Properties

Label 2304.3.h.l.2177.7
Level $2304$
Weight $3$
Character 2304.2177
Analytic conductor $62.779$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2304,3,Mod(2177,2304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2304, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2304.2177");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2304.h (of order \(2\), degree \(1\), not 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: \(62.7794529086\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.959512576.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 7x^{4} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: no (minimal twist has level 1152)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2177.7
Root \(-1.52616 + 0.819051i\) of defining polynomial
Character \(\chi\) \(=\) 2304.2177
Dual form 2304.3.h.l.2177.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+8.04746 q^{5} +2.00000 q^{7} +O(q^{10})\) \(q+8.04746 q^{5} +2.00000 q^{7} +21.7518 q^{11} -17.3808i q^{13} +11.8523i q^{17} -10.7617i q^{19} +35.0183i q^{23} +39.7617 q^{25} +11.7515 q^{29} +35.5233 q^{31} +16.0949 q^{35} +26.0000i q^{37} +2.28985i q^{41} -65.5233i q^{43} -27.2071i q^{47} -45.0000 q^{49} -49.5982 q^{53} +175.047 q^{55} +73.5391 q^{59} +7.52333i q^{61} -139.872i q^{65} +65.2383i q^{67} -84.1787i q^{71} -84.5700 q^{73} +43.5036 q^{77} -75.5233 q^{79} +48.2848 q^{83} +95.3808i q^{85} +146.067i q^{89} -34.7617i q^{91} -86.6041i q^{95} -106.762 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{7} + 168 q^{25} - 16 q^{31} - 360 q^{49} + 800 q^{55} + 224 q^{73} - 304 q^{79} - 704 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
\(5\) 8.04746 1.60949 0.804746 0.593619i \(-0.202301\pi\)
0.804746 + 0.593619i \(0.202301\pi\)
\(6\) 0 0
\(7\) 2.00000 0.285714 0.142857 0.989743i \(-0.454371\pi\)
0.142857 + 0.989743i \(0.454371\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 21.7518 1.97743 0.988717 0.149794i \(-0.0478609\pi\)
0.988717 + 0.149794i \(0.0478609\pi\)
\(12\) 0 0
\(13\) − 17.3808i − 1.33699i −0.743718 0.668494i \(-0.766939\pi\)
0.743718 0.668494i \(-0.233061\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 11.8523i 0.697193i 0.937273 + 0.348597i \(0.113342\pi\)
−0.937273 + 0.348597i \(0.886658\pi\)
\(18\) 0 0
\(19\) − 10.7617i − 0.566403i −0.959060 0.283202i \(-0.908603\pi\)
0.959060 0.283202i \(-0.0913966\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 35.0183i 1.52253i 0.648439 + 0.761267i \(0.275422\pi\)
−0.648439 + 0.761267i \(0.724578\pi\)
\(24\) 0 0
\(25\) 39.7617 1.59047
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 11.7515 0.405225 0.202613 0.979259i \(-0.435057\pi\)
0.202613 + 0.979259i \(0.435057\pi\)
\(30\) 0 0
\(31\) 35.5233 1.14591 0.572957 0.819586i \(-0.305796\pi\)
0.572957 + 0.819586i \(0.305796\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 16.0949 0.459855
\(36\) 0 0
\(37\) 26.0000i 0.702703i 0.936244 + 0.351351i \(0.114278\pi\)
−0.936244 + 0.351351i \(0.885722\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.28985i 0.0558500i 0.999610 + 0.0279250i \(0.00888996\pi\)
−0.999610 + 0.0279250i \(0.991110\pi\)
\(42\) 0 0
\(43\) − 65.5233i − 1.52380i −0.647696 0.761899i \(-0.724267\pi\)
0.647696 0.761899i \(-0.275733\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 27.2071i − 0.578875i −0.957197 0.289437i \(-0.906532\pi\)
0.957197 0.289437i \(-0.0934682\pi\)
\(48\) 0 0
\(49\) −45.0000 −0.918367
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −49.5982 −0.935816 −0.467908 0.883777i \(-0.654992\pi\)
−0.467908 + 0.883777i \(0.654992\pi\)
\(54\) 0 0
\(55\) 175.047 3.18267
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 73.5391 1.24643 0.623213 0.782052i \(-0.285827\pi\)
0.623213 + 0.782052i \(0.285827\pi\)
\(60\) 0 0
\(61\) 7.52333i 0.123333i 0.998097 + 0.0616666i \(0.0196416\pi\)
−0.998097 + 0.0616666i \(0.980358\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 139.872i − 2.15187i
\(66\) 0 0
\(67\) 65.2383i 0.973707i 0.873484 + 0.486853i \(0.161855\pi\)
−0.873484 + 0.486853i \(0.838145\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 84.1787i − 1.18562i −0.805344 0.592808i \(-0.798019\pi\)
