Properties

Label 2304.3.b.s.127.3
Level $2304$
Weight $3$
Character 2304.127
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(127,2304)]
 
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([1, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2304.127");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2304.b (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: 8.0.157351936.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + x^{4} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{18} \)
Twist minimal: no (minimal twist has level 1152)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 127.3
Root \(-1.28897 + 0.581861i\) of defining polynomial
Character \(\chi\) \(=\) 2304.127
Dual form 2304.3.b.s.127.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.46308i q^{5} -5.48331i q^{7} +O(q^{10})\) \(q-2.46308i q^{5} -5.48331i q^{7} +10.5830 q^{11} +8.96663i q^{13} +16.2399 q^{17} -2.96663 q^{19} -1.46141i q^{23} +18.9333 q^{25} -25.0905i q^{29} -10.5167i q^{31} -13.5058 q^{35} +16.9666i q^{37} -29.0150 q^{41} +34.9666 q^{43} +86.1254i q^{47} +18.9333 q^{49} +80.1976i q^{53} -26.0667i q^{55} +66.4208 q^{59} -0.966630i q^{61} +22.0855 q^{65} +113.800 q^{67} -90.5097i q^{71} -51.7998 q^{73} -58.0299i q^{77} -80.4499i q^{79} -79.9267 q^{83} -40.0000i q^{85} -142.694 q^{89} +49.1669 q^{91} +7.30703i q^{95} -45.8665 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 96 q^{19} - 88 q^{25} + 160 q^{43} - 88 q^{49} + 192 q^{67} + 304 q^{73} + 992 q^{91} + 112 q^{97}+O(q^{100}) \) 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
\(5\) − 2.46308i − 0.492615i −0.969192 0.246308i \(-0.920783\pi\)
0.969192 0.246308i \(-0.0792173\pi\)
\(6\) 0 0
\(7\) − 5.48331i − 0.783331i −0.920108 0.391665i \(-0.871899\pi\)
0.920108 0.391665i \(-0.128101\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 10.5830 0.962091 0.481046 0.876696i \(-0.340257\pi\)
0.481046 + 0.876696i \(0.340257\pi\)
\(12\) 0 0
\(13\) 8.96663i 0.689741i 0.938650 + 0.344870i \(0.112077\pi\)
−0.938650 + 0.344870i \(0.887923\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 16.2399 0.955286 0.477643 0.878554i \(-0.341491\pi\)
0.477643 + 0.878554i \(0.341491\pi\)
\(18\) 0 0
\(19\) −2.96663 −0.156138 −0.0780692 0.996948i \(-0.524876\pi\)
−0.0780692 + 0.996948i \(0.524876\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 1.46141i − 0.0635394i −0.999495 0.0317697i \(-0.989886\pi\)
0.999495 0.0317697i \(-0.0101143\pi\)
\(24\) 0 0
\(25\) 18.9333 0.757330
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 25.0905i − 0.865189i −0.901589 0.432595i \(-0.857598\pi\)
0.901589 0.432595i \(-0.142402\pi\)
\(30\) 0 0
\(31\) − 10.5167i − 0.339248i −0.985509 0.169624i \(-0.945745\pi\)
0.985509 0.169624i \(-0.0542553\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −13.5058 −0.385881
\(36\) 0 0
\(37\) 16.9666i 0.458558i 0.973361 + 0.229279i \(0.0736367\pi\)
−0.973361 + 0.229279i \(0.926363\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −29.0150 −0.707682 −0.353841 0.935306i \(-0.615125\pi\)
−0.353841 + 0.935306i \(0.615125\pi\)
\(42\) 0 0
\(43\) 34.9666 0.813177 0.406589 0.913611i \(-0.366718\pi\)
0.406589 + 0.913611i \(0.366718\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 86.1254i 1.83246i 0.400657 + 0.916228i \(0.368782\pi\)
−0.400657 + 0.916228i \(0.631218\pi\)
\(48\) 0 0
\(49\) 18.9333 0.386393
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 80.1976i 1.51316i 0.653900 + 0.756581i \(0.273132\pi\)
−0.653900 + 0.756581i \(0.726868\pi\)
\(54\) 0 0
\(55\) − 26.0667i − 0.473941i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 66.4208 1.12578 0.562889 0.826533i \(-0.309690\pi\)
0.562889 + 0.826533i \(0.309690\pi\)
\(60\) 0 0
\(61\) − 0.966630i − 0.0158464i −0.999969 0.00792319i \(-0.997478\pi\)
0.999969 0.00792319i \(-0.00252206\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 22.0855 0.339777
\(66\) 0 0
\(67\) 113.800 1.69850 0.849252 0.527988i \(-0.177053\pi\)
0.849252 + 0.527988i \(0.177053\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 90.5097i − 1.27478i −0.770540 0.637392i \(-0.780013\pi\)
0.770540 0.637392i \(-0.219987\pi\)
\(72\) 0 0
