Properties

Label 6384.2.a.bs.1.2
Level $6384$
Weight $2$
Character 6384.1
Self dual yes
Analytic conductor $50.976$
Analytic rank $1$
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] = [6384,2,Mod(1,6384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6384, 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("6384.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6384 = 2^{4} \cdot 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6384.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: \(50.9764966504\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3192)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 6384.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +2.73205 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +2.73205 q^{5} -1.00000 q^{7} +1.00000 q^{9} -5.46410 q^{13} +2.73205 q^{15} -1.26795 q^{17} +1.00000 q^{19} -1.00000 q^{21} +2.46410 q^{25} +1.00000 q^{27} -9.66025 q^{29} -2.00000 q^{31} -2.73205 q^{35} -10.0000 q^{37} -5.46410 q^{39} -8.92820 q^{41} -4.92820 q^{43} +2.73205 q^{45} +11.6603 q^{47} +1.00000 q^{49} -1.26795 q^{51} -6.73205 q^{53} +1.00000 q^{57} -8.00000 q^{59} -2.00000 q^{61} -1.00000 q^{63} -14.9282 q^{65} +13.4641 q^{67} -4.73205 q^{71} +10.3923 q^{73} +2.46410 q^{75} -4.00000 q^{79} +1.00000 q^{81} -10.1962 q^{83} -3.46410 q^{85} -9.66025 q^{87} +4.92820 q^{89} +5.46410 q^{91} -2.00000 q^{93} +2.73205 q^{95} +2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} - 4 q^{13} + 2 q^{15} - 6 q^{17} + 2 q^{19} - 2 q^{21} - 2 q^{25} + 2 q^{27} - 2 q^{29} - 4 q^{31} - 2 q^{35} - 20 q^{37} - 4 q^{39} - 4 q^{41} + 4 q^{43} + 2 q^{45} + 6 q^{47} + 2 q^{49} - 6 q^{51} - 10 q^{53} + 2 q^{57} - 16 q^{59} - 4 q^{61} - 2 q^{63} - 16 q^{65} + 20 q^{67} - 6 q^{71} - 2 q^{75} - 8 q^{79} + 2 q^{81} - 10 q^{83} - 2 q^{87} - 4 q^{89} + 4 q^{91} - 4 q^{93} + 2 q^{95} + 4 q^{97}+O(q^{100}) \) 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
\(5\) 2.73205 1.22181 0.610905 0.791704i \(-0.290806\pi\)
0.610905 + 0.791704i \(0.290806\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −5.46410 −1.51547 −0.757735 0.652563i \(-0.773694\pi\)
−0.757735 + 0.652563i \(0.773694\pi\)
\(14\) 0 0
\(15\) 2.73205 0.705412
\(16\) 0 0
\(17\) −1.26795 −0.307523 −0.153761 0.988108i \(-0.549139\pi\)
−0.153761 + 0.988108i \(0.549139\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 2.46410 0.492820
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −9.66025 −1.79386 −0.896932 0.442168i \(-0.854209\pi\)
−0.896932 + 0.442168i \(0.854209\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.73205 −0.461801
\(36\) 0 0
\(37\) −10.0000 −1.64399 −0.821995 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 0 0
\(39\) −5.46410 −0.874957
\(40\) 0 0
\(41\) −8.92820 −1.39435 −0.697176 0.716900i \(-0.745560\pi\)
−0.697176 + 0.716900i \(0.745560\pi\)
\(42\) 0 0
\(43\) −4.92820 −0.751544 −0.375772 0.926712i \(-0.622622\pi\)
−0.375772 + 0.926712i \(0.622622\pi\)
\(44\) 0 0
\(45\) 2.73205 0.407270
\(46\) 0 0
\(47\) 11.6603 1.70082 0.850411 0.526118i \(-0.176353\pi\)
0.850411 + 0.526118i \(0.176353\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −1.26795 −0.177548
\(52\) 0 0
\(53\) −6.73205 −0.924718 −0.462359 0.886693i \(-0.652997\pi\)
−0.462359 + 0.886693i \(0.652997\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) 0 0
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) −14.9282 −1.85162
\(66\) 0 0
\(67\) 13.4641 1.64490 0.822451 0.568836i \(-0.192606\pi\)
0.822451 + 0.568836i \(0.192606\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −4.73205 −0.561591 −0.280796 0.959768i \(-0.590598\pi\)
−0.280796 + 0.959768i \(0.590598\pi\)
\(72\) 0 0
\(73\) 10.3923 1.21633 0.608164 0.793812i \(-0.291906\pi\)
0.608164 + 0.793812i \(0.291906\pi\)
\(74\) 0 0
\(75\) 2.46410 0.284530
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −10.1962 −1.11917 −0.559587 0.828772i \(-0.689040\pi\)
−0.559587 + 0.828772i \(0.689040\pi\)
