Properties

Label 9072.2.a.ca.1.3
Level $9072$
Weight $2$
Character 9072.1
Self dual yes
Analytic conductor $72.440$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9072,2,Mod(1,9072)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9072, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("9072.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9072 = 2^{4} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9072.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: \(72.4402847137\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.53209\) of defining polynomial
Character \(\chi\) \(=\) 9072.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.53209 q^{5} -1.00000 q^{7} +O(q^{10})\) \(q+2.53209 q^{5} -1.00000 q^{7} -0.467911 q^{11} +5.82295 q^{13} +3.87939 q^{17} +2.18479 q^{19} +0.106067 q^{23} +1.41147 q^{25} +8.78106 q^{29} +7.68004 q^{31} -2.53209 q^{35} -7.68004 q^{37} -2.22668 q^{41} -1.22668 q^{43} +5.33275 q^{47} +1.00000 q^{49} -0.716881 q^{53} -1.18479 q^{55} -0.736482 q^{59} +0.958111 q^{61} +14.7442 q^{65} +9.63816 q^{67} -13.2344 q^{71} -10.2686 q^{73} +0.467911 q^{77} +12.6382 q^{79} +2.73143 q^{83} +9.82295 q^{85} -8.11381 q^{89} -5.82295 q^{91} +5.53209 q^{95} -13.6040 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{5} - 3 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{5} - 3 q^{7} - 6 q^{11} - 3 q^{13} + 6 q^{17} + 3 q^{19} - 12 q^{23} - 6 q^{25} + 9 q^{29} + 3 q^{31} - 3 q^{35} - 3 q^{37} + 3 q^{43} - 3 q^{47} + 3 q^{49} + 6 q^{53} + 3 q^{59} + 6 q^{61} + 15 q^{65} + 12 q^{67} - 9 q^{71} - 21 q^{73} + 6 q^{77} + 21 q^{79} + 18 q^{83} + 9 q^{85} + 12 q^{89} + 3 q^{91} + 12 q^{95} - 3 q^{97}+O(q^{100}) \) 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
\(5\) 2.53209 1.13238 0.566192 0.824273i \(-0.308416\pi\)
0.566192 + 0.824273i \(0.308416\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −0.467911 −0.141081 −0.0705403 0.997509i \(-0.522472\pi\)
−0.0705403 + 0.997509i \(0.522472\pi\)
\(12\) 0 0
\(13\) 5.82295 1.61500 0.807498 0.589871i \(-0.200821\pi\)
0.807498 + 0.589871i \(0.200821\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.87939 0.940889 0.470445 0.882430i \(-0.344094\pi\)
0.470445 + 0.882430i \(0.344094\pi\)
\(18\) 0 0
\(19\) 2.18479 0.501226 0.250613 0.968087i \(-0.419368\pi\)
0.250613 + 0.968087i \(0.419368\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.106067 0.0221165 0.0110582 0.999939i \(-0.496480\pi\)
0.0110582 + 0.999939i \(0.496480\pi\)
\(24\) 0 0
\(25\) 1.41147 0.282295
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 8.78106 1.63060 0.815301 0.579038i \(-0.196572\pi\)
0.815301 + 0.579038i \(0.196572\pi\)
\(30\) 0 0
\(31\) 7.68004 1.37938 0.689688 0.724106i \(-0.257748\pi\)
0.689688 + 0.724106i \(0.257748\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.53209 −0.428001
\(36\) 0 0
\(37\) −7.68004 −1.26259 −0.631296 0.775542i \(-0.717477\pi\)
−0.631296 + 0.775542i \(0.717477\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.22668 −0.347749 −0.173875 0.984768i \(-0.555629\pi\)
−0.173875 + 0.984768i \(0.555629\pi\)
\(42\) 0 0
\(43\) −1.22668 −0.187067 −0.0935336 0.995616i \(-0.529816\pi\)
−0.0935336 + 0.995616i \(0.529816\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.33275 0.777861 0.388931 0.921267i \(-0.372845\pi\)
0.388931 + 0.921267i \(0.372845\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −0.716881 −0.0984712 −0.0492356 0.998787i \(-0.515679\pi\)
−0.0492356 + 0.998787i \(0.515679\pi\)
\(54\) 0 0
\(55\) −1.18479 −0.159757
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −0.736482 −0.0958818 −0.0479409 0.998850i \(-0.515266\pi\)
−0.0479409 + 0.998850i \(0.515266\pi\)
\(60\) 0 0
\(61\) 0.958111 0.122674 0.0613368 0.998117i \(-0.480464\pi\)
0.0613368 + 0.998117i \(0.480464\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 14.7442 1.82880
\(66\) 0 0
\(67\) 9.63816 1.17749 0.588744 0.808320i \(-0.299623\pi\)
0.588744 + 0.808320i \(0.299623\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −13.2344 −1.57064 −0.785318 0.619092i \(-0.787501\pi\)
−0.785318 + 0.619092i \(0.787501\pi\)
\(72\) 0 0
\(73\) −10.2686 −1.20185 −0.600923 0.799307i \(-0.705200\pi\)
−0.600923 + 0.799307i \(0.705200\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.467911 0.0533234
