Properties

Label 7728.2.a.bj.1.2
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $0$
Dimension $2$
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] = [7728,2,Mod(1,7728)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7728, 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("7728.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7728.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: \(61.7083906820\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3864)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.37228\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +3.37228 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +3.37228 q^{5} -1.00000 q^{7} +1.00000 q^{9} +2.37228 q^{11} +5.37228 q^{13} +3.37228 q^{15} +4.00000 q^{17} -1.62772 q^{19} -1.00000 q^{21} -1.00000 q^{23} +6.37228 q^{25} +1.00000 q^{27} +5.37228 q^{29} +6.00000 q^{31} +2.37228 q^{33} -3.37228 q^{35} -2.62772 q^{37} +5.37228 q^{39} -5.74456 q^{41} -3.37228 q^{43} +3.37228 q^{45} -3.74456 q^{47} +1.00000 q^{49} +4.00000 q^{51} +7.11684 q^{53} +8.00000 q^{55} -1.62772 q^{57} -3.62772 q^{59} -8.37228 q^{61} -1.00000 q^{63} +18.1168 q^{65} +4.00000 q^{67} -1.00000 q^{69} -1.25544 q^{71} +12.0000 q^{73} +6.37228 q^{75} -2.37228 q^{77} -8.74456 q^{79} +1.00000 q^{81} +8.74456 q^{83} +13.4891 q^{85} +5.37228 q^{87} -7.48913 q^{89} -5.37228 q^{91} +6.00000 q^{93} -5.48913 q^{95} -12.1168 q^{97} +2.37228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + q^{5} - 2 q^{7} + 2 q^{9} - q^{11} + 5 q^{13} + q^{15} + 8 q^{17} - 9 q^{19} - 2 q^{21} - 2 q^{23} + 7 q^{25} + 2 q^{27} + 5 q^{29} + 12 q^{31} - q^{33} - q^{35} - 11 q^{37} + 5 q^{39} - q^{43} + q^{45} + 4 q^{47} + 2 q^{49} + 8 q^{51} - 3 q^{53} + 16 q^{55} - 9 q^{57} - 13 q^{59} - 11 q^{61} - 2 q^{63} + 19 q^{65} + 8 q^{67} - 2 q^{69} - 14 q^{71} + 24 q^{73} + 7 q^{75} + q^{77} - 6 q^{79} + 2 q^{81} + 6 q^{83} + 4 q^{85} + 5 q^{87} + 8 q^{89} - 5 q^{91} + 12 q^{93} + 12 q^{95} - 7 q^{97} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 3.37228 1.50813 0.754065 0.656800i \(-0.228090\pi\)
0.754065 + 0.656800i \(0.228090\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.37228 0.715270 0.357635 0.933862i \(-0.383583\pi\)
0.357635 + 0.933862i \(0.383583\pi\)
\(12\) 0 0
\(13\) 5.37228 1.49000 0.745001 0.667063i \(-0.232449\pi\)
0.745001 + 0.667063i \(0.232449\pi\)
\(14\) 0 0
\(15\) 3.37228 0.870719
\(16\) 0 0
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) −1.62772 −0.373424 −0.186712 0.982415i \(-0.559783\pi\)
−0.186712 + 0.982415i \(0.559783\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 6.37228 1.27446
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 5.37228 0.997608 0.498804 0.866715i \(-0.333773\pi\)
0.498804 + 0.866715i \(0.333773\pi\)
\(30\) 0 0
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 0 0
\(33\) 2.37228 0.412961
\(34\) 0 0
\(35\) −3.37228 −0.570020
\(36\) 0 0
\(37\) −2.62772 −0.431994 −0.215997 0.976394i \(-0.569300\pi\)
−0.215997 + 0.976394i \(0.569300\pi\)
\(38\) 0 0
\(39\) 5.37228 0.860253
\(40\) 0 0
\(41\) −5.74456 −0.897150 −0.448575 0.893745i \(-0.648068\pi\)
−0.448575 + 0.893745i \(0.648068\pi\)
\(42\) 0 0
\(43\) −3.37228 −0.514268 −0.257134 0.966376i \(-0.582778\pi\)
−0.257134 + 0.966376i \(0.582778\pi\)
\(44\) 0 0
\(45\) 3.37228 0.502710
\(46\) 0 0
\(47\) −3.74456 −0.546201 −0.273100 0.961986i \(-0.588049\pi\)
−0.273100 + 0.961986i \(0.588049\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 4.00000 0.560112
\(52\) 0 0
\(53\) 7.11684 0.977574 0.488787 0.872403i \(-0.337440\pi\)
0.488787 + 0.872403i \(0.337440\pi\)
\(54\) 0 0
\(55\) 8.00000 1.07872
\(56\) 0 0
\(57\) −1.62772 −0.215597
\(58\) 0 0
\(59\) −3.62772 −0.472289 −0.236144 0.971718i \(-0.575884\pi\)
−0.236144 + 0.971718i \(0.575884\pi\)
\(60\) 0 0
\(61\) −8.37228 −1.07196 −0.535980 0.844230i \(-0.680058\pi\)
−0.535980 + 0.844230i \(0.680058\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 18.1168 2.24712
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −1.25544 −0.148993 −0.0744965 0.997221i \(-0.523735\pi\)
−0.0744965 + 0.997221i \(0.523735\pi\)
