Properties

Label 2304.3.b.q.127.4
Level $2304$
Weight $3$
Character 2304.127
Analytic conductor $62.779$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [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: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{18} \)
Twist minimal: no (minimal twist has level 384)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 127.4
Root \(0.965926 - 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 2304.127
Dual form 2304.3.b.q.127.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.36433i q^{5} -1.24213i q^{7} +O(q^{10})\) \(q-1.36433i q^{5} -1.24213i q^{7} +5.79796 q^{11} -16.3830i q^{13} +5.01086 q^{17} +26.1835 q^{19} +25.1117i q^{23} +23.1386 q^{25} +32.7743i q^{29} -1.01836i q^{31} -1.69466 q^{35} +14.9948i q^{37} -72.5212 q^{41} -33.4922 q^{43} -66.5640i q^{47} +47.4571 q^{49} +54.6513i q^{53} -7.91030i q^{55} +20.5880 q^{59} -111.026i q^{61} -22.3518 q^{65} +60.9540 q^{67} -80.4576i q^{71} -30.0525 q^{73} -7.20179i q^{77} -80.9441i q^{79} +113.958 q^{83} -6.83644i q^{85} -21.0637 q^{89} -20.3498 q^{91} -35.7228i q^{95} +160.594 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 32 q^{11} - 16 q^{17} + 96 q^{19} + 8 q^{25} + 96 q^{35} + 80 q^{41} + 224 q^{43} - 88 q^{49} + 512 q^{59} + 160 q^{65} + 16 q^{73} + 544 q^{83} - 240 q^{89} - 32 q^{91} + 400 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(5\) − 1.36433i − 0.272865i −0.990649 0.136433i \(-0.956436\pi\)
0.990649 0.136433i \(-0.0435637\pi\)
\(6\) 0 0
\(7\) − 1.24213i − 0.177446i −0.996056 0.0887232i \(-0.971721\pi\)
0.996056 0.0887232i \(-0.0282787\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.79796 0.527087 0.263544 0.964647i \(-0.415109\pi\)
0.263544 + 0.964647i \(0.415109\pi\)
\(12\) 0 0
\(13\) − 16.3830i − 1.26023i −0.776501 0.630116i \(-0.783007\pi\)
0.776501 0.630116i \(-0.216993\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.01086 0.294756 0.147378 0.989080i \(-0.452917\pi\)
0.147378 + 0.989080i \(0.452917\pi\)
\(18\) 0 0
\(19\) 26.1835 1.37808 0.689039 0.724725i \(-0.258033\pi\)
0.689039 + 0.724725i \(0.258033\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 25.1117i 1.09181i 0.837847 + 0.545906i \(0.183814\pi\)
−0.837847 + 0.545906i \(0.816186\pi\)
\(24\) 0 0
\(25\) 23.1386 0.925545
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 32.7743i 1.13015i 0.825040 + 0.565074i \(0.191152\pi\)
−0.825040 + 0.565074i \(0.808848\pi\)
\(30\) 0 0
\(31\) − 1.01836i − 0.0328504i −0.999865 0.0164252i \(-0.994771\pi\)
0.999865 0.0164252i \(-0.00522854\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.69466 −0.0484189
\(36\) 0 0
\(37\) 14.9948i 0.405264i 0.979255 + 0.202632i \(0.0649495\pi\)
−0.979255 + 0.202632i \(0.935050\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −72.5212 −1.76881 −0.884405 0.466720i \(-0.845435\pi\)
−0.884405 + 0.466720i \(0.845435\pi\)
\(42\) 0 0
\(43\) −33.4922 −0.778888 −0.389444 0.921050i \(-0.627333\pi\)
−0.389444 + 0.921050i \(0.627333\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 66.5640i − 1.41626i −0.706085 0.708128i \(-0.749540\pi\)
0.706085 0.708128i \(-0.250460\pi\)
\(48\) 0 0
\(49\) 47.4571 0.968513
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 54.6513i 1.03116i 0.856842 + 0.515579i \(0.172423\pi\)
−0.856842 + 0.515579i \(0.827577\pi\)
\(54\) 0 0
\(55\) − 7.91030i − 0.143824i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 20.5880 0.348949 0.174474 0.984662i \(-0.444177\pi\)
0.174474 + 0.984662i \(0.444177\pi\)
\(60\) 0 0
\(61\) − 111.026i − 1.82010i −0.414499 0.910050i \(-0.636043\pi\)
0.414499 0.910050i \(-0.363957\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −22.3518 −0.343873
\(66\) 0 0
\(67\) 60.9540 0.909762 0.454881 0.890552i \(-0.349682\pi\)
0.454881 + 0.890552i \(0.349682\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 80.4576i − 1.13320i −0.823991 0.566602i \(-0.808258\pi\)
0.823991 0.566602i \(-0.191742\pi\)
\(72\) 0 0
\(73\) −30.0525 −0.411679 −0.205839 0.978586i \(-0.565992\pi\)
−0.205839 + 0.978586i \(0.565992\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 7.20179i − 0.0935298i
\(78\) 0 0
