Properties

Label 2704.2.a.ba.1.2
Level $2704$
Weight $2$
Character 2704.1
Self dual yes
Analytic conductor $21.592$
Analytic rank $1$
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: \(1\)
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 \(-0.395268\pi\)
0.323121 + 0.946358i \(0.395268\pi\)
\(6\) 0 0
\(7\) 2.04892 0.774418 0.387209 0.921992i \(-0.373439\pi\)
0.387209 + 0.921992i \(0.373439\pi\)
\(8\) 0 0
\(9\) −2.69202 −0.897340
\(10\) 0 0
\(11\) −2.55496 −0.770349 −0.385174 0.922844i \(-0.625859\pi\)
−0.385174 + 0.922844i \(0.625859\pi\)
\(12\) 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 \(-0.734195\pi\)
−0.671139 + 0.741331i \(0.734195\pi\)
\(20\) 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 \(-0.625228\pi\)
−0.383345 + 0.923605i \(0.625228\pi\)
\(32\) 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 \(-0.645140\pi\)
−0.440334 + 0.897834i \(0.645140\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1.27413 −0.198985 −0.0994926 0.995038i \(-0.531722\pi\)
−0.0994926 + 0.995038i \(0.531722\pi\)
\(42\) 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 \(-0.569050\pi\)
−0.215230 + 0.976563i \(0.569050\pi\)
\(48\) 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 \(-0.792349\pi\)
−0.794657 + 0.607059i \(0.792349\pi\)
\(60\) 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 \(-0.488784\pi\)
0.0352293 + 0.999379i \(0.488784\pi\)
\(68\) 0 0
\(69\) 1.04892 0.126275
\(70\) 0 0
\(71\) −4.59419 −0.545230 −0.272615 0.962123i \(-0.587888\pi\)
−0.272615 + 0.962123i \(0.587888\pi\)
\(72\) 0 0
\(73\) 10.5526 1.23508 0.617542 0.786538i \(-0.288128\pi\)
0.617542 + 0.786538i \(0.288128\pi\)
\(74\) 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 \(-0.360667\pi\)
0.423881 + 0.905718i \(0.360667\pi\)
\(84\) 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 \(-0.613995\pi\)
−0.350518 + 0.936556i \(0.613995\pi\)
\(90\) 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 \(-0.707026\pi\)
−0.605498 + 0.795847i \(0.707026\pi\)
\(98\) 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 \(-0.501860\pi\)
−0.00584264 + 0.999983i \(0.501860\pi\)
\(110\) 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 \(-0.510775\pi\)
−0.0338432 + 0.999427i \(0.510775\pi\)
\(138\) 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 \(-0.611889\pi\)
−0.344316 + 0.938854i \(0.611889\pi\)
\(150\) 0 0
\(151\) 14.1293 1.14983 0.574913 0.818215i \(-0.305036\pi\)
0.574913 + 0.818215i \(0.305036\pi\)
\(152\) 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 \(-0.389275\pi\)
0.340879 + 0.940107i \(0.389275\pi\)
\(164\) 0 0
\(165\) −2.04892 −0.159508
\(166\) 0 0
\(167\) −23.8538 −1.84587 −0.922933 0.384961i \(-0.874215\pi\)
−0.922933 + 0.384961i \(0.874215\pi\)
\(168\) 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 \(-0.569920\pi\)
−0.217899 + 0.975971i \(0.569920\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 11.4155 0.813321 0.406660 0.913579i \(-0.366693\pi\)
0.406660 + 0.913579i \(0.366693\pi\)
\(198\) 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 \(-0.578984\pi\)
−0.245598 + 0.969372i \(0.578984\pi\)
\(224\) 0 0
\(225\) 7.83877 0.522585
\(226\) 0 0
\(227\) −8.67456 −0.575751 −0.287875 0.957668i \(-0.592949\pi\)
−0.287875 + 0.957668i \(0.592949\pi\)
\(228\) 0 0
\(229\) 13.6866 0.904439 0.452219 0.891907i \(-0.350632\pi\)
0.452219 + 0.891907i \(0.350632\pi\)
\(230\) 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 \(-0.614942\pi\)
−0.353305 + 0.935508i \(0.614942\pi\)
\(240\) 0 0
\(241\) −11.9148 −0.767502 −0.383751 0.923437i \(-0.625368\pi\)
−0.383751 + 0.923437i \(0.625368\pi\)
\(242\) 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 \(-0.519306\pi\)
−0.0606147 + 0.998161i \(0.519306\pi\)
\(272\) 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 \(-0.438162\pi\)
0.193049 + 0.981189i \(0.438162\pi\)
\(282\) 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 \(-0.248277\pi\)
0.710924 + 0.703269i \(0.248277\pi\)
\(294\) 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 \(-0.631550\pi\)
−0.401613 + 0.915809i \(0.631550\pi\)