\(72\) 0 0
\(73\) 12.0000 1.40449 0.702247 0.711934i \(-0.252180\pi\)
0.702247 + 0.711934i \(0.252180\pi\)
\(74\) 0 0
\(75\) 6.37228 0.735808
\(76\) 0 0
\(77\) −2.37228 −0.270347
\(78\) 0 0
\(79\) −8.74456 −0.983840 −0.491920 0.870640i \(-0.663705\pi\)
−0.491920 + 0.870640i \(0.663705\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 8.74456 0.959840 0.479920 0.877312i \(-0.340666\pi\)
0.479920 + 0.877312i \(0.340666\pi\)
\(84\) 0 0
\(85\) 13.4891 1.46310
\(86\) 0 0
\(87\) 5.37228 0.575969
\(88\) 0 0
\(89\) −7.48913 −0.793846 −0.396923 0.917852i \(-0.629922\pi\)
−0.396923 + 0.917852i \(0.629922\pi\)
\(90\) 0 0
\(91\) −5.37228 −0.563168
\(92\) 0 0
\(93\) 6.00000 0.622171
\(94\) 0 0
\(95\) −5.48913 −0.563172
\(96\) 0 0
\(97\) −12.1168 −1.23028 −0.615140 0.788418i \(-0.710900\pi\)
−0.615140 + 0.788418i \(0.710900\pi\)
\(98\) 0 0
\(99\) 2.37228 0.238423
\(100\) 0 0
\(101\) −5.11684 −0.509145 −0.254573 0.967054i \(-0.581935\pi\)
−0.254573 + 0.967054i \(0.581935\pi\)
\(102\) 0 0
\(103\) −7.00000 −0.689730 −0.344865 0.938652i \(-0.612075\pi\)
−0.344865 + 0.938652i \(0.612075\pi\)
\(104\) 0 0
\(105\) −3.37228 −0.329101
\(106\) 0 0
\(107\) −17.4891 −1.69074 −0.845369 0.534183i \(-0.820619\pi\)
−0.845369 + 0.534183i \(0.820619\pi\)
\(108\) 0 0
\(109\) 4.62772 0.443255 0.221628 0.975131i \(-0.428863\pi\)
0.221628 + 0.975131i \(0.428863\pi\)
\(110\) 0 0
\(111\) −2.62772 −0.249412
\(112\) 0 0
\(113\) −13.3723 −1.25796 −0.628979 0.777422i \(-0.716527\pi\)
−0.628979 + 0.777422i \(0.716527\pi\)
\(114\) 0 0
\(115\) −3.37228 −0.314467
\(116\) 0 0
\(117\) 5.37228 0.496668
\(118\) 0 0
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) −5.37228 −0.488389
\(122\) 0 0
\(123\) −5.74456 −0.517970
\(124\) 0 0
\(125\) 4.62772 0.413916
\(126\) 0 0
\(127\) 1.00000 0.0887357 0.0443678 0.999015i \(-0.485873\pi\)
0.0443678 + 0.999015i \(0.485873\pi\)
\(128\) 0 0
\(129\) −3.37228 −0.296913
\(130\) 0 0
\(131\) 13.1168 1.14602 0.573012 0.819547i \(-0.305775\pi\)
0.573012 + 0.819547i \(0.305775\pi\)
\(132\) 0 0
\(133\) 1.62772 0.141141
\(134\) 0 0
\(135\) 3.37228 0.290240
\(136\) 0 0
\(137\) −4.48913 −0.383532 −0.191766 0.981441i \(-0.561421\pi\)
−0.191766 + 0.981441i \(0.561421\pi\)
\(138\) 0 0
\(139\) 11.3723 0.964584 0.482292 0.876010i \(-0.339804\pi\)
0.482292 + 0.876010i \(0.339804\pi\)
\(140\) 0 0
\(141\) −3.74456 −0.315349
\(142\) 0 0
\(143\) 12.7446 1.06575
\(144\) 0 0
\(145\) 18.1168 1.50452
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) 4.88316 0.400044 0.200022 0.979791i \(-0.435899\pi\)
0.200022 + 0.979791i \(0.435899\pi\)
\(150\) 0 0
\(151\) 18.4891 1.50462 0.752312 0.658807i \(-0.228939\pi\)
0.752312 + 0.658807i \(0.228939\pi\)
\(152\) 0 0
\(153\) 4.00000 0.323381
\(154\) 0 0
\(155\) 20.2337 1.62521
\(156\) 0 0
\(157\) 15.8614 1.26588 0.632939 0.774202i \(-0.281848\pi\)
0.632939 + 0.774202i \(0.281848\pi\)
\(158\) 0 0
\(159\) 7.11684 0.564402
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) −2.37228 −0.185811 −0.0929057 0.995675i \(-0.529616\pi\)
−0.0929057 + 0.995675i \(0.529616\pi\)
\(164\) 0 0
\(165\) 8.00000 0.622799
\(166\) 0 0
\(167\) −19.1168 −1.47931 −0.739653 0.672989i \(-0.765010\pi\)
−0.739653 + 0.672989i \(0.765010\pi\)
\(168\) 0 0
\(169\) 15.8614 1.22011
\(170\) 0 0
\(171\) −1.62772 −0.124475
\(172\) 0 0
\(173\) 15.4891 1.17762 0.588808 0.808273i \(-0.299597\pi\)
0.588808 + 0.808273i \(0.299597\pi\)
\(174\) 0 0
\(175\) −6.37228 −0.481699
\(176\) 0 0
\(177\) −3.62772 −0.272676
\(178\) 0 0
\(179\) −2.11684 −0.158220 −0.0791102 0.996866i \(-0.525208\pi\)
−0.0791102 + 0.996866i \(0.525208\pi\)
\(180\) 0 0
\(181\) −6.00000 −0.445976 −0.222988 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 0 0
\(183\) −8.37228 −0.618897
\(184\) 0 0
\(185\) −8.86141 −0.651504
\(186\) 0 0
\(187\) 9.48913 0.693914
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −15.8614 −1.14769 −0.573846 0.818964i \(-0.694549\pi\)
−0.573846 + 0.818964i \(0.694549\pi\)
\(192\) 0 0
\(193\) −7.23369 −0.520692 −0.260346 0.965515i \(-0.583837\pi\)
−0.260346 + 0.965515i \(0.583837\pi\)
\(194\) 0 0
\(195\) 18.1168 1.29737
