Properties

Label 6525.2.a.bv.1.7
Level $6525$
Weight $2$
Character 6525.1
Self dual yes
Analytic conductor $52.102$
Analytic rank $1$
Dimension $7$
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: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 13x^{5} + 12x^{4} + 47x^{3} - 37x^{2} - 35x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1305)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(-2.66072\) 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)
 
\(f(q)\) \(=\) \(q+2.66072 q^{2} +5.07943 q^{4} -4.68271 q^{7} +8.19351 q^{8} +O(q^{10})\) \(q+2.66072 q^{2} +5.07943 q^{4} -4.68271 q^{7} +8.19351 q^{8} -3.50383 q^{11} +4.56233 q^{13} -12.4594 q^{14} +11.6418 q^{16} -6.56233 q^{17} -4.10839 q^{19} -9.32272 q^{22} -6.92951 q^{23} +12.1391 q^{26} -23.7855 q^{28} -1.00000 q^{29} +8.10839 q^{31} +14.5885 q^{32} -17.4605 q^{34} -4.27156 q^{37} -10.9313 q^{38} -5.10898 q^{41} +8.47439 q^{43} -17.7975 q^{44} -18.4375 q^{46} -5.41114 q^{47} +14.9278 q^{49} +23.1741 q^{52} -4.20108 q^{53} -38.3678 q^{56} -2.66072 q^{58} +5.36542 q^{59} +1.27150 q^{61} +21.5741 q^{62} +15.5323 q^{64} -9.96017 q^{67} -33.3329 q^{68} -12.2168 q^{71} -8.37344 q^{73} -11.3654 q^{74} -20.8683 q^{76} +16.4074 q^{77} -11.1161 q^{79} -13.5936 q^{82} -14.6657 q^{83} +22.5480 q^{86} -28.7087 q^{88} +2.20459 q^{89} -21.3641 q^{91} -35.1980 q^{92} -14.3975 q^{94} -5.46673 q^{97} +39.7186 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - q^{2} + 13 q^{4} - 10 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - q^{2} + 13 q^{4} - 10 q^{7} - 3 q^{11} - 6 q^{13} - 9 q^{14} + 21 q^{16} - 8 q^{17} + 10 q^{19} - 9 q^{22} - 11 q^{23} + 3 q^{26} - 25 q^{28} - 7 q^{29} + 18 q^{31} - q^{32} - q^{34} - 13 q^{37} - 12 q^{38} - 13 q^{41} - 9 q^{43} - 37 q^{44} - 8 q^{46} - 2 q^{47} + 21 q^{49} + q^{52} - 5 q^{53} - 30 q^{56} + q^{58} - 8 q^{59} + 14 q^{61} + 8 q^{62} + 8 q^{64} - 14 q^{67} - 27 q^{68} - 8 q^{71} - 3 q^{73} - 34 q^{74} + 4 q^{76} + 28 q^{77} + 4 q^{79} + 20 q^{82} - 17 q^{83} + 4 q^{86} - 26 q^{88} - 20 q^{89} + 12 q^{91} - 60 q^{92} - 21 q^{94} - 13 q^{97} + 20 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.66072 1.88141 0.940707 0.339221i \(-0.110163\pi\)
0.940707 + 0.339221i \(0.110163\pi\)
\(3\) 0 0
\(4\) 5.07943 2.53972
\(5\) 0 0
\(6\) 0 0
\(7\) −4.68271 −1.76990 −0.884949 0.465689i \(-0.845807\pi\)
−0.884949 + 0.465689i \(0.845807\pi\)
\(8\) 8.19351 2.89684
\(9\) 0 0
\(10\) 0 0
\(11\) −3.50383 −1.05645 −0.528223 0.849106i \(-0.677141\pi\)
−0.528223 + 0.849106i \(0.677141\pi\)
\(12\) 0 0
\(13\) 4.56233 1.26536 0.632682 0.774412i \(-0.281954\pi\)
0.632682 + 0.774412i \(0.281954\pi\)
\(14\) −12.4594 −3.32991
\(15\) 0 0
\(16\) 11.6418 2.91044
\(17\) −6.56233 −1.59160 −0.795800 0.605560i \(-0.792949\pi\)
−0.795800 + 0.605560i \(0.792949\pi\)
\(18\) 0 0
\(19\) −4.10839 −0.942529 −0.471264 0.881992i \(-0.656202\pi\)
−0.471264 + 0.881992i \(0.656202\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −9.32272 −1.98761
\(23\) −6.92951 −1.44490 −0.722452 0.691422i \(-0.756985\pi\)
−0.722452 + 0.691422i \(0.756985\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 12.1391 2.38067
\(27\) 0 0
\(28\) −23.7855 −4.49504
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 8.10839 1.45631 0.728155 0.685413i \(-0.240378\pi\)
0.728155 + 0.685413i \(0.240378\pi\)
\(32\) 14.5885 2.57890
\(33\) 0 0
\(34\) −17.4605 −2.99446
\(35\) 0 0
\(36\) 0 0
\(37\) −4.27156 −0.702239 −0.351120 0.936331i \(-0.614199\pi\)
−0.351120 + 0.936331i \(0.614199\pi\)
\(38\) −10.9313 −1.77329
\(39\) 0 0
\(40\) 0 0
\(41\) −5.10898 −0.797889 −0.398944 0.916975i \(-0.630623\pi\)
−0.398944 + 0.916975i \(0.630623\pi\)
\(42\) 0 0
\(43\) 8.47439 1.29233 0.646167 0.763196i \(-0.276371\pi\)
0.646167 + 0.763196i \(0.276371\pi\)
\(44\) −17.7975 −2.68307
\(45\) 0 0
\(46\) −18.4375 −2.71846
\(47\) −5.41114 −0.789295 −0.394648 0.918833i \(-0.629133\pi\)
−0.394648 + 0.918833i \(0.629133\pi\)
\(48\) 0 0
\(49\) 14.9278 2.13254
\(50\) 0 0
\(51\) 0 0
\(52\) 23.1741 3.21367
\(53\) −4.20108 −0.577063 −0.288532 0.957470i \(-0.593167\pi\)
−0.288532 + 0.957470i \(0.593167\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −38.3678 −5.12711
\(57\) 0 0
\(58\) −2.66072 −0.349370
\(59\) 5.36542 0.698518 0.349259 0.937026i \(-0.386433\pi\)
0.349259 + 0.937026i \(0.386433\pi\)
