Properties

Label 9216.2.a.o.1.1
Level $9216$
Weight $2$
Character 9216.1
Self dual yes
Analytic conductor $73.590$
Analytic rank $0$
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] = [9216,2,Mod(1,9216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9216, 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("9216.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9216 = 2^{10} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9216.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: \(73.5901305028\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 768)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 9216.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.41421 q^{7} +O(q^{10})\) \(q-1.41421 q^{7} -1.41421 q^{13} +6.00000 q^{17} +6.00000 q^{19} -8.48528 q^{23} -5.00000 q^{25} -8.48528 q^{29} +1.41421 q^{31} -7.07107 q^{37} +6.00000 q^{41} +6.00000 q^{43} +8.48528 q^{47} -5.00000 q^{49} +8.48528 q^{53} +7.07107 q^{61} -4.00000 q^{67} +8.48528 q^{71} +1.41421 q^{79} +12.0000 q^{83} -6.00000 q^{89} +2.00000 q^{91} -12.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 12 q^{17} + 12 q^{19} - 10 q^{25} + 12 q^{41} + 12 q^{43} - 10 q^{49} - 8 q^{67} + 24 q^{83} - 12 q^{89} + 4 q^{91} - 24 q^{97}+O(q^{100}) \) 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
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) −1.41421 −0.534522 −0.267261 0.963624i \(-0.586119\pi\)
−0.267261 + 0.963624i \(0.586119\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −1.41421 −0.392232 −0.196116 0.980581i \(-0.562833\pi\)
−0.196116 + 0.980581i \(0.562833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −8.48528 −1.76930 −0.884652 0.466252i \(-0.845604\pi\)
−0.884652 + 0.466252i \(0.845604\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −8.48528 −1.57568 −0.787839 0.615882i \(-0.788800\pi\)
−0.787839 + 0.615882i \(0.788800\pi\)
\(30\) 0 0
\(31\) 1.41421 0.254000 0.127000 0.991903i \(-0.459465\pi\)
0.127000 + 0.991903i \(0.459465\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −7.07107 −1.16248 −0.581238 0.813733i \(-0.697432\pi\)
−0.581238 + 0.813733i \(0.697432\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 6.00000 0.914991 0.457496 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.48528 1.23771 0.618853 0.785507i \(-0.287598\pi\)
0.618853 + 0.785507i \(0.287598\pi\)
\(48\) 0 0
\(49\) −5.00000 −0.714286
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 8.48528 1.16554 0.582772 0.812636i \(-0.301968\pi\)
0.582772 + 0.812636i \(0.301968\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 7.07107 0.905357 0.452679 0.891674i \(-0.350468\pi\)
0.452679 + 0.891674i \(0.350468\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8.48528 1.00702 0.503509 0.863990i \(-0.332042\pi\)
0.503509 + 0.863990i \(0.332042\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 1.41421 0.159111 0.0795557 0.996830i \(-0.474650\pi\)
0.0795557 + 0.996830i \(0.474650\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −12.0000 −1.21842 −0.609208 0.793011i \(-0.708512\pi\)
−0.609208 + 0.793011i \(0.708512\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −8.48528 −0.844317 −0.422159 0.906522i \(-0.638727\pi\)
−0.422159 + 0.906522i \(0.638727\pi\)
\(102\) 0 0
\(103\) 9.89949 0.975426 0.487713 0.873004i \(-0.337831\pi\)
0.487713 + 0.873004i \(0.337831\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) 18.3848 1.76094 0.880471 0.474100i \(-0.157226\pi\)
0.880471 + 0.474100i \(0.157226\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −8.48528 −0.777844
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −9.89949 −0.878438 −0.439219 0.898380i \(-0.644745\pi\)
−0.439219 + 0.898380i \(0.644745\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) −8.48528 −0.735767
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 18.0000 1.53784 0.768922 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 16.9706 1.39028 0.695141 0.718873i \(-0.255342\pi\)
