35147 −0.684117
\(25\) −4.81089 −0.962178
\(26\) −3.38686 −0.664218
\(27\) −17.5362 −3.37484
\(28\) 3.30614 0.624802
\(29\) −4.63290 −0.860308 −0.430154 0.902755i \(-0.641541\pi\)
−0.430154 + 0.902755i \(0.641541\pi\)
\(30\) −1.45745 −0.266093
\(31\) −5.72971 −1.02909 −0.514543 0.857465i \(-0.672038\pi\)
−0.514543 + 0.857465i \(0.672038\pi\)
\(32\) 1.00000 0.176777
\(33\) 0.728143 0.126753
\(34\) 5.65181 0.969278
\(35\) 1.43774 0.243022
\(36\) 8.23237 1.37206
\(37\) −9.14910 −1.50410 −0.752051 0.659105i \(-0.770935\pi\)
−0.752051 + 0.659105i \(0.770935\pi\)
\(38\) −2.20866 −0.358291
\(39\) 11.3510 1.81761
\(40\) 0.434868 0.0687587
\(41\) 0.00426203 0.000665617 0 0.000332809 1.00000i \(-0.499894\pi\)
0.000332809 1.00000i \(0.499894\pi\)
\(42\) −11.0804 −1.70975
\(43\) −1.55279 −0.236799 −0.118399 0.992966i \(-0.537776\pi\)
−0.118399 + 0.992966i \(0.537776\pi\)
\(44\) −0.217260 −0.0327533
\(45\) 3.58000 0.533675
\(46\) 0.600212 0.0884965
\(47\) 2.85449 0.416370 0.208185 0.978089i \(-0.433244\pi\)
0.208185 + 0.978089i \(0.433244\pi\)
\(48\) −3.35147 −0.483743
\(49\) 3.93056 0.561509
\(50\) −4.81089 −0.680363
\(51\) −18.9419 −2.65240
\(52\) −3.38686 −0.469673
\(53\) −10.1788 −1.39817 −0.699083 0.715041i \(-0.746408\pi\)
−0.699083 + 0.715041i \(0.746408\pi\)
\(54\) −17.5362 −2.38637
\(55\) −0.0944797 −0.0127396
\(56\) 3.30614 0.441802
\(57\) 7.40225 0.980452
\(58\) −4.63290 −0.608330
\(59\) −7.46902 −0.972384 −0.486192 0.873852i \(-0.661614\pi\)
−0.486192 + 0.873852i \(0.661614\pi\)
\(60\) −1.45745 −0.188156
\(61\) 8.79215 1.12572 0.562860 0.826552i \(-0.309701\pi\)
0.562860 + 0.826552i \(0.309701\pi\)
\(62\) −5.72971 −0.727673
\(63\) 27.2174 3.42907
\(64\) 1.00000 0.125000
\(65\) −1.47284 −0.182683
\(66\) 0.728143 0.0896282
\(67\) 2.66054 0.325037 0.162518 0.986706i \(-0.448038\pi\)
0.162518 + 0.986706i \(0.448038\pi\)
\(68\) 5.65181 0.685383
\(69\) −2.01160 −0.242168
\(70\) 1.43774 0.171842
\(71\) −2.21118 −0.262419 −0.131210 0.991355i \(-0.541886\pi\)
−0.131210 + 0.991355i \(0.541886\pi\)
\(72\) 8.23237 0.970194
\(73\) −8.40293 −0.983488 −0.491744 0.870740i \(-0.663640\pi\)
−0.491744 + 0.870740i \(0.663640\pi\)
\(74\) −9.14910 −1.06356
\(75\) 16.1236 1.86179
\(76\) −2.20866 −0.253350
\(77\) −0.718294 −0.0818572
\(78\) 11.3510 1.28524
\(79\) −4.85441 −0.546163 −0.273082 0.961991i \(-0.588043\pi\)
−0.273082 + 0.961991i \(0.588043\pi\)
\(80\) 0.434868 0.0486198
\(81\) 34.0748 3.78609
\(82\) 0.00426203 0.000470662 0
\(83\) −9.12566 −1.00167 −0.500836 0.865542i \(-0.666974\pi\)
−0.500836 + 0.865542i \(0.666974\pi\)
\(84\) −11.0804 −1.20898
\(85\) 2.45779 0.266585
\(86\) −1.55279 −0.167442
\(87\) 15.5270 1.66467
\(88\) −0.217260 −0.0231600
\(89\) 13.1009 1.38869 0.694344 0.719643i \(-0.255695\pi\)
0.694344 + 0.719643i \(0.255695\pi\)
\(90\) 3.58000 0.377365
\(91\) −11.1974 −1.17381
\(92\) 0.600212 0.0625765
\(93\) 19.2030 1.99125
\(94\) 2.85449 0.294418
\(95\) −0.960474 −0.0985426
\(96\) −3.35147 −0.342058
\(97\) 17.7175 1.79894 0.899471 0.436981i \(-0.143952\pi\)
0.899471 + 0.436981i \(0.143952\pi\)
\(98\) 3.93056 0.397047
\(99\) −1.78857 −0.179758
\(100\) −4.81089 −0.481089
\(101\) −12.2341 −1.21734 −0.608668 0.793425i \(-0.708296\pi\)
−0.608668 + 0.793425i \(0.708296\pi\)
\(102\) −18.9419 −1.87553
\(103\) −1.99867 −0.196935 −0.0984676 0.995140i \(-0.531394\pi\)
−0.0984676 + 0.995140i \(0.531394\pi\)
\(104\) −3.38686 −0.332109
\(105\) −4.81853 −0.470241
\(106\) −10.1788 −0.988652
\(107\) −9.21214 −0.890571 −0.445285 0.895389i \(-0.646898\pi\)
−0.445285 + 0.895389i \(0.646898\pi\)
\(108\) −17.5362 −1.68742
\(109\) 7.79859 0.746969 0.373485 0.927636i \(-0.378163\pi\)
0.373485 + 0.927636i \(0.378163\pi\)
\(110\) −0.0944797 −0.00900829
\(111\) 30.6629 2.91040
\(112\) 3.30614 0.312401
\(113\) −6.09730 −0.573586 −0.286793 0.957993i \(-0.592589\pi\)
−0.286793 + 0.957993i \(0.592589\pi\)
\(114\) 7.40225 0.693284
\(115\) 0.261013 0.0243396
\(116\) −4.63290 −0.430154
\(117\) −27.8819 −2.57768
\(118\) −7.46902 −0.687579
\(119\) 18.6857 1.71291
\(120\) −1.45745 −0.133046
\(121\) −10.9528 −0.995709
\(122\) 8.79215 0.796004
\(123\) −0.0142841 −0.00128795
\(124\) −5.72971 −0.514543
\(125\) −4.26645 −0.381602
\(126\) 27.2174 2.42472
\(127\) −14.9342 −1.32520 −0.662598 0.748976i \(-0.730546\pi\)
−0.662598 + 0.748976i \(0.730546\pi\)
\(128\) 1.00000 0.0883883
\(129\) 5.20415 0.458200
\(130\) −1.47284 −0.129176
\(131\) −3.28045 −0.286614 −0.143307 0.989678i \(-0.545774\pi\)
−0.143307 + 0.989678i \(0.545774\pi\)
\(132\) 0.728143 0.0633767
\(133\) −7.30212 −0.633174
\(134\) 2.66054 0.229836
\(135\) −7.62592 −0.656335
\(136\) 5.65181 0.484639
\(137\) 21.9159 1.87240 0.936200 0.351466i \(-0.114317\pi\)
0.936200 + 0.351466i \(0.114317\pi\)
