Properties

Label 6960.2.a.co.1.3
Level $6960$
Weight $2$
Character 6960.1
Self dual yes
Analytic conductor $55.576$
Analytic rank $1$
Dimension $4$
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] = [6960,2,Mod(1,6960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6960, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("6960.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6960 = 2^{4} \cdot 3 \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6960.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: \(55.5758798068\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.2225.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + 2x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 435)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.820249\) of defining polynomial
Character \(\chi\) \(=\) 6960.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} +0.729126 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.00000 q^{5} +0.729126 q^{7} +1.00000 q^{9} -0.729126 q^{11} +3.38351 q^{13} -1.00000 q^{15} -5.74301 q^{17} -6.11263 q^{19} +0.729126 q^{21} +9.48602 q^{23} +1.00000 q^{25} +1.00000 q^{27} -1.00000 q^{29} -5.48602 q^{31} -0.729126 q^{33} -0.729126 q^{35} -10.2949 q^{37} +3.38351 q^{39} -11.3088 q^{41} +10.1404 q^{43} -1.00000 q^{45} +1.89749 q^{47} -6.46838 q^{49} -5.74301 q^{51} +8.14040 q^{53} +0.729126 q^{55} -6.11263 q^{57} -8.68215 q^{59} -15.5709 q^{61} +0.729126 q^{63} -3.38351 q^{65} +2.55187 q^{67} +9.48602 q^{69} +4.83164 q^{71} +6.29488 q^{73} +1.00000 q^{75} -0.531625 q^{77} +5.39363 q^{79} +1.00000 q^{81} -0.0848668 q^{83} +5.74301 q^{85} -1.00000 q^{87} +4.63674 q^{89} +2.46700 q^{91} -5.48602 q^{93} +6.11263 q^{95} +1.30377 q^{97} -0.729126 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 4 q^{5} - 2 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} - 4 q^{5} - 2 q^{7} + 4 q^{9} + 2 q^{11} - 8 q^{13} - 4 q^{15} - 10 q^{17} + 2 q^{19} - 2 q^{21} + 12 q^{23} + 4 q^{25} + 4 q^{27} - 4 q^{29} + 4 q^{31} + 2 q^{33} + 2 q^{35} - 16 q^{37} - 8 q^{39} - 12 q^{41} - 2 q^{43} - 4 q^{45} + 12 q^{47} + 6 q^{49} - 10 q^{51} - 10 q^{53} - 2 q^{55} + 2 q^{57} - 2 q^{59} - 26 q^{61} - 2 q^{63} + 8 q^{65} - 2 q^{67} + 12 q^{69} + 10 q^{71} + 4 q^{75} - 34 q^{77} - 22 q^{79} + 4 q^{81} + 10 q^{83} + 10 q^{85} - 4 q^{87} - 4 q^{89} + 8 q^{91} + 4 q^{93} - 2 q^{95} - 22 q^{97} + 2 q^{99}+O(q^{100}) \) 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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0.729126 0.275584 0.137792 0.990461i \(-0.455999\pi\)
0.137792 + 0.990461i \(0.455999\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −0.729126 −0.219840 −0.109920 0.993940i \(-0.535059\pi\)
−0.109920 + 0.993940i \(0.535059\pi\)
\(12\) 0 0
\(13\) 3.38351 0.938416 0.469208 0.883088i \(-0.344539\pi\)
0.469208 + 0.883088i \(0.344539\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −5.74301 −1.39288 −0.696442 0.717613i \(-0.745235\pi\)
−0.696442 + 0.717613i \(0.745235\pi\)
\(18\) 0 0
\(19\) −6.11263 −1.40233 −0.701167 0.712997i \(-0.747337\pi\)
−0.701167 + 0.712997i \(0.747337\pi\)
\(20\) 0 0
\(21\) 0.729126 0.159108
\(22\) 0 0
\(23\) 9.48602 1.97797 0.988986 0.148010i \(-0.0472866\pi\)
0.988986 + 0.148010i \(0.0472866\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) −5.48602 −0.985318 −0.492659 0.870222i \(-0.663975\pi\)
−0.492659 + 0.870222i \(0.663975\pi\)
\(32\) 0 0
\(33\) −0.729126 −0.126925
\(34\) 0 0
\(35\) −0.729126 −0.123245
\(36\) 0 0
\(37\) −10.2949 −1.69247 −0.846234 0.532811i \(-0.821135\pi\)
−0.846234 + 0.532811i \(0.821135\pi\)
\(38\) 0 0
\(39\) 3.38351 0.541795
\(40\) 0 0
\(41\) −11.3088 −1.76613 −0.883066 0.469249i \(-0.844525\pi\)
−0.883066 + 0.469249i \(0.844525\pi\)
\(42\) 0 0
\(43\) 10.1404 1.54640 0.773198 0.634164i \(-0.218656\pi\)
0.773198 + 0.634164i \(0.218656\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 1.89749 0.276777 0.138389 0.990378i \(-0.455808\pi\)
0.138389 + 0.990378i \(0.455808\pi\)
\(48\) 0 0
\(49\) −6.46838 −0.924054
\(50\) 0 0
\(51\) −5.74301 −0.804182
\(52\) 0 0
\(53\) 8.14040 1.11817 0.559085 0.829110i \(-0.311152\pi\)
0.559085 + 0.829110i \(0.311152\pi\)
\(54\) 0 0
\(55\) 0.729126 0.0983153
\(56\) 0 0
\(57\) −6.11263 −0.809638
\(58\) 0 0