\(73\) −51.7998 −0.709586 −0.354793 0.934945i \(-0.615449\pi\)
−0.354793 + 0.934945i \(0.615449\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 58.0299i − 0.753636i
\(78\) 0 0
\(79\) − 80.4499i − 1.01835i −0.860662 0.509177i \(-0.829950\pi\)
0.860662 0.509177i \(-0.170050\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −79.9267 −0.962972 −0.481486 0.876454i \(-0.659903\pi\)
−0.481486 + 0.876454i \(0.659903\pi\)
\(84\) 0 0
\(85\) − 40.0000i − 0.470588i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −142.694 −1.60330 −0.801652 0.597791i \(-0.796045\pi\)
−0.801652 + 0.597791i \(0.796045\pi\)
\(90\) 0 0
\(91\) 49.1669 0.540295
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 7.30703i 0.0769161i
\(96\) 0 0
\(97\) −45.8665 −0.472851 −0.236425 0.971650i \(-0.575976\pi\)
−0.236425 + 0.971650i \(0.575976\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 83.1204i 0.822975i 0.911415 + 0.411487i \(0.134991\pi\)
−0.911415 + 0.411487i \(0.865009\pi\)
\(102\) 0 0
\(103\) − 106.517i − 1.03414i −0.855942 0.517071i \(-0.827022\pi\)
0.855942 0.517071i \(-0.172978\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 114.598 1.07101 0.535507 0.844531i \(-0.320121\pi\)
0.535507 + 0.844531i \(0.320121\pi\)
\(108\) 0 0
\(109\) − 94.7664i − 0.869417i −0.900571 0.434708i \(-0.856851\pi\)
0.900571 0.434708i \(-0.143149\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 201.808 1.78591 0.892955 0.450146i \(-0.148628\pi\)
0.892955 + 0.450146i \(0.148628\pi\)
\(114\) 0 0
\(115\) −3.59955 −0.0313005
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 89.0483i − 0.748305i
\(120\) 0 0
\(121\) −9.00000 −0.0743802
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 108.211i − 0.865687i
\(126\) 0 0
\(127\) − 171.417i − 1.34974i −0.737938 0.674868i \(-0.764200\pi\)
0.737938 0.674868i \(-0.235800\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −169.328 −1.29258 −0.646290 0.763092i \(-0.723681\pi\)
−0.646290 + 0.763092i \(0.723681\pi\)
\(132\) 0 0
\(133\) 16.2670i 0.122308i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2.38088 −0.0173787 −0.00868935 0.999962i \(-0.502766\pi\)
−0.00868935 + 0.999962i \(0.502766\pi\)
\(138\) 0 0
\(139\) 209.800 1.50935 0.754675 0.656098i \(-0.227794\pi\)
0.754675 + 0.656098i \(0.227794\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 94.8939i 0.663594i
\(144\) 0 0
\(145\) −61.7998 −0.426205
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 51.7246i − 0.347145i −0.984821 0.173572i \(-0.944469\pi\)
0.984821 0.173572i \(-0.0555311\pi\)
\(150\) 0 0
\(151\) − 227.150i − 1.50430i −0.658991 0.752151i \(-0.729016\pi\)
0.658991 0.752151i \(-0.270984\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −25.9034 −0.167119
\(156\) 0 0
\(157\) 110.766i 0.705519i 0.935714 + 0.352759i \(0.114757\pi\)
−0.935714 + 0.352759i \(0.885243\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −8.01335 −0.0497724
\(162\) 0 0
\(163\) 276.766 1.69795 0.848977 0.528430i \(-0.177219\pi\)
0.848977 + 0.528430i \(0.177219\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 80.2798i − 0.480717i −0.970684 0.240359i \(-0.922735\pi\)
0.970684 0.240359i \(-0.0772651\pi\)
\(168\) 0 0
\(169\) 88.5996 0.524258
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 124.533i − 0.719844i −0.932982 0.359922i \(-0.882803\pi\)
0.932982 0.359922i \(-0.117197\pi\)
\(174\) 0 0
\(175\) − 103.817i − 0.593240i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −66.4208 −0.371066 −0.185533 0.982638i \(-0.559401\pi\)
−0.185533 + 0.982638i \(0.559401\pi\)
\(180\) 0 0
\(181\) − 160.967i − 0.889318i −0.895700 0.444659i \(-0.853325\pi\)
0.895700 0.444659i \(-0.146675\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 41.7901 0.225892
\(186\) 0 0
\(187\) 171.867 0.919072
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 83.2026i 0.435616i 0.975992 + 0.217808i \(0.0698906\pi\)
−0.975992 + 0.217808i \(0.930109\pi\)
\(192\) 0 0
\(193\) 131.666 0.682209 0.341104 0.940025i \(-0.389199\pi\)
0.341104 + 0.940025i \(0.389199\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 184.566i 0.936885i 0.883494 + 0.468442i \(0.155185\pi\)
−0.883494 + 0.468442i \(0.844815\pi\)
\(198\) 0 0