\(138\) −2.01160 −0.171238
\(139\) 15.5050 1.31511 0.657557 0.753405i \(-0.271590\pi\)
0.657557 + 0.753405i \(0.271590\pi\)
\(140\) 1.43774 0.121511
\(141\) −9.56674 −0.805665
\(142\) −2.21118 −0.185558
\(143\) 0.735831 0.0615333
\(144\) 8.23237 0.686031
\(145\) −2.01470 −0.167312
\(146\) −8.40293 −0.695431
\(147\) −13.1732 −1.08650
\(148\) −9.14910 −0.752051
\(149\) 18.6163 1.52510 0.762551 0.646928i \(-0.223946\pi\)
0.762551 + 0.646928i \(0.223946\pi\)
\(150\) 16.1236 1.31648
\(151\) −20.3836 −1.65880 −0.829398 0.558658i \(-0.811316\pi\)
−0.829398 + 0.558658i \(0.811316\pi\)
\(152\) −2.20866 −0.179146
\(153\) 46.5278 3.76155
\(154\) −0.718294 −0.0578817
\(155\) −2.49167 −0.200136
\(156\) 11.3510 0.908805
\(157\) 20.2350 1.61493 0.807466 0.589914i \(-0.200838\pi\)
0.807466 + 0.589914i \(0.200838\pi\)
\(158\) −4.85441 −0.386196
\(159\) 34.1140 2.70541
\(160\) 0.434868 0.0343794
\(161\) 1.98439 0.156392
\(162\) 34.0748 2.67717
\(163\) −1.43128 −0.112106 −0.0560532 0.998428i \(-0.517852\pi\)
−0.0560532 + 0.998428i \(0.517852\pi\)
\(164\) 0.00426203 0.000332809 0
\(165\) 0.316646 0.0246509
\(166\) −9.12566 −0.708289
\(167\) 9.12428 0.706058 0.353029 0.935612i \(-0.385152\pi\)
0.353029 + 0.935612i \(0.385152\pi\)
\(168\) −11.0804 −0.854874
\(169\) −1.52917 −0.117629
\(170\) 2.45779 0.188504
\(171\) −18.1825 −1.39045
\(172\) −1.55279 −0.118399
\(173\) 21.3312 1.62178 0.810890 0.585199i \(-0.198984\pi\)
0.810890 + 0.585199i \(0.198984\pi\)
\(174\) 15.5270 1.17710
\(175\) −15.9055 −1.20234
\(176\) −0.217260 −0.0163766
\(177\) 25.0322 1.88154
\(178\) 13.1009 0.981950
\(179\) −18.8002 −1.40519 −0.702595 0.711590i \(-0.747975\pi\)
−0.702595 + 0.711590i \(0.747975\pi\)
\(180\) 3.58000 0.266837
\(181\) −15.9940 −1.18882 −0.594412 0.804160i \(-0.702615\pi\)
−0.594412 + 0.804160i \(0.702615\pi\)
\(182\) −11.1974 −0.830009
\(183\) −29.4667 −2.17824
\(184\) 0.600212 0.0442482
\(185\) −3.97865 −0.292516
\(186\) 19.2030 1.40803
\(187\) −1.22792 −0.0897941
\(188\) 2.85449 0.208185
\(189\) −57.9770 −4.21721
\(190\) −0.960474 −0.0696801
\(191\) −6.14526 −0.444655 −0.222328 0.974972i \(-0.571365\pi\)
−0.222328 + 0.974972i \(0.571365\pi\)
\(192\) −3.35147 −0.241872
\(193\) −20.0206 −1.44111 −0.720557 0.693395i \(-0.756114\pi\)
−0.720557 + 0.693395i \(0.756114\pi\)
\(194\) 17.7175 1.27204
\(195\) 4.93618 0.353487
\(196\) 3.93056 0.280754
\(197\) −12.9034 −0.919328 −0.459664 0.888093i \(-0.652030\pi\)
−0.459664 + 0.888093i \(0.652030\pi\)
\(198\) −1.78857 −0.127108
\(199\) −10.4760 −0.742624 −0.371312 0.928508i \(-0.621092\pi\)
−0.371312 + 0.928508i \(0.621092\pi\)
\(200\) −4.81089 −0.340181
\(201\) −8.91673 −0.628938
\(202\) −12.2341 −0.860786
\(203\) −15.3170 −1.07504
\(204\) −18.9419 −1.32620
\(205\) 0.00185342 0.000129449 0
\(206\) −1.99867 −0.139254
\(207\) 4.94117 0.343435
\(208\) −3.38686 −0.234837
\(209\) 0.479853 0.0331922
\(210\) −4.81853 −0.332510
\(211\) −11.6433 −0.801557 −0.400779 0.916175i \(-0.631260\pi\)
−0.400779 + 0.916175i \(0.631260\pi\)
\(212\) −10.1788 −0.699083
\(213\) 7.41073 0.507775
\(214\) −9.21214 −0.629729
\(215\) −0.675261 −0.0460524
\(216\) −17.5362 −1.19318
\(217\) −18.9432 −1.28595
\(218\) 7.79859 0.528187
\(219\) 28.1622 1.90302
\(220\) −0.0944797 −0.00636982
\(221\) −19.1419 −1.28762
\(222\) 30.6629 2.05796
\(223\) 5.38752 0.360775 0.180387 0.983596i \(-0.442265\pi\)
0.180387 + 0.983596i \(0.442265\pi\)
\(224\) 3.30614 0.220901
\(225\) −39.6050 −2.64034
\(226\) −6.09730 −0.405586
\(227\) 16.7265 1.11018 0.555090 0.831790i \(-0.312684\pi\)
0.555090 + 0.831790i \(0.312684\pi\)
\(228\) 7.40225 0.490226
\(229\) 10.2285 0.675915 0.337958 0.941161i \(-0.390264\pi\)
0.337958 + 0.941161i \(0.390264\pi\)
\(230\) 0.261013 0.0172107
\(231\) 2.40734 0.158391
\(232\) −4.63290 −0.304165
\(233\) 12.9318 0.847190 0.423595 0.905852i \(-0.360768\pi\)
0.423595 + 0.905852i \(0.360768\pi\)
\(234\) −27.8819 −1.82270
\(235\) 1.24133 0.0809752
\(236\) −7.46902 −0.486192
\(237\) 16.2694 1.05681
\(238\) 18.6857 1.21121
\(239\) −8.86048 −0.573137 −0.286568 0.958060i \(-0.592515\pi\)
−0.286568 + 0.958060i \(0.592515\pi\)
\(240\) −1.45745 −0.0940780
\(241\) −19.5027 −1.25628 −0.628139 0.778101i \(-0.716183\pi\)
−0.628139 + 0.778101i \(0.716183\pi\)
\(242\) −10.9528 −0.704073
\(243\) −61.5925 −3.95116
\(244\) 8.79215 0.562860
\(245\) 1.70928 0.109202
\(246\) −0.0142841 −0.000910719 0
\(247\) 7.48041 0.475967
\(248\) −5.72971 −0.363837
\(249\) 30.5844 1.93821
\(250\) −4.26645 −0.269834
\(251\) −15.5243 −0.979887 −0.489944 0.871754i \(-0.662983\pi\)
−0.489944 + 0.871754i \(0.662983\pi\)
\(252\) 27.2174 1.71453
\(253\) −0.130402 −0.00819833
\(254\) −14.9342 −0.937054
\(255\) −8.23723 −0.515835
