Properties

Label 5415.2.a.r.1.2
Level $5415$
Weight $2$
Character 5415.1
Self dual yes
Analytic conductor $43.239$
Analytic rank $1$
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] = [5415,2,Mod(1,5415)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5415, 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("5415.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5415 = 3 \cdot 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5415.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: \(43.2389926945\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 285)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 5415.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.73205 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.73205 q^{6} -2.73205 q^{7} -1.73205 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.73205 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.73205 q^{6} -2.73205 q^{7} -1.73205 q^{8} +1.00000 q^{9} +1.73205 q^{10} +4.73205 q^{11} -1.00000 q^{12} -0.732051 q^{13} -4.73205 q^{14} -1.00000 q^{15} -5.00000 q^{16} +1.73205 q^{18} +1.00000 q^{20} +2.73205 q^{21} +8.19615 q^{22} +3.46410 q^{23} +1.73205 q^{24} +1.00000 q^{25} -1.26795 q^{26} -1.00000 q^{27} -2.73205 q^{28} -8.19615 q^{29} -1.73205 q^{30} -8.92820 q^{31} -5.19615 q^{32} -4.73205 q^{33} -2.73205 q^{35} +1.00000 q^{36} +6.19615 q^{37} +0.732051 q^{39} -1.73205 q^{40} -1.26795 q^{41} +4.73205 q^{42} +4.19615 q^{43} +4.73205 q^{44} +1.00000 q^{45} +6.00000 q^{46} -3.46410 q^{47} +5.00000 q^{48} +0.464102 q^{49} +1.73205 q^{50} -0.732051 q^{52} +9.46410 q^{53} -1.73205 q^{54} +4.73205 q^{55} +4.73205 q^{56} -14.1962 q^{58} -2.53590 q^{59} -1.00000 q^{60} -6.53590 q^{61} -15.4641 q^{62} -2.73205 q^{63} +1.00000 q^{64} -0.732051 q^{65} -8.19615 q^{66} -8.00000 q^{67} -3.46410 q^{69} -4.73205 q^{70} +4.39230 q^{71} -1.73205 q^{72} -16.9282 q^{73} +10.7321 q^{74} -1.00000 q^{75} -12.9282 q^{77} +1.26795 q^{78} +10.9282 q^{79} -5.00000 q^{80} +1.00000 q^{81} -2.19615 q^{82} -12.9282 q^{83} +2.73205 q^{84} +7.26795 q^{86} +8.19615 q^{87} -8.19615 q^{88} -10.7321 q^{89} +1.73205 q^{90} +2.00000 q^{91} +3.46410 q^{92} +8.92820 q^{93} -6.00000 q^{94} +5.19615 q^{96} +6.19615 q^{97} +0.803848 q^{98} +4.73205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{7} + 2 q^{9} + 6 q^{11} - 2 q^{12} + 2 q^{13} - 6 q^{14} - 2 q^{15} - 10 q^{16} + 2 q^{20} + 2 q^{21} + 6 q^{22} + 2 q^{25} - 6 q^{26} - 2 q^{27} - 2 q^{28} - 6 q^{29} - 4 q^{31} - 6 q^{33} - 2 q^{35} + 2 q^{36} + 2 q^{37} - 2 q^{39} - 6 q^{41} + 6 q^{42} - 2 q^{43} + 6 q^{44} + 2 q^{45} + 12 q^{46} + 10 q^{48} - 6 q^{49} + 2 q^{52} + 12 q^{53} + 6 q^{55} + 6 q^{56} - 18 q^{58} - 12 q^{59} - 2 q^{60} - 20 q^{61} - 24 q^{62} - 2 q^{63} + 2 q^{64} + 2 q^{65} - 6 q^{66} - 16 q^{67} - 6 q^{70} - 12 q^{71} - 20 q^{73} + 18 q^{74} - 2 q^{75} - 12 q^{77} + 6 q^{78} + 8 q^{79} - 10 q^{80} + 2 q^{81} + 6 q^{82} - 12 q^{83} + 2 q^{84} + 18 q^{86} + 6 q^{87} - 6 q^{88} - 18 q^{89} + 4 q^{91} + 4 q^{93} - 12 q^{94} + 2 q^{97} + 12 q^{98} + 6 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.73205 1.22474 0.612372 0.790569i \(-0.290215\pi\)
0.612372 + 0.790569i \(0.290215\pi\)
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −1.73205 −0.707107
\(7\) −2.73205 −1.03262 −0.516309 0.856402i \(-0.672694\pi\)
−0.516309 + 0.856402i \(0.672694\pi\)
\(8\) −1.73205 −0.612372
\(9\) 1.00000 0.333333
\(10\) 1.73205 0.547723
\(11\) 4.73205 1.42677 0.713384 0.700774i \(-0.247162\pi\)
0.713384 + 0.700774i \(0.247162\pi\)
\(12\) −1.00000 −0.288675
\(13\) −0.732051 −0.203034 −0.101517 0.994834i \(-0.532370\pi\)
−0.101517 + 0.994834i \(0.532370\pi\)
\(14\) −4.73205 −1.26469
\(15\) −1.00000 −0.258199
\(16\) −5.00000 −1.25000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 1.73205 0.408248
\(19\) 0 0
\(20\) 1.00000 0.223607
\(21\) 2.73205 0.596182
\(22\) 8.19615 1.74743
\(23\) 3.46410 0.722315 0.361158 0.932505i \(-0.382382\pi\)
0.361158 + 0.932505i \(0.382382\pi\)
\(24\) 1.73205 0.353553
\(25\) 1.00000 0.200000
\(26\) −1.26795 −0.248665
\(27\) −1.00000 −0.192450
\(28\) −2.73205 −0.516309
\(29\) −8.19615 −1.52199 −0.760994 0.648759i \(-0.775288\pi\)
−0.760994 + 0.648759i \(0.775288\pi\)
\(30\) −1.73205 −0.316228
\(31\) −8.92820 −1.60355 −0.801776 0.597624i \(-0.796111\pi\)
−0.801776 + 0.597624i \(0.796111\pi\)
\(32\) −5.19615 −0.918559
\(33\) −4.73205 −0.823744
\(34\) 0 0
\(35\) −2.73205 −0.461801
\(36\) 1.00000 0.166667
\(37\) 6.19615 1.01864 0.509321 0.860577i \(-0.329897\pi\)
0.509321 + 0.860577i \(0.329897\pi\)
\(38\) 0 0
\(39\) 0.732051 0.117222
\(40\) −1.73205 −0.273861
\(41\) −1.26795 −0.198020 −0.0990102 0.995086i \(-0.531568\pi\)
−0.0990102 + 0.995086i \(0.531568\pi\)
\(42\) 4.73205 0.730171
\(43\) 4.19615 0.639907 0.319954 0.947433i \(-0.396333\pi\)
0.319954 + 0.947433i \(0.396333\pi\)
\(44\) 4.73205 0.713384
\(45\) 1.00000 0.149071
\(46\) 6.00000 0.884652
\(47\) −3.46410 −0.505291 −0.252646 0.967559i \(-0.581301\pi\)
−0.252646 + 0.967559i \(0.581301\pi\)
