Properties

Label 7728.2.a.cc.1.3
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
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] = [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: \(4\)
Coefficient field: 4.4.75645.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 13x^{2} + 22x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 1.3
Root \(1.20386\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.20386 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.20386 q^{5} +1.00000 q^{7} +1.00000 q^{9} -2.02422 q^{11} -3.90542 q^{13} +1.20386 q^{15} +4.70156 q^{17} +4.02422 q^{19} +1.00000 q^{21} +1.00000 q^{23} -3.55073 q^{25} +1.00000 q^{27} +5.10927 q^{29} +5.10927 q^{31} -2.02422 q^{33} +1.20386 q^{35} -2.70156 q^{37} -3.90542 q^{39} +1.61651 q^{41} +6.55073 q^{43} +1.20386 q^{45} -7.13349 q^{47} +1.00000 q^{49} +4.70156 q^{51} -7.69204 q^{53} -2.43687 q^{55} +4.02422 q^{57} +4.52651 q^{59} +4.58276 q^{61} +1.00000 q^{63} -4.70156 q^{65} +2.96626 q^{67} +1.00000 q^{69} +8.14302 q^{71} +5.34687 q^{73} -3.55073 q^{75} -2.02422 q^{77} +8.46396 q^{79} +1.00000 q^{81} -6.51240 q^{83} +5.66000 q^{85} +5.10927 q^{87} -2.31313 q^{89} -3.90542 q^{91} +5.10927 q^{93} +4.84458 q^{95} +4.46396 q^{97} -2.02422 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} + q^{5} + 4 q^{7} + 4 q^{9} + 7 q^{11} + q^{13} + q^{15} + 6 q^{17} + q^{19} + 4 q^{21} + 4 q^{23} + 7 q^{25} + 4 q^{27} + 7 q^{33} + q^{35} + 2 q^{37} + q^{39} - q^{41} + 5 q^{43} + q^{45} + 7 q^{47} + 4 q^{49} + 6 q^{51} + 4 q^{53} + 9 q^{55} + q^{57} + 12 q^{59} + 4 q^{61} + 4 q^{63} - 6 q^{65} + 5 q^{67} + 4 q^{69} + 19 q^{71} + 4 q^{73} + 7 q^{75} + 7 q^{77} + 18 q^{79} + 4 q^{81} + 20 q^{83} - 19 q^{85} + 15 q^{89} + q^{91} - 7 q^{95} + 2 q^{97} + 7 q^{99}+O(q^{100}) \) 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
\(5\) 1.20386 0.538381 0.269190 0.963087i \(-0.413244\pi\)
0.269190 + 0.963087i \(0.413244\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.02422 −0.610325 −0.305162 0.952300i \(-0.598711\pi\)
−0.305162 + 0.952300i \(0.598711\pi\)
\(12\) 0 0
\(13\) −3.90542 −1.08317 −0.541584 0.840647i \(-0.682175\pi\)
−0.541584 + 0.840647i \(0.682175\pi\)
\(14\) 0 0
\(15\) 1.20386 0.310834
\(16\) 0 0
\(17\) 4.70156 1.14030 0.570148 0.821542i \(-0.306886\pi\)
0.570148 + 0.821542i \(0.306886\pi\)
\(18\) 0 0
\(19\) 4.02422 0.923219 0.461609 0.887083i \(-0.347272\pi\)
0.461609 + 0.887083i \(0.347272\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −3.55073 −0.710146
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 5.10927 0.948768 0.474384 0.880318i \(-0.342671\pi\)
0.474384 + 0.880318i \(0.342671\pi\)
\(30\) 0 0
\(31\) 5.10927 0.917653 0.458826 0.888526i \(-0.348270\pi\)
0.458826 + 0.888526i \(0.348270\pi\)
\(32\) 0 0
\(33\) −2.02422 −0.352371
\(34\) 0 0
\(35\) 1.20386 0.203489
\(36\) 0 0
\(37\) −2.70156 −0.444134 −0.222067 0.975031i \(-0.571280\pi\)
−0.222067 + 0.975031i \(0.571280\pi\)
\(38\) 0 0
\(39\) −3.90542 −0.625367
\(40\) 0 0
\(41\) 1.61651 0.252456 0.126228 0.992001i \(-0.459713\pi\)
0.126228 + 0.992001i \(0.459713\pi\)
\(42\) 0 0
\(43\) 6.55073 0.998977 0.499488 0.866321i \(-0.333521\pi\)
0.499488 + 0.866321i \(0.333521\pi\)
\(44\) 0 0
\(45\) 1.20386 0.179460
\(46\) 0 0
\(47\) −7.13349 −1.04053 −0.520263 0.854006i \(-0.674166\pi\)
−0.520263 + 0.854006i \(0.674166\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 4.70156 0.658350
\(52\) 0 0
\(53\) −7.69204 −1.05658 −0.528291 0.849063i \(-0.677167\pi\)
−0.528291 + 0.849063i \(0.677167\pi\)
\(54\) 0 0
\(55\) −2.43687 −0.328587
\(56\) 0 0
\(57\) 4.02422 0.533021
\(58\) 0 0
\(59\) 4.52651 0.589302 0.294651 0.955605i \(-0.404797\pi\)
0.294651 + 0.955605i \(0.404797\pi\)
\(60\) 0 0
\(61\) 4.58276 0.586763 0.293381 0.955996i \(-0.405219\pi\)
0.293381 + 0.955996i \(0.405219\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) −4.70156 −0.583157
\(66\) 0 0
\(67\) 2.96626 0.362386 0.181193 0.983448i \(-0.442004\pi\)
0.181193 + 0.983448i \(0.442004\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 8.14302 0.966398 0.483199 0.875510i \(-0.339475\pi\)