\(78\) 0 0
\(79\) 12.6382 1.42190 0.710952 0.703241i \(-0.248264\pi\)
0.710952 + 0.703241i \(0.248264\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 2.73143 0.299813 0.149907 0.988700i \(-0.452103\pi\)
0.149907 + 0.988700i \(0.452103\pi\)
\(84\) 0 0
\(85\) 9.82295 1.06545
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −8.11381 −0.860062 −0.430031 0.902814i \(-0.641497\pi\)
−0.430031 + 0.902814i \(0.641497\pi\)
\(90\) 0 0
\(91\) −5.82295 −0.610411
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 5.53209 0.567580
\(96\) 0 0
\(97\) −13.6040 −1.38128 −0.690639 0.723200i \(-0.742671\pi\)
−0.690639 + 0.723200i \(0.742671\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −9.57398 −0.952646 −0.476323 0.879270i \(-0.658031\pi\)
−0.476323 + 0.879270i \(0.658031\pi\)
\(102\) 0 0
\(103\) −3.04189 −0.299726 −0.149863 0.988707i \(-0.547883\pi\)
−0.149863 + 0.988707i \(0.547883\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.51754 0.630074 0.315037 0.949079i \(-0.397983\pi\)
0.315037 + 0.949079i \(0.397983\pi\)
\(108\) 0 0
\(109\) 10.6382 1.01895 0.509475 0.860485i \(-0.329840\pi\)
0.509475 + 0.860485i \(0.329840\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 5.17705 0.487016 0.243508 0.969899i \(-0.421702\pi\)
0.243508 + 0.969899i \(0.421702\pi\)
\(114\) 0 0
\(115\) 0.268571 0.0250443
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −3.87939 −0.355623
\(120\) 0 0
\(121\) −10.7811 −0.980096
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −9.08647 −0.812718
\(126\) 0 0
\(127\) 8.88207 0.788157 0.394078 0.919077i \(-0.371064\pi\)
0.394078 + 0.919077i \(0.371064\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −11.3628 −0.992771 −0.496385 0.868102i \(-0.665340\pi\)
−0.496385 + 0.868102i \(0.665340\pi\)
\(132\) 0 0
\(133\) −2.18479 −0.189446
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −5.72462 −0.489087 −0.244544 0.969638i \(-0.578638\pi\)
−0.244544 + 0.969638i \(0.578638\pi\)
\(138\) 0 0
\(139\) 0.923963 0.0783695 0.0391847 0.999232i \(-0.487524\pi\)
0.0391847 + 0.999232i \(0.487524\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −2.72462 −0.227844
\(144\) 0 0
\(145\) 22.2344 1.84647
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 8.72462 0.714749 0.357374 0.933961i \(-0.383672\pi\)
0.357374 + 0.933961i \(0.383672\pi\)
\(150\) 0 0
\(151\) −18.4270 −1.49956 −0.749782 0.661685i \(-0.769842\pi\)
−0.749782 + 0.661685i \(0.769842\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 19.4466 1.56198
\(156\) 0 0
\(157\) 4.92396 0.392975 0.196488 0.980506i \(-0.437046\pi\)
0.196488 + 0.980506i \(0.437046\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −0.106067 −0.00835924
\(162\) 0 0
\(163\) −7.63816 −0.598267 −0.299133 0.954211i \(-0.596698\pi\)
−0.299133 + 0.954211i \(0.596698\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 5.65539 0.437627 0.218814 0.975767i \(-0.429781\pi\)
0.218814 + 0.975767i \(0.429781\pi\)
\(168\) 0 0
\(169\) 20.9067 1.60821
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 21.0692 1.60186 0.800932 0.598755i \(-0.204338\pi\)
0.800932 + 0.598755i \(0.204338\pi\)
\(174\) 0 0
\(175\) −1.41147 −0.106697
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 5.12061 0.382733 0.191366 0.981519i \(-0.438708\pi\)
0.191366 + 0.981519i \(0.438708\pi\)
\(180\) 0 0
\(181\) −0.319955 −0.0237821 −0.0118910 0.999929i \(-0.503785\pi\)
−0.0118910 + 0.999929i \(0.503785\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −19.4466 −1.42974
\(186\) 0 0
\(187\) −1.81521 −0.132741
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 15.5672 1.12640 0.563200 0.826320i \(-0.309570\pi\)
0.563200 + 0.826320i \(0.309570\pi\)
\(192\) 0 0
\(193\) 6.04189 0.434905 0.217452 0.976071i \(-0.430225\pi\)
0.217452 + 0.976071i \(0.430225\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 25.2344 1.79788 0.898939 0.438074i \(-0.144339\pi\)
0.898939 + 0.438074i \(0.144339\pi\)
\(198\) 0 0
\(199\) −3.04189 −0.215634 −0.107817 0.994171i \(-0.534386\pi\)
−0.107817 + 0.994171i \(0.534386\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −8.78106 −0.616310
\(204\) 0 0
\(205\) −5.63816 −0.393786
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −1.02229 −0.0707132