\(60\) 0 0
\(61\) 1.27150 0.162798 0.0813991 0.996682i \(-0.474061\pi\)
0.0813991 + 0.996682i \(0.474061\pi\)
\(62\) 21.5741 2.73992
\(63\) 0 0
\(64\) 15.5323 1.94154
\(65\) 0 0
\(66\) 0 0
\(67\) −9.96017 −1.21683 −0.608414 0.793619i \(-0.708194\pi\)
−0.608414 + 0.793619i \(0.708194\pi\)
\(68\) −33.3329 −4.04221
\(69\) 0 0
\(70\) 0 0
\(71\) −12.2168 −1.44986 −0.724932 0.688820i \(-0.758129\pi\)
−0.724932 + 0.688820i \(0.758129\pi\)
\(72\) 0 0
\(73\) −8.37344 −0.980037 −0.490018 0.871712i \(-0.663010\pi\)
−0.490018 + 0.871712i \(0.663010\pi\)
\(74\) −11.3654 −1.32120
\(75\) 0 0
\(76\) −20.8683 −2.39375
\(77\) 16.4074 1.86980
\(78\) 0 0
\(79\) −11.1161 −1.25065 −0.625327 0.780363i \(-0.715034\pi\)
−0.625327 + 0.780363i \(0.715034\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −13.5936 −1.50116
\(83\) −14.6657 −1.60977 −0.804884 0.593432i \(-0.797772\pi\)
−0.804884 + 0.593432i \(0.797772\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 22.5480 2.43141
\(87\) 0 0
\(88\) −28.7087 −3.06036
\(89\) 2.20459 0.233686 0.116843 0.993150i \(-0.462723\pi\)
0.116843 + 0.993150i \(0.462723\pi\)
\(90\) 0 0
\(91\) −21.3641 −2.23956
\(92\) −35.1980 −3.66964
\(93\) 0 0
\(94\) −14.3975 −1.48499
\(95\) 0 0
\(96\) 0 0
\(97\) −5.46673 −0.555062 −0.277531 0.960717i \(-0.589516\pi\)
−0.277531 + 0.960717i \(0.589516\pi\)
\(98\) 39.7186 4.01218
\(99\) 0 0
\(100\) 0 0
\(101\) 7.54728 0.750982 0.375491 0.926826i \(-0.377474\pi\)
0.375491 + 0.926826i \(0.377474\pi\)
\(102\) 0 0
\(103\) 5.18595 0.510987 0.255493 0.966811i \(-0.417762\pi\)
0.255493 + 0.966811i \(0.417762\pi\)
\(104\) 37.3815 3.66556
\(105\) 0 0
\(106\) −11.1779 −1.08569
\(107\) 4.39624 0.425001 0.212500 0.977161i \(-0.431839\pi\)
0.212500 + 0.977161i \(0.431839\pi\)
\(108\) 0 0
\(109\) −3.48595 −0.333894 −0.166947 0.985966i \(-0.553391\pi\)
−0.166947 + 0.985966i \(0.553391\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −54.5150 −5.15118
\(113\) 10.6642 1.00321 0.501603 0.865098i \(-0.332744\pi\)
0.501603 + 0.865098i \(0.332744\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −5.07943 −0.471613
\(117\) 0 0
\(118\) 14.2759 1.31420
\(119\) 30.7295 2.81697
\(120\) 0 0
\(121\) 1.27684 0.116077
\(122\) 3.38309 0.306291
\(123\) 0 0
\(124\) 41.1860 3.69861
\(125\) 0 0
\(126\) 0 0
\(127\) 1.46673 0.130151 0.0650756 0.997880i \(-0.479271\pi\)
0.0650756 + 0.997880i \(0.479271\pi\)
\(128\) 12.1502 1.07393
\(129\) 0 0
\(130\) 0 0
\(131\) 5.43157 0.474559 0.237279 0.971441i \(-0.423744\pi\)
0.237279 + 0.971441i \(0.423744\pi\)
\(132\) 0 0
\(133\) 19.2384 1.66818
\(134\) −26.5012 −2.28936
\(135\) 0 0
\(136\) −53.7685 −4.61061
\(137\) −5.27747 −0.450884 −0.225442 0.974257i \(-0.572383\pi\)
−0.225442 + 0.974257i \(0.572383\pi\)
\(138\) 0 0
\(139\) 10.7043 0.907929 0.453964 0.891020i \(-0.350009\pi\)
0.453964 + 0.891020i \(0.350009\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −32.5054 −2.72779
\(143\) −15.9857 −1.33679
\(144\) 0 0
\(145\) 0 0
\(146\) −22.2794 −1.84385
\(147\) 0 0
\(148\) −21.6971 −1.78349
\(149\) 10.7308 0.879104 0.439552 0.898217i \(-0.355137\pi\)
0.439552 + 0.898217i \(0.355137\pi\)
\(150\) 0 0
\(151\) 20.2395 1.64707 0.823534 0.567268i \(-0.191999\pi\)
0.823534 + 0.567268i \(0.191999\pi\)
\(152\) −33.6621 −2.73036
\(153\) 0 0
\(154\) 43.6556 3.51787
\(155\) 0 0
\(156\) 0 0
\(157\) 1.62924 0.130028 0.0650139 0.997884i \(-0.479291\pi\)
0.0650139 + 0.997884i \(0.479291\pi\)
\(158\) −29.5767 −2.35300
\(159\) 0 0
\(160\) 0 0
\(161\) 32.4489 2.55733
\(162\) 0 0
\(163\) 16.6586 1.30480 0.652400 0.757875i \(-0.273762\pi\)
0.652400 + 0.757875i \(0.273762\pi\)
\(164\) −25.9507 −2.02641
\(165\) 0 0
\(166\) −39.0213 −3.02864
\(167\) −10.7214 −0.829646 −0.414823 0.909902i \(-0.636157\pi\)
−0.414823 + 0.909902i \(0.636157\pi\)
\(168\) 0 0
\(169\) 7.81490 0.601146
\(170\) 0 0
\(171\) 0 0
\(172\) 43.0451 3.28216
\(173\) 12.7826 0.971846 0.485923 0.874002i \(-0.338483\pi\)
0.485923 + 0.874002i \(0.338483\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −40.7908 −3.07472
\(177\) 0 0
\(178\) 5.86579 0.439659
\(179\) 3.19306 0.238660 0.119330 0.992855i \(-0.461925\pi\)
0.119330 + 0.992855i \(0.461925\pi\)
\(180\) 0 0
\(181\) −17.8264 −1.32503 −0.662514 0.749049i \(-0.730511\pi\)
−0.662514 + 0.749049i \(0.730511\pi\)
\(182\) −56.8438 −4.21355
\(183\) 0 0
\(184\) −56.7770 −4.18566
