Properties

Label 8820.2.d.a.881.5
Level $8820$
Weight $2$
Character 8820.881
Analytic conductor $70.428$
Analytic rank $0$
Dimension $12$
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] = [8820,2,Mod(881,8820)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8820, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8820.881");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8820 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8820.d (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(70.4280545828\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} - 9 x^{10} + 58 x^{9} - 78 x^{8} - 298 x^{7} + 1341 x^{6} - 2086 x^{5} - 3822 x^{4} + 19894 x^{3} - 21609 x^{2} - 33614 x + 117649 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 1260)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 881.5
Root \(1.75207 + 1.98249i\) of defining polynomial
Character \(\chi\) \(=\) 8820.881
Dual form 8820.2.d.a.881.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{5} +O(q^{10})\) \(q-1.00000 q^{5} -2.23292i q^{11} -4.92905i q^{13} -3.83953 q^{17} +5.96386i q^{19} -4.24918i q^{23} +1.00000 q^{25} +6.73838i q^{29} -1.80933i q^{31} -4.32755 q^{37} -2.50414 q^{41} +4.96414 q^{43} -10.8837 q^{47} +4.50545i q^{53} +2.23292i q^{55} +3.99215 q^{59} +6.25890i q^{61} +4.92905i q^{65} -11.1832 q^{67} -8.87665i q^{71} -5.01716i q^{73} +11.2041 q^{79} -0.295092 q^{83} +3.83953 q^{85} -14.1360 q^{89} -5.96386i q^{95} +1.05218i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{5} + 12 q^{25} + 4 q^{37} + 8 q^{41} + 36 q^{43} - 32 q^{47} - 4 q^{67} - 28 q^{79} - 40 q^{83} - 40 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8820\mathbb{Z}\right)^\times\).

\(n\) \(1081\) \(4411\) \(7057\) \(7841\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 2.23292i − 0.673252i −0.941638 0.336626i \(-0.890714\pi\)
0.941638 0.336626i \(-0.109286\pi\)
\(12\) 0 0
\(13\) − 4.92905i − 1.36707i −0.729917 0.683536i \(-0.760441\pi\)
0.729917 0.683536i \(-0.239559\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.83953 −0.931224 −0.465612 0.884989i \(-0.654166\pi\)
−0.465612 + 0.884989i \(0.654166\pi\)
\(18\) 0 0
\(19\) 5.96386i 1.36820i 0.729386 + 0.684102i \(0.239806\pi\)
−0.729386 + 0.684102i \(0.760194\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 4.24918i − 0.886015i −0.896518 0.443008i \(-0.853911\pi\)
0.896518 0.443008i \(-0.146089\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.73838i 1.25129i 0.780110 + 0.625643i \(0.215163\pi\)
−0.780110 + 0.625643i \(0.784837\pi\)
\(30\) 0 0
\(31\) − 1.80933i − 0.324966i −0.986711 0.162483i \(-0.948050\pi\)
0.986711 0.162483i \(-0.0519502\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −4.32755 −0.711444 −0.355722 0.934592i \(-0.615765\pi\)
−0.355722 + 0.934592i \(0.615765\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.50414 −0.391080 −0.195540 0.980696i \(-0.562646\pi\)
−0.195540 + 0.980696i \(0.562646\pi\)
\(42\) 0 0
\(43\) 4.96414 0.757025 0.378512 0.925596i \(-0.376436\pi\)
0.378512 + 0.925596i \(0.376436\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −10.8837 −1.58755 −0.793773 0.608214i \(-0.791886\pi\)
−0.793773 + 0.608214i \(0.791886\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.50545i 0.618872i 0.950920 + 0.309436i \(0.100140\pi\)
−0.950920 + 0.309436i \(0.899860\pi\)
\(54\) 0 0
\(55\) 2.23292i 0.301088i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.99215 0.519733 0.259867 0.965645i \(-0.416321\pi\)
0.259867 + 0.965645i \(0.416321\pi\)
\(60\) 0 0
\(61\) 6.25890i 0.801369i 0.916216 + 0.400685i \(0.131228\pi\)
−0.916216 + 0.400685i \(0.868772\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.92905i 0.611373i
\(66\) 0 0
\(67\) −11.1832 −1.36625 −0.683123 0.730303i \(-0.739379\pi\)
−0.683123 + 0.730303i \(0.739379\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 8.87665i − 1.05347i −0.850031 0.526733i \(-0.823417\pi\)
0.850031 0.526733i \(-0.176583\pi\)
