Properties

Label 2704.2.a.z.1.1
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 1.1
Root \(1.80194\) 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)
 
\(f(q)\) \(=\) \(q-0.801938 q^{3} -2.80194 q^{5} +2.69202 q^{7} -2.35690 q^{9} +O(q^{10})\) \(q-0.801938 q^{3} -2.80194 q^{5} +2.69202 q^{7} -2.35690 q^{9} +1.19806 q^{11} +2.24698 q^{15} +1.13706 q^{17} -1.93900 q^{19} -2.15883 q^{21} +4.60388 q^{23} +2.85086 q^{25} +4.29590 q^{27} -7.89977 q^{29} -5.89977 q^{31} -0.960771 q^{33} -7.54288 q^{35} +0.951083 q^{37} +3.31767 q^{41} -7.15883 q^{43} +6.60388 q^{45} +7.69202 q^{47} +0.246980 q^{49} -0.911854 q^{51} +5.87263 q^{53} -3.35690 q^{55} +1.55496 q^{57} +0.0120816 q^{59} -8.03684 q^{61} -6.34481 q^{63} +9.25667 q^{67} -3.69202 q^{69} +13.7409 q^{71} +12.8170 q^{73} -2.28621 q^{75} +3.22521 q^{77} -0.807315 q^{79} +3.62565 q^{81} +16.3327 q^{83} -3.18598 q^{85} +6.33513 q^{87} -14.7289 q^{89} +4.73125 q^{93} +5.43296 q^{95} +3.13169 q^{97} -2.82371 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{3} - 4 q^{5} + 3 q^{7} - 3 q^{9}+O(q^{10}) \) 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
\(3\) −0.801938 −0.462999 −0.231499 0.972835i \(-0.574363\pi\)
−0.231499 + 0.972835i \(0.574363\pi\)
\(4\) 0 0
\(5\) −2.80194 −1.25306 −0.626532 0.779395i \(-0.715526\pi\)
−0.626532 + 0.779395i \(0.715526\pi\)
\(6\) 0 0
\(7\) 2.69202 1.01749 0.508744 0.860918i \(-0.330110\pi\)
0.508744 + 0.860918i \(0.330110\pi\)
\(8\) 0 0
\(9\) −2.35690 −0.785632
\(10\) 0 0
\(11\) 1.19806 0.361229 0.180615 0.983554i \(-0.442191\pi\)
0.180615 + 0.983554i \(0.442191\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) 2.24698 0.580168
\(16\) 0 0
\(17\) 1.13706 0.275778 0.137889 0.990448i \(-0.455968\pi\)
0.137889 + 0.990448i \(0.455968\pi\)
\(18\) 0 0
\(19\) −1.93900 −0.444837 −0.222419 0.974951i \(-0.571395\pi\)
−0.222419 + 0.974951i \(0.571395\pi\)
\(20\) 0 0
\(21\) −2.15883 −0.471096
\(22\) 0 0
\(23\) 4.60388 0.959974 0.479987 0.877275i \(-0.340641\pi\)
0.479987 + 0.877275i \(0.340641\pi\)
\(24\) 0 0
\(25\) 2.85086 0.570171
\(26\) 0 0
\(27\) 4.29590 0.826746
\(28\) 0 0
\(29\) −7.89977 −1.46695 −0.733475 0.679716i \(-0.762103\pi\)
−0.733475 + 0.679716i \(0.762103\pi\)
\(30\) 0 0
\(31\) −5.89977 −1.05963 −0.529815 0.848113i \(-0.677739\pi\)
−0.529815 + 0.848113i \(0.677739\pi\)
\(32\) 0 0
\(33\) −0.960771 −0.167249
\(34\) 0 0
\(35\) −7.54288 −1.27498
\(36\) 0 0
\(37\) 0.951083 0.156357 0.0781785 0.996939i \(-0.475090\pi\)
0.0781785 + 0.996939i \(0.475090\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.31767 0.518133 0.259066 0.965860i \(-0.416585\pi\)
0.259066 + 0.965860i \(0.416585\pi\)
\(42\) 0 0
\(43\) −7.15883 −1.09171 −0.545856 0.837879i \(-0.683795\pi\)
−0.545856 + 0.837879i \(0.683795\pi\)
\(44\) 0 0
\(45\) 6.60388 0.984448
\(46\) 0 0
\(47\) 7.69202 1.12200 0.560998 0.827817i \(-0.310417\pi\)
0.560998 + 0.827817i \(0.310417\pi\)
\(48\) 0 0
\(49\) 0.246980 0.0352828
\(50\) 0 0
\(51\) −0.911854 −0.127685
\(52\) 0 0
\(53\) 5.87263 0.806667 0.403334 0.915053i \(-0.367851\pi\)
0.403334 + 0.915053i \(0.367851\pi\)
\(54\) 0 0
\(55\) −3.35690 −0.452644
\(56\) 0 0
\(57\) 1.55496 0.205959
\(58\) 0 0
\(59\) 0.0120816 0.00157289 0.000786444 1.00000i \(-0.499750\pi\)
0.000786444 1.00000i \(0.499750\pi\)
\(60\) 0 0
\(61\) −8.03684 −1.02901 −0.514506 0.857487i \(-0.672025\pi\)
−0.514506 + 0.857487i \(0.672025\pi\)
\(62\) 0 0
\(63\) −6.34481 −0.799371
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 9.25667 1.13088 0.565441 0.824789i \(-0.308706\pi\)
0.565441 + 0.824789i \(0.308706\pi\)
\(68\) 0 0
\(69\) −3.69202 −0.444467
\(70\) 0 0
\(71\) 13.7409 1.63075 0.815375 0.578934i \(-0.196531\pi\)
0.815375 + 0.578934i \(0.196531\pi\)
\(72\) 0 0
\(73\) 12.8170 1.50012 0.750058 0.661372i \(-0.230025\pi\)
0.750058 + 0.661372i \(0.230025\pi\)
\(74\) 0 0
\(75\) −2.28621 −0.263989
\(76\) 0 0
\(77\) 3.22521 0.367547
\(78\) 0 0
\(79\) −0.807315 −0.0908300 −0.0454150 0.998968i \(-0.514461\pi\)
−0.0454150 + 0.998968i \(0.514461\pi\)
\(80\) 0 0
\(81\) 3.62565 0.402850
\(82\) 0 0
\(83\) 16.3327 1.79275 0.896375 0.443296i \(-0.146191\pi\)
0.896375 + 0.443296i \(0.146191\pi\)
\(84\) 0 0
\(85\) −3.18598 −0.345568
\(86\) 0 0
\(87\) 6.33513 0.679197
\(88\) 0 0