0.695141 + 0.718873i \(0.255342\pi\)
\(150\) 0 0
\(151\) −18.3848 −1.49613 −0.748066 0.663624i \(-0.769017\pi\)
−0.748066 + 0.663624i \(0.769017\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −7.07107 −0.564333 −0.282166 0.959366i \(-0.591053\pi\)
−0.282166 + 0.959366i \(0.591053\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 12.0000 0.945732
\(162\) 0 0
\(163\) 6.00000 0.469956 0.234978 0.972001i \(-0.424498\pi\)
0.234978 + 0.972001i \(0.424498\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −16.9706 −1.31322 −0.656611 0.754230i \(-0.728011\pi\)
−0.656611 + 0.754230i \(0.728011\pi\)
\(168\) 0 0
\(169\) −11.0000 −0.846154
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −16.9706 −1.29025 −0.645124 0.764078i \(-0.723194\pi\)
−0.645124 + 0.764078i \(0.723194\pi\)
\(174\) 0 0
\(175\) 7.07107 0.534522
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 24.0000 1.79384 0.896922 0.442189i \(-0.145798\pi\)
0.896922 + 0.442189i \(0.145798\pi\)
\(180\) 0 0
\(181\) −15.5563 −1.15629 −0.578147 0.815933i \(-0.696224\pi\)
−0.578147 + 0.815933i \(0.696224\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 16.9706 1.22795 0.613973 0.789327i \(-0.289570\pi\)
0.613973 + 0.789327i \(0.289570\pi\)
\(192\) 0 0
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 8.48528 0.604551 0.302276 0.953221i \(-0.402254\pi\)
0.302276 + 0.953221i \(0.402254\pi\)
\(198\) 0 0
\(199\) −9.89949 −0.701757 −0.350878 0.936421i \(-0.614117\pi\)
−0.350878 + 0.936421i \(0.614117\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 12.0000 0.842235
\(204\) 0 0
\(205\) 0 0
\(206\) 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
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −2.00000 −0.135769
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −8.48528 −0.570782
\(222\) 0 0
\(223\) 26.8701 1.79935 0.899676 0.436558i \(-0.143803\pi\)
0.899676 + 0.436558i \(0.143803\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) −1.41421 −0.0934539 −0.0467269 0.998908i \(-0.514879\pi\)
−0.0467269 + 0.998908i \(0.514879\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 16.9706 1.09773 0.548867 0.835910i \(-0.315059\pi\)
0.548867 + 0.835910i \(0.315059\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −8.48528 −0.539906
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 24.0000 1.51487 0.757433 0.652913i \(-0.226453\pi\)
0.757433 + 0.652913i \(0.226453\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 0 0
\(259\) 10.0000 0.621370
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 8.48528 0.517357 0.258678 0.965964i \(-0.416713\pi\)
0.258678 + 0.965964i \(0.416713\pi\)
\(270\) 0 0
\(271\) 26.8701 1.63224 0.816120 0.577883i \(-0.196121\pi\)
0.816120 + 0.577883i \(0.196121\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 15.5563 0.934690 0.467345 0.884075i \(-0.345211\pi\)
0.467345 + 0.884075i \(0.345211\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −8.48528 −0.500870
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 8.48528 0.495715 0.247858 0.968796i \(-0.420273\pi\)
0.247858 + 0.968796i \(0.420273\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 12.0000 0.693978
\(300\) 0 0
\(301\) −8.48528 −0.489083
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −16.9706 −0.962312 −0.481156 0.876635i \(-0.659783\pi\)
−0.481156 + 0.876635i \(0.659783\pi\)
\(312\) 0 0
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 8.48528 0.476581 0.238290 0.971194i \(-0.423413\pi\)
0.238290 + 0.971194i \(0.423413\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 36.0000 2.00309
\(324\) 0 0
\(325\) 7.07107 0.392232
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) 28.0000 1.53902 0.769510 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −24.0000 −1.30736 −0.653682 0.756770i \(-0.726776\pi\)
−0.653682 + 0.756770i \(0.726776\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 16.9706 0.916324
