Properties

Label 855.2.a.f.1.2
Level $855$
Weight $2$
Character 855.1
Self dual yes
Analytic conductor $6.827$
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] = [855,2,Mod(1,855)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(855, 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("855.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 855.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: \(6.82720937282\)
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 \(\chi\) \(=\) 855.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.73205 q^{2} +1.00000 q^{4} -1.00000 q^{5} -2.73205 q^{7} -1.73205 q^{8} +O(q^{10})\) \(q+1.73205 q^{2} +1.00000 q^{4} -1.00000 q^{5} -2.73205 q^{7} -1.73205 q^{8} -1.73205 q^{10} -4.73205 q^{11} +0.732051 q^{13} -4.73205 q^{14} -5.00000 q^{16} +1.00000 q^{19} -1.00000 q^{20} -8.19615 q^{22} -3.46410 q^{23} +1.00000 q^{25} +1.26795 q^{26} -2.73205 q^{28} -8.19615 q^{29} +8.92820 q^{31} -5.19615 q^{32} +2.73205 q^{35} -6.19615 q^{37} +1.73205 q^{38} +1.73205 q^{40} -1.26795 q^{41} +4.19615 q^{43} -4.73205 q^{44} -6.00000 q^{46} +3.46410 q^{47} +0.464102 q^{49} +1.73205 q^{50} +0.732051 q^{52} +9.46410 q^{53} +4.73205 q^{55} +4.73205 q^{56} -14.1962 q^{58} -2.53590 q^{59} -6.53590 q^{61} +15.4641 q^{62} +1.00000 q^{64} -0.732051 q^{65} +8.00000 q^{67} +4.73205 q^{70} +4.39230 q^{71} -16.9282 q^{73} -10.7321 q^{74} +1.00000 q^{76} +12.9282 q^{77} -10.9282 q^{79} +5.00000 q^{80} -2.19615 q^{82} +12.9282 q^{83} +7.26795 q^{86} +8.19615 q^{88} -10.7321 q^{89} -2.00000 q^{91} -3.46410 q^{92} +6.00000 q^{94} -1.00000 q^{95} -6.19615 q^{97} +0.803848 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} - 2 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} - 2 q^{5} - 2 q^{7} - 6 q^{11} - 2 q^{13} - 6 q^{14} - 10 q^{16} + 2 q^{19} - 2 q^{20} - 6 q^{22} + 2 q^{25} + 6 q^{26} - 2 q^{28} - 6 q^{29} + 4 q^{31} + 2 q^{35} - 2 q^{37} - 6 q^{41} - 2 q^{43} - 6 q^{44} - 12 q^{46} - 6 q^{49} - 2 q^{52} + 12 q^{53} + 6 q^{55} + 6 q^{56} - 18 q^{58} - 12 q^{59} - 20 q^{61} + 24 q^{62} + 2 q^{64} + 2 q^{65} + 16 q^{67} + 6 q^{70} - 12 q^{71} - 20 q^{73} - 18 q^{74} + 2 q^{76} + 12 q^{77} - 8 q^{79} + 10 q^{80} + 6 q^{82} + 12 q^{83} + 18 q^{86} + 6 q^{88} - 18 q^{89} - 4 q^{91} + 12 q^{94} - 2 q^{95} - 2 q^{97} + 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.73205 1.22474 0.612372 0.790569i \(-0.290215\pi\)
0.612372 + 0.790569i \(0.290215\pi\)
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −2.73205 −1.03262 −0.516309 0.856402i \(-0.672694\pi\)
−0.516309 + 0.856402i \(0.672694\pi\)
\(8\) −1.73205 −0.612372
\(9\) 0 0
\(10\) −1.73205 −0.547723
\(11\) −4.73205 −1.42677 −0.713384 0.700774i \(-0.752838\pi\)
−0.713384 + 0.700774i \(0.752838\pi\)
\(12\) 0 0
\(13\) 0.732051 0.203034 0.101517 0.994834i \(-0.467630\pi\)
0.101517 + 0.994834i \(0.467630\pi\)
\(14\) −4.73205 −1.26469
\(15\) 0 0
\(16\) −5.00000 −1.25000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) −8.19615 −1.74743
\(23\) −3.46410 −0.722315 −0.361158 0.932505i \(-0.617618\pi\)
−0.361158 + 0.932505i \(0.617618\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 1.26795 0.248665
\(27\) 0 0
\(28\) −2.73205 −0.516309
\(29\) −8.19615 −1.52199 −0.760994 0.648759i \(-0.775288\pi\)
−0.760994 + 0.648759i \(0.775288\pi\)
\(30\) 0 0
\(31\) 8.92820 1.60355 0.801776 0.597624i \(-0.203889\pi\)
0.801776 + 0.597624i \(0.203889\pi\)
\(32\) −5.19615 −0.918559
\(33\) 0 0
\(34\) 0 0
\(35\) 2.73205 0.461801
\(36\) 0 0
\(37\) −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 0
\(40\) 1.73205 0.273861
\(41\) −1.26795 −0.198020 −0.0990102 0.995086i \(-0.531568\pi\)
−0.0990102 + 0.995086i \(0.531568\pi\)
\(42\) 0 0
\(43\) 4.19615 0.639907 0.319954 0.947433i \(-0.396333\pi\)
0.319954 + 0.947433i \(0.396333\pi\)
\(44\) −4.73205 −0.713384
\(45\) 0 0
\(46\) −6.00000 −0.884652
\(47\) 3.46410 0.505291 0.252646 0.967559i \(-0.418699\pi\)