0.805344 0.592808i \(-0.201981\pi\)
\(72\) 0 0
\(73\) −84.5700 −1.15849 −0.579246 0.815152i \(-0.696653\pi\)
−0.579246 + 0.815152i \(0.696653\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 43.5036 0.564981
\(78\) 0 0
\(79\) −75.5233 −0.955991 −0.477996 0.878362i \(-0.658637\pi\)
−0.477996 + 0.878362i \(0.658637\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 48.2848 0.581744 0.290872 0.956762i \(-0.406055\pi\)
0.290872 + 0.956762i \(0.406055\pi\)
\(84\) 0 0
\(85\) 95.3808i 1.12213i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 146.067i 1.64120i 0.571501 + 0.820601i \(0.306361\pi\)
−0.571501 + 0.820601i \(0.693639\pi\)
\(90\) 0 0
\(91\) − 34.7617i − 0.381996i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 86.6041i − 0.911622i
\(96\) 0 0
\(97\) −106.762 −1.10064 −0.550318 0.834955i \(-0.685493\pi\)
−0.550318 + 0.834955i \(0.685493\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −144.889 −1.43455 −0.717273 0.696792i \(-0.754610\pi\)
−0.717273 + 0.696792i \(0.754610\pi\)
\(102\) 0 0
\(103\) −26.5700 −0.257961 −0.128980 0.991647i \(-0.541170\pi\)
−0.128980 + 0.991647i \(0.541170\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 71.3848 0.667148 0.333574 0.942724i \(-0.391745\pi\)
0.333574 + 0.942724i \(0.391745\pi\)
\(108\) 0 0
\(109\) − 118.619i − 1.08825i −0.839004 0.544125i \(-0.816862\pi\)
0.839004 0.544125i \(-0.183138\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 86.6701i 0.766992i 0.923543 + 0.383496i \(0.125280\pi\)
−0.923543 + 0.383496i \(0.874720\pi\)
\(114\) 0 0
\(115\) 281.808i 2.45051i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 23.7046i 0.199198i
\(120\) 0 0
\(121\) 352.140 2.91025
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 118.794 0.950352
\(126\) 0 0
\(127\) −69.0467 −0.543674 −0.271837 0.962343i \(-0.587631\pi\)
−0.271837 + 0.962343i \(0.587631\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 5.25382 0.0401055 0.0200527 0.999799i \(-0.493617\pi\)
0.0200527 + 0.999799i \(0.493617\pi\)
\(132\) 0 0
\(133\) − 21.5233i − 0.161830i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 120.139i − 0.876924i −0.898750 0.438462i \(-0.855523\pi\)
0.898750 0.438462i \(-0.144477\pi\)
\(138\) 0 0
\(139\) 242.378i 1.74373i 0.489747 + 0.871864i \(0.337089\pi\)
−0.489747 + 0.871864i \(0.662911\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 378.064i − 2.64380i
\(144\) 0 0
\(145\) 94.5700 0.652207
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 192.500 1.29194 0.645972 0.763361i \(-0.276452\pi\)
0.645972 + 0.763361i \(0.276452\pi\)
\(150\) 0 0
\(151\) 17.4300 0.115431 0.0577153 0.998333i \(-0.481618\pi\)
0.0577153 + 0.998333i \(0.481618\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 285.873 1.84434
\(156\) 0 0
\(157\) − 213.617i − 1.36062i −0.732927 0.680308i \(-0.761846\pi\)
0.732927 0.680308i \(-0.238154\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 70.0366i 0.435010i
\(162\) 0 0
\(163\) 231.047i 1.41746i 0.705478 + 0.708732i \(0.250732\pi\)
−0.705478 + 0.708732i \(0.749268\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 126.202i − 0.755701i −0.925867 0.377850i \(-0.876663\pi\)
0.925867 0.377850i \(-0.123337\pi\)
\(168\) 0 0
\(169\) −133.093 −0.787534
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −199.032 −1.15048 −0.575238 0.817986i \(-0.695090\pi\)
−0.575238 + 0.817986i \(0.695090\pi\)
\(174\) 0 0
\(175\) 79.5233 0.454419
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −170.915 −0.954831 −0.477415 0.878678i \(-0.658426\pi\)
−0.477415 + 0.878678i \(0.658426\pi\)
\(180\) 0 0
\(181\) 38.6192i 0.213366i 0.994293 + 0.106683i \(0.0340229\pi\)
−0.994293 + 0.106683i \(0.965977\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 209.234i 1.13099i
\(186\) 0 0
\(187\) 257.808i 1.37865i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 269.778i − 1.41245i −0.707988 0.706224i \(-0.750397\pi\)
0.707988 0.706224i \(-0.249603\pi\)
\(192\) 0 0
\(193\) −263.140 −1.36342 −0.681710 0.731623i \(-0.738763\pi\)
−0.681710 + 0.731623i \(0.738763\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 80.0368 0.406278 0.203139 0.979150i \(-0.434886\pi\)
0.203139 + 0.979150i \(0.434886\pi\)
