Properties

Label 6480.2.a.bv.1.1
Level $6480$
Weight $2$
Character 6480.1
Self dual yes
Analytic conductor $51.743$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6480,2,Mod(1,6480)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6480, 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("6480.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6480 = 2^{4} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6480.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: \(51.7430605098\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.564.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 45)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.08613\) of defining polynomial
Character \(\chi\) \(=\) 6480.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{5} -4.08613 q^{7} +O(q^{10})\) \(q+1.00000 q^{5} -4.08613 q^{7} +1.35194 q^{11} +0.648061 q^{13} -1.35194 q^{17} -0.648061 q^{19} -4.79001 q^{23} +1.00000 q^{25} +3.87614 q^{29} +7.69646 q^{31} -4.08613 q^{35} +7.52420 q^{37} -0.179679 q^{41} +0.820321 q^{43} -10.9065 q^{47} +9.69646 q^{49} +4.17226 q^{53} +1.35194 q^{55} -4.17226 q^{59} -3.82032 q^{61} +0.648061 q^{65} -8.14195 q^{67} +6.11644 q^{71} -12.3445 q^{73} -5.52420 q^{77} -10.3445 q^{79} +12.2584 q^{83} -1.35194 q^{85} -3.00000 q^{89} -2.64806 q^{91} -0.648061 q^{95} -13.5800 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{5} - 5 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{5} - 5 q^{7} + 2 q^{11} + 4 q^{13} - 2 q^{17} - 4 q^{19} - 3 q^{23} + 3 q^{25} - 7 q^{29} - 8 q^{31} - 5 q^{35} + 6 q^{37} - 13 q^{41} - 10 q^{43} - 13 q^{47} - 2 q^{49} - 2 q^{53} + 2 q^{55} + 2 q^{59} + q^{61} + 4 q^{65} - 11 q^{67} + 10 q^{71} - 8 q^{73} - 2 q^{79} + 15 q^{83} - 2 q^{85} - 9 q^{89} - 10 q^{91} - 4 q^{95} - 18 q^{97}+O(q^{100}) \) 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
\(6\) 0 0
\(7\) −4.08613 −1.54441 −0.772206 0.635372i \(-0.780847\pi\)
−0.772206 + 0.635372i \(0.780847\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.35194 0.407625 0.203813 0.979010i \(-0.434667\pi\)
0.203813 + 0.979010i \(0.434667\pi\)
\(12\) 0 0
\(13\) 0.648061 0.179740 0.0898699 0.995954i \(-0.471355\pi\)
0.0898699 + 0.995954i \(0.471355\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.35194 −0.327893 −0.163947 0.986469i \(-0.552423\pi\)
−0.163947 + 0.986469i \(0.552423\pi\)
\(18\) 0 0
\(19\) −0.648061 −0.148675 −0.0743377 0.997233i \(-0.523684\pi\)
−0.0743377 + 0.997233i \(0.523684\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.79001 −0.998786 −0.499393 0.866376i \(-0.666444\pi\)
−0.499393 + 0.866376i \(0.666444\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.87614 0.719781 0.359890 0.932995i \(-0.382814\pi\)
0.359890 + 0.932995i \(0.382814\pi\)
\(30\) 0 0
\(31\) 7.69646 1.38233 0.691163 0.722699i \(-0.257099\pi\)
0.691163 + 0.722699i \(0.257099\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.08613 −0.690682
\(36\) 0 0
\(37\) 7.52420 1.23697 0.618485 0.785796i \(-0.287747\pi\)
0.618485 + 0.785796i \(0.287747\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −0.179679 −0.0280611 −0.0140306 0.999902i \(-0.504466\pi\)
−0.0140306 + 0.999902i \(0.504466\pi\)
\(42\) 0 0
\(43\) 0.820321 0.125098 0.0625489 0.998042i \(-0.480077\pi\)
0.0625489 + 0.998042i \(0.480077\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −10.9065 −1.59087 −0.795435 0.606039i \(-0.792757\pi\)
−0.795435 + 0.606039i \(0.792757\pi\)
\(48\) 0 0
\(49\) 9.69646 1.38521
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.17226 0.573104 0.286552 0.958065i \(-0.407491\pi\)
0.286552 + 0.958065i \(0.407491\pi\)
\(54\) 0 0
\(55\) 1.35194 0.182295
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.17226 −0.543182 −0.271591 0.962413i \(-0.587550\pi\)
−0.271591 + 0.962413i \(0.587550\pi\)
\(60\) 0 0
\(61\) −3.82032 −0.489142 −0.244571 0.969631i \(-0.578647\pi\)
−0.244571 + 0.969631i \(0.578647\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.648061 0.0803820
\(66\) 0 0
\(67\) −8.14195 −0.994697 −0.497349 0.867551i \(-0.665693\pi\)
−0.497349 + 0.867551i \(0.665693\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 6.11644 0.725888 0.362944 0.931811i \(-0.381772\pi\)
