Properties

Label 6525.2.a.cc.1.5
Level $6525$
Weight $2$
Character 6525.1
Self dual yes
Analytic conductor $52.102$
Analytic rank $0$
Dimension $9$
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] = [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: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 2x^{8} - 12x^{7} + 21x^{6} + 48x^{5} - 68x^{4} - 73x^{3} + 66x^{2} + 40x - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(0.200649\) 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)
 
\(f(q)\) \(=\) \(q+0.200649 q^{2} -1.95974 q^{4} -0.775135 q^{7} -0.794517 q^{8} +O(q^{10})\) \(q+0.200649 q^{2} -1.95974 q^{4} -0.775135 q^{7} -0.794517 q^{8} -4.45447 q^{11} -3.67934 q^{13} -0.155530 q^{14} +3.76006 q^{16} +1.05645 q^{17} +3.76245 q^{19} -0.893784 q^{22} +6.21538 q^{23} -0.738254 q^{26} +1.51906 q^{28} -1.00000 q^{29} -3.96384 q^{31} +2.34348 q^{32} +0.211974 q^{34} -2.64473 q^{37} +0.754931 q^{38} -6.67607 q^{41} -9.96230 q^{43} +8.72961 q^{44} +1.24711 q^{46} -12.6272 q^{47} -6.39917 q^{49} +7.21054 q^{52} -2.87463 q^{53} +0.615858 q^{56} -0.200649 q^{58} +7.92534 q^{59} -2.22149 q^{61} -0.795340 q^{62} -7.04991 q^{64} +4.12604 q^{67} -2.07036 q^{68} -13.6161 q^{71} +8.60138 q^{73} -0.530661 q^{74} -7.37343 q^{76} +3.45282 q^{77} +6.97979 q^{79} -1.33955 q^{82} +2.90876 q^{83} -1.99892 q^{86} +3.53915 q^{88} +4.30975 q^{89} +2.85198 q^{91} -12.1805 q^{92} -2.53364 q^{94} -3.82019 q^{97} -1.28398 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 2 q^{2} + 10 q^{4} - q^{7} + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 2 q^{2} + 10 q^{4} - q^{7} + 9 q^{8} - 2 q^{11} - q^{13} + 3 q^{14} + 4 q^{16} + 12 q^{17} - q^{19} - 3 q^{22} + 16 q^{23} - 6 q^{26} + 4 q^{28} - 9 q^{29} + 5 q^{31} + 20 q^{32} + 3 q^{34} + 30 q^{38} + 10 q^{41} - 3 q^{43} + 13 q^{44} + 4 q^{46} + 26 q^{47} - 8 q^{49} + 9 q^{52} + 22 q^{53} - 22 q^{56} - 2 q^{58} - 4 q^{59} + 7 q^{61} + 28 q^{62} + 9 q^{64} - 5 q^{67} + 39 q^{68} + 10 q^{73} + 34 q^{74} - 2 q^{76} + 34 q^{77} + 10 q^{79} + 8 q^{82} + 46 q^{83} - 28 q^{86} - 2 q^{88} - 4 q^{89} - 21 q^{91} + 20 q^{92} + 5 q^{94} - 7 q^{97} + 51 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.200649 0.141880 0.0709400 0.997481i \(-0.477400\pi\)
0.0709400 + 0.997481i \(0.477400\pi\)
\(3\) 0 0
\(4\) −1.95974 −0.979870
\(5\) 0 0
\(6\) 0 0
\(7\) −0.775135 −0.292974 −0.146487 0.989213i \(-0.546797\pi\)
−0.146487 + 0.989213i \(0.546797\pi\)
\(8\) −0.794517 −0.280904
\(9\) 0 0
\(10\) 0 0
\(11\) −4.45447 −1.34307 −0.671537 0.740971i \(-0.734365\pi\)
−0.671537 + 0.740971i \(0.734365\pi\)
\(12\) 0 0
\(13\) −3.67934 −1.02046 −0.510232 0.860037i \(-0.670441\pi\)
−0.510232 + 0.860037i \(0.670441\pi\)
\(14\) −0.155530 −0.0415671
\(15\) 0 0
\(16\) 3.76006 0.940015
\(17\) 1.05645 0.256226 0.128113 0.991760i \(-0.459108\pi\)
0.128113 + 0.991760i \(0.459108\pi\)
\(18\) 0 0
\(19\) 3.76245 0.863166 0.431583 0.902073i \(-0.357955\pi\)
0.431583 + 0.902073i \(0.357955\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −0.893784 −0.190555
\(23\) 6.21538 1.29600 0.647998 0.761642i \(-0.275607\pi\)
0.647998 + 0.761642i \(0.275607\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −0.738254 −0.144784
\(27\) 0 0
\(28\) 1.51906 0.287076
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) −3.96384 −0.711927 −0.355963 0.934500i \(-0.615847\pi\)
−0.355963 + 0.934500i \(0.615847\pi\)
\(32\) 2.34348 0.414273
\(33\) 0 0
\(34\) 0.211974 0.0363533
\(35\) 0 0
\(36\) 0 0
\(37\) −2.64473 −0.434790 −0.217395 0.976084i \(-0.569756\pi\)
−0.217395 + 0.976084i \(0.569756\pi\)
\(38\) 0.754931 0.122466
\(39\) 0 0
\(40\) 0 0
\(41\) −6.67607 −1.04263 −0.521314 0.853365i \(-0.674558\pi\)
−0.521314 + 0.853365i \(0.674558\pi\)
\(42\) 0 0
\(43\) −9.96230 −1.51924 −0.759618 0.650369i \(-0.774614\pi\)
−0.759618 + 0.650369i \(0.774614\pi\)
\(44\) 8.72961 1.31604
\(45\) 0 0
\(46\) 1.24711 0.183876
\(47\) −12.6272 −1.84187 −0.920936 0.389715i \(-0.872574\pi\)
−0.920936 + 0.389715i \(0.872574\pi\)
\(48\) 0 0
\(49\) −6.39917 −0.914166
\(50\) 0 0
\(51\) 0 0
\(52\) 7.21054 0.999922
\(53\) −2.87463 −0.394860 −0.197430 0.980317i \(-0.563260\pi\)
−0.197430 + 0.980317i \(0.563260\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0.615858 0.0822975
\(57\) 0 0
\(58\) −0.200649 −0.0263465
\(59\) 7.92534 1.03179 0.515896 0.856651i \(-0.327459\pi\)
0.515896 + 0.856651i \(0.327459\pi\)