\(196\) 0 0
\(197\) −18.8614 −1.34382 −0.671910 0.740633i \(-0.734526\pi\)
−0.671910 + 0.740633i \(0.734526\pi\)
\(198\) 0 0
\(199\) −1.00000 −0.0708881 −0.0354441 0.999372i \(-0.511285\pi\)
−0.0354441 + 0.999372i \(0.511285\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 0 0
\(203\) −5.37228 −0.377060
\(204\) 0 0
\(205\) −19.3723 −1.35302
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) −3.86141 −0.267099
\(210\) 0 0
\(211\) 5.62772 0.387428 0.193714 0.981058i \(-0.437947\pi\)
0.193714 + 0.981058i \(0.437947\pi\)
\(212\) 0 0
\(213\) −1.25544 −0.0860212
\(214\) 0 0
\(215\) −11.3723 −0.775583
\(216\) 0 0
\(217\) −6.00000 −0.407307
\(218\) 0 0
\(219\) 12.0000 0.810885
\(220\) 0 0
\(221\) 21.4891 1.44551
\(222\) 0 0
\(223\) −14.0000 −0.937509 −0.468755 0.883328i \(-0.655297\pi\)
−0.468755 + 0.883328i \(0.655297\pi\)
\(224\) 0 0
\(225\) 6.37228 0.424819
\(226\) 0 0
\(227\) −8.11684 −0.538734 −0.269367 0.963038i \(-0.586814\pi\)
−0.269367 + 0.963038i \(0.586814\pi\)
\(228\) 0 0
\(229\) 25.3505 1.67521 0.837605 0.546276i \(-0.183955\pi\)
0.837605 + 0.546276i \(0.183955\pi\)
\(230\) 0 0
\(231\) −2.37228 −0.156085
\(232\) 0 0
\(233\) 14.7446 0.965948 0.482974 0.875635i \(-0.339556\pi\)
0.482974 + 0.875635i \(0.339556\pi\)
\(234\) 0 0
\(235\) −12.6277 −0.823742
\(236\) 0 0
\(237\) −8.74456 −0.568020
\(238\) 0 0
\(239\) −13.4891 −0.872539 −0.436269 0.899816i \(-0.643701\pi\)
−0.436269 + 0.899816i \(0.643701\pi\)
\(240\) 0 0
\(241\) −0.489125 −0.0315073 −0.0157537 0.999876i \(-0.505015\pi\)
−0.0157537 + 0.999876i \(0.505015\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 3.37228 0.215447
\(246\) 0 0
\(247\) −8.74456 −0.556403
\(248\) 0 0
\(249\) 8.74456 0.554164
\(250\) 0 0
\(251\) 18.8614 1.19052 0.595261 0.803533i \(-0.297049\pi\)
0.595261 + 0.803533i \(0.297049\pi\)
\(252\) 0 0
\(253\) −2.37228 −0.149144
\(254\) 0 0
\(255\) 13.4891 0.844722
\(256\) 0 0
\(257\) 10.8832 0.678873 0.339436 0.940629i \(-0.389764\pi\)
0.339436 + 0.940629i \(0.389764\pi\)
\(258\) 0 0
\(259\) 2.62772 0.163278
\(260\) 0 0
\(261\) 5.37228 0.332536
\(262\) 0 0
\(263\) −1.00000 −0.0616626 −0.0308313 0.999525i \(-0.509815\pi\)
−0.0308313 + 0.999525i \(0.509815\pi\)
\(264\) 0 0
\(265\) 24.0000 1.47431
\(266\) 0 0
\(267\) −7.48913 −0.458327
\(268\) 0 0
\(269\) 0.510875 0.0311486 0.0155743 0.999879i \(-0.495042\pi\)
0.0155743 + 0.999879i \(0.495042\pi\)
\(270\) 0 0
\(271\) 1.48913 0.0904579 0.0452290 0.998977i \(-0.485598\pi\)
0.0452290 + 0.998977i \(0.485598\pi\)
\(272\) 0 0
\(273\) −5.37228 −0.325145
\(274\) 0 0
\(275\) 15.1168 0.911580
\(276\) 0 0
\(277\) 0.883156 0.0530637 0.0265319 0.999648i \(-0.491554\pi\)
0.0265319 + 0.999648i \(0.491554\pi\)
\(278\) 0 0
\(279\) 6.00000 0.359211
\(280\) 0 0
\(281\) 13.3723 0.797723 0.398862 0.917011i \(-0.369405\pi\)
0.398862 + 0.917011i \(0.369405\pi\)
\(282\) 0 0
\(283\) −5.25544 −0.312403 −0.156202 0.987725i \(-0.549925\pi\)
−0.156202 + 0.987725i \(0.549925\pi\)
\(284\) 0 0
\(285\) −5.48913 −0.325148
\(286\) 0 0
\(287\) 5.74456 0.339091
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) −12.1168 −0.710302
\(292\) 0 0
\(293\) −12.2337 −0.714700 −0.357350 0.933971i \(-0.616320\pi\)
−0.357350 + 0.933971i \(0.616320\pi\)
\(294\) 0 0
\(295\) −12.2337 −0.712273
\(296\) 0 0
\(297\) 2.37228 0.137654
\(298\) 0 0
\(299\) −5.37228 −0.310687
\(300\) 0 0
\(301\) 3.37228 0.194375
\(302\) 0 0
\(303\) −5.11684 −0.293955
\(304\) 0 0
\(305\) −28.2337 −1.61666
\(306\) 0 0
\(307\) 20.6277 1.17729 0.588643 0.808393i \(-0.299662\pi\)
0.588643 + 0.808393i \(0.299662\pi\)
\(308\) 0 0
\(309\) −7.00000 −0.398216
\(310\) 0 0
\(311\) −22.3723 −1.26862 −0.634308 0.773081i \(-0.718715\pi\)
−0.634308 + 0.773081i \(0.718715\pi\)
\(312\) 0 0
\(313\) 32.0951 1.81412 0.907061 0.420999i \(-0.138320\pi\)
0.907061 + 0.420999i \(0.138320\pi\)
\(314\) 0 0
\(315\) −3.37228 −0.190007
\(316\) 0 0
\(317\) 28.1168 1.57920 0.789600 0.613622i \(-0.210288\pi\)
0.789600 + 0.613622i \(0.210288\pi\)
\(318\) 0 0
\(319\) 12.7446 0.713559
\(320\) 0 0
\(321\) −17.4891 −0.976148
\(322\) 0 0
\(323\) −6.51087 −0.362275
\(324\) 0 0