\(210\) 0 0
\(211\) 5.45336 0.375425 0.187713 0.982224i \(-0.439893\pi\)
0.187713 + 0.982224i \(0.439893\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −3.10607 −0.211832
\(216\) 0 0
\(217\) −7.68004 −0.521355
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 22.5895 1.51953
\(222\) 0 0
\(223\) −14.1925 −0.950402 −0.475201 0.879877i \(-0.657625\pi\)
−0.475201 + 0.879877i \(0.657625\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2.89393 0.192077 0.0960385 0.995378i \(-0.469383\pi\)
0.0960385 + 0.995378i \(0.469383\pi\)
\(228\) 0 0
\(229\) 9.16756 0.605809 0.302905 0.953021i \(-0.402044\pi\)
0.302905 + 0.953021i \(0.402044\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 13.2713 0.869429 0.434715 0.900568i \(-0.356849\pi\)
0.434715 + 0.900568i \(0.356849\pi\)
\(234\) 0 0
\(235\) 13.5030 0.880838
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −9.53714 −0.616906 −0.308453 0.951240i \(-0.599811\pi\)
−0.308453 + 0.951240i \(0.599811\pi\)
\(240\) 0 0
\(241\) −8.95811 −0.577043 −0.288521 0.957473i \(-0.593164\pi\)
−0.288521 + 0.957473i \(0.593164\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.53209 0.161769
\(246\) 0 0
\(247\) 12.7219 0.809477
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 24.9982 1.57788 0.788938 0.614473i \(-0.210631\pi\)
0.788938 + 0.614473i \(0.210631\pi\)
\(252\) 0 0
\(253\) −0.0496299 −0.00312020
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 10.8520 0.676932 0.338466 0.940979i \(-0.390092\pi\)
0.338466 + 0.940979i \(0.390092\pi\)
\(258\) 0 0
\(259\) 7.68004 0.477215
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −26.0874 −1.60862 −0.804309 0.594211i \(-0.797464\pi\)
−0.804309 + 0.594211i \(0.797464\pi\)
\(264\) 0 0
\(265\) −1.81521 −0.111507
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −7.63310 −0.465399 −0.232699 0.972549i \(-0.574756\pi\)
−0.232699 + 0.972549i \(0.574756\pi\)
\(270\) 0 0
\(271\) −3.40373 −0.206762 −0.103381 0.994642i \(-0.532966\pi\)
−0.103381 + 0.994642i \(0.532966\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −0.660444 −0.0398263
\(276\) 0 0
\(277\) −5.72193 −0.343798 −0.171899 0.985115i \(-0.554990\pi\)
−0.171899 + 0.985115i \(0.554990\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 28.3773 1.69285 0.846425 0.532508i \(-0.178751\pi\)
0.846425 + 0.532508i \(0.178751\pi\)
\(282\) 0 0
\(283\) −4.57129 −0.271735 −0.135867 0.990727i \(-0.543382\pi\)
−0.135867 + 0.990727i \(0.543382\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 2.22668 0.131437
\(288\) 0 0
\(289\) −1.95037 −0.114728
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 4.32770 0.252827 0.126413 0.991978i \(-0.459653\pi\)
0.126413 + 0.991978i \(0.459653\pi\)
\(294\) 0 0
\(295\) −1.86484 −0.108575
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0.617622 0.0357180
\(300\) 0 0
\(301\) 1.22668 0.0707048
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 2.42602 0.138914
\(306\) 0 0
\(307\) −12.3773 −0.706411 −0.353206 0.935546i \(-0.614908\pi\)
−0.353206 + 0.935546i \(0.614908\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 21.9855 1.24668 0.623340 0.781951i \(-0.285775\pi\)
0.623340 + 0.781951i \(0.285775\pi\)
\(312\) 0 0
\(313\) −13.8898 −0.785099 −0.392549 0.919731i \(-0.628407\pi\)
−0.392549 + 0.919731i \(0.628407\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −6.18210 −0.347222 −0.173611 0.984814i \(-0.555543\pi\)
−0.173611 + 0.984814i \(0.555543\pi\)
\(318\) 0 0
\(319\) −4.10876 −0.230046
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 8.47565 0.471598
\(324\) 0 0
\(325\) 8.21894 0.455905
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −5.33275 −0.294004
\(330\) 0 0
\(331\) −10.7314 −0.589853 −0.294926 0.955520i \(-0.595295\pi\)
−0.294926 + 0.955520i \(0.595295\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 24.4047 1.33337
\(336\) 0 0
\(337\) −18.5945 −1.01291 −0.506454 0.862267i \(-0.669044\pi\)
−0.506454 + 0.862267i \(0.669044\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −3.59358 −0.194603
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 20.4124 1.09580 0.547898 0.836545i \(-0.315428\pi\)
0.547898 + 0.836545i \(0.315428\pi\)
\(348\) 0 0
\(349\) −3.56212 −0.190676 −0.0953379 0.995445i \(-0.530393\pi\)
