Properties

Label 7569.2.a.bj.1.8
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 1.8
Root \(2.21072\) 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)
 
\(f(q)\) \(=\) \(q+1.21072 q^{2} -0.534161 q^{4} -1.79844 q^{5} +4.27693 q^{7} -3.06816 q^{8} +O(q^{10})\) \(q+1.21072 q^{2} -0.534161 q^{4} -1.79844 q^{5} +4.27693 q^{7} -3.06816 q^{8} -2.17740 q^{10} -1.05626 q^{11} -5.28193 q^{13} +5.17816 q^{14} -2.64635 q^{16} -5.61769 q^{17} +6.41175 q^{19} +0.960655 q^{20} -1.27883 q^{22} -2.04633 q^{23} -1.76563 q^{25} -6.39492 q^{26} -2.28457 q^{28} +3.82471 q^{31} +2.93233 q^{32} -6.80144 q^{34} -7.69178 q^{35} -0.350297 q^{37} +7.76282 q^{38} +5.51788 q^{40} +3.62240 q^{41} +1.74900 q^{43} +0.564212 q^{44} -2.47753 q^{46} -6.98005 q^{47} +11.2921 q^{49} -2.13768 q^{50} +2.82140 q^{52} +7.81869 q^{53} +1.89961 q^{55} -13.1223 q^{56} -0.382668 q^{59} -5.46644 q^{61} +4.63064 q^{62} +8.84292 q^{64} +9.49920 q^{65} +7.81166 q^{67} +3.00075 q^{68} -9.31258 q^{70} +15.6115 q^{71} -2.54289 q^{73} -0.424110 q^{74} -3.42491 q^{76} -4.51754 q^{77} +10.2680 q^{79} +4.75929 q^{80} +4.38571 q^{82} -1.52139 q^{83} +10.1030 q^{85} +2.11755 q^{86} +3.24076 q^{88} +17.9444 q^{89} -22.5904 q^{91} +1.09307 q^{92} -8.45087 q^{94} -11.5311 q^{95} -6.80968 q^{97} +13.6716 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 5 q^{2} + 11 q^{4} + 4 q^{5} + 5 q^{7} - 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 5 q^{2} + 11 q^{4} + 4 q^{5} + 5 q^{7} - 24 q^{8} + q^{11} + q^{13} - 9 q^{14} + 35 q^{16} - 2 q^{17} + 9 q^{19} + 18 q^{20} - 4 q^{22} + 4 q^{23} + q^{25} + 8 q^{26} + 40 q^{28} + 8 q^{31} - 43 q^{32} - 4 q^{34} - 22 q^{35} + 27 q^{37} + 30 q^{38} - 29 q^{40} - 12 q^{41} + 16 q^{43} - 37 q^{44} - 22 q^{46} + 8 q^{47} - 6 q^{49} + 7 q^{50} + 33 q^{52} + 8 q^{53} + 9 q^{55} - 40 q^{56} + 16 q^{59} + 21 q^{61} + 32 q^{62} + 36 q^{64} + 31 q^{65} + 3 q^{67} - 33 q^{68} - 6 q^{70} + 33 q^{71} + 3 q^{73} - 28 q^{74} - 26 q^{76} - 24 q^{77} + 3 q^{79} + 64 q^{80} + 13 q^{82} - 13 q^{83} + 6 q^{85} - 58 q^{86} + 27 q^{88} - 6 q^{89} + q^{91} + 29 q^{92} - 18 q^{94} - 48 q^{95} + 4 q^{97} + 30 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.21072 0.856107 0.428054 0.903753i \(-0.359199\pi\)
0.428054 + 0.903753i \(0.359199\pi\)
\(3\) 0 0
\(4\) −0.534161 −0.267081
\(5\) −1.79844 −0.804285 −0.402142 0.915577i \(-0.631734\pi\)
−0.402142 + 0.915577i \(0.631734\pi\)
\(6\) 0 0
\(7\) 4.27693 1.61653 0.808264 0.588821i \(-0.200408\pi\)
0.808264 + 0.588821i \(0.200408\pi\)
\(8\) −3.06816 −1.08476
\(9\) 0 0
\(10\) −2.17740 −0.688554
\(11\) −1.05626 −0.318474 −0.159237 0.987240i \(-0.550903\pi\)
−0.159237 + 0.987240i \(0.550903\pi\)
\(12\) 0 0
\(13\) −5.28193 −1.46494 −0.732471 0.680798i \(-0.761633\pi\)
−0.732471 + 0.680798i \(0.761633\pi\)
\(14\) 5.17816 1.38392
\(15\) 0 0
\(16\) −2.64635 −0.661587
\(17\) −5.61769 −1.36249 −0.681245 0.732056i \(-0.738561\pi\)
−0.681245 + 0.732056i \(0.738561\pi\)
\(18\) 0 0
\(19\) 6.41175 1.47096 0.735478 0.677548i \(-0.236958\pi\)
0.735478 + 0.677548i \(0.236958\pi\)
\(20\) 0.960655 0.214809
\(21\) 0 0
\(22\) −1.27883 −0.272648
\(23\) −2.04633 −0.426689 −0.213345 0.976977i \(-0.568436\pi\)
−0.213345 + 0.976977i \(0.568436\pi\)
\(24\) 0 0
\(25\) −1.76563 −0.353126
\(26\) −6.39492 −1.25415
\(27\) 0 0
\(28\) −2.28457 −0.431743
\(29\) 0 0
\(30\) 0 0
\(31\) 3.82471 0.686937 0.343469 0.939164i \(-0.388398\pi\)
0.343469 + 0.939164i \(0.388398\pi\)
\(32\) 2.93233 0.518367
\(33\) 0 0
\(34\) −6.80144 −1.16644
\(35\) −7.69178 −1.30015
\(36\) 0 0
\(37\) −0.350297 −0.0575884 −0.0287942 0.999585i \(-0.509167\pi\)
−0.0287942 + 0.999585i \(0.509167\pi\)
\(38\) 7.76282 1.25930
\(39\) 0 0
\(40\) 5.51788 0.872453
\(41\) 3.62240 0.565725 0.282862 0.959161i \(-0.408716\pi\)
0.282862 + 0.959161i \(0.408716\pi\)
\(42\) 0 0
\(43\) 1.74900 0.266721 0.133360 0.991068i \(-0.457423\pi\)
0.133360 + 0.991068i \(0.457423\pi\)
\(44\) 0.564212 0.0850582
\(45\) 0 0
\(46\) −2.47753 −0.365292
\(47\) −6.98005 −1.01814 −0.509072 0.860724i \(-0.670011\pi\)
−0.509072 + 0.860724i \(0.670011\pi\)
\(48\) 0 0
\(49\) 11.2921 1.61316
\(50\) −2.13768 −0.302314
\(51\) 0 0
\(52\) 2.82140 0.391258
\(53\) 7.81869 1.07398 0.536990 0.843589i \(-0.319561\pi\)
0.536990 + 0.843589i \(0.319561\pi\)
\(54\) 0 0
\(55\) 1.89961 0.256144
\(56\) −13.1223 −1.75354
\(57\) 0 0
\(58\) 0 0
\(59\) −0.382668 −0.0498191 −0.0249096 0.999690i \(-0.507930\pi\)
−0.0249096 + 0.999690i \(0.507930\pi\)
\(60\) 0 0
\(61\) −5.46644 −0.699906 −0.349953 0.936767i \(-0.613803\pi\)
