Properties

Label 2704.2.f.o.337.1
Level $2704$
Weight $2$
Character 2704.337
Analytic conductor $21.592$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2704,2,Mod(337,2704)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2704, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2704.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2704 = 2^{4} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2704.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(21.5915487066\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.153664.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 5x^{4} + 6x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 169)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.1
Root \(1.24698i\) of defining polynomial
Character \(\chi\) \(=\) 2704.337
Dual form 2704.2.f.o.337.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.801938 q^{3} -2.80194i q^{5} -2.69202i q^{7} -2.35690 q^{9} +O(q^{10})\) \(q-0.801938 q^{3} -2.80194i q^{5} -2.69202i q^{7} -2.35690 q^{9} -1.19806i q^{11} +2.24698i q^{15} -1.13706 q^{17} -1.93900i q^{19} +2.15883i q^{21} -4.60388 q^{23} -2.85086 q^{25} +4.29590 q^{27} -7.89977 q^{29} -5.89977i q^{31} +0.960771i q^{33} -7.54288 q^{35} -0.951083i q^{37} +3.31767i q^{41} +7.15883 q^{43} +6.60388i q^{45} -7.69202i q^{47} -0.246980 q^{49} +0.911854 q^{51} +5.87263 q^{53} -3.35690 q^{55} +1.55496i q^{57} -0.0120816i q^{59} -8.03684 q^{61} +6.34481i q^{63} +9.25667i q^{67} +3.69202 q^{69} +13.7409i q^{71} -12.8170i q^{73} +2.28621 q^{75} -3.22521 q^{77} -0.807315 q^{79} +3.62565 q^{81} +16.3327i q^{83} +3.18598i q^{85} +6.33513 q^{87} +14.7289i q^{89} +4.73125i q^{93} -5.43296 q^{95} +3.13169i q^{97} +2.82371i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 4 q^{3} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 4 q^{3} - 6 q^{9} + 4 q^{17} - 10 q^{23} + 10 q^{25} - 2 q^{27} - 2 q^{29} - 8 q^{35} + 26 q^{43} + 8 q^{49} - 2 q^{51} + 2 q^{53} - 12 q^{55} + 8 q^{61} + 12 q^{69} + 30 q^{75} - 16 q^{77} + 10 q^{79} - 2 q^{81} + 36 q^{87} + 6 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.801938 −0.462999 −0.231499 0.972835i \(-0.574363\pi\)
−0.231499 + 0.972835i \(0.574363\pi\)
\(4\) 0 0
\(5\) − 2.80194i − 1.25306i −0.779395 0.626532i \(-0.784474\pi\)
0.779395 0.626532i \(-0.215526\pi\)
\(6\) 0 0
\(7\) − 2.69202i − 1.01749i −0.860918 0.508744i \(-0.830110\pi\)
0.860918 0.508744i \(-0.169890\pi\)
\(8\) 0 0
\(9\) −2.35690 −0.785632
\(10\) 0 0
\(11\) − 1.19806i − 0.361229i −0.983554 0.180615i \(-0.942191\pi\)
0.983554 0.180615i \(-0.0578087\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) 2.24698i 0.580168i
\(16\) 0 0
\(17\) −1.13706 −0.275778 −0.137889 0.990448i \(-0.544032\pi\)
−0.137889 + 0.990448i \(0.544032\pi\)
\(18\) 0 0
\(19\) − 1.93900i − 0.444837i −0.974951 0.222419i \(-0.928605\pi\)
0.974951 0.222419i \(-0.0713952\pi\)
\(20\) 0 0
\(21\) 2.15883i 0.471096i
\(22\) 0 0
\(23\) −4.60388 −0.959974 −0.479987 0.877275i \(-0.659359\pi\)
−0.479987 + 0.877275i \(0.659359\pi\)
\(24\) 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
\(31\) − 5.89977i − 1.05963i −0.848113 0.529815i \(-0.822261\pi\)
0.848113 0.529815i \(-0.177739\pi\)
\(32\) 0 0
\(33\) 0.960771i 0.167249i
\(34\) 0 0
\(35\) −7.54288 −1.27498
\(36\) 0 0
\(37\) − 0.951083i − 0.156357i −0.996939 0.0781785i \(-0.975090\pi\)
0.996939 0.0781785i \(-0.0249104\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.31767i 0.518133i 0.965860 + 0.259066i \(0.0834148\pi\)
−0.965860 + 0.259066i \(0.916585\pi\)
\(42\) 0 0
\(43\) 7.15883 1.09171 0.545856 0.837879i \(-0.316205\pi\)
0.545856 + 0.837879i \(0.316205\pi\)
\(44\) 0 0
\(45\) 6.60388i 0.984448i
\(46\) 0 0
\(47\) − 7.69202i − 1.12200i −0.827817 0.560998i \(-0.810417\pi\)
0.827817 0.560998i \(-0.189583\pi\)
\(48\) 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
\(57\) 1.55496i 0.205959i
\(58\) 0 0