\(198\) 0 0
\(199\) 265.047 1.33189 0.665946 0.746000i \(-0.268028\pi\)
0.665946 + 0.746000i \(0.268028\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 23.5031 0.115779
\(204\) 0 0
\(205\) 18.4275i 0.0898902i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 234.085i − 1.12003i
\(210\) 0 0
\(211\) − 159.332i − 0.755126i −0.925984 0.377563i \(-0.876762\pi\)
0.925984 0.377563i \(-0.123238\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) − 527.297i − 2.45254i
\(216\) 0 0
\(217\) 71.0467 0.327404
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 206.003 0.932138
\(222\) 0 0
\(223\) 90.9533 0.407863 0.203931 0.978985i \(-0.434628\pi\)
0.203931 + 0.978985i \(0.434628\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 322.371 1.42014 0.710069 0.704133i \(-0.248664\pi\)
0.710069 + 0.704133i \(0.248664\pi\)
\(228\) 0 0
\(229\) 180.997i 0.790382i 0.918599 + 0.395191i \(0.129322\pi\)
−0.918599 + 0.395191i \(0.870678\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 329.241i − 1.41305i −0.707688 0.706525i \(-0.750262\pi\)
0.707688 0.706525i \(-0.249738\pi\)
\(234\) 0 0
\(235\) − 218.948i − 0.931695i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 197.045i 0.824455i 0.911081 + 0.412227i \(0.135249\pi\)
−0.911081 + 0.412227i \(0.864751\pi\)
\(240\) 0 0
\(241\) 215.332 0.893492 0.446746 0.894661i \(-0.352583\pi\)
0.446746 + 0.894661i \(0.352583\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −362.136 −1.47811
\(246\) 0 0
\(247\) −187.047 −0.757274
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −245.329 −0.977408 −0.488704 0.872450i \(-0.662530\pi\)
−0.488704 + 0.872450i \(0.662530\pi\)
\(252\) 0 0
\(253\) 761.710i 3.01071i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 39.5320i 0.153821i 0.997038 + 0.0769105i \(0.0245056\pi\)
−0.997038 + 0.0769105i \(0.975494\pi\)
\(258\) 0 0
\(259\) 52.0000i 0.200772i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 217.921i 0.828596i 0.910141 + 0.414298i \(0.135973\pi\)
−0.910141 + 0.414298i \(0.864027\pi\)
\(264\) 0 0
\(265\) −399.140 −1.50619
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −205.433 −0.763691 −0.381845 0.924226i \(-0.624711\pi\)
−0.381845 + 0.924226i \(0.624711\pi\)
\(270\) 0 0
\(271\) 441.047 1.62748 0.813739 0.581230i \(-0.197428\pi\)
0.813739 + 0.581230i \(0.197428\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 864.887 3.14504
\(276\) 0 0
\(277\) − 185.951i − 0.671303i −0.941986 0.335651i \(-0.891044\pi\)
0.941986 0.335651i \(-0.108956\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 87.7472i 0.312268i 0.987736 + 0.156134i \(0.0499031\pi\)
−0.987736 + 0.156134i \(0.950097\pi\)
\(282\) 0 0
\(283\) − 46.0933i − 0.162874i −0.996678 0.0814369i \(-0.974049\pi\)
0.996678 0.0814369i \(-0.0259509\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 4.57970i 0.0159571i
\(288\) 0 0
\(289\) 148.523 0.513922
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −118.287 −0.403708 −0.201854 0.979416i \(-0.564697\pi\)
−0.201854 + 0.979416i \(0.564697\pi\)
\(294\) 0 0
\(295\) 591.803 2.00611
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 608.647 2.03561
\(300\) 0 0
\(301\) − 131.047i − 0.435371i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 60.5437i 0.198504i
\(306\) 0 0
\(307\) 116.285i 0.378778i 0.981902 + 0.189389i \(0.0606508\pi\)
−0.981902 + 0.189389i \(0.939349\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 261.564i − 0.841040i −0.907283 0.420520i \(-0.861848\pi\)
0.907283 0.420520i \(-0.138152\pi\)
\(312\) 0 0
\(313\) −310.000 −0.990415 −0.495208 0.868775i \(-0.664908\pi\)
−0.495208 + 0.868775i \(0.664908\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 223.279 0.704350 0.352175 0.935934i \(-0.385442\pi\)
0.352175 + 0.935934i \(0.385442\pi\)
\(318\) 0 0
\(319\) 255.617 0.801306
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 127.550 0.394893
\(324\) 0 0
\(325\) − 691.091i − 2.12643i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 54.4142i − 0.165393i
\(330\) 0 0
\(331\) 353.238i 1.06719i 0.845742 + 0.533593i \(0.179158\pi\)
−0.845742 + 0.533593i \(0.820842\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 525.003i 1.56717i
\(336\) 0 0