\(89\) −14.7289 −1.56126 −0.780628 0.624996i \(-0.785101\pi\)
−0.780628 + 0.624996i \(0.785101\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 4.73125 0.490608
\(94\) 0 0
\(95\) 5.43296 0.557410
\(96\) 0 0
\(97\) 3.13169 0.317975 0.158987 0.987281i \(-0.449177\pi\)
0.158987 + 0.987281i \(0.449177\pi\)
\(98\) 0 0
\(99\) −2.82371 −0.283793
\(100\) 0 0
\(101\) 5.29052 0.526426 0.263213 0.964738i \(-0.415218\pi\)
0.263213 + 0.964738i \(0.415218\pi\)
\(102\) 0 0
\(103\) 13.5308 1.33323 0.666614 0.745403i \(-0.267743\pi\)
0.666614 + 0.745403i \(0.267743\pi\)
\(104\) 0 0
\(105\) 6.04892 0.590314
\(106\) 0 0
\(107\) −5.63102 −0.544371 −0.272186 0.962245i \(-0.587747\pi\)
−0.272186 + 0.962245i \(0.587747\pi\)
\(108\) 0 0
\(109\) 4.17629 0.400016 0.200008 0.979794i \(-0.435903\pi\)
0.200008 + 0.979794i \(0.435903\pi\)
\(110\) 0 0
\(111\) −0.762709 −0.0723931
\(112\) 0 0
\(113\) 7.64310 0.719003 0.359501 0.933145i \(-0.382947\pi\)
0.359501 + 0.933145i \(0.382947\pi\)
\(114\) 0 0
\(115\) −12.8998 −1.20291
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 3.06100 0.280601
\(120\) 0 0
\(121\) −9.56465 −0.869513
\(122\) 0 0
\(123\) −2.66056 −0.239895
\(124\) 0 0
\(125\) 6.02177 0.538604
\(126\) 0 0
\(127\) −6.77777 −0.601430 −0.300715 0.953714i \(-0.597225\pi\)
−0.300715 + 0.953714i \(0.597225\pi\)
\(128\) 0 0
\(129\) 5.74094 0.505461
\(130\) 0 0
\(131\) 13.6799 1.19522 0.597611 0.801786i \(-0.296117\pi\)
0.597611 + 0.801786i \(0.296117\pi\)
\(132\) 0 0
\(133\) −5.21983 −0.452617
\(134\) 0 0
\(135\) −12.0368 −1.03597
\(136\) 0 0
\(137\) 12.9879 1.10963 0.554816 0.831973i \(-0.312789\pi\)
0.554816 + 0.831973i \(0.312789\pi\)
\(138\) 0 0
\(139\) −12.0465 −1.02177 −0.510886 0.859648i \(-0.670683\pi\)
−0.510886 + 0.859648i \(0.670683\pi\)
\(140\) 0 0
\(141\) −6.16852 −0.519483
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 22.1347 1.83818
\(146\) 0 0
\(147\) −0.198062 −0.0163359
\(148\) 0 0
\(149\) −0.740939 −0.0607001 −0.0303500 0.999539i \(-0.509662\pi\)
−0.0303500 + 0.999539i \(0.509662\pi\)
\(150\) 0 0
\(151\) 19.0737 1.55219 0.776097 0.630614i \(-0.217197\pi\)
0.776097 + 0.630614i \(0.217197\pi\)
\(152\) 0 0
\(153\) −2.67994 −0.216660
\(154\) 0 0
\(155\) 16.5308 1.32779
\(156\) 0 0
\(157\) −4.02177 −0.320972 −0.160486 0.987038i \(-0.551306\pi\)
−0.160486 + 0.987038i \(0.551306\pi\)
\(158\) 0 0
\(159\) −4.70948 −0.373486
\(160\) 0 0
\(161\) 12.3937 0.976763
\(162\) 0 0
\(163\) −15.1371 −1.18563 −0.592813 0.805340i \(-0.701983\pi\)
−0.592813 + 0.805340i \(0.701983\pi\)
\(164\) 0 0
\(165\) 2.69202 0.209574
\(166\) 0 0
\(167\) 6.26337 0.484674 0.242337 0.970192i \(-0.422086\pi\)
0.242337 + 0.970192i \(0.422086\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 4.57002 0.349478
\(172\) 0 0
\(173\) 16.3913 1.24621 0.623105 0.782138i \(-0.285871\pi\)
0.623105 + 0.782138i \(0.285871\pi\)
\(174\) 0 0
\(175\) 7.67456 0.580142
\(176\) 0 0
\(177\) −0.00968868 −0.000728246 0
\(178\) 0 0
\(179\) 2.45473 0.183475 0.0917376 0.995783i \(-0.470758\pi\)
0.0917376 + 0.995783i \(0.470758\pi\)
\(180\) 0 0
\(181\) 11.8073 0.877631 0.438815 0.898577i \(-0.355398\pi\)
0.438815 + 0.898577i \(0.355398\pi\)
\(182\) 0 0
\(183\) 6.44504 0.476431
\(184\) 0 0
\(185\) −2.66487 −0.195925
\(186\) 0 0
\(187\) 1.36227 0.0996192
\(188\) 0 0
\(189\) 11.5646 0.841204
\(190\) 0 0
\(191\) 8.99330 0.650732 0.325366 0.945588i \(-0.394512\pi\)
0.325366 + 0.945588i \(0.394512\pi\)
\(192\) 0 0
\(193\) 13.5254 0.973581 0.486790 0.873519i \(-0.338168\pi\)
0.486790 + 0.873519i \(0.338168\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 12.9758 0.924490 0.462245 0.886752i \(-0.347044\pi\)
0.462245 + 0.886752i \(0.347044\pi\)
\(198\) 0 0
\(199\) 13.5864 0.963116 0.481558 0.876414i \(-0.340071\pi\)
0.481558 + 0.876414i \(0.340071\pi\)
\(200\) 0 0
\(201\) −7.42327 −0.523597
\(202\) 0 0
\(203\) −21.2664 −1.49261
\(204\) 0 0
\(205\) −9.29590 −0.649254
\(206\) 0 0
\(207\) −10.8509 −0.754187
\(208\) 0 0
\(209\) −2.32304 −0.160688
\(210\) 0 0
\(211\) −10.4601 −0.720103 −0.360052 0.932932i \(-0.617241\pi\)
−0.360052 + 0.932932i \(0.617241\pi\)
\(212\) 0 0
\(213\) −11.0194 −0.755035
\(214\) 0 0
\(215\) 20.0586 1.36799
\(216\) 0 0
\(217\) −15.8823 −1.07816
\(218\) 0 0
