Properties

Label 1152.3.g.c.127.1
Level $1152$
Weight $3$
Character 1152.127
Analytic conductor $31.390$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1152,3,Mod(127,1152)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1152, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1152.127");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1152.g (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: \(31.3897264543\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{21} \)
Twist minimal: no (minimal twist has level 384)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 127.1
Root \(-0.965926 - 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 1152.127
Dual form 1152.3.g.c.127.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-8.29253 q^{5} -8.55583i q^{7} +O(q^{10})\) \(q-8.29253 q^{5} -8.55583i q^{7} +13.7980i q^{11} -17.0693 q^{13} -20.3246 q^{17} +20.4440i q^{19} -5.51575i q^{23} +43.7660 q^{25} +41.0586 q^{29} -22.2953i q^{31} +70.9495i q^{35} +11.6326 q^{37} -35.9527 q^{41} -66.8648i q^{43} +19.9366i q^{47} -24.2023 q^{49} +17.6329 q^{53} -114.420i q^{55} -62.1572i q^{59} +47.4836 q^{61} +141.548 q^{65} +74.8105i q^{67} +16.9150i q^{71} +101.712 q^{73} +118.053 q^{77} -0.879320i q^{79} -23.2131i q^{83} +168.542 q^{85} +16.3089 q^{89} +146.042i q^{91} -169.532i q^{95} +188.307 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{5} - 48 q^{13} - 16 q^{17} - 8 q^{25} - 80 q^{29} + 16 q^{37} - 80 q^{41} - 88 q^{49} + 176 q^{53} + 272 q^{61} + 160 q^{65} - 16 q^{73} + 320 q^{77} - 32 q^{85} + 240 q^{89} + 400 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(5\) −8.29253 −1.65851 −0.829253 0.558874i \(-0.811234\pi\)
−0.829253 + 0.558874i \(0.811234\pi\)
\(6\) 0 0
\(7\) − 8.55583i − 1.22226i −0.791529 0.611131i \(-0.790715\pi\)
0.791529 0.611131i \(-0.209285\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 13.7980i 1.25436i 0.778874 + 0.627180i \(0.215791\pi\)
−0.778874 + 0.627180i \(0.784209\pi\)
\(12\) 0 0
\(13\) −17.0693 −1.31302 −0.656512 0.754316i \(-0.727969\pi\)
−0.656512 + 0.754316i \(0.727969\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −20.3246 −1.19556 −0.597781 0.801659i \(-0.703951\pi\)
−0.597781 + 0.801659i \(0.703951\pi\)
\(18\) 0 0
\(19\) 20.4440i 1.07600i 0.842946 + 0.537999i \(0.180819\pi\)
−0.842946 + 0.537999i \(0.819181\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 5.51575i − 0.239815i −0.992785 0.119908i \(-0.961740\pi\)
0.992785 0.119908i \(-0.0382598\pi\)
\(24\) 0 0
\(25\) 43.7660 1.75064
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 41.0586 1.41581 0.707906 0.706306i \(-0.249640\pi\)
0.707906 + 0.706306i \(0.249640\pi\)
\(30\) 0 0
\(31\) − 22.2953i − 0.719205i −0.933106 0.359602i \(-0.882912\pi\)
0.933106 0.359602i \(-0.117088\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 70.9495i 2.02713i
\(36\) 0 0
\(37\) 11.6326 0.314396 0.157198 0.987567i \(-0.449754\pi\)
0.157198 + 0.987567i \(0.449754\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −35.9527 −0.876894 −0.438447 0.898757i \(-0.644471\pi\)
−0.438447 + 0.898757i \(0.644471\pi\)
\(42\) 0 0
\(43\) − 66.8648i − 1.55499i −0.628886 0.777497i \(-0.716489\pi\)
0.628886 0.777497i \(-0.283511\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 19.9366i 0.424182i 0.977250 + 0.212091i \(0.0680274\pi\)
−0.977250 + 0.212091i \(0.931973\pi\)
\(48\) 0 0
\(49\) −24.2023 −0.493924
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 17.6329 0.332697 0.166348 0.986067i \(-0.446802\pi\)
0.166348 + 0.986067i \(0.446802\pi\)
\(54\) 0 0
\(55\) − 114.420i − 2.08036i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 62.1572i − 1.05351i −0.850017 0.526756i \(-0.823408\pi\)
0.850017 0.526756i \(-0.176592\pi\)
\(60\) 0 0
\(61\) 47.4836 0.778419 0.389210 0.921149i \(-0.372748\pi\)
0.389210 + 0.921149i \(0.372748\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 141.548 2.17766
\(66\) 0 0
\(67\) 74.8105i 1.11657i 0.829648 + 0.558287i \(0.188541\pi\)
−0.829648 + 0.558287i \(0.811459\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 16.9150i 0.238240i 0.992880 + 0.119120i \(0.0380073\pi\)
−0.992880 + 0.119120i \(0.961993\pi\)