\(256\) 1.00000 0.0625000
\(257\) −14.2727 −0.890307 −0.445153 0.895454i \(-0.646851\pi\)
−0.445153 + 0.895454i \(0.646851\pi\)
\(258\) 5.20415 0.323996
\(259\) −30.2482 −1.87953
\(260\) −1.47284 −0.0913416
\(261\) −38.1398 −2.36079
\(262\) −3.28045 −0.202667
\(263\) −14.4433 −0.890613 −0.445306 0.895378i \(-0.646905\pi\)
−0.445306 + 0.895378i \(0.646905\pi\)
\(264\) 0.728143 0.0448141
\(265\) −4.42644 −0.271914
\(266\) −7.30212 −0.447722
\(267\) −43.9072 −2.68707
\(268\) 2.66054 0.162518
\(269\) 17.1774 1.04733 0.523663 0.851926i \(-0.324565\pi\)
0.523663 + 0.851926i \(0.324565\pi\)
\(270\) −7.62592 −0.464099
\(271\) 18.4436 1.12037 0.560185 0.828368i \(-0.310730\pi\)
0.560185 + 0.828368i \(0.310730\pi\)
\(272\) 5.65181 0.342691
\(273\) 37.5279 2.27129
\(274\) 21.9159 1.32399
\(275\) 1.04522 0.0630289
\(276\) −2.01160 −0.121084
\(277\) −29.9926 −1.80208 −0.901039 0.433738i \(-0.857194\pi\)
−0.901039 + 0.433738i \(0.857194\pi\)
\(278\) 15.5050 0.929926
\(279\) −47.1691 −2.82394
\(280\) 1.43774 0.0859211
\(281\) 2.45136 0.146236 0.0731179 0.997323i \(-0.476705\pi\)
0.0731179 + 0.997323i \(0.476705\pi\)
\(282\) −9.56674 −0.569691
\(283\) −6.68694 −0.397498 −0.198749 0.980050i \(-0.563688\pi\)
−0.198749 + 0.980050i \(0.563688\pi\)
\(284\) −2.21118 −0.131210
\(285\) 3.21900 0.190677
\(286\) 0.735831 0.0435106
\(287\) 0.0140909 0.000831757 0
\(288\) 8.23237 0.485097
\(289\) 14.9430 0.878998
\(290\) −2.01470 −0.118307
\(291\) −59.3798 −3.48091
\(292\) −8.40293 −0.491744
\(293\) −1.60411 −0.0937132 −0.0468566 0.998902i \(-0.514920\pi\)
−0.0468566 + 0.998902i \(0.514920\pi\)
\(294\) −13.1732 −0.768275
\(295\) −3.24804 −0.189108
\(296\) −9.14910 −0.531780
\(297\) 3.80991 0.221074
\(298\) 18.6163 1.07841
\(299\) −2.03284 −0.117562
\(300\) 16.1236 0.930895
\(301\) −5.13376 −0.295905
\(302\) −20.3836 −1.17295
\(303\) 41.0022 2.35551
\(304\) −2.20866 −0.126675
\(305\) 3.82343 0.218929
\(306\) 46.5278 2.65982
\(307\) −28.5303 −1.62831 −0.814154 0.580648i \(-0.802799\pi\)
−0.814154 + 0.580648i \(0.802799\pi\)
\(308\) −0.718294 −0.0409286
\(309\) 6.69850 0.381064
\(310\) −2.49167 −0.141517
\(311\) 5.86377 0.332504 0.166252 0.986083i \(-0.446834\pi\)
0.166252 + 0.986083i \(0.446834\pi\)
\(312\) 11.3510 0.642622
\(313\) −9.05604 −0.511878 −0.255939 0.966693i \(-0.582385\pi\)
−0.255939 + 0.966693i \(0.582385\pi\)
\(314\) 20.2350 1.14193
\(315\) 11.8360 0.666882
\(316\) −4.85441 −0.273082
\(317\) 12.6046 0.707946 0.353973 0.935256i \(-0.384830\pi\)
0.353973 + 0.935256i \(0.384830\pi\)
\(318\) 34.1140 1.91302
\(319\) 1.00655 0.0563558
\(320\) 0.434868 0.0243099
\(321\) 30.8742 1.72323
\(322\) 1.98439 0.110585
\(323\) −12.4829 −0.694567
\(324\) 34.0748 1.89305
\(325\) 16.2938 0.903818
\(326\) −1.43128 −0.0792711
\(327\) −26.1368 −1.44537
\(328\) 0.00426203 0.000235331 0
\(329\) 9.43734 0.520297
\(330\) 0.316646 0.0174308
\(331\) 20.0210 1.10045 0.550227 0.835015i \(-0.314541\pi\)
0.550227 + 0.835015i \(0.314541\pi\)
\(332\) −9.12566 −0.500836
\(333\) −75.3188 −4.12744
\(334\) 9.12428 0.499258
\(335\) 1.15698 0.0632128
\(336\) −11.0804 −0.604488
\(337\) 22.6253 1.23248 0.616239 0.787559i \(-0.288656\pi\)
0.616239 + 0.787559i \(0.288656\pi\)
\(338\) −1.52917 −0.0831760
\(339\) 20.4349 1.10987
\(340\) 2.45779 0.133293
\(341\) 1.24484 0.0674118
\(342\) −18.1825 −0.983195
\(343\) −10.1480 −0.547940
\(344\) −1.55279 −0.0837211
\(345\) −0.874779 −0.0470965
\(346\) 21.3312 1.14677
\(347\) −11.6323 −0.624456 −0.312228 0.950007i \(-0.601075\pi\)
−0.312228 + 0.950007i \(0.601075\pi\)
\(348\) 15.5270 0.832337
\(349\) 10.1951 0.545731 0.272866 0.962052i \(-0.412029\pi\)
0.272866 + 0.962052i \(0.412029\pi\)
\(350\) −15.9055 −0.850183
\(351\) 59.3925 3.17014
\(352\) −0.217260 −0.0115800
\(353\) −7.96016 −0.423677 −0.211838 0.977305i \(-0.567945\pi\)
−0.211838 + 0.977305i \(0.567945\pi\)
\(354\) 25.0322 1.33045
\(355\) −0.961574 −0.0510351
\(356\) 13.1009 0.694344
\(357\) −62.6245 −3.31444
\(358\) −18.8002 −0.993619
\(359\) −13.0352 −0.687971 −0.343985 0.938975i \(-0.611777\pi\)
−0.343985 + 0.938975i \(0.611777\pi\)
\(360\) 3.58000 0.188682
\(361\) −14.1218 −0.743255
\(362\) −15.9940 −0.840626
\(363\) 36.7080 1.92667
\(364\) −11.1974 −0.586905
\(365\) −3.65417 −0.191268
\(366\) −29.4667 −1.54025
\(367\) −21.3991 −1.11702 −0.558511 0.829497i \(-0.688627\pi\)
−0.558511 + 0.829497i \(0.688627\pi\)
\(368\) 0.600212 0.0312882
\(369\) 0.0350866 0.00182654
\(370\) −3.97865 −0.206840
\(371\) −33.6525 −1.74715
\(372\) 19.2030 0.995627
\(373\) 28.2030 1.46029 0.730147 0.683290i \(-0.239451\pi\)
0.730147 + 0.683290i \(0.239451\pi\)
\(374\) −1.22792 −0.0634940
\(375\) 14.2989 0.738391
\(376\) 2.85449 0.147209
\(377\) 15.6910 0.808128
\(378\) −57.9770 −2.98201