\(219\) −10.2784 −0.694553
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 11.4058 0.763790 0.381895 0.924206i \(-0.375272\pi\)
0.381895 + 0.924206i \(0.375272\pi\)
\(224\) 0 0
\(225\) −6.71917 −0.447945
\(226\) 0 0
\(227\) −10.6407 −0.706249 −0.353124 0.935576i \(-0.614881\pi\)
−0.353124 + 0.935576i \(0.614881\pi\)
\(228\) 0 0
\(229\) −1.13946 −0.0752974 −0.0376487 0.999291i \(-0.511987\pi\)
−0.0376487 + 0.999291i \(0.511987\pi\)
\(230\) 0 0
\(231\) −2.58642 −0.170174
\(232\) 0 0
\(233\) −10.8509 −0.710863 −0.355432 0.934702i \(-0.615666\pi\)
−0.355432 + 0.934702i \(0.615666\pi\)
\(234\) 0 0
\(235\) −21.5526 −1.40593
\(236\) 0 0
\(237\) 0.647416 0.0420542
\(238\) 0 0
\(239\) 11.9293 0.771643 0.385822 0.922573i \(-0.373918\pi\)
0.385822 + 0.922573i \(0.373918\pi\)
\(240\) 0 0
\(241\) −3.64848 −0.235019 −0.117510 0.993072i \(-0.537491\pi\)
−0.117510 + 0.993072i \(0.537491\pi\)
\(242\) 0 0
\(243\) −15.7952 −1.01326
\(244\) 0 0
\(245\) −0.692021 −0.0442116
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −13.0978 −0.830042
\(250\) 0 0
\(251\) −1.37329 −0.0866813 −0.0433406 0.999060i \(-0.513800\pi\)
−0.0433406 + 0.999060i \(0.513800\pi\)
\(252\) 0 0
\(253\) 5.51573 0.346771
\(254\) 0 0
\(255\) 2.55496 0.159998
\(256\) 0 0
\(257\) 29.4359 1.83616 0.918082 0.396391i \(-0.129737\pi\)
0.918082 + 0.396391i \(0.129737\pi\)
\(258\) 0 0
\(259\) 2.56033 0.159091
\(260\) 0 0
\(261\) 18.6189 1.15248
\(262\) 0 0
\(263\) −10.6963 −0.659564 −0.329782 0.944057i \(-0.606975\pi\)
−0.329782 + 0.944057i \(0.606975\pi\)
\(264\) 0 0
\(265\) −16.4547 −1.01081
\(266\) 0 0
\(267\) 11.8116 0.722860
\(268\) 0 0
\(269\) −10.1860 −0.621050 −0.310525 0.950565i \(-0.600505\pi\)
−0.310525 + 0.950565i \(0.600505\pi\)
\(270\) 0 0
\(271\) 29.4523 1.78910 0.894551 0.446966i \(-0.147495\pi\)
0.894551 + 0.446966i \(0.147495\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3.41550 0.205963
\(276\) 0 0
\(277\) −10.2446 −0.615538 −0.307769 0.951461i \(-0.599582\pi\)
−0.307769 + 0.951461i \(0.599582\pi\)
\(278\) 0 0
\(279\) 13.9051 0.832480
\(280\) 0 0
\(281\) −11.5646 −0.689889 −0.344944 0.938623i \(-0.612102\pi\)
−0.344944 + 0.938623i \(0.612102\pi\)
\(282\) 0 0
\(283\) 30.7090 1.82546 0.912730 0.408562i \(-0.133970\pi\)
0.912730 + 0.408562i \(0.133970\pi\)
\(284\) 0 0
\(285\) −4.35690 −0.258080
\(286\) 0 0
\(287\) 8.93123 0.527194
\(288\) 0 0
\(289\) −15.7071 −0.923946
\(290\) 0 0
\(291\) −2.51142 −0.147222
\(292\) 0 0
\(293\) −18.6082 −1.08710 −0.543551 0.839376i \(-0.682921\pi\)
−0.543551 + 0.839376i \(0.682921\pi\)
\(294\) 0 0
\(295\) −0.0338518 −0.00197093
\(296\) 0 0
\(297\) 5.14675 0.298645
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −19.2717 −1.11080
\(302\) 0 0
\(303\) −4.24267 −0.243735
\(304\) 0 0
\(305\) 22.5187 1.28942
\(306\) 0 0
\(307\) −8.94438 −0.510483 −0.255241 0.966877i \(-0.582155\pi\)
−0.255241 + 0.966877i \(0.582155\pi\)
\(308\) 0 0
\(309\) −10.8509 −0.617284
\(310\) 0 0
\(311\) −21.0398 −1.19306 −0.596529 0.802591i \(-0.703454\pi\)
−0.596529 + 0.802591i \(0.703454\pi\)
\(312\) 0 0
\(313\) −7.12737 −0.402863 −0.201432 0.979503i \(-0.564559\pi\)
−0.201432 + 0.979503i \(0.564559\pi\)
\(314\) 0 0
\(315\) 17.7778 1.00166
\(316\) 0 0
\(317\) 23.9651 1.34601 0.673007 0.739636i \(-0.265003\pi\)
0.673007 + 0.739636i \(0.265003\pi\)
\(318\) 0 0
\(319\) −9.46442 −0.529906
\(320\) 0 0
\(321\) 4.51573 0.252043
\(322\) 0 0
\(323\) −2.20477 −0.122677
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −3.34913 −0.185207
\(328\) 0 0
\(329\) 20.7071 1.14162
\(330\) 0 0
\(331\) −2.89546 −0.159149 −0.0795745 0.996829i \(-0.525356\pi\)
−0.0795745 + 0.996829i \(0.525356\pi\)
\(332\) 0 0
\(333\) −2.24160 −0.122839
\(334\) 0 0
\(335\) −25.9366 −1.41707
\(336\) 0 0
\(337\) −3.10560 −0.169173 −0.0845865 0.996416i \(-0.526957\pi\)
−0.0845865 + 0.996416i \(0.526957\pi\)
\(338\) 0 0
\(339\) −6.12929 −0.332898
\(340\) 0 0
\(341\) −7.06829 −0.382770
\(342\) 0 0
\(343\) −18.1793 −0.981589
\(344\) 0 0
\(345\) 10.3448 0.556946
\(346\) 0 0
\(347\) 11.3787 0.610839 0.305419 0.952218i \(-0.401203\pi\)
0.305419 + 0.952218i \(0.401203\pi\)
\(348\) 0 0
\(349\) −3.34721 −0.179172 −0.0895859 0.995979i \(-0.528554\pi\)
