Properties

Label 2160.2.h.a.431.2
Level $2160$
Weight $2$
Character 2160.431
Analytic conductor $17.248$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2160,2,Mod(431,2160)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2160, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2160.431");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2160.h (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(17.2476868366\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 431.2
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 2160.431
Dual form 2160.2.h.a.431.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{5} +4.73205i q^{7} +O(q^{10})\) \(q-1.00000i q^{5} +4.73205i q^{7} -4.73205 q^{11} +6.19615 q^{13} -5.19615i q^{17} +0.464102i q^{19} +4.26795 q^{23} -1.00000 q^{25} +8.19615i q^{29} +0.464102i q^{31} +4.73205 q^{35} +2.00000 q^{37} +2.19615i q^{41} +5.66025i q^{43} -9.46410 q^{47} -15.3923 q^{49} +11.1962i q^{53} +4.73205i q^{55} +5.66025 q^{59} -1.00000 q^{61} -6.19615i q^{65} +10.3923i q^{67} -4.73205 q^{71} -4.19615 q^{73} -22.3923i q^{77} +9.92820i q^{79} -5.19615 q^{83} -5.19615 q^{85} +14.1962i q^{89} +29.3205i q^{91} +0.464102 q^{95} -8.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{11} + 4 q^{13} + 24 q^{23} - 4 q^{25} + 12 q^{35} + 8 q^{37} - 24 q^{47} - 20 q^{49} - 12 q^{59} - 4 q^{61} - 12 q^{71} + 4 q^{73} - 12 q^{95} - 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times\).

\(n\) \(271\) \(1297\) \(1621\) \(2081\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) − 1.00000i − 0.447214i
\(6\) 0 0
\(7\) 4.73205i 1.78855i 0.447521 + 0.894274i \(0.352307\pi\)
−0.447521 + 0.894274i \(0.647693\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −4.73205 −1.42677 −0.713384 0.700774i \(-0.752838\pi\)
−0.713384 + 0.700774i \(0.752838\pi\)
\(12\) 0 0
\(13\) 6.19615 1.71850 0.859252 0.511553i \(-0.170930\pi\)
0.859252 + 0.511553i \(0.170930\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 5.19615i − 1.26025i −0.776493 0.630126i \(-0.783003\pi\)
0.776493 0.630126i \(-0.216997\pi\)
\(18\) 0 0
\(19\) 0.464102i 0.106472i 0.998582 + 0.0532361i \(0.0169536\pi\)
−0.998582 + 0.0532361i \(0.983046\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.26795 0.889929 0.444964 0.895548i \(-0.353216\pi\)
0.444964 + 0.895548i \(0.353216\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 8.19615i 1.52199i 0.648759 + 0.760994i \(0.275288\pi\)
−0.648759 + 0.760994i \(0.724712\pi\)
\(30\) 0 0
\(31\) 0.464102i 0.0833551i 0.999131 + 0.0416776i \(0.0132702\pi\)
−0.999131 + 0.0416776i \(0.986730\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4.73205 0.799863
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.19615i 0.342981i 0.985186 + 0.171491i \(0.0548583\pi\)
−0.985186 + 0.171491i \(0.945142\pi\)
\(42\) 0 0
\(43\) 5.66025i 0.863181i 0.902070 + 0.431590i \(0.142047\pi\)
−0.902070 + 0.431590i \(0.857953\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −9.46410 −1.38048 −0.690241 0.723580i \(-0.742495\pi\)
−0.690241 + 0.723580i \(0.742495\pi\)
\(48\) 0 0
\(49\) −15.3923 −2.19890
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 11.1962i 1.53791i 0.639303 + 0.768955i \(0.279223\pi\)
−0.639303 + 0.768955i \(0.720777\pi\)
\(54\) 0 0
\(55\) 4.73205i 0.638070i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 5.66025 0.736902 0.368451 0.929647i \(-0.379888\pi\)
0.368451 + 0.929647i \(0.379888\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037 −0.0640184 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 6.19615i − 0.768538i
\(66\) 0 0
\(67\) 10.3923i 1.26962i 0.772667 + 0.634811i \(0.218922\pi\)
−0.772667 + 0.634811i \(0.781078\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −4.73205 −0.561591 −0.280796 0.959768i \(-0.590598\pi\)
−0.280796 + 0.959768i \(0.590598\pi\)
\(72\) 0 0
\(73\) −4.19615 −0.491122 −0.245561 0.969381i \(-0.578972\pi\)
−0.245561 + 0.969381i \(0.578972\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 22.3923i − 2.55184i
