Properties

Label 7569.2.a.bj.1.2
Level $7569$
Weight $2$
Character 7569.1
Self dual yes
Analytic conductor $60.439$
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] = [7569,2,Mod(1,7569)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7569, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("7569.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7569 = 3^{2} \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7569.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: \(60.4387692899\)
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: \( x^{9} - 4x^{8} - 6x^{7} + 33x^{6} + 6x^{5} - 90x^{4} + 21x^{3} + 84x^{2} - 36x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 87)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.58860\) of defining polynomial
Character \(\chi\) \(=\) 7569.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.58860 q^{2} +4.70087 q^{4} +1.14894 q^{5} +2.76526 q^{7} -6.99148 q^{8} -2.97416 q^{10} +4.35418 q^{11} +5.62914 q^{13} -7.15816 q^{14} +8.69643 q^{16} -2.41516 q^{17} +1.87296 q^{19} +5.40103 q^{20} -11.2712 q^{22} +8.82048 q^{23} -3.67993 q^{25} -14.5716 q^{26} +12.9991 q^{28} +1.56053 q^{31} -8.52865 q^{32} +6.25190 q^{34} +3.17713 q^{35} +7.76330 q^{37} -4.84834 q^{38} -8.03281 q^{40} -3.16072 q^{41} -1.46397 q^{43} +20.4684 q^{44} -22.8327 q^{46} -0.357351 q^{47} +0.646666 q^{49} +9.52588 q^{50} +26.4618 q^{52} +1.63498 q^{53} +5.00270 q^{55} -19.3333 q^{56} +4.85026 q^{59} +1.55510 q^{61} -4.03961 q^{62} +4.68444 q^{64} +6.46755 q^{65} -9.66950 q^{67} -11.3534 q^{68} -8.22432 q^{70} +8.18108 q^{71} +11.3786 q^{73} -20.0961 q^{74} +8.80452 q^{76} +12.0404 q^{77} -8.13075 q^{79} +9.99170 q^{80} +8.18185 q^{82} +0.481835 q^{83} -2.77488 q^{85} +3.78965 q^{86} -30.4421 q^{88} +1.99621 q^{89} +15.5660 q^{91} +41.4639 q^{92} +0.925040 q^{94} +2.15192 q^{95} +4.63516 q^{97} -1.67396 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 5 q^{2} + 11 q^{4} + 4 q^{5} + 5 q^{7} - 24 q^{8} + q^{11} + q^{13} - 9 q^{14} + 35 q^{16} - 2 q^{17} + 9 q^{19} + 18 q^{20} - 4 q^{22} + 4 q^{23} + q^{25} + 8 q^{26} + 40 q^{28} + 8 q^{31} - 43 q^{32}+ \cdots + 30 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.58860 −1.83042 −0.915210 0.402978i \(-0.867975\pi\)
−0.915210 + 0.402978i \(0.867975\pi\)
\(3\) 0 0
\(4\) 4.70087 2.35043
\(5\) 1.14894 0.513823 0.256911 0.966435i \(-0.417295\pi\)
0.256911 + 0.966435i \(0.417295\pi\)
\(6\) 0 0
\(7\) 2.76526 1.04517 0.522585 0.852587i \(-0.324968\pi\)
0.522585 + 0.852587i \(0.324968\pi\)
\(8\) −6.99148 −2.47186
\(9\) 0 0
\(10\) −2.97416 −0.940511
\(11\) 4.35418 1.31283 0.656417 0.754399i \(-0.272071\pi\)
0.656417 + 0.754399i \(0.272071\pi\)
\(12\) 0 0
\(13\) 5.62914 1.56124 0.780621 0.625005i \(-0.214903\pi\)
0.780621 + 0.625005i \(0.214903\pi\)
\(14\) −7.15816 −1.91310
\(15\) 0 0
\(16\) 8.69643 2.17411
\(17\) −2.41516 −0.585763 −0.292881 0.956149i \(-0.594614\pi\)
−0.292881 + 0.956149i \(0.594614\pi\)
\(18\) 0 0
\(19\) 1.87296 0.429686 0.214843 0.976649i \(-0.431076\pi\)
0.214843 + 0.976649i \(0.431076\pi\)
\(20\) 5.40103 1.20771
\(21\) 0 0
\(22\) −11.2712 −2.40304
\(23\) 8.82048 1.83920 0.919598 0.392860i \(-0.128514\pi\)
0.919598 + 0.392860i \(0.128514\pi\)
\(24\) 0 0
\(25\) −3.67993 −0.735986
\(26\) −14.5716 −2.85773
\(27\) 0 0
\(28\) 12.9991 2.45660
\(29\) 0 0
\(30\) 0 0
\(31\) 1.56053 0.280280 0.140140 0.990132i \(-0.455245\pi\)
0.140140 + 0.990132i \(0.455245\pi\)
\(32\) −8.52865 −1.50767
\(33\) 0 0
\(34\) 6.25190 1.07219
\(35\) 3.17713 0.537032
\(36\) 0 0
\(37\) 7.76330 1.27628 0.638139 0.769921i \(-0.279705\pi\)
0.638139 + 0.769921i \(0.279705\pi\)
\(38\) −4.84834 −0.786505
\(39\) 0 0
\(40\) −8.03281 −1.27010
\(41\) −3.16072 −0.493622 −0.246811 0.969064i \(-0.579383\pi\)
−0.246811 + 0.969064i \(0.579383\pi\)
\(42\) 0 0
\(43\) −1.46397 −0.223254 −0.111627 0.993750i \(-0.535606\pi\)
−0.111627 + 0.993750i \(0.535606\pi\)
\(44\) 20.4684 3.08573
\(45\) 0 0
\(46\) −22.8327 −3.36650
\(47\) −0.357351 −0.0521250 −0.0260625 0.999660i \(-0.508297\pi\)
−0.0260625 + 0.999660i \(0.508297\pi\)
\(48\) 0 0
\(49\) 0.646666 0.0923809
\(50\) 9.52588 1.34716
\(51\) 0 0
\(52\) 26.4618 3.66960
\(53\) 1.63498 0.224582 0.112291 0.993675i \(-0.464181\pi\)
0.112291 + 0.993675i \(0.464181\pi\)
\(54\) 0 0
\(55\) 5.00270 0.674563
\(56\) −19.3333 −2.58352
\(57\) 0 0
\(58\) 0 0
\(59\) 4.85026 0.631450 0.315725 0.948851i \(-0.397752\pi\)
0.315725 + 0.948851i \(0.397752\pi\)
\(60\) 0 0
\(61\) 1.55510 0.199110 0.0995551 0.995032i \(-0.468258\pi\)