\(325\) 34.2337 1.89894
\(326\) 0 0
\(327\) 4.62772 0.255913
\(328\) 0 0
\(329\) 3.74456 0.206444
\(330\) 0 0
\(331\) −2.37228 −0.130392 −0.0651962 0.997872i \(-0.520767\pi\)
−0.0651962 + 0.997872i \(0.520767\pi\)
\(332\) 0 0
\(333\) −2.62772 −0.143998
\(334\) 0 0
\(335\) 13.4891 0.736990
\(336\) 0 0
\(337\) −19.2554 −1.04891 −0.524455 0.851438i \(-0.675731\pi\)
−0.524455 + 0.851438i \(0.675731\pi\)
\(338\) 0 0
\(339\) −13.3723 −0.726283
\(340\) 0 0
\(341\) 14.2337 0.770797
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −3.37228 −0.181558
\(346\) 0 0
\(347\) 6.62772 0.355795 0.177897 0.984049i \(-0.443071\pi\)
0.177897 + 0.984049i \(0.443071\pi\)
\(348\) 0 0
\(349\) −31.4891 −1.68557 −0.842787 0.538247i \(-0.819087\pi\)
−0.842787 + 0.538247i \(0.819087\pi\)
\(350\) 0 0
\(351\) 5.37228 0.286751
\(352\) 0 0
\(353\) 28.1168 1.49651 0.748254 0.663412i \(-0.230892\pi\)
0.748254 + 0.663412i \(0.230892\pi\)
\(354\) 0 0
\(355\) −4.23369 −0.224701
\(356\) 0 0
\(357\) −4.00000 −0.211702
\(358\) 0 0
\(359\) 24.8614 1.31213 0.656067 0.754702i \(-0.272219\pi\)
0.656067 + 0.754702i \(0.272219\pi\)
\(360\) 0 0
\(361\) −16.3505 −0.860554
\(362\) 0 0
\(363\) −5.37228 −0.281972
\(364\) 0 0
\(365\) 40.4674 2.11816
\(366\) 0 0
\(367\) −3.51087 −0.183266 −0.0916331 0.995793i \(-0.529209\pi\)
−0.0916331 + 0.995793i \(0.529209\pi\)
\(368\) 0 0
\(369\) −5.74456 −0.299050
\(370\) 0 0
\(371\) −7.11684 −0.369488
\(372\) 0 0
\(373\) −19.4891 −1.00911 −0.504554 0.863380i \(-0.668343\pi\)
−0.504554 + 0.863380i \(0.668343\pi\)
\(374\) 0 0
\(375\) 4.62772 0.238974
\(376\) 0 0
\(377\) 28.8614 1.48644
\(378\) 0 0
\(379\) −14.8614 −0.763379 −0.381690 0.924291i \(-0.624658\pi\)
−0.381690 + 0.924291i \(0.624658\pi\)
\(380\) 0 0
\(381\) 1.00000 0.0512316
\(382\) 0 0
\(383\) −26.2337 −1.34048 −0.670239 0.742145i \(-0.733809\pi\)
−0.670239 + 0.742145i \(0.733809\pi\)
\(384\) 0 0
\(385\) −8.00000 −0.407718
\(386\) 0 0
\(387\) −3.37228 −0.171423
\(388\) 0 0
\(389\) 19.4891 0.988138 0.494069 0.869423i \(-0.335509\pi\)
0.494069 + 0.869423i \(0.335509\pi\)
\(390\) 0 0
\(391\) −4.00000 −0.202289
\(392\) 0 0
\(393\) 13.1168 0.661657
\(394\) 0 0
\(395\) −29.4891 −1.48376
\(396\) 0 0
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) 0 0
\(399\) 1.62772 0.0814879
\(400\) 0 0
\(401\) 13.1168 0.655024 0.327512 0.944847i \(-0.393790\pi\)
0.327512 + 0.944847i \(0.393790\pi\)
\(402\) 0 0
\(403\) 32.2337 1.60567
\(404\) 0 0
\(405\) 3.37228 0.167570
\(406\) 0 0
\(407\) −6.23369 −0.308992
\(408\) 0 0
\(409\) −18.2337 −0.901598 −0.450799 0.892626i \(-0.648861\pi\)
−0.450799 + 0.892626i \(0.648861\pi\)
\(410\) 0 0
\(411\) −4.48913 −0.221432
\(412\) 0 0
\(413\) 3.62772 0.178508
\(414\) 0 0
\(415\) 29.4891 1.44756
\(416\) 0 0
\(417\) 11.3723 0.556903
\(418\) 0 0
\(419\) 23.2554 1.13610 0.568051 0.822993i \(-0.307698\pi\)
0.568051 + 0.822993i \(0.307698\pi\)
\(420\) 0 0
\(421\) 20.6277 1.00533 0.502667 0.864480i \(-0.332352\pi\)
0.502667 + 0.864480i \(0.332352\pi\)
\(422\) 0 0
\(423\) −3.74456 −0.182067
\(424\) 0 0
\(425\) 25.4891 1.23640
\(426\) 0 0
\(427\) 8.37228 0.405163
\(428\) 0 0
\(429\) 12.7446 0.615313
\(430\) 0 0
\(431\) −15.5109 −0.747133 −0.373566 0.927603i \(-0.621865\pi\)
−0.373566 + 0.927603i \(0.621865\pi\)
\(432\) 0 0
\(433\) −28.4891 −1.36910 −0.684550 0.728966i \(-0.740001\pi\)
−0.684550 + 0.728966i \(0.740001\pi\)
\(434\) 0 0
\(435\) 18.1168 0.868636
\(436\) 0 0
\(437\) 1.62772 0.0778643
\(438\) 0 0
\(439\) −18.7446 −0.894629 −0.447315 0.894377i \(-0.647620\pi\)
−0.447315 + 0.894377i \(0.647620\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −22.3505 −1.06191 −0.530953 0.847401i \(-0.678166\pi\)
−0.530953 + 0.847401i \(0.678166\pi\)
\(444\) 0 0
\(445\) −25.2554 −1.19722
\(446\) 0 0
\(447\) 4.88316 0.230965
\(448\) 0 0
\(449\) −38.4674 −1.81539 −0.907694 0.419633i \(-0.862159\pi\)
−0.907694 + 0.419633i \(0.862159\pi\)
\(450\) 0 0
\(451\) −13.6277 −0.641704
\(452\) 0 0
\(453\) 18.4891 0.868695
\(454\) 0 0
\(455\) −18.1168 −0.849331
\(456\) 0 0
\(457\) −22.2337 −1.04005 −0.520024 0.854152i \(-0.674077\pi\)
−0.520024 + 0.854152i \(0.674077\pi\)