−0.0953379 + 0.995445i \(0.530393\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 10.0223 0.533433 0.266716 0.963775i \(-0.414061\pi\)
0.266716 + 0.963775i \(0.414061\pi\)
\(354\) 0 0
\(355\) −33.5107 −1.77857
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −9.48070 −0.500372 −0.250186 0.968198i \(-0.580492\pi\)
−0.250186 + 0.968198i \(0.580492\pi\)
\(360\) 0 0
\(361\) −14.2267 −0.748773
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −26.0009 −1.36095
\(366\) 0 0
\(367\) −16.1334 −0.842157 −0.421079 0.907024i \(-0.638348\pi\)
−0.421079 + 0.907024i \(0.638348\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0.716881 0.0372186
\(372\) 0 0
\(373\) 14.0496 0.727462 0.363731 0.931504i \(-0.381503\pi\)
0.363731 + 0.931504i \(0.381503\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 51.1317 2.63341
\(378\) 0 0
\(379\) −16.0574 −0.824812 −0.412406 0.911000i \(-0.635311\pi\)
−0.412406 + 0.911000i \(0.635311\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 32.0205 1.63617 0.818086 0.575095i \(-0.195035\pi\)
0.818086 + 0.575095i \(0.195035\pi\)
\(384\) 0 0
\(385\) 1.18479 0.0603826
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −30.0428 −1.52323 −0.761616 0.648029i \(-0.775594\pi\)
−0.761616 + 0.648029i \(0.775594\pi\)
\(390\) 0 0
\(391\) 0.411474 0.0208091
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 32.0009 1.61014
\(396\) 0 0
\(397\) −12.3200 −0.618321 −0.309160 0.951010i \(-0.600048\pi\)
−0.309160 + 0.951010i \(0.600048\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 20.9760 1.04749 0.523745 0.851875i \(-0.324535\pi\)
0.523745 + 0.851875i \(0.324535\pi\)
\(402\) 0 0
\(403\) 44.7205 2.22769
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 3.59358 0.178127
\(408\) 0 0
\(409\) 25.6614 1.26887 0.634437 0.772975i \(-0.281232\pi\)
0.634437 + 0.772975i \(0.281232\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0.736482 0.0362399
\(414\) 0 0
\(415\) 6.91622 0.339504
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 1.47977 0.0722915 0.0361458 0.999347i \(-0.488492\pi\)
0.0361458 + 0.999347i \(0.488492\pi\)
\(420\) 0 0
\(421\) 13.1070 0.638796 0.319398 0.947621i \(-0.396519\pi\)
0.319398 + 0.947621i \(0.396519\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 5.47565 0.265608
\(426\) 0 0
\(427\) −0.958111 −0.0463662
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −17.7270 −0.853879 −0.426939 0.904280i \(-0.640408\pi\)
−0.426939 + 0.904280i \(0.640408\pi\)
\(432\) 0 0
\(433\) −5.83843 −0.280577 −0.140289 0.990111i \(-0.544803\pi\)
−0.140289 + 0.990111i \(0.544803\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0.231734 0.0110853
\(438\) 0 0
\(439\) −29.8553 −1.42492 −0.712459 0.701714i \(-0.752418\pi\)
−0.712459 + 0.701714i \(0.752418\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −10.6655 −0.506733 −0.253367 0.967370i \(-0.581538\pi\)
−0.253367 + 0.967370i \(0.581538\pi\)
\(444\) 0 0
\(445\) −20.5449 −0.973921
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 3.55438 0.167741 0.0838707 0.996477i \(-0.473272\pi\)
0.0838707 + 0.996477i \(0.473272\pi\)
\(450\) 0 0
\(451\) 1.04189 0.0490606
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −14.7442 −0.691220
\(456\) 0 0
\(457\) 5.02322 0.234976 0.117488 0.993074i \(-0.462516\pi\)
0.117488 + 0.993074i \(0.462516\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 18.4611 0.859819 0.429910 0.902872i \(-0.358545\pi\)
0.429910 + 0.902872i \(0.358545\pi\)
\(462\) 0 0
\(463\) 14.2344 0.661530 0.330765 0.943713i \(-0.392693\pi\)
0.330765 + 0.943713i \(0.392693\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 3.36865 0.155883 0.0779413 0.996958i \(-0.475165\pi\)
0.0779413 + 0.996958i \(0.475165\pi\)
\(468\) 0 0
\(469\) −9.63816 −0.445049
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0.573978 0.0263915
\(474\) 0 0
\(475\) 3.08378 0.141493
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 36.7665 1.67990 0.839952 0.542660i \(-0.182583\pi\)
0.839952 + 0.542660i \(0.182583\pi\)
\(480\) 0 0
\(481\) −44.7205 −2.03908
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −34.4466 −1.56414
\(486\) 0 0
\(487\) 37.4175 1.69555 0.847773 0.530358i \(-0.177943\pi\)