0.252646 + 0.967559i \(0.418699\pi\)
\(48\) 0 0
\(49\) 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 \(0.274770\pi\)
\(54\) 0 0
\(55\) 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 \(0.552786\pi\)
\(60\) 0 0
\(61\) −6.53590 −0.836836 −0.418418 0.908255i \(-0.637415\pi\)
−0.418418 + 0.908255i \(0.637415\pi\)
\(62\) 15.4641 1.96394
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −0.732051 −0.0907997
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 4.73205 0.565588
\(71\) 4.39230 0.521271 0.260635 0.965437i \(-0.416068\pi\)
0.260635 + 0.965437i \(0.416068\pi\)
\(72\) 0 0
\(73\) −16.9282 −1.98130 −0.990648 0.136441i \(-0.956434\pi\)
−0.990648 + 0.136441i \(0.956434\pi\)
\(74\) −10.7321 −1.24758
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) 12.9282 1.47331
\(78\) 0 0
\(79\) −10.9282 −1.22952 −0.614759 0.788715i \(-0.710747\pi\)
−0.614759 + 0.788715i \(0.710747\pi\)
\(80\) 5.00000 0.559017
\(81\) 0 0
\(82\) −2.19615 −0.242524
\(83\) 12.9282 1.41905 0.709527 0.704678i \(-0.248908\pi\)
0.709527 + 0.704678i \(0.248908\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 7.26795 0.783723
\(87\) 0 0
\(88\) 8.19615 0.873713
\(89\) −10.7321 −1.13760 −0.568798 0.822478i \(-0.692591\pi\)
−0.568798 + 0.822478i \(0.692591\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) −3.46410 −0.361158
\(93\) 0 0
\(94\) 6.00000 0.618853
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) −6.19615 −0.629124 −0.314562 0.949237i \(-0.601858\pi\)
−0.314562 + 0.949237i \(0.601858\pi\)
\(98\) 0.803848 0.0812009
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 10.3923 1.03407 0.517036 0.855963i \(-0.327035\pi\)
0.517036 + 0.855963i \(0.327035\pi\)
\(102\) 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
\(105\) 0 0
\(106\) 16.3923 1.59216
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) −14.3923 −1.37853 −0.689266 0.724508i \(-0.742067\pi\)
−0.689266 + 0.724508i \(0.742067\pi\)
\(110\) 8.19615 0.781472
\(111\) 0 0
\(112\) 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 0
\(118\) −4.39230 −0.404344
\(119\) 0 0
\(120\) 0 0
\(121\) 11.3923 1.03566
\(122\) −11.3205 −1.02491
\(123\) 0 0
\(124\) 8.92820 0.801776
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −4.00000 −0.354943 −0.177471 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) 12.1244 1.07165
\(129\) 0 0
\(130\) −1.26795 −0.111207
\(131\) 9.12436 0.797199 0.398599 0.917125i \(-0.369496\pi\)
0.398599 + 0.917125i \(0.369496\pi\)
\(132\) 0 0
\(133\) −2.73205 −0.236899
\(134\) 13.8564 1.19701
\(135\) 0 0
\(136\) 0 0
\(137\) −19.8564 −1.69645 −0.848224 0.529638i \(-0.822328\pi\)
−0.848224 + 0.529638i \(0.822328\pi\)
\(138\) 0 0
\(139\) −8.39230 −0.711826 −0.355913 0.934519i \(-0.615830\pi\)
−0.355913 + 0.934519i \(0.615830\pi\)
\(140\) 2.73205 0.230900
\(141\) 0 0
\(142\) 7.60770 0.638424
\(143\) −3.46410 −0.289683
\(144\) 0 0
\(145\) 8.19615 0.680653
\(146\) −29.3205 −2.42658
\(147\) 0 0
\(148\) −6.19615 −0.509321
\(149\) 19.8564 1.62670 0.813350 0.581775i \(-0.197641\pi\)
0.813350 + 0.581775i \(0.197641\pi\)
\(150\) 0 0
\(151\) 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 0
\(157\) 6.39230 0.510161 0.255081 0.966920i \(-0.417898\pi\)
0.255081 + 0.966920i \(0.417898\pi\)
\(158\) −18.9282 −1.50585
\(159\) 0 0
\(160\) 5.19615 0.410792
\(161\) 9.46410 0.745876
\(162\) 0 0
\(163\) 9.26795 0.725922 0.362961 0.931804i \(-0.381766\pi\)
0.362961 + 0.931804i \(0.381766\pi\)
\(164\) −1.26795 −0.0990102
\(165\) 0 0
\(166\) 22.3923 1.73798
\(167\) −3.46410 −0.268060 −0.134030 0.990977i \(-0.542792\pi\)
−0.134030 + 0.990977i \(0.542792\pi\)
\(168\) 0 0
\(169\) −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 \(0.584834\pi\)
\(174\) 0 0
\(175\) −2.73205 −0.206524
\(176\) 23.6603 1.78346
\(177\) 0 0
\(178\) −18.5885 −1.39326
\(179\) −23.3205 −1.74306 −0.871528 0.490345i \(-0.836871\pi\)
−0.871528 + 0.490345i \(0.836871\pi\)