\(78\) 0 0
\(79\) 9.92820i 1.11701i 0.829501 + 0.558505i \(0.188625\pi\)
−0.829501 + 0.558505i \(0.811375\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −5.19615 −0.570352 −0.285176 0.958475i \(-0.592052\pi\)
−0.285176 + 0.958475i \(0.592052\pi\)
\(84\) 0 0
\(85\) −5.19615 −0.563602
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 14.1962i 1.50479i 0.658713 + 0.752395i \(0.271101\pi\)
−0.658713 + 0.752395i \(0.728899\pi\)
\(90\) 0 0
\(91\) 29.3205i 3.07362i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0.464102 0.0476158
\(96\) 0 0
\(97\) −8.00000 −0.812277 −0.406138 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 4.39230i − 0.437051i −0.975831 0.218525i \(-0.929875\pi\)
0.975831 0.218525i \(-0.0701246\pi\)
\(102\) 0 0
\(103\) 9.46410i 0.932526i 0.884646 + 0.466263i \(0.154400\pi\)
−0.884646 + 0.466263i \(0.845600\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 18.9282 1.82986 0.914929 0.403614i \(-0.132246\pi\)
0.914929 + 0.403614i \(0.132246\pi\)
\(108\) 0 0
\(109\) 11.3923 1.09118 0.545592 0.838051i \(-0.316305\pi\)
0.545592 + 0.838051i \(0.316305\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 12.0000i − 1.12887i −0.825479 0.564433i \(-0.809095\pi\)
0.825479 0.564433i \(-0.190905\pi\)
\(114\) 0 0
\(115\) − 4.26795i − 0.397988i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 24.5885 2.25402
\(120\) 0 0
\(121\) 11.3923 1.03566
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) − 18.0000i − 1.59724i −0.601834 0.798621i \(-0.705563\pi\)
0.601834 0.798621i \(-0.294437\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −0.928203 −0.0810975 −0.0405487 0.999178i \(-0.512911\pi\)
−0.0405487 + 0.999178i \(0.512911\pi\)
\(132\) 0 0
\(133\) −2.19615 −0.190431
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 5.19615i 0.443937i 0.975054 + 0.221969i \(0.0712483\pi\)
−0.975054 + 0.221969i \(0.928752\pi\)
\(138\) 0 0
\(139\) 19.8564i 1.68420i 0.539323 + 0.842099i \(0.318680\pi\)
−0.539323 + 0.842099i \(0.681320\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −29.3205 −2.45190
\(144\) 0 0
\(145\) 8.19615 0.680653
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 9.80385i − 0.803162i −0.915823 0.401581i \(-0.868461\pi\)
0.915823 0.401581i \(-0.131539\pi\)
\(150\) 0 0
\(151\) − 18.9282i − 1.54036i −0.637829 0.770178i \(-0.720168\pi\)
0.637829 0.770178i \(-0.279832\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0.464102 0.0372775
\(156\) 0 0
\(157\) 16.1962 1.29259 0.646297 0.763086i \(-0.276317\pi\)
0.646297 + 0.763086i \(0.276317\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 20.1962i 1.59168i
\(162\) 0 0
\(163\) 10.3923i 0.813988i 0.913431 + 0.406994i \(0.133423\pi\)
−0.913431 + 0.406994i \(0.866577\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −14.6603 −1.13444 −0.567222 0.823565i \(-0.691982\pi\)
−0.567222 + 0.823565i \(0.691982\pi\)
\(168\) 0 0
\(169\) 25.3923 1.95325
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 17.1962i 1.30740i 0.756754 + 0.653700i \(0.226784\pi\)
−0.756754 + 0.653700i \(0.773216\pi\)
\(174\) 0 0
\(175\) − 4.73205i − 0.357709i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 1.85641 0.138754 0.0693772 0.997591i \(-0.477899\pi\)
0.0693772 + 0.997591i \(0.477899\pi\)
\(180\) 0 0
\(181\) −15.3923 −1.14410 −0.572051 0.820218i \(-0.693852\pi\)
−0.572051 + 0.820218i \(0.693852\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 2.00000i − 0.147043i
\(186\) 0 0
\(187\) 24.5885i 1.79809i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 18.0000 1.30243 0.651217 0.758891i \(-0.274259\pi\)
0.651217 + 0.758891i \(0.274259\pi\)
\(192\) 0 0
\(193\) −1.80385 −0.129844 −0.0649219 0.997890i \(-0.520680\pi\)
−0.0649219 + 0.997890i \(0.520680\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 17.1962i 1.22518i 0.790403 + 0.612588i \(0.209871\pi\)
−0.790403 + 0.612588i \(0.790129\pi\)
\(198\) 0 0
\(199\) − 10.3923i − 0.736691i −0.929689 0.368345i \(-0.879924\pi\)
