Properties

Label 1336.2.a.d.1.8
Level $1336$
Weight $2$
Character 1336.1
Self dual yes
Analytic conductor $10.668$
Analytic rank $0$
Dimension $12$
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] = [1336,2,Mod(1,1336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1336, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1336.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1336 = 2^{3} \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1336.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: \(10.6680137100\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} - 25 x^{10} + 20 x^{9} + 224 x^{8} - 135 x^{7} - 865 x^{6} + 330 x^{5} + 1341 x^{4} - 127 x^{3} - 652 x^{2} - 48 x + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(-0.571891\) of defining polynomial
Character \(\chi\) \(=\) 1336.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.571891 q^{3} -2.42234 q^{5} -3.89487 q^{7} -2.67294 q^{9} +O(q^{10})\) \(q+0.571891 q^{3} -2.42234 q^{5} -3.89487 q^{7} -2.67294 q^{9} +1.06107 q^{11} +5.87406 q^{13} -1.38532 q^{15} +4.69476 q^{17} -4.56178 q^{19} -2.22744 q^{21} +6.88976 q^{23} +0.867754 q^{25} -3.24430 q^{27} -4.73271 q^{29} +5.09371 q^{31} +0.606816 q^{33} +9.43473 q^{35} +7.53085 q^{37} +3.35932 q^{39} -4.59931 q^{41} +7.94427 q^{43} +6.47478 q^{45} +10.0221 q^{47} +8.17004 q^{49} +2.68489 q^{51} +7.90356 q^{53} -2.57028 q^{55} -2.60884 q^{57} -7.93731 q^{59} +12.9263 q^{61} +10.4108 q^{63} -14.2290 q^{65} -6.91109 q^{67} +3.94019 q^{69} -5.67998 q^{71} +14.1491 q^{73} +0.496260 q^{75} -4.13273 q^{77} -5.05349 q^{79} +6.16344 q^{81} -10.2582 q^{83} -11.3723 q^{85} -2.70659 q^{87} +12.7067 q^{89} -22.8787 q^{91} +2.91305 q^{93} +11.0502 q^{95} -10.4437 q^{97} -2.83618 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - q^{3} + 8 q^{5} - 4 q^{7} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - q^{3} + 8 q^{5} - 4 q^{7} + 15 q^{9} + 6 q^{11} + 13 q^{13} + 4 q^{15} + 10 q^{17} - q^{19} + 13 q^{21} + 3 q^{23} + 18 q^{25} - 10 q^{27} + 29 q^{29} - 3 q^{31} + 3 q^{35} + 41 q^{37} + 10 q^{39} + 20 q^{41} - q^{43} + 42 q^{45} + 5 q^{47} + 20 q^{49} + 14 q^{51} + 39 q^{53} + 3 q^{55} + 3 q^{57} + 8 q^{59} + 30 q^{61} - 2 q^{63} + 21 q^{65} - 9 q^{67} + 33 q^{69} + 29 q^{71} + 12 q^{73} + q^{75} + 19 q^{77} - 2 q^{79} + 24 q^{81} - 5 q^{83} + 44 q^{85} - 2 q^{87} + 7 q^{89} - 4 q^{91} + 37 q^{93} + 18 q^{95} - 14 q^{97} + 5 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.571891 0.330181 0.165091 0.986278i \(-0.447208\pi\)
0.165091 + 0.986278i \(0.447208\pi\)
\(4\) 0 0
\(5\) −2.42234 −1.08331 −0.541653 0.840602i \(-0.682201\pi\)
−0.541653 + 0.840602i \(0.682201\pi\)
\(6\) 0 0
\(7\) −3.89487 −1.47212 −0.736062 0.676914i \(-0.763317\pi\)
−0.736062 + 0.676914i \(0.763317\pi\)
\(8\) 0 0
\(9\) −2.67294 −0.890980
\(10\) 0 0
\(11\) 1.06107 0.319924 0.159962 0.987123i \(-0.448863\pi\)
0.159962 + 0.987123i \(0.448863\pi\)
\(12\) 0 0
\(13\) 5.87406 1.62917 0.814585 0.580044i \(-0.196965\pi\)
0.814585 + 0.580044i \(0.196965\pi\)
\(14\) 0 0
\(15\) −1.38532 −0.357687
\(16\) 0 0
\(17\) 4.69476 1.13865 0.569324 0.822114i \(-0.307205\pi\)
0.569324 + 0.822114i \(0.307205\pi\)
\(18\) 0 0
\(19\) −4.56178 −1.04654 −0.523272 0.852166i \(-0.675289\pi\)
−0.523272 + 0.852166i \(0.675289\pi\)
\(20\) 0 0
\(21\) −2.22744 −0.486068
\(22\) 0 0
\(23\) 6.88976 1.43661 0.718307 0.695726i \(-0.244917\pi\)
0.718307 + 0.695726i \(0.244917\pi\)
\(24\) 0 0
\(25\) 0.867754 0.173551
\(26\) 0 0
\(27\) −3.24430 −0.624366
\(28\) 0 0
\(29\) −4.73271 −0.878841 −0.439421 0.898281i \(-0.644816\pi\)
−0.439421 + 0.898281i \(0.644816\pi\)
\(30\) 0 0
\(31\) 5.09371 0.914858 0.457429 0.889246i \(-0.348770\pi\)
0.457429 + 0.889246i \(0.348770\pi\)
\(32\) 0 0
\(33\) 0.606816 0.105633
\(34\) 0 0
\(35\) 9.43473 1.59476
\(36\) 0 0
\(37\) 7.53085 1.23806 0.619032 0.785366i \(-0.287525\pi\)
0.619032 + 0.785366i \(0.287525\pi\)
\(38\) 0 0
\(39\) 3.35932 0.537922
\(40\) 0 0
\(41\) −4.59931 −0.718291 −0.359145 0.933282i \(-0.616932\pi\)
−0.359145 + 0.933282i \(0.616932\pi\)
\(42\) 0 0
\(43\) 7.94427 1.21149 0.605745 0.795659i \(-0.292875\pi\)
0.605745 + 0.795659i \(0.292875\pi\)
\(44\) 0 0
\(45\) 6.47478 0.965204
\(46\) 0 0
