Properties

Label 1336.2.a.c.1.5
Level $1336$
Weight $2$
Character 1336.1
Self dual yes
Analytic conductor $10.668$
Analytic rank $1$
Dimension $9$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [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: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - x^{8} - 13x^{7} + 8x^{6} + 56x^{5} - 15x^{4} - 81x^{3} + 2x^{2} + 13x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-0.0812095\) 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)
 
\(f(q)\) \(=\) \(q+0.0812095 q^{3} +2.29286 q^{5} +0.862825 q^{7} -2.99341 q^{9} +O(q^{10})\) \(q+0.0812095 q^{3} +2.29286 q^{5} +0.862825 q^{7} -2.99341 q^{9} -1.76111 q^{11} -4.62247 q^{13} +0.186202 q^{15} -7.59644 q^{17} -6.30697 q^{19} +0.0700696 q^{21} -6.92950 q^{23} +0.257215 q^{25} -0.486722 q^{27} -0.203942 q^{29} +7.39783 q^{31} -0.143019 q^{33} +1.97834 q^{35} +4.37224 q^{37} -0.375388 q^{39} -4.55508 q^{41} +6.44082 q^{43} -6.86346 q^{45} +12.7488 q^{47} -6.25553 q^{49} -0.616904 q^{51} +0.311880 q^{53} -4.03798 q^{55} -0.512186 q^{57} +3.24784 q^{59} -5.12026 q^{61} -2.58278 q^{63} -10.5987 q^{65} -1.11318 q^{67} -0.562742 q^{69} +7.73760 q^{71} +4.17979 q^{73} +0.0208883 q^{75} -1.51953 q^{77} -0.537021 q^{79} +8.94069 q^{81} -3.11313 q^{83} -17.4176 q^{85} -0.0165620 q^{87} +3.20922 q^{89} -3.98838 q^{91} +0.600775 q^{93} -14.4610 q^{95} +2.73666 q^{97} +5.27171 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - q^{3} - 8 q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - q^{3} - 8 q^{5} + 2 q^{7} - 10 q^{11} - 13 q^{13} + 2 q^{15} - 8 q^{17} - q^{19} - 19 q^{21} - 3 q^{23} + 3 q^{25} - 10 q^{27} - 25 q^{29} - q^{31} - 12 q^{33} - 17 q^{35} - 35 q^{37} - 4 q^{39} - 16 q^{41} + 9 q^{43} - 24 q^{45} - q^{47} - q^{49} - 10 q^{51} - 29 q^{53} + 9 q^{55} - 17 q^{57} - 14 q^{59} - 28 q^{61} + 4 q^{63} - 31 q^{65} + 19 q^{67} - 19 q^{69} - 9 q^{71} - 7 q^{75} - 33 q^{77} - 18 q^{79} - 27 q^{81} - 13 q^{83} - 36 q^{85} + 18 q^{87} - 21 q^{89} + 20 q^{91} - 35 q^{93} - 12 q^{95} + 2 q^{97} - 3 q^{99}+O(q^{100}) \) 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
\(3\) 0.0812095 0.0468864 0.0234432 0.999725i \(-0.492537\pi\)
0.0234432 + 0.999725i \(0.492537\pi\)
\(4\) 0 0
\(5\) 2.29286 1.02540 0.512699 0.858568i \(-0.328646\pi\)
0.512699 + 0.858568i \(0.328646\pi\)
\(6\) 0 0
\(7\) 0.862825 0.326117 0.163059 0.986616i \(-0.447864\pi\)
0.163059 + 0.986616i \(0.447864\pi\)
\(8\) 0 0
\(9\) −2.99341 −0.997802
\(10\) 0 0
\(11\) −1.76111 −0.530994 −0.265497 0.964112i \(-0.585536\pi\)
−0.265497 + 0.964112i \(0.585536\pi\)
\(12\) 0 0
\(13\) −4.62247 −1.28204 −0.641021 0.767524i \(-0.721489\pi\)