0.483199 + 0.875510i \(0.339475\pi\)
\(72\) 0 0
\(73\) 5.34687 0.625804 0.312902 0.949785i \(-0.398699\pi\)
0.312902 + 0.949785i \(0.398699\pi\)
\(74\) 0 0
\(75\) −3.55073 −0.410003
\(76\) 0 0
\(77\) −2.02422 −0.230681
\(78\) 0 0
\(79\) 8.46396 0.952270 0.476135 0.879372i \(-0.342037\pi\)
0.476135 + 0.879372i \(0.342037\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −6.51240 −0.714829 −0.357414 0.933946i \(-0.616342\pi\)
−0.357414 + 0.933946i \(0.616342\pi\)
\(84\) 0 0
\(85\) 5.66000 0.613914
\(86\) 0 0
\(87\) 5.10927 0.547772
\(88\) 0 0
\(89\) −2.31313 −0.245191 −0.122596 0.992457i \(-0.539122\pi\)
−0.122596 + 0.992457i \(0.539122\pi\)
\(90\) 0 0
\(91\) −3.90542 −0.409399
\(92\) 0 0
\(93\) 5.10927 0.529807
\(94\) 0 0
\(95\) 4.84458 0.497043
\(96\) 0 0
\(97\) 4.46396 0.453247 0.226623 0.973982i \(-0.427231\pi\)
0.226623 + 0.973982i \(0.427231\pi\)
\(98\) 0 0
\(99\) −2.02422 −0.203442
\(100\) 0 0
\(101\) −8.87661 −0.883256 −0.441628 0.897198i \(-0.645599\pi\)
−0.441628 + 0.897198i \(0.645599\pi\)
\(102\) 0 0
\(103\) −8.08047 −0.796192 −0.398096 0.917344i \(-0.630329\pi\)
−0.398096 + 0.917344i \(0.630329\pi\)
\(104\) 0 0
\(105\) 1.20386 0.117484
\(106\) 0 0
\(107\) 5.73531 0.554453 0.277226 0.960805i \(-0.410585\pi\)
0.277226 + 0.960805i \(0.410585\pi\)
\(108\) 0 0
\(109\) −6.32094 −0.605437 −0.302718 0.953080i \(-0.597894\pi\)
−0.302718 + 0.953080i \(0.597894\pi\)
\(110\) 0 0
\(111\) −2.70156 −0.256421
\(112\) 0 0
\(113\) 2.84458 0.267596 0.133798 0.991009i \(-0.457283\pi\)
0.133798 + 0.991009i \(0.457283\pi\)
\(114\) 0 0
\(115\) 1.20386 0.112260
\(116\) 0 0
\(117\) −3.90542 −0.361056
\(118\) 0 0
\(119\) 4.70156 0.430991
\(120\) 0 0
\(121\) −6.90254 −0.627504
\(122\) 0 0
\(123\) 1.61651 0.145755
\(124\) 0 0
\(125\) −10.2938 −0.920710
\(126\) 0 0
\(127\) 12.2235 1.08466 0.542329 0.840166i \(-0.317542\pi\)
0.542329 + 0.840166i \(0.317542\pi\)
\(128\) 0 0
\(129\) 6.55073 0.576760
\(130\) 0 0
\(131\) 16.5444 1.44549 0.722747 0.691113i \(-0.242879\pi\)
0.722747 + 0.691113i \(0.242879\pi\)
\(132\) 0 0
\(133\) 4.02422 0.348944
\(134\) 0 0
\(135\) 1.20386 0.103611
\(136\) 0 0
\(137\) 11.5975 0.990837 0.495419 0.868654i \(-0.335015\pi\)
0.495419 + 0.868654i \(0.335015\pi\)
\(138\) 0 0
\(139\) −8.65542 −0.734143 −0.367071 0.930193i \(-0.619640\pi\)
−0.367071 + 0.930193i \(0.619640\pi\)
\(140\) 0 0
\(141\) −7.13349 −0.600748
\(142\) 0 0
\(143\) 7.90542 0.661084
\(144\) 0 0
\(145\) 6.15083 0.510799
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) 9.84287 0.806359 0.403180 0.915121i \(-0.367905\pi\)
0.403180 + 0.915121i \(0.367905\pi\)
\(150\) 0 0
\(151\) 18.4726 1.50328 0.751638 0.659576i \(-0.229264\pi\)
0.751638 + 0.659576i \(0.229264\pi\)
\(152\) 0 0
\(153\) 4.70156 0.380099
\(154\) 0 0
\(155\) 6.15083 0.494047
\(156\) 0 0
\(157\) 20.0614 1.60108 0.800538 0.599282i \(-0.204547\pi\)
0.800538 + 0.599282i \(0.204547\pi\)
\(158\) 0 0
\(159\) −7.69204 −0.610018
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) 7.23589 0.566759 0.283379 0.959008i \(-0.408544\pi\)
0.283379 + 0.959008i \(0.408544\pi\)
\(164\) 0 0
\(165\) −2.43687 −0.189710
\(166\) 0 0
\(167\) 17.1897 1.33018 0.665091 0.746762i \(-0.268393\pi\)
0.665091 + 0.746762i \(0.268393\pi\)
\(168\) 0 0
\(169\) 2.25229 0.173253
\(170\) 0 0
\(171\) 4.02422 0.307740
\(172\) 0 0
\(173\) 7.15771 0.544191 0.272095 0.962270i \(-0.412283\pi\)
0.272095 + 0.962270i \(0.412283\pi\)
\(174\) 0 0
\(175\) −3.55073 −0.268410
\(176\) 0 0
\(177\) 4.52651 0.340233
\(178\) 0 0
\(179\) −4.36938 −0.326583 −0.163291 0.986578i \(-0.552211\pi\)
−0.163291 + 0.986578i \(0.552211\pi\)
\(180\) 0 0
\(181\) −24.7309 −1.83824 −0.919118 0.393981i \(-0.871097\pi\)
−0.919118 + 0.393981i \(0.871097\pi\)
\(182\) 0 0
\(183\) 4.58276 0.338768
\(184\) 0 0
\(185\) −3.25229 −0.239113
\(186\) 0 0
\(187\) −9.51699 −0.695951
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 15.0118 1.08622 0.543108 0.839663i \(-0.317247\pi\)
0.543108 + 0.839663i \(0.317247\pi\)
\(192\) 0 0
\(193\) −3.08506 −0.222067 −0.111034 0.993817i \(-0.535416\pi\)
−0.111034 + 0.993817i \(0.535416\pi\)
\(194\) 0 0
\(195\) −4.70156 −0.336686
\(196\) 0 0
