Properties

Label 5184.2.a.bl.1.1
Level $5184$
Weight $2$
Character 5184.1
Self dual yes
Analytic conductor $41.394$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5184,2,Mod(1,5184)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5184, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("5184.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5184 = 2^{6} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5184.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: \(41.3944484078\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 288)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 5184.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{5} -1.73205 q^{7} +O(q^{10})\) \(q-1.00000 q^{5} -1.73205 q^{7} +1.73205 q^{11} +3.00000 q^{13} -4.00000 q^{17} -6.92820 q^{19} +8.66025 q^{23} -4.00000 q^{25} +1.00000 q^{29} +5.19615 q^{31} +1.73205 q^{35} +8.00000 q^{37} -5.00000 q^{41} +8.66025 q^{43} -12.1244 q^{47} -4.00000 q^{49} -8.00000 q^{53} -1.73205 q^{55} +1.73205 q^{59} +7.00000 q^{61} -3.00000 q^{65} +8.66025 q^{67} -3.46410 q^{71} -12.0000 q^{73} -3.00000 q^{77} -5.19615 q^{79} -8.66025 q^{83} +4.00000 q^{85} +4.00000 q^{89} -5.19615 q^{91} +6.92820 q^{95} -3.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} + 6 q^{13} - 8 q^{17} - 8 q^{25} + 2 q^{29} + 16 q^{37} - 10 q^{41} - 8 q^{49} - 16 q^{53} + 14 q^{61} - 6 q^{65} - 24 q^{73} - 6 q^{77} + 8 q^{85} + 8 q^{89} - 6 q^{97}+O(q^{100}) \) 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
\(3\) 0 0
\(4\) 0 0
\(5\) −1.00000 −0.447214 −0.223607 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) 0 0
\(7\) −1.73205 −0.654654 −0.327327 0.944911i \(-0.606148\pi\)
−0.327327 + 0.944911i \(0.606148\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.73205 0.522233 0.261116 0.965307i \(-0.415909\pi\)
0.261116 + 0.965307i \(0.415909\pi\)
\(12\) 0 0
\(13\) 3.00000 0.832050 0.416025 0.909353i \(-0.363423\pi\)
0.416025 + 0.909353i \(0.363423\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 0 0
\(19\) −6.92820 −1.58944 −0.794719 0.606977i \(-0.792382\pi\)
−0.794719 + 0.606977i \(0.792382\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.66025 1.80579 0.902894 0.429863i \(-0.141438\pi\)
0.902894 + 0.429863i \(0.141438\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.00000 0.185695 0.0928477 0.995680i \(-0.470403\pi\)
0.0928477 + 0.995680i \(0.470403\pi\)
\(30\) 0 0
\(31\) 5.19615 0.933257 0.466628 0.884454i \(-0.345469\pi\)
0.466628 + 0.884454i \(0.345469\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.73205 0.292770
\(36\) 0 0
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −5.00000 −0.780869 −0.390434 0.920631i \(-0.627675\pi\)
−0.390434 + 0.920631i \(0.627675\pi\)
\(42\) 0 0
\(43\) 8.66025 1.32068 0.660338 0.750968i \(-0.270413\pi\)
0.660338 + 0.750968i \(0.270413\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −12.1244 −1.76852 −0.884260 0.466996i \(-0.845336\pi\)
−0.884260 + 0.466996i \(0.845336\pi\)
\(48\) 0 0
\(49\) −4.00000 −0.571429
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −8.00000 −1.09888 −0.549442 0.835532i \(-0.685160\pi\)
−0.549442 + 0.835532i \(0.685160\pi\)
\(54\) 0 0
\(55\) −1.73205 −0.233550
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.73205 0.225494 0.112747 0.993624i \(-0.464035\pi\)
0.112747 + 0.993624i \(0.464035\pi\)
\(60\) 0 0
\(61\) 7.00000 0.896258 0.448129 0.893969i \(-0.352090\pi\)
0.448129 + 0.893969i \(0.352090\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −3.00000 −0.372104
\(66\) 0 0
\(67\) 8.66025 1.05802 0.529009 0.848616i \(-0.322564\pi\)
0.529009 + 0.848616i \(0.322564\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −3.46410 −0.411113 −0.205557 0.978645i \(-0.565900\pi\)
−0.205557 + 0.978645i \(0.565900\pi\)
\(72\) 0 0
\(73\) −12.0000 −1.40449 −0.702247 0.711934i \(-0.747820\pi\)
−0.702247 + 0.711934i \(0.747820\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −3.00000 −0.341882
\(78\) 0 0
\(79\) −5.19615 −0.584613 −0.292306 0.956325i \(-0.594423\pi\)
