Properties

Label 5610.2.a.bx.1.2
Level $5610$
Weight $2$
Character 5610.1
Self dual yes
Analytic conductor $44.796$
Analytic rank $0$
Dimension $2$
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] = [5610,2,Mod(1,5610)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5610, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5610.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5610 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5610.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: \(44.7960755339\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.37228\) of defining polynomial
Character \(\chi\) \(=\) 5610.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +3.37228 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +1.00000 q^{11} +1.00000 q^{12} +4.74456 q^{13} +3.37228 q^{14} +1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} +1.00000 q^{20} +3.37228 q^{21} +1.00000 q^{22} -3.37228 q^{23} +1.00000 q^{24} +1.00000 q^{25} +4.74456 q^{26} +1.00000 q^{27} +3.37228 q^{28} +2.62772 q^{29} +1.00000 q^{30} +4.62772 q^{31} +1.00000 q^{32} +1.00000 q^{33} +1.00000 q^{34} +3.37228 q^{35} +1.00000 q^{36} -10.0000 q^{37} +4.00000 q^{38} +4.74456 q^{39} +1.00000 q^{40} -12.7446 q^{41} +3.37228 q^{42} +7.37228 q^{43} +1.00000 q^{44} +1.00000 q^{45} -3.37228 q^{46} -13.4891 q^{47} +1.00000 q^{48} +4.37228 q^{49} +1.00000 q^{50} +1.00000 q^{51} +4.74456 q^{52} -0.744563 q^{53} +1.00000 q^{54} +1.00000 q^{55} +3.37228 q^{56} +4.00000 q^{57} +2.62772 q^{58} -10.7446 q^{59} +1.00000 q^{60} -2.00000 q^{61} +4.62772 q^{62} +3.37228 q^{63} +1.00000 q^{64} +4.74456 q^{65} +1.00000 q^{66} +4.00000 q^{67} +1.00000 q^{68} -3.37228 q^{69} +3.37228 q^{70} +1.25544 q^{71} +1.00000 q^{72} -6.00000 q^{73} -10.0000 q^{74} +1.00000 q^{75} +4.00000 q^{76} +3.37228 q^{77} +4.74456 q^{78} -13.4891 q^{79} +1.00000 q^{80} +1.00000 q^{81} -12.7446 q^{82} -12.0000 q^{83} +3.37228 q^{84} +1.00000 q^{85} +7.37228 q^{86} +2.62772 q^{87} +1.00000 q^{88} +16.7446 q^{89} +1.00000 q^{90} +16.0000 q^{91} -3.37228 q^{92} +4.62772 q^{93} -13.4891 q^{94} +4.00000 q^{95} +1.00000 q^{96} -9.37228 q^{97} +4.37228 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} + q^{7} + 2 q^{8} + 2 q^{9} + 2 q^{10} + 2 q^{11} + 2 q^{12} - 2 q^{13} + q^{14} + 2 q^{15} + 2 q^{16} + 2 q^{17} + 2 q^{18} + 8 q^{19} + 2 q^{20}+ \cdots + 2 q^{99}+O(q^{100}) \) 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
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 1.00000 0.408248
\(7\) 3.37228 1.27460 0.637301 0.770615i \(-0.280051\pi\)
0.637301 + 0.770615i \(0.280051\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 1.00000 0.301511
\(12\) 1.00000 0.288675
\(13\) 4.74456 1.31590 0.657952 0.753059i \(-0.271423\pi\)
0.657952 + 0.753059i \(0.271423\pi\)
\(14\) 3.37228 0.901280
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 1.00000 0.242536
\(18\) 1.00000 0.235702
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 1.00000 0.223607
\(21\) 3.37228 0.735892
\(22\) 1.00000 0.213201
\(23\) −3.37228 −0.703169 −0.351585 0.936156i \(-0.614357\pi\)
−0.351585 + 0.936156i \(0.614357\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 4.74456 0.930485
\(27\) 1.00000 0.192450
\(28\) 3.37228 0.637301
\(29\) 2.62772 0.487955 0.243978 0.969781i \(-0.421548\pi\)
0.243978 + 0.969781i \(0.421548\pi\)
\(30\) 1.00000 0.182574
\(31\) 4.62772 0.831163 0.415581 0.909556i \(-0.363578\pi\)
0.415581 + 0.909556i \(0.363578\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.00000 0.174078
\(34\) 1.00000 0.171499
\(35\) 3.37228 0.570020
\(36\) 1.00000 0.166667
\(37\) −10.0000 −1.64399 −0.821995 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 4.00000 0.648886
\(39\) 4.74456 0.759738
\(40\) 1.00000 0.158114
\(41\) −12.7446 −1.99037 −0.995183 0.0980332i \(-0.968745\pi\)
−0.995183 + 0.0980332i \(0.968745\pi\)
\(42\) 3.37228 0.520354
\(43\) 7.37228 1.12426 0.562131 0.827048i \(-0.309982\pi\)
0.562131 + 0.827048i \(0.309982\pi\)
\(44\) 1.00000 0.150756
\(45\) 1.00000 0.149071
\(46\) −3.37228 −0.497216
\(47\) −13.4891 −1.96759 −0.983796 0.179294i \(-0.942619\pi\)
−0.983796 + 0.179294i \(0.942619\pi\)
\(48\) 1.00000 0.144338
\(49\) 4.37228 0.624612
\(50\) 1.00000 0.141421
\(51\) 1.00000 0.140028
\(52\) 4.74456 0.657952
\(53\) −0.744563 −0.102274 −0.0511368 0.998692i \(-0.516284\pi\)
−0.0511368 + 0.998692i \(0.516284\pi\)
\(54\) 1.00000 0.136083
\(55\) 1.00000 0.134840
\(56\) 3.37228 0.450640
\(57\) 4.00000 0.529813
\(58\) 2.62772 0.345036
\(59\) −10.7446 −1.39882 −0.699411 0.714719i \(-0.746554\pi\)
−0.699411 + 0.714719i \(0.746554\pi\)
\(60\) 1.00000 0.129099
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 4.62772 0.587721
\(63\) 3.37228 0.424868
\(64\) 1.00000 0.125000
\(65\) 4.74456 0.588491
\(66\) 1.00000 0.123091
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 1.00000 0.121268
\(69\) −3.37228 −0.405975
\(70\) 3.37228 0.403065
\(71\) 1.25544 0.148993 0.0744965 0.997221i \(-0.476265\pi\)
