Properties

Label 6525.2.a.cf.1.8
Level $6525$
Weight $2$
Character 6525.1
Self dual yes
Analytic conductor $52.102$
Analytic rank $0$
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: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 20x^{10} + 148x^{8} - 502x^{6} + 792x^{4} - 496x^{2} + 45 \) 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 1.8
Root \(1.27263\) 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)
 
\(f(q)\) \(=\) \(q+1.27263 q^{2} -0.380419 q^{4} +0.255813 q^{7} -3.02939 q^{8} -4.63446 q^{11} +5.02700 q^{13} +0.325555 q^{14} -3.09444 q^{16} -0.336444 q^{17} -2.91437 q^{19} -5.89795 q^{22} +8.65656 q^{23} +6.39750 q^{26} -0.0973162 q^{28} -1.00000 q^{29} -3.26943 q^{31} +2.12070 q^{32} -0.428167 q^{34} -3.86954 q^{37} -3.70891 q^{38} -5.71649 q^{41} +6.98619 q^{43} +1.76304 q^{44} +11.0166 q^{46} -0.336444 q^{47} -6.93456 q^{49} -1.91237 q^{52} +6.01484 q^{53} -0.774958 q^{56} -1.27263 q^{58} +13.2799 q^{59} -7.77792 q^{61} -4.16077 q^{62} +8.88775 q^{64} -11.2605 q^{67} +0.127989 q^{68} +13.9668 q^{71} +8.46426 q^{73} -4.92448 q^{74} +1.10868 q^{76} -1.18556 q^{77} -15.3102 q^{79} -7.27496 q^{82} +7.60521 q^{83} +8.89082 q^{86} +14.0396 q^{88} +13.0383 q^{89} +1.28597 q^{91} -3.29312 q^{92} -0.428167 q^{94} +11.5445 q^{97} -8.82511 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 16 q^{4} + 12 q^{11} + 16 q^{14} + 16 q^{16} - 20 q^{19} + 56 q^{26} - 12 q^{29} - 16 q^{31} + 4 q^{34} + 32 q^{41} + 68 q^{44} + 20 q^{46} - 4 q^{49} + 76 q^{56} + 44 q^{59} - 52 q^{61} + 36 q^{64}+ \cdots + 4 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.27263 0.899884 0.449942 0.893058i \(-0.351445\pi\)
0.449942 + 0.893058i \(0.351445\pi\)
\(3\) 0 0
\(4\) −0.380419 −0.190209
\(5\) 0 0
\(6\) 0 0
\(7\) 0.255813 0.0966884 0.0483442 0.998831i \(-0.484606\pi\)
0.0483442 + 0.998831i \(0.484606\pi\)
\(8\) −3.02939 −1.07105
\(9\) 0 0
\(10\) 0 0
\(11\) −4.63446 −1.39734 −0.698672 0.715442i \(-0.746225\pi\)
−0.698672 + 0.715442i \(0.746225\pi\)
\(12\) 0 0
\(13\) 5.02700 1.39424 0.697120 0.716955i \(-0.254465\pi\)
0.697120 + 0.716955i \(0.254465\pi\)
\(14\) 0.325555 0.0870083
\(15\) 0 0
\(16\) −3.09444 −0.773611
\(17\) −0.336444 −0.0815995 −0.0407998 0.999167i \(-0.512991\pi\)
−0.0407998 + 0.999167i \(0.512991\pi\)
\(18\) 0 0
\(19\) −2.91437 −0.668602 −0.334301 0.942466i \(-0.608500\pi\)
−0.334301 + 0.942466i \(0.608500\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −5.89795 −1.25745
\(23\) 8.65656 1.80502 0.902509 0.430671i \(-0.141723\pi\)
0.902509 + 0.430671i \(0.141723\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 6.39750 1.25465
\(27\) 0 0
\(28\) −0.0973162 −0.0183910
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) −3.26943 −0.587207 −0.293603 0.955927i \(-0.594854\pi\)
−0.293603 + 0.955927i \(0.594854\pi\)
\(32\) 2.12070 0.374890
\(33\) 0 0
\(34\) −0.428167 −0.0734301
\(35\) 0 0
\(36\) 0 0
\(37\) −3.86954 −0.636148 −0.318074 0.948066i \(-0.603036\pi\)
−0.318074 + 0.948066i \(0.603036\pi\)
\(38\) −3.70891 −0.601664
\(39\) 0 0
\(40\) 0 0
\(41\) −5.71649 −0.892765 −0.446383 0.894842i \(-0.647288\pi\)
−0.446383 + 0.894842i \(0.647288\pi\)
\(42\) 0 0
\(43\) 6.98619 1.06538 0.532692 0.846309i \(-0.321180\pi\)
0.532692 + 0.846309i \(0.321180\pi\)
\(44\) 1.76304 0.265788
\(45\) 0 0
\(46\) 11.0166 1.62431
\(47\) −0.336444 −0.0490753 −0.0245377 0.999699i \(-0.507811\pi\)
−0.0245377 + 0.999699i \(0.507811\pi\)
\(48\) 0 0
\(49\) −6.93456 −0.990651
\(50\) 0 0
\(51\) 0 0
\(52\) −1.91237 −0.265197
\(53\) 6.01484 0.826202 0.413101 0.910685i \(-0.364446\pi\)
0.413101 + 0.910685i \(0.364446\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −0.774958 −0.103558
\(57\) 0 0
\(58\) −1.27263 −0.167104
\(59\) 13.2799 1.72890 0.864451 0.502718i \(-0.167666\pi\)
0.864451 + 0.502718i \(0.167666\pi\)
\(60\) 0 0
\(61\) −7.77792 −0.995861 −0.497930 0.867217i \(-0.665906\pi\)
−0.497930 + 0.867217i \(0.665906\pi\)
\(62\) −4.16077 −0.528418
\(63\) 0 0
\(64\) 8.88775 1.11097
\(65\) 0 0
\(66\) 0 0
\(67\) −11.2605 −1.37569 −0.687847 0.725856i \(-0.741444\pi\)
−0.687847 + 0.725856i \(0.741444\pi\)
\(68\) 0.127989 0.0155210
\(69\) 0 0
\(70\) 0 0
\(71\) 13.9668 1.65756 0.828778 0.559578i \(-0.189037\pi\)
0.828778 + 0.559578i \(0.189037\pi\)
\(72\) 0 0
\(73\) 8.46426 0.990667 0.495333 0.868703i \(-0.335046\pi\)