\(47\) 10.0221 1.46188 0.730940 0.682442i \(-0.239082\pi\)
0.730940 + 0.682442i \(0.239082\pi\)
\(48\) 0 0
\(49\) 8.17004 1.16715
\(50\) 0 0
\(51\) 2.68489 0.375960
\(52\) 0 0
\(53\) 7.90356 1.08564 0.542819 0.839850i \(-0.317357\pi\)
0.542819 + 0.839850i \(0.317357\pi\)
\(54\) 0 0
\(55\) −2.57028 −0.346576
\(56\) 0 0
\(57\) −2.60884 −0.345549
\(58\) 0 0
\(59\) −7.93731 −1.03335 −0.516675 0.856182i \(-0.672830\pi\)
−0.516675 + 0.856182i \(0.672830\pi\)
\(60\) 0 0
\(61\) 12.9263 1.65504 0.827518 0.561439i \(-0.189752\pi\)
0.827518 + 0.561439i \(0.189752\pi\)
\(62\) 0 0
\(63\) 10.4108 1.31163
\(64\) 0 0
\(65\) −14.2290 −1.76489
\(66\) 0 0
\(67\) −6.91109 −0.844324 −0.422162 0.906520i \(-0.638729\pi\)
−0.422162 + 0.906520i \(0.638729\pi\)
\(68\) 0 0
\(69\) 3.94019 0.474343
\(70\) 0 0
\(71\) −5.67998 −0.674090 −0.337045 0.941489i \(-0.609427\pi\)
−0.337045 + 0.941489i \(0.609427\pi\)
\(72\) 0 0
\(73\) 14.1491 1.65603 0.828016 0.560705i \(-0.189470\pi\)
0.828016 + 0.560705i \(0.189470\pi\)
\(74\) 0 0
\(75\) 0.496260 0.0573032
\(76\) 0 0
\(77\) −4.13273 −0.470968
\(78\) 0 0
\(79\) −5.05349 −0.568562 −0.284281 0.958741i \(-0.591755\pi\)
−0.284281 + 0.958741i \(0.591755\pi\)
\(80\) 0 0
\(81\) 6.16344 0.684826
\(82\) 0 0
\(83\) −10.2582 −1.12599 −0.562994 0.826461i \(-0.690351\pi\)
−0.562994 + 0.826461i \(0.690351\pi\)
\(84\) 0 0
\(85\) −11.3723 −1.23350
\(86\) 0 0
\(87\) −2.70659 −0.290177
\(88\) 0 0
\(89\) 12.7067 1.34691 0.673453 0.739230i \(-0.264810\pi\)
0.673453 + 0.739230i \(0.264810\pi\)
\(90\) 0 0
\(91\) −22.8787 −2.39834
\(92\) 0 0
\(93\) 2.91305 0.302069
\(94\) 0 0
\(95\) 11.0502 1.13373
\(96\) 0 0
\(97\) −10.4437 −1.06039 −0.530197 0.847875i \(-0.677882\pi\)
−0.530197 + 0.847875i \(0.677882\pi\)
\(98\) 0 0
\(99\) −2.83618 −0.285046
\(100\) 0 0
\(101\) 13.1024 1.30374 0.651871 0.758330i \(-0.273984\pi\)
0.651871 + 0.758330i \(0.273984\pi\)
\(102\) 0 0
\(103\) −12.0031 −1.18270 −0.591352 0.806413i \(-0.701406\pi\)
−0.591352 + 0.806413i \(0.701406\pi\)
\(104\) 0 0
\(105\) 5.39563 0.526560
\(106\) 0 0
\(107\) 9.54799 0.923039 0.461519 0.887130i \(-0.347304\pi\)
0.461519 + 0.887130i \(0.347304\pi\)
\(108\) 0 0
\(109\) −16.3174 −1.56292 −0.781461 0.623954i \(-0.785525\pi\)
−0.781461 + 0.623954i \(0.785525\pi\)
\(110\) 0 0
\(111\) 4.30682 0.408786
\(112\) 0 0
\(113\) 8.43572 0.793566 0.396783 0.917912i \(-0.370127\pi\)
0.396783 + 0.917912i \(0.370127\pi\)
\(114\) 0 0
\(115\) −16.6894 −1.55629
\(116\) 0 0
\(117\) −15.7010 −1.45156
\(118\) 0 0
\(119\) −18.2855 −1.67623
\(120\) 0 0
\(121\) −9.87413 −0.897648
\(122\) 0 0
\(123\) −2.63030 −0.237166
\(124\) 0 0
\(125\) 10.0097 0.895297
\(126\) 0 0
\(127\) 16.8782 1.49770 0.748849 0.662741i \(-0.230607\pi\)
0.748849 + 0.662741i \(0.230607\pi\)
\(128\) 0 0
\(129\) 4.54325 0.400011
\(130\) 0 0
\(131\) 10.0754 0.880292 0.440146 0.897926i \(-0.354927\pi\)
0.440146 + 0.897926i \(0.354927\pi\)
\(132\) 0 0
\(133\) 17.7676 1.54064
\(134\) 0 0
\(135\) 7.85882 0.676379
\(136\) 0 0
\(137\) −11.6192 −0.992696 −0.496348 0.868124i \(-0.665326\pi\)
−0.496348 + 0.868124i \(0.665326\pi\)
\(138\) 0 0
\(139\) −19.7902 −1.67859 −0.839293 0.543679i \(-0.817031\pi\)
−0.839293 + 0.543679i \(0.817031\pi\)
\(140\) 0 0
\(141\) 5.73157 0.482685
\(142\) 0 0
\(143\) 6.23278 0.521212
\(144\) 0 0
\(145\) 11.4642 0.952054
\(146\) 0 0
\(147\) 4.67237 0.385371
\(148\) 0 0
\(149\) 18.7206 1.53365 0.766825 0.641857i \(-0.221836\pi\)
0.766825 + 0.641857i \(0.221836\pi\)
\(150\) 0 0
\(151\) −7.14926 −0.581799 −0.290899 0.956754i \(-0.593954\pi\)
−0.290899 + 0.956754i \(0.593954\pi\)
\(152\) 0 0
\(153\) −12.5488 −1.01451
\(154\) 0 0
\(155\) −12.3387 −0.991071
\(156\) 0 0
\(157\) −4.73863 −0.378184 −0.189092 0.981959i \(-0.560554\pi\)
−0.189092 + 0.981959i \(0.560554\pi\)
\(158\) 0 0
\(159\) 4.51997 0.358457
\(160\) 0 0
\(161\) −26.8347 −2.11487
\(162\) 0 0
\(163\) 3.04789 0.238729 0.119365 0.992850i \(-0.461914\pi\)
0.119365 + 0.992850i \(0.461914\pi\)
\(164\) 0 0
\(165\) −1.46992 −0.114433
\(166\) 0 0
\(167\) −1.00000 −0.0773823
\(168\) 0 0
\(169\) 21.5046 1.65420
\(170\) 0 0
\(171\) 12.1934 0.932451
\(172\) 0 0
\(173\) 18.6556 1.41836 0.709178 0.705030i \(-0.249066\pi\)
0.709178 + 0.705030i \(0.249066\pi\)
\(174\) 0 0
\(175\) −3.37979 −0.255488
\(176\) 0 0