\(72\) 0 0
\(73\) 101.712 1.39331 0.696657 0.717404i \(-0.254670\pi\)
0.696657 + 0.717404i \(0.254670\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 118.053 1.53316
\(78\) 0 0
\(79\) − 0.879320i − 0.0111306i −0.999985 0.00556532i \(-0.998228\pi\)
0.999985 0.00556532i \(-0.00177151\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 23.2131i − 0.279676i −0.990174 0.139838i \(-0.955342\pi\)
0.990174 0.139838i \(-0.0446582\pi\)
\(84\) 0 0
\(85\) 168.542 1.98285
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 16.3089 0.183246 0.0916231 0.995794i \(-0.470795\pi\)
0.0916231 + 0.995794i \(0.470795\pi\)
\(90\) 0 0
\(91\) 146.042i 1.60486i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 169.532i − 1.78455i
\(96\) 0 0
\(97\) 188.307 1.94131 0.970656 0.240474i \(-0.0773028\pi\)
0.970656 + 0.240474i \(0.0773028\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −66.8468 −0.661849 −0.330925 0.943657i \(-0.607361\pi\)
−0.330925 + 0.943657i \(0.607361\pi\)
\(102\) 0 0
\(103\) 166.313i 1.61469i 0.590083 + 0.807343i \(0.299095\pi\)
−0.590083 + 0.807343i \(0.700905\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 80.1467i − 0.749035i −0.927220 0.374517i \(-0.877808\pi\)
0.927220 0.374517i \(-0.122192\pi\)
\(108\) 0 0
\(109\) −26.8781 −0.246589 −0.123294 0.992370i \(-0.539346\pi\)
−0.123294 + 0.992370i \(0.539346\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 79.0958 0.699963 0.349981 0.936757i \(-0.386188\pi\)
0.349981 + 0.936757i \(0.386188\pi\)
\(114\) 0 0
\(115\) 45.7395i 0.397735i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 173.894i 1.46129i
\(120\) 0 0
\(121\) −69.3837 −0.573419
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −155.618 −1.24494
\(126\) 0 0
\(127\) − 170.725i − 1.34429i −0.740419 0.672145i \(-0.765373\pi\)
0.740419 0.672145i \(-0.234627\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 178.980i 1.36626i 0.730297 + 0.683129i \(0.239381\pi\)
−0.730297 + 0.683129i \(0.760619\pi\)
\(132\) 0 0
\(133\) 174.915 1.31515
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 93.2075 0.680347 0.340173 0.940363i \(-0.389514\pi\)
0.340173 + 0.940363i \(0.389514\pi\)
\(138\) 0 0
\(139\) 222.476i 1.60055i 0.599635 + 0.800274i \(0.295313\pi\)
−0.599635 + 0.800274i \(0.704687\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 235.522i − 1.64700i
\(144\) 0 0
\(145\) −340.479 −2.34813
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 94.0043 0.630901 0.315451 0.948942i \(-0.397844\pi\)
0.315451 + 0.948942i \(0.397844\pi\)
\(150\) 0 0
\(151\) 82.0081i 0.543100i 0.962424 + 0.271550i \(0.0875362\pi\)
−0.962424 + 0.271550i \(0.912464\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 184.885i 1.19281i
\(156\) 0 0
\(157\) 56.7180 0.361261 0.180631 0.983551i \(-0.442186\pi\)
0.180631 + 0.983551i \(0.442186\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −47.1918 −0.293117
\(162\) 0 0
\(163\) 189.588i 1.16312i 0.813504 + 0.581559i \(0.197557\pi\)
−0.813504 + 0.581559i \(0.802443\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 312.777i − 1.87291i −0.350783 0.936457i \(-0.614084\pi\)
0.350783 0.936457i \(-0.385916\pi\)
\(168\) 0 0
\(169\) 122.361 0.724031
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −182.538 −1.05513 −0.527567 0.849514i \(-0.676896\pi\)
−0.527567 + 0.849514i \(0.676896\pi\)
\(174\) 0 0
\(175\) − 374.455i − 2.13974i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 19.7781i − 0.110492i −0.998473 0.0552461i \(-0.982406\pi\)
0.998473 0.0552461i \(-0.0175943\pi\)
\(180\) 0 0
\(181\) −265.750 −1.46823 −0.734117 0.679023i \(-0.762403\pi\)
−0.734117 + 0.679023i \(0.762403\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −96.4640 −0.521427
\(186\) 0 0
\(187\) − 280.438i − 1.49967i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 288.025i − 1.50799i −0.656882 0.753993i \(-0.728125\pi\)
0.656882 0.753993i \(-0.271875\pi\)
\(192\) 0 0
\(193\) 281.010 1.45601 0.728005 0.685572i \(-0.240448\pi\)
0.728005 + 0.685572i \(0.240448\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −33.8330 −0.171741 −0.0858705 0.996306i \(-0.527367\pi\)
−0.0858705 + 0.996306i \(0.527367\pi\)