0.495333 + 0.868703i \(0.335046\pi\)
\(74\) −4.92448 −0.572459
\(75\) 0 0
\(76\) 1.10868 0.127174
\(77\) −1.18556 −0.135107
\(78\) 0 0
\(79\) −15.3102 −1.72253 −0.861267 0.508153i \(-0.830328\pi\)
−0.861267 + 0.508153i \(0.830328\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −7.27496 −0.803385
\(83\) 7.60521 0.834781 0.417390 0.908727i \(-0.362945\pi\)
0.417390 + 0.908727i \(0.362945\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 8.89082 0.958722
\(87\) 0 0
\(88\) 14.0396 1.49662
\(89\) 13.0383 1.38206 0.691029 0.722827i \(-0.257157\pi\)
0.691029 + 0.722827i \(0.257157\pi\)
\(90\) 0 0
\(91\) 1.28597 0.134807
\(92\) −3.29312 −0.343331
\(93\) 0 0
\(94\) −0.428167 −0.0441621
\(95\) 0 0
\(96\) 0 0
\(97\) 11.5445 1.17216 0.586081 0.810253i \(-0.300670\pi\)
0.586081 + 0.810253i \(0.300670\pi\)
\(98\) −8.82511 −0.891471
\(99\) 0 0
\(100\) 0 0
\(101\) 8.19832 0.815763 0.407882 0.913035i \(-0.366268\pi\)
0.407882 + 0.913035i \(0.366268\pi\)
\(102\) 0 0
\(103\) −1.73014 −0.170476 −0.0852380 0.996361i \(-0.527165\pi\)
−0.0852380 + 0.996361i \(0.527165\pi\)
\(104\) −15.2287 −1.49330
\(105\) 0 0
\(106\) 7.65465 0.743486
\(107\) 5.38921 0.520995 0.260497 0.965475i \(-0.416113\pi\)
0.260497 + 0.965475i \(0.416113\pi\)
\(108\) 0 0
\(109\) 14.2391 1.36386 0.681929 0.731418i \(-0.261141\pi\)
0.681929 + 0.731418i \(0.261141\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −0.791600 −0.0747992
\(113\) −9.89466 −0.930811 −0.465406 0.885098i \(-0.654091\pi\)
−0.465406 + 0.885098i \(0.654091\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0.380419 0.0353210
\(117\) 0 0
\(118\) 16.9004 1.55581
\(119\) −0.0860668 −0.00788973
\(120\) 0 0
\(121\) 10.4783 0.952569
\(122\) −9.89840 −0.896159
\(123\) 0 0
\(124\) 1.24375 0.111692
\(125\) 0 0
\(126\) 0 0
\(127\) 12.2226 1.08458 0.542291 0.840191i \(-0.317557\pi\)
0.542291 + 0.840191i \(0.317557\pi\)
\(128\) 7.06940 0.624852
\(129\) 0 0
\(130\) 0 0
\(131\) 12.0464 1.05250 0.526250 0.850330i \(-0.323598\pi\)
0.526250 + 0.850330i \(0.323598\pi\)
\(132\) 0 0
\(133\) −0.745535 −0.0646461
\(134\) −14.3305 −1.23796
\(135\) 0 0
\(136\) 1.01922 0.0873972
\(137\) −5.47512 −0.467771 −0.233886 0.972264i \(-0.575144\pi\)
−0.233886 + 0.972264i \(0.575144\pi\)
\(138\) 0 0
\(139\) 2.71403 0.230201 0.115100 0.993354i \(-0.463281\pi\)
0.115100 + 0.993354i \(0.463281\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 17.7745 1.49161
\(143\) −23.2975 −1.94823
\(144\) 0 0
\(145\) 0 0
\(146\) 10.7719 0.891485
\(147\) 0 0
\(148\) 1.47204 0.121001
\(149\) 3.70395 0.303439 0.151720 0.988424i \(-0.451519\pi\)
0.151720 + 0.988424i \(0.451519\pi\)
\(150\) 0 0
\(151\) 15.3481 1.24901 0.624505 0.781021i \(-0.285301\pi\)
0.624505 + 0.781021i \(0.285301\pi\)
\(152\) 8.82875 0.716107
\(153\) 0 0
\(154\) −1.50877 −0.121580
\(155\) 0 0
\(156\) 0 0
\(157\) 21.8231 1.74168 0.870838 0.491571i \(-0.163577\pi\)
0.870838 + 0.491571i \(0.163577\pi\)
\(158\) −19.4842 −1.55008
\(159\) 0 0
\(160\) 0 0
\(161\) 2.21447 0.174524
\(162\) 0 0
\(163\) −3.08019 −0.241259 −0.120630 0.992698i \(-0.538491\pi\)
−0.120630 + 0.992698i \(0.538491\pi\)
\(164\) 2.17466 0.169812
\(165\) 0 0
\(166\) 9.67861 0.751206
\(167\) 8.55017 0.661632 0.330816 0.943695i \(-0.392676\pi\)
0.330816 + 0.943695i \(0.392676\pi\)
\(168\) 0 0
\(169\) 12.2707 0.943903
\(170\) 0 0
\(171\) 0 0
\(172\) −2.65768 −0.202646
\(173\) 5.76340 0.438183 0.219092 0.975704i \(-0.429691\pi\)
0.219092 + 0.975704i \(0.429691\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 14.3411 1.08100
\(177\) 0 0
\(178\) 16.5929 1.24369
\(179\) −14.1430 −1.05710 −0.528548 0.848903i \(-0.677264\pi\)
−0.528548 + 0.848903i \(0.677264\pi\)
\(180\) 0 0
\(181\) −15.0322 −1.11734 −0.558668 0.829391i \(-0.688688\pi\)
−0.558668 + 0.829391i \(0.688688\pi\)
\(182\) 1.63657 0.121310
\(183\) 0 0
\(184\) −26.2241 −1.93326
\(185\) 0 0
\(186\) 0 0
\(187\) 1.55924 0.114023
\(188\) 0.127989 0.00933459
\(189\) 0 0
\(190\) 0 0
\(191\) 12.2634 0.887346 0.443673 0.896189i \(-0.353675\pi\)
0.443673 + 0.896189i \(0.353675\pi\)
\(192\) 0 0
\(193\) 8.34203 0.600472 0.300236 0.953865i \(-0.402935\pi\)
0.300236 + 0.953865i \(0.402935\pi\)
\(194\) 14.6918 1.05481
\(195\) 0 0
\(196\) 2.63804 0.188431
\(197\) 6.69263 0.476830 0.238415 0.971163i \(-0.423372\pi\)
0.238415 + 0.971163i \(0.423372\pi\)
\(198\) 0 0