0.0744965 + 0.997221i \(0.476265\pi\)
\(72\) 1.00000 0.117851
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) −10.0000 −1.16248
\(75\) 1.00000 0.115470
\(76\) 4.00000 0.458831
\(77\) 3.37228 0.384307
\(78\) 4.74456 0.537216
\(79\) −13.4891 −1.51765 −0.758823 0.651297i \(-0.774225\pi\)
−0.758823 + 0.651297i \(0.774225\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −12.7446 −1.40740
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 3.37228 0.367946
\(85\) 1.00000 0.108465
\(86\) 7.37228 0.794974
\(87\) 2.62772 0.281721
\(88\) 1.00000 0.106600
\(89\) 16.7446 1.77492 0.887460 0.460885i \(-0.152468\pi\)
0.887460 + 0.460885i \(0.152468\pi\)
\(90\) 1.00000 0.105409
\(91\) 16.0000 1.67726
\(92\) −3.37228 −0.351585
\(93\) 4.62772 0.479872
\(94\) −13.4891 −1.39130
\(95\) 4.00000 0.410391
\(96\) 1.00000 0.102062
\(97\) −9.37228 −0.951611 −0.475805 0.879551i \(-0.657843\pi\)
−0.475805 + 0.879551i \(0.657843\pi\)
\(98\) 4.37228 0.441667
\(99\) 1.00000 0.100504
\(100\) 1.00000 0.100000
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 1.00000 0.0990148
\(103\) −3.37228 −0.332281 −0.166140 0.986102i \(-0.553130\pi\)
−0.166140 + 0.986102i \(0.553130\pi\)
\(104\) 4.74456 0.465243
\(105\) 3.37228 0.329101
\(106\) −0.744563 −0.0723183
\(107\) 14.1168 1.36473 0.682363 0.731013i \(-0.260952\pi\)
0.682363 + 0.731013i \(0.260952\pi\)
\(108\) 1.00000 0.0962250
\(109\) −8.74456 −0.837577 −0.418789 0.908084i \(-0.637545\pi\)
−0.418789 + 0.908084i \(0.637545\pi\)
\(110\) 1.00000 0.0953463
\(111\) −10.0000 −0.949158
\(112\) 3.37228 0.318651
\(113\) −11.4891 −1.08081 −0.540403 0.841406i \(-0.681728\pi\)
−0.540403 + 0.841406i \(0.681728\pi\)
\(114\) 4.00000 0.374634
\(115\) −3.37228 −0.314467
\(116\) 2.62772 0.243978
\(117\) 4.74456 0.438635
\(118\) −10.7446 −0.989117
\(119\) 3.37228 0.309137
\(120\) 1.00000 0.0912871
\(121\) 1.00000 0.0909091
\(122\) −2.00000 −0.181071
\(123\) −12.7446 −1.14914
\(124\) 4.62772 0.415581
\(125\) 1.00000 0.0894427
\(126\) 3.37228 0.300427
\(127\) 20.2337 1.79545 0.897725 0.440557i \(-0.145219\pi\)
0.897725 + 0.440557i \(0.145219\pi\)
\(128\) 1.00000 0.0883883
\(129\) 7.37228 0.649093
\(130\) 4.74456 0.416126
\(131\) −9.48913 −0.829069 −0.414534 0.910034i \(-0.636056\pi\)
−0.414534 + 0.910034i \(0.636056\pi\)
\(132\) 1.00000 0.0870388
\(133\) 13.4891 1.16966
\(134\) 4.00000 0.345547
\(135\) 1.00000 0.0860663
\(136\) 1.00000 0.0857493
\(137\) −14.8614 −1.26970 −0.634848 0.772637i \(-0.718937\pi\)
−0.634848 + 0.772637i \(0.718937\pi\)
\(138\) −3.37228 −0.287068
\(139\) 14.1168 1.19738 0.598688 0.800983i \(-0.295689\pi\)
0.598688 + 0.800983i \(0.295689\pi\)
\(140\) 3.37228 0.285010
\(141\) −13.4891 −1.13599
\(142\) 1.25544 0.105354
\(143\) 4.74456 0.396760
\(144\) 1.00000 0.0833333
\(145\) 2.62772 0.218220
\(146\) −6.00000 −0.496564
\(147\) 4.37228 0.360620
\(148\) −10.0000 −0.821995
\(149\) −0.744563 −0.0609969 −0.0304985 0.999535i \(-0.509709\pi\)
−0.0304985 + 0.999535i \(0.509709\pi\)
\(150\) 1.00000 0.0816497
\(151\) 21.4891 1.74876 0.874380 0.485242i \(-0.161268\pi\)
0.874380 + 0.485242i \(0.161268\pi\)
\(152\) 4.00000 0.324443
\(153\) 1.00000 0.0808452
\(154\) 3.37228 0.271746
\(155\) 4.62772 0.371707
\(156\) 4.74456 0.379869
\(157\) 11.4891 0.916932 0.458466 0.888712i \(-0.348399\pi\)
0.458466 + 0.888712i \(0.348399\pi\)
\(158\) −13.4891 −1.07314
\(159\) −0.744563 −0.0590477
\(160\) 1.00000 0.0790569
\(161\) −11.3723 −0.896261
\(162\) 1.00000 0.0785674
\(163\) −7.37228 −0.577442 −0.288721 0.957413i \(-0.593230\pi\)
−0.288721 + 0.957413i \(0.593230\pi\)
\(164\) −12.7446 −0.995183
\(165\) 1.00000 0.0778499
\(166\) −12.0000 −0.931381
\(167\) −12.2337 −0.946671 −0.473336 0.880882i \(-0.656950\pi\)
−0.473336 + 0.880882i \(0.656950\pi\)
\(168\) 3.37228 0.260177
\(169\) 9.51087 0.731606
\(170\) 1.00000 0.0766965
\(171\) 4.00000 0.305888
\(172\) 7.37228 0.562131
\(173\) 11.4891 0.873502 0.436751 0.899582i \(-0.356129\pi\)
0.436751 + 0.899582i \(0.356129\pi\)
\(174\) 2.62772 0.199207
\(175\) 3.37228 0.254921
\(176\) 1.00000 0.0753778
\(177\) −10.7446 −0.807611
\(178\) 16.7446 1.25506
\(179\) −18.7446 −1.40103 −0.700517 0.713636i \(-0.747047\pi\)
−0.700517 + 0.713636i \(0.747047\pi\)
\(180\) 1.00000 0.0745356
\(181\) −14.6277 −1.08727 −0.543635 0.839322i \(-0.682952\pi\)
−0.543635 + 0.839322i \(0.682952\pi\)
\(182\) 16.0000 1.18600
\(183\) −2.00000 −0.147844
\(184\) −3.37228 −0.248608
\(185\) −10.0000 −0.735215
\(186\) 4.62772 0.339321
\(187\) 1.00000 0.0731272
\(188\) −13.4891 −0.983796
\(189\) 3.37228 0.245297
\(190\) 4.00000 0.290191
\(191\) −4.62772 −0.334850 −0.167425 0.985885i \(-0.553545\pi\)
−0.167425 + 0.985885i \(0.553545\pi\)
\(192\) 1.00000 0.0721688
\(193\) −0.510875 −0.0367736 −0.0183868 0.999831i \(-0.505853\pi\)
−0.0183868 + 0.999831i \(0.505853\pi\)
\(194\) −9.37228 −0.672891
\(195\) 4.74456 0.339765
\(196\) 4.37228 0.312306
\(197\) 26.2337 1.86907 0.934536 0.355867i \(-0.115815\pi\)
0.934536 + 0.355867i \(0.115815\pi\)
\(198\) 1.00000 0.0710669