\(79\) − 80.9441i − 1.02461i −0.858804 0.512304i \(-0.828792\pi\)
0.858804 0.512304i \(-0.171208\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 113.958 1.37299 0.686496 0.727134i \(-0.259148\pi\)
0.686496 + 0.727134i \(0.259148\pi\)
\(84\) 0 0
\(85\) − 6.83644i − 0.0804288i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −21.0637 −0.236671 −0.118335 0.992974i \(-0.537756\pi\)
−0.118335 + 0.992974i \(0.537756\pi\)
\(90\) 0 0
\(91\) −20.3498 −0.223624
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 35.7228i − 0.376029i
\(96\) 0 0
\(97\) 160.594 1.65561 0.827806 0.561014i \(-0.189589\pi\)
0.827806 + 0.561014i \(0.189589\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 76.2681i 0.755130i 0.925983 + 0.377565i \(0.123239\pi\)
−0.925983 + 0.377565i \(0.876761\pi\)
\(102\) 0 0
\(103\) 182.763i 1.77440i 0.461383 + 0.887201i \(0.347353\pi\)
−0.461383 + 0.887201i \(0.652647\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 31.8533 0.297694 0.148847 0.988860i \(-0.452444\pi\)
0.148847 + 0.988860i \(0.452444\pi\)
\(108\) 0 0
\(109\) − 11.3289i − 0.103935i −0.998649 0.0519676i \(-0.983451\pi\)
0.998649 0.0519676i \(-0.0165493\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −49.9587 −0.442113 −0.221056 0.975261i \(-0.570950\pi\)
−0.221056 + 0.975261i \(0.570950\pi\)
\(114\) 0 0
\(115\) 34.2605 0.297917
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 6.22412i − 0.0523035i
\(120\) 0 0
\(121\) −87.3837 −0.722179
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 65.6767i − 0.525414i
\(126\) 0 0
\(127\) − 208.236i − 1.63965i −0.572614 0.819825i \(-0.694071\pi\)
0.572614 0.819825i \(-0.305929\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 220.549 1.68358 0.841791 0.539804i \(-0.181502\pi\)
0.841791 + 0.539804i \(0.181502\pi\)
\(132\) 0 0
\(133\) − 32.5231i − 0.244535i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 15.2664 0.111433 0.0557167 0.998447i \(-0.482256\pi\)
0.0557167 + 0.998447i \(0.482256\pi\)
\(138\) 0 0
\(139\) 86.7117 0.623825 0.311912 0.950111i \(-0.399030\pi\)
0.311912 + 0.950111i \(0.399030\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 94.9881i − 0.664252i
\(144\) 0 0
\(145\) 44.7148 0.308378
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 146.849i 0.985561i 0.870154 + 0.492780i \(0.164019\pi\)
−0.870154 + 0.492780i \(0.835981\pi\)
\(150\) 0 0
\(151\) − 195.933i − 1.29757i −0.760972 0.648785i \(-0.775277\pi\)
0.760972 0.648785i \(-0.224723\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −1.38938 −0.00896374
\(156\) 0 0
\(157\) − 4.65454i − 0.0296468i −0.999890 0.0148234i \(-0.995281\pi\)
0.999890 0.0148234i \(-0.00471860\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 31.1918 0.193738
\(162\) 0 0
\(163\) 59.5489 0.365331 0.182665 0.983175i \(-0.441528\pi\)
0.182665 + 0.983175i \(0.441528\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 209.012i − 1.25157i −0.779996 0.625785i \(-0.784779\pi\)
0.779996 0.625785i \(-0.215221\pi\)
\(168\) 0 0
\(169\) −99.4032 −0.588185
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 96.7635i − 0.559326i −0.960098 0.279663i \(-0.909777\pi\)
0.960098 0.279663i \(-0.0902228\pi\)
\(174\) 0 0
\(175\) − 28.7411i − 0.164235i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 49.5039 0.276558 0.138279 0.990393i \(-0.455843\pi\)
0.138279 + 0.990393i \(0.455843\pi\)
\(180\) 0 0
\(181\) 141.417i 0.781310i 0.920537 + 0.390655i \(0.127752\pi\)
−0.920537 + 0.390655i \(0.872248\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 20.4578 0.110582
\(186\) 0 0
\(187\) 29.0528 0.155362
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 116.994i − 0.612533i −0.951946 0.306267i \(-0.900920\pi\)
0.951946 0.306267i \(-0.0990799\pi\)
\(192\) 0 0
\(193\) −90.7357 −0.470133 −0.235067 0.971979i \(-0.575531\pi\)
−0.235067 + 0.971979i \(0.575531\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 380.039i − 1.92913i −0.263840 0.964566i \(-0.584989\pi\)
0.263840 0.964566i \(-0.415011\pi\)
\(198\) 0 0
\(199\) 77.6563i 0.390233i 0.980780 + 0.195116i \(0.0625084\pi\)
−0.980780 + 0.195116i \(0.937492\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 40.7098 0.200541
