Properties

Label 2704.2.a.z.1.2
Level $2704$
Weight $2$
Character 2704.1
Self dual yes
Analytic conductor $21.592$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2704,2,Mod(1,2704)]
 
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, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2704.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2704 = 2^{4} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2704.a (trivial)

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: yes
Analytic conductor: \(21.5915487066\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 169)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.445042\) of defining polynomial
Character \(\chi\) \(=\) 2704.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.554958 q^{3} -1.44504 q^{5} -2.04892 q^{7} -2.69202 q^{9} +O(q^{10})\) \(q+0.554958 q^{3} -1.44504 q^{5} -2.04892 q^{7} -2.69202 q^{9} +2.55496 q^{11} -0.801938 q^{15} -5.29590 q^{17} +5.85086 q^{19} -1.13706 q^{21} +1.89008 q^{23} -2.91185 q^{25} -3.15883 q^{27} +2.26875 q^{29} +4.26875 q^{31} +1.41789 q^{33} +2.96077 q^{35} +5.35690 q^{37} +1.27413 q^{41} -6.13706 q^{43} +3.89008 q^{45} +2.95108 q^{47} -2.80194 q^{49} -2.93900 q^{51} +5.52111 q^{53} -3.69202 q^{55} +3.24698 q^{57} +12.2078 q^{59} +8.56465 q^{61} +5.51573 q^{63} -0.576728 q^{67} +1.04892 q^{69} +4.59419 q^{71} -10.5526 q^{73} -1.61596 q^{75} -5.23490 q^{77} +15.7778 q^{79} +6.32304 q^{81} -7.72348 q^{83} +7.65279 q^{85} +1.25906 q^{87} +6.61356 q^{89} +2.36898 q^{93} -8.45473 q^{95} +11.9269 q^{97} -6.87800 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{3} - 4 q^{5} + 3 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{3} - 4 q^{5} + 3 q^{7} - 3 q^{9} + 8 q^{11} + 2 q^{15} - 2 q^{17} + 4 q^{19} + 2 q^{21} + 5 q^{23} - 5 q^{25} - q^{27} - q^{29} + 5 q^{31} + 10 q^{33} - 4 q^{35} + 12 q^{37} - 7 q^{41} - 13 q^{43} + 11 q^{45} + 18 q^{47} - 4 q^{49} + q^{51} + q^{53} - 6 q^{55} + 5 q^{57} + 19 q^{59} + 4 q^{61} + 4 q^{63} + q^{67} - 6 q^{69} + 27 q^{71} + 9 q^{73} - 15 q^{75} + 8 q^{77} + 5 q^{79} - q^{81} + 7 q^{83} + 5 q^{85} + 18 q^{87} - 11 q^{89} + 22 q^{93} - 3 q^{95} + 7 q^{97} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) −1.44504 −0.646242 −0.323121 0.946358i \(-0.604732\pi\)
−0.323121 + 0.946358i \(0.604732\pi\)
\(6\) 0 0
\(7\) −2.04892 −0.774418 −0.387209 0.921992i \(-0.626561\pi\)
−0.387209 + 0.921992i \(0.626561\pi\)
\(8\) 0 0
\(9\) −2.69202 −0.897340
\(10\) 0 0
\(11\) 2.55496 0.770349 0.385174 0.922844i \(-0.374141\pi\)
0.385174 + 0.922844i \(0.374141\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) −0.801938 −0.207059
\(16\) 0 0
\(17\) −5.29590 −1.28444 −0.642222 0.766519i \(-0.721987\pi\)
−0.642222 + 0.766519i \(0.721987\pi\)
\(18\) 0 0
\(19\) 5.85086 1.34228 0.671139 0.741331i \(-0.265805\pi\)
0.671139 + 0.741331i \(0.265805\pi\)
\(20\) 0 0
\(21\) −1.13706 −0.248128
\(22\) 0 0
\(23\) 1.89008 0.394110 0.197055 0.980392i \(-0.436862\pi\)
0.197055 + 0.980392i \(0.436862\pi\)
\(24\) 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
\(31\) 4.26875 0.766690 0.383345 0.923605i \(-0.374772\pi\)
0.383345 + 0.923605i \(0.374772\pi\)
\(32\) 0 0
\(33\) 1.41789 0.246824
\(34\) 0 0
\(35\) 2.96077 0.500462
\(36\) 0 0
\(37\) 5.35690 0.880668 0.440334 0.897834i \(-0.354860\pi\)
0.440334 + 0.897834i \(0.354860\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.27413 0.198985 0.0994926 0.995038i \(-0.468278\pi\)
0.0994926 + 0.995038i \(0.468278\pi\)
\(42\) 0 0
\(43\) −6.13706 −0.935893 −0.467947 0.883757i \(-0.655006\pi\)
−0.467947 + 0.883757i \(0.655006\pi\)
\(44\) 0 0
\(45\) 3.89008 0.579899
\(46\) 0 0
\(47\) 2.95108 0.430460 0.215230 0.976563i \(-0.430950\pi\)
0.215230 + 0.976563i \(0.430950\pi\)
\(48\) 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
\(57\) 3.24698 0.430073
