Properties

Label 9680.2.a.ca.1.1
Level $9680$
Weight $2$
Character 9680.1
Self dual yes
Analytic conductor $77.295$
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] = [9680,2,Mod(1,9680)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9680, 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("9680.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9680.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: \(77.2951891566\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.404.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 4840)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.86620\) of defining polynomial
Character \(\chi\) \(=\) 9680.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.34889 q^{3} -1.00000 q^{5} -2.86620 q^{7} +2.51730 q^{9} +O(q^{10})\) \(q-2.34889 q^{3} -1.00000 q^{5} -2.86620 q^{7} +2.51730 q^{9} +6.24970 q^{13} +2.34889 q^{15} -3.73240 q^{17} +8.24970 q^{19} +6.73240 q^{21} -8.94749 q^{23} +1.00000 q^{25} +1.13380 q^{27} +8.69779 q^{29} +5.21509 q^{31} +2.86620 q^{35} -2.69779 q^{37} -14.6799 q^{39} +8.21509 q^{41} -7.13380 q^{43} -2.51730 q^{45} -1.31429 q^{47} +1.21509 q^{49} +8.76700 q^{51} -0.180484 q^{53} -19.3777 q^{57} -3.55191 q^{59} +5.94749 q^{61} -7.21509 q^{63} -6.24970 q^{65} -2.27968 q^{67} +21.0167 q^{69} +2.51730 q^{71} +8.36097 q^{73} -2.34889 q^{75} -15.1280 q^{79} -10.2151 q^{81} +3.55191 q^{83} +3.73240 q^{85} -20.4302 q^{87} +0.732397 q^{89} -17.9129 q^{91} -12.2497 q^{93} -8.24970 q^{95} +1.75030 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{3} - 3 q^{5} - q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{3} - 3 q^{5} - q^{7} + 6 q^{9} + 2 q^{13} + q^{15} + 4 q^{17} + 8 q^{19} + 5 q^{21} + 2 q^{23} + 3 q^{25} + 11 q^{27} + 14 q^{29} + 2 q^{31} + q^{35} + 4 q^{37} + 11 q^{41} - 29 q^{43} - 6 q^{45} - q^{47} - 10 q^{49} + 8 q^{51} + 10 q^{53} - 2 q^{57} - 6 q^{59} - 11 q^{61} - 8 q^{63} - 2 q^{65} - 7 q^{67} + 28 q^{69} + 6 q^{71} + 4 q^{73} - q^{75} - 6 q^{79} - 17 q^{81} + 6 q^{83} - 4 q^{85} - 34 q^{87} - 13 q^{89} - 28 q^{91} - 20 q^{93} - 8 q^{95} + 22 q^{97}+O(q^{100}) \) 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
\(3\) −2.34889 −1.35613 −0.678067 0.735000i \(-0.737182\pi\)
−0.678067 + 0.735000i \(0.737182\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −2.86620 −1.08332 −0.541661 0.840597i \(-0.682204\pi\)
−0.541661 + 0.840597i \(0.682204\pi\)
\(8\) 0 0
\(9\) 2.51730 0.839101
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) 6.24970 1.73336 0.866678 0.498869i \(-0.166251\pi\)
0.866678 + 0.498869i \(0.166251\pi\)
\(14\) 0 0
\(15\) 2.34889 0.606482
\(16\) 0 0
\(17\) −3.73240 −0.905239 −0.452620 0.891704i \(-0.649510\pi\)
−0.452620 + 0.891704i \(0.649510\pi\)
\(18\) 0 0
\(19\) 8.24970 1.89261 0.946306 0.323274i \(-0.104783\pi\)
0.946306 + 0.323274i \(0.104783\pi\)
\(20\) 0 0
\(21\) 6.73240 1.46913
\(22\) 0 0
\(23\) −8.94749 −1.86568 −0.932840 0.360290i \(-0.882678\pi\)
−0.932840 + 0.360290i \(0.882678\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.13380 0.218200
\(28\) 0 0
\(29\) 8.69779 1.61514 0.807569 0.589773i \(-0.200783\pi\)
0.807569 + 0.589773i \(0.200783\pi\)
\(30\) 0 0
\(31\) 5.21509 0.936658 0.468329 0.883554i \(-0.344856\pi\)
0.468329 + 0.883554i \(0.344856\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.86620 0.484476
\(36\) 0 0
\(37\) −2.69779 −0.443514 −0.221757 0.975102i \(-0.571179\pi\)
−0.221757 + 0.975102i \(0.571179\pi\)
\(38\) 0 0
\(39\) −14.6799 −2.35066
\(40\) 0 0
\(41\) 8.21509 1.28298 0.641491 0.767131i \(-0.278316\pi\)
0.641491 + 0.767131i \(0.278316\pi\)
\(42\) 0 0
\(43\) −7.13380 −1.08789 −0.543947 0.839119i \(-0.683071\pi\)
−0.543947 + 0.839119i \(0.683071\pi\)
\(44\) 0 0
\(45\) −2.51730 −0.375258
\(46\) 0 0
\(47\) −1.31429 −0.191708 −0.0958542 0.995395i \(-0.530558\pi\)
−0.0958542 + 0.995395i \(0.530558\pi\)
\(48\) 0 0
\(49\) 1.21509 0.173585
\(50\) 0 0
\(51\) 8.76700 1.22763
\(52\) 0 0
\(53\) −0.180484 −0.0247914 −0.0123957 0.999923i \(-0.503946\pi\)
−0.0123957 + 0.999923i \(0.503946\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −19.3777 −2.56664
\(58\) 0 0
\(59\) −3.55191 −0.462420 −0.231210 0.972904i \(-0.574268\pi\)