−0.0895859 + 0.995979i \(0.528554\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0.637727 0.0339428 0.0169714 0.999856i \(-0.494598\pi\)
0.0169714 + 0.999856i \(0.494598\pi\)
\(354\) 0 0
\(355\) −38.5013 −2.04343
\(356\) 0 0
\(357\) −2.45473 −0.129918
\(358\) 0 0
\(359\) 21.4590 1.13256 0.566282 0.824211i \(-0.308381\pi\)
0.566282 + 0.824211i \(0.308381\pi\)
\(360\) 0 0
\(361\) −15.2403 −0.802120
\(362\) 0 0
\(363\) 7.67025 0.402584
\(364\) 0 0
\(365\) −35.9124 −1.87974
\(366\) 0 0
\(367\) 9.38703 0.489999 0.244999 0.969523i \(-0.421212\pi\)
0.244999 + 0.969523i \(0.421212\pi\)
\(368\) 0 0
\(369\) −7.81940 −0.407062
\(370\) 0 0
\(371\) 15.8092 0.820775
\(372\) 0 0
\(373\) 27.7265 1.43562 0.717811 0.696238i \(-0.245144\pi\)
0.717811 + 0.696238i \(0.245144\pi\)
\(374\) 0 0
\(375\) −4.82908 −0.249373
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −35.8702 −1.84253 −0.921265 0.388935i \(-0.872843\pi\)
−0.921265 + 0.388935i \(0.872843\pi\)
\(380\) 0 0
\(381\) 5.43535 0.278462
\(382\) 0 0
\(383\) −4.85517 −0.248087 −0.124044 0.992277i \(-0.539586\pi\)
−0.124044 + 0.992277i \(0.539586\pi\)
\(384\) 0 0
\(385\) −9.03684 −0.460560
\(386\) 0 0
\(387\) 16.8726 0.857684
\(388\) 0 0
\(389\) 2.38537 0.120943 0.0604716 0.998170i \(-0.480740\pi\)
0.0604716 + 0.998170i \(0.480740\pi\)
\(390\) 0 0
\(391\) 5.23490 0.264740
\(392\) 0 0
\(393\) −10.9705 −0.553387
\(394\) 0 0
\(395\) 2.26205 0.113816
\(396\) 0 0
\(397\) −15.2664 −0.766196 −0.383098 0.923708i \(-0.625143\pi\)
−0.383098 + 0.923708i \(0.625143\pi\)
\(398\) 0 0
\(399\) 4.18598 0.209561
\(400\) 0 0
\(401\) 12.7584 0.637124 0.318562 0.947902i \(-0.396800\pi\)
0.318562 + 0.947902i \(0.396800\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −10.1588 −0.504797
\(406\) 0 0
\(407\) 1.13946 0.0564807
\(408\) 0 0
\(409\) −25.3588 −1.25391 −0.626956 0.779054i \(-0.715700\pi\)
−0.626956 + 0.779054i \(0.715700\pi\)
\(410\) 0 0
\(411\) −10.4155 −0.513759
\(412\) 0 0
\(413\) 0.0325239 0.00160040
\(414\) 0 0
\(415\) −45.7633 −2.24643
\(416\) 0 0
\(417\) 9.66056 0.473080
\(418\) 0 0
\(419\) 11.6673 0.569983 0.284992 0.958530i \(-0.408009\pi\)
0.284992 + 0.958530i \(0.408009\pi\)
\(420\) 0 0
\(421\) −8.29291 −0.404172 −0.202086 0.979368i \(-0.564772\pi\)
−0.202086 + 0.979368i \(0.564772\pi\)
\(422\) 0 0
\(423\) −18.1293 −0.881476
\(424\) 0 0
\(425\) 3.24160 0.157241
\(426\) 0 0
\(427\) −21.6353 −1.04701
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −0.932296 −0.0449071 −0.0224536 0.999748i \(-0.507148\pi\)
−0.0224536 + 0.999748i \(0.507148\pi\)
\(432\) 0 0
\(433\) −13.3502 −0.641569 −0.320785 0.947152i \(-0.603947\pi\)
−0.320785 + 0.947152i \(0.603947\pi\)
\(434\) 0 0
\(435\) −17.7506 −0.851077
\(436\) 0 0
\(437\) −8.92692 −0.427032
\(438\) 0 0
\(439\) −13.9922 −0.667813 −0.333906 0.942606i \(-0.608367\pi\)
−0.333906 + 0.942606i \(0.608367\pi\)
\(440\) 0 0
\(441\) −0.582105 −0.0277193
\(442\) 0 0
\(443\) −23.7017 −1.12610 −0.563051 0.826422i \(-0.690373\pi\)
−0.563051 + 0.826422i \(0.690373\pi\)
\(444\) 0 0
\(445\) 41.2693 1.95635
\(446\) 0 0
\(447\) 0.594187 0.0281041
\(448\) 0 0
\(449\) −12.5864 −0.593990 −0.296995 0.954879i \(-0.595984\pi\)
−0.296995 + 0.954879i \(0.595984\pi\)
\(450\) 0 0
\(451\) 3.97477 0.187165
\(452\) 0 0
\(453\) −15.2959 −0.718664
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −33.6383 −1.57353 −0.786767 0.617250i \(-0.788247\pi\)
−0.786767 + 0.617250i \(0.788247\pi\)
\(458\) 0 0
\(459\) 4.88471 0.227999
\(460\) 0 0
\(461\) −1.40283 −0.0653363 −0.0326681 0.999466i \(-0.510400\pi\)
−0.0326681 + 0.999466i \(0.510400\pi\)
\(462\) 0 0
\(463\) 15.2010 0.706453 0.353226 0.935538i \(-0.385085\pi\)
0.353226 + 0.935538i \(0.385085\pi\)
\(464\) 0 0
\(465\) −13.2567 −0.614763
\(466\) 0 0
\(467\) 39.3414 1.82050 0.910250 0.414058i \(-0.135889\pi\)
0.910250 + 0.414058i \(0.135889\pi\)
\(468\) 0 0
\(469\) 24.9191 1.15066
\(470\) 0 0
\(471\) 3.22521 0.148610
\(472\) 0 0
\(473\) −8.57673 −0.394358
\(474\) 0 0
\(475\) −5.52781 −0.253633
\(476\) 0 0
\(477\) −13.8412 −0.633743
\(478\) 0 0
\(479\) 22.3690 1.02206 0.511032 0.859561i \(-0.329263\pi\)
0.511032 + 0.859561i \(0.329263\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) −9.93900 −0.452240
\(484\) 0 0
\(485\) −8.77479 −0.398443
\(486\) 0 0