\(72\) 0 0
\(73\) − 5.01716i − 0.587214i −0.955926 0.293607i \(-0.905144\pi\)
0.955926 0.293607i \(-0.0948557\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 11.2041 1.26056 0.630281 0.776367i \(-0.282939\pi\)
0.630281 + 0.776367i \(0.282939\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −0.295092 −0.0323906 −0.0161953 0.999869i \(-0.505155\pi\)
−0.0161953 + 0.999869i \(0.505155\pi\)
\(84\) 0 0
\(85\) 3.83953 0.416456
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −14.1360 −1.49842 −0.749208 0.662335i \(-0.769565\pi\)
−0.749208 + 0.662335i \(0.769565\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 5.96386i − 0.611879i
\(96\) 0 0
\(97\) 1.05218i 0.106833i 0.998572 + 0.0534165i \(0.0170111\pi\)
−0.998572 + 0.0534165i \(0.982989\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 17.4071 1.73207 0.866034 0.499985i \(-0.166661\pi\)
0.866034 + 0.499985i \(0.166661\pi\)
\(102\) 0 0
\(103\) 12.5384i 1.23544i 0.786397 + 0.617721i \(0.211944\pi\)
−0.786397 + 0.617721i \(0.788056\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.92932i 0.476535i 0.971200 + 0.238268i \(0.0765795\pi\)
−0.971200 + 0.238268i \(0.923420\pi\)
\(108\) 0 0
\(109\) 13.0223 1.24731 0.623656 0.781699i \(-0.285647\pi\)
0.623656 + 0.781699i \(0.285647\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 7.82363i − 0.735985i −0.929829 0.367993i \(-0.880045\pi\)
0.929829 0.367993i \(-0.119955\pi\)
\(114\) 0 0
\(115\) 4.24918i 0.396238i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 6.01405 0.546731
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −6.53736 −0.580097 −0.290048 0.957012i \(-0.593671\pi\)
−0.290048 + 0.957012i \(0.593671\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0.315665 0.0275798 0.0137899 0.999905i \(-0.495610\pi\)
0.0137899 + 0.999905i \(0.495610\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 22.2097i 1.89750i 0.316026 + 0.948751i \(0.397651\pi\)
−0.316026 + 0.948751i \(0.602349\pi\)
\(138\) 0 0
\(139\) 14.8183i 1.25687i 0.777862 + 0.628435i \(0.216304\pi\)
−0.777862 + 0.628435i \(0.783696\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −11.0062 −0.920384
\(144\) 0 0
\(145\) − 6.73838i − 0.559592i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 4.10602i 0.336379i 0.985755 + 0.168189i \(0.0537920\pi\)
−0.985755 + 0.168189i \(0.946208\pi\)
\(150\) 0 0
\(151\) 11.5675 0.941352 0.470676 0.882306i \(-0.344010\pi\)
0.470676 + 0.882306i \(0.344010\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1.80933i 0.145329i
\(156\) 0 0
\(157\) − 1.92814i − 0.153883i −0.997036 0.0769413i \(-0.975485\pi\)
0.997036 0.0769413i \(-0.0245154\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 16.3345 1.27942 0.639709 0.768617i \(-0.279055\pi\)
0.639709 + 0.768617i \(0.279055\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −16.2112 −1.25446 −0.627231 0.778834i \(-0.715812\pi\)
−0.627231 + 0.778834i \(0.715812\pi\)
\(168\) 0 0
\(169\) −11.2955 −0.868885
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 21.5331 1.63713 0.818564 0.574415i \(-0.194770\pi\)
0.818564 + 0.574415i \(0.194770\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 21.7519i − 1.62581i −0.582393 0.812907i \(-0.697884\pi\)
0.582393 0.812907i \(-0.302116\pi\)
\(180\) 0 0
\(181\) 10.4316i 0.775378i 0.921790 + 0.387689i \(0.126726\pi\)
−0.921790 + 0.387689i \(0.873274\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 4.32755 0.318167
\(186\) 0 0
\(187\) 8.57339i 0.626949i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 23.6063i 1.70809i 0.520201 + 0.854044i \(0.325857\pi\)
−0.520201 + 0.854044i \(0.674143\pi\)
\(192\) 0 0
\(193\) 16.3226 1.17493 0.587464 0.809250i \(-0.300126\pi\)
0.587464 + 0.809250i \(0.300126\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 7.03368i − 0.501129i −0.968100 0.250565i \(-0.919384\pi\)
0.968100 0.250565i \(-0.0806162\pi\)
\(198\) 0 0
\(199\) 6.43512i 0.456173i 0.973641 + 0.228087i \(0.0732470\pi\)