\(48\) 5.00000 0.721688
\(49\) 0.464102 0.0663002
\(50\) 1.73205 0.244949
\(51\) 0 0
\(52\) −0.732051 −0.101517
\(53\) 9.46410 1.29999 0.649997 0.759937i \(-0.274770\pi\)
0.649997 + 0.759937i \(0.274770\pi\)
\(54\) −1.73205 −0.235702
\(55\) 4.73205 0.638070
\(56\) 4.73205 0.632347
\(57\) 0 0
\(58\) −14.1962 −1.86405
\(59\) −2.53590 −0.330146 −0.165073 0.986281i \(-0.552786\pi\)
−0.165073 + 0.986281i \(0.552786\pi\)
\(60\) −1.00000 −0.129099
\(61\) −6.53590 −0.836836 −0.418418 0.908255i \(-0.637415\pi\)
−0.418418 + 0.908255i \(0.637415\pi\)
\(62\) −15.4641 −1.96394
\(63\) −2.73205 −0.344206
\(64\) 1.00000 0.125000
\(65\) −0.732051 −0.0907997
\(66\) −8.19615 −1.00888
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) −3.46410 −0.417029
\(70\) −4.73205 −0.565588
\(71\) 4.39230 0.521271 0.260635 0.965437i \(-0.416068\pi\)
0.260635 + 0.965437i \(0.416068\pi\)
\(72\) −1.73205 −0.204124
\(73\) −16.9282 −1.98130 −0.990648 0.136441i \(-0.956434\pi\)
−0.990648 + 0.136441i \(0.956434\pi\)
\(74\) 10.7321 1.24758
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −12.9282 −1.47331
\(78\) 1.26795 0.143567
\(79\) 10.9282 1.22952 0.614759 0.788715i \(-0.289253\pi\)
0.614759 + 0.788715i \(0.289253\pi\)
\(80\) −5.00000 −0.559017
\(81\) 1.00000 0.111111
\(82\) −2.19615 −0.242524
\(83\) −12.9282 −1.41905 −0.709527 0.704678i \(-0.751092\pi\)
−0.709527 + 0.704678i \(0.751092\pi\)
\(84\) 2.73205 0.298091
\(85\) 0 0
\(86\) 7.26795 0.783723
\(87\) 8.19615 0.878720
\(88\) −8.19615 −0.873713
\(89\) −10.7321 −1.13760 −0.568798 0.822478i \(-0.692591\pi\)
−0.568798 + 0.822478i \(0.692591\pi\)
\(90\) 1.73205 0.182574
\(91\) 2.00000 0.209657
\(92\) 3.46410 0.361158
\(93\) 8.92820 0.925812
\(94\) −6.00000 −0.618853
\(95\) 0 0
\(96\) 5.19615 0.530330
\(97\) 6.19615 0.629124 0.314562 0.949237i \(-0.398142\pi\)
0.314562 + 0.949237i \(0.398142\pi\)
\(98\) 0.803848 0.0812009
\(99\) 4.73205 0.475589
\(100\) 1.00000 0.100000
\(101\) −10.3923 −1.03407 −0.517036 0.855963i \(-0.672965\pi\)
−0.517036 + 0.855963i \(0.672965\pi\)
\(102\) 0 0
\(103\) −9.85641 −0.971181 −0.485590 0.874187i \(-0.661395\pi\)
−0.485590 + 0.874187i \(0.661395\pi\)
\(104\) 1.26795 0.124333
\(105\) 2.73205 0.266621
\(106\) 16.3923 1.59216
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 14.3923 1.37853 0.689266 0.724508i \(-0.257933\pi\)
0.689266 + 0.724508i \(0.257933\pi\)
\(110\) 8.19615 0.781472
\(111\) −6.19615 −0.588113
\(112\) 13.6603 1.29077
\(113\) −18.9282 −1.78062 −0.890308 0.455359i \(-0.849511\pi\)
−0.890308 + 0.455359i \(0.849511\pi\)
\(114\) 0 0
\(115\) 3.46410 0.323029
\(116\) −8.19615 −0.760994
\(117\) −0.732051 −0.0676781
\(118\) −4.39230 −0.404344
\(119\) 0 0
\(120\) 1.73205 0.158114
\(121\) 11.3923 1.03566
\(122\) −11.3205 −1.02491
\(123\) 1.26795 0.114327
\(124\) −8.92820 −0.801776
\(125\) 1.00000 0.0894427
\(126\) −4.73205 −0.421565
\(127\) 4.00000 0.354943 0.177471 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) 12.1244 1.07165
\(129\) −4.19615 −0.369451
\(130\) −1.26795 −0.111207
\(131\) −9.12436 −0.797199 −0.398599 0.917125i \(-0.630504\pi\)
−0.398599 + 0.917125i \(0.630504\pi\)
\(132\) −4.73205 −0.411872
\(133\) 0 0
\(134\) −13.8564 −1.19701
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 19.8564 1.69645 0.848224 0.529638i \(-0.177672\pi\)
0.848224 + 0.529638i \(0.177672\pi\)
\(138\) −6.00000 −0.510754
\(139\) −8.39230 −0.711826 −0.355913 0.934519i \(-0.615830\pi\)
−0.355913 + 0.934519i \(0.615830\pi\)
\(140\) −2.73205 −0.230900
\(141\) 3.46410 0.291730
\(142\) 7.60770 0.638424
\(143\) −3.46410 −0.289683
\(144\) −5.00000 −0.416667
\(145\) −8.19615 −0.680653
\(146\) −29.3205 −2.42658
\(147\) −0.464102 −0.0382785
\(148\) 6.19615 0.509321
\(149\) −19.8564 −1.62670 −0.813350 0.581775i \(-0.802359\pi\)
−0.813350 + 0.581775i \(0.802359\pi\)
\(150\) −1.73205 −0.141421
\(151\) −14.0000 −1.13930 −0.569652 0.821886i \(-0.692922\pi\)
−0.569652 + 0.821886i \(0.692922\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −22.3923 −1.80442
\(155\) −8.92820 −0.717131
\(156\) 0.732051 0.0586110
\(157\) 6.39230 0.510161 0.255081 0.966920i \(-0.417898\pi\)
0.255081 + 0.966920i \(0.417898\pi\)
\(158\) 18.9282 1.50585
\(159\) −9.46410 −0.750552
\(160\) −5.19615 −0.410792
\(161\) −9.46410 −0.745876
\(162\) 1.73205 0.136083
\(163\) 9.26795 0.725922 0.362961 0.931804i \(-0.381766\pi\)
0.362961 + 0.931804i \(0.381766\pi\)
\(164\) −1.26795 −0.0990102
\(165\) −4.73205 −0.368390
\(166\) −22.3923 −1.73798
\(167\) −3.46410 −0.268060 −0.134030 0.990977i \(-0.542792\pi\)
−0.134030 + 0.990977i \(0.542792\pi\)
\(168\) −4.73205 −0.365086
\(169\) −12.4641 −0.958777
\(170\) 0 0
\(171\) 0 0
\(172\) 4.19615 0.319954
\(173\) −6.92820 −0.526742 −0.263371 0.964695i \(-0.584834\pi\)
−0.263371 + 0.964695i \(0.584834\pi\)
\(174\) 14.1962 1.07621
\(175\) −2.73205 −0.206524
\(176\) −23.6603 −1.78346
\(177\) 2.53590 0.190610
\(178\) −18.5885 −1.39326
\(179\) −23.3205 −1.74306 −0.871528 0.490345i \(-0.836871\pi\)
−0.871528 + 0.490345i \(0.836871\pi\)