\(487\) −22.9205 −1.03863 −0.519313 0.854584i \(-0.673812\pi\)
−0.519313 + 0.854584i \(0.673812\pi\)
\(488\) 0 0
\(489\) 12.1390 0.548944
\(490\) 0 0
\(491\) −1.84356 −0.0831987 −0.0415993 0.999134i \(-0.513245\pi\)
−0.0415993 + 0.999134i \(0.513245\pi\)
\(492\) 0 0
\(493\) −8.98254 −0.404553
\(494\) 0 0
\(495\) 7.91185 0.355611
\(496\) 0 0
\(497\) 36.9909 1.65927
\(498\) 0 0
\(499\) 12.0344 0.538736 0.269368 0.963037i \(-0.413185\pi\)
0.269368 + 0.963037i \(0.413185\pi\)
\(500\) 0 0
\(501\) −5.02284 −0.224404
\(502\) 0 0
\(503\) 30.5056 1.36018 0.680088 0.733130i \(-0.261942\pi\)
0.680088 + 0.733130i \(0.261942\pi\)
\(504\) 0 0
\(505\) −14.8237 −0.659646
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −1.51142 −0.0669924 −0.0334962 0.999439i \(-0.510664\pi\)
−0.0334962 + 0.999439i \(0.510664\pi\)
\(510\) 0 0
\(511\) 34.5036 1.52635
\(512\) 0 0
\(513\) −8.32975 −0.367767
\(514\) 0 0
\(515\) −37.9124 −1.67062
\(516\) 0 0
\(517\) 9.21552 0.405298
\(518\) 0 0
\(519\) −13.1448 −0.576994
\(520\) 0 0
\(521\) −5.64012 −0.247098 −0.123549 0.992338i \(-0.539428\pi\)
−0.123549 + 0.992338i \(0.539428\pi\)
\(522\) 0 0
\(523\) 31.7506 1.38836 0.694179 0.719802i \(-0.255768\pi\)
0.694179 + 0.719802i \(0.255768\pi\)
\(524\) 0 0
\(525\) −6.15452 −0.268605
\(526\) 0 0
\(527\) −6.70841 −0.292223
\(528\) 0 0
\(529\) −1.80433 −0.0784492
\(530\) 0 0
\(531\) −0.0284750 −0.00123571
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 15.7778 0.682133
\(536\) 0 0
\(537\) −1.96854 −0.0849488
\(538\) 0 0
\(539\) 0.295897 0.0127452
\(540\) 0 0
\(541\) 24.3297 1.04602 0.523009 0.852327i \(-0.324809\pi\)
0.523009 + 0.852327i \(0.324809\pi\)
\(542\) 0 0
\(543\) −9.46873 −0.406342
\(544\) 0 0
\(545\) −11.7017 −0.501246
\(546\) 0 0
\(547\) 8.18896 0.350135 0.175067 0.984556i \(-0.443986\pi\)
0.175067 + 0.984556i \(0.443986\pi\)
\(548\) 0 0
\(549\) 18.9420 0.808424
\(550\) 0 0
\(551\) 15.3177 0.652555
\(552\) 0 0
\(553\) −2.17331 −0.0924185
\(554\) 0 0
\(555\) 2.13706 0.0907133
\(556\) 0 0
\(557\) 25.3327 1.07338 0.536691 0.843779i \(-0.319674\pi\)
0.536691 + 0.843779i \(0.319674\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −1.09246 −0.0461236
\(562\) 0 0
\(563\) 25.3937 1.07022 0.535109 0.844783i \(-0.320270\pi\)
0.535109 + 0.844783i \(0.320270\pi\)
\(564\) 0 0
\(565\) −21.4155 −0.900957
\(566\) 0 0
\(567\) 9.76032 0.409895
\(568\) 0 0
\(569\) −31.1347 −1.30523 −0.652617 0.757688i \(-0.726329\pi\)
−0.652617 + 0.757688i \(0.726329\pi\)
\(570\) 0 0
\(571\) 20.5090 0.858276 0.429138 0.903239i \(-0.358817\pi\)
0.429138 + 0.903239i \(0.358817\pi\)
\(572\) 0 0
\(573\) −7.21206 −0.301288
\(574\) 0 0
\(575\) 13.1250 0.547350
\(576\) 0 0
\(577\) 15.6890 0.653143 0.326572 0.945172i \(-0.394107\pi\)
0.326572 + 0.945172i \(0.394107\pi\)
\(578\) 0 0
\(579\) −10.8465 −0.450767
\(580\) 0 0
\(581\) 43.9681 1.82410
\(582\) 0 0
\(583\) 7.03577 0.291392
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −30.5687 −1.26171 −0.630853 0.775903i \(-0.717295\pi\)
−0.630853 + 0.775903i \(0.717295\pi\)
\(588\) 0 0
\(589\) 11.4397 0.471363
\(590\) 0 0
\(591\) −10.4058 −0.428038
\(592\) 0 0
\(593\) −29.6883 −1.21915 −0.609576 0.792727i \(-0.708660\pi\)
−0.609576 + 0.792727i \(0.708660\pi\)
\(594\) 0 0
\(595\) −8.57673 −0.351612
\(596\) 0 0
\(597\) −10.8955 −0.445922
\(598\) 0 0
\(599\) −24.2325 −0.990113 −0.495057 0.868861i \(-0.664853\pi\)
−0.495057 + 0.868861i \(0.664853\pi\)
\(600\) 0 0
\(601\) 16.4819 0.672310 0.336155 0.941807i \(-0.390873\pi\)
0.336155 + 0.941807i \(0.390873\pi\)
\(602\) 0 0
\(603\) −21.8170 −0.888457
\(604\) 0 0
\(605\) 26.7995 1.08956
\(606\) 0 0
\(607\) −1.43190 −0.0581188 −0.0290594 0.999578i \(-0.509251\pi\)
−0.0290594 + 0.999578i \(0.509251\pi\)
\(608\) 0 0
\(609\) 17.0543 0.691075
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 3.84846 0.155438 0.0777190 0.996975i \(-0.475236\pi\)
0.0777190 + 0.996975i \(0.475236\pi\)
\(614\) 0 0
\(615\) 7.45473 0.300604
\(616\) 0 0
\(617\) −15.0388 −0.605437 −0.302719 0.953080i \(-0.597894\pi\)
−0.302719 + 0.953080i \(0.597894\pi\)
\(618\) 0 0
\(619\) −12.8170 −0.515159 −0.257579 0.966257i \(-0.582925\pi\)
−0.257579 + 0.966257i \(0.582925\pi\)
\(620\) 0 0
\(621\) 19.7778 0.793655
\(622\) 0 0
\(623\) −39.6504 −1.58856
\(624\) 0 0