0.362944 + 0.931811i \(0.381772\pi\)
\(72\) 0 0
\(73\) −12.3445 −1.44482 −0.722408 0.691467i \(-0.756965\pi\)
−0.722408 + 0.691467i \(0.756965\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −5.52420 −0.629541
\(78\) 0 0
\(79\) −10.3445 −1.16385 −0.581925 0.813243i \(-0.697700\pi\)
−0.581925 + 0.813243i \(0.697700\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 12.2584 1.34553 0.672767 0.739855i \(-0.265106\pi\)
0.672767 + 0.739855i \(0.265106\pi\)
\(84\) 0 0
\(85\) −1.35194 −0.146638
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −3.00000 −0.317999 −0.159000 0.987279i \(-0.550827\pi\)
−0.159000 + 0.987279i \(0.550827\pi\)
\(90\) 0 0
\(91\) −2.64806 −0.277592
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −0.648061 −0.0664896
\(96\) 0 0
\(97\) −13.5800 −1.37884 −0.689421 0.724361i \(-0.742135\pi\)
−0.689421 + 0.724361i \(0.742135\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1.46838 −0.146109 −0.0730547 0.997328i \(-0.523275\pi\)
−0.0730547 + 0.997328i \(0.523275\pi\)
\(102\) 0 0
\(103\) −7.52420 −0.741381 −0.370691 0.928756i \(-0.620879\pi\)
−0.370691 + 0.928756i \(0.620879\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1.20999 −0.116974 −0.0584871 0.998288i \(-0.518628\pi\)
−0.0584871 + 0.998288i \(0.518628\pi\)
\(108\) 0 0
\(109\) 14.1042 1.35094 0.675469 0.737388i \(-0.263941\pi\)
0.675469 + 0.737388i \(0.263941\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −11.9245 −1.12177 −0.560883 0.827895i \(-0.689538\pi\)
−0.560883 + 0.827895i \(0.689538\pi\)
\(114\) 0 0
\(115\) −4.79001 −0.446671
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 5.52420 0.506403
\(120\) 0 0
\(121\) −9.17226 −0.833842
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 7.07871 0.628134 0.314067 0.949401i \(-0.398308\pi\)
0.314067 + 0.949401i \(0.398308\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 6.00000 0.524222 0.262111 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) 0 0
\(133\) 2.64806 0.229616
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −7.46838 −0.638067 −0.319033 0.947743i \(-0.603358\pi\)
−0.319033 + 0.947743i \(0.603358\pi\)
\(138\) 0 0
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0.876139 0.0732664
\(144\) 0 0
\(145\) 3.87614 0.321896
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −10.5848 −0.867143 −0.433571 0.901119i \(-0.642747\pi\)
−0.433571 + 0.901119i \(0.642747\pi\)
\(150\) 0 0
\(151\) −17.6965 −1.44012 −0.720059 0.693913i \(-0.755885\pi\)
−0.720059 + 0.693913i \(0.755885\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 7.69646 0.618195
\(156\) 0 0
\(157\) −2.53162 −0.202045 −0.101023 0.994884i \(-0.532211\pi\)
−0.101023 + 0.994884i \(0.532211\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 19.5726 1.54254
\(162\) 0 0
\(163\) −8.47580 −0.663876 −0.331938 0.943301i \(-0.607702\pi\)
−0.331938 + 0.943301i \(0.607702\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −12.7342 −0.985401 −0.492701 0.870199i \(-0.663990\pi\)
−0.492701 + 0.870199i \(0.663990\pi\)
\(168\) 0 0
\(169\) −12.5800 −0.967694
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 23.0484 1.75234 0.876169 0.482005i \(-0.160091\pi\)
0.876169 + 0.482005i \(0.160091\pi\)
\(174\) 0 0
\(175\) −4.08613 −0.308882
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −2.22808 −0.166534 −0.0832672 0.996527i \(-0.526535\pi\)
−0.0832672 + 0.996527i \(0.526535\pi\)
\(180\) 0 0
\(181\) 0.468382 0.0348146 0.0174073 0.999848i \(-0.494459\pi\)
0.0174073 + 0.999848i \(0.494459\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 7.52420 0.553190
\(186\) 0 0
\(187\) −1.82774 −0.133658
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 20.2281 1.46365 0.731826 0.681491i \(-0.238668\pi\)
0.731826 + 0.681491i \(0.238668\pi\)
\(192\) 0 0
\(193\) −19.9293 −1.43455 −0.717273 0.696792i \(-0.754610\pi\)
−0.717273 + 0.696792i \(0.754610\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −15.5800 −1.11003 −0.555015 0.831840i \(-0.687288\pi\)
−0.555015 + 0.831840i \(0.687288\pi\)
\(198\) 0 0
\(199\) −3.58482 −0.254121 −0.127061 0.991895i \(-0.540554\pi\)
−0.127061 + 0.991895i \(0.540554\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −15.8384 −1.11164
