Properties

Label 285.2.a.e.1.1
Level $285$
Weight $2$
Character 285.1
Self dual yes
Analytic conductor $2.276$
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] = [285,2,Mod(1,285)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(285, 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("285.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 285 = 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 285.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: \(2.27573645761\)
Analytic rank: \(0\)
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: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 285.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73205 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.73205 q^{6} -2.73205 q^{7} +1.73205 q^{8} +1.00000 q^{9} -1.73205 q^{10} +4.73205 q^{11} +1.00000 q^{12} +0.732051 q^{13} +4.73205 q^{14} +1.00000 q^{15} -5.00000 q^{16} -1.73205 q^{18} +1.00000 q^{19} +1.00000 q^{20} -2.73205 q^{21} -8.19615 q^{22} +3.46410 q^{23} +1.73205 q^{24} +1.00000 q^{25} -1.26795 q^{26} +1.00000 q^{27} -2.73205 q^{28} +8.19615 q^{29} -1.73205 q^{30} +8.92820 q^{31} +5.19615 q^{32} +4.73205 q^{33} -2.73205 q^{35} +1.00000 q^{36} -6.19615 q^{37} -1.73205 q^{38} +0.732051 q^{39} +1.73205 q^{40} +1.26795 q^{41} +4.73205 q^{42} +4.19615 q^{43} +4.73205 q^{44} +1.00000 q^{45} -6.00000 q^{46} -3.46410 q^{47} -5.00000 q^{48} +0.464102 q^{49} -1.73205 q^{50} +0.732051 q^{52} -9.46410 q^{53} -1.73205 q^{54} +4.73205 q^{55} -4.73205 q^{56} +1.00000 q^{57} -14.1962 q^{58} +2.53590 q^{59} +1.00000 q^{60} -6.53590 q^{61} -15.4641 q^{62} -2.73205 q^{63} +1.00000 q^{64} +0.732051 q^{65} -8.19615 q^{66} +8.00000 q^{67} +3.46410 q^{69} +4.73205 q^{70} -4.39230 q^{71} +1.73205 q^{72} -16.9282 q^{73} +10.7321 q^{74} +1.00000 q^{75} +1.00000 q^{76} -12.9282 q^{77} -1.26795 q^{78} -10.9282 q^{79} -5.00000 q^{80} +1.00000 q^{81} -2.19615 q^{82} -12.9282 q^{83} -2.73205 q^{84} -7.26795 q^{86} +8.19615 q^{87} +8.19615 q^{88} +10.7321 q^{89} -1.73205 q^{90} -2.00000 q^{91} +3.46410 q^{92} +8.92820 q^{93} +6.00000 q^{94} +1.00000 q^{95} +5.19615 q^{96} -6.19615 q^{97} -0.803848 q^{98} +4.73205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{7} + 2 q^{9} + 6 q^{11} + 2 q^{12} - 2 q^{13} + 6 q^{14} + 2 q^{15} - 10 q^{16} + 2 q^{19} + 2 q^{20} - 2 q^{21} - 6 q^{22} + 2 q^{25} - 6 q^{26} + 2 q^{27} - 2 q^{28}+ \cdots + 6 q^{99}+O(q^{100}) \) 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 \(-0.709785\pi\)
−0.612372 + 0.790569i \(0.709785\pi\)
\(3\) 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 \(-0.467630\pi\)
0.101517 + 0.994834i \(0.467630\pi\)
\(14\) 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
\(19\) 1.00000 0.229416
\(20\) 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 \(-0.224712\pi\)
0.760994 + 0.648759i \(0.224712\pi\)
\(30\) −1.73205 −0.316228
\(31\) 8.92820 1.60355 0.801776 0.597624i \(-0.203889\pi\)
0.801776 + 0.597624i \(0.203889\pi\)
\(32\) 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 \(-0.670103\pi\)
−0.509321 + 0.860577i \(0.670103\pi\)
\(38\) −1.73205 −0.280976
\(39\) 0.732051 0.117222
\(40\) 1.73205 0.273861
\(41\) 1.26795 0.198020 0.0990102 0.995086i \(-0.468432\pi\)
0.0990102 + 0.995086i \(0.468432\pi\)
\(42\) 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 \(-0.725230\pi\)
−0.649997 + 0.759937i \(0.725230\pi\)
\(54\) −1.73205 −0.235702
\(55\) 4.73205 0.638070
\(56\) −4.73205 −0.632347
\(57\) 1.00000 0.132453