\(59\) −8.68215 −1.13032 −0.565160 0.824981i \(-0.691186\pi\)
−0.565160 + 0.824981i \(0.691186\pi\)
\(60\) 0 0
\(61\) −15.5709 −1.99365 −0.996824 0.0796378i \(-0.974624\pi\)
−0.996824 + 0.0796378i \(0.974624\pi\)
\(62\) 0 0
\(63\) 0.729126 0.0918612
\(64\) 0 0
\(65\) −3.38351 −0.419673
\(66\) 0 0
\(67\) 2.55187 0.311761 0.155880 0.987776i \(-0.450179\pi\)
0.155880 + 0.987776i \(0.450179\pi\)
\(68\) 0 0
\(69\) 9.48602 1.14198
\(70\) 0 0
\(71\) 4.83164 0.573410 0.286705 0.958019i \(-0.407440\pi\)
0.286705 + 0.958019i \(0.407440\pi\)
\(72\) 0 0
\(73\) 6.29488 0.736760 0.368380 0.929675i \(-0.379913\pi\)
0.368380 + 0.929675i \(0.379913\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) −0.531625 −0.0605843
\(78\) 0 0
\(79\) 5.39363 0.606831 0.303415 0.952858i \(-0.401873\pi\)
0.303415 + 0.952858i \(0.401873\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −0.0848668 −0.00931534 −0.00465767 0.999989i \(-0.501483\pi\)
−0.00465767 + 0.999989i \(0.501483\pi\)
\(84\) 0 0
\(85\) 5.74301 0.622917
\(86\) 0 0
\(87\) −1.00000 −0.107211
\(88\) 0 0
\(89\) 4.63674 0.491493 0.245747 0.969334i \(-0.420967\pi\)
0.245747 + 0.969334i \(0.420967\pi\)
\(90\) 0 0
\(91\) 2.46700 0.258612
\(92\) 0 0
\(93\) −5.48602 −0.568874
\(94\) 0 0
\(95\) 6.11263 0.627143
\(96\) 0 0
\(97\) 1.30377 0.132378 0.0661891 0.997807i \(-0.478916\pi\)
0.0661891 + 0.997807i \(0.478916\pi\)
\(98\) 0 0
\(99\) −0.729126 −0.0732799
\(100\) 0 0
\(101\) −5.35574 −0.532916 −0.266458 0.963847i \(-0.585853\pi\)
−0.266458 + 0.963847i \(0.585853\pi\)
\(102\) 0 0
\(103\) −16.7670 −1.65210 −0.826052 0.563594i \(-0.809418\pi\)
−0.826052 + 0.563594i \(0.809418\pi\)
\(104\) 0 0
\(105\) −0.729126 −0.0711554
\(106\) 0 0
\(107\) −7.30876 −0.706565 −0.353282 0.935517i \(-0.614935\pi\)
−0.353282 + 0.935517i \(0.614935\pi\)
\(108\) 0 0
\(109\) −8.21515 −0.786868 −0.393434 0.919353i \(-0.628713\pi\)
−0.393434 + 0.919353i \(0.628713\pi\)
\(110\) 0 0
\(111\) −10.2949 −0.977147
\(112\) 0 0
\(113\) 8.51003 0.800556 0.400278 0.916394i \(-0.368914\pi\)
0.400278 + 0.916394i \(0.368914\pi\)
\(114\) 0 0
\(115\) −9.48602 −0.884576
\(116\) 0 0
\(117\) 3.38351 0.312805
\(118\) 0 0
\(119\) −4.18738 −0.383856
\(120\) 0 0
\(121\) −10.4684 −0.951670
\(122\) 0 0
\(123\) −11.3088 −1.01968
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −4.51398 −0.400551 −0.200275 0.979740i \(-0.564184\pi\)
−0.200275 + 0.979740i \(0.564184\pi\)
\(128\) 0 0
\(129\) 10.1404 0.892813
\(130\) 0 0
\(131\) 14.7771 1.29108 0.645542 0.763724i \(-0.276631\pi\)
0.645542 + 0.763724i \(0.276631\pi\)
\(132\) 0 0
\(133\) −4.45688 −0.386461
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −5.19114 −0.443509 −0.221754 0.975103i \(-0.571178\pi\)
−0.221754 + 0.975103i \(0.571178\pi\)
\(138\) 0 0
\(139\) 1.15824 0.0982406 0.0491203 0.998793i \(-0.484358\pi\)
0.0491203 + 0.998793i \(0.484358\pi\)
\(140\) 0 0
\(141\) 1.89749 0.159797
\(142\) 0 0
\(143\) −2.46700 −0.206301
\(144\) 0 0
\(145\) 1.00000 0.0830455
\(146\) 0 0
\(147\) −6.46838 −0.533503
\(148\) 0 0
\(149\) 6.68215 0.547423 0.273712 0.961812i \(-0.411749\pi\)
0.273712 + 0.961812i \(0.411749\pi\)
\(150\) 0 0
\(151\) −20.2253 −1.64591 −0.822955 0.568107i \(-0.807676\pi\)
−0.822955 + 0.568107i \(0.807676\pi\)
\(152\) 0 0
\(153\) −5.74301 −0.464295
\(154\) 0 0
\(155\) 5.48602 0.440648
\(156\) 0 0
\(157\) −23.8658 −1.90470 −0.952348 0.305014i \(-0.901339\pi\)
−0.952348 + 0.305014i \(0.901339\pi\)
\(158\) 0 0
\(159\) 8.14040 0.645576
\(160\) 0 0
\(161\) 6.91650 0.545097
\(162\) 0 0
\(163\) 1.93538 0.151591 0.0757953 0.997123i \(-0.475850\pi\)
0.0757953 + 0.997123i \(0.475850\pi\)
\(164\) 0 0
\(165\) 0.729126 0.0567624
\(166\) 0 0
\(167\) 12.4025 0.959736 0.479868 0.877341i \(-0.340685\pi\)
0.479868 + 0.877341i \(0.340685\pi\)
\(168\) 0 0
\(169\) −1.55187 −0.119375
\(170\) 0 0
\(171\) −6.11263 −0.467445
\(172\) 0 0
\(173\) −21.7113 −1.65068 −0.825339 0.564637i \(-0.809016\pi\)
−0.825339 + 0.564637i \(0.809016\pi\)
\(174\) 0 0
\(175\) 0.729126 0.0551167
\(176\) 0 0
\(177\) −8.68215 −0.652590
\(178\) 0 0
\(179\) −6.08487 −0.454804 −0.227402 0.973801i \(-0.573023\pi\)
−0.227402 + 0.973801i \(0.573023\pi\)
\(180\) 0 0
\(181\) 6.77714 0.503741 0.251870 0.967761i \(-0.418954\pi\)
0.251870 + 0.967761i \(0.418954\pi\)