\(185\) 0 0
\(186\) 0 0
\(187\) 22.9933 1.68144
\(188\) −27.4855 −2.00459
\(189\) 0 0
\(190\) 0 0
\(191\) 3.37016 0.243856 0.121928 0.992539i \(-0.461092\pi\)
0.121928 + 0.992539i \(0.461092\pi\)
\(192\) 0 0
\(193\) −11.0542 −0.795699 −0.397849 0.917451i \(-0.630243\pi\)
−0.397849 + 0.917451i \(0.630243\pi\)
\(194\) −14.5454 −1.04430
\(195\) 0 0
\(196\) 75.8245 5.41604
\(197\) 7.99846 0.569866 0.284933 0.958547i \(-0.408029\pi\)
0.284933 + 0.958547i \(0.408029\pi\)
\(198\) 0 0
\(199\) 8.30813 0.588948 0.294474 0.955660i \(-0.404856\pi\)
0.294474 + 0.955660i \(0.404856\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 20.0812 1.41291
\(203\) 4.68271 0.328662
\(204\) 0 0
\(205\) 0 0
\(206\) 13.7984 0.961377
\(207\) 0 0
\(208\) 53.1136 3.68277
\(209\) 14.3951 0.995730
\(210\) 0 0
\(211\) 6.10957 0.420600 0.210300 0.977637i \(-0.432556\pi\)
0.210300 + 0.977637i \(0.432556\pi\)
\(212\) −21.3391 −1.46558
\(213\) 0 0
\(214\) 11.6972 0.799602
\(215\) 0 0
\(216\) 0 0
\(217\) −37.9692 −2.57752
\(218\) −9.27514 −0.628192
\(219\) 0 0
\(220\) 0 0
\(221\) −29.9396 −2.01395
\(222\) 0 0
\(223\) −21.0235 −1.40784 −0.703920 0.710280i \(-0.748568\pi\)
−0.703920 + 0.710280i \(0.748568\pi\)
\(224\) −68.3135 −4.56439
\(225\) 0 0
\(226\) 28.3745 1.88744
\(227\) 5.59548 0.371385 0.185692 0.982608i \(-0.440547\pi\)
0.185692 + 0.982608i \(0.440547\pi\)
\(228\) 0 0
\(229\) 29.3568 1.93995 0.969975 0.243204i \(-0.0781985\pi\)
0.969975 + 0.243204i \(0.0781985\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −8.19351 −0.537930
\(233\) 9.47855 0.620960 0.310480 0.950580i \(-0.399510\pi\)
0.310480 + 0.950580i \(0.399510\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 27.2533 1.77404
\(237\) 0 0
\(238\) 81.7626 5.29988
\(239\) −5.45686 −0.352975 −0.176487 0.984303i \(-0.556473\pi\)
−0.176487 + 0.984303i \(0.556473\pi\)
\(240\) 0 0
\(241\) −0.670684 −0.0432026 −0.0216013 0.999767i \(-0.506876\pi\)
−0.0216013 + 0.999767i \(0.506876\pi\)
\(242\) 3.39733 0.218388
\(243\) 0 0
\(244\) 6.45847 0.413461
\(245\) 0 0
\(246\) 0 0
\(247\) −18.7438 −1.19264
\(248\) 66.4361 4.21870
\(249\) 0 0
\(250\) 0 0
\(251\) −17.3401 −1.09450 −0.547250 0.836969i \(-0.684325\pi\)
−0.547250 + 0.836969i \(0.684325\pi\)
\(252\) 0 0
\(253\) 24.2799 1.52646
\(254\) 3.90256 0.244868
\(255\) 0 0
\(256\) 1.26361 0.0789755
\(257\) −15.8457 −0.988425 −0.494213 0.869341i \(-0.664544\pi\)
−0.494213 + 0.869341i \(0.664544\pi\)
\(258\) 0 0
\(259\) 20.0024 1.24289
\(260\) 0 0
\(261\) 0 0
\(262\) 14.4519 0.892841
\(263\) −9.70623 −0.598512 −0.299256 0.954173i \(-0.596738\pi\)
−0.299256 + 0.954173i \(0.596738\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 51.1879 3.13853
\(267\) 0 0
\(268\) −50.5920 −3.09040
\(269\) −9.34619 −0.569847 −0.284924 0.958550i \(-0.591968\pi\)
−0.284924 + 0.958550i \(0.591968\pi\)
\(270\) 0 0
\(271\) −32.4018 −1.96827 −0.984135 0.177421i \(-0.943225\pi\)
−0.984135 + 0.177421i \(0.943225\pi\)
\(272\) −76.3972 −4.63226
\(273\) 0 0
\(274\) −14.0419 −0.848300
\(275\) 0 0
\(276\) 0 0
\(277\) 21.4844 1.29087 0.645436 0.763815i \(-0.276676\pi\)
0.645436 + 0.763815i \(0.276676\pi\)
\(278\) 28.4812 1.70819
\(279\) 0 0
\(280\) 0 0
\(281\) 3.09329 0.184530 0.0922651 0.995734i \(-0.470589\pi\)
0.0922651 + 0.995734i \(0.470589\pi\)
\(282\) 0 0
\(283\) −5.90145 −0.350805 −0.175402 0.984497i \(-0.556123\pi\)
−0.175402 + 0.984497i \(0.556123\pi\)
\(284\) −62.0543 −3.68224
\(285\) 0 0
\(286\) −42.5334 −2.51505
\(287\) 23.9239 1.41218
\(288\) 0 0
\(289\) 26.0642 1.53319
\(290\) 0 0
\(291\) 0 0
\(292\) −42.5323 −2.48902
\(293\) 2.17061 0.126808 0.0634042 0.997988i \(-0.479804\pi\)
0.0634042 + 0.997988i \(0.479804\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −34.9990 −2.03428
\(297\) 0 0
\(298\) 28.5517 1.65396
\(299\) −31.6148 −1.82833
\(300\) 0 0
\(301\) −39.6831 −2.28730
\(302\) 53.8516 3.09881
\(303\) 0 0
\(304\) −47.8289 −2.74317
\(305\) 0 0
\(306\) 0 0
\(307\) −9.26410 −0.528730 −0.264365 0.964423i \(-0.585162\pi\)
−0.264365 + 0.964423i \(0.585162\pi\)
\(308\) 83.3404 4.74876
\(309\) 0 0
\(310\) 0 0
\(311\) −22.1337 −1.25509 −0.627544 0.778581i \(-0.715940\pi\)
−0.627544 + 0.778581i \(0.715940\pi\)
\(312\) 0 0
\(313\) −0.982472 −0.0555326 −0.0277663 0.999614i \(-0.508839\pi\)
−0.0277663 + 0.999614i \(0.508839\pi\)
\(314\) 4.33496 0.244636