\(199\) − 168.717i − 0.847824i −0.905703 0.423912i \(-0.860657\pi\)
0.905703 0.423912i \(-0.139343\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −137.579 −0.677729
\(204\) 0 0
\(205\) 71.4661i 0.348615i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −31.3959 −0.150219
\(210\) 0 0
\(211\) 197.933 0.938072 0.469036 0.883179i \(-0.344601\pi\)
0.469036 + 0.883179i \(0.344601\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) − 86.1254i − 0.400583i
\(216\) 0 0
\(217\) −57.6663 −0.265743
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 145.617i 0.658900i
\(222\) 0 0
\(223\) − 99.6835i − 0.447011i −0.974703 0.223506i \(-0.928250\pi\)
0.974703 0.223506i \(-0.0717501\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 243.409 1.07229 0.536143 0.844127i \(-0.319881\pi\)
0.536143 + 0.844127i \(0.319881\pi\)
\(228\) 0 0
\(229\) 454.232i 1.98355i 0.128002 + 0.991774i \(0.459144\pi\)
−0.128002 + 0.991774i \(0.540856\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −77.7346 −0.333625 −0.166812 0.985989i \(-0.553347\pi\)
−0.166812 + 0.985989i \(0.553347\pi\)
\(234\) 0 0
\(235\) 212.133 0.902696
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 348.886i − 1.45977i −0.683568 0.729887i \(-0.739573\pi\)
0.683568 0.729887i \(-0.260427\pi\)
\(240\) 0 0
\(241\) 75.7998 0.314522 0.157261 0.987557i \(-0.449734\pi\)
0.157261 + 0.987557i \(0.449734\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 46.6340i − 0.190343i
\(246\) 0 0
\(247\) − 26.6007i − 0.107695i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 131.733 0.524834 0.262417 0.964955i \(-0.415480\pi\)
0.262417 + 0.964955i \(0.415480\pi\)
\(252\) 0 0
\(253\) − 15.4661i − 0.0611307i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −305.093 −1.18713 −0.593565 0.804786i \(-0.702280\pi\)
−0.593565 + 0.804786i \(0.702280\pi\)
\(258\) 0 0
\(259\) 93.0334 0.359202
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 338.656i 1.28767i 0.765166 + 0.643833i \(0.222657\pi\)
−0.765166 + 0.643833i \(0.777343\pi\)
\(264\) 0 0
\(265\) 197.533 0.745407
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 271.234i 1.00830i 0.863615 + 0.504152i \(0.168195\pi\)
−0.863615 + 0.504152i \(0.831805\pi\)
\(270\) 0 0
\(271\) − 406.749i − 1.50092i −0.660916 0.750460i \(-0.729832\pi\)
0.660916 0.750460i \(-0.270168\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 200.371 0.728621
\(276\) 0 0
\(277\) − 249.234i − 0.899760i −0.893089 0.449880i \(-0.851467\pi\)
0.893089 0.449880i \(-0.148533\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 56.9461 0.202655 0.101328 0.994853i \(-0.467691\pi\)
0.101328 + 0.994853i \(0.467691\pi\)
\(282\) 0 0
\(283\) 136.267 0.481509 0.240754 0.970586i \(-0.422605\pi\)
0.240754 + 0.970586i \(0.422605\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 159.098i 0.554349i
\(288\) 0 0
\(289\) −25.2670 −0.0874289
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 445.324i − 1.51988i −0.649996 0.759938i \(-0.725229\pi\)
0.649996 0.759938i \(-0.274771\pi\)
\(294\) 0 0
\(295\) − 163.600i − 0.554575i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 13.1039 0.0438257
\(300\) 0 0
\(301\) − 191.733i − 0.636987i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −2.38088 −0.00780617
\(306\) 0 0
\(307\) 329.533 1.07340 0.536698 0.843774i \(-0.319671\pi\)
0.536698 + 0.843774i \(0.319671\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 359.116i 1.15471i 0.816492 + 0.577357i \(0.195916\pi\)
−0.816492 + 0.577357i \(0.804084\pi\)
\(312\) 0 0
\(313\) −147.666 −0.471777 −0.235889 0.971780i \(-0.575800\pi\)
−0.235889 + 0.971780i \(0.575800\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 233.663i 0.737109i 0.929606 + 0.368554i \(0.120147\pi\)
−0.929606 + 0.368554i \(0.879853\pi\)
\(318\) 0 0
\(319\) − 265.533i − 0.832391i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −48.1776 −0.149157
\(324\) 0 0
\(325\) 169.768i 0.522362i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 472.253 1.43542
\(330\) 0 0
\(331\) −163.600 −0.494258 −0.247129 0.968983i \(-0.579487\pi\)
−0.247129 + 0.968983i \(0.579487\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) − 280.297i − 0.836709i
\(336\) 0 0