−0.231210 + 0.972904i \(0.574268\pi\)
\(60\) 0 0
\(61\) 5.94749 0.761498 0.380749 0.924678i \(-0.375666\pi\)
0.380749 + 0.924678i \(0.375666\pi\)
\(62\) 0 0
\(63\) −7.21509 −0.909016
\(64\) 0 0
\(65\) −6.24970 −0.775180
\(66\) 0 0
\(67\) −2.27968 −0.278507 −0.139253 0.990257i \(-0.544470\pi\)
−0.139253 + 0.990257i \(0.544470\pi\)
\(68\) 0 0
\(69\) 21.0167 2.53011
\(70\) 0 0
\(71\) 2.51730 0.298749 0.149375 0.988781i \(-0.452274\pi\)
0.149375 + 0.988781i \(0.452274\pi\)
\(72\) 0 0
\(73\) 8.36097 0.978577 0.489289 0.872122i \(-0.337256\pi\)
0.489289 + 0.872122i \(0.337256\pi\)
\(74\) 0 0
\(75\) −2.34889 −0.271227
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −15.1280 −1.70203 −0.851015 0.525141i \(-0.824012\pi\)
−0.851015 + 0.525141i \(0.824012\pi\)
\(80\) 0 0
\(81\) −10.2151 −1.13501
\(82\) 0 0
\(83\) 3.55191 0.389873 0.194937 0.980816i \(-0.437550\pi\)
0.194937 + 0.980816i \(0.437550\pi\)
\(84\) 0 0
\(85\) 3.73240 0.404835
\(86\) 0 0
\(87\) −20.4302 −2.19035
\(88\) 0 0
\(89\) 0.732397 0.0776339 0.0388169 0.999246i \(-0.487641\pi\)
0.0388169 + 0.999246i \(0.487641\pi\)
\(90\) 0 0
\(91\) −17.9129 −1.87778
\(92\) 0 0
\(93\) −12.2497 −1.27023
\(94\) 0 0
\(95\) −8.24970 −0.846401
\(96\) 0 0
\(97\) 1.75030 0.177716 0.0888580 0.996044i \(-0.471678\pi\)
0.0888580 + 0.996044i \(0.471678\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −5.69779 −0.566951 −0.283476 0.958979i \(-0.591487\pi\)
−0.283476 + 0.958979i \(0.591487\pi\)
\(102\) 0 0
\(103\) 19.0167 1.87377 0.936886 0.349635i \(-0.113695\pi\)
0.936886 + 0.349635i \(0.113695\pi\)
\(104\) 0 0
\(105\) −6.73240 −0.657015
\(106\) 0 0
\(107\) −7.45272 −0.720481 −0.360241 0.932859i \(-0.617305\pi\)
−0.360241 + 0.932859i \(0.617305\pi\)
\(108\) 0 0
\(109\) −0.551912 −0.0528636 −0.0264318 0.999651i \(-0.508414\pi\)
−0.0264318 + 0.999651i \(0.508414\pi\)
\(110\) 0 0
\(111\) 6.33682 0.601464
\(112\) 0 0
\(113\) −5.46479 −0.514084 −0.257042 0.966400i \(-0.582748\pi\)
−0.257042 + 0.966400i \(0.582748\pi\)
\(114\) 0 0
\(115\) 8.94749 0.834358
\(116\) 0 0
\(117\) 15.7324 1.45446
\(118\) 0 0
\(119\) 10.6978 0.980665
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) −19.2964 −1.73990
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 9.38350 0.832651 0.416326 0.909216i \(-0.363318\pi\)
0.416326 + 0.909216i \(0.363318\pi\)
\(128\) 0 0
\(129\) 16.7565 1.47533
\(130\) 0 0
\(131\) −11.2151 −0.979867 −0.489934 0.871760i \(-0.662979\pi\)
−0.489934 + 0.871760i \(0.662979\pi\)
\(132\) 0 0
\(133\) −23.6453 −2.05031
\(134\) 0 0
\(135\) −1.13380 −0.0975821
\(136\) 0 0
\(137\) −21.3085 −1.82050 −0.910252 0.414054i \(-0.864112\pi\)
−0.910252 + 0.414054i \(0.864112\pi\)
\(138\) 0 0
\(139\) −4.18048 −0.354584 −0.177292 0.984158i \(-0.556734\pi\)
−0.177292 + 0.984158i \(0.556734\pi\)
\(140\) 0 0
\(141\) 3.08712 0.259982
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −8.69779 −0.722312
\(146\) 0 0
\(147\) −2.85412 −0.235404
\(148\) 0 0
\(149\) −15.7491 −1.29022 −0.645108 0.764091i \(-0.723188\pi\)
−0.645108 + 0.764091i \(0.723188\pi\)
\(150\) 0 0
\(151\) −0.0871191 −0.00708965 −0.00354483 0.999994i \(-0.501128\pi\)
−0.00354483 + 0.999994i \(0.501128\pi\)
\(152\) 0 0
\(153\) −9.39558 −0.759587
\(154\) 0 0
\(155\) −5.21509 −0.418886
\(156\) 0 0
\(157\) −17.8950 −1.42817 −0.714087 0.700057i \(-0.753158\pi\)
−0.714087 + 0.700057i \(0.753158\pi\)
\(158\) 0 0
\(159\) 0.423939 0.0336205
\(160\) 0 0
\(161\) 25.6453 2.02113
\(162\) 0 0
\(163\) −1.20302 −0.0942276 −0.0471138 0.998890i \(-0.515002\pi\)
−0.0471138 + 0.998890i \(0.515002\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −1.73822 −0.134508 −0.0672539 0.997736i \(-0.521424\pi\)
−0.0672539 + 0.997736i \(0.521424\pi\)
\(168\) 0 0
\(169\) 26.0588 2.00452
\(170\) 0 0
\(171\) 20.7670 1.58809
\(172\) 0 0
\(173\) −3.64528 −0.277145 −0.138573 0.990352i \(-0.544251\pi\)
−0.138573 + 0.990352i \(0.544251\pi\)
\(174\) 0 0
\(175\) −2.86620 −0.216664
\(176\) 0 0
\(177\) 8.34307 0.627103
\(178\) 0 0
\(179\) −4.96539 −0.371131 −0.185565 0.982632i \(-0.559412\pi\)
−0.185565 + 0.982632i \(0.559412\pi\)
\(180\) 0 0
\(181\) 22.2664 1.65505 0.827524 0.561430i \(-0.189748\pi\)
0.827524 + 0.561430i \(0.189748\pi\)