\(199\) 12.5583 0.890235 0.445117 0.895472i \(-0.353162\pi\)
0.445117 + 0.895472i \(0.353162\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 10.4334 0.734092
\(203\) −0.255813 −0.0179546
\(204\) 0 0
\(205\) 0 0
\(206\) −2.20183 −0.153409
\(207\) 0 0
\(208\) −15.5558 −1.07860
\(209\) 13.5065 0.934267
\(210\) 0 0
\(211\) −13.6479 −0.939563 −0.469782 0.882783i \(-0.655667\pi\)
−0.469782 + 0.882783i \(0.655667\pi\)
\(212\) −2.28816 −0.157151
\(213\) 0 0
\(214\) 6.85846 0.468835
\(215\) 0 0
\(216\) 0 0
\(217\) −0.836364 −0.0567761
\(218\) 18.1211 1.22731
\(219\) 0 0
\(220\) 0 0
\(221\) −1.69130 −0.113769
\(222\) 0 0
\(223\) 13.4721 0.902158 0.451079 0.892484i \(-0.351039\pi\)
0.451079 + 0.892484i \(0.351039\pi\)
\(224\) 0.542503 0.0362475
\(225\) 0 0
\(226\) −12.5922 −0.837622
\(227\) −1.54298 −0.102411 −0.0512057 0.998688i \(-0.516306\pi\)
−0.0512057 + 0.998688i \(0.516306\pi\)
\(228\) 0 0
\(229\) 7.89526 0.521734 0.260867 0.965375i \(-0.415992\pi\)
0.260867 + 0.965375i \(0.415992\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 3.02939 0.198889
\(233\) 28.4782 1.86567 0.932836 0.360301i \(-0.117326\pi\)
0.932836 + 0.360301i \(0.117326\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −5.05194 −0.328853
\(237\) 0 0
\(238\) −0.109531 −0.00709984
\(239\) −8.88508 −0.574728 −0.287364 0.957821i \(-0.592779\pi\)
−0.287364 + 0.957821i \(0.592779\pi\)
\(240\) 0 0
\(241\) −29.8901 −1.92539 −0.962695 0.270589i \(-0.912782\pi\)
−0.962695 + 0.270589i \(0.912782\pi\)
\(242\) 13.3349 0.857202
\(243\) 0 0
\(244\) 2.95887 0.189422
\(245\) 0 0
\(246\) 0 0
\(247\) −14.6505 −0.932192
\(248\) 9.90437 0.628928
\(249\) 0 0
\(250\) 0 0
\(251\) 15.9551 1.00708 0.503539 0.863973i \(-0.332031\pi\)
0.503539 + 0.863973i \(0.332031\pi\)
\(252\) 0 0
\(253\) −40.1185 −2.52223
\(254\) 15.5548 0.975998
\(255\) 0 0
\(256\) −8.77878 −0.548674
\(257\) −17.1593 −1.07037 −0.535185 0.844735i \(-0.679758\pi\)
−0.535185 + 0.844735i \(0.679758\pi\)
\(258\) 0 0
\(259\) −0.989880 −0.0615081
\(260\) 0 0
\(261\) 0 0
\(262\) 15.3306 0.947128
\(263\) 1.72781 0.106542 0.0532708 0.998580i \(-0.483035\pi\)
0.0532708 + 0.998580i \(0.483035\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −0.948789 −0.0581740
\(267\) 0 0
\(268\) 4.28372 0.261670
\(269\) 4.97931 0.303594 0.151797 0.988412i \(-0.451494\pi\)
0.151797 + 0.988412i \(0.451494\pi\)
\(270\) 0 0
\(271\) 13.0881 0.795047 0.397524 0.917592i \(-0.369870\pi\)
0.397524 + 0.917592i \(0.369870\pi\)
\(272\) 1.04111 0.0631263
\(273\) 0 0
\(274\) −6.96779 −0.420939
\(275\) 0 0
\(276\) 0 0
\(277\) 20.8147 1.25063 0.625316 0.780371i \(-0.284970\pi\)
0.625316 + 0.780371i \(0.284970\pi\)
\(278\) 3.45394 0.207154
\(279\) 0 0
\(280\) 0 0
\(281\) −3.93338 −0.234646 −0.117323 0.993094i \(-0.537431\pi\)
−0.117323 + 0.993094i \(0.537431\pi\)
\(282\) 0 0
\(283\) −5.47745 −0.325600 −0.162800 0.986659i \(-0.552053\pi\)
−0.162800 + 0.986659i \(0.552053\pi\)
\(284\) −5.31324 −0.315282
\(285\) 0 0
\(286\) −29.6490 −1.75318
\(287\) −1.46235 −0.0863201
\(288\) 0 0
\(289\) −16.8868 −0.993342
\(290\) 0 0
\(291\) 0 0
\(292\) −3.21996 −0.188434
\(293\) −11.9974 −0.700893 −0.350446 0.936583i \(-0.613970\pi\)
−0.350446 + 0.936583i \(0.613970\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 11.7223 0.681347
\(297\) 0 0
\(298\) 4.71375 0.273060
\(299\) 43.5166 2.51663
\(300\) 0 0
\(301\) 1.78716 0.103010
\(302\) 19.5324 1.12396
\(303\) 0 0
\(304\) 9.01836 0.517238
\(305\) 0 0
\(306\) 0 0
\(307\) −26.4807 −1.51133 −0.755666 0.654957i \(-0.772687\pi\)
−0.755666 + 0.654957i \(0.772687\pi\)
\(308\) 0.451009 0.0256986
\(309\) 0 0
\(310\) 0 0
\(311\) 1.71530 0.0972660 0.0486330 0.998817i \(-0.484514\pi\)
0.0486330 + 0.998817i \(0.484514\pi\)
\(312\) 0 0
\(313\) −33.3782 −1.88665 −0.943325 0.331871i \(-0.892320\pi\)
−0.943325 + 0.331871i \(0.892320\pi\)
\(314\) 27.7727 1.56731
\(315\) 0 0
\(316\) 5.82429 0.327642
\(317\) −17.9712 −1.00936 −0.504682 0.863305i \(-0.668390\pi\)
−0.504682 + 0.863305i \(0.668390\pi\)
\(318\) 0 0
\(319\) 4.63446 0.259480
\(320\) 0 0
\(321\) 0 0
\(322\) 2.81819 0.157052
\(323\) 0.980521 0.0545577
\(324\) 0 0
\(325\) 0 0
\(326\) −3.91994 −0.217105
\(327\) 0 0
\(328\) 17.3175 0.956196
\(329\) −0.0860668 −0.00474501
\(330\) 0 0
\(331\) 6.49393 0.356939 0.178469 0.983945i \(-0.442885\pi\)
0.178469 + 0.983945i \(0.442885\pi\)