\(59\) − 0.0120816i − 0.00157289i −1.00000 0.000786444i \(-0.999750\pi\)
1.00000 0.000786444i \(-0.000250333\pi\)
\(60\) 0 0
\(61\) −8.03684 −1.02901 −0.514506 0.857487i \(-0.672025\pi\)
−0.514506 + 0.857487i \(0.672025\pi\)
\(62\) 0 0
\(63\) 6.34481i 0.799371i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 9.25667i 1.13088i 0.824789 + 0.565441i \(0.191294\pi\)
−0.824789 + 0.565441i \(0.808706\pi\)
\(68\) 0 0
\(69\) 3.69202 0.444467
\(70\) 0 0
\(71\) 13.7409i 1.63075i 0.578934 + 0.815375i \(0.303469\pi\)
−0.578934 + 0.815375i \(0.696531\pi\)
\(72\) 0 0
\(73\) − 12.8170i − 1.50012i −0.661372 0.750058i \(-0.730025\pi\)
0.661372 0.750058i \(-0.269975\pi\)
\(74\) 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
\(83\) 16.3327i 1.79275i 0.443296 + 0.896375i \(0.353809\pi\)
−0.443296 + 0.896375i \(0.646191\pi\)
\(84\) 0 0
\(85\) 3.18598i 0.345568i
\(86\) 0 0
\(87\) 6.33513 0.679197
\(88\) 0 0
\(89\) 14.7289i 1.56126i 0.624996 + 0.780628i \(0.285101\pi\)
−0.624996 + 0.780628i \(0.714899\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 4.73125i 0.490608i
\(94\) 0 0
\(95\) −5.43296 −0.557410
\(96\) 0 0
\(97\) 3.13169i 0.317975i 0.987281 + 0.158987i \(0.0508229\pi\)
−0.987281 + 0.158987i \(0.949177\pi\)
\(98\) 0 0
\(99\) 2.82371i 0.283793i
\(100\) 0 0
\(101\) −5.29052 −0.526426 −0.263213 0.964738i \(-0.584782\pi\)
−0.263213 + 0.964738i \(0.584782\pi\)
\(102\) 0 0
\(103\) −13.5308 −1.33323 −0.666614 0.745403i \(-0.732257\pi\)
−0.666614 + 0.745403i \(0.732257\pi\)
\(104\) 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
\(109\) 4.17629i 0.400016i 0.979794 + 0.200008i \(0.0640969\pi\)
−0.979794 + 0.200008i \(0.935903\pi\)
\(110\) 0 0
\(111\) 0.762709i 0.0723931i
\(112\) 0 0
\(113\) 7.64310 0.719003 0.359501 0.933145i \(-0.382947\pi\)
0.359501 + 0.933145i \(0.382947\pi\)
\(114\) 0 0
\(115\) 12.8998i 1.20291i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 3.06100i 0.280601i
\(120\) 0 0
\(121\) 9.56465 0.869513
\(122\) 0 0
\(123\) − 2.66056i − 0.239895i
\(124\) 0 0
\(125\) − 6.02177i − 0.538604i
\(126\) 0 0
\(127\) 6.77777 0.601430 0.300715 0.953714i \(-0.402775\pi\)
0.300715 + 0.953714i \(0.402775\pi\)
\(128\) 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
\(135\) − 12.0368i − 1.03597i
\(136\) 0 0
\(137\) − 12.9879i − 1.10963i −0.831973 0.554816i \(-0.812789\pi\)
0.831973 0.554816i \(-0.187211\pi\)
\(138\) 0 0
\(139\) −12.0465 −1.02177 −0.510886 0.859648i \(-0.670683\pi\)
−0.510886 + 0.859648i \(0.670683\pi\)
\(140\) 0 0
\(141\) 6.16852i 0.519483i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 22.1347i 1.83818i
\(146\) 0 0
\(147\) 0.198062 0.0163359
\(148\) 0 0
\(149\) − 0.740939i − 0.0607001i −0.999539 0.0303500i \(-0.990338\pi\)
0.999539 0.0303500i \(-0.00966220\pi\)
\(150\) 0 0
\(151\) − 19.0737i − 1.55219i −0.630614 0.776097i \(-0.717197\pi\)
0.630614 0.776097i \(-0.282803\pi\)
\(152\) 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
\(161\) 12.3937i 0.976763i
\(162\) 0 0
\(163\) 15.1371i 1.18563i 0.805340 + 0.592813i \(0.201983\pi\)
−0.805340 + 0.592813i \(0.798017\pi\)
\(164\) 0 0
\(165\) 2.69202 0.209574
\(166\) 0 0
\(167\) − 6.26337i − 0.484674i −0.970192 0.242337i \(-0.922086\pi\)
0.970192 0.242337i \(-0.0779140\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 4.57002i 0.349478i
\(172\) 0 0
\(173\) −16.3913 −1.24621 −0.623105 0.782138i \(-0.714129\pi\)
−0.623105 + 0.782138i \(0.714129\pi\)
\(174\) 0 0
\(175\) 7.67456i 0.580142i
\(176\) 0 0
\(177\) 0.00968868i 0 0.000728246i
\(178\) 0 0
\(179\) −2.45473 −0.183475 −0.0917376 0.995783i \(-0.529242\pi\)
−0.0917376 + 0.995783i \(0.529242\pi\)
\(180\) 0 0
\(181\) −11.8073 −0.877631 −0.438815 0.898577i \(-0.644602\pi\)
−0.438815 + 0.898577i \(0.644602\pi\)
\(182\) 0 0
\(183\) 6.44504 0.476431
\(184\) 0 0
\(185\) −2.66487 −0.195925
\(186\) 0 0
\(187\) 1.36227i 0.0996192i
\(188\) 0 0
\(189\) − 11.5646i − 0.841204i
\(190\) 0 0
\(191\) 8.99330 0.650732 0.325366 0.945588i \(-0.394512\pi\)