\(182\) 0 0
\(183\) −13.9700 −1.03269
\(184\) 0 0
\(185\) 2.69779 0.198345
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −3.24970 −0.236381
\(190\) 0 0
\(191\) −13.3777 −0.967975 −0.483987 0.875075i \(-0.660812\pi\)
−0.483987 + 0.875075i \(0.660812\pi\)
\(192\) 0 0
\(193\) 16.8362 1.21190 0.605949 0.795504i \(-0.292794\pi\)
0.605949 + 0.795504i \(0.292794\pi\)
\(194\) 0 0
\(195\) 14.6799 1.05125
\(196\) 0 0
\(197\) 16.6978 1.18967 0.594834 0.803849i \(-0.297218\pi\)
0.594834 + 0.803849i \(0.297218\pi\)
\(198\) 0 0
\(199\) 14.3610 1.01802 0.509011 0.860760i \(-0.330011\pi\)
0.509011 + 0.860760i \(0.330011\pi\)
\(200\) 0 0
\(201\) 5.35472 0.377693
\(202\) 0 0
\(203\) −24.9296 −1.74971
\(204\) 0 0
\(205\) −8.21509 −0.573767
\(206\) 0 0
\(207\) −22.5236 −1.56549
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 10.7491 0.739999 0.369999 0.929032i \(-0.379358\pi\)
0.369999 + 0.929032i \(0.379358\pi\)
\(212\) 0 0
\(213\) −5.91288 −0.405144
\(214\) 0 0
\(215\) 7.13380 0.486521
\(216\) 0 0
\(217\) −14.9475 −1.01470
\(218\) 0 0
\(219\) −19.6390 −1.32708
\(220\) 0 0
\(221\) −23.3264 −1.56910
\(222\) 0 0
\(223\) −17.7266 −1.18706 −0.593529 0.804812i \(-0.702266\pi\)
−0.593529 + 0.804812i \(0.702266\pi\)
\(224\) 0 0
\(225\) 2.51730 0.167820
\(226\) 0 0
\(227\) 22.6920 1.50612 0.753059 0.657953i \(-0.228577\pi\)
0.753059 + 0.657953i \(0.228577\pi\)
\(228\) 0 0
\(229\) −17.5928 −1.16256 −0.581281 0.813703i \(-0.697448\pi\)
−0.581281 + 0.813703i \(0.697448\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 14.8783 0.974708 0.487354 0.873204i \(-0.337962\pi\)
0.487354 + 0.873204i \(0.337962\pi\)
\(234\) 0 0
\(235\) 1.31429 0.0857346
\(236\) 0 0
\(237\) 35.5340 2.30818
\(238\) 0 0
\(239\) −4.24970 −0.274890 −0.137445 0.990509i \(-0.543889\pi\)
−0.137445 + 0.990509i \(0.543889\pi\)
\(240\) 0 0
\(241\) 4.88873 0.314911 0.157455 0.987526i \(-0.449671\pi\)
0.157455 + 0.987526i \(0.449671\pi\)
\(242\) 0 0
\(243\) 20.5928 1.32103
\(244\) 0 0
\(245\) −1.21509 −0.0776294
\(246\) 0 0
\(247\) 51.5582 3.28057
\(248\) 0 0
\(249\) −8.34307 −0.528720
\(250\) 0 0
\(251\) −0.947489 −0.0598050 −0.0299025 0.999553i \(-0.509520\pi\)
−0.0299025 + 0.999553i \(0.509520\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −8.76700 −0.549011
\(256\) 0 0
\(257\) 22.6799 1.41473 0.707366 0.706847i \(-0.249883\pi\)
0.707366 + 0.706847i \(0.249883\pi\)
\(258\) 0 0
\(259\) 7.73240 0.480468
\(260\) 0 0
\(261\) 21.8950 1.35527
\(262\) 0 0
\(263\) −1.75030 −0.107928 −0.0539640 0.998543i \(-0.517186\pi\)
−0.0539640 + 0.998543i \(0.517186\pi\)
\(264\) 0 0
\(265\) 0.180484 0.0110871
\(266\) 0 0
\(267\) −1.72032 −0.105282
\(268\) 0 0
\(269\) 15.2738 0.931263 0.465632 0.884979i \(-0.345827\pi\)
0.465632 + 0.884979i \(0.345827\pi\)
\(270\) 0 0
\(271\) 20.0934 1.22059 0.610293 0.792176i \(-0.291052\pi\)
0.610293 + 0.792176i \(0.291052\pi\)
\(272\) 0 0
\(273\) 42.0755 2.54652
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −4.55936 −0.273945 −0.136973 0.990575i \(-0.543737\pi\)
−0.136973 + 0.990575i \(0.543737\pi\)
\(278\) 0 0
\(279\) 13.1280 0.785951
\(280\) 0 0
\(281\) −24.7912 −1.47892 −0.739458 0.673203i \(-0.764918\pi\)
−0.739458 + 0.673203i \(0.764918\pi\)
\(282\) 0 0
\(283\) 1.91871 0.114055 0.0570277 0.998373i \(-0.481838\pi\)
0.0570277 + 0.998373i \(0.481838\pi\)
\(284\) 0 0
\(285\) 19.3777 1.14783
\(286\) 0 0
\(287\) −23.5461 −1.38988
\(288\) 0 0
\(289\) −3.06922 −0.180542
\(290\) 0 0
\(291\) −4.11127 −0.241007
\(292\) 0 0
\(293\) 17.9129 1.04648 0.523241 0.852185i \(-0.324723\pi\)
0.523241 + 0.852185i \(0.324723\pi\)
\(294\) 0 0
\(295\) 3.55191 0.206800
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −55.9191 −3.23389
\(300\) 0 0
\(301\) 20.4469 1.17854
\(302\) 0 0
\(303\) 13.3835 0.768862
\(304\) 0 0
\(305\) −5.94749 −0.340552
\(306\) 0 0
\(307\) −0.448088 −0.0255737 −0.0127869 0.999918i \(-0.504070\pi\)
−0.0127869 + 0.999918i \(0.504070\pi\)
\(308\) 0 0
\(309\) −44.6682 −2.54109
\(310\) 0 0
\(311\) −19.7145 −1.11791 −0.558953 0.829199i \(-0.688797\pi\)
−0.558953 + 0.829199i \(0.688797\pi\)
\(312\) 0 0
\(313\) −34.4815 −1.94901 −0.974505 0.224367i \(-0.927969\pi\)