−0.349953 + 0.936767i \(0.613803\pi\)
\(62\) 4.63064 0.588092
\(63\) 0 0
\(64\) 8.84292 1.10537
\(65\) 9.49920 1.17823
\(66\) 0 0
\(67\) 7.81166 0.954346 0.477173 0.878809i \(-0.341662\pi\)
0.477173 + 0.878809i \(0.341662\pi\)
\(68\) 3.00075 0.363895
\(69\) 0 0
\(70\) −9.31258 −1.11307
\(71\) 15.6115 1.85274 0.926370 0.376615i \(-0.122912\pi\)
0.926370 + 0.376615i \(0.122912\pi\)
\(72\) 0 0
\(73\) −2.54289 −0.297623 −0.148811 0.988866i \(-0.547545\pi\)
−0.148811 + 0.988866i \(0.547545\pi\)
\(74\) −0.424110 −0.0493018
\(75\) 0 0
\(76\) −3.42491 −0.392864
\(77\) −4.51754 −0.514822
\(78\) 0 0
\(79\) 10.2680 1.15524 0.577619 0.816306i \(-0.303982\pi\)
0.577619 + 0.816306i \(0.303982\pi\)
\(80\) 4.75929 0.532104
\(81\) 0 0
\(82\) 4.38571 0.484321
\(83\) −1.52139 −0.166994 −0.0834971 0.996508i \(-0.526609\pi\)
−0.0834971 + 0.996508i \(0.526609\pi\)
\(84\) 0 0
\(85\) 10.1030 1.09583
\(86\) 2.11755 0.228341
\(87\) 0 0
\(88\) 3.24076 0.345467
\(89\) 17.9444 1.90210 0.951049 0.309041i \(-0.100008\pi\)
0.951049 + 0.309041i \(0.100008\pi\)
\(90\) 0 0
\(91\) −22.5904 −2.36812
\(92\) 1.09307 0.113960
\(93\) 0 0
\(94\) −8.45087 −0.871641
\(95\) −11.5311 −1.18307
\(96\) 0 0
\(97\) −6.80968 −0.691418 −0.345709 0.938342i \(-0.612362\pi\)
−0.345709 + 0.938342i \(0.612362\pi\)
\(98\) 13.6716 1.38104
\(99\) 0 0
\(100\) 0.943132 0.0943132
\(101\) −4.72573 −0.470228 −0.235114 0.971968i \(-0.575546\pi\)
−0.235114 + 0.971968i \(0.575546\pi\)
\(102\) 0 0
\(103\) 3.49491 0.344364 0.172182 0.985065i \(-0.444918\pi\)
0.172182 + 0.985065i \(0.444918\pi\)
\(104\) 16.2058 1.58911
\(105\) 0 0
\(106\) 9.46624 0.919442
\(107\) 4.92192 0.475820 0.237910 0.971287i \(-0.423538\pi\)
0.237910 + 0.971287i \(0.423538\pi\)
\(108\) 0 0
\(109\) −14.8847 −1.42569 −0.712846 0.701321i \(-0.752594\pi\)
−0.712846 + 0.701321i \(0.752594\pi\)
\(110\) 2.29989 0.219286
\(111\) 0 0
\(112\) −11.3182 −1.06947
\(113\) 6.34278 0.596678 0.298339 0.954460i \(-0.403567\pi\)
0.298339 + 0.954460i \(0.403567\pi\)
\(114\) 0 0
\(115\) 3.68019 0.343180
\(116\) 0 0
\(117\) 0 0
\(118\) −0.463303 −0.0426505
\(119\) −24.0265 −2.20250
\(120\) 0 0
\(121\) −9.88432 −0.898574
\(122\) −6.61832 −0.599195
\(123\) 0 0
\(124\) −2.04301 −0.183468
\(125\) 12.1675 1.08830
\(126\) 0 0
\(127\) −4.46569 −0.396266 −0.198133 0.980175i \(-0.563488\pi\)
−0.198133 + 0.980175i \(0.563488\pi\)
\(128\) 4.84163 0.427944
\(129\) 0 0
\(130\) 11.5009 1.00869
\(131\) −0.00995943 −0.000870160 0 −0.000435080 1.00000i \(-0.500138\pi\)
−0.000435080 1.00000i \(0.500138\pi\)
\(132\) 0 0
\(133\) 27.4226 2.37784
\(134\) 9.45772 0.817022
\(135\) 0 0
\(136\) 17.2359 1.47797
\(137\) −12.2134 −1.04346 −0.521732 0.853109i \(-0.674714\pi\)
−0.521732 + 0.853109i \(0.674714\pi\)
\(138\) 0 0
\(139\) 4.94561 0.419481 0.209741 0.977757i \(-0.432738\pi\)
0.209741 + 0.977757i \(0.432738\pi\)
\(140\) 4.10865 0.347245
\(141\) 0 0
\(142\) 18.9011 1.58614
\(143\) 5.57908 0.466546
\(144\) 0 0
\(145\) 0 0
\(146\) −3.07873 −0.254797
\(147\) 0 0
\(148\) 0.187115 0.0153808
\(149\) −14.0140 −1.14807 −0.574035 0.818831i \(-0.694623\pi\)
−0.574035 + 0.818831i \(0.694623\pi\)
\(150\) 0 0
\(151\) 2.93392 0.238759 0.119379 0.992849i \(-0.461910\pi\)
0.119379 + 0.992849i \(0.461910\pi\)
\(152\) −19.6722 −1.59563
\(153\) 0 0
\(154\) −5.46947 −0.440742
\(155\) −6.87849 −0.552493
\(156\) 0 0
\(157\) 16.1919 1.29226 0.646128 0.763229i \(-0.276387\pi\)
0.646128 + 0.763229i \(0.276387\pi\)
\(158\) 12.4316 0.989008
\(159\) 0 0
\(160\) −5.27360 −0.416915
\(161\) −8.75201 −0.689755
\(162\) 0 0
\(163\) 13.5734 1.06315 0.531575 0.847011i \(-0.321600\pi\)
0.531575 + 0.847011i \(0.321600\pi\)
\(164\) −1.93495 −0.151094
\(165\) 0 0
\(166\) −1.84197 −0.142965
\(167\) 16.3616 1.26610 0.633050 0.774111i \(-0.281803\pi\)
0.633050 + 0.774111i \(0.281803\pi\)
\(168\) 0 0
\(169\) 14.8987 1.14606
\(170\) 12.2319 0.938147
\(171\) 0 0
\(172\) −0.934250 −0.0712359
\(173\) 4.46374 0.339372 0.169686 0.985498i \(-0.445725\pi\)
0.169686 + 0.985498i \(0.445725\pi\)
\(174\) 0 0
\(175\) −7.55148 −0.570838
\(176\) 2.79523 0.210698
\(177\) 0 0
\(178\) 21.7256 1.62840
\(179\) −3.40063 −0.254175 −0.127088 0.991891i \(-0.540563\pi\)
−0.127088 + 0.991891i \(0.540563\pi\)
\(180\) 0 0
\(181\) 6.62134 0.492160 0.246080 0.969250i \(-0.420857\pi\)
0.246080 + 0.969250i \(0.420857\pi\)
\(182\) −27.3506 −2.02736
\(183\) 0 0
\(184\) 6.27846 0.462854
\(185\) 0.629986 0.0463175
\(186\) 0 0
\(187\) 5.93373 0.433917
\(188\) 3.72847 0.271927
\(189\) 0 0
\(190\) −13.9609 −1.01283