\(315\) 0 0
\(316\) −56.4632 −3.17631
\(317\) 5.37998 0.302170 0.151085 0.988521i \(-0.451723\pi\)
0.151085 + 0.988521i \(0.451723\pi\)
\(318\) 0 0
\(319\) 3.50383 0.196177
\(320\) 0 0
\(321\) 0 0
\(322\) 86.3374 4.81139
\(323\) 26.9606 1.50013
\(324\) 0 0
\(325\) 0 0
\(326\) 44.3238 2.45487
\(327\) 0 0
\(328\) −41.8605 −2.31136
\(329\) 25.3388 1.39697
\(330\) 0 0
\(331\) 28.8813 1.58746 0.793729 0.608271i \(-0.208137\pi\)
0.793729 + 0.608271i \(0.208137\pi\)
\(332\) −74.4933 −4.08835
\(333\) 0 0
\(334\) −28.5266 −1.56091
\(335\) 0 0
\(336\) 0 0
\(337\) 17.6796 0.963071 0.481535 0.876427i \(-0.340079\pi\)
0.481535 + 0.876427i \(0.340079\pi\)
\(338\) 20.7933 1.13100
\(339\) 0 0
\(340\) 0 0
\(341\) −28.4104 −1.53851
\(342\) 0 0
\(343\) −37.1233 −2.00447
\(344\) 69.4350 3.74369
\(345\) 0 0
\(346\) 34.0110 1.82844
\(347\) −10.2375 −0.549578 −0.274789 0.961505i \(-0.588608\pi\)
−0.274789 + 0.961505i \(0.588608\pi\)
\(348\) 0 0
\(349\) −7.50163 −0.401553 −0.200777 0.979637i \(-0.564347\pi\)
−0.200777 + 0.979637i \(0.564347\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −51.1156 −2.72447
\(353\) 25.5268 1.35865 0.679327 0.733835i \(-0.262272\pi\)
0.679327 + 0.733835i \(0.262272\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 11.1980 0.593495
\(357\) 0 0
\(358\) 8.49584 0.449019
\(359\) 26.6765 1.40793 0.703967 0.710232i \(-0.251410\pi\)
0.703967 + 0.710232i \(0.251410\pi\)
\(360\) 0 0
\(361\) −2.12116 −0.111640
\(362\) −47.4312 −2.49293
\(363\) 0 0
\(364\) −108.517 −5.68786
\(365\) 0 0
\(366\) 0 0
\(367\) 9.02936 0.471329 0.235664 0.971835i \(-0.424273\pi\)
0.235664 + 0.971835i \(0.424273\pi\)
\(368\) −80.6718 −4.20531
\(369\) 0 0
\(370\) 0 0
\(371\) 19.6724 1.02134
\(372\) 0 0
\(373\) −22.4175 −1.16073 −0.580366 0.814356i \(-0.697091\pi\)
−0.580366 + 0.814356i \(0.697091\pi\)
\(374\) 61.1788 3.16348
\(375\) 0 0
\(376\) −44.3362 −2.28646
\(377\) −4.56233 −0.234972
\(378\) 0 0
\(379\) 9.65684 0.496038 0.248019 0.968755i \(-0.420220\pi\)
0.248019 + 0.968755i \(0.420220\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 8.96706 0.458795
\(383\) 20.0311 1.02354 0.511771 0.859122i \(-0.328990\pi\)
0.511771 + 0.859122i \(0.328990\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −29.4121 −1.49704
\(387\) 0 0
\(388\) −27.7679 −1.40970
\(389\) −19.7468 −1.00120 −0.500602 0.865678i \(-0.666888\pi\)
−0.500602 + 0.865678i \(0.666888\pi\)
\(390\) 0 0
\(391\) 45.4738 2.29971
\(392\) 122.311 6.17762
\(393\) 0 0
\(394\) 21.2817 1.07215
\(395\) 0 0
\(396\) 0 0
\(397\) 1.39536 0.0700310 0.0350155 0.999387i \(-0.488852\pi\)
0.0350155 + 0.999387i \(0.488852\pi\)
\(398\) 22.1056 1.10805
\(399\) 0 0
\(400\) 0 0
\(401\) 25.3242 1.26463 0.632315 0.774711i \(-0.282105\pi\)
0.632315 + 0.774711i \(0.282105\pi\)
\(402\) 0 0
\(403\) 36.9932 1.84276
\(404\) 38.3359 1.90728
\(405\) 0 0
\(406\) 12.4594 0.618348
\(407\) 14.9668 0.741878
\(408\) 0 0
\(409\) −5.44982 −0.269476 −0.134738 0.990881i \(-0.543019\pi\)
−0.134738 + 0.990881i \(0.543019\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 26.3417 1.29776
\(413\) −25.1247 −1.23630
\(414\) 0 0
\(415\) 0 0
\(416\) 66.5575 3.26325
\(417\) 0 0
\(418\) 38.3013 1.87338
\(419\) −26.6684 −1.30284 −0.651419 0.758719i \(-0.725826\pi\)
−0.651419 + 0.758719i \(0.725826\pi\)
\(420\) 0 0
\(421\) 18.7599 0.914301 0.457150 0.889389i \(-0.348870\pi\)
0.457150 + 0.889389i \(0.348870\pi\)
\(422\) 16.2559 0.791323
\(423\) 0 0
\(424\) −34.4216 −1.67166
\(425\) 0 0
\(426\) 0 0
\(427\) −5.95404 −0.288136
\(428\) 22.3304 1.07938
\(429\) 0 0
\(430\) 0 0
\(431\) −21.5170 −1.03644 −0.518219 0.855248i \(-0.673405\pi\)
−0.518219 + 0.855248i \(0.673405\pi\)
\(432\) 0 0
\(433\) −17.5946 −0.845544 −0.422772 0.906236i \(-0.638943\pi\)
−0.422772 + 0.906236i \(0.638943\pi\)
\(434\) −101.025 −4.84938
\(435\) 0 0
\(436\) −17.7067 −0.847995
\(437\) 28.4691 1.36186
\(438\) 0 0
\(439\) −28.3164 −1.35147 −0.675734 0.737146i \(-0.736173\pi\)
−0.675734 + 0.737146i \(0.736173\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −79.6608 −3.78908
\(443\) −21.9916 −1.04485 −0.522427 0.852684i \(-0.674973\pi\)
−0.522427 + 0.852684i \(0.674973\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −55.9377 −2.64873
\(447\) 0 0
\(448\) −72.7332 −3.43632
\(449\) 6.08052 0.286957 0.143479 0.989653i \(-0.454171\pi\)