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 0 0
\(349\) 26.8701 1.43832 0.719161 0.694844i \(-0.244527\pi\)
0.719161 + 0.694844i \(0.244527\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −25.4558 −1.34351 −0.671754 0.740774i \(-0.734459\pi\)
−0.671754 + 0.740774i \(0.734459\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −32.5269 −1.69789 −0.848945 0.528480i \(-0.822762\pi\)
−0.848945 + 0.528480i \(0.822762\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −12.0000 −0.623009
\(372\) 0 0
\(373\) −7.07107 −0.366126 −0.183063 0.983101i \(-0.558601\pi\)
−0.183063 + 0.983101i \(0.558601\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) 30.0000 1.54100 0.770498 0.637442i \(-0.220007\pi\)
0.770498 + 0.637442i \(0.220007\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 33.9411 1.73431 0.867155 0.498038i \(-0.165946\pi\)
0.867155 + 0.498038i \(0.165946\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 16.9706 0.860442 0.430221 0.902724i \(-0.358436\pi\)
0.430221 + 0.902724i \(0.358436\pi\)
\(390\) 0 0
\(391\) −50.9117 −2.57471
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 9.89949 0.496841 0.248421 0.968652i \(-0.420088\pi\)
0.248421 + 0.968652i \(0.420088\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 0 0
\(403\) −2.00000 −0.0996271
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 12.0000 0.593362 0.296681 0.954977i \(-0.404120\pi\)
0.296681 + 0.954977i \(0.404120\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 36.0000 1.75872 0.879358 0.476162i \(-0.157972\pi\)
0.879358 + 0.476162i \(0.157972\pi\)
\(420\) 0 0
\(421\) 15.5563 0.758170 0.379085 0.925362i \(-0.376239\pi\)
0.379085 + 0.925362i \(0.376239\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −30.0000 −1.45521
\(426\) 0 0
\(427\) −10.0000 −0.483934
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 8.48528 0.408722 0.204361 0.978896i \(-0.434488\pi\)
0.204361 + 0.978896i \(0.434488\pi\)
\(432\) 0 0
\(433\) −24.0000 −1.15337 −0.576683 0.816968i \(-0.695653\pi\)
−0.576683 + 0.816968i \(0.695653\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −50.9117 −2.43544
\(438\) 0 0
\(439\) 7.07107 0.337484 0.168742 0.985660i \(-0.446030\pi\)
0.168742 + 0.985660i \(0.446030\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −12.0000 −0.561336 −0.280668 0.959805i \(-0.590556\pi\)
−0.280668 + 0.959805i \(0.590556\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 33.9411 1.58080 0.790398 0.612594i \(-0.209874\pi\)
0.790398 + 0.612594i \(0.209874\pi\)
\(462\) 0 0
\(463\) 7.07107 0.328620 0.164310 0.986409i \(-0.447460\pi\)
0.164310 + 0.986409i \(0.447460\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 0 0
\(469\) 5.65685 0.261209
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −30.0000 −1.37649
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −8.48528 −0.387702 −0.193851 0.981031i \(-0.562098\pi\)
−0.193851 + 0.981031i \(0.562098\pi\)
\(480\) 0 0
\(481\) 10.0000 0.455961
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 18.3848 0.833094 0.416547 0.909114i \(-0.363240\pi\)
0.416547 + 0.909114i \(0.363240\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −24.0000 −1.08310 −0.541552 0.840667i \(-0.682163\pi\)
−0.541552 + 0.840667i \(0.682163\pi\)
\(492\) 0 0
\(493\) −50.9117 −2.29295
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −12.0000 −0.538274
\(498\) 0 0
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −8.48528 −0.378340 −0.189170 0.981944i \(-0.560580\pi\)
−0.189170 + 0.981944i \(0.560580\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 25.4558 1.12831 0.564155 0.825669i \(-0.309202\pi\)
0.564155 + 0.825669i \(0.309202\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −30.0000 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) 0 0
\(523\) −18.0000 −0.787085 −0.393543 0.919306i \(-0.628751\pi\)
−0.393543 + 0.919306i \(0.628751\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 8.48528 0.369625
\(528\) 0 0