\(177\) −4.53927 −0.341193
\(178\) 0 0
\(179\) 3.44709 0.257648 0.128824 0.991667i \(-0.458880\pi\)
0.128824 + 0.991667i \(0.458880\pi\)
\(180\) 0 0
\(181\) 14.9657 1.11239 0.556195 0.831052i \(-0.312261\pi\)
0.556195 + 0.831052i \(0.312261\pi\)
\(182\) 0 0
\(183\) 7.39240 0.546462
\(184\) 0 0
\(185\) −18.2423 −1.34120
\(186\) 0 0
\(187\) 4.98147 0.364281
\(188\) 0 0
\(189\) 12.6361 0.919144
\(190\) 0 0
\(191\) 4.37740 0.316737 0.158369 0.987380i \(-0.449377\pi\)
0.158369 + 0.987380i \(0.449377\pi\)
\(192\) 0 0
\(193\) −9.36494 −0.674103 −0.337052 0.941486i \(-0.609430\pi\)
−0.337052 + 0.941486i \(0.609430\pi\)
\(194\) 0 0
\(195\) −8.13743 −0.582734
\(196\) 0 0
\(197\) −3.30685 −0.235603 −0.117802 0.993037i \(-0.537585\pi\)
−0.117802 + 0.993037i \(0.537585\pi\)
\(198\) 0 0
\(199\) −16.1610 −1.14562 −0.572810 0.819688i \(-0.694147\pi\)
−0.572810 + 0.819688i \(0.694147\pi\)
\(200\) 0 0
\(201\) −3.95239 −0.278780
\(202\) 0 0
\(203\) 18.4333 1.29376
\(204\) 0 0
\(205\) 11.1411 0.778128
\(206\) 0 0
\(207\) −18.4159 −1.28000
\(208\) 0 0
\(209\) −4.84037 −0.334815
\(210\) 0 0
\(211\) 6.54669 0.450693 0.225346 0.974279i \(-0.427649\pi\)
0.225346 + 0.974279i \(0.427649\pi\)
\(212\) 0 0
\(213\) −3.24833 −0.222572
\(214\) 0 0
\(215\) −19.2438 −1.31241
\(216\) 0 0
\(217\) −19.8394 −1.34678
\(218\) 0 0
\(219\) 8.09176 0.546791
\(220\) 0 0
\(221\) 27.5773 1.85505
\(222\) 0 0
\(223\) −20.5139 −1.37371 −0.686857 0.726793i \(-0.741010\pi\)
−0.686857 + 0.726793i \(0.741010\pi\)
\(224\) 0 0
\(225\) −2.31945 −0.154630
\(226\) 0 0
\(227\) −16.3625 −1.08602 −0.543009 0.839727i \(-0.682715\pi\)
−0.543009 + 0.839727i \(0.682715\pi\)
\(228\) 0 0
\(229\) −7.74442 −0.511766 −0.255883 0.966708i \(-0.582366\pi\)
−0.255883 + 0.966708i \(0.582366\pi\)
\(230\) 0 0
\(231\) −2.36347 −0.155505
\(232\) 0 0
\(233\) −7.83490 −0.513281 −0.256641 0.966507i \(-0.582616\pi\)
−0.256641 + 0.966507i \(0.582616\pi\)
\(234\) 0 0
\(235\) −24.2771 −1.58366
\(236\) 0 0
\(237\) −2.89004 −0.187728
\(238\) 0 0
\(239\) 6.60183 0.427037 0.213518 0.976939i \(-0.431508\pi\)
0.213518 + 0.976939i \(0.431508\pi\)
\(240\) 0 0
\(241\) 20.3394 1.31018 0.655089 0.755552i \(-0.272631\pi\)
0.655089 + 0.755552i \(0.272631\pi\)
\(242\) 0 0
\(243\) 13.2577 0.850483
\(244\) 0 0
\(245\) −19.7906 −1.26438
\(246\) 0 0
\(247\) −26.7962 −1.70500
\(248\) 0 0
\(249\) −5.86659 −0.371780
\(250\) 0 0
\(251\) 21.8497 1.37914 0.689572 0.724217i \(-0.257799\pi\)
0.689572 + 0.724217i \(0.257799\pi\)
\(252\) 0 0
\(253\) 7.31052 0.459608
\(254\) 0 0
\(255\) −6.50373 −0.407279
\(256\) 0 0
\(257\) −0.922785 −0.0575617 −0.0287809 0.999586i \(-0.509163\pi\)
−0.0287809 + 0.999586i \(0.509163\pi\)
\(258\) 0 0
\(259\) −29.3317 −1.82258
\(260\) 0 0
\(261\) 12.6502 0.783030
\(262\) 0 0
\(263\) −10.4243 −0.642790 −0.321395 0.946945i \(-0.604152\pi\)
−0.321395 + 0.946945i \(0.604152\pi\)
\(264\) 0 0
\(265\) −19.1452 −1.17608
\(266\) 0 0
\(267\) 7.26684 0.444723
\(268\) 0 0
\(269\) 21.1738 1.29099 0.645495 0.763764i \(-0.276651\pi\)
0.645495 + 0.763764i \(0.276651\pi\)
\(270\) 0 0
\(271\) 6.48917 0.394189 0.197095 0.980384i \(-0.436849\pi\)
0.197095 + 0.980384i \(0.436849\pi\)
\(272\) 0 0
\(273\) −13.0841 −0.791887
\(274\) 0 0
\(275\) 0.920747 0.0555231
\(276\) 0 0
\(277\) 3.08812 0.185547 0.0927736 0.995687i \(-0.470427\pi\)
0.0927736 + 0.995687i \(0.470427\pi\)
\(278\) 0 0
\(279\) −13.6152 −0.815121
\(280\) 0 0
\(281\) 30.5330 1.82145 0.910724 0.413016i \(-0.135525\pi\)
0.910724 + 0.413016i \(0.135525\pi\)
\(282\) 0 0
\(283\) 15.5616 0.925039 0.462519 0.886609i \(-0.346946\pi\)
0.462519 + 0.886609i \(0.346946\pi\)
\(284\) 0 0
\(285\) 6.31951 0.374336
\(286\) 0 0
\(287\) 17.9137 1.05741
\(288\) 0 0
\(289\) 5.04079 0.296517
\(290\) 0 0
\(291\) −5.97264 −0.350122
\(292\) 0 0
\(293\) −12.5838 −0.735151 −0.367575 0.929994i \(-0.619812\pi\)
−0.367575 + 0.929994i \(0.619812\pi\)
\(294\) 0 0
\(295\) 19.2269 1.11943
\(296\) 0 0
\(297\) −3.44243 −0.199750
\(298\) 0 0
\(299\) 40.4709 2.34049
\(300\) 0 0
\(301\) −30.9419 −1.78346
\(302\) 0 0
\(303\) 7.49317 0.430471
\(304\) 0 0
\(305\) −31.3118 −1.79291
\(306\) 0 0
\(307\) 20.5504 1.17287 0.586435 0.809996i \(-0.300531\pi\)
0.586435 + 0.809996i \(0.300531\pi\)
\(308\) 0 0