\(84\) 0 0
\(85\) −3.46410 −0.375735
\(86\) 0 0
\(87\) −9.66025 −1.03569
\(88\) 0 0
\(89\) 4.92820 0.522388 0.261194 0.965286i \(-0.415884\pi\)
0.261194 + 0.965286i \(0.415884\pi\)
\(90\) 0 0
\(91\) 5.46410 0.572793
\(92\) 0 0
\(93\) −2.00000 −0.207390
\(94\) 0 0
\(95\) 2.73205 0.280302
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1.26795 0.126166 0.0630828 0.998008i \(-0.479907\pi\)
0.0630828 + 0.998008i \(0.479907\pi\)
\(102\) 0 0
\(103\) 2.92820 0.288524 0.144262 0.989539i \(-0.453919\pi\)
0.144262 + 0.989539i \(0.453919\pi\)
\(104\) 0 0
\(105\) −2.73205 −0.266621
\(106\) 0 0
\(107\) 4.73205 0.457465 0.228732 0.973489i \(-0.426542\pi\)
0.228732 + 0.973489i \(0.426542\pi\)
\(108\) 0 0
\(109\) 2.39230 0.229141 0.114571 0.993415i \(-0.463451\pi\)
0.114571 + 0.993415i \(0.463451\pi\)
\(110\) 0 0
\(111\) −10.0000 −0.949158
\(112\) 0 0
\(113\) 0.196152 0.0184525 0.00922623 0.999957i \(-0.497063\pi\)
0.00922623 + 0.999957i \(0.497063\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −5.46410 −0.505156
\(118\) 0 0
\(119\) 1.26795 0.116233
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) −8.92820 −0.805029
\(124\) 0 0
\(125\) −6.92820 −0.619677
\(126\) 0 0
\(127\) −20.3923 −1.80952 −0.904762 0.425917i \(-0.859952\pi\)
−0.904762 + 0.425917i \(0.859952\pi\)
\(128\) 0 0
\(129\) −4.92820 −0.433904
\(130\) 0 0
\(131\) −14.5885 −1.27460 −0.637300 0.770616i \(-0.719949\pi\)
−0.637300 + 0.770616i \(0.719949\pi\)
\(132\) 0 0
\(133\) −1.00000 −0.0867110
\(134\) 0 0
\(135\) 2.73205 0.235137
\(136\) 0 0
\(137\) 12.9282 1.10453 0.552265 0.833668i \(-0.313763\pi\)
0.552265 + 0.833668i \(0.313763\pi\)
\(138\) 0 0
\(139\) 15.3205 1.29947 0.649734 0.760161i \(-0.274880\pi\)
0.649734 + 0.760161i \(0.274880\pi\)
\(140\) 0 0
\(141\) 11.6603 0.981971
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −26.3923 −2.19176
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) 14.3923 1.17906 0.589532 0.807745i \(-0.299312\pi\)
0.589532 + 0.807745i \(0.299312\pi\)
\(150\) 0 0
\(151\) 2.53590 0.206368 0.103184 0.994662i \(-0.467097\pi\)
0.103184 + 0.994662i \(0.467097\pi\)
\(152\) 0 0
\(153\) −1.26795 −0.102508
\(154\) 0 0
\(155\) −5.46410 −0.438887
\(156\) 0 0
\(157\) −14.0000 −1.11732 −0.558661 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) 0 0
\(159\) −6.73205 −0.533886
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 7.07180 0.553906 0.276953 0.960883i \(-0.410675\pi\)
0.276953 + 0.960883i \(0.410675\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −11.3205 −0.876007 −0.438004 0.898973i \(-0.644314\pi\)
−0.438004 + 0.898973i \(0.644314\pi\)
\(168\) 0 0
\(169\) 16.8564 1.29665
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 0 0
\(173\) −16.9282 −1.28703 −0.643514 0.765435i \(-0.722524\pi\)
−0.643514 + 0.765435i \(0.722524\pi\)
\(174\) 0 0
\(175\) −2.46410 −0.186269
\(176\) 0 0
\(177\) −8.00000 −0.601317
\(178\) 0 0
\(179\) 14.1962 1.06107 0.530535 0.847663i \(-0.321991\pi\)
0.530535 + 0.847663i \(0.321991\pi\)
\(180\) 0 0
\(181\) 4.92820 0.366310 0.183155 0.983084i \(-0.441369\pi\)
0.183155 + 0.983084i \(0.441369\pi\)
\(182\) 0 0
\(183\) −2.00000 −0.147844
\(184\) 0 0
\(185\) −27.3205 −2.00864
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) 14.9282 1.08017 0.540083 0.841611i \(-0.318393\pi\)
0.540083 + 0.841611i \(0.318393\pi\)
\(192\) 0 0
\(193\) −23.8564 −1.71722 −0.858611 0.512628i \(-0.828672\pi\)
−0.858611 + 0.512628i \(0.828672\pi\)
\(194\) 0 0
\(195\) −14.9282 −1.06903
\(196\) 0 0
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) 12.3923 0.878467 0.439234 0.898373i \(-0.355250\pi\)
0.439234 + 0.898373i \(0.355250\pi\)
\(200\) 0 0
\(201\) 13.4641 0.949685
\(202\) 0 0
\(203\) 9.66025 0.678017
\(204\) 0 0
\(205\) −24.3923 −1.70363
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 0 0
\(213\) −4.73205 −0.324235
\(214\) 0 0
\(215\) −13.4641 −0.918244
\(216\) 0 0
\(217\) 2.00000 0.135769
\(218\) 0 0
