Properties

Label 6160.2.a.bj.1.2
Level $6160$
Weight $2$
Character 6160.1
Self dual yes
Analytic conductor $49.188$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6160,2,Mod(1,6160)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6160, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("6160.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6160.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: \(49.1878476451\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) 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 385)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.17009\) of defining polynomial
Character \(\chi\) \(=\) 6160.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.539189 q^{3} +1.00000 q^{5} -1.00000 q^{7} -2.70928 q^{9} +O(q^{10})\) \(q-0.539189 q^{3} +1.00000 q^{5} -1.00000 q^{7} -2.70928 q^{9} -1.00000 q^{11} +0.539189 q^{13} -0.539189 q^{15} +1.46081 q^{17} +1.41855 q^{19} +0.539189 q^{21} +1.70928 q^{23} +1.00000 q^{25} +3.07838 q^{27} -6.68035 q^{29} +0.248464 q^{31} +0.539189 q^{33} -1.00000 q^{35} +2.04945 q^{37} -0.290725 q^{39} +7.32684 q^{41} +6.97107 q^{43} -2.70928 q^{45} -11.2195 q^{47} +1.00000 q^{49} -0.787653 q^{51} -3.89269 q^{53} -1.00000 q^{55} -0.764867 q^{57} -0.829914 q^{59} +3.35350 q^{61} +2.70928 q^{63} +0.539189 q^{65} -0.630898 q^{67} -0.921622 q^{69} +10.8371 q^{71} -9.46081 q^{73} -0.539189 q^{75} +1.00000 q^{77} -2.63090 q^{79} +6.46800 q^{81} -9.02052 q^{83} +1.46081 q^{85} +3.60197 q^{87} +0.764867 q^{89} -0.539189 q^{91} -0.133969 q^{93} +1.41855 q^{95} -11.2351 q^{97} +2.70928 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{5} - 3 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{5} - 3 q^{7} - q^{9} - 3 q^{11} + 6 q^{17} - 10 q^{19} - 2 q^{23} + 3 q^{25} + 6 q^{27} + 2 q^{29} - 8 q^{31} - 3 q^{35} - 12 q^{37} - 8 q^{39} + 10 q^{41} + 6 q^{43} - q^{45} - 10 q^{47} + 3 q^{49} + 8 q^{51} - 3 q^{55} - 12 q^{57} - 8 q^{59} + q^{63} + 2 q^{67} - 6 q^{69} + 4 q^{71} - 30 q^{73} + 3 q^{77} - 4 q^{79} - 13 q^{81} + 6 q^{83} + 6 q^{85} - 8 q^{87} + 12 q^{89} - 14 q^{93} - 10 q^{95} - 24 q^{97} + q^{99}+O(q^{100}) \) 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
\(3\) −0.539189 −0.311301 −0.155650 0.987812i \(-0.549747\pi\)
−0.155650 + 0.987812i \(0.549747\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) −2.70928 −0.903092
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 0.539189 0.149544 0.0747720 0.997201i \(-0.476177\pi\)
0.0747720 + 0.997201i \(0.476177\pi\)
\(14\) 0 0
\(15\) −0.539189 −0.139218
\(16\) 0 0
\(17\) 1.46081 0.354299 0.177149 0.984184i \(-0.443312\pi\)
0.177149 + 0.984184i \(0.443312\pi\)
\(18\) 0 0
\(19\) 1.41855 0.325438 0.162719 0.986672i \(-0.447974\pi\)
0.162719 + 0.986672i \(0.447974\pi\)
\(20\) 0 0
\(21\) 0.539189 0.117661
\(22\) 0 0
\(23\) 1.70928 0.356409 0.178204 0.983994i \(-0.442971\pi\)
0.178204 + 0.983994i \(0.442971\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 3.07838 0.592434
\(28\) 0 0
\(29\) −6.68035 −1.24051 −0.620255 0.784401i \(-0.712971\pi\)
−0.620255 + 0.784401i \(0.712971\pi\)
\(30\) 0 0
\(31\) 0.248464 0.0446255 0.0223127 0.999751i \(-0.492897\pi\)
0.0223127 + 0.999751i \(0.492897\pi\)
\(32\) 0 0
\(33\) 0.539189 0.0938607
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) 2.04945 0.336927 0.168464 0.985708i \(-0.446119\pi\)
0.168464 + 0.985708i \(0.446119\pi\)
\(38\) 0 0
\(39\) −0.290725 −0.0465532
\(40\) 0 0
\(41\) 7.32684 1.14426 0.572130 0.820163i \(-0.306117\pi\)
0.572130 + 0.820163i \(0.306117\pi\)
\(42\) 0 0
\(43\) 6.97107 1.06308 0.531539 0.847034i \(-0.321614\pi\)
0.531539 + 0.847034i \(0.321614\pi\)
\(44\) 0 0
\(45\) −2.70928 −0.403875
\(46\) 0 0
\(47\) −11.2195 −1.63654 −0.818269 0.574836i \(-0.805066\pi\)
−0.818269 + 0.574836i \(0.805066\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −0.787653 −0.110293
\(52\) 0 0
\(53\) −3.89269 −0.534702 −0.267351 0.963599i \(-0.586148\pi\)
−0.267351 + 0.963599i \(0.586148\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) −0.764867 −0.101309
\(58\) 0 0
\(59\) −0.829914 −0.108046 −0.0540228 0.998540i \(-0.517204\pi\)
−0.0540228 + 0.998540i \(0.517204\pi\)
\(60\) 0 0
\(61\) 3.35350 0.429372 0.214686 0.976683i \(-0.431127\pi\)
0.214686 + 0.976683i \(0.431127\pi\)