\(529\) 49.0000 2.13043
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −8.48528 −0.367538
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −35.3553 −1.52004 −0.760022 0.649897i \(-0.774812\pi\)
−0.760022 + 0.649897i \(0.774812\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 18.0000 0.769624 0.384812 0.922995i \(-0.374266\pi\)
0.384812 + 0.922995i \(0.374266\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −50.9117 −2.16891
\(552\) 0 0
\(553\) −2.00000 −0.0850487
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −16.9706 −0.719066 −0.359533 0.933132i \(-0.617064\pi\)
−0.359533 + 0.933132i \(0.617064\pi\)
\(558\) 0 0
\(559\) −8.48528 −0.358889
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −18.0000 −0.754599 −0.377300 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 42.4264 1.76930
\(576\) 0 0
\(577\) 36.0000 1.49870 0.749350 0.662174i \(-0.230366\pi\)
0.749350 + 0.662174i \(0.230366\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −16.9706 −0.704058
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 8.48528 0.349630
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −18.0000 −0.739171 −0.369586 0.929197i \(-0.620500\pi\)
−0.369586 + 0.929197i \(0.620500\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 25.4558 1.04010 0.520049 0.854137i \(-0.325914\pi\)
0.520049 + 0.854137i \(0.325914\pi\)
\(600\) 0 0
\(601\) 12.0000 0.489490 0.244745 0.969587i \(-0.421296\pi\)
0.244745 + 0.969587i \(0.421296\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 24.0416 0.975820 0.487910 0.872894i \(-0.337759\pi\)
0.487910 + 0.872894i \(0.337759\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −12.0000 −0.485468
\(612\) 0 0
\(613\) 26.8701 1.08527 0.542636 0.839968i \(-0.317426\pi\)
0.542636 + 0.839968i \(0.317426\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 0 0
\(619\) 28.0000 1.12542 0.562708 0.826656i \(-0.309760\pi\)
0.562708 + 0.826656i \(0.309760\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 8.48528 0.339956
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −42.4264 −1.69165
\(630\) 0 0
\(631\) 18.3848 0.731886 0.365943 0.930637i \(-0.380746\pi\)
0.365943 + 0.930637i \(0.380746\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 7.07107 0.280166
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 42.0000 1.65890 0.829450 0.558581i \(-0.188654\pi\)
0.829450 + 0.558581i \(0.188654\pi\)
\(642\) 0 0
\(643\) 18.0000 0.709851 0.354925 0.934895i \(-0.384506\pi\)
0.354925 + 0.934895i \(0.384506\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −8.48528 −0.333591 −0.166795 0.985992i \(-0.553342\pi\)
−0.166795 + 0.985992i \(0.553342\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −16.9706 −0.664109 −0.332055 0.943260i \(-0.607742\pi\)
−0.332055 + 0.943260i \(0.607742\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −9.89949 −0.385046 −0.192523 0.981292i \(-0.561667\pi\)
−0.192523 + 0.981292i \(0.561667\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 72.0000 2.78785
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −24.0000 −0.925132 −0.462566 0.886585i \(-0.653071\pi\)
−0.462566 + 0.886585i \(0.653071\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) 16.9706 0.651270
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) −42.0000 −1.59776 −0.798878 0.601494i \(-0.794573\pi\)
−0.798878 + 0.601494i \(0.794573\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 36.0000 1.36360
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 25.4558 0.961454 0.480727 0.876870i \(-0.340373\pi\)
0.480727 + 0.876870i \(0.340373\pi\)
\(702\) 0 0
\(703\) −42.4264 −1.60014
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 12.0000 0.451306
\(708\) 0 0
\(709\) 1.41421 0.0531119 0.0265560 0.999647i \(-0.491546\pi\)
0.0265560 + 0.999647i \(0.491546\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −12.0000 −0.449404
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −8.48528 −0.316448 −0.158224 0.987403i \(-0.550577\pi\)