−0.292306 + 0.956325i \(0.594423\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −8.66025 −0.950586 −0.475293 0.879827i \(-0.657658\pi\)
−0.475293 + 0.879827i \(0.657658\pi\)
\(84\) 0 0
\(85\) 4.00000 0.433861
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 4.00000 0.423999 0.212000 0.977270i \(-0.432002\pi\)
0.212000 + 0.977270i \(0.432002\pi\)
\(90\) 0 0
\(91\) −5.19615 −0.544705
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 6.92820 0.710819
\(96\) 0 0
\(97\) −3.00000 −0.304604 −0.152302 0.988334i \(-0.548669\pi\)
−0.152302 + 0.988334i \(0.548669\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −13.0000 −1.29355 −0.646774 0.762682i \(-0.723882\pi\)
−0.646774 + 0.762682i \(0.723882\pi\)
\(102\) 0 0
\(103\) 1.73205 0.170664 0.0853320 0.996353i \(-0.472805\pi\)
0.0853320 + 0.996353i \(0.472805\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −6.92820 −0.669775 −0.334887 0.942258i \(-0.608698\pi\)
−0.334887 + 0.942258i \(0.608698\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.00000 0.0940721 0.0470360 0.998893i \(-0.485022\pi\)
0.0470360 + 0.998893i \(0.485022\pi\)
\(114\) 0 0
\(115\) −8.66025 −0.807573
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 6.92820 0.635107
\(120\) 0 0
\(121\) −8.00000 −0.727273
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) −13.8564 −1.22956 −0.614779 0.788700i \(-0.710755\pi\)
−0.614779 + 0.788700i \(0.710755\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 15.5885 1.36197 0.680985 0.732297i \(-0.261552\pi\)
0.680985 + 0.732297i \(0.261552\pi\)
\(132\) 0 0
\(133\) 12.0000 1.04053
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −13.0000 −1.11066 −0.555332 0.831628i \(-0.687409\pi\)
−0.555332 + 0.831628i \(0.687409\pi\)
\(138\) 0 0
\(139\) −1.73205 −0.146911 −0.0734553 0.997299i \(-0.523403\pi\)
−0.0734553 + 0.997299i \(0.523403\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 5.19615 0.434524
\(144\) 0 0
\(145\) −1.00000 −0.0830455
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −11.0000 −0.901155 −0.450578 0.892737i \(-0.648782\pi\)
−0.450578 + 0.892737i \(0.648782\pi\)
\(150\) 0 0
\(151\) 22.5167 1.83238 0.916190 0.400744i \(-0.131248\pi\)
0.916190 + 0.400744i \(0.131248\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −5.19615 −0.417365
\(156\) 0 0
\(157\) 17.0000 1.35675 0.678374 0.734717i \(-0.262685\pi\)
0.678374 + 0.734717i \(0.262685\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −15.0000 −1.18217
\(162\) 0 0
\(163\) 6.92820 0.542659 0.271329 0.962487i \(-0.412537\pi\)
0.271329 + 0.962487i \(0.412537\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −15.5885 −1.20627 −0.603136 0.797639i \(-0.706082\pi\)
−0.603136 + 0.797639i \(0.706082\pi\)
\(168\) 0 0
\(169\) −4.00000 −0.307692
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −13.0000 −0.988372 −0.494186 0.869356i \(-0.664534\pi\)
−0.494186 + 0.869356i \(0.664534\pi\)
\(174\) 0 0
\(175\) 6.92820 0.523723
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 6.92820 0.517838 0.258919 0.965899i \(-0.416634\pi\)
0.258919 + 0.965899i \(0.416634\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −8.00000 −0.588172
\(186\) 0 0
\(187\) −6.92820 −0.506640
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1.73205 0.125327 0.0626634 0.998035i \(-0.480041\pi\)
0.0626634 + 0.998035i \(0.480041\pi\)
\(192\) 0 0
\(193\) −11.0000 −0.791797 −0.395899 0.918294i \(-0.629567\pi\)
−0.395899 + 0.918294i \(0.629567\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −22.0000 −1.56744 −0.783718 0.621117i \(-0.786679\pi\)
−0.783718 + 0.621117i \(0.786679\pi\)
\(198\) 0 0
\(199\) −3.46410 −0.245564 −0.122782 0.992434i \(-0.539182\pi\)
−0.122782 + 0.992434i \(0.539182\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −1.73205 −0.121566
\(204\) 0 0
\(205\) 5.00000 0.349215
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −12.0000 −0.830057
\(210\) 0 0
\(211\) 8.66025 0.596196 0.298098 0.954535i \(-0.403648\pi\)
0.298098 + 0.954535i \(0.403648\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −8.66025 −0.590624