\(458\) 0 0
\(459\) 4.00000 0.186704
\(460\) 0 0
\(461\) 12.9783 0.604457 0.302229 0.953235i \(-0.402269\pi\)
0.302229 + 0.953235i \(0.402269\pi\)
\(462\) 0 0
\(463\) 25.2337 1.17271 0.586354 0.810055i \(-0.300563\pi\)
0.586354 + 0.810055i \(0.300563\pi\)
\(464\) 0 0
\(465\) 20.2337 0.938315
\(466\) 0 0
\(467\) −23.8832 −1.10518 −0.552590 0.833453i \(-0.686361\pi\)
−0.552590 + 0.833453i \(0.686361\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) 15.8614 0.730855
\(472\) 0 0
\(473\) −8.00000 −0.367840
\(474\) 0 0
\(475\) −10.3723 −0.475913
\(476\) 0 0
\(477\) 7.11684 0.325858
\(478\) 0 0
\(479\) −25.7228 −1.17531 −0.587653 0.809113i \(-0.699948\pi\)
−0.587653 + 0.809113i \(0.699948\pi\)
\(480\) 0 0
\(481\) −14.1168 −0.643673
\(482\) 0 0
\(483\) 1.00000 0.0455016
\(484\) 0 0
\(485\) −40.8614 −1.85542
\(486\) 0 0
\(487\) 0.627719 0.0284446 0.0142223 0.999899i \(-0.495473\pi\)
0.0142223 + 0.999899i \(0.495473\pi\)
\(488\) 0 0
\(489\) −2.37228 −0.107278
\(490\) 0 0
\(491\) 5.76631 0.260230 0.130115 0.991499i \(-0.458465\pi\)
0.130115 + 0.991499i \(0.458465\pi\)
\(492\) 0 0
\(493\) 21.4891 0.967822
\(494\) 0 0
\(495\) 8.00000 0.359573
\(496\) 0 0
\(497\) 1.25544 0.0563141
\(498\) 0 0
\(499\) 18.7446 0.839122 0.419561 0.907727i \(-0.362184\pi\)
0.419561 + 0.907727i \(0.362184\pi\)
\(500\) 0 0
\(501\) −19.1168 −0.854078
\(502\) 0 0
\(503\) 36.2337 1.61558 0.807790 0.589470i \(-0.200663\pi\)
0.807790 + 0.589470i \(0.200663\pi\)
\(504\) 0 0
\(505\) −17.2554 −0.767857
\(506\) 0 0
\(507\) 15.8614 0.704430
\(508\) 0 0
\(509\) 31.1168 1.37923 0.689615 0.724176i \(-0.257780\pi\)
0.689615 + 0.724176i \(0.257780\pi\)
\(510\) 0 0
\(511\) −12.0000 −0.530849
\(512\) 0 0
\(513\) −1.62772 −0.0718655
\(514\) 0 0
\(515\) −23.6060 −1.04020
\(516\) 0 0
\(517\) −8.88316 −0.390681
\(518\) 0 0
\(519\) 15.4891 0.679897
\(520\) 0 0
\(521\) −11.4891 −0.503348 −0.251674 0.967812i \(-0.580981\pi\)
−0.251674 + 0.967812i \(0.580981\pi\)
\(522\) 0 0
\(523\) 42.6060 1.86303 0.931514 0.363704i \(-0.118488\pi\)
0.931514 + 0.363704i \(0.118488\pi\)
\(524\) 0 0
\(525\) −6.37228 −0.278109
\(526\) 0 0
\(527\) 24.0000 1.04546
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −3.62772 −0.157430
\(532\) 0 0
\(533\) −30.8614 −1.33676
\(534\) 0 0
\(535\) −58.9783 −2.54985
\(536\) 0 0
\(537\) −2.11684 −0.0913486
\(538\) 0 0
\(539\) 2.37228 0.102181
\(540\) 0 0
\(541\) −30.0951 −1.29389 −0.646945 0.762537i \(-0.723954\pi\)
−0.646945 + 0.762537i \(0.723954\pi\)
\(542\) 0 0
\(543\) −6.00000 −0.257485
\(544\) 0 0
\(545\) 15.6060 0.668486
\(546\) 0 0
\(547\) −9.25544 −0.395734 −0.197867 0.980229i \(-0.563401\pi\)
−0.197867 + 0.980229i \(0.563401\pi\)
\(548\) 0 0
\(549\) −8.37228 −0.357320
\(550\) 0 0
\(551\) −8.74456 −0.372531
\(552\) 0 0
\(553\) 8.74456 0.371857
\(554\) 0 0
\(555\) −8.86141 −0.376146
\(556\) 0 0
\(557\) −38.2337 −1.62001 −0.810007 0.586421i \(-0.800537\pi\)
−0.810007 + 0.586421i \(0.800537\pi\)
\(558\) 0 0
\(559\) −18.1168 −0.766261
\(560\) 0 0
\(561\) 9.48913 0.400631
\(562\) 0 0
\(563\) 16.6277 0.700775 0.350387 0.936605i \(-0.386050\pi\)
0.350387 + 0.936605i \(0.386050\pi\)
\(564\) 0 0
\(565\) −45.0951 −1.89716
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) −1.00000 −0.0419222 −0.0209611 0.999780i \(-0.506673\pi\)
−0.0209611 + 0.999780i \(0.506673\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 0 0
\(573\) −15.8614 −0.662620
\(574\) 0 0
\(575\) −6.37228 −0.265743
\(576\) 0 0
\(577\) −35.4891 −1.47743 −0.738716 0.674017i \(-0.764567\pi\)
−0.738716 + 0.674017i \(0.764567\pi\)
\(578\) 0 0
\(579\) −7.23369 −0.300622
\(580\) 0 0
\(581\) −8.74456 −0.362786
\(582\) 0 0
\(583\) 16.8832 0.699229
\(584\) 0 0
\(585\) 18.1168 0.749039
\(586\) 0 0
\(587\) 37.3505 1.54162 0.770811 0.637064i \(-0.219851\pi\)
0.770811 + 0.637064i \(0.219851\pi\)
\(588\) 0 0
\(589\) −9.76631 −0.402414
\(590\) 0 0
\(591\) −18.8614 −0.775855
\(592\) 0 0
\(593\) 11.8832 0.487983 0.243991 0.969777i \(-0.421543\pi\)
0.243991 + 0.969777i \(0.421543\pi\)
\(594\) 0 0
\(595\) −13.4891 −0.553000
\(596\) 0 0
\(597\) −1.00000 −0.0409273
\(598\) 0 0
