Properties

Label 6525.2.a.ce.1.4
Level $6525$
Weight $2$
Character 6525.1
Self dual yes
Analytic conductor $52.102$
Analytic rank $1$
Dimension $12$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6525,2,Mod(1,6525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6525, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("6525.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6525 = 3^{2} \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6525.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: \(52.1023873189\)
Analytic rank: \(1\)
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: \( x^{12} - 20x^{10} + 148x^{8} - 502x^{6} + 792x^{4} - 496x^{2} + 45 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 1305)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.35513\) of defining polynomial
Character \(\chi\) \(=\) 6525.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.35513 q^{2} -0.163621 q^{4} +1.26969 q^{7} +2.93199 q^{8} +0.474782 q^{11} +0.407685 q^{13} -1.72060 q^{14} -3.64599 q^{16} -2.97132 q^{17} +5.80491 q^{19} -0.643392 q^{22} +3.06516 q^{23} -0.552466 q^{26} -0.207747 q^{28} +1.00000 q^{29} -4.49210 q^{31} -0.923188 q^{32} +4.02653 q^{34} -7.20325 q^{37} -7.86641 q^{38} -12.0668 q^{41} +1.03847 q^{43} -0.0776841 q^{44} -4.15369 q^{46} -2.97132 q^{47} -5.38789 q^{49} -0.0667056 q^{52} -7.87199 q^{53} +3.72271 q^{56} -1.35513 q^{58} +4.45891 q^{59} -1.71609 q^{61} +6.08739 q^{62} +8.54301 q^{64} +15.7754 q^{67} +0.486169 q^{68} -9.00806 q^{71} +0.677990 q^{73} +9.76134 q^{74} -0.949802 q^{76} +0.602826 q^{77} +1.63957 q^{79} +16.3521 q^{82} +15.5423 q^{83} -1.40726 q^{86} +1.39206 q^{88} -13.4175 q^{89} +0.517633 q^{91} -0.501523 q^{92} +4.02653 q^{94} -7.10733 q^{97} +7.30129 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 16 q^{4} - 12 q^{11} - 16 q^{14} + 16 q^{16} - 20 q^{19} - 56 q^{26} + 12 q^{29} - 16 q^{31} + 4 q^{34} - 32 q^{41} - 68 q^{44} + 20 q^{46} - 4 q^{49} - 76 q^{56} - 44 q^{59} - 52 q^{61} + 36 q^{64}+ \cdots + 4 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.35513 −0.958222 −0.479111 0.877754i \(-0.659041\pi\)
−0.479111 + 0.877754i \(0.659041\pi\)
\(3\) 0 0
\(4\) −0.163621 −0.0818103
\(5\) 0 0
\(6\) 0 0
\(7\) 1.26969 0.479897 0.239949 0.970786i \(-0.422869\pi\)
0.239949 + 0.970786i \(0.422869\pi\)
\(8\) 2.93199 1.03661
\(9\) 0 0
\(10\) 0 0
\(11\) 0.474782 0.143152 0.0715761 0.997435i \(-0.477197\pi\)
0.0715761 + 0.997435i \(0.477197\pi\)
\(12\) 0 0
\(13\) 0.407685 0.113071 0.0565357 0.998401i \(-0.481995\pi\)
0.0565357 + 0.998401i \(0.481995\pi\)
\(14\) −1.72060 −0.459848
\(15\) 0 0
\(16\) −3.64599 −0.911497
\(17\) −2.97132 −0.720651 −0.360326 0.932827i \(-0.617334\pi\)
−0.360326 + 0.932827i \(0.617334\pi\)
\(18\) 0 0
\(19\) 5.80491 1.33174 0.665868 0.746069i \(-0.268061\pi\)
0.665868 + 0.746069i \(0.268061\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −0.643392 −0.137172
\(23\) 3.06516 0.639130 0.319565 0.947564i \(-0.396463\pi\)
0.319565 + 0.947564i \(0.396463\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −0.552466 −0.108348
\(27\) 0 0
\(28\) −0.207747 −0.0392605
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −4.49210 −0.806805 −0.403403 0.915023i \(-0.632173\pi\)
−0.403403 + 0.915023i \(0.632173\pi\)
\(32\) −0.923188 −0.163198
\(33\) 0 0
\(34\) 4.02653 0.690544
\(35\) 0 0
\(36\) 0 0
\(37\) −7.20325 −1.18421 −0.592103 0.805862i \(-0.701702\pi\)
−0.592103 + 0.805862i \(0.701702\pi\)
\(38\) −7.86641 −1.27610
\(39\) 0 0
\(40\) 0 0
\(41\) −12.0668 −1.88452 −0.942259 0.334886i \(-0.891302\pi\)
−0.942259 + 0.334886i \(0.891302\pi\)
\(42\) 0 0
\(43\) 1.03847 0.158365 0.0791827 0.996860i \(-0.474769\pi\)
0.0791827 + 0.996860i \(0.474769\pi\)
\(44\) −0.0776841 −0.0117113
\(45\) 0 0
\(46\) −4.15369 −0.612428
\(47\) −2.97132 −0.433412 −0.216706 0.976237i \(-0.569531\pi\)
−0.216706 + 0.976237i \(0.569531\pi\)
\(48\) 0 0
\(49\) −5.38789 −0.769698
\(50\) 0 0
\(51\) 0 0
\(52\) −0.0667056 −0.00925040
\(53\) −7.87199 −1.08130 −0.540651 0.841247i \(-0.681822\pi\)
−0.540651 + 0.841247i \(0.681822\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 3.72271 0.497469
\(57\) 0 0
\(58\) −1.35513 −0.177937
\(59\) 4.45891 0.580501 0.290250 0.956951i \(-0.406261\pi\)
0.290250 + 0.956951i \(0.406261\pi\)
\(60\) 0 0
\(61\) −1.71609 −0.219722 −0.109861 0.993947i \(-0.535041\pi\)
−0.109861 + 0.993947i \(0.535041\pi\)
\(62\) 6.08739 0.773099