\(379\) 14.0002 0.719143 0.359572 0.933117i \(-0.382923\pi\)
0.359572 + 0.933117i \(0.382923\pi\)
\(380\) −0.960474 −0.0492713
\(381\) 50.0515 2.56422
\(382\) −6.14526 −0.314419
\(383\) 30.1483 1.54051 0.770253 0.637738i \(-0.220130\pi\)
0.770253 + 0.637738i \(0.220130\pi\)
\(384\) −3.35147 −0.171029
\(385\) −0.312363 −0.0159195
\(386\) −20.0206 −1.01902
\(387\) −12.7832 −0.649806
\(388\) 17.7175 0.899471
\(389\) −21.1708 −1.07340 −0.536702 0.843772i \(-0.680330\pi\)
−0.536702 + 0.843772i \(0.680330\pi\)
\(390\) 4.93618 0.249953
\(391\) 3.39229 0.171555
\(392\) 3.93056 0.198523
\(393\) 10.9943 0.554590
\(394\) −12.9034 −0.650063
\(395\) −2.11103 −0.106217
\(396\) −1.78857 −0.0898790
\(397\) −5.71182 −0.286668 −0.143334 0.989674i \(-0.545782\pi\)
−0.143334 + 0.989674i \(0.545782\pi\)
\(398\) −10.4760 −0.525114
\(399\) 24.4729 1.22518
\(400\) −4.81089 −0.240544
\(401\) 15.4069 0.769384 0.384692 0.923045i \(-0.374308\pi\)
0.384692 + 0.923045i \(0.374308\pi\)
\(402\) −8.91673 −0.444726
\(403\) 19.4057 0.966668
\(404\) −12.2341 −0.608668
\(405\) 14.8181 0.736316
\(406\) −15.3170 −0.760171
\(407\) 1.98774 0.0985285
\(408\) −18.9419 −0.937763
\(409\) 17.1086 0.845968 0.422984 0.906137i \(-0.360983\pi\)
0.422984 + 0.906137i \(0.360983\pi\)
\(410\) 0.00185342 9.15340e−5 0
\(411\) −73.4505 −3.62305
\(412\) −1.99867 −0.0984676
\(413\) −24.6936 −1.21509
\(414\) 4.94117 0.242845
\(415\) −3.96846 −0.194804
\(416\) −3.38686 −0.166055
\(417\) −51.9645 −2.54471
\(418\) 0.479853 0.0234704
\(419\) 11.2794 0.551035 0.275518 0.961296i \(-0.411151\pi\)
0.275518 + 0.961296i \(0.411151\pi\)
\(420\) −4.81853 −0.235120
\(421\) 10.2858 0.501298 0.250649 0.968078i \(-0.419356\pi\)
0.250649 + 0.968078i \(0.419356\pi\)
\(422\) −11.6433 −0.566786
\(423\) 23.4992 1.14257
\(424\) −10.1788 −0.494326
\(425\) −27.1902 −1.31892
\(426\) 7.41073 0.359051
\(427\) 29.0681 1.40670
\(428\) −9.21214 −0.445285
\(429\) −2.46612 −0.119065
\(430\) −0.675261 −0.0325640
\(431\) 0.224416 0.0108098 0.00540488 0.999985i \(-0.498280\pi\)
0.00540488 + 0.999985i \(0.498280\pi\)
\(432\) −17.5362 −0.843709
\(433\) 19.0708 0.916483 0.458242 0.888828i \(-0.348479\pi\)
0.458242 + 0.888828i \(0.348479\pi\)
\(434\) −18.9432 −0.909303
\(435\) 6.75222 0.323744
\(436\) 7.79859 0.373485
\(437\) −1.32566 −0.0634150
\(438\) 28.1622 1.34564
\(439\) −39.6835 −1.89399 −0.946994 0.321251i \(-0.895897\pi\)
−0.946994 + 0.321251i \(0.895897\pi\)
\(440\) −0.0944797 −0.00450414
\(441\) 32.3578 1.54085
\(442\) −19.1419 −0.910487
\(443\) 8.94594 0.425035 0.212517 0.977157i \(-0.431834\pi\)
0.212517 + 0.977157i \(0.431834\pi\)
\(444\) 30.6629 1.45520
\(445\) 5.69715 0.270071
\(446\) 5.38752 0.255106
\(447\) −62.3919 −2.95103
\(448\) 3.30614 0.156200
\(449\) −23.2518 −1.09732 −0.548660 0.836046i \(-0.684862\pi\)
−0.548660 + 0.836046i \(0.684862\pi\)
\(450\) −39.6050 −1.86700
\(451\) −0.000925971 0 −4.36022e−5 0
\(452\) −6.09730 −0.286793
\(453\) 68.3152 3.20973
\(454\) 16.7265 0.785016
\(455\) −4.86941 −0.228281
\(456\) 7.40225 0.346642
\(457\) 20.4390 0.956095 0.478048 0.878334i \(-0.341345\pi\)
0.478048 + 0.878334i \(0.341345\pi\)
\(458\) 10.2285 0.477944
\(459\) −99.1111 −4.62611
\(460\) 0.261013 0.0121698
\(461\) −20.1844 −0.940080 −0.470040 0.882645i \(-0.655761\pi\)
−0.470040 + 0.882645i \(0.655761\pi\)
\(462\) 2.40734 0.112000
\(463\) −30.3322 −1.40966 −0.704829 0.709378i \(-0.748976\pi\)
−0.704829 + 0.709378i \(0.748976\pi\)
\(464\) −4.63290 −0.215077
\(465\) 8.35076 0.387257
\(466\) 12.9318 0.599054
\(467\) −9.47525 −0.438462 −0.219231 0.975673i \(-0.570355\pi\)
−0.219231 + 0.975673i \(0.570355\pi\)
\(468\) −27.8819 −1.28884
\(469\) 8.79612 0.406167
\(470\) 1.24133 0.0572581
\(471\) −67.8172 −3.12485
\(472\) −7.46902 −0.343790
\(473\) 0.337361 0.0155119
\(474\) 16.2694 0.747279
\(475\) 10.6256 0.487536
\(476\) 18.6857 0.856457
\(477\) −83.7957 −3.83674
\(478\) −8.86048 −0.405269
\(479\) 12.5310 0.572555 0.286278 0.958147i \(-0.407582\pi\)
0.286278 + 0.958147i \(0.407582\pi\)
\(480\) −1.45745 −0.0665232
\(481\) 30.9867 1.41287
\(482\) −19.5027 −0.888323
\(483\) −6.65062 −0.302613
\(484\) −10.9528 −0.497854
\(485\) 7.70479 0.349856
\(486\) −61.5925 −2.79389
\(487\) −0.847982 −0.0384257 −0.0192129 0.999815i \(-0.506116\pi\)
−0.0192129 + 0.999815i \(0.506116\pi\)
\(488\) 8.79215 0.398002
\(489\) 4.79689 0.216923
\(490\) 1.70928 0.0772172
\(491\) 22.6153 1.02061 0.510307 0.859993i \(-0.329532\pi\)
0.510307 + 0.859993i \(0.329532\pi\)
\(492\) −0.0142841 −0.000643976 0
\(493\) −26.1843 −1.17928
\(494\) 7.48041 0.336559
\(495\) −0.777792 −0.0349592
\(496\) −5.72971 −0.257271
\(497\) −7.31049 −0.327920
\(498\) 30.5844 1.37052
\(499\) −18.9290 −0.847379 −0.423690 0.905807i \(-0.639265\pi\)