0.0995551 + 0.995032i \(0.468258\pi\)
\(62\) −4.03961 −0.513030
\(63\) 0 0
\(64\) 4.68444 0.585555
\(65\) 6.46755 0.802201
\(66\) 0 0
\(67\) −9.66950 −1.18132 −0.590659 0.806922i \(-0.701132\pi\)
−0.590659 + 0.806922i \(0.701132\pi\)
\(68\) −11.3534 −1.37680
\(69\) 0 0
\(70\) −8.22432 −0.982994
\(71\) 8.18108 0.970915 0.485458 0.874260i \(-0.338653\pi\)
0.485458 + 0.874260i \(0.338653\pi\)
\(72\) 0 0
\(73\) 11.3786 1.33177 0.665884 0.746055i \(-0.268054\pi\)
0.665884 + 0.746055i \(0.268054\pi\)
\(74\) −20.0961 −2.33612
\(75\) 0 0
\(76\) 8.80452 1.00995
\(77\) 12.0404 1.37213
\(78\) 0 0
\(79\) −8.13075 −0.914781 −0.457391 0.889266i \(-0.651216\pi\)
−0.457391 + 0.889266i \(0.651216\pi\)
\(80\) 9.99170 1.11711
\(81\) 0 0
\(82\) 8.18185 0.903535
\(83\) 0.481835 0.0528882 0.0264441 0.999650i \(-0.491582\pi\)
0.0264441 + 0.999650i \(0.491582\pi\)
\(84\) 0 0
\(85\) −2.77488 −0.300978
\(86\) 3.78965 0.408648
\(87\) 0 0
\(88\) −30.4421 −3.24514
\(89\) 1.99621 0.211598 0.105799 0.994388i \(-0.466260\pi\)
0.105799 + 0.994388i \(0.466260\pi\)
\(90\) 0 0
\(91\) 15.5660 1.63176
\(92\) 41.4639 4.32291
\(93\) 0 0
\(94\) 0.925040 0.0954106
\(95\) 2.15192 0.220782
\(96\) 0 0
\(97\) 4.63516 0.470629 0.235315 0.971919i \(-0.424388\pi\)
0.235315 + 0.971919i \(0.424388\pi\)
\(98\) −1.67396 −0.169096
\(99\) 0 0
\(100\) −17.2989 −1.72989
\(101\) 1.89907 0.188964 0.0944822 0.995527i \(-0.469880\pi\)
0.0944822 + 0.995527i \(0.469880\pi\)
\(102\) 0 0
\(103\) −8.22122 −0.810061 −0.405030 0.914303i \(-0.632739\pi\)
−0.405030 + 0.914303i \(0.632739\pi\)
\(104\) −39.3560 −3.85917
\(105\) 0 0
\(106\) −4.23233 −0.411080
\(107\) −4.45778 −0.430950 −0.215475 0.976509i \(-0.569130\pi\)
−0.215475 + 0.976509i \(0.569130\pi\)
\(108\) 0 0
\(109\) 17.5256 1.67865 0.839324 0.543631i \(-0.182951\pi\)
0.839324 + 0.543631i \(0.182951\pi\)
\(110\) −12.9500 −1.23473
\(111\) 0 0
\(112\) 24.0479 2.27231
\(113\) −7.93117 −0.746102 −0.373051 0.927811i \(-0.621688\pi\)
−0.373051 + 0.927811i \(0.621688\pi\)
\(114\) 0 0
\(115\) 10.1342 0.945021
\(116\) 0 0
\(117\) 0 0
\(118\) −12.5554 −1.15582
\(119\) −6.67855 −0.612222
\(120\) 0 0
\(121\) 7.95884 0.723531
\(122\) −4.02554 −0.364455
\(123\) 0 0
\(124\) 7.33587 0.658780
\(125\) −9.97274 −0.891989
\(126\) 0 0
\(127\) −3.35542 −0.297745 −0.148873 0.988856i \(-0.547564\pi\)
−0.148873 + 0.988856i \(0.547564\pi\)
\(128\) 4.93114 0.435855
\(129\) 0 0
\(130\) −16.7419 −1.46836
\(131\) 8.69320 0.759528 0.379764 0.925083i \(-0.376005\pi\)
0.379764 + 0.925083i \(0.376005\pi\)
\(132\) 0 0
\(133\) 5.17921 0.449095
\(134\) 25.0305 2.16231
\(135\) 0 0
\(136\) 16.8856 1.44792
\(137\) −3.22382 −0.275429 −0.137715 0.990472i \(-0.543976\pi\)
−0.137715 + 0.990472i \(0.543976\pi\)
\(138\) 0 0
\(139\) −22.4884 −1.90744 −0.953721 0.300693i \(-0.902782\pi\)
−0.953721 + 0.300693i \(0.902782\pi\)
\(140\) 14.9352 1.26226
\(141\) 0 0
\(142\) −21.1776 −1.77718
\(143\) 24.5102 2.04965
\(144\) 0 0
\(145\) 0 0
\(146\) −29.4548 −2.43769
\(147\) 0 0
\(148\) 36.4942 2.99981
\(149\) −1.28934 −0.105627 −0.0528133 0.998604i \(-0.516819\pi\)
−0.0528133 + 0.998604i \(0.516819\pi\)
\(150\) 0 0
\(151\) 6.55195 0.533190 0.266595 0.963809i \(-0.414101\pi\)
0.266595 + 0.963809i \(0.414101\pi\)
\(152\) −13.0947 −1.06212
\(153\) 0 0
\(154\) −31.1679 −2.51158
\(155\) 1.79296 0.144014
\(156\) 0 0
\(157\) −24.1178 −1.92481 −0.962407 0.271612i \(-0.912443\pi\)
−0.962407 + 0.271612i \(0.912443\pi\)
\(158\) 21.0473 1.67443
\(159\) 0 0
\(160\) −9.79893 −0.774674
\(161\) 24.3909 1.92227
\(162\) 0 0
\(163\) 16.1434 1.26445 0.632223 0.774786i \(-0.282143\pi\)
0.632223 + 0.774786i \(0.282143\pi\)
\(164\) −14.8581 −1.16023
\(165\) 0 0
\(166\) −1.24728 −0.0968076
\(167\) 5.42045 0.419447 0.209723 0.977761i \(-0.432744\pi\)
0.209723 + 0.977761i \(0.432744\pi\)
\(168\) 0 0
\(169\) 18.6872 1.43747
\(170\) 7.18307 0.550916
\(171\) 0 0
\(172\) −6.88195 −0.524744
\(173\) 11.2923 0.858537 0.429268 0.903177i \(-0.358771\pi\)
0.429268 + 0.903177i \(0.358771\pi\)
\(174\) 0 0
\(175\) −10.1760 −0.769231
\(176\) 37.8658 2.85424
\(177\) 0 0
\(178\) −5.16739 −0.387312
\(179\) 10.6541 0.796323 0.398161 0.917315i \(-0.369648\pi\)
0.398161 + 0.917315i \(0.369648\pi\)
\(180\) 0 0
\(181\) −12.2934 −0.913763 −0.456882 0.889527i \(-0.651034\pi\)
−0.456882 + 0.889527i \(0.651034\pi\)
\(182\) −40.2943 −2.98681
\(183\) 0 0
\(184\) −61.6682 −4.54624
\(185\) 8.91958 0.655781
\(186\) 0 0
\(187\) −10.5160 −0.769009
\(188\) −1.67986 −0.122516
\(189\) 0 0