\(337\) 316.000 0.937685 0.468843 0.883282i \(-0.344671\pi\)
0.468843 + 0.883282i \(0.344671\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 772.696 2.26597
\(342\) 0 0
\(343\) −188.000 −0.548105
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 230.916 0.665465 0.332732 0.943021i \(-0.392029\pi\)
0.332732 + 0.943021i \(0.392029\pi\)
\(348\) 0 0
\(349\) − 189.233i − 0.542216i −0.962549 0.271108i \(-0.912610\pi\)
0.962549 0.271108i \(-0.0873900\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 151.050i 0.427903i 0.976844 + 0.213952i \(0.0686335\pi\)
−0.976844 + 0.213952i \(0.931367\pi\)
\(354\) 0 0
\(355\) − 677.425i − 1.90824i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 217.789i − 0.606654i −0.952886 0.303327i \(-0.901903\pi\)
0.952886 0.303327i \(-0.0980975\pi\)
\(360\) 0 0
\(361\) 245.187 0.679187
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −680.574 −1.86459
\(366\) 0 0
\(367\) 444.093 1.21006 0.605032 0.796201i \(-0.293160\pi\)
0.605032 + 0.796201i \(0.293160\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −99.1965 −0.267376
\(372\) 0 0
\(373\) 124.663i 0.334218i 0.985938 + 0.167109i \(0.0534432\pi\)
−0.985938 + 0.167109i \(0.946557\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 204.251i − 0.541781i
\(378\) 0 0
\(379\) 419.332i 1.10642i 0.833043 + 0.553208i \(0.186597\pi\)
−0.833043 + 0.553208i \(0.813403\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 218.060i 0.569347i 0.958625 + 0.284674i \(0.0918852\pi\)
−0.958625 + 0.284674i \(0.908115\pi\)
\(384\) 0 0
\(385\) 350.093 0.909333
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 214.988 0.552669 0.276334 0.961062i \(-0.410880\pi\)
0.276334 + 0.961062i \(0.410880\pi\)
\(390\) 0 0
\(391\) −415.047 −1.06150
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −607.771 −1.53866
\(396\) 0 0
\(397\) − 506.187i − 1.27503i −0.770438 0.637515i \(-0.779963\pi\)
0.770438 0.637515i \(-0.220037\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 718.820i 1.79257i 0.443480 + 0.896284i \(0.353744\pi\)
−0.443480 + 0.896284i \(0.646256\pi\)
\(402\) 0 0
\(403\) − 617.425i − 1.53207i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 565.546i 1.38955i
\(408\) 0 0
\(409\) 29.2383 0.0714874 0.0357437 0.999361i \(-0.488620\pi\)
0.0357437 + 0.999361i \(0.488620\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 147.078 0.356122
\(414\) 0 0
\(415\) 388.570 0.936313
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 2.22387 0.00530757 0.00265379 0.999996i \(-0.499155\pi\)
0.00265379 + 0.999996i \(0.499155\pi\)
\(420\) 0 0
\(421\) − 117.194i − 0.278371i −0.990266 0.139186i \(-0.955552\pi\)
0.990266 0.139186i \(-0.0444484\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 471.267i 1.10886i
\(426\) 0 0
\(427\) 15.0467i 0.0352381i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 755.315i − 1.75247i −0.481884 0.876235i \(-0.660047\pi\)
0.481884 0.876235i \(-0.339953\pi\)
\(432\) 0 0
\(433\) −433.233 −1.00054 −0.500269 0.865870i \(-0.666766\pi\)
−0.500269 + 0.865870i \(0.666766\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 376.855 0.862368
\(438\) 0 0
\(439\) −231.140 −0.526515 −0.263257 0.964726i \(-0.584797\pi\)
−0.263257 + 0.964726i \(0.584797\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 9.08983 0.0205188 0.0102594 0.999947i \(-0.496734\pi\)
0.0102594 + 0.999947i \(0.496734\pi\)
\(444\) 0 0
\(445\) 1175.47i 2.64150i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 290.921i 0.647932i 0.946069 + 0.323966i \(0.105016\pi\)
−0.946069 + 0.323966i \(0.894984\pi\)
\(450\) 0 0
\(451\) 49.8083i 0.110440i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) − 279.743i − 0.614820i
\(456\) 0 0
\(457\) 333.042 0.728756 0.364378 0.931251i \(-0.381282\pi\)
0.364378 + 0.931251i \(0.381282\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −132.971 −0.288440 −0.144220 0.989546i \(-0.546067\pi\)
−0.144220 + 0.989546i \(0.546067\pi\)
\(462\) 0 0
\(463\) −391.140 −0.844795 −0.422397 0.906411i \(-0.638811\pi\)
−0.422397 + 0.906411i \(0.638811\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −469.310 −1.00495 −0.502473 0.864593i \(-0.667577\pi\)
−0.502473 + 0.864593i \(0.667577\pi\)