0.929689 0.368345i \(-0.120076\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −38.7846 −2.72215
\(204\) 0 0
\(205\) 2.19615 0.153386
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 2.19615i − 0.151911i
\(210\) 0 0
\(211\) − 9.00000i − 0.619586i −0.950804 0.309793i \(-0.899740\pi\)
0.950804 0.309793i \(-0.100260\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 5.66025 0.386026
\(216\) 0 0
\(217\) −2.19615 −0.149085
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 32.1962i − 2.16575i
\(222\) 0 0
\(223\) − 0.928203i − 0.0621571i −0.999517 0.0310785i \(-0.990106\pi\)
0.999517 0.0310785i \(-0.00989420\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 4.26795 0.283274 0.141637 0.989919i \(-0.454763\pi\)
0.141637 + 0.989919i \(0.454763\pi\)
\(228\) 0 0
\(229\) −5.00000 −0.330409 −0.165205 0.986259i \(-0.552828\pi\)
−0.165205 + 0.986259i \(0.552828\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 14.7846i − 0.968572i −0.874910 0.484286i \(-0.839079\pi\)
0.874910 0.484286i \(-0.160921\pi\)
\(234\) 0 0
\(235\) 9.46410i 0.617370i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 29.3205 1.89659 0.948293 0.317396i \(-0.102809\pi\)
0.948293 + 0.317396i \(0.102809\pi\)
\(240\) 0 0
\(241\) −3.39230 −0.218518 −0.109259 0.994013i \(-0.534848\pi\)
−0.109259 + 0.994013i \(0.534848\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 15.3923i 0.983378i
\(246\) 0 0
\(247\) 2.87564i 0.182973i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −11.3205 −0.714544 −0.357272 0.934000i \(-0.616293\pi\)
−0.357272 + 0.934000i \(0.616293\pi\)
\(252\) 0 0
\(253\) −20.1962 −1.26972
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 9.58846i 0.598112i 0.954236 + 0.299056i \(0.0966717\pi\)
−0.954236 + 0.299056i \(0.903328\pi\)
\(258\) 0 0
\(259\) 9.46410i 0.588071i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −11.3205 −0.698052 −0.349026 0.937113i \(-0.613488\pi\)
−0.349026 + 0.937113i \(0.613488\pi\)
\(264\) 0 0
\(265\) 11.1962 0.687774
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 14.7846i − 0.901434i −0.892667 0.450717i \(-0.851168\pi\)
0.892667 0.450717i \(-0.148832\pi\)
\(270\) 0 0
\(271\) 17.5359i 1.06523i 0.846358 + 0.532615i \(0.178791\pi\)
−0.846358 + 0.532615i \(0.821209\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 4.73205 0.285353
\(276\) 0 0
\(277\) −28.1962 −1.69414 −0.847071 0.531479i \(-0.821636\pi\)
−0.847071 + 0.531479i \(0.821636\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 12.5885i 0.750964i 0.926830 + 0.375482i \(0.122523\pi\)
−0.926830 + 0.375482i \(0.877477\pi\)
\(282\) 0 0
\(283\) − 13.2679i − 0.788698i −0.918961 0.394349i \(-0.870970\pi\)
0.918961 0.394349i \(-0.129030\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −10.3923 −0.613438
\(288\) 0 0
\(289\) −10.0000 −0.588235
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 12.8038i − 0.748009i −0.927427 0.374004i \(-0.877984\pi\)
0.927427 0.374004i \(-0.122016\pi\)
\(294\) 0 0
\(295\) − 5.66025i − 0.329553i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 26.4449 1.52935
\(300\) 0 0
\(301\) −26.7846 −1.54384
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1.00000i 0.0572598i
\(306\) 0 0
\(307\) − 19.8564i − 1.13326i −0.823971 0.566632i \(-0.808246\pi\)
0.823971 0.566632i \(-0.191754\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −22.7321 −1.28902 −0.644508 0.764597i \(-0.722938\pi\)
−0.644508 + 0.764597i \(0.722938\pi\)
\(312\) 0 0
\(313\) 8.00000 0.452187 0.226093 0.974106i \(-0.427405\pi\)
0.226093 + 0.974106i \(0.427405\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 18.8038i 1.05613i 0.849204 + 0.528065i \(0.177082\pi\)
−0.849204 + 0.528065i \(0.822918\pi\)
\(318\) 0 0
\(319\) − 38.7846i − 2.17152i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 2.41154 0.134182
\(324\) 0 0
\(325\) −6.19615 −0.343701
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 44.7846i − 2.46906i
\(330\) 0 0
\(331\) − 8.53590i − 0.469175i −0.972095 0.234588i \(-0.924626\pi\)