0.143479 + 0.989653i \(0.454171\pi\)
\(450\) 0 0
\(451\) 17.9010 0.842926
\(452\) 54.1682 2.54786
\(453\) 0 0
\(454\) 14.8880 0.698728
\(455\) 0 0
\(456\) 0 0
\(457\) 27.5955 1.29086 0.645432 0.763818i \(-0.276677\pi\)
0.645432 + 0.763818i \(0.276677\pi\)
\(458\) 78.1102 3.64985
\(459\) 0 0
\(460\) 0 0
\(461\) 20.3657 0.948525 0.474262 0.880384i \(-0.342715\pi\)
0.474262 + 0.880384i \(0.342715\pi\)
\(462\) 0 0
\(463\) −40.1133 −1.86422 −0.932111 0.362172i \(-0.882035\pi\)
−0.932111 + 0.362172i \(0.882035\pi\)
\(464\) −11.6418 −0.540455
\(465\) 0 0
\(466\) 25.2198 1.16828
\(467\) −32.0467 −1.48295 −0.741473 0.670982i \(-0.765873\pi\)
−0.741473 + 0.670982i \(0.765873\pi\)
\(468\) 0 0
\(469\) 46.6406 2.15366
\(470\) 0 0
\(471\) 0 0
\(472\) 43.9616 2.02350
\(473\) −29.6929 −1.36528
\(474\) 0 0
\(475\) 0 0
\(476\) 156.088 7.15430
\(477\) 0 0
\(478\) −14.5192 −0.664091
\(479\) −22.6657 −1.03562 −0.517811 0.855495i \(-0.673253\pi\)
−0.517811 + 0.855495i \(0.673253\pi\)
\(480\) 0 0
\(481\) −19.4883 −0.888589
\(482\) −1.78450 −0.0812819
\(483\) 0 0
\(484\) 6.48565 0.294802
\(485\) 0 0
\(486\) 0 0
\(487\) 25.6034 1.16020 0.580101 0.814545i \(-0.303013\pi\)
0.580101 + 0.814545i \(0.303013\pi\)
\(488\) 10.4180 0.471601
\(489\) 0 0
\(490\) 0 0
\(491\) 3.40078 0.153475 0.0767375 0.997051i \(-0.475550\pi\)
0.0767375 + 0.997051i \(0.475550\pi\)
\(492\) 0 0
\(493\) 6.56233 0.295553
\(494\) −49.8721 −2.24385
\(495\) 0 0
\(496\) 94.3960 4.23850
\(497\) 57.2076 2.56611
\(498\) 0 0
\(499\) 5.58352 0.249953 0.124976 0.992160i \(-0.460114\pi\)
0.124976 + 0.992160i \(0.460114\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −46.1372 −2.05921
\(503\) −7.58471 −0.338185 −0.169093 0.985600i \(-0.554084\pi\)
−0.169093 + 0.985600i \(0.554084\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 64.6019 2.87190
\(507\) 0 0
\(508\) 7.45015 0.330547
\(509\) 9.47234 0.419854 0.209927 0.977717i \(-0.432677\pi\)
0.209927 + 0.977717i \(0.432677\pi\)
\(510\) 0 0
\(511\) 39.2104 1.73456
\(512\) −20.9382 −0.925348
\(513\) 0 0
\(514\) −42.1609 −1.85964
\(515\) 0 0
\(516\) 0 0
\(517\) 18.9597 0.833847
\(518\) 53.2209 2.33839
\(519\) 0 0
\(520\) 0 0
\(521\) −15.9046 −0.696794 −0.348397 0.937347i \(-0.613274\pi\)
−0.348397 + 0.937347i \(0.613274\pi\)
\(522\) 0 0
\(523\) −0.965392 −0.0422137 −0.0211068 0.999777i \(-0.506719\pi\)
−0.0211068 + 0.999777i \(0.506719\pi\)
\(524\) 27.5893 1.20524
\(525\) 0 0
\(526\) −25.8256 −1.12605
\(527\) −53.2099 −2.31786
\(528\) 0 0
\(529\) 25.0181 1.08775
\(530\) 0 0
\(531\) 0 0
\(532\) 97.7200 4.23670
\(533\) −23.3089 −1.00962
\(534\) 0 0
\(535\) 0 0
\(536\) −81.6088 −3.52496
\(537\) 0 0
\(538\) −24.8676 −1.07212
\(539\) −52.3043 −2.25291
\(540\) 0 0
\(541\) −17.0885 −0.734694 −0.367347 0.930084i \(-0.619734\pi\)
−0.367347 + 0.930084i \(0.619734\pi\)
\(542\) −86.2122 −3.70313
\(543\) 0 0
\(544\) −95.7344 −4.10458
\(545\) 0 0
\(546\) 0 0
\(547\) −33.0004 −1.41100 −0.705498 0.708712i \(-0.749277\pi\)
−0.705498 + 0.708712i \(0.749277\pi\)
\(548\) −26.8065 −1.14512
\(549\) 0 0
\(550\) 0 0
\(551\) 4.10839 0.175023
\(552\) 0 0
\(553\) 52.0532 2.21353
\(554\) 57.1639 2.42866
\(555\) 0 0
\(556\) 54.3719 2.30588
\(557\) 34.0021 1.44071 0.720357 0.693603i \(-0.243978\pi\)
0.720357 + 0.693603i \(0.243978\pi\)
\(558\) 0 0
\(559\) 38.6630 1.63527
\(560\) 0 0
\(561\) 0 0
\(562\) 8.23038 0.347178
\(563\) −23.4343 −0.987640 −0.493820 0.869564i \(-0.664400\pi\)
−0.493820 + 0.869564i \(0.664400\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −15.7021 −0.660008
\(567\) 0 0
\(568\) −100.098 −4.20003
\(569\) 1.69285 0.0709681 0.0354841 0.999370i \(-0.488703\pi\)
0.0354841 + 0.999370i \(0.488703\pi\)
\(570\) 0 0
\(571\) −17.8528 −0.747117 −0.373559 0.927607i \(-0.621863\pi\)
−0.373559 + 0.927607i \(0.621863\pi\)
\(572\) −81.1981 −3.39506
\(573\) 0 0
\(574\) 63.6547 2.65690
\(575\) 0 0
\(576\) 0 0
\(577\) 2.78174 0.115805 0.0579027 0.998322i \(-0.481559\pi\)
0.0579027 + 0.998322i \(0.481559\pi\)
\(578\) 69.3497 2.88457
\(579\) 0 0
\(580\) 0 0
\(581\) 68.6751 2.84912
\(582\) 0 0
\(583\) 14.7199 0.609636
\(584\) −68.6078 −2.83901
\(585\) 0 0
\(586\) 5.77539 0.238579
\(587\) −11.8335 −0.488422 −0.244211 0.969722i \(-0.578529\pi\)
−0.244211 + 0.969722i \(0.578529\pi\)
\(588\) 0 0
\(589\) −33.3124 −1.37261
\(590\) 0 0
\(591\) 0 0
\(592\) −49.7285 −2.04383