\(60\) 0 0
\(61\) −2.22149 −0.284432 −0.142216 0.989836i \(-0.545423\pi\)
−0.142216 + 0.989836i \(0.545423\pi\)
\(62\) −0.795340 −0.101008
\(63\) 0 0
\(64\) −7.04991 −0.881238
\(65\) 0 0
\(66\) 0 0
\(67\) 4.12604 0.504077 0.252038 0.967717i \(-0.418899\pi\)
0.252038 + 0.967717i \(0.418899\pi\)
\(68\) −2.07036 −0.251068
\(69\) 0 0
\(70\) 0 0
\(71\) −13.6161 −1.61593 −0.807966 0.589229i \(-0.799432\pi\)
−0.807966 + 0.589229i \(0.799432\pi\)
\(72\) 0 0
\(73\) 8.60138 1.00672 0.503358 0.864078i \(-0.332098\pi\)
0.503358 + 0.864078i \(0.332098\pi\)
\(74\) −0.530661 −0.0616881
\(75\) 0 0
\(76\) −7.37343 −0.845790
\(77\) 3.45282 0.393485
\(78\) 0 0
\(79\) 6.97979 0.785288 0.392644 0.919691i \(-0.371560\pi\)
0.392644 + 0.919691i \(0.371560\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −1.33955 −0.147928
\(83\) 2.90876 0.319278 0.159639 0.987175i \(-0.448967\pi\)
0.159639 + 0.987175i \(0.448967\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1.99892 −0.215549
\(87\) 0 0
\(88\) 3.53915 0.377275
\(89\) 4.30975 0.456833 0.228416 0.973564i \(-0.426645\pi\)
0.228416 + 0.973564i \(0.426645\pi\)
\(90\) 0 0
\(91\) 2.85198 0.298969
\(92\) −12.1805 −1.26991
\(93\) 0 0
\(94\) −2.53364 −0.261325
\(95\) 0 0
\(96\) 0 0
\(97\) −3.82019 −0.387881 −0.193941 0.981013i \(-0.562127\pi\)
−0.193941 + 0.981013i \(0.562127\pi\)
\(98\) −1.28398 −0.129702
\(99\) 0 0
\(100\) 0 0
\(101\) 1.79941 0.179048 0.0895241 0.995985i \(-0.471465\pi\)
0.0895241 + 0.995985i \(0.471465\pi\)
\(102\) 0 0
\(103\) 0.658227 0.0648570 0.0324285 0.999474i \(-0.489676\pi\)
0.0324285 + 0.999474i \(0.489676\pi\)
\(104\) 2.92329 0.286653
\(105\) 0 0
\(106\) −0.576790 −0.0560228
\(107\) −6.24882 −0.604097 −0.302048 0.953293i \(-0.597670\pi\)
−0.302048 + 0.953293i \(0.597670\pi\)
\(108\) 0 0
\(109\) 7.44434 0.713039 0.356519 0.934288i \(-0.383963\pi\)
0.356519 + 0.934288i \(0.383963\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −2.91456 −0.275400
\(113\) 21.0265 1.97801 0.989003 0.147896i \(-0.0472500\pi\)
0.989003 + 0.147896i \(0.0472500\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 1.95974 0.181957
\(117\) 0 0
\(118\) 1.59021 0.146391
\(119\) −0.818888 −0.0750673
\(120\) 0 0
\(121\) 8.84231 0.803847
\(122\) −0.445738 −0.0403552
\(123\) 0 0
\(124\) 7.76810 0.697596
\(125\) 0 0
\(126\) 0 0
\(127\) 11.9855 1.06354 0.531771 0.846888i \(-0.321527\pi\)
0.531771 + 0.846888i \(0.321527\pi\)
\(128\) −6.10152 −0.539304
\(129\) 0 0
\(130\) 0 0
\(131\) 13.6387 1.19162 0.595810 0.803126i \(-0.296831\pi\)
0.595810 + 0.803126i \(0.296831\pi\)
\(132\) 0 0
\(133\) −2.91641 −0.252885
\(134\) 0.827885 0.0715184
\(135\) 0 0
\(136\) −0.839363 −0.0719748
\(137\) 5.55942 0.474974 0.237487 0.971391i \(-0.423676\pi\)
0.237487 + 0.971391i \(0.423676\pi\)
\(138\) 0 0
\(139\) 22.0649 1.87152 0.935759 0.352639i \(-0.114716\pi\)
0.935759 + 0.352639i \(0.114716\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −2.73205 −0.229269
\(143\) 16.3895 1.37056
\(144\) 0 0
\(145\) 0 0
\(146\) 1.72586 0.142833
\(147\) 0 0
\(148\) 5.18298 0.426038
\(149\) 8.92269 0.730975 0.365488 0.930816i \(-0.380902\pi\)
0.365488 + 0.930816i \(0.380902\pi\)
\(150\) 0 0
\(151\) −7.84172 −0.638150 −0.319075 0.947729i \(-0.603372\pi\)
−0.319075 + 0.947729i \(0.603372\pi\)
\(152\) −2.98933 −0.242467
\(153\) 0 0
\(154\) 0.692803 0.0558277
\(155\) 0 0
\(156\) 0 0
\(157\) −17.8114 −1.42150 −0.710751 0.703444i \(-0.751645\pi\)
−0.710751 + 0.703444i \(0.751645\pi\)
\(158\) 1.40049 0.111417
\(159\) 0 0
\(160\) 0 0
\(161\) −4.81776 −0.379693
\(162\) 0 0
\(163\) 12.3040 0.963724 0.481862 0.876247i \(-0.339961\pi\)
0.481862 + 0.876247i \(0.339961\pi\)
\(164\) 13.0834 1.02164
\(165\) 0 0
\(166\) 0.583640 0.0452992
\(167\) 21.6643 1.67643 0.838216 0.545338i \(-0.183599\pi\)
0.838216 + 0.545338i \(0.183599\pi\)
\(168\) 0 0
\(169\) 0.537513 0.0413471
\(170\) 0 0
\(171\) 0 0
\(172\) 19.5235 1.48865
\(173\) 4.25059 0.323167 0.161583 0.986859i \(-0.448340\pi\)
0.161583 + 0.986859i \(0.448340\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −16.7491 −1.26251
\(177\) 0 0
\(178\) 0.864746 0.0648154
\(179\) 23.2124 1.73497 0.867487 0.497460i \(-0.165734\pi\)
0.867487 + 0.497460i \(0.165734\pi\)
\(180\) 0 0
\(181\) 9.68131 0.719606 0.359803 0.933028i \(-0.382844\pi\)
0.359803 + 0.933028i \(0.382844\pi\)
\(182\) 0.572247 0.0424177
\(183\) 0 0
