Properties

Label 7569.2.a.bv.1.6
Level $7569$
Weight $2$
Character 7569.1
Self dual yes
Analytic conductor $60.439$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 31x^{14} + 397x^{12} - 2723x^{10} + 10845x^{8} - 25387x^{6} + 33562x^{4} - 22454x^{2} + 5821 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-1.64329\) 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)
 
\(f(q)\) \(=\) \(q-1.64329 q^{2} +0.700392 q^{4} +1.62862 q^{5} +0.373038 q^{7} +2.13563 q^{8} -2.67628 q^{10} +2.25630 q^{11} +5.41272 q^{13} -0.613008 q^{14} -4.91024 q^{16} +2.79201 q^{17} +2.62779 q^{19} +1.14067 q^{20} -3.70774 q^{22} -3.35111 q^{23} -2.34761 q^{25} -8.89465 q^{26} +0.261273 q^{28} +3.01079 q^{31} +3.79767 q^{32} -4.58807 q^{34} +0.607535 q^{35} +6.20826 q^{37} -4.31822 q^{38} +3.47812 q^{40} -4.29911 q^{41} -0.708513 q^{43} +1.58029 q^{44} +5.50683 q^{46} +10.7372 q^{47} -6.86084 q^{49} +3.85780 q^{50} +3.79102 q^{52} +13.8501 q^{53} +3.67464 q^{55} +0.796670 q^{56} +2.48056 q^{59} +1.78277 q^{61} -4.94760 q^{62} +3.57981 q^{64} +8.81523 q^{65} -3.94023 q^{67} +1.95550 q^{68} -0.998354 q^{70} -14.7180 q^{71} +13.3319 q^{73} -10.2020 q^{74} +1.84049 q^{76} +0.841683 q^{77} +3.49825 q^{79} -7.99688 q^{80} +7.06468 q^{82} +12.4465 q^{83} +4.54711 q^{85} +1.16429 q^{86} +4.81861 q^{88} -2.54336 q^{89} +2.01915 q^{91} -2.34709 q^{92} -17.6442 q^{94} +4.27967 q^{95} +12.1555 q^{97} +11.2743 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 30 q^{4} + 36 q^{10} + 10 q^{13} + 26 q^{16} + 22 q^{19} - 52 q^{22} + 58 q^{25} - 10 q^{28} + 56 q^{31} - 30 q^{34} + 30 q^{37} + 118 q^{40} + 56 q^{43} + 16 q^{46} - 52 q^{49} - 50 q^{52} + 64 q^{55}+ \cdots + 48 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.64329 −1.16198 −0.580990 0.813911i \(-0.697334\pi\)
−0.580990 + 0.813911i \(0.697334\pi\)
\(3\) 0 0
\(4\) 0.700392 0.350196
\(5\) 1.62862 0.728339 0.364169 0.931333i \(-0.381353\pi\)
0.364169 + 0.931333i \(0.381353\pi\)
\(6\) 0 0
\(7\) 0.373038 0.140995 0.0704975 0.997512i \(-0.477541\pi\)
0.0704975 + 0.997512i \(0.477541\pi\)
\(8\) 2.13563 0.755059
\(9\) 0 0
\(10\) −2.67628 −0.846315
\(11\) 2.25630 0.680299 0.340149 0.940371i \(-0.389522\pi\)
0.340149 + 0.940371i \(0.389522\pi\)
\(12\) 0 0
\(13\) 5.41272 1.50122 0.750609 0.660747i \(-0.229760\pi\)
0.750609 + 0.660747i \(0.229760\pi\)
\(14\) −0.613008 −0.163833
\(15\) 0 0
\(16\) −4.91024 −1.22756
\(17\) 2.79201 0.677162 0.338581 0.940937i \(-0.390053\pi\)
0.338581 + 0.940937i \(0.390053\pi\)
\(18\) 0 0
\(19\) 2.62779 0.602857 0.301429 0.953489i \(-0.402536\pi\)
0.301429 + 0.953489i \(0.402536\pi\)
\(20\) 1.14067 0.255061
\(21\) 0 0
\(22\) −3.70774 −0.790493
\(23\) −3.35111 −0.698754 −0.349377 0.936982i \(-0.613607\pi\)
−0.349377 + 0.936982i \(0.613607\pi\)
\(24\) 0 0
\(25\) −2.34761 −0.469522
\(26\) −8.89465 −1.74438
\(27\) 0 0
\(28\) 0.261273 0.0493759
\(29\) 0 0
\(30\) 0 0
\(31\) 3.01079 0.540755 0.270377 0.962754i \(-0.412852\pi\)
0.270377 + 0.962754i \(0.412852\pi\)
\(32\) 3.79767 0.671339
\(33\) 0 0
\(34\) −4.58807 −0.786848
\(35\) 0.607535 0.102692
\(36\) 0 0
\(37\) 6.20826 1.02063 0.510316 0.859987i \(-0.329528\pi\)
0.510316 + 0.859987i \(0.329528\pi\)
\(38\) −4.31822 −0.700508
\(39\) 0 0
\(40\) 3.47812 0.549939
\(41\) −4.29911 −0.671409 −0.335704 0.941967i \(-0.608974\pi\)
−0.335704 + 0.941967i \(0.608974\pi\)
\(42\) 0 0
\(43\) −0.708513 −0.108047 −0.0540236 0.998540i \(-0.517205\pi\)
−0.0540236 + 0.998540i \(0.517205\pi\)
\(44\) 1.58029 0.238238
\(45\) 0 0
\(46\) 5.50683 0.811938
\(47\) 10.7372 1.56618 0.783088 0.621911i \(-0.213643\pi\)
0.783088 + 0.621911i \(0.213643\pi\)
\(48\) 0 0
\(49\) −6.86084 −0.980120
\(50\) 3.85780 0.545575
\(51\) 0 0
\(52\) 3.79102 0.525720
\(53\) 13.8501 1.90246 0.951230 0.308482i \(-0.0998211\pi\)
0.951230 + 0.308482i \(0.0998211\pi\)
\(54\) 0 0
\(55\) 3.67464 0.495488
\(56\) 0.796670 0.106460
\(57\) 0 0
\(58\) 0 0
\(59\) 2.48056 0.322942 0.161471 0.986877i \(-0.448376\pi\)
0.161471 + 0.986877i \(0.448376\pi\)
\(60\) 0 0
\(61\) 1.78277 0.228260 0.114130 0.993466i \(-0.463592\pi\)
0.114130 + 0.993466i \(0.463592\pi\)
\(62\) −4.94760 −0.628346
\(63\) 0 0
\(64\) 3.57981 0.447477
\(65\) 8.81523 1.09340
\(66\) 0 0
\(67\) −3.94023 −0.481375 −0.240688 0.970603i \(-0.577373\pi\)
−0.240688 + 0.970603i \(0.577373\pi\)
\(68\) 1.95550 0.237139
\(69\) 0 0
\(70\) −0.998354 −0.119326
\(71\) −14.7180 −1.74670 −0.873350 0.487092i \(-0.838057\pi\)