0.325366 + 0.945588i \(0.394512\pi\)
\(192\) 0 0
\(193\) − 13.5254i − 0.973581i −0.873519 0.486790i \(-0.838168\pi\)
0.873519 0.486790i \(-0.161832\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 12.9758i 0.924490i 0.886752 + 0.462245i \(0.152956\pi\)
−0.886752 + 0.462245i \(0.847044\pi\)
\(198\) 0 0
\(199\) −13.5864 −0.963116 −0.481558 0.876414i \(-0.659929\pi\)
−0.481558 + 0.876414i \(0.659929\pi\)
\(200\) 0 0
\(201\) − 7.42327i − 0.523597i
\(202\) 0 0
\(203\) 21.2664i 1.49261i
\(204\) 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
\(213\) − 11.0194i − 0.755035i
\(214\) 0 0
\(215\) − 20.0586i − 1.36799i
\(216\) 0 0
\(217\) −15.8823 −1.07816
\(218\) 0 0
\(219\) 10.2784i 0.694553i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 11.4058i 0.763790i 0.924206 + 0.381895i \(0.124728\pi\)
−0.924206 + 0.381895i \(0.875272\pi\)
\(224\) 0 0
\(225\) 6.71917 0.447945
\(226\) 0 0
\(227\) − 10.6407i − 0.706249i −0.935576 0.353124i \(-0.885119\pi\)
0.935576 0.353124i \(-0.114881\pi\)
\(228\) 0 0
\(229\) 1.13946i 0.0752974i 0.999291 + 0.0376487i \(0.0119868\pi\)
−0.999291 + 0.0376487i \(0.988013\pi\)
\(230\) 0 0
\(231\) 2.58642 0.170174
\(232\) 0 0
\(233\) 10.8509 0.710863 0.355432 0.934702i \(-0.384334\pi\)
0.355432 + 0.934702i \(0.384334\pi\)
\(234\) 0 0
\(235\) −21.5526 −1.40593
\(236\) 0 0
\(237\) 0.647416 0.0420542
\(238\) 0 0
\(239\) 11.9293i 0.771643i 0.922573 + 0.385822i \(0.126082\pi\)
−0.922573 + 0.385822i \(0.873918\pi\)
\(240\) 0 0
\(241\) 3.64848i 0.235019i 0.993072 + 0.117510i \(0.0374911\pi\)
−0.993072 + 0.117510i \(0.962509\pi\)
\(242\) 0 0
\(243\) −15.7952 −1.01326
\(244\) 0 0
\(245\) 0.692021i 0.0442116i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) − 13.0978i − 0.830042i
\(250\) 0 0
\(251\) 1.37329 0.0866813 0.0433406 0.999060i \(-0.486200\pi\)
0.0433406 + 0.999060i \(0.486200\pi\)
\(252\) 0 0
\(253\) 5.51573i 0.346771i
\(254\) 0 0
\(255\) − 2.55496i − 0.159998i
\(256\) 0 0
\(257\) −29.4359 −1.83616 −0.918082 0.396391i \(-0.870263\pi\)
−0.918082 + 0.396391i \(0.870263\pi\)
\(258\) 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
\(265\) − 16.4547i − 1.01081i
\(266\) 0 0
\(267\) − 11.8116i − 0.722860i
\(268\) 0 0
\(269\) −10.1860 −0.621050 −0.310525 0.950565i \(-0.600505\pi\)
−0.310525 + 0.950565i \(0.600505\pi\)
\(270\) 0 0
\(271\) − 29.4523i − 1.78910i −0.446966 0.894551i \(-0.647495\pi\)
0.446966 0.894551i \(-0.352505\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3.41550i 0.205963i
\(276\) 0 0
\(277\) 10.2446 0.615538 0.307769 0.951461i \(-0.400418\pi\)
0.307769 + 0.951461i \(0.400418\pi\)
\(278\) 0 0
\(279\) 13.9051i 0.832480i
\(280\) 0 0
\(281\) 11.5646i 0.689889i 0.938623 + 0.344944i \(0.112102\pi\)
−0.938623 + 0.344944i \(0.887898\pi\)
\(282\) 0 0
\(283\) −30.7090 −1.82546 −0.912730 0.408562i \(-0.866030\pi\)
−0.912730 + 0.408562i \(0.866030\pi\)
\(284\) 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
\(291\) − 2.51142i − 0.147222i
\(292\) 0 0
\(293\) 18.6082i 1.08710i 0.839376 + 0.543551i \(0.182921\pi\)
−0.839376 + 0.543551i \(0.817079\pi\)
\(294\) 0 0
\(295\) −0.0338518 −0.00197093
\(296\) 0 0
\(297\) − 5.14675i − 0.298645i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) − 19.2717i − 1.11080i
\(302\) 0 0
\(303\) 4.24267 0.243735
\(304\) 0 0
\(305\) 22.5187i 1.28942i
\(306\) 0 0
\(307\) 8.94438i 0.510483i 0.966877 + 0.255241i \(0.0821549\pi\)
−0.966877 + 0.255241i \(0.917845\pi\)
\(308\) 0 0
\(309\) 10.8509 0.617284
\(310\) 0 0
\(311\) 21.0398 1.19306 0.596529 0.802591i \(-0.296546\pi\)
0.596529 + 0.802591i \(0.296546\pi\)
\(312\) 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
\(317\) 23.9651i 1.34601i 0.739636 + 0.673007i \(0.234997\pi\)
−0.739636 + 0.673007i \(0.765003\pi\)
\(318\) 0 0
\(319\) 9.46442i 0.529906i
\(320\) 0 0
\(321\) 4.51573 0.252043
\(322\) 0 0
\(323\) 2.20477i 0.122677i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 3.34913i − 0.185207i