\(197\) 1.66782 0.118827 0.0594136 0.998233i \(-0.481077\pi\)
0.0594136 + 0.998233i \(0.481077\pi\)
\(198\) 0 0
\(199\) −0.764112 −0.0541664 −0.0270832 0.999633i \(-0.508622\pi\)
−0.0270832 + 0.999633i \(0.508622\pi\)
\(200\) 0 0
\(201\) 2.96626 0.209224
\(202\) 0 0
\(203\) 5.10927 0.358601
\(204\) 0 0
\(205\) 1.94604 0.135917
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −8.14589 −0.563463
\(210\) 0 0
\(211\) −3.95673 −0.272393 −0.136196 0.990682i \(-0.543488\pi\)
−0.136196 + 0.990682i \(0.543488\pi\)
\(212\) 0 0
\(213\) 8.14302 0.557950
\(214\) 0 0
\(215\) 7.88614 0.537830
\(216\) 0 0
\(217\) 5.10927 0.346840
\(218\) 0 0
\(219\) 5.34687 0.361308
\(220\) 0 0
\(221\) −18.3616 −1.23513
\(222\) 0 0
\(223\) 11.6600 0.780812 0.390406 0.920643i \(-0.372335\pi\)
0.390406 + 0.920643i \(0.372335\pi\)
\(224\) 0 0
\(225\) −3.55073 −0.236715
\(226\) 0 0
\(227\) 15.6600 1.03939 0.519696 0.854352i \(-0.326045\pi\)
0.519696 + 0.854352i \(0.326045\pi\)
\(228\) 0 0
\(229\) −20.8419 −1.37727 −0.688637 0.725106i \(-0.741791\pi\)
−0.688637 + 0.725106i \(0.741791\pi\)
\(230\) 0 0
\(231\) −2.02422 −0.133184
\(232\) 0 0
\(233\) 9.48989 0.621703 0.310852 0.950458i \(-0.399386\pi\)
0.310852 + 0.950458i \(0.399386\pi\)
\(234\) 0 0
\(235\) −8.58770 −0.560200
\(236\) 0 0
\(237\) 8.46396 0.549793
\(238\) 0 0
\(239\) −25.0925 −1.62310 −0.811550 0.584283i \(-0.801376\pi\)
−0.811550 + 0.584283i \(0.801376\pi\)
\(240\) 0 0
\(241\) 4.09170 0.263570 0.131785 0.991278i \(-0.457929\pi\)
0.131785 + 0.991278i \(0.457929\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 1.20386 0.0769116
\(246\) 0 0
\(247\) −15.7163 −1.00000
\(248\) 0 0
\(249\) −6.51240 −0.412707
\(250\) 0 0
\(251\) 21.8108 1.37669 0.688344 0.725385i \(-0.258338\pi\)
0.688344 + 0.725385i \(0.258338\pi\)
\(252\) 0 0
\(253\) −2.02422 −0.127261
\(254\) 0 0
\(255\) 5.66000 0.354443
\(256\) 0 0
\(257\) 10.3181 0.643623 0.321812 0.946804i \(-0.395708\pi\)
0.321812 + 0.946804i \(0.395708\pi\)
\(258\) 0 0
\(259\) −2.70156 −0.167867
\(260\) 0 0
\(261\) 5.10927 0.316256
\(262\) 0 0
\(263\) 29.8835 1.84269 0.921347 0.388740i \(-0.127090\pi\)
0.921347 + 0.388740i \(0.127090\pi\)
\(264\) 0 0
\(265\) −9.26011 −0.568844
\(266\) 0 0
\(267\) −2.31313 −0.141561
\(268\) 0 0
\(269\) −7.71625 −0.470468 −0.235234 0.971939i \(-0.575586\pi\)
−0.235234 + 0.971939i \(0.575586\pi\)
\(270\) 0 0
\(271\) 5.45937 0.331633 0.165817 0.986157i \(-0.446974\pi\)
0.165817 + 0.986157i \(0.446974\pi\)
\(272\) 0 0
\(273\) −3.90542 −0.236367
\(274\) 0 0
\(275\) 7.18745 0.433420
\(276\) 0 0
\(277\) 7.62661 0.458239 0.229119 0.973398i \(-0.426415\pi\)
0.229119 + 0.973398i \(0.426415\pi\)
\(278\) 0 0
\(279\) 5.10927 0.305884
\(280\) 0 0
\(281\) −20.5202 −1.22413 −0.612067 0.790806i \(-0.709662\pi\)
−0.612067 + 0.790806i \(0.709662\pi\)
\(282\) 0 0
\(283\) 1.31636 0.0782493 0.0391246 0.999234i \(-0.487543\pi\)
0.0391246 + 0.999234i \(0.487543\pi\)
\(284\) 0 0
\(285\) 4.84458 0.286968
\(286\) 0 0
\(287\) 1.61651 0.0954193
\(288\) 0 0
\(289\) 5.10469 0.300276
\(290\) 0 0
\(291\) 4.46396 0.261682
\(292\) 0 0
\(293\) −16.1545 −0.943755 −0.471878 0.881664i \(-0.656424\pi\)
−0.471878 + 0.881664i \(0.656424\pi\)
\(294\) 0 0
\(295\) 5.44927 0.317269
\(296\) 0 0
\(297\) −2.02422 −0.117457
\(298\) 0 0
\(299\) −3.90542 −0.225856
\(300\) 0 0
\(301\) 6.55073 0.377578
\(302\) 0 0
\(303\) −8.87661 −0.509948
\(304\) 0 0
\(305\) 5.51699 0.315902
\(306\) 0 0
\(307\) −1.23095 −0.0702541 −0.0351271 0.999383i \(-0.511184\pi\)
−0.0351271 + 0.999383i \(0.511184\pi\)
\(308\) 0 0
\(309\) −8.08047 −0.459682
\(310\) 0 0
\(311\) −27.2483 −1.54511 −0.772554 0.634949i \(-0.781021\pi\)
−0.772554 + 0.634949i \(0.781021\pi\)
\(312\) 0 0
\(313\) 1.27422 0.0720232 0.0360116 0.999351i \(-0.488535\pi\)
0.0360116 + 0.999351i \(0.488535\pi\)
\(314\) 0 0
\(315\) 1.20386 0.0678296
\(316\) 0 0
\(317\) −10.3694 −0.582402 −0.291201 0.956662i \(-0.594055\pi\)
−0.291201 + 0.956662i \(0.594055\pi\)
\(318\) 0 0
\(319\) −10.3423 −0.579057
\(320\) 0 0
\(321\) 5.73531 0.320114
\(322\) 0 0
\(323\) 18.9201 1.05274
\(324\) 0 0
\(325\) 13.8671 0.769208
\(326\) 0 0
\(327\) −6.32094 −0.349549