\(182\) 0 0
\(183\) −15.5709 −1.15103
\(184\) 0 0
\(185\) 10.2949 0.756895
\(186\) 0 0
\(187\) 4.18738 0.306211
\(188\) 0 0
\(189\) 0.729126 0.0530361
\(190\) 0 0
\(191\) 18.9645 1.37222 0.686112 0.727496i \(-0.259316\pi\)
0.686112 + 0.727496i \(0.259316\pi\)
\(192\) 0 0
\(193\) 3.69760 0.266159 0.133079 0.991105i \(-0.457513\pi\)
0.133079 + 0.991105i \(0.457513\pi\)
\(194\) 0 0
\(195\) −3.38351 −0.242298
\(196\) 0 0
\(197\) 20.4783 1.45902 0.729509 0.683971i \(-0.239748\pi\)
0.729509 + 0.683971i \(0.239748\pi\)
\(198\) 0 0
\(199\) 10.8418 0.768552 0.384276 0.923218i \(-0.374451\pi\)
0.384276 + 0.923218i \(0.374451\pi\)
\(200\) 0 0
\(201\) 2.55187 0.179995
\(202\) 0 0
\(203\) −0.729126 −0.0511746
\(204\) 0 0
\(205\) 11.3088 0.789838
\(206\) 0 0
\(207\) 9.48602 0.659324
\(208\) 0 0
\(209\) 4.45688 0.308289
\(210\) 0 0
\(211\) −2.00000 −0.137686 −0.0688428 0.997628i \(-0.521931\pi\)
−0.0688428 + 0.997628i \(0.521931\pi\)
\(212\) 0 0
\(213\) 4.83164 0.331058
\(214\) 0 0
\(215\) −10.1404 −0.691570
\(216\) 0 0
\(217\) −4.00000 −0.271538
\(218\) 0 0
\(219\) 6.29488 0.425369
\(220\) 0 0
\(221\) −19.4315 −1.30711
\(222\) 0 0
\(223\) 4.21515 0.282267 0.141134 0.989991i \(-0.454925\pi\)
0.141134 + 0.989991i \(0.454925\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 15.4214 1.02355 0.511777 0.859118i \(-0.328987\pi\)
0.511777 + 0.859118i \(0.328987\pi\)
\(228\) 0 0
\(229\) −24.2530 −1.60269 −0.801343 0.598205i \(-0.795881\pi\)
−0.801343 + 0.598205i \(0.795881\pi\)
\(230\) 0 0
\(231\) −0.531625 −0.0349783
\(232\) 0 0
\(233\) −5.33653 −0.349608 −0.174804 0.984603i \(-0.555929\pi\)
−0.174804 + 0.984603i \(0.555929\pi\)
\(234\) 0 0
\(235\) −1.89749 −0.123779
\(236\) 0 0
\(237\) 5.39363 0.350354
\(238\) 0 0
\(239\) −16.0202 −1.03626 −0.518132 0.855301i \(-0.673372\pi\)
−0.518132 + 0.855301i \(0.673372\pi\)
\(240\) 0 0
\(241\) −6.60741 −0.425620 −0.212810 0.977094i \(-0.568262\pi\)
−0.212810 + 0.977094i \(0.568262\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 6.46838 0.413249
\(246\) 0 0
\(247\) −20.6821 −1.31597
\(248\) 0 0
\(249\) −0.0848668 −0.00537821
\(250\) 0 0
\(251\) −25.6834 −1.62112 −0.810560 0.585655i \(-0.800837\pi\)
−0.810560 + 0.585655i \(0.800837\pi\)
\(252\) 0 0
\(253\) −6.91650 −0.434837
\(254\) 0 0
\(255\) 5.74301 0.359641
\(256\) 0 0
\(257\) −7.93538 −0.494995 −0.247498 0.968888i \(-0.579608\pi\)
−0.247498 + 0.968888i \(0.579608\pi\)
\(258\) 0 0
\(259\) −7.50627 −0.466417
\(260\) 0 0
\(261\) −1.00000 −0.0618984
\(262\) 0 0
\(263\) −23.5618 −1.45288 −0.726441 0.687228i \(-0.758827\pi\)
−0.726441 + 0.687228i \(0.758827\pi\)
\(264\) 0 0
\(265\) −8.14040 −0.500061
\(266\) 0 0
\(267\) 4.63674 0.283764
\(268\) 0 0
\(269\) −10.6645 −0.650226 −0.325113 0.945675i \(-0.605402\pi\)
−0.325113 + 0.945675i \(0.605402\pi\)
\(270\) 0 0
\(271\) 3.30876 0.200993 0.100497 0.994937i \(-0.467957\pi\)
0.100497 + 0.994937i \(0.467957\pi\)
\(272\) 0 0
\(273\) 2.46700 0.149310
\(274\) 0 0
\(275\) −0.729126 −0.0439680
\(276\) 0 0
\(277\) −0.0469761 −0.00282252 −0.00141126 0.999999i \(-0.500449\pi\)
−0.00141126 + 0.999999i \(0.500449\pi\)
\(278\) 0 0
\(279\) −5.48602 −0.328439
\(280\) 0 0
\(281\) 29.8037 1.77794 0.888969 0.457967i \(-0.151422\pi\)
0.888969 + 0.457967i \(0.151422\pi\)
\(282\) 0 0
\(283\) −5.79498 −0.344476 −0.172238 0.985055i \(-0.555100\pi\)
−0.172238 + 0.985055i \(0.555100\pi\)
\(284\) 0 0
\(285\) 6.11263 0.362081
\(286\) 0 0
\(287\) −8.24552 −0.486717
\(288\) 0 0
\(289\) 15.9822 0.940127
\(290\) 0 0
\(291\) 1.30377 0.0764286
\(292\) 0 0
\(293\) 2.99624 0.175042 0.0875211 0.996163i \(-0.472105\pi\)
0.0875211 + 0.996163i \(0.472105\pi\)
\(294\) 0 0
\(295\) 8.68215 0.505494
\(296\) 0 0
\(297\) −0.729126 −0.0423082
\(298\) 0 0
\(299\) 32.0960 1.85616
\(300\) 0 0
\(301\) 7.39363 0.426162
\(302\) 0 0
\(303\) −5.35574 −0.307679
\(304\) 0 0
\(305\) 15.5709 0.891586
\(306\) 0 0
\(307\) −4.05554 −0.231462 −0.115731 0.993281i \(-0.536921\pi\)
−0.115731 + 0.993281i \(0.536921\pi\)
\(308\) 0 0
\(309\) −16.7670 −0.953842
\(310\) 0 0
\(311\) −13.6734 −0.775347 −0.387674 0.921797i \(-0.626721\pi\)
−0.387674 + 0.921797i \(0.626721\pi\)
\(312\) 0 0
\(313\) −4.25200 −0.240337 −0.120169 0.992753i \(-0.538344\pi\)
−0.120169 + 0.992753i \(0.538344\pi\)
\(314\) 0 0