−0.423690 + 0.905807i \(0.639265\pi\)
\(500\) −4.26645 −0.190801
\(501\) −30.5798 −1.36620
\(502\) −15.5243 −0.692885
\(503\) 4.79194 0.213662 0.106831 0.994277i \(-0.465930\pi\)
0.106831 + 0.994277i \(0.465930\pi\)
\(504\) 27.2174 1.21236
\(505\) −5.32021 −0.236746
\(506\) −0.130402 −0.00579709
\(507\) 5.12498 0.227608
\(508\) −14.9342 −0.662598
\(509\) −11.6694 −0.517237 −0.258619 0.965979i \(-0.583267\pi\)
−0.258619 + 0.965979i \(0.583267\pi\)
\(510\) −8.23723 −0.364751
\(511\) −27.7813 −1.22897
\(512\) 1.00000 0.0441942
\(513\) 38.7313 1.71003
\(514\) −14.2727 −0.629542
\(515\) −0.869160 −0.0382998
\(516\) 5.20415 0.229100
\(517\) −0.620167 −0.0272749
\(518\) −30.2482 −1.32903
\(519\) −71.4909 −3.13810
\(520\) −1.47284 −0.0645882
\(521\) −30.3473 −1.32954 −0.664770 0.747048i \(-0.731470\pi\)
−0.664770 + 0.747048i \(0.731470\pi\)
\(522\) −38.1398 −1.66933
\(523\) 17.0560 0.745808 0.372904 0.927870i \(-0.378362\pi\)
0.372904 + 0.927870i \(0.378362\pi\)
\(524\) −3.28045 −0.143307
\(525\) 53.3068 2.32650
\(526\) −14.4433 −0.629758
\(527\) −32.3832 −1.41064
\(528\) 0.728143 0.0316883
\(529\) −22.6397 −0.984337
\(530\) −4.42644 −0.192272
\(531\) −61.4878 −2.66834
\(532\) −7.30212 −0.316587
\(533\) −0.0144349 −0.000625245 0
\(534\) −43.9072 −1.90005
\(535\) −4.00607 −0.173197
\(536\) 2.66054 0.114918
\(537\) 63.0082 2.71901
\(538\) 17.1774 0.740571
\(539\) −0.853956 −0.0367825
\(540\) −7.62592 −0.328167
\(541\) −21.9122 −0.942077 −0.471039 0.882113i \(-0.656121\pi\)
−0.471039 + 0.882113i \(0.656121\pi\)
\(542\) 18.4436 0.792221
\(543\) 53.6035 2.30034
\(544\) 5.65181 0.242319
\(545\) 3.39136 0.145270
\(546\) 37.5279 1.60605
\(547\) 41.8157 1.78791 0.893956 0.448155i \(-0.147919\pi\)
0.893956 + 0.448155i \(0.147919\pi\)
\(548\) 21.9159 0.936200
\(549\) 72.3802 3.08911
\(550\) 1.04522 0.0445682
\(551\) 10.2325 0.435918
\(552\) −2.01160 −0.0856192
\(553\) −16.0493 −0.682488
\(554\) −29.9926 −1.27426
\(555\) 13.3343 0.566011
\(556\) 15.5050 0.657557
\(557\) 4.41538 0.187086 0.0935429 0.995615i \(-0.470181\pi\)
0.0935429 + 0.995615i \(0.470181\pi\)
\(558\) −47.1691 −1.99683
\(559\) 5.25910 0.222436
\(560\) 1.43774 0.0607554
\(561\) 4.11533 0.173749
\(562\) 2.45136 0.103404
\(563\) −20.4225 −0.860707 −0.430353 0.902661i \(-0.641611\pi\)
−0.430353 + 0.902661i \(0.641611\pi\)
\(564\) −9.56674 −0.402832
\(565\) −2.65152 −0.111550
\(566\) −6.68694 −0.281073
\(567\) 112.656 4.73112
\(568\) −2.21118 −0.0927792
\(569\) 20.8290 0.873198 0.436599 0.899656i \(-0.356183\pi\)
0.436599 + 0.899656i \(0.356183\pi\)
\(570\) 3.21900 0.134829
\(571\) 4.49847 0.188255 0.0941276 0.995560i \(-0.469994\pi\)
0.0941276 + 0.995560i \(0.469994\pi\)
\(572\) 0.735831 0.0307666
\(573\) 20.5957 0.860396
\(574\) 0.0140909 0.000588141 0
\(575\) −2.88756 −0.120419
\(576\) 8.23237 0.343016
\(577\) −10.8655 −0.452335 −0.226168 0.974088i \(-0.572620\pi\)
−0.226168 + 0.974088i \(0.572620\pi\)
\(578\) 14.9430 0.621546
\(579\) 67.0985 2.78852
\(580\) −2.01470 −0.0836560
\(581\) −30.1707 −1.25169
\(582\) −59.3798 −2.46137
\(583\) 2.21145 0.0915889
\(584\) −8.40293 −0.347716
\(585\) −12.1250 −0.501305
\(586\) −1.60411 −0.0662652
\(587\) 12.1632 0.502031 0.251015 0.967983i \(-0.419236\pi\)
0.251015 + 0.967983i \(0.419236\pi\)
\(588\) −13.1732 −0.543252
\(589\) 12.6549 0.521438
\(590\) −3.24804 −0.133720
\(591\) 43.2453 1.77887
\(592\) −9.14910 −0.376026
\(593\) 15.3029 0.628416 0.314208 0.949354i \(-0.398261\pi\)
0.314208 + 0.949354i \(0.398261\pi\)
\(594\) 3.80991 0.156323
\(595\) 8.12581 0.333126
\(596\) 18.6163 0.762551
\(597\) 35.1100 1.43696
\(598\) −2.03284 −0.0831288
\(599\) −28.3164 −1.15697 −0.578487 0.815691i \(-0.696357\pi\)
−0.578487 + 0.815691i \(0.696357\pi\)
\(600\) 16.1236 0.658242
\(601\) 11.1384 0.454344 0.227172 0.973855i \(-0.427052\pi\)
0.227172 + 0.973855i \(0.427052\pi\)
\(602\) −5.13376 −0.209236
\(603\) 21.9026 0.891941
\(604\) −20.3836 −0.829398
\(605\) −4.76302 −0.193645
\(606\) 41.0022 1.66560
\(607\) 3.03254 0.123087 0.0615436 0.998104i \(-0.480398\pi\)
0.0615436 + 0.998104i \(0.480398\pi\)
\(608\) −2.20866 −0.0895728
\(609\) 51.3346 2.08018
\(610\) 3.82343 0.154806
\(611\) −9.66775 −0.391115
\(612\) 46.5278 1.88078
\(613\) 28.0600 1.13333 0.566667 0.823947i \(-0.308233\pi\)
0.566667 + 0.823947i \(0.308233\pi\)
\(614\) −28.5303 −1.15139
\(615\) −0.00621169 −0.000250480 0
\(616\) −0.718294 −0.0289409
\(617\) −18.0963 −0.728528 −0.364264 0.931296i \(-0.618679\pi\)
−0.364264 + 0.931296i \(0.618679\pi\)
\(618\) 6.69850 0.269453
\(619\) −5.14305 −0.206717 −0.103358 0.994644i \(-0.532959\pi\)
−0.103358 + 0.994644i \(0.532959\pi\)
\(620\) −2.49167 −0.100068
\(621\) −10.5254 −0.422370
\(622\) 5.86377 0.235116