\(219\) 10.3923 0.702247
\(220\) 0 0
\(221\) 6.92820 0.466041
\(222\) 0 0
\(223\) 20.9282 1.40146 0.700728 0.713428i \(-0.252859\pi\)
0.700728 + 0.713428i \(0.252859\pi\)
\(224\) 0 0
\(225\) 2.46410 0.164273
\(226\) 0 0
\(227\) −16.3923 −1.08800 −0.543998 0.839087i \(-0.683090\pi\)
−0.543998 + 0.839087i \(0.683090\pi\)
\(228\) 0 0
\(229\) 28.2487 1.86673 0.933364 0.358932i \(-0.116859\pi\)
0.933364 + 0.358932i \(0.116859\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −2.39230 −0.156725 −0.0783626 0.996925i \(-0.524969\pi\)
−0.0783626 + 0.996925i \(0.524969\pi\)
\(234\) 0 0
\(235\) 31.8564 2.07808
\(236\) 0 0
\(237\) −4.00000 −0.259828
\(238\) 0 0
\(239\) 4.39230 0.284115 0.142057 0.989858i \(-0.454628\pi\)
0.142057 + 0.989858i \(0.454628\pi\)
\(240\) 0 0
\(241\) −12.9282 −0.832779 −0.416389 0.909186i \(-0.636705\pi\)
−0.416389 + 0.909186i \(0.636705\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 2.73205 0.174544
\(246\) 0 0
\(247\) −5.46410 −0.347672
\(248\) 0 0
\(249\) −10.1962 −0.646155
\(250\) 0 0
\(251\) −0.339746 −0.0214446 −0.0107223 0.999943i \(-0.503413\pi\)
−0.0107223 + 0.999943i \(0.503413\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −3.46410 −0.216930
\(256\) 0 0
\(257\) 11.4641 0.715111 0.357556 0.933892i \(-0.383610\pi\)
0.357556 + 0.933892i \(0.383610\pi\)
\(258\) 0 0
\(259\) 10.0000 0.621370
\(260\) 0 0
\(261\) −9.66025 −0.597955
\(262\) 0 0
\(263\) 19.3205 1.19135 0.595677 0.803224i \(-0.296884\pi\)
0.595677 + 0.803224i \(0.296884\pi\)
\(264\) 0 0
\(265\) −18.3923 −1.12983
\(266\) 0 0
\(267\) 4.92820 0.301601
\(268\) 0 0
\(269\) 31.1769 1.90089 0.950445 0.310893i \(-0.100628\pi\)
0.950445 + 0.310893i \(0.100628\pi\)
\(270\) 0 0
\(271\) 28.7846 1.74854 0.874270 0.485440i \(-0.161340\pi\)
0.874270 + 0.485440i \(0.161340\pi\)
\(272\) 0 0
\(273\) 5.46410 0.330702
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 12.3923 0.744581 0.372291 0.928116i \(-0.378572\pi\)
0.372291 + 0.928116i \(0.378572\pi\)
\(278\) 0 0
\(279\) −2.00000 −0.119737
\(280\) 0 0
\(281\) −4.19615 −0.250321 −0.125161 0.992136i \(-0.539945\pi\)
−0.125161 + 0.992136i \(0.539945\pi\)
\(282\) 0 0
\(283\) 20.3923 1.21220 0.606098 0.795390i \(-0.292734\pi\)
0.606098 + 0.795390i \(0.292734\pi\)
\(284\) 0 0
\(285\) 2.73205 0.161833
\(286\) 0 0
\(287\) 8.92820 0.527015
\(288\) 0 0
\(289\) −15.3923 −0.905430
\(290\) 0 0
\(291\) 2.00000 0.117242
\(292\) 0 0
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 0 0
\(295\) −21.8564 −1.27253
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 4.92820 0.284057
\(302\) 0 0
\(303\) 1.26795 0.0728418
\(304\) 0 0
\(305\) −5.46410 −0.312874
\(306\) 0 0
\(307\) −18.7846 −1.07209 −0.536047 0.844188i \(-0.680083\pi\)
−0.536047 + 0.844188i \(0.680083\pi\)
\(308\) 0 0
\(309\) 2.92820 0.166580
\(310\) 0 0
\(311\) −18.5885 −1.05405 −0.527027 0.849848i \(-0.676693\pi\)
−0.527027 + 0.849848i \(0.676693\pi\)
\(312\) 0 0
\(313\) −2.00000 −0.113047 −0.0565233 0.998401i \(-0.518002\pi\)
−0.0565233 + 0.998401i \(0.518002\pi\)
\(314\) 0 0
\(315\) −2.73205 −0.153934
\(316\) 0 0
\(317\) 12.1962 0.685004 0.342502 0.939517i \(-0.388726\pi\)
0.342502 + 0.939517i \(0.388726\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 4.73205 0.264117
\(322\) 0 0
\(323\) −1.26795 −0.0705506
\(324\) 0 0
\(325\) −13.4641 −0.746854
\(326\) 0 0
\(327\) 2.39230 0.132295
\(328\) 0 0
\(329\) −11.6603 −0.642851
\(330\) 0 0
\(331\) −22.2487 −1.22290 −0.611450 0.791283i \(-0.709413\pi\)
−0.611450 + 0.791283i \(0.709413\pi\)
\(332\) 0 0
\(333\) −10.0000 −0.547997
\(334\) 0 0
\(335\) 36.7846 2.00976
\(336\) 0 0
\(337\) 30.3923 1.65557 0.827787 0.561042i \(-0.189599\pi\)
0.827787 + 0.561042i \(0.189599\pi\)
\(338\) 0 0
\(339\) 0.196152 0.0106535
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 16.3923 0.879985 0.439993 0.898001i \(-0.354981\pi\)
0.439993 + 0.898001i \(0.354981\pi\)
\(348\) 0 0
\(349\) −19.0718 −1.02089 −0.510445 0.859910i \(-0.670519\pi\)