−0.158224 + 0.987403i \(0.550577\pi\)
\(720\) 0 0
\(721\) −14.0000 −0.521387
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 42.4264 1.57568
\(726\) 0 0
\(727\) −32.5269 −1.20636 −0.603178 0.797606i \(-0.706099\pi\)
−0.603178 + 0.797606i \(0.706099\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 36.0000 1.33151
\(732\) 0 0
\(733\) −49.4975 −1.82823 −0.914115 0.405454i \(-0.867113\pi\)
−0.914115 + 0.405454i \(0.867113\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −16.9706 −0.622590 −0.311295 0.950313i \(-0.600763\pi\)
−0.311295 + 0.950313i \(0.600763\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −16.9706 −0.620091
\(750\) 0 0
\(751\) 49.4975 1.80619 0.903094 0.429442i \(-0.141290\pi\)
0.903094 + 0.429442i \(0.141290\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 1.41421 0.0514005 0.0257002 0.999670i \(-0.491818\pi\)
0.0257002 + 0.999670i \(0.491818\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 0 0
\(763\) −26.0000 −0.941263
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −25.4558 −0.915583 −0.457792 0.889060i \(-0.651359\pi\)
−0.457792 + 0.889060i \(0.651359\pi\)
\(774\) 0 0
\(775\) −7.07107 −0.254000
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 36.0000 1.28983
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 18.0000 0.641631 0.320815 0.947142i \(-0.396043\pi\)
0.320815 + 0.947142i \(0.396043\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −8.48528 −0.301702
\(792\) 0 0
\(793\) −10.0000 −0.355110
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −16.9706 −0.601128 −0.300564 0.953762i \(-0.597175\pi\)
−0.300564 + 0.953762i \(0.597175\pi\)
\(798\) 0 0
\(799\) 50.9117 1.80113
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 18.0000 0.632846 0.316423 0.948618i \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) 0 0
\(811\) −6.00000 −0.210688 −0.105344 0.994436i \(-0.533594\pi\)
−0.105344 + 0.994436i \(0.533594\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 36.0000 1.25948
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −8.48528 −0.296138 −0.148069 0.988977i \(-0.547306\pi\)
−0.148069 + 0.988977i \(0.547306\pi\)
\(822\) 0 0
\(823\) −32.5269 −1.13382 −0.566908 0.823781i \(-0.691861\pi\)
−0.566908 + 0.823781i \(0.691861\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 0 0
\(829\) −15.5563 −0.540294 −0.270147 0.962819i \(-0.587072\pi\)
−0.270147 + 0.962819i \(0.587072\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −30.0000 −1.03944
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 25.4558 0.878833 0.439417 0.898283i \(-0.355185\pi\)
0.439417 + 0.898283i \(0.355185\pi\)
\(840\) 0 0
\(841\) 43.0000 1.48276
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 15.5563 0.534522
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 60.0000 2.05677
\(852\) 0 0
\(853\) 43.8406 1.50107 0.750537 0.660828i \(-0.229795\pi\)
0.750537 + 0.660828i \(0.229795\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −42.0000 −1.43469 −0.717346 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) 0 0
\(859\) −6.00000 −0.204717 −0.102359 0.994748i \(-0.532639\pi\)
−0.102359 + 0.994748i \(0.532639\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 5.65685 0.191675
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 7.07107 0.238773 0.119386 0.992848i \(-0.461907\pi\)
0.119386 + 0.992848i \(0.461907\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 0 0
\(883\) 30.0000 1.00958 0.504790 0.863242i \(-0.331570\pi\)
0.504790 + 0.863242i \(0.331570\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 16.9706 0.569816 0.284908 0.958555i \(-0.408037\pi\)
0.284908 + 0.958555i \(0.408037\pi\)
\(888\) 0 0
\(889\) 14.0000 0.469545
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 50.9117 1.70369
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −12.0000 −0.400222
\(900\) 0 0
\(901\) 50.9117 1.69611
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −6.00000 −0.199227 −0.0996134 0.995026i \(-0.531761\pi\)