\(328\) 0 0
\(329\) −20.7071 −1.14162
\(330\) 0 0
\(331\) − 2.89546i − 0.159149i −0.996829 0.0795745i \(-0.974644\pi\)
0.996829 0.0795745i \(-0.0253561\pi\)
\(332\) 0 0
\(333\) 2.24160i 0.122839i
\(334\) 0 0
\(335\) 25.9366 1.41707
\(336\) 0 0
\(337\) 3.10560 0.169173 0.0845865 0.996416i \(-0.473043\pi\)
0.0845865 + 0.996416i \(0.473043\pi\)
\(338\) 0 0
\(339\) −6.12929 −0.332898
\(340\) 0 0
\(341\) −7.06829 −0.382770
\(342\) 0 0
\(343\) − 18.1793i − 0.981589i
\(344\) 0 0
\(345\) − 10.3448i − 0.556946i
\(346\) 0 0
\(347\) 11.3787 0.610839 0.305419 0.952218i \(-0.401203\pi\)
0.305419 + 0.952218i \(0.401203\pi\)
\(348\) 0 0
\(349\) 3.34721i 0.179172i 0.995979 + 0.0895859i \(0.0285544\pi\)
−0.995979 + 0.0895859i \(0.971446\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0.637727i 0.0339428i 0.999856 + 0.0169714i \(0.00540242\pi\)
−0.999856 + 0.0169714i \(0.994598\pi\)
\(354\) 0 0
\(355\) 38.5013 2.04343
\(356\) 0 0
\(357\) − 2.45473i − 0.129918i
\(358\) 0 0
\(359\) − 21.4590i − 1.13256i −0.824211 0.566282i \(-0.808381\pi\)
0.824211 0.566282i \(-0.191619\pi\)
\(360\) 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
\(369\) − 7.81940i − 0.407062i
\(370\) 0 0
\(371\) − 15.8092i − 0.820775i
\(372\) 0 0
\(373\) 27.7265 1.43562 0.717811 0.696238i \(-0.245144\pi\)
0.717811 + 0.696238i \(0.245144\pi\)
\(374\) 0 0
\(375\) 4.82908i 0.249373i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) − 35.8702i − 1.84253i −0.388935 0.921265i \(-0.627157\pi\)
0.388935 0.921265i \(-0.372843\pi\)
\(380\) 0 0
\(381\) −5.43535 −0.278462
\(382\) 0 0
\(383\) − 4.85517i − 0.248087i −0.992277 0.124044i \(-0.960414\pi\)
0.992277 0.124044i \(-0.0395863\pi\)
\(384\) 0 0
\(385\) 9.03684i 0.460560i
\(386\) 0 0
\(387\) −16.8726 −0.857684
\(388\) 0 0
\(389\) −2.38537 −0.120943 −0.0604716 0.998170i \(-0.519260\pi\)
−0.0604716 + 0.998170i \(0.519260\pi\)
\(390\) 0 0
\(391\) 5.23490 0.264740
\(392\) 0 0
\(393\) −10.9705 −0.553387
\(394\) 0 0
\(395\) 2.26205i 0.113816i
\(396\) 0 0
\(397\) 15.2664i 0.766196i 0.923708 + 0.383098i \(0.125143\pi\)
−0.923708 + 0.383098i \(0.874857\pi\)
\(398\) 0 0
\(399\) 4.18598 0.209561
\(400\) 0 0
\(401\) − 12.7584i − 0.637124i −0.947902 0.318562i \(-0.896800\pi\)
0.947902 0.318562i \(-0.103200\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) − 10.1588i − 0.504797i
\(406\) 0 0
\(407\) −1.13946 −0.0564807
\(408\) 0 0
\(409\) − 25.3588i − 1.25391i −0.779054 0.626956i \(-0.784300\pi\)
0.779054 0.626956i \(-0.215700\pi\)
\(410\) 0 0
\(411\) 10.4155i 0.513759i
\(412\) 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
\(421\) − 8.29291i − 0.404172i −0.979368 0.202086i \(-0.935228\pi\)
0.979368 0.202086i \(-0.0647720\pi\)
\(422\) 0 0
\(423\) 18.1293i 0.881476i
\(424\) 0 0
\(425\) 3.24160 0.157241
\(426\) 0 0
\(427\) 21.6353i 1.04701i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 0.932296i − 0.0449071i −0.999748 0.0224536i \(-0.992852\pi\)
0.999748 0.0224536i \(-0.00714779\pi\)
\(432\) 0 0
\(433\) 13.3502 0.641569 0.320785 0.947152i \(-0.396053\pi\)
0.320785 + 0.947152i \(0.396053\pi\)
\(434\) 0 0
\(435\) − 17.7506i − 0.851077i
\(436\) 0 0
\(437\) 8.92692i 0.427032i
\(438\) 0 0
\(439\) 13.9922 0.667813 0.333906 0.942606i \(-0.391633\pi\)
0.333906 + 0.942606i \(0.391633\pi\)
\(440\) 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
\(447\) 0.594187i 0.0281041i
\(448\) 0 0
\(449\) 12.5864i 0.593990i 0.954879 + 0.296995i \(0.0959844\pi\)
−0.954879 + 0.296995i \(0.904016\pi\)
\(450\) 0 0
\(451\) 3.97477 0.187165
\(452\) 0 0
\(453\) 15.2959i 0.718664i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 33.6383i − 1.57353i −0.617250 0.786767i \(-0.711753\pi\)
0.617250 0.786767i \(-0.288247\pi\)
\(458\) 0 0
\(459\) −4.88471 −0.227999
\(460\) 0 0
\(461\) − 1.40283i − 0.0653363i −0.999466 0.0326681i \(-0.989600\pi\)
0.999466 0.0326681i \(-0.0104004\pi\)
\(462\) 0 0
