Properties

Label 1456.2.a.v.1.4
Level $1456$
Weight $2$
Character 1456.1
Self dual yes
Analytic conductor $11.626$
Analytic rank $0$
Dimension $4$
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] = [1456,2,Mod(1,1456)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1456, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1456.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1456.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: \(11.6262185343\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.64268.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 12x^{2} + 6x + 32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 728)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-2.65932\) of defining polynomial
Character \(\chi\) \(=\) 1456.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.65932 q^{3} -2.96141 q^{5} +1.00000 q^{7} +4.07200 q^{9} +O(q^{10})\) \(q+2.65932 q^{3} -2.96141 q^{5} +1.00000 q^{7} +4.07200 q^{9} +4.80333 q^{11} -1.00000 q^{13} -7.87534 q^{15} +0.268672 q^{17} +6.50125 q^{19} +2.65932 q^{21} -2.30208 q^{23} +3.76992 q^{25} +2.85081 q^{27} +6.54873 q^{29} -4.03341 q^{31} +12.7736 q^{33} -2.96141 q^{35} +2.51532 q^{37} -2.65932 q^{39} +4.46784 q^{41} -10.6927 q^{43} -12.0589 q^{45} +8.63757 q^{47} +1.00000 q^{49} +0.714485 q^{51} -3.37409 q^{53} -14.2246 q^{55} +17.2889 q^{57} +11.7788 q^{59} +9.13882 q^{61} +4.07200 q^{63} +2.96141 q^{65} -13.7093 q^{67} -6.12198 q^{69} +16.2583 q^{71} +6.03341 q^{73} +10.0254 q^{75} +4.80333 q^{77} +5.76474 q^{79} -4.63479 q^{81} +1.56557 q^{83} -0.795646 q^{85} +17.4152 q^{87} +6.64526 q^{89} -1.00000 q^{91} -10.7261 q^{93} -19.2528 q^{95} -2.74789 q^{97} +19.5592 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{3} + 2 q^{5} + 4 q^{7} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{3} + 2 q^{5} + 4 q^{7} + 13 q^{9} + q^{11} - 4 q^{13} - 10 q^{15} + 16 q^{17} + 10 q^{19} - q^{21} - 7 q^{23} + 14 q^{25} - 13 q^{27} + 4 q^{29} + q^{31} - q^{33} + 2 q^{35} + 5 q^{37} + q^{39} + 19 q^{41} - 14 q^{43} + 10 q^{45} + 13 q^{47} + 4 q^{49} - 16 q^{51} - 8 q^{53} - 2 q^{55} + 12 q^{57} + 26 q^{59} - q^{61} + 13 q^{63} - 2 q^{65} - 5 q^{67} + 17 q^{69} + 18 q^{71} + 7 q^{73} - q^{75} + q^{77} - 9 q^{79} - 4 q^{81} - 12 q^{83} + 14 q^{85} + 46 q^{87} + 4 q^{89} - 4 q^{91} + 3 q^{93} + 26 q^{95} + 25 q^{97} + 48 q^{99}+O(q^{100}) \) 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
\(3\) 2.65932 1.53536 0.767681 0.640833i \(-0.221411\pi\)
0.767681 + 0.640833i \(0.221411\pi\)
\(4\) 0 0
\(5\) −2.96141 −1.32438 −0.662190 0.749336i \(-0.730373\pi\)
−0.662190 + 0.749336i \(0.730373\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 4.07200 1.35733
\(10\) 0 0
\(11\) 4.80333 1.44826 0.724130 0.689664i \(-0.242242\pi\)
0.724130 + 0.689664i \(0.242242\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) −7.87534 −2.03340
\(16\) 0 0
\(17\) 0.268672 0.0651625 0.0325812 0.999469i \(-0.489627\pi\)
0.0325812 + 0.999469i \(0.489627\pi\)
\(18\) 0 0
\(19\) 6.50125 1.49149 0.745745 0.666232i \(-0.232094\pi\)
0.745745 + 0.666232i \(0.232094\pi\)
\(20\) 0 0
\(21\) 2.65932 0.580312
\(22\) 0 0
\(23\) −2.30208 −0.480017 −0.240009 0.970771i \(-0.577150\pi\)
−0.240009 + 0.970771i \(0.577150\pi\)
\(24\) 0 0
\(25\) 3.76992 0.753985
\(26\) 0 0
\(27\) 2.85081 0.548638
\(28\) 0 0
\(29\) 6.54873 1.21607 0.608034 0.793911i \(-0.291958\pi\)
0.608034 + 0.793911i \(0.291958\pi\)
\(30\) 0 0
\(31\) −4.03341 −0.724422 −0.362211 0.932096i \(-0.617978\pi\)
−0.362211 + 0.932096i \(0.617978\pi\)
\(32\) 0 0
\(33\) 12.7736 2.22360
\(34\) 0 0
\(35\) −2.96141 −0.500569
\(36\) 0 0
\(37\) 2.51532 0.413515 0.206758 0.978392i \(-0.433709\pi\)
0.206758 + 0.978392i \(0.433709\pi\)
\(38\) 0 0
\(39\) −2.65932 −0.425833
\(40\) 0 0
\(41\) 4.46784 0.697760 0.348880 0.937167i \(-0.386562\pi\)
0.348880 + 0.937167i \(0.386562\pi\)
\(42\) 0 0
\(43\) −10.6927 −1.63063 −0.815313 0.579020i \(-0.803435\pi\)
−0.815313 + 0.579020i \(0.803435\pi\)
\(44\) 0 0
\(45\) −12.0589 −1.79763
\(46\) 0 0
\(47\) 8.63757 1.25992 0.629960 0.776628i \(-0.283071\pi\)
0.629960 + 0.776628i \(0.283071\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0.714485 0.100048
\(52\) 0 0
\(53\) −3.37409 −0.463466 −0.231733 0.972779i \(-0.574440\pi\)
