Properties

Label 7728.2.a.bx.1.1
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
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] = [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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.837.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 6x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3864)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.36147\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -2.36147 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -2.36147 q^{5} -1.00000 q^{7} +1.00000 q^{9} -5.93800 q^{11} +6.29947 q^{13} -2.36147 q^{15} +0.784934 q^{17} +1.93800 q^{19} -1.00000 q^{21} +1.00000 q^{23} +0.576535 q^{25} +1.00000 q^{27} +5.00000 q^{29} -7.93800 q^{31} -5.93800 q^{33} +2.36147 q^{35} -7.72294 q^{37} +6.29947 q^{39} +6.15307 q^{41} +13.0224 q^{43} -2.36147 q^{45} -0.215066 q^{47} +1.00000 q^{49} +0.784934 q^{51} +2.14640 q^{53} +14.0224 q^{55} +1.93800 q^{57} -11.7296 q^{59} -4.51454 q^{61} -1.00000 q^{63} -14.8760 q^{65} -0.993333 q^{67} +1.00000 q^{69} +4.51454 q^{71} -2.35480 q^{73} +0.576535 q^{75} +5.93800 q^{77} -0.492128 q^{79} +1.00000 q^{81} -9.07774 q^{83} -1.85360 q^{85} +5.00000 q^{87} -4.51454 q^{89} -6.29947 q^{91} -7.93800 q^{93} -4.57653 q^{95} +5.59894 q^{97} -5.93800 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{3} - 3 q^{7} + 3 q^{9} - 6 q^{11} - 6 q^{19} - 3 q^{21} + 3 q^{23} - 3 q^{25} + 3 q^{27} + 15 q^{29} - 12 q^{31} - 6 q^{33} - 9 q^{37} + 9 q^{41} + 6 q^{43} - 3 q^{47} + 3 q^{49} - 3 q^{53} + 9 q^{55} - 6 q^{57} - 21 q^{59} + 3 q^{61} - 3 q^{63} - 21 q^{65} - 3 q^{67} + 3 q^{69} - 3 q^{71} - 3 q^{75} + 6 q^{77} - 18 q^{79} + 3 q^{81} - 6 q^{83} - 15 q^{85} + 15 q^{87} + 3 q^{89} - 12 q^{93} - 9 q^{95} - 21 q^{97} - 6 q^{99}+O(q^{100}) \) 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
\(5\) −2.36147 −1.05608 −0.528040 0.849219i \(-0.677073\pi\)
−0.528040 + 0.849219i \(0.677073\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.93800 −1.79038 −0.895188 0.445689i \(-0.852959\pi\)
−0.895188 + 0.445689i \(0.852959\pi\)
\(12\) 0 0
\(13\) 6.29947 1.74716 0.873580 0.486681i \(-0.161793\pi\)
0.873580 + 0.486681i \(0.161793\pi\)
\(14\) 0 0
\(15\) −2.36147 −0.609729
\(16\) 0 0
\(17\) 0.784934 0.190374 0.0951872 0.995459i \(-0.469655\pi\)
0.0951872 + 0.995459i \(0.469655\pi\)
\(18\) 0 0
\(19\) 1.93800 0.444608 0.222304 0.974977i \(-0.428642\pi\)
0.222304 + 0.974977i \(0.428642\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 0.576535 0.115307
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 0 0
\(31\) −7.93800 −1.42571 −0.712854 0.701313i \(-0.752598\pi\)
−0.712854 + 0.701313i \(0.752598\pi\)
\(32\) 0 0
\(33\) −5.93800 −1.03367
\(34\) 0 0
\(35\) 2.36147 0.399161
\(36\) 0 0
\(37\) −7.72294 −1.26964 −0.634822 0.772659i \(-0.718926\pi\)
−0.634822 + 0.772659i \(0.718926\pi\)
\(38\) 0 0
\(39\) 6.29947 1.00872
\(40\) 0 0
\(41\) 6.15307 0.960948 0.480474 0.877009i \(-0.340465\pi\)
0.480474 + 0.877009i \(0.340465\pi\)
\(42\) 0 0
\(43\) 13.0224 1.98590 0.992949 0.118539i \(-0.0378210\pi\)
0.992949 + 0.118539i \(0.0378210\pi\)
\(44\) 0 0
\(45\) −2.36147 −0.352027
\(46\) 0 0
\(47\) −0.215066 −0.0313706 −0.0156853 0.999877i \(-0.504993\pi\)
−0.0156853 + 0.999877i \(0.504993\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0.784934 0.109913
\(52\) 0 0
\(53\) 2.14640 0.294831 0.147416 0.989075i \(-0.452905\pi\)
0.147416 + 0.989075i \(0.452905\pi\)
\(54\) 0 0
\(55\) 14.0224 1.89078
\(56\) 0 0
\(57\) 1.93800 0.256695
\(58\) 0 0
\(59\) −11.7296 −1.52706 −0.763532 0.645770i \(-0.776537\pi\)
−0.763532 + 0.645770i \(0.776537\pi\)
\(60\) 0 0
\(61\) −4.51454 −0.578027 −0.289014 0.957325i \(-0.593327\pi\)
−0.289014 + 0.957325i \(0.593327\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) −14.8760 −1.84514
\(66\) 0 0
\(67\) −0.993333 −0.121355 −0.0606775 0.998157i \(-0.519326\pi\)
−0.0606775 + 0.998157i \(0.519326\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 4.51454 0.535777 0.267889 0.963450i \(-0.413674\pi\)
0.267889 + 0.963450i \(0.413674\pi\)
\(72\) 0 0