\(180\) 0 0
\(181\) −2.39230 −0.177819 −0.0889093 0.996040i \(-0.528338\pi\)
−0.0889093 + 0.996040i \(0.528338\pi\)
\(182\) −3.46410 −0.256776
\(183\) 0 0
\(184\) 6.00000 0.442326
\(185\) 6.19615 0.455550
\(186\) 0 0
\(187\) 0 0
\(188\) 3.46410 0.252646
\(189\) 0 0
\(190\) −1.73205 −0.125656
\(191\) −0.339746 −0.0245832 −0.0122916 0.999924i \(-0.503913\pi\)
−0.0122916 + 0.999924i \(0.503913\pi\)
\(192\) 0 0
\(193\) 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 0
\(196\) 0.464102 0.0331501
\(197\) 24.0000 1.70993 0.854965 0.518686i \(-0.173579\pi\)
0.854965 + 0.518686i \(0.173579\pi\)
\(198\) 0 0
\(199\) −15.3205 −1.08604 −0.543021 0.839719i \(-0.682720\pi\)
−0.543021 + 0.839719i \(0.682720\pi\)
\(200\) −1.73205 −0.122474
\(201\) 0 0
\(202\) 18.0000 1.26648
\(203\) 22.3923 1.57163
\(204\) 0 0
\(205\) 1.26795 0.0885574
\(206\) 17.0718 1.18945
\(207\) 0 0
\(208\) −3.66025 −0.253793
\(209\) −4.73205 −0.327323
\(210\) 0 0
\(211\) 1.07180 0.0737855 0.0368928 0.999319i \(-0.488254\pi\)
0.0368928 + 0.999319i \(0.488254\pi\)
\(212\) 9.46410 0.649997
\(213\) 0 0
\(214\) 0 0
\(215\) −4.19615 −0.286175
\(216\) 0 0
\(217\) −24.3923 −1.65586
\(218\) −24.9282 −1.68835
\(219\) 0 0
\(220\) 4.73205 0.319035
\(221\) 0 0
\(222\) 0 0
\(223\) −17.8564 −1.19575 −0.597877 0.801588i \(-0.703989\pi\)
−0.597877 + 0.801588i \(0.703989\pi\)
\(224\) 14.1962 0.948520
\(225\) 0 0
\(226\) −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
\(231\) 0 0
\(232\) 14.1962 0.932023
\(233\) 7.85641 0.514690 0.257345 0.966320i \(-0.417152\pi\)
0.257345 + 0.966320i \(0.417152\pi\)
\(234\) 0 0
\(235\) −3.46410 −0.225973
\(236\) −2.53590 −0.165073
\(237\) 0 0
\(238\) 0 0
\(239\) 9.80385 0.634158 0.317079 0.948399i \(-0.397298\pi\)
0.317079 + 0.948399i \(0.397298\pi\)
\(240\) 0 0
\(241\) −3.07180 −0.197872 −0.0989359 0.995094i \(-0.531544\pi\)
−0.0989359 + 0.995094i \(0.531544\pi\)
\(242\) 19.7321 1.26842
\(243\) 0 0
\(244\) −6.53590 −0.418418
\(245\) −0.464102 −0.0296504
\(246\) 0 0
\(247\) 0.732051 0.0465793
\(248\) −15.4641 −0.981971
\(249\) 0 0
\(250\) −1.73205 −0.109545
\(251\) −28.0526 −1.77066 −0.885331 0.464961i \(-0.846068\pi\)
−0.885331 + 0.464961i \(0.846068\pi\)
\(252\) 0 0
\(253\) 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 \(0.230755\pi\)
\(258\) 0 0
\(259\) 16.9282 1.05187
\(260\) −0.732051 −0.0453999
\(261\) 0 0
\(262\) 15.8038 0.976365
\(263\) −6.00000 −0.369976 −0.184988 0.982741i \(-0.559225\pi\)
−0.184988 + 0.982741i \(0.559225\pi\)
\(264\) 0 0
\(265\) −9.46410 −0.581375
\(266\) −4.73205 −0.290141
\(267\) 0 0
\(268\) 8.00000 0.488678
\(269\) −0.588457 −0.0358789 −0.0179394 0.999839i \(-0.505711\pi\)
−0.0179394 + 0.999839i \(0.505711\pi\)
\(270\) 0 0
\(271\) 0.392305 0.0238308 0.0119154 0.999929i \(-0.496207\pi\)
0.0119154 + 0.999929i \(0.496207\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −34.3923 −2.07772
\(275\) −4.73205 −0.285353
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) −14.5359 −0.871805
\(279\) 0 0
\(280\) −4.73205 −0.282794
\(281\) −1.26795 −0.0756395 −0.0378198 0.999285i \(-0.512041\pi\)
−0.0378198 + 0.999285i \(0.512041\pi\)
\(282\) 0 0
\(283\) 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
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 14.1962 0.833627
\(291\) 0 0
\(292\) −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 0
\(295\) 2.53590 0.147646
\(296\) 10.7321 0.623788
\(297\) 0 0
\(298\) 34.3923 1.99229
\(299\) −2.53590 −0.146655
\(300\) 0 0
\(301\) −11.4641 −0.660780
\(302\) 24.2487 1.39536
\(303\) 0 0
\(304\) −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
\(309\) 0 0
\(310\) −15.4641 −0.878302
\(311\) −32.4449 −1.83978 −0.919890 0.392177i \(-0.871722\pi\)
−0.919890 + 0.392177i \(0.871722\pi\)
\(312\) 0 0
\(313\) 6.39230 0.361314 0.180657 0.983546i \(-0.442178\pi\)
0.180657 + 0.983546i \(0.442178\pi\)
\(314\) 11.0718 0.624818
\(315\) 0 0
\(316\) −10.9282 −0.614759