\(204\) 0 0
\(205\) 98.9425i 0.482646i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 151.811 0.726367
\(210\) 0 0
\(211\) 191.446 0.907325 0.453662 0.891174i \(-0.350117\pi\)
0.453662 + 0.891174i \(0.350117\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 45.6943i 0.212531i
\(216\) 0 0
\(217\) −1.26493 −0.00582919
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 82.0930i − 0.371462i
\(222\) 0 0
\(223\) − 168.451i − 0.755387i −0.925931 0.377693i \(-0.876717\pi\)
0.925931 0.377693i \(-0.123283\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −113.516 −0.500071 −0.250036 0.968237i \(-0.580442\pi\)
−0.250036 + 0.968237i \(0.580442\pi\)
\(228\) 0 0
\(229\) 117.618i 0.513615i 0.966463 + 0.256808i \(0.0826707\pi\)
−0.966463 + 0.256808i \(0.917329\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 277.085 1.18921 0.594604 0.804019i \(-0.297309\pi\)
0.594604 + 0.804019i \(0.297309\pi\)
\(234\) 0 0
\(235\) −90.8150 −0.386447
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 343.072i − 1.43545i −0.696327 0.717724i \(-0.745184\pi\)
0.696327 0.717724i \(-0.254816\pi\)
\(240\) 0 0
\(241\) −328.140 −1.36157 −0.680787 0.732481i \(-0.738362\pi\)
−0.680787 + 0.732481i \(0.738362\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 64.7470i − 0.264273i
\(246\) 0 0
\(247\) − 428.964i − 1.73670i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 452.914 1.80444 0.902219 0.431279i \(-0.141937\pi\)
0.902219 + 0.431279i \(0.141937\pi\)
\(252\) 0 0
\(253\) 145.596i 0.575480i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 346.830 1.34953 0.674767 0.738031i \(-0.264244\pi\)
0.674767 + 0.738031i \(0.264244\pi\)
\(258\) 0 0
\(259\) 18.6254 0.0719127
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 402.440i − 1.53019i −0.643917 0.765095i \(-0.722692\pi\)
0.643917 0.765095i \(-0.277308\pi\)
\(264\) 0 0
\(265\) 74.5622 0.281367
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 321.562i 1.19540i 0.801721 + 0.597699i \(0.203918\pi\)
−0.801721 + 0.597699i \(0.796082\pi\)
\(270\) 0 0
\(271\) 456.902i 1.68599i 0.537924 + 0.842993i \(0.319209\pi\)
−0.537924 + 0.842993i \(0.680791\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 134.157 0.487843
\(276\) 0 0
\(277\) 329.543i 1.18969i 0.803842 + 0.594843i \(0.202786\pi\)
−0.803842 + 0.594843i \(0.797214\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −175.064 −0.623005 −0.311503 0.950245i \(-0.600832\pi\)
−0.311503 + 0.950245i \(0.600832\pi\)
\(282\) 0 0
\(283\) 150.298 0.531087 0.265544 0.964099i \(-0.414449\pi\)
0.265544 + 0.964099i \(0.414449\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 90.0804i 0.313869i
\(288\) 0 0
\(289\) −263.891 −0.913119
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 160.435i − 0.547561i −0.961792 0.273781i \(-0.911726\pi\)
0.961792 0.273781i \(-0.0882742\pi\)
\(294\) 0 0
\(295\) − 28.0887i − 0.0952159i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 411.405 1.37594
\(300\) 0 0
\(301\) 41.6015i 0.138211i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −151.476 −0.496642
\(306\) 0 0
\(307\) −168.120 −0.547621 −0.273811 0.961784i \(-0.588284\pi\)
−0.273811 + 0.961784i \(0.588284\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 470.376i 1.51246i 0.654305 + 0.756231i \(0.272961\pi\)
−0.654305 + 0.756231i \(0.727039\pi\)
\(312\) 0 0
\(313\) −19.4378 −0.0621016 −0.0310508 0.999518i \(-0.509885\pi\)
−0.0310508 + 0.999518i \(0.509885\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 242.195i − 0.764021i −0.924158 0.382011i \(-0.875232\pi\)
0.924158 0.382011i \(-0.124768\pi\)
\(318\) 0 0
\(319\) 190.024i 0.595686i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 131.202 0.406197
\(324\) 0 0
\(325\) − 379.080i − 1.16640i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −82.6808 −0.251309
\(330\) 0 0
\(331\) 440.951 1.33218 0.666090 0.745872i \(-0.267967\pi\)
0.666090 + 0.745872i \(0.267967\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) − 83.1612i − 0.248242i
\(336\) 0 0
\(337\) −250.841 −0.744335 −0.372167 0.928166i \(-0.621385\pi\)
−0.372167 + 0.928166i \(0.621385\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 5.90443i − 0.0173150i