\(191\) −17.6196 −1.27491 −0.637454 0.770489i \(-0.720012\pi\)
−0.637454 + 0.770489i \(0.720012\pi\)
\(192\) 0 0
\(193\) −13.3986 −0.964454 −0.482227 0.876046i \(-0.660172\pi\)
−0.482227 + 0.876046i \(0.660172\pi\)
\(194\) −8.24460 −0.591928
\(195\) 0 0
\(196\) −6.03182 −0.430844
\(197\) 8.20476 0.584565 0.292282 0.956332i \(-0.405585\pi\)
0.292282 + 0.956332i \(0.405585\pi\)
\(198\) 0 0
\(199\) 24.4954 1.73643 0.868216 0.496187i \(-0.165267\pi\)
0.868216 + 0.496187i \(0.165267\pi\)
\(200\) 5.41723 0.383056
\(201\) 0 0
\(202\) −5.72153 −0.402565
\(203\) 0 0
\(204\) 0 0
\(205\) −6.51466 −0.455004
\(206\) 4.23135 0.294812
\(207\) 0 0
\(208\) 13.9778 0.969187
\(209\) −6.77246 −0.468461
\(210\) 0 0
\(211\) −2.21143 −0.152241 −0.0761205 0.997099i \(-0.524253\pi\)
−0.0761205 + 0.997099i \(0.524253\pi\)
\(212\) −4.17644 −0.286839
\(213\) 0 0
\(214\) 5.95905 0.407353
\(215\) −3.14547 −0.214519
\(216\) 0 0
\(217\) 16.3580 1.11045
\(218\) −18.0211 −1.22054
\(219\) 0 0
\(220\) −1.01470 −0.0684110
\(221\) 29.6722 1.99597
\(222\) 0 0
\(223\) 2.70437 0.181098 0.0905491 0.995892i \(-0.471138\pi\)
0.0905491 + 0.995892i \(0.471138\pi\)
\(224\) 12.5414 0.837955
\(225\) 0 0
\(226\) 7.67932 0.510820
\(227\) −14.4225 −0.957258 −0.478629 0.878017i \(-0.658866\pi\)
−0.478629 + 0.878017i \(0.658866\pi\)
\(228\) 0 0
\(229\) 25.7467 1.70139 0.850694 0.525662i \(-0.176182\pi\)
0.850694 + 0.525662i \(0.176182\pi\)
\(230\) 4.45567 0.293798
\(231\) 0 0
\(232\) 0 0
\(233\) 22.6039 1.48083 0.740417 0.672148i \(-0.234628\pi\)
0.740417 + 0.672148i \(0.234628\pi\)
\(234\) 0 0
\(235\) 12.5532 0.818878
\(236\) 0.204406 0.0133057
\(237\) 0 0
\(238\) −29.0893 −1.88558
\(239\) −23.4233 −1.51513 −0.757565 0.652760i \(-0.773611\pi\)
−0.757565 + 0.652760i \(0.773611\pi\)
\(240\) 0 0
\(241\) 22.3826 1.44179 0.720896 0.693043i \(-0.243730\pi\)
0.720896 + 0.693043i \(0.243730\pi\)
\(242\) −11.9671 −0.769276
\(243\) 0 0
\(244\) 2.91996 0.186932
\(245\) −20.3082 −1.29744
\(246\) 0 0
\(247\) −33.8664 −2.15487
\(248\) −11.7348 −0.745160
\(249\) 0 0
\(250\) 14.7315 0.931700
\(251\) 5.95981 0.376180 0.188090 0.982152i \(-0.439770\pi\)
0.188090 + 0.982152i \(0.439770\pi\)
\(252\) 0 0
\(253\) 2.16145 0.135889
\(254\) −5.40669 −0.339246
\(255\) 0 0
\(256\) −11.8240 −0.739000
\(257\) 15.0723 0.940185 0.470092 0.882617i \(-0.344221\pi\)
0.470092 + 0.882617i \(0.344221\pi\)
\(258\) 0 0
\(259\) −1.49819 −0.0930932
\(260\) −5.07411 −0.314683
\(261\) 0 0
\(262\) −0.0120581 −0.000744950 0
\(263\) 18.7252 1.15465 0.577324 0.816515i \(-0.304097\pi\)
0.577324 + 0.816515i \(0.304097\pi\)
\(264\) 0 0
\(265\) −14.0614 −0.863786
\(266\) 33.2010 2.03569
\(267\) 0 0
\(268\) −4.17269 −0.254887
\(269\) 23.1259 1.41001 0.705006 0.709202i \(-0.250944\pi\)
0.705006 + 0.709202i \(0.250944\pi\)
\(270\) 0 0
\(271\) 21.3469 1.29673 0.648365 0.761330i \(-0.275453\pi\)
0.648365 + 0.761330i \(0.275453\pi\)
\(272\) 14.8664 0.901406
\(273\) 0 0
\(274\) −14.7870 −0.893317
\(275\) 1.86496 0.112461
\(276\) 0 0
\(277\) −8.32181 −0.500009 −0.250004 0.968245i \(-0.580432\pi\)
−0.250004 + 0.968245i \(0.580432\pi\)
\(278\) 5.98774 0.359121
\(279\) 0 0
\(280\) 23.5996 1.41034
\(281\) 28.9202 1.72524 0.862618 0.505856i \(-0.168823\pi\)
0.862618 + 0.505856i \(0.168823\pi\)
\(282\) 0 0
\(283\) 29.9799 1.78212 0.891060 0.453885i \(-0.149962\pi\)
0.891060 + 0.453885i \(0.149962\pi\)
\(284\) −8.33904 −0.494831
\(285\) 0 0
\(286\) 6.75469 0.399413
\(287\) 15.4928 0.914509
\(288\) 0 0
\(289\) 14.5584 0.856378
\(290\) 0 0
\(291\) 0 0
\(292\) 1.35831 0.0794893
\(293\) −8.75406 −0.511418 −0.255709 0.966754i \(-0.582309\pi\)
−0.255709 + 0.966754i \(0.582309\pi\)
\(294\) 0 0
\(295\) 0.688204 0.0400688
\(296\) 1.07476 0.0624694
\(297\) 0 0
\(298\) −16.9670 −0.982871
\(299\) 10.8086 0.625075
\(300\) 0 0
\(301\) 7.48036 0.431161
\(302\) 3.55215 0.204403
\(303\) 0 0
\(304\) −16.9677 −0.973166
\(305\) 9.83105 0.562924
\(306\) 0 0
\(307\) −33.9161 −1.93570 −0.967848 0.251536i \(-0.919064\pi\)
−0.967848 + 0.251536i \(0.919064\pi\)
\(308\) 2.41310 0.137499
\(309\) 0 0
\(310\) −8.32791 −0.472993
\(311\) −8.12574 −0.460769 −0.230384 0.973100i \(-0.573998\pi\)
−0.230384 + 0.973100i \(0.573998\pi\)
\(312\) 0 0
\(313\) 2.52738 0.142856 0.0714280 0.997446i \(-0.477244\pi\)
0.0714280 + 0.997446i \(0.477244\pi\)
\(314\) 19.6039 1.10631
\(315\) 0 0
\(316\) −5.48476 −0.308542
\(317\) −16.6383 −0.934499 −0.467250 0.884125i \(-0.654755\pi\)
−0.467250 + 0.884125i \(0.654755\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −15.9034 −0.889028