\(184\) −4.93822 −0.364050
\(185\) 0 0
\(186\) 0 0
\(187\) −4.70590 −0.344130
\(188\) 24.7461 1.80479
\(189\) 0 0
\(190\) 0 0
\(191\) −21.1664 −1.53155 −0.765775 0.643109i \(-0.777644\pi\)
−0.765775 + 0.643109i \(0.777644\pi\)
\(192\) 0 0
\(193\) 17.6231 1.26854 0.634269 0.773112i \(-0.281301\pi\)
0.634269 + 0.773112i \(0.281301\pi\)
\(194\) −0.766516 −0.0550326
\(195\) 0 0
\(196\) 12.5407 0.895764
\(197\) 12.6217 0.899257 0.449628 0.893216i \(-0.351556\pi\)
0.449628 + 0.893216i \(0.351556\pi\)
\(198\) 0 0
\(199\) −20.9741 −1.48681 −0.743406 0.668841i \(-0.766791\pi\)
−0.743406 + 0.668841i \(0.766791\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0.361050 0.0254034
\(203\) 0.775135 0.0544038
\(204\) 0 0
\(205\) 0 0
\(206\) 0.132072 0.00920191
\(207\) 0 0
\(208\) −13.8345 −0.959252
\(209\) −16.7597 −1.15930
\(210\) 0 0
\(211\) −25.1750 −1.73312 −0.866558 0.499077i \(-0.833672\pi\)
−0.866558 + 0.499077i \(0.833672\pi\)
\(212\) 5.63352 0.386912
\(213\) 0 0
\(214\) −1.25382 −0.0857093
\(215\) 0 0
\(216\) 0 0
\(217\) 3.07251 0.208576
\(218\) 1.49370 0.101166
\(219\) 0 0
\(220\) 0 0
\(221\) −3.88702 −0.261469
\(222\) 0 0
\(223\) −0.225647 −0.0151104 −0.00755521 0.999971i \(-0.502405\pi\)
−0.00755521 + 0.999971i \(0.502405\pi\)
\(224\) −1.81652 −0.121371
\(225\) 0 0
\(226\) 4.21894 0.280640
\(227\) 26.3772 1.75072 0.875359 0.483474i \(-0.160625\pi\)
0.875359 + 0.483474i \(0.160625\pi\)
\(228\) 0 0
\(229\) −3.12235 −0.206331 −0.103165 0.994664i \(-0.532897\pi\)
−0.103165 + 0.994664i \(0.532897\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0.794517 0.0521626
\(233\) 12.4808 0.817644 0.408822 0.912614i \(-0.365940\pi\)
0.408822 + 0.912614i \(0.365940\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −15.5316 −1.01102
\(237\) 0 0
\(238\) −0.164309 −0.0106506
\(239\) 12.6664 0.819321 0.409660 0.912238i \(-0.365647\pi\)
0.409660 + 0.912238i \(0.365647\pi\)
\(240\) 0 0
\(241\) −0.738241 −0.0475543 −0.0237771 0.999717i \(-0.507569\pi\)
−0.0237771 + 0.999717i \(0.507569\pi\)
\(242\) 1.77420 0.114050
\(243\) 0 0
\(244\) 4.35354 0.278707
\(245\) 0 0
\(246\) 0 0
\(247\) −13.8433 −0.880830
\(248\) 3.14934 0.199983
\(249\) 0 0
\(250\) 0 0
\(251\) −12.3549 −0.779832 −0.389916 0.920850i \(-0.627496\pi\)
−0.389916 + 0.920850i \(0.627496\pi\)
\(252\) 0 0
\(253\) −27.6862 −1.74062
\(254\) 2.40488 0.150895
\(255\) 0 0
\(256\) 12.8755 0.804722
\(257\) 17.5823 1.09675 0.548377 0.836231i \(-0.315246\pi\)
0.548377 + 0.836231i \(0.315246\pi\)
\(258\) 0 0
\(259\) 2.05002 0.127382
\(260\) 0 0
\(261\) 0 0
\(262\) 2.73659 0.169067
\(263\) 11.3491 0.699816 0.349908 0.936784i \(-0.386213\pi\)
0.349908 + 0.936784i \(0.386213\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −0.585174 −0.0358793
\(267\) 0 0
\(268\) −8.08597 −0.493930
\(269\) −14.3872 −0.877201 −0.438601 0.898682i \(-0.644526\pi\)
−0.438601 + 0.898682i \(0.644526\pi\)
\(270\) 0 0
\(271\) 6.32554 0.384249 0.192124 0.981371i \(-0.438462\pi\)
0.192124 + 0.981371i \(0.438462\pi\)
\(272\) 3.97230 0.240856
\(273\) 0 0
\(274\) 1.11549 0.0673893
\(275\) 0 0
\(276\) 0 0
\(277\) 18.3726 1.10390 0.551952 0.833876i \(-0.313883\pi\)
0.551952 + 0.833876i \(0.313883\pi\)
\(278\) 4.42729 0.265531
\(279\) 0 0
\(280\) 0 0
\(281\) 17.1207 1.02133 0.510667 0.859778i \(-0.329398\pi\)
0.510667 + 0.859778i \(0.329398\pi\)
\(282\) 0 0
\(283\) −28.6804 −1.70487 −0.852435 0.522833i \(-0.824875\pi\)
−0.852435 + 0.522833i \(0.824875\pi\)
\(284\) 26.6840 1.58340
\(285\) 0 0
\(286\) 3.28853 0.194455
\(287\) 5.17486 0.305462
\(288\) 0 0
\(289\) −15.8839 −0.934348
\(290\) 0 0
\(291\) 0 0
\(292\) −16.8565 −0.986450
\(293\) 9.71695 0.567670 0.283835 0.958873i \(-0.408393\pi\)
0.283835 + 0.958873i \(0.408393\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 2.10128 0.122134
\(297\) 0 0
\(298\) 1.79033 0.103711
\(299\) −22.8685 −1.32252
\(300\) 0 0
\(301\) 7.72213 0.445096
\(302\) −1.57343 −0.0905408
\(303\) 0 0
\(304\) 14.1471 0.811389
\(305\) 0 0
\(306\) 0 0
\(307\) 8.56349 0.488744 0.244372 0.969682i \(-0.421418\pi\)
0.244372 + 0.969682i \(0.421418\pi\)
\(308\) −6.76663 −0.385564
\(309\) 0 0
\(310\) 0 0
\(311\) −17.1647 −0.973324 −0.486662 0.873590i \(-0.661786\pi\)
−0.486662 + 0.873590i \(0.661786\pi\)
\(312\) 0 0
\(313\) −3.40045 −0.192205 −0.0961025 0.995371i \(-0.530638\pi\)
−0.0961025 + 0.995371i \(0.530638\pi\)
\(314\) −3.57383 −0.201683
\(315\) 0 0