\(625\) −31.1269 −1.24508
\(626\) 0 0
\(627\) 1.86294 0.0743985
\(628\) 0 0
\(629\) 1.08144 0.0431199
\(630\) 0 0
\(631\) −25.7517 −1.02516 −0.512579 0.858640i \(-0.671310\pi\)
−0.512579 + 0.858640i \(0.671310\pi\)
\(632\) 0 0
\(633\) 8.38835 0.333407
\(634\) 0 0
\(635\) 18.9909 0.753631
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −32.3860 −1.28117
\(640\) 0 0
\(641\) 24.4571 0.965998 0.482999 0.875621i \(-0.339547\pi\)
0.482999 + 0.875621i \(0.339547\pi\)
\(642\) 0 0
\(643\) 9.97344 0.393314 0.196657 0.980472i \(-0.436991\pi\)
0.196657 + 0.980472i \(0.436991\pi\)
\(644\) 0 0
\(645\) −16.0858 −0.633376
\(646\) 0 0
\(647\) −11.8431 −0.465600 −0.232800 0.972525i \(-0.574789\pi\)
−0.232800 + 0.972525i \(0.574789\pi\)
\(648\) 0 0
\(649\) 0.0144745 0.000568173 0
\(650\) 0 0
\(651\) 12.7366 0.499188
\(652\) 0 0
\(653\) −7.47411 −0.292484 −0.146242 0.989249i \(-0.546718\pi\)
−0.146242 + 0.989249i \(0.546718\pi\)
\(654\) 0 0
\(655\) −38.3303 −1.49769
\(656\) 0 0
\(657\) −30.2083 −1.17854
\(658\) 0 0
\(659\) −34.1739 −1.33123 −0.665613 0.746297i \(-0.731830\pi\)
−0.665613 + 0.746297i \(0.731830\pi\)
\(660\) 0 0
\(661\) 33.6088 1.30723 0.653615 0.756827i \(-0.273252\pi\)
0.653615 + 0.756827i \(0.273252\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 14.6256 0.567158
\(666\) 0 0
\(667\) −36.3696 −1.40824
\(668\) 0 0
\(669\) −9.14675 −0.353634
\(670\) 0 0
\(671\) −9.62863 −0.371709
\(672\) 0 0
\(673\) 48.0320 1.85150 0.925750 0.378137i \(-0.123435\pi\)
0.925750 + 0.378137i \(0.123435\pi\)
\(674\) 0 0
\(675\) 12.2470 0.471386
\(676\) 0 0
\(677\) −33.6582 −1.29359 −0.646794 0.762665i \(-0.723891\pi\)
−0.646794 + 0.762665i \(0.723891\pi\)
\(678\) 0 0
\(679\) 8.43057 0.323535
\(680\) 0 0
\(681\) 8.53319 0.326992
\(682\) 0 0
\(683\) −15.9041 −0.608553 −0.304276 0.952584i \(-0.598415\pi\)
−0.304276 + 0.952584i \(0.598415\pi\)
\(684\) 0 0
\(685\) −36.3913 −1.39044
\(686\) 0 0
\(687\) 0.913773 0.0348626
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −33.1903 −1.26262 −0.631309 0.775531i \(-0.717482\pi\)
−0.631309 + 0.775531i \(0.717482\pi\)
\(692\) 0 0
\(693\) −7.60148 −0.288756
\(694\) 0 0
\(695\) 33.7536 1.28035
\(696\) 0 0
\(697\) 3.77240 0.142890
\(698\) 0 0
\(699\) 8.70171 0.329129
\(700\) 0 0
\(701\) −14.9129 −0.563253 −0.281627 0.959524i \(-0.590874\pi\)
−0.281627 + 0.959524i \(0.590874\pi\)
\(702\) 0 0
\(703\) −1.84415 −0.0695534
\(704\) 0 0
\(705\) 17.2838 0.650946
\(706\) 0 0
\(707\) 14.2422 0.535633
\(708\) 0 0
\(709\) 38.4312 1.44331 0.721656 0.692252i \(-0.243381\pi\)
0.721656 + 0.692252i \(0.243381\pi\)
\(710\) 0 0
\(711\) 1.90276 0.0713589
\(712\) 0 0
\(713\) −27.1618 −1.01722
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −9.56657 −0.357270
\(718\) 0 0
\(719\) 11.4373 0.426538 0.213269 0.976993i \(-0.431589\pi\)
0.213269 + 0.976993i \(0.431589\pi\)
\(720\) 0 0
\(721\) 36.4252 1.35654
\(722\) 0 0
\(723\) 2.92585 0.108814
\(724\) 0 0
\(725\) −22.5211 −0.836413
\(726\) 0 0
\(727\) −3.63640 −0.134867 −0.0674333 0.997724i \(-0.521481\pi\)
−0.0674333 + 0.997724i \(0.521481\pi\)
\(728\) 0 0
\(729\) 1.78986 0.0662910
\(730\) 0 0
\(731\) −8.14005 −0.301071
\(732\) 0 0
\(733\) −3.52217 −0.130094 −0.0650472 0.997882i \(-0.520720\pi\)
−0.0650472 + 0.997882i \(0.520720\pi\)
\(734\) 0 0
\(735\) 0.554958 0.0204699
\(736\) 0 0
\(737\) 11.0901 0.408508
\(738\) 0 0
\(739\) −0.420288 −0.0154605 −0.00773027 0.999970i \(-0.502461\pi\)
−0.00773027 + 0.999970i \(0.502461\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 25.3623 0.930452 0.465226 0.885192i \(-0.345973\pi\)
0.465226 + 0.885192i \(0.345973\pi\)
\(744\) 0 0
\(745\) 2.07606 0.0760611
\(746\) 0 0
\(747\) −38.4946 −1.40844
\(748\) 0 0
\(749\) −15.1588 −0.553892
\(750\) 0 0
\(751\) −0.650874 −0.0237507 −0.0118754 0.999929i \(-0.503780\pi\)
−0.0118754 + 0.999929i \(0.503780\pi\)
\(752\) 0 0
\(753\) 1.10129 0.0401333
\(754\) 0 0
\(755\) −53.4432 −1.94500
\(756\) 0 0
\(757\) −16.7909 −0.610276 −0.305138 0.952308i \(-0.598703\pi\)
−0.305138 + 0.952308i \(0.598703\pi\)
\(758\) 0 0
\(759\) −4.42327 −0.160555
\(760\) 0 0
\(761\) 30.9221 1.12093 0.560463 0.828179i \(-0.310623\pi\)
0.560463 + 0.828179i \(0.310623\pi\)
\(762\) 0 0
\(763\) 11.2427 0.407012
\(764\) 0 0