\(315\) −0.729126 −0.0410816
\(316\) 0 0
\(317\) −4.28476 −0.240656 −0.120328 0.992734i \(-0.538395\pi\)
−0.120328 + 0.992734i \(0.538395\pi\)
\(318\) 0 0
\(319\) 0.729126 0.0408232
\(320\) 0 0
\(321\) −7.30876 −0.407935
\(322\) 0 0
\(323\) 35.1049 1.95329
\(324\) 0 0
\(325\) 3.38351 0.187683
\(326\) 0 0
\(327\) −8.21515 −0.454299
\(328\) 0 0
\(329\) 1.38351 0.0762753
\(330\) 0 0
\(331\) −3.61772 −0.198848 −0.0994240 0.995045i \(-0.531700\pi\)
−0.0994240 + 0.995045i \(0.531700\pi\)
\(332\) 0 0
\(333\) −10.2949 −0.564156
\(334\) 0 0
\(335\) −2.55187 −0.139424
\(336\) 0 0
\(337\) 23.4442 1.27709 0.638543 0.769586i \(-0.279538\pi\)
0.638543 + 0.769586i \(0.279538\pi\)
\(338\) 0 0
\(339\) 8.51003 0.462201
\(340\) 0 0
\(341\) 4.00000 0.216612
\(342\) 0 0
\(343\) −9.82014 −0.530238
\(344\) 0 0
\(345\) −9.48602 −0.510710
\(346\) 0 0
\(347\) −20.8226 −1.11781 −0.558907 0.829231i \(-0.688779\pi\)
−0.558907 + 0.829231i \(0.688779\pi\)
\(348\) 0 0
\(349\) −18.0555 −0.966491 −0.483245 0.875485i \(-0.660542\pi\)
−0.483245 + 0.875485i \(0.660542\pi\)
\(350\) 0 0
\(351\) 3.38351 0.180598
\(352\) 0 0
\(353\) −14.3645 −0.764545 −0.382272 0.924050i \(-0.624858\pi\)
−0.382272 + 0.924050i \(0.624858\pi\)
\(354\) 0 0
\(355\) −4.83164 −0.256437
\(356\) 0 0
\(357\) −4.18738 −0.221620
\(358\) 0 0
\(359\) −6.20502 −0.327489 −0.163744 0.986503i \(-0.552357\pi\)
−0.163744 + 0.986503i \(0.552357\pi\)
\(360\) 0 0
\(361\) 18.3643 0.966542
\(362\) 0 0
\(363\) −10.4684 −0.549447
\(364\) 0 0
\(365\) −6.29488 −0.329489
\(366\) 0 0
\(367\) −18.1975 −0.949902 −0.474951 0.880012i \(-0.657534\pi\)
−0.474951 + 0.880012i \(0.657534\pi\)
\(368\) 0 0
\(369\) −11.3088 −0.588711
\(370\) 0 0
\(371\) 5.93538 0.308150
\(372\) 0 0
\(373\) 34.8606 1.80501 0.902506 0.430677i \(-0.141725\pi\)
0.902506 + 0.430677i \(0.141725\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −3.38351 −0.174260
\(378\) 0 0
\(379\) 6.41005 0.329262 0.164631 0.986355i \(-0.447357\pi\)
0.164631 + 0.986355i \(0.447357\pi\)
\(380\) 0 0
\(381\) −4.51398 −0.231258
\(382\) 0 0
\(383\) −20.7202 −1.05875 −0.529376 0.848387i \(-0.677574\pi\)
−0.529376 + 0.848387i \(0.677574\pi\)
\(384\) 0 0
\(385\) 0.531625 0.0270941
\(386\) 0 0
\(387\) 10.1404 0.515466
\(388\) 0 0
\(389\) −24.7581 −1.25528 −0.627642 0.778502i \(-0.715980\pi\)
−0.627642 + 0.778502i \(0.715980\pi\)
\(390\) 0 0
\(391\) −54.4783 −2.75509
\(392\) 0 0
\(393\) 14.7771 0.745408
\(394\) 0 0
\(395\) −5.39363 −0.271383
\(396\) 0 0
\(397\) −12.7468 −0.639742 −0.319871 0.947461i \(-0.603640\pi\)
−0.319871 + 0.947461i \(0.603640\pi\)
\(398\) 0 0
\(399\) −4.45688 −0.223123
\(400\) 0 0
\(401\) 28.1959 1.40804 0.704019 0.710181i \(-0.251387\pi\)
0.704019 + 0.710181i \(0.251387\pi\)
\(402\) 0 0
\(403\) −18.5620 −0.924639
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 7.50627 0.372072
\(408\) 0 0
\(409\) −24.0112 −1.18728 −0.593638 0.804732i \(-0.702309\pi\)
−0.593638 + 0.804732i \(0.702309\pi\)
\(410\) 0 0
\(411\) −5.19114 −0.256060
\(412\) 0 0
\(413\) −6.33038 −0.311498
\(414\) 0 0
\(415\) 0.0848668 0.00416595
\(416\) 0 0
\(417\) 1.15824 0.0567192
\(418\) 0 0
\(419\) −14.7670 −0.721416 −0.360708 0.932679i \(-0.617465\pi\)
−0.360708 + 0.932679i \(0.617465\pi\)
\(420\) 0 0
\(421\) −12.7302 −0.620430 −0.310215 0.950666i \(-0.600401\pi\)
−0.310215 + 0.950666i \(0.600401\pi\)
\(422\) 0 0
\(423\) 1.89749 0.0922591
\(424\) 0 0
\(425\) −5.74301 −0.278577
\(426\) 0 0
\(427\) −11.3531 −0.549417
\(428\) 0 0
\(429\) −2.46700 −0.119108
\(430\) 0 0
\(431\) −9.66327 −0.465464 −0.232732 0.972541i \(-0.574766\pi\)
−0.232732 + 0.972541i \(0.574766\pi\)
\(432\) 0 0
\(433\) 13.1190 0.630459 0.315229 0.949016i \(-0.397919\pi\)
0.315229 + 0.949016i \(0.397919\pi\)
\(434\) 0 0
\(435\) 1.00000 0.0479463
\(436\) 0 0
\(437\) −57.9846 −2.77378
\(438\) 0 0
\(439\) 19.9253 0.950981 0.475490 0.879721i \(-0.342271\pi\)
0.475490 + 0.879721i \(0.342271\pi\)
\(440\) 0 0
\(441\) −6.46838 −0.308018
\(442\) 0 0
\(443\) −1.94304 −0.0923167 −0.0461583 0.998934i \(-0.514698\pi\)
−0.0461583 + 0.998934i \(0.514698\pi\)
\(444\) 0 0
\(445\) −4.63674 −0.219802
\(446\) 0 0
\(447\) 6.68215 0.316055
\(448\) 0 0
\(449\) 17.0215 0.803293 0.401647 0.915795i \(-0.368438\pi\)
0.401647 + 0.915795i \(0.368438\pi\)
\(450\) 0 0