\(316\) −13.6786 −0.769480
\(317\) −23.0876 −1.29673 −0.648365 0.761329i \(-0.724547\pi\)
−0.648365 + 0.761329i \(0.724547\pi\)
\(318\) 0 0
\(319\) 4.45447 0.249403
\(320\) 0 0
\(321\) 0 0
\(322\) −0.966677 −0.0538708
\(323\) 3.97482 0.221165
\(324\) 0 0
\(325\) 0 0
\(326\) 2.46878 0.136733
\(327\) 0 0
\(328\) 5.30425 0.292878
\(329\) 9.78781 0.539620
\(330\) 0 0
\(331\) −15.3051 −0.841245 −0.420623 0.907236i \(-0.638188\pi\)
−0.420623 + 0.907236i \(0.638188\pi\)
\(332\) −5.70042 −0.312851
\(333\) 0 0
\(334\) 4.34691 0.237852
\(335\) 0 0
\(336\) 0 0
\(337\) −2.93351 −0.159799 −0.0798994 0.996803i \(-0.525460\pi\)
−0.0798994 + 0.996803i \(0.525460\pi\)
\(338\) 0.107851 0.00586634
\(339\) 0 0
\(340\) 0 0
\(341\) 17.6568 0.956170
\(342\) 0 0
\(343\) 10.3862 0.560800
\(344\) 7.91521 0.426760
\(345\) 0 0
\(346\) 0.852876 0.0458509
\(347\) 31.0176 1.66511 0.832555 0.553942i \(-0.186877\pi\)
0.832555 + 0.553942i \(0.186877\pi\)
\(348\) 0 0
\(349\) −8.15838 −0.436708 −0.218354 0.975870i \(-0.570069\pi\)
−0.218354 + 0.975870i \(0.570069\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −10.4390 −0.556400
\(353\) 1.25800 0.0669568 0.0334784 0.999439i \(-0.489342\pi\)
0.0334784 + 0.999439i \(0.489342\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −8.44599 −0.447637
\(357\) 0 0
\(358\) 4.65753 0.246158
\(359\) −3.64864 −0.192568 −0.0962838 0.995354i \(-0.530696\pi\)
−0.0962838 + 0.995354i \(0.530696\pi\)
\(360\) 0 0
\(361\) −4.84396 −0.254945
\(362\) 1.94254 0.102098
\(363\) 0 0
\(364\) −5.58915 −0.292951
\(365\) 0 0
\(366\) 0 0
\(367\) 13.1323 0.685503 0.342751 0.939426i \(-0.388641\pi\)
0.342751 + 0.939426i \(0.388641\pi\)
\(368\) 23.3702 1.21826
\(369\) 0 0
\(370\) 0 0
\(371\) 2.22822 0.115684
\(372\) 0 0
\(373\) −7.58444 −0.392707 −0.196354 0.980533i \(-0.562910\pi\)
−0.196354 + 0.980533i \(0.562910\pi\)
\(374\) −0.944233 −0.0488251
\(375\) 0 0
\(376\) 10.0325 0.517389
\(377\) 3.67934 0.189495
\(378\) 0 0
\(379\) −27.8949 −1.43287 −0.716434 0.697655i \(-0.754227\pi\)
−0.716434 + 0.697655i \(0.754227\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −4.24702 −0.217296
\(383\) 12.0440 0.615418 0.307709 0.951481i \(-0.400438\pi\)
0.307709 + 0.951481i \(0.400438\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 3.53605 0.179980
\(387\) 0 0
\(388\) 7.48658 0.380073
\(389\) 29.0575 1.47327 0.736636 0.676290i \(-0.236413\pi\)
0.736636 + 0.676290i \(0.236413\pi\)
\(390\) 0 0
\(391\) 6.56620 0.332067
\(392\) 5.08424 0.256793
\(393\) 0 0
\(394\) 2.53252 0.127587
\(395\) 0 0
\(396\) 0 0
\(397\) −11.1050 −0.557346 −0.278673 0.960386i \(-0.589895\pi\)
−0.278673 + 0.960386i \(0.589895\pi\)
\(398\) −4.20842 −0.210949
\(399\) 0 0
\(400\) 0 0
\(401\) 14.9928 0.748703 0.374352 0.927287i \(-0.377865\pi\)
0.374352 + 0.927287i \(0.377865\pi\)
\(402\) 0 0
\(403\) 14.5843 0.726496
\(404\) −3.52638 −0.175444
\(405\) 0 0
\(406\) 0.155530 0.00771882
\(407\) 11.7809 0.583955
\(408\) 0 0
\(409\) 26.3994 1.30537 0.652684 0.757631i \(-0.273643\pi\)
0.652684 + 0.757631i \(0.273643\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −1.28995 −0.0635514
\(413\) −6.14321 −0.302288
\(414\) 0 0
\(415\) 0 0
\(416\) −8.62247 −0.422751
\(417\) 0 0
\(418\) −3.36282 −0.164481
\(419\) −15.1863 −0.741901 −0.370950 0.928653i \(-0.620968\pi\)
−0.370950 + 0.928653i \(0.620968\pi\)
\(420\) 0 0
\(421\) 4.51805 0.220196 0.110098 0.993921i \(-0.464883\pi\)
0.110098 + 0.993921i \(0.464883\pi\)
\(422\) −5.05132 −0.245894
\(423\) 0 0
\(424\) 2.28394 0.110918
\(425\) 0 0
\(426\) 0 0
\(427\) 1.72195 0.0833311
\(428\) 12.2461 0.591936
\(429\) 0 0
\(430\) 0 0
\(431\) 3.10385 0.149507 0.0747536 0.997202i \(-0.476183\pi\)
0.0747536 + 0.997202i \(0.476183\pi\)
\(432\) 0 0
\(433\) −8.92413 −0.428866 −0.214433 0.976739i \(-0.568790\pi\)
−0.214433 + 0.976739i \(0.568790\pi\)
\(434\) 0.616496 0.0295927
\(435\) 0 0
\(436\) −14.5890 −0.698685
\(437\) 23.3851 1.11866
\(438\) 0 0
\(439\) −32.6862 −1.56003 −0.780013 0.625764i \(-0.784787\pi\)
−0.780013 + 0.625764i \(0.784787\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −0.779925 −0.0370972
\(443\) 14.9791 0.711677 0.355838 0.934548i \(-0.384195\pi\)
0.355838 + 0.934548i \(0.384195\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −0.0452757 −0.00214387
\(447\) 0 0
\(448\) 5.46463 0.258180
\(449\) −34.4130 −1.62405 −0.812024 0.583623i \(-0.801634\pi\)
