Properties

Label 3249.2.a.o.1.2
Level $3249$
Weight $2$
Character 3249.1
Self dual yes
Analytic conductor $25.943$
Analytic rank $1$
Dimension $2$
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] = [3249,2,Mod(1,3249)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3249, 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("3249.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3249 = 3^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3249.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: \(25.9433956167\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 361)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 3249.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.61803 q^{2} +0.618034 q^{4} -3.23607 q^{5} +3.00000 q^{7} -2.23607 q^{8} +O(q^{10})\) \(q+1.61803 q^{2} +0.618034 q^{4} -3.23607 q^{5} +3.00000 q^{7} -2.23607 q^{8} -5.23607 q^{10} +1.61803 q^{11} +1.00000 q^{13} +4.85410 q^{14} -4.85410 q^{16} -0.763932 q^{17} -2.00000 q^{20} +2.61803 q^{22} -5.38197 q^{23} +5.47214 q^{25} +1.61803 q^{26} +1.85410 q^{28} -3.61803 q^{29} +8.85410 q^{31} -3.38197 q^{32} -1.23607 q^{34} -9.70820 q^{35} -8.85410 q^{37} +7.23607 q^{40} -3.00000 q^{41} -0.145898 q^{43} +1.00000 q^{44} -8.70820 q^{46} -3.00000 q^{47} +2.00000 q^{49} +8.85410 q^{50} +0.618034 q^{52} -6.32624 q^{53} -5.23607 q^{55} -6.70820 q^{56} -5.85410 q^{58} +0.326238 q^{59} -10.2361 q^{61} +14.3262 q^{62} +4.23607 q^{64} -3.23607 q^{65} +7.00000 q^{67} -0.472136 q^{68} -15.7082 q^{70} -7.47214 q^{71} -2.70820 q^{73} -14.3262 q^{74} +4.85410 q^{77} -13.4164 q^{79} +15.7082 q^{80} -4.85410 q^{82} -8.47214 q^{83} +2.47214 q^{85} -0.236068 q^{86} -3.61803 q^{88} -7.76393 q^{89} +3.00000 q^{91} -3.32624 q^{92} -4.85410 q^{94} -13.8541 q^{97} +3.23607 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{4} - 2 q^{5} + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - q^{4} - 2 q^{5} + 6 q^{7} - 6 q^{10} + q^{11} + 2 q^{13} + 3 q^{14} - 3 q^{16} - 6 q^{17} - 4 q^{20} + 3 q^{22} - 13 q^{23} + 2 q^{25} + q^{26} - 3 q^{28} - 5 q^{29} + 11 q^{31} - 9 q^{32} + 2 q^{34} - 6 q^{35} - 11 q^{37} + 10 q^{40} - 6 q^{41} - 7 q^{43} + 2 q^{44} - 4 q^{46} - 6 q^{47} + 4 q^{49} + 11 q^{50} - q^{52} + 3 q^{53} - 6 q^{55} - 5 q^{58} - 15 q^{59} - 16 q^{61} + 13 q^{62} + 4 q^{64} - 2 q^{65} + 14 q^{67} + 8 q^{68} - 18 q^{70} - 6 q^{71} + 8 q^{73} - 13 q^{74} + 3 q^{77} + 18 q^{80} - 3 q^{82} - 8 q^{83} - 4 q^{85} + 4 q^{86} - 5 q^{88} - 20 q^{89} + 6 q^{91} + 9 q^{92} - 3 q^{94} - 21 q^{97} + 2 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.61803 1.14412 0.572061 0.820211i \(-0.306144\pi\)
0.572061 + 0.820211i \(0.306144\pi\)
\(3\) 0 0
\(4\) 0.618034 0.309017
\(5\) −3.23607 −1.44721 −0.723607 0.690212i \(-0.757517\pi\)
−0.723607 + 0.690212i \(0.757517\pi\)
\(6\) 0 0
\(7\) 3.00000 1.13389 0.566947 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) −2.23607 −0.790569
\(9\) 0 0
\(10\) −5.23607 −1.65579
\(11\) 1.61803 0.487856 0.243928 0.969793i \(-0.421564\pi\)
0.243928 + 0.969793i \(0.421564\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 4.85410 1.29731
\(15\) 0 0
\(16\) −4.85410 −1.21353
\(17\) −0.763932 −0.185281 −0.0926404 0.995700i \(-0.529531\pi\)
−0.0926404 + 0.995700i \(0.529531\pi\)
\(18\) 0 0
\(19\) 0 0
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) 2.61803 0.558167
\(23\) −5.38197 −1.12222 −0.561109 0.827742i \(-0.689625\pi\)
−0.561109 + 0.827742i \(0.689625\pi\)
\(24\) 0 0
\(25\) 5.47214 1.09443
\(26\) 1.61803 0.317323
\(27\) 0 0
\(28\) 1.85410 0.350392
\(29\) −3.61803 −0.671852 −0.335926 0.941888i \(-0.609049\pi\)
−0.335926 + 0.941888i \(0.609049\pi\)
\(30\) 0 0
\(31\) 8.85410 1.59024 0.795122 0.606450i \(-0.207407\pi\)
0.795122 + 0.606450i \(0.207407\pi\)
\(32\) −3.38197 −0.597853
\(33\) 0 0
\(34\) −1.23607 −0.211984
\(35\) −9.70820 −1.64099
\(36\) 0 0
\(37\) −8.85410 −1.45561 −0.727803 0.685787i \(-0.759458\pi\)
−0.727803 + 0.685787i \(0.759458\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 7.23607 1.14412
\(41\) −3.00000 −0.468521 −0.234261 0.972174i \(-0.575267\pi\)
−0.234261 + 0.972174i \(0.575267\pi\)
\(42\) 0 0
\(43\) −0.145898 −0.0222492 −0.0111246 0.999938i \(-0.503541\pi\)
−0.0111246 + 0.999938i \(0.503541\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) −8.70820 −1.28395
\(47\) −3.00000 −0.437595 −0.218797 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) 8.85410 1.25216
\(51\) 0 0
\(52\) 0.618034 0.0857059
\(53\) −6.32624 −0.868976 −0.434488 0.900678i \(-0.643071\pi\)
−0.434488 + 0.900678i \(0.643071\pi\)
\(54\) 0 0
\(55\) −5.23607 −0.706031
\(56\) −6.70820 −0.896421
\(57\) 0 0
\(58\) −5.85410 −0.768681