\(451\) 8.24552 0.388266
\(452\) 0 0
\(453\) −20.2253 −0.950266
\(454\) 0 0
\(455\) −2.46700 −0.115655
\(456\) 0 0
\(457\) 15.9633 0.746731 0.373366 0.927684i \(-0.378204\pi\)
0.373366 + 0.927684i \(0.378204\pi\)
\(458\) 0 0
\(459\) −5.74301 −0.268061
\(460\) 0 0
\(461\) −17.6910 −0.823954 −0.411977 0.911194i \(-0.635162\pi\)
−0.411977 + 0.911194i \(0.635162\pi\)
\(462\) 0 0
\(463\) 2.60741 0.121176 0.0605882 0.998163i \(-0.480702\pi\)
0.0605882 + 0.998163i \(0.480702\pi\)
\(464\) 0 0
\(465\) 5.48602 0.254408
\(466\) 0 0
\(467\) 8.00000 0.370196 0.185098 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 0 0
\(469\) 1.86064 0.0859162
\(470\) 0 0
\(471\) −23.8658 −1.09968
\(472\) 0 0
\(473\) −7.39363 −0.339960
\(474\) 0 0
\(475\) −6.11263 −0.280467
\(476\) 0 0
\(477\) 8.14040 0.372723
\(478\) 0 0
\(479\) −20.6833 −0.945045 −0.472523 0.881319i \(-0.656657\pi\)
−0.472523 + 0.881319i \(0.656657\pi\)
\(480\) 0 0
\(481\) −34.8328 −1.58824
\(482\) 0 0
\(483\) 6.91650 0.314712
\(484\) 0 0
\(485\) −1.30377 −0.0592013
\(486\) 0 0
\(487\) 32.1035 1.45475 0.727375 0.686240i \(-0.240740\pi\)
0.727375 + 0.686240i \(0.240740\pi\)
\(488\) 0 0
\(489\) 1.93538 0.0875209
\(490\) 0 0
\(491\) 40.6933 1.83646 0.918232 0.396044i \(-0.129617\pi\)
0.918232 + 0.396044i \(0.129617\pi\)
\(492\) 0 0
\(493\) 5.74301 0.258652
\(494\) 0 0
\(495\) 0.729126 0.0327718
\(496\) 0 0
\(497\) 3.52287 0.158022
\(498\) 0 0
\(499\) −10.8418 −0.485344 −0.242672 0.970108i \(-0.578024\pi\)
−0.242672 + 0.970108i \(0.578024\pi\)
\(500\) 0 0
\(501\) 12.4025 0.554104
\(502\) 0 0
\(503\) 2.47727 0.110456 0.0552280 0.998474i \(-0.482411\pi\)
0.0552280 + 0.998474i \(0.482411\pi\)
\(504\) 0 0
\(505\) 5.35574 0.238327
\(506\) 0 0
\(507\) −1.55187 −0.0689210
\(508\) 0 0
\(509\) −24.3101 −1.07753 −0.538764 0.842457i \(-0.681109\pi\)
−0.538764 + 0.842457i \(0.681109\pi\)
\(510\) 0 0
\(511\) 4.58976 0.203039
\(512\) 0 0
\(513\) −6.11263 −0.269879
\(514\) 0 0
\(515\) 16.7670 0.738843
\(516\) 0 0
\(517\) −1.38351 −0.0608466
\(518\) 0 0
\(519\) −21.7113 −0.953020
\(520\) 0 0
\(521\) 25.2997 1.10840 0.554200 0.832384i \(-0.313024\pi\)
0.554200 + 0.832384i \(0.313024\pi\)
\(522\) 0 0
\(523\) −39.2277 −1.71531 −0.857653 0.514228i \(-0.828078\pi\)
−0.857653 + 0.514228i \(0.828078\pi\)
\(524\) 0 0
\(525\) 0.729126 0.0318217
\(526\) 0 0
\(527\) 31.5063 1.37243
\(528\) 0 0
\(529\) 66.9846 2.91237
\(530\) 0 0
\(531\) −8.68215 −0.376773
\(532\) 0 0
\(533\) −38.2633 −1.65737
\(534\) 0 0
\(535\) 7.30876 0.315985
\(536\) 0 0
\(537\) −6.08487 −0.262581
\(538\) 0 0
\(539\) 4.71626 0.203144
\(540\) 0 0
\(541\) 6.28080 0.270033 0.135016 0.990843i \(-0.456891\pi\)
0.135016 + 0.990843i \(0.456891\pi\)
\(542\) 0 0
\(543\) 6.77714 0.290835
\(544\) 0 0
\(545\) 8.21515 0.351898
\(546\) 0 0
\(547\) 20.2809 0.867151 0.433575 0.901117i \(-0.357252\pi\)
0.433575 + 0.901117i \(0.357252\pi\)
\(548\) 0 0
\(549\) −15.5709 −0.664549
\(550\) 0 0
\(551\) 6.11263 0.260407
\(552\) 0 0
\(553\) 3.93264 0.167233
\(554\) 0 0
\(555\) 10.2949 0.436993
\(556\) 0 0
\(557\) 16.3252 0.691720 0.345860 0.938286i \(-0.387587\pi\)
0.345860 + 0.938286i \(0.387587\pi\)
\(558\) 0 0
\(559\) 34.3101 1.45116
\(560\) 0 0
\(561\) 4.18738 0.176791
\(562\) 0 0
\(563\) 12.8037 0.539613 0.269806 0.962915i \(-0.413040\pi\)
0.269806 + 0.962915i \(0.413040\pi\)
\(564\) 0 0
\(565\) −8.51003 −0.358020
\(566\) 0 0
\(567\) 0.729126 0.0306204
\(568\) 0 0
\(569\) 9.35574 0.392213 0.196107 0.980583i \(-0.437170\pi\)
0.196107 + 0.980583i \(0.437170\pi\)
\(570\) 0 0
\(571\) 27.0125 1.13044 0.565220 0.824940i \(-0.308791\pi\)
0.565220 + 0.824940i \(0.308791\pi\)
\(572\) 0 0
\(573\) 18.9645 0.792254
\(574\) 0 0
\(575\) 9.48602 0.395594
\(576\) 0 0
\(577\) −25.9859 −1.08181 −0.540904 0.841084i \(-0.681918\pi\)
−0.540904 + 0.841084i \(0.681918\pi\)
\(578\) 0 0
\(579\) 3.69760 0.153667
\(580\) 0 0
\(581\) −0.0618786 −0.00256716
\(582\) 0 0
\(583\) −5.93538 −0.245818
\(584\) 0 0
\(585\) −3.38351 −0.139891
\(586\) 0 0
\(587\) −19.6467 −0.810905 −0.405452 0.914116i \(-0.632886\pi\)
−0.405452 + 0.914116i \(0.632886\pi\)
\(588\) 0 0
\(589\) 33.5340 1.38175
\(590\) 0 0
\(591\) 20.4783 0.842365
\(592\) 0 0
\(593\) 26.1682 1.07460 0.537299 0.843392i \(-0.319445\pi\)
0.537299 + 0.843392i \(0.319445\pi\)