−0.873350 + 0.487092i \(0.838057\pi\)
\(72\) 0 0
\(73\) 13.3319 1.56038 0.780191 0.625541i \(-0.215122\pi\)
0.780191 + 0.625541i \(0.215122\pi\)
\(74\) −10.2020 −1.18595
\(75\) 0 0
\(76\) 1.84049 0.211118
\(77\) 0.841683 0.0959187
\(78\) 0 0
\(79\) 3.49825 0.393584 0.196792 0.980445i \(-0.436948\pi\)
0.196792 + 0.980445i \(0.436948\pi\)
\(80\) −7.99688 −0.894079
\(81\) 0 0
\(82\) 7.06468 0.780163
\(83\) 12.4465 1.36618 0.683089 0.730335i \(-0.260636\pi\)
0.683089 + 0.730335i \(0.260636\pi\)
\(84\) 0 0
\(85\) 4.54711 0.493203
\(86\) 1.16429 0.125549
\(87\) 0 0
\(88\) 4.81861 0.513665
\(89\) −2.54336 −0.269595 −0.134798 0.990873i \(-0.543038\pi\)
−0.134798 + 0.990873i \(0.543038\pi\)
\(90\) 0 0
\(91\) 2.01915 0.211664
\(92\) −2.34709 −0.244701
\(93\) 0 0
\(94\) −17.6442 −1.81986
\(95\) 4.27967 0.439085
\(96\) 0 0
\(97\) 12.1555 1.23420 0.617100 0.786884i \(-0.288307\pi\)
0.617100 + 0.786884i \(0.288307\pi\)
\(98\) 11.2743 1.13888
\(99\) 0 0
\(100\) −1.64425 −0.164425
\(101\) −0.0790358 −0.00786435 −0.00393218 0.999992i \(-0.501252\pi\)
−0.00393218 + 0.999992i \(0.501252\pi\)
\(102\) 0 0
\(103\) 17.0482 1.67981 0.839907 0.542731i \(-0.182610\pi\)
0.839907 + 0.542731i \(0.182610\pi\)
\(104\) 11.5596 1.13351
\(105\) 0 0
\(106\) −22.7597 −2.21062
\(107\) −16.3525 −1.58086 −0.790429 0.612554i \(-0.790142\pi\)
−0.790429 + 0.612554i \(0.790142\pi\)
\(108\) 0 0
\(109\) 2.68290 0.256975 0.128487 0.991711i \(-0.458988\pi\)
0.128487 + 0.991711i \(0.458988\pi\)
\(110\) −6.03848 −0.575747
\(111\) 0 0
\(112\) −1.83170 −0.173080
\(113\) −13.8065 −1.29881 −0.649403 0.760444i \(-0.724981\pi\)
−0.649403 + 0.760444i \(0.724981\pi\)
\(114\) 0 0
\(115\) −5.45766 −0.508930
\(116\) 0 0
\(117\) 0 0
\(118\) −4.07628 −0.375252
\(119\) 1.04153 0.0954765
\(120\) 0 0
\(121\) −5.90913 −0.537194
\(122\) −2.92960 −0.265234
\(123\) 0 0
\(124\) 2.10874 0.189370
\(125\) −11.9664 −1.07031
\(126\) 0 0
\(127\) 17.9165 1.58984 0.794918 0.606717i \(-0.207514\pi\)
0.794918 + 0.606717i \(0.207514\pi\)
\(128\) −13.4780 −1.19130
\(129\) 0 0
\(130\) −14.4860 −1.27050
\(131\) −10.5180 −0.918961 −0.459481 0.888188i \(-0.651964\pi\)
−0.459481 + 0.888188i \(0.651964\pi\)
\(132\) 0 0
\(133\) 0.980267 0.0849999
\(134\) 6.47492 0.559348
\(135\) 0 0
\(136\) 5.96270 0.511297
\(137\) −19.3331 −1.65174 −0.825868 0.563864i \(-0.809314\pi\)
−0.825868 + 0.563864i \(0.809314\pi\)
\(138\) 0 0
\(139\) 12.9103 1.09504 0.547520 0.836792i \(-0.315572\pi\)
0.547520 + 0.836792i \(0.315572\pi\)
\(140\) 0.425513 0.0359624
\(141\) 0 0
\(142\) 24.1858 2.02963
\(143\) 12.2127 1.02128
\(144\) 0 0
\(145\) 0 0
\(146\) −21.9082 −1.81313
\(147\) 0 0
\(148\) 4.34822 0.357421
\(149\) −20.6557 −1.69218 −0.846091 0.533038i \(-0.821050\pi\)
−0.846091 + 0.533038i \(0.821050\pi\)
\(150\) 0 0
\(151\) 9.84913 0.801511 0.400756 0.916185i \(-0.368748\pi\)
0.400756 + 0.916185i \(0.368748\pi\)
\(152\) 5.61199 0.455193
\(153\) 0 0
\(154\) −1.38313 −0.111456
\(155\) 4.90343 0.393853
\(156\) 0 0
\(157\) −7.01656 −0.559983 −0.279991 0.960003i \(-0.590332\pi\)
−0.279991 + 0.960003i \(0.590332\pi\)
\(158\) −5.74863 −0.457337
\(159\) 0 0
\(160\) 6.18494 0.488962
\(161\) −1.25009 −0.0985208
\(162\) 0 0
\(163\) 12.3424 0.966728 0.483364 0.875419i \(-0.339415\pi\)
0.483364 + 0.875419i \(0.339415\pi\)
\(164\) −3.01107 −0.235125
\(165\) 0 0
\(166\) −20.4531 −1.58747
\(167\) −10.0491 −0.777626 −0.388813 0.921317i \(-0.627115\pi\)
−0.388813 + 0.921317i \(0.627115\pi\)
\(168\) 0 0
\(169\) 16.2975 1.25365
\(170\) −7.47221 −0.573092
\(171\) 0 0
\(172\) −0.496237 −0.0378377
\(173\) −13.6711 −1.03940 −0.519699 0.854349i \(-0.673956\pi\)
−0.519699 + 0.854349i \(0.673956\pi\)
\(174\) 0 0
\(175\) −0.875748 −0.0662003
\(176\) −11.0789 −0.835106
\(177\) 0 0
\(178\) 4.17947 0.313264
\(179\) −1.14662 −0.0857021 −0.0428511 0.999081i \(-0.513644\pi\)
−0.0428511 + 0.999081i \(0.513644\pi\)
\(180\) 0 0
\(181\) −10.1064 −0.751203 −0.375602 0.926781i \(-0.622564\pi\)
−0.375602 + 0.926781i \(0.622564\pi\)
\(182\) −3.31804 −0.245949
\(183\) 0 0
\(184\) −7.15672 −0.527600
\(185\) 10.1109 0.743366
\(186\) 0 0
\(187\) 6.29960 0.460672
\(188\) 7.52022 0.548469
\(189\) 0 0
\(190\) −7.03272 −0.510207
\(191\) 21.3388 1.54402 0.772011 0.635609i \(-0.219251\pi\)
0.772011 + 0.635609i \(0.219251\pi\)
\(192\) 0 0
\(193\) −4.89472 −0.352330 −0.176165 0.984361i \(-0.556369\pi\)
−0.176165 + 0.984361i \(0.556369\pi\)
\(194\) −19.9749 −1.43412
\(195\) 0 0
\(196\) −4.80528 −0.343234