\(204\) 0 0
\(205\) −0.179679 −0.0125493
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −0.876139 −0.0606038
\(210\) 0 0
\(211\) −14.9926 −1.03213 −0.516066 0.856549i \(-0.672604\pi\)
−0.516066 + 0.856549i \(0.672604\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0.820321 0.0559454
\(216\) 0 0
\(217\) −31.4487 −2.13488
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −0.876139 −0.0589355
\(222\) 0 0
\(223\) 26.8310 1.79674 0.898368 0.439244i \(-0.144754\pi\)
0.898368 + 0.439244i \(0.144754\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −1.35194 −0.0897314 −0.0448657 0.998993i \(-0.514286\pi\)
−0.0448657 + 0.998993i \(0.514286\pi\)
\(228\) 0 0
\(229\) −8.23550 −0.544217 −0.272108 0.962267i \(-0.587721\pi\)
−0.272108 + 0.962267i \(0.587721\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 8.58744 0.562582 0.281291 0.959623i \(-0.409237\pi\)
0.281291 + 0.959623i \(0.409237\pi\)
\(234\) 0 0
\(235\) −10.9065 −0.711458
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 23.9245 1.54755 0.773775 0.633461i \(-0.218366\pi\)
0.773775 + 0.633461i \(0.218366\pi\)
\(240\) 0 0
\(241\) −6.24030 −0.401973 −0.200987 0.979594i \(-0.564415\pi\)
−0.200987 + 0.979594i \(0.564415\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 9.69646 0.619484
\(246\) 0 0
\(247\) −0.419983 −0.0267229
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 28.5726 1.80349 0.901743 0.432272i \(-0.142288\pi\)
0.901743 + 0.432272i \(0.142288\pi\)
\(252\) 0 0
\(253\) −6.47580 −0.407130
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) −30.7449 −1.91039
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −31.8687 −1.96511 −0.982555 0.185974i \(-0.940456\pi\)
−0.982555 + 0.185974i \(0.940456\pi\)
\(264\) 0 0
\(265\) 4.17226 0.256300
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −31.4971 −1.92041 −0.960207 0.279289i \(-0.909901\pi\)
−0.960207 + 0.279289i \(0.909901\pi\)
\(270\) 0 0
\(271\) 3.24030 0.196834 0.0984172 0.995145i \(-0.468622\pi\)
0.0984172 + 0.995145i \(0.468622\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.35194 0.0815250
\(276\) 0 0
\(277\) −5.58482 −0.335560 −0.167780 0.985824i \(-0.553660\pi\)
−0.167780 + 0.985824i \(0.553660\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −24.1042 −1.43794 −0.718969 0.695043i \(-0.755385\pi\)
−0.718969 + 0.695043i \(0.755385\pi\)
\(282\) 0 0
\(283\) 10.5423 0.626674 0.313337 0.949642i \(-0.398553\pi\)
0.313337 + 0.949642i \(0.398553\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0.734191 0.0433379
\(288\) 0 0
\(289\) −15.1723 −0.892486
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 18.9926 1.10956 0.554779 0.831998i \(-0.312803\pi\)
0.554779 + 0.831998i \(0.312803\pi\)
\(294\) 0 0
\(295\) −4.17226 −0.242918
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −3.10422 −0.179521
\(300\) 0 0
\(301\) −3.35194 −0.193203
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −3.82032 −0.218751
\(306\) 0 0
\(307\) −29.4791 −1.68246 −0.841229 0.540679i \(-0.818167\pi\)
−0.841229 + 0.540679i \(0.818167\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 9.41256 0.533738 0.266869 0.963733i \(-0.414011\pi\)
0.266869 + 0.963733i \(0.414011\pi\)
\(312\) 0 0
\(313\) 11.6210 0.656858 0.328429 0.944529i \(-0.393481\pi\)
0.328429 + 0.944529i \(0.393481\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −9.17968 −0.515582 −0.257791 0.966201i \(-0.582995\pi\)
−0.257791 + 0.966201i \(0.582995\pi\)
\(318\) 0 0
\(319\) 5.24030 0.293401
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0.876139 0.0487497
\(324\) 0 0
\(325\) 0.648061 0.0359479
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 44.5652 2.45696
\(330\) 0 0
\(331\) 7.22066 0.396883 0.198442 0.980113i \(-0.436412\pi\)
0.198442 + 0.980113i \(0.436412\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −8.14195 −0.444842
\(336\) 0 0
\(337\) 2.28390 0.124412 0.0622059 0.998063i \(-0.480186\pi\)
0.0622059 + 0.998063i \(0.480186\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 10.4051 0.563470
\(342\) 0 0
\(343\) −11.0181 −0.594921
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0.708686 0.0380443 0.0190221 0.999819i \(-0.493945\pi\)
0.0190221 + 0.999819i \(0.493945\pi\)