\(328\) 0 0
\(329\) −7.13349 −0.393282
\(330\) 0 0
\(331\) −4.12890 −0.226945 −0.113473 0.993541i \(-0.536197\pi\)
−0.113473 + 0.993541i \(0.536197\pi\)
\(332\) 0 0
\(333\) −2.70156 −0.148045
\(334\) 0 0
\(335\) 3.57095 0.195102
\(336\) 0 0
\(337\) −11.8864 −0.647492 −0.323746 0.946144i \(-0.604942\pi\)
−0.323746 + 0.946144i \(0.604942\pi\)
\(338\) 0 0
\(339\) 2.84458 0.154496
\(340\) 0 0
\(341\) −10.3423 −0.560066
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 1.20386 0.0648134
\(346\) 0 0
\(347\) 18.8717 1.01308 0.506542 0.862215i \(-0.330923\pi\)
0.506542 + 0.862215i \(0.330923\pi\)
\(348\) 0 0
\(349\) −16.0776 −0.860614 −0.430307 0.902683i \(-0.641595\pi\)
−0.430307 + 0.902683i \(0.641595\pi\)
\(350\) 0 0
\(351\) −3.90542 −0.208456
\(352\) 0 0
\(353\) 18.5124 0.985316 0.492658 0.870223i \(-0.336025\pi\)
0.492658 + 0.870223i \(0.336025\pi\)
\(354\) 0 0
\(355\) 9.80302 0.520290
\(356\) 0 0
\(357\) 4.70156 0.248833
\(358\) 0 0
\(359\) 13.1462 0.693832 0.346916 0.937896i \(-0.387229\pi\)
0.346916 + 0.937896i \(0.387229\pi\)
\(360\) 0 0
\(361\) −2.80567 −0.147667
\(362\) 0 0
\(363\) −6.90254 −0.362289
\(364\) 0 0
\(365\) 6.43687 0.336921
\(366\) 0 0
\(367\) −28.8013 −1.50342 −0.751708 0.659496i \(-0.770770\pi\)
−0.751708 + 0.659496i \(0.770770\pi\)
\(368\) 0 0
\(369\) 1.61651 0.0841519
\(370\) 0 0
\(371\) −7.69204 −0.399351
\(372\) 0 0
\(373\) −21.9830 −1.13824 −0.569119 0.822255i \(-0.692715\pi\)
−0.569119 + 0.822255i \(0.692715\pi\)
\(374\) 0 0
\(375\) −10.2938 −0.531572
\(376\) 0 0
\(377\) −19.9539 −1.02768
\(378\) 0 0
\(379\) 21.4516 1.10189 0.550946 0.834541i \(-0.314267\pi\)
0.550946 + 0.834541i \(0.314267\pi\)
\(380\) 0 0
\(381\) 12.2235 0.626228
\(382\) 0 0
\(383\) −19.9731 −1.02058 −0.510290 0.860003i \(-0.670462\pi\)
−0.510290 + 0.860003i \(0.670462\pi\)
\(384\) 0 0
\(385\) −2.43687 −0.124194
\(386\) 0 0
\(387\) 6.55073 0.332992
\(388\) 0 0
\(389\) 0.881549 0.0446963 0.0223482 0.999750i \(-0.492886\pi\)
0.0223482 + 0.999750i \(0.492886\pi\)
\(390\) 0 0
\(391\) 4.70156 0.237768
\(392\) 0 0
\(393\) 16.5444 0.834556
\(394\) 0 0
\(395\) 10.1894 0.512684
\(396\) 0 0
\(397\) 14.4561 0.725533 0.362767 0.931880i \(-0.381832\pi\)
0.362767 + 0.931880i \(0.381832\pi\)
\(398\) 0 0
\(399\) 4.02422 0.201463
\(400\) 0 0
\(401\) 5.72901 0.286093 0.143046 0.989716i \(-0.454310\pi\)
0.143046 + 0.989716i \(0.454310\pi\)
\(402\) 0 0
\(403\) −19.9539 −0.993972
\(404\) 0 0
\(405\) 1.20386 0.0598201
\(406\) 0 0
\(407\) 5.46855 0.271066
\(408\) 0 0
\(409\) 23.8187 1.17776 0.588878 0.808222i \(-0.299570\pi\)
0.588878 + 0.808222i \(0.299570\pi\)
\(410\) 0 0
\(411\) 11.5975 0.572060
\(412\) 0 0
\(413\) 4.52651 0.222735
\(414\) 0 0
\(415\) −7.83999 −0.384850
\(416\) 0 0
\(417\) −8.65542 −0.423858
\(418\) 0 0
\(419\) 23.3583 1.14113 0.570565 0.821253i \(-0.306724\pi\)
0.570565 + 0.821253i \(0.306724\pi\)
\(420\) 0 0
\(421\) −26.9302 −1.31250 −0.656249 0.754544i \(-0.727858\pi\)
−0.656249 + 0.754544i \(0.727858\pi\)
\(422\) 0 0
\(423\) −7.13349 −0.346842
\(424\) 0 0
\(425\) −16.6940 −0.809777
\(426\) 0 0
\(427\) 4.58276 0.221775
\(428\) 0 0
\(429\) 7.90542 0.381677
\(430\) 0 0
\(431\) −17.2765 −0.832180 −0.416090 0.909323i \(-0.636600\pi\)
−0.416090 + 0.909323i \(0.636600\pi\)
\(432\) 0 0
\(433\) −6.01298 −0.288965 −0.144483 0.989507i \(-0.546152\pi\)
−0.144483 + 0.989507i \(0.546152\pi\)
\(434\) 0 0
\(435\) 6.15083 0.294910
\(436\) 0 0
\(437\) 4.02422 0.192504
\(438\) 0 0
\(439\) −1.63291 −0.0779345 −0.0389673 0.999240i \(-0.512407\pi\)
−0.0389673 + 0.999240i \(0.512407\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −22.9031 −1.08816 −0.544080 0.839033i \(-0.683121\pi\)
−0.544080 + 0.839033i \(0.683121\pi\)
\(444\) 0 0
\(445\) −2.78468 −0.132006
\(446\) 0 0
\(447\) 9.84287 0.465552
\(448\) 0 0
\(449\) 15.4865 0.730852 0.365426 0.930840i \(-0.380923\pi\)
0.365426 + 0.930840i \(0.380923\pi\)
\(450\) 0 0
\(451\) −3.27216 −0.154080
\(452\) 0 0
\(453\) 18.4726 0.867916
\(454\) 0 0
\(455\) −4.70156 −0.220413
\(456\) 0 0
\(457\) −27.0925 −1.26733 −0.633667 0.773606i \(-0.718451\pi\)
−0.633667 + 0.773606i \(0.718451\pi\)