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 1.00000 0.0707107
\(201\) 4.00000 0.282138
\(202\) 6.00000 0.422159
\(203\) 8.86141 0.621949
\(204\) 1.00000 0.0700140
\(205\) −12.7446 −0.890119
\(206\) −3.37228 −0.234958
\(207\) −3.37228 −0.234390
\(208\) 4.74456 0.328976
\(209\) 4.00000 0.276686
\(210\) 3.37228 0.232710
\(211\) −14.1168 −0.971844 −0.485922 0.874002i \(-0.661516\pi\)
−0.485922 + 0.874002i \(0.661516\pi\)
\(212\) −0.744563 −0.0511368
\(213\) 1.25544 0.0860212
\(214\) 14.1168 0.965008
\(215\) 7.37228 0.502785
\(216\) 1.00000 0.0680414
\(217\) 15.6060 1.05940
\(218\) −8.74456 −0.592257
\(219\) −6.00000 −0.405442
\(220\) 1.00000 0.0674200
\(221\) 4.74456 0.319154
\(222\) −10.0000 −0.671156
\(223\) −11.3723 −0.761544 −0.380772 0.924669i \(-0.624342\pi\)
−0.380772 + 0.924669i \(0.624342\pi\)
\(224\) 3.37228 0.225320
\(225\) 1.00000 0.0666667
\(226\) −11.4891 −0.764245
\(227\) 6.11684 0.405989 0.202995 0.979180i \(-0.434933\pi\)
0.202995 + 0.979180i \(0.434933\pi\)
\(228\) 4.00000 0.264906
\(229\) −3.25544 −0.215125 −0.107563 0.994198i \(-0.534305\pi\)
−0.107563 + 0.994198i \(0.534305\pi\)
\(230\) −3.37228 −0.222362
\(231\) 3.37228 0.221880
\(232\) 2.62772 0.172518
\(233\) −14.8614 −0.973603 −0.486802 0.873513i \(-0.661837\pi\)
−0.486802 + 0.873513i \(0.661837\pi\)
\(234\) 4.74456 0.310162
\(235\) −13.4891 −0.879934
\(236\) −10.7446 −0.699411
\(237\) −13.4891 −0.876213
\(238\) 3.37228 0.218593
\(239\) 20.2337 1.30881 0.654404 0.756145i \(-0.272919\pi\)
0.654404 + 0.756145i \(0.272919\pi\)
\(240\) 1.00000 0.0645497
\(241\) −9.37228 −0.603722 −0.301861 0.953352i \(-0.597608\pi\)
−0.301861 + 0.953352i \(0.597608\pi\)
\(242\) 1.00000 0.0642824
\(243\) 1.00000 0.0641500
\(244\) −2.00000 −0.128037
\(245\) 4.37228 0.279335
\(246\) −12.7446 −0.812564
\(247\) 18.9783 1.20756
\(248\) 4.62772 0.293860
\(249\) −12.0000 −0.760469
\(250\) 1.00000 0.0632456
\(251\) −6.51087 −0.410963 −0.205481 0.978661i \(-0.565876\pi\)
−0.205481 + 0.978661i \(0.565876\pi\)
\(252\) 3.37228 0.212434
\(253\) −3.37228 −0.212014
\(254\) 20.2337 1.26957
\(255\) 1.00000 0.0626224
\(256\) 1.00000 0.0625000
\(257\) 13.3723 0.834140 0.417070 0.908874i \(-0.363057\pi\)
0.417070 + 0.908874i \(0.363057\pi\)
\(258\) 7.37228 0.458978
\(259\) −33.7228 −2.09543
\(260\) 4.74456 0.294245
\(261\) 2.62772 0.162652
\(262\) −9.48913 −0.586240
\(263\) 0.861407 0.0531166 0.0265583 0.999647i \(-0.491545\pi\)
0.0265583 + 0.999647i \(0.491545\pi\)
\(264\) 1.00000 0.0615457
\(265\) −0.744563 −0.0457381
\(266\) 13.4891 0.827071
\(267\) 16.7446 1.02475
\(268\) 4.00000 0.244339
\(269\) 14.0000 0.853595 0.426798 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) 1.00000 0.0608581
\(271\) −2.11684 −0.128589 −0.0642946 0.997931i \(-0.520480\pi\)
−0.0642946 + 0.997931i \(0.520480\pi\)
\(272\) 1.00000 0.0606339
\(273\) 16.0000 0.968364
\(274\) −14.8614 −0.897810
\(275\) 1.00000 0.0603023
\(276\) −3.37228 −0.202987
\(277\) 22.0000 1.32185 0.660926 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) 14.1168 0.846672
\(279\) 4.62772 0.277054
\(280\) 3.37228 0.201532
\(281\) 18.8614 1.12518 0.562589 0.826737i \(-0.309806\pi\)
0.562589 + 0.826737i \(0.309806\pi\)
\(282\) −13.4891 −0.803266
\(283\) 18.7446 1.11425 0.557124 0.830429i \(-0.311905\pi\)
0.557124 + 0.830429i \(0.311905\pi\)
\(284\) 1.25544 0.0744965
\(285\) 4.00000 0.236940
\(286\) 4.74456 0.280552
\(287\) −42.9783 −2.53693
\(288\) 1.00000 0.0589256
\(289\) 1.00000 0.0588235
\(290\) 2.62772 0.154305
\(291\) −9.37228 −0.549413
\(292\) −6.00000 −0.351123
\(293\) 30.8614 1.80294 0.901471 0.432839i \(-0.142488\pi\)
0.901471 + 0.432839i \(0.142488\pi\)
\(294\) 4.37228 0.254997
\(295\) −10.7446 −0.625573
\(296\) −10.0000 −0.581238
\(297\) 1.00000 0.0580259
\(298\) −0.744563 −0.0431314
\(299\) −16.0000 −0.925304
\(300\) 1.00000 0.0577350
\(301\) 24.8614 1.43299
\(302\) 21.4891 1.23656
\(303\) 6.00000 0.344691
\(304\) 4.00000 0.229416
\(305\) −2.00000 −0.114520
\(306\) 1.00000 0.0571662
\(307\) 4.00000 0.228292 0.114146 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) 3.37228 0.192154
\(309\) −3.37228 −0.191842
\(310\) 4.62772 0.262837
\(311\) −5.48913 −0.311260 −0.155630 0.987815i \(-0.549741\pi\)
−0.155630 + 0.987815i \(0.549741\pi\)
\(312\) 4.74456 0.268608
\(313\) 18.8614 1.06611 0.533055 0.846081i \(-0.321044\pi\)
0.533055 + 0.846081i \(0.321044\pi\)
\(314\) 11.4891 0.648369
\(315\) 3.37228 0.190007
\(316\) −13.4891 −0.758823
\(317\) 0.116844 0.00656261 0.00328131 0.999995i \(-0.498956\pi\)
0.00328131 + 0.999995i \(0.498956\pi\)
\(318\) −0.744563 −0.0417530
\(319\) 2.62772 0.147124
\(320\) 1.00000 0.0559017
\(321\) 14.1168 0.787925
\(322\) −11.3723 −0.633752
\(323\) 4.00000 0.222566
\(324\) 1.00000 0.0555556
\(325\) 4.74456 0.263181
\(326\) −7.37228 −0.408313
\(327\) −8.74456 −0.483575
\(328\) −12.7446 −0.703701
\(329\) −45.4891 −2.50790
\(330\) 1.00000 0.0550482
\(331\) 18.3505 1.00864 0.504318 0.863518i \(-0.331744\pi\)
0.504318 + 0.863518i \(0.331744\pi\)