\(197\) −4.46002 −0.317763 −0.158882 0.987298i \(-0.550789\pi\)
−0.158882 + 0.987298i \(0.550789\pi\)
\(198\) 0 0
\(199\) −11.2018 −0.794073 −0.397036 0.917803i \(-0.629961\pi\)
−0.397036 + 0.917803i \(0.629961\pi\)
\(200\) −5.01363 −0.354517
\(201\) 0 0
\(202\) 0.129878 0.00913822
\(203\) 0 0
\(204\) 0 0
\(205\) −7.00160 −0.489013
\(206\) −28.0152 −1.95191
\(207\) 0 0
\(208\) −26.5777 −1.84283
\(209\) 5.92908 0.410123
\(210\) 0 0
\(211\) 3.16651 0.217992 0.108996 0.994042i \(-0.465236\pi\)
0.108996 + 0.994042i \(0.465236\pi\)
\(212\) 9.70051 0.666234
\(213\) 0 0
\(214\) 26.8719 1.83692
\(215\) −1.15389 −0.0786950
\(216\) 0 0
\(217\) 1.12314 0.0762437
\(218\) −4.40877 −0.298599
\(219\) 0 0
\(220\) 2.57369 0.173518
\(221\) 15.1124 1.01657
\(222\) 0 0
\(223\) −17.1423 −1.14794 −0.573968 0.818878i \(-0.694597\pi\)
−0.573968 + 0.818878i \(0.694597\pi\)
\(224\) 1.41667 0.0946555
\(225\) 0 0
\(226\) 22.6880 1.50919
\(227\) 10.7698 0.714816 0.357408 0.933948i \(-0.383661\pi\)
0.357408 + 0.933948i \(0.383661\pi\)
\(228\) 0 0
\(229\) −3.98851 −0.263568 −0.131784 0.991278i \(-0.542071\pi\)
−0.131784 + 0.991278i \(0.542071\pi\)
\(230\) 8.96850 0.591366
\(231\) 0 0
\(232\) 0 0
\(233\) 25.5903 1.67647 0.838237 0.545306i \(-0.183587\pi\)
0.838237 + 0.545306i \(0.183587\pi\)
\(234\) 0 0
\(235\) 17.4867 1.14071
\(236\) 1.73737 0.113093
\(237\) 0 0
\(238\) −1.71153 −0.110942
\(239\) 13.7321 0.888254 0.444127 0.895964i \(-0.353514\pi\)
0.444127 + 0.895964i \(0.353514\pi\)
\(240\) 0 0
\(241\) 16.5978 1.06916 0.534581 0.845117i \(-0.320470\pi\)
0.534581 + 0.845117i \(0.320470\pi\)
\(242\) 9.71040 0.624208
\(243\) 0 0
\(244\) 1.24864 0.0799358
\(245\) −11.1737 −0.713860
\(246\) 0 0
\(247\) 14.2235 0.905020
\(248\) 6.42994 0.408302
\(249\) 0 0
\(250\) 19.6643 1.24368
\(251\) 17.5574 1.10821 0.554106 0.832446i \(-0.313060\pi\)
0.554106 + 0.832446i \(0.313060\pi\)
\(252\) 0 0
\(253\) −7.56108 −0.475361
\(254\) −29.4420 −1.84736
\(255\) 0 0
\(256\) 14.9886 0.936787
\(257\) 5.75740 0.359137 0.179568 0.983745i \(-0.442530\pi\)
0.179568 + 0.983745i \(0.442530\pi\)
\(258\) 0 0
\(259\) 2.31592 0.143904
\(260\) 6.17412 0.382903
\(261\) 0 0
\(262\) 17.2841 1.06781
\(263\) 5.22043 0.321905 0.160953 0.986962i \(-0.448543\pi\)
0.160953 + 0.986962i \(0.448543\pi\)
\(264\) 0 0
\(265\) 22.5565 1.38564
\(266\) −1.61086 −0.0987681
\(267\) 0 0
\(268\) −2.75970 −0.168576
\(269\) −18.9198 −1.15356 −0.576779 0.816900i \(-0.695691\pi\)
−0.576779 + 0.816900i \(0.695691\pi\)
\(270\) 0 0
\(271\) −22.0038 −1.33663 −0.668317 0.743877i \(-0.732985\pi\)
−0.668317 + 0.743877i \(0.732985\pi\)
\(272\) −13.7094 −0.831256
\(273\) 0 0
\(274\) 31.7698 1.91928
\(275\) −5.29691 −0.319415
\(276\) 0 0
\(277\) −13.2254 −0.794636 −0.397318 0.917681i \(-0.630059\pi\)
−0.397318 + 0.917681i \(0.630059\pi\)
\(278\) −21.2154 −1.27241
\(279\) 0 0
\(280\) 1.29747 0.0775386
\(281\) −15.0461 −0.897575 −0.448788 0.893638i \(-0.648144\pi\)
−0.448788 + 0.893638i \(0.648144\pi\)
\(282\) 0 0
\(283\) −30.7504 −1.82792 −0.913960 0.405804i \(-0.866991\pi\)
−0.913960 + 0.405804i \(0.866991\pi\)
\(284\) −10.3083 −0.611688
\(285\) 0 0
\(286\) −20.0690 −1.18670
\(287\) −1.60373 −0.0946653
\(288\) 0 0
\(289\) −9.20468 −0.541452
\(290\) 0 0
\(291\) 0 0
\(292\) 9.33757 0.546440
\(293\) 4.27848 0.249952 0.124976 0.992160i \(-0.460115\pi\)
0.124976 + 0.992160i \(0.460115\pi\)
\(294\) 0 0
\(295\) 4.03988 0.235211
\(296\) 13.2585 0.770637
\(297\) 0 0
\(298\) 33.9433 1.96628
\(299\) −18.1386 −1.04898
\(300\) 0 0
\(301\) −0.264302 −0.0152341
\(302\) −16.1850 −0.931339
\(303\) 0 0
\(304\) −12.9031 −0.740043
\(305\) 2.90345 0.166251
\(306\) 0 0
\(307\) 25.9080 1.47865 0.739324 0.673350i \(-0.235145\pi\)
0.739324 + 0.673350i \(0.235145\pi\)
\(308\) 0.589508 0.0335904
\(309\) 0 0
\(310\) −8.05774 −0.457649
\(311\) −27.7460 −1.57333 −0.786665 0.617379i \(-0.788194\pi\)
−0.786665 + 0.617379i \(0.788194\pi\)
\(312\) 0 0
\(313\) 26.4201 1.49335 0.746676 0.665188i \(-0.231649\pi\)
0.746676 + 0.665188i \(0.231649\pi\)
\(314\) 11.5302 0.650688
\(315\) 0 0
\(316\) 2.45015 0.137832
\(317\) 6.19862 0.348149 0.174075 0.984732i \(-0.444307\pi\)
0.174075 + 0.984732i \(0.444307\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 5.83014 0.325915
\(321\) 0 0
\(322\) 2.05425 0.114479
\(323\) 7.33683 0.408232
\(324\) 0 0
\(325\) −12.7070 −0.704855
\(326\) −20.2820 −1.12332
\(327\) 0 0
\(328\) −9.18131 −0.506953