\(348\) 0 0
\(349\) −21.3445 −1.14255 −0.571273 0.820760i \(-0.693550\pi\)
−0.571273 + 0.820760i \(0.693550\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −10.0968 −0.537398 −0.268699 0.963224i \(-0.586594\pi\)
−0.268699 + 0.963224i \(0.586594\pi\)
\(354\) 0 0
\(355\) 6.11644 0.324627
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −30.5578 −1.61278 −0.806388 0.591386i \(-0.798581\pi\)
−0.806388 + 0.591386i \(0.798581\pi\)
\(360\) 0 0
\(361\) −18.5800 −0.977896
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −12.3445 −0.646142
\(366\) 0 0
\(367\) 7.17968 0.374776 0.187388 0.982286i \(-0.439998\pi\)
0.187388 + 0.982286i \(0.439998\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −17.0484 −0.885109
\(372\) 0 0
\(373\) −21.9245 −1.13521 −0.567605 0.823301i \(-0.692130\pi\)
−0.567605 + 0.823301i \(0.692130\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2.51197 0.129373
\(378\) 0 0
\(379\) 17.3929 0.893414 0.446707 0.894680i \(-0.352597\pi\)
0.446707 + 0.894680i \(0.352597\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −0.475800 −0.0243123 −0.0121561 0.999926i \(-0.503870\pi\)
−0.0121561 + 0.999926i \(0.503870\pi\)
\(384\) 0 0
\(385\) −5.52420 −0.281539
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −5.58744 −0.283294 −0.141647 0.989917i \(-0.545240\pi\)
−0.141647 + 0.989917i \(0.545240\pi\)
\(390\) 0 0
\(391\) 6.47580 0.327495
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −10.3445 −0.520489
\(396\) 0 0
\(397\) 3.75228 0.188321 0.0941607 0.995557i \(-0.469983\pi\)
0.0941607 + 0.995557i \(0.469983\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 23.5652 1.17679 0.588394 0.808574i \(-0.299760\pi\)
0.588394 + 0.808574i \(0.299760\pi\)
\(402\) 0 0
\(403\) 4.98777 0.248459
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 10.1723 0.504220
\(408\) 0 0
\(409\) 1.04840 0.0518400 0.0259200 0.999664i \(-0.491748\pi\)
0.0259200 + 0.999664i \(0.491748\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 17.0484 0.838897
\(414\) 0 0
\(415\) 12.2584 0.601741
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 25.9197 1.26626 0.633131 0.774045i \(-0.281769\pi\)
0.633131 + 0.774045i \(0.281769\pi\)
\(420\) 0 0
\(421\) 7.64064 0.372382 0.186191 0.982514i \(-0.440386\pi\)
0.186191 + 0.982514i \(0.440386\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −1.35194 −0.0655787
\(426\) 0 0
\(427\) 15.6103 0.755437
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −7.98516 −0.384632 −0.192316 0.981333i \(-0.561600\pi\)
−0.192316 + 0.981333i \(0.561600\pi\)
\(432\) 0 0
\(433\) −12.5120 −0.601287 −0.300644 0.953737i \(-0.597201\pi\)
−0.300644 + 0.953737i \(0.597201\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3.10422 0.148495
\(438\) 0 0
\(439\) 8.76450 0.418307 0.209153 0.977883i \(-0.432929\pi\)
0.209153 + 0.977883i \(0.432929\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −3.67095 −0.174412 −0.0872062 0.996190i \(-0.527794\pi\)
−0.0872062 + 0.996190i \(0.527794\pi\)
\(444\) 0 0
\(445\) −3.00000 −0.142214
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 28.1723 1.32953 0.664766 0.747052i \(-0.268531\pi\)
0.664766 + 0.747052i \(0.268531\pi\)
\(450\) 0 0
\(451\) −0.242915 −0.0114384
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −2.64806 −0.124143
\(456\) 0 0
\(457\) 35.2616 1.64947 0.824735 0.565519i \(-0.191324\pi\)
0.824735 + 0.565519i \(0.191324\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 34.6768 1.61506 0.807530 0.589826i \(-0.200804\pi\)
0.807530 + 0.589826i \(0.200804\pi\)
\(462\) 0 0
\(463\) −7.44874 −0.346172 −0.173086 0.984907i \(-0.555374\pi\)
−0.173086 + 0.984907i \(0.555374\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −29.9655 −1.38664 −0.693319 0.720630i \(-0.743852\pi\)
−0.693319 + 0.720630i \(0.743852\pi\)
\(468\) 0 0
\(469\) 33.2691 1.53622
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1.10902 0.0509930
\(474\) 0 0
\(475\) −0.648061 −0.0297351
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 7.98516 0.364851 0.182426 0.983220i \(-0.441605\pi\)
0.182426 + 0.983220i \(0.441605\pi\)
\(480\) 0 0
\(481\) 4.87614 0.222333
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −13.5800 −0.616637