\(59\) 0.326238 0.0424726 0.0212363 0.999774i \(-0.493240\pi\)
0.0212363 + 0.999774i \(0.493240\pi\)
\(60\) 0 0
\(61\) −10.2361 −1.31059 −0.655297 0.755371i \(-0.727457\pi\)
−0.655297 + 0.755371i \(0.727457\pi\)
\(62\) 14.3262 1.81943
\(63\) 0 0
\(64\) 4.23607 0.529508
\(65\) −3.23607 −0.401385
\(66\) 0 0
\(67\) 7.00000 0.855186 0.427593 0.903971i \(-0.359362\pi\)
0.427593 + 0.903971i \(0.359362\pi\)
\(68\) −0.472136 −0.0572549
\(69\) 0 0
\(70\) −15.7082 −1.87749
\(71\) −7.47214 −0.886779 −0.443390 0.896329i \(-0.646224\pi\)
−0.443390 + 0.896329i \(0.646224\pi\)
\(72\) 0 0
\(73\) −2.70820 −0.316971 −0.158486 0.987361i \(-0.550661\pi\)
−0.158486 + 0.987361i \(0.550661\pi\)
\(74\) −14.3262 −1.66539
\(75\) 0 0
\(76\) 0 0
\(77\) 4.85410 0.553176
\(78\) 0 0
\(79\) −13.4164 −1.50946 −0.754732 0.656033i \(-0.772233\pi\)
−0.754732 + 0.656033i \(0.772233\pi\)
\(80\) 15.7082 1.75623
\(81\) 0 0
\(82\) −4.85410 −0.536046
\(83\) −8.47214 −0.929938 −0.464969 0.885327i \(-0.653934\pi\)
−0.464969 + 0.885327i \(0.653934\pi\)
\(84\) 0 0
\(85\) 2.47214 0.268141
\(86\) −0.236068 −0.0254559
\(87\) 0 0
\(88\) −3.61803 −0.385684
\(89\) −7.76393 −0.822975 −0.411488 0.911415i \(-0.634991\pi\)
−0.411488 + 0.911415i \(0.634991\pi\)
\(90\) 0 0
\(91\) 3.00000 0.314485
\(92\) −3.32624 −0.346784
\(93\) 0 0
\(94\) −4.85410 −0.500662
\(95\) 0 0
\(96\) 0 0
\(97\) −13.8541 −1.40667 −0.703335 0.710858i \(-0.748307\pi\)
−0.703335 + 0.710858i \(0.748307\pi\)
\(98\) 3.23607 0.326892
\(99\) 0 0
\(100\) 3.38197 0.338197
\(101\) 9.18034 0.913478 0.456739 0.889601i \(-0.349017\pi\)
0.456739 + 0.889601i \(0.349017\pi\)
\(102\) 0 0
\(103\) −14.3262 −1.41161 −0.705803 0.708408i \(-0.749414\pi\)
−0.705803 + 0.708408i \(0.749414\pi\)
\(104\) −2.23607 −0.219265
\(105\) 0 0
\(106\) −10.2361 −0.994215
\(107\) 16.4164 1.58703 0.793517 0.608548i \(-0.208248\pi\)
0.793517 + 0.608548i \(0.208248\pi\)
\(108\) 0 0
\(109\) 3.29180 0.315297 0.157648 0.987495i \(-0.449609\pi\)
0.157648 + 0.987495i \(0.449609\pi\)
\(110\) −8.47214 −0.807786
\(111\) 0 0
\(112\) −14.5623 −1.37601
\(113\) 6.76393 0.636297 0.318149 0.948041i \(-0.396939\pi\)
0.318149 + 0.948041i \(0.396939\pi\)
\(114\) 0 0
\(115\) 17.4164 1.62409
\(116\) −2.23607 −0.207614
\(117\) 0 0
\(118\) 0.527864 0.0485938
\(119\) −2.29180 −0.210089
\(120\) 0 0
\(121\) −8.38197 −0.761997
\(122\) −16.5623 −1.49948
\(123\) 0 0
\(124\) 5.47214 0.491412
\(125\) −1.52786 −0.136656
\(126\) 0 0
\(127\) −5.76393 −0.511466 −0.255733 0.966747i \(-0.582317\pi\)
−0.255733 + 0.966747i \(0.582317\pi\)
\(128\) 13.6180 1.20368
\(129\) 0 0
\(130\) −5.23607 −0.459234
\(131\) −15.0902 −1.31843 −0.659217 0.751953i \(-0.729112\pi\)
−0.659217 + 0.751953i \(0.729112\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 11.3262 0.978438
\(135\) 0 0
\(136\) 1.70820 0.146477
\(137\) −7.47214 −0.638388 −0.319194 0.947689i \(-0.603412\pi\)
−0.319194 + 0.947689i \(0.603412\pi\)
\(138\) 0 0
\(139\) 14.7984 1.25518 0.627591 0.778543i \(-0.284041\pi\)
0.627591 + 0.778543i \(0.284041\pi\)
\(140\) −6.00000 −0.507093
\(141\) 0 0
\(142\) −12.0902 −1.01458
\(143\) 1.61803 0.135307
\(144\) 0 0
\(145\) 11.7082 0.972313
\(146\) −4.38197 −0.362654
\(147\) 0 0
\(148\) −5.47214 −0.449807
\(149\) 1.90983 0.156459 0.0782297 0.996935i \(-0.475073\pi\)
0.0782297 + 0.996935i \(0.475073\pi\)
\(150\) 0 0
\(151\) 21.0902 1.71629 0.858147 0.513404i \(-0.171616\pi\)
0.858147 + 0.513404i \(0.171616\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 7.85410 0.632902
\(155\) −28.6525 −2.30142
\(156\) 0 0
\(157\) −17.8541 −1.42491 −0.712456 0.701717i \(-0.752417\pi\)
−0.712456 + 0.701717i \(0.752417\pi\)
\(158\) −21.7082 −1.72701
\(159\) 0 0
\(160\) 10.9443 0.865221
\(161\) −16.1459 −1.27248
\(162\) 0 0
\(163\) 1.76393 0.138162 0.0690809 0.997611i \(-0.477993\pi\)
0.0690809 + 0.997611i \(0.477993\pi\)
\(164\) −1.85410 −0.144781
\(165\) 0 0
\(166\) −13.7082 −1.06396
\(167\) 20.2361 1.56591 0.782957 0.622076i \(-0.213710\pi\)
0.782957 + 0.622076i \(0.213710\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 4.00000 0.306786
\(171\) 0 0
\(172\) −0.0901699 −0.00687539
\(173\) −0.472136 −0.0358958 −0.0179479 0.999839i \(-0.505713\pi\)
−0.0179479 + 0.999839i \(0.505713\pi\)
\(174\) 0 0
\(175\) 16.4164 1.24096
\(176\) −7.85410 −0.592025
\(177\) 0 0
\(178\) −12.5623 −0.941585
\(179\) 12.2361 0.914567 0.457283 0.889321i \(-0.348823\pi\)
0.457283 + 0.889321i \(0.348823\pi\)
\(180\) 0 0
\(181\) −12.0000 −0.891953 −0.445976 0.895045i \(-0.647144\pi\)
−0.445976 + 0.895045i \(0.647144\pi\)
\(182\) 4.85410 0.359810
\(183\) 0 0
\(184\) 12.0344 0.887191
\(185\) 28.6525 2.10657
\(186\) 0 0
\(187\) −1.23607 −0.0903902