\(190\) −5.57047 −0.404124
\(191\) 1.68311 0.121786 0.0608929 0.998144i \(-0.480605\pi\)
0.0608929 + 0.998144i \(0.480605\pi\)
\(192\) 0 0
\(193\) 10.5761 0.761284 0.380642 0.924722i \(-0.375703\pi\)
0.380642 + 0.924722i \(0.375703\pi\)
\(194\) −11.9986 −0.861449
\(195\) 0 0
\(196\) 3.03989 0.217135
\(197\) 12.1150 0.863161 0.431581 0.902074i \(-0.357956\pi\)
0.431581 + 0.902074i \(0.357956\pi\)
\(198\) 0 0
\(199\) 4.99592 0.354151 0.177076 0.984197i \(-0.443336\pi\)
0.177076 + 0.984197i \(0.443336\pi\)
\(200\) 25.7282 1.81926
\(201\) 0 0
\(202\) −4.91594 −0.345884
\(203\) 0 0
\(204\) 0 0
\(205\) −3.63149 −0.253634
\(206\) 21.2815 1.48275
\(207\) 0 0
\(208\) 48.9534 3.39431
\(209\) 8.15518 0.564106
\(210\) 0 0
\(211\) −19.2108 −1.32253 −0.661263 0.750154i \(-0.729979\pi\)
−0.661263 + 0.750154i \(0.729979\pi\)
\(212\) 7.68585 0.527866
\(213\) 0 0
\(214\) 11.5394 0.788819
\(215\) −1.68202 −0.114713
\(216\) 0 0
\(217\) 4.31528 0.292941
\(218\) −45.3668 −3.07263
\(219\) 0 0
\(220\) 23.5170 1.58552
\(221\) −13.5953 −0.914517
\(222\) 0 0
\(223\) −21.2442 −1.42262 −0.711308 0.702881i \(-0.751897\pi\)
−0.711308 + 0.702881i \(0.751897\pi\)
\(224\) −23.5840 −1.57577
\(225\) 0 0
\(226\) 20.5307 1.36568
\(227\) −19.7559 −1.31125 −0.655624 0.755088i \(-0.727594\pi\)
−0.655624 + 0.755088i \(0.727594\pi\)
\(228\) 0 0
\(229\) 20.7139 1.36881 0.684405 0.729102i \(-0.260062\pi\)
0.684405 + 0.729102i \(0.260062\pi\)
\(230\) −26.2335 −1.72978
\(231\) 0 0
\(232\) 0 0
\(233\) −20.7745 −1.36098 −0.680492 0.732756i \(-0.738234\pi\)
−0.680492 + 0.732756i \(0.738234\pi\)
\(234\) 0 0
\(235\) −0.410576 −0.0267830
\(236\) 22.8004 1.48418
\(237\) 0 0
\(238\) 17.2881 1.12062
\(239\) 2.27481 0.147145 0.0735727 0.997290i \(-0.476560\pi\)
0.0735727 + 0.997290i \(0.476560\pi\)
\(240\) 0 0
\(241\) 11.7783 0.758706 0.379353 0.925252i \(-0.376147\pi\)
0.379353 + 0.925252i \(0.376147\pi\)
\(242\) −20.6023 −1.32437
\(243\) 0 0
\(244\) 7.31032 0.467995
\(245\) 0.742982 0.0474674
\(246\) 0 0
\(247\) 10.5431 0.670843
\(248\) −10.9104 −0.692814
\(249\) 0 0
\(250\) 25.8155 1.63271
\(251\) 1.42714 0.0900805 0.0450402 0.998985i \(-0.485658\pi\)
0.0450402 + 0.998985i \(0.485658\pi\)
\(252\) 0 0
\(253\) 38.4059 2.41456
\(254\) 8.68585 0.544999
\(255\) 0 0
\(256\) −22.1337 −1.38335
\(257\) 28.6450 1.78683 0.893414 0.449234i \(-0.148303\pi\)
0.893414 + 0.449234i \(0.148303\pi\)
\(258\) 0 0
\(259\) 21.4675 1.33393
\(260\) 30.4031 1.88552
\(261\) 0 0
\(262\) −22.5032 −1.39025
\(263\) −18.5023 −1.14090 −0.570451 0.821332i \(-0.693231\pi\)
−0.570451 + 0.821332i \(0.693231\pi\)
\(264\) 0 0
\(265\) 1.87850 0.115396
\(266\) −13.4069 −0.822032
\(267\) 0 0
\(268\) −45.4550 −2.77661
\(269\) 28.7074 1.75032 0.875160 0.483834i \(-0.160756\pi\)
0.875160 + 0.483834i \(0.160756\pi\)
\(270\) 0 0
\(271\) 1.80441 0.109610 0.0548050 0.998497i \(-0.482546\pi\)
0.0548050 + 0.998497i \(0.482546\pi\)
\(272\) −21.0033 −1.27351
\(273\) 0 0
\(274\) 8.34518 0.504151
\(275\) −16.0231 −0.966227
\(276\) 0 0
\(277\) −14.1949 −0.852890 −0.426445 0.904513i \(-0.640234\pi\)
−0.426445 + 0.904513i \(0.640234\pi\)
\(278\) 58.2136 3.49142
\(279\) 0 0
\(280\) −22.2128 −1.32747
\(281\) −18.8216 −1.12280 −0.561400 0.827545i \(-0.689737\pi\)
−0.561400 + 0.827545i \(0.689737\pi\)
\(282\) 0 0
\(283\) 21.8832 1.30082 0.650410 0.759584i \(-0.274597\pi\)
0.650410 + 0.759584i \(0.274597\pi\)
\(284\) 38.4582 2.28207
\(285\) 0 0
\(286\) −63.4473 −3.75172
\(287\) −8.74022 −0.515919
\(288\) 0 0
\(289\) −11.1670 −0.656882
\(290\) 0 0
\(291\) 0 0
\(292\) 53.4895 3.13023
\(293\) −0.00776592 −0.000453690 0 −0.000226845 1.00000i \(-0.500072\pi\)
−0.000226845 1.00000i \(0.500072\pi\)
\(294\) 0 0
\(295\) 5.57267 0.324453
\(296\) −54.2769 −3.15478
\(297\) 0 0
\(298\) 3.33758 0.193341
\(299\) 49.6517 2.87143
\(300\) 0 0
\(301\) −4.04827 −0.233338
\(302\) −16.9604 −0.975961
\(303\) 0 0
\(304\) 16.2880 0.934183
\(305\) 1.78672 0.102307
\(306\) 0 0
\(307\) 0.214941 0.0122673 0.00613367 0.999981i \(-0.498048\pi\)
0.00613367 + 0.999981i \(0.498048\pi\)
\(308\) 56.6005 3.22511
\(309\) 0 0
\(310\) −4.64127 −0.263607
\(311\) −28.2266 −1.60058 −0.800291 0.599612i \(-0.795322\pi\)
−0.800291 + 0.599612i \(0.795322\pi\)
\(312\) 0 0
\(313\) −10.2302 −0.578244 −0.289122 0.957292i \(-0.593363\pi\)
−0.289122 + 0.957292i \(0.593363\pi\)
\(314\) 62.4316 3.52322
\(315\) 0 0
\(316\) −38.2216 −2.15013
\(317\) −23.7789 −1.33555 −0.667777 0.744361i \(-0.732754\pi\)
−0.667777 + 0.744361i \(0.732754\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 5.38216 0.300872