\(216\) 0 0
\(217\) −9.00000 −0.610960
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) 0 0
\(223\) −1.73205 −0.115987 −0.0579934 0.998317i \(-0.518470\pi\)
−0.0579934 + 0.998317i \(0.518470\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −12.1244 −0.804722 −0.402361 0.915481i \(-0.631810\pi\)
−0.402361 + 0.915481i \(0.631810\pi\)
\(228\) 0 0
\(229\) −15.0000 −0.991228 −0.495614 0.868543i \(-0.665057\pi\)
−0.495614 + 0.868543i \(0.665057\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −10.0000 −0.655122 −0.327561 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) 0 0
\(235\) 12.1244 0.790906
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −19.0526 −1.23241 −0.616204 0.787587i \(-0.711330\pi\)
−0.616204 + 0.787587i \(0.711330\pi\)
\(240\) 0 0
\(241\) −9.00000 −0.579741 −0.289870 0.957066i \(-0.593612\pi\)
−0.289870 + 0.957066i \(0.593612\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 4.00000 0.255551
\(246\) 0 0
\(247\) −20.7846 −1.32249
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −20.7846 −1.31191 −0.655956 0.754799i \(-0.727735\pi\)
−0.655956 + 0.754799i \(0.727735\pi\)
\(252\) 0 0
\(253\) 15.0000 0.943042
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −31.0000 −1.93373 −0.966863 0.255294i \(-0.917828\pi\)
−0.966863 + 0.255294i \(0.917828\pi\)
\(258\) 0 0
\(259\) −13.8564 −0.860995
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 19.0526 1.17483 0.587416 0.809285i \(-0.300145\pi\)
0.587416 + 0.809285i \(0.300145\pi\)
\(264\) 0 0
\(265\) 8.00000 0.491436
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −16.0000 −0.975537 −0.487769 0.872973i \(-0.662189\pi\)
−0.487769 + 0.872973i \(0.662189\pi\)
\(270\) 0 0
\(271\) 13.8564 0.841717 0.420858 0.907126i \(-0.361729\pi\)
0.420858 + 0.907126i \(0.361729\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −6.92820 −0.417786
\(276\) 0 0
\(277\) −27.0000 −1.62227 −0.811136 0.584857i \(-0.801151\pi\)
−0.811136 + 0.584857i \(0.801151\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 19.0000 1.13344 0.566722 0.823909i \(-0.308211\pi\)
0.566722 + 0.823909i \(0.308211\pi\)
\(282\) 0 0
\(283\) −29.4449 −1.75032 −0.875158 0.483838i \(-0.839242\pi\)
−0.875158 + 0.483838i \(0.839242\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 8.66025 0.511199
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 7.00000 0.408944 0.204472 0.978872i \(-0.434452\pi\)
0.204472 + 0.978872i \(0.434452\pi\)
\(294\) 0 0
\(295\) −1.73205 −0.100844
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 25.9808 1.50251
\(300\) 0 0
\(301\) −15.0000 −0.864586
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −7.00000 −0.400819
\(306\) 0 0
\(307\) 6.92820 0.395413 0.197707 0.980261i \(-0.436651\pi\)
0.197707 + 0.980261i \(0.436651\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −5.19615 −0.294647 −0.147323 0.989088i \(-0.547066\pi\)
−0.147323 + 0.989088i \(0.547066\pi\)
\(312\) 0 0
\(313\) −1.00000 −0.0565233 −0.0282617 0.999601i \(-0.508997\pi\)
−0.0282617 + 0.999601i \(0.508997\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 17.0000 0.954815 0.477408 0.878682i \(-0.341577\pi\)
0.477408 + 0.878682i \(0.341577\pi\)
\(318\) 0 0
\(319\) 1.73205 0.0969762
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 27.7128 1.54198
\(324\) 0 0
\(325\) −12.0000 −0.665640
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 21.0000 1.15777
\(330\) 0 0
\(331\) 12.1244 0.666415 0.333207 0.942854i \(-0.391869\pi\)
0.333207 + 0.942854i \(0.391869\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −8.66025 −0.473160
\(336\) 0 0
\(337\) 15.0000 0.817102 0.408551 0.912735i \(-0.366034\pi\)
0.408551 + 0.912735i \(0.366034\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 9.00000 0.487377
\(342\) 0 0
\(343\) 19.0526 1.02874
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −22.5167 −1.20876 −0.604379 0.796697i \(-0.706579\pi\)
−0.604379 + 0.796697i \(0.706579\pi\)
\(348\) 0 0
\(349\) −11.0000 −0.588817 −0.294408 0.955680i \(-0.595123\pi\)