\(321\) 0 0
\(322\) −10.5962 −0.590504
\(323\) −36.0192 −2.00416
\(324\) 0 0
\(325\) 9.32593 0.517309
\(326\) 16.4336 0.910170
\(327\) 0 0
\(328\) −11.1141 −0.613673
\(329\) −29.8532 −1.64586
\(330\) 0 0
\(331\) 12.9286 0.710619 0.355309 0.934749i \(-0.384375\pi\)
0.355309 + 0.934749i \(0.384375\pi\)
\(332\) 0.812667 0.0446009
\(333\) 0 0
\(334\) 19.8093 1.08392
\(335\) −14.0488 −0.767566
\(336\) 0 0
\(337\) −5.56273 −0.303021 −0.151511 0.988456i \(-0.548414\pi\)
−0.151511 + 0.988456i \(0.548414\pi\)
\(338\) 18.0382 0.981148
\(339\) 0 0
\(340\) −5.39666 −0.292675
\(341\) −4.03988 −0.218772
\(342\) 0 0
\(343\) 18.3571 0.991192
\(344\) −5.36621 −0.289327
\(345\) 0 0
\(346\) 5.40433 0.290539
\(347\) 8.78686 0.471703 0.235852 0.971789i \(-0.424212\pi\)
0.235852 + 0.971789i \(0.424212\pi\)
\(348\) 0 0
\(349\) 15.1943 0.813332 0.406666 0.913577i \(-0.366691\pi\)
0.406666 + 0.913577i \(0.366691\pi\)
\(350\) −9.14271 −0.488698
\(351\) 0 0
\(352\) −3.09729 −0.165086
\(353\) 12.3667 0.658214 0.329107 0.944293i \(-0.393252\pi\)
0.329107 + 0.944293i \(0.393252\pi\)
\(354\) 0 0
\(355\) −28.0762 −1.49013
\(356\) −9.58518 −0.508014
\(357\) 0 0
\(358\) −4.11721 −0.217601
\(359\) 16.0164 0.845312 0.422656 0.906290i \(-0.361098\pi\)
0.422656 + 0.906290i \(0.361098\pi\)
\(360\) 0 0
\(361\) 22.1105 1.16371
\(362\) 8.01657 0.421342
\(363\) 0 0
\(364\) 12.0669 0.632479
\(365\) 4.57323 0.239374
\(366\) 0 0
\(367\) −22.2447 −1.16116 −0.580581 0.814203i \(-0.697174\pi\)
−0.580581 + 0.814203i \(0.697174\pi\)
\(368\) 5.41530 0.282292
\(369\) 0 0
\(370\) 0.762735 0.0396527
\(371\) 33.4400 1.73612
\(372\) 0 0
\(373\) 6.93029 0.358837 0.179418 0.983773i \(-0.442578\pi\)
0.179418 + 0.983773i \(0.442578\pi\)
\(374\) 7.18407 0.371480
\(375\) 0 0
\(376\) 21.4159 1.10444
\(377\) 0 0
\(378\) 0 0
\(379\) −7.78848 −0.400067 −0.200034 0.979789i \(-0.564105\pi\)
−0.200034 + 0.979789i \(0.564105\pi\)
\(380\) 6.15948 0.315975
\(381\) 0 0
\(382\) −21.3323 −1.09146
\(383\) −25.4625 −1.30107 −0.650537 0.759475i \(-0.725456\pi\)
−0.650537 + 0.759475i \(0.725456\pi\)
\(384\) 0 0
\(385\) 8.12451 0.414063
\(386\) −16.2220 −0.825676
\(387\) 0 0
\(388\) 3.63747 0.184664
\(389\) −10.9838 −0.556902 −0.278451 0.960450i \(-0.589821\pi\)
−0.278451 + 0.960450i \(0.589821\pi\)
\(390\) 0 0
\(391\) 11.4956 0.581359
\(392\) −34.6460 −1.74989
\(393\) 0 0
\(394\) 9.93365 0.500450
\(395\) −18.4663 −0.929140
\(396\) 0 0
\(397\) 5.10047 0.255985 0.127993 0.991775i \(-0.459147\pi\)
0.127993 + 0.991775i \(0.459147\pi\)
\(398\) 29.6570 1.48657
\(399\) 0 0
\(400\) 4.67247 0.233624
\(401\) 19.5421 0.975886 0.487943 0.872875i \(-0.337747\pi\)
0.487943 + 0.872875i \(0.337747\pi\)
\(402\) 0 0
\(403\) −20.2018 −1.00632
\(404\) 2.52430 0.125589
\(405\) 0 0
\(406\) 0 0
\(407\) 0.370004 0.0183404
\(408\) 0 0
\(409\) −17.9244 −0.886305 −0.443152 0.896446i \(-0.646140\pi\)
−0.443152 + 0.896446i \(0.646140\pi\)
\(410\) −7.88742 −0.389532
\(411\) 0 0
\(412\) −1.86685 −0.0919730
\(413\) −1.63664 −0.0805340
\(414\) 0 0
\(415\) 2.73612 0.134311
\(416\) −15.4883 −0.759378
\(417\) 0 0
\(418\) −8.19954 −0.401053
\(419\) −3.73821 −0.182623 −0.0913116 0.995822i \(-0.529106\pi\)
−0.0913116 + 0.995822i \(0.529106\pi\)
\(420\) 0 0
\(421\) 8.28779 0.403922 0.201961 0.979394i \(-0.435269\pi\)
0.201961 + 0.979394i \(0.435269\pi\)
\(422\) −2.67742 −0.130335
\(423\) 0 0
\(424\) −23.9890 −1.16501
\(425\) 9.91876 0.481131
\(426\) 0 0
\(427\) −23.3796 −1.13142
\(428\) −2.62910 −0.127082
\(429\) 0 0
\(430\) −3.80828 −0.183651
\(431\) −19.1294 −0.921432 −0.460716 0.887548i \(-0.652407\pi\)
−0.460716 + 0.887548i \(0.652407\pi\)
\(432\) 0 0
\(433\) −16.1509 −0.776163 −0.388081 0.921625i \(-0.626862\pi\)
−0.388081 + 0.921625i \(0.626862\pi\)
\(434\) 19.8049 0.950667
\(435\) 0 0
\(436\) 7.95081 0.380775
\(437\) −13.1206 −0.627641
\(438\) 0 0
\(439\) 2.96836 0.141672 0.0708360 0.997488i \(-0.477433\pi\)
0.0708360 + 0.997488i \(0.477433\pi\)
\(440\) −5.82830 −0.277853
\(441\) 0 0
\(442\) 35.9247 1.70876
\(443\) −10.4916 −0.498471 −0.249235 0.968443i \(-0.580179\pi\)
−0.249235 + 0.968443i \(0.580179\pi\)
\(444\) 0 0
\(445\) −32.2718 −1.52983
\(446\) 3.27423 0.155039
\(447\) 0 0
\(448\) 37.8206 1.78685
\(449\) 4.40325 0.207802 0.103901 0.994588i \(-0.466867\pi\)
0.103901 + 0.994588i \(0.466867\pi\)
\(450\) 0 0
\(451\) −3.82619 −0.180168
\(452\) −3.38807 −0.159361
\(453\) 0 0
\(454\) −17.4616 −0.819515
\(455\) 40.6274 1.90464
\(456\) 0 0
\(457\) 11.3485 0.530860 0.265430 0.964130i \(-0.414486\pi\)