\(623\) 43.3132 1.73531
\(624\) 11.3510 0.454403
\(625\) 22.1991 0.887964
\(626\) −9.05604 −0.361952
\(627\) −1.60822 −0.0642260
\(628\) 20.2350 0.807466
\(629\) −51.7090 −2.06177
\(630\) 11.8360 0.471557
\(631\) −34.0365 −1.35497 −0.677486 0.735536i \(-0.736930\pi\)
−0.677486 + 0.735536i \(0.736930\pi\)
\(632\) −4.85441 −0.193098
\(633\) 39.0222 1.55099
\(634\) 12.6046 0.500594
\(635\) −6.49441 −0.257723
\(636\) 34.1140 1.35271
\(637\) −13.3123 −0.527451
\(638\) 1.00655 0.0398496
\(639\) −18.2033 −0.720111
\(640\) 0.434868 0.0171897
\(641\) −1.45353 −0.0574110 −0.0287055 0.999588i \(-0.509139\pi\)
−0.0287055 + 0.999588i \(0.509139\pi\)
\(642\) 30.8742 1.21851
\(643\) 1.37521 0.0542330 0.0271165 0.999632i \(-0.491367\pi\)
0.0271165 + 0.999632i \(0.491367\pi\)
\(644\) 1.98439 0.0781958
\(645\) 2.26312 0.0891103
\(646\) −12.4829 −0.491133
\(647\) −8.41504 −0.330829 −0.165415 0.986224i \(-0.552896\pi\)
−0.165415 + 0.986224i \(0.552896\pi\)
\(648\) 34.0748 1.33859
\(649\) 1.62272 0.0636975
\(650\) 16.2938 0.639096
\(651\) 63.4877 2.48828
\(652\) −1.43128 −0.0560532
\(653\) −37.3567 −1.46188 −0.730940 0.682442i \(-0.760918\pi\)
−0.730940 + 0.682442i \(0.760918\pi\)
\(654\) −26.1368 −1.02203
\(655\) −1.42656 −0.0557404
\(656\) 0.00426203 0.000166404 0
\(657\) −69.1760 −2.69881
\(658\) 9.43734 0.367906
\(659\) 24.6868 0.961661 0.480831 0.876814i \(-0.340335\pi\)
0.480831 + 0.876814i \(0.340335\pi\)
\(660\) 0.316646 0.0123254
\(661\) 3.10382 0.120725 0.0603623 0.998177i \(-0.480774\pi\)
0.0603623 + 0.998177i \(0.480774\pi\)
\(662\) 20.0210 0.778138
\(663\) 64.1536 2.49152
\(664\) −9.12566 −0.354144
\(665\) −3.17546 −0.123139
\(666\) −75.3188 −2.91854
\(667\) −2.78073 −0.107670
\(668\) 9.12428 0.353029
\(669\) −18.0561 −0.698090
\(670\) 1.15698 0.0446982
\(671\) −1.91019 −0.0737419
\(672\) −11.0804 −0.427437
\(673\) 25.1937 0.971145 0.485573 0.874196i \(-0.338611\pi\)
0.485573 + 0.874196i \(0.338611\pi\)
\(674\) 22.6253 0.871493
\(675\) 84.3645 3.24719
\(676\) −1.52917 −0.0588143
\(677\) 13.6488 0.524566 0.262283 0.964991i \(-0.415525\pi\)
0.262283 + 0.964991i \(0.415525\pi\)
\(678\) 20.4349 0.784799
\(679\) 58.5766 2.24796
\(680\) 2.45779 0.0942521
\(681\) −56.0586 −2.14817
\(682\) 1.24484 0.0476673
\(683\) 24.2634 0.928413 0.464207 0.885727i \(-0.346339\pi\)
0.464207 + 0.885727i \(0.346339\pi\)
\(684\) −18.1825 −0.695224
\(685\) 9.53053 0.364143
\(686\) −10.1480 −0.387452
\(687\) −34.2804 −1.30788
\(688\) −1.55279 −0.0591997
\(689\) 34.4742 1.31336
\(690\) −0.874779 −0.0333023
\(691\) 21.3447 0.811992 0.405996 0.913875i \(-0.366925\pi\)
0.405996 + 0.913875i \(0.366925\pi\)
\(692\) 21.3312 0.810890
\(693\) −5.91326 −0.224626
\(694\) −11.6323 −0.441557
\(695\) 6.74262 0.255762
\(696\) 15.5270 0.588551
\(697\) 0.0240882 0.000912405 0
\(698\) 10.1951 0.385890
\(699\) −43.3406 −1.63929
\(700\) −15.9055 −0.601170
\(701\) 0.169561 0.00640425 0.00320212 0.999995i \(-0.498981\pi\)
0.00320212 + 0.999995i \(0.498981\pi\)
\(702\) 59.3925 2.24163
\(703\) 20.2072 0.762129
\(704\) −0.217260 −0.00818831
\(705\) −4.16027 −0.156685
\(706\) −7.96016 −0.299585
\(707\) −40.4475 −1.52119
\(708\) 25.0322 0.940769
\(709\) −6.51741 −0.244766 −0.122383 0.992483i \(-0.539054\pi\)
−0.122383 + 0.992483i \(0.539054\pi\)
\(710\) −0.961574 −0.0360872
\(711\) −39.9633 −1.49874
\(712\) 13.1009 0.490975
\(713\) −3.43904 −0.128793
\(714\) −62.6245 −2.34366
\(715\) 0.319990 0.0119669
\(716\) −18.8002 −0.702595
\(717\) 29.6957 1.10901
\(718\) −13.0352 −0.486469
\(719\) −17.0819 −0.637047 −0.318523 0.947915i \(-0.603187\pi\)
−0.318523 + 0.947915i \(0.603187\pi\)
\(720\) 3.58000 0.133419
\(721\) −6.60790 −0.246091
\(722\) −14.1218 −0.525561
\(723\) 65.3627 2.43087
\(724\) −15.9940 −0.594412
\(725\) 22.2884 0.827770
\(726\) 36.7080 1.36236
\(727\) 33.6111 1.24657 0.623283 0.781996i \(-0.285798\pi\)
0.623283 + 0.781996i \(0.285798\pi\)
\(728\) −11.1974 −0.415005
\(729\) 104.201 3.85929
\(730\) −3.65417 −0.135247
\(731\) −8.77610 −0.324596
\(732\) −29.4667 −1.08912
\(733\) −38.8574 −1.43523 −0.717615 0.696440i \(-0.754766\pi\)
−0.717615 + 0.696440i \(0.754766\pi\)
\(734\) −21.3991 −0.789854
\(735\) −5.72859 −0.211302
\(736\) 0.600212 0.0221241
\(737\) −0.578030 −0.0212920
\(738\) 0.0350866 0.00129156
\(739\) 40.0712 1.47404 0.737022 0.675868i \(-0.236231\pi\)
0.737022 + 0.675868i \(0.236231\pi\)
\(740\) −3.97865 −0.146258
\(741\) −25.0704 −0.920984
\(742\) −33.6525 −1.23542
\(743\) −51.6743 −1.89575 −0.947874 0.318647i \(-0.896772\pi\)
−0.947874 + 0.318647i \(0.896772\pi\)
\(744\) 19.2030 0.704015
\(745\) 8.09562 0.296600
\(746\) 28.2030 1.03258
\(747\) −75.1259 −2.74871
\(748\) −1.22792 −0.0448970
\(749\) −30.4566 −1.11286
\(750\) 14.2989 0.522121