\(458\) 0 0
\(459\) 4.70156 0.219450
\(460\) 0 0
\(461\) −14.9022 −0.694064 −0.347032 0.937853i \(-0.612811\pi\)
−0.347032 + 0.937853i \(0.612811\pi\)
\(462\) 0 0
\(463\) −18.2519 −0.848240 −0.424120 0.905606i \(-0.639416\pi\)
−0.424120 + 0.905606i \(0.639416\pi\)
\(464\) 0 0
\(465\) 6.15083 0.285238
\(466\) 0 0
\(467\) −22.0216 −1.01904 −0.509518 0.860460i \(-0.670176\pi\)
−0.509518 + 0.860460i \(0.670176\pi\)
\(468\) 0 0
\(469\) 2.96626 0.136969
\(470\) 0 0
\(471\) 20.0614 0.924381
\(472\) 0 0
\(473\) −13.2601 −0.609700
\(474\) 0 0
\(475\) −14.2889 −0.655620
\(476\) 0 0
\(477\) −7.69204 −0.352194
\(478\) 0 0
\(479\) −10.3324 −0.472100 −0.236050 0.971741i \(-0.575853\pi\)
−0.236050 + 0.971741i \(0.575853\pi\)
\(480\) 0 0
\(481\) 10.5507 0.481072
\(482\) 0 0
\(483\) 1.00000 0.0455016
\(484\) 0 0
\(485\) 5.37397 0.244019
\(486\) 0 0
\(487\) 11.8480 0.536886 0.268443 0.963296i \(-0.413491\pi\)
0.268443 + 0.963296i \(0.413491\pi\)
\(488\) 0 0
\(489\) 7.23589 0.327218
\(490\) 0 0
\(491\) 17.4821 0.788955 0.394478 0.918906i \(-0.370926\pi\)
0.394478 + 0.918906i \(0.370926\pi\)
\(492\) 0 0
\(493\) 24.0216 1.08188
\(494\) 0 0
\(495\) −2.43687 −0.109529
\(496\) 0 0
\(497\) 8.14302 0.365264
\(498\) 0 0
\(499\) 2.32437 0.104053 0.0520265 0.998646i \(-0.483432\pi\)
0.0520265 + 0.998646i \(0.483432\pi\)
\(500\) 0 0
\(501\) 17.1897 0.767981
\(502\) 0 0
\(503\) 24.0135 1.07071 0.535355 0.844627i \(-0.320178\pi\)
0.535355 + 0.844627i \(0.320178\pi\)
\(504\) 0 0
\(505\) −10.6862 −0.475528
\(506\) 0 0
\(507\) 2.25229 0.100028
\(508\) 0 0
\(509\) 4.26963 0.189248 0.0946240 0.995513i \(-0.469835\pi\)
0.0946240 + 0.995513i \(0.469835\pi\)
\(510\) 0 0
\(511\) 5.34687 0.236532
\(512\) 0 0
\(513\) 4.02422 0.177674
\(514\) 0 0
\(515\) −9.72772 −0.428655
\(516\) 0 0
\(517\) 14.4397 0.635059
\(518\) 0 0
\(519\) 7.15771 0.314189
\(520\) 0 0
\(521\) 7.47636 0.327545 0.163773 0.986498i \(-0.447634\pi\)
0.163773 + 0.986498i \(0.447634\pi\)
\(522\) 0 0
\(523\) −15.2742 −0.667895 −0.333947 0.942592i \(-0.608381\pi\)
−0.333947 + 0.942592i \(0.608381\pi\)
\(524\) 0 0
\(525\) −3.55073 −0.154967
\(526\) 0 0
\(527\) 24.0216 1.04640
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 4.52651 0.196434
\(532\) 0 0
\(533\) −6.31313 −0.273452
\(534\) 0 0
\(535\) 6.90448 0.298507
\(536\) 0 0
\(537\) −4.36938 −0.188553
\(538\) 0 0
\(539\) −2.02422 −0.0871892
\(540\) 0 0
\(541\) 21.4974 0.924243 0.462122 0.886817i \(-0.347088\pi\)
0.462122 + 0.886817i \(0.347088\pi\)
\(542\) 0 0
\(543\) −24.7309 −1.06131
\(544\) 0 0
\(545\) −7.60951 −0.325956
\(546\) 0 0
\(547\) 1.84917 0.0790647 0.0395324 0.999218i \(-0.487413\pi\)
0.0395324 + 0.999218i \(0.487413\pi\)
\(548\) 0 0
\(549\) 4.58276 0.195588
\(550\) 0 0
\(551\) 20.5608 0.875921
\(552\) 0 0
\(553\) 8.46396 0.359924
\(554\) 0 0
\(555\) −3.25229 −0.138052
\(556\) 0 0
\(557\) 38.3448 1.62472 0.812361 0.583155i \(-0.198182\pi\)
0.812361 + 0.583155i \(0.198182\pi\)
\(558\) 0 0
\(559\) −25.5833 −1.08206
\(560\) 0 0
\(561\) −9.51699 −0.401808
\(562\) 0 0
\(563\) −30.4394 −1.28287 −0.641434 0.767179i \(-0.721660\pi\)
−0.641434 + 0.767179i \(0.721660\pi\)
\(564\) 0 0
\(565\) 3.42447 0.144068
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −23.0084 −0.964562 −0.482281 0.876017i \(-0.660192\pi\)
−0.482281 + 0.876017i \(0.660192\pi\)
\(570\) 0 0
\(571\) −42.4876 −1.77805 −0.889025 0.457858i \(-0.848617\pi\)
−0.889025 + 0.457858i \(0.848617\pi\)
\(572\) 0 0
\(573\) 15.0118 0.627128
\(574\) 0 0
\(575\) −3.55073 −0.148076
\(576\) 0 0
\(577\) −12.4279 −0.517381 −0.258691 0.965960i \(-0.583291\pi\)
−0.258691 + 0.965960i \(0.583291\pi\)
\(578\) 0 0
\(579\) −3.08506 −0.128211
\(580\) 0 0
\(581\) −6.51240 −0.270180
\(582\) 0 0
\(583\) 15.5704 0.644858
\(584\) 0 0
\(585\) −4.70156 −0.194386
\(586\) 0 0
\(587\) 45.2384 1.86719 0.933594 0.358331i \(-0.116654\pi\)
0.933594 + 0.358331i \(0.116654\pi\)
\(588\) 0 0
\(589\) 20.5608 0.847194
\(590\) 0 0
\(591\) 1.66782 0.0686049
\(592\) 0 0
\(593\) 7.00459 0.287644 0.143822 0.989604i \(-0.454061\pi\)
0.143822 + 0.989604i \(0.454061\pi\)
\(594\) 0 0
\(595\) 5.66000 0.232038
\(596\) 0 0