\(58\) 0 0
\(59\) 12.2078 1.58931 0.794657 0.607059i \(-0.207651\pi\)
0.794657 + 0.607059i \(0.207651\pi\)
\(60\) 0 0
\(61\) 8.56465 1.09659 0.548295 0.836285i \(-0.315277\pi\)
0.548295 + 0.836285i \(0.315277\pi\)
\(62\) 0 0
\(63\) 5.51573 0.694917
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −0.576728 −0.0704586 −0.0352293 0.999379i \(-0.511216\pi\)
−0.0352293 + 0.999379i \(0.511216\pi\)
\(68\) 0 0
\(69\) 1.04892 0.126275
\(70\) 0 0
\(71\) 4.59419 0.545230 0.272615 0.962123i \(-0.412112\pi\)
0.272615 + 0.962123i \(0.412112\pi\)
\(72\) 0 0
\(73\) −10.5526 −1.23508 −0.617542 0.786538i \(-0.711872\pi\)
−0.617542 + 0.786538i \(0.711872\pi\)
\(74\) 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
\(83\) −7.72348 −0.847762 −0.423881 0.905718i \(-0.639333\pi\)
−0.423881 + 0.905718i \(0.639333\pi\)
\(84\) 0 0
\(85\) 7.65279 0.830062
\(86\) 0 0
\(87\) 1.25906 0.134986
\(88\) 0 0
\(89\) 6.61356 0.701036 0.350518 0.936556i \(-0.386005\pi\)
0.350518 + 0.936556i \(0.386005\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 2.36898 0.245652
\(94\) 0 0
\(95\) −8.45473 −0.867437
\(96\) 0 0
\(97\) 11.9269 1.21100 0.605498 0.795847i \(-0.292974\pi\)
0.605498 + 0.795847i \(0.292974\pi\)
\(98\) 0 0
\(99\) −6.87800 −0.691265
\(100\) 0 0
\(101\) 13.0640 1.29991 0.649957 0.759971i \(-0.274787\pi\)
0.649957 + 0.759971i \(0.274787\pi\)
\(102\) 0 0
\(103\) −9.16852 −0.903401 −0.451701 0.892170i \(-0.649182\pi\)
−0.451701 + 0.892170i \(0.649182\pi\)
\(104\) 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
\(109\) 0.121998 0.0116853 0.00584264 0.999983i \(-0.498140\pi\)
0.00584264 + 0.999983i \(0.498140\pi\)
\(110\) 0 0
\(111\) 2.97285 0.282171
\(112\) 0 0
\(113\) 7.30798 0.687477 0.343738 0.939065i \(-0.388307\pi\)
0.343738 + 0.939065i \(0.388307\pi\)
\(114\) 0 0
\(115\) −2.73125 −0.254690
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 10.8509 0.994696
\(120\) 0 0
\(121\) −4.47219 −0.406563
\(122\) 0 0
\(123\) 0.707087 0.0637559
\(124\) 0 0
\(125\) 11.4330 1.02260
\(126\) 0 0
\(127\) 18.9705 1.68336 0.841678 0.539980i \(-0.181568\pi\)
0.841678 + 0.539980i \(0.181568\pi\)
\(128\) 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
\(135\) 4.56465 0.392862
\(136\) 0 0
\(137\) 0.792249 0.0676864 0.0338432 0.999427i \(-0.489225\pi\)
0.0338432 + 0.999427i \(0.489225\pi\)
\(138\) 0 0
\(139\) 11.3394 0.961799 0.480899 0.876776i \(-0.340310\pi\)
0.480899 + 0.876776i \(0.340310\pi\)
\(140\) 0 0
\(141\) 1.63773 0.137922
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −3.27844 −0.272260
\(146\) 0 0
\(147\) −1.55496 −0.128251
\(148\) 0 0
\(149\) 8.40581 0.688631 0.344316 0.938854i \(-0.388111\pi\)
0.344316 + 0.938854i \(0.388111\pi\)
\(150\) 0 0
\(151\) −14.1293 −1.14983 −0.574913 0.818215i \(-0.694964\pi\)
−0.574913 + 0.818215i \(0.694964\pi\)
\(152\) 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
\(161\) −3.87263 −0.305206
\(162\) 0 0
\(163\) −8.70410 −0.681758 −0.340879 0.940107i \(-0.610725\pi\)
−0.340879 + 0.940107i \(0.610725\pi\)
\(164\) 0 0
\(165\) −2.04892 −0.159508
\(166\) 0 0
\(167\) 23.8538 1.84587 0.922933 0.384961i \(-0.125785\pi\)
0.922933 + 0.384961i \(0.125785\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) −15.7506 −1.20448
\(172\) 0 0
\(173\) −18.8552 −1.43353 −0.716766 0.697314i \(-0.754378\pi\)
−0.716766 + 0.697314i \(0.754378\pi\)
\(174\) 0 0
\(175\) 5.96615 0.450998
\(176\) 0 0
\(177\) 6.77479 0.509224
\(178\) 0 0
\(179\) −6.02177 −0.450088 −0.225044 0.974349i \(-0.572253\pi\)
−0.225044 + 0.974349i \(0.572253\pi\)
\(180\) 0 0
\(181\) −4.77777 −0.355129 −0.177565 0.984109i \(-0.556822\pi\)
−0.177565 + 0.984109i \(0.556822\pi\)
\(182\) 0 0
\(183\) 4.75302 0.351353
\(184\) 0 0
\(185\) −7.74094 −0.569125
\(186\) 0 0
\(187\) −13.5308 −0.989470
\(188\) 0 0
\(189\) 6.47219 0.470782
\(190\) 0 0