\(317\) 11.3205 0.635823 0.317912 0.948120i \(-0.397018\pi\)
0.317912 + 0.948120i \(0.397018\pi\)
\(318\) 0 0
\(319\) 38.7846 2.17152
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) 16.3923 0.913507
\(323\) 0 0
\(324\) 0 0
\(325\) 0.732051 0.0406069
\(326\) 16.0526 0.889069
\(327\) 0 0
\(328\) 2.19615 0.121262
\(329\) −9.46410 −0.521773
\(330\) 0 0
\(331\) −25.7128 −1.41330 −0.706652 0.707561i \(-0.749795\pi\)
−0.706652 + 0.707561i \(0.749795\pi\)
\(332\) 12.9282 0.709527
\(333\) 0 0
\(334\) −6.00000 −0.328305
\(335\) −8.00000 −0.437087
\(336\) 0 0
\(337\) 5.12436 0.279141 0.139571 0.990212i \(-0.455428\pi\)
0.139571 + 0.990212i \(0.455428\pi\)
\(338\) −21.5885 −1.17426
\(339\) 0 0
\(340\) 0 0
\(341\) −42.2487 −2.28790
\(342\) 0 0
\(343\) 17.8564 0.964155
\(344\) −7.26795 −0.391862
\(345\) 0 0
\(346\) −12.0000 −0.645124
\(347\) 0.928203 0.0498286 0.0249143 0.999690i \(-0.492069\pi\)
0.0249143 + 0.999690i \(0.492069\pi\)
\(348\) 0 0
\(349\) −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 0
\(352\) 24.5885 1.31057
\(353\) 14.7846 0.786905 0.393453 0.919345i \(-0.371281\pi\)
0.393453 + 0.919345i \(0.371281\pi\)
\(354\) 0 0
\(355\) −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 \(-0.502854\pi\)
−0.00896555 + 0.999960i \(0.502854\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −4.14359 −0.217782
\(363\) 0 0
\(364\) −2.00000 −0.104828
\(365\) 16.9282 0.886063
\(366\) 0 0
\(367\) 16.1962 0.845432 0.422716 0.906262i \(-0.361077\pi\)
0.422716 + 0.906262i \(0.361077\pi\)
\(368\) 17.3205 0.902894
\(369\) 0 0
\(370\) 10.7321 0.557933
\(371\) −25.8564 −1.34240
\(372\) 0 0
\(373\) −6.19615 −0.320825 −0.160412 0.987050i \(-0.551282\pi\)
−0.160412 + 0.987050i \(0.551282\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −6.00000 −0.309426
\(377\) −6.00000 −0.309016
\(378\) 0 0
\(379\) 20.9282 1.07501 0.537505 0.843261i \(-0.319367\pi\)
0.537505 + 0.843261i \(0.319367\pi\)
\(380\) −1.00000 −0.0512989
\(381\) 0 0
\(382\) −0.588457 −0.0301081
\(383\) 17.0718 0.872328 0.436164 0.899867i \(-0.356337\pi\)
0.436164 + 0.899867i \(0.356337\pi\)
\(384\) 0 0
\(385\) −12.9282 −0.658882
\(386\) 29.6603 1.50967
\(387\) 0 0
\(388\) −6.19615 −0.314562
\(389\) −7.85641 −0.398336 −0.199168 0.979965i \(-0.563824\pi\)
−0.199168 + 0.979965i \(0.563824\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −0.803848 −0.0406004
\(393\) 0 0
\(394\) 41.5692 2.09423
\(395\) 10.9282 0.549858
\(396\) 0 0
\(397\) 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 \(0.176450\pi\)
\(402\) 0 0
\(403\) 6.53590 0.325576
\(404\) 10.3923 0.517036
\(405\) 0 0
\(406\) 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
\(411\) 0 0
\(412\) 9.85641 0.485590
\(413\) 6.92820 0.340915
\(414\) 0 0
\(415\) −12.9282 −0.634621
\(416\) −3.80385 −0.186499
\(417\) 0 0
\(418\) −8.19615 −0.400887
\(419\) 28.0526 1.37046 0.685229 0.728328i \(-0.259702\pi\)
0.685229 + 0.728328i \(0.259702\pi\)
\(420\) 0 0
\(421\) −18.7846 −0.915506 −0.457753 0.889079i \(-0.651346\pi\)
−0.457753 + 0.889079i \(0.651346\pi\)
\(422\) 1.85641 0.0903685
\(423\) 0 0
\(424\) −16.3923 −0.796081
\(425\) 0 0
\(426\) 0 0
\(427\) 17.8564 0.864132
\(428\) 0 0
\(429\) 0 0
\(430\) −7.26795 −0.350492
\(431\) 11.3205 0.545290 0.272645 0.962115i \(-0.412102\pi\)
0.272645 + 0.962115i \(0.412102\pi\)
\(432\) 0 0
\(433\) −10.5885 −0.508849 −0.254424 0.967093i \(-0.581886\pi\)
−0.254424 + 0.967093i \(0.581886\pi\)
\(434\) −42.2487 −2.02800
\(435\) 0 0
\(436\) −14.3923 −0.689266
\(437\) −3.46410 −0.165710
\(438\) 0 0
\(439\) 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 0
\(442\) 0 0
\(443\) 5.32051 0.252785 0.126392 0.991980i \(-0.459660\pi\)
0.126392 + 0.991980i \(0.459660\pi\)
\(444\) 0 0
\(445\) 10.7321 0.508748
\(446\) −30.9282 −1.46449
\(447\) 0 0
\(448\) −2.73205 −0.129077
\(449\) 5.66025 0.267124 0.133562 0.991040i \(-0.457358\pi\)
0.133562 + 0.991040i \(0.457358\pi\)