\(180\) 1.00000 0.0745356
\(181\) 2.39230 0.177819 0.0889093 0.996040i \(-0.471662\pi\)
0.0889093 + 0.996040i \(0.471662\pi\)
\(182\) 3.46410 0.256776
\(183\) 6.53590 0.483148
\(184\) −6.00000 −0.442326
\(185\) 6.19615 0.455550
\(186\) 15.4641 1.13388
\(187\) 0 0
\(188\) −3.46410 −0.252646
\(189\) 2.73205 0.198727
\(190\) 0 0
\(191\) 0.339746 0.0245832 0.0122916 0.999924i \(-0.496087\pi\)
0.0122916 + 0.999924i \(0.496087\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −17.1244 −1.23264 −0.616319 0.787497i \(-0.711377\pi\)
−0.616319 + 0.787497i \(0.711377\pi\)
\(194\) 10.7321 0.770516
\(195\) 0.732051 0.0524232
\(196\) 0.464102 0.0331501
\(197\) −24.0000 −1.70993 −0.854965 0.518686i \(-0.826421\pi\)
−0.854965 + 0.518686i \(0.826421\pi\)
\(198\) 8.19615 0.582475
\(199\) −15.3205 −1.08604 −0.543021 0.839719i \(-0.682720\pi\)
−0.543021 + 0.839719i \(0.682720\pi\)
\(200\) −1.73205 −0.122474
\(201\) 8.00000 0.564276
\(202\) −18.0000 −1.26648
\(203\) 22.3923 1.57163
\(204\) 0 0
\(205\) −1.26795 −0.0885574
\(206\) −17.0718 −1.18945
\(207\) 3.46410 0.240772
\(208\) 3.66025 0.253793
\(209\) 0 0
\(210\) 4.73205 0.326543
\(211\) −1.07180 −0.0737855 −0.0368928 0.999319i \(-0.511746\pi\)
−0.0368928 + 0.999319i \(0.511746\pi\)
\(212\) 9.46410 0.649997
\(213\) −4.39230 −0.300956
\(214\) 0 0
\(215\) 4.19615 0.286175
\(216\) 1.73205 0.117851
\(217\) 24.3923 1.65586
\(218\) 24.9282 1.68835
\(219\) 16.9282 1.14390
\(220\) 4.73205 0.319035
\(221\) 0 0
\(222\) −10.7321 −0.720288
\(223\) 17.8564 1.19575 0.597877 0.801588i \(-0.296011\pi\)
0.597877 + 0.801588i \(0.296011\pi\)
\(224\) 14.1962 0.948520
\(225\) 1.00000 0.0666667
\(226\) −32.7846 −2.18080
\(227\) −10.3923 −0.689761 −0.344881 0.938647i \(-0.612081\pi\)
−0.344881 + 0.938647i \(0.612081\pi\)
\(228\) 0 0
\(229\) −18.5359 −1.22489 −0.612443 0.790515i \(-0.709813\pi\)
−0.612443 + 0.790515i \(0.709813\pi\)
\(230\) 6.00000 0.395628
\(231\) 12.9282 0.850613
\(232\) 14.1962 0.932023
\(233\) −7.85641 −0.514690 −0.257345 0.966320i \(-0.582848\pi\)
−0.257345 + 0.966320i \(0.582848\pi\)
\(234\) −1.26795 −0.0828884
\(235\) −3.46410 −0.225973
\(236\) −2.53590 −0.165073
\(237\) −10.9282 −0.709863
\(238\) 0 0
\(239\) −9.80385 −0.634158 −0.317079 0.948399i \(-0.602702\pi\)
−0.317079 + 0.948399i \(0.602702\pi\)
\(240\) 5.00000 0.322749
\(241\) 3.07180 0.197872 0.0989359 0.995094i \(-0.468456\pi\)
0.0989359 + 0.995094i \(0.468456\pi\)
\(242\) 19.7321 1.26842
\(243\) −1.00000 −0.0641500
\(244\) −6.53590 −0.418418
\(245\) 0.464102 0.0296504
\(246\) 2.19615 0.140022
\(247\) 0 0
\(248\) 15.4641 0.981971
\(249\) 12.9282 0.819292
\(250\) 1.73205 0.109545
\(251\) 28.0526 1.77066 0.885331 0.464961i \(-0.153932\pi\)
0.885331 + 0.464961i \(0.153932\pi\)
\(252\) −2.73205 −0.172103
\(253\) 16.3923 1.03058
\(254\) 6.92820 0.434714
\(255\) 0 0
\(256\) 19.0000 1.18750
\(257\) 24.0000 1.49708 0.748539 0.663090i \(-0.230755\pi\)
0.748539 + 0.663090i \(0.230755\pi\)
\(258\) −7.26795 −0.452483
\(259\) −16.9282 −1.05187
\(260\) −0.732051 −0.0453999
\(261\) −8.19615 −0.507329
\(262\) −15.8038 −0.976365
\(263\) 6.00000 0.369976 0.184988 0.982741i \(-0.440775\pi\)
0.184988 + 0.982741i \(0.440775\pi\)
\(264\) 8.19615 0.504438
\(265\) 9.46410 0.581375
\(266\) 0 0
\(267\) 10.7321 0.656791
\(268\) −8.00000 −0.488678
\(269\) −0.588457 −0.0358789 −0.0179394 0.999839i \(-0.505711\pi\)
−0.0179394 + 0.999839i \(0.505711\pi\)
\(270\) −1.73205 −0.105409
\(271\) 0.392305 0.0238308 0.0119154 0.999929i \(-0.496207\pi\)
0.0119154 + 0.999929i \(0.496207\pi\)
\(272\) 0 0
\(273\) −2.00000 −0.121046
\(274\) 34.3923 2.07772
\(275\) 4.73205 0.285353
\(276\) −3.46410 −0.208514
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) −14.5359 −0.871805
\(279\) −8.92820 −0.534518
\(280\) 4.73205 0.282794
\(281\) −1.26795 −0.0756395 −0.0378198 0.999285i \(-0.512041\pi\)
−0.0378198 + 0.999285i \(0.512041\pi\)
\(282\) 6.00000 0.357295
\(283\) 24.9808 1.48495 0.742476 0.669873i \(-0.233651\pi\)
0.742476 + 0.669873i \(0.233651\pi\)
\(284\) 4.39230 0.260635
\(285\) 0 0
\(286\) −6.00000 −0.354787
\(287\) 3.46410 0.204479
\(288\) −5.19615 −0.306186
\(289\) −17.0000 −1.00000
\(290\) −14.1962 −0.833627
\(291\) −6.19615 −0.363225
\(292\) −16.9282 −0.990648
\(293\) −27.7128 −1.61900 −0.809500 0.587120i \(-0.800262\pi\)
−0.809500 + 0.587120i \(0.800262\pi\)
\(294\) −0.803848 −0.0468813
\(295\) −2.53590 −0.147646
\(296\) −10.7321 −0.623788
\(297\) −4.73205 −0.274581
\(298\) −34.3923 −1.99229
\(299\) −2.53590 −0.146655
\(300\) −1.00000 −0.0577350
\(301\) −11.4641 −0.660780
\(302\) −24.2487 −1.39536
\(303\) 10.3923 0.597022
\(304\) 0 0
\(305\) −6.53590 −0.374244
\(306\) 0 0
\(307\) 32.3923 1.84873 0.924363 0.381514i \(-0.124597\pi\)
0.924363 + 0.381514i \(0.124597\pi\)
\(308\) −12.9282 −0.736653
\(309\) 9.85641 0.560711
\(310\) −15.4641 −0.878302
\(311\) 32.4449 1.83978 0.919890 0.392177i \(-0.128278\pi\)
0.919890 + 0.392177i \(0.128278\pi\)
\(312\) −1.26795 −0.0717835
\(313\) 6.39230 0.361314 0.180657 0.983546i \(-0.442178\pi\)