−0.641021 + 0.767524i \(0.721489\pi\)
\(14\) 0 0
\(15\) 0.186202 0.0480772
\(16\) 0 0
\(17\) −7.59644 −1.84241 −0.921204 0.389080i \(-0.872793\pi\)
−0.921204 + 0.389080i \(0.872793\pi\)
\(18\) 0 0
\(19\) −6.30697 −1.44692 −0.723459 0.690368i \(-0.757449\pi\)
−0.723459 + 0.690368i \(0.757449\pi\)
\(20\) 0 0
\(21\) 0.0700696 0.0152904
\(22\) 0 0
\(23\) −6.92950 −1.44490 −0.722450 0.691423i \(-0.756984\pi\)
−0.722450 + 0.691423i \(0.756984\pi\)
\(24\) 0 0
\(25\) 0.257215 0.0514431
\(26\) 0 0
\(27\) −0.486722 −0.0936696
\(28\) 0 0
\(29\) −0.203942 −0.0378710 −0.0189355 0.999821i \(-0.506028\pi\)
−0.0189355 + 0.999821i \(0.506028\pi\)
\(30\) 0 0
\(31\) 7.39783 1.32869 0.664345 0.747426i \(-0.268711\pi\)
0.664345 + 0.747426i \(0.268711\pi\)
\(32\) 0 0
\(33\) −0.143019 −0.0248964
\(34\) 0 0
\(35\) 1.97834 0.334400
\(36\) 0 0
\(37\) 4.37224 0.718791 0.359396 0.933185i \(-0.382983\pi\)
0.359396 + 0.933185i \(0.382983\pi\)
\(38\) 0 0
\(39\) −0.375388 −0.0601102
\(40\) 0 0
\(41\) −4.55508 −0.711383 −0.355692 0.934603i \(-0.615755\pi\)
−0.355692 + 0.934603i \(0.615755\pi\)
\(42\) 0 0
\(43\) 6.44082 0.982217 0.491108 0.871099i \(-0.336592\pi\)
0.491108 + 0.871099i \(0.336592\pi\)
\(44\) 0 0
\(45\) −6.86346 −1.02314
\(46\) 0 0
\(47\) 12.7488 1.85961 0.929803 0.368057i \(-0.119977\pi\)
0.929803 + 0.368057i \(0.119977\pi\)
\(48\) 0 0
\(49\) −6.25553 −0.893648
\(50\) 0 0
\(51\) −0.616904 −0.0863838
\(52\) 0 0
\(53\) 0.311880 0.0428400 0.0214200 0.999771i \(-0.493181\pi\)
0.0214200 + 0.999771i \(0.493181\pi\)
\(54\) 0 0
\(55\) −4.03798 −0.544481
\(56\) 0 0
\(57\) −0.512186 −0.0678407
\(58\) 0 0
\(59\) 3.24784 0.422833 0.211416 0.977396i \(-0.432192\pi\)
0.211416 + 0.977396i \(0.432192\pi\)
\(60\) 0 0
\(61\) −5.12026 −0.655582 −0.327791 0.944750i \(-0.606304\pi\)
−0.327791 + 0.944750i \(0.606304\pi\)
\(62\) 0 0
\(63\) −2.58278 −0.325400
\(64\) 0 0
\(65\) −10.5987 −1.31460
\(66\) 0 0
\(67\) −1.11318 −0.135997 −0.0679986 0.997685i \(-0.521661\pi\)
−0.0679986 + 0.997685i \(0.521661\pi\)
\(68\) 0 0
\(69\) −0.562742 −0.0677461
\(70\) 0 0
\(71\) 7.73760 0.918284 0.459142 0.888363i \(-0.348157\pi\)
0.459142 + 0.888363i \(0.348157\pi\)
\(72\) 0 0
\(73\) 4.17979 0.489207 0.244604 0.969623i \(-0.421342\pi\)
0.244604 + 0.969623i \(0.421342\pi\)
\(74\) 0 0
\(75\) 0.0208883 0.00241198
\(76\) 0 0
\(77\) −1.51953 −0.173166
\(78\) 0 0
\(79\) −0.537021 −0.0604196 −0.0302098 0.999544i \(-0.509618\pi\)
−0.0302098 + 0.999544i \(0.509618\pi\)
\(80\) 0 0
\(81\) 8.94069 0.993410
\(82\) 0 0
\(83\) −3.11313 −0.341711 −0.170855 0.985296i \(-0.554653\pi\)
−0.170855 + 0.985296i \(0.554653\pi\)
\(84\) 0 0
\(85\) −17.4176 −1.88920
\(86\) 0 0
\(87\) −0.0165620 −0.00177563