−0.812024 + 0.583623i \(0.801634\pi\)
\(450\) 0 0
\(451\) 29.7384 1.40033
\(452\) −41.2065 −1.93819
\(453\) 0 0
\(454\) 5.29256 0.248392
\(455\) 0 0
\(456\) 0 0
\(457\) −29.7364 −1.39101 −0.695506 0.718520i \(-0.744820\pi\)
−0.695506 + 0.718520i \(0.744820\pi\)
\(458\) −0.626496 −0.0292742
\(459\) 0 0
\(460\) 0 0
\(461\) 5.01736 0.233682 0.116841 0.993151i \(-0.462723\pi\)
0.116841 + 0.993151i \(0.462723\pi\)
\(462\) 0 0
\(463\) 16.0818 0.747384 0.373692 0.927553i \(-0.378092\pi\)
0.373692 + 0.927553i \(0.378092\pi\)
\(464\) −3.76006 −0.174556
\(465\) 0 0
\(466\) 2.50425 0.116007
\(467\) 21.5438 0.996928 0.498464 0.866910i \(-0.333898\pi\)
0.498464 + 0.866910i \(0.333898\pi\)
\(468\) 0 0
\(469\) −3.19824 −0.147681
\(470\) 0 0
\(471\) 0 0
\(472\) −6.29681 −0.289834
\(473\) 44.3768 2.04045
\(474\) 0 0
\(475\) 0 0
\(476\) 1.60481 0.0735562
\(477\) 0 0
\(478\) 2.54149 0.116245
\(479\) 14.2131 0.649414 0.324707 0.945815i \(-0.394734\pi\)
0.324707 + 0.945815i \(0.394734\pi\)
\(480\) 0 0
\(481\) 9.73084 0.443688
\(482\) −0.148127 −0.00674700
\(483\) 0 0
\(484\) −17.3286 −0.787665
\(485\) 0 0
\(486\) 0 0
\(487\) 19.1443 0.867512 0.433756 0.901030i \(-0.357188\pi\)
0.433756 + 0.901030i \(0.357188\pi\)
\(488\) 1.76501 0.0798981
\(489\) 0 0
\(490\) 0 0
\(491\) −32.2694 −1.45630 −0.728148 0.685420i \(-0.759619\pi\)
−0.728148 + 0.685420i \(0.759619\pi\)
\(492\) 0 0
\(493\) −1.05645 −0.0475799
\(494\) −2.77764 −0.124972
\(495\) 0 0
\(496\) −14.9043 −0.669222
\(497\) 10.5543 0.473426
\(498\) 0 0
\(499\) 20.4860 0.917078 0.458539 0.888674i \(-0.348373\pi\)
0.458539 + 0.888674i \(0.348373\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −2.47899 −0.110643
\(503\) −3.42204 −0.152581 −0.0762905 0.997086i \(-0.524308\pi\)
−0.0762905 + 0.997086i \(0.524308\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −5.55520 −0.246959
\(507\) 0 0
\(508\) −23.4885 −1.04213
\(509\) 21.5512 0.955238 0.477619 0.878567i \(-0.341500\pi\)
0.477619 + 0.878567i \(0.341500\pi\)
\(510\) 0 0
\(511\) −6.66723 −0.294941
\(512\) 14.7865 0.653478
\(513\) 0 0
\(514\) 3.52787 0.155608
\(515\) 0 0
\(516\) 0 0
\(517\) 56.2476 2.47377
\(518\) 0.411334 0.0180730
\(519\) 0 0
\(520\) 0 0
\(521\) 31.7978 1.39309 0.696544 0.717514i \(-0.254720\pi\)
0.696544 + 0.717514i \(0.254720\pi\)
\(522\) 0 0
\(523\) 39.5953 1.73138 0.865691 0.500579i \(-0.166879\pi\)
0.865691 + 0.500579i \(0.166879\pi\)
\(524\) −26.7283 −1.16763
\(525\) 0 0
\(526\) 2.27718 0.0992900
\(527\) −4.18758 −0.182414
\(528\) 0 0
\(529\) 15.6309 0.679605
\(530\) 0 0
\(531\) 0 0
\(532\) 5.71540 0.247794
\(533\) 24.5635 1.06396
\(534\) 0 0
\(535\) 0 0
\(536\) −3.27821 −0.141597
\(537\) 0 0
\(538\) −2.88677 −0.124457
\(539\) 28.5049 1.22779
\(540\) 0 0
\(541\) −25.5906 −1.10023 −0.550113 0.835090i \(-0.685415\pi\)
−0.550113 + 0.835090i \(0.685415\pi\)
\(542\) 1.26921 0.0545173
\(543\) 0 0
\(544\) 2.47576 0.106147
\(545\) 0 0
\(546\) 0 0
\(547\) 5.96642 0.255106 0.127553 0.991832i \(-0.459288\pi\)
0.127553 + 0.991832i \(0.459288\pi\)
\(548\) −10.8950 −0.465412
\(549\) 0 0
\(550\) 0 0
\(551\) −3.76245 −0.160286
\(552\) 0 0
\(553\) −5.41028 −0.230069
\(554\) 3.68645 0.156622
\(555\) 0 0
\(556\) −43.2414 −1.83385
\(557\) 15.5091 0.657143 0.328572 0.944479i \(-0.393433\pi\)
0.328572 + 0.944479i \(0.393433\pi\)
\(558\) 0 0
\(559\) 36.6547 1.55033
\(560\) 0 0
\(561\) 0 0
\(562\) 3.43524 0.144907
\(563\) 8.67282 0.365516 0.182758 0.983158i \(-0.441498\pi\)
0.182758 + 0.983158i \(0.441498\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −5.75468 −0.241887
\(567\) 0 0
\(568\) 10.8182 0.453922
\(569\) −2.05132 −0.0859959 −0.0429980 0.999075i \(-0.513691\pi\)
−0.0429980 + 0.999075i \(0.513691\pi\)
\(570\) 0 0
\(571\) −2.72574 −0.114069 −0.0570344 0.998372i \(-0.518164\pi\)
−0.0570344 + 0.998372i \(0.518164\pi\)
\(572\) −32.1192 −1.34297
\(573\) 0 0
\(574\) 1.03833 0.0433390
\(575\) 0 0
\(576\) 0 0
\(577\) −24.3626 −1.01423 −0.507115 0.861879i \(-0.669288\pi\)
−0.507115 + 0.861879i \(0.669288\pi\)
\(578\) −3.18709 −0.132565
\(579\) 0 0
\(580\) 0 0
\(581\) −2.25469 −0.0935401
\(582\) 0 0
\(583\) 12.8049 0.530326
\(584\) −6.83394 −0.282790
\(585\) 0 0
\(586\) 1.94969 0.0805411
\(587\) 4.29753 0.177378 0.0886890 0.996059i \(-0.471732\pi\)
0.0886890 + 0.996059i \(0.471732\pi\)
\(588\) 0 0
\(589\) −14.9138 −0.614511
\(590\) 0 0
\(591\) 0 0
\(592\) −9.94433 −0.408710