\(191\) −18.4306 −1.33359 −0.666795 0.745242i \(-0.732334\pi\)
−0.666795 + 0.745242i \(0.732334\pi\)
\(192\) 0 0
\(193\) 6.05429 0.435798 0.217899 0.975971i \(-0.430080\pi\)
0.217899 + 0.975971i \(0.430080\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −11.4155 −0.813321 −0.406660 0.913579i \(-0.633307\pi\)
−0.406660 + 0.913579i \(0.633307\pi\)
\(198\) 0 0
\(199\) 13.9051 0.985710 0.492855 0.870111i \(-0.335953\pi\)
0.492855 + 0.870111i \(0.335953\pi\)
\(200\) 0 0
\(201\) −0.320060 −0.0225753
\(202\) 0 0
\(203\) −4.64848 −0.326259
\(204\) 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
\(213\) 2.54958 0.174694
\(214\) 0 0
\(215\) 8.86831 0.604814
\(216\) 0 0
\(217\) −8.74632 −0.593739
\(218\) 0 0
\(219\) −5.85623 −0.395727
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 7.33513 0.491196 0.245598 0.969372i \(-0.421016\pi\)
0.245598 + 0.969372i \(0.421016\pi\)
\(224\) 0 0
\(225\) 7.83877 0.522585
\(226\) 0 0
\(227\) 8.67456 0.575751 0.287875 0.957668i \(-0.407051\pi\)
0.287875 + 0.957668i \(0.407051\pi\)
\(228\) 0 0
\(229\) −13.6866 −0.904439 −0.452219 0.891907i \(-0.649368\pi\)
−0.452219 + 0.891907i \(0.649368\pi\)
\(230\) 0 0
\(231\) −2.90515 −0.191145
\(232\) 0 0
\(233\) −5.08815 −0.333336 −0.166668 0.986013i \(-0.553301\pi\)
−0.166668 + 0.986013i \(0.553301\pi\)
\(234\) 0 0
\(235\) −4.26444 −0.278181
\(236\) 0 0
\(237\) 8.75600 0.568764
\(238\) 0 0
\(239\) 10.9239 0.706611 0.353305 0.935508i \(-0.385058\pi\)
0.353305 + 0.935508i \(0.385058\pi\)
\(240\) 0 0
\(241\) 11.9148 0.767502 0.383751 0.923437i \(-0.374632\pi\)
0.383751 + 0.923437i \(0.374632\pi\)
\(242\) 0 0
\(243\) 12.9855 0.833022
\(244\) 0 0
\(245\) 4.04892 0.258676
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −4.28621 −0.271627
\(250\) 0 0
\(251\) −22.3478 −1.41058 −0.705290 0.708919i \(-0.749183\pi\)
−0.705290 + 0.708919i \(0.749183\pi\)
\(252\) 0 0
\(253\) 4.82908 0.303602
\(254\) 0 0
\(255\) 4.24698 0.265956
\(256\) 0 0
\(257\) −18.6601 −1.16398 −0.581992 0.813194i \(-0.697727\pi\)
−0.581992 + 0.813194i \(0.697727\pi\)
\(258\) 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
\(265\) −7.97823 −0.490099
\(266\) 0 0
\(267\) 3.67025 0.224616
\(268\) 0 0
\(269\) 0.652793 0.0398015 0.0199007 0.999802i \(-0.493665\pi\)
0.0199007 + 0.999802i \(0.493665\pi\)
\(270\) 0 0
\(271\) 1.99569 0.121229 0.0606147 0.998161i \(-0.480694\pi\)
0.0606147 + 0.998161i \(0.480694\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −7.43967 −0.448629
\(276\) 0 0
\(277\) 11.7845 0.708061 0.354030 0.935234i \(-0.384811\pi\)
0.354030 + 0.935234i \(0.384811\pi\)
\(278\) 0 0
\(279\) −11.4916 −0.687982
\(280\) 0 0
\(281\) −6.47219 −0.386098 −0.193049 0.981189i \(-0.561838\pi\)
−0.193049 + 0.981189i \(0.561838\pi\)
\(282\) 0 0
\(283\) −6.58104 −0.391202 −0.195601 0.980684i \(-0.562666\pi\)
−0.195601 + 0.980684i \(0.562666\pi\)
\(284\) 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
\(291\) 6.61894 0.388009
\(292\) 0 0
\(293\) −24.3381 −1.42185 −0.710924 0.703269i \(-0.751723\pi\)
−0.710924 + 0.703269i \(0.751723\pi\)
\(294\) 0 0
\(295\) −17.6407 −1.02708
\(296\) 0 0
\(297\) −8.07069 −0.468309
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 12.5743 0.724773
\(302\) 0 0
\(303\) 7.24996 0.416500
\(304\) 0 0
\(305\) −12.3763 −0.708663
\(306\) 0 0
\(307\) 14.0737 0.803227 0.401613 0.915809i \(-0.368450\pi\)
0.401613 + 0.915809i \(0.368450\pi\)
\(308\) 0 0
\(309\) −5.08815 −0.289455
\(310\) 0 0
\(311\) 29.7700 1.68810 0.844051 0.536263i \(-0.180164\pi\)
0.844051 + 0.536263i \(0.180164\pi\)
\(312\) 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
\(317\) 30.0301 1.68666 0.843330 0.537396i \(-0.180592\pi\)
0.843330 + 0.537396i \(0.180592\pi\)
\(318\) 0 0
\(319\) 5.79656 0.324545
\(320\) 0 0
\(321\) 3.82908 0.213719
\(322\) 0 0
\(323\) −30.9855 −1.72408