\(463\) − 15.2010i − 0.706453i −0.935538 0.353226i \(-0.885085\pi\)
0.935538 0.353226i \(-0.114915\pi\)
\(464\) 0 0
\(465\) 13.2567 0.614763
\(466\) 0 0
\(467\) −39.3414 −1.82050 −0.910250 0.414058i \(-0.864111\pi\)
−0.910250 + 0.414058i \(0.864111\pi\)
\(468\) 0 0
\(469\) 24.9191 1.15066
\(470\) 0 0
\(471\) 3.22521 0.148610
\(472\) 0 0
\(473\) − 8.57673i − 0.394358i
\(474\) 0 0
\(475\) 5.52781i 0.253633i
\(476\) 0 0
\(477\) −13.8412 −0.633743
\(478\) 0 0
\(479\) − 22.3690i − 1.02206i −0.859561 0.511032i \(-0.829263\pi\)
0.859561 0.511032i \(-0.170737\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) − 9.93900i − 0.452240i
\(484\) 0 0
\(485\) 8.77479 0.398443
\(486\) 0 0
\(487\) − 22.9205i − 1.03863i −0.854584 0.519313i \(-0.826188\pi\)
0.854584 0.519313i \(-0.173812\pi\)
\(488\) 0 0
\(489\) − 12.1390i − 0.548944i
\(490\) 0 0
\(491\) 1.84356 0.0831987 0.0415993 0.999134i \(-0.486755\pi\)
0.0415993 + 0.999134i \(0.486755\pi\)
\(492\) 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
\(499\) 12.0344i 0.538736i 0.963037 + 0.269368i \(0.0868147\pi\)
−0.963037 + 0.269368i \(0.913185\pi\)
\(500\) 0 0
\(501\) 5.02284i 0.224404i
\(502\) 0 0
\(503\) 30.5056 1.36018 0.680088 0.733130i \(-0.261942\pi\)
0.680088 + 0.733130i \(0.261942\pi\)
\(504\) 0 0
\(505\) 14.8237i 0.659646i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 1.51142i − 0.0669924i −0.999439 0.0334962i \(-0.989336\pi\)
0.999439 0.0334962i \(-0.0106642\pi\)
\(510\) 0 0
\(511\) −34.5036 −1.52635
\(512\) 0 0
\(513\) − 8.32975i − 0.367767i
\(514\) 0 0
\(515\) 37.9124i 1.67062i
\(516\) 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
\(525\) − 6.15452i − 0.268605i
\(526\) 0 0
\(527\) 6.70841i 0.292223i
\(528\) 0 0
\(529\) −1.80433 −0.0784492
\(530\) 0 0
\(531\) 0.0284750i 0.00123571i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 15.7778i 0.682133i
\(536\) 0 0
\(537\) 1.96854 0.0849488
\(538\) 0 0
\(539\) 0.295897i 0.0127452i
\(540\) 0 0
\(541\) − 24.3297i − 1.04602i −0.852327 0.523009i \(-0.824809\pi\)
0.852327 0.523009i \(-0.175191\pi\)
\(542\) 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
\(551\) 15.3177i 0.652555i
\(552\) 0 0
\(553\) 2.17331i 0.0924185i
\(554\) 0 0
\(555\) 2.13706 0.0907133
\(556\) 0 0
\(557\) − 25.3327i − 1.07338i −0.843779 0.536691i \(-0.819674\pi\)
0.843779 0.536691i \(-0.180326\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) − 1.09246i − 0.0461236i
\(562\) 0 0
\(563\) −25.3937 −1.07022 −0.535109 0.844783i \(-0.679730\pi\)
−0.535109 + 0.844783i \(0.679730\pi\)
\(564\) 0 0
\(565\) − 21.4155i − 0.900957i
\(566\) 0 0
\(567\) − 9.76032i − 0.409895i
\(568\) 0 0
\(569\) 31.1347 1.30523 0.652617 0.757688i \(-0.273671\pi\)
0.652617 + 0.757688i \(0.273671\pi\)
\(570\) 0 0
\(571\) −20.5090 −0.858276 −0.429138 0.903239i \(-0.641183\pi\)
−0.429138 + 0.903239i \(0.641183\pi\)
\(572\) 0 0
\(573\) −7.21206 −0.301288
\(574\) 0 0
\(575\) 13.1250 0.547350
\(576\) 0 0
\(577\) 15.6890i 0.653143i 0.945172 + 0.326572i \(0.105893\pi\)
−0.945172 + 0.326572i \(0.894107\pi\)
\(578\) 0 0
\(579\) 10.8465i 0.450767i
\(580\) 0 0
\(581\) 43.9681 1.82410
\(582\) 0 0
\(583\) − 7.03577i − 0.291392i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 30.5687i − 1.26171i −0.775903 0.630853i \(-0.782705\pi\)
0.775903 0.630853i \(-0.217295\pi\)
\(588\) 0 0
\(589\) −11.4397 −0.471363
\(590\) 0 0
\(591\) − 10.4058i − 0.428038i
\(592\) 0 0
\(593\) 29.6883i 1.21915i 0.792727 + 0.609576i \(0.208660\pi\)
−0.792727 + 0.609576i \(0.791340\pi\)
\(594\) 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
\(603\) − 21.8170i − 0.888457i
\(604\) 0 0
\(605\) − 26.7995i − 1.08956i
\(606\) 0 0
\(607\) −1.43190 −0.0581188 −0.0290594 0.999578i \(-0.509251\pi\)
−0.0290594 + 0.999578i \(0.509251\pi\)
\(608\) 0 0
\(609\) − 17.0543i − 0.691075i
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 3.84846i 0.155438i 0.996975 + 0.0777190i \(0.0247637\pi\)