−0.973641 + 0.228087i \(0.926753\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 2.50414 0.174896
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 13.3169 0.921146
\(210\) 0 0
\(211\) −1.59463 −0.109779 −0.0548895 0.998492i \(-0.517481\pi\)
−0.0548895 + 0.998492i \(0.517481\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −4.96414 −0.338552
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 18.9252i 1.27305i
\(222\) 0 0
\(223\) − 20.8137i − 1.39379i −0.717174 0.696894i \(-0.754565\pi\)
0.717174 0.696894i \(-0.245435\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −9.89554 −0.656790 −0.328395 0.944540i \(-0.606508\pi\)
−0.328395 + 0.944540i \(0.606508\pi\)
\(228\) 0 0
\(229\) − 20.2219i − 1.33630i −0.744025 0.668152i \(-0.767086\pi\)
0.744025 0.668152i \(-0.232914\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 8.65346i 0.566907i 0.958986 + 0.283454i \(0.0914802\pi\)
−0.958986 + 0.283454i \(0.908520\pi\)
\(234\) 0 0
\(235\) 10.8837 0.709972
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 17.9691i 1.16233i 0.813787 + 0.581163i \(0.197402\pi\)
−0.813787 + 0.581163i \(0.802598\pi\)
\(240\) 0 0
\(241\) 26.0955i 1.68096i 0.541846 + 0.840478i \(0.317726\pi\)
−0.541846 + 0.840478i \(0.682274\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 29.3962 1.87043
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 3.97721 0.251039 0.125520 0.992091i \(-0.459940\pi\)
0.125520 + 0.992091i \(0.459940\pi\)
\(252\) 0 0
\(253\) −9.48810 −0.596512
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −6.31927 −0.394185 −0.197093 0.980385i \(-0.563150\pi\)
−0.197093 + 0.980385i \(0.563150\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 18.4861i 1.13990i 0.821678 + 0.569952i \(0.193038\pi\)
−0.821678 + 0.569952i \(0.806962\pi\)
\(264\) 0 0
\(265\) − 4.50545i − 0.276768i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −14.5950 −0.889872 −0.444936 0.895562i \(-0.646774\pi\)
−0.444936 + 0.895562i \(0.646774\pi\)
\(270\) 0 0
\(271\) 20.6470i 1.25422i 0.778931 + 0.627109i \(0.215762\pi\)
−0.778931 + 0.627109i \(0.784238\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 2.23292i − 0.134650i
\(276\) 0 0
\(277\) 24.3484 1.46295 0.731477 0.681866i \(-0.238831\pi\)
0.731477 + 0.681866i \(0.238831\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 30.6625i − 1.82917i −0.404393 0.914585i \(-0.632517\pi\)
0.404393 0.914585i \(-0.367483\pi\)
\(282\) 0 0
\(283\) 4.44821i 0.264419i 0.991222 + 0.132209i \(0.0422071\pi\)
−0.991222 + 0.132209i \(0.957793\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −2.25797 −0.132822
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −32.7139 −1.91117 −0.955583 0.294721i \(-0.904773\pi\)
−0.955583 + 0.294721i \(0.904773\pi\)
\(294\) 0 0
\(295\) −3.99215 −0.232432
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −20.9444 −1.21125
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 6.25890i − 0.358383i
\(306\) 0 0
\(307\) 2.37826i 0.135735i 0.997694 + 0.0678673i \(0.0216195\pi\)
−0.997694 + 0.0678673i \(0.978381\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −7.47133 −0.423660 −0.211830 0.977306i \(-0.567942\pi\)
−0.211830 + 0.977306i \(0.567942\pi\)
\(312\) 0 0
\(313\) 19.3497i 1.09371i 0.837228 + 0.546854i \(0.184175\pi\)
−0.837228 + 0.546854i \(0.815825\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 6.38007i 0.358341i 0.983818 + 0.179170i \(0.0573413\pi\)
−0.983818 + 0.179170i \(0.942659\pi\)
\(318\) 0 0
\(319\) 15.0463 0.842431
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 22.8985i − 1.27410i
\(324\) 0 0
\(325\) − 4.92905i − 0.273414i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0.654466 0.0359727 0.0179863 0.999838i \(-0.494274\pi\)
0.0179863 + 0.999838i \(0.494274\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 11.1832 0.611004
\(336\) 0 0
\(337\) −20.4752 −1.11536 −0.557679 0.830057i \(-0.688308\pi\)
−0.557679 + 0.830057i \(0.688308\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −4.04010 −0.218784