\(597\) −0.764112 −0.0312730
\(598\) 0 0
\(599\) 2.42563 0.0991086 0.0495543 0.998771i \(-0.484220\pi\)
0.0495543 + 0.998771i \(0.484220\pi\)
\(600\) 0 0
\(601\) 31.3301 1.27798 0.638991 0.769214i \(-0.279352\pi\)
0.638991 + 0.769214i \(0.279352\pi\)
\(602\) 0 0
\(603\) 2.96626 0.120795
\(604\) 0 0
\(605\) −8.30967 −0.337836
\(606\) 0 0
\(607\) −0.517922 −0.0210218 −0.0105109 0.999945i \(-0.503346\pi\)
−0.0105109 + 0.999945i \(0.503346\pi\)
\(608\) 0 0
\(609\) 5.10927 0.207038
\(610\) 0 0
\(611\) 27.8593 1.12707
\(612\) 0 0
\(613\) 14.8526 0.599892 0.299946 0.953956i \(-0.403031\pi\)
0.299946 + 0.953956i \(0.403031\pi\)
\(614\) 0 0
\(615\) 1.94604 0.0784719
\(616\) 0 0
\(617\) −1.32759 −0.0534469 −0.0267234 0.999643i \(-0.508507\pi\)
−0.0267234 + 0.999643i \(0.508507\pi\)
\(618\) 0 0
\(619\) −16.5992 −0.667177 −0.333588 0.942719i \(-0.608260\pi\)
−0.333588 + 0.942719i \(0.608260\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) −2.31313 −0.0926736
\(624\) 0 0
\(625\) 5.36134 0.214453
\(626\) 0 0
\(627\) −8.14589 −0.325316
\(628\) 0 0
\(629\) −12.7016 −0.506444
\(630\) 0 0
\(631\) 34.2500 1.36347 0.681735 0.731599i \(-0.261226\pi\)
0.681735 + 0.731599i \(0.261226\pi\)
\(632\) 0 0
\(633\) −3.95673 −0.157266
\(634\) 0 0
\(635\) 14.7153 0.583960
\(636\) 0 0
\(637\) −3.90542 −0.154738
\(638\) 0 0
\(639\) 8.14302 0.322133
\(640\) 0 0
\(641\) 12.0919 0.477603 0.238801 0.971068i \(-0.423246\pi\)
0.238801 + 0.971068i \(0.423246\pi\)
\(642\) 0 0
\(643\) −11.8826 −0.468603 −0.234301 0.972164i \(-0.575280\pi\)
−0.234301 + 0.972164i \(0.575280\pi\)
\(644\) 0 0
\(645\) 7.88614 0.310516
\(646\) 0 0
\(647\) 8.93803 0.351390 0.175695 0.984445i \(-0.443783\pi\)
0.175695 + 0.984445i \(0.443783\pi\)
\(648\) 0 0
\(649\) −9.16265 −0.359665
\(650\) 0 0
\(651\) 5.10927 0.200248
\(652\) 0 0
\(653\) −9.45133 −0.369859 −0.184930 0.982752i \(-0.559206\pi\)
−0.184930 + 0.982752i \(0.559206\pi\)
\(654\) 0 0
\(655\) 19.9171 0.778226
\(656\) 0 0
\(657\) 5.34687 0.208601
\(658\) 0 0
\(659\) 9.85721 0.383982 0.191991 0.981397i \(-0.438505\pi\)
0.191991 + 0.981397i \(0.438505\pi\)
\(660\) 0 0
\(661\) 14.0917 0.548104 0.274052 0.961715i \(-0.411636\pi\)
0.274052 + 0.961715i \(0.411636\pi\)
\(662\) 0 0
\(663\) −18.3616 −0.713104
\(664\) 0 0
\(665\) 4.84458 0.187865
\(666\) 0 0
\(667\) 5.10927 0.197832
\(668\) 0 0
\(669\) 11.6600 0.450802
\(670\) 0 0
\(671\) −9.27651 −0.358116
\(672\) 0 0
\(673\) 21.0674 0.812087 0.406044 0.913854i \(-0.366908\pi\)
0.406044 + 0.913854i \(0.366908\pi\)
\(674\) 0 0
\(675\) −3.55073 −0.136668
\(676\) 0 0
\(677\) 4.88520 0.187754 0.0938768 0.995584i \(-0.470074\pi\)
0.0938768 + 0.995584i \(0.470074\pi\)
\(678\) 0 0
\(679\) 4.46396 0.171311
\(680\) 0 0
\(681\) 15.6600 0.600093
\(682\) 0 0
\(683\) −0.604690 −0.0231378 −0.0115689 0.999933i \(-0.503683\pi\)
−0.0115689 + 0.999933i \(0.503683\pi\)
\(684\) 0 0
\(685\) 13.9617 0.533448
\(686\) 0 0
\(687\) −20.8419 −0.795169
\(688\) 0 0
\(689\) 30.0406 1.14446
\(690\) 0 0
\(691\) 1.84781 0.0702938 0.0351469 0.999382i \(-0.488810\pi\)
0.0351469 + 0.999382i \(0.488810\pi\)
\(692\) 0 0
\(693\) −2.02422 −0.0768937
\(694\) 0 0
\(695\) −10.4199 −0.395248
\(696\) 0 0
\(697\) 7.60010 0.287874
\(698\) 0 0
\(699\) 9.48989 0.358941
\(700\) 0 0
\(701\) 36.5403 1.38011 0.690053 0.723758i \(-0.257587\pi\)
0.690053 + 0.723758i \(0.257587\pi\)
\(702\) 0 0
\(703\) −10.8717 −0.410033
\(704\) 0 0
\(705\) −8.58770 −0.323431
\(706\) 0 0
\(707\) −8.87661 −0.333839
\(708\) 0 0
\(709\) −46.3524 −1.74080 −0.870400 0.492345i \(-0.836140\pi\)
−0.870400 + 0.492345i \(0.836140\pi\)
\(710\) 0 0
\(711\) 8.46396 0.317423
\(712\) 0 0
\(713\) 5.10927 0.191344
\(714\) 0 0
\(715\) 9.51699 0.355915
\(716\) 0 0
\(717\) −25.0925 −0.937097
\(718\) 0 0
\(719\) −41.3088 −1.54056 −0.770279 0.637707i \(-0.779883\pi\)
−0.770279 + 0.637707i \(0.779883\pi\)
\(720\) 0 0
\(721\) −8.08047 −0.300932
\(722\) 0 0
\(723\) 4.09170 0.152172
\(724\) 0 0
\(725\) −18.1417 −0.673764
\(726\) 0 0
\(727\) 19.5398 0.724693 0.362346 0.932044i \(-0.381976\pi\)
0.362346 + 0.932044i \(0.381976\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 30.7987 1.13913