\(593\) 28.2961 1.16198 0.580990 0.813910i \(-0.302665\pi\)
0.580990 + 0.813910i \(0.302665\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 54.5065 2.23267
\(597\) 0 0
\(598\) −84.1180 −3.43984
\(599\) −7.78454 −0.318068 −0.159034 0.987273i \(-0.550838\pi\)
−0.159034 + 0.987273i \(0.550838\pi\)
\(600\) 0 0
\(601\) −20.5514 −0.838311 −0.419155 0.907915i \(-0.637674\pi\)
−0.419155 + 0.907915i \(0.637674\pi\)
\(602\) −105.586 −4.30335
\(603\) 0 0
\(604\) 102.805 4.18308
\(605\) 0 0
\(606\) 0 0
\(607\) −17.4256 −0.707281 −0.353641 0.935381i \(-0.615056\pi\)
−0.353641 + 0.935381i \(0.615056\pi\)
\(608\) −59.9351 −2.43069
\(609\) 0 0
\(610\) 0 0
\(611\) −24.6874 −0.998746
\(612\) 0 0
\(613\) 35.2749 1.42474 0.712369 0.701805i \(-0.247622\pi\)
0.712369 + 0.701805i \(0.247622\pi\)
\(614\) −24.6492 −0.994760
\(615\) 0 0
\(616\) 134.434 5.41651
\(617\) 40.3548 1.62462 0.812311 0.583225i \(-0.198209\pi\)
0.812311 + 0.583225i \(0.198209\pi\)
\(618\) 0 0
\(619\) 15.3790 0.618134 0.309067 0.951040i \(-0.399983\pi\)
0.309067 + 0.951040i \(0.399983\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −58.8916 −2.36134
\(623\) −10.3234 −0.413599
\(624\) 0 0
\(625\) 0 0
\(626\) −2.61408 −0.104480
\(627\) 0 0
\(628\) 8.27564 0.330234
\(629\) 28.0314 1.11768
\(630\) 0 0
\(631\) −23.7917 −0.947131 −0.473565 0.880759i \(-0.657033\pi\)
−0.473565 + 0.880759i \(0.657033\pi\)
\(632\) −91.0795 −3.62295
\(633\) 0 0
\(634\) 14.3146 0.568506
\(635\) 0 0
\(636\) 0 0
\(637\) 68.1054 2.69843
\(638\) 9.32272 0.369090
\(639\) 0 0
\(640\) 0 0
\(641\) −30.4524 −1.20280 −0.601398 0.798949i \(-0.705390\pi\)
−0.601398 + 0.798949i \(0.705390\pi\)
\(642\) 0 0
\(643\) 26.8637 1.05940 0.529700 0.848185i \(-0.322304\pi\)
0.529700 + 0.848185i \(0.322304\pi\)
\(644\) 164.822 6.49489
\(645\) 0 0
\(646\) 71.7346 2.82236
\(647\) −7.95411 −0.312708 −0.156354 0.987701i \(-0.549974\pi\)
−0.156354 + 0.987701i \(0.549974\pi\)
\(648\) 0 0
\(649\) −18.7995 −0.737946
\(650\) 0 0
\(651\) 0 0
\(652\) 84.6161 3.31382
\(653\) 22.9890 0.899627 0.449814 0.893122i \(-0.351490\pi\)
0.449814 + 0.893122i \(0.351490\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −59.4776 −2.32221
\(657\) 0 0
\(658\) 67.4194 2.62828
\(659\) −3.03695 −0.118303 −0.0591515 0.998249i \(-0.518839\pi\)
−0.0591515 + 0.998249i \(0.518839\pi\)
\(660\) 0 0
\(661\) −21.2414 −0.826194 −0.413097 0.910687i \(-0.635553\pi\)
−0.413097 + 0.910687i \(0.635553\pi\)
\(662\) 76.8450 2.98667
\(663\) 0 0
\(664\) −120.163 −4.66324
\(665\) 0 0
\(666\) 0 0
\(667\) 6.92951 0.268312
\(668\) −54.4586 −2.10707
\(669\) 0 0
\(670\) 0 0
\(671\) −4.45511 −0.171987
\(672\) 0 0
\(673\) 41.7638 1.60988 0.804938 0.593359i \(-0.202198\pi\)
0.804938 + 0.593359i \(0.202198\pi\)
\(674\) 47.0406 1.81193
\(675\) 0 0
\(676\) 39.6953 1.52674
\(677\) 16.1894 0.622210 0.311105 0.950375i \(-0.399301\pi\)
0.311105 + 0.950375i \(0.399301\pi\)
\(678\) 0 0
\(679\) 25.5991 0.982403
\(680\) 0 0
\(681\) 0 0
\(682\) −75.5922 −2.89458
\(683\) −36.0012 −1.37755 −0.688774 0.724977i \(-0.741851\pi\)
−0.688774 + 0.724977i \(0.741851\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −98.7748 −3.77124
\(687\) 0 0
\(688\) 98.6569 3.76126
\(689\) −19.1667 −0.730195
\(690\) 0 0
\(691\) −18.5707 −0.706464 −0.353232 0.935536i \(-0.614917\pi\)
−0.353232 + 0.935536i \(0.614917\pi\)
\(692\) 64.9286 2.46821
\(693\) 0 0
\(694\) −27.2391 −1.03398
\(695\) 0 0
\(696\) 0 0
\(697\) 33.5268 1.26992
\(698\) −19.9597 −0.755488
\(699\) 0 0
\(700\) 0 0
\(701\) −30.3600 −1.14668 −0.573342 0.819316i \(-0.694353\pi\)
−0.573342 + 0.819316i \(0.694353\pi\)
\(702\) 0 0
\(703\) 17.5492 0.661881
\(704\) −54.4226 −2.05113
\(705\) 0 0
\(706\) 67.9197 2.55619
\(707\) −35.3417 −1.32916
\(708\) 0 0
\(709\) −31.8654 −1.19673 −0.598365 0.801224i \(-0.704183\pi\)
−0.598365 + 0.801224i \(0.704183\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 18.0633 0.676950
\(713\) −56.1872 −2.10423
\(714\) 0 0
\(715\) 0 0
\(716\) 16.2189 0.606130
\(717\) 0 0
\(718\) 70.9788 2.64891
\(719\) −29.9357 −1.11641 −0.558206 0.829702i \(-0.688510\pi\)
−0.558206 + 0.829702i \(0.688510\pi\)
\(720\) 0 0
\(721\) −24.2843 −0.904394
\(722\) −5.64381 −0.210041
\(723\) 0 0
\(724\) −90.5482 −3.36520
\(725\) 0 0
\(726\) 0 0
\(727\) −36.9479 −1.37032 −0.685161 0.728392i \(-0.740268\pi\)
−0.685161 + 0.728392i \(0.740268\pi\)
