Properties

Label 2704.2.f.o.337.4
Level $2704$
Weight $2$
Character 2704.337
Analytic conductor $21.592$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2704,2,Mod(337,2704)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2704, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2704.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2704 = 2^{4} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2704.f (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: \(21.5915487066\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.153664.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 5x^{4} + 6x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 169)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.4
Root \(1.80194i\) of defining polynomial
Character \(\chi\) \(=\) 2704.337
Dual form 2704.2.f.o.337.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.554958 q^{3} +1.44504i q^{5} -2.04892i q^{7} -2.69202 q^{9} +O(q^{10})\) \(q+0.554958 q^{3} +1.44504i q^{5} -2.04892i q^{7} -2.69202 q^{9} +2.55496i q^{11} +0.801938i q^{15} +5.29590 q^{17} -5.85086i q^{19} -1.13706i q^{21} -1.89008 q^{23} +2.91185 q^{25} -3.15883 q^{27} +2.26875 q^{29} -4.26875i q^{31} +1.41789i q^{33} +2.96077 q^{35} +5.35690i q^{37} -1.27413i q^{41} +6.13706 q^{43} -3.89008i q^{45} +2.95108i q^{47} +2.80194 q^{49} +2.93900 q^{51} +5.52111 q^{53} -3.69202 q^{55} -3.24698i q^{57} +12.2078i q^{59} +8.56465 q^{61} +5.51573i q^{63} +0.576728i q^{67} -1.04892 q^{69} -4.59419i q^{71} -10.5526i q^{73} +1.61596 q^{75} +5.23490 q^{77} +15.7778 q^{79} +6.32304 q^{81} +7.72348i q^{83} +7.65279i q^{85} +1.25906 q^{87} +6.61356i q^{89} -2.36898i q^{93} +8.45473 q^{95} -11.9269i q^{97} -6.87800i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 4 q^{3} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 4 q^{3} - 6 q^{9} + 4 q^{17} - 10 q^{23} + 10 q^{25} - 2 q^{27} - 2 q^{29} - 8 q^{35} + 26 q^{43} + 8 q^{49} - 2 q^{51} + 2 q^{53} - 12 q^{55} + 8 q^{61} + 12 q^{69} + 30 q^{75} - 16 q^{77} + 10 q^{79} - 2 q^{81} + 36 q^{87} + 6 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2704\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(1185\) \(2367\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 0.554958 0.320405 0.160203 0.987084i \(-0.448785\pi\)
0.160203 + 0.987084i \(0.448785\pi\)
\(4\) 0 0
\(5\) 1.44504i 0.646242i 0.946358 + 0.323121i \(0.104732\pi\)
−0.946358 + 0.323121i \(0.895268\pi\)
\(6\) 0 0
\(7\) − 2.04892i − 0.774418i −0.921992 0.387209i \(-0.873439\pi\)
0.921992 0.387209i \(-0.126561\pi\)
\(8\) 0 0
\(9\) −2.69202 −0.897340
\(10\) 0 0
\(11\) 2.55496i 0.770349i 0.922844 + 0.385174i \(0.125859\pi\)
−0.922844 + 0.385174i \(0.874141\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) 0.801938i 0.207059i
\(16\) 0 0
\(17\) 5.29590 1.28444 0.642222 0.766519i \(-0.278013\pi\)
0.642222 + 0.766519i \(0.278013\pi\)
\(18\) 0 0
\(19\) − 5.85086i − 1.34228i −0.741331 0.671139i \(-0.765805\pi\)
0.741331 0.671139i \(-0.234195\pi\)
\(20\) 0 0
\(21\) − 1.13706i − 0.248128i
\(22\) 0 0
\(23\) −1.89008 −0.394110 −0.197055 0.980392i \(-0.563138\pi\)
−0.197055 + 0.980392i \(0.563138\pi\)
\(24\) 0 0
\(25\) 2.91185 0.582371
\(26\) 0 0
\(27\) −3.15883 −0.607918
\(28\) 0 0
\(29\) 2.26875 0.421296 0.210648 0.977562i \(-0.432443\pi\)
0.210648 + 0.977562i \(0.432443\pi\)
\(30\) 0 0
\(31\) − 4.26875i − 0.766690i −0.923605 0.383345i \(-0.874772\pi\)
0.923605 0.383345i \(-0.125228\pi\)
\(32\) 0 0
\(33\) 1.41789i 0.246824i
\(34\) 0 0
\(35\) 2.96077 0.500462
\(36\) 0 0
\(37\) 5.35690i 0.880668i 0.897834 + 0.440334i \(0.145140\pi\)
−0.897834 + 0.440334i \(0.854860\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 1.27413i − 0.198985i −0.995038 0.0994926i \(-0.968278\pi\)
0.995038 0.0994926i \(-0.0317220\pi\)
\(42\) 0 0
\(43\) 6.13706 0.935893 0.467947 0.883757i \(-0.344994\pi\)
0.467947 + 0.883757i \(0.344994\pi\)
\(44\) 0 0
\(45\) − 3.89008i − 0.579899i
\(46\) 0 0
\(47\) 2.95108i 0.430460i 0.976563 + 0.215230i \(0.0690501\pi\)
−0.976563 + 0.215230i \(0.930950\pi\)
\(48\) 0 0
\(49\) 2.80194 0.400277
\(50\) 0 0
\(51\) 2.93900 0.411542
\(52\) 0 0
\(53\) 5.52111 0.758382 0.379191 0.925318i \(-0.376202\pi\)
0.379191 + 0.925318i \(0.376202\pi\)
\(54\) 0 0
\(55\) −3.69202 −0.497832
\(56\) 0 0
\(57\) − 3.24698i − 0.430073i
\(58\) 0 0
\(59\) 12.2078i 1.58931i 0.607059 + 0.794657i \(0.292349\pi\)
−0.607059 + 0.794657i \(0.707651\pi\)
\(60\) 0 0
\(61\) 8.56465 1.09659 0.548295 0.836285i \(-0.315277\pi\)