−0.510445 + 0.859910i \(0.670519\pi\)
\(350\) 0 0
\(351\) −5.46410 −0.291652
\(352\) 0 0
\(353\) 16.1962 0.862034 0.431017 0.902344i \(-0.358155\pi\)
0.431017 + 0.902344i \(0.358155\pi\)
\(354\) 0 0
\(355\) −12.9282 −0.686158
\(356\) 0 0
\(357\) 1.26795 0.0671070
\(358\) 0 0
\(359\) −30.9282 −1.63233 −0.816164 0.577820i \(-0.803903\pi\)
−0.816164 + 0.577820i \(0.803903\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −11.0000 −0.577350
\(364\) 0 0
\(365\) 28.3923 1.48612
\(366\) 0 0
\(367\) −18.5359 −0.967566 −0.483783 0.875188i \(-0.660738\pi\)
−0.483783 + 0.875188i \(0.660738\pi\)
\(368\) 0 0
\(369\) −8.92820 −0.464784
\(370\) 0 0
\(371\) 6.73205 0.349511
\(372\) 0 0
\(373\) 0.928203 0.0480605 0.0240303 0.999711i \(-0.492350\pi\)
0.0240303 + 0.999711i \(0.492350\pi\)
\(374\) 0 0
\(375\) −6.92820 −0.357771
\(376\) 0 0
\(377\) 52.7846 2.71855
\(378\) 0 0
\(379\) 15.3205 0.786962 0.393481 0.919333i \(-0.371271\pi\)
0.393481 + 0.919333i \(0.371271\pi\)
\(380\) 0 0
\(381\) −20.3923 −1.04473
\(382\) 0 0
\(383\) 11.3205 0.578451 0.289225 0.957261i \(-0.406602\pi\)
0.289225 + 0.957261i \(0.406602\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −4.92820 −0.250515
\(388\) 0 0
\(389\) −20.5359 −1.04121 −0.520606 0.853797i \(-0.674294\pi\)
−0.520606 + 0.853797i \(0.674294\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −14.5885 −0.735890
\(394\) 0 0
\(395\) −10.9282 −0.549858
\(396\) 0 0
\(397\) −21.7128 −1.08973 −0.544867 0.838522i \(-0.683420\pi\)
−0.544867 + 0.838522i \(0.683420\pi\)
\(398\) 0 0
\(399\) −1.00000 −0.0500626
\(400\) 0 0
\(401\) 0.196152 0.00979538 0.00489769 0.999988i \(-0.498441\pi\)
0.00489769 + 0.999988i \(0.498441\pi\)
\(402\) 0 0
\(403\) 10.9282 0.544373
\(404\) 0 0
\(405\) 2.73205 0.135757
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −28.3923 −1.40391 −0.701955 0.712222i \(-0.747689\pi\)
−0.701955 + 0.712222i \(0.747689\pi\)
\(410\) 0 0
\(411\) 12.9282 0.637701
\(412\) 0 0
\(413\) 8.00000 0.393654
\(414\) 0 0
\(415\) −27.8564 −1.36742
\(416\) 0 0
\(417\) 15.3205 0.750249
\(418\) 0 0
\(419\) −20.7321 −1.01283 −0.506413 0.862291i \(-0.669029\pi\)
−0.506413 + 0.862291i \(0.669029\pi\)
\(420\) 0 0
\(421\) −32.2487 −1.57171 −0.785853 0.618413i \(-0.787776\pi\)
−0.785853 + 0.618413i \(0.787776\pi\)
\(422\) 0 0
\(423\) 11.6603 0.566941
\(424\) 0 0
\(425\) −3.12436 −0.151554
\(426\) 0 0
\(427\) 2.00000 0.0967868
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −0.0525589 −0.00253167 −0.00126584 0.999999i \(-0.500403\pi\)
−0.00126584 + 0.999999i \(0.500403\pi\)
\(432\) 0 0
\(433\) −15.8564 −0.762010 −0.381005 0.924573i \(-0.624422\pi\)
−0.381005 + 0.924573i \(0.624422\pi\)
\(434\) 0 0
\(435\) −26.3923 −1.26541
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −35.8564 −1.71133 −0.855666 0.517528i \(-0.826852\pi\)
−0.855666 + 0.517528i \(0.826852\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 34.5359 1.64085 0.820425 0.571754i \(-0.193737\pi\)
0.820425 + 0.571754i \(0.193737\pi\)
\(444\) 0 0
\(445\) 13.4641 0.638260
\(446\) 0 0
\(447\) 14.3923 0.680733
\(448\) 0 0
\(449\) 5.26795 0.248610 0.124305 0.992244i \(-0.460330\pi\)
0.124305 + 0.992244i \(0.460330\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 2.53590 0.119147
\(454\) 0 0
\(455\) 14.9282 0.699845
\(456\) 0 0
\(457\) −20.3923 −0.953912 −0.476956 0.878927i \(-0.658260\pi\)
−0.476956 + 0.878927i \(0.658260\pi\)
\(458\) 0 0
\(459\) −1.26795 −0.0591828
\(460\) 0 0
\(461\) 14.3397 0.667869 0.333934 0.942596i \(-0.391624\pi\)
0.333934 + 0.942596i \(0.391624\pi\)
\(462\) 0 0
\(463\) −21.8564 −1.01575 −0.507877 0.861430i \(-0.669569\pi\)
−0.507877 + 0.861430i \(0.669569\pi\)
\(464\) 0 0
\(465\) −5.46410 −0.253392
\(466\) 0 0
\(467\) 0.0525589 0.00243214 0.00121607 0.999999i \(-0.499613\pi\)
0.00121607 + 0.999999i \(0.499613\pi\)
\(468\) 0 0
\(469\) −13.4641 −0.621714
\(470\) 0 0
\(471\) −14.0000 −0.645086