\(332\) −2.89317 −0.158783
\(333\) 0 0
\(334\) 10.8812 0.595392
\(335\) 0 0
\(336\) 0 0
\(337\) 6.36156 0.346536 0.173268 0.984875i \(-0.444567\pi\)
0.173268 + 0.984875i \(0.444567\pi\)
\(338\) 15.6161 0.849403
\(339\) 0 0
\(340\) 0 0
\(341\) 15.1521 0.820530
\(342\) 0 0
\(343\) −3.56465 −0.192473
\(344\) −21.1639 −1.14108
\(345\) 0 0
\(346\) 7.33466 0.394314
\(347\) 21.6036 1.15974 0.579871 0.814708i \(-0.303103\pi\)
0.579871 + 0.814708i \(0.303103\pi\)
\(348\) 0 0
\(349\) −4.12603 −0.220861 −0.110431 0.993884i \(-0.535223\pi\)
−0.110431 + 0.993884i \(0.535223\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −9.82830 −0.523850
\(353\) 7.07724 0.376684 0.188342 0.982104i \(-0.439689\pi\)
0.188342 + 0.982104i \(0.439689\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −4.96002 −0.262881
\(357\) 0 0
\(358\) −17.9988 −0.951264
\(359\) 14.4359 0.761900 0.380950 0.924596i \(-0.375597\pi\)
0.380950 + 0.924596i \(0.375597\pi\)
\(360\) 0 0
\(361\) −10.5064 −0.552971
\(362\) −19.1304 −1.00547
\(363\) 0 0
\(364\) −0.489209 −0.0256415
\(365\) 0 0
\(366\) 0 0
\(367\) 15.4145 0.804630 0.402315 0.915501i \(-0.368206\pi\)
0.402315 + 0.915501i \(0.368206\pi\)
\(368\) −26.7873 −1.39638
\(369\) 0 0
\(370\) 0 0
\(371\) 1.53868 0.0798841
\(372\) 0 0
\(373\) 29.8574 1.54596 0.772979 0.634432i \(-0.218766\pi\)
0.772979 + 0.634432i \(0.218766\pi\)
\(374\) 1.98433 0.102607
\(375\) 0 0
\(376\) 1.01922 0.0525621
\(377\) −5.02700 −0.258904
\(378\) 0 0
\(379\) 37.3451 1.91829 0.959144 0.282920i \(-0.0913030\pi\)
0.959144 + 0.282920i \(0.0913030\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 15.6067 0.798508
\(383\) −25.8159 −1.31913 −0.659566 0.751647i \(-0.729260\pi\)
−0.659566 + 0.751647i \(0.729260\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 10.6163 0.540355
\(387\) 0 0
\(388\) −4.39173 −0.222956
\(389\) −20.6935 −1.04920 −0.524600 0.851349i \(-0.675785\pi\)
−0.524600 + 0.851349i \(0.675785\pi\)
\(390\) 0 0
\(391\) −2.91245 −0.147289
\(392\) 21.0075 1.06104
\(393\) 0 0
\(394\) 8.51723 0.429092
\(395\) 0 0
\(396\) 0 0
\(397\) 2.38045 0.119471 0.0597357 0.998214i \(-0.480974\pi\)
0.0597357 + 0.998214i \(0.480974\pi\)
\(398\) 15.9820 0.801108
\(399\) 0 0
\(400\) 0 0
\(401\) 32.0936 1.60268 0.801339 0.598211i \(-0.204121\pi\)
0.801339 + 0.598211i \(0.204121\pi\)
\(402\) 0 0
\(403\) −16.4354 −0.818707
\(404\) −3.11879 −0.155166
\(405\) 0 0
\(406\) −0.325555 −0.0161570
\(407\) 17.9332 0.888918
\(408\) 0 0
\(409\) −31.5011 −1.55763 −0.778815 0.627253i \(-0.784179\pi\)
−0.778815 + 0.627253i \(0.784179\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0.658179 0.0324261
\(413\) 3.39719 0.167165
\(414\) 0 0
\(415\) 0 0
\(416\) 10.6608 0.522686
\(417\) 0 0
\(418\) 17.1888 0.840732
\(419\) 3.15481 0.154123 0.0770613 0.997026i \(-0.475446\pi\)
0.0770613 + 0.997026i \(0.475446\pi\)
\(420\) 0 0
\(421\) −11.9631 −0.583048 −0.291524 0.956564i \(-0.594162\pi\)
−0.291524 + 0.956564i \(0.594162\pi\)
\(422\) −17.3687 −0.845497
\(423\) 0 0
\(424\) −18.2213 −0.884904
\(425\) 0 0
\(426\) 0 0
\(427\) −1.98970 −0.0962882
\(428\) −2.05016 −0.0990981
\(429\) 0 0
\(430\) 0 0
\(431\) 31.2658 1.50602 0.753011 0.658008i \(-0.228601\pi\)
0.753011 + 0.658008i \(0.228601\pi\)
\(432\) 0 0
\(433\) −30.6459 −1.47275 −0.736373 0.676575i \(-0.763463\pi\)
−0.736373 + 0.676575i \(0.763463\pi\)
\(434\) −1.06438 −0.0510919
\(435\) 0 0
\(436\) −5.41682 −0.259419
\(437\) −25.2284 −1.20684
\(438\) 0 0
\(439\) 0.717425 0.0342408 0.0171204 0.999853i \(-0.494550\pi\)
0.0171204 + 0.999853i \(0.494550\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −2.15240 −0.102379
\(443\) −27.0649 −1.28589 −0.642947 0.765911i \(-0.722288\pi\)
−0.642947 + 0.765911i \(0.722288\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 17.1450 0.811838
\(447\) 0 0
\(448\) 2.27361 0.107418
\(449\) 30.3663 1.43307 0.716536 0.697550i \(-0.245727\pi\)
0.716536 + 0.697550i \(0.245727\pi\)
\(450\) 0 0
\(451\) 26.4929 1.24750
\(452\) 3.76411 0.177049
\(453\) 0 0
\(454\) −1.96364 −0.0921584
\(455\) 0 0
\(456\) 0 0
\(457\) −2.43167 −0.113749 −0.0568743 0.998381i \(-0.518113\pi\)
−0.0568743 + 0.998381i \(0.518113\pi\)
\(458\) 10.0477 0.469500
\(459\) 0 0
\(460\) 0 0
\(461\) −8.28439 −0.385842 −0.192921 0.981214i \(-0.561796\pi\)
−0.192921 + 0.981214i \(0.561796\pi\)
\(462\) 0 0