\(751\) 13.5297 0.493705 0.246852 0.969053i \(-0.420604\pi\)
0.246852 + 0.969053i \(0.420604\pi\)
\(752\) 2.85449 0.104092
\(753\) 52.0294 1.89606
\(754\) 15.6910 0.571432
\(755\) −8.86419 −0.322601
\(756\) −57.9770 −2.10860
\(757\) 21.7540 0.790661 0.395331 0.918539i \(-0.370630\pi\)
0.395331 + 0.918539i \(0.370630\pi\)
\(758\) 14.0002 0.508511
\(759\) 0.437040 0.0158636
\(760\) −0.960474 −0.0348401
\(761\) −0.211788 −0.00767729 −0.00383865 0.999993i \(-0.501222\pi\)
−0.00383865 + 0.999993i \(0.501222\pi\)
\(762\) 50.0515 1.81318
\(763\) 25.7832 0.933415
\(764\) −6.14526 −0.222328
\(765\) 20.2335 0.731543
\(766\) 30.1483 1.08930
\(767\) 25.2965 0.913405
\(768\) −3.35147 −0.120936
\(769\) 13.0480 0.470522 0.235261 0.971932i \(-0.424406\pi\)
0.235261 + 0.971932i \(0.424406\pi\)
\(770\) −0.312363 −0.0112568
\(771\) 47.8346 1.72272
\(772\) −20.0206 −0.720557
\(773\) −2.11599 −0.0761070 −0.0380535 0.999276i \(-0.512116\pi\)
−0.0380535 + 0.999276i \(0.512116\pi\)
\(774\) −12.7832 −0.459482
\(775\) 27.5650 0.990163
\(776\) 17.7175 0.636022
\(777\) 101.376 3.63684
\(778\) −21.1708 −0.759011
\(779\) −0.00941335 −0.000337268 0
\(780\) 4.93618 0.176744
\(781\) 0.480403 0.0171902
\(782\) 3.39229 0.121308
\(783\) 81.2433 2.90340
\(784\) 3.93056 0.140377
\(785\) 8.79958 0.314070
\(786\) 10.9943 0.392155
\(787\) −21.3347 −0.760500 −0.380250 0.924884i \(-0.624162\pi\)
−0.380250 + 0.924884i \(0.624162\pi\)
\(788\) −12.9034 −0.459664
\(789\) 48.4064 1.72331
\(790\) −2.11103 −0.0751070
\(791\) −20.1585 −0.716755
\(792\) −1.78857 −0.0635540
\(793\) −29.7778 −1.05744
\(794\) −5.71182 −0.202705
\(795\) 14.8351 0.526146
\(796\) −10.4760 −0.371312
\(797\) 15.6134 0.553054 0.276527 0.961006i \(-0.410816\pi\)
0.276527 + 0.961006i \(0.410816\pi\)
\(798\) 24.4729 0.866330
\(799\) 16.1330 0.570745
\(800\) −4.81089 −0.170091
\(801\) 107.851 3.81073
\(802\) 15.4069 0.544036
\(803\) 1.82562 0.0644249
\(804\) −8.91673 −0.314469
\(805\) 0.862947 0.0304149
\(806\) 19.4057 0.683537
\(807\) −57.5697 −2.02655
\(808\) −12.2341 −0.430393
\(809\) 34.6120 1.21689 0.608446 0.793595i \(-0.291793\pi\)
0.608446 + 0.793595i \(0.291793\pi\)
\(810\) 14.8181 0.520654
\(811\) 27.7945 0.975997 0.487998 0.872845i \(-0.337727\pi\)
0.487998 + 0.872845i \(0.337727\pi\)
\(812\) −15.3170 −0.537522
\(813\) −61.8132 −2.16789
\(814\) 1.98774 0.0696701
\(815\) −0.622417 −0.0218023
\(816\) −18.9419 −0.663099
\(817\) 3.42959 0.119986
\(818\) 17.1086 0.598189
\(819\) −92.1815 −3.22108
\(820\) 0.00185342 6.47243e−5 0
\(821\) 54.8556 1.91447 0.957236 0.289307i \(-0.0934247\pi\)
0.957236 + 0.289307i \(0.0934247\pi\)
\(822\) −73.4505 −2.56188
\(823\) 2.80963 0.0979374 0.0489687 0.998800i \(-0.484407\pi\)
0.0489687 + 0.998800i \(0.484407\pi\)
\(824\) −1.99867 −0.0696271
\(825\) −3.50301 −0.121959
\(826\) −24.6936 −0.859201
\(827\) −43.2228 −1.50300 −0.751502 0.659731i \(-0.770670\pi\)
−0.751502 + 0.659731i \(0.770670\pi\)
\(828\) 4.94117 0.171718
\(829\) 2.97614 0.103366 0.0516828 0.998664i \(-0.483542\pi\)
0.0516828 + 0.998664i \(0.483542\pi\)
\(830\) −3.96846 −0.137747
\(831\) 100.519 3.48697
\(832\) −3.38686 −0.117418
\(833\) 22.2148 0.769697
\(834\) −51.9645 −1.79938
\(835\) 3.96786 0.137313
\(836\) 0.479853 0.0165961
\(837\) 100.477 3.47299
\(838\) 11.2794 0.389641
\(839\) −39.3598 −1.35885 −0.679425 0.733745i \(-0.737771\pi\)
−0.679425 + 0.733745i \(0.737771\pi\)
\(840\) −4.81853 −0.166255
\(841\) −7.53621 −0.259869
\(842\) 10.2858 0.354471
\(843\) −8.21567 −0.282963
\(844\) −11.6433 −0.400779
\(845\) −0.664988 −0.0228763
\(846\) 23.4992 0.807919
\(847\) −36.2115 −1.24424
\(848\) −10.1788 −0.349541
\(849\) 22.4111 0.769147
\(850\) −27.1902 −0.932617
\(851\) −5.49140 −0.188243
\(852\) 7.41073 0.253887
\(853\) −9.30954 −0.318753 −0.159376 0.987218i \(-0.550948\pi\)
−0.159376 + 0.987218i \(0.550948\pi\)
\(854\) 29.0681 0.994689
\(855\) −7.90698 −0.270413
\(856\) −9.21214 −0.314864
\(857\) 34.5400 1.17986 0.589932 0.807453i \(-0.299155\pi\)
0.589932 + 0.807453i \(0.299155\pi\)
\(858\) −2.46612 −0.0841919
\(859\) 33.1808 1.13211 0.566056 0.824367i \(-0.308468\pi\)
0.566056 + 0.824367i \(0.308468\pi\)
\(860\) −0.675261 −0.0230262
\(861\) −0.0472252 −0.00160943
\(862\) 0.224416 0.00764365
\(863\) −53.1590 −1.80955 −0.904777 0.425886i \(-0.859962\pi\)
−0.904777 + 0.425886i \(0.859962\pi\)
\(864\) −17.5362 −0.596592
\(865\) 9.27625 0.315402
\(866\) 19.0708 0.648051
\(867\) −50.0810 −1.70084
\(868\) −18.9432 −0.642974
\(869\) 1.05467 0.0357773
\(870\) 6.75222 0.228922
\(871\) −9.01088 −0.305322
\(872\) 7.79859 0.264093
\(873\) 145.857 4.93652
\(874\) −1.32566 −0.0448412
\(875\) −14.1055 −0.476852
\(876\) 28.1622 0.951512
\(877\) 42.7438 1.44335 0.721677 0.692230i \(-0.243371\pi\)