−0.294408 + 0.955680i \(0.595123\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −29.0000 −1.54351 −0.771757 0.635917i \(-0.780622\pi\)
−0.771757 + 0.635917i \(0.780622\pi\)
\(354\) 0 0
\(355\) 3.46410 0.183855
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 29.0000 1.52632
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 12.0000 0.628109
\(366\) 0 0
\(367\) −32.9090 −1.71783 −0.858917 0.512115i \(-0.828862\pi\)
−0.858917 + 0.512115i \(0.828862\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 13.8564 0.719389
\(372\) 0 0
\(373\) 19.0000 0.983783 0.491891 0.870657i \(-0.336306\pi\)
0.491891 + 0.870657i \(0.336306\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 3.00000 0.154508
\(378\) 0 0
\(379\) 10.3923 0.533817 0.266908 0.963722i \(-0.413998\pi\)
0.266908 + 0.963722i \(0.413998\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 25.9808 1.32755 0.663777 0.747930i \(-0.268952\pi\)
0.663777 + 0.747930i \(0.268952\pi\)
\(384\) 0 0
\(385\) 3.00000 0.152894
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −29.0000 −1.47036 −0.735179 0.677873i \(-0.762902\pi\)
−0.735179 + 0.677873i \(0.762902\pi\)
\(390\) 0 0
\(391\) −34.6410 −1.75187
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 5.19615 0.261447
\(396\) 0 0
\(397\) −8.00000 −0.401508 −0.200754 0.979642i \(-0.564339\pi\)
−0.200754 + 0.979642i \(0.564339\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −13.0000 −0.649189 −0.324595 0.945853i \(-0.605228\pi\)
−0.324595 + 0.945853i \(0.605228\pi\)
\(402\) 0 0
\(403\) 15.5885 0.776516
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 13.8564 0.686837
\(408\) 0 0
\(409\) −9.00000 −0.445021 −0.222511 0.974930i \(-0.571425\pi\)
−0.222511 + 0.974930i \(0.571425\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −3.00000 −0.147620
\(414\) 0 0
\(415\) 8.66025 0.425115
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −22.5167 −1.10001 −0.550005 0.835161i \(-0.685374\pi\)
−0.550005 + 0.835161i \(0.685374\pi\)
\(420\) 0 0
\(421\) 15.0000 0.731055 0.365528 0.930800i \(-0.380889\pi\)
0.365528 + 0.930800i \(0.380889\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 16.0000 0.776114
\(426\) 0 0
\(427\) −12.1244 −0.586739
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 27.7128 1.33488 0.667440 0.744664i \(-0.267390\pi\)
0.667440 + 0.744664i \(0.267390\pi\)
\(432\) 0 0
\(433\) 4.00000 0.192228 0.0961139 0.995370i \(-0.469359\pi\)
0.0961139 + 0.995370i \(0.469359\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −60.0000 −2.87019
\(438\) 0 0
\(439\) −5.19615 −0.247999 −0.123999 0.992282i \(-0.539572\pi\)
−0.123999 + 0.992282i \(0.539572\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 32.9090 1.56355 0.781776 0.623559i \(-0.214314\pi\)
0.781776 + 0.623559i \(0.214314\pi\)
\(444\) 0 0
\(445\) −4.00000 −0.189618
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −20.0000 −0.943858 −0.471929 0.881636i \(-0.656442\pi\)
−0.471929 + 0.881636i \(0.656442\pi\)
\(450\) 0 0
\(451\) −8.66025 −0.407795
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 5.19615 0.243599
\(456\) 0 0
\(457\) 21.0000 0.982339 0.491169 0.871064i \(-0.336570\pi\)
0.491169 + 0.871064i \(0.336570\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 1.00000 0.0465746 0.0232873 0.999729i \(-0.492587\pi\)
0.0232873 + 0.999729i \(0.492587\pi\)
\(462\) 0 0
\(463\) 32.9090 1.52941 0.764705 0.644381i \(-0.222885\pi\)
0.764705 + 0.644381i \(0.222885\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 6.92820 0.320599 0.160300 0.987068i \(-0.448754\pi\)
0.160300 + 0.987068i \(0.448754\pi\)
\(468\) 0 0
\(469\) −15.0000 −0.692636
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 15.0000 0.689701
\(474\) 0 0
\(475\) 27.7128 1.27155
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −8.66025 −0.395697 −0.197849 0.980233i \(-0.563395\pi\)
−0.197849 + 0.980233i \(0.563395\pi\)
\(480\) 0 0
\(481\) 24.0000 1.09431
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 3.00000 0.136223
\(486\) 0 0