\(486\) 0 0
\(487\) 11.9442 0.541243 0.270621 0.962686i \(-0.412771\pi\)
0.270621 + 0.962686i \(0.412771\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 9.22066 0.416123 0.208061 0.978116i \(-0.433285\pi\)
0.208061 + 0.978116i \(0.433285\pi\)
\(492\) 0 0
\(493\) −5.24030 −0.236011
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −24.9926 −1.12107
\(498\) 0 0
\(499\) −30.1723 −1.35070 −0.675348 0.737499i \(-0.736007\pi\)
−0.675348 + 0.737499i \(0.736007\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 10.5981 0.472546 0.236273 0.971687i \(-0.424074\pi\)
0.236273 + 0.971687i \(0.424074\pi\)
\(504\) 0 0
\(505\) −1.46838 −0.0653421
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 28.7523 1.27442 0.637211 0.770689i \(-0.280088\pi\)
0.637211 + 0.770689i \(0.280088\pi\)
\(510\) 0 0
\(511\) 50.4413 2.23139
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −7.52420 −0.331556
\(516\) 0 0
\(517\) −14.7449 −0.648478
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −36.0942 −1.58132 −0.790658 0.612259i \(-0.790261\pi\)
−0.790658 + 0.612259i \(0.790261\pi\)
\(522\) 0 0
\(523\) −11.1297 −0.486669 −0.243334 0.969942i \(-0.578241\pi\)
−0.243334 + 0.969942i \(0.578241\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −10.4051 −0.453255
\(528\) 0 0
\(529\) −0.0558176 −0.00242685
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −0.116443 −0.00504370
\(534\) 0 0
\(535\) −1.20999 −0.0523125
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 13.1090 0.564646
\(540\) 0 0
\(541\) −34.7374 −1.49348 −0.746740 0.665116i \(-0.768382\pi\)
−0.746740 + 0.665116i \(0.768382\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 14.1042 0.604158
\(546\) 0 0
\(547\) 2.71455 0.116066 0.0580328 0.998315i \(-0.481517\pi\)
0.0580328 + 0.998315i \(0.481517\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −2.51197 −0.107014
\(552\) 0 0
\(553\) 42.2691 1.79746
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −8.93676 −0.378663 −0.189331 0.981913i \(-0.560632\pi\)
−0.189331 + 0.981913i \(0.560632\pi\)
\(558\) 0 0
\(559\) 0.531618 0.0224850
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 9.36261 0.394587 0.197293 0.980344i \(-0.436785\pi\)
0.197293 + 0.980344i \(0.436785\pi\)
\(564\) 0 0
\(565\) −11.9245 −0.501669
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 35.8735 1.50390 0.751948 0.659222i \(-0.229114\pi\)
0.751948 + 0.659222i \(0.229114\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −4.79001 −0.199757
\(576\) 0 0
\(577\) 1.35675 0.0564821 0.0282411 0.999601i \(-0.491009\pi\)
0.0282411 + 0.999601i \(0.491009\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −50.0894 −2.07806
\(582\) 0 0
\(583\) 5.64064 0.233612
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −28.7900 −1.18829 −0.594145 0.804358i \(-0.702510\pi\)
−0.594145 + 0.804358i \(0.702510\pi\)
\(588\) 0 0
\(589\) −4.98777 −0.205518
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 30.9171 1.26961 0.634807 0.772671i \(-0.281080\pi\)
0.634807 + 0.772671i \(0.281080\pi\)
\(594\) 0 0
\(595\) 5.52420 0.226470
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −1.39292 −0.0569132 −0.0284566 0.999595i \(-0.509059\pi\)
−0.0284566 + 0.999595i \(0.509059\pi\)
\(600\) 0 0
\(601\) 8.82513 0.359985 0.179992 0.983668i \(-0.442393\pi\)
0.179992 + 0.983668i \(0.442393\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −9.17226 −0.372905
\(606\) 0 0
\(607\) 2.15678 0.0875412 0.0437706 0.999042i \(-0.486063\pi\)
0.0437706 + 0.999042i \(0.486063\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −7.06804 −0.285942
\(612\) 0 0
\(613\) −9.57521 −0.386739 −0.193370 0.981126i \(-0.561942\pi\)
−0.193370 + 0.981126i \(0.561942\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −37.6768 −1.51681 −0.758406 0.651783i \(-0.774021\pi\)
−0.758406 + 0.651783i \(0.774021\pi\)
\(618\) 0 0
\(619\) −17.1042 −0.687477 −0.343738 0.939065i \(-0.611693\pi\)
−0.343738 + 0.939065i \(0.611693\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 12.2584 0.491122
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −10.1723 −0.405595
\(630\) 0 0
\(631\) −33.1090 −1.31805 −0.659025 0.752121i \(-0.729031\pi\)