0.180657 + 0.983546i \(0.442178\pi\)
\(314\) 11.0718 0.624818
\(315\) −2.73205 −0.153934
\(316\) 10.9282 0.614759
\(317\) 11.3205 0.635823 0.317912 0.948120i \(-0.397018\pi\)
0.317912 + 0.948120i \(0.397018\pi\)
\(318\) −16.3923 −0.919235
\(319\) −38.7846 −2.17152
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) −16.3923 −0.913507
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) −0.732051 −0.0406069
\(326\) 16.0526 0.889069
\(327\) −14.3923 −0.795896
\(328\) 2.19615 0.121262
\(329\) 9.46410 0.521773
\(330\) −8.19615 −0.451183
\(331\) 25.7128 1.41330 0.706652 0.707561i \(-0.250205\pi\)
0.706652 + 0.707561i \(0.250205\pi\)
\(332\) −12.9282 −0.709527
\(333\) 6.19615 0.339547
\(334\) −6.00000 −0.328305
\(335\) −8.00000 −0.437087
\(336\) −13.6603 −0.745228
\(337\) −5.12436 −0.279141 −0.139571 0.990212i \(-0.544572\pi\)
−0.139571 + 0.990212i \(0.544572\pi\)
\(338\) −21.5885 −1.17426
\(339\) 18.9282 1.02804
\(340\) 0 0
\(341\) −42.2487 −2.28790
\(342\) 0 0
\(343\) 17.8564 0.964155
\(344\) −7.26795 −0.391862
\(345\) −3.46410 −0.186501
\(346\) −12.0000 −0.645124
\(347\) −0.928203 −0.0498286 −0.0249143 0.999690i \(-0.507931\pi\)
−0.0249143 + 0.999690i \(0.507931\pi\)
\(348\) 8.19615 0.439360
\(349\) −22.0000 −1.17763 −0.588817 0.808267i \(-0.700406\pi\)
−0.588817 + 0.808267i \(0.700406\pi\)
\(350\) −4.73205 −0.252939
\(351\) 0.732051 0.0390740
\(352\) −24.5885 −1.31057
\(353\) −14.7846 −0.786905 −0.393453 0.919345i \(-0.628719\pi\)
−0.393453 + 0.919345i \(0.628719\pi\)
\(354\) 4.39230 0.233448
\(355\) 4.39230 0.233119
\(356\) −10.7321 −0.568798
\(357\) 0 0
\(358\) −40.3923 −2.13480
\(359\) 0.339746 0.0179311 0.00896555 0.999960i \(-0.497146\pi\)
0.00896555 + 0.999960i \(0.497146\pi\)
\(360\) −1.73205 −0.0912871
\(361\) 0 0
\(362\) 4.14359 0.217782
\(363\) −11.3923 −0.597941
\(364\) 2.00000 0.104828
\(365\) −16.9282 −0.886063
\(366\) 11.3205 0.591732
\(367\) 16.1962 0.845432 0.422716 0.906262i \(-0.361077\pi\)
0.422716 + 0.906262i \(0.361077\pi\)
\(368\) −17.3205 −0.902894
\(369\) −1.26795 −0.0660068
\(370\) 10.7321 0.557933
\(371\) −25.8564 −1.34240
\(372\) 8.92820 0.462906
\(373\) 6.19615 0.320825 0.160412 0.987050i \(-0.448718\pi\)
0.160412 + 0.987050i \(0.448718\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 6.00000 0.309426
\(377\) 6.00000 0.309016
\(378\) 4.73205 0.243390
\(379\) −20.9282 −1.07501 −0.537505 0.843261i \(-0.680633\pi\)
−0.537505 + 0.843261i \(0.680633\pi\)
\(380\) 0 0
\(381\) −4.00000 −0.204926
\(382\) 0.588457 0.0301081
\(383\) 17.0718 0.872328 0.436164 0.899867i \(-0.356337\pi\)
0.436164 + 0.899867i \(0.356337\pi\)
\(384\) −12.1244 −0.618718
\(385\) −12.9282 −0.658882
\(386\) −29.6603 −1.50967
\(387\) 4.19615 0.213302
\(388\) 6.19615 0.314562
\(389\) 7.85641 0.398336 0.199168 0.979965i \(-0.436176\pi\)
0.199168 + 0.979965i \(0.436176\pi\)
\(390\) 1.26795 0.0642051
\(391\) 0 0
\(392\) −0.803848 −0.0406004
\(393\) 9.12436 0.460263
\(394\) −41.5692 −2.09423
\(395\) 10.9282 0.549858
\(396\) 4.73205 0.237795
\(397\) 8.92820 0.448094 0.224047 0.974578i \(-0.428073\pi\)
0.224047 + 0.974578i \(0.428073\pi\)
\(398\) −26.5359 −1.33012
\(399\) 0 0
\(400\) −5.00000 −0.250000
\(401\) 34.0526 1.70050 0.850252 0.526376i \(-0.176450\pi\)
0.850252 + 0.526376i \(0.176450\pi\)
\(402\) 13.8564 0.691095
\(403\) 6.53590 0.325576
\(404\) −10.3923 −0.517036
\(405\) 1.00000 0.0496904
\(406\) 38.7846 1.92485
\(407\) 29.3205 1.45336
\(408\) 0 0
\(409\) 26.3923 1.30502 0.652508 0.757782i \(-0.273717\pi\)
0.652508 + 0.757782i \(0.273717\pi\)
\(410\) −2.19615 −0.108460
\(411\) −19.8564 −0.979444
\(412\) −9.85641 −0.485590
\(413\) 6.92820 0.340915
\(414\) 6.00000 0.294884
\(415\) −12.9282 −0.634621
\(416\) 3.80385 0.186499
\(417\) 8.39230 0.410973
\(418\) 0 0
\(419\) −28.0526 −1.37046 −0.685229 0.728328i \(-0.740298\pi\)
−0.685229 + 0.728328i \(0.740298\pi\)
\(420\) 2.73205 0.133310
\(421\) 18.7846 0.915506 0.457753 0.889079i \(-0.348654\pi\)
0.457753 + 0.889079i \(0.348654\pi\)
\(422\) −1.85641 −0.0903685
\(423\) −3.46410 −0.168430
\(424\) −16.3923 −0.796081
\(425\) 0 0
\(426\) −7.60770 −0.368594
\(427\) 17.8564 0.864132
\(428\) 0 0
\(429\) 3.46410 0.167248
\(430\) 7.26795 0.350492
\(431\) 11.3205 0.545290 0.272645 0.962115i \(-0.412102\pi\)
0.272645 + 0.962115i \(0.412102\pi\)
\(432\) 5.00000 0.240563
\(433\) 10.5885 0.508849 0.254424 0.967093i \(-0.418114\pi\)
0.254424 + 0.967093i \(0.418114\pi\)
\(434\) 42.2487 2.02800
\(435\) 8.19615 0.392975
\(436\) 14.3923 0.689266
\(437\) 0 0
\(438\) 29.3205 1.40099
\(439\) −26.9282 −1.28521 −0.642607 0.766196i \(-0.722147\pi\)
−0.642607 + 0.766196i \(0.722147\pi\)
\(440\) −8.19615 −0.390736
\(441\) 0.464102 0.0221001
\(442\) 0 0
\(443\) −5.32051 −0.252785 −0.126392 0.991980i \(-0.540340\pi\)
−0.126392 + 0.991980i \(0.540340\pi\)
\(444\) −6.19615 −0.294056
\(445\) −10.7321 −0.508748
\(446\) 30.9282 1.46449
\(447\) 19.8564 0.939176
\(448\) −2.73205 −0.129077
\(449\) 5.66025 0.267124 0.133562 0.991040i \(-0.457358\pi\)
0.133562 + 0.991040i \(0.457358\pi\)