0.972095 0.234588i \(-0.0753740\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 10.3923 0.567792
\(336\) 0 0
\(337\) −2.58846 −0.141002 −0.0705011 0.997512i \(-0.522460\pi\)
−0.0705011 + 0.997512i \(0.522460\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 2.19615i − 0.118928i
\(342\) 0 0
\(343\) − 39.7128i − 2.14429i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 10.3923 0.557888 0.278944 0.960307i \(-0.410016\pi\)
0.278944 + 0.960307i \(0.410016\pi\)
\(348\) 0 0
\(349\) 12.6077 0.674874 0.337437 0.941348i \(-0.390440\pi\)
0.337437 + 0.941348i \(0.390440\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 22.3923i − 1.19182i −0.803050 0.595911i \(-0.796791\pi\)
0.803050 0.595911i \(-0.203209\pi\)
\(354\) 0 0
\(355\) 4.73205i 0.251151i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 21.7128 1.14596 0.572979 0.819570i \(-0.305788\pi\)
0.572979 + 0.819570i \(0.305788\pi\)
\(360\) 0 0
\(361\) 18.7846 0.988664
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 4.19615i 0.219637i
\(366\) 0 0
\(367\) − 6.58846i − 0.343915i −0.985104 0.171957i \(-0.944991\pi\)
0.985104 0.171957i \(-0.0550091\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −52.9808 −2.75062
\(372\) 0 0
\(373\) −8.39230 −0.434537 −0.217269 0.976112i \(-0.569715\pi\)
−0.217269 + 0.976112i \(0.569715\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 50.7846i 2.61554i
\(378\) 0 0
\(379\) − 27.0000i − 1.38690i −0.720506 0.693448i \(-0.756091\pi\)
0.720506 0.693448i \(-0.243909\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 3.33975 0.170653 0.0853265 0.996353i \(-0.472807\pi\)
0.0853265 + 0.996353i \(0.472807\pi\)
\(384\) 0 0
\(385\) −22.3923 −1.14122
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 6.58846i − 0.334048i −0.985953 0.167024i \(-0.946584\pi\)
0.985953 0.167024i \(-0.0534157\pi\)
\(390\) 0 0
\(391\) − 22.1769i − 1.12153i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 9.92820 0.499542
\(396\) 0 0
\(397\) 26.5885 1.33444 0.667218 0.744862i \(-0.267485\pi\)
0.667218 + 0.744862i \(0.267485\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 10.3923i − 0.518967i −0.965748 0.259483i \(-0.916448\pi\)
0.965748 0.259483i \(-0.0835523\pi\)
\(402\) 0 0
\(403\) 2.87564i 0.143246i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −9.46410 −0.469118
\(408\) 0 0
\(409\) 3.78461 0.187137 0.0935685 0.995613i \(-0.470173\pi\)
0.0935685 + 0.995613i \(0.470173\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 26.7846i 1.31798i
\(414\) 0 0
\(415\) 5.19615i 0.255069i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 34.9808 1.70892 0.854461 0.519515i \(-0.173888\pi\)
0.854461 + 0.519515i \(0.173888\pi\)
\(420\) 0 0
\(421\) 32.1769 1.56821 0.784103 0.620630i \(-0.213123\pi\)
0.784103 + 0.620630i \(0.213123\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 5.19615i 0.252050i
\(426\) 0 0
\(427\) − 4.73205i − 0.229000i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 15.1244 0.728515 0.364257 0.931298i \(-0.381323\pi\)
0.364257 + 0.931298i \(0.381323\pi\)
\(432\) 0 0
\(433\) 26.5885 1.27776 0.638880 0.769307i \(-0.279398\pi\)
0.638880 + 0.769307i \(0.279398\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.98076i 0.0947527i
\(438\) 0 0
\(439\) − 30.7128i − 1.46584i −0.680313 0.732921i \(-0.738156\pi\)
0.680313 0.732921i \(-0.261844\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −16.5167 −0.784730 −0.392365 0.919810i \(-0.628343\pi\)
−0.392365 + 0.919810i \(0.628343\pi\)
\(444\) 0 0
\(445\) 14.1962 0.672962
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 28.9808i 1.36769i 0.729629 + 0.683843i \(0.239693\pi\)
−0.729629 + 0.683843i \(0.760307\pi\)
\(450\) 0 0
\(451\) − 10.3923i − 0.489355i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 29.3205 1.37457
\(456\) 0 0
\(457\) 0.784610 0.0367025 0.0183512 0.999832i \(-0.494158\pi\)
0.0183512 + 0.999832i \(0.494158\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 22.3923i 1.04291i 0.853278 + 0.521457i \(0.174611\pi\)
−0.853278 + 0.521457i \(0.825389\pi\)
\(462\) 0 0