\(58\) −14.1962 −1.86405
\(59\) 2.53590 0.330146 0.165073 0.986281i \(-0.447214\pi\)
0.165073 + 0.986281i \(0.447214\pi\)
\(60\) 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 \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 0 0
\(69\) 3.46410 0.417029
\(70\) 4.73205 0.565588
\(71\) −4.39230 −0.521271 −0.260635 0.965437i \(-0.583932\pi\)
−0.260635 + 0.965437i \(0.583932\pi\)
\(72\) 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
\(76\) 1.00000 0.114708
\(77\) −12.9282 −1.47331
\(78\) −1.26795 −0.143567
\(79\) −10.9282 −1.22952 −0.614759 0.788715i \(-0.710747\pi\)
−0.614759 + 0.788715i \(0.710747\pi\)
\(80\) −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 \(-0.307409\pi\)
0.568798 + 0.822478i \(0.307409\pi\)
\(90\) −1.73205 −0.182574
\(91\) −2.00000 −0.209657
\(92\) 3.46410 0.361158
\(93\) 8.92820 0.925812
\(94\) 6.00000 0.618853
\(95\) 1.00000 0.102598
\(96\) 5.19615 0.530330
\(97\) −6.19615 −0.629124 −0.314562 0.949237i \(-0.601858\pi\)
−0.314562 + 0.949237i \(0.601858\pi\)
\(98\) −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 \(-0.338605\pi\)
0.485590 + 0.874187i \(0.338605\pi\)
\(104\) 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 \(-0.742067\pi\)
−0.689266 + 0.724508i \(0.742067\pi\)
\(110\) −8.19615 −0.781472
\(111\) −6.19615 −0.588113
\(112\) 13.6603 1.29077
\(113\) 18.9282 1.78062 0.890308 0.455359i \(-0.150489\pi\)
0.890308 + 0.455359i \(0.150489\pi\)
\(114\) −1.73205 −0.162221
\(115\) 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 \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) −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
\(133\) −2.73205 −0.236899
\(134\) −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 \(-0.307078\pi\)
0.569652 + 0.821886i \(0.307078\pi\)
\(152\) 1.73205 0.140488
\(153\) 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 \(-0.457208\pi\)
0.134030 + 0.990977i \(0.457208\pi\)
\(168\) −4.73205 −0.365086
\(169\) −12.4641 −0.958777
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 4.19615 0.319954
\(173\) 6.92820 0.526742 0.263371 0.964695i \(-0.415166\pi\)
0.263371 + 0.964695i \(0.415166\pi\)
\(174\) −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 \(-0.163129\pi\)
0.871528 + 0.490345i \(0.163129\pi\)
\(180\) 1.00000 0.0745356
\(181\) −2.39230 −0.177819 −0.0889093 0.996040i \(-0.528338\pi\)
−0.0889093 + 0.996040i \(0.528338\pi\)
\(182\) 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
\(190\) −1.73205 −0.125656
\(191\) 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 \(-0.288623\pi\)
0.616319 + 0.787497i \(0.288623\pi\)
\(194\) 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
\(209\) 4.73205 0.327323
\(210\) 4.73205 0.326543
\(211\) 1.07180 0.0737855 0.0368928 0.999319i \(-0.488254\pi\)
0.0368928 + 0.999319i \(0.488254\pi\)
\(212\) −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 \(-0.703989\pi\)
−0.597877 + 0.801588i \(0.703989\pi\)
\(224\) −14.1962 −0.948520
\(225\) 1.00000 0.0666667
\(226\) −32.7846 −2.18080
\(227\) 10.3923 0.689761 0.344881 0.938647i \(-0.387919\pi\)
0.344881 + 0.938647i \(0.387919\pi\)
\(228\) 1.00000 0.0662266
\(229\) −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 \(-0.531544\pi\)
−0.0989359 + 0.995094i \(0.531544\pi\)
\(242\) −19.7321 −1.26842
\(243\) 1.00000 0.0641500
\(244\) −6.53590 −0.418418
\(245\) 0.464102 0.0296504
\(246\) −2.19615 −0.140022
\(247\) 0.732051 0.0465793
\(248\) 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 \(-0.769245\pi\)
