Properties

Label 2304.3.h.j.2177.1
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 2177.1
Root \(-1.52616 + 0.819051i\) of defining polynomial
Character \(\chi\) \(=\) 2304.2177
Dual form 2304.3.h.j.2177.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-8.04746 q^{5} -2.00000 q^{7} +O(q^{10})\) \(q-8.04746 q^{5} -2.00000 q^{7} +21.7518 q^{11} -17.3808i q^{13} -11.8523i q^{17} +10.7617i q^{19} +35.0183i q^{23} +39.7617 q^{25} -11.7515 q^{29} -35.5233 q^{31} +16.0949 q^{35} +26.0000i q^{37} -2.28985i q^{41} +65.5233i q^{43} -27.2071i q^{47} -45.0000 q^{49} +49.5982 q^{53} -175.047 q^{55} +73.5391 q^{59} +7.52333i q^{61} +139.872i q^{65} -65.2383i q^{67} -84.1787i q^{71} -84.5700 q^{73} -43.5036 q^{77} +75.5233 q^{79} +48.2848 q^{83} +95.3808i q^{85} -146.067i q^{89} +34.7617i q^{91} -86.6041i q^{95} -106.762 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{7}+O(q^{10}) \) 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 \(-0.797699\pi\)
−0.804746 + 0.593619i \(0.797699\pi\)
\(6\) 0 0
\(7\) −2.00000 −0.285714 −0.142857 0.989743i \(-0.545629\pi\)
−0.142857 + 0.989743i \(0.545629\pi\)
\(8\) 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 \(-0.886658\pi\)
0.937273 0.348597i \(-0.113342\pi\)
\(18\) 0 0
\(19\) 10.7617i 0.566403i 0.959060 + 0.283202i \(0.0913966\pi\)
−0.959060 + 0.283202i \(0.908603\pi\)
\(20\) 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 \(-0.564943\pi\)
−0.202613 + 0.979259i \(0.564943\pi\)
\(30\) 0 0
\(31\) −35.5233 −1.14591 −0.572957 0.819586i \(-0.694204\pi\)
−0.572957 + 0.819586i \(0.694204\pi\)
\(32\) 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 \(-0.991110\pi\)
0.999610 0.0279250i \(-0.00888996\pi\)
\(42\) 0 0
\(43\) 65.5233i 1.52380i 0.647696 + 0.761899i \(0.275733\pi\)
−0.647696 + 0.761899i \(0.724267\pi\)
\(44\) 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 \(-0.345008\pi\)
0.467908 + 0.883777i \(0.345008\pi\)
\(54\) 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 \(-0.838145\pi\)
0.873484 0.486853i \(-0.161855\pi\)
\(68\) 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 \(-0.341363\pi\)
0.477996 + 0.878362i \(0.341363\pi\)
\(80\) 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 \(-0.693639\pi\)
0.571501 0.820601i \(-0.306361\pi\)
\(90\) 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 \(-0.245390\pi\)
0.717273 + 0.696792i \(0.245390\pi\)
\(102\) 0 0
\(103\) 26.5700 0.257961 0.128980 0.991647i \(-0.458830\pi\)
0.128980 + 0.991647i \(0.458830\pi\)
\(104\) 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 \(-0.874720\pi\)
0.923543 0.383496i \(-0.125280\pi\)
\(114\) 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 \(-0.412369\pi\)
0.271837 + 0.962343i \(0.412369\pi\)
\(128\) 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 \(0.144477\pi\)
−0.898750 + 0.438462i \(0.855523\pi\)
\(138\) 0 0
\(139\) − 242.378i − 1.74373i −0.489747 0.871864i \(-0.662911\pi\)
0.489747 0.871864i \(-0.337089\pi\)
\(140\) 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 \(-0.723548\pi\)
−0.645972 + 0.763361i \(0.723548\pi\)
\(150\) 0 0
\(151\) −17.4300 −0.115431 −0.0577153 0.998333i \(-0.518382\pi\)
−0.0577153 + 0.998333i \(0.518382\pi\)
\(152\) 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 \(-0.749268\pi\)
0.705478 0.708732i \(-0.250732\pi\)
\(164\) 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 \(-0.304910\pi\)
0.575238 + 0.817986i \(0.304910\pi\)
\(174\) 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 \(-0.565114\pi\)
−0.203139 + 0.979150i \(0.565114\pi\)
\(198\) 0 0
\(199\) −265.047 −1.33189 −0.665946 0.746000i \(-0.731972\pi\)
−0.665946 + 0.746000i \(0.731972\pi\)
\(200\) 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 \(0.123238\pi\)
−0.925984 + 0.377563i \(0.876762\pi\)
\(212\) 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 \(-0.565372\pi\)
−0.203931 + 0.978985i \(0.565372\pi\)
\(224\) 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 \(0.249738\pi\)
−0.707688 + 0.706525i \(0.750262\pi\)
\(234\) 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 \(-0.975494\pi\)