−0.974505 + 0.224367i \(0.927969\pi\)
\(314\) 0 0
\(315\) 7.21509 0.406524
\(316\) 0 0
\(317\) 32.2559 1.81167 0.905837 0.423626i \(-0.139243\pi\)
0.905837 + 0.423626i \(0.139243\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 17.5056 0.977070
\(322\) 0 0
\(323\) −30.7912 −1.71327
\(324\) 0 0
\(325\) 6.24970 0.346671
\(326\) 0 0
\(327\) 1.29638 0.0716902
\(328\) 0 0
\(329\) 3.76700 0.207682
\(330\) 0 0
\(331\) −6.05131 −0.332610 −0.166305 0.986074i \(-0.553184\pi\)
−0.166305 + 0.986074i \(0.553184\pi\)
\(332\) 0 0
\(333\) −6.79115 −0.372153
\(334\) 0 0
\(335\) 2.27968 0.124552
\(336\) 0 0
\(337\) −17.0409 −0.928274 −0.464137 0.885763i \(-0.653636\pi\)
−0.464137 + 0.885763i \(0.653636\pi\)
\(338\) 0 0
\(339\) 12.8362 0.697168
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 16.5807 0.895273
\(344\) 0 0
\(345\) −21.0167 −1.13150
\(346\) 0 0
\(347\) −20.7370 −1.11322 −0.556611 0.830773i \(-0.687899\pi\)
−0.556611 + 0.830773i \(0.687899\pi\)
\(348\) 0 0
\(349\) 4.33682 0.232145 0.116072 0.993241i \(-0.462970\pi\)
0.116072 + 0.993241i \(0.462970\pi\)
\(350\) 0 0
\(351\) 7.08592 0.378219
\(352\) 0 0
\(353\) −3.48270 −0.185365 −0.0926826 0.995696i \(-0.529544\pi\)
−0.0926826 + 0.995696i \(0.529544\pi\)
\(354\) 0 0
\(355\) −2.51730 −0.133605
\(356\) 0 0
\(357\) −25.1280 −1.32991
\(358\) 0 0
\(359\) −17.9129 −0.945406 −0.472703 0.881222i \(-0.656722\pi\)
−0.472703 + 0.881222i \(0.656722\pi\)
\(360\) 0 0
\(361\) 49.0576 2.58198
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −8.36097 −0.437633
\(366\) 0 0
\(367\) 6.91751 0.361091 0.180546 0.983567i \(-0.442214\pi\)
0.180546 + 0.983567i \(0.442214\pi\)
\(368\) 0 0
\(369\) 20.6799 1.07655
\(370\) 0 0
\(371\) 0.517304 0.0268571
\(372\) 0 0
\(373\) −22.4123 −1.16046 −0.580232 0.814451i \(-0.697038\pi\)
−0.580232 + 0.814451i \(0.697038\pi\)
\(374\) 0 0
\(375\) 2.34889 0.121296
\(376\) 0 0
\(377\) 54.3586 2.79961
\(378\) 0 0
\(379\) 20.1626 1.03568 0.517841 0.855477i \(-0.326736\pi\)
0.517841 + 0.855477i \(0.326736\pi\)
\(380\) 0 0
\(381\) −22.0409 −1.12919
\(382\) 0 0
\(383\) 33.0409 1.68831 0.844154 0.536100i \(-0.180103\pi\)
0.844154 + 0.536100i \(0.180103\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −17.9579 −0.912854
\(388\) 0 0
\(389\) 28.9370 1.46717 0.733583 0.679600i \(-0.237847\pi\)
0.733583 + 0.679600i \(0.237847\pi\)
\(390\) 0 0
\(391\) 33.3956 1.68889
\(392\) 0 0
\(393\) 26.3431 1.32883
\(394\) 0 0
\(395\) 15.1280 0.761171
\(396\) 0 0
\(397\) −2.96539 −0.148829 −0.0744144 0.997227i \(-0.523709\pi\)
−0.0744144 + 0.997227i \(0.523709\pi\)
\(398\) 0 0
\(399\) 55.5403 2.78049
\(400\) 0 0
\(401\) 0.0704139 0.00351630 0.00175815 0.999998i \(-0.499440\pi\)
0.00175815 + 0.999998i \(0.499440\pi\)
\(402\) 0 0
\(403\) 32.5928 1.62356
\(404\) 0 0
\(405\) 10.2151 0.507592
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 4.14588 0.205000 0.102500 0.994733i \(-0.467316\pi\)
0.102500 + 0.994733i \(0.467316\pi\)
\(410\) 0 0
\(411\) 50.0513 2.46885
\(412\) 0 0
\(413\) 10.1805 0.500949
\(414\) 0 0
\(415\) −3.55191 −0.174357
\(416\) 0 0
\(417\) 9.81952 0.480864
\(418\) 0 0
\(419\) 26.2738 1.28356 0.641781 0.766888i \(-0.278196\pi\)
0.641781 + 0.766888i \(0.278196\pi\)
\(420\) 0 0
\(421\) 14.7503 0.718886 0.359443 0.933167i \(-0.382967\pi\)
0.359443 + 0.933167i \(0.382967\pi\)
\(422\) 0 0
\(423\) −3.30846 −0.160863
\(424\) 0 0
\(425\) −3.73240 −0.181048
\(426\) 0 0
\(427\) −17.0467 −0.824947
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −7.26641 −0.350010 −0.175005 0.984568i \(-0.555994\pi\)
−0.175005 + 0.984568i \(0.555994\pi\)
\(432\) 0 0
\(433\) 16.2497 0.780911 0.390455 0.920622i \(-0.372318\pi\)
0.390455 + 0.920622i \(0.372318\pi\)
\(434\) 0 0
\(435\) 20.4302 0.979552
\(436\) 0 0
\(437\) −73.8141 −3.53101
\(438\) 0 0
\(439\) 20.4543 0.976232 0.488116 0.872779i \(-0.337684\pi\)
0.488116 + 0.872779i \(0.337684\pi\)
\(440\) 0 0
\(441\) 3.05876 0.145655
\(442\) 0 0
\(443\) 16.5628 0.786922 0.393461 0.919341i \(-0.371278\pi\)
0.393461 + 0.919341i \(0.371278\pi\)
\(444\) 0 0
\(445\) −0.732397 −0.0347189
\(446\) 0 0
\(447\) 36.9930 1.74971
\(448\) 0 0
\(449\) −0.308458 −0.0145570 −0.00727851 0.999974i \(-0.502317\pi\)