\(450\) 1.73205 0.0816497
\(451\) −6.00000 −0.282529
\(452\) −18.9282 −0.890308
\(453\) 14.0000 0.657777
\(454\) −18.0000 −0.844782
\(455\) 2.00000 0.0937614
\(456\) 0 0
\(457\) 4.53590 0.212180 0.106090 0.994357i \(-0.466167\pi\)
0.106090 + 0.994357i \(0.466167\pi\)
\(458\) −32.1051 −1.50017
\(459\) 0 0
\(460\) 3.46410 0.161515
\(461\) 6.00000 0.279448 0.139724 0.990190i \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(462\) 22.3923 1.04178
\(463\) −35.5167 −1.65060 −0.825300 0.564695i \(-0.808994\pi\)
−0.825300 + 0.564695i \(0.808994\pi\)
\(464\) 40.9808 1.90248
\(465\) 8.92820 0.414036
\(466\) −13.6077 −0.630364
\(467\) 20.5359 0.950288 0.475144 0.879908i \(-0.342396\pi\)
0.475144 + 0.879908i \(0.342396\pi\)
\(468\) −0.732051 −0.0338391
\(469\) 21.8564 1.00924
\(470\) −6.00000 −0.276759
\(471\) −6.39230 −0.294542
\(472\) 4.39230 0.202172
\(473\) 19.8564 0.912999
\(474\) −18.9282 −0.869401
\(475\) 0 0
\(476\) 0 0
\(477\) 9.46410 0.433331
\(478\) −16.9808 −0.776682
\(479\) 25.5167 1.16589 0.582943 0.812513i \(-0.301901\pi\)
0.582943 + 0.812513i \(0.301901\pi\)
\(480\) 5.19615 0.237171
\(481\) −4.53590 −0.206819
\(482\) 5.32051 0.242343
\(483\) 9.46410 0.430632
\(484\) 11.3923 0.517832
\(485\) 6.19615 0.281353
\(486\) −1.73205 −0.0785674
\(487\) 32.3923 1.46784 0.733918 0.679238i \(-0.237690\pi\)
0.733918 + 0.679238i \(0.237690\pi\)
\(488\) 11.3205 0.512455
\(489\) −9.26795 −0.419111
\(490\) 0.803848 0.0363141
\(491\) 16.0526 0.724442 0.362221 0.932092i \(-0.382019\pi\)
0.362221 + 0.932092i \(0.382019\pi\)
\(492\) 1.26795 0.0571636
\(493\) 0 0
\(494\) 0 0
\(495\) 4.73205 0.212690
\(496\) 44.6410 2.00444
\(497\) −12.0000 −0.538274
\(498\) 22.3923 1.00342
\(499\) 10.5359 0.471652 0.235826 0.971795i \(-0.424221\pi\)
0.235826 + 0.971795i \(0.424221\pi\)
\(500\) 1.00000 0.0447214
\(501\) 3.46410 0.154765
\(502\) 48.5885 2.16861
\(503\) −23.0718 −1.02872 −0.514360 0.857574i \(-0.671971\pi\)
−0.514360 + 0.857574i \(0.671971\pi\)
\(504\) 4.73205 0.210782
\(505\) −10.3923 −0.462451
\(506\) 28.3923 1.26219
\(507\) 12.4641 0.553550
\(508\) 4.00000 0.177471
\(509\) 10.0526 0.445572 0.222786 0.974867i \(-0.428485\pi\)
0.222786 + 0.974867i \(0.428485\pi\)
\(510\) 0 0
\(511\) 46.2487 2.04592
\(512\) 8.66025 0.382733
\(513\) 0 0
\(514\) 41.5692 1.83354
\(515\) −9.85641 −0.434325
\(516\) −4.19615 −0.184725
\(517\) −16.3923 −0.720933
\(518\) −29.3205 −1.28827
\(519\) 6.92820 0.304114
\(520\) 1.26795 0.0556033
\(521\) −37.2679 −1.63274 −0.816369 0.577530i \(-0.804017\pi\)
−0.816369 + 0.577530i \(0.804017\pi\)
\(522\) −14.1962 −0.621349
\(523\) −8.67949 −0.379528 −0.189764 0.981830i \(-0.560772\pi\)
−0.189764 + 0.981830i \(0.560772\pi\)
\(524\) −9.12436 −0.398599
\(525\) 2.73205 0.119236
\(526\) 10.3923 0.453126
\(527\) 0 0
\(528\) 23.6603 1.02968
\(529\) −11.0000 −0.478261
\(530\) 16.3923 0.712036
\(531\) −2.53590 −0.110049
\(532\) 0 0
\(533\) 0.928203 0.0402049
\(534\) 18.5885 0.804401
\(535\) 0 0
\(536\) 13.8564 0.598506
\(537\) 23.3205 1.00635
\(538\) −1.01924 −0.0439425
\(539\) 2.19615 0.0945950
\(540\) −1.00000 −0.0430331
\(541\) 41.7128 1.79337 0.896687 0.442665i \(-0.145967\pi\)
0.896687 + 0.442665i \(0.145967\pi\)
\(542\) 0.679492 0.0291867
\(543\) −2.39230 −0.102664
\(544\) 0 0
\(545\) 14.3923 0.616499
\(546\) −3.46410 −0.148250
\(547\) −43.3205 −1.85225 −0.926126 0.377215i \(-0.876882\pi\)
−0.926126 + 0.377215i \(0.876882\pi\)
\(548\) 19.8564 0.848224
\(549\) −6.53590 −0.278945
\(550\) 8.19615 0.349485
\(551\) 0 0
\(552\) 6.00000 0.255377
\(553\) −29.8564 −1.26962
\(554\) 3.46410 0.147176
\(555\) −6.19615 −0.263012
\(556\) −8.39230 −0.355913
\(557\) 0.928203 0.0393292 0.0196646 0.999807i \(-0.493740\pi\)
0.0196646 + 0.999807i \(0.493740\pi\)
\(558\) −15.4641 −0.654648
\(559\) −3.07180 −0.129923
\(560\) 13.6603 0.577251
\(561\) 0 0
\(562\) −2.19615 −0.0926391
\(563\) 27.4641 1.15747 0.578737 0.815514i \(-0.303546\pi\)
0.578737 + 0.815514i \(0.303546\pi\)
\(564\) 3.46410 0.145865
\(565\) −18.9282 −0.796315
\(566\) 43.2679 1.81869
\(567\) −2.73205 −0.114735
\(568\) −7.60770 −0.319212
\(569\) −22.0526 −0.924491 −0.462246 0.886752i \(-0.652956\pi\)
−0.462246 + 0.886752i \(0.652956\pi\)
\(570\) 0 0
\(571\) −34.2487 −1.43326 −0.716632 0.697452i \(-0.754317\pi\)
−0.716632 + 0.697452i \(0.754317\pi\)
\(572\) −3.46410 −0.144841
\(573\) −0.339746 −0.0141931
\(574\) 6.00000 0.250435
\(575\) 3.46410 0.144463
\(576\) 1.00000 0.0416667
\(577\) 15.1769 0.631823 0.315912 0.948789i \(-0.397690\pi\)
0.315912 + 0.948789i \(0.397690\pi\)
\(578\) −29.4449 −1.22474
\(579\) 17.1244 0.711664
\(580\) −8.19615 −0.340327
\(581\) 35.3205 1.46534
\(582\) −10.7321 −0.444858
\(583\) 44.7846 1.85479
\(584\) 29.3205 1.21329
\(585\) −0.732051 −0.0302666
\(586\) −48.0000 −1.98286
\(587\) −3.46410 −0.142979 −0.0714894 0.997441i \(-0.522775\pi\)
−0.0714894 + 0.997441i \(0.522775\pi\)
\(588\) −0.464102 −0.0191392
\(589\) 0 0
\(590\) −4.39230 −0.180828
\(591\) 24.0000 0.987228
\(592\) −30.9808 −1.27330