\(62\) 0 0
\(63\) 2.70928 0.341337
\(64\) 0 0
\(65\) 0.539189 0.0668781
\(66\) 0 0
\(67\) −0.630898 −0.0770764 −0.0385382 0.999257i \(-0.512270\pi\)
−0.0385382 + 0.999257i \(0.512270\pi\)
\(68\) 0 0
\(69\) −0.921622 −0.110950
\(70\) 0 0
\(71\) 10.8371 1.28613 0.643064 0.765813i \(-0.277663\pi\)
0.643064 + 0.765813i \(0.277663\pi\)
\(72\) 0 0
\(73\) −9.46081 −1.10730 −0.553652 0.832748i \(-0.686766\pi\)
−0.553652 + 0.832748i \(0.686766\pi\)
\(74\) 0 0
\(75\) −0.539189 −0.0622602
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) −2.63090 −0.295999 −0.148000 0.988987i \(-0.547283\pi\)
−0.148000 + 0.988987i \(0.547283\pi\)
\(80\) 0 0
\(81\) 6.46800 0.718667
\(82\) 0 0
\(83\) −9.02052 −0.990131 −0.495065 0.868856i \(-0.664856\pi\)
−0.495065 + 0.868856i \(0.664856\pi\)
\(84\) 0 0
\(85\) 1.46081 0.158447
\(86\) 0 0
\(87\) 3.60197 0.386172
\(88\) 0 0
\(89\) 0.764867 0.0810757 0.0405379 0.999178i \(-0.487093\pi\)
0.0405379 + 0.999178i \(0.487093\pi\)
\(90\) 0 0
\(91\) −0.539189 −0.0565224
\(92\) 0 0
\(93\) −0.133969 −0.0138920
\(94\) 0 0
\(95\) 1.41855 0.145540
\(96\) 0 0
\(97\) −11.2351 −1.14075 −0.570377 0.821383i \(-0.693203\pi\)
−0.570377 + 0.821383i \(0.693203\pi\)
\(98\) 0 0
\(99\) 2.70928 0.272292
\(100\) 0 0
\(101\) −11.4257 −1.13690 −0.568452 0.822717i \(-0.692457\pi\)
−0.568452 + 0.822717i \(0.692457\pi\)
\(102\) 0 0
\(103\) −4.90602 −0.483405 −0.241702 0.970350i \(-0.577706\pi\)
−0.241702 + 0.970350i \(0.577706\pi\)
\(104\) 0 0
\(105\) 0.539189 0.0526194
\(106\) 0 0
\(107\) −16.0989 −1.55634 −0.778170 0.628054i \(-0.783852\pi\)
−0.778170 + 0.628054i \(0.783852\pi\)
\(108\) 0 0
\(109\) 16.4391 1.57458 0.787289 0.616585i \(-0.211484\pi\)
0.787289 + 0.616585i \(0.211484\pi\)
\(110\) 0 0
\(111\) −1.10504 −0.104886
\(112\) 0 0
\(113\) −5.23513 −0.492480 −0.246240 0.969209i \(-0.579195\pi\)
−0.246240 + 0.969209i \(0.579195\pi\)
\(114\) 0 0
\(115\) 1.70928 0.159391
\(116\) 0 0
\(117\) −1.46081 −0.135052
\(118\) 0 0
\(119\) −1.46081 −0.133912
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −3.95055 −0.356209
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −9.17727 −0.814351 −0.407176 0.913350i \(-0.633486\pi\)
−0.407176 + 0.913350i \(0.633486\pi\)
\(128\) 0 0
\(129\) −3.75872 −0.330937
\(130\) 0 0
\(131\) −20.9939 −1.83424 −0.917121 0.398609i \(-0.869493\pi\)
−0.917121 + 0.398609i \(0.869493\pi\)
\(132\) 0 0
\(133\) −1.41855 −0.123004
\(134\) 0 0
\(135\) 3.07838 0.264945
\(136\) 0 0
\(137\) 8.20620 0.701103 0.350552 0.936543i \(-0.385994\pi\)
0.350552 + 0.936543i \(0.385994\pi\)
\(138\) 0 0
\(139\) −15.9155 −1.34993 −0.674967 0.737848i \(-0.735842\pi\)
−0.674967 + 0.737848i \(0.735842\pi\)
\(140\) 0 0
\(141\) 6.04945 0.509455
\(142\) 0 0
\(143\) −0.539189 −0.0450892
\(144\) 0 0
\(145\) −6.68035 −0.554773
\(146\) 0 0
\(147\) −0.539189 −0.0444715
\(148\) 0 0
\(149\) 5.02052 0.411297 0.205648 0.978626i \(-0.434070\pi\)
0.205648 + 0.978626i \(0.434070\pi\)
\(150\) 0 0
\(151\) −12.9711 −1.05557 −0.527785 0.849378i \(-0.676977\pi\)
−0.527785 + 0.849378i \(0.676977\pi\)
\(152\) 0 0
\(153\) −3.95774 −0.319964
\(154\) 0 0
\(155\) 0.248464 0.0199571
\(156\) 0 0
\(157\) 0.313511 0.0250209 0.0125105 0.999922i \(-0.496018\pi\)
0.0125105 + 0.999922i \(0.496018\pi\)
\(158\) 0 0
\(159\) 2.09890 0.166453
\(160\) 0 0
\(161\) −1.70928 −0.134710
\(162\) 0 0
\(163\) 21.9916 1.72251 0.861257 0.508169i \(-0.169678\pi\)
0.861257 + 0.508169i \(0.169678\pi\)
\(164\) 0 0
\(165\) 0.539189 0.0419758
\(166\) 0 0
\(167\) 18.1978 1.40819 0.704094 0.710107i \(-0.251353\pi\)
0.704094 + 0.710107i \(0.251353\pi\)
\(168\) 0 0
\(169\) −12.7093 −0.977637
\(170\) 0 0
\(171\) −3.84324 −0.293900
\(172\) 0 0
\(173\) 9.64423 0.733237 0.366619 0.930371i \(-0.380515\pi\)
0.366619 + 0.930371i \(0.380515\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 0.447480 0.0336347
\(178\) 0 0
\(179\) −18.6225 −1.39191 −0.695955 0.718085i \(-0.745019\pi\)
−0.695955 + 0.718085i \(0.745019\pi\)
\(180\) 0 0
\(181\) −7.10504 −0.528113 −0.264057 0.964507i \(-0.585061\pi\)
−0.264057 + 0.964507i \(0.585061\pi\)
\(182\) 0 0
\(183\) −1.80817 −0.133664
\(184\) 0 0
\(185\) 2.04945 0.150678
\(186\) 0 0
\(187\) −1.46081 −0.106825
\(188\) 0 0
\(189\) −3.07838 −0.223919
\(190\) 0 0