−0.231733 + 0.972779i \(0.574440\pi\)
\(54\) 0 0
\(55\) −14.2246 −1.91805
\(56\) 0 0
\(57\) 17.2889 2.28998
\(58\) 0 0
\(59\) 11.7788 1.53347 0.766735 0.641964i \(-0.221880\pi\)
0.766735 + 0.641964i \(0.221880\pi\)
\(60\) 0 0
\(61\) 9.13882 1.17011 0.585053 0.810995i \(-0.301074\pi\)
0.585053 + 0.810995i \(0.301074\pi\)
\(62\) 0 0
\(63\) 4.07200 0.513024
\(64\) 0 0
\(65\) 2.96141 0.367317
\(66\) 0 0
\(67\) −13.7093 −1.67486 −0.837429 0.546546i \(-0.815942\pi\)
−0.837429 + 0.546546i \(0.815942\pi\)
\(68\) 0 0
\(69\) −6.12198 −0.737000
\(70\) 0 0
\(71\) 16.2583 1.92951 0.964753 0.263158i \(-0.0847640\pi\)
0.964753 + 0.263158i \(0.0847640\pi\)
\(72\) 0 0
\(73\) 6.03341 0.706157 0.353079 0.935594i \(-0.385135\pi\)
0.353079 + 0.935594i \(0.385135\pi\)
\(74\) 0 0
\(75\) 10.0254 1.15764
\(76\) 0 0
\(77\) 4.80333 0.547391
\(78\) 0 0
\(79\) 5.76474 0.648584 0.324292 0.945957i \(-0.394874\pi\)
0.324292 + 0.945957i \(0.394874\pi\)
\(80\) 0 0
\(81\) −4.63479 −0.514977
\(82\) 0 0
\(83\) 1.56557 0.171843 0.0859217 0.996302i \(-0.472616\pi\)
0.0859217 + 0.996302i \(0.472616\pi\)
\(84\) 0 0
\(85\) −0.795646 −0.0862999
\(86\) 0 0
\(87\) 17.4152 1.86710
\(88\) 0 0
\(89\) 6.64526 0.704396 0.352198 0.935925i \(-0.385434\pi\)
0.352198 + 0.935925i \(0.385434\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) −10.7261 −1.11225
\(94\) 0 0
\(95\) −19.2528 −1.97530
\(96\) 0 0
\(97\) −2.74789 −0.279006 −0.139503 0.990222i \(-0.544551\pi\)
−0.139503 + 0.990222i \(0.544551\pi\)
\(98\) 0 0
\(99\) 19.5592 1.96577
\(100\) 0 0
\(101\) −5.59501 −0.556724 −0.278362 0.960476i \(-0.589791\pi\)
−0.278362 + 0.960476i \(0.589791\pi\)
\(102\) 0 0
\(103\) 8.78130 0.865248 0.432624 0.901575i \(-0.357588\pi\)
0.432624 + 0.901575i \(0.357588\pi\)
\(104\) 0 0
\(105\) −7.87534 −0.768554
\(106\) 0 0
\(107\) −17.3855 −1.68072 −0.840358 0.542031i \(-0.817655\pi\)
−0.840358 + 0.542031i \(0.817655\pi\)
\(108\) 0 0
\(109\) −9.04998 −0.866831 −0.433415 0.901194i \(-0.642692\pi\)
−0.433415 + 0.901194i \(0.642692\pi\)
\(110\) 0 0
\(111\) 6.68904 0.634895
\(112\) 0 0
\(113\) −10.9394 −1.02909 −0.514545 0.857463i \(-0.672039\pi\)
−0.514545 + 0.857463i \(0.672039\pi\)
\(114\) 0 0
\(115\) 6.81740 0.635726
\(116\) 0 0
\(117\) −4.07200 −0.376457
\(118\) 0 0
\(119\) 0.268672 0.0246291
\(120\) 0 0
\(121\) 12.0720 1.09745
\(122\) 0 0
\(123\) 11.8814 1.07131
\(124\) 0 0
\(125\) 3.64276 0.325818
\(126\) 0 0
\(127\) −18.9176 −1.67867 −0.839334 0.543616i \(-0.817055\pi\)
−0.839334 + 0.543616i \(0.817055\pi\)
\(128\) 0 0
\(129\) −28.4354 −2.50360
\(130\) 0 0
\(131\) −7.75067 −0.677180 −0.338590 0.940934i \(-0.609950\pi\)
−0.338590 + 0.940934i \(0.609950\pi\)
\(132\) 0 0
\(133\) 6.50125 0.563730
\(134\) 0 0
\(135\) −8.44240 −0.726606
\(136\) 0 0
\(137\) −3.95253 −0.337687 −0.168844 0.985643i \(-0.554003\pi\)
−0.168844 + 0.985643i \(0.554003\pi\)
\(138\) 0 0
\(139\) −7.04998 −0.597971 −0.298986 0.954258i \(-0.596648\pi\)
−0.298986 + 0.954258i \(0.596648\pi\)
\(140\) 0 0
\(141\) 22.9701 1.93443
\(142\) 0 0
\(143\) −4.80333 −0.401675
\(144\) 0 0
\(145\) −19.3934 −1.61054
\(146\) 0 0
\(147\) 2.65932 0.219337
\(148\) 0 0
\(149\) −13.3738 −1.09563 −0.547813 0.836601i \(-0.684539\pi\)
−0.547813 + 0.836601i \(0.684539\pi\)
\(150\) 0 0
\(151\) −18.5126 −1.50654 −0.753268 0.657713i \(-0.771524\pi\)
−0.753268 + 0.657713i \(0.771524\pi\)
\(152\) 0 0
\(153\) 1.09403 0.0884473
\(154\) 0 0
\(155\) 11.9446 0.959410
\(156\) 0 0
\(157\) −17.7568 −1.41714 −0.708572 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(158\) 0 0
\(159\) −8.97279 −0.711588
\(160\) 0 0
\(161\) −2.30208 −0.181429
\(162\) 0 0
\(163\) −16.6710 −1.30577 −0.652886 0.757456i \(-0.726442\pi\)
−0.652886 + 0.757456i \(0.726442\pi\)
\(164\) 0 0
\(165\) −37.8279 −2.94490
\(166\) 0 0
\(167\) −17.2054 −1.33139 −0.665696 0.746223i \(-0.731865\pi\)
−0.665696 + 0.746223i \(0.731865\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 26.4731 2.02445
\(172\) 0 0
\(173\) −9.58732 −0.728910 −0.364455 0.931221i \(-0.618745\pi\)
−0.364455 + 0.931221i \(0.618745\pi\)
\(174\) 0 0
\(175\) 3.76992 0.284979
\(176\) 0 0
\(177\) 31.3237 2.35443
\(178\) 0 0
\(179\) 10.3766 0.775583 0.387791 0.921747i \(-0.373238\pi\)
0.387791 + 0.921747i \(0.373238\pi\)
\(180\) 0 0