\(309\) −6.86449 −0.390507
\(310\) 0 0
\(311\) −32.9297 −1.86727 −0.933636 0.358222i \(-0.883383\pi\)
−0.933636 + 0.358222i \(0.883383\pi\)
\(312\) 0 0
\(313\) 9.25293 0.523006 0.261503 0.965203i \(-0.415782\pi\)
0.261503 + 0.965203i \(0.415782\pi\)
\(314\) 0 0
\(315\) −25.2185 −1.42090
\(316\) 0 0
\(317\) 22.7853 1.27975 0.639875 0.768479i \(-0.278986\pi\)
0.639875 + 0.768479i \(0.278986\pi\)
\(318\) 0 0
\(319\) −5.02173 −0.281163
\(320\) 0 0
\(321\) 5.46041 0.304770
\(322\) 0 0
\(323\) −21.4165 −1.19164
\(324\) 0 0
\(325\) 5.09724 0.282744
\(326\) 0 0
\(327\) −9.33176 −0.516048
\(328\) 0 0
\(329\) −39.0350 −2.15207
\(330\) 0 0
\(331\) 11.6285 0.639162 0.319581 0.947559i \(-0.396458\pi\)
0.319581 + 0.947559i \(0.396458\pi\)
\(332\) 0 0
\(333\) −20.1295 −1.10309
\(334\) 0 0
\(335\) 16.7410 0.914660
\(336\) 0 0
\(337\) −14.5453 −0.792331 −0.396165 0.918179i \(-0.629659\pi\)
−0.396165 + 0.918179i \(0.629659\pi\)
\(338\) 0 0
\(339\) 4.82431 0.262021
\(340\) 0 0
\(341\) 5.40478 0.292685
\(342\) 0 0
\(343\) −4.55715 −0.246063
\(344\) 0 0
\(345\) −9.54450 −0.513859
\(346\) 0 0
\(347\) 6.30707 0.338581 0.169291 0.985566i \(-0.445852\pi\)
0.169291 + 0.985566i \(0.445852\pi\)
\(348\) 0 0
\(349\) −19.5992 −1.04912 −0.524561 0.851373i \(-0.675770\pi\)
−0.524561 + 0.851373i \(0.675770\pi\)
\(350\) 0 0
\(351\) −19.0572 −1.01720
\(352\) 0 0
\(353\) −8.14946 −0.433752 −0.216876 0.976199i \(-0.569587\pi\)
−0.216876 + 0.976199i \(0.569587\pi\)
\(354\) 0 0
\(355\) 13.7589 0.730245
\(356\) 0 0
\(357\) −10.4573 −0.553460
\(358\) 0 0
\(359\) 24.0437 1.26898 0.634490 0.772931i \(-0.281210\pi\)
0.634490 + 0.772931i \(0.281210\pi\)
\(360\) 0 0
\(361\) 1.80985 0.0952554
\(362\) 0 0
\(363\) −5.64692 −0.296387
\(364\) 0 0
\(365\) −34.2741 −1.79399
\(366\) 0 0
\(367\) −11.6955 −0.610499 −0.305250 0.952272i \(-0.598740\pi\)
−0.305250 + 0.952272i \(0.598740\pi\)
\(368\) 0 0
\(369\) 12.2937 0.639983
\(370\) 0 0
\(371\) −30.7834 −1.59819
\(372\) 0 0
\(373\) −4.35308 −0.225394 −0.112697 0.993629i \(-0.535949\pi\)
−0.112697 + 0.993629i \(0.535949\pi\)
\(374\) 0 0
\(375\) 5.72447 0.295610
\(376\) 0 0
\(377\) −27.8002 −1.43178
\(378\) 0 0
\(379\) 16.7673 0.861277 0.430638 0.902525i \(-0.358288\pi\)
0.430638 + 0.902525i \(0.358288\pi\)
\(380\) 0 0
\(381\) 9.65248 0.494512
\(382\) 0 0
\(383\) 7.78673 0.397883 0.198942 0.980011i \(-0.436250\pi\)
0.198942 + 0.980011i \(0.436250\pi\)
\(384\) 0 0
\(385\) 10.0109 0.510203
\(386\) 0 0
\(387\) −21.2346 −1.07941
\(388\) 0 0
\(389\) −5.32989 −0.270236 −0.135118 0.990830i \(-0.543141\pi\)
−0.135118 + 0.990830i \(0.543141\pi\)
\(390\) 0 0
\(391\) 32.3458 1.63580
\(392\) 0 0
\(393\) 5.76203 0.290656
\(394\) 0 0
\(395\) 12.2413 0.615926
\(396\) 0 0
\(397\) 19.4029 0.973805 0.486903 0.873456i \(-0.338127\pi\)
0.486903 + 0.873456i \(0.338127\pi\)
\(398\) 0 0
\(399\) 10.1611 0.508691
\(400\) 0 0
\(401\) 23.5610 1.17658 0.588291 0.808650i \(-0.299801\pi\)
0.588291 + 0.808650i \(0.299801\pi\)
\(402\) 0 0
\(403\) 29.9208 1.49046
\(404\) 0 0
\(405\) −14.9300 −0.741876
\(406\) 0 0
\(407\) 7.99076 0.396087
\(408\) 0 0
\(409\) −22.0973 −1.09264 −0.546321 0.837576i \(-0.683972\pi\)
−0.546321 + 0.837576i \(0.683972\pi\)
\(410\) 0 0
\(411\) −6.64492 −0.327770
\(412\) 0 0
\(413\) 30.9148 1.52122
\(414\) 0 0
\(415\) 24.8490 1.21979
\(416\) 0 0
\(417\) −11.3179 −0.554238
\(418\) 0 0
\(419\) 38.5514 1.88336 0.941679 0.336514i \(-0.109248\pi\)
0.941679 + 0.336514i \(0.109248\pi\)
\(420\) 0 0
\(421\) −27.3205 −1.33152 −0.665759 0.746167i \(-0.731892\pi\)
−0.665759 + 0.746167i \(0.731892\pi\)
\(422\) 0 0
\(423\) −26.7886 −1.30251
\(424\) 0 0
\(425\) 4.07390 0.197613
\(426\) 0 0
\(427\) −50.3461 −2.43642
\(428\) 0 0
\(429\) 3.56447 0.172094
\(430\) 0 0
\(431\) −4.18336 −0.201505 −0.100753 0.994912i \(-0.532125\pi\)
−0.100753 + 0.994912i \(0.532125\pi\)
\(432\) 0 0
\(433\) 1.43297 0.0688641 0.0344320 0.999407i \(-0.489038\pi\)
0.0344320 + 0.999407i \(0.489038\pi\)
\(434\) 0 0
\(435\) 6.55630 0.314350
\(436\) 0 0
\(437\) −31.4296 −1.50348
\(438\) 0 0
\(439\) −24.5235 −1.17044 −0.585221 0.810874i \(-0.698992\pi\)
−0.585221 + 0.810874i \(0.698992\pi\)
\(440\) 0 0
\(441\) −21.8380 −1.03991
\(442\) 0 0
\(443\) −20.2089 −0.960156 −0.480078 0.877226i \(-0.659392\pi\)