\(468\) 0 0
\(469\) 130.477i 0.278202i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 1425.25i − 3.01321i
\(474\) 0 0
\(475\) − 427.902i − 0.900846i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 355.708i − 0.742605i −0.928512 0.371302i \(-0.878911\pi\)
0.928512 0.371302i \(-0.121089\pi\)
\(480\) 0 0
\(481\) 451.902 0.939504
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −859.161 −1.77147
\(486\) 0 0
\(487\) −839.897 −1.72463 −0.862317 0.506369i \(-0.830987\pi\)
−0.862317 + 0.506369i \(0.830987\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −388.572 −0.791388 −0.395694 0.918382i \(-0.629496\pi\)
−0.395694 + 0.918382i \(0.629496\pi\)
\(492\) 0 0
\(493\) 139.282i 0.282520i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 168.357i − 0.338747i
\(498\) 0 0
\(499\) 488.373i 0.978704i 0.872086 + 0.489352i \(0.162767\pi\)
−0.872086 + 0.489352i \(0.837233\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 533.488i − 1.06061i −0.847806 0.530307i \(-0.822077\pi\)
0.847806 0.530307i \(-0.177923\pi\)
\(504\) 0 0
\(505\) −1165.99 −2.30889
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −179.094 −0.351855 −0.175927 0.984403i \(-0.556292\pi\)
−0.175927 + 0.984403i \(0.556292\pi\)
\(510\) 0 0
\(511\) −169.140 −0.330998
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −213.821 −0.415186
\(516\) 0 0
\(517\) − 591.803i − 1.14469i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 397.456i 0.762872i 0.924395 + 0.381436i \(0.124570\pi\)
−0.924395 + 0.381436i \(0.875430\pi\)
\(522\) 0 0
\(523\) 987.135i 1.88745i 0.330735 + 0.943724i \(0.392703\pi\)
−0.330735 + 0.943724i \(0.607297\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 421.033i 0.798923i
\(528\) 0 0
\(529\) −697.280 −1.31811
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 39.7995 0.0746707
\(534\) 0 0
\(535\) 574.467 1.07377
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −978.830 −1.81601
\(540\) 0 0
\(541\) − 214.992i − 0.397398i −0.980061 0.198699i \(-0.936328\pi\)
0.980061 0.198699i \(-0.0636716\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 954.583i − 1.75153i
\(546\) 0 0
\(547\) 381.710i 0.697824i 0.937155 + 0.348912i \(0.113449\pi\)
−0.937155 + 0.348912i \(0.886551\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) − 126.466i − 0.229521i
\(552\) 0 0
\(553\) −151.047 −0.273140
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −924.388 −1.65958 −0.829792 0.558073i \(-0.811541\pi\)
−0.829792 + 0.558073i \(0.811541\pi\)
\(558\) 0 0
\(559\) −1138.85 −2.03730
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −70.5092 −0.125238 −0.0626191 0.998037i \(-0.519945\pi\)
−0.0626191 + 0.998037i \(0.519945\pi\)
\(564\) 0 0
\(565\) 697.474i 1.23447i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 607.844i 1.06827i 0.845400 + 0.534134i \(0.179362\pi\)
−0.845400 + 0.534134i \(0.820638\pi\)
\(570\) 0 0
\(571\) − 455.430i − 0.797601i −0.917038 0.398800i \(-0.869427\pi\)
0.917038 0.398800i \(-0.130573\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1392.39i 2.42154i
\(576\) 0 0
\(577\) 501.233 0.868688 0.434344 0.900747i \(-0.356980\pi\)
0.434344 + 0.900747i \(0.356980\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 96.5696 0.166213
\(582\) 0 0
\(583\) −1078.85 −1.85051
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 178.059 0.303337 0.151669 0.988431i \(-0.451535\pi\)
0.151669 + 0.988431i \(0.451535\pi\)
\(588\) 0 0
\(589\) − 382.290i − 0.649049i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 960.244i − 1.61930i −0.586914 0.809649i \(-0.699657\pi\)
0.586914 0.809649i \(-0.300343\pi\)
\(594\) 0 0
\(595\) 190.762i 0.320608i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 680.706i 1.13640i 0.822889 + 0.568202i \(0.192361\pi\)
−0.822889 + 0.568202i \(0.807639\pi\)
\(600\) 0 0
\(601\) −239.327 −0.398214 −0.199107 0.979978i \(-0.563804\pi\)
−0.199107 + 0.979978i \(0.563804\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 2833.83 4.68402
\(606\) 0 0
\(607\) −1177.80 −1.94037 −0.970184 0.242370i \(-0.922075\pi\)
−0.970184 + 0.242370i \(0.922075\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −472.882 −0.773948
\(612\) 0 0