\(329\) 4.00537 0.220823
\(330\) 0 0
\(331\) 26.6349 1.46399 0.731993 0.681312i \(-0.238590\pi\)
0.731993 + 0.681312i \(0.238590\pi\)
\(332\) 8.71741 0.478430
\(333\) 0 0
\(334\) 16.5136 0.903585
\(335\) −6.41711 −0.350604
\(336\) 0 0
\(337\) −16.1020 −0.877134 −0.438567 0.898698i \(-0.644514\pi\)
−0.438567 + 0.898698i \(0.644514\pi\)
\(338\) −26.7815 −1.45672
\(339\) 0 0
\(340\) 3.18476 0.172718
\(341\) 6.79324 0.367875
\(342\) 0 0
\(343\) −5.17062 −0.279187
\(344\) −1.51312 −0.0815820
\(345\) 0 0
\(346\) 22.4656 1.20776
\(347\) −11.9111 −0.639422 −0.319711 0.947515i \(-0.603586\pi\)
−0.319711 + 0.947515i \(0.603586\pi\)
\(348\) 0 0
\(349\) 7.50331 0.401643 0.200822 0.979628i \(-0.435639\pi\)
0.200822 + 0.979628i \(0.435639\pi\)
\(350\) 1.43911 0.0769234
\(351\) 0 0
\(352\) 8.56866 0.456711
\(353\) 20.3746 1.08443 0.542216 0.840239i \(-0.317585\pi\)
0.542216 + 0.840239i \(0.317585\pi\)
\(354\) 0 0
\(355\) −23.9699 −1.27219
\(356\) −1.78135 −0.0944113
\(357\) 0 0
\(358\) 1.88422 0.0995841
\(359\) −19.1003 −1.00808 −0.504038 0.863681i \(-0.668153\pi\)
−0.504038 + 0.863681i \(0.668153\pi\)
\(360\) 0 0
\(361\) −12.0947 −0.636563
\(362\) 16.6077 0.872883
\(363\) 0 0
\(364\) 1.41420 0.0741240
\(365\) 21.7126 1.13649
\(366\) 0 0
\(367\) −31.3448 −1.63619 −0.818093 0.575086i \(-0.804968\pi\)
−0.818093 + 0.575086i \(0.804968\pi\)
\(368\) 16.4547 0.857761
\(369\) 0 0
\(370\) −16.6151 −0.863776
\(371\) 5.16662 0.268237
\(372\) 0 0
\(373\) −11.0294 −0.571083 −0.285541 0.958366i \(-0.592173\pi\)
−0.285541 + 0.958366i \(0.592173\pi\)
\(374\) −10.3521 −0.535292
\(375\) 0 0
\(376\) 22.9306 1.18256
\(377\) 0 0
\(378\) 0 0
\(379\) 23.6969 1.21723 0.608614 0.793466i \(-0.291726\pi\)
0.608614 + 0.793466i \(0.291726\pi\)
\(380\) 2.99744 0.153766
\(381\) 0 0
\(382\) −35.0658 −1.79412
\(383\) −24.9756 −1.27619 −0.638097 0.769956i \(-0.720278\pi\)
−0.638097 + 0.769956i \(0.720278\pi\)
\(384\) 0 0
\(385\) 1.37078 0.0698613
\(386\) 8.04343 0.409400
\(387\) 0 0
\(388\) 8.51359 0.432212
\(389\) 15.4660 0.784155 0.392078 0.919932i \(-0.371756\pi\)
0.392078 + 0.919932i \(0.371756\pi\)
\(390\) 0 0
\(391\) −9.35632 −0.473170
\(392\) −14.6522 −0.740049
\(393\) 0 0
\(394\) 7.32909 0.369234
\(395\) 5.69731 0.286663
\(396\) 0 0
\(397\) −8.26306 −0.414711 −0.207356 0.978266i \(-0.566486\pi\)
−0.207356 + 0.978266i \(0.566486\pi\)
\(398\) 18.4077 0.922696
\(399\) 0 0
\(400\) 11.5273 0.576366
\(401\) −24.7673 −1.23682 −0.618409 0.785857i \(-0.712222\pi\)
−0.618409 + 0.785857i \(0.712222\pi\)
\(402\) 0 0
\(403\) 16.2966 0.811791
\(404\) −0.0553560 −0.00275407
\(405\) 0 0
\(406\) 0 0
\(407\) 14.0077 0.694335
\(408\) 0 0
\(409\) −5.10464 −0.252408 −0.126204 0.992004i \(-0.540279\pi\)
−0.126204 + 0.992004i \(0.540279\pi\)
\(410\) 11.5056 0.568223
\(411\) 0 0
\(412\) 11.9405 0.588264
\(413\) 0.925343 0.0455332
\(414\) 0 0
\(415\) 20.2705 0.995040
\(416\) 20.5557 1.00783
\(417\) 0 0
\(418\) −9.74318 −0.476555
\(419\) 24.8882 1.21587 0.607935 0.793987i \(-0.291998\pi\)
0.607935 + 0.793987i \(0.291998\pi\)
\(420\) 0 0
\(421\) 34.2350 1.66851 0.834257 0.551376i \(-0.185897\pi\)
0.834257 + 0.551376i \(0.185897\pi\)
\(422\) −5.20349 −0.253302
\(423\) 0 0
\(424\) 29.5787 1.43647
\(425\) −6.55456 −0.317943
\(426\) 0 0
\(427\) 0.665040 0.0321836
\(428\) −11.4532 −0.553610
\(429\) 0 0
\(430\) 1.89618 0.0914419
\(431\) 8.32784 0.401138 0.200569 0.979680i \(-0.435721\pi\)
0.200569 + 0.979680i \(0.435721\pi\)
\(432\) 0 0
\(433\) −3.43346 −0.165002 −0.0825008 0.996591i \(-0.526291\pi\)
−0.0825008 + 0.996591i \(0.526291\pi\)
\(434\) −1.84564 −0.0885936
\(435\) 0 0
\(436\) 1.87908 0.0899916
\(437\) −8.80602 −0.421249
\(438\) 0 0
\(439\) −22.8754 −1.09179 −0.545893 0.837855i \(-0.683809\pi\)
−0.545893 + 0.837855i \(0.683809\pi\)
\(440\) 7.84766 0.374123
\(441\) 0 0
\(442\) −24.8340 −1.18123
\(443\) 32.8455 1.56054 0.780268 0.625446i \(-0.215083\pi\)
0.780268 + 0.625446i \(0.215083\pi\)
\(444\) 0 0
\(445\) −4.14215 −0.196357
\(446\) 28.1698 1.33388
\(447\) 0 0
\(448\) 1.33541 0.0630920
\(449\) −26.3567 −1.24385 −0.621924 0.783078i \(-0.713649\pi\)
−0.621924 + 0.783078i \(0.713649\pi\)
\(450\) 0 0
\(451\) −9.70007 −0.456758
\(452\) −9.66996 −0.454837
\(453\) 0 0
\(454\) −17.6978 −0.830601
\(455\) 3.28842 0.154163
\(456\) 0 0
\(457\) 15.5928 0.729402 0.364701 0.931125i \(-0.381171\pi\)
0.364701 + 0.931125i \(0.381171\pi\)
\(458\) 6.55426 0.306261
\(459\) 0 0
\(460\) −3.82250 −0.178225