\(63\) 0 0
\(64\) 8.54301 1.06788
\(65\) 0 0
\(66\) 0 0
\(67\) 15.7754 1.92727 0.963636 0.267218i \(-0.0861043\pi\)
0.963636 + 0.267218i \(0.0861043\pi\)
\(68\) 0.486169 0.0589567
\(69\) 0 0
\(70\) 0 0
\(71\) −9.00806 −1.06906 −0.534530 0.845149i \(-0.679511\pi\)
−0.534530 + 0.845149i \(0.679511\pi\)
\(72\) 0 0
\(73\) 0.677990 0.0793527 0.0396763 0.999213i \(-0.487367\pi\)
0.0396763 + 0.999213i \(0.487367\pi\)
\(74\) 9.76134 1.13473
\(75\) 0 0
\(76\) −0.949802 −0.108950
\(77\) 0.602826 0.0686984
\(78\) 0 0
\(79\) 1.63957 0.184466 0.0922329 0.995737i \(-0.470600\pi\)
0.0922329 + 0.995737i \(0.470600\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 16.3521 1.80579
\(83\) 15.5423 1.70599 0.852995 0.521919i \(-0.174784\pi\)
0.852995 + 0.521919i \(0.174784\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1.40726 −0.151749
\(87\) 0 0
\(88\) 1.39206 0.148394
\(89\) −13.4175 −1.42225 −0.711126 0.703065i \(-0.751814\pi\)
−0.711126 + 0.703065i \(0.751814\pi\)
\(90\) 0 0
\(91\) 0.517633 0.0542627
\(92\) −0.501523 −0.0522874
\(93\) 0 0
\(94\) 4.02653 0.415305
\(95\) 0 0
\(96\) 0 0
\(97\) −7.10733 −0.721640 −0.360820 0.932636i \(-0.617503\pi\)
−0.360820 + 0.932636i \(0.617503\pi\)
\(98\) 7.30129 0.737542
\(99\) 0 0
\(100\) 0 0
\(101\) −4.68433 −0.466108 −0.233054 0.972464i \(-0.574872\pi\)
−0.233054 + 0.972464i \(0.574872\pi\)
\(102\) 0 0
\(103\) 15.5180 1.52904 0.764519 0.644601i \(-0.222976\pi\)
0.764519 + 0.644601i \(0.222976\pi\)
\(104\) 1.19533 0.117212
\(105\) 0 0
\(106\) 10.6676 1.03613
\(107\) −0.676108 −0.0653618 −0.0326809 0.999466i \(-0.510405\pi\)
−0.0326809 + 0.999466i \(0.510405\pi\)
\(108\) 0 0
\(109\) −7.44734 −0.713326 −0.356663 0.934233i \(-0.616086\pi\)
−0.356663 + 0.934233i \(0.616086\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −4.62927 −0.437425
\(113\) −14.1017 −1.32658 −0.663290 0.748363i \(-0.730840\pi\)
−0.663290 + 0.748363i \(0.730840\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −0.163621 −0.0151918
\(117\) 0 0
\(118\) −6.04241 −0.556249
\(119\) −3.77265 −0.345839
\(120\) 0 0
\(121\) −10.7746 −0.979507
\(122\) 2.32552 0.210543
\(123\) 0 0
\(124\) 0.735000 0.0660050
\(125\) 0 0
\(126\) 0 0
\(127\) −11.5508 −1.02497 −0.512486 0.858696i \(-0.671275\pi\)
−0.512486 + 0.858696i \(0.671275\pi\)
\(128\) −9.73053 −0.860065
\(129\) 0 0
\(130\) 0 0
\(131\) −10.2432 −0.894954 −0.447477 0.894296i \(-0.647677\pi\)
−0.447477 + 0.894296i \(0.647677\pi\)
\(132\) 0 0
\(133\) 7.37043 0.639097
\(134\) −21.3777 −1.84676
\(135\) 0 0
\(136\) −8.71188 −0.747038
\(137\) −2.06577 −0.176491 −0.0882455 0.996099i \(-0.528126\pi\)
−0.0882455 + 0.996099i \(0.528126\pi\)
\(138\) 0 0
\(139\) 3.48237 0.295370 0.147685 0.989034i \(-0.452818\pi\)
0.147685 + 0.989034i \(0.452818\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 12.2071 1.02440
\(143\) 0.193562 0.0161864
\(144\) 0 0
\(145\) 0 0
\(146\) −0.918765 −0.0760375
\(147\) 0 0
\(148\) 1.17860 0.0968802
\(149\) −10.8558 −0.889343 −0.444672 0.895694i \(-0.646680\pi\)
−0.444672 + 0.895694i \(0.646680\pi\)
\(150\) 0 0
\(151\) 11.3555 0.924097 0.462048 0.886855i \(-0.347115\pi\)
0.462048 + 0.886855i \(0.347115\pi\)
\(152\) 17.0199 1.38050
\(153\) 0 0
\(154\) −0.816908 −0.0658283
\(155\) 0 0
\(156\) 0 0
\(157\) 16.9813 1.35526 0.677628 0.735405i \(-0.263008\pi\)
0.677628 + 0.735405i \(0.263008\pi\)
\(158\) −2.22183 −0.176759
\(159\) 0 0
\(160\) 0 0
\(161\) 3.89180 0.306717
\(162\) 0 0
\(163\) 7.78532 0.609793 0.304897 0.952385i \(-0.401378\pi\)
0.304897 + 0.952385i \(0.401378\pi\)
\(164\) 1.97438 0.154173
\(165\) 0 0
\(166\) −21.0619 −1.63472
\(167\) 2.06389 0.159709 0.0798544 0.996807i \(-0.474554\pi\)
0.0798544 + 0.996807i \(0.474554\pi\)
\(168\) 0 0
\(169\) −12.8338 −0.987215
\(170\) 0 0
\(171\) 0 0
\(172\) −0.169915 −0.0129559
\(173\) 0.522120 0.0396960 0.0198480 0.999803i \(-0.493682\pi\)
0.0198480 + 0.999803i \(0.493682\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1.73105 −0.130483
\(177\) 0 0
\(178\) 18.1825 1.36283
\(179\) −10.4805 −0.783346 −0.391673 0.920104i \(-0.628104\pi\)
−0.391673 + 0.920104i \(0.628104\pi\)
\(180\) 0 0
\(181\) 12.2724 0.912197 0.456099 0.889929i \(-0.349246\pi\)
0.456099 + 0.889929i \(0.349246\pi\)
\(182\) −0.701461 −0.0519957
\(183\) 0 0
\(184\) 8.98701 0.662531
\(185\) 0 0
\(186\) 0 0
\(187\) −1.41073 −0.103163
\(188\) 0.486169 0.0354575
\(189\) 0 0
\(190\) 0 0
\(191\) −9.69478 −0.701490 −0.350745 0.936471i \(-0.614072\pi\)