\(593\) −38.7846 −1.59269 −0.796347 0.604841i \(-0.793237\pi\)
−0.796347 + 0.604841i \(0.793237\pi\)
\(594\) −8.19615 −0.336292
\(595\) 0 0
\(596\) −19.8564 −0.813350
\(597\) 15.3205 0.627027
\(598\) −4.39230 −0.179615
\(599\) 13.8564 0.566157 0.283079 0.959097i \(-0.408644\pi\)
0.283079 + 0.959097i \(0.408644\pi\)
\(600\) 1.73205 0.0707107
\(601\) 47.1769 1.92439 0.962193 0.272368i \(-0.0878067\pi\)
0.962193 + 0.272368i \(0.0878067\pi\)
\(602\) −19.8564 −0.809287
\(603\) −8.00000 −0.325785
\(604\) −14.0000 −0.569652
\(605\) 11.3923 0.463163
\(606\) 18.0000 0.731200
\(607\) 11.6077 0.471142 0.235571 0.971857i \(-0.424304\pi\)
0.235571 + 0.971857i \(0.424304\pi\)
\(608\) 0 0
\(609\) −22.3923 −0.907382
\(610\) −11.3205 −0.458354
\(611\) 2.53590 0.102591
\(612\) 0 0
\(613\) 42.3923 1.71221 0.856105 0.516803i \(-0.172878\pi\)
0.856105 + 0.516803i \(0.172878\pi\)
\(614\) 56.1051 2.26422
\(615\) 1.26795 0.0511286
\(616\) 22.3923 0.902212
\(617\) 27.7128 1.11568 0.557838 0.829950i \(-0.311631\pi\)
0.557838 + 0.829950i \(0.311631\pi\)
\(618\) 17.0718 0.686728
\(619\) −15.3205 −0.615783 −0.307892 0.951421i \(-0.599623\pi\)
−0.307892 + 0.951421i \(0.599623\pi\)
\(620\) −8.92820 −0.358565
\(621\) −3.46410 −0.139010
\(622\) 56.1962 2.25326
\(623\) 29.3205 1.17470
\(624\) −3.66025 −0.146527
\(625\) 1.00000 0.0400000
\(626\) 11.0718 0.442518
\(627\) 0 0
\(628\) 6.39230 0.255081
\(629\) 0 0
\(630\) −4.73205 −0.188529
\(631\) −34.9282 −1.39047 −0.695235 0.718783i \(-0.744700\pi\)
−0.695235 + 0.718783i \(0.744700\pi\)
\(632\) −18.9282 −0.752923
\(633\) 1.07180 0.0426001
\(634\) 19.6077 0.778721
\(635\) 4.00000 0.158735
\(636\) −9.46410 −0.375276
\(637\) −0.339746 −0.0134612
\(638\) −67.1769 −2.65956
\(639\) 4.39230 0.173757
\(640\) 12.1244 0.479257
\(641\) −48.5885 −1.91913 −0.959564 0.281489i \(-0.909172\pi\)
−0.959564 + 0.281489i \(0.909172\pi\)
\(642\) 0 0
\(643\) −12.1962 −0.480969 −0.240485 0.970653i \(-0.577306\pi\)
−0.240485 + 0.970653i \(0.577306\pi\)
\(644\) −9.46410 −0.372938
\(645\) −4.19615 −0.165223
\(646\) 0 0
\(647\) −4.14359 −0.162901 −0.0814507 0.996677i \(-0.525955\pi\)
−0.0814507 + 0.996677i \(0.525955\pi\)
\(648\) −1.73205 −0.0680414
\(649\) −12.0000 −0.471041
\(650\) −1.26795 −0.0497331
\(651\) −24.3923 −0.956010
\(652\) 9.26795 0.362961
\(653\) 17.0718 0.668071 0.334036 0.942560i \(-0.391589\pi\)
0.334036 + 0.942560i \(0.391589\pi\)
\(654\) −24.9282 −0.974770
\(655\) −9.12436 −0.356518
\(656\) 6.33975 0.247525
\(657\) −16.9282 −0.660432
\(658\) 16.3923 0.639039
\(659\) −5.07180 −0.197569 −0.0987846 0.995109i \(-0.531495\pi\)
−0.0987846 + 0.995109i \(0.531495\pi\)
\(660\) −4.73205 −0.184195
\(661\) −39.1769 −1.52381 −0.761903 0.647692i \(-0.775735\pi\)
−0.761903 + 0.647692i \(0.775735\pi\)
\(662\) 44.5359 1.73094
\(663\) 0 0
\(664\) 22.3923 0.868990
\(665\) 0 0
\(666\) 10.7321 0.415859
\(667\) −28.3923 −1.09935
\(668\) −3.46410 −0.134030
\(669\) −17.8564 −0.690369
\(670\) −13.8564 −0.535320
\(671\) −30.9282 −1.19397
\(672\) −14.1962 −0.547628
\(673\) −17.1244 −0.660095 −0.330048 0.943964i \(-0.607065\pi\)
−0.330048 + 0.943964i \(0.607065\pi\)
\(674\) −8.87564 −0.341877
\(675\) −1.00000 −0.0384900
\(676\) −12.4641 −0.479389
\(677\) −0.679492 −0.0261150 −0.0130575 0.999915i \(-0.504156\pi\)
−0.0130575 + 0.999915i \(0.504156\pi\)
\(678\) 32.7846 1.25909
\(679\) −16.9282 −0.649645
\(680\) 0 0
\(681\) 10.3923 0.398234
\(682\) −73.1769 −2.80209
\(683\) 5.07180 0.194067 0.0970335 0.995281i \(-0.469065\pi\)
0.0970335 + 0.995281i \(0.469065\pi\)
\(684\) 0 0
\(685\) 19.8564 0.758674
\(686\) 30.9282 1.18084
\(687\) 18.5359 0.707189
\(688\) −20.9808 −0.799884
\(689\) −6.92820 −0.263944
\(690\) −6.00000 −0.228416
\(691\) 12.3923 0.471425 0.235713 0.971823i \(-0.424258\pi\)
0.235713 + 0.971823i \(0.424258\pi\)
\(692\) −6.92820 −0.263371
\(693\) −12.9282 −0.491102
\(694\) −1.60770 −0.0610273
\(695\) −8.39230 −0.318338
\(696\) −14.1962 −0.538104
\(697\) 0 0
\(698\) −38.1051 −1.44230
\(699\) 7.85641 0.297157
\(700\) −2.73205 −0.103262
\(701\) −33.7128 −1.27332 −0.636658 0.771147i \(-0.719684\pi\)
−0.636658 + 0.771147i \(0.719684\pi\)
\(702\) 1.26795 0.0478557
\(703\) 0 0
\(704\) 4.73205 0.178346
\(705\) 3.46410 0.130466
\(706\) −25.6077 −0.963758
\(707\) 28.3923 1.06780
\(708\) 2.53590 0.0953049
\(709\) −29.1769 −1.09576 −0.547881 0.836556i \(-0.684565\pi\)
−0.547881 + 0.836556i \(0.684565\pi\)
\(710\) 7.60770 0.285512
\(711\) 10.9282 0.409840
\(712\) 18.5885 0.696632
\(713\) −30.9282 −1.15827
\(714\) 0 0
\(715\) −3.46410 −0.129550
\(716\) −23.3205 −0.871528
\(717\) 9.80385 0.366131
\(718\) 0.588457 0.0219610
\(719\) −11.6603 −0.434854 −0.217427 0.976077i \(-0.569766\pi\)
−0.217427 + 0.976077i \(0.569766\pi\)
\(720\) −5.00000 −0.186339
\(721\) 26.9282 1.00286
\(722\) 0 0
\(723\) −3.07180 −0.114241
\(724\) 2.39230 0.0889093
\(725\) −8.19615 −0.304397
\(726\) −19.7321 −0.732325
\(727\) 25.6603 0.951686 0.475843 0.879530i \(-0.342143\pi\)
0.475843 + 0.879530i \(0.342143\pi\)
\(728\) −3.46410 −0.128388
\(729\) 1.00000 0.0370370