−0.659025 + 0.752121i \(0.729031\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 7.07871 0.280910
\(636\) 0 0
\(637\) 6.28390 0.248977
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −23.1526 −0.914473 −0.457237 0.889345i \(-0.651161\pi\)
−0.457237 + 0.889345i \(0.651161\pi\)
\(642\) 0 0
\(643\) −43.0639 −1.69827 −0.849137 0.528173i \(-0.822877\pi\)
−0.849137 + 0.528173i \(0.822877\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 20.6439 0.811595 0.405798 0.913963i \(-0.366994\pi\)
0.405798 + 0.913963i \(0.366994\pi\)
\(648\) 0 0
\(649\) −5.64064 −0.221415
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 6.83516 0.267480 0.133740 0.991016i \(-0.457301\pi\)
0.133740 + 0.991016i \(0.457301\pi\)
\(654\) 0 0
\(655\) 6.00000 0.234439
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 26.8613 1.04637 0.523184 0.852220i \(-0.324744\pi\)
0.523184 + 0.852220i \(0.324744\pi\)
\(660\) 0 0
\(661\) −2.12125 −0.0825071 −0.0412535 0.999149i \(-0.513135\pi\)
−0.0412535 + 0.999149i \(0.513135\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 2.64806 0.102687
\(666\) 0 0
\(667\) −18.5667 −0.718907
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −5.16484 −0.199387
\(672\) 0 0
\(673\) 34.8203 1.34222 0.671112 0.741356i \(-0.265817\pi\)
0.671112 + 0.741356i \(0.265817\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −24.6842 −0.948692 −0.474346 0.880338i \(-0.657315\pi\)
−0.474346 + 0.880338i \(0.657315\pi\)
\(678\) 0 0
\(679\) 55.4897 2.12950
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 38.4610 1.47167 0.735834 0.677162i \(-0.236790\pi\)
0.735834 + 0.677162i \(0.236790\pi\)
\(684\) 0 0
\(685\) −7.46838 −0.285352
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 2.70388 0.103010
\(690\) 0 0
\(691\) −0.480608 −0.0182832 −0.00914159 0.999958i \(-0.502910\pi\)
−0.00914159 + 0.999958i \(0.502910\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −8.00000 −0.303457
\(696\) 0 0
\(697\) 0.242915 0.00920105
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −18.1797 −0.686637 −0.343318 0.939219i \(-0.611551\pi\)
−0.343318 + 0.939219i \(0.611551\pi\)
\(702\) 0 0
\(703\) −4.87614 −0.183907
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 6.00000 0.225653
\(708\) 0 0
\(709\) 7.18710 0.269917 0.134959 0.990851i \(-0.456910\pi\)
0.134959 + 0.990851i \(0.456910\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −36.8661 −1.38065
\(714\) 0 0
\(715\) 0.876139 0.0327657
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −12.5168 −0.466797 −0.233399 0.972381i \(-0.574985\pi\)
−0.233399 + 0.972381i \(0.574985\pi\)
\(720\) 0 0
\(721\) 30.7449 1.14500
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 3.87614 0.143956
\(726\) 0 0
\(727\) −8.42584 −0.312497 −0.156249 0.987718i \(-0.549940\pi\)
−0.156249 + 0.987718i \(0.549940\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −1.10902 −0.0410187
\(732\) 0 0
\(733\) −22.0000 −0.812589 −0.406294 0.913742i \(-0.633179\pi\)
−0.406294 + 0.913742i \(0.633179\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −11.0074 −0.405463
\(738\) 0 0
\(739\) 1.81290 0.0666887 0.0333444 0.999444i \(-0.489384\pi\)
0.0333444 + 0.999444i \(0.489384\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −20.1371 −0.738760 −0.369380 0.929278i \(-0.620430\pi\)
−0.369380 + 0.929278i \(0.620430\pi\)
\(744\) 0 0
\(745\) −10.5848 −0.387798
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 4.94418 0.180656
\(750\) 0 0
\(751\) −12.2132 −0.445668 −0.222834 0.974856i \(-0.571531\pi\)
−0.222834 + 0.974856i \(0.571531\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −17.6965 −0.644040
\(756\) 0 0
\(757\) 52.9533 1.92462 0.962310 0.271955i \(-0.0876701\pi\)
0.962310 + 0.271955i \(0.0876701\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −18.4535 −0.668940 −0.334470 0.942406i \(-0.608557\pi\)
−0.334470 + 0.942406i \(0.608557\pi\)
\(762\) 0 0
\(763\) −57.6317 −2.08641
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −2.70388 −0.0976314
\(768\) 0 0
\(769\) −4.45355 −0.160599 −0.0802995 0.996771i \(-0.525588\pi\)
−0.0802995 + 0.996771i \(0.525588\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 38.9368 1.40046 0.700229 0.713918i \(-0.253081\pi\)