\(463\) −7.05654 −0.327945 −0.163973 0.986465i \(-0.552431\pi\)
−0.163973 + 0.986465i \(0.552431\pi\)
\(464\) 3.09444 0.143656
\(465\) 0 0
\(466\) 36.2422 1.67889
\(467\) −6.07165 −0.280963 −0.140481 0.990083i \(-0.544865\pi\)
−0.140481 + 0.990083i \(0.544865\pi\)
\(468\) 0 0
\(469\) −2.88060 −0.133014
\(470\) 0 0
\(471\) 0 0
\(472\) −40.2301 −1.85174
\(473\) −32.3773 −1.48871
\(474\) 0 0
\(475\) 0 0
\(476\) 0.0327414 0.00150070
\(477\) 0 0
\(478\) −11.3074 −0.517188
\(479\) 29.7675 1.36011 0.680055 0.733161i \(-0.261956\pi\)
0.680055 + 0.733161i \(0.261956\pi\)
\(480\) 0 0
\(481\) −19.4522 −0.886943
\(482\) −38.0390 −1.73263
\(483\) 0 0
\(484\) −3.98613 −0.181188
\(485\) 0 0
\(486\) 0 0
\(487\) −41.9050 −1.89890 −0.949448 0.313923i \(-0.898357\pi\)
−0.949448 + 0.313923i \(0.898357\pi\)
\(488\) 23.5623 1.06662
\(489\) 0 0
\(490\) 0 0
\(491\) 7.22435 0.326030 0.163015 0.986624i \(-0.447878\pi\)
0.163015 + 0.986624i \(0.447878\pi\)
\(492\) 0 0
\(493\) 0.336444 0.0151527
\(494\) −18.6447 −0.838864
\(495\) 0 0
\(496\) 10.1171 0.454270
\(497\) 3.57290 0.160266
\(498\) 0 0
\(499\) −15.5607 −0.696593 −0.348296 0.937384i \(-0.613240\pi\)
−0.348296 + 0.937384i \(0.613240\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 20.3049 0.906253
\(503\) 38.5096 1.71706 0.858529 0.512765i \(-0.171379\pi\)
0.858529 + 0.512765i \(0.171379\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −51.0560 −2.26971
\(507\) 0 0
\(508\) −4.64971 −0.206298
\(509\) −31.6646 −1.40351 −0.701754 0.712419i \(-0.747600\pi\)
−0.701754 + 0.712419i \(0.747600\pi\)
\(510\) 0 0
\(511\) 2.16527 0.0957860
\(512\) −25.3109 −1.11860
\(513\) 0 0
\(514\) −21.8375 −0.963209
\(515\) 0 0
\(516\) 0 0
\(517\) 1.55924 0.0685751
\(518\) −1.25975 −0.0553502
\(519\) 0 0
\(520\) 0 0
\(521\) −20.2023 −0.885079 −0.442539 0.896749i \(-0.645922\pi\)
−0.442539 + 0.896749i \(0.645922\pi\)
\(522\) 0 0
\(523\) 12.1052 0.529325 0.264663 0.964341i \(-0.414739\pi\)
0.264663 + 0.964341i \(0.414739\pi\)
\(524\) −4.58268 −0.200195
\(525\) 0 0
\(526\) 2.19886 0.0958751
\(527\) 1.09998 0.0479158
\(528\) 0 0
\(529\) 51.9361 2.25809
\(530\) 0 0
\(531\) 0 0
\(532\) 0.283615 0.0122963
\(533\) −28.7368 −1.24473
\(534\) 0 0
\(535\) 0 0
\(536\) 34.1125 1.47344
\(537\) 0 0
\(538\) 6.33681 0.273199
\(539\) 32.1380 1.38428
\(540\) 0 0
\(541\) −14.4279 −0.620304 −0.310152 0.950687i \(-0.600380\pi\)
−0.310152 + 0.950687i \(0.600380\pi\)
\(542\) 16.6563 0.715450
\(543\) 0 0
\(544\) −0.713495 −0.0305909
\(545\) 0 0
\(546\) 0 0
\(547\) −41.9178 −1.79228 −0.896139 0.443774i \(-0.853639\pi\)
−0.896139 + 0.443774i \(0.853639\pi\)
\(548\) 2.08284 0.0889744
\(549\) 0 0
\(550\) 0 0
\(551\) 2.91437 0.124156
\(552\) 0 0
\(553\) −3.91656 −0.166549
\(554\) 26.4893 1.12542
\(555\) 0 0
\(556\) −1.03247 −0.0437863
\(557\) 3.60324 0.152674 0.0763372 0.997082i \(-0.475677\pi\)
0.0763372 + 0.997082i \(0.475677\pi\)
\(558\) 0 0
\(559\) 35.1196 1.48540
\(560\) 0 0
\(561\) 0 0
\(562\) −5.00573 −0.211154
\(563\) 15.5829 0.656741 0.328370 0.944549i \(-0.393501\pi\)
0.328370 + 0.944549i \(0.393501\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −6.97075 −0.293003
\(567\) 0 0
\(568\) −42.3109 −1.77532
\(569\) −36.9718 −1.54994 −0.774970 0.631998i \(-0.782235\pi\)
−0.774970 + 0.631998i \(0.782235\pi\)
\(570\) 0 0
\(571\) 29.9692 1.25417 0.627086 0.778950i \(-0.284248\pi\)
0.627086 + 0.778950i \(0.284248\pi\)
\(572\) 8.86279 0.370572
\(573\) 0 0
\(574\) −1.86103 −0.0776780
\(575\) 0 0
\(576\) 0 0
\(577\) −7.12327 −0.296546 −0.148273 0.988946i \(-0.547371\pi\)
−0.148273 + 0.988946i \(0.547371\pi\)
\(578\) −21.4906 −0.893892
\(579\) 0 0
\(580\) 0 0
\(581\) 1.94552 0.0807136
\(582\) 0 0
\(583\) −27.8756 −1.15449
\(584\) −25.6415 −1.06105
\(585\) 0 0
\(586\) −15.2682 −0.630722
\(587\) −7.87237 −0.324928 −0.162464 0.986715i \(-0.551944\pi\)
−0.162464 + 0.986715i \(0.551944\pi\)
\(588\) 0 0
\(589\) 9.52833 0.392608
\(590\) 0 0
\(591\) 0 0
\(592\) 11.9741 0.492131
\(593\) −17.9188 −0.735836 −0.367918 0.929858i \(-0.619929\pi\)
−0.367918 + 0.929858i \(0.619929\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −1.40905 −0.0577170
\(597\) 0 0
\(598\) 55.3804 2.26467
\(599\) −20.1934 −0.825078 −0.412539 0.910940i \(-0.635358\pi\)
−0.412539 + 0.910940i \(0.635358\pi\)
\(600\) 0 0
\(601\) 23.2527 0.948496 0.474248 0.880391i \(-0.342720\pi\)
0.474248 + 0.880391i \(0.342720\pi\)