\(613\) 1106.56i 1.80515i 0.430528 + 0.902577i \(0.358327\pi\)
−0.430528 + 0.902577i \(0.641673\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 271.185i 0.439522i 0.975554 + 0.219761i \(0.0705277\pi\)
−0.975554 + 0.219761i \(0.929472\pi\)
\(618\) 0 0
\(619\) − 447.430i − 0.722827i −0.932406 0.361414i \(-0.882294\pi\)
0.932406 0.361414i \(-0.117706\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 292.134i 0.468915i
\(624\) 0 0
\(625\) −38.0517 −0.0608828
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −308.159 −0.489920
\(630\) 0 0
\(631\) 369.047 0.584860 0.292430 0.956287i \(-0.405536\pi\)
0.292430 + 0.956287i \(0.405536\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −555.650 −0.875040
\(636\) 0 0
\(637\) 782.137i 1.22785i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 319.198i 0.497970i 0.968507 + 0.248985i \(0.0800969\pi\)
−0.968507 + 0.248985i \(0.919903\pi\)
\(642\) 0 0
\(643\) 41.3266i 0.0642715i 0.999484 + 0.0321357i \(0.0102309\pi\)
−0.999484 + 0.0321357i \(0.989769\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 160.011i 0.247313i 0.992325 + 0.123656i \(0.0394620\pi\)
−0.992325 + 0.123656i \(0.960538\pi\)
\(648\) 0 0
\(649\) 1599.61 2.46472
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −1111.54 −1.70220 −0.851100 0.525003i \(-0.824064\pi\)
−0.851100 + 0.525003i \(0.824064\pi\)
\(654\) 0 0
\(655\) 42.2799 0.0645495
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −800.313 −1.21444 −0.607218 0.794536i \(-0.707714\pi\)
−0.607218 + 0.794536i \(0.707714\pi\)
\(660\) 0 0
\(661\) − 1142.18i − 1.72795i −0.503533 0.863976i \(-0.667967\pi\)
0.503533 0.863976i \(-0.332033\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 173.208i − 0.260463i
\(666\) 0 0
\(667\) 411.518i 0.616969i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 163.646i 0.243883i
\(672\) 0 0
\(673\) −169.047 −0.251184 −0.125592 0.992082i \(-0.540083\pi\)
−0.125592 + 0.992082i \(0.540083\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −533.801 −0.788481 −0.394240 0.919007i \(-0.628992\pi\)
−0.394240 + 0.919007i \(0.628992\pi\)
\(678\) 0 0
\(679\) −213.523 −0.314467
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −408.711 −0.598406 −0.299203 0.954189i \(-0.596721\pi\)
−0.299203 + 0.954189i \(0.596721\pi\)
\(684\) 0 0
\(685\) − 966.811i − 1.41140i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 862.059i 1.25117i
\(690\) 0 0
\(691\) − 331.233i − 0.479353i −0.970853 0.239677i \(-0.922959\pi\)
0.970853 0.239677i \(-0.0770415\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1950.53i 2.80652i
\(696\) 0 0
\(697\) −27.1400 −0.0389382
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 116.814 0.166638 0.0833192 0.996523i \(-0.473448\pi\)
0.0833192 + 0.996523i \(0.473448\pi\)
\(702\) 0 0
\(703\) 279.803 0.398013
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −289.778 −0.409870
\(708\) 0 0
\(709\) − 737.469i − 1.04015i −0.854119 0.520077i \(-0.825903\pi\)
0.854119 0.520077i \(-0.174097\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1243.97i 1.74469i
\(714\) 0 0
\(715\) − 3042.46i − 4.25518i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) − 1017.28i − 1.41486i −0.706785 0.707428i \(-0.749855\pi\)
0.706785 0.707428i \(-0.250145\pi\)
\(720\) 0 0
\(721\) −53.1400 −0.0737031
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 467.260 0.644497
\(726\) 0 0
\(727\) 277.430 0.381609 0.190805 0.981628i \(-0.438890\pi\)
0.190805 + 0.981628i \(0.438890\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 776.601 1.06238
\(732\) 0 0
\(733\) − 405.371i − 0.553030i −0.961010 0.276515i \(-0.910821\pi\)
0.961010 0.276515i \(-0.0891795\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1419.05i 1.92544i
\(738\) 0 0
\(739\) − 295.036i − 0.399237i −0.979874 0.199619i \(-0.936030\pi\)
0.979874 0.199619i \(-0.0639704\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 486.350i 0.654576i 0.944925 + 0.327288i \(0.106135\pi\)
−0.944925 + 0.327288i \(0.893865\pi\)
\(744\) 0 0
\(745\) 1549.13 2.07938
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 142.770 0.190614
\(750\) 0 0
\(751\) −705.430 −0.939321 −0.469660 0.882847i \(-0.655624\pi\)