\(88\) 0 0
\(89\) 3.20922 0.340177 0.170088 0.985429i \(-0.445595\pi\)
0.170088 + 0.985429i \(0.445595\pi\)
\(90\) 0 0
\(91\) −3.98838 −0.418096
\(92\) 0 0
\(93\) 0.600775 0.0622974
\(94\) 0 0
\(95\) −14.4610 −1.48367
\(96\) 0 0
\(97\) 2.73666 0.277866 0.138933 0.990302i \(-0.455633\pi\)
0.138933 + 0.990302i \(0.455633\pi\)
\(98\) 0 0
\(99\) 5.27171 0.529827
\(100\) 0 0
\(101\) 14.7368 1.46637 0.733183 0.680032i \(-0.238034\pi\)
0.733183 + 0.680032i \(0.238034\pi\)
\(102\) 0 0
\(103\) −3.20154 −0.315457 −0.157729 0.987483i \(-0.550417\pi\)
−0.157729 + 0.987483i \(0.550417\pi\)
\(104\) 0 0
\(105\) 0.160660 0.0156788
\(106\) 0 0
\(107\) 5.82728 0.563344 0.281672 0.959511i \(-0.409111\pi\)
0.281672 + 0.959511i \(0.409111\pi\)
\(108\) 0 0
\(109\) −2.67970 −0.256669 −0.128335 0.991731i \(-0.540963\pi\)
−0.128335 + 0.991731i \(0.540963\pi\)
\(110\) 0 0
\(111\) 0.355067 0.0337015
\(112\) 0 0
\(113\) −9.41423 −0.885616 −0.442808 0.896616i \(-0.646018\pi\)
−0.442808 + 0.896616i \(0.646018\pi\)
\(114\) 0 0
\(115\) −15.8884 −1.48160
\(116\) 0 0
\(117\) 13.8369 1.27922
\(118\) 0 0
\(119\) −6.55440 −0.600841
\(120\) 0 0
\(121\) −7.89850 −0.718045
\(122\) 0 0
\(123\) −0.369916 −0.0333542
\(124\) 0 0
\(125\) −10.8745 −0.972649
\(126\) 0 0
\(127\) 11.9736 1.06248 0.531241 0.847220i \(-0.321726\pi\)
0.531241 + 0.847220i \(0.321726\pi\)
\(128\) 0 0
\(129\) 0.523056 0.0460525
\(130\) 0 0
\(131\) −2.57207 −0.224723 −0.112361 0.993667i \(-0.535841\pi\)
−0.112361 + 0.993667i \(0.535841\pi\)
\(132\) 0 0
\(133\) −5.44181 −0.471864
\(134\) 0 0
\(135\) −1.11599 −0.0960487
\(136\) 0 0
\(137\) −22.5342 −1.92523 −0.962613 0.270880i \(-0.912685\pi\)
−0.962613 + 0.270880i \(0.912685\pi\)
\(138\) 0 0
\(139\) −11.4030 −0.967189 −0.483594 0.875292i \(-0.660669\pi\)
−0.483594 + 0.875292i \(0.660669\pi\)
\(140\) 0 0
\(141\) 1.03533 0.0871902
\(142\) 0 0
\(143\) 8.14066 0.680756
\(144\) 0 0
\(145\) −0.467610 −0.0388329
\(146\) 0 0
\(147\) −0.508009 −0.0418999
\(148\) 0 0
\(149\) −8.55792 −0.701092 −0.350546 0.936546i \(-0.614004\pi\)
−0.350546 + 0.936546i \(0.614004\pi\)
\(150\) 0 0
\(151\) −13.5694 −1.10427 −0.552133 0.833756i \(-0.686186\pi\)
−0.552133 + 0.833756i \(0.686186\pi\)
\(152\) 0 0
\(153\) 22.7392 1.83836
\(154\) 0 0
\(155\) 16.9622 1.36244
\(156\) 0 0
\(157\) 11.3501 0.905840 0.452920 0.891551i \(-0.350382\pi\)
0.452920 + 0.891551i \(0.350382\pi\)
\(158\) 0 0
\(159\) 0.0253276 0.00200861
\(160\) 0 0
\(161\) −5.97895 −0.471207
\(162\) 0 0
\(163\) 4.33221 0.339325 0.169663 0.985502i \(-0.445732\pi\)
0.169663 + 0.985502i \(0.445732\pi\)
\(164\) 0 0
\(165\) −0.327922 −0.0255287
\(166\) 0 0
\(167\) 1.00000 0.0773823