\(599\) −0.744563 −0.0304220 −0.0152110 0.999884i \(-0.504842\pi\)
−0.0152110 + 0.999884i \(0.504842\pi\)
\(600\) 0 0
\(601\) 30.9783 1.26363 0.631815 0.775120i \(-0.282310\pi\)
0.631815 + 0.775120i \(0.282310\pi\)
\(602\) 0 0
\(603\) 4.00000 0.162893
\(604\) 0 0
\(605\) −18.1168 −0.736554
\(606\) 0 0
\(607\) −5.25544 −0.213312 −0.106656 0.994296i \(-0.534014\pi\)
−0.106656 + 0.994296i \(0.534014\pi\)
\(608\) 0 0
\(609\) −5.37228 −0.217696
\(610\) 0 0
\(611\) −20.1168 −0.813840
\(612\) 0 0
\(613\) −32.3505 −1.30663 −0.653313 0.757088i \(-0.726621\pi\)
−0.653313 + 0.757088i \(0.726621\pi\)
\(614\) 0 0
\(615\) −19.3723 −0.781166
\(616\) 0 0
\(617\) 28.9783 1.16662 0.583310 0.812249i \(-0.301757\pi\)
0.583310 + 0.812249i \(0.301757\pi\)
\(618\) 0 0
\(619\) 21.7228 0.873114 0.436557 0.899677i \(-0.356198\pi\)
0.436557 + 0.899677i \(0.356198\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 7.48913 0.300045
\(624\) 0 0
\(625\) −16.2554 −0.650217
\(626\) 0 0
\(627\) −3.86141 −0.154210
\(628\) 0 0
\(629\) −10.5109 −0.419096
\(630\) 0 0
\(631\) 27.7228 1.10363 0.551814 0.833967i \(-0.313936\pi\)
0.551814 + 0.833967i \(0.313936\pi\)
\(632\) 0 0
\(633\) 5.62772 0.223682
\(634\) 0 0
\(635\) 3.37228 0.133825
\(636\) 0 0
\(637\) 5.37228 0.212858
\(638\) 0 0
\(639\) −1.25544 −0.0496643
\(640\) 0 0
\(641\) 9.51087 0.375657 0.187828 0.982202i \(-0.439855\pi\)
0.187828 + 0.982202i \(0.439855\pi\)
\(642\) 0 0
\(643\) −17.1168 −0.675022 −0.337511 0.941322i \(-0.609585\pi\)
−0.337511 + 0.941322i \(0.609585\pi\)
\(644\) 0 0
\(645\) −11.3723 −0.447783
\(646\) 0 0
\(647\) 20.4674 0.804656 0.402328 0.915496i \(-0.368201\pi\)
0.402328 + 0.915496i \(0.368201\pi\)
\(648\) 0 0
\(649\) −8.60597 −0.337814
\(650\) 0 0
\(651\) −6.00000 −0.235159
\(652\) 0 0
\(653\) −25.8832 −1.01289 −0.506443 0.862273i \(-0.669040\pi\)
−0.506443 + 0.862273i \(0.669040\pi\)
\(654\) 0 0
\(655\) 44.2337 1.72835
\(656\) 0 0
\(657\) 12.0000 0.468165
\(658\) 0 0
\(659\) −16.0000 −0.623272 −0.311636 0.950202i \(-0.600877\pi\)
−0.311636 + 0.950202i \(0.600877\pi\)
\(660\) 0 0
\(661\) −18.8832 −0.734470 −0.367235 0.930128i \(-0.619695\pi\)
−0.367235 + 0.930128i \(0.619695\pi\)
\(662\) 0 0
\(663\) 21.4891 0.834568
\(664\) 0 0
\(665\) 5.48913 0.212859
\(666\) 0 0
\(667\) −5.37228 −0.208016
\(668\) 0 0
\(669\) −14.0000 −0.541271
\(670\) 0 0
\(671\) −19.8614 −0.766741
\(672\) 0 0
\(673\) 15.7446 0.606908 0.303454 0.952846i \(-0.401860\pi\)
0.303454 + 0.952846i \(0.401860\pi\)
\(674\) 0 0
\(675\) 6.37228 0.245269
\(676\) 0 0
\(677\) 22.7446 0.874145 0.437072 0.899426i \(-0.356015\pi\)
0.437072 + 0.899426i \(0.356015\pi\)
\(678\) 0 0
\(679\) 12.1168 0.465002
\(680\) 0 0
\(681\) −8.11684 −0.311038
\(682\) 0 0
\(683\) 14.9783 0.573127 0.286563 0.958061i \(-0.407487\pi\)
0.286563 + 0.958061i \(0.407487\pi\)
\(684\) 0 0
\(685\) −15.1386 −0.578416
\(686\) 0 0
\(687\) 25.3505 0.967183
\(688\) 0 0
\(689\) 38.2337 1.45659
\(690\) 0 0
\(691\) 43.6060 1.65885 0.829425 0.558619i \(-0.188668\pi\)
0.829425 + 0.558619i \(0.188668\pi\)
\(692\) 0 0
\(693\) −2.37228 −0.0901155
\(694\) 0 0
\(695\) 38.3505 1.45472
\(696\) 0 0
\(697\) −22.9783 −0.870363
\(698\) 0 0
\(699\) 14.7446 0.557691
\(700\) 0 0
\(701\) −9.35053 −0.353165 −0.176582 0.984286i \(-0.556504\pi\)
−0.176582 + 0.984286i \(0.556504\pi\)
\(702\) 0 0
\(703\) 4.27719 0.161317
\(704\) 0 0
\(705\) −12.6277 −0.475587
\(706\) 0 0
\(707\) 5.11684 0.192439
\(708\) 0 0
\(709\) −19.4891 −0.731929 −0.365965 0.930629i \(-0.619261\pi\)
−0.365965 + 0.930629i \(0.619261\pi\)
\(710\) 0 0
\(711\) −8.74456 −0.327947
\(712\) 0 0
\(713\) −6.00000 −0.224702
\(714\) 0 0
\(715\) 42.9783 1.60730
\(716\) 0 0
\(717\) −13.4891 −0.503761
\(718\) 0 0
\(719\) 3.37228 0.125765 0.0628824 0.998021i \(-0.479971\pi\)
0.0628824 + 0.998021i \(0.479971\pi\)
\(720\) 0 0
\(721\) 7.00000 0.260694
\(722\) 0 0
\(723\) −0.489125 −0.0181908
\(724\) 0 0
\(725\) 34.2337 1.27141
\(726\) 0 0
\(727\) −33.3505 −1.23690 −0.618451 0.785823i \(-0.712240\pi\)
−0.618451 + 0.785823i \(0.712240\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −13.4891 −0.498913
\(732\) 0 0