−0.00727851 + 0.999974i \(0.502317\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0.204634 0.00961452
\(454\) 0 0
\(455\) 17.9129 0.839769
\(456\) 0 0
\(457\) −32.0397 −1.49875 −0.749376 0.662145i \(-0.769646\pi\)
−0.749376 + 0.662145i \(0.769646\pi\)
\(458\) 0 0
\(459\) −4.23180 −0.197523
\(460\) 0 0
\(461\) −34.7324 −1.61765 −0.808824 0.588050i \(-0.799896\pi\)
−0.808824 + 0.588050i \(0.799896\pi\)
\(462\) 0 0
\(463\) 34.0634 1.58306 0.791530 0.611130i \(-0.209285\pi\)
0.791530 + 0.611130i \(0.209285\pi\)
\(464\) 0 0
\(465\) 12.2497 0.568066
\(466\) 0 0
\(467\) −22.0213 −1.01903 −0.509513 0.860463i \(-0.670174\pi\)
−0.509513 + 0.860463i \(0.670174\pi\)
\(468\) 0 0
\(469\) 6.53401 0.301713
\(470\) 0 0
\(471\) 42.0334 1.93680
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 8.24970 0.378522
\(476\) 0 0
\(477\) −0.454334 −0.0208025
\(478\) 0 0
\(479\) 11.6274 0.531268 0.265634 0.964074i \(-0.414419\pi\)
0.265634 + 0.964074i \(0.414419\pi\)
\(480\) 0 0
\(481\) −16.8604 −0.768767
\(482\) 0 0
\(483\) −60.2380 −2.74093
\(484\) 0 0
\(485\) −1.75030 −0.0794770
\(486\) 0 0
\(487\) 35.9129 1.62737 0.813684 0.581307i \(-0.197459\pi\)
0.813684 + 0.581307i \(0.197459\pi\)
\(488\) 0 0
\(489\) 2.82576 0.127785
\(490\) 0 0
\(491\) −6.23180 −0.281237 −0.140619 0.990064i \(-0.544909\pi\)
−0.140619 + 0.990064i \(0.544909\pi\)
\(492\) 0 0
\(493\) −32.4636 −1.46209
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −7.21509 −0.323641
\(498\) 0 0
\(499\) 9.13963 0.409146 0.204573 0.978851i \(-0.434419\pi\)
0.204573 + 0.978851i \(0.434419\pi\)
\(500\) 0 0
\(501\) 4.08291 0.182411
\(502\) 0 0
\(503\) 26.7191 1.19135 0.595673 0.803227i \(-0.296885\pi\)
0.595673 + 0.803227i \(0.296885\pi\)
\(504\) 0 0
\(505\) 5.69779 0.253548
\(506\) 0 0
\(507\) −61.2093 −2.71840
\(508\) 0 0
\(509\) −0.870829 −0.0385988 −0.0192994 0.999814i \(-0.506144\pi\)
−0.0192994 + 0.999814i \(0.506144\pi\)
\(510\) 0 0
\(511\) −23.9642 −1.06011
\(512\) 0 0
\(513\) 9.35352 0.412968
\(514\) 0 0
\(515\) −19.0167 −0.837976
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 8.56237 0.375846
\(520\) 0 0
\(521\) −28.6966 −1.25722 −0.628610 0.777720i \(-0.716376\pi\)
−0.628610 + 0.777720i \(0.716376\pi\)
\(522\) 0 0
\(523\) 0.354723 0.0155109 0.00775547 0.999970i \(-0.497531\pi\)
0.00775547 + 0.999970i \(0.497531\pi\)
\(524\) 0 0
\(525\) 6.73240 0.293826
\(526\) 0 0
\(527\) −19.4648 −0.847900
\(528\) 0 0
\(529\) 57.0576 2.48076
\(530\) 0 0
\(531\) −8.94124 −0.388017
\(532\) 0 0
\(533\) 51.3419 2.22386
\(534\) 0 0
\(535\) 7.45272 0.322209
\(536\) 0 0
\(537\) 11.6632 0.503303
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −41.9296 −1.80269 −0.901347 0.433098i \(-0.857420\pi\)
−0.901347 + 0.433098i \(0.857420\pi\)
\(542\) 0 0
\(543\) −52.3014 −2.24447
\(544\) 0 0
\(545\) 0.551912 0.0236413
\(546\) 0 0
\(547\) 14.5173 0.620715 0.310358 0.950620i \(-0.399551\pi\)
0.310358 + 0.950620i \(0.399551\pi\)
\(548\) 0 0
\(549\) 14.9716 0.638974
\(550\) 0 0
\(551\) 71.7542 3.05683
\(552\) 0 0
\(553\) 43.3598 1.84385
\(554\) 0 0
\(555\) −6.33682 −0.268983
\(556\) 0 0
\(557\) 18.1626 0.769573 0.384787 0.923006i \(-0.374275\pi\)
0.384787 + 0.923006i \(0.374275\pi\)
\(558\) 0 0
\(559\) −44.5841 −1.88571
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 6.62274 0.279115 0.139558 0.990214i \(-0.455432\pi\)
0.139558 + 0.990214i \(0.455432\pi\)
\(564\) 0 0
\(565\) 5.46479 0.229906
\(566\) 0 0
\(567\) 29.2785 1.22958
\(568\) 0 0
\(569\) 37.9475 1.59084 0.795421 0.606058i \(-0.207250\pi\)
0.795421 + 0.606058i \(0.207250\pi\)
\(570\) 0 0
\(571\) −11.2664 −0.471484 −0.235742 0.971816i \(-0.575752\pi\)
−0.235742 + 0.971816i \(0.575752\pi\)
\(572\) 0 0
\(573\) 31.4227 1.31270
\(574\) 0 0
\(575\) −8.94749 −0.373136
\(576\) 0 0
\(577\) 1.69034 0.0703700 0.0351850 0.999381i \(-0.488798\pi\)
0.0351850 + 0.999381i \(0.488798\pi\)
\(578\) 0 0
\(579\) −39.5465 −1.64350
\(580\) 0 0
\(581\) −10.1805 −0.422358
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −15.7324 −0.650455
\(586\) 0 0
\(587\) 8.76118 0.361612 0.180806 0.983519i \(-0.442129\pi\)
0.180806 + 0.983519i \(0.442129\pi\)
\(588\) 0 0
\(589\) 43.0230 1.77273
\(590\) 0 0
\(591\) −39.2213 −1.61335