\(168\) 0 0
\(169\) 8.36718 0.643630
\(170\) 0 0
\(171\) 18.8793 1.44374
\(172\) 0 0
\(173\) −20.4544 −1.55512 −0.777559 0.628811i \(-0.783542\pi\)
−0.777559 + 0.628811i \(0.783542\pi\)
\(174\) 0 0
\(175\) 0.221932 0.0167765
\(176\) 0 0
\(177\) 0.263756 0.0198251
\(178\) 0 0
\(179\) −0.241097 −0.0180204 −0.00901022 0.999959i \(-0.502868\pi\)
−0.00901022 + 0.999959i \(0.502868\pi\)
\(180\) 0 0
\(181\) −9.39871 −0.698600 −0.349300 0.937011i \(-0.613581\pi\)
−0.349300 + 0.937011i \(0.613581\pi\)
\(182\) 0 0
\(183\) −0.415814 −0.0307379
\(184\) 0 0
\(185\) 10.0249 0.737048
\(186\) 0 0
\(187\) 13.3782 0.978308
\(188\) 0 0
\(189\) −0.419956 −0.0305473
\(190\) 0 0
\(191\) −10.3388 −0.748087 −0.374044 0.927411i \(-0.622029\pi\)
−0.374044 + 0.927411i \(0.622029\pi\)
\(192\) 0 0
\(193\) 12.9283 0.930599 0.465300 0.885153i \(-0.345947\pi\)
0.465300 + 0.885153i \(0.345947\pi\)
\(194\) 0 0
\(195\) −0.860713 −0.0616370
\(196\) 0 0
\(197\) −10.8318 −0.771735 −0.385868 0.922554i \(-0.626098\pi\)
−0.385868 + 0.922554i \(0.626098\pi\)
\(198\) 0 0
\(199\) 6.55955 0.464994 0.232497 0.972597i \(-0.425310\pi\)
0.232497 + 0.972597i \(0.425310\pi\)
\(200\) 0 0
\(201\) −0.0904012 −0.00637641
\(202\) 0 0
\(203\) −0.175966 −0.0123504
\(204\) 0 0
\(205\) −10.4442 −0.729452
\(206\) 0 0
\(207\) 20.7428 1.44172
\(208\) 0 0
\(209\) 11.1073 0.768305
\(210\) 0 0
\(211\) 15.3104 1.05401 0.527007 0.849861i \(-0.323314\pi\)
0.527007 + 0.849861i \(0.323314\pi\)
\(212\) 0 0
\(213\) 0.628367 0.0430550
\(214\) 0 0
\(215\) 14.7679 1.00716
\(216\) 0 0
\(217\) 6.38303 0.433309
\(218\) 0 0
\(219\) 0.339439 0.0229371
\(220\) 0 0
\(221\) 35.1143 2.36204
\(222\) 0 0
\(223\) 7.22753 0.483991 0.241996 0.970277i \(-0.422198\pi\)
0.241996 + 0.970277i \(0.422198\pi\)
\(224\) 0 0
\(225\) −0.769949 −0.0513300
\(226\) 0 0
\(227\) −22.7927 −1.51281 −0.756403 0.654105i \(-0.773045\pi\)
−0.756403 + 0.654105i \(0.773045\pi\)
\(228\) 0 0
\(229\) −14.5332 −0.960380 −0.480190 0.877164i \(-0.659432\pi\)
−0.480190 + 0.877164i \(0.659432\pi\)
\(230\) 0 0
\(231\) −0.123400 −0.00811914
\(232\) 0 0
\(233\) −13.6521 −0.894381 −0.447191 0.894439i \(-0.647575\pi\)
−0.447191 + 0.894439i \(0.647575\pi\)
\(234\) 0 0
\(235\) 29.2313 1.90684
\(236\) 0 0
\(237\) −0.0436112 −0.00283285
\(238\) 0 0
\(239\) −21.5820 −1.39602 −0.698010 0.716088i \(-0.745931\pi\)
−0.698010 + 0.716088i \(0.745931\pi\)
\(240\) 0 0
\(241\) 5.66976 0.365222 0.182611 0.983185i \(-0.441545\pi\)
0.182611 + 0.983185i \(0.441545\pi\)
\(242\) 0 0
\(243\) 2.18623 0.140247
\(244\) 0 0
\(245\) −14.3431 −0.916345
\(246\) 0 0
\(247\) 29.1537 1.85501
\(248\) 0 0
\(249\) −0.252816 −0.0160216
\(250\) 0 0