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0.0677037 0.00374402
\(328\) 0 0
\(329\) −6.04652 −0.333356
\(330\) 0 0
\(331\) 15.7168 0.863872 0.431936 0.901904i \(-0.357831\pi\)
0.431936 + 0.901904i \(0.357831\pi\)
\(332\) 0 0
\(333\) −14.4209 −0.790259
\(334\) 0 0
\(335\) 0.833397 0.0455333
\(336\) 0 0
\(337\) 1.95407 0.106445 0.0532224 0.998583i \(-0.483051\pi\)
0.0532224 + 0.998583i \(0.483051\pi\)
\(338\) 0 0
\(339\) 4.05562 0.220271
\(340\) 0 0
\(341\) 10.9065 0.590619
\(342\) 0 0
\(343\) 20.0834 1.08440
\(344\) 0 0
\(345\) −1.51573 −0.0816041
\(346\) 0 0
\(347\) 17.1250 0.919317 0.459659 0.888096i \(-0.347972\pi\)
0.459659 + 0.888096i \(0.347972\pi\)
\(348\) 0 0
\(349\) −10.4668 −0.560276 −0.280138 0.959960i \(-0.590380\pi\)
−0.280138 + 0.959960i \(0.590380\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 15.5308 0.826621 0.413310 0.910590i \(-0.364372\pi\)
0.413310 + 0.910590i \(0.364372\pi\)
\(354\) 0 0
\(355\) −6.63879 −0.352351
\(356\) 0 0
\(357\) 6.02177 0.318706
\(358\) 0 0
\(359\) 21.4263 1.13083 0.565417 0.824805i \(-0.308715\pi\)
0.565417 + 0.824805i \(0.308715\pi\)
\(360\) 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
\(369\) −3.42998 −0.178557
\(370\) 0 0
\(371\) −11.3123 −0.587305
\(372\) 0 0
\(373\) −12.5961 −0.652202 −0.326101 0.945335i \(-0.605735\pi\)
−0.326101 + 0.945335i \(0.605735\pi\)
\(374\) 0 0
\(375\) 6.34481 0.327645
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −16.5386 −0.849529 −0.424765 0.905304i \(-0.639643\pi\)
−0.424765 + 0.905304i \(0.639643\pi\)
\(380\) 0 0
\(381\) 10.5278 0.539356
\(382\) 0 0
\(383\) −7.53617 −0.385080 −0.192540 0.981289i \(-0.561673\pi\)
−0.192540 + 0.981289i \(0.561673\pi\)
\(384\) 0 0
\(385\) 7.56465 0.385530
\(386\) 0 0
\(387\) 16.5211 0.839815
\(388\) 0 0
\(389\) 35.5555 1.80274 0.901369 0.433052i \(-0.142563\pi\)
0.901369 + 0.433052i \(0.142563\pi\)
\(390\) 0 0
\(391\) −10.0097 −0.506212
\(392\) 0 0
\(393\) −1.80731 −0.0911670
\(394\) 0 0
\(395\) −22.7995 −1.14717
\(396\) 0 0
\(397\) 1.35152 0.0678308 0.0339154 0.999425i \(-0.489202\pi\)
0.0339154 + 0.999425i \(0.489202\pi\)
\(398\) 0 0
\(399\) −6.65279 −0.333056
\(400\) 0 0
\(401\) 0.579121 0.0289199 0.0144600 0.999895i \(-0.495397\pi\)
0.0144600 + 0.999895i \(0.495397\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −9.13706 −0.454024
\(406\) 0 0
\(407\) 13.6866 0.678422
\(408\) 0 0
\(409\) −15.1575 −0.749490 −0.374745 0.927128i \(-0.622270\pi\)
−0.374745 + 0.927128i \(0.622270\pi\)
\(410\) 0 0
\(411\) 0.439665 0.0216871
\(412\) 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
\(421\) −35.0465 −1.70806 −0.854032 0.520221i \(-0.825849\pi\)
−0.854032 + 0.520221i \(0.825849\pi\)
\(422\) 0 0
\(423\) −7.94438 −0.386269
\(424\) 0 0
\(425\) 15.4209 0.748022
\(426\) 0 0
\(427\) −17.5483 −0.849220
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 34.2814 1.65128 0.825639 0.564199i \(-0.190815\pi\)
0.825639 + 0.564199i \(0.190815\pi\)
\(432\) 0 0
\(433\) 13.7385 0.660232 0.330116 0.943940i \(-0.392912\pi\)
0.330116 + 0.943940i \(0.392912\pi\)
\(434\) 0 0
\(435\) −1.81940 −0.0872334
\(436\) 0 0
\(437\) 11.0586 0.529005
\(438\) 0 0
\(439\) −10.2403 −0.488742 −0.244371 0.969682i \(-0.578581\pi\)
−0.244371 + 0.969682i \(0.578581\pi\)
\(440\) 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
\(447\) 4.66487 0.220641
\(448\) 0 0
\(449\) −12.9051 −0.609032 −0.304516 0.952507i \(-0.598495\pi\)
−0.304516 + 0.952507i \(0.598495\pi\)
\(450\) 0 0
\(451\) 3.25534 0.153288
\(452\) 0 0
\(453\) −7.84117 −0.368410
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 4.65710 0.217850 0.108925 0.994050i \(-0.465259\pi\)
0.108925 + 0.994050i \(0.465259\pi\)
\(458\) 0 0
\(459\) 16.7289 0.780836
\(460\) 0 0
\(461\) −31.5405 −1.46899 −0.734493 0.678616i \(-0.762580\pi\)
