Properties

Label 1323.2.a.ba.1.3
Level $1323$
Weight $2$
Character 1323.1
Self dual yes
Analytic conductor $10.564$
Analytic rank $0$
Dimension $3$
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] = [1323,2,Mod(1,1323)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1323, 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("1323.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1323.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: \(10.5642081874\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.321.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 189)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.69963\) of defining polynomial
Character \(\chi\) \(=\) 1323.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.69963 q^{2} +5.28799 q^{4} -1.58836 q^{5} +8.87636 q^{8} +O(q^{10})\) \(q+2.69963 q^{2} +5.28799 q^{4} -1.58836 q^{5} +8.87636 q^{8} -4.28799 q^{10} +0.300372 q^{11} +2.81089 q^{13} +13.3869 q^{16} +5.87636 q^{17} -2.28799 q^{19} -8.39926 q^{20} +0.810892 q^{22} +1.88874 q^{23} -2.47710 q^{25} +7.58836 q^{26} +2.52290 q^{29} -4.81089 q^{31} +18.3869 q^{32} +15.8640 q^{34} -4.47710 q^{37} -6.17673 q^{38} -14.0989 q^{40} +8.90978 q^{41} -9.09888 q^{43} +1.58836 q^{44} +5.09888 q^{46} -3.21015 q^{47} -6.68725 q^{50} +14.8640 q^{52} +2.01238 q^{53} -0.477100 q^{55} +6.81089 q^{58} -4.88874 q^{59} +7.57598 q^{61} -12.9876 q^{62} +22.8640 q^{64} -4.46472 q^{65} +0.712008 q^{67} +31.0741 q^{68} -12.8640 q^{71} -11.6749 q^{73} -12.0865 q^{74} -12.0989 q^{76} +1.66621 q^{79} -21.2632 q^{80} +24.0531 q^{82} -5.43268 q^{83} -9.33379 q^{85} -24.5636 q^{86} +2.66621 q^{88} -9.35346 q^{89} +9.98762 q^{92} -8.66621 q^{94} +3.63416 q^{95} -12.5760 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{2} + 4 q^{4} + q^{5} + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{2} + 4 q^{4} + q^{5} + 9 q^{8} - q^{10} + 7 q^{11} + 2 q^{13} + 10 q^{16} + 5 q^{19} - 13 q^{20} - 4 q^{22} + 6 q^{23} - 2 q^{25} + 17 q^{26} + 13 q^{29} - 8 q^{31} + 25 q^{32} + 12 q^{34} - 8 q^{37} - 7 q^{38} - 24 q^{40} + 2 q^{41} - 9 q^{43} - q^{44} - 3 q^{46} + 9 q^{47} + 4 q^{50} + 9 q^{52} + 24 q^{53} + 4 q^{55} + 14 q^{58} - 15 q^{59} - q^{61} - 21 q^{62} + 33 q^{64} + 10 q^{65} + 14 q^{67} + 39 q^{68} - 3 q^{71} + 7 q^{73} - 18 q^{76} + 6 q^{79} - 16 q^{80} + 43 q^{82} + 3 q^{83} - 27 q^{85} - 32 q^{86} + 9 q^{88} - 5 q^{89} + 12 q^{92} - 27 q^{94} + 16 q^{95} - 14 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.69963 1.90893 0.954463 0.298330i \(-0.0964297\pi\)
0.954463 + 0.298330i \(0.0964297\pi\)
\(3\) 0 0
\(4\) 5.28799 2.64400
\(5\) −1.58836 −0.710338 −0.355169 0.934802i \(-0.615577\pi\)
−0.355169 + 0.934802i \(0.615577\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 8.87636 3.13827
\(9\) 0 0
\(10\) −4.28799 −1.35598
\(11\) 0.300372 0.0905655 0.0452828 0.998974i \(-0.485581\pi\)
0.0452828 + 0.998974i \(0.485581\pi\)
\(12\) 0 0
\(13\) 2.81089 0.779601 0.389801 0.920899i \(-0.372544\pi\)
0.389801 + 0.920899i \(0.372544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 13.3869 3.34672
\(17\) 5.87636 1.42523 0.712613 0.701557i \(-0.247512\pi\)
0.712613 + 0.701557i \(0.247512\pi\)
\(18\) 0 0
\(19\) −2.28799 −0.524901 −0.262451 0.964945i \(-0.584531\pi\)
−0.262451 + 0.964945i \(0.584531\pi\)
\(20\) −8.39926 −1.87813
\(21\) 0 0
\(22\) 0.810892 0.172883
\(23\) 1.88874 0.393829 0.196914 0.980421i \(-0.436908\pi\)
0.196914 + 0.980421i \(0.436908\pi\)
\(24\) 0 0
\(25\) −2.47710 −0.495420
\(26\) 7.58836 1.48820
\(27\) 0 0
\(28\) 0 0
\(29\) 2.52290 0.468491 0.234245 0.972178i \(-0.424738\pi\)
0.234245 + 0.972178i \(0.424738\pi\)
\(30\) 0 0
\(31\) −4.81089 −0.864062 −0.432031 0.901859i \(-0.642203\pi\)
−0.432031 + 0.901859i \(0.642203\pi\)
\(32\) 18.3869 3.25037
\(33\) 0 0
\(34\) 15.8640 2.72065
\(35\) 0 0
\(36\) 0 0
\(37\) −4.47710 −0.736031 −0.368015 0.929820i \(-0.619963\pi\)
−0.368015 + 0.929820i \(0.619963\pi\)
\(38\) −6.17673 −1.00200
\(39\) 0 0
\(40\) −14.0989 −2.22923
\(41\) 8.90978 1.39147 0.695737 0.718297i \(-0.255078\pi\)
0.695737 + 0.718297i \(0.255078\pi\)
\(42\) 0 0
\(43\) −9.09888 −1.38757 −0.693783 0.720184i \(-0.744058\pi\)
−0.693783 + 0.720184i \(0.744058\pi\)
\(44\) 1.58836 0.239455
\(45\) 0 0
\(46\) 5.09888 0.751789
\(47\) −3.21015 −0.468248 −0.234124 0.972207i \(-0.575222\pi\)
−0.234124 + 0.972207i \(0.575222\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −6.68725 −0.945720
\(51\) 0 0
\(52\) 14.8640 2.06126
\(53\) 2.01238 0.276422 0.138211 0.990403i \(-0.455865\pi\)