0.847773 + 0.530358i \(0.177943\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 26.6705 1.20363 0.601813 0.798637i \(-0.294445\pi\)
0.601813 + 0.798637i \(0.294445\pi\)
\(492\) 0 0
\(493\) 34.0651 1.53422
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 13.2344 0.593645
\(498\) 0 0
\(499\) −33.7452 −1.51064 −0.755320 0.655356i \(-0.772519\pi\)
−0.755320 + 0.655356i \(0.772519\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 32.0401 1.42860 0.714299 0.699840i \(-0.246745\pi\)
0.714299 + 0.699840i \(0.246745\pi\)
\(504\) 0 0
\(505\) −24.2422 −1.07876
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −7.93851 −0.351868 −0.175934 0.984402i \(-0.556295\pi\)
−0.175934 + 0.984402i \(0.556295\pi\)
\(510\) 0 0
\(511\) 10.2686 0.454255
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −7.70233 −0.339405
\(516\) 0 0
\(517\) −2.49525 −0.109741
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −14.6750 −0.642923 −0.321462 0.946923i \(-0.604174\pi\)
−0.321462 + 0.946923i \(0.604174\pi\)
\(522\) 0 0
\(523\) −28.3432 −1.23936 −0.619680 0.784854i \(-0.712738\pi\)
−0.619680 + 0.784854i \(0.712738\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 29.7939 1.29784
\(528\) 0 0
\(529\) −22.9887 −0.999511
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −12.9659 −0.561613
\(534\) 0 0
\(535\) 16.5030 0.713487
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −0.467911 −0.0201544
\(540\) 0 0
\(541\) 11.2858 0.485215 0.242607 0.970125i \(-0.421997\pi\)
0.242607 + 0.970125i \(0.421997\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 26.9368 1.15384
\(546\) 0 0
\(547\) 29.2404 1.25023 0.625115 0.780533i \(-0.285052\pi\)
0.625115 + 0.780533i \(0.285052\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 19.1848 0.817300
\(552\) 0 0
\(553\) −12.6382 −0.537429
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −0.775682 −0.0328667 −0.0164334 0.999865i \(-0.505231\pi\)
−0.0164334 + 0.999865i \(0.505231\pi\)
\(558\) 0 0
\(559\) −7.14290 −0.302113
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −24.9522 −1.05161 −0.525806 0.850605i \(-0.676236\pi\)
−0.525806 + 0.850605i \(0.676236\pi\)
\(564\) 0 0
\(565\) 13.1088 0.551489
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −24.8033 −1.03981 −0.519905 0.854224i \(-0.674033\pi\)
−0.519905 + 0.854224i \(0.674033\pi\)
\(570\) 0 0
\(571\) −8.79654 −0.368124 −0.184062 0.982915i \(-0.558925\pi\)
−0.184062 + 0.982915i \(0.558925\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0.149711 0.00624336
\(576\) 0 0
\(577\) −12.8743 −0.535965 −0.267983 0.963424i \(-0.586357\pi\)
−0.267983 + 0.963424i \(0.586357\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −2.73143 −0.113319
\(582\) 0 0
\(583\) 0.335437 0.0138924
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −44.8631 −1.85170 −0.925849 0.377894i \(-0.876648\pi\)
−0.925849 + 0.377894i \(0.876648\pi\)
\(588\) 0 0
\(589\) 16.7793 0.691379
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 3.76053 0.154426 0.0772131 0.997015i \(-0.475398\pi\)
0.0772131 + 0.997015i \(0.475398\pi\)
\(594\) 0 0
\(595\) −9.82295 −0.402702
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 3.69047 0.150789 0.0753943 0.997154i \(-0.475978\pi\)
0.0753943 + 0.997154i \(0.475978\pi\)
\(600\) 0 0
\(601\) −21.8571 −0.891570 −0.445785 0.895140i \(-0.647075\pi\)
−0.445785 + 0.895140i \(0.647075\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −27.2986 −1.10985
\(606\) 0 0
\(607\) −24.3946 −0.990145 −0.495072 0.868852i \(-0.664858\pi\)
−0.495072 + 0.868852i \(0.664858\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 31.0523 1.25624
\(612\) 0 0
\(613\) 42.0215 1.69723 0.848616 0.529010i \(-0.177437\pi\)
0.848616 + 0.529010i \(0.177437\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 46.4097 1.86838 0.934192 0.356769i \(-0.116122\pi\)
0.934192 + 0.356769i \(0.116122\pi\)
\(618\) 0 0
\(619\) 27.2094 1.09364 0.546820 0.837250i \(-0.315838\pi\)
0.546820 + 0.837250i \(0.315838\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 8.11381 0.325073
\(624\) 0 0
\(625\) −30.0651 −1.20260
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −29.7939 −1.18796
\(630\) 0 0
\(631\) 29.6023 1.17845 0.589224 0.807970i \(-0.299434\pi\)
0.589224 + 0.807970i \(0.299434\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 22.4902 0.892496