\(251\) −6.58019 −0.415338 −0.207669 0.978199i \(-0.566588\pi\)
−0.207669 + 0.978199i \(0.566588\pi\)
\(252\) 0 0
\(253\) 12.2036 0.767234
\(254\) 0 0
\(255\) −1.41447 −0.0885779
\(256\) 0 0
\(257\) 27.8355 1.73633 0.868165 0.496276i \(-0.165300\pi\)
0.868165 + 0.496276i \(0.165300\pi\)
\(258\) 0 0
\(259\) 3.77248 0.234410
\(260\) 0 0
\(261\) 0.610480 0.0377878
\(262\) 0 0
\(263\) 8.19313 0.505210 0.252605 0.967570i \(-0.418713\pi\)
0.252605 + 0.967570i \(0.418713\pi\)
\(264\) 0 0
\(265\) 0.715097 0.0439281
\(266\) 0 0
\(267\) 0.260619 0.0159496
\(268\) 0 0
\(269\) 27.5537 1.67998 0.839989 0.542603i \(-0.182561\pi\)
0.839989 + 0.542603i \(0.182561\pi\)
\(270\) 0 0
\(271\) −1.62325 −0.0986056 −0.0493028 0.998784i \(-0.515700\pi\)
−0.0493028 + 0.998784i \(0.515700\pi\)
\(272\) 0 0
\(273\) −0.323894 −0.0196030
\(274\) 0 0
\(275\) −0.452984 −0.0273160
\(276\) 0 0
\(277\) −5.50841 −0.330968 −0.165484 0.986212i \(-0.552919\pi\)
−0.165484 + 0.986212i \(0.552919\pi\)
\(278\) 0 0
\(279\) −22.1447 −1.32577
\(280\) 0 0
\(281\) −18.3226 −1.09304 −0.546518 0.837447i \(-0.684047\pi\)
−0.546518 + 0.837447i \(0.684047\pi\)
\(282\) 0 0
\(283\) −16.8520 −1.00175 −0.500874 0.865520i \(-0.666988\pi\)
−0.500874 + 0.865520i \(0.666988\pi\)
\(284\) 0 0
\(285\) −1.17437 −0.0695637
\(286\) 0 0
\(287\) −3.93023 −0.231994
\(288\) 0 0
\(289\) 40.7060 2.39447
\(290\) 0 0
\(291\) 0.222243 0.0130281
\(292\) 0 0
\(293\) −10.5638 −0.617142 −0.308571 0.951201i \(-0.599851\pi\)
−0.308571 + 0.951201i \(0.599851\pi\)
\(294\) 0 0
\(295\) 7.44685 0.433572
\(296\) 0 0
\(297\) 0.857170 0.0497380
\(298\) 0 0
\(299\) 32.0314 1.85242
\(300\) 0 0
\(301\) 5.55730 0.320318
\(302\) 0 0
\(303\) 1.19677 0.0687525
\(304\) 0 0
\(305\) −11.7400 −0.672233
\(306\) 0 0
\(307\) −30.9725 −1.76769 −0.883846 0.467777i \(-0.845055\pi\)
−0.883846 + 0.467777i \(0.845055\pi\)
\(308\) 0 0
\(309\) −0.259996 −0.0147906
\(310\) 0 0
\(311\) −7.64222 −0.433350 −0.216675 0.976244i \(-0.569521\pi\)
−0.216675 + 0.976244i \(0.569521\pi\)
\(312\) 0 0
\(313\) 1.86882 0.105632 0.0528158 0.998604i \(-0.483180\pi\)
0.0528158 + 0.998604i \(0.483180\pi\)
\(314\) 0 0
\(315\) −5.92197 −0.333665
\(316\) 0 0
\(317\) −24.2502 −1.36203 −0.681015 0.732270i \(-0.738461\pi\)
−0.681015 + 0.732270i \(0.738461\pi\)
\(318\) 0 0
\(319\) 0.359163 0.0201093
\(320\) 0 0
\(321\) 0.473231 0.0264132
\(322\) 0 0
\(323\) 47.9105 2.66581
\(324\) 0 0
\(325\) −1.18897 −0.0659521
\(326\) 0 0
\(327\) −0.217618 −0.0120343
\(328\) 0 0
\(329\) 11.0000 0.606449
\(330\) 0 0
\(331\) 19.4463 1.06886 0.534432 0.845212i \(-0.320526\pi\)
0.534432 + 0.845212i \(0.320526\pi\)
\(332\) 0 0
\(333\) −13.0879 −0.717211
\(334\) 0 0
\(335\) −2.55238 −0.139451