\(593\) 20.7059 0.850291 0.425146 0.905125i \(-0.360223\pi\)
0.425146 + 0.905125i \(0.360223\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −17.4862 −0.716261
\(597\) 0 0
\(598\) −4.58853 −0.187639
\(599\) −2.58646 −0.105680 −0.0528399 0.998603i \(-0.516827\pi\)
−0.0528399 + 0.998603i \(0.516827\pi\)
\(600\) 0 0
\(601\) 25.6870 1.04779 0.523897 0.851782i \(-0.324478\pi\)
0.523897 + 0.851782i \(0.324478\pi\)
\(602\) 1.54944 0.0631503
\(603\) 0 0
\(604\) 15.3677 0.625304
\(605\) 0 0
\(606\) 0 0
\(607\) 18.6725 0.757893 0.378947 0.925419i \(-0.376286\pi\)
0.378947 + 0.925419i \(0.376286\pi\)
\(608\) 8.81725 0.357587
\(609\) 0 0
\(610\) 0 0
\(611\) 46.4598 1.87956
\(612\) 0 0
\(613\) −28.5152 −1.15172 −0.575858 0.817549i \(-0.695332\pi\)
−0.575858 + 0.817549i \(0.695332\pi\)
\(614\) 1.71825 0.0693430
\(615\) 0 0
\(616\) −2.74332 −0.110532
\(617\) −31.7077 −1.27651 −0.638253 0.769827i \(-0.720343\pi\)
−0.638253 + 0.769827i \(0.720343\pi\)
\(618\) 0 0
\(619\) 33.0039 1.32654 0.663269 0.748381i \(-0.269169\pi\)
0.663269 + 0.748381i \(0.269169\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −3.44408 −0.138095
\(623\) −3.34064 −0.133840
\(624\) 0 0
\(625\) 0 0
\(626\) −0.682297 −0.0272701
\(627\) 0 0
\(628\) 34.9057 1.39289
\(629\) −2.79401 −0.111404
\(630\) 0 0
\(631\) −11.7586 −0.468102 −0.234051 0.972224i \(-0.575198\pi\)
−0.234051 + 0.972224i \(0.575198\pi\)
\(632\) −5.54556 −0.220591
\(633\) 0 0
\(634\) −4.63250 −0.183980
\(635\) 0 0
\(636\) 0 0
\(637\) 23.5447 0.932874
\(638\) 0.893784 0.0353852
\(639\) 0 0
\(640\) 0 0
\(641\) −3.63965 −0.143758 −0.0718788 0.997413i \(-0.522899\pi\)
−0.0718788 + 0.997413i \(0.522899\pi\)
\(642\) 0 0
\(643\) 33.1571 1.30759 0.653794 0.756673i \(-0.273176\pi\)
0.653794 + 0.756673i \(0.273176\pi\)
\(644\) 9.44155 0.372049
\(645\) 0 0
\(646\) 0.797543 0.0313789
\(647\) 47.8411 1.88083 0.940413 0.340034i \(-0.110439\pi\)
0.940413 + 0.340034i \(0.110439\pi\)
\(648\) 0 0
\(649\) −35.3032 −1.38577
\(650\) 0 0
\(651\) 0 0
\(652\) −24.1127 −0.944325
\(653\) −10.3879 −0.406512 −0.203256 0.979126i \(-0.565152\pi\)
−0.203256 + 0.979126i \(0.565152\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −25.1024 −0.980086
\(657\) 0 0
\(658\) 1.96391 0.0765612
\(659\) 33.5433 1.30666 0.653331 0.757073i \(-0.273371\pi\)
0.653331 + 0.757073i \(0.273371\pi\)
\(660\) 0 0
\(661\) −18.6291 −0.724589 −0.362294 0.932064i \(-0.618006\pi\)
−0.362294 + 0.932064i \(0.618006\pi\)
\(662\) −3.07095 −0.119356
\(663\) 0 0
\(664\) −2.31106 −0.0896866
\(665\) 0 0
\(666\) 0 0
\(667\) −6.21538 −0.240660
\(668\) −42.4564 −1.64269
\(669\) 0 0
\(670\) 0 0
\(671\) 9.89555 0.382013
\(672\) 0 0
\(673\) −29.3062 −1.12967 −0.564835 0.825204i \(-0.691060\pi\)
−0.564835 + 0.825204i \(0.691060\pi\)
\(674\) −0.588606 −0.0226723
\(675\) 0 0
\(676\) −1.05339 −0.0405148
\(677\) −20.8134 −0.799925 −0.399962 0.916532i \(-0.630977\pi\)
−0.399962 + 0.916532i \(0.630977\pi\)
\(678\) 0 0
\(679\) 2.96116 0.113639
\(680\) 0 0
\(681\) 0 0
\(682\) 3.54282 0.135661
\(683\) −26.6885 −1.02121 −0.510604 0.859816i \(-0.670578\pi\)
−0.510604 + 0.859816i \(0.670578\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 2.08397 0.0795664
\(687\) 0 0
\(688\) −37.4589 −1.42811
\(689\) 10.5767 0.402941
\(690\) 0 0
\(691\) −34.2486 −1.30288 −0.651439 0.758701i \(-0.725834\pi\)
−0.651439 + 0.758701i \(0.725834\pi\)
\(692\) −8.33006 −0.316661
\(693\) 0 0
\(694\) 6.22363 0.236246
\(695\) 0 0
\(696\) 0 0
\(697\) −7.05290 −0.267148
\(698\) −1.63697 −0.0619601
\(699\) 0 0
\(700\) 0 0
\(701\) −28.4457 −1.07438 −0.537190 0.843461i \(-0.680514\pi\)
−0.537190 + 0.843461i \(0.680514\pi\)
\(702\) 0 0
\(703\) −9.95066 −0.375296
\(704\) 31.4036 1.18357
\(705\) 0 0
\(706\) 0.252417 0.00949983
\(707\) −1.39479 −0.0524564
\(708\) 0 0
\(709\) −37.4260 −1.40556 −0.702782 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −3.42417 −0.128326
\(713\) −24.6368 −0.922654
\(714\) 0 0
\(715\) 0 0
\(716\) −45.4902 −1.70005
\(717\) 0 0
\(718\) −0.732094 −0.0273215
\(719\) −28.3986 −1.05909 −0.529545 0.848282i \(-0.677637\pi\)
−0.529545 + 0.848282i \(0.677637\pi\)
\(720\) 0 0
\(721\) −0.510215 −0.0190014
\(722\) −0.971933 −0.0361716
\(723\) 0 0
\(724\) −18.9729 −0.705120
\(725\) 0 0
\(726\) 0 0
\(727\) −3.50646 −0.130047 −0.0650236 0.997884i \(-0.520712\pi\)
−0.0650236 + 0.997884i \(0.520712\pi\)
\(728\) −2.26595 −0.0839816