0.138211 + 0.990403i \(0.455865\pi\)
\(54\) 0 0
\(55\) −0.477100 −0.0643321
\(56\) 0 0
\(57\) 0 0
\(58\) 6.81089 0.894314
\(59\) −4.88874 −0.636459 −0.318230 0.948014i \(-0.603088\pi\)
−0.318230 + 0.948014i \(0.603088\pi\)
\(60\) 0 0
\(61\) 7.57598 0.970005 0.485003 0.874513i \(-0.338819\pi\)
0.485003 + 0.874513i \(0.338819\pi\)
\(62\) −12.9876 −1.64943
\(63\) 0 0
\(64\) 22.8640 2.85800
\(65\) −4.46472 −0.553780
\(66\) 0 0
\(67\) 0.712008 0.0869856 0.0434928 0.999054i \(-0.486151\pi\)
0.0434928 + 0.999054i \(0.486151\pi\)
\(68\) 31.0741 3.76829
\(69\) 0 0
\(70\) 0 0
\(71\) −12.8640 −1.52667 −0.763337 0.646001i \(-0.776440\pi\)
−0.763337 + 0.646001i \(0.776440\pi\)
\(72\) 0 0
\(73\) −11.6749 −1.36644 −0.683220 0.730213i \(-0.739421\pi\)
−0.683220 + 0.730213i \(0.739421\pi\)
\(74\) −12.0865 −1.40503
\(75\) 0 0
\(76\) −12.0989 −1.38784
\(77\) 0 0
\(78\) 0 0
\(79\) 1.66621 0.187463 0.0937315 0.995598i \(-0.470120\pi\)
0.0937315 + 0.995598i \(0.470120\pi\)
\(80\) −21.2632 −2.37730
\(81\) 0 0
\(82\) 24.0531 2.65622
\(83\) −5.43268 −0.596314 −0.298157 0.954517i \(-0.596372\pi\)
−0.298157 + 0.954517i \(0.596372\pi\)
\(84\) 0 0
\(85\) −9.33379 −1.01239
\(86\) −24.5636 −2.64876
\(87\) 0 0
\(88\) 2.66621 0.284219
\(89\) −9.35346 −0.991464 −0.495732 0.868475i \(-0.665100\pi\)
−0.495732 + 0.868475i \(0.665100\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 9.98762 1.04128
\(93\) 0 0
\(94\) −8.66621 −0.893851
\(95\) 3.63416 0.372857
\(96\) 0 0
\(97\) −12.5760 −1.27690 −0.638449 0.769664i \(-0.720424\pi\)
−0.638449 + 0.769664i \(0.720424\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −13.0989 −1.30989
\(101\) −14.6094 −1.45369 −0.726845 0.686801i \(-0.759014\pi\)
−0.726845 + 0.686801i \(0.759014\pi\)
\(102\) 0 0
\(103\) 9.19777 0.906283 0.453142 0.891439i \(-0.350303\pi\)
0.453142 + 0.891439i \(0.350303\pi\)
\(104\) 24.9505 2.44660
\(105\) 0 0
\(106\) 5.43268 0.527668
\(107\) 12.5105 1.20944 0.604719 0.796439i \(-0.293285\pi\)
0.604719 + 0.796439i \(0.293285\pi\)
\(108\) 0 0
\(109\) −3.04580 −0.291735 −0.145867 0.989304i \(-0.546597\pi\)
−0.145867 + 0.989304i \(0.546597\pi\)
\(110\) −1.28799 −0.122805
\(111\) 0 0
\(112\) 0 0
\(113\) 9.95420 0.936412 0.468206 0.883619i \(-0.344900\pi\)
0.468206 + 0.883619i \(0.344900\pi\)
\(114\) 0 0
\(115\) −3.00000 −0.279751
\(116\) 13.3411 1.23869
\(117\) 0 0
\(118\) −13.1978 −1.21495
\(119\) 0 0
\(120\) 0 0
\(121\) −10.9098 −0.991798
\(122\) 20.4523 1.85167
\(123\) 0 0
\(124\) −25.4400 −2.28458
\(125\) 11.8764 1.06225
\(126\) 0 0
\(127\) −13.4400 −1.19260 −0.596302 0.802760i \(-0.703364\pi\)
−0.596302 + 0.802760i \(0.703364\pi\)
\(128\) 24.9505 2.20533
\(129\) 0 0
\(130\) −12.0531 −1.05713
\(131\) 11.2756 0.985155 0.492577 0.870269i \(-0.336055\pi\)
0.492577 + 0.870269i \(0.336055\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 1.92216 0.166049
\(135\) 0 0
\(136\) 52.1606 4.47274
\(137\) 3.77747 0.322731 0.161366 0.986895i \(-0.448410\pi\)
0.161366 + 0.986895i \(0.448410\pi\)
\(138\) 0 0
\(139\) 16.7193 1.41811 0.709056 0.705152i \(-0.249121\pi\)
0.709056 + 0.705152i \(0.249121\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −34.7280 −2.91431
\(143\) 0.844313 0.0706050
\(144\) 0 0
\(145\) −4.00728 −0.332787
\(146\) −31.5178 −2.60843
\(147\) 0 0
\(148\) −23.6749 −1.94606
\(149\) 10.8887 0.892040 0.446020 0.895023i \(-0.352841\pi\)
0.446020 + 0.895023i \(0.352841\pi\)
\(150\) 0 0
\(151\) −12.0989 −0.984593 −0.492297 0.870427i \(-0.663842\pi\)
−0.492297 + 0.870427i \(0.663842\pi\)
\(152\) −20.3090 −1.64728
\(153\) 0 0
\(154\) 0 0
\(155\) 7.64145 0.613776
\(156\) 0 0
\(157\) −0.189108 −0.0150924 −0.00754622 0.999972i \(-0.502402\pi\)
−0.00754622 + 0.999972i \(0.502402\pi\)
\(158\) 4.49814 0.357853
\(159\) 0 0
\(160\) −29.2051 −2.30886
\(161\) 0 0
\(162\) 0 0
\(163\) −8.95558 −0.701455 −0.350727 0.936478i \(-0.614066\pi\)
−0.350727 + 0.936478i \(0.614066\pi\)
\(164\) 47.1148 3.67905
\(165\) 0 0
\(166\) −14.6662 −1.13832
\(167\) −4.34108 −0.335923 −0.167961 0.985794i \(-0.553718\pi\)
−0.167961 + 0.985794i \(0.553718\pi\)
\(168\) 0 0
\(169\) −5.09888 −0.392222
\(170\) −25.1978 −1.93258
\(171\) 0 0
\(172\) −48.1148 −3.66872
\(173\) −6.30037 −0.479008 −0.239504 0.970895i \(-0.576985\pi\)
−0.239504 + 0.970895i \(0.576985\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 4.02104 0.303097
\(177\) 0 0
\(178\) −25.2509 −1.89263
\(179\) −13.9418 −1.04206 −0.521030 0.853538i \(-0.674452\pi\)
−0.521030 + 0.853538i \(0.674452\pi\)
\(180\) 0 0
\(181\) 18.3411 1.36328 0.681641 0.731687i \(-0.261267\pi\)