\(191\) −14.0722 −1.01823 −0.509116 0.860698i \(-0.670027\pi\)
−0.509116 + 0.860698i \(0.670027\pi\)
\(192\) 0 0
\(193\) −9.15449 −0.658954 −0.329477 0.944164i \(-0.606872\pi\)
−0.329477 + 0.944164i \(0.606872\pi\)
\(194\) 0 0
\(195\) −0.290725 −0.0208192
\(196\) 0 0
\(197\) −20.3051 −1.44668 −0.723339 0.690493i \(-0.757394\pi\)
−0.723339 + 0.690493i \(0.757394\pi\)
\(198\) 0 0
\(199\) −14.8443 −1.05228 −0.526142 0.850397i \(-0.676362\pi\)
−0.526142 + 0.850397i \(0.676362\pi\)
\(200\) 0 0
\(201\) 0.340173 0.0239940
\(202\) 0 0
\(203\) 6.68035 0.468868
\(204\) 0 0
\(205\) 7.32684 0.511729
\(206\) 0 0
\(207\) −4.63090 −0.321870
\(208\) 0 0
\(209\) −1.41855 −0.0981232
\(210\) 0 0
\(211\) −4.18342 −0.287998 −0.143999 0.989578i \(-0.545996\pi\)
−0.143999 + 0.989578i \(0.545996\pi\)
\(212\) 0 0
\(213\) −5.84324 −0.400373
\(214\) 0 0
\(215\) 6.97107 0.475423
\(216\) 0 0
\(217\) −0.248464 −0.0168669
\(218\) 0 0
\(219\) 5.10116 0.344705
\(220\) 0 0
\(221\) 0.787653 0.0529833
\(222\) 0 0
\(223\) 21.8154 1.46086 0.730432 0.682985i \(-0.239319\pi\)
0.730432 + 0.682985i \(0.239319\pi\)
\(224\) 0 0
\(225\) −2.70928 −0.180618
\(226\) 0 0
\(227\) 14.8371 0.984773 0.492387 0.870377i \(-0.336125\pi\)
0.492387 + 0.870377i \(0.336125\pi\)
\(228\) 0 0
\(229\) −18.8638 −1.24655 −0.623276 0.782002i \(-0.714199\pi\)
−0.623276 + 0.782002i \(0.714199\pi\)
\(230\) 0 0
\(231\) −0.539189 −0.0354760
\(232\) 0 0
\(233\) −1.18568 −0.0776768 −0.0388384 0.999246i \(-0.512366\pi\)
−0.0388384 + 0.999246i \(0.512366\pi\)
\(234\) 0 0
\(235\) −11.2195 −0.731882
\(236\) 0 0
\(237\) 1.41855 0.0921448
\(238\) 0 0
\(239\) 1.17727 0.0761516 0.0380758 0.999275i \(-0.487877\pi\)
0.0380758 + 0.999275i \(0.487877\pi\)
\(240\) 0 0
\(241\) 29.2690 1.88538 0.942690 0.333669i \(-0.108287\pi\)
0.942690 + 0.333669i \(0.108287\pi\)
\(242\) 0 0
\(243\) −12.7226 −0.816156
\(244\) 0 0
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) 0.764867 0.0486673
\(248\) 0 0
\(249\) 4.86376 0.308228
\(250\) 0 0
\(251\) −14.0072 −0.884126 −0.442063 0.896984i \(-0.645753\pi\)
−0.442063 + 0.896984i \(0.645753\pi\)
\(252\) 0 0
\(253\) −1.70928 −0.107461
\(254\) 0 0
\(255\) −0.787653 −0.0493248
\(256\) 0 0
\(257\) −12.4124 −0.774265 −0.387132 0.922024i \(-0.626534\pi\)
−0.387132 + 0.922024i \(0.626534\pi\)
\(258\) 0 0
\(259\) −2.04945 −0.127347
\(260\) 0 0
\(261\) 18.0989 1.12029
\(262\) 0 0
\(263\) −12.8638 −0.793214 −0.396607 0.917989i \(-0.629812\pi\)
−0.396607 + 0.917989i \(0.629812\pi\)
\(264\) 0 0
\(265\) −3.89269 −0.239126
\(266\) 0 0
\(267\) −0.412408 −0.0252389
\(268\) 0 0
\(269\) −10.5236 −0.641635 −0.320817 0.947141i \(-0.603958\pi\)
−0.320817 + 0.947141i \(0.603958\pi\)
\(270\) 0 0
\(271\) −11.2351 −0.682486 −0.341243 0.939975i \(-0.610848\pi\)
−0.341243 + 0.939975i \(0.610848\pi\)
\(272\) 0 0
\(273\) 0.290725 0.0175955
\(274\) 0 0
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) −28.1217 −1.68967 −0.844834 0.535028i \(-0.820301\pi\)
−0.844834 + 0.535028i \(0.820301\pi\)
\(278\) 0 0
\(279\) −0.673158 −0.0403009
\(280\) 0 0
\(281\) 5.47187 0.326425 0.163212 0.986591i \(-0.447814\pi\)
0.163212 + 0.986591i \(0.447814\pi\)
\(282\) 0 0
\(283\) 10.7382 0.638320 0.319160 0.947701i \(-0.396599\pi\)
0.319160 + 0.947701i \(0.396599\pi\)
\(284\) 0 0
\(285\) −0.764867 −0.0453068
\(286\) 0 0
\(287\) −7.32684 −0.432490
\(288\) 0 0
\(289\) −14.8660 −0.874472
\(290\) 0 0
\(291\) 6.05786 0.355118
\(292\) 0 0
\(293\) 24.0300 1.40385 0.701923 0.712253i \(-0.252325\pi\)
0.701923 + 0.712253i \(0.252325\pi\)
\(294\) 0 0
\(295\) −0.829914 −0.0483194
\(296\) 0 0
\(297\) −3.07838 −0.178626
\(298\) 0 0
\(299\) 0.921622 0.0532988
\(300\) 0 0
\(301\) −6.97107 −0.401806
\(302\) 0 0
\(303\) 6.16063 0.353919
\(304\) 0 0
\(305\) 3.35350 0.192021
\(306\) 0 0
\(307\) 23.9421 1.36645 0.683225 0.730208i \(-0.260577\pi\)
0.683225 + 0.730208i \(0.260577\pi\)
\(308\) 0 0
\(309\) 2.64527 0.150484
\(310\) 0 0
\(311\) −7.32684 −0.415467 −0.207734 0.978185i \(-0.566609\pi\)
−0.207734 + 0.978185i \(0.566609\pi\)
\(312\) 0 0
\(313\) −14.5814 −0.824192 −0.412096 0.911140i \(-0.635203\pi\)
−0.412096 + 0.911140i \(0.635203\pi\)
\(314\) 0 0
\(315\) 2.70928 0.152650
\(316\) 0 0
\(317\) 21.8843 1.22914 0.614572 0.788861i \(-0.289329\pi\)
0.614572 + 0.788861i \(0.289329\pi\)