\(602\) 2.27439 0.0926973
\(603\) 0 0
\(604\) −5.83870 −0.237573
\(605\) 0 0
\(606\) 0 0
\(607\) −17.6177 −0.715082 −0.357541 0.933897i \(-0.616385\pi\)
−0.357541 + 0.933897i \(0.616385\pi\)
\(608\) −6.18050 −0.250652
\(609\) 0 0
\(610\) 0 0
\(611\) −1.69130 −0.0684228
\(612\) 0 0
\(613\) 13.4161 0.541873 0.270937 0.962597i \(-0.412667\pi\)
0.270937 + 0.962597i \(0.412667\pi\)
\(614\) −33.7001 −1.36002
\(615\) 0 0
\(616\) 3.59151 0.144706
\(617\) 7.99166 0.321732 0.160866 0.986976i \(-0.448571\pi\)
0.160866 + 0.986976i \(0.448571\pi\)
\(618\) 0 0
\(619\) −15.0770 −0.605996 −0.302998 0.952991i \(-0.597988\pi\)
−0.302998 + 0.952991i \(0.597988\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 2.18294 0.0875281
\(623\) 3.33538 0.133629
\(624\) 0 0
\(625\) 0 0
\(626\) −42.4781 −1.69776
\(627\) 0 0
\(628\) −8.30193 −0.331283
\(629\) 1.30188 0.0519094
\(630\) 0 0
\(631\) 10.5154 0.418611 0.209306 0.977850i \(-0.432880\pi\)
0.209306 + 0.977850i \(0.432880\pi\)
\(632\) 46.3805 1.84492
\(633\) 0 0
\(634\) −22.8707 −0.908311
\(635\) 0 0
\(636\) 0 0
\(637\) −34.8600 −1.38121
\(638\) 5.89795 0.233502
\(639\) 0 0
\(640\) 0 0
\(641\) 27.2479 1.07623 0.538113 0.842873i \(-0.319137\pi\)
0.538113 + 0.842873i \(0.319137\pi\)
\(642\) 0 0
\(643\) 36.3023 1.43162 0.715812 0.698293i \(-0.246057\pi\)
0.715812 + 0.698293i \(0.246057\pi\)
\(644\) −0.842424 −0.0331962
\(645\) 0 0
\(646\) 1.24784 0.0490955
\(647\) 8.77892 0.345135 0.172567 0.984998i \(-0.444794\pi\)
0.172567 + 0.984998i \(0.444794\pi\)
\(648\) 0 0
\(649\) −61.5454 −2.41587
\(650\) 0 0
\(651\) 0 0
\(652\) 1.17176 0.0458898
\(653\) 44.1466 1.72759 0.863794 0.503845i \(-0.168082\pi\)
0.863794 + 0.503845i \(0.168082\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 17.6894 0.690653
\(657\) 0 0
\(658\) −0.109531 −0.00426996
\(659\) −22.7653 −0.886810 −0.443405 0.896321i \(-0.646230\pi\)
−0.443405 + 0.896321i \(0.646230\pi\)
\(660\) 0 0
\(661\) 29.3474 1.14148 0.570740 0.821131i \(-0.306656\pi\)
0.570740 + 0.821131i \(0.306656\pi\)
\(662\) 8.26435 0.321203
\(663\) 0 0
\(664\) −23.0391 −0.894092
\(665\) 0 0
\(666\) 0 0
\(667\) −8.65656 −0.335183
\(668\) −3.25264 −0.125849
\(669\) 0 0
\(670\) 0 0
\(671\) 36.0465 1.39156
\(672\) 0 0
\(673\) −42.2267 −1.62772 −0.813860 0.581060i \(-0.802638\pi\)
−0.813860 + 0.581060i \(0.802638\pi\)
\(674\) 8.09590 0.311842
\(675\) 0 0
\(676\) −4.66802 −0.179539
\(677\) −19.9803 −0.767907 −0.383953 0.923353i \(-0.625438\pi\)
−0.383953 + 0.923353i \(0.625438\pi\)
\(678\) 0 0
\(679\) 2.95323 0.113334
\(680\) 0 0
\(681\) 0 0
\(682\) 19.2829 0.738381
\(683\) 23.0491 0.881950 0.440975 0.897519i \(-0.354633\pi\)
0.440975 + 0.897519i \(0.354633\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −4.53647 −0.173203
\(687\) 0 0
\(688\) −21.6184 −0.824193
\(689\) 30.2366 1.15192
\(690\) 0 0
\(691\) 4.01208 0.152627 0.0763134 0.997084i \(-0.475685\pi\)
0.0763134 + 0.997084i \(0.475685\pi\)
\(692\) −2.19250 −0.0833465
\(693\) 0 0
\(694\) 27.4933 1.04363
\(695\) 0 0
\(696\) 0 0
\(697\) 1.92328 0.0728493
\(698\) −5.25090 −0.198750
\(699\) 0 0
\(700\) 0 0
\(701\) −4.49887 −0.169920 −0.0849599 0.996384i \(-0.527076\pi\)
−0.0849599 + 0.996384i \(0.527076\pi\)
\(702\) 0 0
\(703\) 11.2773 0.425330
\(704\) −41.1900 −1.55240
\(705\) 0 0
\(706\) 9.00670 0.338971
\(707\) 2.09724 0.0788748
\(708\) 0 0
\(709\) 36.7777 1.38121 0.690607 0.723230i \(-0.257343\pi\)
0.690607 + 0.723230i \(0.257343\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −39.4981 −1.48025
\(713\) −28.3020 −1.05992
\(714\) 0 0
\(715\) 0 0
\(716\) 5.38026 0.201070
\(717\) 0 0
\(718\) 18.3716 0.685621
\(719\) 25.4279 0.948300 0.474150 0.880444i \(-0.342755\pi\)
0.474150 + 0.880444i \(0.342755\pi\)
\(720\) 0 0
\(721\) −0.442594 −0.0164831
\(722\) −13.3708 −0.497609
\(723\) 0 0
\(724\) 5.71854 0.212528
\(725\) 0 0
\(726\) 0 0
\(727\) 11.6265 0.431205 0.215602 0.976481i \(-0.430829\pi\)
0.215602 + 0.976481i \(0.430829\pi\)
\(728\) −3.89571 −0.144385
\(729\) 0 0
\(730\) 0 0
\(731\) −2.35046 −0.0869349
\(732\) 0 0
\(733\) 25.6926 0.948978 0.474489 0.880261i \(-0.342633\pi\)
0.474489 + 0.880261i \(0.342633\pi\)
\(734\) 19.6169 0.724073
\(735\) 0 0
\(736\) 18.3580 0.676683
\(737\) 52.1866 1.92232
\(738\) 0 0
\(739\) −33.3612 −1.22721 −0.613606 0.789612i \(-0.710282\pi\)