\(636\) 0 0
\(637\) 5.82295 0.230714
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −0.279000 −0.0110198 −0.00550991 0.999985i \(-0.501754\pi\)
−0.00550991 + 0.999985i \(0.501754\pi\)
\(642\) 0 0
\(643\) 18.2439 0.719470 0.359735 0.933055i \(-0.382867\pi\)
0.359735 + 0.933055i \(0.382867\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −22.4570 −0.882875 −0.441438 0.897292i \(-0.645531\pi\)
−0.441438 + 0.897292i \(0.645531\pi\)
\(648\) 0 0
\(649\) 0.344608 0.0135270
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −50.5313 −1.97744 −0.988721 0.149771i \(-0.952146\pi\)
−0.988721 + 0.149771i \(0.952146\pi\)
\(654\) 0 0
\(655\) −28.7716 −1.12420
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 2.67263 0.104111 0.0520554 0.998644i \(-0.483423\pi\)
0.0520554 + 0.998644i \(0.483423\pi\)
\(660\) 0 0
\(661\) −34.6100 −1.34617 −0.673086 0.739564i \(-0.735032\pi\)
−0.673086 + 0.739564i \(0.735032\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −5.53209 −0.214525
\(666\) 0 0
\(667\) 0.931379 0.0360631
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −0.448311 −0.0173068
\(672\) 0 0
\(673\) 16.5125 0.636510 0.318255 0.948005i \(-0.396903\pi\)
0.318255 + 0.948005i \(0.396903\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −43.7579 −1.68175 −0.840877 0.541226i \(-0.817960\pi\)
−0.840877 + 0.541226i \(0.817960\pi\)
\(678\) 0 0
\(679\) 13.6040 0.522074
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −28.2412 −1.08062 −0.540310 0.841466i \(-0.681693\pi\)
−0.540310 + 0.841466i \(0.681693\pi\)
\(684\) 0 0
\(685\) −14.4953 −0.553835
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −4.17436 −0.159031
\(690\) 0 0
\(691\) 29.0651 1.10569 0.552844 0.833284i \(-0.313542\pi\)
0.552844 + 0.833284i \(0.313542\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 2.33956 0.0887444
\(696\) 0 0
\(697\) −8.63816 −0.327193
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −1.10876 −0.0418771 −0.0209386 0.999781i \(-0.506665\pi\)
−0.0209386 + 0.999781i \(0.506665\pi\)
\(702\) 0 0
\(703\) −16.7793 −0.632843
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 9.57398 0.360066
\(708\) 0 0
\(709\) −18.4688 −0.693612 −0.346806 0.937937i \(-0.612734\pi\)
−0.346806 + 0.937937i \(0.612734\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0.814598 0.0305069
\(714\) 0 0
\(715\) −6.89899 −0.258007
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 33.7769 1.25967 0.629834 0.776730i \(-0.283123\pi\)
0.629834 + 0.776730i \(0.283123\pi\)
\(720\) 0 0
\(721\) 3.04189 0.113286
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 12.3942 0.460310
\(726\) 0 0
\(727\) −16.8043 −0.623236 −0.311618 0.950207i \(-0.600871\pi\)
−0.311618 + 0.950207i \(0.600871\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −4.75877 −0.176009
\(732\) 0 0
\(733\) −13.6364 −0.503672 −0.251836 0.967770i \(-0.581034\pi\)
−0.251836 + 0.967770i \(0.581034\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −4.50980 −0.166121
\(738\) 0 0
\(739\) 32.0419 1.17868 0.589340 0.807885i \(-0.299388\pi\)
0.589340 + 0.807885i \(0.299388\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −33.7529 −1.23827 −0.619137 0.785283i \(-0.712517\pi\)
−0.619137 + 0.785283i \(0.712517\pi\)
\(744\) 0 0
\(745\) 22.0915 0.809371
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −6.51754 −0.238146
\(750\) 0 0
\(751\) −26.1165 −0.953004 −0.476502 0.879173i \(-0.658096\pi\)
−0.476502 + 0.879173i \(0.658096\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −46.6587 −1.69808
\(756\) 0 0
\(757\) 35.6536 1.29585 0.647927 0.761703i \(-0.275636\pi\)
0.647927 + 0.761703i \(0.275636\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 40.7648 1.47772 0.738861 0.673858i \(-0.235364\pi\)
0.738861 + 0.673858i \(0.235364\pi\)
\(762\) 0 0
\(763\) −10.6382 −0.385127
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −4.28850 −0.154849
\(768\) 0 0
\(769\) 39.4270 1.42177 0.710886 0.703307i \(-0.248294\pi\)
0.710886 + 0.703307i \(0.248294\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 24.9026 0.895685 0.447842 0.894113i \(-0.352193\pi\)
0.447842 + 0.894113i \(0.352193\pi\)
\(774\) 0 0
\(775\) 10.8402 0.389391
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −4.86484 −0.174301