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 15.6077i 0.837863i 0.908018 + 0.418931i \(0.137595\pi\)
−0.908018 + 0.418931i \(0.862405\pi\)
\(348\) 0 0
\(349\) 22.4320i 1.20076i 0.799715 + 0.600380i \(0.204984\pi\)
−0.799715 + 0.600380i \(0.795016\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −32.5208 −1.73091 −0.865453 0.500990i \(-0.832969\pi\)
−0.865453 + 0.500990i \(0.832969\pi\)
\(354\) 0 0
\(355\) 8.87665i 0.471124i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 15.6380i 0.825344i 0.910880 + 0.412672i \(0.135405\pi\)
−0.910880 + 0.412672i \(0.864595\pi\)
\(360\) 0 0
\(361\) −16.5676 −0.871981
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 5.01716i 0.262610i
\(366\) 0 0
\(367\) − 3.05166i − 0.159295i −0.996823 0.0796476i \(-0.974620\pi\)
0.996823 0.0796476i \(-0.0253795\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 11.9475 0.618619 0.309310 0.950961i \(-0.399902\pi\)
0.309310 + 0.950961i \(0.399902\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 33.2138 1.71060
\(378\) 0 0
\(379\) −34.8181 −1.78848 −0.894242 0.447584i \(-0.852285\pi\)
−0.894242 + 0.447584i \(0.852285\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 1.45814 0.0745076 0.0372538 0.999306i \(-0.488139\pi\)
0.0372538 + 0.999306i \(0.488139\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 20.1509i 1.02169i 0.859673 + 0.510845i \(0.170667\pi\)
−0.859673 + 0.510845i \(0.829333\pi\)
\(390\) 0 0
\(391\) 16.3149i 0.825079i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −11.2041 −0.563741
\(396\) 0 0
\(397\) 19.3115i 0.969215i 0.874732 + 0.484607i \(0.161038\pi\)
−0.874732 + 0.484607i \(0.838962\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 28.3691i − 1.41669i −0.705868 0.708343i \(-0.749443\pi\)
0.705868 0.708343i \(-0.250557\pi\)
\(402\) 0 0
\(403\) −8.91828 −0.444251
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 9.66308i 0.478981i
\(408\) 0 0
\(409\) 15.8762i 0.785026i 0.919747 + 0.392513i \(0.128394\pi\)
−0.919747 + 0.392513i \(0.871606\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0.295092 0.0144855
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −1.98073 −0.0967652 −0.0483826 0.998829i \(-0.515407\pi\)
−0.0483826 + 0.998829i \(0.515407\pi\)
\(420\) 0 0
\(421\) −31.1448 −1.51790 −0.758952 0.651147i \(-0.774288\pi\)
−0.758952 + 0.651147i \(0.774288\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −3.83953 −0.186245
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 25.5894i 1.23260i 0.787512 + 0.616300i \(0.211369\pi\)
−0.787512 + 0.616300i \(0.788631\pi\)
\(432\) 0 0
\(433\) 6.90795i 0.331975i 0.986128 + 0.165987i \(0.0530811\pi\)
−0.986128 + 0.165987i \(0.946919\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 25.3415 1.21225
\(438\) 0 0
\(439\) 18.5663i 0.886120i 0.896492 + 0.443060i \(0.146107\pi\)
−0.896492 + 0.443060i \(0.853893\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 12.0525i 0.572633i 0.958135 + 0.286317i \(0.0924309\pi\)
−0.958135 + 0.286317i \(0.907569\pi\)
\(444\) 0 0
\(445\) 14.1360 0.670112
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 18.2147i − 0.859604i −0.902923 0.429802i \(-0.858583\pi\)
0.902923 0.429802i \(-0.141417\pi\)
\(450\) 0 0
\(451\) 5.59155i 0.263296i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −0.886209 −0.0414551 −0.0207276 0.999785i \(-0.506598\pi\)
−0.0207276 + 0.999785i \(0.506598\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −1.24276 −0.0578811 −0.0289405 0.999581i \(-0.509213\pi\)
−0.0289405 + 0.999581i \(0.509213\pi\)
\(462\) 0 0
\(463\) 35.8986 1.66835 0.834176 0.551499i \(-0.185944\pi\)
0.834176 + 0.551499i \(0.185944\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 17.5952 0.814208 0.407104 0.913382i \(-0.366539\pi\)
0.407104 + 0.913382i \(0.366539\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 11.0846i − 0.509668i
\(474\) 0 0