\(730\) −29.3205 −1.08520
\(731\) 0 0
\(732\) 6.53590 0.241574
\(733\) −18.7846 −0.693825 −0.346913 0.937897i \(-0.612770\pi\)
−0.346913 + 0.937897i \(0.612770\pi\)
\(734\) 28.0526 1.03544
\(735\) −0.464102 −0.0171186
\(736\) −18.0000 −0.663489
\(737\) −37.8564 −1.39446
\(738\) −2.19615 −0.0808415
\(739\) 6.14359 0.225996 0.112998 0.993595i \(-0.463955\pi\)
0.112998 + 0.993595i \(0.463955\pi\)
\(740\) 6.19615 0.227775
\(741\) 0 0
\(742\) −44.7846 −1.64409
\(743\) 3.21539 0.117961 0.0589806 0.998259i \(-0.481215\pi\)
0.0589806 + 0.998259i \(0.481215\pi\)
\(744\) −15.4641 −0.566941
\(745\) −19.8564 −0.727482
\(746\) 10.7321 0.392928
\(747\) −12.9282 −0.473018
\(748\) 0 0
\(749\) 0 0
\(750\) −1.73205 −0.0632456
\(751\) −26.0000 −0.948753 −0.474377 0.880322i \(-0.657327\pi\)
−0.474377 + 0.880322i \(0.657327\pi\)
\(752\) 17.3205 0.631614
\(753\) −28.0526 −1.02229
\(754\) 10.3923 0.378465
\(755\) −14.0000 −0.509512
\(756\) 2.73205 0.0993637
\(757\) 32.2487 1.17210 0.586050 0.810275i \(-0.300682\pi\)
0.586050 + 0.810275i \(0.300682\pi\)
\(758\) −36.2487 −1.31661
\(759\) −16.3923 −0.595003
\(760\) 0 0
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) −6.92820 −0.250982
\(763\) −39.3205 −1.42350
\(764\) 0.339746 0.0122916
\(765\) 0 0
\(766\) 29.5692 1.06838
\(767\) 1.85641 0.0670310
\(768\) −19.0000 −0.685603
\(769\) −20.6410 −0.744334 −0.372167 0.928166i \(-0.621385\pi\)
−0.372167 + 0.928166i \(0.621385\pi\)
\(770\) −22.3923 −0.806963
\(771\) −24.0000 −0.864339
\(772\) −17.1244 −0.616319
\(773\) 25.1769 0.905551 0.452775 0.891625i \(-0.350434\pi\)
0.452775 + 0.891625i \(0.350434\pi\)
\(774\) 7.26795 0.261241
\(775\) −8.92820 −0.320711
\(776\) −10.7321 −0.385258
\(777\) 16.9282 0.607296
\(778\) 13.6077 0.487860
\(779\) 0 0
\(780\) 0.732051 0.0262116
\(781\) 20.7846 0.743732
\(782\) 0 0
\(783\) 8.19615 0.292907
\(784\) −2.32051 −0.0828753
\(785\) 6.39230 0.228151
\(786\) 15.8038 0.563705
\(787\) −8.67949 −0.309390 −0.154695 0.987962i \(-0.549440\pi\)
−0.154695 + 0.987962i \(0.549440\pi\)
\(788\) −24.0000 −0.854965
\(789\) −6.00000 −0.213606
\(790\) 18.9282 0.673435
\(791\) 51.7128 1.83870
\(792\) −8.19615 −0.291238
\(793\) 4.78461 0.169906
\(794\) 15.4641 0.548800
\(795\) −9.46410 −0.335657
\(796\) −15.3205 −0.543021
\(797\) −44.7846 −1.58635 −0.793176 0.608992i \(-0.791574\pi\)
−0.793176 + 0.608992i \(0.791574\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −5.19615 −0.183712
\(801\) −10.7321 −0.379198
\(802\) 58.9808 2.08268
\(803\) −80.1051 −2.82685
\(804\) 8.00000 0.282138
\(805\) −9.46410 −0.333566
\(806\) 11.3205 0.398748
\(807\) 0.588457 0.0207147
\(808\) 18.0000 0.633238
\(809\) 14.7846 0.519799 0.259900 0.965636i \(-0.416311\pi\)
0.259900 + 0.965636i \(0.416311\pi\)
\(810\) 1.73205 0.0608581
\(811\) −37.5692 −1.31923 −0.659617 0.751602i \(-0.729281\pi\)
−0.659617 + 0.751602i \(0.729281\pi\)
\(812\) 22.3923 0.785816
\(813\) −0.392305 −0.0137587
\(814\) 50.7846 1.78000
\(815\) 9.26795 0.324642
\(816\) 0 0
\(817\) 0 0
\(818\) 45.7128 1.59831
\(819\) 2.00000 0.0698857
\(820\) −1.26795 −0.0442787
\(821\) −32.5359 −1.13551 −0.567755 0.823197i \(-0.692188\pi\)
−0.567755 + 0.823197i \(0.692188\pi\)
\(822\) −34.3923 −1.19957
\(823\) 12.9808 0.452481 0.226240 0.974071i \(-0.427356\pi\)
0.226240 + 0.974071i \(0.427356\pi\)
\(824\) 17.0718 0.594724
\(825\) −4.73205 −0.164749
\(826\) 12.0000 0.417533
\(827\) 5.32051 0.185012 0.0925061 0.995712i \(-0.470512\pi\)
0.0925061 + 0.995712i \(0.470512\pi\)
\(828\) 3.46410 0.120386
\(829\) −34.1051 −1.18452 −0.592260 0.805747i \(-0.701764\pi\)
−0.592260 + 0.805747i \(0.701764\pi\)
\(830\) −22.3923 −0.777248
\(831\) −2.00000 −0.0693792
\(832\) −0.732051 −0.0253793
\(833\) 0 0
\(834\) 14.5359 0.503337
\(835\) −3.46410 −0.119880
\(836\) 0 0
\(837\) 8.92820 0.308604
\(838\) −48.5885 −1.67846
\(839\) 19.6077 0.676933 0.338466 0.940978i \(-0.390092\pi\)
0.338466 + 0.940978i \(0.390092\pi\)
\(840\) −4.73205 −0.163271
\(841\) 38.1769 1.31645
\(842\) 32.5359 1.12126
\(843\) 1.26795 0.0436705
\(844\) −1.07180 −0.0368928
\(845\) −12.4641 −0.428778
\(846\) −6.00000 −0.206284
\(847\) −31.1244 −1.06945
\(848\) −47.3205 −1.62499
\(849\) −24.9808 −0.857338
\(850\) 0 0
\(851\) 21.4641 0.735780
\(852\) −4.39230 −0.150478
\(853\) 27.1769 0.930520 0.465260 0.885174i \(-0.345961\pi\)
0.465260 + 0.885174i \(0.345961\pi\)
\(854\) 30.9282 1.05834
\(855\) 0 0
\(856\) 0 0
\(857\) 42.2487 1.44319 0.721594 0.692316i \(-0.243410\pi\)
0.721594 + 0.692316i \(0.243410\pi\)
\(858\) 6.00000 0.204837
\(859\) 32.0000 1.09183 0.545913 0.837842i \(-0.316183\pi\)
0.545913 + 0.837842i \(0.316183\pi\)
\(860\) 4.19615 0.143088
\(861\) −3.46410 −0.118056
\(862\) 19.6077 0.667841
\(863\) 12.0000 0.408485 0.204242 0.978920i \(-0.434527\pi\)
0.204242 + 0.978920i \(0.434527\pi\)
\(864\) 5.19615 0.176777
\(865\) −6.92820 −0.235566
\(866\) 18.3397 0.623210
\(867\) 17.0000 0.577350
\(868\) 24.3923 0.827929
\(869\) 51.7128 1.75424
\(870\) 14.1962 0.481295
\(871\) 5.85641 0.198437
\(872\) −24.9282 −0.844175