0.265430 + 0.964130i \(0.414486\pi\)
\(458\) 31.1720 1.45657
\(459\) 0 0
\(460\) −1.96582 −0.0916566
\(461\) −19.7092 −0.917947 −0.458973 0.888450i \(-0.651783\pi\)
−0.458973 + 0.888450i \(0.651783\pi\)
\(462\) 0 0
\(463\) 0.753315 0.0350095 0.0175048 0.999847i \(-0.494428\pi\)
0.0175048 + 0.999847i \(0.494428\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 27.3670 1.26775
\(467\) −10.4450 −0.483339 −0.241670 0.970359i \(-0.577695\pi\)
−0.241670 + 0.970359i \(0.577695\pi\)
\(468\) 0 0
\(469\) 33.4099 1.54273
\(470\) 15.1983 0.701047
\(471\) 0 0
\(472\) 1.17408 0.0540416
\(473\) −1.84740 −0.0849435
\(474\) 0 0
\(475\) −11.3208 −0.519433
\(476\) 12.8340 0.588246
\(477\) 0 0
\(478\) −28.3591 −1.29711
\(479\) 13.9781 0.638673 0.319337 0.947641i \(-0.396540\pi\)
0.319337 + 0.947641i \(0.396540\pi\)
\(480\) 0 0
\(481\) 1.85024 0.0843637
\(482\) 27.0991 1.23433
\(483\) 0 0
\(484\) 5.27982 0.239992
\(485\) 12.2468 0.556097
\(486\) 0 0
\(487\) −1.46028 −0.0661718 −0.0330859 0.999453i \(-0.510533\pi\)
−0.0330859 + 0.999453i \(0.510533\pi\)
\(488\) 16.7719 0.759228
\(489\) 0 0
\(490\) −24.5875 −1.11075
\(491\) 12.1522 0.548421 0.274210 0.961670i \(-0.411584\pi\)
0.274210 + 0.961670i \(0.411584\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) −41.0027 −1.84480
\(495\) 0 0
\(496\) −10.1215 −0.454469
\(497\) 66.7691 2.99501
\(498\) 0 0
\(499\) −35.9679 −1.61014 −0.805071 0.593178i \(-0.797873\pi\)
−0.805071 + 0.593178i \(0.797873\pi\)
\(500\) −6.49943 −0.290664
\(501\) 0 0
\(502\) 7.21565 0.322050
\(503\) 14.3546 0.640041 0.320020 0.947411i \(-0.396310\pi\)
0.320020 + 0.947411i \(0.396310\pi\)
\(504\) 0 0
\(505\) 8.49892 0.378197
\(506\) 2.61691 0.116336
\(507\) 0 0
\(508\) 2.38540 0.105835
\(509\) 31.0049 1.37427 0.687135 0.726530i \(-0.258868\pi\)
0.687135 + 0.726530i \(0.258868\pi\)
\(510\) 0 0
\(511\) −10.8758 −0.481116
\(512\) −23.9988 −1.06061
\(513\) 0 0
\(514\) 18.2483 0.804899
\(515\) −6.28537 −0.276967
\(516\) 0 0
\(517\) 7.37273 0.324252
\(518\) −1.81389 −0.0796978
\(519\) 0 0
\(520\) −29.1450 −1.27809
\(521\) 14.9084 0.653150 0.326575 0.945171i \(-0.394105\pi\)
0.326575 + 0.945171i \(0.394105\pi\)
\(522\) 0 0
\(523\) −12.9505 −0.566284 −0.283142 0.959078i \(-0.591377\pi\)
−0.283142 + 0.959078i \(0.591377\pi\)
\(524\) 0.00531995 0.000232403 0
\(525\) 0 0
\(526\) 22.6710 0.988502
\(527\) −21.4860 −0.935945
\(528\) 0 0
\(529\) −18.8125 −0.817936
\(530\) −17.0244 −0.739493
\(531\) 0 0
\(532\) −14.6481 −0.635076
\(533\) −19.1333 −0.828754
\(534\) 0 0
\(535\) −8.85175 −0.382694
\(536\) −23.9674 −1.03523
\(537\) 0 0
\(538\) 27.9990 1.20712
\(539\) −11.9274 −0.513750
\(540\) 0 0
\(541\) 10.8732 0.467475 0.233737 0.972300i \(-0.424904\pi\)
0.233737 + 0.972300i \(0.424904\pi\)
\(542\) 25.8450 1.11014
\(543\) 0 0
\(544\) −16.4729 −0.706270
\(545\) 26.7691 1.14666
\(546\) 0 0
\(547\) −23.1301 −0.988970 −0.494485 0.869186i \(-0.664643\pi\)
−0.494485 + 0.869186i \(0.664643\pi\)
\(548\) 6.52394 0.278689
\(549\) 0 0
\(550\) 2.25794 0.0962790
\(551\) 0 0
\(552\) 0 0
\(553\) 43.9154 1.86747
\(554\) −10.0754 −0.428061
\(555\) 0 0
\(556\) −2.64175 −0.112035
\(557\) 43.1297 1.82746 0.913731 0.406319i \(-0.133188\pi\)
0.913731 + 0.406319i \(0.133188\pi\)
\(558\) 0 0
\(559\) −9.23811 −0.390730
\(560\) 20.3551 0.860162
\(561\) 0 0
\(562\) 35.0142 1.47699
\(563\) −34.1036 −1.43729 −0.718647 0.695375i \(-0.755238\pi\)
−0.718647 + 0.695375i \(0.755238\pi\)
\(564\) 0 0
\(565\) −11.4071 −0.479899
\(566\) 36.2972 1.52569
\(567\) 0 0
\(568\) −47.8984 −2.00977
\(569\) 19.0810 0.799917 0.399958 0.916533i \(-0.369025\pi\)
0.399958 + 0.916533i \(0.369025\pi\)
\(570\) 0 0
\(571\) −6.56641 −0.274796 −0.137398 0.990516i \(-0.543874\pi\)
−0.137398 + 0.990516i \(0.543874\pi\)
\(572\) −2.98013 −0.124605
\(573\) 0 0
\(574\) 18.7574 0.782918
\(575\) 3.61306 0.150675
\(576\) 0 0
\(577\) −20.5602 −0.855934 −0.427967 0.903794i \(-0.640770\pi\)
−0.427967 + 0.903794i \(0.640770\pi\)
\(578\) 17.6261 0.733151
\(579\) 0 0
\(580\) 0 0
\(581\) −6.50687 −0.269951
\(582\) 0 0
\(583\) −8.25856 −0.342035
\(584\) 7.80199 0.322849
\(585\) 0 0
\(586\) −10.5987 −0.437828
\(587\) 9.69142 0.400008 0.200004 0.979795i \(-0.435905\pi\)
0.200004 + 0.979795i \(0.435905\pi\)
\(588\) 0 0
\(589\) 24.5231 1.01046
\(590\) 0.833221 0.0343032
\(591\) 0 0
\(592\) 0.927007 0.0380997
\(593\) 13.1810 0.541279 0.270639 0.962681i \(-0.412765\pi\)
0.270639 + 0.962681i \(0.412765\pi\)
\(594\) 0 0
\(595\) 43.2100 1.77144
\(596\) 7.48573 0.306627
\(597\) 0 0