\(318\) 0 0
\(319\) 6.68035 0.374028
\(320\) 0 0
\(321\) 8.68035 0.484490
\(322\) 0 0
\(323\) 2.07223 0.115302
\(324\) 0 0
\(325\) 0.539189 0.0299088
\(326\) 0 0
\(327\) −8.86376 −0.490167
\(328\) 0 0
\(329\) 11.2195 0.618553
\(330\) 0 0
\(331\) 25.3751 1.39474 0.697370 0.716711i \(-0.254353\pi\)
0.697370 + 0.716711i \(0.254353\pi\)
\(332\) 0 0
\(333\) −5.55252 −0.304276
\(334\) 0 0
\(335\) −0.630898 −0.0344696
\(336\) 0 0
\(337\) 12.9854 0.707362 0.353681 0.935366i \(-0.384930\pi\)
0.353681 + 0.935366i \(0.384930\pi\)
\(338\) 0 0
\(339\) 2.82273 0.153309
\(340\) 0 0
\(341\) −0.248464 −0.0134551
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −0.921622 −0.0496185
\(346\) 0 0
\(347\) 24.0183 1.28937 0.644684 0.764449i \(-0.276989\pi\)
0.644684 + 0.764449i \(0.276989\pi\)
\(348\) 0 0
\(349\) −25.4791 −1.36386 −0.681931 0.731416i \(-0.738860\pi\)
−0.681931 + 0.731416i \(0.738860\pi\)
\(350\) 0 0
\(351\) 1.65983 0.0885950
\(352\) 0 0
\(353\) 14.4124 0.767095 0.383548 0.923521i \(-0.374702\pi\)
0.383548 + 0.923521i \(0.374702\pi\)
\(354\) 0 0
\(355\) 10.8371 0.575174
\(356\) 0 0
\(357\) 0.787653 0.0416870
\(358\) 0 0
\(359\) −17.3691 −0.916706 −0.458353 0.888770i \(-0.651560\pi\)
−0.458353 + 0.888770i \(0.651560\pi\)
\(360\) 0 0
\(361\) −16.9877 −0.894090
\(362\) 0 0
\(363\) −0.539189 −0.0283001
\(364\) 0 0
\(365\) −9.46081 −0.495201
\(366\) 0 0
\(367\) 33.0772 1.72661 0.863307 0.504680i \(-0.168389\pi\)
0.863307 + 0.504680i \(0.168389\pi\)
\(368\) 0 0
\(369\) −19.8504 −1.03337
\(370\) 0 0
\(371\) 3.89269 0.202099
\(372\) 0 0
\(373\) −20.0905 −1.04025 −0.520123 0.854091i \(-0.674114\pi\)
−0.520123 + 0.854091i \(0.674114\pi\)
\(374\) 0 0
\(375\) −0.539189 −0.0278436
\(376\) 0 0
\(377\) −3.60197 −0.185511
\(378\) 0 0
\(379\) 14.8904 0.764870 0.382435 0.923982i \(-0.375086\pi\)
0.382435 + 0.923982i \(0.375086\pi\)
\(380\) 0 0
\(381\) 4.94828 0.253508
\(382\) 0 0
\(383\) −11.1929 −0.571929 −0.285965 0.958240i \(-0.592314\pi\)
−0.285965 + 0.958240i \(0.592314\pi\)
\(384\) 0 0
\(385\) 1.00000 0.0509647
\(386\) 0 0
\(387\) −18.8865 −0.960057
\(388\) 0 0
\(389\) −20.2413 −1.02627 −0.513137 0.858307i \(-0.671517\pi\)
−0.513137 + 0.858307i \(0.671517\pi\)
\(390\) 0 0
\(391\) 2.49693 0.126275
\(392\) 0 0
\(393\) 11.3197 0.571001
\(394\) 0 0
\(395\) −2.63090 −0.132375
\(396\) 0 0
\(397\) −9.57531 −0.480571 −0.240285 0.970702i \(-0.577241\pi\)
−0.240285 + 0.970702i \(0.577241\pi\)
\(398\) 0 0
\(399\) 0.764867 0.0382912
\(400\) 0 0
\(401\) −21.8843 −1.09285 −0.546424 0.837508i \(-0.684011\pi\)
−0.546424 + 0.837508i \(0.684011\pi\)
\(402\) 0 0
\(403\) 0.133969 0.00667348
\(404\) 0 0
\(405\) 6.46800 0.321397
\(406\) 0 0
\(407\) −2.04945 −0.101587
\(408\) 0 0
\(409\) −19.3002 −0.954332 −0.477166 0.878813i \(-0.658336\pi\)
−0.477166 + 0.878813i \(0.658336\pi\)
\(410\) 0 0
\(411\) −4.42469 −0.218254
\(412\) 0 0
\(413\) 0.829914 0.0408374
\(414\) 0 0
\(415\) −9.02052 −0.442800
\(416\) 0 0
\(417\) 8.58145 0.420235
\(418\) 0 0
\(419\) 32.2895 1.57745 0.788723 0.614749i \(-0.210743\pi\)
0.788723 + 0.614749i \(0.210743\pi\)
\(420\) 0 0
\(421\) −20.6309 −1.00549 −0.502744 0.864435i \(-0.667676\pi\)
−0.502744 + 0.864435i \(0.667676\pi\)
\(422\) 0 0
\(423\) 30.3968 1.47794
\(424\) 0 0
\(425\) 1.46081 0.0708597
\(426\) 0 0
\(427\) −3.35350 −0.162287
\(428\) 0 0
\(429\) 0.290725 0.0140363
\(430\) 0 0
\(431\) −13.1012 −0.631061 −0.315530 0.948915i \(-0.602182\pi\)
−0.315530 + 0.948915i \(0.602182\pi\)
\(432\) 0 0
\(433\) −9.11942 −0.438251 −0.219126 0.975697i \(-0.570320\pi\)
−0.219126 + 0.975697i \(0.570320\pi\)
\(434\) 0 0
\(435\) 3.60197 0.172701
\(436\) 0 0
\(437\) 2.42469 0.115989
\(438\) 0 0
\(439\) −21.1773 −1.01074 −0.505368 0.862904i \(-0.668643\pi\)
−0.505368 + 0.862904i \(0.668643\pi\)
\(440\) 0 0
\(441\) −2.70928 −0.129013
\(442\) 0 0
\(443\) −16.0183 −0.761050 −0.380525 0.924771i \(-0.624257\pi\)
−0.380525 + 0.924771i \(0.624257\pi\)
\(444\) 0 0
\(445\) 0.764867 0.0362582
\(446\) 0 0
\(447\) −2.70701 −0.128037
\(448\) 0 0
\(449\) −9.79380 −0.462198 −0.231099 0.972930i \(-0.574232\pi\)
−0.231099 + 0.972930i \(0.574232\pi\)
\(450\) 0 0
\(451\) −7.32684 −0.345008
\(452\) 0 0
\(453\) 6.99386 0.328600
\(454\) 0 0
\(455\) −0.539189 −0.0252776
\(456\) 0 0