\(337\) −35.9333 −0.106627 −0.0533134 0.998578i \(-0.516978\pi\)
−0.0533134 + 0.998578i \(0.516978\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 111.298i − 0.326387i
\(342\) 0 0
\(343\) − 372.499i − 1.08600i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −584.282 −1.68381 −0.841905 0.539626i \(-0.818565\pi\)
−0.841905 + 0.539626i \(0.818565\pi\)
\(348\) 0 0
\(349\) 230.766i 0.661222i 0.943767 + 0.330611i \(0.107255\pi\)
−0.943767 + 0.330611i \(0.892745\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −50.0166 −0.141690 −0.0708450 0.997487i \(-0.522570\pi\)
−0.0708450 + 0.997487i \(0.522570\pi\)
\(354\) 0 0
\(355\) −222.932 −0.627978
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 253.992i 0.707499i 0.935340 + 0.353749i \(0.115093\pi\)
−0.935340 + 0.353749i \(0.884907\pi\)
\(360\) 0 0
\(361\) −352.199 −0.975621
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 127.587i 0.349553i
\(366\) 0 0
\(367\) 348.682i 0.950088i 0.879962 + 0.475044i \(0.157568\pi\)
−0.879962 + 0.475044i \(0.842432\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 439.749 1.18531
\(372\) 0 0
\(373\) − 646.499i − 1.73324i −0.498967 0.866621i \(-0.666287\pi\)
0.498967 0.866621i \(-0.333713\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 224.977 0.596756
\(378\) 0 0
\(379\) 274.433 0.724097 0.362048 0.932159i \(-0.382078\pi\)
0.362048 + 0.932159i \(0.382078\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 272.990i − 0.712769i −0.934339 0.356384i \(-0.884009\pi\)
0.934339 0.356384i \(-0.115991\pi\)
\(384\) 0 0
\(385\) −142.932 −0.371252
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 412.679i − 1.06087i −0.847725 0.530436i \(-0.822028\pi\)
0.847725 0.530436i \(-0.177972\pi\)
\(390\) 0 0
\(391\) − 23.7330i − 0.0606983i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −198.154 −0.501656
\(396\) 0 0
\(397\) − 384.700i − 0.969017i −0.874787 0.484508i \(-0.838999\pi\)
0.874787 0.484508i \(-0.161001\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −561.466 −1.40016 −0.700082 0.714063i \(-0.746853\pi\)
−0.700082 + 0.714063i \(0.746853\pi\)
\(402\) 0 0
\(403\) 94.2992 0.233993
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 179.558i 0.441174i
\(408\) 0 0
\(409\) 253.333 0.619395 0.309698 0.950835i \(-0.399772\pi\)
0.309698 + 0.950835i \(0.399772\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 364.206i − 0.881856i
\(414\) 0 0
\(415\) 196.865i 0.474374i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 622.985 1.48684 0.743418 0.668827i \(-0.233203\pi\)
0.743418 + 0.668827i \(0.233203\pi\)
\(420\) 0 0
\(421\) 487.300i 1.15748i 0.815511 + 0.578741i \(0.196456\pi\)
−0.815511 + 0.578741i \(0.803544\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 307.473 0.723467
\(426\) 0 0
\(427\) −5.30033 −0.0124130
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 176.635i 0.409826i 0.978780 + 0.204913i \(0.0656912\pi\)
−0.978780 + 0.204913i \(0.934309\pi\)
\(432\) 0 0
\(433\) −257.066 −0.593685 −0.296843 0.954926i \(-0.595934\pi\)
−0.296843 + 0.954926i \(0.595934\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 4.33545i 0.00992094i
\(438\) 0 0
\(439\) 638.482i 1.45440i 0.686425 + 0.727201i \(0.259179\pi\)
−0.686425 + 0.727201i \(0.740821\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 104.722 0.236392 0.118196 0.992990i \(-0.462289\pi\)
0.118196 + 0.992990i \(0.462289\pi\)
\(444\) 0 0
\(445\) 351.466i 0.789811i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 219.132 0.488043 0.244022 0.969770i \(-0.421533\pi\)
0.244022 + 0.969770i \(0.421533\pi\)
\(450\) 0 0
\(451\) −307.066 −0.680855
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) − 121.102i − 0.266158i
\(456\) 0 0
\(457\) 46.4004 0.101533 0.0507664 0.998711i \(-0.483834\pi\)
0.0507664 + 0.998711i \(0.483834\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 517.377i 1.12229i 0.827717 + 0.561146i \(0.189639\pi\)
−0.827717 + 0.561146i \(0.810361\pi\)
\(462\) 0 0
\(463\) − 111.016i − 0.239776i −0.992787 0.119888i \(-0.961747\pi\)
0.992787 0.119888i \(-0.0382535\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −233.934 −0.500930 −0.250465 0.968126i \(-0.580584\pi\)
−0.250465 + 0.968126i \(0.580584\pi\)