0.681641 + 0.731687i \(0.261267\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 16.7651 1.23594
\(185\) 7.11126 0.522831
\(186\) 0 0
\(187\) 1.76509 0.129076
\(188\) −16.9752 −1.23805
\(189\) 0 0
\(190\) 9.81089 0.711757
\(191\) 7.60803 0.550498 0.275249 0.961373i \(-0.411240\pi\)
0.275249 + 0.961373i \(0.411240\pi\)
\(192\) 0 0
\(193\) 7.66621 0.551826 0.275913 0.961183i \(-0.411020\pi\)
0.275913 + 0.961183i \(0.411020\pi\)
\(194\) −33.9505 −2.43750
\(195\) 0 0
\(196\) 0 0
\(197\) 18.3869 1.31001 0.655005 0.755624i \(-0.272666\pi\)
0.655005 + 0.755624i \(0.272666\pi\)
\(198\) 0 0
\(199\) 5.95420 0.422082 0.211041 0.977477i \(-0.432315\pi\)
0.211041 + 0.977477i \(0.432315\pi\)
\(200\) −21.9876 −1.55476
\(201\) 0 0
\(202\) −39.4400 −2.77499
\(203\) 0 0
\(204\) 0 0
\(205\) −14.1520 −0.988416
\(206\) 24.8306 1.73003
\(207\) 0 0
\(208\) 37.6291 2.60911
\(209\) −0.687248 −0.0475380
\(210\) 0 0
\(211\) 6.67487 0.459517 0.229758 0.973248i \(-0.426206\pi\)
0.229758 + 0.973248i \(0.426206\pi\)
\(212\) 10.6414 0.730858
\(213\) 0 0
\(214\) 33.7738 2.30873
\(215\) 14.4523 0.985641
\(216\) 0 0
\(217\) 0 0
\(218\) −8.22253 −0.556900
\(219\) 0 0
\(220\) −2.52290 −0.170094
\(221\) 16.5178 1.11111
\(222\) 0 0
\(223\) 6.34108 0.424630 0.212315 0.977201i \(-0.431900\pi\)
0.212315 + 0.977201i \(0.431900\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 26.8726 1.78754
\(227\) −27.0865 −1.79779 −0.898897 0.438160i \(-0.855630\pi\)
−0.898897 + 0.438160i \(0.855630\pi\)
\(228\) 0 0
\(229\) 8.19639 0.541633 0.270816 0.962631i \(-0.412706\pi\)
0.270816 + 0.962631i \(0.412706\pi\)
\(230\) −8.09888 −0.534025
\(231\) 0 0
\(232\) 22.3942 1.47025
\(233\) 27.4203 1.79636 0.898182 0.439624i \(-0.144888\pi\)
0.898182 + 0.439624i \(0.144888\pi\)
\(234\) 0 0
\(235\) 5.09888 0.332615
\(236\) −25.8516 −1.68280
\(237\) 0 0
\(238\) 0 0
\(239\) −27.8640 −1.80237 −0.901185 0.433434i \(-0.857302\pi\)
−0.901185 + 0.433434i \(0.857302\pi\)
\(240\) 0 0
\(241\) −22.5302 −1.45130 −0.725648 0.688066i \(-0.758460\pi\)
−0.725648 + 0.688066i \(0.758460\pi\)
\(242\) −29.4523 −1.89327
\(243\) 0 0
\(244\) 40.0617 2.56469
\(245\) 0 0
\(246\) 0 0
\(247\) −6.43130 −0.409214
\(248\) −42.7032 −2.71166
\(249\) 0 0
\(250\) 32.0617 2.02776
\(251\) 31.2509 1.97254 0.986268 0.165152i \(-0.0528114\pi\)
0.986268 + 0.165152i \(0.0528114\pi\)
\(252\) 0 0
\(253\) 0.567323 0.0356673
\(254\) −36.2829 −2.27659
\(255\) 0 0
\(256\) 21.6291 1.35182
\(257\) −10.2015 −0.636351 −0.318176 0.948032i \(-0.603070\pi\)
−0.318176 + 0.948032i \(0.603070\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −23.6094 −1.46419
\(261\) 0 0
\(262\) 30.4400 1.88059
\(263\) 28.4610 1.75498 0.877490 0.479594i \(-0.159216\pi\)
0.877490 + 0.479594i \(0.159216\pi\)
\(264\) 0 0
\(265\) −3.19639 −0.196353
\(266\) 0 0
\(267\) 0 0
\(268\) 3.76509 0.229990
\(269\) 14.8764 0.907027 0.453514 0.891249i \(-0.350170\pi\)
0.453514 + 0.891249i \(0.350170\pi\)
\(270\) 0 0
\(271\) 0.0444229 0.00269850 0.00134925 0.999999i \(-0.499571\pi\)
0.00134925 + 0.999999i \(0.499571\pi\)
\(272\) 78.6661 4.76983
\(273\) 0 0
\(274\) 10.1978 0.616070
\(275\) −0.744051 −0.0448680
\(276\) 0 0
\(277\) 15.6749 0.941812 0.470906 0.882184i \(-0.343927\pi\)
0.470906 + 0.882184i \(0.343927\pi\)
\(278\) 45.1359 2.70707
\(279\) 0 0
\(280\) 0 0
\(281\) 11.9098 0.710478 0.355239 0.934776i \(-0.384400\pi\)
0.355239 + 0.934776i \(0.384400\pi\)
\(282\) 0 0
\(283\) 6.00728 0.357096 0.178548 0.983931i \(-0.442860\pi\)
0.178548 + 0.983931i \(0.442860\pi\)
\(284\) −68.0246 −4.03652
\(285\) 0 0
\(286\) 2.27933 0.134780
\(287\) 0 0
\(288\) 0 0
\(289\) 17.5316 1.03127
\(290\) −10.8182 −0.635265
\(291\) 0 0
\(292\) −61.7366 −3.61286
\(293\) −4.95558 −0.289508 −0.144754 0.989468i \(-0.546239\pi\)
−0.144754 + 0.989468i \(0.546239\pi\)
\(294\) 0 0
\(295\) 7.76509 0.452101
\(296\) −39.7403 −2.30986
\(297\) 0 0
\(298\) 29.3955 1.70284
\(299\) 5.30903 0.307029
\(300\) 0 0
\(301\) 0 0
\(302\) −32.6625 −1.87952
\(303\) 0 0
\(304\) −30.6291 −1.75670
\(305\) −12.0334 −0.689032
\(306\) 0 0
\(307\) −22.0531 −1.25864 −0.629318 0.777148i \(-0.716666\pi\)
−0.629318 + 0.777148i \(0.716666\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 20.6291 1.17165
\(311\) 7.97524 0.452234 0.226117 0.974100i \(-0.427397\pi\)
0.226117 + 0.974100i \(0.427397\pi\)
\(312\) 0 0
\(313\) −22.5302 −1.27348 −0.636741 0.771078i \(-0.719718\pi\)
−0.636741 + 0.771078i \(0.719718\pi\)
\(314\) −0.510520 −0.0288103
\(315\) 0 0
\(316\) 8.81089 0.495651
\(317\) −19.9381 −1.11984 −0.559918 0.828548i \(-0.689167\pi\)