\(457\) −23.1422 −1.08255 −0.541273 0.840847i \(-0.682058\pi\)
−0.541273 + 0.840847i \(0.682058\pi\)
\(458\) 0 0
\(459\) 4.49693 0.209899
\(460\) 0 0
\(461\) 25.0544 1.16690 0.583449 0.812150i \(-0.301703\pi\)
0.583449 + 0.812150i \(0.301703\pi\)
\(462\) 0 0
\(463\) 22.8865 1.06363 0.531814 0.846861i \(-0.321511\pi\)
0.531814 + 0.846861i \(0.321511\pi\)
\(464\) 0 0
\(465\) −0.133969 −0.00621267
\(466\) 0 0
\(467\) 5.70209 0.263861 0.131931 0.991259i \(-0.457882\pi\)
0.131931 + 0.991259i \(0.457882\pi\)
\(468\) 0 0
\(469\) 0.630898 0.0291321
\(470\) 0 0
\(471\) −0.169042 −0.00778903
\(472\) 0 0
\(473\) −6.97107 −0.320530
\(474\) 0 0
\(475\) 1.41855 0.0650876
\(476\) 0 0
\(477\) 10.5464 0.482885
\(478\) 0 0
\(479\) −32.3545 −1.47832 −0.739159 0.673531i \(-0.764777\pi\)
−0.739159 + 0.673531i \(0.764777\pi\)
\(480\) 0 0
\(481\) 1.10504 0.0503855
\(482\) 0 0
\(483\) 0.921622 0.0419353
\(484\) 0 0
\(485\) −11.2351 −0.510161
\(486\) 0 0
\(487\) −38.0060 −1.72221 −0.861107 0.508423i \(-0.830229\pi\)
−0.861107 + 0.508423i \(0.830229\pi\)
\(488\) 0 0
\(489\) −11.8576 −0.536220
\(490\) 0 0
\(491\) 18.3174 0.826652 0.413326 0.910583i \(-0.364367\pi\)
0.413326 + 0.910583i \(0.364367\pi\)
\(492\) 0 0
\(493\) −9.75872 −0.439511
\(494\) 0 0
\(495\) 2.70928 0.121773
\(496\) 0 0
\(497\) −10.8371 −0.486110
\(498\) 0 0
\(499\) −9.88882 −0.442684 −0.221342 0.975196i \(-0.571044\pi\)
−0.221342 + 0.975196i \(0.571044\pi\)
\(500\) 0 0
\(501\) −9.81205 −0.438370
\(502\) 0 0
\(503\) −26.1834 −1.16746 −0.583730 0.811948i \(-0.698408\pi\)
−0.583730 + 0.811948i \(0.698408\pi\)
\(504\) 0 0
\(505\) −11.4257 −0.508439
\(506\) 0 0
\(507\) 6.85270 0.304339
\(508\) 0 0
\(509\) 22.5958 1.00154 0.500771 0.865580i \(-0.333050\pi\)
0.500771 + 0.865580i \(0.333050\pi\)
\(510\) 0 0
\(511\) 9.46081 0.418522
\(512\) 0 0
\(513\) 4.36683 0.192800
\(514\) 0 0
\(515\) −4.90602 −0.216185
\(516\) 0 0
\(517\) 11.2195 0.493435
\(518\) 0 0
\(519\) −5.20006 −0.228257
\(520\) 0 0
\(521\) −28.9770 −1.26951 −0.634753 0.772715i \(-0.718898\pi\)
−0.634753 + 0.772715i \(0.718898\pi\)
\(522\) 0 0
\(523\) −5.41855 −0.236937 −0.118468 0.992958i \(-0.537798\pi\)
−0.118468 + 0.992958i \(0.537798\pi\)
\(524\) 0 0
\(525\) 0.539189 0.0235321
\(526\) 0 0
\(527\) 0.362959 0.0158108
\(528\) 0 0
\(529\) −20.0784 −0.872973
\(530\) 0 0
\(531\) 2.24846 0.0975750
\(532\) 0 0
\(533\) 3.95055 0.171117
\(534\) 0 0
\(535\) −16.0989 −0.696016
\(536\) 0 0
\(537\) 10.0410 0.433303
\(538\) 0 0
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) 17.5031 0.752516 0.376258 0.926515i \(-0.377211\pi\)
0.376258 + 0.926515i \(0.377211\pi\)
\(542\) 0 0
\(543\) 3.83096 0.164402
\(544\) 0 0
\(545\) 16.4391 0.704172
\(546\) 0 0
\(547\) 0.711543 0.0304234 0.0152117 0.999884i \(-0.495158\pi\)
0.0152117 + 0.999884i \(0.495158\pi\)
\(548\) 0 0
\(549\) −9.08557 −0.387762
\(550\) 0 0
\(551\) −9.47641 −0.403709
\(552\) 0 0
\(553\) 2.63090 0.111877
\(554\) 0 0
\(555\) −1.10504 −0.0469063
\(556\) 0 0
\(557\) −21.0700 −0.892763 −0.446382 0.894843i \(-0.647288\pi\)
−0.446382 + 0.894843i \(0.647288\pi\)
\(558\) 0 0
\(559\) 3.75872 0.158977
\(560\) 0 0
\(561\) 0.787653 0.0332547
\(562\) 0 0
\(563\) 18.9672 0.799372 0.399686 0.916652i \(-0.369119\pi\)
0.399686 + 0.916652i \(0.369119\pi\)
\(564\) 0 0
\(565\) −5.23513 −0.220244
\(566\) 0 0
\(567\) −6.46800 −0.271630
\(568\) 0 0
\(569\) 13.1194 0.549995 0.274997 0.961445i \(-0.411323\pi\)
0.274997 + 0.961445i \(0.411323\pi\)
\(570\) 0 0
\(571\) −5.97334 −0.249976 −0.124988 0.992158i \(-0.539889\pi\)
−0.124988 + 0.992158i \(0.539889\pi\)
\(572\) 0 0
\(573\) 7.58759 0.316976
\(574\) 0 0
\(575\) 1.70928 0.0712817
\(576\) 0 0
\(577\) 3.02052 0.125746 0.0628729 0.998022i \(-0.479974\pi\)
0.0628729 + 0.998022i \(0.479974\pi\)
\(578\) 0 0
\(579\) 4.93600 0.205133
\(580\) 0 0
\(581\) 9.02052 0.374234
\(582\) 0 0
\(583\) 3.89269 0.161219
\(584\) 0 0
\(585\) −1.46081 −0.0603971
\(586\) 0 0
\(587\) −12.3558 −0.509977 −0.254989 0.966944i \(-0.582072\pi\)
−0.254989 + 0.966944i \(0.582072\pi\)
\(588\) 0 0
\(589\) 0.352459 0.0145228
\(590\) 0 0
\(591\) 10.9483 0.450352
\(592\) 0 0
\(593\) 36.2544 1.48879 0.744395 0.667739i \(-0.232738\pi\)
0.744395 + 0.667739i \(0.232738\pi\)
\(594\) 0 0
\(595\) −1.46081 −0.0598874
\(596\) 0 0
\(597\) 8.00388 0.327577