\(198\) 0 0
\(199\) 84.9700i 0.426985i 0.976945 + 0.213493i \(0.0684839\pi\)
−0.976945 + 0.213493i \(0.931516\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 351.290i − 1.73049i
\(204\) 0 0
\(205\) 298.138 1.45433
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −282.085 −1.34969
\(210\) 0 0
\(211\) − 140.700i − 0.666826i −0.942781 0.333413i \(-0.891800\pi\)
0.942781 0.333413i \(-0.108200\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 554.478i 2.57897i
\(216\) 0 0
\(217\) −190.755 −0.879057
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 346.926 1.56980
\(222\) 0 0
\(223\) 247.529i 1.11000i 0.831851 + 0.554999i \(0.187281\pi\)
−0.831851 + 0.554999i \(0.812719\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 129.752i 0.571593i 0.958290 + 0.285797i \(0.0922582\pi\)
−0.958290 + 0.285797i \(0.907742\pi\)
\(228\) 0 0
\(229\) −294.304 −1.28517 −0.642586 0.766214i \(-0.722138\pi\)
−0.642586 + 0.766214i \(0.722138\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 239.948 1.02982 0.514911 0.857244i \(-0.327825\pi\)
0.514911 + 0.857244i \(0.327825\pi\)
\(234\) 0 0
\(235\) − 165.325i − 0.703509i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 425.346i 1.77969i 0.456261 + 0.889846i \(0.349188\pi\)
−0.456261 + 0.889846i \(0.650812\pi\)
\(240\) 0 0
\(241\) 264.493 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 200.698 0.819176
\(246\) 0 0
\(247\) − 348.964i − 1.41281i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 221.125i − 0.880976i −0.897759 0.440488i \(-0.854805\pi\)
0.897759 0.440488i \(-0.145195\pi\)
\(252\) 0 0
\(253\) 76.1061 0.300815
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −87.0655 −0.338776 −0.169388 0.985549i \(-0.554179\pi\)
−0.169388 + 0.985549i \(0.554179\pi\)
\(258\) 0 0
\(259\) − 99.5270i − 0.384274i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 277.324i 1.05447i 0.849721 + 0.527233i \(0.176770\pi\)
−0.849721 + 0.527233i \(0.823230\pi\)
\(264\) 0 0
\(265\) −146.222 −0.551780
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −49.8005 −0.185132 −0.0925659 0.995707i \(-0.529507\pi\)
−0.0925659 + 0.995707i \(0.529507\pi\)
\(270\) 0 0
\(271\) − 31.3145i − 0.115552i −0.998330 0.0577758i \(-0.981599\pi\)
0.998330 0.0577758i \(-0.0184009\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 603.882i 2.19593i
\(276\) 0 0
\(277\) 79.1431 0.285715 0.142858 0.989743i \(-0.454371\pi\)
0.142858 + 0.989743i \(0.454371\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 106.190 0.377902 0.188951 0.981987i \(-0.439491\pi\)
0.188951 + 0.981987i \(0.439491\pi\)
\(282\) 0 0
\(283\) − 270.251i − 0.954949i −0.878646 0.477475i \(-0.841552\pi\)
0.878646 0.477475i \(-0.158448\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 307.605i 1.07179i
\(288\) 0 0
\(289\) 124.088 0.429371
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 24.2487 0.0827600 0.0413800 0.999143i \(-0.486825\pi\)
0.0413800 + 0.999143i \(0.486825\pi\)
\(294\) 0 0
\(295\) 515.440i 1.74726i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 94.1500i 0.314883i
\(300\) 0 0
\(301\) −572.084 −1.90061
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −393.759 −1.29101
\(306\) 0 0
\(307\) 122.865i 0.400211i 0.979774 + 0.200106i \(0.0641285\pi\)
−0.979774 + 0.200106i \(0.935871\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 108.808i 0.349865i 0.984580 + 0.174933i \(0.0559707\pi\)
−0.984580 + 0.174933i \(0.944029\pi\)
\(312\) 0 0
\(313\) −52.2216 −0.166842 −0.0834211 0.996514i \(-0.526585\pi\)
−0.0834211 + 0.996514i \(0.526585\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 94.3251 0.297555 0.148778 0.988871i \(-0.452466\pi\)
0.148778 + 0.988871i \(0.452466\pi\)
\(318\) 0 0
\(319\) 566.524i 1.77594i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 415.515i − 1.28642i
\(324\) 0 0
\(325\) −747.056 −2.29863
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 170.574 0.518462
\(330\) 0 0
\(331\) − 406.107i − 1.22691i −0.789730 0.613454i \(-0.789780\pi\)
0.789730 0.613454i \(-0.210220\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) − 620.368i − 1.85184i
\(336\) 0 0