−0.559918 + 0.828548i \(0.689167\pi\)
\(318\) 0 0
\(319\) 0.757808 0.0424291
\(320\) −36.3163 −2.03014
\(321\) 0 0
\(322\) 0 0
\(323\) −13.4451 −0.748103
\(324\) 0 0
\(325\) −6.96286 −0.386230
\(326\) −24.1767 −1.33903
\(327\) 0 0
\(328\) 79.0864 4.36681
\(329\) 0 0
\(330\) 0 0
\(331\) 23.6304 1.29885 0.649423 0.760427i \(-0.275010\pi\)
0.649423 + 0.760427i \(0.275010\pi\)
\(332\) −28.7280 −1.57665
\(333\) 0 0
\(334\) −11.7193 −0.641251
\(335\) −1.13093 −0.0617892
\(336\) 0 0
\(337\) −12.3855 −0.674681 −0.337341 0.941383i \(-0.609527\pi\)
−0.337341 + 0.941383i \(0.609527\pi\)
\(338\) −13.7651 −0.748722
\(339\) 0 0
\(340\) −49.3570 −2.67676
\(341\) −1.44506 −0.0782542
\(342\) 0 0
\(343\) 0 0
\(344\) −80.7649 −4.35455
\(345\) 0 0
\(346\) −17.0087 −0.914391
\(347\) −3.84431 −0.206374 −0.103187 0.994662i \(-0.532904\pi\)
−0.103187 + 0.994662i \(0.532904\pi\)
\(348\) 0 0
\(349\) −12.1891 −0.652468 −0.326234 0.945289i \(-0.605780\pi\)
−0.326234 + 0.945289i \(0.605780\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 5.52290 0.294372
\(353\) −3.56360 −0.189672 −0.0948358 0.995493i \(-0.530233\pi\)
−0.0948358 + 0.995493i \(0.530233\pi\)
\(354\) 0 0
\(355\) 20.4327 1.08445
\(356\) −49.4610 −2.62143
\(357\) 0 0
\(358\) −37.6377 −1.98922
\(359\) −0.967957 −0.0510868 −0.0255434 0.999674i \(-0.508132\pi\)
−0.0255434 + 0.999674i \(0.508132\pi\)
\(360\) 0 0
\(361\) −13.7651 −0.724479
\(362\) 49.5141 2.60240
\(363\) 0 0
\(364\) 0 0
\(365\) 18.5439 0.970634
\(366\) 0 0
\(367\) 20.9542 1.09380 0.546900 0.837198i \(-0.315808\pi\)
0.546900 + 0.837198i \(0.315808\pi\)
\(368\) 25.2843 1.31803
\(369\) 0 0
\(370\) 19.1978 0.998044
\(371\) 0 0
\(372\) 0 0
\(373\) 2.00000 0.103556 0.0517780 0.998659i \(-0.483511\pi\)
0.0517780 + 0.998659i \(0.483511\pi\)
\(374\) 4.76509 0.246397
\(375\) 0 0
\(376\) −28.4944 −1.46949
\(377\) 7.09160 0.365236
\(378\) 0 0
\(379\) −7.14331 −0.366927 −0.183464 0.983027i \(-0.558731\pi\)
−0.183464 + 0.983027i \(0.558731\pi\)
\(380\) 19.2174 0.985833
\(381\) 0 0
\(382\) 20.5388 1.05086
\(383\) −11.9766 −0.611977 −0.305988 0.952035i \(-0.598987\pi\)
−0.305988 + 0.952035i \(0.598987\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 20.6959 1.05339
\(387\) 0 0
\(388\) −66.5017 −3.37611
\(389\) 13.7788 0.698615 0.349308 0.937008i \(-0.386417\pi\)
0.349308 + 0.937008i \(0.386417\pi\)
\(390\) 0 0
\(391\) 11.0989 0.561295
\(392\) 0 0
\(393\) 0 0
\(394\) 49.6377 2.50071
\(395\) −2.64654 −0.133162
\(396\) 0 0
\(397\) 8.38688 0.420925 0.210463 0.977602i \(-0.432503\pi\)
0.210463 + 0.977602i \(0.432503\pi\)
\(398\) 16.0741 0.805723
\(399\) 0 0
\(400\) −33.1606 −1.65803
\(401\) 27.6167 1.37911 0.689556 0.724233i \(-0.257806\pi\)
0.689556 + 0.724233i \(0.257806\pi\)
\(402\) 0 0
\(403\) −13.5229 −0.673624
\(404\) −77.2544 −3.84355
\(405\) 0 0
\(406\) 0 0
\(407\) −1.34479 −0.0666590
\(408\) 0 0
\(409\) 30.5316 1.50969 0.754844 0.655904i \(-0.227712\pi\)
0.754844 + 0.655904i \(0.227712\pi\)
\(410\) −38.2051 −1.88681
\(411\) 0 0
\(412\) 48.6377 2.39621
\(413\) 0 0
\(414\) 0 0
\(415\) 8.62907 0.423584
\(416\) 51.6835 2.53399
\(417\) 0 0
\(418\) −1.85532 −0.0907464
\(419\) 21.0000 1.02592 0.512959 0.858413i \(-0.328549\pi\)
0.512959 + 0.858413i \(0.328549\pi\)
\(420\) 0 0
\(421\) 34.7293 1.69260 0.846302 0.532703i \(-0.178824\pi\)
0.846302 + 0.532703i \(0.178824\pi\)
\(422\) 18.0197 0.877183
\(423\) 0 0
\(424\) 17.8626 0.867484
\(425\) −14.5563 −0.706085
\(426\) 0 0
\(427\) 0 0
\(428\) 66.1555 3.19775
\(429\) 0 0
\(430\) 39.0159 1.88152
\(431\) 31.6043 1.52233 0.761163 0.648561i \(-0.224629\pi\)
0.761163 + 0.648561i \(0.224629\pi\)
\(432\) 0 0
\(433\) −6.48576 −0.311686 −0.155843 0.987782i \(-0.549809\pi\)
−0.155843 + 0.987782i \(0.549809\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −16.1062 −0.771346
\(437\) −4.32141 −0.206721
\(438\) 0 0
\(439\) −15.5760 −0.743401 −0.371701 0.928353i \(-0.621225\pi\)
−0.371701 + 0.928353i \(0.621225\pi\)
\(440\) −4.23491 −0.201891
\(441\) 0 0
\(442\) 44.5919 2.12102
\(443\) −25.0741 −1.19131 −0.595654 0.803241i \(-0.703107\pi\)
−0.595654 + 0.803241i \(0.703107\pi\)
\(444\) 0 0
\(445\) 14.8567 0.704275
\(446\) 17.1185 0.810587
\(447\) 0 0
\(448\) 0 0
\(449\) −15.2967 −0.721894 −0.360947 0.932586i \(-0.617546\pi\)
−0.360947 + 0.932586i \(0.617546\pi\)
\(450\) 0 0
\(451\) 2.67625 0.126020
\(452\) 52.6377 2.47587
\(453\) 0 0
\(454\) −73.1235 −3.43186
\(455\) 0 0
\(456\) 0 0
\(457\) 20.4400 0.956141 0.478071 0.878321i \(-0.341336\pi\)
0.478071 + 0.878321i \(0.341336\pi\)