\(468\) 0 0
\(469\) − 624.000i − 1.33049i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 370.052 0.782351
\(474\) 0 0
\(475\) −56.1680 −0.118248
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 610.185i − 1.27387i −0.770916 0.636937i \(-0.780201\pi\)
0.770916 0.636937i \(-0.219799\pi\)
\(480\) 0 0
\(481\) −152.133 −0.316286
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 112.973i 0.232933i
\(486\) 0 0
\(487\) 104.183i 0.213928i 0.994263 + 0.106964i \(0.0341130\pi\)
−0.994263 + 0.106964i \(0.965887\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −423.320 −0.862159 −0.431080 0.902314i \(-0.641867\pi\)
−0.431080 + 0.902314i \(0.641867\pi\)
\(492\) 0 0
\(493\) − 407.466i − 0.826503i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −496.293 −0.998577
\(498\) 0 0
\(499\) 126.932 0.254373 0.127187 0.991879i \(-0.459405\pi\)
0.127187 + 0.991879i \(0.459405\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 274.452i − 0.545630i −0.962067 0.272815i \(-0.912045\pi\)
0.962067 0.272815i \(-0.0879547\pi\)
\(504\) 0 0
\(505\) 204.732 0.405410
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 750.416i 1.47429i 0.675732 + 0.737147i \(0.263828\pi\)
−0.675732 + 0.737147i \(0.736172\pi\)
\(510\) 0 0
\(511\) 284.034i 0.555840i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −262.359 −0.509434
\(516\) 0 0
\(517\) 911.466i 1.76299i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −822.387 −1.57848 −0.789239 0.614086i \(-0.789525\pi\)
−0.789239 + 0.614086i \(0.789525\pi\)
\(522\) 0 0
\(523\) −184.366 −0.352516 −0.176258 0.984344i \(-0.556399\pi\)
−0.176258 + 0.984344i \(0.556399\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 170.789i − 0.324079i
\(528\) 0 0
\(529\) 526.864 0.995963
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 260.167i − 0.488117i
\(534\) 0 0
\(535\) − 282.265i − 0.527598i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 200.371 0.371745
\(540\) 0 0
\(541\) 432.967i 0.800308i 0.916448 + 0.400154i \(0.131043\pi\)
−0.916448 + 0.400154i \(0.868957\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −233.417 −0.428288
\(546\) 0 0
\(547\) −557.567 −1.01932 −0.509659 0.860376i \(-0.670229\pi\)
−0.509659 + 0.860376i \(0.670229\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 74.4342i 0.135089i
\(552\) 0 0
\(553\) −441.132 −0.797708
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 656.984i − 1.17950i −0.807584 0.589752i \(-0.799226\pi\)
0.807584 0.589752i \(-0.200774\pi\)
\(558\) 0 0
\(559\) 313.533i 0.560882i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −204.706 −0.363599 −0.181799 0.983336i \(-0.558192\pi\)
−0.181799 + 0.983336i \(0.558192\pi\)
\(564\) 0 0
\(565\) − 497.068i − 0.879766i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −251.611 −0.442199 −0.221100 0.975251i \(-0.570965\pi\)
−0.221100 + 0.975251i \(0.570965\pi\)
\(570\) 0 0
\(571\) −832.198 −1.45744 −0.728720 0.684812i \(-0.759884\pi\)
−0.728720 + 0.684812i \(0.759884\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 27.6692i − 0.0481203i
\(576\) 0 0
\(577\) 539.132 0.934372 0.467186 0.884159i \(-0.345268\pi\)
0.467186 + 0.884159i \(0.345268\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 438.263i 0.754325i
\(582\) 0 0
\(583\) 848.732i 1.45580i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 84.6640 0.144232 0.0721159 0.997396i \(-0.477025\pi\)
0.0721159 + 0.997396i \(0.477025\pi\)
\(588\) 0 0
\(589\) 31.1991i 0.0529696i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 923.717 1.55770 0.778851 0.627209i \(-0.215803\pi\)
0.778851 + 0.627209i \(0.215803\pi\)
\(594\) 0 0
\(595\) −219.333 −0.368626
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 538.674i 0.899288i 0.893208 + 0.449644i \(0.148449\pi\)
−0.893208 + 0.449644i \(0.851551\pi\)
\(600\) 0 0
\(601\) 164.334 0.273434 0.136717 0.990610i \(-0.456345\pi\)
0.136717 + 0.990610i \(0.456345\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 22.1677i 0.0366408i
\(606\) 0 0
\(607\) 368.984i 0.607881i 0.952691 + 0.303941i \(0.0983024\pi\)
−0.952691 + 0.303941i \(0.901698\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −772.255 −1.26392
\(612\) 0 0