\(461\) 28.3944 1.32246 0.661229 0.750184i \(-0.270035\pi\)
0.661229 + 0.750184i \(0.270035\pi\)
\(462\) 0 0
\(463\) 3.69723 0.171825 0.0859125 0.996303i \(-0.472619\pi\)
0.0859125 + 0.996303i \(0.472619\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −42.0522 −1.94803
\(467\) −0.293937 −0.0136018 −0.00680089 0.999977i \(-0.502165\pi\)
−0.00680089 + 0.999977i \(0.502165\pi\)
\(468\) 0 0
\(469\) −1.46985 −0.0678715
\(470\) −28.7357 −1.32548
\(471\) 0 0
\(472\) 5.29756 0.243840
\(473\) −1.59861 −0.0735044
\(474\) 0 0
\(475\) −6.16904 −0.283055
\(476\) 0.729476 0.0334355
\(477\) 0 0
\(478\) −22.5657 −1.03213
\(479\) −18.3071 −0.836472 −0.418236 0.908339i \(-0.637351\pi\)
−0.418236 + 0.908339i \(0.637351\pi\)
\(480\) 0 0
\(481\) 33.6036 1.53219
\(482\) −27.2750 −1.24234
\(483\) 0 0
\(484\) −4.13871 −0.188123
\(485\) 19.7966 0.898917
\(486\) 0 0
\(487\) 5.78702 0.262235 0.131118 0.991367i \(-0.458143\pi\)
0.131118 + 0.991367i \(0.458143\pi\)
\(488\) 3.80733 0.172350
\(489\) 0 0
\(490\) 18.3616 0.829490
\(491\) −21.7469 −0.981423 −0.490712 0.871322i \(-0.663263\pi\)
−0.490712 + 0.871322i \(0.663263\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) −23.3733 −1.05161
\(495\) 0 0
\(496\) −14.7837 −0.663808
\(497\) −5.49036 −0.246276
\(498\) 0 0
\(499\) 19.3340 0.865510 0.432755 0.901512i \(-0.357542\pi\)
0.432755 + 0.901512i \(0.357542\pi\)
\(500\) −8.38120 −0.374818
\(501\) 0 0
\(502\) −28.8518 −1.28772
\(503\) −38.2983 −1.70764 −0.853819 0.520569i \(-0.825720\pi\)
−0.853819 + 0.520569i \(0.825720\pi\)
\(504\) 0 0
\(505\) −0.128719 −0.00572791
\(506\) 12.4250 0.552360
\(507\) 0 0
\(508\) 12.5486 0.556754
\(509\) 30.3390 1.34475 0.672376 0.740210i \(-0.265274\pi\)
0.672376 + 0.740210i \(0.265274\pi\)
\(510\) 0 0
\(511\) 4.97331 0.220006
\(512\) 2.32544 0.102771
\(513\) 0 0
\(514\) −9.46107 −0.417310
\(515\) 27.7650 1.22347
\(516\) 0 0
\(517\) 24.2262 1.06547
\(518\) −3.80572 −0.167214
\(519\) 0 0
\(520\) 18.8261 0.825578
\(521\) 1.15099 0.0504259 0.0252129 0.999682i \(-0.491974\pi\)
0.0252129 + 0.999682i \(0.491974\pi\)
\(522\) 0 0
\(523\) −7.09010 −0.310028 −0.155014 0.987912i \(-0.549542\pi\)
−0.155014 + 0.987912i \(0.549542\pi\)
\(524\) −7.36672 −0.321817
\(525\) 0 0
\(526\) −8.57866 −0.374047
\(527\) 8.40617 0.366179
\(528\) 0 0
\(529\) −11.7701 −0.511743
\(530\) −37.0668 −1.61008
\(531\) 0 0
\(532\) 0.686571 0.0297666
\(533\) −23.2699 −1.00793
\(534\) 0 0
\(535\) −26.6320 −1.15140
\(536\) −8.41486 −0.363467
\(537\) 0 0
\(538\) 31.0906 1.34041
\(539\) −15.4801 −0.666775
\(540\) 0 0
\(541\) −23.9272 −1.02871 −0.514355 0.857578i \(-0.671968\pi\)
−0.514355 + 0.857578i \(0.671968\pi\)
\(542\) 36.1585 1.55314
\(543\) 0 0
\(544\) 10.6031 0.454605
\(545\) 4.36941 0.187165
\(546\) 0 0
\(547\) −1.76378 −0.0754138 −0.0377069 0.999289i \(-0.512005\pi\)
−0.0377069 + 0.999289i \(0.512005\pi\)
\(548\) −13.5407 −0.578431
\(549\) 0 0
\(550\) 8.70434 0.371154
\(551\) 0 0
\(552\) 0 0
\(553\) 1.30498 0.0554934
\(554\) 21.7331 0.923351
\(555\) 0 0
\(556\) 9.04230 0.383479
\(557\) 40.1163 1.69978 0.849891 0.526958i \(-0.176668\pi\)
0.849891 + 0.526958i \(0.176668\pi\)
\(558\) 0 0
\(559\) −3.83498 −0.162202
\(560\) −2.98314 −0.126061
\(561\) 0 0
\(562\) 24.7251 1.04296
\(563\) 16.9855 0.715854 0.357927 0.933750i \(-0.383484\pi\)
0.357927 + 0.933750i \(0.383484\pi\)
\(564\) 0 0
\(565\) −22.4855 −0.945971
\(566\) 50.5317 2.12401
\(567\) 0 0
\(568\) −31.4321 −1.31886
\(569\) −29.7390 −1.24672 −0.623361 0.781934i \(-0.714233\pi\)
−0.623361 + 0.781934i \(0.714233\pi\)
\(570\) 0 0
\(571\) 26.0295 1.08930 0.544650 0.838664i \(-0.316663\pi\)
0.544650 + 0.838664i \(0.316663\pi\)
\(572\) 8.55367 0.357647
\(573\) 0 0
\(574\) 2.63539 0.109999
\(575\) 7.86710 0.328081
\(576\) 0 0
\(577\) 3.64937 0.151925 0.0759626 0.997111i \(-0.475797\pi\)
0.0759626 + 0.997111i \(0.475797\pi\)
\(578\) 15.1259 0.629156
\(579\) 0 0
\(580\) 0 0
\(581\) 4.64300 0.192624
\(582\) 0 0
\(583\) 31.2500 1.29424
\(584\) 28.4720 1.17818
\(585\) 0 0
\(586\) −7.03077 −0.290439
\(587\) 2.47754 0.102259 0.0511296 0.998692i \(-0.483718\pi\)
0.0511296 + 0.998692i \(0.483718\pi\)
\(588\) 0 0
\(589\) 7.91175 0.325998
\(590\) −6.63868 −0.273310
\(591\) 0 0
\(592\) −30.4840 −1.25289
\(593\) −15.8763 −0.651962 −0.325981 0.945376i \(-0.605695\pi\)
−0.325981 + 0.945376i \(0.605695\pi\)
\(594\) 0 0
\(595\) 1.69624 0.0695392
\(596\) −14.4671 −0.592596
\(597\) 0 0
\(598\) 29.8069 1.21889