\(728\) −175.047 −6.48766
\(729\) 0 0
\(730\) 0 0
\(731\) −55.6118 −2.05688
\(732\) 0 0
\(733\) 39.5741 1.46170 0.730851 0.682537i \(-0.239123\pi\)
0.730851 + 0.682537i \(0.239123\pi\)
\(734\) 24.0246 0.886764
\(735\) 0 0
\(736\) −101.091 −3.72626
\(737\) 34.8988 1.28551
\(738\) 0 0
\(739\) 6.40567 0.235636 0.117818 0.993035i \(-0.462410\pi\)
0.117818 + 0.993035i \(0.462410\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 52.3429 1.92157
\(743\) 1.37536 0.0504570 0.0252285 0.999682i \(-0.491969\pi\)
0.0252285 + 0.999682i \(0.491969\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −59.6466 −2.18382
\(747\) 0 0
\(748\) 116.793 4.27038
\(749\) −20.5863 −0.752207
\(750\) 0 0
\(751\) 27.5712 1.00609 0.503044 0.864261i \(-0.332213\pi\)
0.503044 + 0.864261i \(0.332213\pi\)
\(752\) −62.9952 −2.29720
\(753\) 0 0
\(754\) −12.1391 −0.442080
\(755\) 0 0
\(756\) 0 0
\(757\) 31.8451 1.15743 0.578714 0.815530i \(-0.303555\pi\)
0.578714 + 0.815530i \(0.303555\pi\)
\(758\) 25.6941 0.933253
\(759\) 0 0
\(760\) 0 0
\(761\) 20.2493 0.734038 0.367019 0.930213i \(-0.380378\pi\)
0.367019 + 0.930213i \(0.380378\pi\)
\(762\) 0 0
\(763\) 16.3237 0.590957
\(764\) 17.1185 0.619326
\(765\) 0 0
\(766\) 53.2971 1.92570
\(767\) 24.4788 0.883879
\(768\) 0 0
\(769\) 14.9837 0.540325 0.270162 0.962815i \(-0.412923\pi\)
0.270162 + 0.962815i \(0.412923\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −56.1490 −2.02085
\(773\) 10.7077 0.385131 0.192565 0.981284i \(-0.438319\pi\)
0.192565 + 0.981284i \(0.438319\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −44.7917 −1.60793
\(777\) 0 0
\(778\) −52.5407 −1.88368
\(779\) 20.9897 0.752033
\(780\) 0 0
\(781\) 42.8055 1.53170
\(782\) 120.993 4.32670
\(783\) 0 0
\(784\) 173.785 6.20662
\(785\) 0 0
\(786\) 0 0
\(787\) 9.72906 0.346804 0.173402 0.984851i \(-0.444524\pi\)
0.173402 + 0.984851i \(0.444524\pi\)
\(788\) 40.6276 1.44730
\(789\) 0 0
\(790\) 0 0
\(791\) −49.9374 −1.77557
\(792\) 0 0
\(793\) 5.80099 0.205999
\(794\) 3.71266 0.131757
\(795\) 0 0
\(796\) 42.2006 1.49576
\(797\) 9.52385 0.337352 0.168676 0.985672i \(-0.446051\pi\)
0.168676 + 0.985672i \(0.446051\pi\)
\(798\) 0 0
\(799\) 35.5097 1.25624
\(800\) 0 0
\(801\) 0 0
\(802\) 67.3806 2.37929
\(803\) 29.3391 1.03536
\(804\) 0 0
\(805\) 0 0
\(806\) 98.4285 3.46700
\(807\) 0 0
\(808\) 61.8387 2.17548
\(809\) −43.2976 −1.52226 −0.761131 0.648598i \(-0.775356\pi\)
−0.761131 + 0.648598i \(0.775356\pi\)
\(810\) 0 0
\(811\) −12.4910 −0.438619 −0.219310 0.975655i \(-0.570381\pi\)
−0.219310 + 0.975655i \(0.570381\pi\)
\(812\) 23.7855 0.834707
\(813\) 0 0
\(814\) 39.8225 1.39578
\(815\) 0 0
\(816\) 0 0
\(817\) −34.8161 −1.21806
\(818\) −14.5005 −0.506996
\(819\) 0 0
\(820\) 0 0
\(821\) 16.3447 0.570434 0.285217 0.958463i \(-0.407934\pi\)
0.285217 + 0.958463i \(0.407934\pi\)
\(822\) 0 0
\(823\) −24.4971 −0.853916 −0.426958 0.904271i \(-0.640415\pi\)
−0.426958 + 0.904271i \(0.640415\pi\)
\(824\) 42.4911 1.48025
\(825\) 0 0
\(826\) −66.8497 −2.32600
\(827\) 44.6342 1.55208 0.776042 0.630681i \(-0.217224\pi\)
0.776042 + 0.630681i \(0.217224\pi\)
\(828\) 0 0
\(829\) 19.8536 0.689543 0.344772 0.938687i \(-0.387956\pi\)
0.344772 + 0.938687i \(0.387956\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 70.8636 2.45675
\(833\) −97.9609 −3.39414
\(834\) 0 0
\(835\) 0 0
\(836\) 73.1189 2.52887
\(837\) 0 0
\(838\) −70.9572 −2.45118
\(839\) −1.62403 −0.0560679 −0.0280339 0.999607i \(-0.508925\pi\)
−0.0280339 + 0.999607i \(0.508925\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 49.9148 1.72018
\(843\) 0 0
\(844\) 31.0332 1.06821
\(845\) 0 0
\(846\) 0 0
\(847\) −5.97909 −0.205444
\(848\) −48.9080 −1.67951
\(849\) 0 0
\(850\) 0 0
\(851\) 29.5998 1.01467
\(852\) 0 0
\(853\) −51.1148 −1.75014 −0.875069 0.483998i \(-0.839184\pi\)
−0.875069 + 0.483998i \(0.839184\pi\)
\(854\) −15.8420 −0.542103
\(855\) 0 0
\(856\) 36.0206 1.23116
\(857\) 33.4832 1.14376 0.571882 0.820336i \(-0.306214\pi\)
0.571882 + 0.820336i \(0.306214\pi\)
\(858\) 0 0
\(859\) 20.6161 0.703414 0.351707 0.936110i \(-0.385601\pi\)
0.351707 + 0.936110i \(0.385601\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −57.2507 −1.94997
\(863\) −5.87941 −0.200137 −0.100069 0.994981i \(-0.531906\pi\)
−0.100069 + 0.994981i \(0.531906\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −46.8144 −1.59082