\(73\) −2.35480 −0.275609 −0.137804 0.990459i \(-0.544005\pi\)
−0.137804 + 0.990459i \(0.544005\pi\)
\(74\) 0 0
\(75\) 0.576535 0.0665725
\(76\) 0 0
\(77\) 5.93800 0.676698
\(78\) 0 0
\(79\) −0.492128 −0.0553688 −0.0276844 0.999617i \(-0.508813\pi\)
−0.0276844 + 0.999617i \(0.508813\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −9.07774 −0.996411 −0.498206 0.867059i \(-0.666008\pi\)
−0.498206 + 0.867059i \(0.666008\pi\)
\(84\) 0 0
\(85\) −1.85360 −0.201051
\(86\) 0 0
\(87\) 5.00000 0.536056
\(88\) 0 0
\(89\) −4.51454 −0.478540 −0.239270 0.970953i \(-0.576908\pi\)
−0.239270 + 0.970953i \(0.576908\pi\)
\(90\) 0 0
\(91\) −6.29947 −0.660364
\(92\) 0 0
\(93\) −7.93800 −0.823133
\(94\) 0 0
\(95\) −4.57653 −0.469543
\(96\) 0 0
\(97\) 5.59894 0.568487 0.284243 0.958752i \(-0.408258\pi\)
0.284243 + 0.958752i \(0.408258\pi\)
\(98\) 0 0
\(99\) −5.93800 −0.596792
\(100\) 0 0
\(101\) −1.29947 −0.129302 −0.0646512 0.997908i \(-0.520593\pi\)
−0.0646512 + 0.997908i \(0.520593\pi\)
\(102\) 0 0
\(103\) 8.38388 0.826088 0.413044 0.910711i \(-0.364466\pi\)
0.413044 + 0.910711i \(0.364466\pi\)
\(104\) 0 0
\(105\) 2.36147 0.230456
\(106\) 0 0
\(107\) −9.36147 −0.905007 −0.452504 0.891763i \(-0.649469\pi\)
−0.452504 + 0.891763i \(0.649469\pi\)
\(108\) 0 0
\(109\) −2.06866 −0.198142 −0.0990710 0.995080i \(-0.531587\pi\)
−0.0990710 + 0.995080i \(0.531587\pi\)
\(110\) 0 0
\(111\) −7.72294 −0.733029
\(112\) 0 0
\(113\) −16.9604 −1.59550 −0.797751 0.602987i \(-0.793977\pi\)
−0.797751 + 0.602987i \(0.793977\pi\)
\(114\) 0 0
\(115\) −2.36147 −0.220208
\(116\) 0 0
\(117\) 6.29947 0.582386
\(118\) 0 0
\(119\) −0.784934 −0.0719548
\(120\) 0 0
\(121\) 24.2599 2.20544
\(122\) 0 0
\(123\) 6.15307 0.554804
\(124\) 0 0
\(125\) 10.4459 0.934307
\(126\) 0 0
\(127\) 16.5923 1.47233 0.736163 0.676804i \(-0.236636\pi\)
0.736163 + 0.676804i \(0.236636\pi\)
\(128\) 0 0
\(129\) 13.0224 1.14656
\(130\) 0 0
\(131\) −13.2151 −1.15461 −0.577303 0.816530i \(-0.695895\pi\)
−0.577303 + 0.816530i \(0.695895\pi\)
\(132\) 0 0
\(133\) −1.93800 −0.168046
\(134\) 0 0
\(135\) −2.36147 −0.203243
\(136\) 0 0
\(137\) −13.0000 −1.11066 −0.555332 0.831628i \(-0.687409\pi\)
−0.555332 + 0.831628i \(0.687409\pi\)
\(138\) 0 0
\(139\) −1.22173 −0.103626 −0.0518130 0.998657i \(-0.516500\pi\)
−0.0518130 + 0.998657i \(0.516500\pi\)
\(140\) 0 0
\(141\) −0.215066 −0.0181118
\(142\) 0 0
\(143\) −37.4063 −3.12807
\(144\) 0 0
\(145\) −11.8073 −0.980547
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) −6.72294 −0.550765 −0.275382 0.961335i \(-0.588804\pi\)
−0.275382 + 0.961335i \(0.588804\pi\)
\(150\) 0 0
\(151\) −17.2308 −1.40222 −0.701112 0.713051i \(-0.747313\pi\)
−0.701112 + 0.713051i \(0.747313\pi\)
\(152\) 0 0
\(153\) 0.784934 0.0634582
\(154\) 0 0
\(155\) 18.7453 1.50566
\(156\) 0 0
\(157\) 22.5989 1.80359 0.901796 0.432162i \(-0.142249\pi\)
0.901796 + 0.432162i \(0.142249\pi\)
\(158\) 0 0
\(159\) 2.14640 0.170221
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) −4.70053 −0.368174 −0.184087 0.982910i \(-0.558933\pi\)
−0.184087 + 0.982910i \(0.558933\pi\)
\(164\) 0 0
\(165\) 14.0224 1.09164
\(166\) 0 0
\(167\) −22.1068 −1.71068 −0.855338 0.518070i \(-0.826651\pi\)
−0.855338 + 0.518070i \(0.826651\pi\)
\(168\) 0 0
\(169\) 26.6834 2.05257
\(170\) 0 0
\(171\) 1.93800 0.148203
\(172\) 0 0
\(173\) 17.8007 1.35336 0.676680 0.736277i \(-0.263418\pi\)
0.676680 + 0.736277i \(0.263418\pi\)
\(174\) 0 0
\(175\) −0.576535 −0.0435819
\(176\) 0 0
\(177\) −11.7296 −0.881651
\(178\) 0 0
\(179\) −0.915594 −0.0684347 −0.0342173 0.999414i \(-0.510894\pi\)
−0.0342173 + 0.999414i \(0.510894\pi\)
\(180\) 0 0
\(181\) −19.9828 −1.48531 −0.742656 0.669673i \(-0.766434\pi\)
−0.742656 + 0.669673i \(0.766434\pi\)
\(182\) 0 0
\(183\) −4.51454 −0.333724
\(184\) 0 0
\(185\) 18.2375 1.34085
\(186\) 0 0
\(187\) −4.66094 −0.340842
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −25.0291 −1.81104 −0.905520 0.424303i \(-0.860519\pi\)
−0.905520 + 0.424303i \(0.860519\pi\)
\(192\) 0 0
\(193\) −9.78493 −0.704335 −0.352167 0.935937i \(-0.614555\pi\)
−0.352167 + 0.935937i \(0.614555\pi\)
\(194\) 0 0
\(195\) −14.8760 −1.06529
\(196\) 0 0