\(321\) 0 0
\(322\) −63.1384 −3.51857
\(323\) −4.52349 −0.251694
\(324\) 0 0
\(325\) −20.7148 −1.14905
\(326\) −41.7888 −2.31447
\(327\) 0 0
\(328\) 22.0981 1.22016
\(329\) −0.988168 −0.0544795
\(330\) 0 0
\(331\) −7.07856 −0.389073 −0.194536 0.980895i \(-0.562320\pi\)
−0.194536 + 0.980895i \(0.562320\pi\)
\(332\) 2.26504 0.124310
\(333\) 0 0
\(334\) −14.0314 −0.767763
\(335\) −11.1097 −0.606987
\(336\) 0 0
\(337\) −7.02402 −0.382623 −0.191311 0.981529i \(-0.561274\pi\)
−0.191311 + 0.981529i \(0.561274\pi\)
\(338\) −48.3737 −2.63118
\(339\) 0 0
\(340\) −13.0444 −0.707429
\(341\) 6.79484 0.367961
\(342\) 0 0
\(343\) −17.5686 −0.948617
\(344\) 10.2353 0.551852
\(345\) 0 0
\(346\) −29.2313 −1.57148
\(347\) −10.0795 −0.541094 −0.270547 0.962707i \(-0.587205\pi\)
−0.270547 + 0.962707i \(0.587205\pi\)
\(348\) 0 0
\(349\) −1.64246 −0.0879187 −0.0439593 0.999033i \(-0.513997\pi\)
−0.0439593 + 0.999033i \(0.513997\pi\)
\(350\) 26.3416 1.40802
\(351\) 0 0
\(352\) −37.1353 −1.97932
\(353\) −24.1232 −1.28395 −0.641975 0.766726i \(-0.721885\pi\)
−0.641975 + 0.766726i \(0.721885\pi\)
\(354\) 0 0
\(355\) 9.39959 0.498878
\(356\) 9.38391 0.497346
\(357\) 0 0
\(358\) −27.5792 −1.45760
\(359\) 12.8958 0.680612 0.340306 0.940315i \(-0.389469\pi\)
0.340306 + 0.940315i \(0.389469\pi\)
\(360\) 0 0
\(361\) −15.4920 −0.815370
\(362\) 31.8228 1.67257
\(363\) 0 0
\(364\) 73.1739 3.83535
\(365\) 13.0734 0.684293
\(366\) 0 0
\(367\) −31.7420 −1.65692 −0.828459 0.560050i \(-0.810782\pi\)
−0.828459 + 0.560050i \(0.810782\pi\)
\(368\) 76.7067 3.99861
\(369\) 0 0
\(370\) −23.0893 −1.20035
\(371\) 4.52116 0.234727
\(372\) 0 0
\(373\) 4.75511 0.246210 0.123105 0.992394i \(-0.460715\pi\)
0.123105 + 0.992394i \(0.460715\pi\)
\(374\) 27.2218 1.40761
\(375\) 0 0
\(376\) 2.49841 0.128846
\(377\) 0 0
\(378\) 0 0
\(379\) 2.11960 0.108877 0.0544383 0.998517i \(-0.482663\pi\)
0.0544383 + 0.998517i \(0.482663\pi\)
\(380\) 10.1159 0.518934
\(381\) 0 0
\(382\) −4.35691 −0.222919
\(383\) −14.6407 −0.748103 −0.374051 0.927408i \(-0.622032\pi\)
−0.374051 + 0.927408i \(0.622032\pi\)
\(384\) 0 0
\(385\) 13.8338 0.705034
\(386\) −27.3773 −1.39347
\(387\) 0 0
\(388\) 21.7893 1.10618
\(389\) −1.93401 −0.0980583 −0.0490292 0.998797i \(-0.515613\pi\)
−0.0490292 + 0.998797i \(0.515613\pi\)
\(390\) 0 0
\(391\) −21.3029 −1.07733
\(392\) −4.52115 −0.228353
\(393\) 0 0
\(394\) −31.3611 −1.57995
\(395\) −9.34177 −0.470035
\(396\) 0 0
\(397\) −30.0191 −1.50662 −0.753309 0.657667i \(-0.771543\pi\)
−0.753309 + 0.657667i \(0.771543\pi\)
\(398\) −12.9324 −0.648245
\(399\) 0 0
\(400\) −32.0023 −1.60011
\(401\) −15.7085 −0.784444 −0.392222 0.919871i \(-0.628293\pi\)
−0.392222 + 0.919871i \(0.628293\pi\)
\(402\) 0 0
\(403\) 8.78446 0.437585
\(404\) 8.92728 0.444149
\(405\) 0 0
\(406\) 0 0
\(407\) 33.8027 1.67554
\(408\) 0 0
\(409\) −4.87216 −0.240913 −0.120456 0.992719i \(-0.538436\pi\)
−0.120456 + 0.992719i \(0.538436\pi\)
\(410\) 9.40048 0.464257
\(411\) 0 0
\(412\) −38.6469 −1.90399
\(413\) 13.4122 0.659973
\(414\) 0 0
\(415\) 0.553600 0.0271752
\(416\) −48.0090 −2.35383
\(417\) 0 0
\(418\) −21.1105 −1.03255
\(419\) −19.0131 −0.928853 −0.464426 0.885612i \(-0.653739\pi\)
−0.464426 + 0.885612i \(0.653739\pi\)
\(420\) 0 0
\(421\) 12.5171 0.610047 0.305023 0.952345i \(-0.401336\pi\)
0.305023 + 0.952345i \(0.401336\pi\)
\(422\) 49.7292 2.42078
\(423\) 0 0
\(424\) −11.4310 −0.555136
\(425\) 8.88763 0.431113
\(426\) 0 0
\(427\) 4.30026 0.208104
\(428\) −20.9554 −1.01292
\(429\) 0 0
\(430\) 4.35409 0.209973
\(431\) −35.3970 −1.70501 −0.852507 0.522716i \(-0.824919\pi\)
−0.852507 + 0.522716i \(0.824919\pi\)
\(432\) 0 0
\(433\) 5.85494 0.281371 0.140685 0.990054i \(-0.455069\pi\)
0.140685 + 0.990054i \(0.455069\pi\)
\(434\) −11.1706 −0.536204
\(435\) 0 0
\(436\) 82.3856 3.94555
\(437\) 16.5204 0.790277
\(438\) 0 0
\(439\) 24.8297 1.18506 0.592530 0.805549i \(-0.298129\pi\)
0.592530 + 0.805549i \(0.298129\pi\)
\(440\) −34.9762 −1.66743
\(441\) 0 0
\(442\) 35.1928 1.67395
\(443\) −23.7488 −1.12834 −0.564170 0.825659i \(-0.690804\pi\)
−0.564170 + 0.825659i \(0.690804\pi\)
\(444\) 0 0
\(445\) 2.29353 0.108724
\(446\) 54.9928 2.60398
\(447\) 0 0
\(448\) 12.9537 0.612005
\(449\) −6.44854 −0.304325 −0.152163 0.988355i \(-0.548624\pi\)
−0.152163 + 0.988355i \(0.548624\pi\)
\(450\) 0 0
\(451\) −13.7623 −0.648043
\(452\) −37.2834 −1.75366
\(453\) 0 0
\(454\) 51.1403 2.40013
\(455\) 17.8845 0.838437
\(456\) 0 0