−0.734493 + 0.678616i \(0.762580\pi\)
\(462\) 0 0
\(463\) −17.6504 −0.820284 −0.410142 0.912022i \(-0.634521\pi\)
−0.410142 + 0.912022i \(0.634521\pi\)
\(464\) 0 0
\(465\) −3.42327 −0.158750
\(466\) 0 0
\(467\) 32.1726 1.48877 0.744385 0.667751i \(-0.232743\pi\)
0.744385 + 0.667751i \(0.232743\pi\)
\(468\) 0 0
\(469\) 1.18167 0.0545644
\(470\) 0 0
\(471\) −5.23490 −0.241211
\(472\) 0 0
\(473\) −15.6799 −0.720964
\(474\) 0 0
\(475\) −17.0368 −0.781704
\(476\) 0 0
\(477\) −14.8629 −0.680527
\(478\) 0 0
\(479\) 34.8998 1.59461 0.797306 0.603576i \(-0.206258\pi\)
0.797306 + 0.603576i \(0.206258\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) −2.14914 −0.0977895
\(484\) 0 0
\(485\) −17.2349 −0.782596
\(486\) 0 0
\(487\) −41.8351 −1.89573 −0.947864 0.318676i \(-0.896762\pi\)
−0.947864 + 0.318676i \(0.896762\pi\)
\(488\) 0 0
\(489\) −4.83041 −0.218439
\(490\) 0 0
\(491\) −21.8455 −0.985873 −0.492936 0.870065i \(-0.664076\pi\)
−0.492936 + 0.870065i \(0.664076\pi\)
\(492\) 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
\(499\) −23.5472 −1.05412 −0.527058 0.849829i \(-0.676705\pi\)
−0.527058 + 0.849829i \(0.676705\pi\)
\(500\) 0 0
\(501\) 13.2379 0.591425
\(502\) 0 0
\(503\) 7.08682 0.315986 0.157993 0.987440i \(-0.449498\pi\)
0.157993 + 0.987440i \(0.449498\pi\)
\(504\) 0 0
\(505\) −18.8780 −0.840060
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 7.61894 0.337704 0.168852 0.985641i \(-0.445994\pi\)
0.168852 + 0.985641i \(0.445994\pi\)
\(510\) 0 0
\(511\) 21.6213 0.956471
\(512\) 0 0
\(513\) −18.4819 −0.815995
\(514\) 0 0
\(515\) 13.2489 0.583816
\(516\) 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
\(525\) 3.31096 0.144502
\(526\) 0 0
\(527\) −22.6069 −0.984770
\(528\) 0 0
\(529\) −19.4276 −0.844678
\(530\) 0 0
\(531\) −32.8635 −1.42616
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −9.97046 −0.431061
\(536\) 0 0
\(537\) −3.34183 −0.144211
\(538\) 0 0
\(539\) −7.15883 −0.308353
\(540\) 0 0
\(541\) 34.4819 1.48249 0.741246 0.671234i \(-0.234235\pi\)
0.741246 + 0.671234i \(0.234235\pi\)
\(542\) 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
\(551\) 13.2741 0.565497
\(552\) 0 0
\(553\) −32.3274 −1.37470
\(554\) 0 0
\(555\) −4.29590 −0.182351
\(556\) 0 0
\(557\) 1.27652 0.0540879 0.0270439 0.999634i \(-0.491391\pi\)
0.0270439 + 0.999634i \(0.491391\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −7.50902 −0.317031
\(562\) 0 0
\(563\) 9.12737 0.384673 0.192336 0.981329i \(-0.438393\pi\)
0.192336 + 0.981329i \(0.438393\pi\)
\(564\) 0 0
\(565\) −10.5603 −0.444277
\(566\) 0 0
\(567\) −12.9554 −0.544075
\(568\) 0 0
\(569\) −5.72156 −0.239860 −0.119930 0.992782i \(-0.538267\pi\)
−0.119930 + 0.992782i \(0.538267\pi\)
\(570\) 0 0
\(571\) −7.60148 −0.318112 −0.159056 0.987270i \(-0.550845\pi\)
−0.159056 + 0.987270i \(0.550845\pi\)
\(572\) 0 0
\(573\) −10.2282 −0.427289
\(574\) 0 0
\(575\) −5.50365 −0.229518
\(576\) 0 0
\(577\) 45.1564 1.87989 0.939944 0.341330i \(-0.110877\pi\)
0.939944 + 0.341330i \(0.110877\pi\)
\(578\) 0 0
\(579\) 3.35988 0.139632
\(580\) 0 0
\(581\) 15.8248 0.656522
\(582\) 0 0
\(583\) 14.1062 0.584219
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 32.4040 1.33746 0.668728 0.743507i \(-0.266839\pi\)
0.668728 + 0.743507i \(0.266839\pi\)
\(588\) 0 0
\(589\) 24.9758 1.02911
\(590\) 0 0
\(591\) −6.33513 −0.260592
\(592\) 0 0
\(593\) 36.6848 1.50647 0.753233 0.657754i \(-0.228493\pi\)
0.753233 + 0.657754i \(0.228493\pi\)
\(594\) 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
\(603\) 1.55257 0.0632253
\(604\) 0 0
\(605\) 6.46250 0.262738
\(606\) 0 0
\(607\) −11.2161 −0.455248 −0.227624 0.973749i \(-0.573096\pi\)
−0.227624 + 0.973749i \(0.573096\pi\)
\(608\) 0 0
\(609\) −2.57971 −0.104535
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −20.8944 −0.843917 −0.421958 0.906615i \(-0.638657\pi\)