\(592\) 0 0
\(593\) −16.0421 −0.658768 −0.329384 0.944196i \(-0.606841\pi\)
−0.329384 + 0.944196i \(0.606841\pi\)
\(594\) 0 0
\(595\) −10.6978 −0.438567
\(596\) 0 0
\(597\) −33.7324 −1.38058
\(598\) 0 0
\(599\) 8.97164 0.366571 0.183286 0.983060i \(-0.441327\pi\)
0.183286 + 0.983060i \(0.441327\pi\)
\(600\) 0 0
\(601\) −6.94124 −0.283139 −0.141570 0.989928i \(-0.545215\pi\)
−0.141570 + 0.989928i \(0.545215\pi\)
\(602\) 0 0
\(603\) −5.73864 −0.233696
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 34.9358 1.41800 0.709001 0.705208i \(-0.249146\pi\)
0.709001 + 0.705208i \(0.249146\pi\)
\(608\) 0 0
\(609\) 58.5570 2.37285
\(610\) 0 0
\(611\) −8.21389 −0.332299
\(612\) 0 0
\(613\) −7.52475 −0.303922 −0.151961 0.988387i \(-0.548559\pi\)
−0.151961 + 0.988387i \(0.548559\pi\)
\(614\) 0 0
\(615\) 19.2964 0.778105
\(616\) 0 0
\(617\) −17.1551 −0.690640 −0.345320 0.938485i \(-0.612230\pi\)
−0.345320 + 0.938485i \(0.612230\pi\)
\(618\) 0 0
\(619\) 4.03581 0.162213 0.0811064 0.996705i \(-0.474155\pi\)
0.0811064 + 0.996705i \(0.474155\pi\)
\(620\) 0 0
\(621\) −10.1447 −0.407092
\(622\) 0 0
\(623\) −2.09919 −0.0841024
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 10.0692 0.401486
\(630\) 0 0
\(631\) −27.4290 −1.09193 −0.545965 0.837808i \(-0.683837\pi\)
−0.545965 + 0.837808i \(0.683837\pi\)
\(632\) 0 0
\(633\) −25.2485 −1.00354
\(634\) 0 0
\(635\) −9.38350 −0.372373
\(636\) 0 0
\(637\) 7.59396 0.300884
\(638\) 0 0
\(639\) 6.33682 0.250681
\(640\) 0 0
\(641\) 2.89618 0.114392 0.0571960 0.998363i \(-0.481784\pi\)
0.0571960 + 0.998363i \(0.481784\pi\)
\(642\) 0 0
\(643\) 39.6666 1.56430 0.782149 0.623091i \(-0.214123\pi\)
0.782149 + 0.623091i \(0.214123\pi\)
\(644\) 0 0
\(645\) −16.7565 −0.659788
\(646\) 0 0
\(647\) 21.4014 0.841376 0.420688 0.907205i \(-0.361789\pi\)
0.420688 + 0.907205i \(0.361789\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 35.1101 1.37607
\(652\) 0 0
\(653\) −29.9191 −1.17083 −0.585413 0.810735i \(-0.699068\pi\)
−0.585413 + 0.810735i \(0.699068\pi\)
\(654\) 0 0
\(655\) 11.2151 0.438210
\(656\) 0 0
\(657\) 21.0471 0.821126
\(658\) 0 0
\(659\) 19.9883 0.778635 0.389318 0.921104i \(-0.372711\pi\)
0.389318 + 0.921104i \(0.372711\pi\)
\(660\) 0 0
\(661\) −17.6107 −0.684976 −0.342488 0.939522i \(-0.611270\pi\)
−0.342488 + 0.939522i \(0.611270\pi\)
\(662\) 0 0
\(663\) 54.7912 2.12791
\(664\) 0 0
\(665\) 23.6453 0.916925
\(666\) 0 0
\(667\) −77.8234 −3.01333
\(668\) 0 0
\(669\) 41.6378 1.60981
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 24.4364 0.941955 0.470978 0.882145i \(-0.343901\pi\)
0.470978 + 0.882145i \(0.343901\pi\)
\(674\) 0 0
\(675\) 1.13380 0.0436400
\(676\) 0 0
\(677\) 45.3956 1.74469 0.872347 0.488887i \(-0.162597\pi\)
0.872347 + 0.488887i \(0.162597\pi\)
\(678\) 0 0
\(679\) −5.01671 −0.192523
\(680\) 0 0
\(681\) −53.3010 −2.04250
\(682\) 0 0
\(683\) −31.1914 −1.19350 −0.596752 0.802426i \(-0.703542\pi\)
−0.596752 + 0.802426i \(0.703542\pi\)
\(684\) 0 0
\(685\) 21.3085 0.814154
\(686\) 0 0
\(687\) 41.3235 1.57659
\(688\) 0 0
\(689\) −1.12797 −0.0429724
\(690\) 0 0
\(691\) 47.6336 1.81207 0.906034 0.423205i \(-0.139095\pi\)
0.906034 + 0.423205i \(0.139095\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 4.18048 0.158575
\(696\) 0 0
\(697\) −30.6620 −1.16141
\(698\) 0 0
\(699\) −34.9475 −1.32184
\(700\) 0 0
\(701\) −6.59277 −0.249005 −0.124503 0.992219i \(-0.539734\pi\)
−0.124503 + 0.992219i \(0.539734\pi\)
\(702\) 0 0
\(703\) −22.2559 −0.839399
\(704\) 0 0
\(705\) −3.08712 −0.116268
\(706\) 0 0
\(707\) 16.3310 0.614190
\(708\) 0 0
\(709\) 40.2213 1.51054 0.755272 0.655411i \(-0.227505\pi\)
0.755272 + 0.655411i \(0.227505\pi\)
\(710\) 0 0
\(711\) −38.0817 −1.42818
\(712\) 0 0
\(713\) −46.6620 −1.74750
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 9.98210 0.372788
\(718\) 0 0
\(719\) 19.2727 0.718749 0.359374 0.933193i \(-0.382990\pi\)
0.359374 + 0.933193i \(0.382990\pi\)
\(720\) 0 0
\(721\) −54.5056 −2.02990
\(722\) 0 0
\(723\) −11.4831 −0.427062
\(724\) 0 0
\(725\) 8.69779 0.323028
\(726\) 0 0
\(727\) −26.8662 −0.996412 −0.498206 0.867059i \(-0.666008\pi\)
−0.498206 + 0.867059i \(0.666008\pi\)
\(728\) 0 0