\(463\) − 0.928203i − 0.0431373i −0.999767 0.0215686i \(-0.993134\pi\)
0.999767 0.0215686i \(-0.00686604\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −31.7321 −1.46838 −0.734192 0.678942i \(-0.762439\pi\)
−0.734192 + 0.678942i \(0.762439\pi\)
\(468\) 0 0
\(469\) −49.1769 −2.27078
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 26.7846i − 1.23156i
\(474\) 0 0
\(475\) − 0.464102i − 0.0212944i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −21.8038 −0.996243 −0.498122 0.867107i \(-0.665977\pi\)
−0.498122 + 0.867107i \(0.665977\pi\)
\(480\) 0 0
\(481\) 12.3923 0.565040
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 8.00000i 0.363261i
\(486\) 0 0
\(487\) − 16.9808i − 0.769472i −0.923027 0.384736i \(-0.874293\pi\)
0.923027 0.384736i \(-0.125707\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −2.78461 −0.125668 −0.0628338 0.998024i \(-0.520014\pi\)
−0.0628338 + 0.998024i \(0.520014\pi\)
\(492\) 0 0
\(493\) 42.5885 1.91809
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 22.3923i − 1.00443i
\(498\) 0 0
\(499\) 35.5359i 1.59081i 0.606081 + 0.795403i \(0.292741\pi\)
−0.606081 + 0.795403i \(0.707259\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 2.41154 0.107525 0.0537627 0.998554i \(-0.482879\pi\)
0.0537627 + 0.998554i \(0.482879\pi\)
\(504\) 0 0
\(505\) −4.39230 −0.195455
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 4.39230i 0.194685i 0.995251 + 0.0973427i \(0.0310343\pi\)
−0.995251 + 0.0973427i \(0.968966\pi\)
\(510\) 0 0
\(511\) − 19.8564i − 0.878396i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 9.46410 0.417038
\(516\) 0 0
\(517\) 44.7846 1.96962
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 6.58846i − 0.288646i −0.989531 0.144323i \(-0.953900\pi\)
0.989531 0.144323i \(-0.0461004\pi\)
\(522\) 0 0
\(523\) 23.6603i 1.03459i 0.855807 + 0.517295i \(0.173061\pi\)
−0.855807 + 0.517295i \(0.826939\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2.41154 0.105048
\(528\) 0 0
\(529\) −4.78461 −0.208027
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 13.6077i 0.589415i
\(534\) 0 0
\(535\) − 18.9282i − 0.818338i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 72.8372 3.13732
\(540\) 0 0
\(541\) 16.0000 0.687894 0.343947 0.938989i \(-0.388236\pi\)
0.343947 + 0.938989i \(0.388236\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 11.3923i − 0.487993i
\(546\) 0 0
\(547\) − 22.7321i − 0.971952i −0.873972 0.485976i \(-0.838464\pi\)
0.873972 0.485976i \(-0.161536\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −3.80385 −0.162049
\(552\) 0 0
\(553\) −46.9808 −1.99783
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 34.3923i − 1.45725i −0.684914 0.728624i \(-0.740160\pi\)
0.684914 0.728624i \(-0.259840\pi\)
\(558\) 0 0
\(559\) 35.0718i 1.48338i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −1.85641 −0.0782382 −0.0391191 0.999235i \(-0.512455\pi\)
−0.0391191 + 0.999235i \(0.512455\pi\)
\(564\) 0 0
\(565\) −12.0000 −0.504844
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 32.1962i − 1.34973i −0.737940 0.674866i \(-0.764201\pi\)
0.737940 0.674866i \(-0.235799\pi\)
\(570\) 0 0
\(571\) 0.464102i 0.0194220i 0.999953 + 0.00971102i \(0.00309116\pi\)
−0.999953 + 0.00971102i \(0.996909\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −4.26795 −0.177986
\(576\) 0 0
\(577\) −4.58846 −0.191020 −0.0955100 0.995428i \(-0.530448\pi\)
−0.0955100 + 0.995428i \(0.530448\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 24.5885i − 1.02010i
\(582\) 0 0
\(583\) − 52.9808i − 2.19424i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 5.19615 0.214468 0.107234 0.994234i \(-0.465801\pi\)
0.107234 + 0.994234i \(0.465801\pi\)
\(588\) 0 0
\(589\) −0.215390 −0.00887500
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 23.1962i 0.952552i 0.879296 + 0.476276i \(0.158014\pi\)
−0.879296 + 0.476276i \(0.841986\pi\)
\(594\) 0 0
\(595\) − 24.5885i − 1.00803i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −11.4115 −0.466263 −0.233131 0.972445i \(-0.574897\pi\)