−0.469660 + 0.882847i \(0.655624\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 140.267 0.185785
\(756\) 0 0
\(757\) 1362.89i 1.80039i 0.435489 + 0.900194i \(0.356576\pi\)
−0.435489 + 0.900194i \(0.643424\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 625.746i − 0.822268i −0.911575 0.411134i \(-0.865133\pi\)
0.911575 0.411134i \(-0.134867\pi\)
\(762\) 0 0
\(763\) − 237.238i − 0.310928i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 1278.17i − 1.66645i
\(768\) 0 0
\(769\) −637.813 −0.829406 −0.414703 0.909957i \(-0.636115\pi\)
−0.414703 + 0.909957i \(0.636115\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 536.282 0.693767 0.346884 0.937908i \(-0.387240\pi\)
0.346884 + 0.937908i \(0.387240\pi\)
\(774\) 0 0
\(775\) 1412.47 1.82254
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 24.6426 0.0316336
\(780\) 0 0
\(781\) − 1831.04i − 2.34448i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 1719.07i − 2.18990i
\(786\) 0 0
\(787\) − 769.425i − 0.977668i −0.872377 0.488834i \(-0.837422\pi\)
0.872377 0.488834i \(-0.162578\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 173.340i 0.219140i
\(792\) 0 0
\(793\) 130.762 0.164895
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 765.593 0.960594 0.480297 0.877106i \(-0.340529\pi\)
0.480297 + 0.877106i \(0.340529\pi\)
\(798\) 0 0
\(799\) 322.467 0.403588
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −1839.55 −2.29084
\(804\) 0 0
\(805\) 563.617i 0.700145i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 865.488i − 1.06982i −0.844908 0.534912i \(-0.820345\pi\)
0.844908 0.534912i \(-0.179655\pi\)
\(810\) 0 0
\(811\) 292.472i 0.360631i 0.983609 + 0.180315i \(0.0577118\pi\)
−0.983609 + 0.180315i \(0.942288\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 1859.34i 2.28140i
\(816\) 0 0
\(817\) −705.140 −0.863084
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 343.616 0.418533 0.209266 0.977859i \(-0.432892\pi\)
0.209266 + 0.977859i \(0.432892\pi\)
\(822\) 0 0
\(823\) −773.430 −0.939769 −0.469885 0.882728i \(-0.655704\pi\)
−0.469885 + 0.882728i \(0.655704\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 854.310 1.03302 0.516511 0.856280i \(-0.327230\pi\)
0.516511 + 0.856280i \(0.327230\pi\)
\(828\) 0 0
\(829\) 85.5674i 0.103218i 0.998667 + 0.0516088i \(0.0164349\pi\)
−0.998667 + 0.0516088i \(0.983565\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 533.353i − 0.640280i
\(834\) 0 0
\(835\) − 1015.61i − 1.21630i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 1653.28i 1.97054i 0.171011 + 0.985269i \(0.445297\pi\)
−0.171011 + 0.985269i \(0.554703\pi\)
\(840\) 0 0
\(841\) −702.902 −0.835793
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −1071.06 −1.26753
\(846\) 0 0
\(847\) 704.280 0.831499
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −910.475 −1.06989
\(852\) 0 0
\(853\) 1045.62i 1.22581i 0.790156 + 0.612905i \(0.209999\pi\)
−0.790156 + 0.612905i \(0.790001\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 939.437i 1.09619i 0.836415 + 0.548096i \(0.184647\pi\)
−0.836415 + 0.548096i \(0.815353\pi\)
\(858\) 0 0
\(859\) − 1249.52i − 1.45463i −0.686306 0.727313i \(-0.740769\pi\)
0.686306 0.727313i \(-0.259231\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1470.64i 1.70411i 0.523456 + 0.852053i \(0.324642\pi\)
−0.523456 + 0.852053i \(0.675358\pi\)
\(864\) 0 0
\(865\) −1601.70 −1.85168
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −1642.77 −1.89041
\(870\) 0 0
\(871\) 1133.90 1.30183
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 237.588 0.271529
\(876\) 0 0
\(877\) − 195.720i − 0.223170i −0.993755 0.111585i \(-0.964407\pi\)
0.993755 0.111585i \(-0.0355927\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 533.686i − 0.605773i −0.953027 0.302887i \(-0.902050\pi\)
0.953027 0.302887i \(-0.0979504\pi\)
\(882\) 0 0
\(883\) − 963.803i − 1.09151i −0.837945 0.545755i \(-0.816243\pi\)
0.837945 0.545755i \(-0.183757\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 507.769i − 0.572456i −0.958162 0.286228i \(-0.907598\pi\)
0.958162 0.286228i \(-0.0924015\pi\)
\(888\) 0 0
\(889\) −138.093 −0.155336
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −292.794 −0.327877