\(732\) 0 0
\(733\) 11.7048 0.432326 0.216163 0.976357i \(-0.430646\pi\)
0.216163 + 0.976357i \(0.430646\pi\)
\(734\) 0 0
\(735\) 1.20386 0.0444049
\(736\) 0 0
\(737\) −6.00435 −0.221173
\(738\) 0 0
\(739\) −50.6184 −1.86203 −0.931015 0.364981i \(-0.881075\pi\)
−0.931015 + 0.364981i \(0.881075\pi\)
\(740\) 0 0
\(741\) −15.7163 −0.577351
\(742\) 0 0
\(743\) 1.68216 0.0617126 0.0308563 0.999524i \(-0.490177\pi\)
0.0308563 + 0.999524i \(0.490177\pi\)
\(744\) 0 0
\(745\) 11.8494 0.434128
\(746\) 0 0
\(747\) −6.51240 −0.238276
\(748\) 0 0
\(749\) 5.73531 0.209563
\(750\) 0 0
\(751\) −38.9125 −1.41994 −0.709969 0.704233i \(-0.751291\pi\)
−0.709969 + 0.704233i \(0.751291\pi\)
\(752\) 0 0
\(753\) 21.8108 0.794831
\(754\) 0 0
\(755\) 22.2383 0.809334
\(756\) 0 0
\(757\) 13.5719 0.493278 0.246639 0.969107i \(-0.420674\pi\)
0.246639 + 0.969107i \(0.420674\pi\)
\(758\) 0 0
\(759\) −2.02422 −0.0734745
\(760\) 0 0
\(761\) 28.3959 1.02935 0.514675 0.857385i \(-0.327913\pi\)
0.514675 + 0.857385i \(0.327913\pi\)
\(762\) 0 0
\(763\) −6.32094 −0.228834
\(764\) 0 0
\(765\) 5.66000 0.204638
\(766\) 0 0
\(767\) −17.6779 −0.638313
\(768\) 0 0
\(769\) 16.1026 0.580676 0.290338 0.956924i \(-0.406232\pi\)
0.290338 + 0.956924i \(0.406232\pi\)
\(770\) 0 0
\(771\) 10.3181 0.371596
\(772\) 0 0
\(773\) 21.0452 0.756944 0.378472 0.925613i \(-0.376450\pi\)
0.378472 + 0.925613i \(0.376450\pi\)
\(774\) 0 0
\(775\) −18.1417 −0.651668
\(776\) 0 0
\(777\) −2.70156 −0.0969180
\(778\) 0 0
\(779\) 6.50517 0.233072
\(780\) 0 0
\(781\) −16.4832 −0.589817
\(782\) 0 0
\(783\) 5.10927 0.182591
\(784\) 0 0
\(785\) 24.1511 0.861988
\(786\) 0 0
\(787\) −35.4487 −1.26361 −0.631805 0.775128i \(-0.717685\pi\)
−0.631805 + 0.775128i \(0.717685\pi\)
\(788\) 0 0
\(789\) 29.8835 1.06388
\(790\) 0 0
\(791\) 2.84458 0.101142
\(792\) 0 0
\(793\) −17.8976 −0.635563
\(794\) 0 0
\(795\) −9.26011 −0.328422
\(796\) 0 0
\(797\) −20.6327 −0.730848 −0.365424 0.930841i \(-0.619076\pi\)
−0.365424 + 0.930841i \(0.619076\pi\)
\(798\) 0 0
\(799\) −33.5386 −1.18651
\(800\) 0 0
\(801\) −2.31313 −0.0817305
\(802\) 0 0
\(803\) −10.8232 −0.381944
\(804\) 0 0
\(805\) 1.20386 0.0424304
\(806\) 0 0
\(807\) −7.71625 −0.271625
\(808\) 0 0
\(809\) 20.8910 0.734487 0.367243 0.930125i \(-0.380302\pi\)
0.367243 + 0.930125i \(0.380302\pi\)
\(810\) 0 0
\(811\) −10.1715 −0.357169 −0.178584 0.983925i \(-0.557152\pi\)
−0.178584 + 0.983925i \(0.557152\pi\)
\(812\) 0 0
\(813\) 5.45937 0.191469
\(814\) 0 0
\(815\) 8.71097 0.305132
\(816\) 0 0
\(817\) 26.3616 0.922274
\(818\) 0 0
\(819\) −3.90542 −0.136466
\(820\) 0 0
\(821\) 5.04385 0.176032 0.0880158 0.996119i \(-0.471947\pi\)
0.0880158 + 0.996119i \(0.471947\pi\)
\(822\) 0 0
\(823\) −35.9205 −1.25211 −0.626054 0.779779i \(-0.715331\pi\)
−0.626054 + 0.779779i \(0.715331\pi\)
\(824\) 0 0
\(825\) 7.18745 0.250235
\(826\) 0 0
\(827\) 28.6078 0.994789 0.497395 0.867524i \(-0.334290\pi\)
0.497395 + 0.867524i \(0.334290\pi\)
\(828\) 0 0
\(829\) −45.1104 −1.56675 −0.783375 0.621549i \(-0.786504\pi\)
−0.783375 + 0.621549i \(0.786504\pi\)
\(830\) 0 0
\(831\) 7.62661 0.264564
\(832\) 0 0
\(833\) 4.70156 0.162899
\(834\) 0 0
\(835\) 20.6940 0.716145
\(836\) 0 0
\(837\) 5.10927 0.176602
\(838\) 0 0
\(839\) −19.1869 −0.662404 −0.331202 0.943560i \(-0.607454\pi\)
−0.331202 + 0.943560i \(0.607454\pi\)
\(840\) 0 0
\(841\) −2.89531 −0.0998384
\(842\) 0 0
\(843\) −20.5202 −0.706754
\(844\) 0 0
\(845\) 2.71144 0.0932762
\(846\) 0 0
\(847\) −6.90254 −0.237174
\(848\) 0 0
\(849\) 1.31636 0.0451772
\(850\) 0 0
\(851\) −2.70156 −0.0926084
\(852\) 0 0
\(853\) 33.1759 1.13592 0.567960 0.823056i \(-0.307733\pi\)
0.567960 + 0.823056i \(0.307733\pi\)
\(854\) 0 0
\(855\) 4.84458 0.165681
\(856\) 0 0
\(857\) 40.2735 1.37572 0.687859 0.725845i \(-0.258551\pi\)
0.687859 + 0.725845i \(0.258551\pi\)
\(858\) 0 0
\(859\) 16.4527 0.561360 0.280680 0.959801i \(-0.409440\pi\)
0.280680 + 0.959801i \(0.409440\pi\)
\(860\) 0 0
\(861\) 1.61651 0.0550904
\(862\) 0 0
\(863\) −8.71280 −0.296587 −0.148294 0.988943i \(-0.547378\pi\)
−0.148294 + 0.988943i \(0.547378\pi\)
\(864\) 0 0
\(865\) 8.61685 0.292982