\(613\) 968.433i 1.57982i 0.613220 + 0.789912i \(0.289874\pi\)
−0.613220 + 0.789912i \(0.710126\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −53.2682 −0.0863342 −0.0431671 0.999068i \(-0.513745\pi\)
−0.0431671 + 0.999068i \(0.513745\pi\)
\(618\) 0 0
\(619\) −1066.80 −1.72342 −0.861711 0.507399i \(-0.830607\pi\)
−0.861711 + 0.507399i \(0.830607\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 782.436i 1.25592i
\(624\) 0 0
\(625\) 206.800 0.330880
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 275.536i 0.438054i
\(630\) 0 0
\(631\) 468.049i 0.741758i 0.928681 + 0.370879i \(0.120944\pi\)
−0.928681 + 0.370879i \(0.879056\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −422.212 −0.664901
\(636\) 0 0
\(637\) 169.768i 0.266511i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 603.043 0.940784 0.470392 0.882458i \(-0.344113\pi\)
0.470392 + 0.882458i \(0.344113\pi\)
\(642\) 0 0
\(643\) −352.633 −0.548418 −0.274209 0.961670i \(-0.588416\pi\)
−0.274209 + 0.961670i \(0.588416\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 897.790i − 1.38762i −0.720158 0.693810i \(-0.755931\pi\)
0.720158 0.693810i \(-0.244069\pi\)
\(648\) 0 0
\(649\) 702.932 1.08310
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 169.459i − 0.259508i −0.991546 0.129754i \(-0.958581\pi\)
0.991546 0.129754i \(-0.0414188\pi\)
\(654\) 0 0
\(655\) 417.068i 0.636745i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 36.4864 0.0553663 0.0276832 0.999617i \(-0.491187\pi\)
0.0276832 + 0.999617i \(0.491187\pi\)
\(660\) 0 0
\(661\) − 1038.77i − 1.57151i −0.618539 0.785754i \(-0.712275\pi\)
0.618539 0.785754i \(-0.287725\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 40.0668 0.0602508
\(666\) 0 0
\(667\) −36.6674 −0.0549736
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) − 10.2298i − 0.0152457i
\(672\) 0 0
\(673\) −593.600 −0.882020 −0.441010 0.897502i \(-0.645380\pi\)
−0.441010 + 0.897502i \(0.645380\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 296.455i 0.437895i 0.975737 + 0.218948i \(0.0702624\pi\)
−0.975737 + 0.218948i \(0.929738\pi\)
\(678\) 0 0
\(679\) 251.501i 0.370398i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1169.67 1.71255 0.856275 0.516520i \(-0.172773\pi\)
0.856275 + 0.516520i \(0.172773\pi\)
\(684\) 0 0
\(685\) 5.86429i 0.00856101i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −719.102 −1.04369
\(690\) 0 0
\(691\) −1242.70 −1.79841 −0.899204 0.437530i \(-0.855853\pi\)
−0.899204 + 0.437530i \(0.855853\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 516.753i − 0.743529i
\(696\) 0 0
\(697\) −471.199 −0.676039
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 1362.28i − 1.94333i −0.236358 0.971666i \(-0.575954\pi\)
0.236358 0.971666i \(-0.424046\pi\)
\(702\) 0 0
\(703\) − 50.3337i − 0.0715984i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 455.776 0.644661
\(708\) 0 0
\(709\) 1261.97i 1.77992i 0.456036 + 0.889962i \(0.349269\pi\)
−0.456036 + 0.889962i \(0.650731\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −15.3692 −0.0215556
\(714\) 0 0
\(715\) 233.731 0.326896
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 951.764i 1.32373i 0.749622 + 0.661867i \(0.230235\pi\)
−0.749622 + 0.661867i \(0.769765\pi\)
\(720\) 0 0
\(721\) −584.065 −0.810076
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 475.045i − 0.655234i
\(726\) 0 0
\(727\) 982.383i 1.35128i 0.737230 + 0.675642i \(0.236133\pi\)
−0.737230 + 0.675642i \(0.763867\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 567.853 0.776817
\(732\) 0 0
\(733\) − 165.699i − 0.226055i −0.993592 0.113028i \(-0.963945\pi\)
0.993592 0.113028i \(-0.0360549\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1204.34 1.63412
\(738\) 0 0
\(739\) 248.999 0.336940 0.168470 0.985707i \(-0.446117\pi\)
0.168470 + 0.985707i \(0.446117\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 538.674i − 0.724998i −0.931984 0.362499i \(-0.881924\pi\)
0.931984 0.362499i \(-0.118076\pi\)
\(744\) 0 0
\(745\) −127.402 −0.171009
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 628.380i − 0.838958i
\(750\) 0 0
\(751\) 979.882i 1.30477i 0.757888 + 0.652385i \(0.226231\pi\)