\(487\) 27.7128 1.25579 0.627894 0.778299i \(-0.283917\pi\)
0.627894 + 0.778299i \(0.283917\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 1.73205 0.0781664 0.0390832 0.999236i \(-0.487556\pi\)
0.0390832 + 0.999236i \(0.487556\pi\)
\(492\) 0 0
\(493\) −4.00000 −0.180151
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 6.00000 0.269137
\(498\) 0 0
\(499\) −1.73205 −0.0775372 −0.0387686 0.999248i \(-0.512344\pi\)
−0.0387686 + 0.999248i \(0.512344\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −24.2487 −1.08120 −0.540598 0.841281i \(-0.681802\pi\)
−0.540598 + 0.841281i \(0.681802\pi\)
\(504\) 0 0
\(505\) 13.0000 0.578492
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 37.0000 1.64000 0.819998 0.572366i \(-0.193974\pi\)
0.819998 + 0.572366i \(0.193974\pi\)
\(510\) 0 0
\(511\) 20.7846 0.919457
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −1.73205 −0.0763233
\(516\) 0 0
\(517\) −21.0000 −0.923579
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 26.0000 1.13908 0.569540 0.821963i \(-0.307121\pi\)
0.569540 + 0.821963i \(0.307121\pi\)
\(522\) 0 0
\(523\) −10.3923 −0.454424 −0.227212 0.973845i \(-0.572961\pi\)
−0.227212 + 0.973845i \(0.572961\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −20.7846 −0.905392
\(528\) 0 0
\(529\) 52.0000 2.26087
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −15.0000 −0.649722
\(534\) 0 0
\(535\) 6.92820 0.299532
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −6.92820 −0.298419
\(540\) 0 0
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −15.5885 −0.666514 −0.333257 0.942836i \(-0.608148\pi\)
−0.333257 + 0.942836i \(0.608148\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −6.92820 −0.295151
\(552\) 0 0
\(553\) 9.00000 0.382719
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 16.0000 0.677942 0.338971 0.940797i \(-0.389921\pi\)
0.338971 + 0.940797i \(0.389921\pi\)
\(558\) 0 0
\(559\) 25.9808 1.09887
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −22.5167 −0.948964 −0.474482 0.880265i \(-0.657365\pi\)
−0.474482 + 0.880265i \(0.657365\pi\)
\(564\) 0 0
\(565\) −1.00000 −0.0420703
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −31.0000 −1.29959 −0.649794 0.760111i \(-0.725145\pi\)
−0.649794 + 0.760111i \(0.725145\pi\)
\(570\) 0 0
\(571\) 8.66025 0.362420 0.181210 0.983444i \(-0.441999\pi\)
0.181210 + 0.983444i \(0.441999\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −34.6410 −1.44463
\(576\) 0 0
\(577\) −4.00000 −0.166522 −0.0832611 0.996528i \(-0.526534\pi\)
−0.0832611 + 0.996528i \(0.526534\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 15.0000 0.622305
\(582\) 0 0
\(583\) −13.8564 −0.573874
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 5.19615 0.214468 0.107234 0.994234i \(-0.465801\pi\)
0.107234 + 0.994234i \(0.465801\pi\)
\(588\) 0 0
\(589\) −36.0000 −1.48335
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 20.0000 0.821302 0.410651 0.911793i \(-0.365302\pi\)
0.410651 + 0.911793i \(0.365302\pi\)
\(594\) 0 0
\(595\) −6.92820 −0.284029
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −1.73205 −0.0707697 −0.0353848 0.999374i \(-0.511266\pi\)
−0.0353848 + 0.999374i \(0.511266\pi\)
\(600\) 0 0
\(601\) −11.0000 −0.448699 −0.224350 0.974509i \(-0.572026\pi\)
−0.224350 + 0.974509i \(0.572026\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 8.00000 0.325246
\(606\) 0 0
\(607\) −12.1244 −0.492112 −0.246056 0.969256i \(-0.579135\pi\)
−0.246056 + 0.969256i \(0.579135\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −36.3731 −1.47150
\(612\) 0 0
\(613\) −8.00000 −0.323117 −0.161558 0.986863i \(-0.551652\pi\)
−0.161558 + 0.986863i \(0.551652\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −5.00000 −0.201292 −0.100646 0.994922i \(-0.532091\pi\)
−0.100646 + 0.994922i \(0.532091\pi\)
\(618\) 0 0
\(619\) −15.5885 −0.626553 −0.313276 0.949662i \(-0.601427\pi\)
−0.313276 + 0.949662i \(0.601427\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −6.92820 −0.277573
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −32.0000 −1.27592