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 2.46410 0.113061
\(476\) 0 0
\(477\) −6.73205 −0.308239
\(478\) 0 0
\(479\) −27.6603 −1.26383 −0.631915 0.775038i \(-0.717731\pi\)
−0.631915 + 0.775038i \(0.717731\pi\)
\(480\) 0 0
\(481\) 54.6410 2.49142
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 5.46410 0.248112
\(486\) 0 0
\(487\) −25.8564 −1.17167 −0.585833 0.810432i \(-0.699232\pi\)
−0.585833 + 0.810432i \(0.699232\pi\)
\(488\) 0 0
\(489\) 7.07180 0.319798
\(490\) 0 0
\(491\) −19.3205 −0.871922 −0.435961 0.899965i \(-0.643591\pi\)
−0.435961 + 0.899965i \(0.643591\pi\)
\(492\) 0 0
\(493\) 12.2487 0.551654
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 4.73205 0.212261
\(498\) 0 0
\(499\) 9.85641 0.441233 0.220617 0.975361i \(-0.429193\pi\)
0.220617 + 0.975361i \(0.429193\pi\)
\(500\) 0 0
\(501\) −11.3205 −0.505763
\(502\) 0 0
\(503\) −4.73205 −0.210992 −0.105496 0.994420i \(-0.533643\pi\)
−0.105496 + 0.994420i \(0.533643\pi\)
\(504\) 0 0
\(505\) 3.46410 0.154150
\(506\) 0 0
\(507\) 16.8564 0.748619
\(508\) 0 0
\(509\) −8.24871 −0.365618 −0.182809 0.983148i \(-0.558519\pi\)
−0.182809 + 0.983148i \(0.558519\pi\)
\(510\) 0 0
\(511\) −10.3923 −0.459728
\(512\) 0 0
\(513\) 1.00000 0.0441511
\(514\) 0 0
\(515\) 8.00000 0.352522
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −16.9282 −0.743066
\(520\) 0 0
\(521\) 36.2487 1.58808 0.794042 0.607862i \(-0.207973\pi\)
0.794042 + 0.607862i \(0.207973\pi\)
\(522\) 0 0
\(523\) 22.0000 0.961993 0.480996 0.876723i \(-0.340275\pi\)
0.480996 + 0.876723i \(0.340275\pi\)
\(524\) 0 0
\(525\) −2.46410 −0.107542
\(526\) 0 0
\(527\) 2.53590 0.110465
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) −8.00000 −0.347170
\(532\) 0 0
\(533\) 48.7846 2.11310
\(534\) 0 0
\(535\) 12.9282 0.558935
\(536\) 0 0
\(537\) 14.1962 0.612609
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −22.0000 −0.945854 −0.472927 0.881102i \(-0.656803\pi\)
−0.472927 + 0.881102i \(0.656803\pi\)
\(542\) 0 0
\(543\) 4.92820 0.211489
\(544\) 0 0
\(545\) 6.53590 0.279967
\(546\) 0 0
\(547\) 11.7128 0.500804 0.250402 0.968142i \(-0.419437\pi\)
0.250402 + 0.968142i \(0.419437\pi\)
\(548\) 0 0
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) −9.66025 −0.411541
\(552\) 0 0
\(553\) 4.00000 0.170097
\(554\) 0 0
\(555\) −27.3205 −1.15969
\(556\) 0 0
\(557\) 20.5359 0.870134 0.435067 0.900398i \(-0.356725\pi\)
0.435067 + 0.900398i \(0.356725\pi\)
\(558\) 0 0
\(559\) 26.9282 1.13894
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 25.4641 1.07318 0.536592 0.843842i \(-0.319711\pi\)
0.536592 + 0.843842i \(0.319711\pi\)
\(564\) 0 0
\(565\) 0.535898 0.0225454
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 28.5885 1.19849 0.599245 0.800566i \(-0.295467\pi\)
0.599245 + 0.800566i \(0.295467\pi\)
\(570\) 0 0
\(571\) 25.8564 1.08206 0.541028 0.841004i \(-0.318035\pi\)
0.541028 + 0.841004i \(0.318035\pi\)
\(572\) 0 0
\(573\) 14.9282 0.623635
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −10.0000 −0.416305 −0.208153 0.978096i \(-0.566745\pi\)
−0.208153 + 0.978096i \(0.566745\pi\)
\(578\) 0 0
\(579\) −23.8564 −0.991438
\(580\) 0 0
\(581\) 10.1962 0.423008
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −14.9282 −0.617205
\(586\) 0 0
\(587\) −25.9090 −1.06938 −0.534689 0.845049i \(-0.679571\pi\)
−0.534689 + 0.845049i \(0.679571\pi\)
\(588\) 0 0
\(589\) −2.00000 −0.0824086
\(590\) 0 0
\(591\) −18.0000 −0.740421
\(592\) 0 0
\(593\) 27.8038 1.14177 0.570884 0.821031i \(-0.306601\pi\)
0.570884 + 0.821031i \(0.306601\pi\)
\(594\) 0 0
\(595\) 3.46410 0.142014
\(596\) 0 0
\(597\) 12.3923 0.507183
\(598\) 0 0
\(599\) 7.26795 0.296960 0.148480 0.988915i \(-0.452562\pi\)
0.148480 + 0.988915i \(0.452562\pi\)
\(600\) 0 0
\(601\) 34.0000 1.38689 0.693444 0.720510i \(-0.256092\pi\)
0.693444 + 0.720510i \(0.256092\pi\)
\(602\) 0 0
\(603\) 13.4641 0.548301
\(604\) 0 0
\(605\) −30.0526 −1.22181
\(606\) 0 0
\(607\) 13.0718 0.530568 0.265284 0.964170i \(-0.414534\pi\)