\(765\) 7.50902 0.271489
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −43.7689 −1.57835 −0.789174 0.614169i \(-0.789491\pi\)
−0.789174 + 0.614169i \(0.789491\pi\)
\(770\) 0 0
\(771\) −23.6058 −0.850142
\(772\) 0 0
\(773\) −42.4209 −1.52577 −0.762886 0.646532i \(-0.776218\pi\)
−0.762886 + 0.646532i \(0.776218\pi\)
\(774\) 0 0
\(775\) −16.8194 −0.604171
\(776\) 0 0
\(777\) −2.05323 −0.0736592
\(778\) 0 0
\(779\) −6.43296 −0.230485
\(780\) 0 0
\(781\) 16.4625 0.589075
\(782\) 0 0
\(783\) −33.9366 −1.21280
\(784\) 0 0
\(785\) 11.2687 0.402199
\(786\) 0 0
\(787\) 36.0116 1.28368 0.641838 0.766841i \(-0.278172\pi\)
0.641838 + 0.766841i \(0.278172\pi\)
\(788\) 0 0
\(789\) 8.57779 0.305378
\(790\) 0 0
\(791\) 20.5754 0.731577
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 13.1957 0.468002
\(796\) 0 0
\(797\) −31.7101 −1.12323 −0.561614 0.827399i \(-0.689819\pi\)
−0.561614 + 0.827399i \(0.689819\pi\)
\(798\) 0 0
\(799\) 8.74632 0.309422
\(800\) 0 0
\(801\) 34.7144 1.22657
\(802\) 0 0
\(803\) 15.3556 0.541886
\(804\) 0 0
\(805\) −34.7265 −1.22395
\(806\) 0 0
\(807\) 8.16852 0.287546
\(808\) 0 0
\(809\) −45.2814 −1.59201 −0.796005 0.605290i \(-0.793057\pi\)
−0.796005 + 0.605290i \(0.793057\pi\)
\(810\) 0 0
\(811\) 42.8635 1.50514 0.752571 0.658511i \(-0.228813\pi\)
0.752571 + 0.658511i \(0.228813\pi\)
\(812\) 0 0
\(813\) −23.6189 −0.828352
\(814\) 0 0
\(815\) 42.4131 1.48567
\(816\) 0 0
\(817\) 13.8810 0.485634
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −7.82776 −0.273191 −0.136595 0.990627i \(-0.543616\pi\)
−0.136595 + 0.990627i \(0.543616\pi\)
\(822\) 0 0
\(823\) 36.7754 1.28191 0.640955 0.767579i \(-0.278539\pi\)
0.640955 + 0.767579i \(0.278539\pi\)
\(824\) 0 0
\(825\) −2.73902 −0.0953604
\(826\) 0 0
\(827\) −47.3293 −1.64580 −0.822900 0.568186i \(-0.807645\pi\)
−0.822900 + 0.568186i \(0.807645\pi\)
\(828\) 0 0
\(829\) 25.2687 0.877620 0.438810 0.898580i \(-0.355400\pi\)
0.438810 + 0.898580i \(0.355400\pi\)
\(830\) 0 0
\(831\) 8.21552 0.284993
\(832\) 0 0
\(833\) 0.280831 0.00973023
\(834\) 0 0
\(835\) −17.5496 −0.607328
\(836\) 0 0
\(837\) −25.3448 −0.876045
\(838\) 0 0
\(839\) 37.6883 1.30114 0.650572 0.759444i \(-0.274529\pi\)
0.650572 + 0.759444i \(0.274529\pi\)
\(840\) 0 0
\(841\) 33.4064 1.15194
\(842\) 0 0
\(843\) 9.27413 0.319418
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −25.7482 −0.884720
\(848\) 0 0
\(849\) −24.6267 −0.845187
\(850\) 0 0
\(851\) 4.37867 0.150099
\(852\) 0 0
\(853\) −31.0121 −1.06183 −0.530917 0.847424i \(-0.678152\pi\)
−0.530917 + 0.847424i \(0.678152\pi\)
\(854\) 0 0
\(855\) −12.8049 −0.437919
\(856\) 0 0
\(857\) 12.4692 0.425940 0.212970 0.977059i \(-0.431686\pi\)
0.212970 + 0.977059i \(0.431686\pi\)
\(858\) 0 0
\(859\) 17.3163 0.590826 0.295413 0.955370i \(-0.404543\pi\)
0.295413 + 0.955370i \(0.404543\pi\)
\(860\) 0 0
\(861\) −7.16229 −0.244090
\(862\) 0 0
\(863\) 3.46383 0.117910 0.0589550 0.998261i \(-0.481223\pi\)
0.0589550 + 0.998261i \(0.481223\pi\)
\(864\) 0 0
\(865\) −45.9275 −1.56158
\(866\) 0 0
\(867\) 12.5961 0.427786
\(868\) 0 0
\(869\) −0.967213 −0.0328105
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −7.38106 −0.249811
\(874\) 0 0
\(875\) 16.2107 0.548023
\(876\) 0 0
\(877\) 57.2549 1.93336 0.966680 0.255989i \(-0.0824010\pi\)
0.966680 + 0.255989i \(0.0824010\pi\)
\(878\) 0 0
\(879\) 14.9226 0.503327
\(880\) 0 0
\(881\) −43.1782 −1.45471 −0.727355 0.686261i \(-0.759251\pi\)
−0.727355 + 0.686261i \(0.759251\pi\)
\(882\) 0 0
\(883\) −49.9560 −1.68115 −0.840576 0.541693i \(-0.817784\pi\)
−0.840576 + 0.541693i \(0.817784\pi\)
\(884\) 0 0
\(885\) 0.0271471 0.000912539 0
\(886\) 0 0
\(887\) −17.6746 −0.593454 −0.296727 0.954962i \(-0.595895\pi\)
−0.296727 + 0.954962i \(0.595895\pi\)
\(888\) 0 0
\(889\) −18.2459 −0.611948
\(890\) 0 0
\(891\) 4.34375 0.145521
\(892\) 0 0
\(893\) −14.9148 −0.499106
\(894\) 0 0
\(895\) −6.87800 −0.229906
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 46.6069 1.55443
\(900\) 0 0
\(901\) 6.67755 0.222461
\(902\) 0 0
\(903\) 15.4547 0.514301
\(904\) 0 0
\(905\) −33.0834 −1.09973
\(906\) 0 0
\(907\) −7.73423 −0.256811 −0.128406 0.991722i \(-0.540986\pi\)
−0.128406 + 0.991722i \(0.540986\pi\)
\(908\) 0 0
\(909\) −12.4692 −0.413577
\(910\) 0 0