\(458\) 22.1272 1.03394
\(459\) 0 0
\(460\) −15.8640 −0.739662
\(461\) −40.8182 −1.90109 −0.950546 0.310584i \(-0.899475\pi\)
−0.950546 + 0.310584i \(0.899475\pi\)
\(462\) 0 0
\(463\) 9.90840 0.460482 0.230241 0.973134i \(-0.426048\pi\)
0.230241 + 0.973134i \(0.426048\pi\)
\(464\) 33.7738 1.56791
\(465\) 0 0
\(466\) 74.0246 3.42912
\(467\) −17.7390 −0.820861 −0.410430 0.911892i \(-0.634622\pi\)
−0.410430 + 0.911892i \(0.634622\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 13.7651 0.634936
\(471\) 0 0
\(472\) −43.3942 −1.99738
\(473\) −2.73305 −0.125666
\(474\) 0 0
\(475\) 5.66758 0.260047
\(476\) 0 0
\(477\) 0 0
\(478\) −75.2224 −3.44059
\(479\) −23.6094 −1.07874 −0.539371 0.842068i \(-0.681338\pi\)
−0.539371 + 0.842068i \(0.681338\pi\)
\(480\) 0 0
\(481\) −12.5846 −0.573810
\(482\) −60.8231 −2.77042
\(483\) 0 0
\(484\) −57.6908 −2.62231
\(485\) 19.9752 0.907029
\(486\) 0 0
\(487\) 34.8726 1.58023 0.790115 0.612959i \(-0.210021\pi\)
0.790115 + 0.612959i \(0.210021\pi\)
\(488\) 67.2471 3.04413
\(489\) 0 0
\(490\) 0 0
\(491\) 23.2051 1.04723 0.523615 0.851955i \(-0.324583\pi\)
0.523615 + 0.851955i \(0.324583\pi\)
\(492\) 0 0
\(493\) 14.8255 0.667705
\(494\) −17.3621 −0.781159
\(495\) 0 0
\(496\) −64.4028 −2.89177
\(497\) 0 0
\(498\) 0 0
\(499\) −5.32513 −0.238386 −0.119193 0.992871i \(-0.538031\pi\)
−0.119193 + 0.992871i \(0.538031\pi\)
\(500\) 62.8021 2.80859
\(501\) 0 0
\(502\) 84.3657 3.76542
\(503\) −10.4313 −0.465109 −0.232554 0.972583i \(-0.574708\pi\)
−0.232554 + 0.972583i \(0.574708\pi\)
\(504\) 0 0
\(505\) 23.2051 1.03261
\(506\) 1.53156 0.0680862
\(507\) 0 0
\(508\) −71.0704 −3.15324
\(509\) 1.50186 0.0665687 0.0332844 0.999446i \(-0.489403\pi\)
0.0332844 + 0.999446i \(0.489403\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 8.48948 0.375186
\(513\) 0 0
\(514\) −27.5402 −1.21475
\(515\) −14.6094 −0.643767
\(516\) 0 0
\(517\) −0.964238 −0.0424072
\(518\) 0 0
\(519\) 0 0
\(520\) −39.6304 −1.73791
\(521\) 37.7417 1.65349 0.826747 0.562574i \(-0.190189\pi\)
0.826747 + 0.562574i \(0.190189\pi\)
\(522\) 0 0
\(523\) −17.9556 −0.785143 −0.392571 0.919722i \(-0.628414\pi\)
−0.392571 + 0.919722i \(0.628414\pi\)
\(524\) 59.6253 2.60475
\(525\) 0 0
\(526\) 76.8341 3.35013
\(527\) −28.2705 −1.23148
\(528\) 0 0
\(529\) −19.4327 −0.844899
\(530\) −8.62907 −0.374823
\(531\) 0 0
\(532\) 0 0
\(533\) 25.0444 1.08479
\(534\) 0 0
\(535\) −19.8713 −0.859109
\(536\) 6.32004 0.272984
\(537\) 0 0
\(538\) 40.1606 1.73145
\(539\) 0 0
\(540\) 0 0
\(541\) 8.37093 0.359894 0.179947 0.983676i \(-0.442407\pi\)
0.179947 + 0.983676i \(0.442407\pi\)
\(542\) 0.119925 0.00515123
\(543\) 0 0
\(544\) 108.048 4.63251
\(545\) 4.83784 0.207230
\(546\) 0 0
\(547\) 30.1075 1.28731 0.643653 0.765318i \(-0.277418\pi\)
0.643653 + 0.765318i \(0.277418\pi\)
\(548\) 19.9752 0.853300
\(549\) 0 0
\(550\) −2.00866 −0.0856496
\(551\) −5.77238 −0.245911
\(552\) 0 0
\(553\) 0 0
\(554\) 42.3163 1.79785
\(555\) 0 0
\(556\) 88.4115 3.74948
\(557\) 16.5353 0.700622 0.350311 0.936633i \(-0.386076\pi\)
0.350311 + 0.936633i \(0.386076\pi\)
\(558\) 0 0
\(559\) −25.5760 −1.08175
\(560\) 0 0
\(561\) 0 0
\(562\) 32.1520 1.35625
\(563\) −22.2312 −0.936933 −0.468466 0.883481i \(-0.655193\pi\)
−0.468466 + 0.883481i \(0.655193\pi\)
\(564\) 0 0
\(565\) −15.8109 −0.665169
\(566\) 16.2174 0.681670
\(567\) 0 0
\(568\) −114.185 −4.79111
\(569\) 2.31275 0.0969556 0.0484778 0.998824i \(-0.484563\pi\)
0.0484778 + 0.998824i \(0.484563\pi\)
\(570\) 0 0
\(571\) −6.09888 −0.255230 −0.127615 0.991824i \(-0.540732\pi\)
−0.127615 + 0.991824i \(0.540732\pi\)
\(572\) 4.46472 0.186679
\(573\) 0 0
\(574\) 0 0
\(575\) −4.67859 −0.195111
\(576\) 0 0
\(577\) 20.2064 0.841205 0.420602 0.907245i \(-0.361819\pi\)
0.420602 + 0.907245i \(0.361819\pi\)
\(578\) 47.3287 1.96861
\(579\) 0 0
\(580\) −21.1905 −0.879887
\(581\) 0 0
\(582\) 0 0
\(583\) 0.604462 0.0250343
\(584\) −103.630 −4.28825
\(585\) 0 0
\(586\) −13.3782 −0.552649
\(587\) 38.6822 1.59658 0.798292 0.602271i \(-0.205737\pi\)
0.798292 + 0.602271i \(0.205737\pi\)
\(588\) 0 0
\(589\) 11.0073 0.453547
\(590\) 20.9629 0.863027
\(591\) 0 0
\(592\) −59.9344 −2.46329
\(593\) 23.7156 0.973882 0.486941 0.873435i \(-0.338113\pi\)
0.486941 + 0.873435i \(0.338113\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 57.5795 2.35855
\(597\) 0 0
\(598\) 14.3324 0.586096
\(599\) −34.4807 −1.40884 −0.704421 0.709783i \(-0.748793\pi\)
−0.704421 + 0.709783i \(0.748793\pi\)
\(600\) 0 0
\(601\) −7.28071 −0.296986 −0.148493 0.988913i \(-0.547442\pi\)