\(188\) −1.85410 −0.135224
\(189\) 0 0
\(190\) 0 0
\(191\) −14.2361 −1.03009 −0.515043 0.857164i \(-0.672224\pi\)
−0.515043 + 0.857164i \(0.672224\pi\)
\(192\) 0 0
\(193\) −5.05573 −0.363919 −0.181960 0.983306i \(-0.558244\pi\)
−0.181960 + 0.983306i \(0.558244\pi\)
\(194\) −22.4164 −1.60940
\(195\) 0 0
\(196\) 1.23607 0.0882906
\(197\) −3.00000 −0.213741 −0.106871 0.994273i \(-0.534083\pi\)
−0.106871 + 0.994273i \(0.534083\pi\)
\(198\) 0 0
\(199\) −13.4164 −0.951064 −0.475532 0.879698i \(-0.657744\pi\)
−0.475532 + 0.879698i \(0.657744\pi\)
\(200\) −12.2361 −0.865221
\(201\) 0 0
\(202\) 14.8541 1.04513
\(203\) −10.8541 −0.761809
\(204\) 0 0
\(205\) 9.70820 0.678050
\(206\) −23.1803 −1.61505
\(207\) 0 0
\(208\) −4.85410 −0.336571
\(209\) 0 0
\(210\) 0 0
\(211\) 3.85410 0.265327 0.132664 0.991161i \(-0.457647\pi\)
0.132664 + 0.991161i \(0.457647\pi\)
\(212\) −3.90983 −0.268528
\(213\) 0 0
\(214\) 26.5623 1.81576
\(215\) 0.472136 0.0321994
\(216\) 0 0
\(217\) 26.5623 1.80317
\(218\) 5.32624 0.360738
\(219\) 0 0
\(220\) −3.23607 −0.218176
\(221\) −0.763932 −0.0513876
\(222\) 0 0
\(223\) 11.6525 0.780307 0.390154 0.920750i \(-0.372422\pi\)
0.390154 + 0.920750i \(0.372422\pi\)
\(224\) −10.1459 −0.677901
\(225\) 0 0
\(226\) 10.9443 0.728002
\(227\) 16.4164 1.08960 0.544798 0.838568i \(-0.316606\pi\)
0.544798 + 0.838568i \(0.316606\pi\)
\(228\) 0 0
\(229\) −13.6180 −0.899905 −0.449953 0.893052i \(-0.648559\pi\)
−0.449953 + 0.893052i \(0.648559\pi\)
\(230\) 28.1803 1.85816
\(231\) 0 0
\(232\) 8.09017 0.531146
\(233\) −4.52786 −0.296630 −0.148315 0.988940i \(-0.547385\pi\)
−0.148315 + 0.988940i \(0.547385\pi\)
\(234\) 0 0
\(235\) 9.70820 0.633293
\(236\) 0.201626 0.0131247
\(237\) 0 0
\(238\) −3.70820 −0.240367
\(239\) 0.326238 0.0211026 0.0105513 0.999944i \(-0.496641\pi\)
0.0105513 + 0.999944i \(0.496641\pi\)
\(240\) 0 0
\(241\) −3.18034 −0.204864 −0.102432 0.994740i \(-0.532662\pi\)
−0.102432 + 0.994740i \(0.532662\pi\)
\(242\) −13.5623 −0.871818
\(243\) 0 0
\(244\) −6.32624 −0.404996
\(245\) −6.47214 −0.413490
\(246\) 0 0
\(247\) 0 0
\(248\) −19.7984 −1.25720
\(249\) 0 0
\(250\) −2.47214 −0.156352
\(251\) 25.3607 1.60075 0.800376 0.599498i \(-0.204633\pi\)
0.800376 + 0.599498i \(0.204633\pi\)
\(252\) 0 0
\(253\) −8.70820 −0.547480
\(254\) −9.32624 −0.585180
\(255\) 0 0
\(256\) 13.5623 0.847644
\(257\) 20.3607 1.27006 0.635032 0.772486i \(-0.280987\pi\)
0.635032 + 0.772486i \(0.280987\pi\)
\(258\) 0 0
\(259\) −26.5623 −1.65050
\(260\) −2.00000 −0.124035
\(261\) 0 0
\(262\) −24.4164 −1.50845
\(263\) 14.9443 0.921503 0.460752 0.887529i \(-0.347580\pi\)
0.460752 + 0.887529i \(0.347580\pi\)
\(264\) 0 0
\(265\) 20.4721 1.25759
\(266\) 0 0
\(267\) 0 0
\(268\) 4.32624 0.264267
\(269\) 30.3262 1.84902 0.924512 0.381154i \(-0.124473\pi\)
0.924512 + 0.381154i \(0.124473\pi\)
\(270\) 0 0
\(271\) 1.14590 0.0696083 0.0348042 0.999394i \(-0.488919\pi\)
0.0348042 + 0.999394i \(0.488919\pi\)
\(272\) 3.70820 0.224843
\(273\) 0 0
\(274\) −12.0902 −0.730394
\(275\) 8.85410 0.533922
\(276\) 0 0
\(277\) 11.4164 0.685945 0.342973 0.939345i \(-0.388566\pi\)
0.342973 + 0.939345i \(0.388566\pi\)
\(278\) 23.9443 1.43608
\(279\) 0 0
\(280\) 21.7082 1.29731
\(281\) −29.5066 −1.76021 −0.880107 0.474775i \(-0.842530\pi\)
−0.880107 + 0.474775i \(0.842530\pi\)
\(282\) 0 0
\(283\) 26.0344 1.54759 0.773793 0.633438i \(-0.218357\pi\)
0.773793 + 0.633438i \(0.218357\pi\)
\(284\) −4.61803 −0.274030
\(285\) 0 0
\(286\) 2.61803 0.154808
\(287\) −9.00000 −0.531253
\(288\) 0 0
\(289\) −16.4164 −0.965671
\(290\) 18.9443 1.11245
\(291\) 0 0
\(292\) −1.67376 −0.0979495
\(293\) −10.1459 −0.592730 −0.296365 0.955075i \(-0.595774\pi\)
−0.296365 + 0.955075i \(0.595774\pi\)
\(294\) 0 0
\(295\) −1.05573 −0.0614669
\(296\) 19.7984 1.15076
\(297\) 0 0
\(298\) 3.09017 0.179009
\(299\) −5.38197 −0.311247
\(300\) 0 0
\(301\) −0.437694 −0.0252283
\(302\) 34.1246 1.96365
\(303\) 0 0
\(304\) 0 0
\(305\) 33.1246 1.89671
\(306\) 0 0
\(307\) 17.3262 0.988861 0.494430 0.869217i \(-0.335377\pi\)
0.494430 + 0.869217i \(0.335377\pi\)
\(308\) 3.00000 0.170941
\(309\) 0 0
\(310\) −46.3607 −2.63311
\(311\) −12.6525 −0.717456 −0.358728 0.933442i \(-0.616789\pi\)
−0.358728 + 0.933442i \(0.616789\pi\)
\(312\) 0 0
\(313\) −11.3262 −0.640197 −0.320098 0.947384i \(-0.603716\pi\)
−0.320098 + 0.947384i \(0.603716\pi\)
\(314\) −28.8885 −1.63027
\(315\) 0 0
\(316\) −8.29180 −0.466450
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 0 0
\(319\) −5.85410 −0.327767
\(320\) −13.7082 −0.766312
\(321\) 0 0
\(322\) −26.1246 −1.45587
\(323\) 0 0
\(324\) 0 0
\(325\) 5.47214 0.303539
\(326\) 2.85410 0.158074
\(327\) 0 0
\(328\) 6.70820 0.370399