0.721677 + 0.692230i \(0.243371\pi\)
\(878\) −39.6835 −1.33925
\(879\) 5.37613 0.181333
\(880\) −0.0944797 −0.00318491
\(881\) −1.21105 −0.0408012 −0.0204006 0.999792i \(-0.506494\pi\)
−0.0204006 + 0.999792i \(0.506494\pi\)
\(882\) 32.3578 1.08955
\(883\) 26.7723 0.900961 0.450480 0.892786i \(-0.351253\pi\)
0.450480 + 0.892786i \(0.351253\pi\)
\(884\) −19.1419 −0.643812
\(885\) 10.8857 0.365920
\(886\) 8.94594 0.300545
\(887\) −40.2904 −1.35282 −0.676410 0.736526i \(-0.736465\pi\)
−0.676410 + 0.736526i \(0.736465\pi\)
\(888\) 30.6629 1.02898
\(889\) −49.3745 −1.65597
\(890\) 5.69715 0.190969
\(891\) −7.40312 −0.248014
\(892\) 5.38752 0.180387
\(893\) −6.30458 −0.210975
\(894\) −62.3919 −2.08670
\(895\) −8.17559 −0.273280
\(896\) 3.30614 0.110450
\(897\) 6.81299 0.227479
\(898\) −23.2518 −0.775922
\(899\) 26.5452 0.885331
\(900\) −39.6050 −1.32017
\(901\) −57.5286 −1.91656
\(902\) −0.000925971 0 −3.08314e−5 0
\(903\) 17.2056 0.572568
\(904\) −6.09730 −0.202793
\(905\) −6.95528 −0.231201
\(906\) 68.3152 2.26962
\(907\) −9.64937 −0.320402 −0.160201 0.987084i \(-0.551214\pi\)
−0.160201 + 0.987084i \(0.551214\pi\)
\(908\) 16.7265 0.555090
\(909\) −100.715 −3.34052
\(910\) −4.86941 −0.161419
\(911\) −34.3560 −1.13827 −0.569133 0.822246i \(-0.692721\pi\)
−0.569133 + 0.822246i \(0.692721\pi\)
\(912\) 7.40225 0.245113
\(913\) 1.98265 0.0656160
\(914\) 20.4390 0.676061
\(915\) −12.8141 −0.423622
\(916\) 10.2285 0.337958
\(917\) −10.8456 −0.358154
\(918\) −99.1111 −3.27115
\(919\) 56.4235 1.86124 0.930620 0.365988i \(-0.119269\pi\)
0.930620 + 0.365988i \(0.119269\pi\)
\(920\) 0.261013 0.00860535
\(921\) 95.6185 3.15074
\(922\) −20.1844 −0.664737
\(923\) 7.48897 0.246503
\(924\) 2.40734 0.0791957
\(925\) 44.0153 1.44721
\(926\) −30.3322 −0.996778
\(927\) −16.4538 −0.540415
\(928\) −4.63290 −0.152082
\(929\) −9.41055 −0.308750 −0.154375 0.988012i \(-0.549336\pi\)
−0.154375 + 0.988012i \(0.549336\pi\)
\(930\) 8.35076 0.273832
\(931\) −8.68125 −0.284517
\(932\) 12.9318 0.423595
\(933\) −19.6523 −0.643386
\(934\) −9.47525 −0.310040
\(935\) −0.533981 −0.0174631
\(936\) −27.8819 −0.911349
\(937\) −4.82246 −0.157543 −0.0787714 0.996893i \(-0.525100\pi\)
−0.0787714 + 0.996893i \(0.525100\pi\)
\(938\) 8.79612 0.287203
\(939\) 30.3511 0.990470
\(940\) 1.24133 0.0404876
\(941\) 25.9614 0.846319 0.423159 0.906055i \(-0.360921\pi\)
0.423159 + 0.906055i \(0.360921\pi\)
\(942\) −67.8172 −2.20960
\(943\) 0.00255812 8.33039e−5 0
\(944\) −7.46902 −0.243096
\(945\) −25.2124 −0.820158
\(946\) 0.337361 0.0109686
\(947\) 0.571460 0.0185700 0.00928498 0.999957i \(-0.497044\pi\)
0.00928498 + 0.999957i \(0.497044\pi\)
\(948\) 16.2694 0.528406
\(949\) 28.4595 0.923836
\(950\) 10.6256 0.344740
\(951\) −42.2441 −1.36986
\(952\) 18.6857 0.605606
\(953\) −35.9909 −1.16586 −0.582929 0.812523i \(-0.698094\pi\)
−0.582929 + 0.812523i \(0.698094\pi\)
\(954\) −83.7957 −2.71298
\(955\) −2.67238 −0.0864761
\(956\) −8.86048 −0.286568
\(957\) −3.37341 −0.109047
\(958\) 12.5310 0.404858
\(959\) 72.4570 2.33976
\(960\) −1.45745 −0.0470390
\(961\) 1.82953 0.0590172
\(962\) 30.9867 0.999052
\(963\) −75.8377 −2.44384
\(964\) −19.5027 −0.628139
\(965\) −8.70633 −0.280267
\(966\) −6.65062 −0.213980
\(967\) −8.34128 −0.268238 −0.134119 0.990965i \(-0.542820\pi\)
−0.134119 + 0.990965i \(0.542820\pi\)
\(968\) −10.9528 −0.352036
\(969\) 41.8361 1.34397
\(970\) 7.70479 0.247386
\(971\) −26.8460 −0.861531 −0.430765 0.902464i \(-0.641756\pi\)
−0.430765 + 0.902464i \(0.641756\pi\)
\(972\) −61.5925 −1.97558
\(973\) 51.2616 1.64337
\(974\) −0.847982 −0.0271711
\(975\) −54.6083 −1.74886
\(976\) 8.79215 0.281430
\(977\) 35.5455 1.13720 0.568600 0.822614i \(-0.307485\pi\)
0.568600 + 0.822614i \(0.307485\pi\)
\(978\) 4.79689 0.153388
\(979\) −2.84630 −0.0909681
\(980\) 1.70928 0.0546008
\(981\) 64.2009 2.04978
\(982\) 22.6153 0.721683
\(983\) −39.2267 −1.25114 −0.625569 0.780169i \(-0.715133\pi\)
−0.625569 + 0.780169i \(0.715133\pi\)
\(984\) −0.0142841 −0.000455360 0
\(985\) −5.61127 −0.178790
\(986\) −26.1843 −0.833878
\(987\) −31.6290 −1.00676
\(988\) 7.48041 0.237983
\(989\) −0.932007 −0.0296361
\(990\) −0.777792 −0.0247199
\(991\) 62.8115 1.99527 0.997636 0.0687227i \(-0.0218924\pi\)
0.997636 + 0.0687227i \(0.0218924\pi\)
\(992\) −5.72971 −0.181918
\(993\) −67.0998 −2.12935
\(994\) −7.31049 −0.231875
\(995\) −4.55568 −0.144425
\(996\) 30.5844 0.969104
\(997\) −3.46020 −0.109586 −0.0547928 0.998498i \(-0.517450\pi\)
−0.0547928 + 0.998498i \(0.517450\pi\)
\(998\) −18.9290 −0.599188
\(999\) 160.440 5.07610
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 6046.2.a.e.1.1 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6046.2.a.e.1.1 56 1.1 even 1 trivial