0.265284 + 0.964170i \(0.414534\pi\)
\(608\) 0 0
\(609\) 9.66025 0.391453
\(610\) 0 0
\(611\) −63.7128 −2.57754
\(612\) 0 0
\(613\) −18.2487 −0.737059 −0.368529 0.929616i \(-0.620139\pi\)
−0.368529 + 0.929616i \(0.620139\pi\)
\(614\) 0 0
\(615\) −24.3923 −0.983593
\(616\) 0 0
\(617\) −24.6410 −0.992010 −0.496005 0.868320i \(-0.665200\pi\)
−0.496005 + 0.868320i \(0.665200\pi\)
\(618\) 0 0
\(619\) 13.1769 0.529625 0.264812 0.964300i \(-0.414690\pi\)
0.264812 + 0.964300i \(0.414690\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −4.92820 −0.197444
\(624\) 0 0
\(625\) −31.2487 −1.24995
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 12.6795 0.505564
\(630\) 0 0
\(631\) −40.9282 −1.62933 −0.814663 0.579935i \(-0.803078\pi\)
−0.814663 + 0.579935i \(0.803078\pi\)
\(632\) 0 0
\(633\) −4.00000 −0.158986
\(634\) 0 0
\(635\) −55.7128 −2.21090
\(636\) 0 0
\(637\) −5.46410 −0.216496
\(638\) 0 0
\(639\) −4.73205 −0.187197
\(640\) 0 0
\(641\) −6.33975 −0.250405 −0.125202 0.992131i \(-0.539958\pi\)
−0.125202 + 0.992131i \(0.539958\pi\)
\(642\) 0 0
\(643\) −20.3923 −0.804194 −0.402097 0.915597i \(-0.631719\pi\)
−0.402097 + 0.915597i \(0.631719\pi\)
\(644\) 0 0
\(645\) −13.4641 −0.530148
\(646\) 0 0
\(647\) 3.66025 0.143899 0.0719497 0.997408i \(-0.477078\pi\)
0.0719497 + 0.997408i \(0.477078\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 2.00000 0.0783862
\(652\) 0 0
\(653\) −42.7846 −1.67429 −0.837146 0.546980i \(-0.815777\pi\)
−0.837146 + 0.546980i \(0.815777\pi\)
\(654\) 0 0
\(655\) −39.8564 −1.55732
\(656\) 0 0
\(657\) 10.3923 0.405442
\(658\) 0 0
\(659\) −32.0526 −1.24859 −0.624295 0.781189i \(-0.714614\pi\)
−0.624295 + 0.781189i \(0.714614\pi\)
\(660\) 0 0
\(661\) −6.53590 −0.254217 −0.127108 0.991889i \(-0.540570\pi\)
−0.127108 + 0.991889i \(0.540570\pi\)
\(662\) 0 0
\(663\) 6.92820 0.269069
\(664\) 0 0
\(665\) −2.73205 −0.105944
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 20.9282 0.809131
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −7.07180 −0.272598 −0.136299 0.990668i \(-0.543521\pi\)
−0.136299 + 0.990668i \(0.543521\pi\)
\(674\) 0 0
\(675\) 2.46410 0.0948433
\(676\) 0 0
\(677\) 29.3205 1.12688 0.563439 0.826157i \(-0.309478\pi\)
0.563439 + 0.826157i \(0.309478\pi\)
\(678\) 0 0
\(679\) −2.00000 −0.0767530
\(680\) 0 0
\(681\) −16.3923 −0.628154
\(682\) 0 0
\(683\) −4.73205 −0.181067 −0.0905334 0.995893i \(-0.528857\pi\)
−0.0905334 + 0.995893i \(0.528857\pi\)
\(684\) 0 0
\(685\) 35.3205 1.34953
\(686\) 0 0
\(687\) 28.2487 1.07776
\(688\) 0 0
\(689\) 36.7846 1.40138
\(690\) 0 0
\(691\) 8.78461 0.334182 0.167091 0.985941i \(-0.446563\pi\)
0.167091 + 0.985941i \(0.446563\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 41.8564 1.58770
\(696\) 0 0
\(697\) 11.3205 0.428795
\(698\) 0 0
\(699\) −2.39230 −0.0904853
\(700\) 0 0
\(701\) 2.00000 0.0755390 0.0377695 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) 0 0
\(703\) −10.0000 −0.377157
\(704\) 0 0
\(705\) 31.8564 1.19978
\(706\) 0 0
\(707\) −1.26795 −0.0476861
\(708\) 0 0
\(709\) 0.392305 0.0147333 0.00736666 0.999973i \(-0.497655\pi\)
0.00736666 + 0.999973i \(0.497655\pi\)
\(710\) 0 0
\(711\) −4.00000 −0.150012
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 4.39230 0.164034
\(718\) 0 0
\(719\) −46.9808 −1.75209 −0.876043 0.482232i \(-0.839826\pi\)
−0.876043 + 0.482232i \(0.839826\pi\)
\(720\) 0 0
\(721\) −2.92820 −0.109052
\(722\) 0 0
\(723\) −12.9282 −0.480805
\(724\) 0 0
\(725\) −23.8038 −0.884053
\(726\) 0 0
\(727\) −18.9282 −0.702008 −0.351004 0.936374i \(-0.614160\pi\)
−0.351004 + 0.936374i \(0.614160\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 6.24871 0.231117
\(732\) 0 0
\(733\) 4.24871 0.156930 0.0784649 0.996917i \(-0.474998\pi\)
0.0784649 + 0.996917i \(0.474998\pi\)
\(734\) 0 0
\(735\) 2.73205 0.100773
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 20.1436 0.740994 0.370497 0.928834i \(-0.379187\pi\)