−0.480078 + 0.877226i \(0.659392\pi\)
\(444\) 0 0
\(445\) −30.7800 −1.45911
\(446\) 0 0
\(447\) 10.7061 0.506382
\(448\) 0 0
\(449\) 5.98049 0.282237 0.141118 0.989993i \(-0.454930\pi\)
0.141118 + 0.989993i \(0.454930\pi\)
\(450\) 0 0
\(451\) −4.88018 −0.229799
\(452\) 0 0
\(453\) −4.08860 −0.192099
\(454\) 0 0
\(455\) 55.4201 2.59814
\(456\) 0 0
\(457\) −8.13632 −0.380601 −0.190300 0.981726i \(-0.560946\pi\)
−0.190300 + 0.981726i \(0.560946\pi\)
\(458\) 0 0
\(459\) −15.2312 −0.710933
\(460\) 0 0
\(461\) −11.6806 −0.544022 −0.272011 0.962294i \(-0.587689\pi\)
−0.272011 + 0.962294i \(0.587689\pi\)
\(462\) 0 0
\(463\) 3.62768 0.168592 0.0842962 0.996441i \(-0.473136\pi\)
0.0842962 + 0.996441i \(0.473136\pi\)
\(464\) 0 0
\(465\) −7.05641 −0.327233
\(466\) 0 0
\(467\) 15.3107 0.708496 0.354248 0.935152i \(-0.384737\pi\)
0.354248 + 0.935152i \(0.384737\pi\)
\(468\) 0 0
\(469\) 26.9178 1.24295
\(470\) 0 0
\(471\) −2.70998 −0.124869
\(472\) 0 0
\(473\) 8.42942 0.387585
\(474\) 0 0
\(475\) −3.95850 −0.181629
\(476\) 0 0
\(477\) −21.1258 −0.967282
\(478\) 0 0
\(479\) 36.8901 1.68555 0.842777 0.538264i \(-0.180920\pi\)
0.842777 + 0.538264i \(0.180920\pi\)
\(480\) 0 0
\(481\) 44.2367 2.01702
\(482\) 0 0
\(483\) −15.3465 −0.698292
\(484\) 0 0
\(485\) 25.2982 1.14873
\(486\) 0 0
\(487\) −13.4148 −0.607883 −0.303941 0.952691i \(-0.598303\pi\)
−0.303941 + 0.952691i \(0.598303\pi\)
\(488\) 0 0
\(489\) 1.74306 0.0788239
\(490\) 0 0
\(491\) −11.0176 −0.497218 −0.248609 0.968604i \(-0.579973\pi\)
−0.248609 + 0.968604i \(0.579973\pi\)
\(492\) 0 0
\(493\) −22.2189 −1.00069
\(494\) 0 0
\(495\) 6.87020 0.308792
\(496\) 0 0
\(497\) 22.1228 0.992343
\(498\) 0 0
\(499\) −29.2887 −1.31114 −0.655570 0.755134i \(-0.727572\pi\)
−0.655570 + 0.755134i \(0.727572\pi\)
\(500\) 0 0
\(501\) −0.571891 −0.0255502
\(502\) 0 0
\(503\) −44.1648 −1.96921 −0.984606 0.174788i \(-0.944076\pi\)
−0.984606 + 0.174788i \(0.944076\pi\)
\(504\) 0 0
\(505\) −31.7386 −1.41235
\(506\) 0 0
\(507\) 12.2983 0.546185
\(508\) 0 0
\(509\) 37.9725 1.68310 0.841550 0.540179i \(-0.181643\pi\)
0.841550 + 0.540179i \(0.181643\pi\)
\(510\) 0 0
\(511\) −55.1091 −2.43788
\(512\) 0 0
\(513\) 14.7998 0.653427
\(514\) 0 0
\(515\) 29.0757 1.28123
\(516\) 0 0
\(517\) 10.6342 0.467691
\(518\) 0 0
\(519\) 10.6689 0.468314
\(520\) 0 0
\(521\) 9.46873 0.414833 0.207416 0.978253i \(-0.433495\pi\)
0.207416 + 0.978253i \(0.433495\pi\)
\(522\) 0 0
\(523\) −17.0224 −0.744337 −0.372169 0.928165i \(-0.621386\pi\)
−0.372169 + 0.928165i \(0.621386\pi\)
\(524\) 0 0
\(525\) −1.93287 −0.0843574
\(526\) 0 0
\(527\) 23.9138 1.04170
\(528\) 0 0
\(529\) 24.4688 1.06386
\(530\) 0 0
\(531\) 21.2160 0.920694
\(532\) 0 0
\(533\) −27.0166 −1.17022
\(534\) 0 0
\(535\) −23.1285 −0.999933
\(536\) 0 0
\(537\) 1.97136 0.0850705
\(538\) 0 0
\(539\) 8.66898 0.373399
\(540\) 0 0
\(541\) −6.15422 −0.264591 −0.132295 0.991210i \(-0.542235\pi\)
−0.132295 + 0.991210i \(0.542235\pi\)
\(542\) 0 0
\(543\) 8.55873 0.367290
\(544\) 0 0
\(545\) 39.5263 1.69312
\(546\) 0 0
\(547\) 31.7274 1.35656 0.678282 0.734801i \(-0.262725\pi\)
0.678282 + 0.734801i \(0.262725\pi\)
\(548\) 0 0
\(549\) −34.5511 −1.47461
\(550\) 0 0
\(551\) 21.5896 0.919747
\(552\) 0 0
\(553\) 19.6827 0.836993
\(554\) 0 0
\(555\) −10.4326 −0.442840
\(556\) 0 0
\(557\) 26.8039 1.13572 0.567858 0.823127i \(-0.307772\pi\)
0.567858 + 0.823127i \(0.307772\pi\)
\(558\) 0 0
\(559\) 46.6651 1.97372
\(560\) 0 0
\(561\) 2.84886 0.120279
\(562\) 0 0
\(563\) −4.48928 −0.189201 −0.0946003 0.995515i \(-0.530157\pi\)
−0.0946003 + 0.995515i \(0.530157\pi\)
\(564\) 0 0
\(565\) −20.4342 −0.859674
\(566\) 0 0
\(567\) −24.0058 −1.00815
\(568\) 0 0
\(569\) −17.5245 −0.734667 −0.367333 0.930089i \(-0.619729\pi\)
−0.367333 + 0.930089i \(0.619729\pi\)
\(570\) 0 0
\(571\) −5.05790 −0.211667 −0.105833 0.994384i \(-0.533751\pi\)
−0.105833 + 0.994384i \(0.533751\pi\)
\(572\) 0 0
\(573\) 2.50339 0.104581
\(574\) 0 0
\(575\) 5.97862 0.249326
\(576\) 0 0
\(577\) −5.91260 −0.246145 −0.123072 0.992398i \(-0.539275\pi\)
−0.123072 + 0.992398i \(0.539275\pi\)
\(578\) 0 0
\(579\) −5.35572 −0.222576
\(580\) 0 0
\(581\) 39.9545 1.65759
\(582\) 0 0
\(583\) 8.38623 0.347322
\(584\) 0 0
\(585\) 38.0333 1.57248
\(586\) 0 0