\(594\) 0 0
\(595\) 4.18738 0.171666
\(596\) 0 0
\(597\) 10.8418 0.443724
\(598\) 0 0
\(599\) 3.45565 0.141194 0.0705970 0.997505i \(-0.477510\pi\)
0.0705970 + 0.997505i \(0.477510\pi\)
\(600\) 0 0
\(601\) 13.5063 0.550932 0.275466 0.961311i \(-0.411168\pi\)
0.275466 + 0.961311i \(0.411168\pi\)
\(602\) 0 0
\(603\) 2.55187 0.103920
\(604\) 0 0
\(605\) 10.4684 0.425600
\(606\) 0 0
\(607\) −12.0773 −0.490204 −0.245102 0.969497i \(-0.578822\pi\)
−0.245102 + 0.969497i \(0.578822\pi\)
\(608\) 0 0
\(609\) −0.729126 −0.0295457
\(610\) 0 0
\(611\) 6.42017 0.259732
\(612\) 0 0
\(613\) −19.9935 −0.807531 −0.403765 0.914863i \(-0.632299\pi\)
−0.403765 + 0.914863i \(0.632299\pi\)
\(614\) 0 0
\(615\) 11.3088 0.456013
\(616\) 0 0
\(617\) −31.9784 −1.28740 −0.643701 0.765277i \(-0.722602\pi\)
−0.643701 + 0.765277i \(0.722602\pi\)
\(618\) 0 0
\(619\) 33.0862 1.32985 0.664924 0.746911i \(-0.268464\pi\)
0.664924 + 0.746911i \(0.268464\pi\)
\(620\) 0 0
\(621\) 9.48602 0.380661
\(622\) 0 0
\(623\) 3.38077 0.135448
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 4.45688 0.177991
\(628\) 0 0
\(629\) 59.1236 2.35741
\(630\) 0 0
\(631\) 3.02654 0.120485 0.0602423 0.998184i \(-0.480813\pi\)
0.0602423 + 0.998184i \(0.480813\pi\)
\(632\) 0 0
\(633\) −2.00000 −0.0794929
\(634\) 0 0
\(635\) 4.51398 0.179132
\(636\) 0 0
\(637\) −21.8858 −0.867147
\(638\) 0 0
\(639\) 4.83164 0.191137
\(640\) 0 0
\(641\) 14.0747 0.555919 0.277959 0.960593i \(-0.410342\pi\)
0.277959 + 0.960593i \(0.410342\pi\)
\(642\) 0 0
\(643\) −4.25044 −0.167621 −0.0838104 0.996482i \(-0.526709\pi\)
−0.0838104 + 0.996482i \(0.526709\pi\)
\(644\) 0 0
\(645\) −10.1404 −0.399278
\(646\) 0 0
\(647\) 43.0185 1.69123 0.845616 0.533792i \(-0.179234\pi\)
0.845616 + 0.533792i \(0.179234\pi\)
\(648\) 0 0
\(649\) 6.33038 0.248489
\(650\) 0 0
\(651\) −4.00000 −0.156772
\(652\) 0 0
\(653\) 18.5175 0.724648 0.362324 0.932052i \(-0.381983\pi\)
0.362324 + 0.932052i \(0.381983\pi\)
\(654\) 0 0
\(655\) −14.7771 −0.577391
\(656\) 0 0
\(657\) 6.29488 0.245587
\(658\) 0 0
\(659\) −6.19736 −0.241415 −0.120707 0.992688i \(-0.538516\pi\)
−0.120707 + 0.992688i \(0.538516\pi\)
\(660\) 0 0
\(661\) −29.0936 −1.13161 −0.565805 0.824539i \(-0.691435\pi\)
−0.565805 + 0.824539i \(0.691435\pi\)
\(662\) 0 0
\(663\) −19.4315 −0.754658
\(664\) 0 0
\(665\) 4.45688 0.172830
\(666\) 0 0
\(667\) −9.48602 −0.367300
\(668\) 0 0
\(669\) 4.21515 0.162967
\(670\) 0 0
\(671\) 11.3531 0.438283
\(672\) 0 0
\(673\) −36.6288 −1.41194 −0.705969 0.708243i \(-0.749488\pi\)
−0.705969 + 0.708243i \(0.749488\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 25.9330 0.996686 0.498343 0.866980i \(-0.333942\pi\)
0.498343 + 0.866980i \(0.333942\pi\)
\(678\) 0 0
\(679\) 0.950615 0.0364813
\(680\) 0 0
\(681\) 15.4214 0.590949
\(682\) 0 0
\(683\) −22.4403 −0.858653 −0.429327 0.903149i \(-0.641249\pi\)
−0.429327 + 0.903149i \(0.641249\pi\)
\(684\) 0 0
\(685\) 5.19114 0.198343
\(686\) 0 0
\(687\) −24.2530 −0.925311
\(688\) 0 0
\(689\) 27.5431 1.04931
\(690\) 0 0
\(691\) −16.3000 −0.620082 −0.310041 0.950723i \(-0.600343\pi\)
−0.310041 + 0.950723i \(0.600343\pi\)
\(692\) 0 0
\(693\) −0.531625 −0.0201948
\(694\) 0 0
\(695\) −1.15824 −0.0439345
\(696\) 0 0
\(697\) 64.9463 2.46002
\(698\) 0 0
\(699\) −5.33653 −0.201846
\(700\) 0 0
\(701\) −10.9811 −0.414751 −0.207376 0.978261i \(-0.566492\pi\)
−0.207376 + 0.978261i \(0.566492\pi\)
\(702\) 0 0
\(703\) 62.9288 2.37341
\(704\) 0 0
\(705\) −1.89749 −0.0714636
\(706\) 0 0
\(707\) −3.90501 −0.146863
\(708\) 0 0
\(709\) −33.0681 −1.24190 −0.620949 0.783851i \(-0.713252\pi\)
−0.620949 + 0.783851i \(0.713252\pi\)
\(710\) 0 0
\(711\) 5.39363 0.202277
\(712\) 0 0
\(713\) −52.0405 −1.94893
\(714\) 0 0
\(715\) 2.46700 0.0922607
\(716\) 0 0
\(717\) −16.0202 −0.598287
\(718\) 0 0
\(719\) 44.0289 1.64200 0.821001 0.570926i \(-0.193416\pi\)
0.821001 + 0.570926i \(0.193416\pi\)
\(720\) 0 0
\(721\) −12.2253 −0.455293
\(722\) 0 0
\(723\) −6.60741 −0.245732
\(724\) 0 0
\(725\) −1.00000 −0.0371391
\(726\) 0 0
\(727\) −20.2708 −0.751803 −0.375902 0.926660i \(-0.622667\pi\)
−0.375902 + 0.926660i \(0.622667\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −58.2364 −2.15395
\(732\) 0 0
\(733\) 45.9138 1.69586 0.847932 0.530105i \(-0.177847\pi\)