0.370497 + 0.928834i \(0.379187\pi\)
\(740\) 0 0
\(741\) −5.46410 −0.200729
\(742\) 0 0
\(743\) 8.73205 0.320348 0.160174 0.987089i \(-0.448794\pi\)
0.160174 + 0.987089i \(0.448794\pi\)
\(744\) 0 0
\(745\) 39.3205 1.44059
\(746\) 0 0
\(747\) −10.1962 −0.373058
\(748\) 0 0
\(749\) −4.73205 −0.172905
\(750\) 0 0
\(751\) 49.1769 1.79449 0.897246 0.441532i \(-0.145565\pi\)
0.897246 + 0.441532i \(0.145565\pi\)
\(752\) 0 0
\(753\) −0.339746 −0.0123810
\(754\) 0 0
\(755\) 6.92820 0.252143
\(756\) 0 0
\(757\) −22.7846 −0.828121 −0.414060 0.910249i \(-0.635890\pi\)
−0.414060 + 0.910249i \(0.635890\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 33.6603 1.22018 0.610092 0.792331i \(-0.291133\pi\)
0.610092 + 0.792331i \(0.291133\pi\)
\(762\) 0 0
\(763\) −2.39230 −0.0866073
\(764\) 0 0
\(765\) −3.46410 −0.125245
\(766\) 0 0
\(767\) 43.7128 1.57838
\(768\) 0 0
\(769\) 0.143594 0.00517812 0.00258906 0.999997i \(-0.499176\pi\)
0.00258906 + 0.999997i \(0.499176\pi\)
\(770\) 0 0
\(771\) 11.4641 0.412870
\(772\) 0 0
\(773\) −14.0000 −0.503545 −0.251773 0.967786i \(-0.581013\pi\)
−0.251773 + 0.967786i \(0.581013\pi\)
\(774\) 0 0
\(775\) −4.92820 −0.177026
\(776\) 0 0
\(777\) 10.0000 0.358748
\(778\) 0 0
\(779\) −8.92820 −0.319886
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −9.66025 −0.345229
\(784\) 0 0
\(785\) −38.2487 −1.36516
\(786\) 0 0
\(787\) −27.8564 −0.992974 −0.496487 0.868044i \(-0.665377\pi\)
−0.496487 + 0.868044i \(0.665377\pi\)
\(788\) 0 0
\(789\) 19.3205 0.687828
\(790\) 0 0
\(791\) −0.196152 −0.00697438
\(792\) 0 0
\(793\) 10.9282 0.388072
\(794\) 0 0
\(795\) −18.3923 −0.652308
\(796\) 0 0
\(797\) −10.7846 −0.382010 −0.191005 0.981589i \(-0.561175\pi\)
−0.191005 + 0.981589i \(0.561175\pi\)
\(798\) 0 0
\(799\) −14.7846 −0.523042
\(800\) 0 0
\(801\) 4.92820 0.174129
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 31.1769 1.09748
\(808\) 0 0
\(809\) −10.7846 −0.379167 −0.189583 0.981865i \(-0.560714\pi\)
−0.189583 + 0.981865i \(0.560714\pi\)
\(810\) 0 0
\(811\) −4.00000 −0.140459 −0.0702295 0.997531i \(-0.522373\pi\)
−0.0702295 + 0.997531i \(0.522373\pi\)
\(812\) 0 0
\(813\) 28.7846 1.00952
\(814\) 0 0
\(815\) 19.3205 0.676768
\(816\) 0 0
\(817\) −4.92820 −0.172416
\(818\) 0 0
\(819\) 5.46410 0.190931
\(820\) 0 0
\(821\) 35.4641 1.23771 0.618853 0.785507i \(-0.287598\pi\)
0.618853 + 0.785507i \(0.287598\pi\)
\(822\) 0 0
\(823\) −20.6410 −0.719501 −0.359750 0.933049i \(-0.617138\pi\)
−0.359750 + 0.933049i \(0.617138\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 53.9090 1.87460 0.937299 0.348526i \(-0.113318\pi\)
0.937299 + 0.348526i \(0.113318\pi\)
\(828\) 0 0
\(829\) −37.7128 −1.30982 −0.654910 0.755707i \(-0.727294\pi\)
−0.654910 + 0.755707i \(0.727294\pi\)
\(830\) 0 0
\(831\) 12.3923 0.429884
\(832\) 0 0
\(833\) −1.26795 −0.0439318
\(834\) 0 0
\(835\) −30.9282 −1.07031
\(836\) 0 0
\(837\) −2.00000 −0.0691301
\(838\) 0 0
\(839\) 28.7846 0.993755 0.496878 0.867821i \(-0.334480\pi\)
0.496878 + 0.867821i \(0.334480\pi\)
\(840\) 0 0
\(841\) 64.3205 2.21795
\(842\) 0 0
\(843\) −4.19615 −0.144523
\(844\) 0 0
\(845\) 46.0526 1.58426
\(846\) 0 0
\(847\) 11.0000 0.377964
\(848\) 0 0
\(849\) 20.3923 0.699862
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −16.1436 −0.552746 −0.276373 0.961050i \(-0.589133\pi\)
−0.276373 + 0.961050i \(0.589133\pi\)
\(854\) 0 0
\(855\) 2.73205 0.0934342
\(856\) 0 0
\(857\) 16.9282 0.578256 0.289128 0.957290i \(-0.406635\pi\)
0.289128 + 0.957290i \(0.406635\pi\)
\(858\) 0 0
\(859\) −45.8564 −1.56460 −0.782300 0.622902i \(-0.785954\pi\)
−0.782300 + 0.622902i \(0.785954\pi\)
\(860\) 0 0
\(861\) 8.92820 0.304272
\(862\) 0 0
\(863\) 42.9808 1.46308 0.731541 0.681797i \(-0.238801\pi\)
0.731541 + 0.681797i \(0.238801\pi\)
\(864\) 0 0
\(865\) −46.2487 −1.57250
\(866\) 0 0
\(867\) −15.3923 −0.522750
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −73.5692 −2.49280
\(872\) 0 0
\(873\) 2.00000 0.0676897