\(337\) −22.4140 −0.0665105 −0.0332552 0.999447i \(-0.510587\pi\)
−0.0332552 + 0.999447i \(0.510587\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 307.630 0.902142
\(342\) 0 0
\(343\) − 212.165i − 0.618557i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 458.377i − 1.32097i −0.750839 0.660486i \(-0.770350\pi\)
0.750839 0.660486i \(-0.229650\pi\)
\(348\) 0 0
\(349\) −282.960 −0.810774 −0.405387 0.914145i \(-0.632863\pi\)
−0.405387 + 0.914145i \(0.632863\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 175.820 0.498073 0.249036 0.968494i \(-0.419886\pi\)
0.249036 + 0.968494i \(0.419886\pi\)
\(354\) 0 0
\(355\) − 140.268i − 0.395122i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 53.5791i − 0.149246i −0.997212 0.0746228i \(-0.976225\pi\)
0.997212 0.0746228i \(-0.0237753\pi\)
\(360\) 0 0
\(361\) −56.9552 −0.157771
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −843.449 −2.31082
\(366\) 0 0
\(367\) − 62.4258i − 0.170097i −0.996377 0.0850487i \(-0.972895\pi\)
0.996377 0.0850487i \(-0.0271046\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 150.864i − 0.406643i
\(372\) 0 0
\(373\) 370.552 0.993437 0.496719 0.867912i \(-0.334538\pi\)
0.496719 + 0.867912i \(0.334538\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −700.841 −1.85900
\(378\) 0 0
\(379\) − 68.7286i − 0.181342i −0.995881 0.0906709i \(-0.971099\pi\)
0.995881 0.0906709i \(-0.0289012\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 155.858i − 0.406939i −0.979081 0.203470i \(-0.934778\pi\)
0.979081 0.203470i \(-0.0652218\pi\)
\(384\) 0 0
\(385\) −978.958 −2.54275
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −481.792 −1.23854 −0.619269 0.785179i \(-0.712571\pi\)
−0.619269 + 0.785179i \(0.712571\pi\)
\(390\) 0 0
\(391\) 112.105i 0.286714i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 7.29179i 0.0184602i
\(396\) 0 0
\(397\) 422.315 1.06377 0.531883 0.846818i \(-0.321485\pi\)
0.531883 + 0.846818i \(0.321485\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 641.025 1.59857 0.799283 0.600955i \(-0.205213\pi\)
0.799283 + 0.600955i \(0.205213\pi\)
\(402\) 0 0
\(403\) 380.566i 0.944333i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 160.507i 0.394365i
\(408\) 0 0
\(409\) −278.562 −0.681081 −0.340541 0.940230i \(-0.610610\pi\)
−0.340541 + 0.940230i \(0.610610\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −531.807 −1.28767
\(414\) 0 0
\(415\) 192.496i 0.463845i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 505.611i 1.20671i 0.797473 + 0.603354i \(0.206169\pi\)
−0.797473 + 0.603354i \(0.793831\pi\)
\(420\) 0 0
\(421\) 179.846 0.427189 0.213594 0.976922i \(-0.431483\pi\)
0.213594 + 0.976922i \(0.431483\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −889.526 −2.09300
\(426\) 0 0
\(427\) − 406.262i − 0.951432i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 580.645i 1.34720i 0.739094 + 0.673602i \(0.235254\pi\)
−0.739094 + 0.673602i \(0.764746\pi\)
\(432\) 0 0
\(433\) −425.621 −0.982959 −0.491479 0.870889i \(-0.663544\pi\)
−0.491479 + 0.870889i \(0.663544\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 112.764 0.258041
\(438\) 0 0
\(439\) 129.991i 0.296107i 0.988979 + 0.148053i \(0.0473008\pi\)
−0.988979 + 0.148053i \(0.952699\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 205.564i − 0.464027i −0.972713 0.232014i \(-0.925469\pi\)
0.972713 0.232014i \(-0.0745314\pi\)
\(444\) 0 0
\(445\) −135.242 −0.303915
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −415.190 −0.924698 −0.462349 0.886698i \(-0.652993\pi\)
−0.462349 + 0.886698i \(0.652993\pi\)
\(450\) 0 0
\(451\) − 496.073i − 1.09994i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) − 1211.06i − 2.66167i
\(456\) 0 0
\(457\) −598.927 −1.31056 −0.655281 0.755385i \(-0.727450\pi\)
−0.655281 + 0.755385i \(0.727450\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 376.804 0.817361 0.408681 0.912677i \(-0.365989\pi\)
0.408681 + 0.912677i \(0.365989\pi\)
\(462\) 0 0
\(463\) 218.577i 0.472088i 0.971742 + 0.236044i \(0.0758510\pi\)
−0.971742 + 0.236044i \(0.924149\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 443.608i − 0.949911i −0.880010 0.474955i \(-0.842464\pi\)