\(729\) 0 0
\(730\) 0 0
\(731\) −10.5246 −0.389267
\(732\) 0 0
\(733\) 36.8907 1.36259 0.681295 0.732009i \(-0.261417\pi\)
0.681295 + 0.732009i \(0.261417\pi\)
\(734\) 2.63499 0.0972592
\(735\) 0 0
\(736\) 14.5656 0.536897
\(737\) −18.3793 −0.677012
\(738\) 0 0
\(739\) −12.5756 −0.462603 −0.231301 0.972882i \(-0.574298\pi\)
−0.231301 + 0.972882i \(0.574298\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0.447090 0.0164132
\(743\) 36.2579 1.33017 0.665086 0.746767i \(-0.268395\pi\)
0.665086 + 0.746767i \(0.268395\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −1.52181 −0.0557173
\(747\) 0 0
\(748\) 9.22235 0.337202
\(749\) 4.84368 0.176984
\(750\) 0 0
\(751\) 22.2839 0.813151 0.406576 0.913617i \(-0.366723\pi\)
0.406576 + 0.913617i \(0.366723\pi\)
\(752\) −47.4792 −1.73139
\(753\) 0 0
\(754\) 0.738254 0.0268856
\(755\) 0 0
\(756\) 0 0
\(757\) 22.1631 0.805530 0.402765 0.915303i \(-0.368049\pi\)
0.402765 + 0.915303i \(0.368049\pi\)
\(758\) −5.59708 −0.203295
\(759\) 0 0
\(760\) 0 0
\(761\) −43.6078 −1.58078 −0.790391 0.612603i \(-0.790123\pi\)
−0.790391 + 0.612603i \(0.790123\pi\)
\(762\) 0 0
\(763\) −5.77037 −0.208901
\(764\) 41.4807 1.50072
\(765\) 0 0
\(766\) 2.41660 0.0873155
\(767\) −29.1600 −1.05291
\(768\) 0 0
\(769\) −5.84025 −0.210605 −0.105302 0.994440i \(-0.533581\pi\)
−0.105302 + 0.994440i \(0.533581\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −34.5367 −1.24300
\(773\) −19.9643 −0.718067 −0.359034 0.933325i \(-0.616894\pi\)
−0.359034 + 0.933325i \(0.616894\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 3.03520 0.108957
\(777\) 0 0
\(778\) 5.83034 0.209028
\(779\) −25.1184 −0.899960
\(780\) 0 0
\(781\) 60.6525 2.17032
\(782\) 1.31750 0.0471137
\(783\) 0 0
\(784\) −24.0613 −0.859331
\(785\) 0 0
\(786\) 0 0
\(787\) −26.1482 −0.932081 −0.466041 0.884763i \(-0.654320\pi\)
−0.466041 + 0.884763i \(0.654320\pi\)
\(788\) −24.7352 −0.881155
\(789\) 0 0
\(790\) 0 0
\(791\) −16.2984 −0.579503
\(792\) 0 0
\(793\) 8.17359 0.290253
\(794\) −2.22821 −0.0790763
\(795\) 0 0
\(796\) 41.1037 1.45688
\(797\) 43.5293 1.54189 0.770944 0.636903i \(-0.219785\pi\)
0.770944 + 0.636903i \(0.219785\pi\)
\(798\) 0 0
\(799\) −13.3400 −0.471934
\(800\) 0 0
\(801\) 0 0
\(802\) 3.00828 0.106226
\(803\) −38.3146 −1.35209
\(804\) 0 0
\(805\) 0 0
\(806\) 2.92632 0.103075
\(807\) 0 0
\(808\) −1.42966 −0.0502954
\(809\) −43.8436 −1.54146 −0.770729 0.637164i \(-0.780108\pi\)
−0.770729 + 0.637164i \(0.780108\pi\)
\(810\) 0 0
\(811\) 15.2295 0.534778 0.267389 0.963589i \(-0.413839\pi\)
0.267389 + 0.963589i \(0.413839\pi\)
\(812\) −1.51906 −0.0533087
\(813\) 0 0
\(814\) 2.36381 0.0828516
\(815\) 0 0
\(816\) 0 0
\(817\) −37.4827 −1.31135
\(818\) 5.29701 0.185206
\(819\) 0 0
\(820\) 0 0
\(821\) 14.3792 0.501837 0.250919 0.968008i \(-0.419267\pi\)
0.250919 + 0.968008i \(0.419267\pi\)
\(822\) 0 0
\(823\) 9.68581 0.337626 0.168813 0.985648i \(-0.446007\pi\)
0.168813 + 0.985648i \(0.446007\pi\)
\(824\) −0.522972 −0.0182186
\(825\) 0 0
\(826\) −1.23263 −0.0428886
\(827\) 30.2089 1.05047 0.525233 0.850958i \(-0.323978\pi\)
0.525233 + 0.850958i \(0.323978\pi\)
\(828\) 0 0
\(829\) 38.9917 1.35424 0.677119 0.735874i \(-0.263228\pi\)
0.677119 + 0.735874i \(0.263228\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 25.9390 0.899272
\(833\) −6.76037 −0.234233
\(834\) 0 0
\(835\) 0 0
\(836\) 32.8447 1.13596
\(837\) 0 0
\(838\) −3.04712 −0.105261
\(839\) −9.74245 −0.336347 −0.168173 0.985757i \(-0.553787\pi\)
−0.168173 + 0.985757i \(0.553787\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0.906541 0.0312415
\(843\) 0 0
\(844\) 49.3364 1.69823
\(845\) 0 0
\(846\) 0 0
\(847\) −6.85399 −0.235506
\(848\) −10.8088 −0.371175
\(849\) 0 0
\(850\) 0 0
\(851\) −16.4380 −0.563486
\(852\) 0 0
\(853\) −51.4478 −1.76154 −0.880769 0.473545i \(-0.842974\pi\)
−0.880769 + 0.473545i \(0.842974\pi\)
\(854\) 0.345507 0.0118230
\(855\) 0 0
\(856\) 4.96479 0.169693
\(857\) −48.8308 −1.66803 −0.834014 0.551743i \(-0.813963\pi\)
−0.834014 + 0.551743i \(0.813963\pi\)
\(858\) 0 0
\(859\) −16.6720 −0.568841 −0.284421 0.958700i \(-0.591801\pi\)
−0.284421 + 0.958700i \(0.591801\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0.622783 0.0212121
\(863\) 2.90194 0.0987831 0.0493915 0.998779i \(-0.484272\pi\)
0.0493915 + 0.998779i \(0.484272\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −1.79061 −0.0608476