\(181\) 0.121981 0.00906675 0.00453338 0.999990i \(-0.498557\pi\)
0.00453338 + 0.999990i \(0.498557\pi\)
\(182\) 0 0
\(183\) 24.3031 1.79654
\(184\) 0 0
\(185\) −7.44887 −0.547652
\(186\) 0 0
\(187\) 1.29052 0.0943721
\(188\) 0 0
\(189\) 2.85081 0.207366
\(190\) 0 0
\(191\) 13.3855 0.968538 0.484269 0.874919i \(-0.339086\pi\)
0.484269 + 0.874919i \(0.339086\pi\)
\(192\) 0 0
\(193\) −15.6067 −1.12339 −0.561696 0.827344i \(-0.689851\pi\)
−0.561696 + 0.827344i \(0.689851\pi\)
\(194\) 0 0
\(195\) 7.87534 0.563965
\(196\) 0 0
\(197\) 10.2798 0.732404 0.366202 0.930535i \(-0.380658\pi\)
0.366202 + 0.930535i \(0.380658\pi\)
\(198\) 0 0
\(199\) −1.66451 −0.117994 −0.0589969 0.998258i \(-0.518790\pi\)
−0.0589969 + 0.998258i \(0.518790\pi\)
\(200\) 0 0
\(201\) −36.4575 −2.57151
\(202\) 0 0
\(203\) 6.54873 0.459630
\(204\) 0 0
\(205\) −13.2311 −0.924099
\(206\) 0 0
\(207\) −9.37409 −0.651544
\(208\) 0 0
\(209\) 31.2277 2.16006
\(210\) 0 0
\(211\) 17.2197 1.18545 0.592727 0.805404i \(-0.298051\pi\)
0.592727 + 0.805404i \(0.298051\pi\)
\(212\) 0 0
\(213\) 43.2361 2.96249
\(214\) 0 0
\(215\) 31.6655 2.15957
\(216\) 0 0
\(217\) −4.03341 −0.273806
\(218\) 0 0
\(219\) 16.0448 1.08421
\(220\) 0 0
\(221\) −0.268672 −0.0180728
\(222\) 0 0
\(223\) −20.1002 −1.34601 −0.673005 0.739638i \(-0.734997\pi\)
−0.673005 + 0.739638i \(0.734997\pi\)
\(224\) 0 0
\(225\) 15.3511 1.02341
\(226\) 0 0
\(227\) −25.9456 −1.72207 −0.861034 0.508547i \(-0.830183\pi\)
−0.861034 + 0.508547i \(0.830183\pi\)
\(228\) 0 0
\(229\) 9.24146 0.610693 0.305346 0.952241i \(-0.401228\pi\)
0.305346 + 0.952241i \(0.401228\pi\)
\(230\) 0 0
\(231\) 12.7736 0.840442
\(232\) 0 0
\(233\) 23.6539 1.54962 0.774808 0.632197i \(-0.217846\pi\)
0.774808 + 0.632197i \(0.217846\pi\)
\(234\) 0 0
\(235\) −25.5794 −1.66861
\(236\) 0 0
\(237\) 15.3303 0.995810
\(238\) 0 0
\(239\) −2.68135 −0.173442 −0.0867211 0.996233i \(-0.527639\pi\)
−0.0867211 + 0.996233i \(0.527639\pi\)
\(240\) 0 0
\(241\) 30.6680 1.97550 0.987751 0.156041i \(-0.0498733\pi\)
0.987751 + 0.156041i \(0.0498733\pi\)
\(242\) 0 0
\(243\) −20.8778 −1.33931
\(244\) 0 0
\(245\) −2.96141 −0.189197
\(246\) 0 0
\(247\) −6.50125 −0.413665
\(248\) 0 0
\(249\) 4.16335 0.263842
\(250\) 0 0
\(251\) −9.90078 −0.624932 −0.312466 0.949929i \(-0.601155\pi\)
−0.312466 + 0.949929i \(0.601155\pi\)
\(252\) 0 0
\(253\) −11.0577 −0.695189
\(254\) 0 0
\(255\) −2.11588 −0.132502
\(256\) 0 0
\(257\) 8.98316 0.560354 0.280177 0.959948i \(-0.409607\pi\)
0.280177 + 0.959948i \(0.409607\pi\)
\(258\) 0 0
\(259\) 2.51532 0.156294
\(260\) 0 0
\(261\) 26.6664 1.65061
\(262\) 0 0
\(263\) 9.77243 0.602594 0.301297 0.953530i \(-0.402581\pi\)
0.301297 + 0.953530i \(0.402581\pi\)
\(264\) 0 0
\(265\) 9.99204 0.613806
\(266\) 0 0
\(267\) 17.6719 1.08150
\(268\) 0 0
\(269\) −19.4213 −1.18414 −0.592068 0.805888i \(-0.701688\pi\)
−0.592068 + 0.805888i \(0.701688\pi\)
\(270\) 0 0
\(271\) 6.56279 0.398661 0.199331 0.979932i \(-0.436123\pi\)
0.199331 + 0.979932i \(0.436123\pi\)
\(272\) 0 0
\(273\) −2.65932 −0.160950
\(274\) 0 0
\(275\) 18.1082 1.09197
\(276\) 0 0
\(277\) −5.61304 −0.337255 −0.168628 0.985680i \(-0.553934\pi\)
−0.168628 + 0.985680i \(0.553934\pi\)
\(278\) 0 0
\(279\) −16.4241 −0.983283
\(280\) 0 0
\(281\) −6.65664 −0.397102 −0.198551 0.980091i \(-0.563624\pi\)
−0.198551 + 0.980091i \(0.563624\pi\)
\(282\) 0 0
\(283\) −2.46784 −0.146698 −0.0733490 0.997306i \(-0.523369\pi\)
−0.0733490 + 0.997306i \(0.523369\pi\)
\(284\) 0 0
\(285\) −51.1995 −3.03280
\(286\) 0 0
\(287\) 4.46784 0.263728
\(288\) 0 0
\(289\) −16.9278 −0.995754
\(290\) 0 0
\(291\) −7.30754 −0.428376
\(292\) 0 0
\(293\) −0.00796359 −0.000465238 0 −0.000232619 1.00000i \(-0.500074\pi\)
−0.000232619 1.00000i \(0.500074\pi\)
\(294\) 0 0
\(295\) −34.8818 −2.03090
\(296\) 0 0
\(297\) 13.6934 0.794570
\(298\) 0 0
\(299\) 2.30208 0.133133
\(300\) 0 0
\(301\) −10.6927 −0.616319
\(302\) 0 0
\(303\) −14.8789 −0.854773
\(304\) 0 0
\(305\) −27.0638 −1.54967
\(306\) 0 0
\(307\) −10.7224 −0.611962 −0.305981 0.952038i \(-0.598984\pi\)
−0.305981 + 0.952038i \(0.598984\pi\)
\(308\) 0 0
\(309\) 23.3523 1.32847
\(310\) 0 0
\(311\) −2.11429 −0.119891 −0.0599453 0.998202i \(-0.519093\pi\)
−0.0599453 + 0.998202i \(0.519093\pi\)
\(312\) 0 0
\(313\) 23.2608 1.31478 0.657389 0.753551i \(-0.271661\pi\)