−0.757888 + 0.652385i \(0.773769\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −559.487 −0.741042
\(756\) 0 0
\(757\) − 1151.90i − 1.52166i −0.648950 0.760831i \(-0.724791\pi\)
0.648950 0.760831i \(-0.275209\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −268.064 −0.352253 −0.176126 0.984368i \(-0.556357\pi\)
−0.176126 + 0.984368i \(0.556357\pi\)
\(762\) 0 0
\(763\) −519.634 −0.681041
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 595.571i 0.776494i
\(768\) 0 0
\(769\) 428.467 0.557174 0.278587 0.960411i \(-0.410134\pi\)
0.278587 + 0.960411i \(0.410134\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1118.14i 1.44649i 0.690592 + 0.723245i \(0.257350\pi\)
−0.690592 + 0.723245i \(0.742650\pi\)
\(774\) 0 0
\(775\) − 199.115i − 0.256923i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 86.0767 0.110496
\(780\) 0 0
\(781\) − 957.864i − 1.22646i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 272.826 0.347549
\(786\) 0 0
\(787\) −711.832 −0.904488 −0.452244 0.891894i \(-0.649376\pi\)
−0.452244 + 0.891894i \(0.649376\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) − 1106.58i − 1.39896i
\(792\) 0 0
\(793\) 8.66741 0.0109299
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 209.033i 0.262274i 0.991364 + 0.131137i \(0.0418628\pi\)
−0.991364 + 0.131137i \(0.958137\pi\)
\(798\) 0 0
\(799\) 1398.67i 1.75052i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −548.197 −0.682687
\(804\) 0 0
\(805\) 19.7375i 0.0245186i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −581.170 −0.718381 −0.359190 0.933264i \(-0.616947\pi\)
−0.359190 + 0.933264i \(0.616947\pi\)
\(810\) 0 0
\(811\) 171.432 0.211383 0.105691 0.994399i \(-0.466294\pi\)
0.105691 + 0.994399i \(0.466294\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) − 681.697i − 0.836437i
\(816\) 0 0
\(817\) −103.733 −0.126968
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1134.49i 1.38184i 0.722931 + 0.690921i \(0.242795\pi\)
−0.722931 + 0.690921i \(0.757205\pi\)
\(822\) 0 0
\(823\) 379.249i 0.460812i 0.973095 + 0.230406i \(0.0740055\pi\)
−0.973095 + 0.230406i \(0.925995\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1177.33 1.42362 0.711809 0.702373i \(-0.247876\pi\)
0.711809 + 0.702373i \(0.247876\pi\)
\(828\) 0 0
\(829\) 1278.56i 1.54230i 0.636656 + 0.771148i \(0.280317\pi\)
−0.636656 + 0.771148i \(0.719683\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 307.473 0.369116
\(834\) 0 0
\(835\) −197.735 −0.236809
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 1015.97i 1.21093i 0.795873 + 0.605464i \(0.207012\pi\)
−0.795873 + 0.605464i \(0.792988\pi\)
\(840\) 0 0
\(841\) 211.467 0.251447
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 218.227i − 0.258257i
\(846\) 0 0
\(847\) 49.3498i 0.0582643i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 24.7951 0.0291365
\(852\) 0 0
\(853\) − 1509.43i − 1.76956i −0.466012 0.884778i \(-0.654310\pi\)
0.466012 0.884778i \(-0.345690\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −844.260 −0.985134 −0.492567 0.870275i \(-0.663941\pi\)
−0.492567 + 0.870275i \(0.663941\pi\)
\(858\) 0 0
\(859\) −989.231 −1.15161 −0.575804 0.817588i \(-0.695311\pi\)
−0.575804 + 0.817588i \(0.695311\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1680.23i 1.94696i 0.228774 + 0.973480i \(0.426528\pi\)
−0.228774 + 0.973480i \(0.573472\pi\)
\(864\) 0 0
\(865\) −306.734 −0.354606
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 851.402i − 0.979749i
\(870\) 0 0
\(871\) 1020.40i 1.17153i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −593.355 −0.678120
\(876\) 0 0
\(877\) − 591.899i − 0.674913i −0.941341 0.337457i \(-0.890433\pi\)
0.941341 0.337457i \(-0.109567\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 797.050 0.904711 0.452355 0.891838i \(-0.350584\pi\)
0.452355 + 0.891838i \(0.350584\pi\)
\(882\) 0 0
\(883\) −276.964 −0.313663 −0.156831 0.987625i \(-0.550128\pi\)
−0.156831 + 0.987625i \(0.550128\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 910.845i 1.02688i 0.858125 + 0.513441i \(0.171630\pi\)
−0.858125 + 0.513441i \(0.828370\pi\)
\(888\) 0 0
\(889\) −939.931 −1.05729
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 255.502i − 0.286117i