\(630\) 0 0
\(631\) −41.5692 −1.65484 −0.827422 0.561580i \(-0.810194\pi\)
−0.827422 + 0.561580i \(0.810194\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 13.8564 0.549875
\(636\) 0 0
\(637\) −12.0000 −0.475457
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −7.00000 −0.276483 −0.138242 0.990399i \(-0.544145\pi\)
−0.138242 + 0.990399i \(0.544145\pi\)
\(642\) 0 0
\(643\) −5.19615 −0.204916 −0.102458 0.994737i \(-0.532671\pi\)
−0.102458 + 0.994737i \(0.532671\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 13.8564 0.544752 0.272376 0.962191i \(-0.412191\pi\)
0.272376 + 0.962191i \(0.412191\pi\)
\(648\) 0 0
\(649\) 3.00000 0.117760
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −1.00000 −0.0391330 −0.0195665 0.999809i \(-0.506229\pi\)
−0.0195665 + 0.999809i \(0.506229\pi\)
\(654\) 0 0
\(655\) −15.5885 −0.609091
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −39.8372 −1.55184 −0.775918 0.630834i \(-0.782713\pi\)
−0.775918 + 0.630834i \(0.782713\pi\)
\(660\) 0 0
\(661\) 17.0000 0.661223 0.330612 0.943767i \(-0.392745\pi\)
0.330612 + 0.943767i \(0.392745\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −12.0000 −0.465340
\(666\) 0 0
\(667\) 8.66025 0.335326
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 12.1244 0.468056
\(672\) 0 0
\(673\) 5.00000 0.192736 0.0963679 0.995346i \(-0.469277\pi\)
0.0963679 + 0.995346i \(0.469277\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 11.0000 0.422764 0.211382 0.977403i \(-0.432204\pi\)
0.211382 + 0.977403i \(0.432204\pi\)
\(678\) 0 0
\(679\) 5.19615 0.199410
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 34.6410 1.32550 0.662751 0.748840i \(-0.269389\pi\)
0.662751 + 0.748840i \(0.269389\pi\)
\(684\) 0 0
\(685\) 13.0000 0.496704
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −24.0000 −0.914327
\(690\) 0 0
\(691\) −19.0526 −0.724793 −0.362397 0.932024i \(-0.618041\pi\)
−0.362397 + 0.932024i \(0.618041\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1.73205 0.0657004
\(696\) 0 0
\(697\) 20.0000 0.757554
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 16.0000 0.604312 0.302156 0.953259i \(-0.402294\pi\)
0.302156 + 0.953259i \(0.402294\pi\)
\(702\) 0 0
\(703\) −55.4256 −2.09042
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 22.5167 0.846826
\(708\) 0 0
\(709\) −27.0000 −1.01401 −0.507003 0.861944i \(-0.669247\pi\)
−0.507003 + 0.861944i \(0.669247\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 45.0000 1.68526
\(714\) 0 0
\(715\) −5.19615 −0.194325
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 41.5692 1.55027 0.775135 0.631795i \(-0.217682\pi\)
0.775135 + 0.631795i \(0.217682\pi\)
\(720\) 0 0
\(721\) −3.00000 −0.111726
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −4.00000 −0.148556
\(726\) 0 0
\(727\) 22.5167 0.835097 0.417548 0.908655i \(-0.362889\pi\)
0.417548 + 0.908655i \(0.362889\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −34.6410 −1.28124
\(732\) 0 0
\(733\) −21.0000 −0.775653 −0.387826 0.921732i \(-0.626774\pi\)
−0.387826 + 0.921732i \(0.626774\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 15.0000 0.552532
\(738\) 0 0
\(739\) 3.46410 0.127429 0.0637145 0.997968i \(-0.479705\pi\)
0.0637145 + 0.997968i \(0.479705\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −1.73205 −0.0635428 −0.0317714 0.999495i \(-0.510115\pi\)
−0.0317714 + 0.999495i \(0.510115\pi\)
\(744\) 0 0
\(745\) 11.0000 0.403009
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 12.0000 0.438470
\(750\) 0 0
\(751\) −25.9808 −0.948051 −0.474026 0.880511i \(-0.657200\pi\)
−0.474026 + 0.880511i \(0.657200\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −22.5167 −0.819465
\(756\) 0 0
\(757\) −6.00000 −0.218074 −0.109037 0.994038i \(-0.534777\pi\)
−0.109037 + 0.994038i \(0.534777\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 43.0000 1.55875 0.779374 0.626559i \(-0.215537\pi\)
0.779374 + 0.626559i \(0.215537\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 5.19615 0.187622
\(768\) 0 0