\(336\) 0 0
\(337\) −5.55152 −0.302411 −0.151205 0.988502i \(-0.548315\pi\)
−0.151205 + 0.988502i \(0.548315\pi\)
\(338\) 0 0
\(339\) −0.764525 −0.0415233
\(340\) 0 0
\(341\) −13.0284 −0.705527
\(342\) 0 0
\(343\) −11.4372 −0.617551
\(344\) 0 0
\(345\) −1.29029 −0.0694668
\(346\) 0 0
\(347\) −5.70726 −0.306382 −0.153191 0.988197i \(-0.548955\pi\)
−0.153191 + 0.988197i \(0.548955\pi\)
\(348\) 0 0
\(349\) 6.24638 0.334361 0.167181 0.985926i \(-0.446534\pi\)
0.167181 + 0.985926i \(0.446534\pi\)
\(350\) 0 0
\(351\) 2.24985 0.120088
\(352\) 0 0
\(353\) −6.51596 −0.346810 −0.173405 0.984851i \(-0.555477\pi\)
−0.173405 + 0.984851i \(0.555477\pi\)
\(354\) 0 0
\(355\) 17.7412 0.941607
\(356\) 0 0
\(357\) −0.532280 −0.0281712
\(358\) 0 0
\(359\) 12.8370 0.677510 0.338755 0.940875i \(-0.389994\pi\)
0.338755 + 0.940875i \(0.389994\pi\)
\(360\) 0 0
\(361\) 20.7778 1.09357
\(362\) 0 0
\(363\) −0.641433 −0.0336665
\(364\) 0 0
\(365\) 9.58368 0.501633
\(366\) 0 0
\(367\) 20.2602 1.05758 0.528788 0.848754i \(-0.322647\pi\)
0.528788 + 0.848754i \(0.322647\pi\)
\(368\) 0 0
\(369\) 13.6352 0.709820
\(370\) 0 0
\(371\) 0.269098 0.0139709
\(372\) 0 0
\(373\) −33.9950 −1.76020 −0.880098 0.474792i \(-0.842523\pi\)
−0.880098 + 0.474792i \(0.842523\pi\)
\(374\) 0 0
\(375\) −0.883117 −0.0456040
\(376\) 0 0
\(377\) 0.942713 0.0485522
\(378\) 0 0
\(379\) −10.8625 −0.557971 −0.278986 0.960295i \(-0.589998\pi\)
−0.278986 + 0.960295i \(0.589998\pi\)
\(380\) 0 0
\(381\) 0.972368 0.0498159
\(382\) 0 0
\(383\) 21.5261 1.09993 0.549966 0.835187i \(-0.314641\pi\)
0.549966 + 0.835187i \(0.314641\pi\)
\(384\) 0 0
\(385\) −3.48407 −0.177565
\(386\) 0 0
\(387\) −19.2800 −0.980057
\(388\) 0 0
\(389\) −26.4022 −1.33864 −0.669322 0.742972i \(-0.733416\pi\)
−0.669322 + 0.742972i \(0.733416\pi\)
\(390\) 0 0
\(391\) 52.6396 2.66210
\(392\) 0 0
\(393\) −0.208877 −0.0105364
\(394\) 0 0
\(395\) −1.23132 −0.0619542
\(396\) 0 0
\(397\) −29.9273 −1.50201 −0.751003 0.660298i \(-0.770430\pi\)
−0.751003 + 0.660298i \(0.770430\pi\)
\(398\) 0 0
\(399\) −0.441927 −0.0221240
\(400\) 0 0
\(401\) 30.1862 1.50743 0.753713 0.657204i \(-0.228261\pi\)
0.753713 + 0.657204i \(0.228261\pi\)
\(402\) 0 0
\(403\) −34.1962 −1.70344
\(404\) 0 0
\(405\) 20.4998 1.01864
\(406\) 0 0
\(407\) −7.69998 −0.381674
\(408\) 0 0
\(409\) 27.8008 1.37466 0.687330 0.726346i \(-0.258783\pi\)
0.687330 + 0.726346i \(0.258783\pi\)
\(410\) 0 0
\(411\) −1.82999 −0.0902668
\(412\) 0 0
\(413\) 2.80232 0.137893
\(414\) 0 0
\(415\) −7.13798 −0.350390
\(416\) 0 0
\(417\) −0.926031 −0.0453479
\(418\) 0 0
\(419\) −37.3619 −1.82525 −0.912624 0.408799i \(-0.865948\pi\)
−0.912624 + 0.408799i \(0.865948\pi\)