−0.233131 + 0.972445i \(0.574897\pi\)
\(600\) 0 0
\(601\) −32.1769 −1.31252 −0.656262 0.754533i \(-0.727863\pi\)
−0.656262 + 0.754533i \(0.727863\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 11.3923i − 0.463163i
\(606\) 0 0
\(607\) − 22.7321i − 0.922665i −0.887227 0.461333i \(-0.847371\pi\)
0.887227 0.461333i \(-0.152629\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −58.6410 −2.37236
\(612\) 0 0
\(613\) 33.1769 1.34000 0.670001 0.742360i \(-0.266293\pi\)
0.670001 + 0.742360i \(0.266293\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 3.58846i − 0.144466i −0.997388 0.0722329i \(-0.976988\pi\)
0.997388 0.0722329i \(-0.0230125\pi\)
\(618\) 0 0
\(619\) − 8.53590i − 0.343087i −0.985177 0.171543i \(-0.945125\pi\)
0.985177 0.171543i \(-0.0548754\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −67.1769 −2.69139
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 10.3923i − 0.414368i
\(630\) 0 0
\(631\) − 47.7846i − 1.90228i −0.308767 0.951138i \(-0.599916\pi\)
0.308767 0.951138i \(-0.400084\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −18.0000 −0.714308
\(636\) 0 0
\(637\) −95.3731 −3.77882
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 16.3923i 0.647457i 0.946150 + 0.323729i \(0.104936\pi\)
−0.946150 + 0.323729i \(0.895064\pi\)
\(642\) 0 0
\(643\) − 6.58846i − 0.259823i −0.991526 0.129912i \(-0.958531\pi\)
0.991526 0.129912i \(-0.0414694\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −7.05256 −0.277265 −0.138632 0.990344i \(-0.544271\pi\)
−0.138632 + 0.990344i \(0.544271\pi\)
\(648\) 0 0
\(649\) −26.7846 −1.05139
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 2.41154i − 0.0943710i −0.998886 0.0471855i \(-0.984975\pi\)
0.998886 0.0471855i \(-0.0150252\pi\)
\(654\) 0 0
\(655\) 0.928203i 0.0362679i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 24.5885 0.957830 0.478915 0.877861i \(-0.341030\pi\)
0.478915 + 0.877861i \(0.341030\pi\)
\(660\) 0 0
\(661\) −4.00000 −0.155582 −0.0777910 0.996970i \(-0.524787\pi\)
−0.0777910 + 0.996970i \(0.524787\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 2.19615i 0.0851631i
\(666\) 0 0
\(667\) 34.9808i 1.35446i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 4.73205 0.182679
\(672\) 0 0
\(673\) 14.5885 0.562344 0.281172 0.959657i \(-0.409277\pi\)
0.281172 + 0.959657i \(0.409277\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) − 37.8564i − 1.45280i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −32.6603 −1.24971 −0.624855 0.780741i \(-0.714842\pi\)
−0.624855 + 0.780741i \(0.714842\pi\)
\(684\) 0 0
\(685\) 5.19615 0.198535
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 69.3731i 2.64290i
\(690\) 0 0
\(691\) 27.0000i 1.02713i 0.858051 + 0.513564i \(0.171675\pi\)
−0.858051 + 0.513564i \(0.828325\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 19.8564 0.753196
\(696\) 0 0
\(697\) 11.4115 0.432243
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 6.00000i − 0.226617i −0.993560 0.113308i \(-0.963855\pi\)
0.993560 0.113308i \(-0.0361448\pi\)
\(702\) 0 0
\(703\) 0.928203i 0.0350078i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 20.7846 0.781686
\(708\) 0 0
\(709\) 28.0000 1.05156 0.525781 0.850620i \(-0.323773\pi\)
0.525781 + 0.850620i \(0.323773\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1.98076i 0.0741801i
\(714\) 0 0
\(715\) 29.3205i 1.09652i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 36.9282 1.37719 0.688595 0.725146i \(-0.258228\pi\)
0.688595 + 0.725146i \(0.258228\pi\)
\(720\) 0 0
\(721\) −44.7846 −1.66787
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 8.19615i − 0.304397i
\(726\) 0 0
\(727\) − 39.7128i − 1.47287i −0.676510 0.736433i \(-0.736508\pi\)
0.676510 0.736433i \(-0.263492\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 29.4115 1.08783
\(732\) 0 0
\(733\) −36.3923 −1.34418 −0.672090 0.740469i \(-0.734603\pi\)
−0.672090 + 0.740469i \(0.734603\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 49.1769i − 1.81145i