\(332\) −12.0000 −0.658586
\(333\) −10.0000 −0.547997
\(334\) −12.2337 −0.669398
\(335\) 4.00000 0.218543
\(336\) 3.37228 0.183973
\(337\) −24.9783 −1.36065 −0.680326 0.732910i \(-0.738162\pi\)
−0.680326 + 0.732910i \(0.738162\pi\)
\(338\) 9.51087 0.517323
\(339\) −11.4891 −0.624004
\(340\) 1.00000 0.0542326
\(341\) 4.62772 0.250605
\(342\) 4.00000 0.216295
\(343\) −8.86141 −0.478471
\(344\) 7.37228 0.397487
\(345\) −3.37228 −0.181558
\(346\) 11.4891 0.617659
\(347\) 22.9783 1.23354 0.616769 0.787145i \(-0.288441\pi\)
0.616769 + 0.787145i \(0.288441\pi\)
\(348\) 2.62772 0.140861
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 3.37228 0.180256
\(351\) 4.74456 0.253246
\(352\) 1.00000 0.0533002
\(353\) −22.8614 −1.21679 −0.608395 0.793634i \(-0.708186\pi\)
−0.608395 + 0.793634i \(0.708186\pi\)
\(354\) −10.7446 −0.571067
\(355\) 1.25544 0.0666317
\(356\) 16.7446 0.887460
\(357\) 3.37228 0.178480
\(358\) −18.7446 −0.990681
\(359\) −28.2337 −1.49012 −0.745059 0.666999i \(-0.767578\pi\)
−0.745059 + 0.666999i \(0.767578\pi\)
\(360\) 1.00000 0.0527046
\(361\) −3.00000 −0.157895
\(362\) −14.6277 −0.768816
\(363\) 1.00000 0.0524864
\(364\) 16.0000 0.838628
\(365\) −6.00000 −0.314054
\(366\) −2.00000 −0.104542
\(367\) −22.7446 −1.18726 −0.593628 0.804739i \(-0.702305\pi\)
−0.593628 + 0.804739i \(0.702305\pi\)
\(368\) −3.37228 −0.175792
\(369\) −12.7446 −0.663455
\(370\) −10.0000 −0.519875
\(371\) −2.51087 −0.130358
\(372\) 4.62772 0.239936
\(373\) −23.4891 −1.21622 −0.608110 0.793852i \(-0.708072\pi\)
−0.608110 + 0.793852i \(0.708072\pi\)
\(374\) 1.00000 0.0517088
\(375\) 1.00000 0.0516398
\(376\) −13.4891 −0.695649
\(377\) 12.4674 0.642103
\(378\) 3.37228 0.173451
\(379\) 28.0000 1.43826 0.719132 0.694874i \(-0.244540\pi\)
0.719132 + 0.694874i \(0.244540\pi\)
\(380\) 4.00000 0.205196
\(381\) 20.2337 1.03660
\(382\) −4.62772 −0.236775
\(383\) 2.51087 0.128300 0.0641499 0.997940i \(-0.479566\pi\)
0.0641499 + 0.997940i \(0.479566\pi\)
\(384\) 1.00000 0.0510310
\(385\) 3.37228 0.171867
\(386\) −0.510875 −0.0260028
\(387\) 7.37228 0.374754
\(388\) −9.37228 −0.475805
\(389\) −34.4674 −1.74757 −0.873783 0.486317i \(-0.838340\pi\)
−0.873783 + 0.486317i \(0.838340\pi\)
\(390\) 4.74456 0.240250
\(391\) −3.37228 −0.170544
\(392\) 4.37228 0.220834
\(393\) −9.48913 −0.478663
\(394\) 26.2337 1.32163
\(395\) −13.4891 −0.678712
\(396\) 1.00000 0.0502519
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) −8.00000 −0.401004
\(399\) 13.4891 0.675301
\(400\) 1.00000 0.0500000
\(401\) −9.37228 −0.468029 −0.234015 0.972233i \(-0.575186\pi\)
−0.234015 + 0.972233i \(0.575186\pi\)
\(402\) 4.00000 0.199502
\(403\) 21.9565 1.09373
\(404\) 6.00000 0.298511
\(405\) 1.00000 0.0496904
\(406\) 8.86141 0.439784
\(407\) −10.0000 −0.495682
\(408\) 1.00000 0.0495074
\(409\) 30.2337 1.49496 0.747480 0.664285i \(-0.231264\pi\)
0.747480 + 0.664285i \(0.231264\pi\)
\(410\) −12.7446 −0.629409
\(411\) −14.8614 −0.733059
\(412\) −3.37228 −0.166140
\(413\) −36.2337 −1.78294
\(414\) −3.37228 −0.165739
\(415\) −12.0000 −0.589057
\(416\) 4.74456 0.232621
\(417\) 14.1168 0.691305
\(418\) 4.00000 0.195646
\(419\) 24.6277 1.20314 0.601571 0.798819i \(-0.294542\pi\)
0.601571 + 0.798819i \(0.294542\pi\)
\(420\) 3.37228 0.164550
\(421\) −16.7446 −0.816080 −0.408040 0.912964i \(-0.633788\pi\)
−0.408040 + 0.912964i \(0.633788\pi\)
\(422\) −14.1168 −0.687197
\(423\) −13.4891 −0.655864
\(424\) −0.744563 −0.0361592
\(425\) 1.00000 0.0485071
\(426\) 1.25544 0.0608261
\(427\) −6.74456 −0.326392
\(428\) 14.1168 0.682363
\(429\) 4.74456 0.229070
\(430\) 7.37228 0.355523
\(431\) 2.11684 0.101965 0.0509824 0.998700i \(-0.483765\pi\)
0.0509824 + 0.998700i \(0.483765\pi\)
\(432\) 1.00000 0.0481125
\(433\) 27.2554 1.30981 0.654906 0.755710i \(-0.272708\pi\)
0.654906 + 0.755710i \(0.272708\pi\)
\(434\) 15.6060 0.749110
\(435\) 2.62772 0.125989
\(436\) −8.74456 −0.418789
\(437\) −13.4891 −0.645272
\(438\) −6.00000 −0.286691
\(439\) −28.2337 −1.34752 −0.673760 0.738950i \(-0.735322\pi\)
−0.673760 + 0.738950i \(0.735322\pi\)
\(440\) 1.00000 0.0476731
\(441\) 4.37228 0.208204
\(442\) 4.74456 0.225676
\(443\) −22.1168 −1.05080 −0.525401 0.850854i \(-0.676085\pi\)
−0.525401 + 0.850854i \(0.676085\pi\)
\(444\) −10.0000 −0.474579
\(445\) 16.7446 0.793768
\(446\) −11.3723 −0.538493
\(447\) −0.744563 −0.0352166
\(448\) 3.37228 0.159325
\(449\) 29.3723 1.38616 0.693082 0.720859i \(-0.256252\pi\)
0.693082 + 0.720859i \(0.256252\pi\)
\(450\) 1.00000 0.0471405
\(451\) −12.7446 −0.600118
\(452\) −11.4891 −0.540403
\(453\) 21.4891 1.00965
\(454\) 6.11684 0.287078
\(455\) 16.0000 0.750092
\(456\) 4.00000 0.187317
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) −3.25544 −0.152117
\(459\) 1.00000 0.0466760
\(460\) −3.37228 −0.157233
\(461\) 0.510875 0.0237938 0.0118969 0.999929i \(-0.496213\pi\)
0.0118969 + 0.999929i \(0.496213\pi\)
\(462\) 3.37228 0.156893
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) 2.62772 0.121989