−0.350745 + 0.936471i \(0.614072\pi\)
\(192\) 0 0
\(193\) 11.0857 0.797963 0.398982 0.916959i \(-0.369364\pi\)
0.398982 + 0.916959i \(0.369364\pi\)
\(194\) 9.63136 0.691491
\(195\) 0 0
\(196\) 0.881569 0.0629692
\(197\) 2.00431 0.142801 0.0714007 0.997448i \(-0.477253\pi\)
0.0714007 + 0.997448i \(0.477253\pi\)
\(198\) 0 0
\(199\) −18.1170 −1.28428 −0.642140 0.766588i \(-0.721953\pi\)
−0.642140 + 0.766588i \(0.721953\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 6.34788 0.446635
\(203\) 1.26969 0.0891147
\(204\) 0 0
\(205\) 0 0
\(206\) −21.0290 −1.46516
\(207\) 0 0
\(208\) −1.48641 −0.103064
\(209\) 2.75607 0.190641
\(210\) 0 0
\(211\) 18.4829 1.27241 0.636206 0.771519i \(-0.280503\pi\)
0.636206 + 0.771519i \(0.280503\pi\)
\(212\) 1.28802 0.0884615
\(213\) 0 0
\(214\) 0.916215 0.0626312
\(215\) 0 0
\(216\) 0 0
\(217\) −5.70357 −0.387184
\(218\) 10.0921 0.683525
\(219\) 0 0
\(220\) 0 0
\(221\) −1.21136 −0.0814851
\(222\) 0 0
\(223\) 21.6226 1.44796 0.723978 0.689823i \(-0.242312\pi\)
0.723978 + 0.689823i \(0.242312\pi\)
\(224\) −1.17216 −0.0783184
\(225\) 0 0
\(226\) 19.1097 1.27116
\(227\) −19.5621 −1.29838 −0.649191 0.760625i \(-0.724892\pi\)
−0.649191 + 0.760625i \(0.724892\pi\)
\(228\) 0 0
\(229\) 10.7779 0.712220 0.356110 0.934444i \(-0.384103\pi\)
0.356110 + 0.934444i \(0.384103\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 2.93199 0.192494
\(233\) 4.25222 0.278572 0.139286 0.990252i \(-0.455519\pi\)
0.139286 + 0.990252i \(0.455519\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −0.729569 −0.0474909
\(237\) 0 0
\(238\) 5.11244 0.331390
\(239\) −16.8385 −1.08919 −0.544596 0.838699i \(-0.683317\pi\)
−0.544596 + 0.838699i \(0.683317\pi\)
\(240\) 0 0
\(241\) 21.8396 1.40681 0.703405 0.710789i \(-0.251662\pi\)
0.703405 + 0.710789i \(0.251662\pi\)
\(242\) 14.6010 0.938586
\(243\) 0 0
\(244\) 0.280787 0.0179755
\(245\) 0 0
\(246\) 0 0
\(247\) 2.36657 0.150581
\(248\) −13.1708 −0.836346
\(249\) 0 0
\(250\) 0 0
\(251\) −2.24264 −0.141554 −0.0707772 0.997492i \(-0.522548\pi\)
−0.0707772 + 0.997492i \(0.522548\pi\)
\(252\) 0 0
\(253\) 1.45528 0.0914929
\(254\) 15.6529 0.982150
\(255\) 0 0
\(256\) −3.89989 −0.243743
\(257\) −1.27685 −0.0796479 −0.0398239 0.999207i \(-0.512680\pi\)
−0.0398239 + 0.999207i \(0.512680\pi\)
\(258\) 0 0
\(259\) −9.14589 −0.568298
\(260\) 0 0
\(261\) 0 0
\(262\) 13.8809 0.857564
\(263\) 19.6767 1.21332 0.606658 0.794963i \(-0.292510\pi\)
0.606658 + 0.794963i \(0.292510\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −9.98789 −0.612397
\(267\) 0 0
\(268\) −2.58118 −0.157671
\(269\) 11.7353 0.715516 0.357758 0.933814i \(-0.383541\pi\)
0.357758 + 0.933814i \(0.383541\pi\)
\(270\) 0 0
\(271\) −9.92348 −0.602809 −0.301404 0.953496i \(-0.597455\pi\)
−0.301404 + 0.953496i \(0.597455\pi\)
\(272\) 10.8334 0.656871
\(273\) 0 0
\(274\) 2.79940 0.169118
\(275\) 0 0
\(276\) 0 0
\(277\) −1.50995 −0.0907242 −0.0453621 0.998971i \(-0.514444\pi\)
−0.0453621 + 0.998971i \(0.514444\pi\)
\(278\) −4.71906 −0.283031
\(279\) 0 0
\(280\) 0 0
\(281\) −25.3285 −1.51097 −0.755487 0.655164i \(-0.772600\pi\)
−0.755487 + 0.655164i \(0.772600\pi\)
\(282\) 0 0
\(283\) −2.09287 −0.124408 −0.0622041 0.998063i \(-0.519813\pi\)
−0.0622041 + 0.998063i \(0.519813\pi\)
\(284\) 1.47390 0.0874601
\(285\) 0 0
\(286\) −0.262301 −0.0155102
\(287\) −15.3211 −0.904375
\(288\) 0 0
\(289\) −8.17125 −0.480662
\(290\) 0 0
\(291\) 0 0
\(292\) −0.110933 −0.00649186
\(293\) 10.8526 0.634013 0.317007 0.948423i \(-0.397322\pi\)
0.317007 + 0.948423i \(0.397322\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −21.1198 −1.22757
\(297\) 0 0
\(298\) 14.7111 0.852189
\(299\) 1.24962 0.0722674
\(300\) 0 0
\(301\) 1.31854 0.0759991
\(302\) −15.3882 −0.885490
\(303\) 0 0
\(304\) −21.1646 −1.21387
\(305\) 0 0
\(306\) 0 0
\(307\) −16.8956 −0.964281 −0.482141 0.876094i \(-0.660141\pi\)
−0.482141 + 0.876094i \(0.660141\pi\)
\(308\) −0.0986347 −0.00562023
\(309\) 0 0
\(310\) 0 0
\(311\) 10.6643 0.604717 0.302359 0.953194i \(-0.402226\pi\)
0.302359 + 0.953194i \(0.402226\pi\)
\(312\) 0 0
\(313\) 13.8702 0.783991 0.391995 0.919967i \(-0.371785\pi\)
0.391995 + 0.919967i \(0.371785\pi\)
\(314\) −23.0119 −1.29864
\(315\) 0 0
\(316\) −0.268267 −0.0150912
\(317\) 1.33953 0.0752355 0.0376178 0.999292i \(-0.488023\pi\)
0.0376178 + 0.999292i \(0.488023\pi\)
\(318\) 0 0
\(319\) 0.474782 0.0265827
\(320\) 0 0