\(475\) 5.96386i 0.273641i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 23.5746 1.07715 0.538575 0.842578i \(-0.318963\pi\)
0.538575 + 0.842578i \(0.318963\pi\)
\(480\) 0 0
\(481\) 21.3307i 0.972595i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 1.05218i − 0.0477772i
\(486\) 0 0
\(487\) 3.54793 0.160772 0.0803861 0.996764i \(-0.474385\pi\)
0.0803861 + 0.996764i \(0.474385\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 41.3330i 1.86533i 0.360739 + 0.932667i \(0.382525\pi\)
−0.360739 + 0.932667i \(0.617475\pi\)
\(492\) 0 0
\(493\) − 25.8722i − 1.16523i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 23.7292 1.06226 0.531132 0.847289i \(-0.321767\pi\)
0.531132 + 0.847289i \(0.321767\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −24.4784 −1.09144 −0.545719 0.837968i \(-0.683743\pi\)
−0.545719 + 0.837968i \(0.683743\pi\)
\(504\) 0 0
\(505\) −17.4071 −0.774604
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −43.6140 −1.93316 −0.966579 0.256369i \(-0.917474\pi\)
−0.966579 + 0.256369i \(0.917474\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 12.5384i − 0.552507i
\(516\) 0 0
\(517\) 24.3024i 1.06882i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 10.1104 0.442947 0.221473 0.975166i \(-0.428913\pi\)
0.221473 + 0.975166i \(0.428913\pi\)
\(522\) 0 0
\(523\) 19.1213i 0.836118i 0.908420 + 0.418059i \(0.137289\pi\)
−0.908420 + 0.418059i \(0.862711\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 6.94699i 0.302616i
\(528\) 0 0
\(529\) 4.94447 0.214977
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 12.3430i 0.534635i
\(534\) 0 0
\(535\) − 4.92932i − 0.213113i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 16.3125 0.701331 0.350665 0.936501i \(-0.385955\pi\)
0.350665 + 0.936501i \(0.385955\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −13.0223 −0.557815
\(546\) 0 0
\(547\) 21.0498 0.900023 0.450012 0.893023i \(-0.351420\pi\)
0.450012 + 0.893023i \(0.351420\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −40.1868 −1.71201
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 21.7778i − 0.922754i −0.887204 0.461377i \(-0.847356\pi\)
0.887204 0.461377i \(-0.152644\pi\)
\(558\) 0 0
\(559\) − 24.4685i − 1.03491i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −18.8316 −0.793656 −0.396828 0.917893i \(-0.629889\pi\)
−0.396828 + 0.917893i \(0.629889\pi\)
\(564\) 0 0
\(565\) 7.82363i 0.329143i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 31.9438i 1.33915i 0.742742 + 0.669577i \(0.233525\pi\)
−0.742742 + 0.669577i \(0.766475\pi\)
\(570\) 0 0
\(571\) −19.7088 −0.824789 −0.412394 0.911005i \(-0.635307\pi\)
−0.412394 + 0.911005i \(0.635307\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 4.24918i − 0.177203i
\(576\) 0 0
\(577\) 15.7729i 0.656633i 0.944568 + 0.328317i \(0.106481\pi\)
−0.944568 + 0.328317i \(0.893519\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 10.0603 0.416657
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −5.89811 −0.243441 −0.121721 0.992564i \(-0.538841\pi\)
−0.121721 + 0.992564i \(0.538841\pi\)
\(588\) 0 0
\(589\) 10.7906 0.444619
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −2.36373 −0.0970669 −0.0485335 0.998822i \(-0.515455\pi\)
−0.0485335 + 0.998822i \(0.515455\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 16.9286i 0.691684i 0.938293 + 0.345842i \(0.112407\pi\)
−0.938293 + 0.345842i \(0.887593\pi\)
\(600\) 0 0
\(601\) − 20.7753i − 0.847443i −0.905793 0.423721i \(-0.860724\pi\)
0.905793 0.423721i \(-0.139276\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −6.01405 −0.244506
\(606\) 0 0
\(607\) 2.75183i 0.111693i 0.998439 + 0.0558466i \(0.0177858\pi\)
−0.998439 + 0.0558466i \(0.982214\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 53.6461i 2.17029i
\(612\) 0 0
\(613\) 15.5628 0.628575 0.314287 0.949328i \(-0.398234\pi\)
0.314287 + 0.949328i \(0.398234\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.50012i 0.0603925i 0.999544 + 0.0301963i \(0.00961323\pi\)