0.548295 + 0.836285i \(0.315277\pi\)
\(62\) 0 0
\(63\) 5.51573i 0.694917i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0.576728i 0.0704586i 0.999379 + 0.0352293i \(0.0112162\pi\)
−0.999379 + 0.0352293i \(0.988784\pi\)
\(68\) 0 0
\(69\) −1.04892 −0.126275
\(70\) 0 0
\(71\) − 4.59419i − 0.545230i −0.962123 0.272615i \(-0.912112\pi\)
0.962123 0.272615i \(-0.0878885\pi\)
\(72\) 0 0
\(73\) − 10.5526i − 1.23508i −0.786538 0.617542i \(-0.788128\pi\)
0.786538 0.617542i \(-0.211872\pi\)
\(74\) 0 0
\(75\) 1.61596 0.186595
\(76\) 0 0
\(77\) 5.23490 0.596572
\(78\) 0 0
\(79\) 15.7778 1.77514 0.887569 0.460674i \(-0.152392\pi\)
0.887569 + 0.460674i \(0.152392\pi\)
\(80\) 0 0
\(81\) 6.32304 0.702560
\(82\) 0 0
\(83\) 7.72348i 0.847762i 0.905718 + 0.423881i \(0.139333\pi\)
−0.905718 + 0.423881i \(0.860667\pi\)
\(84\) 0 0
\(85\) 7.65279i 0.830062i
\(86\) 0 0
\(87\) 1.25906 0.134986
\(88\) 0 0
\(89\) 6.61356i 0.701036i 0.936556 + 0.350518i \(0.113995\pi\)
−0.936556 + 0.350518i \(0.886005\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) − 2.36898i − 0.245652i
\(94\) 0 0
\(95\) 8.45473 0.867437
\(96\) 0 0
\(97\) − 11.9269i − 1.21100i −0.795847 0.605498i \(-0.792974\pi\)
0.795847 0.605498i \(-0.207026\pi\)
\(98\) 0 0
\(99\) − 6.87800i − 0.691265i
\(100\) 0 0
\(101\) −13.0640 −1.29991 −0.649957 0.759971i \(-0.725213\pi\)
−0.649957 + 0.759971i \(0.725213\pi\)
\(102\) 0 0
\(103\) 9.16852 0.903401 0.451701 0.892170i \(-0.350818\pi\)
0.451701 + 0.892170i \(0.350818\pi\)
\(104\) 0 0
\(105\) 1.64310 0.160351
\(106\) 0 0
\(107\) 6.89977 0.667026 0.333513 0.942745i \(-0.391766\pi\)
0.333513 + 0.942745i \(0.391766\pi\)
\(108\) 0 0
\(109\) − 0.121998i − 0.0116853i −0.999983 0.00584264i \(-0.998140\pi\)
0.999983 0.00584264i \(-0.00185978\pi\)
\(110\) 0 0
\(111\) 2.97285i 0.282171i
\(112\) 0 0
\(113\) 7.30798 0.687477 0.343738 0.939065i \(-0.388307\pi\)
0.343738 + 0.939065i \(0.388307\pi\)
\(114\) 0 0
\(115\) − 2.73125i − 0.254690i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 10.8509i − 0.994696i
\(120\) 0 0
\(121\) 4.47219 0.406563
\(122\) 0 0
\(123\) − 0.707087i − 0.0637559i
\(124\) 0 0
\(125\) 11.4330i 1.02260i
\(126\) 0 0
\(127\) −18.9705 −1.68336 −0.841678 0.539980i \(-0.818432\pi\)
−0.841678 + 0.539980i \(0.818432\pi\)
\(128\) 0 0
\(129\) 3.40581 0.299865
\(130\) 0 0
\(131\) −3.25667 −0.284536 −0.142268 0.989828i \(-0.545440\pi\)
−0.142268 + 0.989828i \(0.545440\pi\)
\(132\) 0 0
\(133\) −11.9879 −1.03948
\(134\) 0 0
\(135\) − 4.56465i − 0.392862i
\(136\) 0 0
\(137\) 0.792249i 0.0676864i 0.999427 + 0.0338432i \(0.0107747\pi\)
−0.999427 + 0.0338432i \(0.989225\pi\)
\(138\) 0 0
\(139\) 11.3394 0.961799 0.480899 0.876776i \(-0.340310\pi\)
0.480899 + 0.876776i \(0.340310\pi\)
\(140\) 0 0
\(141\) 1.63773i 0.137922i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 3.27844i 0.272260i
\(146\) 0 0
\(147\) 1.55496 0.128251
\(148\) 0 0
\(149\) − 8.40581i − 0.688631i −0.938854 0.344316i \(-0.888111\pi\)
0.938854 0.344316i \(-0.111889\pi\)
\(150\) 0 0
\(151\) − 14.1293i − 1.14983i −0.818215 0.574913i \(-0.805036\pi\)
0.818215 0.574913i \(-0.194964\pi\)
\(152\) 0 0
\(153\) −14.2567 −1.15258
\(154\) 0 0
\(155\) 6.16852 0.495468
\(156\) 0 0
\(157\) −9.43296 −0.752832 −0.376416 0.926451i \(-0.622844\pi\)
−0.376416 + 0.926451i \(0.622844\pi\)
\(158\) 0 0
\(159\) 3.06398 0.242990
\(160\) 0 0
\(161\) 3.87263i 0.305206i
\(162\) 0 0
\(163\) − 8.70410i − 0.681758i −0.940107 0.340879i \(-0.889275\pi\)
0.940107 0.340879i \(-0.110725\pi\)
\(164\) 0 0
\(165\) −2.04892 −0.159508
\(166\) 0 0
\(167\) 23.8538i 1.84587i 0.384961 + 0.922933i \(0.374215\pi\)
−0.384961 + 0.922933i \(0.625785\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 15.7506i 1.20448i
\(172\) 0 0
\(173\) 18.8552 1.43353 0.716766 0.697314i \(-0.245622\pi\)
0.716766 + 0.697314i \(0.245622\pi\)
\(174\) 0 0
\(175\) − 5.96615i − 0.450998i
\(176\) 0 0
\(177\) 6.77479i 0.509224i
\(178\) 0 0
\(179\) 6.02177 0.450088 0.225044 0.974349i \(-0.427747\pi\)
0.225044 + 0.974349i \(0.427747\pi\)
\(180\) 0 0
\(181\) 4.77777 0.355129 0.177565 0.984109i \(-0.443178\pi\)
0.177565 + 0.984109i \(0.443178\pi\)
\(182\) 0 0
\(183\) 4.75302 0.351353
\(184\) 0 0
\(185\) −7.74094 −0.569125
\(186\) 0 0
\(187\) 13.5308i 0.989470i
\(188\) 0 0
\(189\) 6.47219i 0.470782i
\(190\) 0 0
\(191\) −18.4306 −1.33359 −0.666795 0.745242i \(-0.732334\pi\)
−0.666795 + 0.745242i \(0.732334\pi\)
\(192\) 0 0