\(457\) −2.62569 −0.122824 −0.0614122 0.998112i \(-0.519560\pi\)
−0.0614122 + 0.998112i \(0.519560\pi\)
\(458\) −53.6200 −2.50550
\(459\) 0 0
\(460\) 47.6396 2.22121
\(461\) 7.15794 0.333378 0.166689 0.986009i \(-0.446692\pi\)
0.166689 + 0.986009i \(0.446692\pi\)
\(462\) 0 0
\(463\) −18.2411 −0.847737 −0.423868 0.905724i \(-0.639328\pi\)
−0.423868 + 0.905724i \(0.639328\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 53.7770 2.49117
\(467\) −35.4677 −1.64125 −0.820624 0.571468i \(-0.806374\pi\)
−0.820624 + 0.571468i \(0.806374\pi\)
\(468\) 0 0
\(469\) −26.7387 −1.23468
\(470\) 1.06282 0.0490241
\(471\) 0 0
\(472\) −33.9105 −1.56086
\(473\) −6.37440 −0.293095
\(474\) 0 0
\(475\) −6.89235 −0.316243
\(476\) −31.3950 −1.43899
\(477\) 0 0
\(478\) −5.88859 −0.269338
\(479\) −26.1004 −1.19256 −0.596280 0.802777i \(-0.703355\pi\)
−0.596280 + 0.802777i \(0.703355\pi\)
\(480\) 0 0
\(481\) 43.7006 1.99258
\(482\) −30.4893 −1.38875
\(483\) 0 0
\(484\) 37.4135 1.70061
\(485\) 5.32553 0.241820
\(486\) 0 0
\(487\) 22.1493 1.00368 0.501839 0.864961i \(-0.332657\pi\)
0.501839 + 0.864961i \(0.332657\pi\)
\(488\) −10.8724 −0.492173
\(489\) 0 0
\(490\) −1.92329 −0.0868852
\(491\) 11.7483 0.530193 0.265096 0.964222i \(-0.414596\pi\)
0.265096 + 0.964222i \(0.414596\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) −27.2920 −1.22792
\(495\) 0 0
\(496\) 13.5711 0.609359
\(497\) 22.6228 1.01477
\(498\) 0 0
\(499\) −16.3753 −0.733060 −0.366530 0.930406i \(-0.619454\pi\)
−0.366530 + 0.930406i \(0.619454\pi\)
\(500\) −46.8805 −2.09656
\(501\) 0 0
\(502\) −3.69431 −0.164885
\(503\) −14.8354 −0.661476 −0.330738 0.943723i \(-0.607298\pi\)
−0.330738 + 0.943723i \(0.607298\pi\)
\(504\) 0 0
\(505\) 2.18192 0.0970942
\(506\) −99.4177 −4.41965
\(507\) 0 0
\(508\) −15.7734 −0.699831
\(509\) −31.5733 −1.39946 −0.699732 0.714406i \(-0.746697\pi\)
−0.699732 + 0.714406i \(0.746697\pi\)
\(510\) 0 0
\(511\) 31.4649 1.39192
\(512\) 47.4330 2.09626
\(513\) 0 0
\(514\) −74.1507 −3.27065
\(515\) −9.44571 −0.416228
\(516\) 0 0
\(517\) −1.55597 −0.0684314
\(518\) −55.5709 −2.44165
\(519\) 0 0
\(520\) −45.2178 −1.98293
\(521\) 8.03875 0.352184 0.176092 0.984374i \(-0.443654\pi\)
0.176092 + 0.984374i \(0.443654\pi\)
\(522\) 0 0
\(523\) −1.48633 −0.0649927 −0.0324963 0.999472i \(-0.510346\pi\)
−0.0324963 + 0.999472i \(0.510346\pi\)
\(524\) 40.8656 1.78522
\(525\) 0 0
\(526\) 47.8951 2.08833
\(527\) −3.76894 −0.164178
\(528\) 0 0
\(529\) 54.8008 2.38264
\(530\) −4.86270 −0.211222
\(531\) 0 0
\(532\) 24.3468 1.05557
\(533\) −17.7921 −0.770663
\(534\) 0 0
\(535\) −5.12173 −0.221432
\(536\) 67.6041 2.92005
\(537\) 0 0
\(538\) −74.3120 −3.20382
\(539\) 2.81570 0.121281
\(540\) 0 0
\(541\) 38.8306 1.66946 0.834728 0.550662i \(-0.185625\pi\)
0.834728 + 0.550662i \(0.185625\pi\)
\(542\) −4.67090 −0.200632
\(543\) 0 0
\(544\) 20.5981 0.883135
\(545\) 20.1359 0.862528
\(546\) 0 0
\(547\) −17.3495 −0.741809 −0.370905 0.928671i \(-0.620952\pi\)
−0.370905 + 0.928671i \(0.620952\pi\)
\(548\) −15.1547 −0.647378
\(549\) 0 0
\(550\) 41.4774 1.76860
\(551\) 0 0
\(552\) 0 0
\(553\) −22.4837 −0.956102
\(554\) 36.7450 1.56115
\(555\) 0 0
\(556\) −105.715 −4.48332
\(557\) −30.4259 −1.28919 −0.644593 0.764526i \(-0.722973\pi\)
−0.644593 + 0.764526i \(0.722973\pi\)
\(558\) 0 0
\(559\) −8.24091 −0.348553
\(560\) 27.6296 1.16757
\(561\) 0 0
\(562\) 48.7215 2.05519
\(563\) 12.1911 0.513793 0.256897 0.966439i \(-0.417300\pi\)
0.256897 + 0.966439i \(0.417300\pi\)
\(564\) 0 0
\(565\) −9.11246 −0.383364
\(566\) −56.6469 −2.38104
\(567\) 0 0
\(568\) −57.1978 −2.39997
\(569\) 3.15785 0.132384 0.0661920 0.997807i \(-0.478915\pi\)
0.0661920 + 0.997807i \(0.478915\pi\)
\(570\) 0 0
\(571\) 24.3414 1.01865 0.509327 0.860573i \(-0.329894\pi\)
0.509327 + 0.860573i \(0.329894\pi\)
\(572\) 115.219 4.81757
\(573\) 0 0
\(574\) 22.6250 0.944347
\(575\) −32.4588 −1.35362
\(576\) 0 0
\(577\) 7.99079 0.332661 0.166330 0.986070i \(-0.446808\pi\)
0.166330 + 0.986070i \(0.446808\pi\)
\(578\) 28.9069 1.20237
\(579\) 0 0
\(580\) 0 0
\(581\) 1.33240 0.0552772
\(582\) 0 0
\(583\) 7.11901 0.294839
\(584\) −79.5535 −3.29195
\(585\) 0 0
\(586\) 0.0201029 0.000830443 0
\(587\) 42.3655 1.74861 0.874305 0.485377i \(-0.161318\pi\)
0.874305 + 0.485377i \(0.161318\pi\)
\(588\) 0 0
\(589\) 2.92281 0.120432
\(590\) −14.4254 −0.593886
\(591\) 0 0
\(592\) 67.5130 2.77477
\(593\) 26.3761 1.08314 0.541569 0.840656i \(-0.317831\pi\)
0.541569 + 0.840656i \(0.317831\pi\)
\(594\) 0 0
\(595\) −7.67327 −0.314573