\(769\) −1.00000 −0.0360609 −0.0180305 0.999837i \(-0.505740\pi\)
−0.0180305 + 0.999837i \(0.505740\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 40.0000 1.43870 0.719350 0.694648i \(-0.244440\pi\)
0.719350 + 0.694648i \(0.244440\pi\)
\(774\) 0 0
\(775\) −20.7846 −0.746605
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 34.6410 1.24114
\(780\) 0 0
\(781\) −6.00000 −0.214697
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −17.0000 −0.606756
\(786\) 0 0
\(787\) 25.9808 0.926114 0.463057 0.886328i \(-0.346752\pi\)
0.463057 + 0.886328i \(0.346752\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −1.73205 −0.0615846
\(792\) 0 0
\(793\) 21.0000 0.745732
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 29.0000 1.02723 0.513616 0.858020i \(-0.328305\pi\)
0.513616 + 0.858020i \(0.328305\pi\)
\(798\) 0 0
\(799\) 48.4974 1.71572
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −20.7846 −0.733473
\(804\) 0 0
\(805\) 15.0000 0.528681
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 4.00000 0.140633 0.0703163 0.997525i \(-0.477599\pi\)
0.0703163 + 0.997525i \(0.477599\pi\)
\(810\) 0 0
\(811\) −20.7846 −0.729846 −0.364923 0.931038i \(-0.618905\pi\)
−0.364923 + 0.931038i \(0.618905\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −6.92820 −0.242684
\(816\) 0 0
\(817\) −60.0000 −2.09913
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −7.00000 −0.244302 −0.122151 0.992512i \(-0.538979\pi\)
−0.122151 + 0.992512i \(0.538979\pi\)
\(822\) 0 0
\(823\) 46.7654 1.63014 0.815069 0.579364i \(-0.196699\pi\)
0.815069 + 0.579364i \(0.196699\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 6.92820 0.240917 0.120459 0.992718i \(-0.461563\pi\)
0.120459 + 0.992718i \(0.461563\pi\)
\(828\) 0 0
\(829\) −24.0000 −0.833554 −0.416777 0.909009i \(-0.636840\pi\)
−0.416777 + 0.909009i \(0.636840\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 16.0000 0.554367
\(834\) 0 0
\(835\) 15.5885 0.539461
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −25.9808 −0.896956 −0.448478 0.893794i \(-0.648034\pi\)
−0.448478 + 0.893794i \(0.648034\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 4.00000 0.137604
\(846\) 0 0
\(847\) 13.8564 0.476112
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 69.2820 2.37496
\(852\) 0 0
\(853\) 23.0000 0.787505 0.393753 0.919216i \(-0.371177\pi\)
0.393753 + 0.919216i \(0.371177\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 43.0000 1.46885 0.734426 0.678689i \(-0.237451\pi\)
0.734426 + 0.678689i \(0.237451\pi\)
\(858\) 0 0
\(859\) 36.3731 1.24103 0.620517 0.784193i \(-0.286923\pi\)
0.620517 + 0.784193i \(0.286923\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −55.4256 −1.88671 −0.943355 0.331785i \(-0.892349\pi\)
−0.943355 + 0.331785i \(0.892349\pi\)
\(864\) 0 0
\(865\) 13.0000 0.442013
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −9.00000 −0.305304
\(870\) 0 0
\(871\) 25.9808 0.880325
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −15.5885 −0.526986
\(876\) 0 0
\(877\) 43.0000 1.45201 0.726003 0.687691i \(-0.241376\pi\)
0.726003 + 0.687691i \(0.241376\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −52.0000 −1.75192 −0.875962 0.482380i \(-0.839773\pi\)
−0.875962 + 0.482380i \(0.839773\pi\)
\(882\) 0 0
\(883\) 20.7846 0.699458 0.349729 0.936851i \(-0.386274\pi\)
0.349729 + 0.936851i \(0.386274\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −5.19615 −0.174470 −0.0872349 0.996188i \(-0.527803\pi\)
−0.0872349 + 0.996188i \(0.527803\pi\)
\(888\) 0 0
\(889\) 24.0000 0.804934
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 84.0000 2.81095
\(894\) 0 0
\(895\) −6.92820 −0.231584
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 5.19615 0.173301
\(900\) 0 0
\(901\) 32.0000 1.06607
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −32.9090 −1.09272 −0.546362 0.837549i \(-0.683988\pi\)
−0.546362 + 0.837549i \(0.683988\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 32.9090 1.09032 0.545161 0.838331i \(-0.316468\pi\)
0.545161 + 0.838331i \(0.316468\pi\)