\(867\) 0 0
\(868\) −6.02133 −0.204377
\(869\) −31.0913 −1.05470
\(870\) 0 0
\(871\) −15.1811 −0.514392
\(872\) −5.91465 −0.200295
\(873\) 0 0
\(874\) 4.69218 0.158715
\(875\) 0 0
\(876\) 0 0
\(877\) 37.4418 1.26432 0.632160 0.774838i \(-0.282169\pi\)
0.632160 + 0.774838i \(0.282169\pi\)
\(878\) −6.55843 −0.221336
\(879\) 0 0
\(880\) 0 0
\(881\) 31.8351 1.07255 0.536275 0.844043i \(-0.319831\pi\)
0.536275 + 0.844043i \(0.319831\pi\)
\(882\) 0 0
\(883\) 16.4903 0.554942 0.277471 0.960734i \(-0.410504\pi\)
0.277471 + 0.960734i \(0.410504\pi\)
\(884\) 7.61754 0.256206
\(885\) 0 0
\(886\) 3.00553 0.100973
\(887\) −8.78785 −0.295067 −0.147534 0.989057i \(-0.547133\pi\)
−0.147534 + 0.989057i \(0.547133\pi\)
\(888\) 0 0
\(889\) −9.29039 −0.311590
\(890\) 0 0
\(891\) 0 0
\(892\) 0.442209 0.0148062
\(893\) −47.5094 −1.58984
\(894\) 0 0
\(895\) 0 0
\(896\) 4.72951 0.158002
\(897\) 0 0
\(898\) −6.90492 −0.230420
\(899\) 3.96384 0.132202
\(900\) 0 0
\(901\) −3.03688 −0.101173
\(902\) 5.96697 0.198678
\(903\) 0 0
\(904\) −16.7059 −0.555630
\(905\) 0 0
\(906\) 0 0
\(907\) −6.51166 −0.216216 −0.108108 0.994139i \(-0.534479\pi\)
−0.108108 + 0.994139i \(0.534479\pi\)
\(908\) −51.6925 −1.71548
\(909\) 0 0
\(910\) 0 0
\(911\) 22.3746 0.741304 0.370652 0.928772i \(-0.379134\pi\)
0.370652 + 0.928772i \(0.379134\pi\)
\(912\) 0 0
\(913\) −12.9570 −0.428814
\(914\) −5.96658 −0.197357
\(915\) 0 0
\(916\) 6.11900 0.202178
\(917\) −10.5718 −0.349113
\(918\) 0 0
\(919\) 22.2078 0.732568 0.366284 0.930503i \(-0.380630\pi\)
0.366284 + 0.930503i \(0.380630\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 1.00673 0.0331548
\(923\) 50.0982 1.64900
\(924\) 0 0
\(925\) 0 0
\(926\) 3.22679 0.106039
\(927\) 0 0
\(928\) −2.34348 −0.0769287
\(929\) 35.0239 1.14910 0.574548 0.818471i \(-0.305178\pi\)
0.574548 + 0.818471i \(0.305178\pi\)
\(930\) 0 0
\(931\) −24.0766 −0.789077
\(932\) −24.4591 −0.801184
\(933\) 0 0
\(934\) 4.32274 0.141444
\(935\) 0 0
\(936\) 0 0
\(937\) −56.4983 −1.84572 −0.922860 0.385136i \(-0.874155\pi\)
−0.922860 + 0.385136i \(0.874155\pi\)
\(938\) −0.641723 −0.0209530
\(939\) 0 0
\(940\) 0 0
\(941\) 39.3736 1.28354 0.641771 0.766896i \(-0.278200\pi\)
0.641771 + 0.766896i \(0.278200\pi\)
\(942\) 0 0
\(943\) −41.4943 −1.35124
\(944\) 29.7998 0.969900
\(945\) 0 0
\(946\) 8.90414 0.289499
\(947\) −58.9377 −1.91522 −0.957610 0.288069i \(-0.906987\pi\)
−0.957610 + 0.288069i \(0.906987\pi\)
\(948\) 0 0
\(949\) −31.6474 −1.02732
\(950\) 0 0
\(951\) 0 0
\(952\) 0.650620 0.0210867
\(953\) 23.3111 0.755120 0.377560 0.925985i \(-0.376763\pi\)
0.377560 + 0.925985i \(0.376763\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −24.8228 −0.802828
\(957\) 0 0
\(958\) 2.85184 0.0921389
\(959\) −4.30930 −0.139155
\(960\) 0 0
\(961\) −15.2880 −0.493160
\(962\) 1.95248 0.0629505
\(963\) 0 0
\(964\) 1.44676 0.0465970
\(965\) 0 0
\(966\) 0 0
\(967\) −13.7227 −0.441293 −0.220646 0.975354i \(-0.570817\pi\)
−0.220646 + 0.975354i \(0.570817\pi\)
\(968\) −7.02537 −0.225804
\(969\) 0 0
\(970\) 0 0
\(971\) 59.1078 1.89686 0.948430 0.316985i \(-0.102671\pi\)
0.948430 + 0.316985i \(0.102671\pi\)
\(972\) 0 0
\(973\) −17.1033 −0.548306
\(974\) 3.84128 0.123083
\(975\) 0 0
\(976\) −8.35292 −0.267371
\(977\) 25.0687 0.802018 0.401009 0.916074i \(-0.368660\pi\)
0.401009 + 0.916074i \(0.368660\pi\)
\(978\) 0 0
\(979\) −19.1977 −0.613560
\(980\) 0 0
\(981\) 0 0
\(982\) −6.47480 −0.206619
\(983\) −28.3475 −0.904146 −0.452073 0.891981i \(-0.649315\pi\)
−0.452073 + 0.891981i \(0.649315\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −0.211974 −0.00675064
\(987\) 0 0
\(988\) 27.1293 0.863099
\(989\) −61.9195 −1.96892
\(990\) 0 0
\(991\) −4.81196 −0.152857 −0.0764285 0.997075i \(-0.524352\pi\)
−0.0764285 + 0.997075i \(0.524352\pi\)
\(992\) −9.28920 −0.294932
\(993\) 0 0
\(994\) 2.11771 0.0671696
\(995\) 0 0
\(996\) 0 0
\(997\) −18.9930 −0.601514 −0.300757 0.953701i \(-0.597239\pi\)
−0.300757 + 0.953701i \(0.597239\pi\)
\(998\) 4.11048 0.130115
\(999\) 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.cc.1.5 yes 9
3.2 odd 2 6525.2.a.ca.1.5 9
5.4 even 2 6525.2.a.cb.1.5 yes 9
15.14 odd 2 6525.2.a.cd.1.5 yes 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6525.2.a.ca.1.5 9 3.2 odd 2
6525.2.a.cb.1.5 yes 9 5.4 even 2
6525.2.a.cc.1.5 yes 9 1.1 even 1 trivial
6525.2.a.cd.1.5 yes 9 15.14 odd 2