0.997038 0.0769105i \(-0.0245056\pi\)
\(258\) 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 \(-0.375289\pi\)
0.381845 + 0.924226i \(0.375289\pi\)
\(270\) 0 0
\(271\) −441.047 −1.62748 −0.813739 0.581230i \(-0.802572\pi\)
−0.813739 + 0.581230i \(0.802572\pi\)
\(272\) 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 \(-0.950097\pi\)
0.987736 0.156134i \(-0.0499031\pi\)
\(282\) 0 0
\(283\) 46.0933i 0.162874i 0.996678 + 0.0814369i \(0.0259509\pi\)
−0.996678 + 0.0814369i \(0.974049\pi\)
\(284\) 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 \(-0.435303\pi\)
0.201854 + 0.979416i \(0.435303\pi\)
\(294\) 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 \(-0.939349\pi\)
0.981902 0.189389i \(-0.0606508\pi\)
\(308\) 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 \(-0.614558\pi\)
−0.352175 + 0.935934i \(0.614558\pi\)
\(318\) 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 \(-0.820842\pi\)
0.845742 0.533593i \(-0.179158\pi\)
\(332\) 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 \(-0.931367\pi\)
0.976844 0.213952i \(-0.0686335\pi\)
\(354\) 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 \(-0.706840\pi\)
−0.605032 + 0.796201i \(0.706840\pi\)
\(368\) 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 \(-0.813403\pi\)
0.833043 0.553208i \(-0.186597\pi\)
\(380\) 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 \(-0.589120\pi\)
−0.276334 + 0.961062i \(0.589120\pi\)
\(390\) 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 \(-0.646256\pi\)
0.443480 0.896284i \(-0.353744\pi\)
\(402\) 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 \(-0.415203\pi\)
0.263257 + 0.964726i \(0.415203\pi\)
\(440\) 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 \(-0.894984\pi\)
0.946069 0.323966i \(-0.105016\pi\)
\(450\) 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 \(-0.453933\pi\)
0.144220 + 0.989546i \(0.453933\pi\)
\(462\) 0 0
\(463\) 391.140 0.844795 0.422397 0.906411i \(-0.361189\pi\)
0.422397 + 0.906411i \(0.361189\pi\)
\(464\) 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 \(-0.169013\pi\)
0.862317 + 0.506369i \(0.169013\pi\)
\(488\) 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 \(-0.837233\pi\)
0.872086 0.489352i \(-0.162767\pi\)
\(500\) 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 \(-0.443708\pi\)
0.175927 + 0.984403i \(0.443708\pi\)
\(510\) 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 \(-0.875430\pi\)
0.924395 0.381436i \(-0.124570\pi\)
\(522\) 0 0
\(523\) − 987.135i − 1.88745i −0.330735 0.943724i \(-0.607297\pi\)
0.330735 0.943724i \(-0.392703\pi\)
\(524\) 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 \(-0.886551\pi\)
0.937155 0.348912i \(-0.113449\pi\)
\(548\) 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 \(-0.188459\pi\)
0.829792 + 0.558073i \(0.188459\pi\)
\(558\) 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 \(-0.820638\pi\)
0.845400 0.534134i \(-0.179362\pi\)
\(570\) 0 0
\(571\) 455.430i 0.797601i 0.917038 + 0.398800i \(0.130573\pi\)
−0.917038 + 0.398800i \(0.869427\pi\)
\(572\) 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 \(0.300343\pi\)
−0.586914 + 0.809649i \(0.699657\pi\)
\(594\) 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 \(-0.0779248\pi\)
0.970184 + 0.242370i \(0.0779248\pi\)
\(608\) 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 \(-0.929472\pi\)
0.975554 0.219761i \(-0.0705277\pi\)
\(618\) 0 0
\(619\) 447.430i 0.722827i 0.932406 + 0.361414i \(0.117706\pi\)
−0.932406 + 0.361414i \(0.882294\pi\)
\(620\) 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 \(-0.594464\pi\)
−0.292430 + 0.956287i \(0.594464\pi\)
\(632\) 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 \(-0.919903\pi\)
0.968507 0.248985i \(-0.0800969\pi\)
\(642\) 0 0
\(643\) − 41.3266i − 0.0642715i −0.999484 0.0321357i \(-0.989769\pi\)
0.999484 0.0321357i \(-0.0102309\pi\)
\(644\) 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 \(-0.175936\pi\)
0.851100 + 0.525003i \(0.175936\pi\)
\(654\) 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 \(-0.371008\pi\)
0.394240 + 0.919007i \(0.371008\pi\)