\(912\) 0 0
\(913\) −15.0000 −0.496428
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −27.0000 −0.891619
\(918\) 0 0
\(919\) −13.8564 −0.457081 −0.228540 0.973534i \(-0.573395\pi\)
−0.228540 + 0.973534i \(0.573395\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −10.3923 −0.342067
\(924\) 0 0
\(925\) −32.0000 −1.05215
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 35.0000 1.14831 0.574156 0.818746i \(-0.305330\pi\)
0.574156 + 0.818746i \(0.305330\pi\)
\(930\) 0 0
\(931\) 27.7128 0.908251
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 6.92820 0.226576
\(936\) 0 0
\(937\) 4.00000 0.130674 0.0653372 0.997863i \(-0.479188\pi\)
0.0653372 + 0.997863i \(0.479188\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 7.00000 0.228193 0.114097 0.993470i \(-0.463603\pi\)
0.114097 + 0.993470i \(0.463603\pi\)
\(942\) 0 0
\(943\) −43.3013 −1.41008
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 15.5885 0.506557 0.253278 0.967393i \(-0.418491\pi\)
0.253278 + 0.967393i \(0.418491\pi\)
\(948\) 0 0
\(949\) −36.0000 −1.16861
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 44.0000 1.42530 0.712650 0.701520i \(-0.247495\pi\)
0.712650 + 0.701520i \(0.247495\pi\)
\(954\) 0 0
\(955\) −1.73205 −0.0560478
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 22.5167 0.727101
\(960\) 0 0
\(961\) −4.00000 −0.129032
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 11.0000 0.354103
\(966\) 0 0
\(967\) 32.9090 1.05828 0.529140 0.848534i \(-0.322514\pi\)
0.529140 + 0.848534i \(0.322514\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −38.1051 −1.22285 −0.611426 0.791302i \(-0.709404\pi\)
−0.611426 + 0.791302i \(0.709404\pi\)
\(972\) 0 0
\(973\) 3.00000 0.0961756
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 25.0000 0.799821 0.399910 0.916554i \(-0.369041\pi\)
0.399910 + 0.916554i \(0.369041\pi\)
\(978\) 0 0
\(979\) 6.92820 0.221426
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −8.66025 −0.276219 −0.138110 0.990417i \(-0.544103\pi\)
−0.138110 + 0.990417i \(0.544103\pi\)
\(984\) 0 0
\(985\) 22.0000 0.700978
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 75.0000 2.38486
\(990\) 0 0
\(991\) −55.4256 −1.76065 −0.880327 0.474368i \(-0.842677\pi\)
−0.880327 + 0.474368i \(0.842677\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 3.46410 0.109819
\(996\) 0 0
\(997\) 13.0000 0.411714 0.205857 0.978582i \(-0.434002\pi\)
0.205857 + 0.978582i \(0.434002\pi\)
\(998\) 0 0
\(999\) 0 0
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 5184.2.a.bl.1.1 2
3.2 odd 2 5184.2.a.bx.1.1 2
4.3 odd 2 inner 5184.2.a.bl.1.2 2
8.3 odd 2 2592.2.a.p.1.2 2
8.5 even 2 2592.2.a.p.1.1 2
9.2 odd 6 576.2.i.k.193.2 4
9.4 even 3 1728.2.i.l.1153.2 4
9.5 odd 6 576.2.i.k.385.2 4
9.7 even 3 1728.2.i.l.577.2 4
12.11 even 2 5184.2.a.bx.1.2 2
24.5 odd 2 2592.2.a.l.1.1 2
24.11 even 2 2592.2.a.l.1.2 2
36.7 odd 6 1728.2.i.l.577.1 4
36.11 even 6 576.2.i.k.193.1 4
36.23 even 6 576.2.i.k.385.1 4
36.31 odd 6 1728.2.i.l.1153.1 4
72.5 odd 6 288.2.i.d.97.1 4
72.11 even 6 288.2.i.d.193.2 yes 4
72.13 even 6 864.2.i.d.289.2 4
72.29 odd 6 288.2.i.d.193.1 yes 4
72.43 odd 6 864.2.i.d.577.1 4
72.59 even 6 288.2.i.d.97.2 yes 4
72.61 even 6 864.2.i.d.577.2 4
72.67 odd 6 864.2.i.d.289.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
288.2.i.d.97.1 4 72.5 odd 6
288.2.i.d.97.2 yes 4 72.59 even 6
288.2.i.d.193.1 yes 4 72.29 odd 6
288.2.i.d.193.2 yes 4 72.11 even 6
576.2.i.k.193.1 4 36.11 even 6
576.2.i.k.193.2 4 9.2 odd 6
576.2.i.k.385.1 4 36.23 even 6
576.2.i.k.385.2 4 9.5 odd 6
864.2.i.d.289.1 4 72.67 odd 6
864.2.i.d.289.2 4 72.13 even 6
864.2.i.d.577.1 4 72.43 odd 6
864.2.i.d.577.2 4 72.61 even 6
1728.2.i.l.577.1 4 36.7 odd 6
1728.2.i.l.577.2 4 9.7 even 3
1728.2.i.l.1153.1 4 36.31 odd 6
1728.2.i.l.1153.2 4 9.4 even 3
2592.2.a.l.1.1 2 24.5 odd 2
2592.2.a.l.1.2 2 24.11 even 2
2592.2.a.p.1.1 2 8.5 even 2
2592.2.a.p.1.2 2 8.3 odd 2
5184.2.a.bl.1.1 2 1.1 even 1 trivial
5184.2.a.bl.1.2 2 4.3 odd 2 inner
5184.2.a.bx.1.1 2 3.2 odd 2
5184.2.a.bx.1.2 2 12.11 even 2