−0.999544 + 0.0301963i \(0.990387\pi\)
\(618\) 0 0
\(619\) − 10.4623i − 0.420513i −0.977646 0.210257i \(-0.932570\pi\)
0.977646 0.210257i \(-0.0674300\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 16.6158 0.662514
\(630\) 0 0
\(631\) 42.9480 1.70973 0.854866 0.518849i \(-0.173639\pi\)
0.854866 + 0.518849i \(0.173639\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 6.53736 0.259427
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 43.1330i 1.70365i 0.523826 + 0.851825i \(0.324504\pi\)
−0.523826 + 0.851825i \(0.675496\pi\)
\(642\) 0 0
\(643\) 11.6402i 0.459045i 0.973303 + 0.229523i \(0.0737165\pi\)
−0.973303 + 0.229523i \(0.926284\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 37.0432 1.45632 0.728159 0.685408i \(-0.240376\pi\)
0.728159 + 0.685408i \(0.240376\pi\)
\(648\) 0 0
\(649\) − 8.91416i − 0.349912i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 31.7308i − 1.24172i −0.783921 0.620861i \(-0.786783\pi\)
0.783921 0.620861i \(-0.213217\pi\)
\(654\) 0 0
\(655\) −0.315665 −0.0123341
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 1.01969i − 0.0397216i −0.999803 0.0198608i \(-0.993678\pi\)
0.999803 0.0198608i \(-0.00632231\pi\)
\(660\) 0 0
\(661\) − 40.9457i − 1.59260i −0.604899 0.796302i \(-0.706787\pi\)
0.604899 0.796302i \(-0.293213\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 28.6326 1.10866
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 13.9756 0.539524
\(672\) 0 0
\(673\) −16.2023 −0.624552 −0.312276 0.949992i \(-0.601091\pi\)
−0.312276 + 0.949992i \(0.601091\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −8.48419 −0.326074 −0.163037 0.986620i \(-0.552129\pi\)
−0.163037 + 0.986620i \(0.552129\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 32.8128i − 1.25555i −0.778396 0.627773i \(-0.783967\pi\)
0.778396 0.627773i \(-0.216033\pi\)
\(684\) 0 0
\(685\) − 22.2097i − 0.848588i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 22.2076 0.846042
\(690\) 0 0
\(691\) − 34.5459i − 1.31419i −0.753809 0.657094i \(-0.771786\pi\)
0.753809 0.657094i \(-0.228214\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 14.8183i − 0.562089i
\(696\) 0 0
\(697\) 9.61472 0.364183
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 14.0470i 0.530547i 0.964173 + 0.265274i \(0.0854623\pi\)
−0.964173 + 0.265274i \(0.914538\pi\)
\(702\) 0 0
\(703\) − 25.8089i − 0.973401i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −42.3730 −1.59135 −0.795676 0.605722i \(-0.792884\pi\)
−0.795676 + 0.605722i \(0.792884\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −7.68818 −0.287924
\(714\) 0 0
\(715\) 11.0062 0.411608
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 28.3385 1.05685 0.528424 0.848981i \(-0.322783\pi\)
0.528424 + 0.848981i \(0.322783\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 6.73838i 0.250257i
\(726\) 0 0
\(727\) 14.2471i 0.528395i 0.964469 + 0.264197i \(0.0851070\pi\)
−0.964469 + 0.264197i \(0.914893\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −19.0600 −0.704959
\(732\) 0 0
\(733\) 28.7599i 1.06227i 0.847287 + 0.531136i \(0.178234\pi\)
−0.847287 + 0.531136i \(0.821766\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 24.9713i 0.919828i
\(738\) 0 0
\(739\) 21.3948 0.787022 0.393511 0.919320i \(-0.371260\pi\)
0.393511 + 0.919320i \(0.371260\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 11.2869i − 0.414077i −0.978333 0.207039i \(-0.933617\pi\)
0.978333 0.207039i \(-0.0663826\pi\)
\(744\) 0 0
\(745\) − 4.10602i − 0.150433i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −1.85438 −0.0676672 −0.0338336 0.999427i \(-0.510772\pi\)
−0.0338336 + 0.999427i \(0.510772\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −11.5675 −0.420986
\(756\) 0 0
\(757\) 25.3114 0.919958 0.459979 0.887930i \(-0.347857\pi\)
0.459979 + 0.887930i \(0.347857\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 6.45702 0.234067 0.117033 0.993128i \(-0.462662\pi\)