\(867\) 0 0
\(868\) −192.862 −6.54616
\(869\) 38.9488 1.32125
\(870\) 0 0
\(871\) −45.4416 −1.53973
\(872\) −28.5622 −0.967237
\(873\) 0 0
\(874\) 75.7484 2.56223
\(875\) 0 0
\(876\) 0 0
\(877\) 42.2191 1.42564 0.712818 0.701349i \(-0.247418\pi\)
0.712818 + 0.701349i \(0.247418\pi\)
\(878\) −75.3420 −2.54267
\(879\) 0 0
\(880\) 0 0
\(881\) −34.8813 −1.17518 −0.587590 0.809159i \(-0.699923\pi\)
−0.587590 + 0.809159i \(0.699923\pi\)
\(882\) 0 0
\(883\) 44.5605 1.49958 0.749790 0.661675i \(-0.230154\pi\)
0.749790 + 0.661675i \(0.230154\pi\)
\(884\) −152.076 −5.11487
\(885\) 0 0
\(886\) −58.5136 −1.96580
\(887\) −24.0830 −0.808627 −0.404313 0.914620i \(-0.632490\pi\)
−0.404313 + 0.914620i \(0.632490\pi\)
\(888\) 0 0
\(889\) −6.86826 −0.230354
\(890\) 0 0
\(891\) 0 0
\(892\) −106.788 −3.57551
\(893\) 22.2310 0.743933
\(894\) 0 0
\(895\) 0 0
\(896\) −56.8957 −1.90075
\(897\) 0 0
\(898\) 16.1786 0.539885
\(899\) −8.10839 −0.270430
\(900\) 0 0
\(901\) 27.5689 0.918454
\(902\) 47.6296 1.58589
\(903\) 0 0
\(904\) 87.3774 2.90613
\(905\) 0 0
\(906\) 0 0
\(907\) 22.3174 0.741037 0.370519 0.928825i \(-0.379180\pi\)
0.370519 + 0.928825i \(0.379180\pi\)
\(908\) 28.4218 0.943212
\(909\) 0 0
\(910\) 0 0
\(911\) −45.7050 −1.51427 −0.757137 0.653256i \(-0.773403\pi\)
−0.757137 + 0.653256i \(0.773403\pi\)
\(912\) 0 0
\(913\) 51.3861 1.70063
\(914\) 73.4240 2.42865
\(915\) 0 0
\(916\) 149.116 4.92692
\(917\) −25.4345 −0.839920
\(918\) 0 0
\(919\) −29.5423 −0.974509 −0.487255 0.873260i \(-0.662002\pi\)
−0.487255 + 0.873260i \(0.662002\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 54.1874 1.78457
\(923\) −55.7370 −1.83461
\(924\) 0 0
\(925\) 0 0
\(926\) −106.730 −3.50737
\(927\) 0 0
\(928\) −14.5885 −0.478890
\(929\) −55.6718 −1.82653 −0.913267 0.407362i \(-0.866449\pi\)
−0.913267 + 0.407362i \(0.866449\pi\)
\(930\) 0 0
\(931\) −61.3290 −2.00998
\(932\) 48.1456 1.57706
\(933\) 0 0
\(934\) −85.2674 −2.79004
\(935\) 0 0
\(936\) 0 0
\(937\) −36.7113 −1.19931 −0.599654 0.800260i \(-0.704695\pi\)
−0.599654 + 0.800260i \(0.704695\pi\)
\(938\) 124.098 4.05193
\(939\) 0 0
\(940\) 0 0
\(941\) −23.3426 −0.760948 −0.380474 0.924792i \(-0.624239\pi\)
−0.380474 + 0.924792i \(0.624239\pi\)
\(942\) 0 0
\(943\) 35.4027 1.15287
\(944\) 62.4629 2.03299
\(945\) 0 0
\(946\) −79.0044 −2.56865
\(947\) −51.6216 −1.67748 −0.838739 0.544534i \(-0.816707\pi\)
−0.838739 + 0.544534i \(0.816707\pi\)
\(948\) 0 0
\(949\) −38.2024 −1.24010
\(950\) 0 0
\(951\) 0 0
\(952\) 251.782 8.16031
\(953\) −37.5233 −1.21550 −0.607750 0.794129i \(-0.707928\pi\)
−0.607750 + 0.794129i \(0.707928\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −27.7177 −0.896456
\(957\) 0 0
\(958\) −60.3070 −1.94843
\(959\) 24.7128 0.798019
\(960\) 0 0
\(961\) 34.7459 1.12084
\(962\) −51.8528 −1.67180
\(963\) 0 0
\(964\) −3.40670 −0.109722
\(965\) 0 0
\(966\) 0 0
\(967\) −51.5532 −1.65784 −0.828919 0.559369i \(-0.811044\pi\)
−0.828919 + 0.559369i \(0.811044\pi\)
\(968\) 10.4618 0.336256
\(969\) 0 0
\(970\) 0 0
\(971\) 19.8516 0.637068 0.318534 0.947911i \(-0.396809\pi\)
0.318534 + 0.947911i \(0.396809\pi\)
\(972\) 0 0
\(973\) −50.1252 −1.60694
\(974\) 68.1235 2.18282
\(975\) 0 0
\(976\) 14.8024 0.473815
\(977\) −8.90236 −0.284812 −0.142406 0.989808i \(-0.545484\pi\)
−0.142406 + 0.989808i \(0.545484\pi\)
\(978\) 0 0
\(979\) −7.72450 −0.246876
\(980\) 0 0
\(981\) 0 0
\(982\) 9.04853 0.288750
\(983\) 3.44153 0.109768 0.0548838 0.998493i \(-0.482521\pi\)
0.0548838 + 0.998493i \(0.482521\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 17.4605 0.556057
\(987\) 0 0
\(988\) −95.2080 −3.02897
\(989\) −58.7234 −1.86730
\(990\) 0 0
\(991\) −26.7962 −0.851209 −0.425605 0.904909i \(-0.639939\pi\)
−0.425605 + 0.904909i \(0.639939\pi\)
\(992\) 118.289 3.75568
\(993\) 0 0
\(994\) 152.213 4.82791
\(995\) 0 0
\(996\) 0 0
\(997\) −0.913058 −0.0289168 −0.0144584 0.999895i \(-0.504602\pi\)
−0.0144584 + 0.999895i \(0.504602\pi\)
\(998\) 14.8562 0.470265
\(999\) 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.bv.1.7 7
3.2 odd 2 6525.2.a.bw.1.1 7
5.4 even 2 1305.2.a.t.1.1 yes 7
15.14 odd 2 1305.2.a.s.1.7 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1305.2.a.s.1.7 7 15.14 odd 2
1305.2.a.t.1.1 yes 7 5.4 even 2
6525.2.a.bv.1.7 7 1.1 even 1 trivial
6525.2.a.bw.1.1 7 3.2 odd 2