\(420\) 0 0
\(421\) −1.20756 −0.0588530 −0.0294265 0.999567i \(-0.509368\pi\)
−0.0294265 + 0.999567i \(0.509368\pi\)
\(422\) 0 0
\(423\) −38.1624 −1.85552
\(424\) 0 0
\(425\) −1.95392 −0.0947791
\(426\) 0 0
\(427\) −4.41789 −0.213797
\(428\) 0 0
\(429\) 0.661099 0.0319182
\(430\) 0 0
\(431\) −32.1609 −1.54914 −0.774568 0.632490i \(-0.782033\pi\)
−0.774568 + 0.632490i \(0.782033\pi\)
\(432\) 0 0
\(433\) 15.3933 0.739756 0.369878 0.929080i \(-0.379399\pi\)
0.369878 + 0.929080i \(0.379399\pi\)
\(434\) 0 0
\(435\) −0.0379744 −0.00182073
\(436\) 0 0
\(437\) 43.7041 2.09065
\(438\) 0 0
\(439\) −21.1790 −1.01082 −0.505409 0.862880i \(-0.668658\pi\)
−0.505409 + 0.862880i \(0.668658\pi\)
\(440\) 0 0
\(441\) 18.7253 0.891683
\(442\) 0 0
\(443\) 18.9637 0.900993 0.450497 0.892778i \(-0.351247\pi\)
0.450497 + 0.892778i \(0.351247\pi\)
\(444\) 0 0
\(445\) 7.35829 0.348817
\(446\) 0 0
\(447\) −0.694985 −0.0328716
\(448\) 0 0
\(449\) −1.60577 −0.0757809 −0.0378905 0.999282i \(-0.512064\pi\)
−0.0378905 + 0.999282i \(0.512064\pi\)
\(450\) 0 0
\(451\) 8.02198 0.377740
\(452\) 0 0
\(453\) −1.10197 −0.0517750
\(454\) 0 0
\(455\) −9.14480 −0.428715
\(456\) 0 0
\(457\) 41.9837 1.96391 0.981957 0.189104i \(-0.0605585\pi\)
0.981957 + 0.189104i \(0.0605585\pi\)
\(458\) 0 0
\(459\) 3.69735 0.172578
\(460\) 0 0
\(461\) −16.8844 −0.786383 −0.393191 0.919457i \(-0.628629\pi\)
−0.393191 + 0.919457i \(0.628629\pi\)
\(462\) 0 0
\(463\) 2.50068 0.116216 0.0581081 0.998310i \(-0.481493\pi\)
0.0581081 + 0.998310i \(0.481493\pi\)
\(464\) 0 0
\(465\) 1.37749 0.0638797
\(466\) 0 0
\(467\) −40.3886 −1.86896 −0.934481 0.356014i \(-0.884136\pi\)
−0.934481 + 0.356014i \(0.884136\pi\)
\(468\) 0 0
\(469\) −0.960483 −0.0443510
\(470\) 0 0
\(471\) 0.921740 0.0424715
\(472\) 0 0
\(473\) −11.3430 −0.521551
\(474\) 0 0
\(475\) −1.62225 −0.0744338
\(476\) 0 0
\(477\) −0.933583 −0.0427458
\(478\) 0 0
\(479\) −28.8312 −1.31733 −0.658665 0.752436i \(-0.728879\pi\)
−0.658665 + 0.752436i \(0.728879\pi\)
\(480\) 0 0
\(481\) −20.2105 −0.921520
\(482\) 0 0
\(483\) −0.485547 −0.0220932
\(484\) 0 0
\(485\) 6.27478 0.284923
\(486\) 0 0
\(487\) 33.6060 1.52284 0.761418 0.648262i \(-0.224504\pi\)
0.761418 + 0.648262i \(0.224504\pi\)
\(488\) 0 0
\(489\) 0.351817 0.0159097
\(490\) 0 0
\(491\) 10.6015 0.478438 0.239219 0.970966i \(-0.423109\pi\)
0.239219 + 0.970966i \(0.423109\pi\)
\(492\) 0 0
\(493\) 1.54923 0.0697739
\(494\) 0 0
\(495\) 12.0873 0.543284
\(496\) 0 0
\(497\) 6.67619 0.299468
\(498\) 0 0
\(499\) −37.8999 −1.69663 −0.848317 0.529489i \(-0.822384\pi\)
−0.848317 + 0.529489i \(0.822384\pi\)
\(500\) 0 0
\(501\) 0.0812095 0.00362817
\(502\) 0 0