\(308\) 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 \(-0.819408\pi\)
−0.843330 + 0.537396i \(0.819408\pi\)
\(318\) 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 \(-0.642169\pi\)
−0.431936 + 0.901904i \(0.642169\pi\)
\(332\) 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 \(-0.409620\pi\)
0.280138 + 0.959960i \(0.409620\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −15.5308 −0.826621 −0.413310 0.910590i \(-0.635628\pi\)
−0.413310 + 0.910590i \(0.635628\pi\)
\(354\) 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 \(-0.691285\pi\)
−0.565417 + 0.824805i \(0.691285\pi\)
\(360\) 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 \(-0.360357\pi\)
0.424765 + 0.905304i \(0.360357\pi\)
\(380\) 0 0
\(381\) 10.5278 0.539356
\(382\) 0 0
\(383\) 7.53617 0.385080 0.192540 0.981289i \(-0.438327\pi\)
0.192540 + 0.981289i \(0.438327\pi\)
\(384\) 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 \(-0.510798\pi\)
−0.0339154 + 0.999425i \(0.510798\pi\)
\(398\) 0 0
\(399\) −6.65279 −0.333056
\(400\) 0 0
\(401\) −0.579121 −0.0289199 −0.0144600 0.999895i \(-0.504603\pi\)
−0.0144600 + 0.999895i \(0.504603\pi\)
\(402\) 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 \(-0.377730\pi\)
0.374745 + 0.927128i \(0.377730\pi\)
\(410\) 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 \(-0.174151\pi\)
0.854032 + 0.520221i \(0.174151\pi\)
\(422\) 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 \(-0.809185\pi\)
−0.825639 + 0.564199i \(0.809185\pi\)
\(432\) 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 \(-0.401505\pi\)
0.304516 + 0.952507i \(0.401505\pi\)
\(450\) 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 \(-0.534741\pi\)
−0.108925 + 0.994050i \(0.534741\pi\)
\(458\) 0 0
\(459\) 16.7289 0.780836
\(460\) 0 0
\(461\) 31.5405 1.46899 0.734493 0.678616i \(-0.237420\pi\)
0.734493 + 0.678616i \(0.237420\pi\)
\(462\) 0 0
\(463\) 17.6504 0.820284 0.410142 0.912022i \(-0.365479\pi\)
0.410142 + 0.912022i \(0.365479\pi\)
\(464\) 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 \(-0.793742\pi\)
−0.797306 + 0.603576i \(0.793742\pi\)
\(480\) 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 \(-0.103238\pi\)
0.947864 + 0.318676i \(0.103238\pi\)
\(488\) 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 \(-0.323295\pi\)
0.527058 + 0.849829i \(0.323295\pi\)
\(500\) 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 \(-0.554006\pi\)
−0.168852 + 0.985641i \(0.554006\pi\)
\(510\) 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 \(-0.765765\pi\)
−0.741246 + 0.671234i \(0.765765\pi\)
\(542\) 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 \(-0.508609\pi\)
−0.0270439 + 0.999634i \(0.508609\pi\)
\(558\) 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 \(-0.889123\pi\)
−0.939944 + 0.341330i \(0.889123\pi\)
\(578\) 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 \(-0.733161\pi\)
−0.668728 + 0.743507i \(0.733161\pi\)
\(588\) 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 \(-0.771507\pi\)
−0.753233 + 0.657754i \(0.771507\pi\)
\(594\) 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 \(-0.361343\pi\)
0.421958 + 0.906615i \(0.361343\pi\)
\(614\) 0 0
\(615\) −1.02177 −0.0412018
\(616\) 0 0
\(617\) −12.0992 −0.487094 −0.243547 0.969889i \(-0.578311\pi\)
−0.243547 + 0.969889i \(0.578311\pi\)
\(618\) 0 0
\(619\) −10.5526 −0.424143 −0.212072 0.977254i \(-0.568021\pi\)
−0.212072 + 0.977254i \(0.568021\pi\)
\(620\) 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 \(-0.588913\pi\)
−0.275709 + 0.961241i \(0.588913\pi\)
\(632\) 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 \(-0.271172\pi\)
0.658545 + 0.752541i \(0.271172\pi\)
\(644\) 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 \(-0.413192\pi\)
0.269347 + 0.963043i \(0.413192\pi\)
\(662\) 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 \(-0.412554\pi\)
0.271277 + 0.962501i \(0.412554\pi\)
\(684\) 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 \(-0.300799\pi\)
0.585753 + 0.810490i \(0.300799\pi\)