0.657389 + 0.753551i \(0.271661\pi\)
\(314\) 0 0
\(315\) −12.0589 −0.679440
\(316\) 0 0
\(317\) 15.8870 0.892303 0.446151 0.894958i \(-0.352794\pi\)
0.446151 + 0.894958i \(0.352794\pi\)
\(318\) 0 0
\(319\) 31.4557 1.76118
\(320\) 0 0
\(321\) −46.2336 −2.58051
\(322\) 0 0
\(323\) 1.74670 0.0971891
\(324\) 0 0
\(325\) −3.76992 −0.209118
\(326\) 0 0
\(327\) −24.0668 −1.33090
\(328\) 0 0
\(329\) 8.63757 0.476205
\(330\) 0 0
\(331\) 17.1566 0.943011 0.471506 0.881863i \(-0.343711\pi\)
0.471506 + 0.881863i \(0.343711\pi\)
\(332\) 0 0
\(333\) 10.2424 0.561279
\(334\) 0 0
\(335\) 40.5988 2.21815
\(336\) 0 0
\(337\) −10.9394 −0.595906 −0.297953 0.954581i \(-0.596304\pi\)
−0.297953 + 0.954581i \(0.596304\pi\)
\(338\) 0 0
\(339\) −29.0913 −1.58003
\(340\) 0 0
\(341\) −19.3738 −1.04915
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 18.1297 0.976068
\(346\) 0 0
\(347\) −9.71699 −0.521635 −0.260818 0.965388i \(-0.583992\pi\)
−0.260818 + 0.965388i \(0.583992\pi\)
\(348\) 0 0
\(349\) −24.4137 −1.30684 −0.653418 0.756998i \(-0.726665\pi\)
−0.653418 + 0.756998i \(0.726665\pi\)
\(350\) 0 0
\(351\) −2.85081 −0.152165
\(352\) 0 0
\(353\) 37.3160 1.98613 0.993064 0.117573i \(-0.0375115\pi\)
0.993064 + 0.117573i \(0.0375115\pi\)
\(354\) 0 0
\(355\) −48.1474 −2.55540
\(356\) 0 0
\(357\) 0.714485 0.0378146
\(358\) 0 0
\(359\) −11.9703 −0.631767 −0.315884 0.948798i \(-0.602301\pi\)
−0.315884 + 0.948798i \(0.602301\pi\)
\(360\) 0 0
\(361\) 23.2663 1.22454
\(362\) 0 0
\(363\) 32.1034 1.68499
\(364\) 0 0
\(365\) −17.8674 −0.935221
\(366\) 0 0
\(367\) −5.84165 −0.304932 −0.152466 0.988309i \(-0.548721\pi\)
−0.152466 + 0.988309i \(0.548721\pi\)
\(368\) 0 0
\(369\) 18.1931 0.947093
\(370\) 0 0
\(371\) −3.37409 −0.175174
\(372\) 0 0
\(373\) −2.31615 −0.119925 −0.0599627 0.998201i \(-0.519098\pi\)
−0.0599627 + 0.998201i \(0.519098\pi\)
\(374\) 0 0
\(375\) 9.68727 0.500249
\(376\) 0 0
\(377\) −6.54873 −0.337277
\(378\) 0 0
\(379\) −29.9318 −1.53749 −0.768746 0.639554i \(-0.779119\pi\)
−0.768746 + 0.639554i \(0.779119\pi\)
\(380\) 0 0
\(381\) −50.3081 −2.57736
\(382\) 0 0
\(383\) 33.3932 1.70631 0.853155 0.521657i \(-0.174686\pi\)
0.853155 + 0.521657i \(0.174686\pi\)
\(384\) 0 0
\(385\) −14.2246 −0.724954
\(386\) 0 0
\(387\) −43.5409 −2.21331
\(388\) 0 0
\(389\) 35.9174 1.82109 0.910543 0.413413i \(-0.135664\pi\)
0.910543 + 0.413413i \(0.135664\pi\)
\(390\) 0 0
\(391\) −0.618504 −0.0312791
\(392\) 0 0
\(393\) −20.6116 −1.03972
\(394\) 0 0
\(395\) −17.0717 −0.858972
\(396\) 0 0
\(397\) 25.4603 1.27781 0.638907 0.769284i \(-0.279387\pi\)
0.638907 + 0.769284i \(0.279387\pi\)
\(398\) 0 0
\(399\) 17.2889 0.865529
\(400\) 0 0
\(401\) −16.3212 −0.815039 −0.407520 0.913196i \(-0.633606\pi\)
−0.407520 + 0.913196i \(0.633606\pi\)
\(402\) 0 0
\(403\) 4.03341 0.200918
\(404\) 0 0
\(405\) 13.7255 0.682026
\(406\) 0 0
\(407\) 12.0819 0.598877
\(408\) 0 0
\(409\) 22.2029 1.09786 0.548931 0.835868i \(-0.315035\pi\)
0.548931 + 0.835868i \(0.315035\pi\)
\(410\) 0 0
\(411\) −10.5110 −0.518472
\(412\) 0 0
\(413\) 11.7788 0.579597
\(414\) 0 0
\(415\) −4.63628 −0.227586
\(416\) 0 0
\(417\) −18.7482 −0.918102
\(418\) 0 0
\(419\) −21.3634 −1.04367 −0.521836 0.853046i \(-0.674753\pi\)
−0.521836 + 0.853046i \(0.674753\pi\)
\(420\) 0 0
\(421\) −10.9370 −0.533035 −0.266518 0.963830i \(-0.585873\pi\)
−0.266518 + 0.963830i \(0.585873\pi\)
\(422\) 0 0
\(423\) 35.1722 1.71013
\(424\) 0 0
\(425\) 1.01287 0.0491315
\(426\) 0 0
\(427\) 9.13882 0.442259
\(428\) 0 0
\(429\) −12.7736 −0.616716
\(430\) 0 0
\(431\) 17.7451 0.854752 0.427376 0.904074i \(-0.359438\pi\)
0.427376 + 0.904074i \(0.359438\pi\)
\(432\) 0 0
\(433\) −6.46516 −0.310696 −0.155348 0.987860i \(-0.549650\pi\)
−0.155348 + 0.987860i \(0.549650\pi\)
\(434\) 0 0
\(435\) −51.5734 −2.47276
\(436\) 0 0
\(437\) −14.9664 −0.715940
\(438\) 0 0
\(439\) −6.85849 −0.327338 −0.163669 0.986515i \(-0.552333\pi\)
−0.163669 + 0.986515i \(0.552333\pi\)
\(440\) 0 0
\(441\) 4.07200 0.193905
\(442\) 0 0
\(443\) −5.05794 −0.240310 −0.120155 0.992755i \(-0.538339\pi\)
−0.120155 + 0.992755i \(0.538339\pi\)
\(444\) 0 0
\(445\) −19.6793 −0.932889
\(446\) 0 0
\(447\) −35.5653 −1.68218
\(448\) 0 0
\(449\) −21.7734 −1.02755 −0.513776 0.857924i \(-0.671754\pi\)