−0.748539 + 0.663090i \(0.769245\pi\)
\(258\) −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
\(266\) 4.73205 0.290141
\(267\) 10.7321 0.656791
\(268\) 8.00000 0.488678
\(269\) 0.588457 0.0358789 0.0179394 0.999839i \(-0.494289\pi\)
0.0179394 + 0.999839i \(0.494289\pi\)
\(270\) −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 \(-0.487959\pi\)
0.0378198 + 0.999285i \(0.487959\pi\)
\(282\) 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
\(285\) 1.00000 0.0592349
\(286\) −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 \(-0.199738\pi\)
0.809500 + 0.587120i \(0.199738\pi\)
\(294\) −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
\(304\) −5.00000 −0.286770
\(305\) −6.53590 −0.374244
\(306\) 0 0
\(307\) −32.3923 −1.84873 −0.924363 0.381514i \(-0.875403\pi\)
−0.924363 + 0.381514i \(0.875403\pi\)
\(308\) −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 \(-0.602982\pi\)
−0.317912 + 0.948120i \(0.602982\pi\)
\(318\) 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 \(-0.749795\pi\)
−0.706652 + 0.707561i \(0.749795\pi\)
\(332\) −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 \(-0.455428\pi\)
0.139571 + 0.990212i \(0.455428\pi\)
\(338\) 21.5885 1.17426
\(339\) 18.9282 1.02804
\(340\) 0 0
\(341\) 42.2487 2.28790
\(342\) −1.73205 −0.0936586
\(343\) 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
\(361\) 1.00000 0.0526316
\(362\) 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 \(-0.551282\pi\)
−0.160412 + 0.987050i \(0.551282\pi\)
\(374\) 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 \(-0.319367\pi\)
0.537505 + 0.843261i \(0.319367\pi\)
\(380\) 1.00000 0.0512989
\(381\) −4.00000 −0.204926
\(382\) −0.588457 −0.0301081
\(383\) −17.0718 −0.872328 −0.436164 0.899867i \(-0.643663\pi\)
−0.436164 + 0.899867i \(0.643663\pi\)
\(384\) −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
\(399\) −2.73205 −0.136774
\(400\) −5.00000 −0.250000
\(401\) −34.0526 −1.70050 −0.850252 0.526376i \(-0.823550\pi\)
−0.850252 + 0.526376i \(0.823550\pi\)
\(402\) −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 \(-0.726283\pi\)
−0.652508 + 0.757782i \(0.726283\pi\)
\(410\) −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
\(418\) −8.19615 −0.400887
\(419\) −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 \(-0.651346\pi\)
−0.457753 + 0.889079i \(0.651346\pi\)
\(422\) −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 \(-0.587898\pi\)
−0.272645 + 0.962115i \(0.587898\pi\)
\(432\) −5.00000 −0.240563
\(433\) −10.5885 −0.508849 −0.254424 0.967093i \(-0.581886\pi\)
−0.254424 + 0.967093i \(0.581886\pi\)
\(434\) 42.2487 2.02800
\(435\) 8.19615 0.392975
\(436\) −14.3923 −0.689266
\(437\) 3.46410 0.165710
\(438\) 29.3205 1.40099
\(439\) 26.9282 1.28521 0.642607 0.766196i \(-0.277853\pi\)
0.642607 + 0.766196i \(0.277853\pi\)
\(440\) 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 \(-0.542642\pi\)
−0.133562 + 0.991040i \(0.542642\pi\)
\(450\) −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
\(456\) 1.73205 0.0811107
\(457\) 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
\(475\) 1.00000 0.0458831
\(476\) 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 \(-0.762310\pi\)
−0.733918 + 0.679238i \(0.762310\pi\)
\(488\) −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
\(494\) −1.26795 −0.0570477
\(495\) 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 \(-0.571515\pi\)
−0.222786 + 0.974867i \(0.571515\pi\)
\(510\) 0 0
\(511\) 46.2487 2.04592