\(729\) −17.7250 −0.656480
\(730\) 0 0
\(731\) 26.6262 0.984805
\(732\) 0 0
\(733\) 3.45553 0.127633 0.0638165 0.997962i \(-0.479673\pi\)
0.0638165 + 0.997962i \(0.479673\pi\)
\(734\) 0 0
\(735\) 2.85412 0.105276
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −21.8708 −0.804531 −0.402266 0.915523i \(-0.631777\pi\)
−0.402266 + 0.915523i \(0.631777\pi\)
\(740\) 0 0
\(741\) −121.105 −4.44889
\(742\) 0 0
\(743\) −21.6511 −0.794302 −0.397151 0.917753i \(-0.630001\pi\)
−0.397151 + 0.917753i \(0.630001\pi\)
\(744\) 0 0
\(745\) 15.7491 0.577002
\(746\) 0 0
\(747\) 8.94124 0.327143
\(748\) 0 0
\(749\) 21.3610 0.780513
\(750\) 0 0
\(751\) 22.9988 0.839238 0.419619 0.907700i \(-0.362164\pi\)
0.419619 + 0.907700i \(0.362164\pi\)
\(752\) 0 0
\(753\) 2.22555 0.0811036
\(754\) 0 0
\(755\) 0.0871191 0.00317059
\(756\) 0 0
\(757\) −0.947489 −0.0344371 −0.0172185 0.999852i \(-0.505481\pi\)
−0.0172185 + 0.999852i \(0.505481\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 17.3264 0.628080 0.314040 0.949410i \(-0.398317\pi\)
0.314040 + 0.949410i \(0.398317\pi\)
\(762\) 0 0
\(763\) 1.58189 0.0572682
\(764\) 0 0
\(765\) 9.39558 0.339698
\(766\) 0 0
\(767\) −22.1984 −0.801537
\(768\) 0 0
\(769\) −35.2664 −1.27174 −0.635870 0.771797i \(-0.719358\pi\)
−0.635870 + 0.771797i \(0.719358\pi\)
\(770\) 0 0
\(771\) −53.2727 −1.91857
\(772\) 0 0
\(773\) −3.65693 −0.131531 −0.0657654 0.997835i \(-0.520949\pi\)
−0.0657654 + 0.997835i \(0.520949\pi\)
\(774\) 0 0
\(775\) 5.21509 0.187332
\(776\) 0 0
\(777\) −18.1626 −0.651579
\(778\) 0 0
\(779\) 67.7721 2.42819
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 9.86157 0.352424
\(784\) 0 0
\(785\) 17.8950 0.638699
\(786\) 0 0
\(787\) −20.3489 −0.725360 −0.362680 0.931914i \(-0.618138\pi\)
−0.362680 + 0.931914i \(0.618138\pi\)
\(788\) 0 0
\(789\) 4.11127 0.146365
\(790\) 0 0
\(791\) 15.6632 0.556919
\(792\) 0 0
\(793\) 37.1700 1.31995
\(794\) 0 0
\(795\) −0.423939 −0.0150356
\(796\) 0 0
\(797\) −1.32938 −0.0470889 −0.0235445 0.999723i \(-0.507495\pi\)
−0.0235445 + 0.999723i \(0.507495\pi\)
\(798\) 0 0
\(799\) 4.90544 0.173542
\(800\) 0 0
\(801\) 1.84366 0.0651427
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −25.6453 −0.903877
\(806\) 0 0
\(807\) −35.8767 −1.26292
\(808\) 0 0
\(809\) 49.8350 1.75211 0.876053 0.482215i \(-0.160167\pi\)
0.876053 + 0.482215i \(0.160167\pi\)
\(810\) 0 0
\(811\) 48.8183 1.71424 0.857121 0.515114i \(-0.172251\pi\)
0.857121 + 0.515114i \(0.172251\pi\)
\(812\) 0 0
\(813\) −47.1972 −1.65528
\(814\) 0 0
\(815\) 1.20302 0.0421399
\(816\) 0 0
\(817\) −58.8517 −2.05896
\(818\) 0 0
\(819\) −45.0922 −1.57565
\(820\) 0 0
\(821\) −35.6441 −1.24399 −0.621993 0.783022i \(-0.713677\pi\)
−0.621993 + 0.783022i \(0.713677\pi\)
\(822\) 0 0
\(823\) 27.5702 0.961038 0.480519 0.876984i \(-0.340448\pi\)
0.480519 + 0.876984i \(0.340448\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −29.1517 −1.01370 −0.506852 0.862033i \(-0.669191\pi\)
−0.506852 + 0.862033i \(0.669191\pi\)
\(828\) 0 0
\(829\) 7.21629 0.250632 0.125316 0.992117i \(-0.460006\pi\)
0.125316 + 0.992117i \(0.460006\pi\)
\(830\) 0 0
\(831\) 10.7094 0.371507
\(832\) 0 0
\(833\) −4.53521 −0.157136
\(834\) 0 0
\(835\) 1.73822 0.0601538
\(836\) 0 0
\(837\) 5.91288 0.204379
\(838\) 0 0
\(839\) −2.63783 −0.0910681 −0.0455341 0.998963i \(-0.514499\pi\)
−0.0455341 + 0.998963i \(0.514499\pi\)
\(840\) 0 0
\(841\) 46.6515 1.60867
\(842\) 0 0
\(843\) 58.2318 2.00561
\(844\) 0 0
\(845\) −26.0588 −0.896449
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −4.50685 −0.154675
\(850\) 0 0
\(851\) 24.1384 0.827455
\(852\) 0 0
\(853\) 3.32636 0.113892 0.0569462 0.998377i \(-0.481864\pi\)
0.0569462 + 0.998377i \(0.481864\pi\)
\(854\) 0 0
\(855\) −20.7670 −0.710217
\(856\) 0 0
\(857\) −26.0934 −0.891332 −0.445666 0.895199i \(-0.647033\pi\)
−0.445666 + 0.895199i \(0.647033\pi\)
\(858\) 0 0
\(859\) 15.2727 0.521096 0.260548 0.965461i \(-0.416097\pi\)
0.260548 + 0.965461i \(0.416097\pi\)
\(860\) 0 0
\(861\) 55.3073 1.88487
\(862\) 0 0
\(863\) −1.41931 −0.0483138 −0.0241569 0.999708i \(-0.507690\pi\)
−0.0241569 + 0.999708i \(0.507690\pi\)
\(864\) 0 0
\(865\) 3.64528 0.123943
\(866\) 0 0
\(867\) 7.20926 0.244839