\(596\) −6.06100 −0.248268
\(597\) 0 0
\(598\) −128.528 −5.25592
\(599\) −6.35768 −0.259768 −0.129884 0.991529i \(-0.541460\pi\)
−0.129884 + 0.991529i \(0.541460\pi\)
\(600\) 0 0
\(601\) −16.9016 −0.689430 −0.344715 0.938707i \(-0.612024\pi\)
−0.344715 + 0.938707i \(0.612024\pi\)
\(602\) 10.4794 0.427107
\(603\) 0 0
\(604\) 30.7999 1.25323
\(605\) 9.14425 0.371767
\(606\) 0 0
\(607\) −7.41550 −0.300986 −0.150493 0.988611i \(-0.548086\pi\)
−0.150493 + 0.988611i \(0.548086\pi\)
\(608\) −15.9738 −0.647823
\(609\) 0 0
\(610\) −4.62511 −0.187265
\(611\) −2.01158 −0.0813797
\(612\) 0 0
\(613\) −28.9255 −1.16829 −0.584145 0.811649i \(-0.698570\pi\)
−0.584145 + 0.811649i \(0.698570\pi\)
\(614\) −0.556398 −0.0224544
\(615\) 0 0
\(616\) −84.1804 −3.39173
\(617\) 34.9128 1.40554 0.702768 0.711419i \(-0.251947\pi\)
0.702768 + 0.711419i \(0.251947\pi\)
\(618\) 0 0
\(619\) 45.1342 1.81410 0.907048 0.421028i \(-0.138331\pi\)
0.907048 + 0.421028i \(0.138331\pi\)
\(620\) 8.42849 0.338496
\(621\) 0 0
\(622\) 73.0674 2.92974
\(623\) 5.52003 0.221155
\(624\) 0 0
\(625\) 6.94155 0.277662
\(626\) 26.4819 1.05843
\(627\) 0 0
\(628\) −113.375 −4.52415
\(629\) −18.7496 −0.747596
\(630\) 0 0
\(631\) 29.2602 1.16483 0.582415 0.812892i \(-0.302108\pi\)
0.582415 + 0.812892i \(0.302108\pi\)
\(632\) 56.8460 2.26121
\(633\) 0 0
\(634\) 61.5540 2.44462
\(635\) −3.85518 −0.152988
\(636\) 0 0
\(637\) 3.64017 0.144229
\(638\) 0 0
\(639\) 0 0
\(640\) 5.66560 0.223952
\(641\) −17.7234 −0.700031 −0.350016 0.936744i \(-0.613824\pi\)
−0.350016 + 0.936744i \(0.613824\pi\)
\(642\) 0 0
\(643\) 38.4443 1.51609 0.758047 0.652200i \(-0.226154\pi\)
0.758047 + 0.652200i \(0.226154\pi\)
\(644\) 114.659 4.51818
\(645\) 0 0
\(646\) 11.7095 0.460705
\(647\) −40.5910 −1.59580 −0.797899 0.602792i \(-0.794055\pi\)
−0.797899 + 0.602792i \(0.794055\pi\)
\(648\) 0 0
\(649\) 21.1189 0.828989
\(650\) 53.6225 2.10325
\(651\) 0 0
\(652\) 75.8879 2.97200
\(653\) 26.6169 1.04160 0.520800 0.853679i \(-0.325634\pi\)
0.520800 + 0.853679i \(0.325634\pi\)
\(654\) 0 0
\(655\) 9.98798 0.390263
\(656\) −27.4870 −1.07319
\(657\) 0 0
\(658\) 2.55798 0.0997203
\(659\) 23.1105 0.900259 0.450130 0.892963i \(-0.351378\pi\)
0.450130 + 0.892963i \(0.351378\pi\)
\(660\) 0 0
\(661\) 23.4613 0.912537 0.456269 0.889842i \(-0.349186\pi\)
0.456269 + 0.889842i \(0.349186\pi\)
\(662\) 18.3236 0.712166
\(663\) 0 0
\(664\) −3.36874 −0.130732
\(665\) 5.95062 0.230755
\(666\) 0 0
\(667\) 0 0
\(668\) 25.4808 0.985882
\(669\) 0 0
\(670\) 28.7586 1.11104
\(671\) 6.77118 0.261398
\(672\) 0 0
\(673\) −26.9621 −1.03931 −0.519656 0.854376i \(-0.673940\pi\)
−0.519656 + 0.854376i \(0.673940\pi\)
\(674\) 18.1824 0.700360
\(675\) 0 0
\(676\) 87.8460 3.37869
\(677\) −33.5314 −1.28872 −0.644358 0.764724i \(-0.722875\pi\)
−0.644358 + 0.764724i \(0.722875\pi\)
\(678\) 0 0
\(679\) 12.8174 0.491888
\(680\) 19.4005 0.743976
\(681\) 0 0
\(682\) −17.5891 −0.673523
\(683\) 12.3343 0.471957 0.235979 0.971758i \(-0.424170\pi\)
0.235979 + 0.971758i \(0.424170\pi\)
\(684\) 0 0
\(685\) −3.70398 −0.141522
\(686\) 45.4782 1.73637
\(687\) 0 0
\(688\) −12.7313 −0.485378
\(689\) 9.20355 0.350627
\(690\) 0 0
\(691\) 2.02123 0.0768913 0.0384457 0.999261i \(-0.487759\pi\)
0.0384457 + 0.999261i \(0.487759\pi\)
\(692\) 53.0836 2.01793
\(693\) 0 0
\(694\) 26.0918 0.990429
\(695\) −25.8379 −0.980087
\(696\) 0 0
\(697\) 7.63365 0.289145
\(698\) 4.25167 0.160928
\(699\) 0 0
\(700\) −47.8359 −1.80803
\(701\) 33.6980 1.27275 0.636377 0.771378i \(-0.280432\pi\)
0.636377 + 0.771378i \(0.280432\pi\)
\(702\) 0 0
\(703\) 14.5403 0.548398
\(704\) 20.3969 0.768737
\(705\) 0 0
\(706\) 62.4455 2.35017
\(707\) 5.25142 0.197500
\(708\) 0 0
\(709\) 12.6531 0.475198 0.237599 0.971363i \(-0.423640\pi\)
0.237599 + 0.971363i \(0.423640\pi\)
\(710\) −24.3318 −0.913157
\(711\) 0 0
\(712\) −13.9564 −0.523040
\(713\) 13.7647 0.515490
\(714\) 0 0
\(715\) 28.1609 1.05316
\(716\) 50.0834 1.87170
\(717\) 0 0
\(718\) −33.3820 −1.24581
\(719\) −13.3377 −0.497411 −0.248706 0.968579i \(-0.580005\pi\)
−0.248706 + 0.968579i \(0.580005\pi\)
\(720\) 0 0
\(721\) −22.7338 −0.846651
\(722\) 40.1027 1.49247
\(723\) 0 0
\(724\) −57.7898 −2.14774
\(725\) 0 0
\(726\) 0 0
\(727\) −41.7683 −1.54910 −0.774551 0.632512i \(-0.782024\pi\)
−0.774551 + 0.632512i \(0.782024\pi\)
\(728\) −108.830 −4.03349
\(729\) 0 0
\(730\) −33.8418 −1.25254
\(731\) 3.53573 0.130774
\(732\) 0 0
\(733\) −42.5635 −1.57212 −0.786059 0.618152i \(-0.787882\pi\)
−0.786059 + 0.618152i \(0.787882\pi\)