0.880010 0.474955i \(-0.157536\pi\)
\(468\) 0 0
\(469\) 640.066 1.36475
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 922.597 1.95052
\(474\) 0 0
\(475\) 894.751i 1.88369i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 222.133i − 0.463743i −0.972746 0.231871i \(-0.925515\pi\)
0.972746 0.231871i \(-0.0744849\pi\)
\(480\) 0 0
\(481\) −198.561 −0.412809
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −1561.54 −3.21968
\(486\) 0 0
\(487\) 403.795i 0.829148i 0.910016 + 0.414574i \(0.136069\pi\)
−0.910016 + 0.414574i \(0.863931\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 808.347i 1.64633i 0.567803 + 0.823164i \(0.307793\pi\)
−0.567803 + 0.823164i \(0.692207\pi\)
\(492\) 0 0
\(493\) −834.498 −1.69269
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 144.722 0.291192
\(498\) 0 0
\(499\) 132.172i 0.264874i 0.991191 + 0.132437i \(0.0422802\pi\)
−0.991191 + 0.132437i \(0.957720\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 92.8360i − 0.184565i −0.995733 0.0922823i \(-0.970584\pi\)
0.995733 0.0922823i \(-0.0294162\pi\)
\(504\) 0 0
\(505\) 554.329 1.09768
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 927.417 1.82204 0.911018 0.412366i \(-0.135297\pi\)
0.911018 + 0.412366i \(0.135297\pi\)
\(510\) 0 0
\(511\) − 870.231i − 1.70300i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 1379.15i − 2.67796i
\(516\) 0 0
\(517\) −275.084 −0.532077
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −527.145 −1.01179 −0.505897 0.862594i \(-0.668838\pi\)
−0.505897 + 0.862594i \(0.668838\pi\)
\(522\) 0 0
\(523\) 972.249i 1.85898i 0.368843 + 0.929492i \(0.379754\pi\)
−0.368843 + 0.929492i \(0.620246\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 453.143i 0.859854i
\(528\) 0 0
\(529\) 498.577 0.942489
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 613.687 1.15138
\(534\) 0 0
\(535\) 664.619i 1.24228i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 333.942i − 0.619559i
\(540\) 0 0
\(541\) −4.66591 −0.00862461 −0.00431231 0.999991i \(-0.501373\pi\)
−0.00431231 + 0.999991i \(0.501373\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 222.888 0.408968
\(546\) 0 0
\(547\) − 386.796i − 0.707123i −0.935411 0.353561i \(-0.884971\pi\)
0.935411 0.353561i \(-0.115029\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 839.399i 1.52341i
\(552\) 0 0
\(553\) −7.52332 −0.0136046
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 219.630 0.394308 0.197154 0.980373i \(-0.436830\pi\)
0.197154 + 0.980373i \(0.436830\pi\)
\(558\) 0 0
\(559\) 1141.34i 2.04175i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 266.807i 0.473902i 0.971522 + 0.236951i \(0.0761480\pi\)
−0.971522 + 0.236951i \(0.923852\pi\)
\(564\) 0 0
\(565\) −655.904 −1.16089
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 972.046 1.70834 0.854170 0.519993i \(-0.174066\pi\)
0.854170 + 0.519993i \(0.174066\pi\)
\(570\) 0 0
\(571\) − 340.194i − 0.595786i −0.954599 0.297893i \(-0.903716\pi\)
0.954599 0.297893i \(-0.0962838\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 241.402i − 0.419830i
\(576\) 0 0
\(577\) 65.0584 0.112753 0.0563764 0.998410i \(-0.482045\pi\)
0.0563764 + 0.998410i \(0.482045\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −198.608 −0.341838
\(582\) 0 0
\(583\) 243.299i 0.417322i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 162.144i 0.276226i 0.990417 + 0.138113i \(0.0441036\pi\)
−0.990417 + 0.138113i \(0.955896\pi\)
\(588\) 0 0
\(589\) 455.805 0.773862
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 793.227 1.33765 0.668825 0.743420i \(-0.266797\pi\)
0.668825 + 0.743420i \(0.266797\pi\)
\(594\) 0 0
\(595\) − 1442.02i − 2.42356i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 432.194i 0.721525i 0.932658 + 0.360763i \(0.117484\pi\)
−0.932658 + 0.360763i \(0.882516\pi\)
\(600\) 0 0
\(601\) 936.310 1.55792 0.778960 0.627074i \(-0.215748\pi\)
0.778960 + 0.627074i \(0.215748\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 575.366 0.951018
\(606\) 0 0
\(607\) − 223.972i − 0.368982i −0.982834 0.184491i \(-0.940936\pi\)
0.982834 0.184491i \(-0.0590637\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 340.304i − 0.556962i