\(780\) 0 0
\(781\) 6.19253 0.221586
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 12.4679 0.444999
\(786\) 0 0
\(787\) 30.7050 1.09452 0.547258 0.836964i \(-0.315672\pi\)
0.547258 + 0.836964i \(0.315672\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −5.17705 −0.184075
\(792\) 0 0
\(793\) 5.57903 0.198117
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −11.0137 −0.390126 −0.195063 0.980791i \(-0.562491\pi\)
−0.195063 + 0.980791i \(0.562491\pi\)
\(798\) 0 0
\(799\) 20.6878 0.731881
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 4.80478 0.169557
\(804\) 0 0
\(805\) −0.268571 −0.00946587
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 16.9881 0.597271 0.298636 0.954367i \(-0.403468\pi\)
0.298636 + 0.954367i \(0.403468\pi\)
\(810\) 0 0
\(811\) −37.9796 −1.33364 −0.666822 0.745217i \(-0.732346\pi\)
−0.666822 + 0.745217i \(0.732346\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −19.3405 −0.677468
\(816\) 0 0
\(817\) −2.68004 −0.0937629
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 8.27868 0.288928 0.144464 0.989510i \(-0.453854\pi\)
0.144464 + 0.989510i \(0.453854\pi\)
\(822\) 0 0
\(823\) −54.5526 −1.90158 −0.950792 0.309829i \(-0.899728\pi\)
−0.950792 + 0.309829i \(0.899728\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 31.8708 1.10826 0.554129 0.832431i \(-0.313052\pi\)
0.554129 + 0.832431i \(0.313052\pi\)
\(828\) 0 0
\(829\) −0.352349 −0.0122376 −0.00611879 0.999981i \(-0.501948\pi\)
−0.00611879 + 0.999981i \(0.501948\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 3.87939 0.134413
\(834\) 0 0
\(835\) 14.3200 0.495562
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −25.0155 −0.863630 −0.431815 0.901962i \(-0.642127\pi\)
−0.431815 + 0.901962i \(0.642127\pi\)
\(840\) 0 0
\(841\) 48.1070 1.65886
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 52.9377 1.82111
\(846\) 0 0
\(847\) 10.7811 0.370442
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −0.814598 −0.0279241
\(852\) 0 0
\(853\) 39.1908 1.34187 0.670933 0.741518i \(-0.265894\pi\)
0.670933 + 0.741518i \(0.265894\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 16.4074 0.560465 0.280232 0.959932i \(-0.409589\pi\)
0.280232 + 0.959932i \(0.409589\pi\)
\(858\) 0 0
\(859\) 26.8324 0.915511 0.457756 0.889078i \(-0.348653\pi\)
0.457756 + 0.889078i \(0.348653\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −14.5057 −0.493779 −0.246890 0.969044i \(-0.579408\pi\)
−0.246890 + 0.969044i \(0.579408\pi\)
\(864\) 0 0
\(865\) 53.3492 1.81393
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −5.91353 −0.200603
\(870\) 0 0
\(871\) 56.1225 1.90164
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 9.08647 0.307179
\(876\) 0 0
\(877\) 18.9145 0.638696 0.319348 0.947637i \(-0.396536\pi\)
0.319348 + 0.947637i \(0.396536\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 53.8976 1.81585 0.907927 0.419128i \(-0.137664\pi\)
0.907927 + 0.419128i \(0.137664\pi\)
\(882\) 0 0
\(883\) −43.4252 −1.46137 −0.730687 0.682712i \(-0.760800\pi\)
−0.730687 + 0.682712i \(0.760800\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 38.9600 1.30815 0.654074 0.756431i \(-0.273058\pi\)
0.654074 + 0.756431i \(0.273058\pi\)
\(888\) 0 0
\(889\) −8.88207 −0.297895
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 11.6509 0.389884
\(894\) 0 0
\(895\) 12.9659 0.433401
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 67.4389 2.24921
\(900\) 0 0
\(901\) −2.78106 −0.0926505
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −0.810155 −0.0269305
\(906\) 0 0
\(907\) −34.5276 −1.14647 −0.573236 0.819390i \(-0.694312\pi\)
−0.573236 + 0.819390i \(0.694312\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −46.5262 −1.54148 −0.770741 0.637148i \(-0.780114\pi\)
−0.770741 + 0.637148i \(0.780114\pi\)
\(912\) 0 0
\(913\) −1.27807 −0.0422978
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 11.3628 0.375232
\(918\) 0 0
\(919\) 9.95636 0.328430 0.164215 0.986425i \(-0.447491\pi\)
0.164215 + 0.986425i \(0.447491\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −77.0634 −2.53657
\(924\) 0 0
\(925\) −10.8402 −0.356423
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 9.04601 0.296790 0.148395 0.988928i \(-0.452589\pi\)