\(321\) 0 0
\(322\) −5.27390 −0.293903
\(323\) −17.2482 −0.959718
\(324\) 0 0
\(325\) 0 0
\(326\) −10.5501 −0.584317
\(327\) 0 0
\(328\) −35.3797 −1.95352
\(329\) −3.77265 −0.207993
\(330\) 0 0
\(331\) −3.00072 −0.164934 −0.0824672 0.996594i \(-0.526280\pi\)
−0.0824672 + 0.996594i \(0.526280\pi\)
\(332\) −2.54304 −0.139567
\(333\) 0 0
\(334\) −2.79685 −0.153037
\(335\) 0 0
\(336\) 0 0
\(337\) −24.2763 −1.32241 −0.661207 0.750203i \(-0.729956\pi\)
−0.661207 + 0.750203i \(0.729956\pi\)
\(338\) 17.3915 0.945971
\(339\) 0 0
\(340\) 0 0
\(341\) −2.13277 −0.115496
\(342\) 0 0
\(343\) −15.7288 −0.849274
\(344\) 3.04479 0.164164
\(345\) 0 0
\(346\) −0.707540 −0.0380376
\(347\) 14.0792 0.755809 0.377904 0.925845i \(-0.376645\pi\)
0.377904 + 0.925845i \(0.376645\pi\)
\(348\) 0 0
\(349\) −17.9641 −0.961597 −0.480798 0.876831i \(-0.659653\pi\)
−0.480798 + 0.876831i \(0.659653\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −0.438313 −0.0233622
\(353\) 9.49061 0.505134 0.252567 0.967579i \(-0.418725\pi\)
0.252567 + 0.967579i \(0.418725\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 2.19538 0.116355
\(357\) 0 0
\(358\) 14.2024 0.750620
\(359\) 4.09130 0.215931 0.107965 0.994155i \(-0.465566\pi\)
0.107965 + 0.994155i \(0.465566\pi\)
\(360\) 0 0
\(361\) 14.6969 0.773523
\(362\) −16.6307 −0.874088
\(363\) 0 0
\(364\) −0.0846954 −0.00443925
\(365\) 0 0
\(366\) 0 0
\(367\) −18.8383 −0.983351 −0.491676 0.870778i \(-0.663615\pi\)
−0.491676 + 0.870778i \(0.663615\pi\)
\(368\) −11.1755 −0.582565
\(369\) 0 0
\(370\) 0 0
\(371\) −9.99498 −0.518914
\(372\) 0 0
\(373\) −7.23999 −0.374872 −0.187436 0.982277i \(-0.560018\pi\)
−0.187436 + 0.982277i \(0.560018\pi\)
\(374\) 1.91172 0.0988529
\(375\) 0 0
\(376\) −8.71188 −0.449281
\(377\) 0.407685 0.0209968
\(378\) 0 0
\(379\) 20.2420 1.03976 0.519881 0.854239i \(-0.325976\pi\)
0.519881 + 0.854239i \(0.325976\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 13.1377 0.672183
\(383\) −4.34201 −0.221866 −0.110933 0.993828i \(-0.535384\pi\)
−0.110933 + 0.993828i \(0.535384\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −15.0225 −0.764626
\(387\) 0 0
\(388\) 1.16290 0.0590375
\(389\) 0.508687 0.0257915 0.0128957 0.999917i \(-0.495895\pi\)
0.0128957 + 0.999917i \(0.495895\pi\)
\(390\) 0 0
\(391\) −9.10757 −0.460590
\(392\) −15.7972 −0.797881
\(393\) 0 0
\(394\) −2.71611 −0.136835
\(395\) 0 0
\(396\) 0 0
\(397\) −31.2135 −1.56656 −0.783280 0.621669i \(-0.786455\pi\)
−0.783280 + 0.621669i \(0.786455\pi\)
\(398\) 24.5509 1.23063
\(399\) 0 0
\(400\) 0 0
\(401\) 5.60957 0.280129 0.140064 0.990142i \(-0.455269\pi\)
0.140064 + 0.990142i \(0.455269\pi\)
\(402\) 0 0
\(403\) −1.83136 −0.0912267
\(404\) 0.766452 0.0381324
\(405\) 0 0
\(406\) −1.72060 −0.0853917
\(407\) −3.41997 −0.169522
\(408\) 0 0
\(409\) −9.68080 −0.478685 −0.239342 0.970935i \(-0.576932\pi\)
−0.239342 + 0.970935i \(0.576932\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −2.53907 −0.125091
\(413\) 5.66143 0.278581
\(414\) 0 0
\(415\) 0 0
\(416\) −0.376370 −0.0184530
\(417\) 0 0
\(418\) −3.73483 −0.182677
\(419\) 18.2788 0.892979 0.446489 0.894789i \(-0.352674\pi\)
0.446489 + 0.894789i \(0.352674\pi\)
\(420\) 0 0
\(421\) 26.0517 1.26968 0.634842 0.772642i \(-0.281065\pi\)
0.634842 + 0.772642i \(0.281065\pi\)
\(422\) −25.0467 −1.21925
\(423\) 0 0
\(424\) −23.0806 −1.12089
\(425\) 0 0
\(426\) 0 0
\(427\) −2.17890 −0.105444
\(428\) 0.110625 0.00534727
\(429\) 0 0
\(430\) 0 0
\(431\) −6.19278 −0.298296 −0.149148 0.988815i \(-0.547653\pi\)
−0.149148 + 0.988815i \(0.547653\pi\)
\(432\) 0 0
\(433\) −35.1538 −1.68938 −0.844692 0.535253i \(-0.820216\pi\)
−0.844692 + 0.535253i \(0.820216\pi\)
\(434\) 7.72909 0.371008
\(435\) 0 0
\(436\) 1.21854 0.0583574
\(437\) 17.7930 0.851153
\(438\) 0 0
\(439\) −20.1018 −0.959408 −0.479704 0.877430i \(-0.659256\pi\)
−0.479704 + 0.877430i \(0.659256\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 1.64156 0.0780808
\(443\) −20.8513 −0.990676 −0.495338 0.868700i \(-0.664956\pi\)
−0.495338 + 0.868700i \(0.664956\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −29.3014 −1.38746
\(447\) 0 0
\(448\) 10.8470 0.512471
\(449\) −39.5199 −1.86506 −0.932529 0.361094i \(-0.882403\pi\)
−0.932529 + 0.361094i \(0.882403\pi\)
\(450\) 0 0
\(451\) −5.72910 −0.269773
\(452\) 2.30733 0.108528
\(453\) 0 0
\(454\) 26.5092 1.24414
\(455\) 0 0
\(456\) 0 0