\(612\) 0 0
\(613\) 1010.40 1.64828 0.824141 0.566385i \(-0.191658\pi\)
0.824141 + 0.566385i \(0.191658\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 400.206 0.648633 0.324316 0.945949i \(-0.394866\pi\)
0.324316 + 0.945949i \(0.394866\pi\)
\(618\) 0 0
\(619\) − 417.349i − 0.674231i −0.941463 0.337116i \(-0.890549\pi\)
0.941463 0.337116i \(-0.109451\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 139.536i − 0.223975i
\(624\) 0 0
\(625\) 196.315 0.314104
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −236.428 −0.375880
\(630\) 0 0
\(631\) − 718.033i − 1.13793i −0.822362 0.568965i \(-0.807344\pi\)
0.822362 0.568965i \(-0.192656\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 1415.74i 2.22951i
\(636\) 0 0
\(637\) 413.116 0.648534
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −139.926 −0.218293 −0.109146 0.994026i \(-0.534812\pi\)
−0.109146 + 0.994026i \(0.534812\pi\)
\(642\) 0 0
\(643\) 777.990i 1.20994i 0.796249 + 0.604969i \(0.206814\pi\)
−0.796249 + 0.604969i \(0.793186\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 673.336i 1.04070i 0.853952 + 0.520352i \(0.174199\pi\)
−0.853952 + 0.520352i \(0.825801\pi\)
\(648\) 0 0
\(649\) 857.642 1.32148
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 223.410 0.342128 0.171064 0.985260i \(-0.445279\pi\)
0.171064 + 0.985260i \(0.445279\pi\)
\(654\) 0 0
\(655\) − 1484.20i − 2.26595i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 166.367i 0.252453i 0.992001 + 0.126227i \(0.0402866\pi\)
−0.992001 + 0.126227i \(0.959713\pi\)
\(660\) 0 0
\(661\) 655.773 0.992092 0.496046 0.868296i \(-0.334785\pi\)
0.496046 + 0.868296i \(0.334785\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −1450.49 −2.18119
\(666\) 0 0
\(667\) − 226.469i − 0.339533i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 655.176i 0.976418i
\(672\) 0 0
\(673\) −863.477 −1.28303 −0.641513 0.767112i \(-0.721693\pi\)
−0.641513 + 0.767112i \(0.721693\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 831.520 1.22824 0.614121 0.789212i \(-0.289511\pi\)
0.614121 + 0.789212i \(0.289511\pi\)
\(678\) 0 0
\(679\) − 1611.13i − 2.37279i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 308.273i − 0.451351i −0.974202 0.225676i \(-0.927541\pi\)
0.974202 0.225676i \(-0.0724590\pi\)
\(684\) 0 0
\(685\) −772.926 −1.12836
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −300.982 −0.436839
\(690\) 0 0
\(691\) 666.811i 0.964994i 0.875898 + 0.482497i \(0.160270\pi\)
−0.875898 + 0.482497i \(0.839730\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 1844.89i − 2.65452i
\(696\) 0 0
\(697\) 730.722 1.04838
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 242.205 0.345514 0.172757 0.984964i \(-0.444732\pi\)
0.172757 + 0.984964i \(0.444732\pi\)
\(702\) 0 0
\(703\) 237.817i 0.338289i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 571.930i 0.808953i
\(708\) 0 0
\(709\) −1380.24 −1.94674 −0.973371 0.229237i \(-0.926377\pi\)
−0.973371 + 0.229237i \(0.926377\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −122.976 −0.172476
\(714\) 0 0
\(715\) 1953.07i 2.73157i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 968.365i 1.34682i 0.739268 + 0.673411i \(0.235172\pi\)
−0.739268 + 0.673411i \(0.764828\pi\)
\(720\) 0 0
\(721\) 1422.94 1.97357
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1796.97 2.47858
\(726\) 0 0
\(727\) − 1017.52i − 1.39962i −0.714329 0.699810i \(-0.753268\pi\)
0.714329 0.699810i \(-0.246732\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1359.00i 1.85909i
\(732\) 0 0
\(733\) 1026.22 1.40003 0.700016 0.714127i \(-0.253176\pi\)
0.700016 + 0.714127i \(0.253176\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −1032.23 −1.40059
\(738\) 0 0
\(739\) − 923.444i − 1.24959i −0.780791 0.624793i \(-0.785183\pi\)
0.780791 0.624793i \(-0.214817\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 1267.12i − 1.70542i −0.522387 0.852708i \(-0.674958\pi\)
0.522387 0.852708i \(-0.325042\pi\)
\(744\) 0 0
\(745\) −779.533 −1.04635
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −685.722 −0.915517
\(750\) 0 0
\(751\) 791.442i 1.05385i 0.849912 + 0.526925i \(0.176655\pi\)