\(465\) 4.62772 0.214605
\(466\) −14.8614 −0.688441
\(467\) 30.9783 1.43350 0.716751 0.697329i \(-0.245628\pi\)
0.716751 + 0.697329i \(0.245628\pi\)
\(468\) 4.74456 0.219317
\(469\) 13.4891 0.622870
\(470\) −13.4891 −0.622207
\(471\) 11.4891 0.529391
\(472\) −10.7446 −0.494559
\(473\) 7.37228 0.338978
\(474\) −13.4891 −0.619576
\(475\) 4.00000 0.183533
\(476\) 3.37228 0.154568
\(477\) −0.744563 −0.0340912
\(478\) 20.2337 0.925467
\(479\) 35.8397 1.63756 0.818778 0.574110i \(-0.194652\pi\)
0.818778 + 0.574110i \(0.194652\pi\)
\(480\) 1.00000 0.0456435
\(481\) −47.4456 −2.16333
\(482\) −9.37228 −0.426896
\(483\) −11.3723 −0.517457
\(484\) 1.00000 0.0454545
\(485\) −9.37228 −0.425573
\(486\) 1.00000 0.0453609
\(487\) 1.25544 0.0568893 0.0284446 0.999595i \(-0.490945\pi\)
0.0284446 + 0.999595i \(0.490945\pi\)
\(488\) −2.00000 −0.0905357
\(489\) −7.37228 −0.333386
\(490\) 4.37228 0.197520
\(491\) −6.51087 −0.293832 −0.146916 0.989149i \(-0.546935\pi\)
−0.146916 + 0.989149i \(0.546935\pi\)
\(492\) −12.7446 −0.574569
\(493\) 2.62772 0.118346
\(494\) 18.9783 0.853872
\(495\) 1.00000 0.0449467
\(496\) 4.62772 0.207791
\(497\) 4.23369 0.189907
\(498\) −12.0000 −0.537733
\(499\) 1.48913 0.0666624 0.0333312 0.999444i \(-0.489388\pi\)
0.0333312 + 0.999444i \(0.489388\pi\)
\(500\) 1.00000 0.0447214
\(501\) −12.2337 −0.546561
\(502\) −6.51087 −0.290595
\(503\) 2.97825 0.132794 0.0663968 0.997793i \(-0.478850\pi\)
0.0663968 + 0.997793i \(0.478850\pi\)
\(504\) 3.37228 0.150213
\(505\) 6.00000 0.266996
\(506\) −3.37228 −0.149916
\(507\) 9.51087 0.422393
\(508\) 20.2337 0.897725
\(509\) 11.4891 0.509247 0.254623 0.967040i \(-0.418049\pi\)
0.254623 + 0.967040i \(0.418049\pi\)
\(510\) 1.00000 0.0442807
\(511\) −20.2337 −0.895086
\(512\) 1.00000 0.0441942
\(513\) 4.00000 0.176604
\(514\) 13.3723 0.589826
\(515\) −3.37228 −0.148600
\(516\) 7.37228 0.324547
\(517\) −13.4891 −0.593251
\(518\) −33.7228 −1.48170
\(519\) 11.4891 0.504317
\(520\) 4.74456 0.208063
\(521\) 20.9783 0.919074 0.459537 0.888159i \(-0.348015\pi\)
0.459537 + 0.888159i \(0.348015\pi\)
\(522\) 2.62772 0.115012
\(523\) −24.6277 −1.07689 −0.538447 0.842659i \(-0.680989\pi\)
−0.538447 + 0.842659i \(0.680989\pi\)
\(524\) −9.48913 −0.414534
\(525\) 3.37228 0.147178
\(526\) 0.861407 0.0375591
\(527\) 4.62772 0.201587
\(528\) 1.00000 0.0435194
\(529\) −11.6277 −0.505553
\(530\) −0.744563 −0.0323417
\(531\) −10.7446 −0.466274
\(532\) 13.4891 0.584828
\(533\) −60.4674 −2.61913
\(534\) 16.7446 0.724608
\(535\) 14.1168 0.610324
\(536\) 4.00000 0.172774
\(537\) −18.7446 −0.808888
\(538\) 14.0000 0.603583
\(539\) 4.37228 0.188327
\(540\) 1.00000 0.0430331
\(541\) −6.23369 −0.268007 −0.134004 0.990981i \(-0.542783\pi\)
−0.134004 + 0.990981i \(0.542783\pi\)
\(542\) −2.11684 −0.0909262
\(543\) −14.6277 −0.627735
\(544\) 1.00000 0.0428746
\(545\) −8.74456 −0.374576
\(546\) 16.0000 0.684737
\(547\) 8.23369 0.352047 0.176024 0.984386i \(-0.443677\pi\)
0.176024 + 0.984386i \(0.443677\pi\)
\(548\) −14.8614 −0.634848
\(549\) −2.00000 −0.0853579
\(550\) 1.00000 0.0426401
\(551\) 10.5109 0.447778
\(552\) −3.37228 −0.143534
\(553\) −45.4891 −1.93439
\(554\) 22.0000 0.934690
\(555\) −10.0000 −0.424476
\(556\) 14.1168 0.598688
\(557\) −24.3505 −1.03177 −0.515883 0.856659i \(-0.672536\pi\)
−0.515883 + 0.856659i \(0.672536\pi\)
\(558\) 4.62772 0.195907
\(559\) 34.9783 1.47942
\(560\) 3.37228 0.142505
\(561\) 1.00000 0.0422200
\(562\) 18.8614 0.795620
\(563\) −12.0000 −0.505740 −0.252870 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) −13.4891 −0.567995
\(565\) −11.4891 −0.483351
\(566\) 18.7446 0.787893
\(567\) 3.37228 0.141623
\(568\) 1.25544 0.0526770
\(569\) 20.9783 0.879454 0.439727 0.898131i \(-0.355075\pi\)
0.439727 + 0.898131i \(0.355075\pi\)
\(570\) 4.00000 0.167542
\(571\) 22.9783 0.961610 0.480805 0.876828i \(-0.340345\pi\)
0.480805 + 0.876828i \(0.340345\pi\)
\(572\) 4.74456 0.198380
\(573\) −4.62772 −0.193326
\(574\) −42.9783 −1.79388
\(575\) −3.37228 −0.140634
\(576\) 1.00000 0.0416667
\(577\) 47.4891 1.97700 0.988499 0.151227i \(-0.0483224\pi\)
0.988499 + 0.151227i \(0.0483224\pi\)
\(578\) 1.00000 0.0415945
\(579\) −0.510875 −0.0212312
\(580\) 2.62772 0.109110
\(581\) −40.4674 −1.67887
\(582\) −9.37228 −0.388494
\(583\) −0.744563 −0.0308366
\(584\) −6.00000 −0.248282
\(585\) 4.74456 0.196164
\(586\) 30.8614 1.27487
\(587\) 25.0951 1.03579 0.517893 0.855446i \(-0.326717\pi\)
0.517893 + 0.855446i \(0.326717\pi\)
\(588\) 4.37228 0.180310
\(589\) 18.5109 0.762727
\(590\) −10.7446 −0.442347
\(591\) 26.2337 1.07911
\(592\) −10.0000 −0.410997
\(593\) −0.510875 −0.0209791 −0.0104896 0.999945i \(-0.503339\pi\)
−0.0104896 + 0.999945i \(0.503339\pi\)
\(594\) 1.00000 0.0410305
\(595\) 3.37228 0.138250
\(596\) −0.744563 −0.0304985
\(597\) −8.00000 −0.327418
\(598\) −16.0000 −0.654289
\(599\) −30.3505 −1.24009 −0.620045 0.784567i \(-0.712885\pi\)
−0.620045 + 0.784567i \(0.712885\pi\)
\(600\) 1.00000 0.0408248
\(601\) −8.51087 −0.347166 −0.173583 0.984819i \(-0.555534\pi\)