\(733\) −45.2119 −1.66994 −0.834971 0.550295i \(-0.814515\pi\)
−0.834971 + 0.550295i \(0.814515\pi\)
\(734\) 0 0
\(735\) 3.37228 0.124388
\(736\) 0 0
\(737\) 9.48913 0.349536
\(738\) 0 0
\(739\) 13.2554 0.487609 0.243805 0.969824i \(-0.421604\pi\)
0.243805 + 0.969824i \(0.421604\pi\)
\(740\) 0 0
\(741\) −8.74456 −0.321240
\(742\) 0 0
\(743\) −13.6277 −0.499953 −0.249976 0.968252i \(-0.580423\pi\)
−0.249976 + 0.968252i \(0.580423\pi\)
\(744\) 0 0
\(745\) 16.4674 0.603318
\(746\) 0 0
\(747\) 8.74456 0.319947
\(748\) 0 0
\(749\) 17.4891 0.639039
\(750\) 0 0
\(751\) 22.9783 0.838488 0.419244 0.907874i \(-0.362295\pi\)
0.419244 + 0.907874i \(0.362295\pi\)
\(752\) 0 0
\(753\) 18.8614 0.687348
\(754\) 0 0
\(755\) 62.3505 2.26917
\(756\) 0 0
\(757\) −20.2337 −0.735406 −0.367703 0.929943i \(-0.619856\pi\)
−0.367703 + 0.929943i \(0.619856\pi\)
\(758\) 0 0
\(759\) −2.37228 −0.0861084
\(760\) 0 0
\(761\) 28.0951 1.01845 0.509223 0.860634i \(-0.329933\pi\)
0.509223 + 0.860634i \(0.329933\pi\)
\(762\) 0 0
\(763\) −4.62772 −0.167535
\(764\) 0 0
\(765\) 13.4891 0.487700
\(766\) 0 0
\(767\) −19.4891 −0.703712
\(768\) 0 0
\(769\) 20.3505 0.733859 0.366929 0.930249i \(-0.380409\pi\)
0.366929 + 0.930249i \(0.380409\pi\)
\(770\) 0 0
\(771\) 10.8832 0.391947
\(772\) 0 0
\(773\) −28.6277 −1.02967 −0.514834 0.857290i \(-0.672146\pi\)
−0.514834 + 0.857290i \(0.672146\pi\)
\(774\) 0 0
\(775\) 38.2337 1.37339
\(776\) 0 0
\(777\) 2.62772 0.0942689
\(778\) 0 0
\(779\) 9.35053 0.335018
\(780\) 0 0
\(781\) −2.97825 −0.106570
\(782\) 0 0
\(783\) 5.37228 0.191990
\(784\) 0 0
\(785\) 53.4891 1.90911
\(786\) 0 0
\(787\) −26.8397 −0.956730 −0.478365 0.878161i \(-0.658770\pi\)
−0.478365 + 0.878161i \(0.658770\pi\)
\(788\) 0 0
\(789\) −1.00000 −0.0356009
\(790\) 0 0
\(791\) 13.3723 0.475464
\(792\) 0 0
\(793\) −44.9783 −1.59722
\(794\) 0 0
\(795\) 24.0000 0.851192
\(796\) 0 0
\(797\) 29.6060 1.04870 0.524349 0.851504i \(-0.324309\pi\)
0.524349 + 0.851504i \(0.324309\pi\)
\(798\) 0 0
\(799\) −14.9783 −0.529892
\(800\) 0 0
\(801\) −7.48913 −0.264615
\(802\) 0 0
\(803\) 28.4674 1.00459
\(804\) 0 0
\(805\) 3.37228 0.118857
\(806\) 0 0
\(807\) 0.510875 0.0179836
\(808\) 0 0
\(809\) 23.2554 0.817618 0.408809 0.912620i \(-0.365944\pi\)
0.408809 + 0.912620i \(0.365944\pi\)
\(810\) 0 0
\(811\) −10.8614 −0.381396 −0.190698 0.981649i \(-0.561075\pi\)
−0.190698 + 0.981649i \(0.561075\pi\)
\(812\) 0 0
\(813\) 1.48913 0.0522259
\(814\) 0 0
\(815\) −8.00000 −0.280228
\(816\) 0 0
\(817\) 5.48913 0.192040
\(818\) 0 0
\(819\) −5.37228 −0.187723
\(820\) 0 0
\(821\) −44.2337 −1.54377 −0.771883 0.635764i \(-0.780685\pi\)
−0.771883 + 0.635764i \(0.780685\pi\)
\(822\) 0 0
\(823\) 55.2337 1.92533 0.962663 0.270704i \(-0.0872564\pi\)
0.962663 + 0.270704i \(0.0872564\pi\)
\(824\) 0 0
\(825\) 15.1168 0.526301
\(826\) 0 0
\(827\) 30.6060 1.06427 0.532137 0.846658i \(-0.321389\pi\)
0.532137 + 0.846658i \(0.321389\pi\)
\(828\) 0 0
\(829\) −26.4674 −0.919250 −0.459625 0.888113i \(-0.652016\pi\)
−0.459625 + 0.888113i \(0.652016\pi\)
\(830\) 0 0
\(831\) 0.883156 0.0306363
\(832\) 0 0
\(833\) 4.00000 0.138592
\(834\) 0 0
\(835\) −64.4674 −2.23099
\(836\) 0 0
\(837\) 6.00000 0.207390
\(838\) 0 0
\(839\) 42.0000 1.45000 0.725001 0.688748i \(-0.241839\pi\)
0.725001 + 0.688748i \(0.241839\pi\)
\(840\) 0 0
\(841\) −0.138593 −0.00477908
\(842\) 0 0
\(843\) 13.3723 0.460566
\(844\) 0 0
\(845\) 53.4891 1.84008
\(846\) 0 0
\(847\) 5.37228 0.184594
\(848\) 0 0
\(849\) −5.25544 −0.180366
\(850\) 0 0
\(851\) 2.62772 0.0900770
\(852\) 0 0
\(853\) 14.3505 0.491353 0.245676 0.969352i \(-0.420990\pi\)
0.245676 + 0.969352i \(0.420990\pi\)
\(854\) 0 0
\(855\) −5.48913 −0.187724
\(856\) 0 0
\(857\) −4.25544 −0.145363 −0.0726815 0.997355i \(-0.523156\pi\)
−0.0726815 + 0.997355i \(0.523156\pi\)
\(858\) 0 0
\(859\) −18.1168 −0.618139 −0.309069 0.951039i \(-0.600018\pi\)
−0.309069 + 0.951039i \(0.600018\pi\)
\(860\) 0 0
\(861\) 5.74456 0.195774
\(862\) 0 0
\(863\) −31.7228 −1.07986 −0.539929 0.841711i \(-0.681549\pi\)
−0.539929 + 0.841711i \(0.681549\pi\)
\(864\) 0 0
\(865\) 52.2337 1.77600
\(866\) 0 0
\(867\) −1.00000 −0.0339618