\(512\) −8.66025 −0.382733
\(513\) 1.00000 0.0441511
\(514\) 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 \(-0.195983\pi\)
0.816369 + 0.577530i \(0.195983\pi\)
\(522\) −14.1962 −0.621349
\(523\) 8.67949 0.379528 0.189764 0.981830i \(-0.439228\pi\)
0.189764 + 0.981830i \(0.439228\pi\)
\(524\) −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
\(532\) −2.73205 −0.118449
\(533\) 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 \(-0.123118\pi\)
0.926126 + 0.377215i \(0.123118\pi\)
\(548\) 19.8564 0.848224
\(549\) −6.53590 −0.278945
\(550\) −8.19615 −0.349485
\(551\) 8.19615 0.349168
\(552\) 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 \(-0.696454\pi\)
−0.578737 + 0.815514i \(0.696454\pi\)
\(564\) −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 \(-0.347044\pi\)
0.462246 + 0.886752i \(0.347044\pi\)
\(570\) −1.73205 −0.0725476
\(571\) −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
\(589\) 8.92820 0.367880
\(590\) −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 \(-0.591356\pi\)
−0.283079 + 0.959097i \(0.591356\pi\)
\(600\) 1.73205 0.0707107
\(601\) −47.1769 −1.92439 −0.962193 0.272368i \(-0.912193\pi\)
−0.962193 + 0.272368i \(0.912193\pi\)
\(602\) 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 \(-0.575696\pi\)
−0.235571 + 0.971857i \(0.575696\pi\)
\(608\) 5.19615 0.210732
\(609\) −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
\(627\) 4.73205 0.188980
\(628\) 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 \(-0.0908284\pi\)
0.959564 + 0.281489i \(0.0908284\pi\)
\(642\) 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 \(-0.468505\pi\)
0.0987846 + 0.995109i \(0.468505\pi\)
\(660\) 4.73205 0.184195
\(661\) 39.1769 1.52381 0.761903 0.647692i \(-0.224265\pi\)
0.761903 + 0.647692i \(0.224265\pi\)
\(662\) 44.5359 1.73094
\(663\) 0 0
\(664\) −22.3923 −0.868990
\(665\) −2.73205 −0.105944
\(666\) 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 \(-0.392935\pi\)
0.330048 + 0.943964i \(0.392935\pi\)
\(674\) −8.87564 −0.341877
\(675\) 1.00000 0.0384900
\(676\) −12.4641 −0.479389
\(677\) 0.679492 0.0261150 0.0130575 0.999915i \(-0.495844\pi\)
0.0130575 + 0.999915i \(0.495844\pi\)
\(678\) −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 \(-0.530935\pi\)
−0.0970335 + 0.995281i \(0.530935\pi\)
\(684\) 1.00000 0.0382360
\(685\) 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
\(703\) −6.19615 −0.233692
\(704\) 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
\(722\) −1.73205 −0.0644603
\(723\) −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
\(741\) 0.732051 0.0268926
\(742\) −44.7846 −1.64409
\(743\) −3.21539 −0.117961 −0.0589806 0.998259i \(-0.518785\pi\)
−0.0589806 + 0.998259i \(0.518785\pi\)
\(744\) 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 \(-0.342673\pi\)
0.474377 + 0.880322i \(0.342673\pi\)
\(752\) 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
\(760\) 1.73205 0.0628281
\(761\) 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 \(-0.649566\pi\)
−0.452775 + 0.891625i \(0.649566\pi\)
\(774\) −7.26795 −0.261241
\(775\) 8.92820 0.320711
\(776\) −10.7321 −0.385258
\(777\) 16.9282 0.607296
\(778\) −13.6077 −0.487860
\(779\) 1.26795 0.0454290
\(780\) 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 \(-0.450560\pi\)
0.154695 + 0.987962i \(0.450560\pi\)
\(788\) −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 \(-0.208426\pi\)
0.793176 + 0.608992i \(0.208426\pi\)
\(798\) 4.73205 0.167513