−0.173583 + 0.984819i \(0.555534\pi\)
\(602\) 24.8614 1.01328
\(603\) 4.00000 0.162893
\(604\) 21.4891 0.874380
\(605\) 1.00000 0.0406558
\(606\) 6.00000 0.243733
\(607\) 16.0000 0.649420 0.324710 0.945814i \(-0.394733\pi\)
0.324710 + 0.945814i \(0.394733\pi\)
\(608\) 4.00000 0.162221
\(609\) 8.86141 0.359082
\(610\) −2.00000 −0.0809776
\(611\) −64.0000 −2.58916
\(612\) 1.00000 0.0404226
\(613\) −47.9565 −1.93694 −0.968472 0.249121i \(-0.919858\pi\)
−0.968472 + 0.249121i \(0.919858\pi\)
\(614\) 4.00000 0.161427
\(615\) −12.7446 −0.513910
\(616\) 3.37228 0.135873
\(617\) 12.5109 0.503669 0.251834 0.967770i \(-0.418966\pi\)
0.251834 + 0.967770i \(0.418966\pi\)
\(618\) −3.37228 −0.135653
\(619\) 16.2337 0.652487 0.326244 0.945286i \(-0.394217\pi\)
0.326244 + 0.945286i \(0.394217\pi\)
\(620\) 4.62772 0.185854
\(621\) −3.37228 −0.135325
\(622\) −5.48913 −0.220094
\(623\) 56.4674 2.26232
\(624\) 4.74456 0.189935
\(625\) 1.00000 0.0400000
\(626\) 18.8614 0.753853
\(627\) 4.00000 0.159745
\(628\) 11.4891 0.458466
\(629\) −10.0000 −0.398726
\(630\) 3.37228 0.134355
\(631\) −12.2337 −0.487015 −0.243508 0.969899i \(-0.578298\pi\)
−0.243508 + 0.969899i \(0.578298\pi\)
\(632\) −13.4891 −0.536569
\(633\) −14.1168 −0.561094
\(634\) 0.116844 0.00464047
\(635\) 20.2337 0.802949
\(636\) −0.744563 −0.0295238
\(637\) 20.7446 0.821929
\(638\) 2.62772 0.104032
\(639\) 1.25544 0.0496643
\(640\) 1.00000 0.0395285
\(641\) −21.1386 −0.834924 −0.417462 0.908694i \(-0.637080\pi\)
−0.417462 + 0.908694i \(0.637080\pi\)
\(642\) 14.1168 0.557147
\(643\) 10.3505 0.408185 0.204093 0.978952i \(-0.434576\pi\)
0.204093 + 0.978952i \(0.434576\pi\)
\(644\) −11.3723 −0.448131
\(645\) 7.37228 0.290283
\(646\) 4.00000 0.157378
\(647\) −25.7228 −1.01127 −0.505634 0.862748i \(-0.668741\pi\)
−0.505634 + 0.862748i \(0.668741\pi\)
\(648\) 1.00000 0.0392837
\(649\) −10.7446 −0.421761
\(650\) 4.74456 0.186097
\(651\) 15.6060 0.611646
\(652\) −7.37228 −0.288721
\(653\) 16.1168 0.630701 0.315350 0.948975i \(-0.397878\pi\)
0.315350 + 0.948975i \(0.397878\pi\)
\(654\) −8.74456 −0.341939
\(655\) −9.48913 −0.370771
\(656\) −12.7446 −0.497592
\(657\) −6.00000 −0.234082
\(658\) −45.4891 −1.77335
\(659\) 31.3723 1.22209 0.611045 0.791596i \(-0.290749\pi\)
0.611045 + 0.791596i \(0.290749\pi\)
\(660\) 1.00000 0.0389249
\(661\) −5.76631 −0.224284 −0.112142 0.993692i \(-0.535771\pi\)
−0.112142 + 0.993692i \(0.535771\pi\)
\(662\) 18.3505 0.713214
\(663\) 4.74456 0.184264
\(664\) −12.0000 −0.465690
\(665\) 13.4891 0.523086
\(666\) −10.0000 −0.387492
\(667\) −8.86141 −0.343115
\(668\) −12.2337 −0.473336
\(669\) −11.3723 −0.439678
\(670\) 4.00000 0.154533
\(671\) −2.00000 −0.0772091
\(672\) 3.37228 0.130089
\(673\) −18.2337 −0.702857 −0.351429 0.936215i \(-0.614304\pi\)
−0.351429 + 0.936215i \(0.614304\pi\)
\(674\) −24.9783 −0.962126
\(675\) 1.00000 0.0384900
\(676\) 9.51087 0.365803
\(677\) −10.0000 −0.384331 −0.192166 0.981363i \(-0.561551\pi\)
−0.192166 + 0.981363i \(0.561551\pi\)
\(678\) −11.4891 −0.441237
\(679\) −31.6060 −1.21293
\(680\) 1.00000 0.0383482
\(681\) 6.11684 0.234398
\(682\) 4.62772 0.177205
\(683\) −6.51087 −0.249132 −0.124566 0.992211i \(-0.539754\pi\)
−0.124566 + 0.992211i \(0.539754\pi\)
\(684\) 4.00000 0.152944
\(685\) −14.8614 −0.567825
\(686\) −8.86141 −0.338330
\(687\) −3.25544 −0.124203
\(688\) 7.37228 0.281066
\(689\) −3.53262 −0.134582
\(690\) −3.37228 −0.128381
\(691\) 46.9783 1.78714 0.893568 0.448927i \(-0.148194\pi\)
0.893568 + 0.448927i \(0.148194\pi\)
\(692\) 11.4891 0.436751
\(693\) 3.37228 0.128102
\(694\) 22.9783 0.872242
\(695\) 14.1168 0.535482
\(696\) 2.62772 0.0996034
\(697\) −12.7446 −0.482735
\(698\) −2.00000 −0.0757011
\(699\) −14.8614 −0.562110
\(700\) 3.37228 0.127460
\(701\) 11.4891 0.433938 0.216969 0.976178i \(-0.430383\pi\)
0.216969 + 0.976178i \(0.430383\pi\)
\(702\) 4.74456 0.179072
\(703\) −40.0000 −1.50863
\(704\) 1.00000 0.0376889
\(705\) −13.4891 −0.508030
\(706\) −22.8614 −0.860400
\(707\) 20.2337 0.760966
\(708\) −10.7446 −0.403805
\(709\) 30.4674 1.14423 0.572113 0.820175i \(-0.306124\pi\)
0.572113 + 0.820175i \(0.306124\pi\)
\(710\) 1.25544 0.0471157
\(711\) −13.4891 −0.505882
\(712\) 16.7446 0.627529
\(713\) −15.6060 −0.584448
\(714\) 3.37228 0.126204
\(715\) 4.74456 0.177437
\(716\) −18.7446 −0.700517
\(717\) 20.2337 0.755641
\(718\) −28.2337 −1.05367
\(719\) −49.7228 −1.85435 −0.927174 0.374631i \(-0.877769\pi\)
−0.927174 + 0.374631i \(0.877769\pi\)
\(720\) 1.00000 0.0372678
\(721\) −11.3723 −0.423526
\(722\) −3.00000 −0.111648
\(723\) −9.37228 −0.348559
\(724\) −14.6277 −0.543635
\(725\) 2.62772 0.0975910
\(726\) 1.00000 0.0371135
\(727\) −14.3505 −0.532232 −0.266116 0.963941i \(-0.585740\pi\)
−0.266116 + 0.963941i \(0.585740\pi\)
\(728\) 16.0000 0.592999
\(729\) 1.00000 0.0370370
\(730\) −6.00000 −0.222070
\(731\) 7.37228 0.272674
\(732\) −2.00000 −0.0739221
\(733\) 2.23369 0.0825031 0.0412516 0.999149i \(-0.486865\pi\)
0.0412516 + 0.999149i \(0.486865\pi\)