\(598\) 0 0
\(599\) 3.73206 0.152488 0.0762440 0.997089i \(-0.475707\pi\)
0.0762440 + 0.997089i \(0.475707\pi\)
\(600\) 0 0
\(601\) 9.75154 0.397773 0.198887 0.980022i \(-0.436267\pi\)
0.198887 + 0.980022i \(0.436267\pi\)
\(602\) 0 0
\(603\) 1.70928 0.0696071
\(604\) 0 0
\(605\) 1.00000 0.0406558
\(606\) 0 0
\(607\) −22.6270 −0.918402 −0.459201 0.888332i \(-0.651864\pi\)
−0.459201 + 0.888332i \(0.651864\pi\)
\(608\) 0 0
\(609\) −3.60197 −0.145959
\(610\) 0 0
\(611\) −6.04945 −0.244734
\(612\) 0 0
\(613\) 36.3195 1.46693 0.733465 0.679727i \(-0.237902\pi\)
0.733465 + 0.679727i \(0.237902\pi\)
\(614\) 0 0
\(615\) −3.95055 −0.159302
\(616\) 0 0
\(617\) −43.6248 −1.75627 −0.878133 0.478416i \(-0.841211\pi\)
−0.878133 + 0.478416i \(0.841211\pi\)
\(618\) 0 0
\(619\) 31.5246 1.26708 0.633541 0.773709i \(-0.281601\pi\)
0.633541 + 0.773709i \(0.281601\pi\)
\(620\) 0 0
\(621\) 5.26180 0.211149
\(622\) 0 0
\(623\) −0.764867 −0.0306437
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0.764867 0.0305458
\(628\) 0 0
\(629\) 2.99386 0.119373
\(630\) 0 0
\(631\) 9.04718 0.360163 0.180081 0.983652i \(-0.442364\pi\)
0.180081 + 0.983652i \(0.442364\pi\)
\(632\) 0 0
\(633\) 2.25565 0.0896541
\(634\) 0 0
\(635\) −9.17727 −0.364189
\(636\) 0 0
\(637\) 0.539189 0.0213634
\(638\) 0 0
\(639\) −29.3607 −1.16149
\(640\) 0 0
\(641\) 37.0265 1.46246 0.731229 0.682132i \(-0.238947\pi\)
0.731229 + 0.682132i \(0.238947\pi\)
\(642\) 0 0
\(643\) 23.4341 0.924153 0.462076 0.886840i \(-0.347105\pi\)
0.462076 + 0.886840i \(0.347105\pi\)
\(644\) 0 0
\(645\) −3.75872 −0.148000
\(646\) 0 0
\(647\) −40.0833 −1.57584 −0.787919 0.615779i \(-0.788841\pi\)
−0.787919 + 0.615779i \(0.788841\pi\)
\(648\) 0 0
\(649\) 0.829914 0.0325770
\(650\) 0 0
\(651\) 0.133969 0.00525066
\(652\) 0 0
\(653\) 33.2351 1.30059 0.650296 0.759681i \(-0.274645\pi\)
0.650296 + 0.759681i \(0.274645\pi\)
\(654\) 0 0
\(655\) −20.9939 −0.820298
\(656\) 0 0
\(657\) 25.6319 0.999997
\(658\) 0 0
\(659\) −28.0228 −1.09161 −0.545806 0.837911i \(-0.683777\pi\)
−0.545806 + 0.837911i \(0.683777\pi\)
\(660\) 0 0
\(661\) 0.863763 0.0335965 0.0167983 0.999859i \(-0.494653\pi\)
0.0167983 + 0.999859i \(0.494653\pi\)
\(662\) 0 0
\(663\) −0.424694 −0.0164937
\(664\) 0 0
\(665\) −1.41855 −0.0550090
\(666\) 0 0
\(667\) −11.4186 −0.442128
\(668\) 0 0
\(669\) −11.7626 −0.454768
\(670\) 0 0
\(671\) −3.35350 −0.129461
\(672\) 0 0
\(673\) 26.6309 1.02655 0.513273 0.858225i \(-0.328433\pi\)
0.513273 + 0.858225i \(0.328433\pi\)
\(674\) 0 0
\(675\) 3.07838 0.118487
\(676\) 0 0
\(677\) −26.8794 −1.03306 −0.516529 0.856270i \(-0.672776\pi\)
−0.516529 + 0.856270i \(0.672776\pi\)
\(678\) 0 0
\(679\) 11.2351 0.431165
\(680\) 0 0
\(681\) −8.00000 −0.306561
\(682\) 0 0
\(683\) −8.46186 −0.323784 −0.161892 0.986808i \(-0.551760\pi\)
−0.161892 + 0.986808i \(0.551760\pi\)
\(684\) 0 0
\(685\) 8.20620 0.313543
\(686\) 0 0
\(687\) 10.1711 0.388053
\(688\) 0 0
\(689\) −2.09890 −0.0799616
\(690\) 0 0
\(691\) −33.4836 −1.27378 −0.636888 0.770956i \(-0.719779\pi\)
−0.636888 + 0.770956i \(0.719779\pi\)
\(692\) 0 0
\(693\) −2.70928 −0.102917
\(694\) 0 0
\(695\) −15.9155 −0.603709
\(696\) 0 0
\(697\) 10.7031 0.405410
\(698\) 0 0
\(699\) 0.639308 0.0241809
\(700\) 0 0
\(701\) −34.0677 −1.28672 −0.643360 0.765564i \(-0.722460\pi\)
−0.643360 + 0.765564i \(0.722460\pi\)
\(702\) 0 0
\(703\) 2.90725 0.109649
\(704\) 0 0
\(705\) 6.04945 0.227835
\(706\) 0 0
\(707\) 11.4257 0.429709
\(708\) 0 0
\(709\) −0.523590 −0.0196639 −0.00983193 0.999952i \(-0.503130\pi\)
−0.00983193 + 0.999952i \(0.503130\pi\)
\(710\) 0 0
\(711\) 7.12783 0.267314
\(712\) 0 0
\(713\) 0.424694 0.0159049
\(714\) 0 0
\(715\) −0.539189 −0.0201645
\(716\) 0 0
\(717\) −0.634773 −0.0237060
\(718\) 0 0
\(719\) −45.0856 −1.68141 −0.840704 0.541495i \(-0.817858\pi\)
−0.840704 + 0.541495i \(0.817858\pi\)
\(720\) 0 0
\(721\) 4.90602 0.182710
\(722\) 0 0
\(723\) −15.7815 −0.586921
\(724\) 0 0
\(725\) −6.68035 −0.248102
\(726\) 0 0
\(727\) 5.33072 0.197705 0.0988527 0.995102i \(-0.468483\pi\)
0.0988527 + 0.995102i \(0.468483\pi\)
\(728\) 0 0
\(729\) −12.5441 −0.464597
\(730\) 0 0
\(731\) 10.1834 0.376647
\(732\) 0 0
\(733\) 20.2979 0.749721 0.374860 0.927081i \(-0.377691\pi\)
0.374860 + 0.927081i \(0.377691\pi\)