\(894\) 0 0
\(895\) 163.600i 0.182793i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −263.869 −0.293514
\(900\) 0 0
\(901\) 1302.40i 1.44550i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −396.473 −0.438092
\(906\) 0 0
\(907\) −1472.83 −1.62385 −0.811924 0.583763i \(-0.801580\pi\)
−0.811924 + 0.583763i \(0.801580\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 36.5352i 0.0401045i 0.999799 + 0.0200522i \(0.00638325\pi\)
−0.999799 + 0.0200522i \(0.993617\pi\)
\(912\) 0 0
\(913\) −845.864 −0.926467
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 928.479i 1.01252i
\(918\) 0 0
\(919\) − 467.150i − 0.508324i −0.967162 0.254162i \(-0.918200\pi\)
0.967162 0.254162i \(-0.0817996\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 811.567 0.879270
\(924\) 0 0
\(925\) 321.234i 0.347280i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −762.190 −0.820441 −0.410220 0.911986i \(-0.634548\pi\)
−0.410220 + 0.911986i \(0.634548\pi\)
\(930\) 0 0
\(931\) −56.1680 −0.0603308
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) − 423.320i − 0.452749i
\(936\) 0 0
\(937\) −123.335 −0.131627 −0.0658137 0.997832i \(-0.520964\pi\)
−0.0658137 + 0.997832i \(0.520964\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 735.638i 0.781762i 0.920441 + 0.390881i \(0.127830\pi\)
−0.920441 + 0.390881i \(0.872170\pi\)
\(942\) 0 0
\(943\) 42.4027i 0.0449657i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −549.610 −0.580370 −0.290185 0.956971i \(-0.593717\pi\)
−0.290185 + 0.956971i \(0.593717\pi\)
\(948\) 0 0
\(949\) − 464.469i − 0.489430i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1217.33 −1.27737 −0.638684 0.769469i \(-0.720521\pi\)
−0.638684 + 0.769469i \(0.720521\pi\)
\(954\) 0 0
\(955\) 204.934 0.214591
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 13.0551i 0.0136133i
\(960\) 0 0
\(961\) 850.399 0.884911
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 324.304i − 0.336066i
\(966\) 0 0
\(967\) 514.616i 0.532178i 0.963949 + 0.266089i \(0.0857314\pi\)
−0.963949 + 0.266089i \(0.914269\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1248.49 1.28578 0.642889 0.765959i \(-0.277736\pi\)
0.642889 + 0.765959i \(0.277736\pi\)
\(972\) 0 0
\(973\) − 1150.40i − 1.18232i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 126.028 0.128995 0.0644973 0.997918i \(-0.479456\pi\)
0.0644973 + 0.997918i \(0.479456\pi\)
\(978\) 0 0
\(979\) −1510.13 −1.54252
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 1018.89i − 1.03651i −0.855226 0.518256i \(-0.826581\pi\)
0.855226 0.518256i \(-0.173419\pi\)
\(984\) 0 0
\(985\) 454.601 0.461524
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 51.1005i − 0.0516688i
\(990\) 0 0
\(991\) − 859.981i − 0.867791i −0.900963 0.433895i \(-0.857139\pi\)
0.900963 0.433895i \(-0.142861\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −415.562 −0.417651
\(996\) 0 0
\(997\) 1104.43i 1.10776i 0.832598 + 0.553878i \(0.186853\pi\)
−0.832598 + 0.553878i \(0.813147\pi\)
\(998\) 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.b.s.127.3 8
3.2 odd 2 inner 2304.3.b.s.127.5 8
4.3 odd 2 2304.3.b.r.127.4 8
8.3 odd 2 inner 2304.3.b.s.127.6 8
8.5 even 2 2304.3.b.r.127.5 8
12.11 even 2 2304.3.b.r.127.6 8
16.3 odd 4 1152.3.g.e.127.3 yes 8
16.5 even 4 1152.3.g.d.127.6 yes 8
16.11 odd 4 1152.3.g.d.127.5 yes 8
16.13 even 4 1152.3.g.e.127.4 yes 8
24.5 odd 2 2304.3.b.r.127.3 8
24.11 even 2 inner 2304.3.b.s.127.4 8
48.5 odd 4 1152.3.g.d.127.4 yes 8
48.11 even 4 1152.3.g.d.127.3 8
48.29 odd 4 1152.3.g.e.127.6 yes 8
48.35 even 4 1152.3.g.e.127.5 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1152.3.g.d.127.3 8 48.11 even 4
1152.3.g.d.127.4 yes 8 48.5 odd 4
1152.3.g.d.127.5 yes 8 16.11 odd 4
1152.3.g.d.127.6 yes 8 16.5 even 4
1152.3.g.e.127.3 yes 8 16.3 odd 4
1152.3.g.e.127.4 yes 8 16.13 even 4
1152.3.g.e.127.5 yes 8 48.35 even 4
1152.3.g.e.127.6 yes 8 48.29 odd 4
2304.3.b.r.127.3 8 24.5 odd 2
2304.3.b.r.127.4 8 4.3 odd 2
2304.3.b.r.127.5 8 8.5 even 2
2304.3.b.r.127.6 8 12.11 even 2
2304.3.b.s.127.3 8 1.1 even 1 trivial
2304.3.b.s.127.4 8 24.11 even 2 inner
2304.3.b.s.127.5 8 3.2 odd 2 inner
2304.3.b.s.127.6 8 8.3 odd 2 inner