\(734\) −22.7446 −0.839517
\(735\) 4.37228 0.161274
\(736\) −3.37228 −0.124304
\(737\) 4.00000 0.147342
\(738\) −12.7446 −0.469134
\(739\) 10.7446 0.395245 0.197623 0.980278i \(-0.436678\pi\)
0.197623 + 0.980278i \(0.436678\pi\)
\(740\) −10.0000 −0.367607
\(741\) 18.9783 0.697183
\(742\) −2.51087 −0.0921771
\(743\) 5.48913 0.201376 0.100688 0.994918i \(-0.467896\pi\)
0.100688 + 0.994918i \(0.467896\pi\)
\(744\) 4.62772 0.169660
\(745\) −0.744563 −0.0272787
\(746\) −23.4891 −0.859998
\(747\) −12.0000 −0.439057
\(748\) 1.00000 0.0365636
\(749\) 47.6060 1.73948
\(750\) 1.00000 0.0365148
\(751\) −20.6277 −0.752716 −0.376358 0.926474i \(-0.622824\pi\)
−0.376358 + 0.926474i \(0.622824\pi\)
\(752\) −13.4891 −0.491898
\(753\) −6.51087 −0.237269
\(754\) 12.4674 0.454035
\(755\) 21.4891 0.782069
\(756\) 3.37228 0.122649
\(757\) 19.8832 0.722666 0.361333 0.932437i \(-0.382322\pi\)
0.361333 + 0.932437i \(0.382322\pi\)
\(758\) 28.0000 1.01701
\(759\) −3.37228 −0.122406
\(760\) 4.00000 0.145095
\(761\) −44.3505 −1.60771 −0.803853 0.594828i \(-0.797220\pi\)
−0.803853 + 0.594828i \(0.797220\pi\)
\(762\) 20.2337 0.732989
\(763\) −29.4891 −1.06758
\(764\) −4.62772 −0.167425
\(765\) 1.00000 0.0361551
\(766\) 2.51087 0.0907216
\(767\) −50.9783 −1.84072
\(768\) 1.00000 0.0360844
\(769\) 15.4891 0.558552 0.279276 0.960211i \(-0.409906\pi\)
0.279276 + 0.960211i \(0.409906\pi\)
\(770\) 3.37228 0.121529
\(771\) 13.3723 0.481591
\(772\) −0.510875 −0.0183868
\(773\) 15.2554 0.548700 0.274350 0.961630i \(-0.411537\pi\)
0.274350 + 0.961630i \(0.411537\pi\)
\(774\) 7.37228 0.264991
\(775\) 4.62772 0.166233
\(776\) −9.37228 −0.336445
\(777\) −33.7228 −1.20980
\(778\) −34.4674 −1.23572
\(779\) −50.9783 −1.82649
\(780\) 4.74456 0.169883
\(781\) 1.25544 0.0449231
\(782\) −3.37228 −0.120593
\(783\) 2.62772 0.0939070
\(784\) 4.37228 0.156153
\(785\) 11.4891 0.410064
\(786\) −9.48913 −0.338466
\(787\) −25.4891 −0.908589 −0.454295 0.890852i \(-0.650109\pi\)
−0.454295 + 0.890852i \(0.650109\pi\)
\(788\) 26.2337 0.934536
\(789\) 0.861407 0.0306669
\(790\) −13.4891 −0.479922
\(791\) −38.7446 −1.37760
\(792\) 1.00000 0.0355335
\(793\) −9.48913 −0.336969
\(794\) −2.00000 −0.0709773
\(795\) −0.744563 −0.0264069
\(796\) −8.00000 −0.283552
\(797\) 39.2554 1.39050 0.695249 0.718769i \(-0.255294\pi\)
0.695249 + 0.718769i \(0.255294\pi\)
\(798\) 13.4891 0.477510
\(799\) −13.4891 −0.477211
\(800\) 1.00000 0.0353553
\(801\) 16.7446 0.591640
\(802\) −9.37228 −0.330947
\(803\) −6.00000 −0.211735
\(804\) 4.00000 0.141069
\(805\) −11.3723 −0.400820
\(806\) 21.9565 0.773385
\(807\) 14.0000 0.492823
\(808\) 6.00000 0.211079
\(809\) −48.9783 −1.72198 −0.860992 0.508619i \(-0.830156\pi\)
−0.860992 + 0.508619i \(0.830156\pi\)
\(810\) 1.00000 0.0351364
\(811\) −1.48913 −0.0522903 −0.0261451 0.999658i \(-0.508323\pi\)
−0.0261451 + 0.999658i \(0.508323\pi\)
\(812\) 8.86141 0.310974
\(813\) −2.11684 −0.0742410
\(814\) −10.0000 −0.350500
\(815\) −7.37228 −0.258240
\(816\) 1.00000 0.0350070
\(817\) 29.4891 1.03169
\(818\) 30.2337 1.05710
\(819\) 16.0000 0.559085
\(820\) −12.7446 −0.445059
\(821\) 17.3723 0.606297 0.303148 0.952943i \(-0.401962\pi\)
0.303148 + 0.952943i \(0.401962\pi\)
\(822\) −14.8614 −0.518351
\(823\) 17.2554 0.601487 0.300743 0.953705i \(-0.402765\pi\)
0.300743 + 0.953705i \(0.402765\pi\)
\(824\) −3.37228 −0.117479
\(825\) 1.00000 0.0348155
\(826\) −36.2337 −1.26073
\(827\) −6.11684 −0.212704 −0.106352 0.994329i \(-0.533917\pi\)
−0.106352 + 0.994329i \(0.533917\pi\)
\(828\) −3.37228 −0.117195
\(829\) 24.9783 0.867531 0.433765 0.901026i \(-0.357185\pi\)
0.433765 + 0.901026i \(0.357185\pi\)
\(830\) −12.0000 −0.416526
\(831\) 22.0000 0.763172
\(832\) 4.74456 0.164488
\(833\) 4.37228 0.151491
\(834\) 14.1168 0.488826
\(835\) −12.2337 −0.423364
\(836\) 4.00000 0.138343
\(837\) 4.62772 0.159957
\(838\) 24.6277 0.850750
\(839\) 14.7446 0.509039 0.254519 0.967068i \(-0.418083\pi\)
0.254519 + 0.967068i \(0.418083\pi\)
\(840\) 3.37228 0.116355
\(841\) −22.0951 −0.761900
\(842\) −16.7446 −0.577056
\(843\) 18.8614 0.649621
\(844\) −14.1168 −0.485922
\(845\) 9.51087 0.327184
\(846\) −13.4891 −0.463766
\(847\) 3.37228 0.115873
\(848\) −0.744563 −0.0255684
\(849\) 18.7446 0.643312
\(850\) 1.00000 0.0342997
\(851\) 33.7228 1.15600
\(852\) 1.25544 0.0430106
\(853\) −34.8614 −1.19363 −0.596816 0.802378i \(-0.703568\pi\)
−0.596816 + 0.802378i \(0.703568\pi\)
\(854\) −6.74456 −0.230794
\(855\) 4.00000 0.136797
\(856\) 14.1168 0.482504
\(857\) 10.3940 0.355053 0.177527 0.984116i \(-0.443190\pi\)
0.177527 + 0.984116i \(0.443190\pi\)
\(858\) 4.74456 0.161977
\(859\) −15.3723 −0.524495 −0.262248 0.965001i \(-0.584464\pi\)
−0.262248 + 0.965001i \(0.584464\pi\)
\(860\) 7.37228 0.251393
\(861\) −42.9783 −1.46469
\(862\) 2.11684 0.0721000
\(863\) −10.9783 −0.373704 −0.186852 0.982388i \(-0.559828\pi\)
−0.186852 + 0.982388i \(0.559828\pi\)
\(864\) 1.00000 0.0340207
\(865\) 11.4891 0.390642
\(866\) 27.2554 0.926177