\(193\) 6.05429i 0.435798i 0.975971 + 0.217899i \(0.0699203\pi\)
−0.975971 + 0.217899i \(0.930080\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 11.4155i 0.813321i 0.913579 + 0.406660i \(0.133307\pi\)
−0.913579 + 0.406660i \(0.866693\pi\)
\(198\) 0 0
\(199\) −13.9051 −0.985710 −0.492855 0.870111i \(-0.664047\pi\)
−0.492855 + 0.870111i \(0.664047\pi\)
\(200\) 0 0
\(201\) 0.320060i 0.0225753i
\(202\) 0 0
\(203\) − 4.64848i − 0.326259i
\(204\) 0 0
\(205\) 1.84117 0.128593
\(206\) 0 0
\(207\) 5.08815 0.353651
\(208\) 0 0
\(209\) 14.9487 1.03402
\(210\) 0 0
\(211\) 13.2446 0.911795 0.455897 0.890032i \(-0.349318\pi\)
0.455897 + 0.890032i \(0.349318\pi\)
\(212\) 0 0
\(213\) − 2.54958i − 0.174694i
\(214\) 0 0
\(215\) 8.86831i 0.604814i
\(216\) 0 0
\(217\) −8.74632 −0.593739
\(218\) 0 0
\(219\) − 5.85623i − 0.395727i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) − 7.33513i − 0.491196i −0.969372 0.245598i \(-0.921016\pi\)
0.969372 0.245598i \(-0.0789844\pi\)
\(224\) 0 0
\(225\) −7.83877 −0.522585
\(226\) 0 0
\(227\) − 8.67456i − 0.575751i −0.957668 0.287875i \(-0.907051\pi\)
0.957668 0.287875i \(-0.0929489\pi\)
\(228\) 0 0
\(229\) − 13.6866i − 0.904439i −0.891907 0.452219i \(-0.850632\pi\)
0.891907 0.452219i \(-0.149368\pi\)
\(230\) 0 0
\(231\) 2.90515 0.191145
\(232\) 0 0
\(233\) 5.08815 0.333336 0.166668 0.986013i \(-0.446699\pi\)
0.166668 + 0.986013i \(0.446699\pi\)
\(234\) 0 0
\(235\) −4.26444 −0.278181
\(236\) 0 0
\(237\) 8.75600 0.568764
\(238\) 0 0
\(239\) − 10.9239i − 0.706611i −0.935508 0.353305i \(-0.885058\pi\)
0.935508 0.353305i \(-0.114942\pi\)
\(240\) 0 0
\(241\) 11.9148i 0.767502i 0.923437 + 0.383751i \(0.125368\pi\)
−0.923437 + 0.383751i \(0.874632\pi\)
\(242\) 0 0
\(243\) 12.9855 0.833022
\(244\) 0 0
\(245\) 4.04892i 0.258676i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 4.28621i 0.271627i
\(250\) 0 0
\(251\) 22.3478 1.41058 0.705290 0.708919i \(-0.250817\pi\)
0.705290 + 0.708919i \(0.250817\pi\)
\(252\) 0 0
\(253\) − 4.82908i − 0.303602i
\(254\) 0 0
\(255\) 4.24698i 0.265956i
\(256\) 0 0
\(257\) 18.6601 1.16398 0.581992 0.813194i \(-0.302273\pi\)
0.581992 + 0.813194i \(0.302273\pi\)
\(258\) 0 0
\(259\) 10.9758 0.682005
\(260\) 0 0
\(261\) −6.10752 −0.378046
\(262\) 0 0
\(263\) −14.3991 −0.887887 −0.443944 0.896055i \(-0.646421\pi\)
−0.443944 + 0.896055i \(0.646421\pi\)
\(264\) 0 0
\(265\) 7.97823i 0.490099i
\(266\) 0 0
\(267\) 3.67025i 0.224616i
\(268\) 0 0
\(269\) 0.652793 0.0398015 0.0199007 0.999802i \(-0.493665\pi\)
0.0199007 + 0.999802i \(0.493665\pi\)
\(270\) 0 0
\(271\) 1.99569i 0.121229i 0.998161 + 0.0606147i \(0.0193061\pi\)
−0.998161 + 0.0606147i \(0.980694\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 7.43967i 0.448629i
\(276\) 0 0
\(277\) −11.7845 −0.708061 −0.354030 0.935234i \(-0.615189\pi\)
−0.354030 + 0.935234i \(0.615189\pi\)
\(278\) 0 0
\(279\) 11.4916i 0.687982i
\(280\) 0 0
\(281\) − 6.47219i − 0.386098i −0.981189 0.193049i \(-0.938162\pi\)
0.981189 0.193049i \(-0.0618377\pi\)
\(282\) 0 0
\(283\) 6.58104 0.391202 0.195601 0.980684i \(-0.437334\pi\)
0.195601 + 0.980684i \(0.437334\pi\)
\(284\) 0 0
\(285\) 4.69202 0.277931
\(286\) 0 0
\(287\) −2.61058 −0.154098
\(288\) 0 0
\(289\) 11.0465 0.649796
\(290\) 0 0
\(291\) − 6.61894i − 0.388009i
\(292\) 0 0
\(293\) − 24.3381i − 1.42185i −0.703269 0.710924i \(-0.748277\pi\)
0.703269 0.710924i \(-0.251723\pi\)
\(294\) 0 0
\(295\) −17.6407 −1.02708
\(296\) 0 0
\(297\) − 8.07069i − 0.468309i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) − 12.5743i − 0.724773i
\(302\) 0 0
\(303\) −7.24996 −0.416500
\(304\) 0 0
\(305\) 12.3763i 0.708663i
\(306\) 0 0
\(307\) 14.0737i 0.803227i 0.915809 + 0.401613i \(0.131550\pi\)
−0.915809 + 0.401613i \(0.868450\pi\)
\(308\) 0 0
\(309\) 5.08815 0.289455
\(310\) 0 0
\(311\) −29.7700 −1.68810 −0.844051 0.536263i \(-0.819836\pi\)
−0.844051 + 0.536263i \(0.819836\pi\)
\(312\) 0 0
\(313\) −7.47889 −0.422732 −0.211366 0.977407i \(-0.567791\pi\)
−0.211366 + 0.977407i \(0.567791\pi\)
\(314\) 0 0
\(315\) −7.97046 −0.449085
\(316\) 0 0
\(317\) − 30.0301i − 1.68666i −0.537396 0.843330i \(-0.680592\pi\)
0.537396 0.843330i \(-0.319408\pi\)
\(318\) 0 0
\(319\) 5.79656i 0.324545i
\(320\) 0 0
\(321\) 3.82908 0.213719
\(322\) 0 0
\(323\) − 30.9855i − 1.72408i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 0.0677037i − 0.00374402i
\(328\) 0 0
\(329\) 6.04652 0.333356
\(330\) 0 0
\(331\) − 15.7168i − 0.863872i −0.901904 0.431936i \(-0.857831\pi\)