0.117033 + 0.993128i \(0.462662\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 19.6775i − 0.710513i
\(768\) 0 0
\(769\) − 28.0248i − 1.01060i −0.862943 0.505301i \(-0.831382\pi\)
0.862943 0.505301i \(-0.168618\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 11.2519 0.404703 0.202352 0.979313i \(-0.435142\pi\)
0.202352 + 0.979313i \(0.435142\pi\)
\(774\) 0 0
\(775\) − 1.80933i − 0.0649931i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 14.9343i − 0.535078i
\(780\) 0 0
\(781\) −19.8209 −0.709248
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 1.92814i 0.0688184i
\(786\) 0 0
\(787\) 45.7781i 1.63181i 0.578185 + 0.815906i \(0.303761\pi\)
−0.578185 + 0.815906i \(0.696239\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 30.8504 1.09553
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 5.50164 0.194878 0.0974390 0.995241i \(-0.468935\pi\)
0.0974390 + 0.995241i \(0.468935\pi\)
\(798\) 0 0
\(799\) 41.7882 1.47836
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −11.2029 −0.395343
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 28.9201i − 1.01678i −0.861128 0.508389i \(-0.830241\pi\)
0.861128 0.508389i \(-0.169759\pi\)
\(810\) 0 0
\(811\) 39.1672i 1.37535i 0.726021 + 0.687673i \(0.241368\pi\)
−0.726021 + 0.687673i \(0.758632\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −16.3345 −0.572173
\(816\) 0 0
\(817\) 29.6055i 1.03576i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 19.5583i − 0.682589i −0.939956 0.341295i \(-0.889135\pi\)
0.939956 0.341295i \(-0.110865\pi\)
\(822\) 0 0
\(823\) 51.2087 1.78502 0.892511 0.451026i \(-0.148942\pi\)
0.892511 + 0.451026i \(0.148942\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 21.2391i − 0.738555i −0.929319 0.369277i \(-0.879605\pi\)
0.929319 0.369277i \(-0.120395\pi\)
\(828\) 0 0
\(829\) − 22.5901i − 0.784585i −0.919841 0.392292i \(-0.871682\pi\)
0.919841 0.392292i \(-0.128318\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 16.2112 0.561012
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −7.06207 −0.243810 −0.121905 0.992542i \(-0.538900\pi\)
−0.121905 + 0.992542i \(0.538900\pi\)
\(840\) 0 0
\(841\) −16.4058 −0.565716
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 11.2955 0.388577
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 18.3885i 0.630350i
\(852\) 0 0
\(853\) − 32.2255i − 1.10338i −0.834050 0.551689i \(-0.813983\pi\)
0.834050 0.551689i \(-0.186017\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −16.4202 −0.560902 −0.280451 0.959868i \(-0.590484\pi\)
−0.280451 + 0.959868i \(0.590484\pi\)
\(858\) 0 0
\(859\) 28.1679i 0.961076i 0.876974 + 0.480538i \(0.159559\pi\)
−0.876974 + 0.480538i \(0.840441\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 34.1063i 1.16099i 0.814264 + 0.580495i \(0.197141\pi\)
−0.814264 + 0.580495i \(0.802859\pi\)
\(864\) 0 0
\(865\) −21.5331 −0.732146
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 25.0180i − 0.848677i
\(870\) 0 0
\(871\) 55.1225i 1.86776i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −34.7964 −1.17499 −0.587495 0.809227i \(-0.699886\pi\)
−0.587495 + 0.809227i \(0.699886\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 39.6838 1.33698 0.668491 0.743721i \(-0.266941\pi\)
0.668491 + 0.743721i \(0.266941\pi\)
\(882\) 0 0
\(883\) 4.50323 0.151546 0.0757728 0.997125i \(-0.475858\pi\)
0.0757728 + 0.997125i \(0.475858\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −26.2075 −0.879961 −0.439981 0.898007i \(-0.645015\pi\)
−0.439981 + 0.898007i \(0.645015\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 64.9087i − 2.17209i
\(894\) 0 0
\(895\) 21.7519i 0.727086i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 12.1920 0.406625
\(900\) 0 0
\(901\) − 17.2988i − 0.576308i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 10.4316i − 0.346759i
\(906\) 0 0
\(907\) −45.1884 −1.50046 −0.750229 0.661178i \(-0.770057\pi\)