\(587\) 14.8702 0.613757 0.306878 0.951749i \(-0.400715\pi\)
0.306878 + 0.951749i \(0.400715\pi\)
\(588\) 0 0
\(589\) −23.2364 −0.957440
\(590\) 0 0
\(591\) −1.89116 −0.0777919
\(592\) 0 0
\(593\) −19.7408 −0.810659 −0.405329 0.914171i \(-0.632843\pi\)
−0.405329 + 0.914171i \(0.632843\pi\)
\(594\) 0 0
\(595\) 44.2938 1.81587
\(596\) 0 0
\(597\) −9.24230 −0.378262
\(598\) 0 0
\(599\) −22.7370 −0.929010 −0.464505 0.885570i \(-0.653768\pi\)
−0.464505 + 0.885570i \(0.653768\pi\)
\(600\) 0 0
\(601\) 16.0388 0.654238 0.327119 0.944983i \(-0.393922\pi\)
0.327119 + 0.944983i \(0.393922\pi\)
\(602\) 0 0
\(603\) 18.4729 0.752276
\(604\) 0 0
\(605\) 23.9186 0.972427
\(606\) 0 0
\(607\) 6.35650 0.258002 0.129001 0.991644i \(-0.458823\pi\)
0.129001 + 0.991644i \(0.458823\pi\)
\(608\) 0 0
\(609\) 10.5418 0.427176
\(610\) 0 0
\(611\) 58.8707 2.38165
\(612\) 0 0
\(613\) 41.1061 1.66026 0.830130 0.557570i \(-0.188266\pi\)
0.830130 + 0.557570i \(0.188266\pi\)
\(614\) 0 0
\(615\) 6.37149 0.256923
\(616\) 0 0
\(617\) −9.00777 −0.362639 −0.181320 0.983424i \(-0.558037\pi\)
−0.181320 + 0.983424i \(0.558037\pi\)
\(618\) 0 0
\(619\) 21.0370 0.845547 0.422774 0.906235i \(-0.361057\pi\)
0.422774 + 0.906235i \(0.361057\pi\)
\(620\) 0 0
\(621\) −22.3525 −0.896974
\(622\) 0 0
\(623\) −49.4909 −1.98281
\(624\) 0 0
\(625\) −28.5858 −1.14343
\(626\) 0 0
\(627\) −2.76816 −0.110550
\(628\) 0 0
\(629\) 35.3556 1.40972
\(630\) 0 0
\(631\) 36.8742 1.46794 0.733969 0.679182i \(-0.237666\pi\)
0.733969 + 0.679182i \(0.237666\pi\)
\(632\) 0 0
\(633\) 3.74399 0.148810
\(634\) 0 0
\(635\) −40.8848 −1.62246
\(636\) 0 0
\(637\) 47.9913 1.90148
\(638\) 0 0
\(639\) 15.1823 0.600601
\(640\) 0 0
\(641\) −38.7992 −1.53248 −0.766239 0.642556i \(-0.777874\pi\)
−0.766239 + 0.642556i \(0.777874\pi\)
\(642\) 0 0
\(643\) 9.42249 0.371587 0.185793 0.982589i \(-0.440514\pi\)
0.185793 + 0.982589i \(0.440514\pi\)
\(644\) 0 0
\(645\) −11.0053 −0.433334
\(646\) 0 0
\(647\) −19.3331 −0.760064 −0.380032 0.924973i \(-0.624087\pi\)
−0.380032 + 0.924973i \(0.624087\pi\)
\(648\) 0 0
\(649\) −8.42204 −0.330594
\(650\) 0 0
\(651\) −11.3460 −0.444683
\(652\) 0 0
\(653\) 11.3499 0.444157 0.222078 0.975029i \(-0.428716\pi\)
0.222078 + 0.975029i \(0.428716\pi\)
\(654\) 0 0
\(655\) −24.4061 −0.953625
\(656\) 0 0
\(657\) −37.8198 −1.47549
\(658\) 0 0
\(659\) 1.09087 0.0424944 0.0212472 0.999774i \(-0.493236\pi\)
0.0212472 + 0.999774i \(0.493236\pi\)
\(660\) 0 0
\(661\) 7.28977 0.283539 0.141770 0.989900i \(-0.454721\pi\)
0.141770 + 0.989900i \(0.454721\pi\)
\(662\) 0 0
\(663\) 15.7712 0.612503
\(664\) 0 0
\(665\) −43.0392 −1.66899
\(666\) 0 0
\(667\) −32.6072 −1.26256
\(668\) 0 0
\(669\) −11.7317 −0.453575
\(670\) 0 0
\(671\) 13.7156 0.529487
\(672\) 0 0
\(673\) −17.1157 −0.659760 −0.329880 0.944023i \(-0.607008\pi\)
−0.329880 + 0.944023i \(0.607008\pi\)
\(674\) 0 0
\(675\) −2.81526 −0.108359
\(676\) 0 0
\(677\) 24.2655 0.932599 0.466299 0.884627i \(-0.345587\pi\)
0.466299 + 0.884627i \(0.345587\pi\)
\(678\) 0 0
\(679\) 40.6768 1.56103
\(680\) 0 0
\(681\) −9.35756 −0.358583
\(682\) 0 0
\(683\) 31.6640 1.21159 0.605794 0.795621i \(-0.292855\pi\)
0.605794 + 0.795621i \(0.292855\pi\)
\(684\) 0 0
\(685\) 28.1457 1.07539
\(686\) 0 0
\(687\) −4.42896 −0.168975
\(688\) 0 0
\(689\) 46.4260 1.76869
\(690\) 0 0
\(691\) −21.4115 −0.814531 −0.407265 0.913310i \(-0.633518\pi\)
−0.407265 + 0.913310i \(0.633518\pi\)
\(692\) 0 0
\(693\) 11.0465 0.419624
\(694\) 0 0
\(695\) 47.9388 1.81842
\(696\) 0 0
\(697\) −21.5926 −0.817880
\(698\) 0 0
\(699\) −4.48071 −0.169476
\(700\) 0 0
\(701\) −6.17654 −0.233285 −0.116642 0.993174i \(-0.537213\pi\)
−0.116642 + 0.993174i \(0.537213\pi\)
\(702\) 0 0
\(703\) −34.3541 −1.29569
\(704\) 0 0
\(705\) −13.8838 −0.522896
\(706\) 0 0
\(707\) −51.0324 −1.91927
\(708\) 0 0
\(709\) 28.8554 1.08369 0.541844 0.840479i \(-0.317727\pi\)
0.541844 + 0.840479i \(0.317727\pi\)
\(710\) 0 0
\(711\) 13.5077 0.506577
\(712\) 0 0
\(713\) 35.0945 1.31430
\(714\) 0 0
\(715\) −15.0980 −0.564631
\(716\) 0 0
\(717\) 3.77552 0.141000
\(718\) 0 0
\(719\) −28.0992 −1.04792 −0.523962 0.851742i \(-0.675547\pi\)
−0.523962 + 0.851742i \(0.675547\pi\)
\(720\) 0 0
\(721\) 46.7507 1.74109
\(722\) 0 0
\(723\) 11.6319 0.432596