\(450\) 0 0
\(451\) 6.00000 0.282529
\(452\) −18.9282 −0.890308
\(453\) 0 0
\(454\) −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 \(-0.544622\pi\)
−0.139724 + 0.990190i \(0.544622\pi\)
\(462\) 0 0
\(463\) −35.5167 −1.65060 −0.825300 0.564695i \(-0.808994\pi\)
−0.825300 + 0.564695i \(0.808994\pi\)
\(464\) 40.9808 1.90248
\(465\) 0 0
\(466\) 13.6077 0.630364
\(467\) −20.5359 −0.950288 −0.475144 0.879908i \(-0.657604\pi\)
−0.475144 + 0.879908i \(0.657604\pi\)
\(468\) 0 0
\(469\) −21.8564 −1.00924
\(470\) −6.00000 −0.276759
\(471\) 0 0
\(472\) 4.39230 0.202172
\(473\) −19.8564 −0.912999
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) 0 0
\(478\) 16.9808 0.776682
\(479\) −25.5167 −1.16589 −0.582943 0.812513i \(-0.698099\pi\)
−0.582943 + 0.812513i \(0.698099\pi\)
\(480\) 0 0
\(481\) −4.53590 −0.206819
\(482\) −5.32051 −0.242343
\(483\) 0 0
\(484\) 11.3923 0.517832
\(485\) 6.19615 0.281353
\(486\) 0 0
\(487\) −32.3923 −1.46784 −0.733918 0.679238i \(-0.762310\pi\)
−0.733918 + 0.679238i \(0.762310\pi\)
\(488\) 11.3205 0.512455
\(489\) 0 0
\(490\) −0.803848 −0.0363141
\(491\) −16.0526 −0.724442 −0.362221 0.932092i \(-0.617981\pi\)
−0.362221 + 0.932092i \(0.617981\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 1.26795 0.0570477
\(495\) 0 0
\(496\) −44.6410 −2.00444
\(497\) −12.0000 −0.538274
\(498\) 0 0
\(499\) 10.5359 0.471652 0.235826 0.971795i \(-0.424221\pi\)
0.235826 + 0.971795i \(0.424221\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) −48.5885 −2.16861
\(503\) 23.0718 1.02872 0.514360 0.857574i \(-0.328029\pi\)
0.514360 + 0.857574i \(0.328029\pi\)
\(504\) 0 0
\(505\) −10.3923 −0.462451
\(506\) 28.3923 1.26219
\(507\) 0 0
\(508\) −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
\(516\) 0 0
\(517\) −16.3923 −0.720933
\(518\) 29.3205 1.28827
\(519\) 0 0
\(520\) 1.26795 0.0556033
\(521\) −37.2679 −1.63274 −0.816369 0.577530i \(-0.804017\pi\)
−0.816369 + 0.577530i \(0.804017\pi\)
\(522\) 0 0
\(523\) 8.67949 0.379528 0.189764 0.981830i \(-0.439228\pi\)
0.189764 + 0.981830i \(0.439228\pi\)
\(524\) 9.12436 0.398599
\(525\) 0 0
\(526\) −10.3923 −0.453126
\(527\) 0 0
\(528\) 0 0
\(529\) −11.0000 −0.478261
\(530\) −16.3923 −0.712036
\(531\) 0 0
\(532\) −2.73205 −0.118449
\(533\) −0.928203 −0.0402049
\(534\) 0 0
\(535\) 0 0
\(536\) −13.8564 −0.598506
\(537\) 0 0
\(538\) −1.01924 −0.0439425
\(539\) −2.19615 −0.0945950
\(540\) 0 0
\(541\) 41.7128 1.79337 0.896687 0.442665i \(-0.145967\pi\)
0.896687 + 0.442665i \(0.145967\pi\)
\(542\) 0.679492 0.0291867
\(543\) 0 0
\(544\) 0 0
\(545\) 14.3923 0.616499
\(546\) 0 0
\(547\) 43.3205 1.85225 0.926126 0.377215i \(-0.123118\pi\)
0.926126 + 0.377215i \(0.123118\pi\)
\(548\) −19.8564 −0.848224
\(549\) 0 0
\(550\) −8.19615 −0.349485
\(551\) −8.19615 −0.349168
\(552\) 0 0
\(553\) 29.8564 1.26962
\(554\) 3.46410 0.147176
\(555\) 0 0
\(556\) −8.39230 −0.355913
\(557\) −0.928203 −0.0393292 −0.0196646 0.999807i \(-0.506260\pi\)
−0.0196646 + 0.999807i \(0.506260\pi\)
\(558\) 0 0
\(559\) 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 \(0.303546\pi\)
\(564\) 0 0
\(565\) 18.9282 0.796315
\(566\) 43.2679 1.81869
\(567\) 0 0
\(568\) −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 0
\(574\) 6.00000 0.250435
\(575\) −3.46410 −0.144463
\(576\) 0 0
\(577\) 15.1769 0.631823 0.315912 0.948789i \(-0.397690\pi\)
0.315912 + 0.948789i \(0.397690\pi\)
\(578\) −29.4449 −1.22474
\(579\) 0 0
\(580\) 8.19615 0.340327
\(581\) −35.3205 −1.46534
\(582\) 0 0
\(583\) −44.7846 −1.85479
\(584\) 29.3205 1.21329
\(585\) 0 0
\(586\) −48.0000 −1.98286
\(587\) 3.46410 0.142979 0.0714894 0.997441i \(-0.477225\pi\)
0.0714894 + 0.997441i \(0.477225\pi\)
\(588\) 0 0
\(589\) 8.92820 0.367880
\(590\) 4.39230 0.180828
\(591\) 0 0
\(592\) 30.9808 1.27330
\(593\) 38.7846 1.59269 0.796347 0.604841i \(-0.206763\pi\)
0.796347 + 0.604841i \(0.206763\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 19.8564 0.813350