\(867\) 1.00000 0.0339618
\(868\) 15.6060 0.529701
\(869\) −13.4891 −0.457587
\(870\) 2.62772 0.0890880
\(871\) 18.9783 0.643053
\(872\) −8.74456 −0.296128
\(873\) −9.37228 −0.317204
\(874\) −13.4891 −0.456276
\(875\) 3.37228 0.114004
\(876\) −6.00000 −0.202721
\(877\) −42.8614 −1.44733 −0.723663 0.690153i \(-0.757543\pi\)
−0.723663 + 0.690153i \(0.757543\pi\)
\(878\) −28.2337 −0.952841
\(879\) 30.8614 1.04093
\(880\) 1.00000 0.0337100
\(881\) 17.6060 0.593160 0.296580 0.955008i \(-0.404154\pi\)
0.296580 + 0.955008i \(0.404154\pi\)
\(882\) 4.37228 0.147222
\(883\) −33.9565 −1.14273 −0.571364 0.820697i \(-0.693585\pi\)
−0.571364 + 0.820697i \(0.693585\pi\)
\(884\) 4.74456 0.159577
\(885\) −10.7446 −0.361175
\(886\) −22.1168 −0.743030
\(887\) −59.4456 −1.99599 −0.997994 0.0633023i \(-0.979837\pi\)
−0.997994 + 0.0633023i \(0.979837\pi\)
\(888\) −10.0000 −0.335578
\(889\) 68.2337 2.28848
\(890\) 16.7446 0.561279
\(891\) 1.00000 0.0335013
\(892\) −11.3723 −0.380772
\(893\) −53.9565 −1.80559
\(894\) −0.744563 −0.0249019
\(895\) −18.7446 −0.626562
\(896\) 3.37228 0.112660
\(897\) −16.0000 −0.534224
\(898\) 29.3723 0.980166
\(899\) 12.1603 0.405570
\(900\) 1.00000 0.0333333
\(901\) −0.744563 −0.0248050
\(902\) −12.7446 −0.424348
\(903\) 24.8614 0.827336
\(904\) −11.4891 −0.382123
\(905\) −14.6277 −0.486242
\(906\) 21.4891 0.713928
\(907\) −30.5842 −1.01553 −0.507766 0.861495i \(-0.669529\pi\)
−0.507766 + 0.861495i \(0.669529\pi\)
\(908\) 6.11684 0.202995
\(909\) 6.00000 0.199007
\(910\) 16.0000 0.530395
\(911\) −5.02175 −0.166378 −0.0831890 0.996534i \(-0.526511\pi\)
−0.0831890 + 0.996534i \(0.526511\pi\)
\(912\) 4.00000 0.132453
\(913\) −12.0000 −0.397142
\(914\) 10.0000 0.330771
\(915\) −2.00000 −0.0661180
\(916\) −3.25544 −0.107563
\(917\) −32.0000 −1.05673
\(918\) 1.00000 0.0330049
\(919\) 15.1386 0.499376 0.249688 0.968326i \(-0.419672\pi\)
0.249688 + 0.968326i \(0.419672\pi\)
\(920\) −3.37228 −0.111181
\(921\) 4.00000 0.131804
\(922\) 0.510875 0.0168248
\(923\) 5.95650 0.196061
\(924\) 3.37228 0.110940
\(925\) −10.0000 −0.328798
\(926\) 16.0000 0.525793
\(927\) −3.37228 −0.110760
\(928\) 2.62772 0.0862591
\(929\) −2.62772 −0.0862127 −0.0431063 0.999070i \(-0.513725\pi\)
−0.0431063 + 0.999070i \(0.513725\pi\)
\(930\) 4.62772 0.151749
\(931\) 17.4891 0.573183
\(932\) −14.8614 −0.486802
\(933\) −5.48913 −0.179706
\(934\) 30.9783 1.01364
\(935\) 1.00000 0.0327035
\(936\) 4.74456 0.155081
\(937\) 10.0000 0.326686 0.163343 0.986569i \(-0.447772\pi\)
0.163343 + 0.986569i \(0.447772\pi\)
\(938\) 13.4891 0.440436
\(939\) 18.8614 0.615519
\(940\) −13.4891 −0.439967
\(941\) −2.00000 −0.0651981 −0.0325991 0.999469i \(-0.510378\pi\)
−0.0325991 + 0.999469i \(0.510378\pi\)
\(942\) 11.4891 0.374336
\(943\) 42.9783 1.39956
\(944\) −10.7446 −0.349706
\(945\) 3.37228 0.109700
\(946\) 7.37228 0.239694
\(947\) 29.2554 0.950674 0.475337 0.879804i \(-0.342326\pi\)
0.475337 + 0.879804i \(0.342326\pi\)
\(948\) −13.4891 −0.438106
\(949\) −28.4674 −0.924090
\(950\) 4.00000 0.129777
\(951\) 0.116844 0.00378893
\(952\) 3.37228 0.109296
\(953\) −24.5109 −0.793985 −0.396993 0.917822i \(-0.629946\pi\)
−0.396993 + 0.917822i \(0.629946\pi\)
\(954\) −0.744563 −0.0241061
\(955\) −4.62772 −0.149749
\(956\) 20.2337 0.654404
\(957\) 2.62772 0.0849421
\(958\) 35.8397 1.15793
\(959\) −50.1168 −1.61836
\(960\) 1.00000 0.0322749
\(961\) −9.58422 −0.309168
\(962\) −47.4456 −1.52971
\(963\) 14.1168 0.454909
\(964\) −9.37228 −0.301861
\(965\) −0.510875 −0.0164456
\(966\) −11.3723 −0.365897
\(967\) −30.7446 −0.988678 −0.494339 0.869269i \(-0.664590\pi\)
−0.494339 + 0.869269i \(0.664590\pi\)
\(968\) 1.00000 0.0321412
\(969\) 4.00000 0.128499
\(970\) −9.37228 −0.300926
\(971\) −44.4674 −1.42703 −0.713513 0.700642i \(-0.752897\pi\)
−0.713513 + 0.700642i \(0.752897\pi\)
\(972\) 1.00000 0.0320750
\(973\) 47.6060 1.52618
\(974\) 1.25544 0.0402268
\(975\) 4.74456 0.151948
\(976\) −2.00000 −0.0640184
\(977\) −14.0000 −0.447900 −0.223950 0.974601i \(-0.571895\pi\)
−0.223950 + 0.974601i \(0.571895\pi\)
\(978\) −7.37228 −0.235740
\(979\) 16.7446 0.535159
\(980\) 4.37228 0.139667
\(981\) −8.74456 −0.279192
\(982\) −6.51087 −0.207770
\(983\) −8.39403 −0.267728 −0.133864 0.991000i \(-0.542739\pi\)
−0.133864 + 0.991000i \(0.542739\pi\)
\(984\) −12.7446 −0.406282
\(985\) 26.2337 0.835875
\(986\) 2.62772 0.0836836
\(987\) −45.4891 −1.44793
\(988\) 18.9783 0.603779
\(989\) −24.8614 −0.790547
\(990\) 1.00000 0.0317821
\(991\) 35.8397 1.13848 0.569242 0.822170i \(-0.307237\pi\)
0.569242 + 0.822170i \(0.307237\pi\)
\(992\) 4.62772 0.146930
\(993\) 18.3505 0.582337
\(994\) 4.23369 0.134284
\(995\) −8.00000 −0.253617
\(996\) −12.0000 −0.380235
\(997\) −28.1168 −0.890469 −0.445235 0.895414i \(-0.646880\pi\)
−0.445235 + 0.895414i \(0.646880\pi\)
\(998\) 1.48913 0.0471374
\(999\) −10.0000 −0.316386
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 5610.2.a.bx.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5610.2.a.bx.1.2 2 1.1 even 1 trivial