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −14.2473 −0.482752
\(872\) 0 0
\(873\) 4.40604 0.149122
\(874\) 0 0
\(875\) 2.86620 0.0968952
\(876\) 0 0
\(877\) −32.5656 −1.09966 −0.549831 0.835276i \(-0.685308\pi\)
−0.549831 + 0.835276i \(0.685308\pi\)
\(878\) 0 0
\(879\) −42.0755 −1.41917
\(880\) 0 0
\(881\) −1.76076 −0.0593215 −0.0296607 0.999560i \(-0.509443\pi\)
−0.0296607 + 0.999560i \(0.509443\pi\)
\(882\) 0 0
\(883\) −37.2727 −1.25432 −0.627162 0.778889i \(-0.715784\pi\)
−0.627162 + 0.778889i \(0.715784\pi\)
\(884\) 0 0
\(885\) −8.34307 −0.280449
\(886\) 0 0
\(887\) −15.5881 −0.523398 −0.261699 0.965149i \(-0.584283\pi\)
−0.261699 + 0.965149i \(0.584283\pi\)
\(888\) 0 0
\(889\) −26.8950 −0.902029
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −10.8425 −0.362829
\(894\) 0 0
\(895\) 4.96539 0.165975
\(896\) 0 0
\(897\) 131.348 4.38559
\(898\) 0 0
\(899\) 45.3598 1.51283
\(900\) 0 0
\(901\) 0.673639 0.0224422
\(902\) 0 0
\(903\) −48.0276 −1.59826
\(904\) 0 0
\(905\) −22.2664 −0.740160
\(906\) 0 0
\(907\) 41.3897 1.37432 0.687162 0.726504i \(-0.258856\pi\)
0.687162 + 0.726504i \(0.258856\pi\)
\(908\) 0 0
\(909\) −14.3431 −0.475729
\(910\) 0 0
\(911\) 25.5940 0.847966 0.423983 0.905670i \(-0.360632\pi\)
0.423983 + 0.905670i \(0.360632\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 13.9700 0.461835
\(916\) 0 0
\(917\) 32.1447 1.06151
\(918\) 0 0
\(919\) −25.0167 −0.825225 −0.412612 0.910907i \(-0.635384\pi\)
−0.412612 + 0.910907i \(0.635384\pi\)
\(920\) 0 0
\(921\) 1.05251 0.0346814
\(922\) 0 0
\(923\) 15.7324 0.517838
\(924\) 0 0
\(925\) −2.69779 −0.0887027
\(926\) 0 0
\(927\) 47.8708 1.57228
\(928\) 0 0
\(929\) 1.50060 0.0492331 0.0246165 0.999697i \(-0.492164\pi\)
0.0246165 + 0.999697i \(0.492164\pi\)
\(930\) 0 0
\(931\) 10.0241 0.328528
\(932\) 0 0
\(933\) 46.3073 1.51603
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −9.01046 −0.294359 −0.147179 0.989110i \(-0.547019\pi\)
−0.147179 + 0.989110i \(0.547019\pi\)
\(938\) 0 0
\(939\) 80.9934 2.64312
\(940\) 0 0
\(941\) −5.96539 −0.194466 −0.0972331 0.995262i \(-0.530999\pi\)
−0.0972331 + 0.995262i \(0.530999\pi\)
\(942\) 0 0
\(943\) −73.5044 −2.39363
\(944\) 0 0
\(945\) 3.24970 0.105713
\(946\) 0 0
\(947\) 1.17929 0.0383217 0.0191608 0.999816i \(-0.493901\pi\)
0.0191608 + 0.999816i \(0.493901\pi\)
\(948\) 0 0
\(949\) 52.2536 1.69622
\(950\) 0 0
\(951\) −75.7658 −2.45687
\(952\) 0 0
\(953\) 49.9883 1.61928 0.809641 0.586926i \(-0.199662\pi\)
0.809641 + 0.586926i \(0.199662\pi\)
\(954\) 0 0
\(955\) 13.3777 0.432891
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 61.0743 1.97219
\(960\) 0 0
\(961\) −3.80281 −0.122671
\(962\) 0 0
\(963\) −18.7608 −0.604557
\(964\) 0 0
\(965\) −16.8362 −0.541977
\(966\) 0 0
\(967\) 26.2046 0.842684 0.421342 0.906902i \(-0.361559\pi\)
0.421342 + 0.906902i \(0.361559\pi\)
\(968\) 0 0
\(969\) 72.3252 2.32342
\(970\) 0 0
\(971\) −15.5940 −0.500434 −0.250217 0.968190i \(-0.580502\pi\)
−0.250217 + 0.968190i \(0.580502\pi\)
\(972\) 0 0
\(973\) 11.9821 0.384128
\(974\) 0 0
\(975\) −14.6799 −0.470133
\(976\) 0 0
\(977\) −37.6632 −1.20495 −0.602476 0.798137i \(-0.705819\pi\)
−0.602476 + 0.798137i \(0.705819\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −1.38933 −0.0443579
\(982\) 0 0
\(983\) −30.3823 −0.969045 −0.484523 0.874779i \(-0.661007\pi\)
−0.484523 + 0.874779i \(0.661007\pi\)
\(984\) 0 0
\(985\) −16.6978 −0.532036
\(986\) 0 0
\(987\) −8.84830 −0.281644
\(988\) 0 0
\(989\) 63.8296 2.02966
\(990\) 0 0
\(991\) −12.2139 −0.387987 −0.193994 0.981003i \(-0.562144\pi\)
−0.193994 + 0.981003i \(0.562144\pi\)
\(992\) 0 0
\(993\) 14.2139 0.451064
\(994\) 0 0
\(995\) −14.3610 −0.455273
\(996\) 0 0
\(997\) 27.2034 0.861541 0.430771 0.902461i \(-0.358242\pi\)
0.430771 + 0.902461i \(0.358242\pi\)
\(998\) 0 0
\(999\) −3.05876 −0.0967748
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 9680.2.a.ca.1.1 3
4.3 odd 2 4840.2.a.u.1.3 yes 3
11.10 odd 2 9680.2.a.cc.1.1 3
44.43 even 2 4840.2.a.t.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4840.2.a.t.1.3 3 44.43 even 2
4840.2.a.u.1.3 yes 3 4.3 odd 2
9680.2.a.ca.1.1 3 1.1 even 1 trivial
9680.2.a.cc.1.1 3 11.10 odd 2