−0.996975 + 0.0777190i \(0.975236\pi\)
\(614\) 0 0
\(615\) −7.45473 −0.300604
\(616\) 0 0
\(617\) − 15.0388i − 0.605437i −0.953080 0.302719i \(-0.902106\pi\)
0.953080 0.302719i \(-0.0978943\pi\)
\(618\) 0 0
\(619\) 12.8170i 0.515159i 0.966257 + 0.257579i \(0.0829249\pi\)
−0.966257 + 0.257579i \(0.917075\pi\)
\(620\) 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
\(629\) 1.08144i 0.0431199i
\(630\) 0 0
\(631\) 25.7517i 1.02516i 0.858640 + 0.512579i \(0.171310\pi\)
−0.858640 + 0.512579i \(0.828690\pi\)
\(632\) 0 0
\(633\) 8.38835 0.333407
\(634\) 0 0
\(635\) − 18.9909i − 0.753631i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) − 32.3860i − 1.28117i
\(640\) 0 0
\(641\) −24.4571 −0.965998 −0.482999 0.875621i \(-0.660453\pi\)
−0.482999 + 0.875621i \(0.660453\pi\)
\(642\) 0 0
\(643\) 9.97344i 0.393314i 0.980472 + 0.196657i \(0.0630086\pi\)
−0.980472 + 0.196657i \(0.936991\pi\)
\(644\) 0 0
\(645\) 16.0858i 0.633376i
\(646\) 0 0
\(647\) 11.8431 0.465600 0.232800 0.972525i \(-0.425211\pi\)
0.232800 + 0.972525i \(0.425211\pi\)
\(648\) 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
\(655\) − 38.3303i − 1.49769i
\(656\) 0 0
\(657\) 30.2083i 1.17854i
\(658\) 0 0
\(659\) −34.1739 −1.33123 −0.665613 0.746297i \(-0.731830\pi\)
−0.665613 + 0.746297i \(0.731830\pi\)
\(660\) 0 0
\(661\) − 33.6088i − 1.30723i −0.756827 0.653615i \(-0.773252\pi\)
0.756827 0.653615i \(-0.226748\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 14.6256i 0.567158i
\(666\) 0 0
\(667\) 36.3696 1.40824
\(668\) 0 0
\(669\) − 9.14675i − 0.353634i
\(670\) 0 0
\(671\) 9.62863i 0.371709i
\(672\) 0 0
\(673\) −48.0320 −1.85150 −0.925750 0.378137i \(-0.876565\pi\)
−0.925750 + 0.378137i \(0.876565\pi\)
\(674\) 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
\(681\) 8.53319i 0.326992i
\(682\) 0 0
\(683\) 15.9041i 0.608553i 0.952584 + 0.304276i \(0.0984147\pi\)
−0.952584 + 0.304276i \(0.901585\pi\)
\(684\) 0 0
\(685\) −36.3913 −1.39044
\(686\) 0 0
\(687\) − 0.913773i − 0.0348626i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) − 33.1903i − 1.26262i −0.775531 0.631309i \(-0.782518\pi\)
0.775531 0.631309i \(-0.217482\pi\)
\(692\) 0 0
\(693\) 7.60148 0.288756
\(694\) 0 0
\(695\) 33.7536i 1.28035i
\(696\) 0 0
\(697\) − 3.77240i − 0.142890i
\(698\) 0 0
\(699\) −8.70171 −0.329129
\(700\) 0 0
\(701\) 14.9129 0.563253 0.281627 0.959524i \(-0.409126\pi\)
0.281627 + 0.959524i \(0.409126\pi\)
\(702\) 0 0
\(703\) −1.84415 −0.0695534
\(704\) 0 0
\(705\) 17.2838 0.650946
\(706\) 0 0
\(707\) 14.2422i 0.535633i
\(708\) 0 0
\(709\) − 38.4312i − 1.44331i −0.692252 0.721656i \(-0.743381\pi\)
0.692252 0.721656i \(-0.256619\pi\)
\(710\) 0 0
\(711\) 1.90276 0.0713589
\(712\) 0 0
\(713\) 27.1618i 1.01722i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 9.56657i − 0.357270i
\(718\) 0 0
\(719\) −11.4373 −0.426538 −0.213269 0.976993i \(-0.568411\pi\)
−0.213269 + 0.976993i \(0.568411\pi\)
\(720\) 0 0
\(721\) 36.4252i 1.35654i
\(722\) 0 0
\(723\) − 2.92585i − 0.108814i
\(724\) 0 0
\(725\) 22.5211 0.836413
\(726\) 0 0
\(727\) 3.63640 0.134867 0.0674333 0.997724i \(-0.478519\pi\)
0.0674333 + 0.997724i \(0.478519\pi\)
\(728\) 0 0
\(729\) 1.78986 0.0662910
\(730\) 0 0
\(731\) −8.14005 −0.301071
\(732\) 0 0
\(733\) − 3.52217i − 0.130094i −0.997882 0.0650472i \(-0.979280\pi\)
0.997882 0.0650472i \(-0.0207198\pi\)
\(734\) 0 0
\(735\) − 0.554958i − 0.0204699i
\(736\) 0 0
\(737\) 11.0901 0.408508
\(738\) 0 0
\(739\) 0.420288i 0.0154605i 0.999970 + 0.00773027i \(0.00246064\pi\)
−0.999970 + 0.00773027i \(0.997539\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 25.3623i 0.930452i 0.885192 + 0.465226i \(0.154027\pi\)
−0.885192 + 0.465226i \(0.845973\pi\)
\(744\) 0 0
\(745\) −2.07606 −0.0760611
\(746\) 0 0
\(747\) − 38.4946i − 1.40844i
\(748\) 0 0
\(749\) 15.1588i 0.553892i
\(750\) 0 0
\(751\) 0.650874 0.0237507 0.0118754 0.999929i \(-0.496220\pi\)