\(457\) −25.6115 −1.19806 −0.599028 0.800728i \(-0.704446\pi\)
−0.599028 + 0.800728i \(0.704446\pi\)
\(458\) −14.6054 −0.682465
\(459\) 0 0
\(460\) 0 0
\(461\) 0.911675 0.0424609 0.0212305 0.999775i \(-0.493242\pi\)
0.0212305 + 0.999775i \(0.493242\pi\)
\(462\) 0 0
\(463\) −25.9744 −1.20713 −0.603567 0.797312i \(-0.706254\pi\)
−0.603567 + 0.797312i \(0.706254\pi\)
\(464\) −3.64599 −0.169261
\(465\) 0 0
\(466\) −5.76231 −0.266934
\(467\) 15.8506 0.733479 0.366739 0.930324i \(-0.380474\pi\)
0.366739 + 0.930324i \(0.380474\pi\)
\(468\) 0 0
\(469\) 20.0299 0.924893
\(470\) 0 0
\(471\) 0 0
\(472\) 13.0735 0.601756
\(473\) 0.493048 0.0226704
\(474\) 0 0
\(475\) 0 0
\(476\) 0.617284 0.0282932
\(477\) 0 0
\(478\) 22.8184 1.04369
\(479\) −16.3087 −0.745162 −0.372581 0.928000i \(-0.621527\pi\)
−0.372581 + 0.928000i \(0.621527\pi\)
\(480\) 0 0
\(481\) −2.93666 −0.133900
\(482\) −29.5955 −1.34804
\(483\) 0 0
\(484\) 1.76294 0.0801338
\(485\) 0 0
\(486\) 0 0
\(487\) −9.93950 −0.450402 −0.225201 0.974312i \(-0.572304\pi\)
−0.225201 + 0.974312i \(0.572304\pi\)
\(488\) −5.03155 −0.227767
\(489\) 0 0
\(490\) 0 0
\(491\) −38.0638 −1.71779 −0.858897 0.512149i \(-0.828850\pi\)
−0.858897 + 0.512149i \(0.828850\pi\)
\(492\) 0 0
\(493\) −2.97132 −0.133822
\(494\) −3.20702 −0.144290
\(495\) 0 0
\(496\) 16.3781 0.735401
\(497\) −11.4374 −0.513039
\(498\) 0 0
\(499\) 44.3795 1.98670 0.993349 0.115143i \(-0.0367327\pi\)
0.993349 + 0.115143i \(0.0367327\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 3.03908 0.135641
\(503\) 3.18073 0.141822 0.0709108 0.997483i \(-0.477409\pi\)
0.0709108 + 0.997483i \(0.477409\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −1.97210 −0.0876705
\(507\) 0 0
\(508\) 1.88995 0.0838532
\(509\) −30.5882 −1.35580 −0.677899 0.735155i \(-0.737109\pi\)
−0.677899 + 0.735155i \(0.737109\pi\)
\(510\) 0 0
\(511\) 0.860836 0.0380811
\(512\) 24.7459 1.09363
\(513\) 0 0
\(514\) 1.73030 0.0763203
\(515\) 0 0
\(516\) 0 0
\(517\) −1.41073 −0.0620439
\(518\) 12.3939 0.544555
\(519\) 0 0
\(520\) 0 0
\(521\) −28.7858 −1.26113 −0.630564 0.776138i \(-0.717176\pi\)
−0.630564 + 0.776138i \(0.717176\pi\)
\(522\) 0 0
\(523\) −27.2018 −1.18945 −0.594726 0.803928i \(-0.702739\pi\)
−0.594726 + 0.803928i \(0.702739\pi\)
\(524\) 1.67600 0.0732164
\(525\) 0 0
\(526\) −26.6645 −1.16263
\(527\) 13.3475 0.581425
\(528\) 0 0
\(529\) −13.6048 −0.591513
\(530\) 0 0
\(531\) 0 0
\(532\) −1.20595 −0.0522847
\(533\) −4.91945 −0.213085
\(534\) 0 0
\(535\) 0 0
\(536\) 46.2533 1.99784
\(537\) 0 0
\(538\) −15.9029 −0.685623
\(539\) −2.55807 −0.110184
\(540\) 0 0
\(541\) 5.84584 0.251332 0.125666 0.992073i \(-0.459893\pi\)
0.125666 + 0.992073i \(0.459893\pi\)
\(542\) 13.4476 0.577625
\(543\) 0 0
\(544\) 2.74309 0.117609
\(545\) 0 0
\(546\) 0 0
\(547\) 18.9324 0.809489 0.404745 0.914430i \(-0.367360\pi\)
0.404745 + 0.914430i \(0.367360\pi\)
\(548\) 0.338003 0.0144388
\(549\) 0 0
\(550\) 0 0
\(551\) 5.80491 0.247297
\(552\) 0 0
\(553\) 2.08174 0.0885247
\(554\) 2.04618 0.0869340
\(555\) 0 0
\(556\) −0.569787 −0.0241643
\(557\) 29.6529 1.25643 0.628217 0.778038i \(-0.283785\pi\)
0.628217 + 0.778038i \(0.283785\pi\)
\(558\) 0 0
\(559\) 0.423369 0.0179066
\(560\) 0 0
\(561\) 0 0
\(562\) 34.3235 1.44785
\(563\) −19.3966 −0.817471 −0.408736 0.912653i \(-0.634030\pi\)
−0.408736 + 0.912653i \(0.634030\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 2.83611 0.119211
\(567\) 0 0
\(568\) −26.4115 −1.10820
\(569\) 6.00700 0.251826 0.125913 0.992041i \(-0.459814\pi\)
0.125913 + 0.992041i \(0.459814\pi\)
\(570\) 0 0
\(571\) −28.1469 −1.17791 −0.588957 0.808165i \(-0.700461\pi\)
−0.588957 + 0.808165i \(0.700461\pi\)
\(572\) −0.0316706 −0.00132422
\(573\) 0 0
\(574\) 20.7621 0.866592
\(575\) 0 0
\(576\) 0 0
\(577\) −32.9966 −1.37367 −0.686834 0.726815i \(-0.741000\pi\)
−0.686834 + 0.726815i \(0.741000\pi\)
\(578\) 11.0731 0.460581
\(579\) 0 0
\(580\) 0 0
\(581\) 19.7339 0.818700
\(582\) 0 0
\(583\) −3.73748 −0.154791
\(584\) 1.98786 0.0822581
\(585\) 0 0
\(586\) −14.7066 −0.607526
\(587\) 7.81241 0.322453 0.161226 0.986917i \(-0.448455\pi\)
0.161226 + 0.986917i \(0.448455\pi\)
\(588\) 0 0
\(589\) −26.0762 −1.07445
\(590\) 0 0
\(591\) 0 0
\(592\) 26.2629 1.07940
\(593\) −33.5131 −1.37622 −0.688108 0.725608i \(-0.741559\pi\)
−0.688108 + 0.725608i \(0.741559\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 1.77623 0.0727574
\(597\) 0 0
\(598\) −1.69340 −0.0692482