0.901904 0.431936i \(-0.142169\pi\)
\(332\) 0 0
\(333\) − 14.4209i − 0.790259i
\(334\) 0 0
\(335\) −0.833397 −0.0455333
\(336\) 0 0
\(337\) −1.95407 −0.106445 −0.0532224 0.998583i \(-0.516949\pi\)
−0.0532224 + 0.998583i \(0.516949\pi\)
\(338\) 0 0
\(339\) 4.05562 0.220271
\(340\) 0 0
\(341\) 10.9065 0.590619
\(342\) 0 0
\(343\) − 20.0834i − 1.08440i
\(344\) 0 0
\(345\) − 1.51573i − 0.0816041i
\(346\) 0 0
\(347\) 17.1250 0.919317 0.459659 0.888096i \(-0.347972\pi\)
0.459659 + 0.888096i \(0.347972\pi\)
\(348\) 0 0
\(349\) − 10.4668i − 0.560276i −0.959960 0.280138i \(-0.909620\pi\)
0.959960 0.280138i \(-0.0903802\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 15.5308i − 0.826621i −0.910590 0.413310i \(-0.864372\pi\)
0.910590 0.413310i \(-0.135628\pi\)
\(354\) 0 0
\(355\) 6.63879 0.352351
\(356\) 0 0
\(357\) − 6.02177i − 0.318706i
\(358\) 0 0
\(359\) 21.4263i 1.13083i 0.824805 + 0.565417i \(0.191285\pi\)
−0.824805 + 0.565417i \(0.808715\pi\)
\(360\) 0 0
\(361\) −15.2325 −0.801711
\(362\) 0 0
\(363\) 2.48188 0.130265
\(364\) 0 0
\(365\) 15.2489 0.798164
\(366\) 0 0
\(367\) −34.3032 −1.79061 −0.895306 0.445452i \(-0.853043\pi\)
−0.895306 + 0.445452i \(0.853043\pi\)
\(368\) 0 0
\(369\) 3.42998i 0.178557i
\(370\) 0 0
\(371\) − 11.3123i − 0.587305i
\(372\) 0 0
\(373\) −12.5961 −0.652202 −0.326101 0.945335i \(-0.605735\pi\)
−0.326101 + 0.945335i \(0.605735\pi\)
\(374\) 0 0
\(375\) 6.34481i 0.327645i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 16.5386i 0.849529i 0.905304 + 0.424765i \(0.139643\pi\)
−0.905304 + 0.424765i \(0.860357\pi\)
\(380\) 0 0
\(381\) −10.5278 −0.539356
\(382\) 0 0
\(383\) 7.53617i 0.385080i 0.981289 + 0.192540i \(0.0616726\pi\)
−0.981289 + 0.192540i \(0.938327\pi\)
\(384\) 0 0
\(385\) 7.56465i 0.385530i
\(386\) 0 0
\(387\) −16.5211 −0.839815
\(388\) 0 0
\(389\) −35.5555 −1.80274 −0.901369 0.433052i \(-0.857437\pi\)
−0.901369 + 0.433052i \(0.857437\pi\)
\(390\) 0 0
\(391\) −10.0097 −0.506212
\(392\) 0 0
\(393\) −1.80731 −0.0911670
\(394\) 0 0
\(395\) 22.7995i 1.14717i
\(396\) 0 0
\(397\) 1.35152i 0.0678308i 0.999425 + 0.0339154i \(0.0107977\pi\)
−0.999425 + 0.0339154i \(0.989202\pi\)
\(398\) 0 0
\(399\) −6.65279 −0.333056
\(400\) 0 0
\(401\) 0.579121i 0.0289199i 0.999895 + 0.0144600i \(0.00460291\pi\)
−0.999895 + 0.0144600i \(0.995397\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 9.13706i 0.454024i
\(406\) 0 0
\(407\) −13.6866 −0.678422
\(408\) 0 0
\(409\) 15.1575i 0.749490i 0.927128 + 0.374745i \(0.122270\pi\)
−0.927128 + 0.374745i \(0.877730\pi\)
\(410\) 0 0
\(411\) 0.439665i 0.0216871i
\(412\) 0 0
\(413\) 25.0127 1.23079
\(414\) 0 0
\(415\) −11.1608 −0.547860
\(416\) 0 0
\(417\) 6.29291 0.308165
\(418\) 0 0
\(419\) 35.7235 1.74521 0.872603 0.488430i \(-0.162430\pi\)
0.872603 + 0.488430i \(0.162430\pi\)
\(420\) 0 0
\(421\) 35.0465i 1.70806i 0.520221 + 0.854032i \(0.325849\pi\)
−0.520221 + 0.854032i \(0.674151\pi\)
\(422\) 0 0
\(423\) − 7.94438i − 0.386269i
\(424\) 0 0
\(425\) 15.4209 0.748022
\(426\) 0 0
\(427\) − 17.5483i − 0.849220i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 34.2814i − 1.65128i −0.564199 0.825639i \(-0.690815\pi\)
0.564199 0.825639i \(-0.309185\pi\)
\(432\) 0 0
\(433\) −13.7385 −0.660232 −0.330116 0.943940i \(-0.607088\pi\)
−0.330116 + 0.943940i \(0.607088\pi\)
\(434\) 0 0
\(435\) 1.81940i 0.0872334i
\(436\) 0 0
\(437\) 11.0586i 0.529005i
\(438\) 0 0
\(439\) 10.2403 0.488742 0.244371 0.969682i \(-0.421419\pi\)
0.244371 + 0.969682i \(0.421419\pi\)
\(440\) 0 0
\(441\) −7.54288 −0.359185
\(442\) 0 0
\(443\) −12.1763 −0.578513 −0.289257 0.957252i \(-0.593408\pi\)
−0.289257 + 0.957252i \(0.593408\pi\)
\(444\) 0 0
\(445\) −9.55688 −0.453039
\(446\) 0 0
\(447\) − 4.66487i − 0.220641i
\(448\) 0 0
\(449\) − 12.9051i − 0.609032i −0.952507 0.304516i \(-0.901505\pi\)
0.952507 0.304516i \(-0.0984947\pi\)
\(450\) 0 0
\(451\) 3.25534 0.153288
\(452\) 0 0
\(453\) − 7.84117i − 0.368410i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 4.65710i − 0.217850i −0.994050 0.108925i \(-0.965259\pi\)
0.994050 0.108925i \(-0.0347409\pi\)
\(458\) 0 0
\(459\) −16.7289 −0.780836
\(460\) 0 0
\(461\) 31.5405i 1.46899i 0.678616 + 0.734493i \(0.262580\pi\)
−0.678616 + 0.734493i \(0.737420\pi\)
\(462\) 0 0
\(463\) − 17.6504i − 0.820284i −0.912022 0.410142i \(-0.865479\pi\)
0.912022 0.410142i \(-0.134521\pi\)
\(464\) 0 0
\(465\) 3.42327 0.158750
\(466\) 0 0
\(467\) −32.1726 −1.48877 −0.744385 0.667751i \(-0.767257\pi\)