\(329\) −9.00000 −0.496186
\(330\) 0 0
\(331\) −10.9443 −0.601552 −0.300776 0.953695i \(-0.597246\pi\)
−0.300776 + 0.953695i \(0.597246\pi\)
\(332\) −5.23607 −0.287367
\(333\) 0 0
\(334\) 32.7426 1.79160
\(335\) −22.6525 −1.23764
\(336\) 0 0
\(337\) 17.1246 0.932837 0.466419 0.884564i \(-0.345544\pi\)
0.466419 + 0.884564i \(0.345544\pi\)
\(338\) −19.4164 −1.05611
\(339\) 0 0
\(340\) 1.52786 0.0828601
\(341\) 14.3262 0.775809
\(342\) 0 0
\(343\) −15.0000 −0.809924
\(344\) 0.326238 0.0175896
\(345\) 0 0
\(346\) −0.763932 −0.0410692
\(347\) 25.4164 1.36442 0.682212 0.731154i \(-0.261018\pi\)
0.682212 + 0.731154i \(0.261018\pi\)
\(348\) 0 0
\(349\) −20.9787 −1.12296 −0.561482 0.827489i \(-0.689769\pi\)
−0.561482 + 0.827489i \(0.689769\pi\)
\(350\) 26.5623 1.41981
\(351\) 0 0
\(352\) −5.47214 −0.291666
\(353\) −24.4508 −1.30139 −0.650694 0.759340i \(-0.725522\pi\)
−0.650694 + 0.759340i \(0.725522\pi\)
\(354\) 0 0
\(355\) 24.1803 1.28336
\(356\) −4.79837 −0.254313
\(357\) 0 0
\(358\) 19.7984 1.04638
\(359\) −7.03444 −0.371264 −0.185632 0.982619i \(-0.559433\pi\)
−0.185632 + 0.982619i \(0.559433\pi\)
\(360\) 0 0
\(361\) 0 0
\(362\) −19.4164 −1.02050
\(363\) 0 0
\(364\) 1.85410 0.0971813
\(365\) 8.76393 0.458725
\(366\) 0 0
\(367\) −15.9443 −0.832284 −0.416142 0.909300i \(-0.636618\pi\)
−0.416142 + 0.909300i \(0.636618\pi\)
\(368\) 26.1246 1.36184
\(369\) 0 0
\(370\) 46.3607 2.41018
\(371\) −18.9787 −0.985326
\(372\) 0 0
\(373\) 5.47214 0.283336 0.141668 0.989914i \(-0.454753\pi\)
0.141668 + 0.989914i \(0.454753\pi\)
\(374\) −2.00000 −0.103418
\(375\) 0 0
\(376\) 6.70820 0.345949
\(377\) −3.61803 −0.186338
\(378\) 0 0
\(379\) 15.1246 0.776899 0.388450 0.921470i \(-0.373011\pi\)
0.388450 + 0.921470i \(0.373011\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −23.0344 −1.17854
\(383\) 0.381966 0.0195176 0.00975878 0.999952i \(-0.496894\pi\)
0.00975878 + 0.999952i \(0.496894\pi\)
\(384\) 0 0
\(385\) −15.7082 −0.800564
\(386\) −8.18034 −0.416368
\(387\) 0 0
\(388\) −8.56231 −0.434685
\(389\) −9.27051 −0.470034 −0.235017 0.971991i \(-0.575515\pi\)
−0.235017 + 0.971991i \(0.575515\pi\)
\(390\) 0 0
\(391\) 4.11146 0.207925
\(392\) −4.47214 −0.225877
\(393\) 0 0
\(394\) −4.85410 −0.244546
\(395\) 43.4164 2.18452
\(396\) 0 0
\(397\) −11.4721 −0.575770 −0.287885 0.957665i \(-0.592952\pi\)
−0.287885 + 0.957665i \(0.592952\pi\)
\(398\) −21.7082 −1.08813
\(399\) 0 0
\(400\) −26.5623 −1.32812
\(401\) −35.8885 −1.79219 −0.896094 0.443864i \(-0.853607\pi\)
−0.896094 + 0.443864i \(0.853607\pi\)
\(402\) 0 0
\(403\) 8.85410 0.441054
\(404\) 5.67376 0.282280
\(405\) 0 0
\(406\) −17.5623 −0.871603
\(407\) −14.3262 −0.710125
\(408\) 0 0
\(409\) 8.29180 0.410003 0.205001 0.978762i \(-0.434280\pi\)
0.205001 + 0.978762i \(0.434280\pi\)
\(410\) 15.7082 0.775773
\(411\) 0 0
\(412\) −8.85410 −0.436210
\(413\) 0.978714 0.0481594
\(414\) 0 0
\(415\) 27.4164 1.34582
\(416\) −3.38197 −0.165815
\(417\) 0 0
\(418\) 0 0
\(419\) 8.94427 0.436956 0.218478 0.975842i \(-0.429891\pi\)
0.218478 + 0.975842i \(0.429891\pi\)
\(420\) 0 0
\(421\) 27.4721 1.33891 0.669455 0.742853i \(-0.266528\pi\)
0.669455 + 0.742853i \(0.266528\pi\)
\(422\) 6.23607 0.303567
\(423\) 0 0
\(424\) 14.1459 0.686986
\(425\) −4.18034 −0.202776
\(426\) 0 0
\(427\) −30.7082 −1.48607
\(428\) 10.1459 0.490420
\(429\) 0 0
\(430\) 0.763932 0.0368401
\(431\) 27.6525 1.33197 0.665986 0.745964i \(-0.268011\pi\)
0.665986 + 0.745964i \(0.268011\pi\)
\(432\) 0 0
\(433\) 3.43769 0.165205 0.0826025 0.996583i \(-0.473677\pi\)
0.0826025 + 0.996583i \(0.473677\pi\)
\(434\) 42.9787 2.06304
\(435\) 0 0
\(436\) 2.03444 0.0974321
\(437\) 0 0
\(438\) 0 0
\(439\) 34.5967 1.65121 0.825606 0.564247i \(-0.190833\pi\)
0.825606 + 0.564247i \(0.190833\pi\)
\(440\) 11.7082 0.558167
\(441\) 0 0
\(442\) −1.23607 −0.0587938
\(443\) 7.58359 0.360307 0.180154 0.983638i \(-0.442340\pi\)
0.180154 + 0.983638i \(0.442340\pi\)
\(444\) 0 0
\(445\) 25.1246 1.19102
\(446\) 18.8541 0.892768
\(447\) 0 0
\(448\) 12.7082 0.600406
\(449\) 2.88854 0.136319 0.0681594 0.997674i \(-0.478287\pi\)
0.0681594 + 0.997674i \(0.478287\pi\)
\(450\) 0 0
\(451\) −4.85410 −0.228571
\(452\) 4.18034 0.196627
\(453\) 0 0
\(454\) 26.5623 1.24663
\(455\) −9.70820 −0.455128
\(456\) 0 0
\(457\) 19.7082 0.921911 0.460955 0.887423i \(-0.347507\pi\)
0.460955 + 0.887423i \(0.347507\pi\)
\(458\) −22.0344 −1.02960
\(459\) 0 0
\(460\) 10.7639 0.501871
\(461\) −20.9443 −0.975472 −0.487736 0.872991i \(-0.662177\pi\)
−0.487736 + 0.872991i \(0.662177\pi\)
\(462\) 0 0
\(463\) 28.2705 1.31384 0.656921 0.753959i \(-0.271858\pi\)
0.656921 + 0.753959i \(0.271858\pi\)
\(464\) 17.5623 0.815310