0.700229 + 0.713918i \(0.253081\pi\)
\(774\) 0 0
\(775\) 7.69646 0.276465
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0.116443 0.00417200
\(780\) 0 0
\(781\) 8.26906 0.295890
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −2.53162 −0.0903573
\(786\) 0 0
\(787\) −34.2281 −1.22010 −0.610050 0.792363i \(-0.708850\pi\)
−0.610050 + 0.792363i \(0.708850\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 48.7252 1.73247
\(792\) 0 0
\(793\) −2.47580 −0.0879183
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 23.9655 0.848902 0.424451 0.905451i \(-0.360467\pi\)
0.424451 + 0.905451i \(0.360467\pi\)
\(798\) 0 0
\(799\) 14.7449 0.521636
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −16.6890 −0.588943
\(804\) 0 0
\(805\) 19.5726 0.689843
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −0.283896 −0.00998124 −0.00499062 0.999988i \(-0.501589\pi\)
−0.00499062 + 0.999988i \(0.501589\pi\)
\(810\) 0 0
\(811\) −32.4413 −1.13917 −0.569584 0.821933i \(-0.692896\pi\)
−0.569584 + 0.821933i \(0.692896\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −8.47580 −0.296894
\(816\) 0 0
\(817\) −0.531618 −0.0185990
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 41.6694 1.45427 0.727136 0.686493i \(-0.240851\pi\)
0.727136 + 0.686493i \(0.240851\pi\)
\(822\) 0 0
\(823\) 19.3626 0.674938 0.337469 0.941337i \(-0.390429\pi\)
0.337469 + 0.941337i \(0.390429\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −18.8097 −0.654076 −0.327038 0.945011i \(-0.606050\pi\)
−0.327038 + 0.945011i \(0.606050\pi\)
\(828\) 0 0
\(829\) −33.1016 −1.14967 −0.574833 0.818271i \(-0.694933\pi\)
−0.574833 + 0.818271i \(0.694933\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −13.1090 −0.454201
\(834\) 0 0
\(835\) −12.7342 −0.440685
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 39.6965 1.37047 0.685237 0.728320i \(-0.259699\pi\)
0.685237 + 0.728320i \(0.259699\pi\)
\(840\) 0 0
\(841\) −13.9755 −0.481915
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −12.5800 −0.432766
\(846\) 0 0
\(847\) 37.4791 1.28780
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −36.0410 −1.23547
\(852\) 0 0
\(853\) −4.11644 −0.140944 −0.0704722 0.997514i \(-0.522451\pi\)
−0.0704722 + 0.997514i \(0.522451\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 14.7449 0.503675 0.251837 0.967770i \(-0.418965\pi\)
0.251837 + 0.967770i \(0.418965\pi\)
\(858\) 0 0
\(859\) 37.8539 1.29156 0.645779 0.763524i \(-0.276533\pi\)
0.645779 + 0.763524i \(0.276533\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −26.7704 −0.911274 −0.455637 0.890166i \(-0.650588\pi\)
−0.455637 + 0.890166i \(0.650588\pi\)
\(864\) 0 0
\(865\) 23.0484 0.783669
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −13.9852 −0.474414
\(870\) 0 0
\(871\) −5.27648 −0.178787
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −4.08613 −0.138136
\(876\) 0 0
\(877\) −46.9681 −1.58600 −0.793001 0.609221i \(-0.791482\pi\)
−0.793001 + 0.609221i \(0.791482\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −19.8055 −0.667264 −0.333632 0.942703i \(-0.608274\pi\)
−0.333632 + 0.942703i \(0.608274\pi\)
\(882\) 0 0
\(883\) 6.20257 0.208733 0.104367 0.994539i \(-0.466718\pi\)
0.104367 + 0.994539i \(0.466718\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −13.4274 −0.450848 −0.225424 0.974261i \(-0.572377\pi\)
−0.225424 + 0.974261i \(0.572377\pi\)
\(888\) 0 0
\(889\) −28.9245 −0.970098
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 7.06804 0.236523
\(894\) 0 0
\(895\) −2.22808 −0.0744764
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 29.8325 0.994971
\(900\) 0 0
\(901\) −5.64064 −0.187917
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0.468382 0.0155695
\(906\) 0 0
\(907\) −0.673566 −0.0223654 −0.0111827 0.999937i \(-0.503560\pi\)
−0.0111827 + 0.999937i \(0.503560\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 7.90970 0.262060 0.131030 0.991378i \(-0.458172\pi\)
0.131030 + 0.991378i \(0.458172\pi\)
\(912\) 0 0
\(913\) 16.5726 0.548473
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −24.5168 −0.809615
\(918\) 0 0
\(919\) 8.58263 0.283115 0.141557 0.989930i \(-0.454789\pi\)