−0.849912 + 0.526925i \(0.823345\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 680.054i − 0.900734i
\(756\) 0 0
\(757\) 333.990 0.441202 0.220601 0.975364i \(-0.429198\pi\)
0.220601 + 0.975364i \(0.429198\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −617.122 −0.810936 −0.405468 0.914109i \(-0.632891\pi\)
−0.405468 + 0.914109i \(0.632891\pi\)
\(762\) 0 0
\(763\) 229.965i 0.301396i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1060.98i 1.38329i
\(768\) 0 0
\(769\) −657.604 −0.855142 −0.427571 0.903982i \(-0.640631\pi\)
−0.427571 + 0.903982i \(0.640631\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −1314.59 −1.70064 −0.850319 0.526268i \(-0.823591\pi\)
−0.850319 + 0.526268i \(0.823591\pi\)
\(774\) 0 0
\(775\) − 975.779i − 1.25907i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 735.015i − 0.943536i
\(780\) 0 0
\(781\) −233.393 −0.298839
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −470.336 −0.599154
\(786\) 0 0
\(787\) − 301.018i − 0.382488i −0.981542 0.191244i \(-0.938748\pi\)
0.981542 0.191244i \(-0.0612522\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) − 676.731i − 0.855538i
\(792\) 0 0
\(793\) −810.512 −1.02208
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −476.193 −0.597481 −0.298741 0.954334i \(-0.596567\pi\)
−0.298741 + 0.954334i \(0.596567\pi\)
\(798\) 0 0
\(799\) − 405.202i − 0.507137i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 1403.42i 1.74772i
\(804\) 0 0
\(805\) 391.340 0.486136
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 692.874 0.856457 0.428228 0.903671i \(-0.359138\pi\)
0.428228 + 0.903671i \(0.359138\pi\)
\(810\) 0 0
\(811\) − 785.074i − 0.968032i −0.875059 0.484016i \(-0.839178\pi\)
0.875059 0.484016i \(-0.160822\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) − 1572.17i − 1.92904i
\(816\) 0 0
\(817\) 1366.98 1.67317
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 527.681 0.642729 0.321365 0.946956i \(-0.395859\pi\)
0.321365 + 0.946956i \(0.395859\pi\)
\(822\) 0 0
\(823\) 228.552i 0.277706i 0.990313 + 0.138853i \(0.0443416\pi\)
−0.990313 + 0.138853i \(0.955658\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 608.409i 0.735683i 0.929889 + 0.367841i \(0.119903\pi\)
−0.929889 + 0.367841i \(0.880097\pi\)
\(828\) 0 0
\(829\) 151.637 0.182915 0.0914577 0.995809i \(-0.470847\pi\)
0.0914577 + 0.995809i \(0.470847\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 491.901 0.590518
\(834\) 0 0
\(835\) 2593.71i 3.10624i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) − 653.351i − 0.778725i −0.921085 0.389363i \(-0.872695\pi\)
0.921085 0.389363i \(-0.127305\pi\)
\(840\) 0 0
\(841\) 844.805 1.00452
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −1014.68 −1.20081
\(846\) 0 0
\(847\) 593.635i 0.700868i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 64.1628i − 0.0753969i
\(852\) 0 0
\(853\) 1372.25 1.60873 0.804367 0.594132i \(-0.202505\pi\)
0.804367 + 0.594132i \(0.202505\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1453.61 1.69617 0.848083 0.529863i \(-0.177757\pi\)
0.848083 + 0.529863i \(0.177757\pi\)
\(858\) 0 0
\(859\) − 962.466i − 1.12045i −0.828341 0.560225i \(-0.810715\pi\)
0.828341 0.560225i \(-0.189285\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 695.890i 0.806362i 0.915120 + 0.403181i \(0.132095\pi\)
−0.915120 + 0.403181i \(0.867905\pi\)
\(864\) 0 0
\(865\) 1513.70 1.74994
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 12.1328 0.0139618
\(870\) 0 0
\(871\) − 1276.96i − 1.46609i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 1331.44i 1.52165i
\(876\) 0 0
\(877\) 338.191 0.385622 0.192811 0.981236i \(-0.438240\pi\)
0.192811 + 0.981236i \(0.438240\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 459.985 0.522117 0.261058 0.965323i \(-0.415928\pi\)
0.261058 + 0.965323i \(0.415928\pi\)
\(882\) 0 0
\(883\) 1208.31i 1.36842i 0.729286 + 0.684209i \(0.239853\pi\)
−0.729286 + 0.684209i \(0.760147\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 1500.73i − 1.69192i −0.533250 0.845958i \(-0.679029\pi\)
0.533250 0.845958i \(-0.320971\pi\)
\(888\) 0 0
\(889\) −1460.69 −1.64307