\(465\) 0 0
\(466\) −7.32624 −0.339381
\(467\) 15.9443 0.737813 0.368906 0.929467i \(-0.379732\pi\)
0.368906 + 0.929467i \(0.379732\pi\)
\(468\) 0 0
\(469\) 21.0000 0.969690
\(470\) 15.7082 0.724565
\(471\) 0 0
\(472\) −0.729490 −0.0335775
\(473\) −0.236068 −0.0108544
\(474\) 0 0
\(475\) 0 0
\(476\) −1.41641 −0.0649209
\(477\) 0 0
\(478\) 0.527864 0.0241439
\(479\) −23.0902 −1.05502 −0.527508 0.849550i \(-0.676874\pi\)
−0.527508 + 0.849550i \(0.676874\pi\)
\(480\) 0 0
\(481\) −8.85410 −0.403712
\(482\) −5.14590 −0.234389
\(483\) 0 0
\(484\) −5.18034 −0.235470
\(485\) 44.8328 2.03575
\(486\) 0 0
\(487\) −4.18034 −0.189429 −0.0947146 0.995504i \(-0.530194\pi\)
−0.0947146 + 0.995504i \(0.530194\pi\)
\(488\) 22.8885 1.03612
\(489\) 0 0
\(490\) −10.4721 −0.473083
\(491\) 21.2148 0.957410 0.478705 0.877976i \(-0.341106\pi\)
0.478705 + 0.877976i \(0.341106\pi\)
\(492\) 0 0
\(493\) 2.76393 0.124481
\(494\) 0 0
\(495\) 0 0
\(496\) −42.9787 −1.92980
\(497\) −22.4164 −1.00551
\(498\) 0 0
\(499\) 25.1246 1.12473 0.562366 0.826888i \(-0.309891\pi\)
0.562366 + 0.826888i \(0.309891\pi\)
\(500\) −0.944272 −0.0422291
\(501\) 0 0
\(502\) 41.0344 1.83146
\(503\) −35.8328 −1.59771 −0.798853 0.601526i \(-0.794560\pi\)
−0.798853 + 0.601526i \(0.794560\pi\)
\(504\) 0 0
\(505\) −29.7082 −1.32200
\(506\) −14.0902 −0.626384
\(507\) 0 0
\(508\) −3.56231 −0.158052
\(509\) 2.03444 0.0901750 0.0450875 0.998983i \(-0.485643\pi\)
0.0450875 + 0.998983i \(0.485643\pi\)
\(510\) 0 0
\(511\) −8.12461 −0.359412
\(512\) −5.29180 −0.233867
\(513\) 0 0
\(514\) 32.9443 1.45311
\(515\) 46.3607 2.04290
\(516\) 0 0
\(517\) −4.85410 −0.213483
\(518\) −42.9787 −1.88838
\(519\) 0 0
\(520\) 7.23607 0.317323
\(521\) 6.27051 0.274716 0.137358 0.990521i \(-0.456139\pi\)
0.137358 + 0.990521i \(0.456139\pi\)
\(522\) 0 0
\(523\) 4.41641 0.193116 0.0965580 0.995327i \(-0.469217\pi\)
0.0965580 + 0.995327i \(0.469217\pi\)
\(524\) −9.32624 −0.407419
\(525\) 0 0
\(526\) 24.1803 1.05431
\(527\) −6.76393 −0.294642
\(528\) 0 0
\(529\) 5.96556 0.259372
\(530\) 33.1246 1.43884
\(531\) 0 0
\(532\) 0 0
\(533\) −3.00000 −0.129944
\(534\) 0 0
\(535\) −53.1246 −2.29678
\(536\) −15.6525 −0.676084
\(537\) 0 0
\(538\) 49.0689 2.11551
\(539\) 3.23607 0.139387
\(540\) 0 0
\(541\) −29.8328 −1.28261 −0.641306 0.767285i \(-0.721607\pi\)
−0.641306 + 0.767285i \(0.721607\pi\)
\(542\) 1.85410 0.0796405
\(543\) 0 0
\(544\) 2.58359 0.110771
\(545\) −10.6525 −0.456302
\(546\) 0 0
\(547\) 26.9230 1.15114 0.575572 0.817751i \(-0.304779\pi\)
0.575572 + 0.817751i \(0.304779\pi\)
\(548\) −4.61803 −0.197273
\(549\) 0 0
\(550\) 14.3262 0.610873
\(551\) 0 0
\(552\) 0 0
\(553\) −40.2492 −1.71157
\(554\) 18.4721 0.784806
\(555\) 0 0
\(556\) 9.14590 0.387872
\(557\) 0.819660 0.0347301 0.0173651 0.999849i \(-0.494472\pi\)
0.0173651 + 0.999849i \(0.494472\pi\)
\(558\) 0 0
\(559\) −0.145898 −0.00617083
\(560\) 47.1246 1.99138
\(561\) 0 0
\(562\) −47.7426 −2.01390
\(563\) −32.8328 −1.38374 −0.691869 0.722023i \(-0.743212\pi\)
−0.691869 + 0.722023i \(0.743212\pi\)
\(564\) 0 0
\(565\) −21.8885 −0.920858
\(566\) 42.1246 1.77063
\(567\) 0 0
\(568\) 16.7082 0.701061
\(569\) 16.9098 0.708897 0.354448 0.935076i \(-0.384669\pi\)
0.354448 + 0.935076i \(0.384669\pi\)
\(570\) 0 0
\(571\) 6.67376 0.279288 0.139644 0.990202i \(-0.455404\pi\)
0.139644 + 0.990202i \(0.455404\pi\)
\(572\) 1.00000 0.0418121
\(573\) 0 0
\(574\) −14.5623 −0.607819
\(575\) −29.4508 −1.22819
\(576\) 0 0
\(577\) −12.1246 −0.504754 −0.252377 0.967629i \(-0.581212\pi\)
−0.252377 + 0.967629i \(0.581212\pi\)
\(578\) −26.5623 −1.10485
\(579\) 0 0
\(580\) 7.23607 0.300461
\(581\) −25.4164 −1.05445
\(582\) 0 0
\(583\) −10.2361 −0.423935
\(584\) 6.05573 0.250588
\(585\) 0 0
\(586\) −16.4164 −0.678156
\(587\) 2.12461 0.0876921 0.0438461 0.999038i \(-0.486039\pi\)
0.0438461 + 0.999038i \(0.486039\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) −1.70820 −0.0703256
\(591\) 0 0
\(592\) 42.9787 1.76641
\(593\) −0.708204 −0.0290824 −0.0145412 0.999894i \(-0.504629\pi\)
−0.0145412 + 0.999894i \(0.504629\pi\)
\(594\) 0 0
\(595\) 7.41641 0.304043
\(596\) 1.18034 0.0483486
\(597\) 0 0
\(598\) −8.70820 −0.356105
\(599\) 28.4164 1.16106 0.580531 0.814238i \(-0.302845\pi\)
0.580531 + 0.814238i \(0.302845\pi\)
\(600\) 0 0
\(601\) −20.2918 −0.827720 −0.413860 0.910341i \(-0.635820\pi\)
−0.413860 + 0.910341i \(0.635820\pi\)
\(602\) −0.708204 −0.0288642
\(603\) 0 0
\(604\) 13.0344 0.530364
\(605\) 27.1246 1.10277
\(606\) 0 0
\(607\) 6.27051 0.254512 0.127256 0.991870i \(-0.459383\pi\)
0.127256 + 0.991870i \(0.459383\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 53.5967 2.17007
\(611\) −3.00000 −0.121367