\(874\) 0 0
\(875\) 6.92820 0.234216
\(876\) 0 0
\(877\) −53.7128 −1.81375 −0.906876 0.421397i \(-0.861540\pi\)
−0.906876 + 0.421397i \(0.861540\pi\)
\(878\) 0 0
\(879\) −6.00000 −0.202375
\(880\) 0 0
\(881\) −2.73205 −0.0920451 −0.0460226 0.998940i \(-0.514655\pi\)
−0.0460226 + 0.998940i \(0.514655\pi\)
\(882\) 0 0
\(883\) −9.07180 −0.305290 −0.152645 0.988281i \(-0.548779\pi\)
−0.152645 + 0.988281i \(0.548779\pi\)
\(884\) 0 0
\(885\) −21.8564 −0.734695
\(886\) 0 0
\(887\) −42.2487 −1.41857 −0.709286 0.704920i \(-0.750983\pi\)
−0.709286 + 0.704920i \(0.750983\pi\)
\(888\) 0 0
\(889\) 20.3923 0.683936
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 11.6603 0.390196
\(894\) 0 0
\(895\) 38.7846 1.29643
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 19.3205 0.644375
\(900\) 0 0
\(901\) 8.53590 0.284372
\(902\) 0 0
\(903\) 4.92820 0.164000
\(904\) 0 0
\(905\) 13.4641 0.447562
\(906\) 0 0
\(907\) −14.6410 −0.486147 −0.243073 0.970008i \(-0.578156\pi\)
−0.243073 + 0.970008i \(0.578156\pi\)
\(908\) 0 0
\(909\) 1.26795 0.0420552
\(910\) 0 0
\(911\) 12.7321 0.421832 0.210916 0.977504i \(-0.432355\pi\)
0.210916 + 0.977504i \(0.432355\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −5.46410 −0.180638
\(916\) 0 0
\(917\) 14.5885 0.481753
\(918\) 0 0
\(919\) 29.8564 0.984872 0.492436 0.870349i \(-0.336107\pi\)
0.492436 + 0.870349i \(0.336107\pi\)
\(920\) 0 0
\(921\) −18.7846 −0.618974
\(922\) 0 0
\(923\) 25.8564 0.851074
\(924\) 0 0
\(925\) −24.6410 −0.810192
\(926\) 0 0
\(927\) 2.92820 0.0961748
\(928\) 0 0
\(929\) −4.98076 −0.163414 −0.0817068 0.996656i \(-0.526037\pi\)
−0.0817068 + 0.996656i \(0.526037\pi\)
\(930\) 0 0
\(931\) 1.00000 0.0327737
\(932\) 0 0
\(933\) −18.5885 −0.608559
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 48.6410 1.58903 0.794516 0.607243i \(-0.207724\pi\)
0.794516 + 0.607243i \(0.207724\pi\)
\(938\) 0 0
\(939\) −2.00000 −0.0652675
\(940\) 0 0
\(941\) −51.4641 −1.67768 −0.838841 0.544377i \(-0.816766\pi\)
−0.838841 + 0.544377i \(0.816766\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −2.73205 −0.0888736
\(946\) 0 0
\(947\) −9.07180 −0.294794 −0.147397 0.989077i \(-0.547089\pi\)
−0.147397 + 0.989077i \(0.547089\pi\)
\(948\) 0 0
\(949\) −56.7846 −1.84331
\(950\) 0 0
\(951\) 12.1962 0.395487
\(952\) 0 0
\(953\) 40.1962 1.30208 0.651041 0.759043i \(-0.274333\pi\)
0.651041 + 0.759043i \(0.274333\pi\)
\(954\) 0 0
\(955\) 40.7846 1.31976
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −12.9282 −0.417473
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) 4.73205 0.152488
\(964\) 0 0
\(965\) −65.1769 −2.09812
\(966\) 0 0
\(967\) 36.6410 1.17830 0.589148 0.808025i \(-0.299464\pi\)
0.589148 + 0.808025i \(0.299464\pi\)
\(968\) 0 0
\(969\) −1.26795 −0.0407324
\(970\) 0 0
\(971\) −4.28719 −0.137582 −0.0687912 0.997631i \(-0.521914\pi\)
−0.0687912 + 0.997631i \(0.521914\pi\)
\(972\) 0 0
\(973\) −15.3205 −0.491153
\(974\) 0 0
\(975\) −13.4641 −0.431196
\(976\) 0 0
\(977\) −28.3013 −0.905438 −0.452719 0.891653i \(-0.649546\pi\)
−0.452719 + 0.891653i \(0.649546\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 2.39230 0.0763804
\(982\) 0 0
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) 0 0
\(985\) −49.1769 −1.56691
\(986\) 0 0
\(987\) −11.6603 −0.371150
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 12.7846 0.406117 0.203058 0.979167i \(-0.434912\pi\)
0.203058 + 0.979167i \(0.434912\pi\)
\(992\) 0 0
\(993\) −22.2487 −0.706042
\(994\) 0 0
\(995\) 33.8564 1.07332
\(996\) 0 0
\(997\) −19.8564 −0.628859 −0.314429 0.949281i \(-0.601813\pi\)
−0.314429 + 0.949281i \(0.601813\pi\)
\(998\) 0 0
\(999\) −10.0000 −0.316386
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 6384.2.a.bs.1.2 2
4.3 odd 2 3192.2.a.q.1.2 2
12.11 even 2 9576.2.a.bj.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3192.2.a.q.1.2 2 4.3 odd 2
6384.2.a.bs.1.2 2 1.1 even 1 trivial
9576.2.a.bj.1.1 2 12.11 even 2