\(738\) 0 0
\(739\) − 19.3923i − 0.713357i −0.934227 0.356679i \(-0.883909\pi\)
0.934227 0.356679i \(-0.116091\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −50.1051 −1.83818 −0.919089 0.394049i \(-0.871074\pi\)
−0.919089 + 0.394049i \(0.871074\pi\)
\(744\) 0 0
\(745\) −9.80385 −0.359185
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 89.5692i 3.27279i
\(750\) 0 0
\(751\) 48.7128i 1.77756i 0.458338 + 0.888778i \(0.348445\pi\)
−0.458338 + 0.888778i \(0.651555\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −18.9282 −0.688868
\(756\) 0 0
\(757\) 34.0000 1.23575 0.617876 0.786276i \(-0.287994\pi\)
0.617876 + 0.786276i \(0.287994\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 45.3731i 1.64477i 0.568930 + 0.822386i \(0.307358\pi\)
−0.568930 + 0.822386i \(0.692642\pi\)
\(762\) 0 0
\(763\) 53.9090i 1.95164i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 35.0718 1.26637
\(768\) 0 0
\(769\) −13.0000 −0.468792 −0.234396 0.972141i \(-0.575311\pi\)
−0.234396 + 0.972141i \(0.575311\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 23.1962i − 0.834308i −0.908836 0.417154i \(-0.863028\pi\)
0.908836 0.417154i \(-0.136972\pi\)
\(774\) 0 0
\(775\) − 0.464102i − 0.0166710i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −1.01924 −0.0365180
\(780\) 0 0
\(781\) 22.3923 0.801260
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 16.1962i − 0.578065i
\(786\) 0 0
\(787\) − 15.1244i − 0.539125i −0.962983 0.269563i \(-0.913121\pi\)
0.962983 0.269563i \(-0.0868791\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 56.7846 2.01903
\(792\) 0 0
\(793\) −6.19615 −0.220032
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 40.7654i − 1.44398i −0.691902 0.721992i \(-0.743227\pi\)
0.691902 0.721992i \(-0.256773\pi\)
\(798\) 0 0
\(799\) 49.1769i 1.73975i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 19.8564 0.700717
\(804\) 0 0
\(805\) 20.1962 0.711821
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 5.41154i 0.190260i 0.995465 + 0.0951299i \(0.0303266\pi\)
−0.995465 + 0.0951299i \(0.969673\pi\)
\(810\) 0 0
\(811\) − 17.0718i − 0.599472i −0.954022 0.299736i \(-0.903101\pi\)
0.954022 0.299736i \(-0.0968986\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 10.3923 0.364027
\(816\) 0 0
\(817\) −2.62693 −0.0919048
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 10.3923i − 0.362694i −0.983419 0.181347i \(-0.941954\pi\)
0.983419 0.181347i \(-0.0580457\pi\)
\(822\) 0 0
\(823\) 3.80385i 0.132594i 0.997800 + 0.0662969i \(0.0211184\pi\)
−0.997800 + 0.0662969i \(0.978882\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −31.7321 −1.10343 −0.551716 0.834032i \(-0.686027\pi\)
−0.551716 + 0.834032i \(0.686027\pi\)
\(828\) 0 0
\(829\) 29.1769 1.01336 0.506678 0.862135i \(-0.330873\pi\)
0.506678 + 0.862135i \(0.330873\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 79.9808i 2.77117i
\(834\) 0 0
\(835\) 14.6603i 0.507339i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 18.9282 0.653474 0.326737 0.945115i \(-0.394051\pi\)
0.326737 + 0.945115i \(0.394051\pi\)
\(840\) 0 0
\(841\) −38.1769 −1.31645
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 25.3923i − 0.873522i
\(846\) 0 0
\(847\) 53.9090i 1.85233i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 8.53590 0.292607
\(852\) 0 0
\(853\) −0.784610 −0.0268645 −0.0134323 0.999910i \(-0.504276\pi\)
−0.0134323 + 0.999910i \(0.504276\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 12.8038i − 0.437371i −0.975795 0.218686i \(-0.929823\pi\)
0.975795 0.218686i \(-0.0701769\pi\)
\(858\) 0 0
\(859\) 3.24871i 0.110845i 0.998463 + 0.0554223i \(0.0176505\pi\)
−0.998463 + 0.0554223i \(0.982349\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −41.1962 −1.40233 −0.701167 0.712997i \(-0.747337\pi\)
−0.701167 + 0.712997i \(0.747337\pi\)
\(864\) 0 0
\(865\) 17.1962 0.584687
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 46.9808i − 1.59371i
\(870\) 0 0
\(871\) 64.3923i 2.18185i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −4.73205 −0.159973