\(599\) −27.1213 −1.10815 −0.554074 0.832468i \(-0.686927\pi\)
−0.554074 + 0.832468i \(0.686927\pi\)
\(600\) 0 0
\(601\) −17.4641 −0.712376 −0.356188 0.934414i \(-0.615924\pi\)
−0.356188 + 0.934414i \(0.615924\pi\)
\(602\) −1.78679 −0.0728240
\(603\) 0 0
\(604\) −1.85799 −0.0756006
\(605\) 0 0
\(606\) 0 0
\(607\) 32.9273 1.33648 0.668239 0.743947i \(-0.267049\pi\)
0.668239 + 0.743947i \(0.267049\pi\)
\(608\) −5.35902 −0.217337
\(609\) 0 0
\(610\) 0 0
\(611\) −1.21136 −0.0490065
\(612\) 0 0
\(613\) 2.97228 0.120049 0.0600245 0.998197i \(-0.480882\pi\)
0.0600245 + 0.998197i \(0.480882\pi\)
\(614\) 22.8957 0.923996
\(615\) 0 0
\(616\) 1.76748 0.0712138
\(617\) −0.0642578 −0.00258692 −0.00129346 0.999999i \(-0.500412\pi\)
−0.00129346 + 0.999999i \(0.500412\pi\)
\(618\) 0 0
\(619\) 7.45588 0.299677 0.149839 0.988710i \(-0.452125\pi\)
0.149839 + 0.988710i \(0.452125\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −14.4515 −0.579454
\(623\) −17.0360 −0.682535
\(624\) 0 0
\(625\) 0 0
\(626\) −18.7960 −0.751237
\(627\) 0 0
\(628\) −2.77849 −0.110874
\(629\) 21.4032 0.853400
\(630\) 0 0
\(631\) −38.0735 −1.51568 −0.757841 0.652439i \(-0.773746\pi\)
−0.757841 + 0.652439i \(0.773746\pi\)
\(632\) 4.80720 0.191220
\(633\) 0 0
\(634\) −1.81524 −0.0720924
\(635\) 0 0
\(636\) 0 0
\(637\) −2.19656 −0.0870309
\(638\) −0.643392 −0.0254721
\(639\) 0 0
\(640\) 0 0
\(641\) 4.49665 0.177607 0.0888035 0.996049i \(-0.471696\pi\)
0.0888035 + 0.996049i \(0.471696\pi\)
\(642\) 0 0
\(643\) −20.9607 −0.826610 −0.413305 0.910593i \(-0.635626\pi\)
−0.413305 + 0.910593i \(0.635626\pi\)
\(644\) −0.636778 −0.0250926
\(645\) 0 0
\(646\) 23.3736 0.919623
\(647\) 37.8398 1.48764 0.743819 0.668381i \(-0.233013\pi\)
0.743819 + 0.668381i \(0.233013\pi\)
\(648\) 0 0
\(649\) 2.11701 0.0831000
\(650\) 0 0
\(651\) 0 0
\(652\) −1.27384 −0.0498873
\(653\) −48.0703 −1.88114 −0.940568 0.339606i \(-0.889706\pi\)
−0.940568 + 0.339606i \(0.889706\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 43.9954 1.71773
\(657\) 0 0
\(658\) 5.11244 0.199304
\(659\) −26.0861 −1.01617 −0.508084 0.861307i \(-0.669646\pi\)
−0.508084 + 0.861307i \(0.669646\pi\)
\(660\) 0 0
\(661\) 5.00120 0.194524 0.0972620 0.995259i \(-0.468992\pi\)
0.0972620 + 0.995259i \(0.468992\pi\)
\(662\) 4.06636 0.158044
\(663\) 0 0
\(664\) 45.5699 1.76845
\(665\) 0 0
\(666\) 0 0
\(667\) 3.06516 0.118683
\(668\) −0.337695 −0.0130658
\(669\) 0 0
\(670\) 0 0
\(671\) −0.814768 −0.0314538
\(672\) 0 0
\(673\) 26.1113 1.00652 0.503258 0.864136i \(-0.332134\pi\)
0.503258 + 0.864136i \(0.332134\pi\)
\(674\) 32.8976 1.26717
\(675\) 0 0
\(676\) 2.09987 0.0807643
\(677\) −32.0992 −1.23367 −0.616836 0.787092i \(-0.711586\pi\)
−0.616836 + 0.787092i \(0.711586\pi\)
\(678\) 0 0
\(679\) −9.02410 −0.346313
\(680\) 0 0
\(681\) 0 0
\(682\) 2.89018 0.110671
\(683\) 4.70468 0.180020 0.0900098 0.995941i \(-0.471310\pi\)
0.0900098 + 0.995941i \(0.471310\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 21.3145 0.813793
\(687\) 0 0
\(688\) −3.78625 −0.144349
\(689\) −3.20929 −0.122264
\(690\) 0 0
\(691\) 0.881335 0.0335276 0.0167638 0.999859i \(-0.494664\pi\)
0.0167638 + 0.999859i \(0.494664\pi\)
\(692\) −0.0854295 −0.00324754
\(693\) 0 0
\(694\) −19.0791 −0.724233
\(695\) 0 0
\(696\) 0 0
\(697\) 35.8543 1.35808
\(698\) 24.3437 0.921423
\(699\) 0 0
\(700\) 0 0
\(701\) 45.1606 1.70569 0.852847 0.522161i \(-0.174874\pi\)
0.852847 + 0.522161i \(0.174874\pi\)
\(702\) 0 0
\(703\) −41.8142 −1.57705
\(704\) 4.05607 0.152869
\(705\) 0 0
\(706\) −12.8610 −0.484031
\(707\) −5.94764 −0.223684
\(708\) 0 0
\(709\) 18.8061 0.706277 0.353139 0.935571i \(-0.385114\pi\)
0.353139 + 0.935571i \(0.385114\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −39.3399 −1.47433
\(713\) −13.7690 −0.515653
\(714\) 0 0
\(715\) 0 0
\(716\) 1.71482 0.0640857
\(717\) 0 0
\(718\) −5.54425 −0.206910
\(719\) 18.7384 0.698824 0.349412 0.936969i \(-0.386381\pi\)
0.349412 + 0.936969i \(0.386381\pi\)
\(720\) 0 0
\(721\) 19.7031 0.733782
\(722\) −19.9163 −0.741207
\(723\) 0 0
\(724\) −2.00801 −0.0746271
\(725\) 0 0
\(726\) 0 0
\(727\) 20.2155 0.749752 0.374876 0.927075i \(-0.377685\pi\)
0.374876 + 0.927075i \(0.377685\pi\)
\(728\) 1.51769 0.0562495
\(729\) 0 0
\(730\) 0 0
\(731\) −3.08563 −0.114126
\(732\) 0 0
\(733\) −45.1963 −1.66936 −0.834682 0.550732i \(-0.814349\pi\)
−0.834682 + 0.550732i \(0.814349\pi\)
\(734\) 25.5284 0.942269
\(735\) 0 0