−0.744385 + 0.667751i \(0.767257\pi\)
\(468\) 0 0
\(469\) 1.18167 0.0545644
\(470\) 0 0
\(471\) −5.23490 −0.241211
\(472\) 0 0
\(473\) 15.6799i 0.720964i
\(474\) 0 0
\(475\) − 17.0368i − 0.781704i
\(476\) 0 0
\(477\) −14.8629 −0.680527
\(478\) 0 0
\(479\) 34.8998i 1.59461i 0.603576 + 0.797306i \(0.293742\pi\)
−0.603576 + 0.797306i \(0.706258\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 2.14914i 0.0977895i
\(484\) 0 0
\(485\) 17.2349 0.782596
\(486\) 0 0
\(487\) 41.8351i 1.89573i 0.318676 + 0.947864i \(0.396762\pi\)
−0.318676 + 0.947864i \(0.603238\pi\)
\(488\) 0 0
\(489\) − 4.83041i − 0.218439i
\(490\) 0 0
\(491\) 21.8455 0.985873 0.492936 0.870065i \(-0.335924\pi\)
0.492936 + 0.870065i \(0.335924\pi\)
\(492\) 0 0
\(493\) 12.0151 0.541131
\(494\) 0 0
\(495\) 9.93900 0.446725
\(496\) 0 0
\(497\) −9.41311 −0.422236
\(498\) 0 0
\(499\) 23.5472i 1.05412i 0.849829 + 0.527058i \(0.176705\pi\)
−0.849829 + 0.527058i \(0.823295\pi\)
\(500\) 0 0
\(501\) 13.2379i 0.591425i
\(502\) 0 0
\(503\) 7.08682 0.315986 0.157993 0.987440i \(-0.449498\pi\)
0.157993 + 0.987440i \(0.449498\pi\)
\(504\) 0 0
\(505\) − 18.8780i − 0.840060i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 7.61894i − 0.337704i −0.985641 0.168852i \(-0.945994\pi\)
0.985641 0.168852i \(-0.0540059\pi\)
\(510\) 0 0
\(511\) −21.6213 −0.956471
\(512\) 0 0
\(513\) 18.4819i 0.815995i
\(514\) 0 0
\(515\) 13.2489i 0.583816i
\(516\) 0 0
\(517\) −7.53989 −0.331604
\(518\) 0 0
\(519\) 10.4638 0.459311
\(520\) 0 0
\(521\) −39.5133 −1.73111 −0.865555 0.500813i \(-0.833034\pi\)
−0.865555 + 0.500813i \(0.833034\pi\)
\(522\) 0 0
\(523\) 15.8194 0.691734 0.345867 0.938284i \(-0.387585\pi\)
0.345867 + 0.938284i \(0.387585\pi\)
\(524\) 0 0
\(525\) − 3.31096i − 0.144502i
\(526\) 0 0
\(527\) − 22.6069i − 0.984770i
\(528\) 0 0
\(529\) −19.4276 −0.844678
\(530\) 0 0
\(531\) − 32.8635i − 1.42616i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 9.97046i 0.431061i
\(536\) 0 0
\(537\) 3.34183 0.144211
\(538\) 0 0
\(539\) 7.15883i 0.308353i
\(540\) 0 0
\(541\) 34.4819i 1.48249i 0.671234 + 0.741246i \(0.265765\pi\)
−0.671234 + 0.741246i \(0.734235\pi\)
\(542\) 0 0
\(543\) 2.65146 0.113785
\(544\) 0 0
\(545\) 0.176292 0.00755152
\(546\) 0 0
\(547\) −36.8582 −1.57594 −0.787970 0.615713i \(-0.788868\pi\)
−0.787970 + 0.615713i \(0.788868\pi\)
\(548\) 0 0
\(549\) −23.0562 −0.984015
\(550\) 0 0
\(551\) − 13.2741i − 0.565497i
\(552\) 0 0
\(553\) − 32.3274i − 1.37470i
\(554\) 0 0
\(555\) −4.29590 −0.182351
\(556\) 0 0
\(557\) 1.27652i 0.0540879i 0.999634 + 0.0270439i \(0.00860940\pi\)
−0.999634 + 0.0270439i \(0.991391\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 7.50902i 0.317031i
\(562\) 0 0
\(563\) −9.12737 −0.384673 −0.192336 0.981329i \(-0.561607\pi\)
−0.192336 + 0.981329i \(0.561607\pi\)
\(564\) 0 0
\(565\) 10.5603i 0.444277i
\(566\) 0 0
\(567\) − 12.9554i − 0.544075i
\(568\) 0 0
\(569\) 5.72156 0.239860 0.119930 0.992782i \(-0.461733\pi\)
0.119930 + 0.992782i \(0.461733\pi\)
\(570\) 0 0
\(571\) 7.60148 0.318112 0.159056 0.987270i \(-0.449155\pi\)
0.159056 + 0.987270i \(0.449155\pi\)
\(572\) 0 0
\(573\) −10.2282 −0.427289
\(574\) 0 0
\(575\) −5.50365 −0.229518
\(576\) 0 0
\(577\) − 45.1564i − 1.87989i −0.341330 0.939944i \(-0.610877\pi\)
0.341330 0.939944i \(-0.389123\pi\)
\(578\) 0 0
\(579\) 3.35988i 0.139632i
\(580\) 0 0
\(581\) 15.8248 0.656522
\(582\) 0 0
\(583\) 14.1062i 0.584219i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 32.4040i − 1.33746i −0.743507 0.668728i \(-0.766839\pi\)
0.743507 0.668728i \(-0.233161\pi\)
\(588\) 0 0
\(589\) −24.9758 −1.02911
\(590\) 0 0
\(591\) 6.33513i 0.260592i
\(592\) 0 0
\(593\) 36.6848i 1.50647i 0.657754 + 0.753233i \(0.271507\pi\)
−0.657754 + 0.753233i \(0.728493\pi\)
\(594\) 0 0
\(595\) 15.6799 0.642815
\(596\) 0 0
\(597\) −7.71678 −0.315827
\(598\) 0 0
\(599\) 9.99223 0.408271 0.204136 0.978943i \(-0.434562\pi\)
0.204136 + 0.978943i \(0.434562\pi\)
\(600\) 0 0
\(601\) −1.81163 −0.0738978 −0.0369489 0.999317i \(-0.511764\pi\)
−0.0369489 + 0.999317i \(0.511764\pi\)
\(602\) 0 0
\(603\) − 1.55257i − 0.0632253i
\(604\) 0 0
\(605\) 6.46250i 0.262738i
\(606\) 0 0
\(607\) −11.2161 −0.455248 −0.227624 0.973749i \(-0.573096\pi\)
−0.227624 + 0.973749i \(0.573096\pi\)
\(608\) 0 0
\(609\) − 2.57971i − 0.104535i
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 20.8944i 0.843917i 0.906615 + 0.421958i \(0.138657\pi\)
−0.906615 + 0.421958i \(0.861343\pi\)