\(868\) 0 0
\(869\) −20.7446 −0.703711
\(870\) 0 0
\(871\) 21.4891 0.728131
\(872\) 0 0
\(873\) −12.1168 −0.410093
\(874\) 0 0
\(875\) −4.62772 −0.156445
\(876\) 0 0
\(877\) 1.35053 0.0456042 0.0228021 0.999740i \(-0.492741\pi\)
0.0228021 + 0.999740i \(0.492741\pi\)
\(878\) 0 0
\(879\) −12.2337 −0.412632
\(880\) 0 0
\(881\) −23.4891 −0.791369 −0.395684 0.918387i \(-0.629493\pi\)
−0.395684 + 0.918387i \(0.629493\pi\)
\(882\) 0 0
\(883\) −25.7228 −0.865642 −0.432821 0.901480i \(-0.642482\pi\)
−0.432821 + 0.901480i \(0.642482\pi\)
\(884\) 0 0
\(885\) −12.2337 −0.411231
\(886\) 0 0
\(887\) 45.9565 1.54307 0.771534 0.636188i \(-0.219490\pi\)
0.771534 + 0.636188i \(0.219490\pi\)
\(888\) 0 0
\(889\) −1.00000 −0.0335389
\(890\) 0 0
\(891\) 2.37228 0.0794744
\(892\) 0 0
\(893\) 6.09509 0.203965
\(894\) 0 0
\(895\) −7.13859 −0.238617
\(896\) 0 0
\(897\) −5.37228 −0.179375
\(898\) 0 0
\(899\) 32.2337 1.07505
\(900\) 0 0
\(901\) 28.4674 0.948386
\(902\) 0 0
\(903\) 3.37228 0.112222
\(904\) 0 0
\(905\) −20.2337 −0.672591
\(906\) 0 0
\(907\) 4.86141 0.161420 0.0807102 0.996738i \(-0.474281\pi\)
0.0807102 + 0.996738i \(0.474281\pi\)
\(908\) 0 0
\(909\) −5.11684 −0.169715
\(910\) 0 0
\(911\) −34.1168 −1.13034 −0.565171 0.824974i \(-0.691190\pi\)
−0.565171 + 0.824974i \(0.691190\pi\)
\(912\) 0 0
\(913\) 20.7446 0.686545
\(914\) 0 0
\(915\) −28.2337 −0.933377
\(916\) 0 0
\(917\) −13.1168 −0.433156
\(918\) 0 0
\(919\) 46.2337 1.52511 0.762554 0.646924i \(-0.223945\pi\)
0.762554 + 0.646924i \(0.223945\pi\)
\(920\) 0 0
\(921\) 20.6277 0.679706
\(922\) 0 0
\(923\) −6.74456 −0.222000
\(924\) 0 0
\(925\) −16.7446 −0.550558
\(926\) 0 0
\(927\) −7.00000 −0.229910
\(928\) 0 0
\(929\) −8.11684 −0.266305 −0.133153 0.991096i \(-0.542510\pi\)
−0.133153 + 0.991096i \(0.542510\pi\)
\(930\) 0 0
\(931\) −1.62772 −0.0533463
\(932\) 0 0
\(933\) −22.3723 −0.732436
\(934\) 0 0
\(935\) 32.0000 1.04651
\(936\) 0 0
\(937\) −32.2554 −1.05374 −0.526870 0.849946i \(-0.676634\pi\)
−0.526870 + 0.849946i \(0.676634\pi\)
\(938\) 0 0
\(939\) 32.0951 1.04738
\(940\) 0 0
\(941\) −9.09509 −0.296492 −0.148246 0.988951i \(-0.547363\pi\)
−0.148246 + 0.988951i \(0.547363\pi\)
\(942\) 0 0
\(943\) 5.74456 0.187069
\(944\) 0 0
\(945\) −3.37228 −0.109700
\(946\) 0 0
\(947\) 41.0951 1.33541 0.667706 0.744425i \(-0.267277\pi\)
0.667706 + 0.744425i \(0.267277\pi\)
\(948\) 0 0
\(949\) 64.4674 2.09270
\(950\) 0 0
\(951\) 28.1168 0.911751
\(952\) 0 0
\(953\) −3.62772 −0.117513 −0.0587567 0.998272i \(-0.518714\pi\)
−0.0587567 + 0.998272i \(0.518714\pi\)
\(954\) 0 0
\(955\) −53.4891 −1.73087
\(956\) 0 0
\(957\) 12.7446 0.411973
\(958\) 0 0
\(959\) 4.48913 0.144961
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 0 0
\(963\) −17.4891 −0.563579
\(964\) 0 0
\(965\) −24.3940 −0.785272
\(966\) 0 0
\(967\) −4.00000 −0.128631 −0.0643157 0.997930i \(-0.520486\pi\)
−0.0643157 + 0.997930i \(0.520486\pi\)
\(968\) 0 0
\(969\) −6.51087 −0.209159
\(970\) 0 0
\(971\) 62.2337 1.99717 0.998587 0.0531405i \(-0.0169231\pi\)
0.998587 + 0.0531405i \(0.0169231\pi\)
\(972\) 0 0
\(973\) −11.3723 −0.364579
\(974\) 0 0
\(975\) 34.2337 1.09636
\(976\) 0 0
\(977\) 15.5109 0.496237 0.248118 0.968730i \(-0.420188\pi\)
0.248118 + 0.968730i \(0.420188\pi\)
\(978\) 0 0
\(979\) −17.7663 −0.567814
\(980\) 0 0
\(981\) 4.62772 0.147752
\(982\) 0 0
\(983\) −19.4891 −0.621607 −0.310803 0.950474i \(-0.600598\pi\)
−0.310803 + 0.950474i \(0.600598\pi\)
\(984\) 0 0
\(985\) −63.6060 −2.02665
\(986\) 0 0
\(987\) 3.74456 0.119191
\(988\) 0 0
\(989\) 3.37228 0.107232
\(990\) 0 0
\(991\) −57.3505 −1.82180 −0.910900 0.412628i \(-0.864611\pi\)
−0.910900 + 0.412628i \(0.864611\pi\)
\(992\) 0 0
\(993\) −2.37228 −0.0752821
\(994\) 0 0
\(995\) −3.37228 −0.106909
\(996\) 0 0
\(997\) 4.23369 0.134082 0.0670411 0.997750i \(-0.478644\pi\)
0.0670411 + 0.997750i \(0.478644\pi\)
\(998\) 0 0
\(999\) −2.62772 −0.0831373
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 7728.2.a.bj.1.2 2
4.3 odd 2 3864.2.a.i.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3864.2.a.i.1.2 2 4.3 odd 2
7728.2.a.bj.1.2 2 1.1 even 1 trivial