\(612\) 0 0
\(613\) −19.9443 −0.805542 −0.402771 0.915301i \(-0.631953\pi\)
−0.402771 + 0.915301i \(0.631953\pi\)
\(614\) 28.0344 1.13138
\(615\) 0 0
\(616\) −10.8541 −0.437324
\(617\) −7.67376 −0.308934 −0.154467 0.987998i \(-0.549366\pi\)
−0.154467 + 0.987998i \(0.549366\pi\)
\(618\) 0 0
\(619\) 10.1246 0.406943 0.203471 0.979081i \(-0.434778\pi\)
0.203471 + 0.979081i \(0.434778\pi\)
\(620\) −17.7082 −0.711179
\(621\) 0 0
\(622\) −20.4721 −0.820858
\(623\) −23.2918 −0.933166
\(624\) 0 0
\(625\) −22.4164 −0.896656
\(626\) −18.3262 −0.732464
\(627\) 0 0
\(628\) −11.0344 −0.440322
\(629\) 6.76393 0.269696
\(630\) 0 0
\(631\) 29.3607 1.16883 0.584415 0.811455i \(-0.301324\pi\)
0.584415 + 0.811455i \(0.301324\pi\)
\(632\) 30.0000 1.19334
\(633\) 0 0
\(634\) 29.1246 1.15669
\(635\) 18.6525 0.740201
\(636\) 0 0
\(637\) 2.00000 0.0792429
\(638\) −9.47214 −0.375005
\(639\) 0 0
\(640\) −44.0689 −1.74198
\(641\) −39.5066 −1.56042 −0.780208 0.625520i \(-0.784887\pi\)
−0.780208 + 0.625520i \(0.784887\pi\)
\(642\) 0 0
\(643\) −24.2918 −0.957975 −0.478987 0.877822i \(-0.658996\pi\)
−0.478987 + 0.877822i \(0.658996\pi\)
\(644\) −9.97871 −0.393216
\(645\) 0 0
\(646\) 0 0
\(647\) −7.47214 −0.293760 −0.146880 0.989154i \(-0.546923\pi\)
−0.146880 + 0.989154i \(0.546923\pi\)
\(648\) 0 0
\(649\) 0.527864 0.0207205
\(650\) 8.85410 0.347286
\(651\) 0 0
\(652\) 1.09017 0.0426944
\(653\) 23.5623 0.922064 0.461032 0.887383i \(-0.347479\pi\)
0.461032 + 0.887383i \(0.347479\pi\)
\(654\) 0 0
\(655\) 48.8328 1.90806
\(656\) 14.5623 0.568563
\(657\) 0 0
\(658\) −14.5623 −0.567698
\(659\) −25.7771 −1.00413 −0.502066 0.864829i \(-0.667427\pi\)
−0.502066 + 0.864829i \(0.667427\pi\)
\(660\) 0 0
\(661\) −5.41641 −0.210674 −0.105337 0.994437i \(-0.533592\pi\)
−0.105337 + 0.994437i \(0.533592\pi\)
\(662\) −17.7082 −0.688249
\(663\) 0 0
\(664\) 18.9443 0.735180
\(665\) 0 0
\(666\) 0 0
\(667\) 19.4721 0.753964
\(668\) 12.5066 0.483894
\(669\) 0 0
\(670\) −36.6525 −1.41601
\(671\) −16.5623 −0.639381
\(672\) 0 0
\(673\) −34.1246 −1.31541 −0.657704 0.753277i \(-0.728472\pi\)
−0.657704 + 0.753277i \(0.728472\pi\)
\(674\) 27.7082 1.06728
\(675\) 0 0
\(676\) −7.41641 −0.285246
\(677\) −30.7426 −1.18154 −0.590768 0.806842i \(-0.701175\pi\)
−0.590768 + 0.806842i \(0.701175\pi\)
\(678\) 0 0
\(679\) −41.5623 −1.59501
\(680\) −5.52786 −0.211984
\(681\) 0 0
\(682\) 23.1803 0.887621
\(683\) 9.65248 0.369342 0.184671 0.982800i \(-0.440878\pi\)
0.184671 + 0.982800i \(0.440878\pi\)
\(684\) 0 0
\(685\) 24.1803 0.923883
\(686\) −24.2705 −0.926652
\(687\) 0 0
\(688\) 0.708204 0.0270000
\(689\) −6.32624 −0.241010
\(690\) 0 0
\(691\) −39.1803 −1.49049 −0.745245 0.666791i \(-0.767668\pi\)
−0.745245 + 0.666791i \(0.767668\pi\)
\(692\) −0.291796 −0.0110924
\(693\) 0 0
\(694\) 41.1246 1.56107
\(695\) −47.8885 −1.81652
\(696\) 0 0
\(697\) 2.29180 0.0868080
\(698\) −33.9443 −1.28481
\(699\) 0 0
\(700\) 10.1459 0.383479
\(701\) 39.6312 1.49685 0.748425 0.663220i \(-0.230811\pi\)
0.748425 + 0.663220i \(0.230811\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 6.85410 0.258324
\(705\) 0 0
\(706\) −39.5623 −1.48895
\(707\) 27.5410 1.03579
\(708\) 0 0
\(709\) 16.5836 0.622810 0.311405 0.950277i \(-0.399200\pi\)
0.311405 + 0.950277i \(0.399200\pi\)
\(710\) 39.1246 1.46832
\(711\) 0 0
\(712\) 17.3607 0.650619
\(713\) −47.6525 −1.78460
\(714\) 0 0
\(715\) −5.23607 −0.195818
\(716\) 7.56231 0.282617
\(717\) 0 0
\(718\) −11.3820 −0.424771
\(719\) −47.0344 −1.75409 −0.877044 0.480409i \(-0.840488\pi\)
−0.877044 + 0.480409i \(0.840488\pi\)
\(720\) 0 0
\(721\) −42.9787 −1.60061
\(722\) 0 0
\(723\) 0 0
\(724\) −7.41641 −0.275629
\(725\) −19.7984 −0.735293
\(726\) 0 0
\(727\) −16.0689 −0.595962 −0.297981 0.954572i \(-0.596313\pi\)
−0.297981 + 0.954572i \(0.596313\pi\)
\(728\) −6.70820 −0.248623
\(729\) 0 0
\(730\) 14.1803 0.524838
\(731\) 0.111456 0.00412236
\(732\) 0 0
\(733\) −52.9574 −1.95603 −0.978014 0.208541i \(-0.933129\pi\)
−0.978014 + 0.208541i \(0.933129\pi\)
\(734\) −25.7984 −0.952235
\(735\) 0 0
\(736\) 18.2016 0.670921
\(737\) 11.3262 0.417207
\(738\) 0 0
\(739\) −25.0000 −0.919640 −0.459820 0.888012i \(-0.652086\pi\)
−0.459820 + 0.888012i \(0.652086\pi\)
\(740\) 17.7082 0.650967
\(741\) 0 0
\(742\) −30.7082 −1.12733
\(743\) −3.36068 −0.123291 −0.0616457 0.998098i \(-0.519635\pi\)
−0.0616457 + 0.998098i \(0.519635\pi\)
\(744\) 0 0
\(745\) −6.18034 −0.226430
\(746\) 8.85410 0.324172
\(747\) 0 0
\(748\) −0.763932 −0.0279321
\(749\) 49.2492 1.79953
\(750\) 0 0
\(751\) 17.1459 0.625663 0.312831 0.949809i \(-0.398723\pi\)
0.312831 + 0.949809i \(0.398723\pi\)
\(752\) 14.5623 0.531033
\(753\) 0 0