\(598\) 13.0861 0.535131
\(599\) 14.4397 0.589991 0.294995 0.955499i \(-0.404682\pi\)
0.294995 + 0.955499i \(0.404682\pi\)
\(600\) 0 0
\(601\) 4.88823 0.199395 0.0996976 0.995018i \(-0.468212\pi\)
0.0996976 + 0.995018i \(0.468212\pi\)
\(602\) 9.05661 0.369120
\(603\) 0 0
\(604\) −1.56718 −0.0637678
\(605\) 17.7763 0.722710
\(606\) 0 0
\(607\) −12.5548 −0.509583 −0.254792 0.966996i \(-0.582007\pi\)
−0.254792 + 0.966996i \(0.582007\pi\)
\(608\) 18.8014 0.762496
\(609\) 0 0
\(610\) 11.9026 0.481923
\(611\) 36.8681 1.49152
\(612\) 0 0
\(613\) 3.49761 0.141267 0.0706336 0.997502i \(-0.477498\pi\)
0.0706336 + 0.997502i \(0.477498\pi\)
\(614\) −41.0629 −1.65716
\(615\) 0 0
\(616\) 13.8605 0.558456
\(617\) 27.2896 1.09864 0.549318 0.835613i \(-0.314887\pi\)
0.549318 + 0.835613i \(0.314887\pi\)
\(618\) 0 0
\(619\) −25.5410 −1.02658 −0.513289 0.858216i \(-0.671573\pi\)
−0.513289 + 0.858216i \(0.671573\pi\)
\(620\) 3.67422 0.147560
\(621\) 0 0
\(622\) −9.83798 −0.394467
\(623\) 76.7467 3.07479
\(624\) 0 0
\(625\) −13.0544 −0.522176
\(626\) 3.05995 0.122300
\(627\) 0 0
\(628\) −8.64911 −0.345137
\(629\) 1.96786 0.0784636
\(630\) 0 0
\(631\) 20.7956 0.827858 0.413929 0.910309i \(-0.364156\pi\)
0.413929 + 0.910309i \(0.364156\pi\)
\(632\) −31.5038 −1.25315
\(633\) 0 0
\(634\) −20.1443 −0.800031
\(635\) 8.03126 0.318711
\(636\) 0 0
\(637\) −59.6442 −2.36319
\(638\) 0 0
\(639\) 0 0
\(640\) −8.70736 −0.344188
\(641\) 14.1393 0.558468 0.279234 0.960223i \(-0.409920\pi\)
0.279234 + 0.960223i \(0.409920\pi\)
\(642\) 0 0
\(643\) 35.4943 1.39976 0.699878 0.714262i \(-0.253238\pi\)
0.699878 + 0.714262i \(0.253238\pi\)
\(644\) 4.67498 0.184220
\(645\) 0 0
\(646\) −43.6091 −1.71578
\(647\) 20.0626 0.788743 0.394371 0.918951i \(-0.370962\pi\)
0.394371 + 0.918951i \(0.370962\pi\)
\(648\) 0 0
\(649\) 0.404196 0.0158661
\(650\) 11.2911 0.442872
\(651\) 0 0
\(652\) −7.25038 −0.283947
\(653\) −13.3537 −0.522569 −0.261285 0.965262i \(-0.584146\pi\)
−0.261285 + 0.965262i \(0.584146\pi\)
\(654\) 0 0
\(655\) 0.0179114 0.000699856 0
\(656\) −9.58615 −0.374276
\(657\) 0 0
\(658\) −36.1438 −1.40903
\(659\) −11.3791 −0.443267 −0.221633 0.975130i \(-0.571139\pi\)
−0.221633 + 0.975130i \(0.571139\pi\)
\(660\) 0 0
\(661\) −37.0298 −1.44029 −0.720145 0.693823i \(-0.755925\pi\)
−0.720145 + 0.693823i \(0.755925\pi\)
\(662\) 15.6529 0.608366
\(663\) 0 0
\(664\) 4.66786 0.181148
\(665\) −49.3178 −1.91246
\(666\) 0 0
\(667\) 0 0
\(668\) −8.73974 −0.338151
\(669\) 0 0
\(670\) −17.0091 −0.657119
\(671\) 5.77398 0.222902
\(672\) 0 0
\(673\) 37.2436 1.43563 0.717817 0.696232i \(-0.245141\pi\)
0.717817 + 0.696232i \(0.245141\pi\)
\(674\) −6.73490 −0.259419
\(675\) 0 0
\(676\) −7.95833 −0.306090
\(677\) −44.9350 −1.72699 −0.863496 0.504356i \(-0.831730\pi\)
−0.863496 + 0.504356i \(0.831730\pi\)
\(678\) 0 0
\(679\) −29.1245 −1.11770
\(680\) −30.9977 −1.18871
\(681\) 0 0
\(682\) −4.89115 −0.187292
\(683\) −22.6022 −0.864848 −0.432424 0.901670i \(-0.642342\pi\)
−0.432424 + 0.901670i \(0.642342\pi\)
\(684\) 0 0
\(685\) 21.9651 0.839242
\(686\) 22.2253 0.848567
\(687\) 0 0
\(688\) −4.62847 −0.176459
\(689\) −41.2978 −1.57332
\(690\) 0 0
\(691\) −38.8727 −1.47879 −0.739394 0.673273i \(-0.764888\pi\)
−0.739394 + 0.673273i \(0.764888\pi\)
\(692\) −2.38436 −0.0906397
\(693\) 0 0
\(694\) 10.6384 0.403828
\(695\) −8.89436 −0.337382
\(696\) 0 0
\(697\) −20.3495 −0.770794
\(698\) 18.3960 0.696299
\(699\) 0 0
\(700\) 4.03371 0.152460
\(701\) 39.4427 1.48973 0.744865 0.667215i \(-0.232514\pi\)
0.744865 + 0.667215i \(0.232514\pi\)
\(702\) 0 0
\(703\) −2.24601 −0.0847100
\(704\) −9.34041 −0.352030
\(705\) 0 0
\(706\) 14.9726 0.563501
\(707\) −20.2116 −0.760136
\(708\) 0 0
\(709\) −4.07382 −0.152996 −0.0764978 0.997070i \(-0.524374\pi\)
−0.0764978 + 0.997070i \(0.524374\pi\)
\(710\) −33.9924 −1.27571
\(711\) 0 0
\(712\) −55.0561 −2.06331
\(713\) −7.82661 −0.293109
\(714\) 0 0
\(715\) −10.0336 −0.375236
\(716\) 1.81649 0.0678853
\(717\) 0 0
\(718\) 19.3913 0.723677
\(719\) 18.4303 0.687335 0.343667 0.939091i \(-0.388331\pi\)
0.343667 + 0.939091i \(0.388331\pi\)
\(720\) 0 0
\(721\) 14.9475 0.556674
\(722\) 26.7696 0.996263
\(723\) 0 0
\(724\) −3.53686 −0.131446
\(725\) 0 0
\(726\) 0 0
\(727\) 17.6150 0.653306 0.326653 0.945144i \(-0.394079\pi\)
0.326653 + 0.945144i \(0.394079\pi\)
\(728\) 69.3109 2.56883
\(729\) 0 0
\(730\) 5.53689 0.204929
\(731\) −9.82535 −0.363404
\(732\) 0 0
\(733\) −8.22724 −0.303880 −0.151940 0.988390i \(-0.548552\pi\)
−0.151940 + 0.988390i \(0.548552\pi\)