0.148395 + 0.988928i \(0.452589\pi\)
\(930\) 0 0
\(931\) 2.18479 0.0716037
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −4.59627 −0.150314
\(936\) 0 0
\(937\) −24.3928 −0.796878 −0.398439 0.917195i \(-0.630448\pi\)
−0.398439 + 0.917195i \(0.630448\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −59.5381 −1.94089 −0.970443 0.241331i \(-0.922416\pi\)
−0.970443 + 0.241331i \(0.922416\pi\)
\(942\) 0 0
\(943\) −0.236177 −0.00769098
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −8.64858 −0.281041 −0.140521 0.990078i \(-0.544878\pi\)
−0.140521 + 0.990078i \(0.544878\pi\)
\(948\) 0 0
\(949\) −59.7934 −1.94097
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 3.78249 0.122527 0.0612634 0.998122i \(-0.480487\pi\)
0.0612634 + 0.998122i \(0.480487\pi\)
\(954\) 0 0
\(955\) 39.4175 1.27552
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 5.72462 0.184858
\(960\) 0 0
\(961\) 27.9831 0.902680
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 15.2986 0.492479
\(966\) 0 0
\(967\) 32.9489 1.05957 0.529783 0.848133i \(-0.322273\pi\)
0.529783 + 0.848133i \(0.322273\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 55.4570 1.77970 0.889850 0.456254i \(-0.150809\pi\)
0.889850 + 0.456254i \(0.150809\pi\)
\(972\) 0 0
\(973\) −0.923963 −0.0296209
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 56.5485 1.80915 0.904573 0.426318i \(-0.140189\pi\)
0.904573 + 0.426318i \(0.140189\pi\)
\(978\) 0 0
\(979\) 3.79654 0.121338
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −28.9973 −0.924871 −0.462435 0.886653i \(-0.653024\pi\)
−0.462435 + 0.886653i \(0.653024\pi\)
\(984\) 0 0
\(985\) 63.8958 2.03589
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −0.130110 −0.00413726
\(990\) 0 0
\(991\) −6.80922 −0.216302 −0.108151 0.994134i \(-0.534493\pi\)
−0.108151 + 0.994134i \(0.534493\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −7.70233 −0.244180
\(996\) 0 0
\(997\) −38.9377 −1.23317 −0.616585 0.787289i \(-0.711484\pi\)
−0.616585 + 0.787289i \(0.711484\pi\)
\(998\) 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 9072.2.a.ca.1.3 3
3.2 odd 2 9072.2.a.bs.1.1 3
4.3 odd 2 567.2.a.h.1.2 3
9.2 odd 6 3024.2.r.k.1009.3 6
9.4 even 3 1008.2.r.h.673.1 6
9.5 odd 6 3024.2.r.k.2017.3 6
9.7 even 3 1008.2.r.h.337.1 6
12.11 even 2 567.2.a.c.1.2 3
28.27 even 2 3969.2.a.q.1.2 3
36.7 odd 6 63.2.f.a.22.2 6
36.11 even 6 189.2.f.b.64.2 6
36.23 even 6 189.2.f.b.127.2 6
36.31 odd 6 63.2.f.a.43.2 yes 6
84.83 odd 2 3969.2.a.l.1.2 3
252.11 even 6 1323.2.h.c.226.2 6
252.23 even 6 1323.2.h.c.802.2 6
252.31 even 6 441.2.g.b.79.2 6
252.47 odd 6 1323.2.g.e.361.2 6
252.59 odd 6 1323.2.g.e.667.2 6
252.67 odd 6 441.2.g.c.79.2 6
252.79 odd 6 441.2.g.c.67.2 6
252.83 odd 6 1323.2.f.d.442.2 6
252.95 even 6 1323.2.g.d.667.2 6
252.103 even 6 441.2.h.e.214.2 6
252.115 even 6 441.2.h.e.373.2 6
252.131 odd 6 1323.2.h.b.802.2 6
252.139 even 6 441.2.f.c.295.2 6
252.151 odd 6 441.2.h.d.373.2 6
252.167 odd 6 1323.2.f.d.883.2 6
252.187 even 6 441.2.g.b.67.2 6
252.191 even 6 1323.2.g.d.361.2 6
252.223 even 6 441.2.f.c.148.2 6
252.227 odd 6 1323.2.h.b.226.2 6
252.247 odd 6 441.2.h.d.214.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.a.22.2 6 36.7 odd 6
63.2.f.a.43.2 yes 6 36.31 odd 6
189.2.f.b.64.2 6 36.11 even 6
189.2.f.b.127.2 6 36.23 even 6
441.2.f.c.148.2 6 252.223 even 6
441.2.f.c.295.2 6 252.139 even 6
441.2.g.b.67.2 6 252.187 even 6
441.2.g.b.79.2 6 252.31 even 6
441.2.g.c.67.2 6 252.79 odd 6
441.2.g.c.79.2 6 252.67 odd 6
441.2.h.d.214.2 6 252.247 odd 6
441.2.h.d.373.2 6 252.151 odd 6
441.2.h.e.214.2 6 252.103 even 6
441.2.h.e.373.2 6 252.115 even 6
567.2.a.c.1.2 3 12.11 even 2
567.2.a.h.1.2 3 4.3 odd 2
1008.2.r.h.337.1 6 9.7 even 3
1008.2.r.h.673.1 6 9.4 even 3
1323.2.f.d.442.2 6 252.83 odd 6
1323.2.f.d.883.2 6 252.167 odd 6
1323.2.g.d.361.2 6 252.191 even 6
1323.2.g.d.667.2 6 252.95 even 6
1323.2.g.e.361.2 6 252.47 odd 6
1323.2.g.e.667.2 6 252.59 odd 6
1323.2.h.b.226.2 6 252.227 odd 6
1323.2.h.b.802.2 6 252.131 odd 6
1323.2.h.c.226.2 6 252.11 even 6
1323.2.h.c.802.2 6 252.23 even 6
3024.2.r.k.1009.3 6 9.2 odd 6
3024.2.r.k.2017.3 6 9.5 odd 6
3969.2.a.l.1.2 3 84.83 odd 2
3969.2.a.q.1.2 3 28.27 even 2
9072.2.a.bs.1.1 3 3.2 odd 2
9072.2.a.ca.1.3 3 1.1 even 1 trivial