−0.513776 + 0.857924i \(0.671754\pi\)
\(450\) 0 0
\(451\) 21.4605 1.01054
\(452\) 0 0
\(453\) −49.2311 −2.31308
\(454\) 0 0
\(455\) 2.96141 0.138833
\(456\) 0 0
\(457\) 32.0267 1.49815 0.749074 0.662487i \(-0.230499\pi\)
0.749074 + 0.662487i \(0.230499\pi\)
\(458\) 0 0
\(459\) 0.765931 0.0357506
\(460\) 0 0
\(461\) 31.6729 1.47516 0.737578 0.675262i \(-0.235970\pi\)
0.737578 + 0.675262i \(0.235970\pi\)
\(462\) 0 0
\(463\) 39.3221 1.82745 0.913726 0.406332i \(-0.133192\pi\)
0.913726 + 0.406332i \(0.133192\pi\)
\(464\) 0 0
\(465\) 31.7645 1.47304
\(466\) 0 0
\(467\) −35.7210 −1.65297 −0.826484 0.562960i \(-0.809663\pi\)
−0.826484 + 0.562960i \(0.809663\pi\)
\(468\) 0 0
\(469\) −13.7093 −0.633037
\(470\) 0 0
\(471\) −47.2210 −2.17583
\(472\) 0 0
\(473\) −51.3608 −2.36157
\(474\) 0 0
\(475\) 24.5092 1.12456
\(476\) 0 0
\(477\) −13.7393 −0.629079
\(478\) 0 0
\(479\) −28.3365 −1.29473 −0.647364 0.762181i \(-0.724129\pi\)
−0.647364 + 0.762181i \(0.724129\pi\)
\(480\) 0 0
\(481\) −2.51532 −0.114689
\(482\) 0 0
\(483\) −6.12198 −0.278560
\(484\) 0 0
\(485\) 8.13763 0.369511
\(486\) 0 0
\(487\) −35.3364 −1.60125 −0.800623 0.599169i \(-0.795498\pi\)
−0.800623 + 0.599169i \(0.795498\pi\)
\(488\) 0 0
\(489\) −44.3335 −2.00483
\(490\) 0 0
\(491\) 14.9975 0.676828 0.338414 0.940997i \(-0.390110\pi\)
0.338414 + 0.940997i \(0.390110\pi\)
\(492\) 0 0
\(493\) 1.75946 0.0792420
\(494\) 0 0
\(495\) −57.9227 −2.60343
\(496\) 0 0
\(497\) 16.2583 0.729285
\(498\) 0 0
\(499\) −29.9008 −1.33854 −0.669271 0.743018i \(-0.733394\pi\)
−0.669271 + 0.743018i \(0.733394\pi\)
\(500\) 0 0
\(501\) −45.7546 −2.04417
\(502\) 0 0
\(503\) 25.0975 1.11904 0.559520 0.828817i \(-0.310986\pi\)
0.559520 + 0.828817i \(0.310986\pi\)
\(504\) 0 0
\(505\) 16.5691 0.737315
\(506\) 0 0
\(507\) 2.65932 0.118105
\(508\) 0 0
\(509\) −25.5111 −1.13076 −0.565381 0.824830i \(-0.691271\pi\)
−0.565381 + 0.824830i \(0.691271\pi\)
\(510\) 0 0
\(511\) 6.03341 0.266902
\(512\) 0 0
\(513\) 18.5338 0.818288
\(514\) 0 0
\(515\) −26.0050 −1.14592
\(516\) 0 0
\(517\) 41.4891 1.82469
\(518\) 0 0
\(519\) −25.4958 −1.11914
\(520\) 0 0
\(521\) 6.17714 0.270625 0.135313 0.990803i \(-0.456796\pi\)
0.135313 + 0.990803i \(0.456796\pi\)
\(522\) 0 0
\(523\) 15.6756 0.685447 0.342723 0.939436i \(-0.388651\pi\)
0.342723 + 0.939436i \(0.388651\pi\)
\(524\) 0 0
\(525\) 10.0254 0.437546
\(526\) 0 0
\(527\) −1.08366 −0.0472051
\(528\) 0 0
\(529\) −17.7004 −0.769584
\(530\) 0 0
\(531\) 47.9633 2.08143
\(532\) 0 0
\(533\) −4.46784 −0.193524
\(534\) 0 0
\(535\) 51.4854 2.22591
\(536\) 0 0
\(537\) 27.5947 1.19080
\(538\) 0 0
\(539\) 4.80333 0.206894
\(540\) 0 0
\(541\) −17.3815 −0.747289 −0.373644 0.927572i \(-0.621892\pi\)
−0.373644 + 0.927572i \(0.621892\pi\)
\(542\) 0 0
\(543\) 0.324386 0.0139207
\(544\) 0 0
\(545\) 26.8007 1.14801
\(546\) 0 0
\(547\) 27.1579 1.16119 0.580594 0.814193i \(-0.302820\pi\)
0.580594 + 0.814193i \(0.302820\pi\)
\(548\) 0 0
\(549\) 37.2133 1.58823
\(550\) 0 0
\(551\) 42.5749 1.81375
\(552\) 0 0
\(553\) 5.76474 0.245142
\(554\) 0 0
\(555\) −19.8090 −0.840843
\(556\) 0 0
\(557\) 30.3699 1.28681 0.643407 0.765524i \(-0.277520\pi\)
0.643407 + 0.765524i \(0.277520\pi\)
\(558\) 0 0
\(559\) 10.6927 0.452254
\(560\) 0 0
\(561\) 3.43191 0.144895
\(562\) 0 0
\(563\) 11.6321 0.490235 0.245118 0.969493i \(-0.421173\pi\)
0.245118 + 0.969493i \(0.421173\pi\)
\(564\) 0 0
\(565\) 32.3959 1.36291
\(566\) 0 0
\(567\) −4.63479 −0.194643
\(568\) 0 0
\(569\) 23.6979 0.993468 0.496734 0.867903i \(-0.334532\pi\)
0.496734 + 0.867903i \(0.334532\pi\)
\(570\) 0 0
\(571\) −31.0751 −1.30045 −0.650227 0.759740i \(-0.725326\pi\)
−0.650227 + 0.759740i \(0.725326\pi\)
\(572\) 0 0
\(573\) 35.5963 1.48706
\(574\) 0 0
\(575\) −8.67867 −0.361926
\(576\) 0 0
\(577\) −10.2880 −0.428296 −0.214148 0.976801i \(-0.568697\pi\)
−0.214148 + 0.976801i \(0.568697\pi\)
\(578\) 0 0
\(579\) −41.5032 −1.72481
\(580\) 0 0
\(581\) 1.56557 0.0649507
\(582\) 0 0
\(583\) −16.2069 −0.671219
\(584\) 0 0
\(585\) 12.0589 0.498572
\(586\) 0 0
\(587\) 32.2262 1.33012 0.665058 0.746791i \(-0.268407\pi\)
0.665058 + 0.746791i \(0.268407\pi\)
\(588\) 0 0
\(589\) −26.2222 −1.08047
\(590\) 0 0
\(591\) 27.3373 1.12450
\(592\) 0 0
\(593\) −21.3340 −0.876082 −0.438041 0.898955i \(-0.644328\pi\)