\(342\) 0 0
\(343\) − 119.812i − 0.349306i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −16.1029 −0.0464060 −0.0232030 0.999731i \(-0.507386\pi\)
−0.0232030 + 0.999731i \(0.507386\pi\)
\(348\) 0 0
\(349\) 274.843i 0.787516i 0.919214 + 0.393758i \(0.128825\pi\)
−0.919214 + 0.393758i \(0.871175\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −165.428 −0.468634 −0.234317 0.972160i \(-0.575285\pi\)
−0.234317 + 0.972160i \(0.575285\pi\)
\(354\) 0 0
\(355\) −109.770 −0.309212
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 688.519i − 1.91788i −0.283607 0.958941i \(-0.591531\pi\)
0.283607 0.958941i \(-0.408469\pi\)
\(360\) 0 0
\(361\) 324.574 0.899096
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 41.0015i 0.112333i
\(366\) 0 0
\(367\) 102.170i 0.278393i 0.990265 + 0.139196i \(0.0444519\pi\)
−0.990265 + 0.139196i \(0.955548\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 67.8838 0.182975
\(372\) 0 0
\(373\) − 294.317i − 0.789052i −0.918885 0.394526i \(-0.870909\pi\)
0.918885 0.394526i \(-0.129091\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 536.942 1.42425
\(378\) 0 0
\(379\) 81.1923 0.214228 0.107114 0.994247i \(-0.465839\pi\)
0.107114 + 0.994247i \(0.465839\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 198.838i 0.519160i 0.965722 + 0.259580i \(0.0835841\pi\)
−0.965722 + 0.259580i \(0.916416\pi\)
\(384\) 0 0
\(385\) −9.82559 −0.0255210
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 368.767i − 0.947987i −0.880528 0.473993i \(-0.842812\pi\)
0.880528 0.473993i \(-0.157188\pi\)
\(390\) 0 0
\(391\) 125.831i 0.321819i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −110.434 −0.279580
\(396\) 0 0
\(397\) 114.315i 0.287947i 0.989582 + 0.143973i \(0.0459880\pi\)
−0.989582 + 0.143973i \(0.954012\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −39.9083 −0.0995218 −0.0497609 0.998761i \(-0.515846\pi\)
−0.0497609 + 0.998761i \(0.515846\pi\)
\(402\) 0 0
\(403\) −16.6839 −0.0413992
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 86.9391i 0.213610i
\(408\) 0 0
\(409\) 269.868 0.659825 0.329912 0.944012i \(-0.392981\pi\)
0.329912 + 0.944012i \(0.392981\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 25.5728i − 0.0619197i
\(414\) 0 0
\(415\) − 155.476i − 0.374641i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −20.3559 −0.0485821 −0.0242910 0.999705i \(-0.507733\pi\)
−0.0242910 + 0.999705i \(0.507733\pi\)
\(420\) 0 0
\(421\) − 557.905i − 1.32519i −0.748978 0.662595i \(-0.769455\pi\)
0.748978 0.662595i \(-0.230545\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 115.944 0.272810
\(426\) 0 0
\(427\) −137.908 −0.322970
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 376.569i 0.873710i 0.899532 + 0.436855i \(0.143908\pi\)
−0.899532 + 0.436855i \(0.856092\pi\)
\(432\) 0 0
\(433\) 602.876 1.39232 0.696162 0.717885i \(-0.254890\pi\)
0.696162 + 0.717885i \(0.254890\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 657.510i 1.50460i
\(438\) 0 0
\(439\) − 381.087i − 0.868080i −0.900894 0.434040i \(-0.857088\pi\)
0.900894 0.434040i \(-0.142912\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −599.838 −1.35404 −0.677018 0.735966i \(-0.736728\pi\)
−0.677018 + 0.735966i \(0.736728\pi\)
\(444\) 0 0
\(445\) 28.7377i 0.0645791i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −814.240 −1.81345 −0.906726 0.421720i \(-0.861427\pi\)
−0.906726 + 0.421720i \(0.861427\pi\)
\(450\) 0 0
\(451\) −420.475 −0.932317
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 27.7637i 0.0610191i
\(456\) 0 0
\(457\) −111.281 −0.243502 −0.121751 0.992561i \(-0.538851\pi\)
−0.121751 + 0.992561i \(0.538851\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 507.833i 1.10159i 0.834641 + 0.550795i \(0.185675\pi\)
−0.834641 + 0.550795i \(0.814325\pi\)
\(462\) 0 0
\(463\) − 397.302i − 0.858103i −0.903280 0.429052i \(-0.858848\pi\)
0.903280 0.429052i \(-0.141152\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −830.195 −1.77772 −0.888860 0.458179i \(-0.848502\pi\)
−0.888860 + 0.458179i \(0.848502\pi\)
\(468\) 0 0