\(724\) 0 0
\(725\) −4.10682 −0.152524
\(726\) 0 0
\(727\) 38.6298 1.43270 0.716350 0.697741i \(-0.245811\pi\)
0.716350 + 0.697741i \(0.245811\pi\)
\(728\) 0 0
\(729\) −10.9083 −0.404013
\(730\) 0 0
\(731\) 37.2964 1.37946
\(732\) 0 0
\(733\) −43.6843 −1.61352 −0.806759 0.590881i \(-0.798780\pi\)
−0.806759 + 0.590881i \(0.798780\pi\)
\(734\) 0 0
\(735\) −11.3181 −0.417474
\(736\) 0 0
\(737\) −7.33314 −0.270120
\(738\) 0 0
\(739\) −43.7643 −1.60990 −0.804949 0.593345i \(-0.797807\pi\)
−0.804949 + 0.593345i \(0.797807\pi\)
\(740\) 0 0
\(741\) −15.3245 −0.562959
\(742\) 0 0
\(743\) 22.8756 0.839223 0.419612 0.907704i \(-0.362166\pi\)
0.419612 + 0.907704i \(0.362166\pi\)
\(744\) 0 0
\(745\) −45.3477 −1.66141
\(746\) 0 0
\(747\) 27.4197 1.00323
\(748\) 0 0
\(749\) −37.1882 −1.35883
\(750\) 0 0
\(751\) 43.4421 1.58523 0.792613 0.609726i \(-0.208720\pi\)
0.792613 + 0.609726i \(0.208720\pi\)
\(752\) 0 0
\(753\) 12.4957 0.455367
\(754\) 0 0
\(755\) 17.3180 0.630266
\(756\) 0 0
\(757\) 29.0801 1.05693 0.528467 0.848954i \(-0.322767\pi\)
0.528467 + 0.848954i \(0.322767\pi\)
\(758\) 0 0
\(759\) 4.18082 0.151754
\(760\) 0 0
\(761\) −24.2027 −0.877349 −0.438674 0.898646i \(-0.644552\pi\)
−0.438674 + 0.898646i \(0.644552\pi\)
\(762\) 0 0
\(763\) 63.5541 2.30081
\(764\) 0 0
\(765\) 30.3976 1.09903
\(766\) 0 0
\(767\) −46.6242 −1.68350
\(768\) 0 0
\(769\) −19.7954 −0.713842 −0.356921 0.934135i \(-0.616173\pi\)
−0.356921 + 0.934135i \(0.616173\pi\)
\(770\) 0 0
\(771\) −0.527732 −0.0190058
\(772\) 0 0
\(773\) −5.49477 −0.197633 −0.0988166 0.995106i \(-0.531506\pi\)
−0.0988166 + 0.995106i \(0.531506\pi\)
\(774\) 0 0
\(775\) 4.42009 0.158774
\(776\) 0 0
\(777\) −16.7745 −0.601783
\(778\) 0 0
\(779\) 20.9810 0.751723
\(780\) 0 0
\(781\) −6.02685 −0.215658
\(782\) 0 0
\(783\) 15.3543 0.548719
\(784\) 0 0
\(785\) 11.4786 0.409689
\(786\) 0 0
\(787\) −10.8772 −0.387731 −0.193865 0.981028i \(-0.562103\pi\)
−0.193865 + 0.981028i \(0.562103\pi\)
\(788\) 0 0
\(789\) −5.96156 −0.212237
\(790\) 0 0
\(791\) −32.8561 −1.16823
\(792\) 0 0
\(793\) 75.9296 2.69634
\(794\) 0 0
\(795\) −10.9489 −0.388319
\(796\) 0 0
\(797\) −7.62805 −0.270199 −0.135100 0.990832i \(-0.543135\pi\)
−0.135100 + 0.990832i \(0.543135\pi\)
\(798\) 0 0
\(799\) 47.0516 1.66457
\(800\) 0 0
\(801\) −33.9642 −1.20007
\(802\) 0 0
\(803\) 15.0132 0.529805
\(804\) 0 0
\(805\) 65.0030 2.29106
\(806\) 0 0
\(807\) 12.1091 0.426261
\(808\) 0 0
\(809\) 46.5151 1.63539 0.817693 0.575655i \(-0.195253\pi\)
0.817693 + 0.575655i \(0.195253\pi\)
\(810\) 0 0
\(811\) −47.6127 −1.67191 −0.835954 0.548799i \(-0.815085\pi\)
−0.835954 + 0.548799i \(0.815085\pi\)
\(812\) 0 0
\(813\) 3.71110 0.130154
\(814\) 0 0
\(815\) −7.38304 −0.258617
\(816\) 0 0
\(817\) −36.2400 −1.26788
\(818\) 0 0
\(819\) 61.1535 2.13687
\(820\) 0 0
\(821\) −9.54536 −0.333135 −0.166568 0.986030i \(-0.553268\pi\)
−0.166568 + 0.986030i \(0.553268\pi\)
\(822\) 0 0
\(823\) 47.1124 1.64223 0.821117 0.570760i \(-0.193351\pi\)
0.821117 + 0.570760i \(0.193351\pi\)
\(824\) 0 0
\(825\) 0.526567 0.0183327
\(826\) 0 0
\(827\) 13.6412 0.474352 0.237176 0.971467i \(-0.423778\pi\)
0.237176 + 0.971467i \(0.423778\pi\)
\(828\) 0 0
\(829\) −23.4729 −0.815247 −0.407623 0.913150i \(-0.633642\pi\)
−0.407623 + 0.913150i \(0.633642\pi\)
\(830\) 0 0
\(831\) 1.76607 0.0612642
\(832\) 0 0
\(833\) 38.3564 1.32897
\(834\) 0 0
\(835\) 2.42234 0.0838287
\(836\) 0 0
\(837\) −16.5255 −0.571207
\(838\) 0 0
\(839\) −12.4753 −0.430694 −0.215347 0.976538i \(-0.569088\pi\)
−0.215347 + 0.976538i \(0.569088\pi\)
\(840\) 0 0
\(841\) −6.60149 −0.227638
\(842\) 0 0
\(843\) 17.4616 0.601408
\(844\) 0 0
\(845\) −52.0915 −1.79200
\(846\) 0 0
\(847\) 38.4585 1.32145
\(848\) 0 0
\(849\) 8.89951 0.305431
\(850\) 0 0
\(851\) 51.8858 1.77862
\(852\) 0 0
\(853\) −31.5347 −1.07973 −0.539863 0.841753i \(-0.681524\pi\)
−0.539863 + 0.841753i \(0.681524\pi\)
\(854\) 0 0
\(855\) −29.5366 −1.01013
\(856\) 0 0
\(857\) 1.19401 0.0407868 0.0203934 0.999792i \(-0.493508\pi\)
0.0203934 + 0.999792i \(0.493508\pi\)
\(858\) 0 0
\(859\) −1.35552 −0.0462496 −0.0231248 0.999733i \(-0.507362\pi\)
−0.0231248 + 0.999733i \(0.507362\pi\)
\(860\) 0 0
\(861\) 10.2447 0.349138
\(862\) 0 0