\(734\) 82.1674 3.03285
\(735\) 0 0
\(736\) −75.2268 −2.77290
\(737\) −42.1027 −1.55087
\(738\) 0 0
\(739\) 22.2405 0.818129 0.409064 0.912506i \(-0.365855\pi\)
0.409064 + 0.912506i \(0.365855\pi\)
\(740\) 41.9298 1.54137
\(741\) 0 0
\(742\) −11.7035 −0.429648
\(743\) 28.0714 1.02984 0.514920 0.857238i \(-0.327822\pi\)
0.514920 + 0.857238i \(0.327822\pi\)
\(744\) 0 0
\(745\) −1.48137 −0.0542733
\(746\) −12.3091 −0.450668
\(747\) 0 0
\(748\) −49.4345 −1.80750
\(749\) −12.3269 −0.450416
\(750\) 0 0
\(751\) 28.5813 1.04295 0.521473 0.853268i \(-0.325383\pi\)
0.521473 + 0.853268i \(0.325383\pi\)
\(752\) −3.10768 −0.113325
\(753\) 0 0
\(754\) 0 0
\(755\) 7.52781 0.273965
\(756\) 0 0
\(757\) −21.4521 −0.779689 −0.389844 0.920881i \(-0.627471\pi\)
−0.389844 + 0.920881i \(0.627471\pi\)
\(758\) −5.48680 −0.199290
\(759\) 0 0
\(760\) −15.0451 −0.545743
\(761\) 21.4222 0.776553 0.388277 0.921543i \(-0.373071\pi\)
0.388277 + 0.921543i \(0.373071\pi\)
\(762\) 0 0
\(763\) 48.4629 1.75447
\(764\) 7.91209 0.286249
\(765\) 0 0
\(766\) 37.8989 1.36934
\(767\) 27.3028 0.985846
\(768\) 0 0
\(769\) −22.7152 −0.819130 −0.409565 0.912281i \(-0.634320\pi\)
−0.409565 + 0.912281i \(0.634320\pi\)
\(770\) −35.8101 −1.29051
\(771\) 0 0
\(772\) 49.7168 1.78935
\(773\) 0.180228 0.00648235 0.00324118 0.999995i \(-0.498968\pi\)
0.00324118 + 0.999995i \(0.498968\pi\)
\(774\) 0 0
\(775\) −5.74266 −0.206282
\(776\) −32.4066 −1.16333
\(777\) 0 0
\(778\) 5.00639 0.179488
\(779\) −5.91989 −0.212102
\(780\) 0 0
\(781\) 35.6219 1.27465
\(782\) 55.1447 1.97197
\(783\) 0 0
\(784\) 5.62369 0.200846
\(785\) −27.7100 −0.989013
\(786\) 0 0
\(787\) 44.4923 1.58598 0.792989 0.609236i \(-0.208524\pi\)
0.792989 + 0.609236i \(0.208524\pi\)
\(788\) 56.9513 2.02880
\(789\) 0 0
\(790\) 24.1821 0.860362
\(791\) −21.9318 −0.779803
\(792\) 0 0
\(793\) 8.75387 0.310859
\(794\) 77.7076 2.75774
\(795\) 0 0
\(796\) 23.4851 0.832409
\(797\) 2.28072 0.0807872 0.0403936 0.999184i \(-0.487139\pi\)
0.0403936 + 0.999184i \(0.487139\pi\)
\(798\) 0 0
\(799\) 0.863060 0.0305329
\(800\) 31.3849 1.10962
\(801\) 0 0
\(802\) 40.6630 1.43586
\(803\) 49.5446 1.74839
\(804\) 0 0
\(805\) 28.0238 0.987708
\(806\) −22.7395 −0.800964
\(807\) 0 0
\(808\) −13.2773 −0.467094
\(809\) −11.4774 −0.403523 −0.201761 0.979435i \(-0.564667\pi\)
−0.201761 + 0.979435i \(0.564667\pi\)
\(810\) 0 0
\(811\) 27.4524 0.963985 0.481993 0.876175i \(-0.339913\pi\)
0.481993 + 0.876175i \(0.339913\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −87.5019 −3.06694
\(815\) 18.5478 0.649701
\(816\) 0 0
\(817\) −2.74196 −0.0959290
\(818\) 12.6121 0.440971
\(819\) 0 0
\(820\) −17.0711 −0.596150
\(821\) 25.1785 0.878734 0.439367 0.898308i \(-0.355203\pi\)
0.439367 + 0.898308i \(0.355203\pi\)
\(822\) 0 0
\(823\) 2.12913 0.0742169 0.0371085 0.999311i \(-0.488185\pi\)
0.0371085 + 0.999311i \(0.488185\pi\)
\(824\) 57.4785 2.00236
\(825\) 0 0
\(826\) −34.7190 −1.20803
\(827\) 18.3427 0.637837 0.318918 0.947782i \(-0.396680\pi\)
0.318918 + 0.947782i \(0.396680\pi\)
\(828\) 0 0
\(829\) 52.9373 1.83859 0.919295 0.393570i \(-0.128760\pi\)
0.919295 + 0.393570i \(0.128760\pi\)
\(830\) −1.43305 −0.0497419
\(831\) 0 0
\(832\) 26.3694 0.914194
\(833\) −1.56180 −0.0541133
\(834\) 0 0
\(835\) 6.22778 0.215521
\(836\) 38.3364 1.32589
\(837\) 0 0
\(838\) 49.2175 1.70019
\(839\) 26.8505 0.926984 0.463492 0.886101i \(-0.346596\pi\)
0.463492 + 0.886101i \(0.346596\pi\)
\(840\) 0 0
\(841\) 0 0
\(842\) −32.4018 −1.11664
\(843\) 0 0
\(844\) −90.3075 −3.10851
\(845\) 21.4705 0.738607
\(846\) 0 0
\(847\) 22.0083 0.756213
\(848\) 14.2185 0.488266
\(849\) 0 0
\(850\) −23.0065 −0.789118
\(851\) 68.4760 2.34733
\(852\) 0 0
\(853\) 39.8554 1.36462 0.682311 0.731062i \(-0.260975\pi\)
0.682311 + 0.731062i \(0.260975\pi\)
\(854\) −11.1317 −0.380918
\(855\) 0 0
\(856\) 31.1665 1.06525
\(857\) −50.9827 −1.74154 −0.870768 0.491694i \(-0.836378\pi\)
−0.870768 + 0.491694i \(0.836378\pi\)
\(858\) 0 0
\(859\) 14.8714 0.507404 0.253702 0.967282i \(-0.418352\pi\)
0.253702 + 0.967282i \(0.418352\pi\)
\(860\) −7.90696 −0.269625
\(861\) 0 0
\(862\) 91.6288 3.12089
\(863\) 20.4603 0.696478 0.348239 0.937406i \(-0.386780\pi\)
0.348239 + 0.937406i \(0.386780\pi\)
\(864\) 0 0
\(865\) 12.9742 0.441136
\(866\) −15.1561 −0.515026
\(867\) 0 0
\(868\) 20.2856 0.688538
\(869\) −35.4027 −1.20096
\(870\) 0 0
\(871\) −54.4309 −1.84432
\(872\) −122.530 −4.14939
\(873\) 0 0
\(874\) −42.7647 −1.44654
\(875\) −27.5772 −0.932280
\(876\) 0 0