\(469\) − 75.7126i − 0.161434i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −194.186 −0.410542
\(474\) 0 0
\(475\) 605.849 1.27547
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 146.251i 0.305325i 0.988278 + 0.152662i \(0.0487847\pi\)
−0.988278 + 0.152662i \(0.951215\pi\)
\(480\) 0 0
\(481\) 245.660 0.510727
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 219.103i − 0.451759i
\(486\) 0 0
\(487\) 177.070i 0.363593i 0.983336 + 0.181797i \(0.0581913\pi\)
−0.983336 + 0.181797i \(0.941809\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −94.9463 −0.193373 −0.0966866 0.995315i \(-0.530824\pi\)
−0.0966866 + 0.995315i \(0.530824\pi\)
\(492\) 0 0
\(493\) 164.227i 0.333118i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −99.9384 −0.201083
\(498\) 0 0
\(499\) −744.720 −1.49243 −0.746213 0.665707i \(-0.768130\pi\)
−0.746213 + 0.665707i \(0.768130\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 578.757i − 1.15061i −0.817939 0.575305i \(-0.804883\pi\)
0.817939 0.575305i \(-0.195117\pi\)
\(504\) 0 0
\(505\) 104.055 0.206049
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 323.101i − 0.634777i −0.948296 0.317388i \(-0.897194\pi\)
0.948296 0.317388i \(-0.102806\pi\)
\(510\) 0 0
\(511\) 37.3290i 0.0730509i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 249.349 0.484172
\(516\) 0 0
\(517\) − 385.935i − 0.746490i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −582.929 −1.11887 −0.559433 0.828875i \(-0.688981\pi\)
−0.559433 + 0.828875i \(0.688981\pi\)
\(522\) 0 0
\(523\) 227.111 0.434247 0.217124 0.976144i \(-0.430332\pi\)
0.217124 + 0.976144i \(0.430332\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 5.10288i − 0.00968288i
\(528\) 0 0
\(529\) −101.596 −0.192053
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 1188.12i 2.22911i
\(534\) 0 0
\(535\) − 43.4582i − 0.0812303i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 275.154 0.510491
\(540\) 0 0
\(541\) − 551.391i − 1.01921i −0.860409 0.509603i \(-0.829792\pi\)
0.860409 0.509603i \(-0.170208\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −15.4564 −0.0283603
\(546\) 0 0
\(547\) 745.659 1.36318 0.681590 0.731735i \(-0.261289\pi\)
0.681590 + 0.731735i \(0.261289\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 858.144i 1.55743i
\(552\) 0 0
\(553\) −100.543 −0.181813
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 755.207i 1.35585i 0.735132 + 0.677924i \(0.237120\pi\)
−0.735132 + 0.677924i \(0.762880\pi\)
\(558\) 0 0
\(559\) 548.703i 0.981580i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 699.309 1.24211 0.621056 0.783766i \(-0.286704\pi\)
0.621056 + 0.783766i \(0.286704\pi\)
\(564\) 0 0
\(565\) 68.1600i 0.120637i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 35.8709 0.0630419 0.0315210 0.999503i \(-0.489965\pi\)
0.0315210 + 0.999503i \(0.489965\pi\)
\(570\) 0 0
\(571\) −828.429 −1.45084 −0.725420 0.688307i \(-0.758354\pi\)
−0.725420 + 0.688307i \(0.758354\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 581.049i 1.01052i
\(576\) 0 0
\(577\) −471.333 −0.816867 −0.408434 0.912788i \(-0.633925\pi\)
−0.408434 + 0.912788i \(0.633925\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 141.550i − 0.243632i
\(582\) 0 0
\(583\) 316.866i 0.543510i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −645.149 −1.09906 −0.549531 0.835473i \(-0.685193\pi\)
−0.549531 + 0.835473i \(0.685193\pi\)
\(588\) 0 0
\(589\) − 26.6643i − 0.0452704i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −203.619 −0.343370 −0.171685 0.985152i \(-0.554921\pi\)
−0.171685 + 0.985152i \(0.554921\pi\)
\(594\) 0 0
\(595\) −8.49172 −0.0142718
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 603.605i 1.00769i 0.863795 + 0.503844i \(0.168081\pi\)
−0.863795 + 0.503844i \(0.831919\pi\)
\(600\) 0 0
\(601\) 626.271 1.04205 0.521024 0.853542i \(-0.325550\pi\)
0.521024 + 0.853542i \(0.325550\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 119.220i 0.197057i
\(606\) 0 0
\(607\) 421.012i 0.693595i 0.937940 + 0.346797i \(0.112731\pi\)
−0.937940 + 0.346797i \(0.887269\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −1090.52 −1.78481
\(612\) 0 0
\(613\) 12.9743i 0.0211652i 0.999944 + 0.0105826i \(0.00336861\pi\)