\(894\) 0 0
\(895\) −1375.43 −1.53679
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 417.453 0.464353
\(900\) 0 0
\(901\) − 587.852i − 0.652444i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 310.786i 0.343410i
\(906\) 0 0
\(907\) − 48.3834i − 0.0533444i −0.999644 0.0266722i \(-0.991509\pi\)
0.999644 0.0266722i \(-0.00849103\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 318.257i 0.349349i 0.984626 + 0.174674i \(0.0558873\pi\)
−0.984626 + 0.174674i \(0.944113\pi\)
\(912\) 0 0
\(913\) 1050.28 1.15036
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 10.5076 0.0114587
\(918\) 0 0
\(919\) −583.513 −0.634944 −0.317472 0.948268i \(-0.602834\pi\)
−0.317472 + 0.948268i \(0.602834\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −1463.10 −1.58515
\(924\) 0 0
\(925\) 1033.80i 1.11763i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 322.847i − 0.347521i −0.984788 0.173761i \(-0.944408\pi\)
0.984788 0.173761i \(-0.0555919\pi\)
\(930\) 0 0
\(931\) 484.275i 0.520166i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 2074.70i 2.21893i
\(936\) 0 0
\(937\) −468.467 −0.499964 −0.249982 0.968250i \(-0.580425\pi\)
−0.249982 + 0.968250i \(0.580425\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −469.964 −0.499430 −0.249715 0.968319i \(-0.580337\pi\)
−0.249715 + 0.968319i \(0.580337\pi\)
\(942\) 0 0
\(943\) −80.1866 −0.0850335
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −200.686 −0.211918 −0.105959 0.994370i \(-0.533791\pi\)
−0.105959 + 0.994370i \(0.533791\pi\)
\(948\) 0 0
\(949\) 1469.90i 1.54889i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 493.081i 0.517398i 0.965958 + 0.258699i \(0.0832939\pi\)
−0.965958 + 0.258699i \(0.916706\pi\)
\(954\) 0 0
\(955\) − 2171.03i − 2.27333i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 240.277i − 0.250550i
\(960\) 0 0
\(961\) 300.907 0.313118
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −2117.61 −2.19441
\(966\) 0 0
\(967\) 204.467 0.211444 0.105722 0.994396i \(-0.466285\pi\)
0.105722 + 0.994396i \(0.466285\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 250.986 0.258482 0.129241 0.991613i \(-0.458746\pi\)
0.129241 + 0.991613i \(0.458746\pi\)
\(972\) 0 0
\(973\) 484.757i 0.498208i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 520.823i 0.533084i 0.963823 + 0.266542i \(0.0858811\pi\)
−0.963823 + 0.266542i \(0.914119\pi\)
\(978\) 0 0
\(979\) 3177.22i 3.24537i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 569.174i − 0.579017i −0.957175 0.289508i \(-0.906508\pi\)
0.957175 0.289508i \(-0.0934918\pi\)
\(984\) 0 0
\(985\) 644.093 0.653902
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 2294.51 2.32003
\(990\) 0 0
\(991\) 97.2434 0.0981266 0.0490633 0.998796i \(-0.484376\pi\)
0.0490633 + 0.998796i \(0.484376\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 2132.95 2.14367
\(996\) 0 0
\(997\) − 112.290i − 0.112628i −0.998413 0.0563140i \(-0.982065\pi\)
0.998413 0.0563140i \(-0.0179348\pi\)
\(998\) 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 2304.3.h.l.2177.7 8
3.2 odd 2 inner 2304.3.h.l.2177.1 8
4.3 odd 2 2304.3.h.j.2177.7 8
8.3 odd 2 2304.3.h.j.2177.2 8
8.5 even 2 inner 2304.3.h.l.2177.2 8
12.11 even 2 2304.3.h.j.2177.1 8
16.3 odd 4 1152.3.e.g.1025.1 yes 4
16.5 even 4 1152.3.e.a.1025.4 yes 4
16.11 odd 4 1152.3.e.e.1025.4 yes 4
16.13 even 4 1152.3.e.c.1025.1 yes 4
24.5 odd 2 inner 2304.3.h.l.2177.8 8
24.11 even 2 2304.3.h.j.2177.8 8
48.5 odd 4 1152.3.e.a.1025.1 4
48.11 even 4 1152.3.e.e.1025.1 yes 4
48.29 odd 4 1152.3.e.c.1025.4 yes 4
48.35 even 4 1152.3.e.g.1025.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1152.3.e.a.1025.1 4 48.5 odd 4
1152.3.e.a.1025.4 yes 4 16.5 even 4
1152.3.e.c.1025.1 yes 4 16.13 even 4
1152.3.e.c.1025.4 yes 4 48.29 odd 4
1152.3.e.e.1025.1 yes 4 48.11 even 4
1152.3.e.e.1025.4 yes 4 16.11 odd 4
1152.3.e.g.1025.1 yes 4 16.3 odd 4
1152.3.e.g.1025.4 yes 4 48.35 even 4
2304.3.h.j.2177.1 8 12.11 even 2
2304.3.h.j.2177.2 8 8.3 odd 2
2304.3.h.j.2177.7 8 4.3 odd 2
2304.3.h.j.2177.8 8 24.11 even 2
2304.3.h.l.2177.1 8 3.2 odd 2 inner
2304.3.h.l.2177.2 8 8.5 even 2 inner
2304.3.h.l.2177.7 8 1.1 even 1 trivial
2304.3.h.l.2177.8 8 24.5 odd 2 inner