0.0118754 + 0.999929i \(0.496220\pi\)
\(752\) 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
\(759\) − 4.42327i − 0.160555i
\(760\) 0 0
\(761\) − 30.9221i − 1.12093i −0.828179 0.560463i \(-0.810623\pi\)
0.828179 0.560463i \(-0.189377\pi\)
\(762\) 0 0
\(763\) 11.2427 0.407012
\(764\) 0 0
\(765\) − 7.50902i − 0.271489i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) − 43.7689i − 1.57835i −0.614169 0.789174i \(-0.710509\pi\)
0.614169 0.789174i \(-0.289491\pi\)
\(770\) 0 0
\(771\) 23.6058 0.850142
\(772\) 0 0
\(773\) − 42.4209i − 1.52577i −0.646532 0.762886i \(-0.723782\pi\)
0.646532 0.762886i \(-0.276218\pi\)
\(774\) 0 0
\(775\) 16.8194i 0.604171i
\(776\) 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
\(785\) 11.2687i 0.402199i
\(786\) 0 0
\(787\) − 36.0116i − 1.28368i −0.766841 0.641838i \(-0.778172\pi\)
0.766841 0.641838i \(-0.221828\pi\)
\(788\) 0 0
\(789\) 8.57779 0.305378
\(790\) 0 0
\(791\) − 20.5754i − 0.731577i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 13.1957i 0.468002i
\(796\) 0 0
\(797\) 31.7101 1.12323 0.561614 0.827399i \(-0.310181\pi\)
0.561614 + 0.827399i \(0.310181\pi\)
\(798\) 0 0
\(799\) 8.74632i 0.309422i
\(800\) 0 0
\(801\) − 34.7144i − 1.22657i
\(802\) 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
\(811\) 42.8635i 1.50514i 0.658511 + 0.752571i \(0.271187\pi\)
−0.658511 + 0.752571i \(0.728813\pi\)
\(812\) 0 0
\(813\) 23.6189i 0.828352i
\(814\) 0 0
\(815\) 42.4131 1.48567
\(816\) 0 0
\(817\) − 13.8810i − 0.485634i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 7.82776i − 0.273191i −0.990627 0.136595i \(-0.956384\pi\)
0.990627 0.136595i \(-0.0436160\pi\)
\(822\) 0 0
\(823\) −36.7754 −1.28191 −0.640955 0.767579i \(-0.721461\pi\)
−0.640955 + 0.767579i \(0.721461\pi\)
\(824\) 0 0
\(825\) − 2.73902i − 0.0953604i
\(826\) 0 0
\(827\) 47.3293i 1.64580i 0.568186 + 0.822900i \(0.307645\pi\)
−0.568186 + 0.822900i \(0.692355\pi\)
\(828\) 0 0
\(829\) −25.2687 −0.877620 −0.438810 0.898580i \(-0.644600\pi\)
−0.438810 + 0.898580i \(0.644600\pi\)
\(830\) 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
\(837\) − 25.3448i − 0.876045i
\(838\) 0 0
\(839\) − 37.6883i − 1.30114i −0.759444 0.650572i \(-0.774529\pi\)
0.759444 0.650572i \(-0.225471\pi\)
\(840\) 0 0
\(841\) 33.4064 1.15194
\(842\) 0 0
\(843\) − 9.27413i − 0.319418i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 25.7482i − 0.884720i
\(848\) 0 0
\(849\) 24.6267 0.845187
\(850\) 0 0
\(851\) 4.37867i 0.150099i
\(852\) 0 0
\(853\) 31.0121i 1.06183i 0.847424 + 0.530917i \(0.178152\pi\)
−0.847424 + 0.530917i \(0.821848\pi\)
\(854\) 0 0
\(855\) 12.8049 0.437919
\(856\) 0 0
\(857\) −12.4692 −0.425940 −0.212970 0.977059i \(-0.568314\pi\)
−0.212970 + 0.977059i \(0.568314\pi\)
\(858\) 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
\(863\) 3.46383i 0.117910i 0.998261 + 0.0589550i \(0.0187769\pi\)
−0.998261 + 0.0589550i \(0.981223\pi\)
\(864\) 0 0
\(865\) 45.9275i 1.56158i
\(866\) 0 0
\(867\) 12.5961 0.427786
\(868\) 0 0
\(869\) 0.967213i 0.0328105i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) − 7.38106i − 0.249811i
\(874\) 0 0
\(875\) −16.2107 −0.548023
\(876\) 0 0
\(877\) 57.2549i 1.93336i 0.255989 + 0.966680i \(0.417599\pi\)
−0.255989 + 0.966680i \(0.582401\pi\)
\(878\) 0 0
\(879\) − 14.9226i − 0.503327i
\(880\) 0 0
\(881\) 43.1782 1.45471 0.727355 0.686261i \(-0.240749\pi\)
0.727355 + 0.686261i \(0.240749\pi\)
\(882\) 0 0
\(883\) 49.9560 1.68115 0.840576 0.541693i \(-0.182216\pi\)
0.840576 + 0.541693i \(0.182216\pi\)
\(884\) 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
\(889\) − 18.2459i − 0.611948i
\(890\) 0 0
\(891\) − 4.34375i − 0.145521i
\(892\) 0 0
\(893\) −14.9148 −0.499106
\(894\) 0 0
\(895\) 6.87800i 0.229906i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 46.6069i 1.55443i
\(900\) 0 0
\(901\) −6.67755 −0.222461
\(902\) 0 0
\(903\) 15.4547i 0.514301i
\(904\) 0 0
\(905\) 33.0834i 1.09973i