\(734\) −26.9320 −0.994079
\(735\) 0 0
\(736\) −6.00051 −0.221182
\(737\) −8.25113 −0.303934
\(738\) 0 0
\(739\) 35.8226 1.31776 0.658878 0.752250i \(-0.271031\pi\)
0.658878 + 0.752250i \(0.271031\pi\)
\(740\) −0.336514 −0.0123705
\(741\) 0 0
\(742\) 40.4864 1.48630
\(743\) −9.29461 −0.340986 −0.170493 0.985359i \(-0.554536\pi\)
−0.170493 + 0.985359i \(0.554536\pi\)
\(744\) 0 0
\(745\) 25.2032 0.923375
\(746\) 8.39063 0.307203
\(747\) 0 0
\(748\) −3.16957 −0.115891
\(749\) 21.0507 0.769175
\(750\) 0 0
\(751\) −44.8712 −1.63737 −0.818687 0.574240i \(-0.805298\pi\)
−0.818687 + 0.574240i \(0.805298\pi\)
\(752\) 18.4716 0.673592
\(753\) 0 0
\(754\) 0 0
\(755\) −5.27646 −0.192030
\(756\) 0 0
\(757\) 21.6178 0.785712 0.392856 0.919600i \(-0.371487\pi\)
0.392856 + 0.919600i \(0.371487\pi\)
\(758\) −9.42966 −0.342501
\(759\) 0 0
\(760\) 35.3793 1.28334
\(761\) 39.3758 1.42737 0.713685 0.700466i \(-0.247025\pi\)
0.713685 + 0.700466i \(0.247025\pi\)
\(762\) 0 0
\(763\) −63.6606 −2.30467
\(764\) 9.41169 0.340503
\(765\) 0 0
\(766\) −30.8279 −1.11386
\(767\) 2.02122 0.0729822
\(768\) 0 0
\(769\) 4.29550 0.154900 0.0774499 0.996996i \(-0.475322\pi\)
0.0774499 + 0.996996i \(0.475322\pi\)
\(770\) 9.83649 0.354482
\(771\) 0 0
\(772\) 7.15703 0.257587
\(773\) −27.9002 −1.00350 −0.501751 0.865012i \(-0.667310\pi\)
−0.501751 + 0.865012i \(0.667310\pi\)
\(774\) 0 0
\(775\) −6.75302 −0.242576
\(776\) 20.8932 0.750021
\(777\) 0 0
\(778\) −13.2983 −0.476768
\(779\) 23.2260 0.832156
\(780\) 0 0
\(781\) −16.4897 −0.590049
\(782\) 13.9180 0.497706
\(783\) 0 0
\(784\) −29.8829 −1.06725
\(785\) −29.1201 −1.03934
\(786\) 0 0
\(787\) −5.90689 −0.210558 −0.105279 0.994443i \(-0.533574\pi\)
−0.105279 + 0.994443i \(0.533574\pi\)
\(788\) −4.38266 −0.156126
\(789\) 0 0
\(790\) −22.3575 −0.795444
\(791\) 27.1276 0.964547
\(792\) 0 0
\(793\) 28.8734 1.02532
\(794\) 6.17523 0.219151
\(795\) 0 0
\(796\) −13.0845 −0.463767
\(797\) 7.59899 0.269170 0.134585 0.990902i \(-0.457030\pi\)
0.134585 + 0.990902i \(0.457030\pi\)
\(798\) 0 0
\(799\) 39.2117 1.38721
\(800\) −5.17741 −0.183049
\(801\) 0 0
\(802\) 23.6600 0.835463
\(803\) 2.68595 0.0947851
\(804\) 0 0
\(805\) 15.7399 0.554759
\(806\) −24.4587 −0.861521
\(807\) 0 0
\(808\) 14.4993 0.510083
\(809\) 9.85027 0.346317 0.173159 0.984894i \(-0.444603\pi\)
0.173159 + 0.984894i \(0.444603\pi\)
\(810\) 0 0
\(811\) 46.3250 1.62669 0.813346 0.581780i \(-0.197644\pi\)
0.813346 + 0.581780i \(0.197644\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0.447970 0.0157013
\(815\) −24.4109 −0.855075
\(816\) 0 0
\(817\) 11.2142 0.392334
\(818\) −21.7014 −0.758772
\(819\) 0 0
\(820\) 3.47988 0.121523
\(821\) −18.0554 −0.630136 −0.315068 0.949069i \(-0.602027\pi\)
−0.315068 + 0.949069i \(0.602027\pi\)
\(822\) 0 0
\(823\) −33.6347 −1.17243 −0.586216 0.810155i \(-0.699383\pi\)
−0.586216 + 0.810155i \(0.699383\pi\)
\(824\) −10.7229 −0.373551
\(825\) 0 0
\(826\) −1.98151 −0.0689457
\(827\) −41.1338 −1.43036 −0.715181 0.698939i \(-0.753656\pi\)
−0.715181 + 0.698939i \(0.753656\pi\)
\(828\) 0 0
\(829\) 7.95542 0.276303 0.138152 0.990411i \(-0.455884\pi\)
0.138152 + 0.990411i \(0.455884\pi\)
\(830\) 3.31267 0.114984
\(831\) 0 0
\(832\) −46.7077 −1.61930
\(833\) −63.4357 −2.19792
\(834\) 0 0
\(835\) −29.4253 −1.01830
\(836\) 3.61759 0.125117
\(837\) 0 0
\(838\) −4.52591 −0.156345
\(839\) 11.3228 0.390908 0.195454 0.980713i \(-0.437382\pi\)
0.195454 + 0.980713i \(0.437382\pi\)
\(840\) 0 0
\(841\) 0 0
\(842\) 10.0342 0.345801
\(843\) 0 0
\(844\) 1.18126 0.0406606
\(845\) −26.7944 −0.921756
\(846\) 0 0
\(847\) −42.2745 −1.45257
\(848\) −20.6910 −0.710532
\(849\) 0 0
\(850\) 12.0088 0.411899
\(851\) 0.716822 0.0245723
\(852\) 0 0
\(853\) 14.5146 0.496971 0.248486 0.968636i \(-0.420067\pi\)
0.248486 + 0.968636i \(0.420067\pi\)
\(854\) −28.3061 −0.968615
\(855\) 0 0
\(856\) −15.1012 −0.516148
\(857\) 10.7169 0.366083 0.183042 0.983105i \(-0.441406\pi\)
0.183042 + 0.983105i \(0.441406\pi\)
\(858\) 0 0
\(859\) 15.7139 0.536150 0.268075 0.963398i \(-0.413612\pi\)
0.268075 + 0.963398i \(0.413612\pi\)
\(860\) 1.68019 0.0572939
\(861\) 0 0
\(862\) −23.1603 −0.788844
\(863\) 14.4146 0.490679 0.245340 0.969437i \(-0.421101\pi\)
0.245340 + 0.969437i \(0.421101\pi\)
\(864\) 0 0
\(865\) −8.02775 −0.272952
\(866\) −19.5542 −0.664478
\(867\) 0 0
\(868\) −8.73781 −0.296581
\(869\) −10.8456 −0.367913
\(870\) 0 0
\(871\) −41.2606 −1.39806
\(872\) 45.6684 1.54653
\(873\) 0 0
\(874\) −15.8853 −0.537328
\(875\) 52.0398 1.75926
\(876\) 0 0