\(876\) 0 0
\(877\) 7.41154 0.250270 0.125135 0.992140i \(-0.460064\pi\)
0.125135 + 0.992140i \(0.460064\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 11.4115i − 0.384465i −0.981349 0.192232i \(-0.938427\pi\)
0.981349 0.192232i \(-0.0615727\pi\)
\(882\) 0 0
\(883\) 4.73205i 0.159246i 0.996825 + 0.0796231i \(0.0253717\pi\)
−0.996825 + 0.0796231i \(0.974628\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −34.5167 −1.15896 −0.579478 0.814988i \(-0.696744\pi\)
−0.579478 + 0.814988i \(0.696744\pi\)
\(888\) 0 0
\(889\) 85.1769 2.85674
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 4.39230i − 0.146983i
\(894\) 0 0
\(895\) − 1.85641i − 0.0620528i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −3.80385 −0.126865
\(900\) 0 0
\(901\) 58.1769 1.93815
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 15.3923i 0.511658i
\(906\) 0 0
\(907\) 18.9282i 0.628501i 0.949340 + 0.314250i \(0.101753\pi\)
−0.949340 + 0.314250i \(0.898247\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 7.51666 0.249038 0.124519 0.992217i \(-0.460261\pi\)
0.124519 + 0.992217i \(0.460261\pi\)
\(912\) 0 0
\(913\) 24.5885 0.813759
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 4.39230i − 0.145047i
\(918\) 0 0
\(919\) 21.7128i 0.716240i 0.933676 + 0.358120i \(0.116582\pi\)
−0.933676 + 0.358120i \(0.883418\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −29.3205 −0.965096
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 9.80385i − 0.321654i −0.986983 0.160827i \(-0.948584\pi\)
0.986983 0.160827i \(-0.0514161\pi\)
\(930\) 0 0
\(931\) − 7.14359i − 0.234122i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 24.5885 0.804129
\(936\) 0 0
\(937\) 33.1769 1.08384 0.541921 0.840429i \(-0.317697\pi\)
0.541921 + 0.840429i \(0.317697\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 55.1769i 1.79872i 0.437213 + 0.899358i \(0.355966\pi\)
−0.437213 + 0.899358i \(0.644034\pi\)
\(942\) 0 0
\(943\) 9.37307i 0.305229i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 46.7654 1.51967 0.759835 0.650116i \(-0.225280\pi\)
0.759835 + 0.650116i \(0.225280\pi\)
\(948\) 0 0
\(949\) −26.0000 −0.843996
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 6.00000i − 0.194359i −0.995267 0.0971795i \(-0.969018\pi\)
0.995267 0.0971795i \(-0.0309821\pi\)
\(954\) 0 0
\(955\) − 18.0000i − 0.582466i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −24.5885 −0.794003
\(960\) 0 0
\(961\) 30.7846 0.993052
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 1.80385i 0.0580679i
\(966\) 0 0
\(967\) 29.3205i 0.942884i 0.881897 + 0.471442i \(0.156266\pi\)
−0.881897 + 0.471442i \(0.843734\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −29.3205 −0.940940 −0.470470 0.882416i \(-0.655916\pi\)
−0.470470 + 0.882416i \(0.655916\pi\)
\(972\) 0 0
\(973\) −93.9615 −3.01227
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 31.1769i − 0.997438i −0.866764 0.498719i \(-0.833804\pi\)
0.866764 0.498719i \(-0.166196\pi\)
\(978\) 0 0
\(979\) − 67.1769i − 2.14698i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 13.7321 0.437984 0.218992 0.975727i \(-0.429723\pi\)
0.218992 + 0.975727i \(0.429723\pi\)
\(984\) 0 0
\(985\) 17.1962 0.547915
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 24.1577i 0.768169i
\(990\) 0 0
\(991\) − 44.0718i − 1.39999i −0.714149 0.699993i \(-0.753186\pi\)
0.714149 0.699993i \(-0.246814\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −10.3923 −0.329458
\(996\) 0 0
\(997\) −48.1962 −1.52639 −0.763194 0.646170i \(-0.776370\pi\)
−0.763194 + 0.646170i \(0.776370\pi\)
\(998\) 0 0
\(999\) 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 2160.2.h.a.431.2 4
3.2 odd 2 2160.2.h.f.431.4 yes 4
4.3 odd 2 2160.2.h.f.431.1 yes 4
12.11 even 2 inner 2160.2.h.a.431.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2160.2.h.a.431.2 4 1.1 even 1 trivial
2160.2.h.a.431.3 yes 4 12.11 even 2 inner
2160.2.h.f.431.1 yes 4 4.3 odd 2
2160.2.h.f.431.4 yes 4 3.2 odd 2