\(597\) 0 0
\(598\) −4.39230 −0.179615
\(599\) 13.8564 0.566157 0.283079 0.959097i \(-0.408644\pi\)
0.283079 + 0.959097i \(0.408644\pi\)
\(600\) 0 0
\(601\) −47.1769 −1.92439 −0.962193 0.272368i \(-0.912193\pi\)
−0.962193 + 0.272368i \(0.912193\pi\)
\(602\) −19.8564 −0.809287
\(603\) 0 0
\(604\) 14.0000 0.569652
\(605\) −11.3923 −0.463163
\(606\) 0 0
\(607\) −11.6077 −0.471142 −0.235571 0.971857i \(-0.575696\pi\)
−0.235571 + 0.971857i \(0.575696\pi\)
\(608\) −5.19615 −0.210732
\(609\) 0 0
\(610\) 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
\(615\) 0 0
\(616\) −22.3923 −0.902212
\(617\) −27.7128 −1.11568 −0.557838 0.829950i \(-0.688369\pi\)
−0.557838 + 0.829950i \(0.688369\pi\)
\(618\) 0 0
\(619\) −15.3205 −0.615783 −0.307892 0.951421i \(-0.599623\pi\)
−0.307892 + 0.951421i \(0.599623\pi\)
\(620\) −8.92820 −0.358565
\(621\) 0 0
\(622\) −56.1962 −2.25326
\(623\) 29.3205 1.17470
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 11.0718 0.442518
\(627\) 0 0
\(628\) 6.39230 0.255081
\(629\) 0 0
\(630\) 0 0
\(631\) −34.9282 −1.39047 −0.695235 0.718783i \(-0.744700\pi\)
−0.695235 + 0.718783i \(0.744700\pi\)
\(632\) 18.9282 0.752923
\(633\) 0 0
\(634\) 19.6077 0.778721
\(635\) 4.00000 0.158735
\(636\) 0 0
\(637\) 0.339746 0.0134612
\(638\) 67.1769 2.65956
\(639\) 0 0
\(640\) −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
\(645\) 0 0
\(646\) 0 0
\(647\) 4.14359 0.162901 0.0814507 0.996677i \(-0.474045\pi\)
0.0814507 + 0.996677i \(0.474045\pi\)
\(648\) 0 0
\(649\) 12.0000 0.471041
\(650\) 1.26795 0.0497331
\(651\) 0 0
\(652\) 9.26795 0.362961
\(653\) −17.0718 −0.668071 −0.334036 0.942560i \(-0.608411\pi\)
−0.334036 + 0.942560i \(0.608411\pi\)
\(654\) 0 0
\(655\) −9.12436 −0.356518
\(656\) 6.33975 0.247525
\(657\) 0 0
\(658\) −16.3923 −0.639039
\(659\) −5.07180 −0.197569 −0.0987846 0.995109i \(-0.531495\pi\)
−0.0987846 + 0.995109i \(0.531495\pi\)
\(660\) 0 0
\(661\) 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
\(666\) 0 0
\(667\) 28.3923 1.09935
\(668\) −3.46410 −0.134030
\(669\) 0 0
\(670\) −13.8564 −0.535320
\(671\) 30.9282 1.19397
\(672\) 0 0
\(673\) 17.1244 0.660095 0.330048 0.943964i \(-0.392935\pi\)
0.330048 + 0.943964i \(0.392935\pi\)
\(674\) 8.87564 0.341877
\(675\) 0 0
\(676\) −12.4641 −0.479389
\(677\) −0.679492 −0.0261150 −0.0130575 0.999915i \(-0.504156\pi\)
−0.0130575 + 0.999915i \(0.504156\pi\)
\(678\) 0 0
\(679\) 16.9282 0.649645
\(680\) 0 0
\(681\) 0 0
\(682\) −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
\(687\) 0 0
\(688\) −20.9808 −0.799884
\(689\) 6.92820 0.263944
\(690\) 0 0
\(691\) 12.3923 0.471425 0.235713 0.971823i \(-0.424258\pi\)
0.235713 + 0.971823i \(0.424258\pi\)
\(692\) −6.92820 −0.263371
\(693\) 0 0
\(694\) 1.60770 0.0610273
\(695\) 8.39230 0.318338
\(696\) 0 0
\(697\) 0 0
\(698\) −38.1051 −1.44230
\(699\) 0 0
\(700\) −2.73205 −0.103262
\(701\) 33.7128 1.27332 0.636658 0.771147i \(-0.280316\pi\)
0.636658 + 0.771147i \(0.280316\pi\)
\(702\) 0 0
\(703\) −6.19615 −0.233692
\(704\) −4.73205 −0.178346
\(705\) 0 0
\(706\) 25.6077 0.963758
\(707\) −28.3923 −1.06780
\(708\) 0 0
\(709\) −29.1769 −1.09576 −0.547881 0.836556i \(-0.684565\pi\)
−0.547881 + 0.836556i \(0.684565\pi\)
\(710\) −7.60770 −0.285512
\(711\) 0 0
\(712\) 18.5885 0.696632
\(713\) −30.9282 −1.15827
\(714\) 0 0
\(715\) 3.46410 0.129550
\(716\) −23.3205 −0.871528
\(717\) 0 0
\(718\) −0.588457 −0.0219610
\(719\) 11.6603 0.434854 0.217427 0.976077i \(-0.430234\pi\)
0.217427 + 0.976077i \(0.430234\pi\)
\(720\) 0 0
\(721\) −26.9282 −1.00286
\(722\) 1.73205 0.0644603
\(723\) 0 0
\(724\) −2.39230 −0.0889093
\(725\) −8.19615 −0.304397
\(726\) 0 0
\(727\) 25.6603 0.951686 0.475843 0.879530i \(-0.342143\pi\)
0.475843 + 0.879530i \(0.342143\pi\)
\(728\) 3.46410 0.128388
\(729\) 0 0
\(730\) 29.3205 1.08520
\(731\) 0 0
\(732\) 0 0
\(733\) −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 0