\(599\) −1.44578 −0.0590732 −0.0295366 0.999564i \(-0.509403\pi\)
−0.0295366 + 0.999564i \(0.509403\pi\)
\(600\) 0 0
\(601\) −22.5274 −0.918912 −0.459456 0.888201i \(-0.651956\pi\)
−0.459456 + 0.888201i \(0.651956\pi\)
\(602\) 0.434324 0.0177017
\(603\) 0 0
\(604\) 6.89825 0.280686
\(605\) −9.62370 −0.391259
\(606\) 0 0
\(607\) −23.6228 −0.958818 −0.479409 0.877592i \(-0.659149\pi\)
−0.479409 + 0.877592i \(0.659149\pi\)
\(608\) 9.97949 0.404722
\(609\) 0 0
\(610\) −4.77119 −0.193180
\(611\) 58.1172 2.35117
\(612\) 0 0
\(613\) 23.8449 0.963085 0.481542 0.876423i \(-0.340077\pi\)
0.481542 + 0.876423i \(0.340077\pi\)
\(614\) −42.5743 −1.71816
\(615\) 0 0
\(616\) 1.79752 0.0724243
\(617\) 18.4046 0.740943 0.370471 0.928844i \(-0.379196\pi\)
0.370471 + 0.928844i \(0.379196\pi\)
\(618\) 0 0
\(619\) 26.9332 1.08254 0.541268 0.840850i \(-0.317945\pi\)
0.541268 + 0.840850i \(0.317945\pi\)
\(620\) 3.43432 0.137926
\(621\) 0 0
\(622\) 45.5946 1.82818
\(623\) −0.948769 −0.0380116
\(624\) 0 0
\(625\) −7.75066 −0.310026
\(626\) −43.4158 −1.73524
\(627\) 0 0
\(628\) −4.91435 −0.196104
\(629\) 17.3335 0.691133
\(630\) 0 0
\(631\) −37.3093 −1.48526 −0.742630 0.669702i \(-0.766422\pi\)
−0.742630 + 0.669702i \(0.766422\pi\)
\(632\) 7.47097 0.297179
\(633\) 0 0
\(634\) −10.1861 −0.404542
\(635\) 29.1791 1.15794
\(636\) 0 0
\(637\) −37.1358 −1.47137
\(638\) 0 0
\(639\) 0 0
\(640\) −21.9505 −0.867668
\(641\) −1.52697 −0.0603117 −0.0301558 0.999545i \(-0.509600\pi\)
−0.0301558 + 0.999545i \(0.509600\pi\)
\(642\) 0 0
\(643\) 26.4450 1.04289 0.521444 0.853286i \(-0.325394\pi\)
0.521444 + 0.853286i \(0.325394\pi\)
\(644\) −0.875552 −0.0345016
\(645\) 0 0
\(646\) −12.0565 −0.474357
\(647\) 15.0249 0.590692 0.295346 0.955390i \(-0.404565\pi\)
0.295346 + 0.955390i \(0.404565\pi\)
\(648\) 0 0
\(649\) 5.59688 0.219697
\(650\) 20.8812 0.819027
\(651\) 0 0
\(652\) 8.64449 0.338544
\(653\) 22.6351 0.885780 0.442890 0.896576i \(-0.353953\pi\)
0.442890 + 0.896576i \(0.353953\pi\)
\(654\) 0 0
\(655\) −17.1298 −0.669315
\(656\) 21.1097 0.824194
\(657\) 0 0
\(658\) −6.58197 −0.256592
\(659\) 19.2087 0.748264 0.374132 0.927375i \(-0.377941\pi\)
0.374132 + 0.927375i \(0.377941\pi\)
\(660\) 0 0
\(661\) 28.9135 1.12460 0.562301 0.826932i \(-0.309916\pi\)
0.562301 + 0.826932i \(0.309916\pi\)
\(662\) −43.7688 −1.70112
\(663\) 0 0
\(664\) 26.5810 1.03154
\(665\) 1.59648 0.0619087
\(666\) 0 0
\(667\) 0 0
\(668\) −7.03834 −0.272322
\(669\) 0 0
\(670\) 10.5452 0.407395
\(671\) 4.02245 0.155285
\(672\) 0 0
\(673\) −8.85259 −0.341242 −0.170621 0.985337i \(-0.554577\pi\)
−0.170621 + 0.985337i \(0.554577\pi\)
\(674\) 26.4603 1.01921
\(675\) 0 0
\(676\) 11.4146 0.439025
\(677\) −34.0025 −1.30682 −0.653412 0.757003i \(-0.726663\pi\)
−0.653412 + 0.757003i \(0.726663\pi\)
\(678\) 0 0
\(679\) 4.53445 0.174016
\(680\) 9.71094 0.372398
\(681\) 0 0
\(682\) −11.1632 −0.427463
\(683\) 6.78343 0.259561 0.129780 0.991543i \(-0.458573\pi\)
0.129780 + 0.991543i \(0.458573\pi\)
\(684\) 0 0
\(685\) −31.4861 −1.20302
\(686\) 8.49681 0.324410
\(687\) 0 0
\(688\) 3.47896 0.132634
\(689\) 74.9668 2.85601
\(690\) 0 0
\(691\) 36.9928 1.40727 0.703637 0.710560i \(-0.251558\pi\)
0.703637 + 0.710560i \(0.251558\pi\)
\(692\) −9.57516 −0.363993
\(693\) 0 0
\(694\) 19.5734 0.742995
\(695\) 21.0260 0.797561
\(696\) 0 0
\(697\) −12.0032 −0.454653
\(698\) −12.3301 −0.466701
\(699\) 0 0
\(700\) −0.613367 −0.0231831
\(701\) −2.60481 −0.0983823 −0.0491912 0.998789i \(-0.515664\pi\)
−0.0491912 + 0.998789i \(0.515664\pi\)
\(702\) 0 0
\(703\) 16.3140 0.615296
\(704\) 8.07711 0.304418
\(705\) 0 0
\(706\) −33.4814 −1.26009
\(707\) −0.0294833 −0.00110883
\(708\) 0 0
\(709\) −10.0044 −0.375723 −0.187862 0.982195i \(-0.560156\pi\)
−0.187862 + 0.982195i \(0.560156\pi\)
\(710\) 39.3894 1.47826
\(711\) 0 0
\(712\) −5.43167 −0.203560
\(713\) −10.0895 −0.377854
\(714\) 0 0
\(715\) 19.8898 0.743835
\(716\) −0.803081 −0.0300125
\(717\) 0 0
\(718\) 31.3873 1.17136
\(719\) 39.6266 1.47782 0.738912 0.673802i \(-0.235340\pi\)
0.738912 + 0.673802i \(0.235340\pi\)
\(720\) 0 0
\(721\) 6.35964 0.236845
\(722\) 19.8751 0.739673
\(723\) 0 0
\(724\) −7.07845 −0.263068
\(725\) 0 0
\(726\) 0 0
\(727\) 40.0756 1.48632 0.743160 0.669114i \(-0.233326\pi\)
0.743160 + 0.669114i \(0.233326\pi\)
\(728\) 4.31215 0.159819
\(729\) 0 0
\(730\) −35.6800 −1.32057
\(731\) −1.97818 −0.0731655
\(732\) 0 0
\(733\) 8.62875 0.318710 0.159355 0.987221i \(-0.449059\pi\)