\(197\) 13.8827 0.989100 0.494550 0.869149i \(-0.335333\pi\)
0.494550 + 0.869149i \(0.335333\pi\)
\(198\) 0 0
\(199\) −12.2241 −0.866546 −0.433273 0.901263i \(-0.642641\pi\)
−0.433273 + 0.901263i \(0.642641\pi\)
\(200\) 0 0
\(201\) −0.993333 −0.0700643
\(202\) 0 0
\(203\) −5.00000 −0.350931
\(204\) 0 0
\(205\) −14.5303 −1.01484
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −11.5079 −0.796016
\(210\) 0 0
\(211\) −4.66094 −0.320872 −0.160436 0.987046i \(-0.551290\pi\)
−0.160436 + 0.987046i \(0.551290\pi\)
\(212\) 0 0
\(213\) 4.51454 0.309331
\(214\) 0 0
\(215\) −30.7520 −2.09727
\(216\) 0 0
\(217\) 7.93800 0.538867
\(218\) 0 0
\(219\) −2.35480 −0.159123
\(220\) 0 0
\(221\) 4.94467 0.332615
\(222\) 0 0
\(223\) −10.8693 −0.727865 −0.363932 0.931425i \(-0.618566\pi\)
−0.363932 + 0.931425i \(0.618566\pi\)
\(224\) 0 0
\(225\) 0.576535 0.0384356
\(226\) 0 0
\(227\) −11.2084 −0.743928 −0.371964 0.928247i \(-0.621315\pi\)
−0.371964 + 0.928247i \(0.621315\pi\)
\(228\) 0 0
\(229\) 6.14640 0.406166 0.203083 0.979162i \(-0.434904\pi\)
0.203083 + 0.979162i \(0.434904\pi\)
\(230\) 0 0
\(231\) 5.93800 0.390692
\(232\) 0 0
\(233\) −12.0224 −0.787614 −0.393807 0.919193i \(-0.628842\pi\)
−0.393807 + 0.919193i \(0.628842\pi\)
\(234\) 0 0
\(235\) 0.507872 0.0331299
\(236\) 0 0
\(237\) −0.492128 −0.0319672
\(238\) 0 0
\(239\) −19.5923 −1.26732 −0.633660 0.773612i \(-0.718448\pi\)
−0.633660 + 0.773612i \(0.718448\pi\)
\(240\) 0 0
\(241\) −11.9051 −0.766874 −0.383437 0.923567i \(-0.625260\pi\)
−0.383437 + 0.923567i \(0.625260\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −2.36147 −0.150869
\(246\) 0 0
\(247\) 12.2084 0.776802
\(248\) 0 0
\(249\) −9.07774 −0.575278
\(250\) 0 0
\(251\) −22.2151 −1.40220 −0.701101 0.713062i \(-0.747308\pi\)
−0.701101 + 0.713062i \(0.747308\pi\)
\(252\) 0 0
\(253\) −5.93800 −0.373319
\(254\) 0 0
\(255\) −1.85360 −0.116077
\(256\) 0 0
\(257\) 1.70719 0.106492 0.0532459 0.998581i \(-0.483043\pi\)
0.0532459 + 0.998581i \(0.483043\pi\)
\(258\) 0 0
\(259\) 7.72294 0.479880
\(260\) 0 0
\(261\) 5.00000 0.309492
\(262\) 0 0
\(263\) 15.4750 0.954226 0.477113 0.878842i \(-0.341683\pi\)
0.477113 + 0.878842i \(0.341683\pi\)
\(264\) 0 0
\(265\) −5.06866 −0.311366
\(266\) 0 0
\(267\) −4.51454 −0.276285
\(268\) 0 0
\(269\) −22.1292 −1.34924 −0.674621 0.738164i \(-0.735693\pi\)
−0.674621 + 0.738164i \(0.735693\pi\)
\(270\) 0 0
\(271\) 16.6609 1.01208 0.506040 0.862510i \(-0.331109\pi\)
0.506040 + 0.862510i \(0.331109\pi\)
\(272\) 0 0
\(273\) −6.29947 −0.381261
\(274\) 0 0
\(275\) −3.42347 −0.206443
\(276\) 0 0
\(277\) 14.5145 0.872094 0.436047 0.899924i \(-0.356378\pi\)
0.436047 + 0.899924i \(0.356378\pi\)
\(278\) 0 0
\(279\) −7.93800 −0.475236
\(280\) 0 0
\(281\) −17.9671 −1.07183 −0.535913 0.844273i \(-0.680033\pi\)
−0.535913 + 0.844273i \(0.680033\pi\)
\(282\) 0 0
\(283\) 9.29947 0.552796 0.276398 0.961043i \(-0.410859\pi\)
0.276398 + 0.961043i \(0.410859\pi\)
\(284\) 0 0
\(285\) −4.57653 −0.271091
\(286\) 0 0
\(287\) −6.15307 −0.363204
\(288\) 0 0
\(289\) −16.3839 −0.963758
\(290\) 0 0
\(291\) 5.59894 0.328216
\(292\) 0 0
\(293\) −24.1821 −1.41274 −0.706368 0.707845i \(-0.749668\pi\)
−0.706368 + 0.707845i \(0.749668\pi\)
\(294\) 0 0
\(295\) 27.6991 1.61270
\(296\) 0 0
\(297\) −5.93800 −0.344558
\(298\) 0 0
\(299\) 6.29947 0.364308
\(300\) 0 0
\(301\) −13.0224 −0.750599
\(302\) 0 0
\(303\) −1.29947 −0.0746527
\(304\) 0 0
\(305\) 10.6609 0.610444
\(306\) 0 0
\(307\) −29.4616 −1.68146 −0.840732 0.541452i \(-0.817875\pi\)
−0.840732 + 0.541452i \(0.817875\pi\)
\(308\) 0 0
\(309\) 8.38388 0.476942
\(310\) 0 0
\(311\) 26.2666 1.48944 0.744720 0.667377i \(-0.232583\pi\)
0.744720 + 0.667377i \(0.232583\pi\)
\(312\) 0 0
\(313\) −26.4883 −1.49721 −0.748603 0.663018i \(-0.769275\pi\)
−0.748603 + 0.663018i \(0.769275\pi\)
\(314\) 0 0
\(315\) 2.36147 0.133054
\(316\) 0 0
\(317\) 9.51454 0.534390 0.267195 0.963643i \(-0.413903\pi\)
0.267195 + 0.963643i \(0.413903\pi\)
\(318\) 0 0
\(319\) −29.6900 −1.66232
\(320\) 0 0
\(321\) −9.36147 −0.522506
\(322\) 0 0
\(323\) 1.52120 0.0846421
\(324\) 0 0
\(325\) 3.63186 0.201460
\(326\) 0 0
\(327\) −2.06866 −0.114397