−0.438041 + 0.898955i \(0.644328\pi\)
\(594\) 0 0
\(595\) −0.795646 −0.0326183
\(596\) 0 0
\(597\) −4.42647 −0.181163
\(598\) 0 0
\(599\) −32.0859 −1.31099 −0.655497 0.755198i \(-0.727541\pi\)
−0.655497 + 0.755198i \(0.727541\pi\)
\(600\) 0 0
\(601\) 11.5626 0.471649 0.235824 0.971796i \(-0.424221\pi\)
0.235824 + 0.971796i \(0.424221\pi\)
\(602\) 0 0
\(603\) −55.8243 −2.27334
\(604\) 0 0
\(605\) −35.7501 −1.45345
\(606\) 0 0
\(607\) 27.4276 1.11325 0.556626 0.830764i \(-0.312096\pi\)
0.556626 + 0.830764i \(0.312096\pi\)
\(608\) 0 0
\(609\) 17.4152 0.705699
\(610\) 0 0
\(611\) −8.63757 −0.349439
\(612\) 0 0
\(613\) 22.8839 0.924274 0.462137 0.886809i \(-0.347083\pi\)
0.462137 + 0.886809i \(0.347083\pi\)
\(614\) 0 0
\(615\) −35.1858 −1.41883
\(616\) 0 0
\(617\) −29.5576 −1.18994 −0.594972 0.803746i \(-0.702837\pi\)
−0.594972 + 0.803746i \(0.702837\pi\)
\(618\) 0 0
\(619\) 31.5730 1.26903 0.634513 0.772913i \(-0.281201\pi\)
0.634513 + 0.772913i \(0.281201\pi\)
\(620\) 0 0
\(621\) −6.56279 −0.263356
\(622\) 0 0
\(623\) 6.64526 0.266237
\(624\) 0 0
\(625\) −29.6373 −1.18549
\(626\) 0 0
\(627\) 83.0445 3.31648
\(628\) 0 0
\(629\) 0.675794 0.0269457
\(630\) 0 0
\(631\) 38.6092 1.53701 0.768503 0.639846i \(-0.221002\pi\)
0.768503 + 0.639846i \(0.221002\pi\)
\(632\) 0 0
\(633\) 45.7928 1.82010
\(634\) 0 0
\(635\) 56.0228 2.22320
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) 0 0
\(639\) 66.2039 2.61898
\(640\) 0 0
\(641\) 22.7463 0.898425 0.449213 0.893425i \(-0.351705\pi\)
0.449213 + 0.893425i \(0.351705\pi\)
\(642\) 0 0
\(643\) 23.9406 0.944124 0.472062 0.881565i \(-0.343510\pi\)
0.472062 + 0.881565i \(0.343510\pi\)
\(644\) 0 0
\(645\) 84.2089 3.31572
\(646\) 0 0
\(647\) −29.5329 −1.16106 −0.580529 0.814240i \(-0.697154\pi\)
−0.580529 + 0.814240i \(0.697154\pi\)
\(648\) 0 0
\(649\) 56.5775 2.22086
\(650\) 0 0
\(651\) −10.7261 −0.420391
\(652\) 0 0
\(653\) −9.57298 −0.374620 −0.187310 0.982301i \(-0.559977\pi\)
−0.187310 + 0.982301i \(0.559977\pi\)
\(654\) 0 0
\(655\) 22.9529 0.896844
\(656\) 0 0
\(657\) 24.5681 0.958492
\(658\) 0 0
\(659\) −21.5947 −0.841211 −0.420606 0.907244i \(-0.638182\pi\)
−0.420606 + 0.907244i \(0.638182\pi\)
\(660\) 0 0
\(661\) −13.9302 −0.541823 −0.270911 0.962604i \(-0.587325\pi\)
−0.270911 + 0.962604i \(0.587325\pi\)
\(662\) 0 0
\(663\) −0.714485 −0.0277483
\(664\) 0 0
\(665\) −19.2528 −0.746593
\(666\) 0 0
\(667\) −15.0757 −0.583733
\(668\) 0 0
\(669\) −53.4530 −2.06661
\(670\) 0 0
\(671\) 43.8968 1.69462
\(672\) 0 0
\(673\) 3.83692 0.147902 0.0739512 0.997262i \(-0.476439\pi\)
0.0739512 + 0.997262i \(0.476439\pi\)
\(674\) 0 0
\(675\) 10.7473 0.413665
\(676\) 0 0
\(677\) −22.3047 −0.857239 −0.428619 0.903485i \(-0.641000\pi\)
−0.428619 + 0.903485i \(0.641000\pi\)
\(678\) 0 0
\(679\) −2.74789 −0.105455
\(680\) 0 0
\(681\) −68.9977 −2.64400
\(682\) 0 0
\(683\) 19.8305 0.758795 0.379397 0.925234i \(-0.376131\pi\)
0.379397 + 0.925234i \(0.376131\pi\)
\(684\) 0 0
\(685\) 11.7050 0.447226
\(686\) 0 0
\(687\) 24.5760 0.937634
\(688\) 0 0
\(689\) 3.37409 0.128542
\(690\) 0 0
\(691\) −30.8996 −1.17548 −0.587738 0.809051i \(-0.699981\pi\)
−0.587738 + 0.809051i \(0.699981\pi\)
\(692\) 0 0
\(693\) 19.5592 0.742992
\(694\) 0 0
\(695\) 20.8778 0.791942
\(696\) 0 0
\(697\) 1.20038 0.0454677
\(698\) 0 0
\(699\) 62.9033 2.37922
\(700\) 0 0
\(701\) −3.62341 −0.136854 −0.0684272 0.997656i \(-0.521798\pi\)
−0.0684272 + 0.997656i \(0.521798\pi\)
\(702\) 0 0
\(703\) 16.3527 0.616754
\(704\) 0 0
\(705\) −68.0238 −2.56192
\(706\) 0 0
\(707\) −5.59501 −0.210422
\(708\) 0 0
\(709\) −42.0602 −1.57960 −0.789801 0.613363i \(-0.789816\pi\)
−0.789801 + 0.613363i \(0.789816\pi\)
\(710\) 0 0
\(711\) 23.4740 0.880345
\(712\) 0 0
\(713\) 9.28524 0.347735
\(714\) 0 0
\(715\) 14.2246 0.531970
\(716\) 0 0
\(717\) −7.13058 −0.266297
\(718\) 0 0
\(719\) −18.9110 −0.705260 −0.352630 0.935763i \(-0.614713\pi\)
−0.352630 + 0.935763i \(0.614713\pi\)
\(720\) 0 0
\(721\) 8.78130 0.327033
\(722\) 0 0
\(723\) 81.5562 3.03311
\(724\) 0 0
\(725\) 24.6882 0.916896
\(726\) 0 0
\(727\) −5.23645 −0.194209 −0.0971047 0.995274i \(-0.530958\pi\)
−0.0971047 + 0.995274i \(0.530958\pi\)
\(728\) 0 0
\(729\) −41.6166 −1.54135
\(730\) 0 0
\(731\) −2.87283 −0.106256