\(866\) 0 0
\(867\) 5.10469 0.173364
\(868\) 0 0
\(869\) −17.1329 −0.581194
\(870\) 0 0
\(871\) −11.5845 −0.392525
\(872\) 0 0
\(873\) 4.46396 0.151082
\(874\) 0 0
\(875\) −10.2938 −0.347996
\(876\) 0 0
\(877\) 1.96797 0.0664536 0.0332268 0.999448i \(-0.489422\pi\)
0.0332268 + 0.999448i \(0.489422\pi\)
\(878\) 0 0
\(879\) −16.1545 −0.544877
\(880\) 0 0
\(881\) 40.8684 1.37689 0.688446 0.725287i \(-0.258293\pi\)
0.688446 + 0.725287i \(0.258293\pi\)
\(882\) 0 0
\(883\) −20.2243 −0.680601 −0.340300 0.940317i \(-0.610529\pi\)
−0.340300 + 0.940317i \(0.610529\pi\)
\(884\) 0 0
\(885\) 5.44927 0.183175
\(886\) 0 0
\(887\) −43.1038 −1.44728 −0.723641 0.690176i \(-0.757533\pi\)
−0.723641 + 0.690176i \(0.757533\pi\)
\(888\) 0 0
\(889\) 12.2235 0.409963
\(890\) 0 0
\(891\) −2.02422 −0.0678139
\(892\) 0 0
\(893\) −28.7067 −0.960634
\(894\) 0 0
\(895\) −5.26011 −0.175826
\(896\) 0 0
\(897\) −3.90542 −0.130398
\(898\) 0 0
\(899\) 26.1047 0.870640
\(900\) 0 0
\(901\) −36.1646 −1.20482
\(902\) 0 0
\(903\) 6.55073 0.217995
\(904\) 0 0
\(905\) −29.7725 −0.989672
\(906\) 0 0
\(907\) −5.08793 −0.168942 −0.0844710 0.996426i \(-0.526920\pi\)
−0.0844710 + 0.996426i \(0.526920\pi\)
\(908\) 0 0
\(909\) −8.87661 −0.294419
\(910\) 0 0
\(911\) 6.27686 0.207962 0.103981 0.994579i \(-0.466842\pi\)
0.103981 + 0.994579i \(0.466842\pi\)
\(912\) 0 0
\(913\) 13.1825 0.436278
\(914\) 0 0
\(915\) 5.51699 0.182386
\(916\) 0 0
\(917\) 16.5444 0.546345
\(918\) 0 0
\(919\) −43.1779 −1.42431 −0.712154 0.702023i \(-0.752280\pi\)
−0.712154 + 0.702023i \(0.752280\pi\)
\(920\) 0 0
\(921\) −1.23095 −0.0405612
\(922\) 0 0
\(923\) −31.8019 −1.04677
\(924\) 0 0
\(925\) 9.59252 0.315400
\(926\) 0 0
\(927\) −8.08047 −0.265397
\(928\) 0 0
\(929\) −38.7142 −1.27017 −0.635086 0.772442i \(-0.719035\pi\)
−0.635086 + 0.772442i \(0.719035\pi\)
\(930\) 0 0
\(931\) 4.02422 0.131888
\(932\) 0 0
\(933\) −27.2483 −0.892069
\(934\) 0 0
\(935\) −11.4571 −0.374687
\(936\) 0 0
\(937\) 51.1459 1.67086 0.835432 0.549594i \(-0.185218\pi\)
0.835432 + 0.549594i \(0.185218\pi\)
\(938\) 0 0
\(939\) 1.27422 0.0415826
\(940\) 0 0
\(941\) −47.8730 −1.56062 −0.780308 0.625395i \(-0.784938\pi\)
−0.780308 + 0.625395i \(0.784938\pi\)
\(942\) 0 0
\(943\) 1.61651 0.0526407
\(944\) 0 0
\(945\) 1.20386 0.0391614
\(946\) 0 0
\(947\) −15.7342 −0.511292 −0.255646 0.966770i \(-0.582288\pi\)
−0.255646 + 0.966770i \(0.582288\pi\)
\(948\) 0 0
\(949\) −20.8818 −0.677851
\(950\) 0 0
\(951\) −10.3694 −0.336250
\(952\) 0 0
\(953\) −28.4524 −0.921663 −0.460831 0.887488i \(-0.652449\pi\)
−0.460831 + 0.887488i \(0.652449\pi\)
\(954\) 0 0
\(955\) 18.0721 0.584798
\(956\) 0 0
\(957\) −10.3423 −0.334319
\(958\) 0 0
\(959\) 11.5975 0.374501
\(960\) 0 0
\(961\) −4.89531 −0.157913
\(962\) 0 0
\(963\) 5.73531 0.184818
\(964\) 0 0
\(965\) −3.71396 −0.119557
\(966\) 0 0
\(967\) −56.7433 −1.82474 −0.912371 0.409363i \(-0.865751\pi\)
−0.912371 + 0.409363i \(0.865751\pi\)
\(968\) 0 0
\(969\) 18.9201 0.607802
\(970\) 0 0
\(971\) 33.9001 1.08791 0.543953 0.839115i \(-0.316927\pi\)
0.543953 + 0.839115i \(0.316927\pi\)
\(972\) 0 0
\(973\) −8.65542 −0.277480
\(974\) 0 0
\(975\) 13.8671 0.444102
\(976\) 0 0
\(977\) 4.43745 0.141967 0.0709834 0.997477i \(-0.477386\pi\)
0.0709834 + 0.997477i \(0.477386\pi\)
\(978\) 0 0
\(979\) 4.68228 0.149646
\(980\) 0 0
\(981\) −6.32094 −0.201812
\(982\) 0 0
\(983\) −56.0517 −1.78777 −0.893885 0.448296i \(-0.852031\pi\)
−0.893885 + 0.448296i \(0.852031\pi\)
\(984\) 0 0
\(985\) 2.00781 0.0639743
\(986\) 0 0
\(987\) −7.13349 −0.227062
\(988\) 0 0
\(989\) 6.55073 0.208301
\(990\) 0 0
\(991\) 25.4218 0.807551 0.403775 0.914858i \(-0.367698\pi\)
0.403775 + 0.914858i \(0.367698\pi\)
\(992\) 0 0
\(993\) −4.12890 −0.131027
\(994\) 0 0
\(995\) −0.919881 −0.0291622
\(996\) 0 0
\(997\) −36.6474 −1.16063 −0.580317 0.814391i \(-0.697071\pi\)
−0.580317 + 0.814391i \(0.697071\pi\)
\(998\) 0 0
\(999\) −2.70156 −0.0854736
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.cc.1.3 4
4.3 odd 2 3864.2.a.s.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3864.2.a.s.1.3 4 4.3 odd 2
7728.2.a.cc.1.3 4 1.1 even 1 trivial