\(863\) −11.6022 −0.394945 −0.197473 0.980308i \(-0.563273\pi\)
−0.197473 + 0.980308i \(0.563273\pi\)
\(864\) 0 0
\(865\) −45.1902 −1.53651
\(866\) 0 0
\(867\) 2.88278 0.0979044
\(868\) 0 0
\(869\) −5.36210 −0.181897
\(870\) 0 0
\(871\) −40.5961 −1.37555
\(872\) 0 0
\(873\) 27.9153 0.944790
\(874\) 0 0
\(875\) −38.9866 −1.31799
\(876\) 0 0
\(877\) −23.1480 −0.781653 −0.390826 0.920464i \(-0.627811\pi\)
−0.390826 + 0.920464i \(0.627811\pi\)
\(878\) 0 0
\(879\) −7.19653 −0.242733
\(880\) 0 0
\(881\) −34.8493 −1.17410 −0.587051 0.809550i \(-0.699711\pi\)
−0.587051 + 0.809550i \(0.699711\pi\)
\(882\) 0 0
\(883\) 27.3131 0.919158 0.459579 0.888137i \(-0.348000\pi\)
0.459579 + 0.888137i \(0.348000\pi\)
\(884\) 0 0
\(885\) 10.9957 0.369616
\(886\) 0 0
\(887\) 30.3880 1.02033 0.510165 0.860077i \(-0.329584\pi\)
0.510165 + 0.860077i \(0.329584\pi\)
\(888\) 0 0
\(889\) −65.7384 −2.20480
\(890\) 0 0
\(891\) 6.53983 0.219093
\(892\) 0 0
\(893\) −45.7188 −1.52992
\(894\) 0 0
\(895\) −8.35005 −0.279111
\(896\) 0 0
\(897\) 23.1449 0.772786
\(898\) 0 0
\(899\) −24.1071 −0.804015
\(900\) 0 0
\(901\) 37.1053 1.23616
\(902\) 0 0
\(903\) −17.6954 −0.588866
\(904\) 0 0
\(905\) −36.2520 −1.20506
\(906\) 0 0
\(907\) 5.45970 0.181287 0.0906433 0.995883i \(-0.471108\pi\)
0.0906433 + 0.995883i \(0.471108\pi\)
\(908\) 0 0
\(909\) −35.0221 −1.16161
\(910\) 0 0
\(911\) −13.2698 −0.439648 −0.219824 0.975540i \(-0.570548\pi\)
−0.219824 + 0.975540i \(0.570548\pi\)
\(912\) 0 0
\(913\) −10.8847 −0.360231
\(914\) 0 0
\(915\) −17.9069 −0.591985
\(916\) 0 0
\(917\) −39.2424 −1.29590
\(918\) 0 0
\(919\) 43.9992 1.45140 0.725699 0.688012i \(-0.241516\pi\)
0.725699 + 0.688012i \(0.241516\pi\)
\(920\) 0 0
\(921\) 11.7526 0.387260
\(922\) 0 0
\(923\) −33.3645 −1.09821
\(924\) 0 0
\(925\) 6.53493 0.214867
\(926\) 0 0
\(927\) 32.0837 1.05377
\(928\) 0 0
\(929\) −50.2648 −1.64913 −0.824567 0.565765i \(-0.808581\pi\)
−0.824567 + 0.565765i \(0.808581\pi\)
\(930\) 0 0
\(931\) −37.2699 −1.22147
\(932\) 0 0
\(933\) −18.8322 −0.616538
\(934\) 0 0
\(935\) −12.0668 −0.394628
\(936\) 0 0
\(937\) −49.8524 −1.62861 −0.814304 0.580439i \(-0.802881\pi\)
−0.814304 + 0.580439i \(0.802881\pi\)
\(938\) 0 0
\(939\) 5.29166 0.172687
\(940\) 0 0
\(941\) 0.194970 0.00635584 0.00317792 0.999995i \(-0.498988\pi\)
0.00317792 + 0.999995i \(0.498988\pi\)
\(942\) 0 0
\(943\) −31.6881 −1.03191
\(944\) 0 0
\(945\) −30.6091 −0.995714
\(946\) 0 0
\(947\) 12.3319 0.400732 0.200366 0.979721i \(-0.435787\pi\)
0.200366 + 0.979721i \(0.435787\pi\)
\(948\) 0 0
\(949\) 83.1129 2.69796
\(950\) 0 0
\(951\) 13.0307 0.422550
\(952\) 0 0
\(953\) −3.29690 −0.106797 −0.0533985 0.998573i \(-0.517005\pi\)
−0.0533985 + 0.998573i \(0.517005\pi\)
\(954\) 0 0
\(955\) −10.6036 −0.343123
\(956\) 0 0
\(957\) −2.87188 −0.0928347
\(958\) 0 0
\(959\) 45.2553 1.46137
\(960\) 0 0
\(961\) −5.05408 −0.163035
\(962\) 0 0
\(963\) −25.5212 −0.822409
\(964\) 0 0
\(965\) 22.6851 0.730260
\(966\) 0 0
\(967\) −4.24763 −0.136594 −0.0682972 0.997665i \(-0.521757\pi\)
−0.0682972 + 0.997665i \(0.521757\pi\)
\(968\) 0 0
\(969\) −12.2479 −0.393459
\(970\) 0 0
\(971\) −48.3992 −1.55320 −0.776602 0.629992i \(-0.783058\pi\)
−0.776602 + 0.629992i \(0.783058\pi\)
\(972\) 0 0
\(973\) 77.0805 2.47109
\(974\) 0 0
\(975\) 2.91506 0.0933567
\(976\) 0 0
\(977\) 24.6611 0.788979 0.394489 0.918900i \(-0.370921\pi\)
0.394489 + 0.918900i \(0.370921\pi\)
\(978\) 0 0
\(979\) 13.4827 0.430908
\(980\) 0 0
\(981\) 43.6154 1.39253
\(982\) 0 0
\(983\) 20.0710 0.640166 0.320083 0.947390i \(-0.396289\pi\)
0.320083 + 0.947390i \(0.396289\pi\)
\(984\) 0 0
\(985\) 8.01034 0.255231
\(986\) 0 0
\(987\) −22.3237 −0.710572
\(988\) 0 0
\(989\) 54.7341 1.74044
\(990\) 0 0
\(991\) −32.5808 −1.03496 −0.517481 0.855695i \(-0.673130\pi\)
−0.517481 + 0.855695i \(0.673130\pi\)
\(992\) 0 0
\(993\) 6.65025 0.211039
\(994\) 0 0
\(995\) 39.1474 1.24106
\(996\) 0 0
\(997\) 54.5379 1.72723 0.863617 0.504149i \(-0.168194\pi\)
0.863617 + 0.504149i \(0.168194\pi\)
\(998\) 0 0
\(999\) −24.4324 −0.773006
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 1336.2.a.d.1.8 12
4.3 odd 2 2672.2.a.p.1.5 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1336.2.a.d.1.8 12 1.1 even 1 trivial
2672.2.a.p.1.5 12 4.3 odd 2