\(732\) 0 0
\(733\) −9.77584 −0.361079 −0.180540 0.983568i \(-0.557784\pi\)
−0.180540 + 0.983568i \(0.557784\pi\)
\(734\) 0 0
\(735\) −7.87534 −0.290486
\(736\) 0 0
\(737\) −65.8503 −2.42563
\(738\) 0 0
\(739\) −23.8318 −0.876668 −0.438334 0.898812i \(-0.644431\pi\)
−0.438334 + 0.898812i \(0.644431\pi\)
\(740\) 0 0
\(741\) −17.2889 −0.635125
\(742\) 0 0
\(743\) 42.5492 1.56098 0.780489 0.625169i \(-0.214970\pi\)
0.780489 + 0.625169i \(0.214970\pi\)
\(744\) 0 0
\(745\) 39.6053 1.45102
\(746\) 0 0
\(747\) 6.37500 0.233249
\(748\) 0 0
\(749\) −17.3855 −0.635251
\(750\) 0 0
\(751\) 10.8903 0.397393 0.198697 0.980061i \(-0.436329\pi\)
0.198697 + 0.980061i \(0.436329\pi\)
\(752\) 0 0
\(753\) −26.3294 −0.959496
\(754\) 0 0
\(755\) 54.8234 1.99523
\(756\) 0 0
\(757\) 4.58186 0.166530 0.0832652 0.996527i \(-0.473465\pi\)
0.0832652 + 0.996527i \(0.473465\pi\)
\(758\) 0 0
\(759\) −29.4059 −1.06737
\(760\) 0 0
\(761\) 41.0537 1.48819 0.744097 0.668071i \(-0.232880\pi\)
0.744097 + 0.668071i \(0.232880\pi\)
\(762\) 0 0
\(763\) −9.04998 −0.327631
\(764\) 0 0
\(765\) −3.23987 −0.117138
\(766\) 0 0
\(767\) −11.7788 −0.425308
\(768\) 0 0
\(769\) −11.6173 −0.418931 −0.209465 0.977816i \(-0.567172\pi\)
−0.209465 + 0.977816i \(0.567172\pi\)
\(770\) 0 0
\(771\) 23.8891 0.860346
\(772\) 0 0
\(773\) 15.9663 0.574268 0.287134 0.957890i \(-0.407297\pi\)
0.287134 + 0.957890i \(0.407297\pi\)
\(774\) 0 0
\(775\) −15.2056 −0.546203
\(776\) 0 0
\(777\) 6.68904 0.239968
\(778\) 0 0
\(779\) 29.0466 1.04070
\(780\) 0 0
\(781\) 78.0940 2.79442
\(782\) 0 0
\(783\) 18.6692 0.667181
\(784\) 0 0
\(785\) 52.5850 1.87684
\(786\) 0 0
\(787\) 36.2256 1.29130 0.645652 0.763632i \(-0.276586\pi\)
0.645652 + 0.763632i \(0.276586\pi\)
\(788\) 0 0
\(789\) 25.9880 0.925199
\(790\) 0 0
\(791\) −10.9394 −0.388959
\(792\) 0 0
\(793\) −9.13882 −0.324529
\(794\) 0 0
\(795\) 26.5721 0.942414
\(796\) 0 0
\(797\) −29.2725 −1.03688 −0.518442 0.855113i \(-0.673488\pi\)
−0.518442 + 0.855113i \(0.673488\pi\)
\(798\) 0 0
\(799\) 2.32067 0.0820994
\(800\) 0 0
\(801\) 27.0595 0.956101
\(802\) 0 0
\(803\) 28.9805 1.02270
\(804\) 0 0
\(805\) 6.81740 0.240282
\(806\) 0 0
\(807\) −51.6475 −1.81808
\(808\) 0 0
\(809\) 6.16631 0.216796 0.108398 0.994108i \(-0.465428\pi\)
0.108398 + 0.994108i \(0.465428\pi\)
\(810\) 0 0
\(811\) −29.9843 −1.05289 −0.526445 0.850209i \(-0.676475\pi\)
−0.526445 + 0.850209i \(0.676475\pi\)
\(812\) 0 0
\(813\) 17.4526 0.612089
\(814\) 0 0
\(815\) 49.3695 1.72934
\(816\) 0 0
\(817\) −69.5161 −2.43206
\(818\) 0 0
\(819\) −4.07200 −0.142287
\(820\) 0 0
\(821\) −11.8119 −0.412239 −0.206120 0.978527i \(-0.566084\pi\)
−0.206120 + 0.978527i \(0.566084\pi\)
\(822\) 0 0
\(823\) −43.5212 −1.51705 −0.758527 0.651641i \(-0.774081\pi\)
−0.758527 + 0.651641i \(0.774081\pi\)
\(824\) 0 0
\(825\) 48.1556 1.67656
\(826\) 0 0
\(827\) −25.2470 −0.877925 −0.438962 0.898505i \(-0.644654\pi\)
−0.438962 + 0.898505i \(0.644654\pi\)
\(828\) 0 0
\(829\) 1.79314 0.0622784 0.0311392 0.999515i \(-0.490086\pi\)
0.0311392 + 0.999515i \(0.490086\pi\)
\(830\) 0 0
\(831\) −14.9269 −0.517808
\(832\) 0 0
\(833\) 0.268672 0.00930892
\(834\) 0 0
\(835\) 50.9521 1.76327
\(836\) 0 0
\(837\) −11.4985 −0.397445
\(838\) 0 0
\(839\) −51.9588 −1.79382 −0.896909 0.442215i \(-0.854193\pi\)
−0.896909 + 0.442215i \(0.854193\pi\)
\(840\) 0 0
\(841\) 13.8858 0.478821
\(842\) 0 0
\(843\) −17.7022 −0.609695
\(844\) 0 0
\(845\) −2.96141 −0.101875
\(846\) 0 0
\(847\) 12.0720 0.414799
\(848\) 0 0
\(849\) −6.56279 −0.225234
\(850\) 0 0
\(851\) −5.79046 −0.198494
\(852\) 0 0
\(853\) 13.9976 0.479269 0.239634 0.970863i \(-0.422972\pi\)
0.239634 + 0.970863i \(0.422972\pi\)
\(854\) 0 0
\(855\) −78.3977 −2.68114
\(856\) 0 0
\(857\) 18.3830 0.627950 0.313975 0.949431i \(-0.398339\pi\)
0.313975 + 0.949431i \(0.398339\pi\)
\(858\) 0 0
\(859\) 27.4496 0.936568 0.468284 0.883578i \(-0.344872\pi\)
0.468284 + 0.883578i \(0.344872\pi\)
\(860\) 0 0
\(861\) 11.8814 0.404918
\(862\) 0 0
\(863\) 13.7170 0.466932 0.233466 0.972365i \(-0.424993\pi\)
0.233466 + 0.972365i \(0.424993\pi\)
\(864\) 0 0
\(865\) 28.3919 0.965355
\(866\) 0 0
\(867\) −45.0165 −1.52884
\(868\) 0 0
\(869\) 27.6900 0.939317
\(870\) 0 0
\(871\) 13.7093 0.464522
\(872\) 0 0
\(873\) −11.1894 −0.378705