\(736\) −2.82972 −0.104305
\(737\) 7.48988 0.275893
\(738\) 0 0
\(739\) 42.2473 1.55409 0.777046 0.629443i \(-0.216717\pi\)
0.777046 + 0.629443i \(0.216717\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 13.5445 0.497235
\(743\) 38.5919 1.41580 0.707900 0.706313i \(-0.249643\pi\)
0.707900 + 0.706313i \(0.249643\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 9.81113 0.359211
\(747\) 0 0
\(748\) 0.230825 0.00843978
\(749\) −0.858447 −0.0313670
\(750\) 0 0
\(751\) −1.50634 −0.0549670 −0.0274835 0.999622i \(-0.508749\pi\)
−0.0274835 + 0.999622i \(0.508749\pi\)
\(752\) 10.8334 0.395053
\(753\) 0 0
\(754\) −0.552466 −0.0201196
\(755\) 0 0
\(756\) 0 0
\(757\) −17.3890 −0.632016 −0.316008 0.948757i \(-0.602343\pi\)
−0.316008 + 0.948757i \(0.602343\pi\)
\(758\) −27.4306 −0.996324
\(759\) 0 0
\(760\) 0 0
\(761\) −9.21457 −0.334028 −0.167014 0.985955i \(-0.553413\pi\)
−0.167014 + 0.985955i \(0.553413\pi\)
\(762\) 0 0
\(763\) −9.45581 −0.342323
\(764\) 1.58626 0.0573890
\(765\) 0 0
\(766\) 5.88399 0.212597
\(767\) 1.81783 0.0656381
\(768\) 0 0
\(769\) 25.6939 0.926547 0.463273 0.886215i \(-0.346675\pi\)
0.463273 + 0.886215i \(0.346675\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −1.81384 −0.0652816
\(773\) 42.9070 1.54326 0.771628 0.636074i \(-0.219443\pi\)
0.771628 + 0.636074i \(0.219443\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −20.8386 −0.748062
\(777\) 0 0
\(778\) −0.689338 −0.0247140
\(779\) −70.0466 −2.50968
\(780\) 0 0
\(781\) −4.27687 −0.153038
\(782\) 12.3420 0.441347
\(783\) 0 0
\(784\) 19.6442 0.701578
\(785\) 0 0
\(786\) 0 0
\(787\) −22.2120 −0.791772 −0.395886 0.918300i \(-0.629562\pi\)
−0.395886 + 0.918300i \(0.629562\pi\)
\(788\) −0.327947 −0.0116826
\(789\) 0 0
\(790\) 0 0
\(791\) −17.9048 −0.636622
\(792\) 0 0
\(793\) −0.699623 −0.0248443
\(794\) 42.2984 1.50111
\(795\) 0 0
\(796\) 2.96431 0.105067
\(797\) −46.2386 −1.63785 −0.818927 0.573897i \(-0.805431\pi\)
−0.818927 + 0.573897i \(0.805431\pi\)
\(798\) 0 0
\(799\) 8.82875 0.312339
\(800\) 0 0
\(801\) 0 0
\(802\) −7.60171 −0.268426
\(803\) 0.321897 0.0113595
\(804\) 0 0
\(805\) 0 0
\(806\) 2.48174 0.0874154
\(807\) 0 0
\(808\) −13.7344 −0.483175
\(809\) −23.9257 −0.841184 −0.420592 0.907250i \(-0.638178\pi\)
−0.420592 + 0.907250i \(0.638178\pi\)
\(810\) 0 0
\(811\) −52.0210 −1.82670 −0.913352 0.407170i \(-0.866516\pi\)
−0.913352 + 0.407170i \(0.866516\pi\)
\(812\) −0.207747 −0.00729050
\(813\) 0 0
\(814\) 4.63451 0.162440
\(815\) 0 0
\(816\) 0 0
\(817\) 6.02823 0.210901
\(818\) 13.1188 0.458686
\(819\) 0 0
\(820\) 0 0
\(821\) −15.0444 −0.525052 −0.262526 0.964925i \(-0.584556\pi\)
−0.262526 + 0.964925i \(0.584556\pi\)
\(822\) 0 0
\(823\) 44.8652 1.56390 0.781952 0.623339i \(-0.214224\pi\)
0.781952 + 0.623339i \(0.214224\pi\)
\(824\) 45.4987 1.58502
\(825\) 0 0
\(826\) −7.67198 −0.266942
\(827\) −41.5569 −1.44508 −0.722538 0.691331i \(-0.757025\pi\)
−0.722538 + 0.691331i \(0.757025\pi\)
\(828\) 0 0
\(829\) −25.3645 −0.880946 −0.440473 0.897766i \(-0.645189\pi\)
−0.440473 + 0.897766i \(0.645189\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 3.48286 0.120746
\(833\) 16.0092 0.554684
\(834\) 0 0
\(835\) 0 0
\(836\) −0.450949 −0.0155964
\(837\) 0 0
\(838\) −24.7702 −0.855672
\(839\) 3.34480 0.115475 0.0577376 0.998332i \(-0.481611\pi\)
0.0577376 + 0.998332i \(0.481611\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) −35.3035 −1.21664
\(843\) 0 0
\(844\) −3.02417 −0.104096
\(845\) 0 0
\(846\) 0 0
\(847\) −13.6804 −0.470063
\(848\) 28.7012 0.985603
\(849\) 0 0
\(850\) 0 0
\(851\) −22.0791 −0.756862
\(852\) 0 0
\(853\) −45.8873 −1.57115 −0.785575 0.618766i \(-0.787633\pi\)
−0.785575 + 0.618766i \(0.787633\pi\)
\(854\) 2.95269 0.101039
\(855\) 0 0
\(856\) −1.98234 −0.0677550
\(857\) 51.8755 1.77203 0.886017 0.463652i \(-0.153461\pi\)
0.886017 + 0.463652i \(0.153461\pi\)
\(858\) 0 0
\(859\) 31.7698 1.08397 0.541987 0.840387i \(-0.317672\pi\)
0.541987 + 0.840387i \(0.317672\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 8.39203 0.285834
\(863\) 39.9472 1.35982 0.679909 0.733296i \(-0.262019\pi\)
0.679909 + 0.733296i \(0.262019\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 47.6380 1.61880
\(867\) 0 0
\(868\) 0.933222 0.0316756
\(869\) 0.778438 0.0264067
\(870\) 0 0
\(871\) 6.43139 0.217920
\(872\) −21.8355 −0.739444
\(873\) 0 0
\(874\) −24.1118 −0.815594
\(875\) 0 0
\(876\) 0 0