\(873\) 6.19615 0.209708
\(874\) 0 0
\(875\) −2.73205 −0.0923602
\(876\) 16.9282 0.571951
\(877\) −53.1244 −1.79388 −0.896941 0.442150i \(-0.854216\pi\)
−0.896941 + 0.442150i \(0.854216\pi\)
\(878\) −46.6410 −1.57406
\(879\) 27.7128 0.934730
\(880\) −23.6603 −0.797587
\(881\) 8.53590 0.287582 0.143791 0.989608i \(-0.454071\pi\)
0.143791 + 0.989608i \(0.454071\pi\)
\(882\) 0.803848 0.0270670
\(883\) 36.9808 1.24450 0.622251 0.782818i \(-0.286218\pi\)
0.622251 + 0.782818i \(0.286218\pi\)
\(884\) 0 0
\(885\) 2.53590 0.0852433
\(886\) −9.21539 −0.309597
\(887\) 24.0000 0.805841 0.402921 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) 10.7321 0.360144
\(889\) −10.9282 −0.366520
\(890\) −18.5885 −0.623087
\(891\) 4.73205 0.158530
\(892\) 17.8564 0.597877
\(893\) 0 0
\(894\) 34.3923 1.15025
\(895\) −23.3205 −0.779519
\(896\) −33.1244 −1.10661
\(897\) 2.53590 0.0846712
\(898\) 9.80385 0.327159
\(899\) 73.1769 2.44059
\(900\) 1.00000 0.0333333
\(901\) 0 0
\(902\) −10.3923 −0.346026
\(903\) 11.4641 0.381501
\(904\) 32.7846 1.09040
\(905\) 2.39230 0.0795229
\(906\) 24.2487 0.805609
\(907\) 32.3923 1.07557 0.537784 0.843082i \(-0.319261\pi\)
0.537784 + 0.843082i \(0.319261\pi\)
\(908\) −10.3923 −0.344881
\(909\) −10.3923 −0.344691
\(910\) 3.46410 0.114834
\(911\) 54.9282 1.81985 0.909926 0.414770i \(-0.136138\pi\)
0.909926 + 0.414770i \(0.136138\pi\)
\(912\) 0 0
\(913\) −61.1769 −2.02466
\(914\) 7.85641 0.259867
\(915\) 6.53590 0.216070
\(916\) −18.5359 −0.612443
\(917\) 24.9282 0.823202
\(918\) 0 0
\(919\) 51.4256 1.69637 0.848187 0.529696i \(-0.177694\pi\)
0.848187 + 0.529696i \(0.177694\pi\)
\(920\) −6.00000 −0.197814
\(921\) −32.3923 −1.06736
\(922\) 10.3923 0.342252
\(923\) −3.21539 −0.105836
\(924\) 12.9282 0.425307
\(925\) 6.19615 0.203728
\(926\) −61.5167 −2.02156
\(927\) −9.85641 −0.323727
\(928\) 42.5885 1.39803
\(929\) −1.60770 −0.0527468 −0.0263734 0.999652i \(-0.508396\pi\)
−0.0263734 + 0.999652i \(0.508396\pi\)
\(930\) 15.4641 0.507088
\(931\) 0 0
\(932\) −7.85641 −0.257345
\(933\) −32.4449 −1.06220
\(934\) 35.5692 1.16386
\(935\) 0 0
\(936\) 1.26795 0.0414442
\(937\) −16.2487 −0.530822 −0.265411 0.964135i \(-0.585508\pi\)
−0.265411 + 0.964135i \(0.585508\pi\)
\(938\) 37.8564 1.23606
\(939\) −6.39230 −0.208605
\(940\) −3.46410 −0.112987
\(941\) −0.588457 −0.0191832 −0.00959158 0.999954i \(-0.503053\pi\)
−0.00959158 + 0.999954i \(0.503053\pi\)
\(942\) −11.0718 −0.360739
\(943\) −4.39230 −0.143033
\(944\) 12.6795 0.412682
\(945\) 2.73205 0.0888736
\(946\) 34.3923 1.11819
\(947\) 28.1436 0.914544 0.457272 0.889327i \(-0.348827\pi\)
0.457272 + 0.889327i \(0.348827\pi\)
\(948\) −10.9282 −0.354932
\(949\) 12.3923 0.402271
\(950\) 0 0
\(951\) −11.3205 −0.367093
\(952\) 0 0
\(953\) −37.8564 −1.22629 −0.613145 0.789971i \(-0.710096\pi\)
−0.613145 + 0.789971i \(0.710096\pi\)
\(954\) 16.3923 0.530720
\(955\) 0.339746 0.0109939
\(956\) −9.80385 −0.317079
\(957\) 38.7846 1.25373
\(958\) 44.1962 1.42791
\(959\) −54.2487 −1.75178
\(960\) −1.00000 −0.0322749
\(961\) 48.7128 1.57138
\(962\) −7.85641 −0.253301
\(963\) 0 0
\(964\) 3.07180 0.0989359
\(965\) −17.1244 −0.551253
\(966\) 16.3923 0.527414
\(967\) 4.87564 0.156790 0.0783951 0.996922i \(-0.475020\pi\)
0.0783951 + 0.996922i \(0.475020\pi\)
\(968\) −19.7321 −0.634212
\(969\) 0 0
\(970\) 10.7321 0.344585
\(971\) 27.7128 0.889346 0.444673 0.895693i \(-0.353320\pi\)
0.444673 + 0.895693i \(0.353320\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 22.9282 0.735044
\(974\) 56.1051 1.79772
\(975\) 0.732051 0.0234444
\(976\) 32.6795 1.04605
\(977\) 39.0333 1.24879 0.624393 0.781110i \(-0.285346\pi\)
0.624393 + 0.781110i \(0.285346\pi\)
\(978\) −16.0526 −0.513304
\(979\) −50.7846 −1.62308
\(980\) 0.464102 0.0148252
\(981\) 14.3923 0.459511
\(982\) 27.8038 0.887256
\(983\) 41.3205 1.31792 0.658960 0.752178i \(-0.270997\pi\)
0.658960 + 0.752178i \(0.270997\pi\)
\(984\) −2.19615 −0.0700108
\(985\) −24.0000 −0.764704
\(986\) 0 0
\(987\) −9.46410 −0.301246
\(988\) 0 0
\(989\) 14.5359 0.462215
\(990\) 8.19615 0.260491
\(991\) −13.0718 −0.415239 −0.207620 0.978210i \(-0.566572\pi\)
−0.207620 + 0.978210i \(0.566572\pi\)
\(992\) 46.3923 1.47296
\(993\) −25.7128 −0.815971
\(994\) −20.7846 −0.659248
\(995\) −15.3205 −0.485693
\(996\) 12.9282 0.409646
\(997\) −17.6077 −0.557641 −0.278821 0.960343i \(-0.589944\pi\)
−0.278821 + 0.960343i \(0.589944\pi\)
\(998\) 18.2487 0.577653
\(999\) −6.19615 −0.196038
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 5415.2.a.r.1.2 2
19.18 odd 2 285.2.a.e.1.1 2
57.56 even 2 855.2.a.f.1.2 2
76.75 even 2 4560.2.a.bh.1.2 2
95.18 even 4 1425.2.c.k.799.4 4
95.37 even 4 1425.2.c.k.799.1 4
95.94 odd 2 1425.2.a.o.1.2 2
285.284 even 2 4275.2.a.t.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.e.1.1 2 19.18 odd 2
855.2.a.f.1.2 2 57.56 even 2
1425.2.a.o.1.2 2 95.94 odd 2
1425.2.c.k.799.1 4 95.37 even 4
1425.2.c.k.799.4 4 95.18 even 4
4275.2.a.t.1.1 2 285.284 even 2
4560.2.a.bh.1.2 2 76.75 even 2
5415.2.a.r.1.2 2 1.1 even 1 trivial