\(328\) 0 0
\(329\) 0.215066 0.0118570
\(330\) 0 0
\(331\) −7.87601 −0.432904 −0.216452 0.976293i \(-0.569449\pi\)
−0.216452 + 0.976293i \(0.569449\pi\)
\(332\) 0 0
\(333\) −7.72294 −0.423214
\(334\) 0 0
\(335\) 2.34573 0.128161
\(336\) 0 0
\(337\) 6.39055 0.348115 0.174058 0.984735i \(-0.444312\pi\)
0.174058 + 0.984735i \(0.444312\pi\)
\(338\) 0 0
\(339\) −16.9604 −0.921163
\(340\) 0 0
\(341\) 47.1359 2.55255
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −2.36147 −0.127137
\(346\) 0 0
\(347\) 25.3510 1.36091 0.680455 0.732790i \(-0.261782\pi\)
0.680455 + 0.732790i \(0.261782\pi\)
\(348\) 0 0
\(349\) −26.3285 −1.40933 −0.704667 0.709538i \(-0.748904\pi\)
−0.704667 + 0.709538i \(0.748904\pi\)
\(350\) 0 0
\(351\) 6.29947 0.336241
\(352\) 0 0
\(353\) −24.9208 −1.32640 −0.663201 0.748441i \(-0.730802\pi\)
−0.663201 + 0.748441i \(0.730802\pi\)
\(354\) 0 0
\(355\) −10.6609 −0.565824
\(356\) 0 0
\(357\) −0.784934 −0.0415431
\(358\) 0 0
\(359\) −5.37721 −0.283798 −0.141899 0.989881i \(-0.545321\pi\)
−0.141899 + 0.989881i \(0.545321\pi\)
\(360\) 0 0
\(361\) −15.2441 −0.802323
\(362\) 0 0
\(363\) 24.2599 1.27331
\(364\) 0 0
\(365\) 5.56079 0.291065
\(366\) 0 0
\(367\) −6.85360 −0.357755 −0.178877 0.983871i \(-0.557247\pi\)
−0.178877 + 0.983871i \(0.557247\pi\)
\(368\) 0 0
\(369\) 6.15307 0.320316
\(370\) 0 0
\(371\) −2.14640 −0.111436
\(372\) 0 0
\(373\) 30.2890 1.56830 0.784152 0.620569i \(-0.213098\pi\)
0.784152 + 0.620569i \(0.213098\pi\)
\(374\) 0 0
\(375\) 10.4459 0.539423
\(376\) 0 0
\(377\) 31.4974 1.62220
\(378\) 0 0
\(379\) 7.79827 0.400570 0.200285 0.979738i \(-0.435813\pi\)
0.200285 + 0.979738i \(0.435813\pi\)
\(380\) 0 0
\(381\) 16.5923 0.850048
\(382\) 0 0
\(383\) −2.92226 −0.149321 −0.0746603 0.997209i \(-0.523787\pi\)
−0.0746603 + 0.997209i \(0.523787\pi\)
\(384\) 0 0
\(385\) −14.0224 −0.714648
\(386\) 0 0
\(387\) 13.0224 0.661966
\(388\) 0 0
\(389\) 34.5369 1.75109 0.875546 0.483134i \(-0.160502\pi\)
0.875546 + 0.483134i \(0.160502\pi\)
\(390\) 0 0
\(391\) 0.784934 0.0396958
\(392\) 0 0
\(393\) −13.2151 −0.666612
\(394\) 0 0
\(395\) 1.16215 0.0584739
\(396\) 0 0
\(397\) 26.0448 1.30715 0.653576 0.756861i \(-0.273268\pi\)
0.653576 + 0.756861i \(0.273268\pi\)
\(398\) 0 0
\(399\) −1.93800 −0.0970215
\(400\) 0 0
\(401\) 1.21507 0.0606775 0.0303387 0.999540i \(-0.490341\pi\)
0.0303387 + 0.999540i \(0.490341\pi\)
\(402\) 0 0
\(403\) −50.0052 −2.49094
\(404\) 0 0
\(405\) −2.36147 −0.117342
\(406\) 0 0
\(407\) 45.8588 2.27314
\(408\) 0 0
\(409\) 9.10441 0.450184 0.225092 0.974338i \(-0.427732\pi\)
0.225092 + 0.974338i \(0.427732\pi\)
\(410\) 0 0
\(411\) −13.0000 −0.641243
\(412\) 0 0
\(413\) 11.7296 0.577176
\(414\) 0 0
\(415\) 21.4368 1.05229
\(416\) 0 0
\(417\) −1.22173 −0.0598285
\(418\) 0 0
\(419\) 25.8984 1.26522 0.632610 0.774470i \(-0.281984\pi\)
0.632610 + 0.774470i \(0.281984\pi\)
\(420\) 0 0
\(421\) 0.605611 0.0295157 0.0147578 0.999891i \(-0.495302\pi\)
0.0147578 + 0.999891i \(0.495302\pi\)
\(422\) 0 0
\(423\) −0.215066 −0.0104569
\(424\) 0 0
\(425\) 0.452542 0.0219515
\(426\) 0 0
\(427\) 4.51454 0.218474
\(428\) 0 0
\(429\) −37.4063 −1.80599
\(430\) 0 0
\(431\) −27.7902 −1.33861 −0.669303 0.742990i \(-0.733407\pi\)
−0.669303 + 0.742990i \(0.733407\pi\)
\(432\) 0 0
\(433\) −14.8007 −0.711275 −0.355638 0.934624i \(-0.615736\pi\)
−0.355638 + 0.934624i \(0.615736\pi\)
\(434\) 0 0
\(435\) −11.8073 −0.566119
\(436\) 0 0
\(437\) 1.93800 0.0927073
\(438\) 0 0
\(439\) −5.63186 −0.268794 −0.134397 0.990928i \(-0.542910\pi\)
−0.134397 + 0.990928i \(0.542910\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −3.36814 −0.160025 −0.0800125 0.996794i \(-0.525496\pi\)
−0.0800125 + 0.996794i \(0.525496\pi\)
\(444\) 0 0
\(445\) 10.6609 0.505377
\(446\) 0 0
\(447\) −6.72294 −0.317984
\(448\) 0 0
\(449\) −3.11733 −0.147116 −0.0735579 0.997291i \(-0.523435\pi\)
−0.0735579 + 0.997291i \(0.523435\pi\)
\(450\) 0 0
\(451\) −36.5369 −1.72046
\(452\) 0 0
\(453\) −17.2308 −0.809574
\(454\) 0 0
\(455\) 14.8760 0.697398
\(456\) 0 0
\(457\) −9.72960 −0.455132 −0.227566 0.973763i \(-0.573077\pi\)
−0.227566 + 0.973763i \(0.573077\pi\)