\(736\) 18.0000 0.663489
\(737\) −37.8564 −1.39446
\(738\) 0 0
\(739\) 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 \(0.481215\pi\)
\(744\) 0 0
\(745\) −19.8564 −0.727482
\(746\) −10.7321 −0.392928
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 26.0000 0.948753 0.474377 0.880322i \(-0.342673\pi\)
0.474377 + 0.880322i \(0.342673\pi\)
\(752\) −17.3205 −0.631614
\(753\) 0 0
\(754\) −10.3923 −0.378465
\(755\) −14.0000 −0.509512
\(756\) 0 0
\(757\) 32.2487 1.17210 0.586050 0.810275i \(-0.300682\pi\)
0.586050 + 0.810275i \(0.300682\pi\)
\(758\) 36.2487 1.31661
\(759\) 0 0
\(760\) 1.73205 0.0628281
\(761\) −6.00000 −0.217500 −0.108750 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) 0 0
\(763\) 39.3205 1.42350
\(764\) −0.339746 −0.0122916
\(765\) 0 0
\(766\) 29.5692 1.06838
\(767\) −1.85641 −0.0670310
\(768\) 0 0
\(769\) −20.6410 −0.744334 −0.372167 0.928166i \(-0.621385\pi\)
−0.372167 + 0.928166i \(0.621385\pi\)
\(770\) −22.3923 −0.806963
\(771\) 0 0
\(772\) 17.1244 0.616319
\(773\) 25.1769 0.905551 0.452775 0.891625i \(-0.350434\pi\)
0.452775 + 0.891625i \(0.350434\pi\)
\(774\) 0 0
\(775\) 8.92820 0.320711
\(776\) 10.7321 0.385258
\(777\) 0 0
\(778\) −13.6077 −0.487860
\(779\) −1.26795 −0.0454290
\(780\) 0 0
\(781\) −20.7846 −0.743732
\(782\) 0 0
\(783\) 0 0
\(784\) −2.32051 −0.0828753
\(785\) −6.39230 −0.228151
\(786\) 0 0
\(787\) 8.67949 0.309390 0.154695 0.987962i \(-0.450560\pi\)
0.154695 + 0.987962i \(0.450560\pi\)
\(788\) 24.0000 0.854965
\(789\) 0 0
\(790\) 18.9282 0.673435
\(791\) 51.7128 1.83870
\(792\) 0 0
\(793\) −4.78461 −0.169906
\(794\) 15.4641 0.548800
\(795\) 0 0
\(796\) −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
\(801\) 0 0
\(802\) 58.9808 2.08268
\(803\) 80.1051 2.82685
\(804\) 0 0
\(805\) −9.46410 −0.333566
\(806\) 11.3205 0.398748
\(807\) 0 0
\(808\) −18.0000 −0.633238
\(809\) −14.7846 −0.519799 −0.259900 0.965636i \(-0.583689\pi\)
−0.259900 + 0.965636i \(0.583689\pi\)
\(810\) 0 0
\(811\) 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 0
\(814\) 50.7846 1.78000
\(815\) −9.26795 −0.324642
\(816\) 0 0
\(817\) 4.19615 0.146805
\(818\) −45.7128 −1.59831
\(819\) 0 0
\(820\) 1.26795 0.0442787
\(821\) 32.5359 1.13551 0.567755 0.823197i \(-0.307812\pi\)
0.567755 + 0.823197i \(0.307812\pi\)
\(822\) 0 0
\(823\) 12.9808 0.452481 0.226240 0.974071i \(-0.427356\pi\)
0.226240 + 0.974071i \(0.427356\pi\)
\(824\) −17.0718 −0.594724
\(825\) 0 0
\(826\) 12.0000 0.417533
\(827\) 5.32051 0.185012 0.0925061 0.995712i \(-0.470512\pi\)
0.0925061 + 0.995712i \(0.470512\pi\)
\(828\) 0 0
\(829\) 34.1051 1.18452 0.592260 0.805747i \(-0.298236\pi\)
0.592260 + 0.805747i \(0.298236\pi\)
\(830\) −22.3923 −0.777248
\(831\) 0 0
\(832\) 0.732051 0.0253793
\(833\) 0 0
\(834\) 0 0
\(835\) 3.46410 0.119880
\(836\) −4.73205 −0.163661
\(837\) 0 0
\(838\) 48.5885 1.67846
\(839\) 19.6077 0.676933 0.338466 0.940978i \(-0.390092\pi\)
0.338466 + 0.940978i \(0.390092\pi\)
\(840\) 0 0
\(841\) 38.1769 1.31645
\(842\) −32.5359 −1.12126
\(843\) 0 0
\(844\) 1.07180 0.0368928
\(845\) 12.4641 0.428778
\(846\) 0 0
\(847\) −31.1244 −1.06945
\(848\) −47.3205 −1.62499
\(849\) 0 0
\(850\) 0 0
\(851\) 21.4641 0.735780
\(852\) 0 0
\(853\) 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 \(0.243410\pi\)
\(858\) 0 0
\(859\) 32.0000 1.09183 0.545913 0.837842i \(-0.316183\pi\)
0.545913 + 0.837842i \(0.316183\pi\)
\(860\) −4.19615 −0.143088
\(861\) 0 0
\(862\) 19.6077 0.667841
\(863\) 12.0000 0.408485 0.204242 0.978920i \(-0.434527\pi\)
0.204242 + 0.978920i \(0.434527\pi\)
\(864\) 0 0
\(865\) 6.92820 0.235566
\(866\) −18.3397 −0.623210
\(867\) 0 0
\(868\) −24.3923 −0.827929
\(869\) 51.7128 1.75424
\(870\) 0 0
\(871\) 5.85641 0.198437
\(872\) 24.9282 0.844175
\(873\) 0 0
\(874\) −6.00000 −0.202953
\(875\) 2.73205 0.0923602
\(876\) 0 0
\(877\) 53.1244 1.79388 0.896941 0.442150i \(-0.145784\pi\)