−0.613606 + 0.789612i \(0.710282\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 1.95816 0.0718864
\(743\) 25.6430 0.940750 0.470375 0.882467i \(-0.344119\pi\)
0.470375 + 0.882467i \(0.344119\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 37.9973 1.39118
\(747\) 0 0
\(748\) −0.593162 −0.0216882
\(749\) 1.37863 0.0503741
\(750\) 0 0
\(751\) 46.6411 1.70196 0.850979 0.525200i \(-0.176009\pi\)
0.850979 + 0.525200i \(0.176009\pi\)
\(752\) 1.04111 0.0379652
\(753\) 0 0
\(754\) −6.39750 −0.232983
\(755\) 0 0
\(756\) 0 0
\(757\) 2.69262 0.0978650 0.0489325 0.998802i \(-0.484418\pi\)
0.0489325 + 0.998802i \(0.484418\pi\)
\(758\) 47.5264 1.72624
\(759\) 0 0
\(760\) 0 0
\(761\) 0.471501 0.0170919 0.00854595 0.999963i \(-0.497280\pi\)
0.00854595 + 0.999963i \(0.497280\pi\)
\(762\) 0 0
\(763\) 3.64255 0.131869
\(764\) −4.66521 −0.168781
\(765\) 0 0
\(766\) −32.8540 −1.18706
\(767\) 66.7583 2.41050
\(768\) 0 0
\(769\) 9.04380 0.326128 0.163064 0.986616i \(-0.447862\pi\)
0.163064 + 0.986616i \(0.447862\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −3.17346 −0.114215
\(773\) −9.91726 −0.356699 −0.178350 0.983967i \(-0.557076\pi\)
−0.178350 + 0.983967i \(0.557076\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −34.9726 −1.25544
\(777\) 0 0
\(778\) −26.3351 −0.944159
\(779\) 16.6600 0.596905
\(780\) 0 0
\(781\) −64.7287 −2.31617
\(782\) −3.70646 −0.132543
\(783\) 0 0
\(784\) 21.4586 0.766379
\(785\) 0 0
\(786\) 0 0
\(787\) 8.53609 0.304279 0.152139 0.988359i \(-0.451384\pi\)
0.152139 + 0.988359i \(0.451384\pi\)
\(788\) −2.54600 −0.0906976
\(789\) 0 0
\(790\) 0 0
\(791\) −2.53119 −0.0899986
\(792\) 0 0
\(793\) −39.0996 −1.38847
\(794\) 3.02943 0.107510
\(795\) 0 0
\(796\) −4.77741 −0.169331
\(797\) −12.5563 −0.444768 −0.222384 0.974959i \(-0.571384\pi\)
−0.222384 + 0.974959i \(0.571384\pi\)
\(798\) 0 0
\(799\) 0.113194 0.00400453
\(800\) 0 0
\(801\) 0 0
\(802\) 40.8432 1.44222
\(803\) −39.2273 −1.38430
\(804\) 0 0
\(805\) 0 0
\(806\) −20.9162 −0.736741
\(807\) 0 0
\(808\) −24.8359 −0.873723
\(809\) −16.4082 −0.576883 −0.288441 0.957498i \(-0.593137\pi\)
−0.288441 + 0.957498i \(0.593137\pi\)
\(810\) 0 0
\(811\) 29.4688 1.03479 0.517394 0.855747i \(-0.326902\pi\)
0.517394 + 0.855747i \(0.326902\pi\)
\(812\) 0.0973162 0.00341513
\(813\) 0 0
\(814\) 22.8223 0.799923
\(815\) 0 0
\(816\) 0 0
\(817\) −20.3603 −0.712318
\(818\) −40.0892 −1.40169
\(819\) 0 0
\(820\) 0 0
\(821\) 28.5209 0.995385 0.497693 0.867353i \(-0.334181\pi\)
0.497693 + 0.867353i \(0.334181\pi\)
\(822\) 0 0
\(823\) 16.7546 0.584028 0.292014 0.956414i \(-0.405675\pi\)
0.292014 + 0.956414i \(0.405675\pi\)
\(824\) 5.24127 0.182588
\(825\) 0 0
\(826\) 4.32336 0.150429
\(827\) 3.38369 0.117662 0.0588312 0.998268i \(-0.481263\pi\)
0.0588312 + 0.998268i \(0.481263\pi\)
\(828\) 0 0
\(829\) −14.4434 −0.501639 −0.250819 0.968034i \(-0.580700\pi\)
−0.250819 + 0.968034i \(0.580700\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 44.6787 1.54896
\(833\) 2.33309 0.0808367
\(834\) 0 0
\(835\) 0 0
\(836\) −5.13814 −0.177706
\(837\) 0 0
\(838\) 4.01490 0.138692
\(839\) −27.6642 −0.955075 −0.477538 0.878611i \(-0.658471\pi\)
−0.477538 + 0.878611i \(0.658471\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) −15.2246 −0.524675
\(843\) 0 0
\(844\) 5.19193 0.178714
\(845\) 0 0
\(846\) 0 0
\(847\) 2.68048 0.0921024
\(848\) −18.6126 −0.639159
\(849\) 0 0
\(850\) 0 0
\(851\) −33.4969 −1.14826
\(852\) 0 0
\(853\) 4.26068 0.145883 0.0729415 0.997336i \(-0.476761\pi\)
0.0729415 + 0.997336i \(0.476761\pi\)
\(854\) −2.53214 −0.0866481
\(855\) 0 0
\(856\) −16.3260 −0.558011
\(857\) −28.7104 −0.980729 −0.490364 0.871518i \(-0.663136\pi\)
−0.490364 + 0.871518i \(0.663136\pi\)
\(858\) 0 0
\(859\) −39.3734 −1.34340 −0.671701 0.740822i \(-0.734436\pi\)
−0.671701 + 0.740822i \(0.734436\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 39.7898 1.35525
\(863\) −22.1235 −0.753094 −0.376547 0.926398i \(-0.622889\pi\)
−0.376547 + 0.926398i \(0.622889\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −39.0008 −1.32530
\(867\) 0 0
\(868\) 0.318168 0.0107993
\(869\) 70.9546 2.40697
\(870\) 0 0
\(871\) −56.6068 −1.91805
\(872\) −43.1357 −1.46076
\(873\) 0 0
\(874\) −32.1064 −1.08602
\(875\) 0 0
\(876\) 0 0
\(877\) −32.3439 −1.09217 −0.546087 0.837728i \(-0.683883\pi\)
−0.546087 + 0.837728i \(0.683883\pi\)