−0.148493 + 0.988913i \(0.547442\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −63.9788 −2.60326
\(605\) 17.3287 0.704512
\(606\) 0 0
\(607\) −25.1593 −1.02118 −0.510591 0.859824i \(-0.670573\pi\)
−0.510591 + 0.859824i \(0.670573\pi\)
\(608\) −42.0690 −1.70612
\(609\) 0 0
\(610\) −32.4858 −1.31531
\(611\) −9.02338 −0.365047
\(612\) 0 0
\(613\) −37.5461 −1.51647 −0.758237 0.651979i \(-0.773939\pi\)
−0.758237 + 0.651979i \(0.773939\pi\)
\(614\) −59.5351 −2.40264
\(615\) 0 0
\(616\) 0 0
\(617\) −13.7280 −0.552667 −0.276333 0.961062i \(-0.589119\pi\)
−0.276333 + 0.961062i \(0.589119\pi\)
\(618\) 0 0
\(619\) 11.8109 0.474720 0.237360 0.971422i \(-0.423718\pi\)
0.237360 + 0.971422i \(0.423718\pi\)
\(620\) 40.4079 1.62282
\(621\) 0 0
\(622\) 21.5302 0.863282
\(623\) 0 0
\(624\) 0 0
\(625\) −6.47848 −0.259139
\(626\) −60.8231 −2.43098
\(627\) 0 0
\(628\) −1.00000 −0.0399043
\(629\) −26.3090 −1.04901
\(630\) 0 0
\(631\) 29.6304 1.17957 0.589785 0.807561i \(-0.299213\pi\)
0.589785 + 0.807561i \(0.299213\pi\)
\(632\) 14.7899 0.588309
\(633\) 0 0
\(634\) −53.8255 −2.13768
\(635\) 21.3475 0.847152
\(636\) 0 0
\(637\) 0 0
\(638\) 2.04580 0.0809940
\(639\) 0 0
\(640\) −39.6304 −1.56653
\(641\) −14.6525 −0.578737 −0.289368 0.957218i \(-0.593445\pi\)
−0.289368 + 0.957218i \(0.593445\pi\)
\(642\) 0 0
\(643\) 2.03714 0.0803369 0.0401685 0.999193i \(-0.487211\pi\)
0.0401685 + 0.999193i \(0.487211\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −36.2967 −1.42807
\(647\) 11.7861 0.463361 0.231680 0.972792i \(-0.425578\pi\)
0.231680 + 0.972792i \(0.425578\pi\)
\(648\) 0 0
\(649\) −1.46844 −0.0576413
\(650\) −18.7971 −0.737284
\(651\) 0 0
\(652\) −47.3570 −1.85464
\(653\) 34.1840 1.33772 0.668862 0.743387i \(-0.266782\pi\)
0.668862 + 0.743387i \(0.266782\pi\)
\(654\) 0 0
\(655\) −17.9098 −0.699793
\(656\) 119.274 4.65687
\(657\) 0 0
\(658\) 0 0
\(659\) −24.8640 −0.968563 −0.484282 0.874912i \(-0.660919\pi\)
−0.484282 + 0.874912i \(0.660919\pi\)
\(660\) 0 0
\(661\) 25.5388 0.993346 0.496673 0.867938i \(-0.334555\pi\)
0.496673 + 0.867938i \(0.334555\pi\)
\(662\) 63.7934 2.47940
\(663\) 0 0
\(664\) −48.2224 −1.87139
\(665\) 0 0
\(666\) 0 0
\(667\) 4.76509 0.184505
\(668\) −22.9556 −0.888178
\(669\) 0 0
\(670\) −3.05308 −0.117951
\(671\) 2.27561 0.0878490
\(672\) 0 0
\(673\) −3.81955 −0.147233 −0.0736165 0.997287i \(-0.523454\pi\)
−0.0736165 + 0.997287i \(0.523454\pi\)
\(674\) −33.4362 −1.28792
\(675\) 0 0
\(676\) −26.9629 −1.03703
\(677\) 17.2225 0.661916 0.330958 0.943646i \(-0.392628\pi\)
0.330958 + 0.943646i \(0.392628\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −82.8501 −3.17715
\(681\) 0 0
\(682\) −3.90112 −0.149381
\(683\) −33.7170 −1.29014 −0.645072 0.764122i \(-0.723172\pi\)
−0.645072 + 0.764122i \(0.723172\pi\)
\(684\) 0 0
\(685\) −6.00000 −0.229248
\(686\) 0 0
\(687\) 0 0
\(688\) −121.806 −4.64380
\(689\) 5.65658 0.215499
\(690\) 0 0
\(691\) −25.4930 −0.969801 −0.484901 0.874569i \(-0.661144\pi\)
−0.484901 + 0.874569i \(0.661144\pi\)
\(692\) −33.3163 −1.26650
\(693\) 0 0
\(694\) −10.3782 −0.393952
\(695\) −26.5563 −1.00734
\(696\) 0 0
\(697\) 52.3570 1.98316
\(698\) −32.9061 −1.24551
\(699\) 0 0
\(700\) 0 0
\(701\) −10.1606 −0.383762 −0.191881 0.981418i \(-0.561459\pi\)
−0.191881 + 0.981418i \(0.561459\pi\)
\(702\) 0 0
\(703\) 10.2436 0.386344
\(704\) 6.86769 0.258836
\(705\) 0 0
\(706\) −9.62041 −0.362069
\(707\) 0 0
\(708\) 0 0
\(709\) −15.9469 −0.598899 −0.299449 0.954112i \(-0.596803\pi\)
−0.299449 + 0.954112i \(0.596803\pi\)
\(710\) 55.1606 2.07014
\(711\) 0 0
\(712\) −83.0246 −3.11148
\(713\) −9.08650 −0.340292
\(714\) 0 0
\(715\) −1.34108 −0.0501534
\(716\) −73.7242 −2.75520
\(717\) 0 0
\(718\) −2.61312 −0.0975209
\(719\) 16.4175 0.612271 0.306136 0.951988i \(-0.400964\pi\)
0.306136 + 0.951988i \(0.400964\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −37.1606 −1.38298
\(723\) 0 0
\(724\) 96.9875 3.60451
\(725\) −6.24948 −0.232100
\(726\) 0 0
\(727\) −35.8282 −1.32879 −0.664397 0.747379i \(-0.731312\pi\)
−0.664397 + 0.747379i \(0.731312\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 50.0617 1.85287
\(731\) −53.4683 −1.97760
\(732\) 0 0
\(733\) 39.2880 1.45114 0.725568 0.688151i \(-0.241577\pi\)
0.725568 + 0.688151i \(0.241577\pi\)
\(734\) 56.5685 2.08798
\(735\) 0 0
\(736\) 34.7280 1.28009
\(737\) 0.213867 0.00787790
\(738\) 0 0
\(739\) 2.09022 0.0768901 0.0384451 0.999261i \(-0.487760\pi\)
0.0384451 + 0.999261i \(0.487760\pi\)
\(740\) 37.6043 1.38236
\(741\) 0 0
\(742\) 0 0