−0.999944 + 0.0105826i \(0.996631\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −423.164 −0.685842 −0.342921 0.939364i \(-0.611416\pi\)
−0.342921 + 0.939364i \(0.611416\pi\)
\(618\) 0 0
\(619\) −625.820 −1.01102 −0.505509 0.862821i \(-0.668695\pi\)
−0.505509 + 0.862821i \(0.668695\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 26.1637i 0.0419963i
\(624\) 0 0
\(625\) 488.861 0.782178
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 75.1367i 0.119454i
\(630\) 0 0
\(631\) 690.848i 1.09485i 0.836856 + 0.547423i \(0.184391\pi\)
−0.836856 + 0.547423i \(0.815609\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −284.101 −0.447403
\(636\) 0 0
\(637\) − 777.491i − 1.22055i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 369.160 0.575913 0.287957 0.957643i \(-0.407024\pi\)
0.287957 + 0.957643i \(0.407024\pi\)
\(642\) 0 0
\(643\) −666.030 −1.03582 −0.517909 0.855436i \(-0.673289\pi\)
−0.517909 + 0.855436i \(0.673289\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 651.886i 1.00755i 0.863835 + 0.503776i \(0.168056\pi\)
−0.863835 + 0.503776i \(0.831944\pi\)
\(648\) 0 0
\(649\) 119.368 0.183926
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 903.324i − 1.38334i −0.722212 0.691672i \(-0.756874\pi\)
0.722212 0.691672i \(-0.243126\pi\)
\(654\) 0 0
\(655\) − 300.901i − 0.459391i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −643.621 −0.976664 −0.488332 0.872658i \(-0.662394\pi\)
−0.488332 + 0.872658i \(0.662394\pi\)
\(660\) 0 0
\(661\) 860.187i 1.30134i 0.759360 + 0.650671i \(0.225512\pi\)
−0.759360 + 0.650671i \(0.774488\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −44.3722 −0.0667250
\(666\) 0 0
\(667\) −823.017 −1.23391
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) − 643.725i − 0.959351i
\(672\) 0 0
\(673\) 866.535 1.28757 0.643785 0.765206i \(-0.277363\pi\)
0.643785 + 0.765206i \(0.277363\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 1307.26i − 1.93095i −0.260489 0.965477i \(-0.583884\pi\)
0.260489 0.965477i \(-0.416116\pi\)
\(678\) 0 0
\(679\) − 199.478i − 0.293783i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 783.569 1.14725 0.573623 0.819120i \(-0.305538\pi\)
0.573623 + 0.819120i \(0.305538\pi\)
\(684\) 0 0
\(685\) − 20.8283i − 0.0304063i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 895.354 1.29950
\(690\) 0 0
\(691\) 1014.95 1.46882 0.734408 0.678708i \(-0.237460\pi\)
0.734408 + 0.678708i \(0.237460\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 118.303i − 0.170220i
\(696\) 0 0
\(697\) −363.394 −0.521368
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 957.527i 1.36595i 0.730444 + 0.682973i \(0.239313\pi\)
−0.730444 + 0.682973i \(0.760687\pi\)
\(702\) 0 0
\(703\) 392.615i 0.558485i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 94.7346 0.133995
\(708\) 0 0
\(709\) 65.7503i 0.0927366i 0.998924 + 0.0463683i \(0.0147648\pi\)
−0.998924 + 0.0463683i \(0.985235\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 25.5728 0.0358665
\(714\) 0 0
\(715\) −129.595 −0.181251
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 573.085i 0.797058i 0.917156 + 0.398529i \(0.130479\pi\)
−0.917156 + 0.398529i \(0.869521\pi\)
\(720\) 0 0
\(721\) 227.015 0.314861
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 758.352i 1.04600i
\(726\) 0 0
\(727\) 249.632i 0.343373i 0.985152 + 0.171686i \(0.0549216\pi\)
−0.985152 + 0.171686i \(0.945078\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −167.825 −0.229582
\(732\) 0 0
\(733\) − 662.187i − 0.903393i −0.892172 0.451696i \(-0.850819\pi\)
0.892172 0.451696i \(-0.149181\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 353.409 0.479524
\(738\) 0 0
\(739\) 98.7372 0.133609 0.0668046 0.997766i \(-0.478720\pi\)
0.0668046 + 0.997766i \(0.478720\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 906.520i 1.22008i 0.792370 + 0.610041i \(0.208847\pi\)
−0.792370 + 0.610041i \(0.791153\pi\)
\(744\) 0 0
\(745\) 200.349 0.268925
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 39.5657i − 0.0528248i
\(750\) 0 0
\(751\) − 286.284i − 0.381204i −0.981667 0.190602i \(-0.938956\pi\)