\(614\) 0 0
\(615\) 1.02177 0.0412018
\(616\) 0 0
\(617\) − 12.0992i − 0.487094i −0.969889 0.243547i \(-0.921689\pi\)
0.969889 0.243547i \(-0.0783110\pi\)
\(618\) 0 0
\(619\) 10.5526i 0.424143i 0.977254 + 0.212072i \(0.0680210\pi\)
−0.977254 + 0.212072i \(0.931979\pi\)
\(620\) 0 0
\(621\) 5.97046 0.239586
\(622\) 0 0
\(623\) 13.5506 0.542895
\(624\) 0 0
\(625\) −1.96184 −0.0784735
\(626\) 0 0
\(627\) 8.29590 0.331306
\(628\) 0 0
\(629\) 28.3696i 1.13117i
\(630\) 0 0
\(631\) 13.8514i 0.551417i 0.961241 + 0.275709i \(0.0889125\pi\)
−0.961241 + 0.275709i \(0.911087\pi\)
\(632\) 0 0
\(633\) 7.35019 0.292144
\(634\) 0 0
\(635\) − 27.4131i − 1.08786i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 12.3676i 0.489257i
\(640\) 0 0
\(641\) −34.9608 −1.38087 −0.690434 0.723396i \(-0.742580\pi\)
−0.690434 + 0.723396i \(0.742580\pi\)
\(642\) 0 0
\(643\) 33.3980i 1.31709i 0.752541 + 0.658545i \(0.228828\pi\)
−0.752541 + 0.658545i \(0.771172\pi\)
\(644\) 0 0
\(645\) 4.92154i 0.193786i
\(646\) 0 0
\(647\) 2.32842 0.0915397 0.0457698 0.998952i \(-0.485426\pi\)
0.0457698 + 0.998952i \(0.485426\pi\)
\(648\) 0 0
\(649\) −31.1903 −1.22433
\(650\) 0 0
\(651\) −4.85384 −0.190237
\(652\) 0 0
\(653\) 14.5714 0.570221 0.285111 0.958495i \(-0.407970\pi\)
0.285111 + 0.958495i \(0.407970\pi\)
\(654\) 0 0
\(655\) − 4.70602i − 0.183879i
\(656\) 0 0
\(657\) 28.4077i 1.10829i
\(658\) 0 0
\(659\) −11.1395 −0.433932 −0.216966 0.976179i \(-0.569616\pi\)
−0.216966 + 0.976179i \(0.569616\pi\)
\(660\) 0 0
\(661\) − 13.8498i − 0.538694i −0.963043 0.269347i \(-0.913192\pi\)
0.963043 0.269347i \(-0.0868079\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 17.3230i − 0.671759i
\(666\) 0 0
\(667\) −4.28813 −0.166037
\(668\) 0 0
\(669\) − 4.07069i − 0.157382i
\(670\) 0 0
\(671\) 21.8823i 0.844757i
\(672\) 0 0
\(673\) 6.52973 0.251703 0.125851 0.992049i \(-0.459834\pi\)
0.125851 + 0.992049i \(0.459834\pi\)
\(674\) 0 0
\(675\) −9.19806 −0.354034
\(676\) 0 0
\(677\) −11.3104 −0.434693 −0.217346 0.976095i \(-0.569740\pi\)
−0.217346 + 0.976095i \(0.569740\pi\)
\(678\) 0 0
\(679\) −24.4373 −0.937816
\(680\) 0 0
\(681\) − 4.81402i − 0.184474i
\(682\) 0 0
\(683\) − 14.1793i − 0.542555i −0.962501 0.271277i \(-0.912554\pi\)
0.962501 0.271277i \(-0.0874461\pi\)
\(684\) 0 0
\(685\) −1.14483 −0.0437418
\(686\) 0 0
\(687\) − 7.59551i − 0.289787i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 30.7952i 1.17151i 0.810490 + 0.585753i \(0.199201\pi\)
−0.810490 + 0.585753i \(0.800799\pi\)
\(692\) 0 0
\(693\) −14.0925 −0.535328
\(694\) 0 0
\(695\) 16.3860i 0.621555i
\(696\) 0 0
\(697\) − 6.74764i − 0.255585i
\(698\) 0 0
\(699\) 2.82371 0.106802
\(700\) 0 0
\(701\) −6.73184 −0.254258 −0.127129 0.991886i \(-0.540576\pi\)
−0.127129 + 0.991886i \(0.540576\pi\)
\(702\) 0 0
\(703\) 31.3424 1.18210
\(704\) 0 0
\(705\) −2.36658 −0.0891307
\(706\) 0 0
\(707\) 26.7670i 1.00668i
\(708\) 0 0
\(709\) − 47.6252i − 1.78860i −0.447467 0.894300i \(-0.647674\pi\)
0.447467 0.894300i \(-0.352326\pi\)
\(710\) 0 0
\(711\) −42.4741 −1.59290
\(712\) 0 0
\(713\) 8.06829i 0.302160i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 6.06233i − 0.226402i
\(718\) 0 0
\(719\) −5.99330 −0.223512 −0.111756 0.993736i \(-0.535648\pi\)
−0.111756 + 0.993736i \(0.535648\pi\)
\(720\) 0 0
\(721\) − 18.7855i − 0.699610i
\(722\) 0 0
\(723\) 6.61224i 0.245912i
\(724\) 0 0
\(725\) 6.60627 0.245351
\(726\) 0 0
\(727\) −24.1226 −0.894657 −0.447329 0.894370i \(-0.647625\pi\)
−0.447329 + 0.894370i \(0.647625\pi\)
\(728\) 0 0
\(729\) −11.7627 −0.435656
\(730\) 0 0
\(731\) 32.5013 1.20210
\(732\) 0 0
\(733\) − 36.0646i − 1.33208i −0.745918 0.666038i \(-0.767989\pi\)
0.745918 0.666038i \(-0.232011\pi\)
\(734\) 0 0
\(735\) 2.24698i 0.0828811i
\(736\) 0 0
\(737\) −1.47352 −0.0542777
\(738\) 0 0
\(739\) − 27.5254i − 1.01254i −0.862375 0.506269i \(-0.831024\pi\)
0.862375 0.506269i \(-0.168976\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 10.4692i − 0.384078i −0.981387 0.192039i \(-0.938490\pi\)
0.981387 0.192039i \(-0.0615100\pi\)
\(744\) 0 0
\(745\) 12.1468 0.445023
\(746\) 0 0
\(747\) − 20.7918i − 0.760731i
\(748\) 0 0
\(749\) − 14.1371i − 0.516557i
\(750\) 0 0
\(751\) 4.06770 0.148433 0.0742163 0.997242i \(-0.476354\pi\)
0.0742163 + 0.997242i \(0.476354\pi\)
\(752\) 0 0
\(753\) 12.4021 0.451957
\(754\) 0 0
\(755\) 20.4174 0.743066
\(756\) 0 0
\(757\) 20.4336 0.742670 0.371335 0.928499i \(-0.378900\pi\)
0.371335 + 0.928499i \(0.378900\pi\)