0.141557 + 0.989930i \(0.454789\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 3.96383 0.130471
\(924\) 0 0
\(925\) 7.52420 0.247394
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 29.6162 0.971676 0.485838 0.874049i \(-0.338515\pi\)
0.485838 + 0.874049i \(0.338515\pi\)
\(930\) 0 0
\(931\) −6.28390 −0.205946
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −1.82774 −0.0597735
\(936\) 0 0
\(937\) −15.2058 −0.496753 −0.248376 0.968664i \(-0.579897\pi\)
−0.248376 + 0.968664i \(0.579897\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 5.65287 0.184278 0.0921391 0.995746i \(-0.470630\pi\)
0.0921391 + 0.995746i \(0.470630\pi\)
\(942\) 0 0
\(943\) 0.860663 0.0280270
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 40.3962 1.31270 0.656350 0.754457i \(-0.272100\pi\)
0.656350 + 0.754457i \(0.272100\pi\)
\(948\) 0 0
\(949\) −8.00000 −0.259691
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −22.9320 −0.742839 −0.371419 0.928465i \(-0.621129\pi\)
−0.371419 + 0.928465i \(0.621129\pi\)
\(954\) 0 0
\(955\) 20.2281 0.654565
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 30.5168 0.985438
\(960\) 0 0
\(961\) 28.2355 0.910822
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −19.9293 −0.641548
\(966\) 0 0
\(967\) −10.3700 −0.333478 −0.166739 0.986001i \(-0.553324\pi\)
−0.166739 + 0.986001i \(0.553324\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −48.0410 −1.54171 −0.770854 0.637012i \(-0.780170\pi\)
−0.770854 + 0.637012i \(0.780170\pi\)
\(972\) 0 0
\(973\) 32.6890 1.04796
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 27.0532 0.865509 0.432754 0.901512i \(-0.357542\pi\)
0.432754 + 0.901512i \(0.357542\pi\)
\(978\) 0 0
\(979\) −4.05582 −0.129624
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 22.4817 0.717054 0.358527 0.933519i \(-0.383279\pi\)
0.358527 + 0.933519i \(0.383279\pi\)
\(984\) 0 0
\(985\) −15.5800 −0.496421
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −3.92935 −0.124946
\(990\) 0 0
\(991\) 26.5316 0.842805 0.421402 0.906874i \(-0.361538\pi\)
0.421402 + 0.906874i \(0.361538\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −3.58482 −0.113647
\(996\) 0 0
\(997\) −28.3659 −0.898356 −0.449178 0.893442i \(-0.648283\pi\)
−0.449178 + 0.893442i \(0.648283\pi\)
\(998\) 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 6480.2.a.bv.1.1 3
3.2 odd 2 6480.2.a.bs.1.1 3
4.3 odd 2 405.2.a.j.1.1 3
9.2 odd 6 2160.2.q.k.1441.3 6
9.4 even 3 720.2.q.i.241.3 6
9.5 odd 6 2160.2.q.k.721.3 6
9.7 even 3 720.2.q.i.481.3 6
12.11 even 2 405.2.a.i.1.3 3
20.3 even 4 2025.2.b.l.649.5 6
20.7 even 4 2025.2.b.l.649.2 6
20.19 odd 2 2025.2.a.n.1.3 3
36.7 odd 6 45.2.e.b.31.3 yes 6
36.11 even 6 135.2.e.b.91.1 6
36.23 even 6 135.2.e.b.46.1 6
36.31 odd 6 45.2.e.b.16.3 6
60.23 odd 4 2025.2.b.m.649.2 6
60.47 odd 4 2025.2.b.m.649.5 6
60.59 even 2 2025.2.a.o.1.1 3
180.7 even 12 225.2.k.b.49.2 12
180.23 odd 12 675.2.k.b.424.5 12
180.43 even 12 225.2.k.b.49.5 12
180.47 odd 12 675.2.k.b.199.5 12
180.59 even 6 675.2.e.b.451.3 6
180.67 even 12 225.2.k.b.124.5 12
180.79 odd 6 225.2.e.b.76.1 6
180.83 odd 12 675.2.k.b.199.2 12
180.103 even 12 225.2.k.b.124.2 12
180.119 even 6 675.2.e.b.226.3 6
180.139 odd 6 225.2.e.b.151.1 6
180.167 odd 12 675.2.k.b.424.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
45.2.e.b.16.3 6 36.31 odd 6
45.2.e.b.31.3 yes 6 36.7 odd 6
135.2.e.b.46.1 6 36.23 even 6
135.2.e.b.91.1 6 36.11 even 6
225.2.e.b.76.1 6 180.79 odd 6
225.2.e.b.151.1 6 180.139 odd 6
225.2.k.b.49.2 12 180.7 even 12
225.2.k.b.49.5 12 180.43 even 12
225.2.k.b.124.2 12 180.103 even 12
225.2.k.b.124.5 12 180.67 even 12
405.2.a.i.1.3 3 12.11 even 2
405.2.a.j.1.1 3 4.3 odd 2
675.2.e.b.226.3 6 180.119 even 6
675.2.e.b.451.3 6 180.59 even 6
675.2.k.b.199.2 12 180.83 odd 12
675.2.k.b.199.5 12 180.47 odd 12
675.2.k.b.424.2 12 180.167 odd 12
675.2.k.b.424.5 12 180.23 odd 12
720.2.q.i.241.3 6 9.4 even 3
720.2.q.i.481.3 6 9.7 even 3
2025.2.a.n.1.3 3 20.19 odd 2
2025.2.a.o.1.1 3 60.59 even 2
2025.2.b.l.649.2 6 20.7 even 4
2025.2.b.l.649.5 6 20.3 even 4
2025.2.b.m.649.2 6 60.23 odd 4
2025.2.b.m.649.5 6 60.47 odd 4
2160.2.q.k.721.3 6 9.5 odd 6
2160.2.q.k.1441.3 6 9.2 odd 6
6480.2.a.bs.1.1 3 3.2 odd 2
6480.2.a.bv.1.1 3 1.1 even 1 trivial