\(906\) 0 0
\(907\) 7.73423 0.256811 0.128406 0.991722i \(-0.459014\pi\)
0.128406 + 0.991722i \(0.459014\pi\)
\(908\) 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
\(915\) − 18.0586i − 0.596999i
\(916\) 0 0
\(917\) − 36.8267i − 1.21612i
\(918\) 0 0
\(919\) −14.6213 −0.482313 −0.241157 0.970486i \(-0.577527\pi\)
−0.241157 + 0.970486i \(0.577527\pi\)
\(920\) 0 0
\(921\) − 7.17283i − 0.236353i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 2.71140i 0.0891502i
\(926\) 0 0
\(927\) 31.8907 1.04743
\(928\) 0 0
\(929\) 3.55735i 0.116713i 0.998296 + 0.0583565i \(0.0185860\pi\)
−0.998296 + 0.0583565i \(0.981414\pi\)
\(930\) 0 0
\(931\) 0.478894i 0.0156951i
\(932\) 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
\(941\) 20.6233i 0.672299i 0.941809 + 0.336149i \(0.109125\pi\)
−0.941809 + 0.336149i \(0.890875\pi\)
\(942\) 0 0
\(943\) − 15.2741i − 0.497394i
\(944\) 0 0
\(945\) −32.4034 −1.05408
\(946\) 0 0
\(947\) 29.4999i 0.958619i 0.877646 + 0.479309i \(0.159113\pi\)
−0.877646 + 0.479309i \(0.840887\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) − 19.2185i − 0.623203i
\(952\) 0 0
\(953\) −26.2389 −0.849963 −0.424981 0.905202i \(-0.639719\pi\)
−0.424981 + 0.905202i \(0.639719\pi\)
\(954\) 0 0
\(955\) − 25.1987i − 0.815409i
\(956\) 0 0
\(957\) − 7.58987i − 0.245346i
\(958\) 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
\(967\) − 17.5176i − 0.563330i −0.959513 0.281665i \(-0.909113\pi\)
0.959513 0.281665i \(-0.0908866\pi\)
\(968\) 0 0
\(969\) − 1.76809i − 0.0567991i
\(970\) 0 0
\(971\) −20.5120 −0.658262 −0.329131 0.944284i \(-0.606756\pi\)
−0.329131 + 0.944284i \(0.606756\pi\)
\(972\) 0 0
\(973\) 32.4295i 1.03964i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 25.4450i 0.814059i 0.913415 + 0.407030i \(0.133435\pi\)
−0.913415 + 0.407030i \(0.866565\pi\)
\(978\) 0 0
\(979\) 17.6461 0.563971
\(980\) 0 0
\(981\) − 9.84309i − 0.314266i
\(982\) 0 0
\(983\) − 39.5244i − 1.26063i −0.776339 0.630316i \(-0.782926\pi\)
0.776339 0.630316i \(-0.217074\pi\)
\(984\) 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
\(993\) 2.32198i 0.0736858i
\(994\) 0 0
\(995\) 38.0683i 1.20685i
\(996\) 0 0
\(997\) −4.93123 −0.156174 −0.0780868 0.996947i \(-0.524881\pi\)
−0.0780868 + 0.996947i \(0.524881\pi\)
\(998\) 0 0
\(999\) − 4.08575i − 0.129268i
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2704.2.f.o.337.1 6
4.3 odd 2 169.2.b.b.168.3 6
12.11 even 2 1521.2.b.l.1351.4 6
13.5 odd 4 2704.2.a.z.1.1 3
13.8 odd 4 2704.2.a.ba.1.1 3
13.12 even 2 inner 2704.2.f.o.337.2 6
52.3 odd 6 169.2.e.b.147.4 12
52.7 even 12 169.2.c.b.146.2 6
52.11 even 12 169.2.c.b.22.2 6
52.15 even 12 169.2.c.c.22.2 6
52.19 even 12 169.2.c.c.146.2 6
52.23 odd 6 169.2.e.b.147.3 12
52.31 even 4 169.2.a.b.1.2 3
52.35 odd 6 169.2.e.b.23.3 12
52.43 odd 6 169.2.e.b.23.4 12
52.47 even 4 169.2.a.c.1.2 yes 3
52.51 odd 2 169.2.b.b.168.4 6
156.47 odd 4 1521.2.a.o.1.2 3
156.83 odd 4 1521.2.a.r.1.2 3
156.155 even 2 1521.2.b.l.1351.3 6
260.99 even 4 4225.2.a.bb.1.2 3
260.239 even 4 4225.2.a.bg.1.2 3
364.83 odd 4 8281.2.a.bf.1.2 3
364.307 odd 4 8281.2.a.bj.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
169.2.a.b.1.2 3 52.31 even 4
169.2.a.c.1.2 yes 3 52.47 even 4
169.2.b.b.168.3 6 4.3 odd 2
169.2.b.b.168.4 6 52.51 odd 2
169.2.c.b.22.2 6 52.11 even 12
169.2.c.b.146.2 6 52.7 even 12
169.2.c.c.22.2 6 52.15 even 12
169.2.c.c.146.2 6 52.19 even 12
169.2.e.b.23.3 12 52.35 odd 6
169.2.e.b.23.4 12 52.43 odd 6
169.2.e.b.147.3 12 52.23 odd 6
169.2.e.b.147.4 12 52.3 odd 6
1521.2.a.o.1.2 3 156.47 odd 4
1521.2.a.r.1.2 3 156.83 odd 4
1521.2.b.l.1351.3 6 156.155 even 2
1521.2.b.l.1351.4 6 12.11 even 2
2704.2.a.z.1.1 3 13.5 odd 4
2704.2.a.ba.1.1 3 13.8 odd 4
2704.2.f.o.337.1 6 1.1 even 1 trivial
2704.2.f.o.337.2 6 13.12 even 2 inner
4225.2.a.bb.1.2 3 260.99 even 4
4225.2.a.bg.1.2 3 260.239 even 4
8281.2.a.bf.1.2 3 364.83 odd 4
8281.2.a.bj.1.2 3 364.307 odd 4