0.847932 + 0.530105i \(0.177847\pi\)
\(734\) 0 0
\(735\) 6.46838 0.238590
\(736\) 0 0
\(737\) −1.86064 −0.0685374
\(738\) 0 0
\(739\) −38.1884 −1.40478 −0.702392 0.711791i \(-0.747885\pi\)
−0.702392 + 0.711791i \(0.747885\pi\)
\(740\) 0 0
\(741\) −20.6821 −0.759778
\(742\) 0 0
\(743\) 34.4773 1.26485 0.632424 0.774622i \(-0.282060\pi\)
0.632424 + 0.774622i \(0.282060\pi\)
\(744\) 0 0
\(745\) −6.68215 −0.244815
\(746\) 0 0
\(747\) −0.0848668 −0.00310511
\(748\) 0 0
\(749\) −5.32901 −0.194718
\(750\) 0 0
\(751\) −41.7113 −1.52207 −0.761033 0.648713i \(-0.775308\pi\)
−0.761033 + 0.648713i \(0.775308\pi\)
\(752\) 0 0
\(753\) −25.6834 −0.935954
\(754\) 0 0
\(755\) 20.2253 0.736073
\(756\) 0 0
\(757\) −13.5659 −0.493061 −0.246530 0.969135i \(-0.579291\pi\)
−0.246530 + 0.969135i \(0.579291\pi\)
\(758\) 0 0
\(759\) −6.91650 −0.251053
\(760\) 0 0
\(761\) 3.27982 0.118893 0.0594467 0.998231i \(-0.481066\pi\)
0.0594467 + 0.998231i \(0.481066\pi\)
\(762\) 0 0
\(763\) −5.98988 −0.216848
\(764\) 0 0
\(765\) 5.74301 0.207639
\(766\) 0 0
\(767\) −29.3761 −1.06071
\(768\) 0 0
\(769\) −35.5527 −1.28206 −0.641032 0.767514i \(-0.721493\pi\)
−0.641032 + 0.767514i \(0.721493\pi\)
\(770\) 0 0
\(771\) −7.93538 −0.285786
\(772\) 0 0
\(773\) −0.965870 −0.0347399 −0.0173700 0.999849i \(-0.505529\pi\)
−0.0173700 + 0.999849i \(0.505529\pi\)
\(774\) 0 0
\(775\) −5.48602 −0.197064
\(776\) 0 0
\(777\) −7.50627 −0.269286
\(778\) 0 0
\(779\) 69.1263 2.47671
\(780\) 0 0
\(781\) −3.52287 −0.126058
\(782\) 0 0
\(783\) −1.00000 −0.0357371
\(784\) 0 0
\(785\) 23.8658 0.851806
\(786\) 0 0
\(787\) −15.0756 −0.537387 −0.268693 0.963226i \(-0.586592\pi\)
−0.268693 + 0.963226i \(0.586592\pi\)
\(788\) 0 0
\(789\) −23.5618 −0.838822
\(790\) 0 0
\(791\) 6.20488 0.220620
\(792\) 0 0
\(793\) −52.6842 −1.87087
\(794\) 0 0
\(795\) −8.14040 −0.288710
\(796\) 0 0
\(797\) 27.9884 0.991399 0.495700 0.868494i \(-0.334912\pi\)
0.495700 + 0.868494i \(0.334912\pi\)
\(798\) 0 0
\(799\) −10.8973 −0.385519
\(800\) 0 0
\(801\) 4.63674 0.163831
\(802\) 0 0
\(803\) −4.58976 −0.161969
\(804\) 0 0
\(805\) −6.91650 −0.243775
\(806\) 0 0
\(807\) −10.6645 −0.375408
\(808\) 0 0
\(809\) 10.0950 0.354921 0.177460 0.984128i \(-0.443212\pi\)
0.177460 + 0.984128i \(0.443212\pi\)
\(810\) 0 0
\(811\) −16.3555 −0.574321 −0.287160 0.957882i \(-0.592711\pi\)
−0.287160 + 0.957882i \(0.592711\pi\)
\(812\) 0 0
\(813\) 3.30876 0.116043
\(814\) 0 0
\(815\) −1.93538 −0.0677934
\(816\) 0 0
\(817\) −61.9846 −2.16857
\(818\) 0 0
\(819\) 2.46700 0.0862041
\(820\) 0 0
\(821\) 20.7468 0.724067 0.362034 0.932165i \(-0.382083\pi\)
0.362034 + 0.932165i \(0.382083\pi\)
\(822\) 0 0
\(823\) −33.4619 −1.16641 −0.583204 0.812326i \(-0.698201\pi\)
−0.583204 + 0.812326i \(0.698201\pi\)
\(824\) 0 0
\(825\) −0.729126 −0.0253849
\(826\) 0 0
\(827\) 20.3363 0.707164 0.353582 0.935404i \(-0.384964\pi\)
0.353582 + 0.935404i \(0.384964\pi\)
\(828\) 0 0
\(829\) 18.4858 0.642039 0.321020 0.947073i \(-0.395974\pi\)
0.321020 + 0.947073i \(0.395974\pi\)
\(830\) 0 0
\(831\) −0.0469761 −0.00162958
\(832\) 0 0
\(833\) 37.1479 1.28710
\(834\) 0 0
\(835\) −12.4025 −0.429207
\(836\) 0 0
\(837\) −5.48602 −0.189625
\(838\) 0 0
\(839\) 22.3571 0.771853 0.385927 0.922529i \(-0.373882\pi\)
0.385927 + 0.922529i \(0.373882\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) 29.8037 1.02649
\(844\) 0 0
\(845\) 1.55187 0.0533860
\(846\) 0 0
\(847\) −7.63277 −0.262265
\(848\) 0 0
\(849\) −5.79498 −0.198883
\(850\) 0 0
\(851\) −97.6574 −3.34765
\(852\) 0 0
\(853\) 16.1950 0.554505 0.277253 0.960797i \(-0.410576\pi\)
0.277253 + 0.960797i \(0.410576\pi\)
\(854\) 0 0
\(855\) 6.11263 0.209048
\(856\) 0 0
\(857\) 11.4785 0.392098 0.196049 0.980594i \(-0.437189\pi\)
0.196049 + 0.980594i \(0.437189\pi\)
\(858\) 0 0
\(859\) 21.6744 0.739522 0.369761 0.929127i \(-0.379440\pi\)
0.369761 + 0.929127i \(0.379440\pi\)
\(860\) 0 0
\(861\) −8.24552 −0.281006
\(862\) 0 0
\(863\) 30.7226 1.04581 0.522905 0.852391i \(-0.324848\pi\)
0.522905 + 0.852391i \(0.324848\pi\)
\(864\) 0 0
\(865\) 21.7113 0.738206
\(866\) 0 0
\(867\) 15.9822 0.542783
\(868\) 0 0
\(869\) −3.93264 −0.133406
\(870\) 0 0
\(871\) 8.63428 0.292561
\(872\) 0 0
\(873\) 1.30377 0.0441260
\(874\) 0 0