0.896941 + 0.442150i \(0.145784\pi\)
\(878\) 46.6410 1.57406
\(879\) 0 0
\(880\) −23.6603 −0.797587
\(881\) −8.53590 −0.287582 −0.143791 0.989608i \(-0.545929\pi\)
−0.143791 + 0.989608i \(0.545929\pi\)
\(882\) 0 0
\(883\) 36.9808 1.24450 0.622251 0.782818i \(-0.286218\pi\)
0.622251 + 0.782818i \(0.286218\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 9.21539 0.309597
\(887\) 24.0000 0.805841 0.402921 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) 0 0
\(889\) 10.9282 0.366520
\(890\) 18.5885 0.623087
\(891\) 0 0
\(892\) −17.8564 −0.597877
\(893\) 3.46410 0.115922
\(894\) 0 0
\(895\) 23.3205 0.779519
\(896\) −33.1244 −1.10661
\(897\) 0 0
\(898\) 9.80385 0.327159
\(899\) −73.1769 −2.44059
\(900\) 0 0
\(901\) 0 0
\(902\) 10.3923 0.346026
\(903\) 0 0
\(904\) 32.7846 1.09040
\(905\) 2.39230 0.0795229
\(906\) 0 0
\(907\) −32.3923 −1.07557 −0.537784 0.843082i \(-0.680739\pi\)
−0.537784 + 0.843082i \(0.680739\pi\)
\(908\) −10.3923 −0.344881
\(909\) 0 0
\(910\) 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
\(915\) 0 0
\(916\) −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
\(921\) 0 0
\(922\) −10.3923 −0.342252
\(923\) 3.21539 0.105836
\(924\) 0 0
\(925\) −6.19615 −0.203728
\(926\) −61.5167 −2.02156
\(927\) 0 0
\(928\) 42.5885 1.39803
\(929\) 1.60770 0.0527468 0.0263734 0.999652i \(-0.491604\pi\)
0.0263734 + 0.999652i \(0.491604\pi\)
\(930\) 0 0
\(931\) 0.464102 0.0152103
\(932\) 7.85641 0.257345
\(933\) 0 0
\(934\) −35.5692 −1.16386
\(935\) 0 0
\(936\) 0 0
\(937\) −16.2487 −0.530822 −0.265411 0.964135i \(-0.585508\pi\)
−0.265411 + 0.964135i \(0.585508\pi\)
\(938\) −37.8564 −1.23606
\(939\) 0 0
\(940\) −3.46410 −0.112987
\(941\) −0.588457 −0.0191832 −0.00959158 0.999954i \(-0.503053\pi\)
−0.00959158 + 0.999954i \(0.503053\pi\)
\(942\) 0 0
\(943\) 4.39230 0.143033
\(944\) 12.6795 0.412682
\(945\) 0 0
\(946\) −34.3923 −1.11819
\(947\) −28.1436 −0.914544 −0.457272 0.889327i \(-0.651173\pi\)
−0.457272 + 0.889327i \(0.651173\pi\)
\(948\) 0 0
\(949\) −12.3923 −0.402271
\(950\) 1.73205 0.0561951
\(951\) 0 0
\(952\) 0 0
\(953\) −37.8564 −1.22629 −0.613145 0.789971i \(-0.710096\pi\)
−0.613145 + 0.789971i \(0.710096\pi\)
\(954\) 0 0
\(955\) 0.339746 0.0109939
\(956\) 9.80385 0.317079
\(957\) 0 0
\(958\) −44.1962 −1.42791
\(959\) 54.2487 1.75178
\(960\) 0 0
\(961\) 48.7128 1.57138
\(962\) −7.85641 −0.253301
\(963\) 0 0
\(964\) −3.07180 −0.0989359
\(965\) −17.1244 −0.551253
\(966\) 0 0
\(967\) 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 \(0.353320\pi\)
\(972\) 0 0
\(973\) 22.9282 0.735044
\(974\) −56.1051 −1.79772
\(975\) 0 0
\(976\) 32.6795 1.04605
\(977\) 39.0333 1.24879 0.624393 0.781110i \(-0.285346\pi\)
0.624393 + 0.781110i \(0.285346\pi\)
\(978\) 0 0
\(979\) 50.7846 1.62308
\(980\) −0.464102 −0.0148252
\(981\) 0 0
\(982\) −27.8038 −0.887256
\(983\) 41.3205 1.31792 0.658960 0.752178i \(-0.270997\pi\)
0.658960 + 0.752178i \(0.270997\pi\)
\(984\) 0 0
\(985\) −24.0000 −0.764704
\(986\) 0 0
\(987\) 0 0
\(988\) 0.732051 0.0232896
\(989\) −14.5359 −0.462215
\(990\) 0 0
\(991\) 13.0718 0.415239 0.207620 0.978210i \(-0.433428\pi\)
0.207620 + 0.978210i \(0.433428\pi\)
\(992\) −46.3923 −1.47296
\(993\) 0 0
\(994\) −20.7846 −0.659248
\(995\) 15.3205 0.485693
\(996\) 0 0
\(997\) −17.6077 −0.557641 −0.278821 0.960343i \(-0.589944\pi\)
−0.278821 + 0.960343i \(0.589944\pi\)
\(998\) 18.2487 0.577653
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 855.2.a.f.1.2 2
3.2 odd 2 285.2.a.e.1.1 2
5.4 even 2 4275.2.a.t.1.1 2
12.11 even 2 4560.2.a.bh.1.2 2
15.2 even 4 1425.2.c.k.799.1 4
15.8 even 4 1425.2.c.k.799.4 4
15.14 odd 2 1425.2.a.o.1.2 2
57.56 even 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 3.2 odd 2
855.2.a.f.1.2 2 1.1 even 1 trivial
1425.2.a.o.1.2 2 15.14 odd 2
1425.2.c.k.799.1 4 15.2 even 4
1425.2.c.k.799.4 4 15.8 even 4
4275.2.a.t.1.1 2 5.4 even 2
4560.2.a.bh.1.2 2 12.11 even 2
5415.2.a.r.1.2 2 57.56 even 2