\(878\) 0.913015 0.0308128
\(879\) 0 0
\(880\) 0 0
\(881\) −54.6109 −1.83989 −0.919944 0.392051i \(-0.871766\pi\)
−0.919944 + 0.392051i \(0.871766\pi\)
\(882\) 0 0
\(883\) 47.5511 1.60022 0.800111 0.599852i \(-0.204774\pi\)
0.800111 + 0.599852i \(0.204774\pi\)
\(884\) 0.643403 0.0216400
\(885\) 0 0
\(886\) −34.4436 −1.15715
\(887\) −10.4630 −0.351312 −0.175656 0.984452i \(-0.556205\pi\)
−0.175656 + 0.984452i \(0.556205\pi\)
\(888\) 0 0
\(889\) 3.12671 0.104866
\(890\) 0 0
\(891\) 0 0
\(892\) −5.12504 −0.171599
\(893\) 0.980521 0.0328119
\(894\) 0 0
\(895\) 0 0
\(896\) 1.80845 0.0604160
\(897\) 0 0
\(898\) 38.6449 1.28960
\(899\) 3.26943 0.109042
\(900\) 0 0
\(901\) −2.02365 −0.0674177
\(902\) 33.7156 1.12261
\(903\) 0 0
\(904\) 29.9747 0.996945
\(905\) 0 0
\(906\) 0 0
\(907\) 10.7211 0.355989 0.177994 0.984032i \(-0.443039\pi\)
0.177994 + 0.984032i \(0.443039\pi\)
\(908\) 0.586980 0.0194796
\(909\) 0 0
\(910\) 0 0
\(911\) −3.57634 −0.118489 −0.0592447 0.998243i \(-0.518869\pi\)
−0.0592447 + 0.998243i \(0.518869\pi\)
\(912\) 0 0
\(913\) −35.2461 −1.16648
\(914\) −3.09461 −0.102361
\(915\) 0 0
\(916\) −3.00351 −0.0992386
\(917\) 3.08163 0.101765
\(918\) 0 0
\(919\) −22.2776 −0.734870 −0.367435 0.930049i \(-0.619764\pi\)
−0.367435 + 0.930049i \(0.619764\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −10.5429 −0.347213
\(923\) 70.2112 2.31103
\(924\) 0 0
\(925\) 0 0
\(926\) −8.98035 −0.295113
\(927\) 0 0
\(928\) −2.12070 −0.0696153
\(929\) −21.1779 −0.694826 −0.347413 0.937712i \(-0.612940\pi\)
−0.347413 + 0.937712i \(0.612940\pi\)
\(930\) 0 0
\(931\) 20.2099 0.662352
\(932\) −10.8337 −0.354868
\(933\) 0 0
\(934\) −7.72695 −0.252834
\(935\) 0 0
\(936\) 0 0
\(937\) −34.4462 −1.12531 −0.562654 0.826693i \(-0.690220\pi\)
−0.562654 + 0.826693i \(0.690220\pi\)
\(938\) −3.66593 −0.119697
\(939\) 0 0
\(940\) 0 0
\(941\) 0.296167 0.00965477 0.00482738 0.999988i \(-0.498463\pi\)
0.00482738 + 0.999988i \(0.498463\pi\)
\(942\) 0 0
\(943\) −49.4851 −1.61146
\(944\) −41.0940 −1.33750
\(945\) 0 0
\(946\) −41.2042 −1.33966
\(947\) −46.9467 −1.52556 −0.762782 0.646656i \(-0.776167\pi\)
−0.762782 + 0.646656i \(0.776167\pi\)
\(948\) 0 0
\(949\) 42.5498 1.38123
\(950\) 0 0
\(951\) 0 0
\(952\) 0.260730 0.00845029
\(953\) 1.91375 0.0619926 0.0309963 0.999519i \(-0.490132\pi\)
0.0309963 + 0.999519i \(0.490132\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 3.38005 0.109319
\(957\) 0 0
\(958\) 37.8829 1.22394
\(959\) −1.40061 −0.0452280
\(960\) 0 0
\(961\) −20.3108 −0.655188
\(962\) −24.7554 −0.798145
\(963\) 0 0
\(964\) 11.3707 0.366227
\(965\) 0 0
\(966\) 0 0
\(967\) 4.01326 0.129058 0.0645288 0.997916i \(-0.479446\pi\)
0.0645288 + 0.997916i \(0.479446\pi\)
\(968\) −31.7427 −1.02025
\(969\) 0 0
\(970\) 0 0
\(971\) −4.54751 −0.145937 −0.0729683 0.997334i \(-0.523247\pi\)
−0.0729683 + 0.997334i \(0.523247\pi\)
\(972\) 0 0
\(973\) 0.694284 0.0222577
\(974\) −53.3295 −1.70879
\(975\) 0 0
\(976\) 24.0683 0.770409
\(977\) 44.2333 1.41515 0.707574 0.706639i \(-0.249789\pi\)
0.707574 + 0.706639i \(0.249789\pi\)
\(978\) 0 0
\(979\) −60.4256 −1.93121
\(980\) 0 0
\(981\) 0 0
\(982\) 9.19391 0.293389
\(983\) 34.2083 1.09107 0.545537 0.838087i \(-0.316326\pi\)
0.545537 + 0.838087i \(0.316326\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0.428167 0.0136356
\(987\) 0 0
\(988\) 5.57334 0.177312
\(989\) 60.4764 1.92304
\(990\) 0 0
\(991\) −1.76770 −0.0561528 −0.0280764 0.999606i \(-0.508938\pi\)
−0.0280764 + 0.999606i \(0.508938\pi\)
\(992\) −6.93347 −0.220138
\(993\) 0 0
\(994\) 4.54697 0.144221
\(995\) 0 0
\(996\) 0 0
\(997\) 23.7398 0.751848 0.375924 0.926650i \(-0.377325\pi\)
0.375924 + 0.926650i \(0.377325\pi\)
\(998\) −19.8030 −0.626853
\(999\) 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.cf.1.8 12
3.2 odd 2 6525.2.a.ce.1.5 12
5.2 odd 4 1305.2.c.l.784.8 yes 12
5.3 odd 4 1305.2.c.l.784.5 yes 12
5.4 even 2 inner 6525.2.a.cf.1.5 12
15.2 even 4 1305.2.c.k.784.5 12
15.8 even 4 1305.2.c.k.784.8 yes 12
15.14 odd 2 6525.2.a.ce.1.8 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1305.2.c.k.784.5 12 15.2 even 4
1305.2.c.k.784.8 yes 12 15.8 even 4
1305.2.c.l.784.5 yes 12 5.3 odd 4
1305.2.c.l.784.8 yes 12 5.2 odd 4
6525.2.a.ce.1.5 12 3.2 odd 2
6525.2.a.ce.1.8 12 15.14 odd 2
6525.2.a.cf.1.5 12 5.4 even 2 inner
6525.2.a.cf.1.8 12 1.1 even 1 trivial