−0.421958 + 0.906615i \(0.638657\pi\)
\(614\) 0 0
\(615\) −1.02177 −0.0412018
\(616\) 0 0
\(617\) 12.0992 0.487094 0.243547 0.969889i \(-0.421689\pi\)
0.243547 + 0.969889i \(0.421689\pi\)
\(618\) 0 0
\(619\) 10.5526 0.424143 0.212072 0.977254i \(-0.431979\pi\)
0.212072 + 0.977254i \(0.431979\pi\)
\(620\) 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
\(629\) −28.3696 −1.13117
\(630\) 0 0
\(631\) 13.8514 0.551417 0.275709 0.961241i \(-0.411087\pi\)
0.275709 + 0.961241i \(0.411087\pi\)
\(632\) 0 0
\(633\) 7.35019 0.292144
\(634\) 0 0
\(635\) −27.4131 −1.08786
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −12.3676 −0.489257
\(640\) 0 0
\(641\) 34.9608 1.38087 0.690434 0.723396i \(-0.257420\pi\)
0.690434 + 0.723396i \(0.257420\pi\)
\(642\) 0 0
\(643\) −33.3980 −1.31709 −0.658545 0.752541i \(-0.728828\pi\)
−0.658545 + 0.752541i \(0.728828\pi\)
\(644\) 0 0
\(645\) 4.92154 0.193786
\(646\) 0 0
\(647\) −2.32842 −0.0915397 −0.0457698 0.998952i \(-0.514574\pi\)
−0.0457698 + 0.998952i \(0.514574\pi\)
\(648\) 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
\(655\) 4.70602 0.183879
\(656\) 0 0
\(657\) 28.4077 1.10829
\(658\) 0 0
\(659\) −11.1395 −0.433932 −0.216966 0.976179i \(-0.569616\pi\)
−0.216966 + 0.976179i \(0.569616\pi\)
\(660\) 0 0
\(661\) −13.8498 −0.538694 −0.269347 0.963043i \(-0.586808\pi\)
−0.269347 + 0.963043i \(0.586808\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 17.3230 0.671759
\(666\) 0 0
\(667\) 4.28813 0.166037
\(668\) 0 0
\(669\) 4.07069 0.157382
\(670\) 0 0
\(671\) 21.8823 0.844757
\(672\) 0 0
\(673\) −6.52973 −0.251703 −0.125851 0.992049i \(-0.540166\pi\)
−0.125851 + 0.992049i \(0.540166\pi\)
\(674\) 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
\(681\) 4.81402 0.184474
\(682\) 0 0
\(683\) −14.1793 −0.542555 −0.271277 0.962501i \(-0.587446\pi\)
−0.271277 + 0.962501i \(0.587446\pi\)
\(684\) 0 0
\(685\) −1.14483 −0.0437418
\(686\) 0 0
\(687\) −7.59551 −0.289787
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −30.7952 −1.17151 −0.585753 0.810490i \(-0.699201\pi\)
−0.585753 + 0.810490i \(0.699201\pi\)
\(692\) 0 0
\(693\) 14.0925 0.535328
\(694\) 0 0
\(695\) −16.3860 −0.621555
\(696\) 0 0
\(697\) −6.74764 −0.255585
\(698\) 0 0
\(699\) −2.82371 −0.106802
\(700\) 0 0
\(701\) 6.73184 0.254258 0.127129 0.991886i \(-0.459424\pi\)
0.127129 + 0.991886i \(0.459424\pi\)
\(702\) 0 0
\(703\) 31.3424 1.18210
\(704\) 0 0
\(705\) −2.36658 −0.0891307
\(706\) 0 0
\(707\) −26.7670 −1.00668
\(708\) 0 0
\(709\) −47.6252 −1.78860 −0.894300 0.447467i \(-0.852326\pi\)
−0.894300 + 0.447467i \(0.852326\pi\)
\(710\) 0 0
\(711\) −42.4741 −1.59290
\(712\) 0 0
\(713\) 8.06829 0.302160
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 6.06233 0.226402
\(718\) 0 0
\(719\) 5.99330 0.223512 0.111756 0.993736i \(-0.464352\pi\)
0.111756 + 0.993736i \(0.464352\pi\)
\(720\) 0 0
\(721\) 18.7855 0.699610
\(722\) 0 0
\(723\) 6.61224 0.245912
\(724\) 0 0
\(725\) −6.60627 −0.245351
\(726\) 0 0
\(727\) 24.1226 0.894657 0.447329 0.894370i \(-0.352375\pi\)
0.447329 + 0.894370i \(0.352375\pi\)
\(728\) 0 0
\(729\) −11.7627 −0.435656
\(730\) 0 0
\(731\) 32.5013 1.20210
\(732\) 0 0
\(733\) 36.0646 1.33208 0.666038 0.745918i \(-0.267989\pi\)
0.666038 + 0.745918i \(0.267989\pi\)
\(734\) 0 0
\(735\) 2.24698 0.0828811
\(736\) 0 0
\(737\) −1.47352 −0.0542777
\(738\) 0 0
\(739\) −27.5254 −1.01254 −0.506269 0.862375i \(-0.668976\pi\)
−0.506269 + 0.862375i \(0.668976\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 10.4692 0.384078 0.192039 0.981387i \(-0.438490\pi\)
0.192039 + 0.981387i \(0.438490\pi\)
\(744\) 0 0
\(745\) −12.1468 −0.445023
\(746\) 0 0
\(747\) 20.7918 0.760731
\(748\) 0 0
\(749\) −14.1371 −0.516557
\(750\) 0 0
\(751\) −4.06770 −0.148433 −0.0742163 0.997242i \(-0.523646\pi\)