\(458\) 0 0
\(459\) 0.784934 0.0366376
\(460\) 0 0
\(461\) 12.7587 0.594231 0.297116 0.954842i \(-0.403975\pi\)
0.297116 + 0.954842i \(0.403975\pi\)
\(462\) 0 0
\(463\) 5.67812 0.263885 0.131942 0.991257i \(-0.457879\pi\)
0.131942 + 0.991257i \(0.457879\pi\)
\(464\) 0 0
\(465\) 18.7453 0.869295
\(466\) 0 0
\(467\) −24.7520 −1.14539 −0.572693 0.819770i \(-0.694101\pi\)
−0.572693 + 0.819770i \(0.694101\pi\)
\(468\) 0 0
\(469\) 0.993333 0.0458679
\(470\) 0 0
\(471\) 22.5989 1.04130
\(472\) 0 0
\(473\) −77.3271 −3.55550
\(474\) 0 0
\(475\) 1.11733 0.0512664
\(476\) 0 0
\(477\) 2.14640 0.0982770
\(478\) 0 0
\(479\) −28.9404 −1.32232 −0.661161 0.750244i \(-0.729936\pi\)
−0.661161 + 0.750244i \(0.729936\pi\)
\(480\) 0 0
\(481\) −48.6504 −2.21827
\(482\) 0 0
\(483\) −1.00000 −0.0455016
\(484\) 0 0
\(485\) −13.2217 −0.600368
\(486\) 0 0
\(487\) −27.0582 −1.22612 −0.613061 0.790036i \(-0.710062\pi\)
−0.613061 + 0.790036i \(0.710062\pi\)
\(488\) 0 0
\(489\) −4.70053 −0.212565
\(490\) 0 0
\(491\) −26.0977 −1.17777 −0.588887 0.808215i \(-0.700434\pi\)
−0.588887 + 0.808215i \(0.700434\pi\)
\(492\) 0 0
\(493\) 3.92467 0.176758
\(494\) 0 0
\(495\) 14.0224 0.630260
\(496\) 0 0
\(497\) −4.51454 −0.202505
\(498\) 0 0
\(499\) 14.9933 0.671194 0.335597 0.942006i \(-0.391062\pi\)
0.335597 + 0.942006i \(0.391062\pi\)
\(500\) 0 0
\(501\) −22.1068 −0.987660
\(502\) 0 0
\(503\) −1.74294 −0.0777137 −0.0388569 0.999245i \(-0.512372\pi\)
−0.0388569 + 0.999245i \(0.512372\pi\)
\(504\) 0 0
\(505\) 3.06866 0.136554
\(506\) 0 0
\(507\) 26.6834 1.18505
\(508\) 0 0
\(509\) 20.7811 0.921106 0.460553 0.887632i \(-0.347651\pi\)
0.460553 + 0.887632i \(0.347651\pi\)
\(510\) 0 0
\(511\) 2.35480 0.104170
\(512\) 0 0
\(513\) 1.93800 0.0855649
\(514\) 0 0
\(515\) −19.7983 −0.872416
\(516\) 0 0
\(517\) 1.27706 0.0561651
\(518\) 0 0
\(519\) 17.8007 0.781363
\(520\) 0 0
\(521\) −0.306139 −0.0134122 −0.00670610 0.999978i \(-0.502135\pi\)
−0.00670610 + 0.999978i \(0.502135\pi\)
\(522\) 0 0
\(523\) −13.7520 −0.601334 −0.300667 0.953729i \(-0.597209\pi\)
−0.300667 + 0.953729i \(0.597209\pi\)
\(524\) 0 0
\(525\) −0.576535 −0.0251620
\(526\) 0 0
\(527\) −6.23081 −0.271418
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −11.7296 −0.509021
\(532\) 0 0
\(533\) 38.7611 1.67893
\(534\) 0 0
\(535\) 22.1068 0.955761
\(536\) 0 0
\(537\) −0.915594 −0.0395108
\(538\) 0 0
\(539\) −5.93800 −0.255768
\(540\) 0 0
\(541\) −45.6147 −1.96113 −0.980564 0.196198i \(-0.937140\pi\)
−0.980564 + 0.196198i \(0.937140\pi\)
\(542\) 0 0
\(543\) −19.9828 −0.857545
\(544\) 0 0
\(545\) 4.88508 0.209254
\(546\) 0 0
\(547\) 36.4392 1.55803 0.779014 0.627007i \(-0.215720\pi\)
0.779014 + 0.627007i \(0.215720\pi\)
\(548\) 0 0
\(549\) −4.51454 −0.192676
\(550\) 0 0
\(551\) 9.69002 0.412809
\(552\) 0 0
\(553\) 0.492128 0.0209274
\(554\) 0 0
\(555\) 18.2375 0.774138
\(556\) 0 0
\(557\) 29.7387 1.26007 0.630034 0.776567i \(-0.283041\pi\)
0.630034 + 0.776567i \(0.283041\pi\)
\(558\) 0 0
\(559\) 82.0343 3.46968
\(560\) 0 0
\(561\) −4.66094 −0.196785
\(562\) 0 0
\(563\) 2.41013 0.101575 0.0507875 0.998709i \(-0.483827\pi\)
0.0507875 + 0.998709i \(0.483827\pi\)
\(564\) 0 0
\(565\) 40.0515 1.68498
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) −23.9537 −1.00419 −0.502097 0.864812i \(-0.667438\pi\)
−0.502097 + 0.864812i \(0.667438\pi\)
\(570\) 0 0
\(571\) −37.5394 −1.57097 −0.785487 0.618879i \(-0.787587\pi\)
−0.785487 + 0.618879i \(0.787587\pi\)
\(572\) 0 0
\(573\) −25.0291 −1.04560
\(574\) 0 0
\(575\) 0.576535 0.0240432
\(576\) 0 0
\(577\) 30.6123 1.27441 0.637203 0.770696i \(-0.280091\pi\)
0.637203 + 0.770696i \(0.280091\pi\)
\(578\) 0 0
\(579\) −9.78493 −0.406648
\(580\) 0 0
\(581\) 9.07774 0.376608
\(582\) 0 0
\(583\) −12.7453 −0.527858
\(584\) 0 0
\(585\) −14.8760 −0.615047
\(586\) 0 0
\(587\) 37.5303 1.54904 0.774520 0.632549i \(-0.217991\pi\)
0.774520 + 0.632549i \(0.217991\pi\)
\(588\) 0 0
\(589\) −15.3839 −0.633882
\(590\) 0 0
\(591\) 13.8827 0.571057
\(592\) 0 0
\(593\) 18.3705 0.754388 0.377194 0.926134i \(-0.376889\pi\)
0.377194 + 0.926134i \(0.376889\pi\)
\(594\) 0 0
\(595\) 1.85360 0.0759901