\(743\) −30.7266 −1.12725 −0.563624 0.826031i \(-0.690593\pi\)
−0.563624 + 0.826031i \(0.690593\pi\)
\(744\) 0 0
\(745\) −17.2953 −0.633650
\(746\) 5.39926 0.197681
\(747\) 0 0
\(748\) 9.33379 0.341277
\(749\) 0 0
\(750\) 0 0
\(751\) 20.7738 0.758045 0.379023 0.925387i \(-0.376260\pi\)
0.379023 + 0.925387i \(0.376260\pi\)
\(752\) −42.9739 −1.56710
\(753\) 0 0
\(754\) 19.1447 0.697208
\(755\) 19.2174 0.699394
\(756\) 0 0
\(757\) 16.9257 0.615176 0.307588 0.951520i \(-0.400478\pi\)
0.307588 + 0.951520i \(0.400478\pi\)
\(758\) −19.2843 −0.700436
\(759\) 0 0
\(760\) 32.2581 1.17013
\(761\) −32.8392 −1.19042 −0.595210 0.803570i \(-0.702931\pi\)
−0.595210 + 0.803570i \(0.702931\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 40.2312 1.45551
\(765\) 0 0
\(766\) −32.3324 −1.16822
\(767\) −13.7417 −0.496184
\(768\) 0 0
\(769\) 31.7293 1.14419 0.572094 0.820188i \(-0.306131\pi\)
0.572094 + 0.820188i \(0.306131\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 40.5388 1.45902
\(773\) −6.37312 −0.229225 −0.114613 0.993410i \(-0.536563\pi\)
−0.114613 + 0.993410i \(0.536563\pi\)
\(774\) 0 0
\(775\) 11.9171 0.428073
\(776\) −111.629 −4.00724
\(777\) 0 0
\(778\) 37.1978 1.33360
\(779\) −20.3855 −0.730386
\(780\) 0 0
\(781\) −3.86398 −0.138264
\(782\) 29.9629 1.07147
\(783\) 0 0
\(784\) 0 0
\(785\) 0.300372 0.0107207
\(786\) 0 0
\(787\) −19.8799 −0.708643 −0.354321 0.935124i \(-0.615288\pi\)
−0.354321 + 0.935124i \(0.615288\pi\)
\(788\) 97.2297 3.46366
\(789\) 0 0
\(790\) −7.14468 −0.254196
\(791\) 0 0
\(792\) 0 0
\(793\) 21.2953 0.756217
\(794\) 22.6414 0.803515
\(795\) 0 0
\(796\) 31.4858 1.11598
\(797\) 5.81955 0.206139 0.103070 0.994674i \(-0.467134\pi\)
0.103070 + 0.994674i \(0.467134\pi\)
\(798\) 0 0
\(799\) −18.8640 −0.667360
\(800\) −45.5461 −1.61030
\(801\) 0 0
\(802\) 74.5548 2.63262
\(803\) −3.50680 −0.123752
\(804\) 0 0
\(805\) 0 0
\(806\) −36.5068 −1.28590
\(807\) 0 0
\(808\) −129.678 −4.56207
\(809\) −2.12227 −0.0746149 −0.0373075 0.999304i \(-0.511878\pi\)
−0.0373075 + 0.999304i \(0.511878\pi\)
\(810\) 0 0
\(811\) −18.4327 −0.647259 −0.323629 0.946184i \(-0.604903\pi\)
−0.323629 + 0.946184i \(0.604903\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −3.63045 −0.127247
\(815\) 14.2247 0.498270
\(816\) 0 0
\(817\) 20.8182 0.728336
\(818\) 82.4239 2.88188
\(819\) 0 0
\(820\) −74.8355 −2.61337
\(821\) 45.1555 1.57594 0.787969 0.615714i \(-0.211132\pi\)
0.787969 + 0.615714i \(0.211132\pi\)
\(822\) 0 0
\(823\) 25.0517 0.873248 0.436624 0.899644i \(-0.356174\pi\)
0.436624 + 0.899644i \(0.356174\pi\)
\(824\) 81.6427 2.84416
\(825\) 0 0
\(826\) 0 0
\(827\) 14.2953 0.497095 0.248548 0.968620i \(-0.420047\pi\)
0.248548 + 0.968620i \(0.420047\pi\)
\(828\) 0 0
\(829\) 31.8255 1.10534 0.552672 0.833399i \(-0.313608\pi\)
0.552672 + 0.833399i \(0.313608\pi\)
\(830\) 23.2953 0.808591
\(831\) 0 0
\(832\) 64.2682 2.22810
\(833\) 0 0
\(834\) 0 0
\(835\) 6.89521 0.238619
\(836\) −3.63416 −0.125690
\(837\) 0 0
\(838\) 56.6922 1.95840
\(839\) −16.4785 −0.568900 −0.284450 0.958691i \(-0.591811\pi\)
−0.284450 + 0.958691i \(0.591811\pi\)
\(840\) 0 0
\(841\) −22.6350 −0.780516
\(842\) 93.7563 3.23105
\(843\) 0 0
\(844\) 35.2967 1.21496
\(845\) 8.09888 0.278610
\(846\) 0 0
\(847\) 0 0
\(848\) 26.9395 0.925106
\(849\) 0 0
\(850\) −39.2967 −1.34786
\(851\) −8.45606 −0.289870
\(852\) 0 0
\(853\) −26.9986 −0.924415 −0.462208 0.886772i \(-0.652943\pi\)
−0.462208 + 0.886772i \(0.652943\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 111.048 3.79554
\(857\) 39.1630 1.33778 0.668891 0.743361i \(-0.266769\pi\)
0.668891 + 0.743361i \(0.266769\pi\)
\(858\) 0 0
\(859\) 9.46844 0.323059 0.161529 0.986868i \(-0.448357\pi\)
0.161529 + 0.986868i \(0.448357\pi\)
\(860\) 76.4239 2.60603
\(861\) 0 0
\(862\) 85.3199 2.90601
\(863\) −2.01238 −0.0685022 −0.0342511 0.999413i \(-0.510905\pi\)
−0.0342511 + 0.999413i \(0.510905\pi\)
\(864\) 0 0
\(865\) 10.0073 0.340258
\(866\) −17.5091 −0.594985
\(867\) 0 0
\(868\) 0 0
\(869\) 0.500482 0.0169777
\(870\) 0 0
\(871\) 2.00138 0.0678141
\(872\) −27.0356 −0.915541
\(873\) 0 0
\(874\) −11.6662 −0.394615
\(875\) 0 0
\(876\) 0 0
\(877\) 44.6995 1.50939 0.754697 0.656073i \(-0.227784\pi\)
0.754697 + 0.656073i \(0.227784\pi\)
\(878\) −42.0494 −1.41910
\(879\) 0 0
\(880\) −6.38688 −0.215302
\(881\) −12.7047 −0.428033 −0.214017 0.976830i \(-0.568655\pi\)
−0.214017 + 0.976830i \(0.568655\pi\)
\(882\) 0 0
\(883\) −17.0014 −0.572142 −0.286071 0.958208i \(-0.592349\pi\)
−0.286071 + 0.958208i \(0.592349\pi\)
\(884\) 87.3460 2.93776
\(885\) 0 0