−0.0742163 + 0.997242i \(0.523646\pi\)
\(752\) 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
\(759\) 2.67994 0.0972757
\(760\) 0 0
\(761\) −27.0237 −0.979608 −0.489804 0.871833i \(-0.662932\pi\)
−0.489804 + 0.871833i \(0.662932\pi\)
\(762\) 0 0
\(763\) −0.249964 −0.00904929
\(764\) 0 0
\(765\) −20.6015 −0.744848
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −37.9407 −1.36818 −0.684088 0.729400i \(-0.739799\pi\)
−0.684088 + 0.729400i \(0.739799\pi\)
\(770\) 0 0
\(771\) −10.3556 −0.372947
\(772\) 0 0
\(773\) −16.3375 −0.587620 −0.293810 0.955864i \(-0.594923\pi\)
−0.293810 + 0.955864i \(0.594923\pi\)
\(774\) 0 0
\(775\) −12.4300 −0.446498
\(776\) 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
\(785\) 13.6310 0.486512
\(786\) 0 0
\(787\) 18.6907 0.666251 0.333126 0.942882i \(-0.391897\pi\)
0.333126 + 0.942882i \(0.391897\pi\)
\(788\) 0 0
\(789\) −7.99090 −0.284484
\(790\) 0 0
\(791\) −14.9734 −0.532394
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −4.42758 −0.157030
\(796\) 0 0
\(797\) 29.2519 1.03615 0.518077 0.855334i \(-0.326648\pi\)
0.518077 + 0.855334i \(0.326648\pi\)
\(798\) 0 0
\(799\) −15.6286 −0.552901
\(800\) 0 0
\(801\) −17.8039 −0.629068
\(802\) 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
\(811\) −3.89200 −0.136667 −0.0683333 0.997663i \(-0.521768\pi\)
−0.0683333 + 0.997663i \(0.521768\pi\)
\(812\) 0 0
\(813\) 1.10752 0.0388425
\(814\) 0 0
\(815\) 12.5778 0.440581
\(816\) 0 0
\(817\) −35.9071 −1.25623
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 45.9982 1.60535 0.802674 0.596418i \(-0.203410\pi\)
0.802674 + 0.596418i \(0.203410\pi\)
\(822\) 0 0
\(823\) −7.95300 −0.277224 −0.138612 0.990347i \(-0.544264\pi\)
−0.138612 + 0.990347i \(0.544264\pi\)
\(824\) 0 0
\(825\) −4.12870 −0.143743
\(826\) 0 0
\(827\) −27.9648 −0.972432 −0.486216 0.873839i \(-0.661623\pi\)
−0.486216 + 0.873839i \(0.661623\pi\)
\(828\) 0 0
\(829\) 27.6310 0.959665 0.479833 0.877360i \(-0.340697\pi\)
0.479833 + 0.877360i \(0.340697\pi\)
\(830\) 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
\(837\) −13.4843 −0.466085
\(838\) 0 0
\(839\) −28.6848 −0.990311 −0.495155 0.868804i \(-0.664889\pi\)
−0.495155 + 0.868804i \(0.664889\pi\)
\(840\) 0 0
\(841\) −23.8528 −0.822509
\(842\) 0 0
\(843\) −3.59179 −0.123708
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 9.16315 0.314849
\(848\) 0 0
\(849\) −3.65220 −0.125343
\(850\) 0 0
\(851\) 10.1250 0.347080
\(852\) 0 0
\(853\) −43.2078 −1.47941 −0.739703 0.672934i \(-0.765034\pi\)
−0.739703 + 0.672934i \(0.765034\pi\)
\(854\) 0 0
\(855\) 22.7603 0.778386
\(856\) 0 0
\(857\) 35.1685 1.20133 0.600667 0.799499i \(-0.294902\pi\)
0.600667 + 0.799499i \(0.294902\pi\)
\(858\) 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
\(863\) 41.3913 1.40898 0.704489 0.709715i \(-0.251176\pi\)
0.704489 + 0.709715i \(0.251176\pi\)
\(864\) 0 0
\(865\) 27.2465 0.926409
\(866\) 0 0
\(867\) 6.13036 0.208198
\(868\) 0 0
\(869\) 40.3116 1.36748
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −32.1075 −1.08668
\(874\) 0 0
\(875\) −23.4252 −0.791916
\(876\) 0 0
\(877\) −24.7472 −0.835653 −0.417826 0.908527i \(-0.637208\pi\)
−0.417826 + 0.908527i \(0.637208\pi\)
\(878\) 0 0
\(879\) −13.5066 −0.455567
\(880\) 0 0
\(881\) −28.5875 −0.963137 −0.481568 0.876409i \(-0.659933\pi\)
−0.481568 + 0.876409i \(0.659933\pi\)
\(882\) 0 0
\(883\) −9.61702 −0.323639 −0.161819 0.986820i \(-0.551736\pi\)
−0.161819 + 0.986820i \(0.551736\pi\)
\(884\) 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
\(889\) −38.8689 −1.30362
\(890\) 0 0
\(891\) 16.1551 0.541217
\(892\) 0 0
\(893\) 17.2664 0.577797
\(894\) 0 0
\(895\) 8.70171 0.290866
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 9.68473 0.323004
\(900\) 0 0
\(901\) −29.2392 −0.974099
\(902\) 0 0
\(903\) 6.97823 0.232221
\(904\) 0 0