\(692\) 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 \(-0.147674\pi\)
0.894300 + 0.447467i \(0.147674\pi\)
\(710\) 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 \(-0.732011\pi\)
−0.666038 + 0.745918i \(0.732011\pi\)
\(734\) 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 \(-0.331024\pi\)
0.506269 + 0.862375i \(0.331024\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −10.4692 −0.384078 −0.192039 0.981387i \(-0.561510\pi\)
−0.192039 + 0.981387i \(0.561510\pi\)
\(744\) 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 \(-0.337068\pi\)
0.489804 + 0.871833i \(0.337068\pi\)
\(762\) 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 \(-0.260201\pi\)
0.684088 + 0.729400i \(0.260201\pi\)
\(770\) 0 0
\(771\) −10.3556 −0.372947
\(772\) 0 0
\(773\) 16.3375 0.587620 0.293810 0.955864i \(-0.405077\pi\)
0.293810 + 0.955864i \(0.405077\pi\)
\(774\) 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 \(-0.608103\pi\)
−0.333126 + 0.942882i \(0.608103\pi\)
\(788\) 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 \(-0.478232\pi\)
0.0683333 + 0.997663i \(0.478232\pi\)
\(812\) 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 \(-0.796590\pi\)
−0.802674 + 0.596418i \(0.796590\pi\)
\(822\) 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 \(-0.338377\pi\)
0.486216 + 0.873839i \(0.338377\pi\)
\(828\) 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 \(-0.335111\pi\)
0.495155 + 0.868804i \(0.335111\pi\)
\(840\) 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 \(-0.234966\pi\)
0.739703 + 0.672934i \(0.234966\pi\)
\(854\) 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 \(-0.748824\pi\)
−0.704489 + 0.709715i \(0.748824\pi\)
\(864\) 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 \(-0.362792\pi\)
0.417826 + 0.908527i \(0.362792\pi\)
\(878\) 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 \(-0.630113\pi\)
−0.397472 + 0.917614i \(0.630113\pi\)
\(930\) 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 \(-0.522539\pi\)
−0.0707482 + 0.997494i \(0.522539\pi\)
\(942\) 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 \(-0.761128\pi\)
−0.731389 + 0.681961i \(0.761128\pi\)
\(948\) 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 \(-0.467729\pi\)
0.101210 + 0.994865i \(0.467729\pi\)
\(968\) 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 \(-0.624063\pi\)
−0.379963 + 0.925002i \(0.624063\pi\)
\(978\) 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 \(-0.848375\pi\)
−0.888678 + 0.458532i \(0.848375\pi\)
\(984\) 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.ba.1.2 3
4.3 odd 2 169.2.a.c.1.3 yes 3
12.11 even 2 1521.2.a.o.1.1 3
13.5 odd 4 2704.2.f.o.337.3 6
13.8 odd 4 2704.2.f.o.337.4 6
13.12 even 2 2704.2.a.z.1.2 3
20.19 odd 2 4225.2.a.bb.1.1 3
28.27 even 2 8281.2.a.bj.1.3 3
52.3 odd 6 169.2.c.b.22.1 6
52.7 even 12 169.2.e.b.23.6 12
52.11 even 12 169.2.e.b.147.1 12
52.15 even 12 169.2.e.b.147.6 12
52.19 even 12 169.2.e.b.23.1 12
52.23 odd 6 169.2.c.c.22.3 6
52.31 even 4 169.2.b.b.168.1 6
52.35 odd 6 169.2.c.b.146.1 6
52.43 odd 6 169.2.c.c.146.3 6
52.47 even 4 169.2.b.b.168.6 6
52.51 odd 2 169.2.a.b.1.1 3
156.47 odd 4 1521.2.b.l.1351.1 6
156.83 odd 4 1521.2.b.l.1351.6 6
156.155 even 2 1521.2.a.r.1.3 3
260.259 odd 2 4225.2.a.bg.1.3 3
364.363 even 2 8281.2.a.bf.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
169.2.a.b.1.1 3 52.51 odd 2
169.2.a.c.1.3 yes 3 4.3 odd 2
169.2.b.b.168.1 6 52.31 even 4
169.2.b.b.168.6 6 52.47 even 4
169.2.c.b.22.1 6 52.3 odd 6
169.2.c.b.146.1 6 52.35 odd 6
169.2.c.c.22.3 6 52.23 odd 6
169.2.c.c.146.3 6 52.43 odd 6
169.2.e.b.23.1 12 52.19 even 12
169.2.e.b.23.6 12 52.7 even 12
169.2.e.b.147.1 12 52.11 even 12
169.2.e.b.147.6 12 52.15 even 12
1521.2.a.o.1.1 3 12.11 even 2
1521.2.a.r.1.3 3 156.155 even 2
1521.2.b.l.1351.1 6 156.47 odd 4
1521.2.b.l.1351.6 6 156.83 odd 4
2704.2.a.z.1.2 3 13.12 even 2
2704.2.a.ba.1.2 3 1.1 even 1 trivial
2704.2.f.o.337.3 6 13.5 odd 4
2704.2.f.o.337.4 6 13.8 odd 4
4225.2.a.bb.1.1 3 20.19 odd 2
4225.2.a.bg.1.3 3 260.259 odd 2
8281.2.a.bf.1.1 3 364.363 even 2
8281.2.a.bj.1.3 3 28.27 even 2