\(754\) −5.85410 −0.213194
\(755\) −68.2492 −2.48384
\(756\) 0 0
\(757\) 26.7426 0.971978 0.485989 0.873965i \(-0.338460\pi\)
0.485989 + 0.873965i \(0.338460\pi\)
\(758\) 24.4721 0.888868
\(759\) 0 0
\(760\) 0 0
\(761\) −4.88854 −0.177210 −0.0886048 0.996067i \(-0.528241\pi\)
−0.0886048 + 0.996067i \(0.528241\pi\)
\(762\) 0 0
\(763\) 9.87539 0.357513
\(764\) −8.79837 −0.318314
\(765\) 0 0
\(766\) 0.618034 0.0223305
\(767\) 0.326238 0.0117798
\(768\) 0 0
\(769\) 36.6312 1.32095 0.660477 0.750846i \(-0.270354\pi\)
0.660477 + 0.750846i \(0.270354\pi\)
\(770\) −25.4164 −0.915944
\(771\) 0 0
\(772\) −3.12461 −0.112457
\(773\) −35.9230 −1.29206 −0.646030 0.763312i \(-0.723572\pi\)
−0.646030 + 0.763312i \(0.723572\pi\)
\(774\) 0 0
\(775\) 48.4508 1.74041
\(776\) 30.9787 1.11207
\(777\) 0 0
\(778\) −15.0000 −0.537776
\(779\) 0 0
\(780\) 0 0
\(781\) −12.0902 −0.432620
\(782\) 6.65248 0.237892
\(783\) 0 0
\(784\) −9.70820 −0.346722
\(785\) 57.7771 2.06215
\(786\) 0 0
\(787\) −38.0000 −1.35455 −0.677277 0.735728i \(-0.736840\pi\)
−0.677277 + 0.735728i \(0.736840\pi\)
\(788\) −1.85410 −0.0660496
\(789\) 0 0
\(790\) 70.2492 2.49936
\(791\) 20.2918 0.721493
\(792\) 0 0
\(793\) −10.2361 −0.363493
\(794\) −18.5623 −0.658752
\(795\) 0 0
\(796\) −8.29180 −0.293895
\(797\) −20.2918 −0.718772 −0.359386 0.933189i \(-0.617014\pi\)
−0.359386 + 0.933189i \(0.617014\pi\)
\(798\) 0 0
\(799\) 2.29180 0.0810779
\(800\) −18.5066 −0.654306
\(801\) 0 0
\(802\) −58.0689 −2.05048
\(803\) −4.38197 −0.154636
\(804\) 0 0
\(805\) 52.2492 1.84154
\(806\) 14.3262 0.504620
\(807\) 0 0
\(808\) −20.5279 −0.722168
\(809\) 24.7984 0.871864 0.435932 0.899980i \(-0.356419\pi\)
0.435932 + 0.899980i \(0.356419\pi\)
\(810\) 0 0
\(811\) −22.9787 −0.806892 −0.403446 0.915004i \(-0.632188\pi\)
−0.403446 + 0.915004i \(0.632188\pi\)
\(812\) −6.70820 −0.235412
\(813\) 0 0
\(814\) −23.1803 −0.812470
\(815\) −5.70820 −0.199950
\(816\) 0 0
\(817\) 0 0
\(818\) 13.4164 0.469094
\(819\) 0 0
\(820\) 6.00000 0.209529
\(821\) −52.0476 −1.81647 −0.908237 0.418457i \(-0.862571\pi\)
−0.908237 + 0.418457i \(0.862571\pi\)
\(822\) 0 0
\(823\) −25.5967 −0.892247 −0.446123 0.894972i \(-0.647196\pi\)
−0.446123 + 0.894972i \(0.647196\pi\)
\(824\) 32.0344 1.11597
\(825\) 0 0
\(826\) 1.58359 0.0551002
\(827\) 32.5967 1.13350 0.566750 0.823890i \(-0.308201\pi\)
0.566750 + 0.823890i \(0.308201\pi\)
\(828\) 0 0
\(829\) −4.67376 −0.162326 −0.0811632 0.996701i \(-0.525864\pi\)
−0.0811632 + 0.996701i \(0.525864\pi\)
\(830\) 44.3607 1.53978
\(831\) 0 0
\(832\) 4.23607 0.146859
\(833\) −1.52786 −0.0529374
\(834\) 0 0
\(835\) −65.4853 −2.26621
\(836\) 0 0
\(837\) 0 0
\(838\) 14.4721 0.499932
\(839\) 15.2016 0.524818 0.262409 0.964957i \(-0.415483\pi\)
0.262409 + 0.964957i \(0.415483\pi\)
\(840\) 0 0
\(841\) −15.9098 −0.548615
\(842\) 44.4508 1.53188
\(843\) 0 0
\(844\) 2.38197 0.0819907
\(845\) 38.8328 1.33589
\(846\) 0 0
\(847\) −25.1459 −0.864023
\(848\) 30.7082 1.05452
\(849\) 0 0
\(850\) −6.76393 −0.232001
\(851\) 47.6525 1.63351
\(852\) 0 0
\(853\) 30.7082 1.05143 0.525714 0.850661i \(-0.323798\pi\)
0.525714 + 0.850661i \(0.323798\pi\)
\(854\) −49.6869 −1.70025
\(855\) 0 0
\(856\) −36.7082 −1.25466
\(857\) −10.6180 −0.362705 −0.181353 0.983418i \(-0.558048\pi\)
−0.181353 + 0.983418i \(0.558048\pi\)
\(858\) 0 0
\(859\) −48.5410 −1.65620 −0.828099 0.560582i \(-0.810578\pi\)
−0.828099 + 0.560582i \(0.810578\pi\)
\(860\) 0.291796 0.00995016
\(861\) 0 0
\(862\) 44.7426 1.52394
\(863\) −27.0557 −0.920988 −0.460494 0.887663i \(-0.652328\pi\)
−0.460494 + 0.887663i \(0.652328\pi\)
\(864\) 0 0
\(865\) 1.52786 0.0519489
\(866\) 5.56231 0.189015
\(867\) 0 0
\(868\) 16.4164 0.557209
\(869\) −21.7082 −0.736400
\(870\) 0 0
\(871\) 7.00000 0.237186
\(872\) −7.36068 −0.249264
\(873\) 0 0
\(874\) 0 0
\(875\) −4.58359 −0.154954
\(876\) 0 0
\(877\) −11.8197 −0.399122 −0.199561 0.979885i \(-0.563952\pi\)
−0.199561 + 0.979885i \(0.563952\pi\)
\(878\) 55.9787 1.88919
\(879\) 0 0
\(880\) 25.4164 0.856787
\(881\) −32.4508 −1.09330 −0.546648 0.837362i \(-0.684097\pi\)
−0.546648 + 0.837362i \(0.684097\pi\)
\(882\) 0 0
\(883\) −50.9230 −1.71369 −0.856847 0.515570i \(-0.827580\pi\)
−0.856847 + 0.515570i \(0.827580\pi\)
\(884\) −0.472136 −0.0158797
\(885\) 0 0
\(886\) 12.2705 0.412236
\(887\) 27.3475 0.918240 0.459120 0.888374i \(-0.348165\pi\)
0.459120 + 0.888374i \(0.348165\pi\)
\(888\) 0 0
\(889\) −17.2918 −0.579948
\(890\) 40.6525 1.36267
\(891\) 0 0
\(892\) 7.20163 0.241128
\(893\) 0 0
\(894\) 0 0
\(895\) −39.5967 −1.32357
\(896\) 40.8541 1.36484
\(897\) 0 0
\(898\) 4.67376 0.155965
\(899\) −32.0344 −1.06841
\(900\) 0 0
\(901\) 4.83282 0.161004