\(799\) 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 \(-0.270719\pi\)
0.659617 + 0.751602i \(0.270719\pi\)
\(812\) −22.3923 −0.785816
\(813\) 0.392305 0.0137587
\(814\) 50.7846 1.78000
\(815\) 9.26795 0.324642
\(816\) 0 0
\(817\) 4.19615 0.146805
\(818\) 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 \(-0.529488\pi\)
−0.0925061 + 0.995712i \(0.529488\pi\)
\(828\) 3.46410 0.120386
\(829\) 34.1051 1.18452 0.592260 0.805747i \(-0.298236\pi\)
0.592260 + 0.805747i \(0.298236\pi\)
\(830\) 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
\(836\) 4.73205 0.163661
\(837\) 8.92820 0.308604
\(838\) 48.5885 1.67846
\(839\) −19.6077 −0.676933 −0.338466 0.940978i \(-0.609908\pi\)
−0.338466 + 0.940978i \(0.609908\pi\)
\(840\) −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
\(855\) 1.00000 0.0341993
\(856\) 0 0
\(857\) −42.2487 −1.44319 −0.721594 0.692316i \(-0.756590\pi\)
−0.721594 + 0.692316i \(0.756590\pi\)
\(858\) −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 \(-0.565473\pi\)
−0.204242 + 0.978920i \(0.565473\pi\)
\(864\) 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
\(874\) −6.00000 −0.202953
\(875\) −2.73205 −0.0923602
\(876\) −16.9282 −0.571951
\(877\) 53.1244 1.79388 0.896941 0.442150i \(-0.145784\pi\)
0.896941 + 0.442150i \(0.145784\pi\)
\(878\) −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 \(-0.632005\pi\)
−0.402921 + 0.915235i \(0.632005\pi\)
\(888\) −10.7321 −0.360144
\(889\) 10.9282 0.366520
\(890\) −18.5885 −0.623087
\(891\) 4.73205 0.158530
\(892\) −17.8564 −0.597877
\(893\) −3.46410 −0.115922
\(894\) 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 \(-0.680739\pi\)
−0.537784 + 0.843082i \(0.680739\pi\)
\(908\) 10.3923 0.344881
\(909\) −10.3923 −0.344691
\(910\) 3.46410 0.114834
\(911\) −54.9282 −1.81985 −0.909926 0.414770i \(-0.863862\pi\)
−0.909926 + 0.414770i \(0.863862\pi\)
\(912\) −5.00000 −0.165567
\(913\) −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
\(931\) 0.464102 0.0152103
\(932\) −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 \(-0.496947\pi\)
0.00959158 + 0.999954i \(0.496947\pi\)
\(942\) −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
\(950\) −1.73205 −0.0561951
\(951\) −11.3205 −0.367093
\(952\) 0 0
\(953\) 37.8564 1.22629 0.613145 0.789971i \(-0.289904\pi\)
0.613145 + 0.789971i \(0.289904\pi\)
\(954\) 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 \(-0.646680\pi\)
−0.444673 + 0.895693i \(0.646680\pi\)
\(972\) 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 \(-0.714654\pi\)
−0.624393 + 0.781110i \(0.714654\pi\)
\(978\) −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 \(-0.729003\pi\)
−0.658960 + 0.752178i \(0.729003\pi\)
\(984\) 2.19615 0.0700108
\(985\) −24.0000 −0.764704
\(986\) 0 0
\(987\) 9.46410 0.301246
\(988\) 0.732051 0.0232896
\(989\) 14.5359 0.462215
\(990\) −8.19615 −0.260491
\(991\) 13.0718 0.415239 0.207620 0.978210i \(-0.433428\pi\)
0.207620 + 0.978210i \(0.433428\pi\)
\(992\) 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 285.2.a.e.1.1 2
3.2 odd 2 855.2.a.f.1.2 2
4.3 odd 2 4560.2.a.bh.1.2 2
5.2 odd 4 1425.2.c.k.799.1 4
5.3 odd 4 1425.2.c.k.799.4 4
5.4 even 2 1425.2.a.o.1.2 2
15.14 odd 2 4275.2.a.t.1.1 2
19.18 odd 2 5415.2.a.r.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.e.1.1 2 1.1 even 1 trivial
855.2.a.f.1.2 2 3.2 odd 2
1425.2.a.o.1.2 2 5.4 even 2
1425.2.c.k.799.1 4 5.2 odd 4
1425.2.c.k.799.4 4 5.3 odd 4
4275.2.a.t.1.1 2 15.14 odd 2
4560.2.a.bh.1.2 2 4.3 odd 2
5415.2.a.r.1.2 2 19.18 odd 2