\(911\) −39.6179 −1.31260 −0.656299 0.754501i \(-0.727879\pi\)
−0.656299 + 0.754501i \(0.727879\pi\)
\(912\) 0 0
\(913\) 19.5676 0.647594
\(914\) 0 0
\(915\) −18.0586 −0.596999
\(916\) 0 0
\(917\) 36.8267 1.21612
\(918\) 0 0
\(919\) −14.6213 −0.482313 −0.241157 0.970486i \(-0.577527\pi\)
−0.241157 + 0.970486i \(0.577527\pi\)
\(920\) 0 0
\(921\) 7.17283 0.236353
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 2.71140 0.0891502
\(926\) 0 0
\(927\) −31.8907 −1.04743
\(928\) 0 0
\(929\) 3.55735 0.116713 0.0583565 0.998296i \(-0.481414\pi\)
0.0583565 + 0.998296i \(0.481414\pi\)
\(930\) 0 0
\(931\) −0.478894 −0.0156951
\(932\) 0 0
\(933\) 16.8726 0.552385
\(934\) 0 0
\(935\) −3.81700 −0.124829
\(936\) 0 0
\(937\) 34.5526 1.12878 0.564392 0.825507i \(-0.309111\pi\)
0.564392 + 0.825507i \(0.309111\pi\)
\(938\) 0 0
\(939\) 5.71571 0.186525
\(940\) 0 0
\(941\) 20.6233 0.672299 0.336149 0.941809i \(-0.390875\pi\)
0.336149 + 0.941809i \(0.390875\pi\)
\(942\) 0 0
\(943\) 15.2741 0.497394
\(944\) 0 0
\(945\) −32.4034 −1.05408
\(946\) 0 0
\(947\) −29.4999 −0.958619 −0.479309 0.877646i \(-0.659113\pi\)
−0.479309 + 0.877646i \(0.659113\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −19.2185 −0.623203
\(952\) 0 0
\(953\) 26.2389 0.849963 0.424981 0.905202i \(-0.360281\pi\)
0.424981 + 0.905202i \(0.360281\pi\)
\(954\) 0 0
\(955\) −25.1987 −0.815409
\(956\) 0 0
\(957\) 7.58987 0.245346
\(958\) 0 0
\(959\) 34.9638 1.12904
\(960\) 0 0
\(961\) 3.80731 0.122817
\(962\) 0 0
\(963\) 13.2717 0.427676
\(964\) 0 0
\(965\) −37.8974 −1.21996
\(966\) 0 0
\(967\) −17.5176 −0.563330 −0.281665 0.959513i \(-0.590887\pi\)
−0.281665 + 0.959513i \(0.590887\pi\)
\(968\) 0 0
\(969\) 1.76809 0.0567991
\(970\) 0 0
\(971\) −20.5120 −0.658262 −0.329131 0.944284i \(-0.606756\pi\)
−0.329131 + 0.944284i \(0.606756\pi\)
\(972\) 0 0
\(973\) −32.4295 −1.03964
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 25.4450 0.814059 0.407030 0.913415i \(-0.366565\pi\)
0.407030 + 0.913415i \(0.366565\pi\)
\(978\) 0 0
\(979\) −17.6461 −0.563971
\(980\) 0 0
\(981\) −9.84309 −0.314266
\(982\) 0 0
\(983\) 39.5244 1.26063 0.630316 0.776339i \(-0.282926\pi\)
0.630316 + 0.776339i \(0.282926\pi\)
\(984\) 0 0
\(985\) −36.3575 −1.15845
\(986\) 0 0
\(987\) −16.6058 −0.528568
\(988\) 0 0
\(989\) −32.9584 −1.04802
\(990\) 0 0
\(991\) 29.8377 0.947826 0.473913 0.880572i \(-0.342841\pi\)
0.473913 + 0.880572i \(0.342841\pi\)
\(992\) 0 0
\(993\) 2.32198 0.0736858
\(994\) 0 0
\(995\) −38.0683 −1.20685
\(996\) 0 0
\(997\) −4.93123 −0.156174 −0.0780868 0.996947i \(-0.524881\pi\)
−0.0780868 + 0.996947i \(0.524881\pi\)
\(998\) 0 0
\(999\) 4.08575 0.129268
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.1.1 3
4.3 odd 2 169.2.a.b.1.2 3
12.11 even 2 1521.2.a.r.1.2 3
13.5 odd 4 2704.2.f.o.337.2 6
13.8 odd 4 2704.2.f.o.337.1 6
13.12 even 2 2704.2.a.ba.1.1 3
20.19 odd 2 4225.2.a.bg.1.2 3
28.27 even 2 8281.2.a.bf.1.2 3
52.3 odd 6 169.2.c.c.22.2 6
52.7 even 12 169.2.e.b.23.3 12
52.11 even 12 169.2.e.b.147.4 12
52.15 even 12 169.2.e.b.147.3 12
52.19 even 12 169.2.e.b.23.4 12
52.23 odd 6 169.2.c.b.22.2 6
52.31 even 4 169.2.b.b.168.4 6
52.35 odd 6 169.2.c.c.146.2 6
52.43 odd 6 169.2.c.b.146.2 6
52.47 even 4 169.2.b.b.168.3 6
52.51 odd 2 169.2.a.c.1.2 yes 3
156.47 odd 4 1521.2.b.l.1351.4 6
156.83 odd 4 1521.2.b.l.1351.3 6
156.155 even 2 1521.2.a.o.1.2 3
260.259 odd 2 4225.2.a.bb.1.2 3
364.363 even 2 8281.2.a.bj.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
169.2.a.b.1.2 3 4.3 odd 2
169.2.a.c.1.2 yes 3 52.51 odd 2
169.2.b.b.168.3 6 52.47 even 4
169.2.b.b.168.4 6 52.31 even 4
169.2.c.b.22.2 6 52.23 odd 6
169.2.c.b.146.2 6 52.43 odd 6
169.2.c.c.22.2 6 52.3 odd 6
169.2.c.c.146.2 6 52.35 odd 6
169.2.e.b.23.3 12 52.7 even 12
169.2.e.b.23.4 12 52.19 even 12
169.2.e.b.147.3 12 52.15 even 12
169.2.e.b.147.4 12 52.11 even 12
1521.2.a.o.1.2 3 156.155 even 2
1521.2.a.r.1.2 3 12.11 even 2
1521.2.b.l.1351.3 6 156.83 odd 4
1521.2.b.l.1351.4 6 156.47 odd 4
2704.2.a.z.1.1 3 1.1 even 1 trivial
2704.2.a.ba.1.1 3 13.12 even 2
2704.2.f.o.337.1 6 13.8 odd 4
2704.2.f.o.337.2 6 13.5 odd 4
4225.2.a.bb.1.2 3 260.259 odd 2
4225.2.a.bg.1.2 3 20.19 odd 2
8281.2.a.bf.1.2 3 28.27 even 2
8281.2.a.bj.1.2 3 364.363 even 2