\(877\) 44.8875 1.51574 0.757872 0.652403i \(-0.226239\pi\)
0.757872 + 0.652403i \(0.226239\pi\)
\(878\) 27.2406 0.919326
\(879\) 0 0
\(880\) 0 0
\(881\) 3.90189 0.131458 0.0657290 0.997838i \(-0.479063\pi\)
0.0657290 + 0.997838i \(0.479063\pi\)
\(882\) 0 0
\(883\) −52.7256 −1.77436 −0.887178 0.461427i \(-0.847338\pi\)
−0.887178 + 0.461427i \(0.847338\pi\)
\(884\) 0.198204 0.00666632
\(885\) 0 0
\(886\) 28.2563 0.949288
\(887\) 20.7833 0.697834 0.348917 0.937154i \(-0.386550\pi\)
0.348917 + 0.937154i \(0.386550\pi\)
\(888\) 0 0
\(889\) −14.6660 −0.491881
\(890\) 0 0
\(891\) 0 0
\(892\) −3.53790 −0.118458
\(893\) −17.2482 −0.577190
\(894\) 0 0
\(895\) 0 0
\(896\) −12.3547 −0.412743
\(897\) 0 0
\(898\) 53.5546 1.78714
\(899\) −4.49210 −0.149820
\(900\) 0 0
\(901\) 23.3902 0.779241
\(902\) 7.76368 0.258502
\(903\) 0 0
\(904\) −41.3461 −1.37515
\(905\) 0 0
\(906\) 0 0
\(907\) 26.2116 0.870342 0.435171 0.900348i \(-0.356688\pi\)
0.435171 + 0.900348i \(0.356688\pi\)
\(908\) 3.20076 0.106221
\(909\) 0 0
\(910\) 0 0
\(911\) −31.4264 −1.04120 −0.520601 0.853800i \(-0.674292\pi\)
−0.520601 + 0.853800i \(0.674292\pi\)
\(912\) 0 0
\(913\) 7.37921 0.244216
\(914\) 34.7070 1.14800
\(915\) 0 0
\(916\) −1.76348 −0.0582669
\(917\) −13.0057 −0.429486
\(918\) 0 0
\(919\) −21.0083 −0.693000 −0.346500 0.938050i \(-0.612630\pi\)
−0.346500 + 0.938050i \(0.612630\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −1.23544 −0.0406870
\(923\) −3.67245 −0.120880
\(924\) 0 0
\(925\) 0 0
\(926\) 35.1988 1.15670
\(927\) 0 0
\(928\) −0.923188 −0.0303051
\(929\) −37.3920 −1.22679 −0.613395 0.789776i \(-0.710197\pi\)
−0.613395 + 0.789776i \(0.710197\pi\)
\(930\) 0 0
\(931\) −31.2762 −1.02504
\(932\) −0.695750 −0.0227901
\(933\) 0 0
\(934\) −21.4796 −0.702836
\(935\) 0 0
\(936\) 0 0
\(937\) 13.8377 0.452057 0.226029 0.974121i \(-0.427426\pi\)
0.226029 + 0.974121i \(0.427426\pi\)
\(938\) −27.1431 −0.886253
\(939\) 0 0
\(940\) 0 0
\(941\) −14.5708 −0.474995 −0.237497 0.971388i \(-0.576327\pi\)
−0.237497 + 0.971388i \(0.576327\pi\)
\(942\) 0 0
\(943\) −36.9867 −1.20445
\(944\) −16.2571 −0.529125
\(945\) 0 0
\(946\) −0.668144 −0.0217232
\(947\) 43.4267 1.41118 0.705589 0.708622i \(-0.250683\pi\)
0.705589 + 0.708622i \(0.250683\pi\)
\(948\) 0 0
\(949\) 0.276406 0.00897252
\(950\) 0 0
\(951\) 0 0
\(952\) −11.0614 −0.358501
\(953\) 51.0500 1.65367 0.826836 0.562442i \(-0.190138\pi\)
0.826836 + 0.562442i \(0.190138\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 2.75512 0.0891071
\(957\) 0 0
\(958\) 22.1004 0.714030
\(959\) −2.62289 −0.0846976
\(960\) 0 0
\(961\) −10.8210 −0.349065
\(962\) 3.97955 0.128306
\(963\) 0 0
\(964\) −3.57340 −0.115091
\(965\) 0 0
\(966\) 0 0
\(967\) −56.0658 −1.80295 −0.901477 0.432828i \(-0.857516\pi\)
−0.901477 + 0.432828i \(0.857516\pi\)
\(968\) −31.5910 −1.01537
\(969\) 0 0
\(970\) 0 0
\(971\) 6.58898 0.211450 0.105725 0.994395i \(-0.466284\pi\)
0.105725 + 0.994395i \(0.466284\pi\)
\(972\) 0 0
\(973\) 4.42152 0.141748
\(974\) 13.4693 0.431585
\(975\) 0 0
\(976\) 6.25683 0.200276
\(977\) 29.3327 0.938437 0.469218 0.883082i \(-0.344536\pi\)
0.469218 + 0.883082i \(0.344536\pi\)
\(978\) 0 0
\(979\) −6.37039 −0.203599
\(980\) 0 0
\(981\) 0 0
\(982\) 51.5814 1.64603
\(983\) 13.6719 0.436064 0.218032 0.975942i \(-0.430036\pi\)
0.218032 + 0.975942i \(0.430036\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 4.02653 0.128231
\(987\) 0 0
\(988\) −0.387220 −0.0123191
\(989\) 3.18308 0.101216
\(990\) 0 0
\(991\) 19.7210 0.626459 0.313230 0.949677i \(-0.398589\pi\)
0.313230 + 0.949677i \(0.398589\pi\)
\(992\) 4.14705 0.131669
\(993\) 0 0
\(994\) 15.4992 0.491606
\(995\) 0 0
\(996\) 0 0
\(997\) −53.4618 −1.69315 −0.846576 0.532268i \(-0.821340\pi\)
−0.846576 + 0.532268i \(0.821340\pi\)
\(998\) −60.1400 −1.90370
\(999\) 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 6525.2.a.ce.1.4 12
3.2 odd 2 6525.2.a.cf.1.9 12
5.2 odd 4 1305.2.c.k.784.4 12
5.3 odd 4 1305.2.c.k.784.9 yes 12
5.4 even 2 inner 6525.2.a.ce.1.9 12
15.2 even 4 1305.2.c.l.784.9 yes 12
15.8 even 4 1305.2.c.l.784.4 yes 12
15.14 odd 2 6525.2.a.cf.1.4 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1305.2.c.k.784.4 12 5.2 odd 4
1305.2.c.k.784.9 yes 12 5.3 odd 4
1305.2.c.l.784.4 yes 12 15.8 even 4
1305.2.c.l.784.9 yes 12 15.2 even 4
6525.2.a.ce.1.4 12 1.1 even 1 trivial
6525.2.a.ce.1.9 12 5.4 even 2 inner
6525.2.a.cf.1.4 12 15.14 odd 2
6525.2.a.cf.1.9 12 3.2 odd 2