\(875\) −0.729126 −0.0246490
\(876\) 0 0
\(877\) 34.2908 1.15792 0.578959 0.815357i \(-0.303459\pi\)
0.578959 + 0.815357i \(0.303459\pi\)
\(878\) 0 0
\(879\) 2.99624 0.101061
\(880\) 0 0
\(881\) 12.8620 0.433332 0.216666 0.976246i \(-0.430482\pi\)
0.216666 + 0.976246i \(0.430482\pi\)
\(882\) 0 0
\(883\) −1.37495 −0.0462707 −0.0231354 0.999732i \(-0.507365\pi\)
−0.0231354 + 0.999732i \(0.507365\pi\)
\(884\) 0 0
\(885\) 8.68215 0.291847
\(886\) 0 0
\(887\) 27.1710 0.912312 0.456156 0.889900i \(-0.349226\pi\)
0.456156 + 0.889900i \(0.349226\pi\)
\(888\) 0 0
\(889\) −3.29126 −0.110385
\(890\) 0 0
\(891\) −0.729126 −0.0244266
\(892\) 0 0
\(893\) −11.5987 −0.388134
\(894\) 0 0
\(895\) 6.08487 0.203395
\(896\) 0 0
\(897\) 32.0960 1.07166
\(898\) 0 0
\(899\) 5.48602 0.182969
\(900\) 0 0
\(901\) −46.7504 −1.55748
\(902\) 0 0
\(903\) 7.39363 0.246045
\(904\) 0 0
\(905\) −6.77714 −0.225280
\(906\) 0 0
\(907\) −0.327640 −0.0108791 −0.00543955 0.999985i \(-0.501731\pi\)
−0.00543955 + 0.999985i \(0.501731\pi\)
\(908\) 0 0
\(909\) −5.35574 −0.177639
\(910\) 0 0
\(911\) −43.0377 −1.42590 −0.712951 0.701214i \(-0.752642\pi\)
−0.712951 + 0.701214i \(0.752642\pi\)
\(912\) 0 0
\(913\) 0.0618786 0.00204788
\(914\) 0 0
\(915\) 15.5709 0.514758
\(916\) 0 0
\(917\) 10.7744 0.355802
\(918\) 0 0
\(919\) 28.2061 0.930432 0.465216 0.885197i \(-0.345977\pi\)
0.465216 + 0.885197i \(0.345977\pi\)
\(920\) 0 0
\(921\) −4.05554 −0.133634
\(922\) 0 0
\(923\) 16.3479 0.538097
\(924\) 0 0
\(925\) −10.2949 −0.338494
\(926\) 0 0
\(927\) −16.7670 −0.550701
\(928\) 0 0
\(929\) −14.4505 −0.474107 −0.237053 0.971497i \(-0.576182\pi\)
−0.237053 + 0.971497i \(0.576182\pi\)
\(930\) 0 0
\(931\) 39.5388 1.29583
\(932\) 0 0
\(933\) −13.6734 −0.447647
\(934\) 0 0
\(935\) −4.18738 −0.136942
\(936\) 0 0
\(937\) 13.9657 0.456241 0.228121 0.973633i \(-0.426742\pi\)
0.228121 + 0.973633i \(0.426742\pi\)
\(938\) 0 0
\(939\) −4.25200 −0.138759
\(940\) 0 0
\(941\) −40.6262 −1.32438 −0.662189 0.749337i \(-0.730372\pi\)
−0.662189 + 0.749337i \(0.730372\pi\)
\(942\) 0 0
\(943\) −107.275 −3.49336
\(944\) 0 0
\(945\) −0.729126 −0.0237185
\(946\) 0 0
\(947\) 20.4495 0.664519 0.332260 0.943188i \(-0.392189\pi\)
0.332260 + 0.943188i \(0.392189\pi\)
\(948\) 0 0
\(949\) 21.2988 0.691388
\(950\) 0 0
\(951\) −4.28476 −0.138943
\(952\) 0 0
\(953\) −51.5527 −1.66996 −0.834978 0.550283i \(-0.814520\pi\)
−0.834978 + 0.550283i \(0.814520\pi\)
\(954\) 0 0
\(955\) −18.9645 −0.613677
\(956\) 0 0
\(957\) 0.729126 0.0235693
\(958\) 0 0
\(959\) −3.78499 −0.122224
\(960\) 0 0
\(961\) −0.903587 −0.0291480
\(962\) 0 0
\(963\) −7.30876 −0.235522
\(964\) 0 0
\(965\) −3.69760 −0.119030
\(966\) 0 0
\(967\) −16.5429 −0.531985 −0.265992 0.963975i \(-0.585700\pi\)
−0.265992 + 0.963975i \(0.585700\pi\)
\(968\) 0 0
\(969\) 35.1049 1.12773
\(970\) 0 0
\(971\) −32.5417 −1.04431 −0.522157 0.852849i \(-0.674873\pi\)
−0.522157 + 0.852849i \(0.674873\pi\)
\(972\) 0 0
\(973\) 0.844503 0.0270735
\(974\) 0 0
\(975\) 3.38351 0.108359
\(976\) 0 0
\(977\) −38.7111 −1.23848 −0.619239 0.785203i \(-0.712559\pi\)
−0.619239 + 0.785203i \(0.712559\pi\)
\(978\) 0 0
\(979\) −3.38077 −0.108050
\(980\) 0 0
\(981\) −8.21515 −0.262289
\(982\) 0 0
\(983\) 31.7315 1.01208 0.506039 0.862510i \(-0.331109\pi\)
0.506039 + 0.862510i \(0.331109\pi\)
\(984\) 0 0
\(985\) −20.4783 −0.652493
\(986\) 0 0
\(987\) 1.38351 0.0440376
\(988\) 0 0
\(989\) 96.1921 3.05873
\(990\) 0 0
\(991\) 22.8620 0.726236 0.363118 0.931743i \(-0.381712\pi\)
0.363118 + 0.931743i \(0.381712\pi\)
\(992\) 0 0
\(993\) −3.61772 −0.114805
\(994\) 0 0
\(995\) −10.8418 −0.343707
\(996\) 0 0
\(997\) 40.9024 1.29539 0.647696 0.761898i \(-0.275733\pi\)
0.647696 + 0.761898i \(0.275733\pi\)
\(998\) 0 0
\(999\) −10.2949 −0.325716
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 6960.2.a.co.1.3 4
4.3 odd 2 435.2.a.j.1.2 4
12.11 even 2 1305.2.a.r.1.3 4
20.3 even 4 2175.2.c.n.349.7 8
20.7 even 4 2175.2.c.n.349.2 8
20.19 odd 2 2175.2.a.v.1.3 4
60.59 even 2 6525.2.a.bi.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.j.1.2 4 4.3 odd 2
1305.2.a.r.1.3 4 12.11 even 2
2175.2.a.v.1.3 4 20.19 odd 2
2175.2.c.n.349.2 8 20.7 even 4
2175.2.c.n.349.7 8 20.3 even 4
6525.2.a.bi.1.2 4 60.59 even 2
6960.2.a.co.1.3 4 1.1 even 1 trivial