\(734\) 0 0
\(735\) −0.539189 −0.0198883
\(736\) 0 0
\(737\) 0.630898 0.0232394
\(738\) 0 0
\(739\) 27.4641 1.01028 0.505142 0.863036i \(-0.331440\pi\)
0.505142 + 0.863036i \(0.331440\pi\)
\(740\) 0 0
\(741\) −0.412408 −0.0151502
\(742\) 0 0
\(743\) −16.9483 −0.621772 −0.310886 0.950447i \(-0.600626\pi\)
−0.310886 + 0.950447i \(0.600626\pi\)
\(744\) 0 0
\(745\) 5.02052 0.183938
\(746\) 0 0
\(747\) 24.4391 0.894179
\(748\) 0 0
\(749\) 16.0989 0.588241
\(750\) 0 0
\(751\) −18.9216 −0.690460 −0.345230 0.938518i \(-0.612199\pi\)
−0.345230 + 0.938518i \(0.612199\pi\)
\(752\) 0 0
\(753\) 7.55252 0.275229
\(754\) 0 0
\(755\) −12.9711 −0.472066
\(756\) 0 0
\(757\) −14.4436 −0.524962 −0.262481 0.964937i \(-0.584541\pi\)
−0.262481 + 0.964937i \(0.584541\pi\)
\(758\) 0 0
\(759\) 0.921622 0.0334528
\(760\) 0 0
\(761\) −37.4934 −1.35914 −0.679568 0.733612i \(-0.737833\pi\)
−0.679568 + 0.733612i \(0.737833\pi\)
\(762\) 0 0
\(763\) −16.4391 −0.595134
\(764\) 0 0
\(765\) −3.95774 −0.143092
\(766\) 0 0
\(767\) −0.447480 −0.0161576
\(768\) 0 0
\(769\) −1.63582 −0.0589891 −0.0294946 0.999565i \(-0.509390\pi\)
−0.0294946 + 0.999565i \(0.509390\pi\)
\(770\) 0 0
\(771\) 6.69263 0.241029
\(772\) 0 0
\(773\) −39.1194 −1.40703 −0.703514 0.710682i \(-0.748387\pi\)
−0.703514 + 0.710682i \(0.748387\pi\)
\(774\) 0 0
\(775\) 0.248464 0.00892510
\(776\) 0 0
\(777\) 1.10504 0.0396431
\(778\) 0 0
\(779\) 10.3935 0.372386
\(780\) 0 0
\(781\) −10.8371 −0.387782
\(782\) 0 0
\(783\) −20.5646 −0.734920
\(784\) 0 0
\(785\) 0.313511 0.0111897
\(786\) 0 0
\(787\) 38.8515 1.38491 0.692453 0.721463i \(-0.256530\pi\)
0.692453 + 0.721463i \(0.256530\pi\)
\(788\) 0 0
\(789\) 6.93600 0.246928
\(790\) 0 0
\(791\) 5.23513 0.186140
\(792\) 0 0
\(793\) 1.80817 0.0642101
\(794\) 0 0
\(795\) 2.09890 0.0744402
\(796\) 0 0
\(797\) −23.3484 −0.827043 −0.413521 0.910494i \(-0.635701\pi\)
−0.413521 + 0.910494i \(0.635701\pi\)
\(798\) 0 0
\(799\) −16.3896 −0.579823
\(800\) 0 0
\(801\) −2.07223 −0.0732188
\(802\) 0 0
\(803\) 9.46081 0.333865
\(804\) 0 0
\(805\) −1.70928 −0.0602440
\(806\) 0 0
\(807\) 5.67420 0.199741
\(808\) 0 0
\(809\) −42.7259 −1.50216 −0.751082 0.660209i \(-0.770468\pi\)
−0.751082 + 0.660209i \(0.770468\pi\)
\(810\) 0 0
\(811\) −7.95652 −0.279391 −0.139696 0.990195i \(-0.544612\pi\)
−0.139696 + 0.990195i \(0.544612\pi\)
\(812\) 0 0
\(813\) 6.05786 0.212458
\(814\) 0 0
\(815\) 21.9916 0.770332
\(816\) 0 0
\(817\) 9.88882 0.345966
\(818\) 0 0
\(819\) 1.46081 0.0510449
\(820\) 0 0
\(821\) −23.9565 −0.836088 −0.418044 0.908427i \(-0.637284\pi\)
−0.418044 + 0.908427i \(0.637284\pi\)
\(822\) 0 0
\(823\) 1.50904 0.0526017 0.0263009 0.999654i \(-0.491627\pi\)
0.0263009 + 0.999654i \(0.491627\pi\)
\(824\) 0 0
\(825\) 0.539189 0.0187721
\(826\) 0 0
\(827\) 35.4368 1.23226 0.616129 0.787645i \(-0.288700\pi\)
0.616129 + 0.787645i \(0.288700\pi\)
\(828\) 0 0
\(829\) 50.3423 1.74846 0.874230 0.485513i \(-0.161367\pi\)
0.874230 + 0.485513i \(0.161367\pi\)
\(830\) 0 0
\(831\) 15.1629 0.525995
\(832\) 0 0
\(833\) 1.46081 0.0506141
\(834\) 0 0
\(835\) 18.1978 0.629761
\(836\) 0 0
\(837\) 0.764867 0.0264377
\(838\) 0 0
\(839\) −20.4585 −0.706307 −0.353154 0.935565i \(-0.614891\pi\)
−0.353154 + 0.935565i \(0.614891\pi\)
\(840\) 0 0
\(841\) 15.6270 0.538863
\(842\) 0 0
\(843\) −2.95037 −0.101616
\(844\) 0 0
\(845\) −12.7093 −0.437212
\(846\) 0 0
\(847\) −1.00000 −0.0343604
\(848\) 0 0
\(849\) −5.78992 −0.198710
\(850\) 0 0
\(851\) 3.50307 0.120084
\(852\) 0 0
\(853\) 28.7903 0.985761 0.492881 0.870097i \(-0.335944\pi\)
0.492881 + 0.870097i \(0.335944\pi\)
\(854\) 0 0
\(855\) −3.84324 −0.131436
\(856\) 0 0
\(857\) 53.4219 1.82486 0.912428 0.409237i \(-0.134205\pi\)
0.912428 + 0.409237i \(0.134205\pi\)
\(858\) 0 0
\(859\) −28.3162 −0.966135 −0.483068 0.875583i \(-0.660477\pi\)
−0.483068 + 0.875583i \(0.660477\pi\)
\(860\) 0 0
\(861\) 3.95055 0.134634
\(862\) 0 0
\(863\) 2.67438 0.0910370 0.0455185 0.998963i \(-0.485506\pi\)
0.0455185 + 0.998963i \(0.485506\pi\)
\(864\) 0 0
\(865\) 9.64423 0.327914
\(866\) 0 0
\(867\) 8.01560 0.272224
\(868\) 0 0
\(869\) 2.63090 0.0892471
\(870\) 0 0
\(871\) −0.340173 −0.0115263
\(872\) 0 0
\(873\) 30.4391 1.03021
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) 15.1278 0.510830 0.255415 0.966831i \(-0.417788\pi\)