\(596\) 0 0
\(597\) −12.2241 −0.500301
\(598\) 0 0
\(599\) 23.9604 0.978996 0.489498 0.872004i \(-0.337180\pi\)
0.489498 + 0.872004i \(0.337180\pi\)
\(600\) 0 0
\(601\) 28.9142 1.17943 0.589717 0.807610i \(-0.299239\pi\)
0.589717 + 0.807610i \(0.299239\pi\)
\(602\) 0 0
\(603\) −0.993333 −0.0404517
\(604\) 0 0
\(605\) −57.2890 −2.32913
\(606\) 0 0
\(607\) 41.5436 1.68620 0.843102 0.537754i \(-0.180727\pi\)
0.843102 + 0.537754i \(0.180727\pi\)
\(608\) 0 0
\(609\) −5.00000 −0.202610
\(610\) 0 0
\(611\) −1.35480 −0.0548094
\(612\) 0 0
\(613\) 39.0582 1.57754 0.788772 0.614686i \(-0.210717\pi\)
0.788772 + 0.614686i \(0.210717\pi\)
\(614\) 0 0
\(615\) −14.5303 −0.585917
\(616\) 0 0
\(617\) 21.1001 0.849460 0.424730 0.905320i \(-0.360369\pi\)
0.424730 + 0.905320i \(0.360369\pi\)
\(618\) 0 0
\(619\) 28.2666 1.13613 0.568064 0.822984i \(-0.307692\pi\)
0.568064 + 0.822984i \(0.307692\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) 4.51454 0.180871
\(624\) 0 0
\(625\) −27.5503 −1.10201
\(626\) 0 0
\(627\) −11.5079 −0.459580
\(628\) 0 0
\(629\) −6.06200 −0.241708
\(630\) 0 0
\(631\) −27.9247 −1.11166 −0.555832 0.831295i \(-0.687600\pi\)
−0.555832 + 0.831295i \(0.687600\pi\)
\(632\) 0 0
\(633\) −4.66094 −0.185256
\(634\) 0 0
\(635\) −39.1821 −1.55490
\(636\) 0 0
\(637\) 6.29947 0.249594
\(638\) 0 0
\(639\) 4.51454 0.178592
\(640\) 0 0
\(641\) −42.6371 −1.68406 −0.842032 0.539428i \(-0.818641\pi\)
−0.842032 + 0.539428i \(0.818641\pi\)
\(642\) 0 0
\(643\) −6.37721 −0.251493 −0.125746 0.992062i \(-0.540133\pi\)
−0.125746 + 0.992062i \(0.540133\pi\)
\(644\) 0 0
\(645\) −30.7520 −1.21086
\(646\) 0 0
\(647\) −27.7611 −1.09140 −0.545701 0.837980i \(-0.683736\pi\)
−0.545701 + 0.837980i \(0.683736\pi\)
\(648\) 0 0
\(649\) 69.6504 2.73402
\(650\) 0 0
\(651\) 7.93800 0.311115
\(652\) 0 0
\(653\) −10.1268 −0.396293 −0.198146 0.980172i \(-0.563492\pi\)
−0.198146 + 0.980172i \(0.563492\pi\)
\(654\) 0 0
\(655\) 31.2070 1.21936
\(656\) 0 0
\(657\) −2.35480 −0.0918696
\(658\) 0 0
\(659\) −38.8879 −1.51486 −0.757429 0.652918i \(-0.773545\pi\)
−0.757429 + 0.652918i \(0.773545\pi\)
\(660\) 0 0
\(661\) −3.98282 −0.154914 −0.0774569 0.996996i \(-0.524680\pi\)
−0.0774569 + 0.996996i \(0.524680\pi\)
\(662\) 0 0
\(663\) 4.94467 0.192035
\(664\) 0 0
\(665\) 4.57653 0.177470
\(666\) 0 0
\(667\) 5.00000 0.193601
\(668\) 0 0
\(669\) −10.8693 −0.420233
\(670\) 0 0
\(671\) 26.8073 1.03489
\(672\) 0 0
\(673\) −5.98426 −0.230676 −0.115338 0.993326i \(-0.536795\pi\)
−0.115338 + 0.993326i \(0.536795\pi\)
\(674\) 0 0
\(675\) 0.576535 0.0221908
\(676\) 0 0
\(677\) −21.3930 −0.822198 −0.411099 0.911591i \(-0.634855\pi\)
−0.411099 + 0.911591i \(0.634855\pi\)
\(678\) 0 0
\(679\) −5.59894 −0.214868
\(680\) 0 0
\(681\) −11.2084 −0.429507
\(682\) 0 0
\(683\) 27.3047 1.04479 0.522393 0.852705i \(-0.325040\pi\)
0.522393 + 0.852705i \(0.325040\pi\)
\(684\) 0 0
\(685\) 30.6991 1.17295
\(686\) 0 0
\(687\) 6.14640 0.234500
\(688\) 0 0
\(689\) 13.5212 0.515117
\(690\) 0 0
\(691\) −25.4974 −0.969965 −0.484983 0.874524i \(-0.661174\pi\)
−0.484983 + 0.874524i \(0.661174\pi\)
\(692\) 0 0
\(693\) 5.93800 0.225566
\(694\) 0 0
\(695\) 2.88508 0.109437
\(696\) 0 0
\(697\) 4.82975 0.182940
\(698\) 0 0
\(699\) −12.0224 −0.454729
\(700\) 0 0
\(701\) 9.85360 0.372165 0.186083 0.982534i \(-0.440421\pi\)
0.186083 + 0.982534i \(0.440421\pi\)
\(702\) 0 0
\(703\) −14.9671 −0.564494
\(704\) 0 0
\(705\) 0.507872 0.0191275
\(706\) 0 0
\(707\) 1.29947 0.0488717
\(708\) 0 0
\(709\) 34.2980 1.28809 0.644045 0.764988i \(-0.277255\pi\)
0.644045 + 0.764988i \(0.277255\pi\)
\(710\) 0 0
\(711\) −0.492128 −0.0184563
\(712\) 0 0
\(713\) −7.93800 −0.297281
\(714\) 0 0
\(715\) 88.3338 3.30350
\(716\) 0 0
\(717\) −19.5923 −0.731687
\(718\) 0 0
\(719\) 24.3219 0.907053 0.453527 0.891243i \(-0.350166\pi\)
0.453527 + 0.891243i \(0.350166\pi\)
\(720\) 0 0
\(721\) −8.38388 −0.312232
\(722\) 0 0
\(723\) −11.9051 −0.442755
\(724\) 0 0
\(725\) 2.88267 0.107060
\(726\) 0 0
\(727\) 31.2599 1.15936 0.579682 0.814842i \(-0.303177\pi\)
0.579682 + 0.814842i \(0.303177\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 10.2217 0.378064