\(877\) −15.4211 −0.520732 −0.260366 0.965510i \(-0.583843\pi\)
−0.260366 + 0.965510i \(0.583843\pi\)
\(878\) −64.2744 −2.16916
\(879\) 0 0
\(880\) 43.5056 1.46657
\(881\) −55.0390 −1.85431 −0.927155 0.374678i \(-0.877753\pi\)
−0.927155 + 0.374678i \(0.877753\pi\)
\(882\) 0 0
\(883\) 37.9954 1.27865 0.639324 0.768938i \(-0.279214\pi\)
0.639324 + 0.768938i \(0.279214\pi\)
\(884\) −63.9096 −2.14951
\(885\) 0 0
\(886\) 61.4762 2.06533
\(887\) −18.2699 −0.613444 −0.306722 0.951799i \(-0.599232\pi\)
−0.306722 + 0.951799i \(0.599232\pi\)
\(888\) 0 0
\(889\) −9.27861 −0.311195
\(890\) −5.93703 −0.199010
\(891\) 0 0
\(892\) −99.8661 −3.34376
\(893\) −0.669303 −0.0223974
\(894\) 0 0
\(895\) 12.2409 0.409169
\(896\) 13.6359 0.455543
\(897\) 0 0
\(898\) 16.6927 0.557043
\(899\) 0 0
\(900\) 0 0
\(901\) −3.94875 −0.131552
\(902\) 35.6252 1.18619
\(903\) 0 0
\(904\) 55.4506 1.84426
\(905\) −14.1244 −0.469512
\(906\) 0 0
\(907\) 25.2735 0.839192 0.419596 0.907711i \(-0.362172\pi\)
0.419596 + 0.907711i \(0.362172\pi\)
\(908\) −92.8701 −3.08200
\(909\) 0 0
\(910\) −46.2958 −1.53469
\(911\) −34.3336 −1.13752 −0.568762 0.822502i \(-0.692578\pi\)
−0.568762 + 0.822502i \(0.692578\pi\)
\(912\) 0 0
\(913\) 2.09799 0.0694334
\(914\) 6.79686 0.224820
\(915\) 0 0
\(916\) 97.3731 3.21730
\(917\) 24.0390 0.793836
\(918\) 0 0
\(919\) −26.8764 −0.886572 −0.443286 0.896380i \(-0.646187\pi\)
−0.443286 + 0.896380i \(0.646187\pi\)
\(920\) −70.8532 −2.33596
\(921\) 0 0
\(922\) −18.5291 −0.610222
\(923\) 46.0524 1.51583
\(924\) 0 0
\(925\) −28.5684 −0.939323
\(926\) 47.2190 1.55171
\(927\) 0 0
\(928\) 0 0
\(929\) 27.4218 0.899679 0.449839 0.893109i \(-0.351481\pi\)
0.449839 + 0.893109i \(0.351481\pi\)
\(930\) 0 0
\(931\) 1.21118 0.0396947
\(932\) −97.6582 −3.19890
\(933\) 0 0
\(934\) 91.8118 3.00417
\(935\) −12.0823 −0.395134
\(936\) 0 0
\(937\) −10.3107 −0.336836 −0.168418 0.985716i \(-0.553866\pi\)
−0.168418 + 0.985716i \(0.553866\pi\)
\(938\) 69.2158 2.25998
\(939\) 0 0
\(940\) −1.93006 −0.0629517
\(941\) −16.9225 −0.551657 −0.275828 0.961207i \(-0.588952\pi\)
−0.275828 + 0.961207i \(0.588952\pi\)
\(942\) 0 0
\(943\) −27.8791 −0.907867
\(944\) 42.1800 1.37284
\(945\) 0 0
\(946\) 16.5008 0.536487
\(947\) 21.2231 0.689659 0.344829 0.938665i \(-0.387937\pi\)
0.344829 + 0.938665i \(0.387937\pi\)
\(948\) 0 0
\(949\) 64.0519 2.07921
\(950\) 17.8416 0.578857
\(951\) 0 0
\(952\) 46.6929 1.51333
\(953\) 8.78444 0.284556 0.142278 0.989827i \(-0.454557\pi\)
0.142278 + 0.989827i \(0.454557\pi\)
\(954\) 0 0
\(955\) 1.93380 0.0625763
\(956\) 10.6936 0.345856
\(957\) 0 0
\(958\) 67.5637 2.18288
\(959\) −8.91469 −0.287870
\(960\) 0 0
\(961\) −28.5647 −0.921443
\(962\) −113.124 −3.64725
\(963\) 0 0
\(964\) 55.3682 1.78329
\(965\) 12.1513 0.391165
\(966\) 0 0
\(967\) −25.9334 −0.833961 −0.416981 0.908915i \(-0.636912\pi\)
−0.416981 + 0.908915i \(0.636912\pi\)
\(968\) −55.6441 −1.78847
\(969\) 0 0
\(970\) −13.7857 −0.442632
\(971\) 58.1391 1.86577 0.932886 0.360172i \(-0.117282\pi\)
0.932886 + 0.360172i \(0.117282\pi\)
\(972\) 0 0
\(973\) −62.1863 −1.99360
\(974\) −57.3357 −1.83715
\(975\) 0 0
\(976\) 13.5238 0.432887
\(977\) −29.9259 −0.957414 −0.478707 0.877975i \(-0.658894\pi\)
−0.478707 + 0.877975i \(0.658894\pi\)
\(978\) 0 0
\(979\) 8.69184 0.277792
\(980\) 3.49266 0.111569
\(981\) 0 0
\(982\) −30.4117 −0.970475
\(983\) 40.2091 1.28247 0.641236 0.767344i \(-0.278422\pi\)
0.641236 + 0.767344i \(0.278422\pi\)
\(984\) 0 0
\(985\) 13.9195 0.443512
\(986\) 0 0
\(987\) 0 0
\(988\) 49.5619 1.57677
\(989\) −12.9129 −0.410608
\(990\) 0 0
\(991\) 39.1707 1.24430 0.622149 0.782898i \(-0.286260\pi\)
0.622149 + 0.782898i \(0.286260\pi\)
\(992\) −13.3093 −0.422569
\(993\) 0 0
\(994\) −58.5615 −1.85746
\(995\) 5.74002 0.181971
\(996\) 0 0
\(997\) 34.4756 1.09185 0.545927 0.837833i \(-0.316177\pi\)
0.545927 + 0.837833i \(0.316177\pi\)
\(998\) 42.3892 1.34181
\(999\) 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 7569.2.a.bj.1.2 9
3.2 odd 2 2523.2.a.r.1.8 9
29.16 even 7 261.2.k.c.82.1 18
29.20 even 7 261.2.k.c.226.1 18
29.28 even 2 7569.2.a.bm.1.8 9
87.20 odd 14 87.2.g.a.52.3 18
87.74 odd 14 87.2.g.a.82.3 yes 18
87.86 odd 2 2523.2.a.o.1.2 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
87.2.g.a.52.3 18 87.20 odd 14
87.2.g.a.82.3 yes 18 87.74 odd 14
261.2.k.c.82.1 18 29.16 even 7
261.2.k.c.226.1 18 29.20 even 7
2523.2.a.o.1.2 9 87.86 odd 2
2523.2.a.r.1.8 9 3.2 odd 2
7569.2.a.bj.1.2 9 1.1 even 1 trivial
7569.2.a.bm.1.8 9 29.28 even 2