0.159355 + 0.987221i \(0.449059\pi\)
\(734\) 51.5085 1.90121
\(735\) 0 0
\(736\) −12.7264 −0.469101
\(737\) −8.89032 −0.327479
\(738\) 0 0
\(739\) 51.4519 1.89269 0.946345 0.323159i \(-0.104745\pi\)
0.946345 + 0.323159i \(0.104745\pi\)
\(740\) 7.08158 0.260324
\(741\) 0 0
\(742\) −8.49024 −0.311686
\(743\) −33.8098 −1.24036 −0.620181 0.784459i \(-0.712941\pi\)
−0.620181 + 0.784459i \(0.712941\pi\)
\(744\) 0 0
\(745\) −33.6402 −1.23248
\(746\) 18.1245 0.663586
\(747\) 0 0
\(748\) 4.41219 0.161326
\(749\) −6.10011 −0.222893
\(750\) 0 0
\(751\) −12.8360 −0.468392 −0.234196 0.972189i \(-0.575246\pi\)
−0.234196 + 0.972189i \(0.575246\pi\)
\(752\) −52.7220 −1.92257
\(753\) 0 0
\(754\) 0 0
\(755\) 16.0404 0.583772
\(756\) 0 0
\(757\) 0.712810 0.0259075 0.0129538 0.999916i \(-0.495877\pi\)
0.0129538 + 0.999916i \(0.495877\pi\)
\(758\) −38.9408 −1.41439
\(759\) 0 0
\(760\) 9.13978 0.331535
\(761\) −28.8108 −1.04439 −0.522196 0.852826i \(-0.674887\pi\)
−0.522196 + 0.852826i \(0.674887\pi\)
\(762\) 0 0
\(763\) 1.00082 0.0362322
\(764\) 14.9455 0.540710
\(765\) 0 0
\(766\) 41.0421 1.48291
\(767\) 13.4266 0.484806
\(768\) 0 0
\(769\) −8.12281 −0.292916 −0.146458 0.989217i \(-0.546787\pi\)
−0.146458 + 0.989217i \(0.546787\pi\)
\(770\) −2.25258 −0.0811774
\(771\) 0 0
\(772\) −3.42822 −0.123384
\(773\) 32.2194 1.15885 0.579425 0.815025i \(-0.303277\pi\)
0.579425 + 0.815025i \(0.303277\pi\)
\(774\) 0 0
\(775\) −7.06818 −0.253896
\(776\) 25.9596 0.931894
\(777\) 0 0
\(778\) −25.4150 −0.911172
\(779\) −11.2972 −0.404764
\(780\) 0 0
\(781\) −33.2081 −1.18828
\(782\) 15.3751 0.549813
\(783\) 0 0
\(784\) 33.6884 1.20316
\(785\) −11.4273 −0.407857
\(786\) 0 0
\(787\) 21.6267 0.770909 0.385454 0.922727i \(-0.374045\pi\)
0.385454 + 0.922727i \(0.374045\pi\)
\(788\) −3.12376 −0.111279
\(789\) 0 0
\(790\) −9.36231 −0.333096
\(791\) −5.15035 −0.183125
\(792\) 0 0
\(793\) 9.64963 0.342668
\(794\) 13.5786 0.481886
\(795\) 0 0
\(796\) −7.84564 −0.278081
\(797\) −29.1287 −1.03179 −0.515896 0.856651i \(-0.672541\pi\)
−0.515896 + 0.856651i \(0.672541\pi\)
\(798\) 0 0
\(799\) 29.9783 1.06056
\(800\) −8.91545 −0.315209
\(801\) 0 0
\(802\) 40.6997 1.43716
\(803\) 30.0807 1.06153
\(804\) 0 0
\(805\) −2.03591 −0.0717565
\(806\) −26.7800 −0.943284
\(807\) 0 0
\(808\) −0.168791 −0.00593805
\(809\) 42.9069 1.50853 0.754263 0.656572i \(-0.227994\pi\)
0.754263 + 0.656572i \(0.227994\pi\)
\(810\) 0 0
\(811\) −22.3136 −0.783535 −0.391768 0.920064i \(-0.628136\pi\)
−0.391768 + 0.920064i \(0.628136\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −23.0186 −0.806803
\(815\) 20.1010 0.704106
\(816\) 0 0
\(817\) −1.86183 −0.0651371
\(818\) 8.38839 0.293293
\(819\) 0 0
\(820\) −4.90387 −0.171250
\(821\) −4.25873 −0.148631 −0.0743153 0.997235i \(-0.523677\pi\)
−0.0743153 + 0.997235i \(0.523677\pi\)
\(822\) 0 0
\(823\) 45.5344 1.58723 0.793615 0.608420i \(-0.208196\pi\)
0.793615 + 0.608420i \(0.208196\pi\)
\(824\) 36.4087 1.26836
\(825\) 0 0
\(826\) −1.52060 −0.0529086
\(827\) 6.33221 0.220193 0.110096 0.993921i \(-0.464884\pi\)
0.110096 + 0.993921i \(0.464884\pi\)
\(828\) 0 0
\(829\) 28.3857 0.985877 0.492938 0.870064i \(-0.335923\pi\)
0.492938 + 0.870064i \(0.335923\pi\)
\(830\) −33.3103 −1.15622
\(831\) 0 0
\(832\) 19.3765 0.671760
\(833\) −19.1555 −0.663700
\(834\) 0 0
\(835\) −16.3662 −0.566375
\(836\) 4.15268 0.143623
\(837\) 0 0
\(838\) −40.8985 −1.41282
\(839\) −32.5963 −1.12535 −0.562675 0.826678i \(-0.690228\pi\)
−0.562675 + 0.826678i \(0.690228\pi\)
\(840\) 0 0
\(841\) 0 0
\(842\) −56.2580 −1.93878
\(843\) 0 0
\(844\) 2.21780 0.0763399
\(845\) 26.5424 0.913085
\(846\) 0 0
\(847\) −2.20433 −0.0757417
\(848\) −68.0073 −2.33538
\(849\) 0 0
\(850\) 10.7710 0.369443
\(851\) −20.8045 −0.713171
\(852\) 0 0
\(853\) −18.5977 −0.636773 −0.318387 0.947961i \(-0.603141\pi\)
−0.318387 + 0.947961i \(0.603141\pi\)
\(854\) −1.09285 −0.0373966
\(855\) 0 0
\(856\) −34.9229 −1.19364
\(857\) 26.3931 0.901570 0.450785 0.892633i \(-0.351144\pi\)
0.450785 + 0.892633i \(0.351144\pi\)
\(858\) 0 0
\(859\) −20.5273 −0.700381 −0.350190 0.936679i \(-0.613883\pi\)
−0.350190 + 0.936679i \(0.613883\pi\)
\(860\) −0.808179 −0.0275587
\(861\) 0 0
\(862\) −13.6850 −0.466114
\(863\) 12.5561 0.427413 0.213707 0.976898i \(-0.431446\pi\)
0.213707 + 0.976898i \(0.431446\pi\)
\(864\) 0 0
\(865\) −22.2650 −0.757034
\(866\) 5.64216 0.191729
\(867\) 0 0
\(868\) 0.786638 0.0267002
\(869\) 7.89309 0.267755