\(678\) 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 \(0.0770415\pi\)
−0.970853 + 0.239677i \(0.922959\pi\)
\(692\) 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 \(-0.526552\pi\)
−0.0833192 + 0.996523i \(0.526552\pi\)
\(702\) 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 \(-0.561110\pi\)
−0.190805 + 0.981628i \(0.561110\pi\)
\(728\) 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 \(0.0639704\pi\)
−0.979874 + 0.199619i \(0.936030\pi\)
\(740\) 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 \(-0.344376\pi\)
0.469660 + 0.882847i \(0.344376\pi\)
\(752\) 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 \(0.134867\pi\)
−0.911575 + 0.411134i \(0.865133\pi\)
\(762\) 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 \(-0.612760\pi\)
−0.346884 + 0.937908i \(0.612760\pi\)
\(774\) 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 \(0.162578\pi\)
−0.872377 + 0.488834i \(0.837422\pi\)
\(788\) 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 \(-0.659471\pi\)
−0.480297 + 0.877106i \(0.659471\pi\)
\(798\) 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 \(0.179655\pi\)
−0.844908 + 0.534912i \(0.820345\pi\)
\(810\) 0 0
\(811\) − 292.472i − 0.360631i −0.983609 0.180315i \(-0.942288\pi\)
0.983609 0.180315i \(-0.0577118\pi\)
\(812\) 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 \(-0.567108\pi\)
−0.209266 + 0.977859i \(0.567108\pi\)
\(822\) 0 0
\(823\) 773.430 0.939769 0.469885 0.882728i \(-0.344296\pi\)
0.469885 + 0.882728i \(0.344296\pi\)
\(824\) 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 \(-0.815353\pi\)
0.836415 0.548096i \(-0.184647\pi\)
\(858\) 0 0
\(859\) 1249.52i 1.45463i 0.686306 + 0.727313i \(0.259231\pi\)
−0.686306 + 0.727313i \(0.740769\pi\)
\(860\) 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 \(0.0979504\pi\)
−0.953027 + 0.302887i \(0.902050\pi\)
\(882\) 0 0
\(883\) 963.803i 1.09151i 0.837945 + 0.545755i \(0.183757\pi\)
−0.837945 + 0.545755i \(0.816243\pi\)
\(884\) 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 \(0.00849103\pi\)
−0.999644 + 0.0266722i \(0.991509\pi\)
\(908\) 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 \(-0.397166\pi\)
0.317472 + 0.948268i \(0.397166\pi\)
\(920\) 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 \(0.0555919\pi\)
−0.984788 + 0.173761i \(0.944408\pi\)
\(930\) 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 \(-0.419663\pi\)
0.249715 + 0.968319i \(0.419663\pi\)
\(942\) 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 \(-0.916706\pi\)
0.965958 0.258699i \(-0.0832939\pi\)
\(954\) 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 \(-0.533715\pi\)
−0.105722 + 0.994396i \(0.533715\pi\)
\(968\) 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 \(-0.914119\pi\)
0.963823 0.266542i \(-0.0858811\pi\)
\(978\) 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 \(-0.515624\pi\)
−0.0490633 + 0.998796i \(0.515624\pi\)
\(992\) 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.j.2177.1 8
3.2 odd 2 inner 2304.3.h.j.2177.7 8
4.3 odd 2 2304.3.h.l.2177.1 8
8.3 odd 2 2304.3.h.l.2177.8 8
8.5 even 2 inner 2304.3.h.j.2177.8 8
12.11 even 2 2304.3.h.l.2177.7 8
16.3 odd 4 1152.3.e.c.1025.4 yes 4
16.5 even 4 1152.3.e.e.1025.1 yes 4
16.11 odd 4 1152.3.e.a.1025.1 4
16.13 even 4 1152.3.e.g.1025.4 yes 4
24.5 odd 2 inner 2304.3.h.j.2177.2 8
24.11 even 2 2304.3.h.l.2177.2 8
48.5 odd 4 1152.3.e.e.1025.4 yes 4
48.11 even 4 1152.3.e.a.1025.4 yes 4
48.29 odd 4 1152.3.e.g.1025.1 yes 4
48.35 even 4 1152.3.e.c.1025.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1152.3.e.a.1025.1 4 16.11 odd 4
1152.3.e.a.1025.4 yes 4 48.11 even 4
1152.3.e.c.1025.1 yes 4 48.35 even 4
1152.3.e.c.1025.4 yes 4 16.3 odd 4
1152.3.e.e.1025.1 yes 4 16.5 even 4
1152.3.e.e.1025.4 yes 4 48.5 odd 4
1152.3.e.g.1025.1 yes 4 48.29 odd 4
1152.3.e.g.1025.4 yes 4 16.13 even 4
2304.3.h.j.2177.1 8 1.1 even 1 trivial
2304.3.h.j.2177.2 8 24.5 odd 2 inner
2304.3.h.j.2177.7 8 3.2 odd 2 inner
2304.3.h.j.2177.8 8 8.5 even 2 inner
2304.3.h.l.2177.1 8 4.3 odd 2
2304.3.h.l.2177.2 8 24.11 even 2
2304.3.h.l.2177.7 8 12.11 even 2
2304.3.h.l.2177.8 8 8.3 odd 2