\(758\) 0 0
\(759\) − 2.67994i − 0.0972757i
\(760\) 0 0
\(761\) − 27.0237i − 0.979608i −0.871833 0.489804i \(-0.837068\pi\)
0.871833 0.489804i \(-0.162932\pi\)
\(762\) 0 0
\(763\) −0.249964 −0.00904929
\(764\) 0 0
\(765\) − 20.6015i − 0.744848i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 37.9407i 1.36818i 0.729400 + 0.684088i \(0.239799\pi\)
−0.729400 + 0.684088i \(0.760201\pi\)
\(770\) 0 0
\(771\) 10.3556 0.372947
\(772\) 0 0
\(773\) 16.3375i 0.587620i 0.955864 + 0.293810i \(0.0949232\pi\)
−0.955864 + 0.293810i \(0.905077\pi\)
\(774\) 0 0
\(775\) − 12.4300i − 0.446498i
\(776\) 0 0
\(777\) 6.09113 0.218518
\(778\) 0 0
\(779\) −7.45473 −0.267093
\(780\) 0 0
\(781\) 11.7380 0.420017
\(782\) 0 0
\(783\) −7.16660 −0.256114
\(784\) 0 0
\(785\) − 13.6310i − 0.486512i
\(786\) 0 0
\(787\) 18.6907i 0.666251i 0.942882 + 0.333126i \(0.108103\pi\)
−0.942882 + 0.333126i \(0.891897\pi\)
\(788\) 0 0
\(789\) −7.99090 −0.284484
\(790\) 0 0
\(791\) − 14.9734i − 0.532394i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 4.42758i 0.157030i
\(796\) 0 0
\(797\) −29.2519 −1.03615 −0.518077 0.855334i \(-0.673352\pi\)
−0.518077 + 0.855334i \(0.673352\pi\)
\(798\) 0 0
\(799\) 15.6286i 0.552901i
\(800\) 0 0
\(801\) − 17.8039i − 0.629068i
\(802\) 0 0
\(803\) 26.9614 0.951446
\(804\) 0 0
\(805\) −5.59611 −0.197237
\(806\) 0 0
\(807\) 0.362273 0.0127526
\(808\) 0 0
\(809\) −6.65087 −0.233832 −0.116916 0.993142i \(-0.537301\pi\)
−0.116916 + 0.993142i \(0.537301\pi\)
\(810\) 0 0
\(811\) 3.89200i 0.136667i 0.997663 + 0.0683333i \(0.0217681\pi\)
−0.997663 + 0.0683333i \(0.978232\pi\)
\(812\) 0 0
\(813\) 1.10752i 0.0388425i
\(814\) 0 0
\(815\) 12.5778 0.440581
\(816\) 0 0
\(817\) − 35.9071i − 1.25623i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 45.9982i − 1.60535i −0.596418 0.802674i \(-0.703410\pi\)
0.596418 0.802674i \(-0.296590\pi\)
\(822\) 0 0
\(823\) 7.95300 0.277224 0.138612 0.990347i \(-0.455736\pi\)
0.138612 + 0.990347i \(0.455736\pi\)
\(824\) 0 0
\(825\) 4.12870i 0.143743i
\(826\) 0 0
\(827\) − 27.9648i − 0.972432i −0.873839 0.486216i \(-0.838377\pi\)
0.873839 0.486216i \(-0.161623\pi\)
\(828\) 0 0
\(829\) −27.6310 −0.959665 −0.479833 0.877360i \(-0.659303\pi\)
−0.479833 + 0.877360i \(0.659303\pi\)
\(830\) 0 0
\(831\) −6.53989 −0.226866
\(832\) 0 0
\(833\) 14.8388 0.514133
\(834\) 0 0
\(835\) −34.4698 −1.19288
\(836\) 0 0
\(837\) 13.4843i 0.466085i
\(838\) 0 0
\(839\) − 28.6848i − 0.990311i −0.868804 0.495155i \(-0.835111\pi\)
0.868804 0.495155i \(-0.164889\pi\)
\(840\) 0 0
\(841\) −23.8528 −0.822509
\(842\) 0 0
\(843\) − 3.59179i − 0.123708i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 9.16315i − 0.314849i
\(848\) 0 0
\(849\) 3.65220 0.125343
\(850\) 0 0
\(851\) − 10.1250i − 0.347080i
\(852\) 0 0
\(853\) − 43.2078i − 1.47941i −0.672934 0.739703i \(-0.734966\pi\)
0.672934 0.739703i \(-0.265034\pi\)
\(854\) 0 0
\(855\) −22.7603 −0.778386
\(856\) 0 0
\(857\) −35.1685 −1.20133 −0.600667 0.799499i \(-0.705098\pi\)
−0.600667 + 0.799499i \(0.705098\pi\)
\(858\) 0 0
\(859\) −27.3793 −0.934168 −0.467084 0.884213i \(-0.654695\pi\)
−0.467084 + 0.884213i \(0.654695\pi\)
\(860\) 0 0
\(861\) −1.44876 −0.0493737
\(862\) 0 0
\(863\) − 41.3913i − 1.40898i −0.709715 0.704489i \(-0.751176\pi\)
0.709715 0.704489i \(-0.248824\pi\)
\(864\) 0 0
\(865\) 27.2465i 0.926409i
\(866\) 0 0
\(867\) 6.13036 0.208198
\(868\) 0 0
\(869\) 40.3116i 1.36748i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 32.1075i 1.08668i
\(874\) 0 0
\(875\) 23.4252 0.791916
\(876\) 0 0
\(877\) 24.7472i 0.835653i 0.908527 + 0.417826i \(0.137208\pi\)
−0.908527 + 0.417826i \(0.862792\pi\)
\(878\) 0 0
\(879\) − 13.5066i − 0.455567i
\(880\) 0 0
\(881\) 28.5875 0.963137 0.481568 0.876409i \(-0.340067\pi\)
0.481568 + 0.876409i \(0.340067\pi\)
\(882\) 0 0
\(883\) 9.61702 0.323639 0.161819 0.986820i \(-0.448264\pi\)
0.161819 + 0.986820i \(0.448264\pi\)
\(884\) 0 0
\(885\) −9.78986 −0.329082
\(886\) 0 0
\(887\) −15.9661 −0.536091 −0.268045 0.963406i \(-0.586378\pi\)
−0.268045 + 0.963406i \(0.586378\pi\)
\(888\) 0 0
\(889\) 38.8689i 1.30362i
\(890\) 0 0
\(891\) 16.1551i 0.541217i
\(892\) 0 0
\(893\) 17.2664 0.577797
\(894\) 0 0
\(895\) 8.70171i 0.290866i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 9.68473i − 0.323004i
\(900\) 0 0
\(901\) 29.2392 0.974099
\(902\) 0 0
\(903\) − 6.97823i − 0.232221i
\(904\) 0 0
\(905\) 6.90408i 0.229500i
\(906\) 0 0