\(870\) 0 0
\(871\) −21.3273 −0.722649
\(872\) 5.72967 0.194031
\(873\) 0 0
\(874\) 14.4708 0.489483
\(875\) −4.46393 −0.150908
\(876\) 0 0
\(877\) 23.8685 0.805983 0.402991 0.915204i \(-0.367970\pi\)
0.402991 + 0.915204i \(0.367970\pi\)
\(878\) 37.5909 1.26863
\(879\) 0 0
\(880\) −18.0433 −0.608241
\(881\) −40.6117 −1.36824 −0.684121 0.729368i \(-0.739814\pi\)
−0.684121 + 0.729368i \(0.739814\pi\)
\(882\) 0 0
\(883\) 28.2345 0.950168 0.475084 0.879941i \(-0.342418\pi\)
0.475084 + 0.879941i \(0.342418\pi\)
\(884\) 10.5846 0.355998
\(885\) 0 0
\(886\) −53.9745 −1.81331
\(887\) 37.5966 1.26237 0.631185 0.775632i \(-0.282569\pi\)
0.631185 + 0.775632i \(0.282569\pi\)
\(888\) 0 0
\(889\) 6.68354 0.224159
\(890\) 6.80674 0.228163
\(891\) 0 0
\(892\) −12.0064 −0.402003
\(893\) 28.2151 0.944181
\(894\) 0 0
\(895\) −1.86740 −0.0624202
\(896\) −5.02780 −0.167967
\(897\) 0 0
\(898\) 43.3116 1.44533
\(899\) 0 0
\(900\) 0 0
\(901\) 38.6697 1.28827
\(902\) 15.9400 0.530744
\(903\) 0 0
\(904\) −29.4856 −0.980675
\(905\) −16.4595 −0.547131
\(906\) 0 0
\(907\) 27.5124 0.913533 0.456766 0.889587i \(-0.349008\pi\)
0.456766 + 0.889587i \(0.349008\pi\)
\(908\) 7.54307 0.250326
\(909\) 0 0
\(910\) −5.40381 −0.179135
\(911\) 30.2567 1.00245 0.501224 0.865318i \(-0.332883\pi\)
0.501224 + 0.865318i \(0.332883\pi\)
\(912\) 0 0
\(913\) 28.0829 0.929409
\(914\) −25.6235 −0.847551
\(915\) 0 0
\(916\) −2.79352 −0.0923005
\(917\) −3.92361 −0.129569
\(918\) 0 0
\(919\) −8.33000 −0.274781 −0.137391 0.990517i \(-0.543872\pi\)
−0.137391 + 0.990517i \(0.543872\pi\)
\(920\) −11.6555 −0.384272
\(921\) 0 0
\(922\) −46.6601 −1.53667
\(923\) −79.6642 −2.62218
\(924\) 0 0
\(925\) −14.5746 −0.479210
\(926\) −6.07561 −0.199657
\(927\) 0 0
\(928\) 0 0
\(929\) −31.2153 −1.02414 −0.512070 0.858944i \(-0.671121\pi\)
−0.512070 + 0.858944i \(0.671121\pi\)
\(930\) 0 0
\(931\) −18.0289 −0.590873
\(932\) 17.9232 0.587095
\(933\) 0 0
\(934\) 0.483022 0.0158050
\(935\) 10.2596 0.335526
\(936\) 0 0
\(937\) −14.5237 −0.474469 −0.237234 0.971452i \(-0.576241\pi\)
−0.237234 + 0.971452i \(0.576241\pi\)
\(938\) 2.41539 0.0788653
\(939\) 0 0
\(940\) 12.2476 0.399471
\(941\) 14.9109 0.486080 0.243040 0.970016i \(-0.421855\pi\)
0.243040 + 0.970016i \(0.421855\pi\)
\(942\) 0 0
\(943\) 14.4068 0.469149
\(944\) −12.1801 −0.396430
\(945\) 0 0
\(946\) 2.62698 0.0854105
\(947\) 28.7646 0.934722 0.467361 0.884066i \(-0.345205\pi\)
0.467361 + 0.884066i \(0.345205\pi\)
\(948\) 0 0
\(949\) 72.1619 2.34247
\(950\) 10.1375 0.328904
\(951\) 0 0
\(952\) 2.22431 0.0720904
\(953\) −30.4381 −0.985987 −0.492994 0.870033i \(-0.664097\pi\)
−0.492994 + 0.870033i \(0.664097\pi\)
\(954\) 0 0
\(955\) 34.7527 1.12457
\(956\) 9.61784 0.311063
\(957\) 0 0
\(958\) 30.0838 0.971963
\(959\) −7.21197 −0.232887
\(960\) 0 0
\(961\) −21.9351 −0.707584
\(962\) −55.2203 −1.78037
\(963\) 0 0
\(964\) 11.6250 0.374416
\(965\) −7.97161 −0.256615
\(966\) 0 0
\(967\) 52.9553 1.70293 0.851464 0.524412i \(-0.175715\pi\)
0.851464 + 0.524412i \(0.175715\pi\)
\(968\) −12.6197 −0.405613
\(969\) 0 0
\(970\) −32.5315 −1.04452
\(971\) −11.4036 −0.365958 −0.182979 0.983117i \(-0.558574\pi\)
−0.182979 + 0.983117i \(0.558574\pi\)
\(972\) 0 0
\(973\) 4.81604 0.154395
\(974\) −9.50974 −0.304712
\(975\) 0 0
\(976\) −8.75382 −0.280203
\(977\) 21.5535 0.689559 0.344780 0.938684i \(-0.387954\pi\)
0.344780 + 0.938684i \(0.387954\pi\)
\(978\) 0 0
\(979\) −5.73857 −0.183405
\(980\) −7.82595 −0.249991
\(981\) 0 0
\(982\) 35.7364 1.14039
\(983\) 13.8780 0.442639 0.221319 0.975201i \(-0.428964\pi\)
0.221319 + 0.975201i \(0.428964\pi\)
\(984\) 0 0
\(985\) −7.26366 −0.231439
\(986\) 0 0
\(987\) 0 0
\(988\) 9.96203 0.316935
\(989\) 2.37430 0.0754984
\(990\) 0 0
\(991\) 16.8339 0.534746 0.267373 0.963593i \(-0.413844\pi\)
0.267373 + 0.963593i \(0.413844\pi\)
\(992\) 11.4340 0.363030
\(993\) 0 0
\(994\) 9.02223 0.286168
\(995\) −18.2434 −0.578354
\(996\) 0 0
\(997\) −40.3768 −1.27875 −0.639373 0.768897i \(-0.720806\pi\)
−0.639373 + 0.768897i \(0.720806\pi\)
\(998\) −31.7713 −1.00570
\(999\) 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.bv.1.6 yes 16
3.2 odd 2 inner 7569.2.a.bv.1.11 yes 16
29.28 even 2 7569.2.a.bu.1.11 yes 16
87.86 odd 2 7569.2.a.bu.1.6 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7569.2.a.bu.1.6 16 87.86 odd 2
7569.2.a.bu.1.11 yes 16 29.28 even 2
7569.2.a.bv.1.6 yes 16 1.1 even 1 trivial
7569.2.a.bv.1.11 yes 16 3.2 odd 2 inner