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −407.582 −0.456419
\(894\) 0 0
\(895\) 164.010i 0.183252i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 915.415i − 1.01826i
\(900\) 0 0
\(901\) −358.382 −0.397760
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 2203.74 2.43507
\(906\) 0 0
\(907\) − 665.128i − 0.733328i −0.930353 0.366664i \(-0.880500\pi\)
0.930353 0.366664i \(-0.119500\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 345.687i 0.379459i 0.981836 + 0.189729i \(0.0607611\pi\)
−0.981836 + 0.189729i \(0.939239\pi\)
\(912\) 0 0
\(913\) 320.294 0.350815
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1531.32 1.66993
\(918\) 0 0
\(919\) 1102.20i 1.19934i 0.800246 + 0.599671i \(0.204702\pi\)
−0.800246 + 0.599671i \(0.795298\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 288.728i − 0.312815i
\(924\) 0 0
\(925\) 509.115 0.550394
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1256.03 1.35202 0.676010 0.736893i \(-0.263708\pi\)
0.676010 + 0.736893i \(0.263708\pi\)
\(930\) 0 0
\(931\) − 494.790i − 0.531461i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 2325.54i 2.48720i
\(936\) 0 0
\(937\) −822.214 −0.877496 −0.438748 0.898610i \(-0.644578\pi\)
−0.438748 + 0.898610i \(0.644578\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 1185.72 1.26006 0.630031 0.776570i \(-0.283042\pi\)
0.630031 + 0.776570i \(0.283042\pi\)
\(942\) 0 0
\(943\) 198.306i 0.210293i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 1335.34i − 1.41007i −0.709170 0.705037i \(-0.750930\pi\)
0.709170 0.705037i \(-0.249070\pi\)
\(948\) 0 0
\(949\) −1736.15 −1.82945
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −663.428 −0.696147 −0.348073 0.937467i \(-0.613164\pi\)
−0.348073 + 0.937467i \(0.613164\pi\)
\(954\) 0 0
\(955\) 2388.46i 2.50100i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 797.468i − 0.831562i
\(960\) 0 0
\(961\) 463.918 0.482745
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −2330.28 −2.41480
\(966\) 0 0
\(967\) 155.996i 0.161320i 0.996742 + 0.0806600i \(0.0257028\pi\)
−0.996742 + 0.0806600i \(0.974297\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 1280.67i − 1.31892i −0.751740 0.659460i \(-0.770785\pi\)
0.751740 0.659460i \(-0.229215\pi\)
\(972\) 0 0
\(973\) 1903.47 1.95629
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 217.470 0.222590 0.111295 0.993787i \(-0.464500\pi\)
0.111295 + 0.993787i \(0.464500\pi\)
\(978\) 0 0
\(979\) 225.030i 0.229857i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 1027.63i − 1.04540i −0.852515 0.522702i \(-0.824924\pi\)
0.852515 0.522702i \(-0.175076\pi\)
\(984\) 0 0
\(985\) 280.561 0.284833
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −368.809 −0.372911
\(990\) 0 0
\(991\) 797.792i 0.805038i 0.915412 + 0.402519i \(0.131865\pi\)
−0.915412 + 0.402519i \(0.868135\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 704.616i − 0.708157i
\(996\) 0 0
\(997\) −1237.58 −1.24131 −0.620654 0.784085i \(-0.713133\pi\)
−0.620654 + 0.784085i \(0.713133\pi\)
\(998\) 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 1152.3.g.c.127.1 8
3.2 odd 2 384.3.g.b.127.4 yes 8
4.3 odd 2 inner 1152.3.g.c.127.2 8
8.3 odd 2 1152.3.g.f.127.8 8
8.5 even 2 1152.3.g.f.127.7 8
12.11 even 2 384.3.g.b.127.8 yes 8
16.3 odd 4 2304.3.b.q.127.8 8
16.5 even 4 2304.3.b.q.127.1 8
16.11 odd 4 2304.3.b.t.127.1 8
16.13 even 4 2304.3.b.t.127.8 8
24.5 odd 2 384.3.g.a.127.5 yes 8
24.11 even 2 384.3.g.a.127.1 8
48.5 odd 4 768.3.b.f.127.4 8
48.11 even 4 768.3.b.e.127.8 8
48.29 odd 4 768.3.b.e.127.5 8
48.35 even 4 768.3.b.f.127.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.3.g.a.127.1 8 24.11 even 2
384.3.g.a.127.5 yes 8 24.5 odd 2
384.3.g.b.127.4 yes 8 3.2 odd 2
384.3.g.b.127.8 yes 8 12.11 even 2
768.3.b.e.127.5 8 48.29 odd 4
768.3.b.e.127.8 8 48.11 even 4
768.3.b.f.127.1 8 48.35 even 4
768.3.b.f.127.4 8 48.5 odd 4
1152.3.g.c.127.1 8 1.1 even 1 trivial
1152.3.g.c.127.2 8 4.3 odd 2 inner
1152.3.g.f.127.7 8 8.5 even 2
1152.3.g.f.127.8 8 8.3 odd 2
2304.3.b.q.127.1 8 16.5 even 4
2304.3.b.q.127.8 8 16.3 odd 4
2304.3.b.t.127.1 8 16.11 odd 4
2304.3.b.t.127.8 8 16.13 even 4