\(732\) 0 0
\(733\) −7.95518 −0.293831 −0.146916 0.989149i \(-0.546935\pi\)
−0.146916 + 0.989149i \(0.546935\pi\)
\(734\) 0 0
\(735\) −2.36147 −0.0871041
\(736\) 0 0
\(737\) 5.89842 0.217271
\(738\) 0 0
\(739\) 17.2151 0.633266 0.316633 0.948548i \(-0.397448\pi\)
0.316633 + 0.948548i \(0.397448\pi\)
\(740\) 0 0
\(741\) 12.2084 0.448487
\(742\) 0 0
\(743\) −12.7453 −0.467581 −0.233791 0.972287i \(-0.575113\pi\)
−0.233791 + 0.972287i \(0.575113\pi\)
\(744\) 0 0
\(745\) 15.8760 0.581652
\(746\) 0 0
\(747\) −9.07774 −0.332137
\(748\) 0 0
\(749\) 9.36147 0.342061
\(750\) 0 0
\(751\) 44.0357 1.60689 0.803444 0.595381i \(-0.202999\pi\)
0.803444 + 0.595381i \(0.202999\pi\)
\(752\) 0 0
\(753\) −22.2151 −0.809562
\(754\) 0 0
\(755\) 40.6900 1.48086
\(756\) 0 0
\(757\) −19.9828 −0.726288 −0.363144 0.931733i \(-0.618297\pi\)
−0.363144 + 0.931733i \(0.618297\pi\)
\(758\) 0 0
\(759\) −5.93800 −0.215536
\(760\) 0 0
\(761\) 39.9075 1.44665 0.723323 0.690510i \(-0.242614\pi\)
0.723323 + 0.690510i \(0.242614\pi\)
\(762\) 0 0
\(763\) 2.06866 0.0748906
\(764\) 0 0
\(765\) −1.85360 −0.0670169
\(766\) 0 0
\(767\) −73.8903 −2.66802
\(768\) 0 0
\(769\) −37.5369 −1.35362 −0.676808 0.736159i \(-0.736637\pi\)
−0.676808 + 0.736159i \(0.736637\pi\)
\(770\) 0 0
\(771\) 1.70719 0.0614831
\(772\) 0 0
\(773\) 49.2718 1.77218 0.886091 0.463510i \(-0.153410\pi\)
0.886091 + 0.463510i \(0.153410\pi\)
\(774\) 0 0
\(775\) −4.57653 −0.164394
\(776\) 0 0
\(777\) 7.72294 0.277059
\(778\) 0 0
\(779\) 11.9247 0.427246
\(780\) 0 0
\(781\) −26.8073 −0.959242
\(782\) 0 0
\(783\) 5.00000 0.178685
\(784\) 0 0
\(785\) −53.3667 −1.90474
\(786\) 0 0
\(787\) 12.1331 0.432497 0.216249 0.976338i \(-0.430618\pi\)
0.216249 + 0.976338i \(0.430618\pi\)
\(788\) 0 0
\(789\) 15.4750 0.550923
\(790\) 0 0
\(791\) 16.9604 0.603043
\(792\) 0 0
\(793\) −28.4392 −1.00991
\(794\) 0 0
\(795\) −5.06866 −0.179767
\(796\) 0 0
\(797\) −36.8273 −1.30449 −0.652246 0.758008i \(-0.726173\pi\)
−0.652246 + 0.758008i \(0.726173\pi\)
\(798\) 0 0
\(799\) −0.168813 −0.00597216
\(800\) 0 0
\(801\) −4.51454 −0.159513
\(802\) 0 0
\(803\) 13.9828 0.493443
\(804\) 0 0
\(805\) 2.36147 0.0832308
\(806\) 0 0
\(807\) −22.1292 −0.778986
\(808\) 0 0
\(809\) −36.6967 −1.29019 −0.645093 0.764104i \(-0.723182\pi\)
−0.645093 + 0.764104i \(0.723182\pi\)
\(810\) 0 0
\(811\) 12.5565 0.440920 0.220460 0.975396i \(-0.429244\pi\)
0.220460 + 0.975396i \(0.429244\pi\)
\(812\) 0 0
\(813\) 16.6609 0.584325
\(814\) 0 0
\(815\) 11.1001 0.388821
\(816\) 0 0
\(817\) 25.2375 0.882948
\(818\) 0 0
\(819\) −6.29947 −0.220121
\(820\) 0 0
\(821\) −10.9051 −0.380590 −0.190295 0.981727i \(-0.560944\pi\)
−0.190295 + 0.981727i \(0.560944\pi\)
\(822\) 0 0
\(823\) −28.8231 −1.00471 −0.502355 0.864662i \(-0.667533\pi\)
−0.502355 + 0.864662i \(0.667533\pi\)
\(824\) 0 0
\(825\) −3.42347 −0.119190
\(826\) 0 0
\(827\) −1.27281 −0.0442598 −0.0221299 0.999755i \(-0.507045\pi\)
−0.0221299 + 0.999755i \(0.507045\pi\)
\(828\) 0 0
\(829\) 12.5236 0.434963 0.217482 0.976064i \(-0.430216\pi\)
0.217482 + 0.976064i \(0.430216\pi\)
\(830\) 0 0
\(831\) 14.5145 0.503504
\(832\) 0 0
\(833\) 0.784934 0.0271964
\(834\) 0 0
\(835\) 52.2046 1.80661
\(836\) 0 0
\(837\) −7.93800 −0.274378
\(838\) 0 0
\(839\) −16.8073 −0.580254 −0.290127 0.956988i \(-0.593698\pi\)
−0.290127 + 0.956988i \(0.593698\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 0 0
\(843\) −17.9671 −0.618819
\(844\) 0 0
\(845\) −63.0119 −2.16768
\(846\) 0 0
\(847\) −24.2599 −0.833580
\(848\) 0 0
\(849\) 9.29947 0.319157
\(850\) 0 0
\(851\) −7.72294 −0.264739
\(852\) 0 0
\(853\) 13.1993 0.451936 0.225968 0.974135i \(-0.427445\pi\)
0.225968 + 0.974135i \(0.427445\pi\)
\(854\) 0 0
\(855\) −4.57653 −0.156514
\(856\) 0 0
\(857\) 0.924670 0.0315861 0.0157931 0.999875i \(-0.494973\pi\)
0.0157931 + 0.999875i \(0.494973\pi\)
\(858\) 0 0
\(859\) −7.24414 −0.247167 −0.123583 0.992334i \(-0.539439\pi\)
−0.123583 + 0.992334i \(0.539439\pi\)
\(860\) 0 0
\(861\) −6.15307 −0.209696
\(862\) 0 0
\(863\) 12.8784 0.438386 0.219193 0.975681i \(-0.429657\pi\)
0.219193 + 0.975681i \(0.429657\pi\)
\(864\) 0 0