\(874\) 0 0
\(875\) 3.64276 0.123148
\(876\) 0 0
\(877\) 40.4100 1.36455 0.682274 0.731096i \(-0.260991\pi\)
0.682274 + 0.731096i \(0.260991\pi\)
\(878\) 0 0
\(879\) −0.0211778 −0.000714308 0
\(880\) 0 0
\(881\) 23.5239 0.792541 0.396270 0.918134i \(-0.370304\pi\)
0.396270 + 0.918134i \(0.370304\pi\)
\(882\) 0 0
\(883\) −17.6502 −0.593975 −0.296988 0.954881i \(-0.595982\pi\)
−0.296988 + 0.954881i \(0.595982\pi\)
\(884\) 0 0
\(885\) −92.7620 −3.11816
\(886\) 0 0
\(887\) 12.8763 0.432342 0.216171 0.976355i \(-0.430643\pi\)
0.216171 + 0.976355i \(0.430643\pi\)
\(888\) 0 0
\(889\) −18.9176 −0.634477
\(890\) 0 0
\(891\) −22.2625 −0.745820
\(892\) 0 0
\(893\) 56.1550 1.87916
\(894\) 0 0
\(895\) −30.7293 −1.02717
\(896\) 0 0
\(897\) 6.12198 0.204407
\(898\) 0 0
\(899\) −26.4137 −0.880946
\(900\) 0 0
\(901\) −0.906521 −0.0302006
\(902\) 0 0
\(903\) −28.4354 −0.946272
\(904\) 0 0
\(905\) −0.361234 −0.0120078
\(906\) 0 0
\(907\) −35.8342 −1.18986 −0.594928 0.803779i \(-0.702820\pi\)
−0.594928 + 0.803779i \(0.702820\pi\)
\(908\) 0 0
\(909\) −22.7829 −0.755661
\(910\) 0 0
\(911\) 12.8214 0.424791 0.212395 0.977184i \(-0.431874\pi\)
0.212395 + 0.977184i \(0.431874\pi\)
\(912\) 0 0
\(913\) 7.51995 0.248874
\(914\) 0 0
\(915\) −71.9713 −2.37930
\(916\) 0 0
\(917\) −7.75067 −0.255950
\(918\) 0 0
\(919\) −28.6352 −0.944587 −0.472294 0.881441i \(-0.656574\pi\)
−0.472294 + 0.881441i \(0.656574\pi\)
\(920\) 0 0
\(921\) −28.5145 −0.939584
\(922\) 0 0
\(923\) −16.2583 −0.535149
\(924\) 0 0
\(925\) 9.48254 0.311784
\(926\) 0 0
\(927\) 35.7575 1.17443
\(928\) 0 0
\(929\) 11.1744 0.366619 0.183309 0.983055i \(-0.441319\pi\)
0.183309 + 0.983055i \(0.441319\pi\)
\(930\) 0 0
\(931\) 6.50125 0.213070
\(932\) 0 0
\(933\) −5.62259 −0.184075
\(934\) 0 0
\(935\) −3.82175 −0.124985
\(936\) 0 0
\(937\) −49.5433 −1.61851 −0.809254 0.587459i \(-0.800128\pi\)
−0.809254 + 0.587459i \(0.800128\pi\)
\(938\) 0 0
\(939\) 61.8580 2.01866
\(940\) 0 0
\(941\) −12.1029 −0.394544 −0.197272 0.980349i \(-0.563208\pi\)
−0.197272 + 0.980349i \(0.563208\pi\)
\(942\) 0 0
\(943\) −10.2853 −0.334937
\(944\) 0 0
\(945\) −8.44240 −0.274631
\(946\) 0 0
\(947\) 21.9228 0.712396 0.356198 0.934411i \(-0.384073\pi\)
0.356198 + 0.934411i \(0.384073\pi\)
\(948\) 0 0
\(949\) −6.03341 −0.195853
\(950\) 0 0
\(951\) 42.2487 1.37001
\(952\) 0 0
\(953\) 22.7105 0.735665 0.367833 0.929892i \(-0.380100\pi\)
0.367833 + 0.929892i \(0.380100\pi\)
\(954\) 0 0
\(955\) −39.6398 −1.28271
\(956\) 0 0
\(957\) 83.6509 2.70405
\(958\) 0 0
\(959\) −3.95253 −0.127634
\(960\) 0 0
\(961\) −14.7316 −0.475213
\(962\) 0 0
\(963\) −70.7937 −2.28130
\(964\) 0 0
\(965\) 46.2177 1.48780
\(966\) 0 0
\(967\) −31.0574 −0.998738 −0.499369 0.866390i \(-0.666435\pi\)
−0.499369 + 0.866390i \(0.666435\pi\)
\(968\) 0 0
\(969\) 4.64505 0.149220
\(970\) 0 0
\(971\) 0.871738 0.0279754 0.0139877 0.999902i \(-0.495547\pi\)
0.0139877 + 0.999902i \(0.495547\pi\)
\(972\) 0 0
\(973\) −7.04998 −0.226012
\(974\) 0 0
\(975\) −10.0254 −0.321071
\(976\) 0 0
\(977\) −23.9090 −0.764917 −0.382459 0.923973i \(-0.624923\pi\)
−0.382459 + 0.923973i \(0.624923\pi\)
\(978\) 0 0
\(979\) 31.9194 1.02015
\(980\) 0 0
\(981\) −36.8515 −1.17658
\(982\) 0 0
\(983\) 42.2466 1.34746 0.673728 0.738979i \(-0.264692\pi\)
0.673728 + 0.738979i \(0.264692\pi\)
\(984\) 0 0
\(985\) −30.4426 −0.969982
\(986\) 0 0
\(987\) 22.9701 0.731147
\(988\) 0 0
\(989\) 24.6155 0.782729
\(990\) 0 0
\(991\) 1.70930 0.0542977 0.0271489 0.999631i \(-0.491357\pi\)
0.0271489 + 0.999631i \(0.491357\pi\)
\(992\) 0 0
\(993\) 45.6249 1.44786
\(994\) 0 0
\(995\) 4.92929 0.156269
\(996\) 0 0
\(997\) 24.7479 0.783773 0.391887 0.920014i \(-0.371823\pi\)
0.391887 + 0.920014i \(0.371823\pi\)
\(998\) 0 0
\(999\) 7.17068 0.226870
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 1456.2.a.v.1.4 4
4.3 odd 2 728.2.a.i.1.1 4
8.3 odd 2 5824.2.a.cb.1.4 4
8.5 even 2 5824.2.a.ce.1.1 4
12.11 even 2 6552.2.a.br.1.4 4
28.27 even 2 5096.2.a.s.1.4 4
52.51 odd 2 9464.2.a.z.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
728.2.a.i.1.1 4 4.3 odd 2
1456.2.a.v.1.4 4 1.1 even 1 trivial
5096.2.a.s.1.4 4 28.27 even 2
5824.2.a.cb.1.4 4 8.3 odd 2
5824.2.a.ce.1.1 4 8.5 even 2
6552.2.a.br.1.4 4 12.11 even 2
9464.2.a.z.1.1 4 52.51 odd 2