0.981667 0.190602i \(-0.0610440\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −267.316 −0.354062
\(756\) 0 0
\(757\) 1162.42i 1.53556i 0.640712 + 0.767781i \(0.278639\pi\)
−0.640712 + 0.767781i \(0.721361\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −994.905 −1.30737 −0.653683 0.756769i \(-0.726777\pi\)
−0.653683 + 0.756769i \(0.726777\pi\)
\(762\) 0 0
\(763\) −14.0720 −0.0184429
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 337.293i − 0.439756i
\(768\) 0 0
\(769\) −614.473 −0.799055 −0.399527 0.916721i \(-0.630826\pi\)
−0.399527 + 0.916721i \(0.630826\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 927.633i − 1.20004i −0.799984 0.600021i \(-0.795159\pi\)
0.799984 0.600021i \(-0.204841\pi\)
\(774\) 0 0
\(775\) − 23.5635i − 0.0304045i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −1898.86 −2.43756
\(780\) 0 0
\(781\) − 466.490i − 0.597298i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −6.35031 −0.00808957
\(786\) 0 0
\(787\) 781.920 0.993545 0.496772 0.867881i \(-0.334518\pi\)
0.496772 + 0.867881i \(0.334518\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 62.0550i 0.0784513i
\(792\) 0 0
\(793\) −1818.94 −2.29375
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 1160.22i − 1.45574i −0.685718 0.727868i \(-0.740511\pi\)
0.685718 0.727868i \(-0.259489\pi\)
\(798\) 0 0
\(799\) − 333.543i − 0.417450i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −174.243 −0.216991
\(804\) 0 0
\(805\) − 42.5558i − 0.0528644i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1512.26 1.86930 0.934651 0.355568i \(-0.115712\pi\)
0.934651 + 0.355568i \(0.115712\pi\)
\(810\) 0 0
\(811\) −1586.92 −1.95674 −0.978371 0.206858i \(-0.933676\pi\)
−0.978371 + 0.206858i \(0.933676\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) − 81.2441i − 0.0996860i
\(816\) 0 0
\(817\) −876.942 −1.07337
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1118.96i 1.36292i 0.731857 + 0.681459i \(0.238654\pi\)
−0.731857 + 0.681459i \(0.761346\pi\)
\(822\) 0 0
\(823\) 1628.26i 1.97844i 0.146438 + 0.989220i \(0.453219\pi\)
−0.146438 + 0.989220i \(0.546781\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −421.552 −0.509736 −0.254868 0.966976i \(-0.582032\pi\)
−0.254868 + 0.966976i \(0.582032\pi\)
\(828\) 0 0
\(829\) 475.263i 0.573297i 0.958036 + 0.286649i \(0.0925412\pi\)
−0.958036 + 0.286649i \(0.907459\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 237.801 0.285475
\(834\) 0 0
\(835\) −285.161 −0.341510
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 653.590i 0.779010i 0.921024 + 0.389505i \(0.127354\pi\)
−0.921024 + 0.389505i \(0.872646\pi\)
\(840\) 0 0
\(841\) −233.154 −0.277234
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 135.618i 0.160495i
\(846\) 0 0
\(847\) 108.541i 0.128148i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −376.544 −0.442472
\(852\) 0 0
\(853\) 140.493i 0.164705i 0.996603 + 0.0823523i \(0.0262433\pi\)
−0.996603 + 0.0823523i \(0.973757\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −562.796 −0.656704 −0.328352 0.944555i \(-0.606493\pi\)
−0.328352 + 0.944555i \(0.606493\pi\)
\(858\) 0 0
\(859\) 228.316 0.265792 0.132896 0.991130i \(-0.457572\pi\)
0.132896 + 0.991130i \(0.457572\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 892.187i 1.03382i 0.856040 + 0.516910i \(0.172918\pi\)
−0.856040 + 0.516910i \(0.827082\pi\)
\(864\) 0 0
\(865\) −132.017 −0.152621
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 469.310i − 0.540058i
\(870\) 0 0
\(871\) − 998.611i − 1.14651i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −81.5787 −0.0932328
\(876\) 0 0
\(877\) 1406.66i 1.60395i 0.597359 + 0.801974i \(0.296217\pi\)
−0.597359 + 0.801974i \(0.703783\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −943.043 −1.07042 −0.535212 0.844718i \(-0.679768\pi\)
−0.535212 + 0.844718i \(0.679768\pi\)
\(882\) 0 0
\(883\) −1146.63 −1.29856 −0.649280 0.760549i \(-0.724930\pi\)
−0.649280 + 0.760549i \(0.724930\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 894.171i 1.00808i 0.863679 + 0.504042i \(0.168154\pi\)
−0.863679 + 0.504042i \(0.831846\pi\)
\(888\) 0 0
\(889\) −258.655 −0.290950