\(877\) 13.5829 0.458662 0.229331 0.973348i \(-0.426346\pi\)
0.229331 + 0.973348i \(0.426346\pi\)
\(878\) 3.59384 0.121286
\(879\) 0 0
\(880\) −5.02703 −0.169461
\(881\) −21.3996 −0.720972 −0.360486 0.932765i \(-0.617389\pi\)
−0.360486 + 0.932765i \(0.617389\pi\)
\(882\) 0 0
\(883\) −11.1221 −0.374288 −0.187144 0.982332i \(-0.559923\pi\)
−0.187144 + 0.982332i \(0.559923\pi\)
\(884\) −15.8498 −0.533085
\(885\) 0 0
\(886\) −12.7024 −0.426744
\(887\) 1.53319 0.0514795 0.0257397 0.999669i \(-0.491806\pi\)
0.0257397 + 0.999669i \(0.491806\pi\)
\(888\) 0 0
\(889\) −19.0995 −0.640575
\(890\) −39.0720 −1.30970
\(891\) 0 0
\(892\) −1.44457 −0.0483678
\(893\) −44.7543 −1.49765
\(894\) 0 0
\(895\) 6.11582 0.204429
\(896\) 20.7073 0.691783
\(897\) 0 0
\(898\) 5.33110 0.177901
\(899\) 0 0
\(900\) 0 0
\(901\) −43.9230 −1.46329
\(902\) −4.63244 −0.154243
\(903\) 0 0
\(904\) −19.4606 −0.647251
\(905\) −11.9080 −0.395837
\(906\) 0 0
\(907\) 7.68948 0.255325 0.127663 0.991818i \(-0.459253\pi\)
0.127663 + 0.991818i \(0.459253\pi\)
\(908\) 7.70396 0.255665
\(909\) 0 0
\(910\) 49.1884 1.63058
\(911\) 29.5727 0.979788 0.489894 0.871782i \(-0.337036\pi\)
0.489894 + 0.871782i \(0.337036\pi\)
\(912\) 0 0
\(913\) 1.60698 0.0531832
\(914\) 13.7398 0.454473
\(915\) 0 0
\(916\) −13.7529 −0.454408
\(917\) −0.0425958 −0.00140664
\(918\) 0 0
\(919\) −34.2295 −1.12913 −0.564564 0.825389i \(-0.690956\pi\)
−0.564564 + 0.825389i \(0.690956\pi\)
\(920\) −11.2914 −0.372266
\(921\) 0 0
\(922\) −23.8622 −0.785861
\(923\) −82.4586 −2.71416
\(924\) 0 0
\(925\) 0.618494 0.0203360
\(926\) 0.912053 0.0299719
\(927\) 0 0
\(928\) 0 0
\(929\) −48.1477 −1.57967 −0.789837 0.613317i \(-0.789835\pi\)
−0.789837 + 0.613317i \(0.789835\pi\)
\(930\) 0 0
\(931\) 72.4023 2.37289
\(932\) −12.0742 −0.395502
\(933\) 0 0
\(934\) −12.6460 −0.413790
\(935\) −10.6714 −0.348993
\(936\) 0 0
\(937\) −13.7375 −0.448784 −0.224392 0.974499i \(-0.572040\pi\)
−0.224392 + 0.974499i \(0.572040\pi\)
\(938\) 40.4500 1.32074
\(939\) 0 0
\(940\) −6.70542 −0.218707
\(941\) 37.2514 1.21436 0.607181 0.794563i \(-0.292300\pi\)
0.607181 + 0.794563i \(0.292300\pi\)
\(942\) 0 0
\(943\) −7.41263 −0.241389
\(944\) 1.01267 0.0329597
\(945\) 0 0
\(946\) −2.23668 −0.0727207
\(947\) −16.4787 −0.535486 −0.267743 0.963490i \(-0.586278\pi\)
−0.267743 + 0.963490i \(0.586278\pi\)
\(948\) 0 0
\(949\) 13.4314 0.436001
\(950\) −13.7063 −0.444690
\(951\) 0 0
\(952\) 73.7169 2.38918
\(953\) 27.7913 0.900247 0.450124 0.892966i \(-0.351380\pi\)
0.450124 + 0.892966i \(0.351380\pi\)
\(954\) 0 0
\(955\) 31.6877 1.02539
\(956\) 12.5118 0.404662
\(957\) 0 0
\(958\) 16.9235 0.546773
\(959\) −52.2360 −1.68679
\(960\) 0 0
\(961\) −16.3716 −0.528117
\(962\) 2.24012 0.0722244
\(963\) 0 0
\(964\) −11.9559 −0.385075
\(965\) 24.0966 0.775696
\(966\) 0 0
\(967\) −51.9876 −1.67181 −0.835904 0.548876i \(-0.815056\pi\)
−0.835904 + 0.548876i \(0.815056\pi\)
\(968\) 30.3266 0.974735
\(969\) 0 0
\(970\) 14.8274 0.476079
\(971\) −26.6145 −0.854099 −0.427049 0.904228i \(-0.640447\pi\)
−0.427049 + 0.904228i \(0.640447\pi\)
\(972\) 0 0
\(973\) 21.1520 0.678103
\(974\) −1.76799 −0.0566502
\(975\) 0 0
\(976\) 14.4661 0.463049
\(977\) −60.8959 −1.94823 −0.974116 0.226050i \(-0.927419\pi\)
−0.974116 + 0.226050i \(0.927419\pi\)
\(978\) 0 0
\(979\) −18.9539 −0.605768
\(980\) 10.8478 0.346521
\(981\) 0 0
\(982\) 14.7129 0.469507
\(983\) −46.9308 −1.49686 −0.748430 0.663214i \(-0.769192\pi\)
−0.748430 + 0.663214i \(0.769192\pi\)
\(984\) 0 0
\(985\) −14.7557 −0.470156
\(986\) 0 0
\(987\) 0 0
\(988\) 18.0901 0.575523
\(989\) −3.57904 −0.113807
\(990\) 0 0
\(991\) 50.7905 1.61341 0.806707 0.590952i \(-0.201248\pi\)
0.806707 + 0.590952i \(0.201248\pi\)
\(992\) 11.2153 0.356086
\(993\) 0 0
\(994\) 80.8386 2.56405
\(995\) −44.0534 −1.39659
\(996\) 0 0
\(997\) −45.8087 −1.45078 −0.725389 0.688339i \(-0.758340\pi\)
−0.725389 + 0.688339i \(0.758340\pi\)
\(998\) −43.5469 −1.37845
\(999\) 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.1.8 9
3.2 odd 2 2523.2.a.r.1.2 9
29.16 even 7 261.2.k.c.82.3 18
29.20 even 7 261.2.k.c.226.3 18
29.28 even 2 7569.2.a.bm.1.2 9
87.20 odd 14 87.2.g.a.52.1 18
87.74 odd 14 87.2.g.a.82.1 yes 18
87.86 odd 2 2523.2.a.o.1.8 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
87.2.g.a.52.1 18 87.20 odd 14
87.2.g.a.82.1 yes 18 87.74 odd 14
261.2.k.c.82.3 18 29.16 even 7
261.2.k.c.226.3 18 29.20 even 7
2523.2.a.o.1.8 9 87.86 odd 2
2523.2.a.r.1.2 9 3.2 odd 2
7569.2.a.bj.1.8 9 1.1 even 1 trivial
7569.2.a.bm.1.2 9 29.28 even 2