\(905\) 6.90408 0.229500
\(906\) 0 0
\(907\) 28.8364 0.957496 0.478748 0.877952i \(-0.341091\pi\)
0.478748 + 0.877952i \(0.341091\pi\)
\(908\) 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
\(915\) −6.86831 −0.227059
\(916\) 0 0
\(917\) 6.67264 0.220350
\(918\) 0 0
\(919\) −8.87502 −0.292760 −0.146380 0.989228i \(-0.546762\pi\)
−0.146380 + 0.989228i \(0.546762\pi\)
\(920\) 0 0
\(921\) 7.81030 0.257358
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −15.5985 −0.512875
\(926\) 0 0
\(927\) 24.6819 0.810659
\(928\) 0 0
\(929\) 24.2295 0.794945 0.397472 0.917614i \(-0.369887\pi\)
0.397472 + 0.917614i \(0.369887\pi\)
\(930\) 0 0
\(931\) −16.3937 −0.537283
\(932\) 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
\(941\) 4.34050 0.141496 0.0707482 0.997494i \(-0.477461\pi\)
0.0707482 + 0.997494i \(0.477461\pi\)
\(942\) 0 0
\(943\) 2.40821 0.0784220
\(944\) 0 0
\(945\) −9.35258 −0.304240
\(946\) 0 0
\(947\) 45.0146 1.46278 0.731389 0.681961i \(-0.238872\pi\)
0.731389 + 0.681961i \(0.238872\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 16.6655 0.540415
\(952\) 0 0
\(953\) −46.8859 −1.51878 −0.759391 0.650634i \(-0.774503\pi\)
−0.759391 + 0.650634i \(0.774503\pi\)
\(954\) 0 0
\(955\) 26.6329 0.861822
\(956\) 0 0
\(957\) 3.21685 0.103986
\(958\) 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
\(967\) −6.29457 −0.202420 −0.101210 0.994865i \(-0.532271\pi\)
−0.101210 + 0.994865i \(0.532271\pi\)
\(968\) 0 0
\(969\) −17.1957 −0.552404
\(970\) 0 0
\(971\) 41.8068 1.34165 0.670823 0.741618i \(-0.265941\pi\)
0.670823 + 0.741618i \(0.265941\pi\)
\(972\) 0 0
\(973\) −23.2336 −0.744834
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 23.7530 0.759926 0.379963 0.925002i \(-0.375937\pi\)
0.379963 + 0.925002i \(0.375937\pi\)
\(978\) 0 0
\(979\) 16.8974 0.540043
\(980\) 0 0
\(981\) −0.328421 −0.0104857
\(982\) 0 0
\(983\) 55.7251 1.77736 0.888678 0.458532i \(-0.151625\pi\)
0.888678 + 0.458532i \(0.151625\pi\)
\(984\) 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
\(993\) 8.72215 0.276789
\(994\) 0 0
\(995\) −20.0935 −0.637007
\(996\) 0 0
\(997\) 6.61058 0.209359 0.104680 0.994506i \(-0.466618\pi\)
0.104680 + 0.994506i \(0.466618\pi\)
\(998\) 0 0
\(999\) −16.9215 −0.535374
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.a.z.1.2 3
4.3 odd 2 169.2.a.b.1.1 3
12.11 even 2 1521.2.a.r.1.3 3
13.5 odd 4 2704.2.f.o.337.4 6
13.8 odd 4 2704.2.f.o.337.3 6
13.12 even 2 2704.2.a.ba.1.2 3
20.19 odd 2 4225.2.a.bg.1.3 3
28.27 even 2 8281.2.a.bf.1.1 3
52.3 odd 6 169.2.c.c.22.3 6
52.7 even 12 169.2.e.b.23.1 12
52.11 even 12 169.2.e.b.147.6 12
52.15 even 12 169.2.e.b.147.1 12
52.19 even 12 169.2.e.b.23.6 12
52.23 odd 6 169.2.c.b.22.1 6
52.31 even 4 169.2.b.b.168.6 6
52.35 odd 6 169.2.c.c.146.3 6
52.43 odd 6 169.2.c.b.146.1 6
52.47 even 4 169.2.b.b.168.1 6
52.51 odd 2 169.2.a.c.1.3 yes 3
156.47 odd 4 1521.2.b.l.1351.6 6
156.83 odd 4 1521.2.b.l.1351.1 6
156.155 even 2 1521.2.a.o.1.1 3
260.259 odd 2 4225.2.a.bb.1.1 3
364.363 even 2 8281.2.a.bj.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
169.2.a.b.1.1 3 4.3 odd 2
169.2.a.c.1.3 yes 3 52.51 odd 2
169.2.b.b.168.1 6 52.47 even 4
169.2.b.b.168.6 6 52.31 even 4
169.2.c.b.22.1 6 52.23 odd 6
169.2.c.b.146.1 6 52.43 odd 6
169.2.c.c.22.3 6 52.3 odd 6
169.2.c.c.146.3 6 52.35 odd 6
169.2.e.b.23.1 12 52.7 even 12
169.2.e.b.23.6 12 52.19 even 12
169.2.e.b.147.1 12 52.15 even 12
169.2.e.b.147.6 12 52.11 even 12
1521.2.a.o.1.1 3 156.155 even 2
1521.2.a.r.1.3 3 12.11 even 2
1521.2.b.l.1351.1 6 156.83 odd 4
1521.2.b.l.1351.6 6 156.47 odd 4
2704.2.a.z.1.2 3 1.1 even 1 trivial
2704.2.a.ba.1.2 3 13.12 even 2
2704.2.f.o.337.3 6 13.8 odd 4
2704.2.f.o.337.4 6 13.5 odd 4
4225.2.a.bb.1.1 3 260.259 odd 2
4225.2.a.bg.1.3 3 20.19 odd 2
8281.2.a.bf.1.1 3 28.27 even 2
8281.2.a.bj.1.3 3 364.363 even 2