\(902\) −7.85410 −0.261513
\(903\) 0 0
\(904\) −15.1246 −0.503037
\(905\) 38.8328 1.29085
\(906\) 0 0
\(907\) 26.4721 0.878993 0.439496 0.898244i \(-0.355157\pi\)
0.439496 + 0.898244i \(0.355157\pi\)
\(908\) 10.1459 0.336703
\(909\) 0 0
\(910\) −15.7082 −0.520722
\(911\) 3.38197 0.112050 0.0560248 0.998429i \(-0.482157\pi\)
0.0560248 + 0.998429i \(0.482157\pi\)
\(912\) 0 0
\(913\) −13.7082 −0.453675
\(914\) 31.8885 1.05478
\(915\) 0 0
\(916\) −8.41641 −0.278086
\(917\) −45.2705 −1.49496
\(918\) 0 0
\(919\) 23.2918 0.768325 0.384163 0.923265i \(-0.374490\pi\)
0.384163 + 0.923265i \(0.374490\pi\)
\(920\) −38.9443 −1.28395
\(921\) 0 0
\(922\) −33.8885 −1.11606
\(923\) −7.47214 −0.245948
\(924\) 0 0
\(925\) −48.4508 −1.59305
\(926\) 45.7426 1.50320
\(927\) 0 0
\(928\) 12.2361 0.401669
\(929\) −16.3820 −0.537475 −0.268737 0.963213i \(-0.586606\pi\)
−0.268737 + 0.963213i \(0.586606\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −2.79837 −0.0916638
\(933\) 0 0
\(934\) 25.7984 0.844149
\(935\) 4.00000 0.130814
\(936\) 0 0
\(937\) 40.5623 1.32511 0.662556 0.749012i \(-0.269471\pi\)
0.662556 + 0.749012i \(0.269471\pi\)
\(938\) 33.9787 1.10944
\(939\) 0 0
\(940\) 6.00000 0.195698
\(941\) −10.6869 −0.348384 −0.174192 0.984712i \(-0.555731\pi\)
−0.174192 + 0.984712i \(0.555731\pi\)
\(942\) 0 0
\(943\) 16.1459 0.525783
\(944\) −1.58359 −0.0515415
\(945\) 0 0
\(946\) −0.381966 −0.0124188
\(947\) 32.6525 1.06106 0.530531 0.847665i \(-0.321992\pi\)
0.530531 + 0.847665i \(0.321992\pi\)
\(948\) 0 0
\(949\) −2.70820 −0.0879120
\(950\) 0 0
\(951\) 0 0
\(952\) 5.12461 0.166090
\(953\) 17.2918 0.560136 0.280068 0.959980i \(-0.409643\pi\)
0.280068 + 0.959980i \(0.409643\pi\)
\(954\) 0 0
\(955\) 46.0689 1.49075
\(956\) 0.201626 0.00652105
\(957\) 0 0
\(958\) −37.3607 −1.20707
\(959\) −22.4164 −0.723864
\(960\) 0 0
\(961\) 47.3951 1.52887
\(962\) −14.3262 −0.461896
\(963\) 0 0
\(964\) −1.96556 −0.0633064
\(965\) 16.3607 0.526669
\(966\) 0 0
\(967\) 6.54102 0.210345 0.105173 0.994454i \(-0.466461\pi\)
0.105173 + 0.994454i \(0.466461\pi\)
\(968\) 18.7426 0.602411
\(969\) 0 0
\(970\) 72.5410 2.32915
\(971\) 23.5066 0.754362 0.377181 0.926140i \(-0.376893\pi\)
0.377181 + 0.926140i \(0.376893\pi\)
\(972\) 0 0
\(973\) 44.3951 1.42324
\(974\) −6.76393 −0.216730
\(975\) 0 0
\(976\) 49.6869 1.59044
\(977\) 10.6393 0.340382 0.170191 0.985411i \(-0.445562\pi\)
0.170191 + 0.985411i \(0.445562\pi\)
\(978\) 0 0
\(979\) −12.5623 −0.401493
\(980\) −4.00000 −0.127775
\(981\) 0 0
\(982\) 34.3262 1.09539
\(983\) 32.6180 1.04035 0.520177 0.854059i \(-0.325866\pi\)
0.520177 + 0.854059i \(0.325866\pi\)
\(984\) 0 0
\(985\) 9.70820 0.309329
\(986\) 4.47214 0.142422
\(987\) 0 0
\(988\) 0 0
\(989\) 0.785218 0.0249685
\(990\) 0 0
\(991\) −7.45085 −0.236684 −0.118342 0.992973i \(-0.537758\pi\)
−0.118342 + 0.992973i \(0.537758\pi\)
\(992\) −29.9443 −0.950732
\(993\) 0 0
\(994\) −36.2705 −1.15043
\(995\) 43.4164 1.37639
\(996\) 0 0
\(997\) 39.7082 1.25757 0.628786 0.777579i \(-0.283552\pi\)
0.628786 + 0.777579i \(0.283552\pi\)
\(998\) 40.6525 1.28683
\(999\) 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 3249.2.a.o.1.2 2
3.2 odd 2 361.2.a.c.1.1 2
12.11 even 2 5776.2.a.bg.1.1 2
15.14 odd 2 9025.2.a.s.1.2 2
19.18 odd 2 3249.2.a.i.1.1 2
57.2 even 18 361.2.e.j.99.1 12
57.5 odd 18 361.2.e.i.234.1 12
57.8 even 6 361.2.c.d.292.1 4
57.11 odd 6 361.2.c.g.292.2 4
57.14 even 18 361.2.e.j.234.2 12
57.17 odd 18 361.2.e.i.99.2 12
57.23 odd 18 361.2.e.i.54.1 12
57.26 odd 6 361.2.c.g.68.2 4
57.29 even 18 361.2.e.j.62.1 12
57.32 even 18 361.2.e.j.245.2 12
57.35 odd 18 361.2.e.i.28.1 12
57.41 even 18 361.2.e.j.28.2 12
57.44 odd 18 361.2.e.i.245.1 12
57.47 odd 18 361.2.e.i.62.2 12
57.50 even 6 361.2.c.d.68.1 4
57.53 even 18 361.2.e.j.54.2 12
57.56 even 2 361.2.a.f.1.2 yes 2
228.227 odd 2 5776.2.a.s.1.2 2
285.284 even 2 9025.2.a.n.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
361.2.a.c.1.1 2 3.2 odd 2
361.2.a.f.1.2 yes 2 57.56 even 2
361.2.c.d.68.1 4 57.50 even 6
361.2.c.d.292.1 4 57.8 even 6
361.2.c.g.68.2 4 57.26 odd 6
361.2.c.g.292.2 4 57.11 odd 6
361.2.e.i.28.1 12 57.35 odd 18
361.2.e.i.54.1 12 57.23 odd 18
361.2.e.i.62.2 12 57.47 odd 18
361.2.e.i.99.2 12 57.17 odd 18
361.2.e.i.234.1 12 57.5 odd 18
361.2.e.i.245.1 12 57.44 odd 18
361.2.e.j.28.2 12 57.41 even 18
361.2.e.j.54.2 12 57.53 even 18
361.2.e.j.62.1 12 57.29 even 18
361.2.e.j.99.1 12 57.2 even 18
361.2.e.j.234.2 12 57.14 even 18
361.2.e.j.245.2 12 57.32 even 18
3249.2.a.i.1.1 2 19.18 odd 2
3249.2.a.o.1.2 2 1.1 even 1 trivial
5776.2.a.s.1.2 2 228.227 odd 2
5776.2.a.bg.1.1 2 12.11 even 2
9025.2.a.n.1.1 2 285.284 even 2
9025.2.a.s.1.2 2 15.14 odd 2