−0.750229 + 0.661178i \(0.770057\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 21.4888i 0.711956i 0.934494 + 0.355978i \(0.115852\pi\)
−0.934494 + 0.355978i \(0.884148\pi\)
\(912\) 0 0
\(913\) 0.658919i 0.0218070i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −9.56919 −0.315659 −0.157829 0.987466i \(-0.550450\pi\)
−0.157829 + 0.987466i \(0.550450\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −43.7535 −1.44016
\(924\) 0 0
\(925\) −4.32755 −0.142289
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 18.2534 0.598874 0.299437 0.954116i \(-0.403201\pi\)
0.299437 + 0.954116i \(0.403201\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) − 8.57339i − 0.280380i
\(936\) 0 0
\(937\) − 30.4878i − 0.995992i −0.867179 0.497996i \(-0.834069\pi\)
0.867179 0.497996i \(-0.165931\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 51.0256 1.66339 0.831694 0.555234i \(-0.187371\pi\)
0.831694 + 0.555234i \(0.187371\pi\)
\(942\) 0 0
\(943\) 10.6405i 0.346503i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 55.5343i − 1.80462i −0.431087 0.902311i \(-0.641870\pi\)
0.431087 0.902311i \(-0.358130\pi\)
\(948\) 0 0
\(949\) −24.7298 −0.802764
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 40.0901i − 1.29865i −0.760512 0.649323i \(-0.775052\pi\)
0.760512 0.649323i \(-0.224948\pi\)
\(954\) 0 0
\(955\) − 23.6063i − 0.763880i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 27.7263 0.894397
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −16.3226 −0.525444
\(966\) 0 0
\(967\) −37.7974 −1.21548 −0.607740 0.794136i \(-0.707924\pi\)
−0.607740 + 0.794136i \(0.707924\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −32.7876 −1.05220 −0.526102 0.850422i \(-0.676347\pi\)
−0.526102 + 0.850422i \(0.676347\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 29.4856i 0.943328i 0.881778 + 0.471664i \(0.156346\pi\)
−0.881778 + 0.471664i \(0.843654\pi\)
\(978\) 0 0
\(979\) 31.5647i 1.00881i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −38.3639 −1.22362 −0.611810 0.791005i \(-0.709558\pi\)
−0.611810 + 0.791005i \(0.709558\pi\)
\(984\) 0 0
\(985\) 7.03368i 0.224112i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 21.0935i − 0.670735i
\(990\) 0 0
\(991\) 9.63794 0.306159 0.153080 0.988214i \(-0.451081\pi\)
0.153080 + 0.988214i \(0.451081\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 6.43512i − 0.204007i
\(996\) 0 0
\(997\) 8.07296i 0.255673i 0.991795 + 0.127837i \(0.0408033\pi\)
−0.991795 + 0.127837i \(0.959197\pi\)
\(998\) 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 8820.2.d.a.881.5 12
3.2 odd 2 8820.2.d.b.881.8 12
7.2 even 3 1260.2.cg.b.521.5 yes 12
7.3 odd 6 1260.2.cg.a.341.5 12
7.6 odd 2 8820.2.d.b.881.5 12
21.2 odd 6 1260.2.cg.a.521.5 yes 12
21.17 even 6 1260.2.cg.b.341.5 yes 12
21.20 even 2 inner 8820.2.d.a.881.8 12
35.2 odd 12 6300.2.dd.b.4049.3 24
35.3 even 12 6300.2.dd.c.1349.3 24
35.9 even 6 6300.2.ch.b.4301.2 12
35.17 even 12 6300.2.dd.c.1349.10 24
35.23 odd 12 6300.2.dd.b.4049.10 24
35.24 odd 6 6300.2.ch.c.1601.2 12
105.2 even 12 6300.2.dd.c.4049.3 24
105.17 odd 12 6300.2.dd.b.1349.10 24
105.23 even 12 6300.2.dd.c.4049.10 24
105.38 odd 12 6300.2.dd.b.1349.3 24
105.44 odd 6 6300.2.ch.c.4301.2 12
105.59 even 6 6300.2.ch.b.1601.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1260.2.cg.a.341.5 12 7.3 odd 6
1260.2.cg.a.521.5 yes 12 21.2 odd 6
1260.2.cg.b.341.5 yes 12 21.17 even 6
1260.2.cg.b.521.5 yes 12 7.2 even 3
6300.2.ch.b.1601.2 12 105.59 even 6
6300.2.ch.b.4301.2 12 35.9 even 6
6300.2.ch.c.1601.2 12 35.24 odd 6
6300.2.ch.c.4301.2 12 105.44 odd 6
6300.2.dd.b.1349.3 24 105.38 odd 12
6300.2.dd.b.1349.10 24 105.17 odd 12
6300.2.dd.b.4049.3 24 35.2 odd 12
6300.2.dd.b.4049.10 24 35.23 odd 12
6300.2.dd.c.1349.3 24 35.3 even 12
6300.2.dd.c.1349.10 24 35.17 even 12
6300.2.dd.c.4049.3 24 105.2 even 12
6300.2.dd.c.4049.10 24 105.23 even 12
8820.2.d.a.881.5 12 1.1 even 1 trivial
8820.2.d.a.881.8 12 21.20 even 2 inner
8820.2.d.b.881.5 12 7.6 odd 2
8820.2.d.b.881.8 12 3.2 odd 2