\(907\) −28.8364 −0.957496 −0.478748 0.877952i \(-0.658909\pi\)
−0.478748 + 0.877952i \(0.658909\pi\)
\(908\) 0 0
\(909\) 35.1685 1.16647
\(910\) 0 0
\(911\) −38.5633 −1.27766 −0.638830 0.769348i \(-0.720581\pi\)
−0.638830 + 0.769348i \(0.720581\pi\)
\(912\) 0 0
\(913\) −19.7332 −0.653073
\(914\) 0 0
\(915\) 6.86831i 0.227059i
\(916\) 0 0
\(917\) 6.67264i 0.220350i
\(918\) 0 0
\(919\) −8.87502 −0.292760 −0.146380 0.989228i \(-0.546762\pi\)
−0.146380 + 0.989228i \(0.546762\pi\)
\(920\) 0 0
\(921\) 7.81030i 0.257358i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 15.5985i 0.512875i
\(926\) 0 0
\(927\) −24.6819 −0.810659
\(928\) 0 0
\(929\) − 24.2295i − 0.794945i −0.917614 0.397472i \(-0.869887\pi\)
0.917614 0.397472i \(-0.130113\pi\)
\(930\) 0 0
\(931\) − 16.3937i − 0.537283i
\(932\) 0 0
\(933\) −16.5211 −0.540877
\(934\) 0 0
\(935\) −19.5526 −0.639437
\(936\) 0 0
\(937\) 17.2644 0.564005 0.282002 0.959414i \(-0.409001\pi\)
0.282002 + 0.959414i \(0.409001\pi\)
\(938\) 0 0
\(939\) −4.15047 −0.135446
\(940\) 0 0
\(941\) − 4.34050i − 0.141496i −0.997494 0.0707482i \(-0.977461\pi\)
0.997494 0.0707482i \(-0.0225387\pi\)
\(942\) 0 0
\(943\) 2.40821i 0.0784220i
\(944\) 0 0
\(945\) −9.35258 −0.304240
\(946\) 0 0
\(947\) 45.0146i 1.46278i 0.681961 + 0.731389i \(0.261128\pi\)
−0.681961 + 0.731389i \(0.738872\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) − 16.6655i − 0.540415i
\(952\) 0 0
\(953\) 46.8859 1.51878 0.759391 0.650634i \(-0.225497\pi\)
0.759391 + 0.650634i \(0.225497\pi\)
\(954\) 0 0
\(955\) − 26.6329i − 0.861822i
\(956\) 0 0
\(957\) 3.21685i 0.103986i
\(958\) 0 0
\(959\) 1.62325 0.0524176
\(960\) 0 0
\(961\) 12.7778 0.412186
\(962\) 0 0
\(963\) −18.5743 −0.598550
\(964\) 0 0
\(965\) −8.74871 −0.281631
\(966\) 0 0
\(967\) 6.29457i 0.202420i 0.994865 + 0.101210i \(0.0322714\pi\)
−0.994865 + 0.101210i \(0.967729\pi\)
\(968\) 0 0
\(969\) − 17.1957i − 0.552404i
\(970\) 0 0
\(971\) 41.8068 1.34165 0.670823 0.741618i \(-0.265941\pi\)
0.670823 + 0.741618i \(0.265941\pi\)
\(972\) 0 0
\(973\) − 23.2336i − 0.744834i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 23.7530i − 0.759926i −0.925002 0.379963i \(-0.875937\pi\)
0.925002 0.379963i \(-0.124063\pi\)
\(978\) 0 0
\(979\) −16.8974 −0.540043
\(980\) 0 0
\(981\) 0.328421i 0.0104857i
\(982\) 0 0
\(983\) 55.7251i 1.77736i 0.458532 + 0.888678i \(0.348375\pi\)
−0.458532 + 0.888678i \(0.651625\pi\)
\(984\) 0 0
\(985\) −16.4959 −0.525602
\(986\) 0 0
\(987\) 3.35557 0.106809
\(988\) 0 0
\(989\) −11.5996 −0.368845
\(990\) 0 0
\(991\) 35.5512 1.12932 0.564661 0.825323i \(-0.309007\pi\)
0.564661 + 0.825323i \(0.309007\pi\)
\(992\) 0 0
\(993\) − 8.72215i − 0.276789i
\(994\) 0 0
\(995\) − 20.0935i − 0.637007i
\(996\) 0 0
\(997\) 6.61058 0.209359 0.104680 0.994506i \(-0.466618\pi\)
0.104680 + 0.994506i \(0.466618\pi\)
\(998\) 0 0
\(999\) − 16.9215i − 0.535374i
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 2704.2.f.o.337.4 6
4.3 odd 2 169.2.b.b.168.6 6
12.11 even 2 1521.2.b.l.1351.1 6
13.5 odd 4 2704.2.a.ba.1.2 3
13.8 odd 4 2704.2.a.z.1.2 3
13.12 even 2 inner 2704.2.f.o.337.3 6
52.3 odd 6 169.2.e.b.147.1 12
52.7 even 12 169.2.c.c.146.3 6
52.11 even 12 169.2.c.c.22.3 6
52.15 even 12 169.2.c.b.22.1 6
52.19 even 12 169.2.c.b.146.1 6
52.23 odd 6 169.2.e.b.147.6 12
52.31 even 4 169.2.a.c.1.3 yes 3
52.35 odd 6 169.2.e.b.23.6 12
52.43 odd 6 169.2.e.b.23.1 12
52.47 even 4 169.2.a.b.1.1 3
52.51 odd 2 169.2.b.b.168.1 6
156.47 odd 4 1521.2.a.r.1.3 3
156.83 odd 4 1521.2.a.o.1.1 3
156.155 even 2 1521.2.b.l.1351.6 6
260.99 even 4 4225.2.a.bg.1.3 3
260.239 even 4 4225.2.a.bb.1.1 3
364.83 odd 4 8281.2.a.bj.1.3 3
364.307 odd 4 8281.2.a.bf.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
169.2.a.b.1.1 3 52.47 even 4
169.2.a.c.1.3 yes 3 52.31 even 4
169.2.b.b.168.1 6 52.51 odd 2
169.2.b.b.168.6 6 4.3 odd 2
169.2.c.b.22.1 6 52.15 even 12
169.2.c.b.146.1 6 52.19 even 12
169.2.c.c.22.3 6 52.11 even 12
169.2.c.c.146.3 6 52.7 even 12
169.2.e.b.23.1 12 52.43 odd 6
169.2.e.b.23.6 12 52.35 odd 6
169.2.e.b.147.1 12 52.3 odd 6
169.2.e.b.147.6 12 52.23 odd 6
1521.2.a.o.1.1 3 156.83 odd 4
1521.2.a.r.1.3 3 156.47 odd 4
1521.2.b.l.1351.1 6 12.11 even 2
1521.2.b.l.1351.6 6 156.155 even 2
2704.2.a.z.1.2 3 13.8 odd 4
2704.2.a.ba.1.2 3 13.5 odd 4
2704.2.f.o.337.3 6 13.12 even 2 inner
2704.2.f.o.337.4 6 1.1 even 1 trivial
4225.2.a.bb.1.1 3 260.239 even 4
4225.2.a.bg.1.3 3 260.99 even 4
8281.2.a.bf.1.1 3 364.307 odd 4
8281.2.a.bj.1.3 3 364.83 odd 4