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 1742.88i − 1.95171i
\(894\) 0 0
\(895\) − 67.5395i − 0.0754631i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 33.3761 0.0371258
\(900\) 0 0
\(901\) 273.850i 0.303940i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 192.939 0.213192
\(906\) 0 0
\(907\) 1358.60 1.49790 0.748950 0.662626i \(-0.230558\pi\)
0.748950 + 0.662626i \(0.230558\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 804.510i 0.883106i 0.897235 + 0.441553i \(0.145572\pi\)
−0.897235 + 0.441553i \(0.854428\pi\)
\(912\) 0 0
\(913\) 660.726 0.723686
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 273.950i − 0.298745i
\(918\) 0 0
\(919\) 1704.73i 1.85498i 0.373849 + 0.927490i \(0.378038\pi\)
−0.373849 + 0.927490i \(0.621962\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −1318.14 −1.42810
\(924\) 0 0
\(925\) 346.958i 0.375090i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1351.05 1.45431 0.727154 0.686475i \(-0.240843\pi\)
0.727154 + 0.686475i \(0.240843\pi\)
\(930\) 0 0
\(931\) 1242.59 1.33469
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) − 39.6374i − 0.0423930i
\(936\) 0 0
\(937\) −672.646 −0.717872 −0.358936 0.933362i \(-0.616860\pi\)
−0.358936 + 0.933362i \(0.616860\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 528.671i 0.561818i 0.959734 + 0.280909i \(0.0906359\pi\)
−0.959734 + 0.280909i \(0.909364\pi\)
\(942\) 0 0
\(943\) − 1821.13i − 1.93121i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 661.066 0.698063 0.349032 0.937111i \(-0.386511\pi\)
0.349032 + 0.937111i \(0.386511\pi\)
\(948\) 0 0
\(949\) 492.351i 0.518811i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1545.41 1.62163 0.810815 0.585303i \(-0.199024\pi\)
0.810815 + 0.585303i \(0.199024\pi\)
\(954\) 0 0
\(955\) −159.618 −0.167139
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 18.9627i − 0.0197735i
\(960\) 0 0
\(961\) 959.963 0.998921
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 123.793i 0.128283i
\(966\) 0 0
\(967\) − 161.279i − 0.166782i −0.996517 0.0833912i \(-0.973425\pi\)
0.996517 0.0833912i \(-0.0265751\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −4.20412 −0.00432969 −0.00216484 0.999998i \(-0.500689\pi\)
−0.00216484 + 0.999998i \(0.500689\pi\)
\(972\) 0 0
\(973\) − 107.707i − 0.110696i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1348.19 1.37993 0.689967 0.723841i \(-0.257625\pi\)
0.689967 + 0.723841i \(0.257625\pi\)
\(978\) 0 0
\(979\) −122.126 −0.124746
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 984.262i − 1.00128i −0.865655 0.500642i \(-0.833097\pi\)
0.865655 0.500642i \(-0.166903\pi\)
\(984\) 0 0
\(985\) −518.497 −0.526393
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 841.045i − 0.850399i
\(990\) 0 0
\(991\) 1013.87i 1.02308i 0.859259 + 0.511541i \(0.170925\pi\)
−0.859259 + 0.511541i \(0.829075\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 105.948 0.106481
\(996\) 0 0
\(997\) 311.310i 0.312246i 0.987738 + 0.156123i \(0.0498997\pi\)
−0.987738 + 0.156123i \(0.950100\pi\)
\(998\) 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.q.127.4 8
3.2 odd 2 768.3.b.f.127.7 8
4.3 odd 2 2304.3.b.t.127.4 8
8.3 odd 2 inner 2304.3.b.q.127.5 8
8.5 even 2 2304.3.b.t.127.5 8
12.11 even 2 768.3.b.e.127.3 8
16.3 odd 4 1152.3.g.c.127.5 8
16.5 even 4 1152.3.g.f.127.4 8
16.11 odd 4 1152.3.g.f.127.3 8
16.13 even 4 1152.3.g.c.127.6 8
24.5 odd 2 768.3.b.e.127.2 8
24.11 even 2 768.3.b.f.127.6 8
48.5 odd 4 384.3.g.a.127.3 8
48.11 even 4 384.3.g.a.127.7 yes 8
48.29 odd 4 384.3.g.b.127.6 yes 8
48.35 even 4 384.3.g.b.127.2 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.3.g.a.127.3 8 48.5 odd 4
384.3.g.a.127.7 yes 8 48.11 even 4
384.3.g.b.127.2 yes 8 48.35 even 4
384.3.g.b.127.6 yes 8 48.29 odd 4
768.3.b.e.127.2 8 24.5 odd 2
768.3.b.e.127.3 8 12.11 even 2
768.3.b.f.127.6 8 24.11 even 2
768.3.b.f.127.7 8 3.2 odd 2
1152.3.g.c.127.5 8 16.3 odd 4
1152.3.g.c.127.6 8 16.13 even 4
1152.3.g.f.127.3 8 16.11 odd 4
1152.3.g.f.127.4 8 16.5 even 4
2304.3.b.q.127.4 8 1.1 even 1 trivial
2304.3.b.q.127.5 8 8.3 odd 2 inner
2304.3.b.t.127.4 8 4.3 odd 2
2304.3.b.t.127.5 8 8.5 even 2