\(865\) −42.0357 −1.42926
\(866\) 0 0
\(867\) −16.3839 −0.556426
\(868\) 0 0
\(869\) 2.92226 0.0991309
\(870\) 0 0
\(871\) −6.25748 −0.212026
\(872\) 0 0
\(873\) 5.59894 0.189496
\(874\) 0 0
\(875\) −10.4459 −0.353135
\(876\) 0 0
\(877\) 46.3958 1.56667 0.783337 0.621597i \(-0.213516\pi\)
0.783337 + 0.621597i \(0.213516\pi\)
\(878\) 0 0
\(879\) −24.1821 −0.815644
\(880\) 0 0
\(881\) −2.07917 −0.0700491 −0.0350246 0.999386i \(-0.511151\pi\)
−0.0350246 + 0.999386i \(0.511151\pi\)
\(882\) 0 0
\(883\) −28.8522 −0.970953 −0.485476 0.874250i \(-0.661354\pi\)
−0.485476 + 0.874250i \(0.661354\pi\)
\(884\) 0 0
\(885\) 27.6991 0.931095
\(886\) 0 0
\(887\) 44.0501 1.47906 0.739528 0.673126i \(-0.235049\pi\)
0.739528 + 0.673126i \(0.235049\pi\)
\(888\) 0 0
\(889\) −16.5923 −0.556487
\(890\) 0 0
\(891\) −5.93800 −0.198931
\(892\) 0 0
\(893\) −0.416799 −0.0139476
\(894\) 0 0
\(895\) 2.16215 0.0722726
\(896\) 0 0
\(897\) 6.29947 0.210333
\(898\) 0 0
\(899\) −39.6900 −1.32374
\(900\) 0 0
\(901\) 1.68478 0.0561283
\(902\) 0 0
\(903\) −13.0224 −0.433359
\(904\) 0 0
\(905\) 47.1888 1.56861
\(906\) 0 0
\(907\) −10.8364 −0.359817 −0.179909 0.983683i \(-0.557580\pi\)
−0.179909 + 0.983683i \(0.557580\pi\)
\(908\) 0 0
\(909\) −1.29947 −0.0431008
\(910\) 0 0
\(911\) −42.2732 −1.40057 −0.700287 0.713861i \(-0.746945\pi\)
−0.700287 + 0.713861i \(0.746945\pi\)
\(912\) 0 0
\(913\) 53.9036 1.78395
\(914\) 0 0
\(915\) 10.6609 0.352440
\(916\) 0 0
\(917\) 13.2151 0.436400
\(918\) 0 0
\(919\) 30.1068 0.993132 0.496566 0.867999i \(-0.334594\pi\)
0.496566 + 0.867999i \(0.334594\pi\)
\(920\) 0 0
\(921\) −29.4616 −0.970793
\(922\) 0 0
\(923\) 28.4392 0.936088
\(924\) 0 0
\(925\) −4.45254 −0.146399
\(926\) 0 0
\(927\) 8.38388 0.275363
\(928\) 0 0
\(929\) −9.76252 −0.320298 −0.160149 0.987093i \(-0.551197\pi\)
−0.160149 + 0.987093i \(0.551197\pi\)
\(930\) 0 0
\(931\) 1.93800 0.0635155
\(932\) 0 0
\(933\) 26.2666 0.859928
\(934\) 0 0
\(935\) 11.0067 0.359956
\(936\) 0 0
\(937\) −27.6438 −0.903082 −0.451541 0.892250i \(-0.649126\pi\)
−0.451541 + 0.892250i \(0.649126\pi\)
\(938\) 0 0
\(939\) −26.4883 −0.864413
\(940\) 0 0
\(941\) 29.9223 0.975438 0.487719 0.873001i \(-0.337829\pi\)
0.487719 + 0.873001i \(0.337829\pi\)
\(942\) 0 0
\(943\) 6.15307 0.200372
\(944\) 0 0
\(945\) 2.36147 0.0768186
\(946\) 0 0
\(947\) 18.8140 0.611373 0.305687 0.952132i \(-0.401114\pi\)
0.305687 + 0.952132i \(0.401114\pi\)
\(948\) 0 0
\(949\) −14.8340 −0.481532
\(950\) 0 0
\(951\) 9.51454 0.308530
\(952\) 0 0
\(953\) −8.34573 −0.270345 −0.135172 0.990822i \(-0.543159\pi\)
−0.135172 + 0.990822i \(0.543159\pi\)
\(954\) 0 0
\(955\) 59.1054 1.91261
\(956\) 0 0
\(957\) −29.6900 −0.959742
\(958\) 0 0
\(959\) 13.0000 0.419792
\(960\) 0 0
\(961\) 32.0119 1.03264
\(962\) 0 0
\(963\) −9.36147 −0.301669
\(964\) 0 0
\(965\) 23.1068 0.743835
\(966\) 0 0
\(967\) 58.5198 1.88187 0.940934 0.338589i \(-0.109950\pi\)
0.940934 + 0.338589i \(0.109950\pi\)
\(968\) 0 0
\(969\) 1.52120 0.0488681
\(970\) 0 0
\(971\) 12.8655 0.412873 0.206437 0.978460i \(-0.433813\pi\)
0.206437 + 0.978460i \(0.433813\pi\)
\(972\) 0 0
\(973\) 1.22173 0.0391670
\(974\) 0 0
\(975\) 3.63186 0.116313
\(976\) 0 0
\(977\) 4.16215 0.133159 0.0665794 0.997781i \(-0.478791\pi\)
0.0665794 + 0.997781i \(0.478791\pi\)
\(978\) 0 0
\(979\) 26.8073 0.856766
\(980\) 0 0
\(981\) −2.06866 −0.0660473
\(982\) 0 0
\(983\) −3.35239 −0.106925 −0.0534624 0.998570i \(-0.517026\pi\)
−0.0534624 + 0.998570i \(0.517026\pi\)
\(984\) 0 0
\(985\) −32.7835 −1.04457
\(986\) 0 0
\(987\) 0.215066 0.00684563
\(988\) 0 0
\(989\) 13.0224 0.414089
\(990\) 0 0
\(991\) −51.1001 −1.62325 −0.811625 0.584179i \(-0.801417\pi\)
−0.811625 + 0.584179i \(0.801417\pi\)
\(992\) 0 0
\(993\) −7.87601 −0.249938
\(994\) 0 0
\(995\) 28.8669 0.915143
\(996\) 0 0
\(997\) −39.9036 −1.26376 −0.631881 0.775066i \(-0.717717\pi\)
−0.631881 + 0.775066i \(0.717717\pi\)
\(998\) 0 0
\(999\) −7.72294 −0.244343
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.bx.1.1 3
4.3 odd 2 3864.2.a.m.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3864.2.a.m.1.1 3 4.3 odd 2
7728.2.a.bx.1.1 3 1.1 even 1 trivial