\(886\) −67.6908 −2.27412
\(887\) −1.20149 −0.0403420 −0.0201710 0.999797i \(-0.506421\pi\)
−0.0201710 + 0.999797i \(0.506421\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 40.1075 1.34441
\(891\) 0 0
\(892\) 33.5316 1.12272
\(893\) 7.34479 0.245784
\(894\) 0 0
\(895\) 22.1447 0.740215
\(896\) 0 0
\(897\) 0 0
\(898\) −41.2953 −1.37804
\(899\) −12.1374 −0.404805
\(900\) 0 0
\(901\) 11.8255 0.393963
\(902\) 7.22487 0.240562
\(903\) 0 0
\(904\) 88.3570 2.93871
\(905\) −29.1323 −0.968391
\(906\) 0 0
\(907\) 17.0631 0.566572 0.283286 0.959036i \(-0.408575\pi\)
0.283286 + 0.959036i \(0.408575\pi\)
\(908\) −143.233 −4.75336
\(909\) 0 0
\(910\) 0 0
\(911\) −22.0014 −0.728938 −0.364469 0.931215i \(-0.618750\pi\)
−0.364469 + 0.931215i \(0.618750\pi\)
\(912\) 0 0
\(913\) −1.63182 −0.0540055
\(914\) 55.1803 1.82520
\(915\) 0 0
\(916\) 43.3425 1.43207
\(917\) 0 0
\(918\) 0 0
\(919\) −29.5906 −0.976102 −0.488051 0.872815i \(-0.662292\pi\)
−0.488051 + 0.872815i \(0.662292\pi\)
\(920\) −26.6291 −0.877934
\(921\) 0 0
\(922\) −110.194 −3.62904
\(923\) −36.1593 −1.19020
\(924\) 0 0
\(925\) 11.0902 0.364644
\(926\) 26.7490 0.879026
\(927\) 0 0
\(928\) 46.3883 1.52277
\(929\) 2.69963 0.0885719 0.0442860 0.999019i \(-0.485899\pi\)
0.0442860 + 0.999019i \(0.485899\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 144.998 4.74958
\(933\) 0 0
\(934\) −47.8886 −1.56696
\(935\) −2.80361 −0.0916878
\(936\) 0 0
\(937\) 39.4472 1.28869 0.644343 0.764737i \(-0.277131\pi\)
0.644343 + 0.764737i \(0.277131\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 26.9629 0.879432
\(941\) 39.7244 1.29498 0.647489 0.762075i \(-0.275819\pi\)
0.647489 + 0.762075i \(0.275819\pi\)
\(942\) 0 0
\(943\) 16.8282 0.548002
\(944\) −65.4449 −2.13005
\(945\) 0 0
\(946\) −7.37822 −0.239886
\(947\) −15.7303 −0.511166 −0.255583 0.966787i \(-0.582267\pi\)
−0.255583 + 0.966787i \(0.582267\pi\)
\(948\) 0 0
\(949\) −32.8168 −1.06528
\(950\) 15.3004 0.496410
\(951\) 0 0
\(952\) 0 0
\(953\) 30.2064 0.978482 0.489241 0.872149i \(-0.337274\pi\)
0.489241 + 0.872149i \(0.337274\pi\)
\(954\) 0 0
\(955\) −12.0843 −0.391039
\(956\) −147.344 −4.76546
\(957\) 0 0
\(958\) −63.7366 −2.05924
\(959\) 0 0
\(960\) 0 0
\(961\) −7.85532 −0.253397
\(962\) −33.9739 −1.09536
\(963\) 0 0
\(964\) −119.139 −3.83722
\(965\) −12.1767 −0.391983
\(966\) 0 0
\(967\) −6.18911 −0.199028 −0.0995141 0.995036i \(-0.531729\pi\)
−0.0995141 + 0.995036i \(0.531729\pi\)
\(968\) −96.8391 −3.11253
\(969\) 0 0
\(970\) 53.9257 1.73145
\(971\) −2.17163 −0.0696910 −0.0348455 0.999393i \(-0.511094\pi\)
−0.0348455 + 0.999393i \(0.511094\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 94.1432 3.01654
\(975\) 0 0
\(976\) 101.419 3.24634
\(977\) −11.4561 −0.366512 −0.183256 0.983065i \(-0.558664\pi\)
−0.183256 + 0.983065i \(0.558664\pi\)
\(978\) 0 0
\(979\) −2.80951 −0.0897925
\(980\) 0 0
\(981\) 0 0
\(982\) 62.6450 1.99908
\(983\) 44.8021 1.42896 0.714482 0.699654i \(-0.246662\pi\)
0.714482 + 0.699654i \(0.246662\pi\)
\(984\) 0 0
\(985\) −29.2051 −0.930550
\(986\) 40.0232 1.27460
\(987\) 0 0
\(988\) −34.0087 −1.08196
\(989\) −17.1854 −0.546464
\(990\) 0 0
\(991\) −24.2953 −0.771765 −0.385882 0.922548i \(-0.626103\pi\)
−0.385882 + 0.922548i \(0.626103\pi\)
\(992\) −88.4573 −2.80852
\(993\) 0 0
\(994\) 0 0
\(995\) −9.45744 −0.299821
\(996\) 0 0
\(997\) −11.3251 −0.358671 −0.179335 0.983788i \(-0.557395\pi\)
−0.179335 + 0.983788i \(0.557395\pi\)
\(998\) −14.3759 −0.455060
\(999\) 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 1323.2.a.ba.1.3 3
3.2 odd 2 1323.2.a.x.1.1 3
7.2 even 3 189.2.e.e.109.1 6
7.4 even 3 189.2.e.e.163.1 yes 6
7.6 odd 2 1323.2.a.z.1.3 3
21.2 odd 6 189.2.e.f.109.3 yes 6
21.11 odd 6 189.2.e.f.163.3 yes 6
21.20 even 2 1323.2.a.y.1.1 3
63.2 odd 6 567.2.g.i.109.3 6
63.4 even 3 567.2.g.h.541.1 6
63.11 odd 6 567.2.h.h.352.1 6
63.16 even 3 567.2.g.h.109.1 6
63.23 odd 6 567.2.h.h.298.1 6
63.25 even 3 567.2.h.i.352.3 6
63.32 odd 6 567.2.g.i.541.3 6
63.58 even 3 567.2.h.i.298.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.e.e.109.1 6 7.2 even 3
189.2.e.e.163.1 yes 6 7.4 even 3
189.2.e.f.109.3 yes 6 21.2 odd 6
189.2.e.f.163.3 yes 6 21.11 odd 6
567.2.g.h.109.1 6 63.16 even 3
567.2.g.h.541.1 6 63.4 even 3
567.2.g.i.109.3 6 63.2 odd 6
567.2.g.i.541.3 6 63.32 odd 6
567.2.h.h.298.1 6 63.23 odd 6
567.2.h.h.352.1 6 63.11 odd 6
567.2.h.i.298.3 6 63.58 even 3
567.2.h.i.352.3 6 63.25 even 3
1323.2.a.x.1.1 3 3.2 odd 2
1323.2.a.y.1.1 3 21.20 even 2
1323.2.a.z.1.3 3 7.6 odd 2
1323.2.a.ba.1.3 3 1.1 even 1 trivial