−0.0996134 + 0.995026i \(0.531761\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −16.9706 −0.562260 −0.281130 0.959670i \(-0.590709\pi\)
−0.281130 + 0.959670i \(0.590709\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 16.9706 0.560417
\(918\) 0 0
\(919\) 1.41421 0.0466506 0.0233253 0.999728i \(-0.492575\pi\)
0.0233253 + 0.999728i \(0.492575\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −12.0000 −0.394985
\(924\) 0 0
\(925\) 35.3553 1.16248
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −6.00000 −0.196854 −0.0984268 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) 0 0
\(931\) −30.0000 −0.983210
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 50.9117 1.65967 0.829837 0.558006i \(-0.188433\pi\)
0.829837 + 0.558006i \(0.188433\pi\)
\(942\) 0 0
\(943\) −50.9117 −1.65791
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 24.0000 0.779895 0.389948 0.920837i \(-0.372493\pi\)
0.389948 + 0.920837i \(0.372493\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −25.4558 −0.822012
\(960\) 0 0
\(961\) −29.0000 −0.935484
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −41.0122 −1.31886 −0.659432 0.751765i \(-0.729203\pi\)
−0.659432 + 0.751765i \(0.729203\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −60.0000 −1.92549 −0.962746 0.270408i \(-0.912841\pi\)
−0.962746 + 0.270408i \(0.912841\pi\)
\(972\) 0 0
\(973\) 5.65685 0.181350
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −6.00000 −0.191957 −0.0959785 0.995383i \(-0.530598\pi\)
−0.0959785 + 0.995383i \(0.530598\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −33.9411 −1.08255 −0.541277 0.840844i \(-0.682059\pi\)
−0.541277 + 0.840844i \(0.682059\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −50.9117 −1.61890
\(990\) 0 0
\(991\) −52.3259 −1.66219 −0.831094 0.556133i \(-0.812285\pi\)
−0.831094 + 0.556133i \(0.812285\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −57.9828 −1.83633 −0.918166 0.396196i \(-0.870330\pi\)
−0.918166 + 0.396196i \(0.870330\pi\)
\(998\) 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 9216.2.a.o.1.1 2
3.2 odd 2 3072.2.a.a.1.1 2
4.3 odd 2 9216.2.a.n.1.2 2
8.3 odd 2 inner 9216.2.a.o.1.2 2
8.5 even 2 9216.2.a.n.1.1 2
12.11 even 2 3072.2.a.g.1.2 2
24.5 odd 2 3072.2.a.g.1.1 2
24.11 even 2 3072.2.a.a.1.2 2
32.3 odd 8 2304.2.k.c.577.1 4
32.5 even 8 2304.2.k.b.1729.1 4
32.11 odd 8 2304.2.k.c.1729.2 4
32.13 even 8 2304.2.k.b.577.2 4
32.19 odd 8 2304.2.k.b.577.1 4
32.21 even 8 2304.2.k.c.1729.1 4
32.27 odd 8 2304.2.k.b.1729.2 4
32.29 even 8 2304.2.k.c.577.2 4
48.5 odd 4 3072.2.d.a.1537.4 4
48.11 even 4 3072.2.d.a.1537.1 4
48.29 odd 4 3072.2.d.a.1537.2 4
48.35 even 4 3072.2.d.a.1537.3 4
96.5 odd 8 768.2.j.b.193.2 yes 4
96.11 even 8 768.2.j.c.193.2 yes 4
96.29 odd 8 768.2.j.c.577.1 yes 4
96.35 even 8 768.2.j.c.577.2 yes 4
96.53 odd 8 768.2.j.c.193.1 yes 4
96.59 even 8 768.2.j.b.193.1 4
96.77 odd 8 768.2.j.b.577.2 yes 4
96.83 even 8 768.2.j.b.577.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
768.2.j.b.193.1 4 96.59 even 8
768.2.j.b.193.2 yes 4 96.5 odd 8
768.2.j.b.577.1 yes 4 96.83 even 8
768.2.j.b.577.2 yes 4 96.77 odd 8
768.2.j.c.193.1 yes 4 96.53 odd 8
768.2.j.c.193.2 yes 4 96.11 even 8
768.2.j.c.577.1 yes 4 96.29 odd 8
768.2.j.c.577.2 yes 4 96.35 even 8
2304.2.k.b.577.1 4 32.19 odd 8
2304.2.k.b.577.2 4 32.13 even 8
2304.2.k.b.1729.1 4 32.5 even 8
2304.2.k.b.1729.2 4 32.27 odd 8
2304.2.k.c.577.1 4 32.3 odd 8
2304.2.k.c.577.2 4 32.29 even 8
2304.2.k.c.1729.1 4 32.21 even 8
2304.2.k.c.1729.2 4 32.11 odd 8
3072.2.a.a.1.1 2 3.2 odd 2
3072.2.a.a.1.2 2 24.11 even 2
3072.2.a.g.1.1 2 24.5 odd 2
3072.2.a.g.1.2 2 12.11 even 2
3072.2.d.a.1537.1 4 48.11 even 4
3072.2.d.a.1537.2 4 48.29 odd 4
3072.2.d.a.1537.3 4 48.35 even 4
3072.2.d.a.1537.4 4 48.5 odd 4
9216.2.a.n.1.1 2 8.5 even 2
9216.2.a.n.1.2 2 4.3 odd 2
9216.2.a.o.1.1 2 1.1 even 1 trivial
9216.2.a.o.1.2 2 8.3 odd 2 inner