0.255415 + 0.966831i \(0.417788\pi\)
\(878\) 0 0
\(879\) −12.9567 −0.437018
\(880\) 0 0
\(881\) 40.0989 1.35097 0.675483 0.737375i \(-0.263935\pi\)
0.675483 + 0.737375i \(0.263935\pi\)
\(882\) 0 0
\(883\) 15.2702 0.513883 0.256942 0.966427i \(-0.417285\pi\)
0.256942 + 0.966427i \(0.417285\pi\)
\(884\) 0 0
\(885\) 0.447480 0.0150419
\(886\) 0 0
\(887\) 25.6430 0.861008 0.430504 0.902589i \(-0.358336\pi\)
0.430504 + 0.902589i \(0.358336\pi\)
\(888\) 0 0
\(889\) 9.17727 0.307796
\(890\) 0 0
\(891\) −6.46800 −0.216686
\(892\) 0 0
\(893\) −15.9155 −0.532591
\(894\) 0 0
\(895\) −18.6225 −0.622481
\(896\) 0 0
\(897\) −0.496928 −0.0165920
\(898\) 0 0
\(899\) −1.65983 −0.0553583
\(900\) 0 0
\(901\) −5.68649 −0.189444
\(902\) 0 0
\(903\) 3.75872 0.125082
\(904\) 0 0
\(905\) −7.10504 −0.236180
\(906\) 0 0
\(907\) 13.3424 0.443028 0.221514 0.975157i \(-0.428900\pi\)
0.221514 + 0.975157i \(0.428900\pi\)
\(908\) 0 0
\(909\) 30.9555 1.02673
\(910\) 0 0
\(911\) −17.6742 −0.585572 −0.292786 0.956178i \(-0.594582\pi\)
−0.292786 + 0.956178i \(0.594582\pi\)
\(912\) 0 0
\(913\) 9.02052 0.298536
\(914\) 0 0
\(915\) −1.80817 −0.0597763
\(916\) 0 0
\(917\) 20.9939 0.693278
\(918\) 0 0
\(919\) −7.81658 −0.257845 −0.128923 0.991655i \(-0.541152\pi\)
−0.128923 + 0.991655i \(0.541152\pi\)
\(920\) 0 0
\(921\) −12.9093 −0.425377
\(922\) 0 0
\(923\) 5.84324 0.192333
\(924\) 0 0
\(925\) 2.04945 0.0673854
\(926\) 0 0
\(927\) 13.2918 0.436559
\(928\) 0 0
\(929\) 29.1461 0.956252 0.478126 0.878291i \(-0.341316\pi\)
0.478126 + 0.878291i \(0.341316\pi\)
\(930\) 0 0
\(931\) 1.41855 0.0464911
\(932\) 0 0
\(933\) 3.95055 0.129335
\(934\) 0 0
\(935\) −1.46081 −0.0477736
\(936\) 0 0
\(937\) −3.62985 −0.118582 −0.0592911 0.998241i \(-0.518884\pi\)
−0.0592911 + 0.998241i \(0.518884\pi\)
\(938\) 0 0
\(939\) 7.86216 0.256572
\(940\) 0 0
\(941\) 51.8648 1.69074 0.845372 0.534178i \(-0.179379\pi\)
0.845372 + 0.534178i \(0.179379\pi\)
\(942\) 0 0
\(943\) 12.5236 0.407824
\(944\) 0 0
\(945\) −3.07838 −0.100140
\(946\) 0 0
\(947\) 19.6514 0.638585 0.319293 0.947656i \(-0.396555\pi\)
0.319293 + 0.947656i \(0.396555\pi\)
\(948\) 0 0
\(949\) −5.10116 −0.165591
\(950\) 0 0
\(951\) −11.7998 −0.382633
\(952\) 0 0
\(953\) 19.2267 0.622815 0.311407 0.950277i \(-0.399200\pi\)
0.311407 + 0.950277i \(0.399200\pi\)
\(954\) 0 0
\(955\) −14.0722 −0.455367
\(956\) 0 0
\(957\) −3.60197 −0.116435
\(958\) 0 0
\(959\) −8.20620 −0.264992
\(960\) 0 0
\(961\) −30.9383 −0.998009
\(962\) 0 0
\(963\) 43.6163 1.40552
\(964\) 0 0
\(965\) −9.15449 −0.294693
\(966\) 0 0
\(967\) −14.7565 −0.474536 −0.237268 0.971444i \(-0.576252\pi\)
−0.237268 + 0.971444i \(0.576252\pi\)
\(968\) 0 0
\(969\) −1.11733 −0.0358937
\(970\) 0 0
\(971\) 52.7331 1.69229 0.846143 0.532956i \(-0.178919\pi\)
0.846143 + 0.532956i \(0.178919\pi\)
\(972\) 0 0
\(973\) 15.9155 0.510227
\(974\) 0 0
\(975\) −0.290725 −0.00931064
\(976\) 0 0
\(977\) −37.1727 −1.18926 −0.594631 0.803999i \(-0.702702\pi\)
−0.594631 + 0.803999i \(0.702702\pi\)
\(978\) 0 0
\(979\) −0.764867 −0.0244452
\(980\) 0 0
\(981\) −44.5380 −1.42199
\(982\) 0 0
\(983\) −30.0566 −0.958658 −0.479329 0.877635i \(-0.659120\pi\)
−0.479329 + 0.877635i \(0.659120\pi\)
\(984\) 0 0
\(985\) −20.3051 −0.646974
\(986\) 0 0
\(987\) −6.04945 −0.192556
\(988\) 0 0
\(989\) 11.9155 0.378890
\(990\) 0 0
\(991\) 6.82273 0.216731 0.108366 0.994111i \(-0.465438\pi\)
0.108366 + 0.994111i \(0.465438\pi\)
\(992\) 0 0
\(993\) −13.6820 −0.434184
\(994\) 0 0
\(995\) −14.8443 −0.470596
\(996\) 0 0
\(997\) 59.7308 1.89169 0.945847 0.324612i \(-0.105234\pi\)
0.945847 + 0.324612i \(0.105234\pi\)
\(998\) 0 0
\(999\) 6.30898 0.199607
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 6160.2.a.bj.1.2 3
4.3 odd 2 385.2.a.g.1.3 3
12.11 even 2 3465.2.a.ba.1.1 3
20.3 even 4 1925.2.b.o.1849.1 6
20.7 even 4 1925.2.b.o.1849.6 6
20.19 odd 2 1925.2.a.u.1.1 3
28.27 even 2 2695.2.a.i.1.3 3
44.43 even 2 4235.2.a.o.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
385.2.a.g.1.3 3 4.3 odd 2
1925.2.a.u.1.1 3 20.19 odd 2
1925.2.b.o.1849.1 6 20.3 even 4
1925.2.b.o.1849.6 6 20.7 even 4
2695.2.a.i.1.3 3 28.27 even 2
3465.2.a.ba.1.1 3 12.11 even 2
4235.2.a.o.1.1 3 44.43 even 2
6160.2.a.bj.1.2 3 1.1 even 1 trivial