\(503\) −15.6201 −0.696466 −0.348233 0.937408i \(-0.613218\pi\)
−0.348233 + 0.937408i \(0.613218\pi\)
\(504\) 0 0
\(505\) 33.7894 1.50361
\(506\) 0 0
\(507\) 0.679495 0.0301774
\(508\) 0 0
\(509\) −14.7163 −0.652286 −0.326143 0.945320i \(-0.605749\pi\)
−0.326143 + 0.945320i \(0.605749\pi\)
\(510\) 0 0
\(511\) 3.60643 0.159539
\(512\) 0 0
\(513\) 3.06974 0.135532
\(514\) 0 0
\(515\) −7.34069 −0.323469
\(516\) 0 0
\(517\) −22.4521 −0.987440
\(518\) 0 0
\(519\) −1.66109 −0.0729138
\(520\) 0 0
\(521\) −30.4380 −1.33351 −0.666757 0.745275i \(-0.732318\pi\)
−0.666757 + 0.745275i \(0.732318\pi\)
\(522\) 0 0
\(523\) 26.1487 1.14340 0.571701 0.820462i \(-0.306284\pi\)
0.571701 + 0.820462i \(0.306284\pi\)
\(524\) 0 0
\(525\) 0.0180230 0.000786587 0
\(526\) 0 0
\(527\) −56.1972 −2.44799
\(528\) 0 0
\(529\) 25.0180 1.08774
\(530\) 0 0
\(531\) −9.72210 −0.421903
\(532\) 0 0
\(533\) 21.0557 0.912023
\(534\) 0 0
\(535\) 13.3611 0.577653
\(536\) 0 0
\(537\) −0.0195794 −0.000844913 0
\(538\) 0 0
\(539\) 11.0167 0.474522
\(540\) 0 0
\(541\) −2.81820 −0.121164 −0.0605818 0.998163i \(-0.519296\pi\)
−0.0605818 + 0.998163i \(0.519296\pi\)
\(542\) 0 0
\(543\) −0.763265 −0.0327548
\(544\) 0 0
\(545\) −6.14419 −0.263188
\(546\) 0 0
\(547\) 44.7573 1.91368 0.956842 0.290610i \(-0.0938581\pi\)
0.956842 + 0.290610i \(0.0938581\pi\)
\(548\) 0 0
\(549\) 15.3270 0.654141
\(550\) 0 0
\(551\) 1.28625 0.0547962
\(552\) 0 0
\(553\) −0.463355 −0.0197039
\(554\) 0 0
\(555\) 0.814121 0.0345575
\(556\) 0 0
\(557\) 31.9026 1.35176 0.675878 0.737013i \(-0.263765\pi\)
0.675878 + 0.737013i \(0.263765\pi\)
\(558\) 0 0
\(559\) −29.7725 −1.25924
\(560\) 0 0
\(561\) 1.08643 0.0458693
\(562\) 0 0
\(563\) −21.6342 −0.911772 −0.455886 0.890038i \(-0.650678\pi\)
−0.455886 + 0.890038i \(0.650678\pi\)
\(564\) 0 0
\(565\) −21.5855 −0.908110
\(566\) 0 0
\(567\) 7.71425 0.323968
\(568\) 0 0
\(569\) 31.2958 1.31199 0.655994 0.754766i \(-0.272250\pi\)
0.655994 + 0.754766i \(0.272250\pi\)
\(570\) 0 0
\(571\) −11.9511 −0.500140 −0.250070 0.968228i \(-0.580454\pi\)
−0.250070 + 0.968228i \(0.580454\pi\)
\(572\) 0 0
\(573\) −0.839607 −0.0350751
\(574\) 0 0
\(575\) −1.78237 −0.0743301
\(576\) 0 0
\(577\) 18.9566 0.789176 0.394588 0.918858i \(-0.370887\pi\)
0.394588 + 0.918858i \(0.370887\pi\)
\(578\) 0 0
\(579\) 1.04990 0.0436324
\(580\) 0 0
\(581\) −2.68609 −0.111438
\(582\) 0 0
\(583\) −0.549254 −0.0227478
\(584\) 0 0
\(585\) 31.7261 1.31171
\(586\) 0 0
\(587\) −24.9488 −1.02975 −0.514874 0.857266i \(-0.672161\pi\)
−0.514874 + 0.857266i \(0.672161\pi\)
\(588\) 0 0
\(589\) −46.6579 −1.92250
\(590\) 0 0
\(591\) −0.879647 −0.0361838
\(592\) 0 0
\(593\) 31.9540 1.31219 0.656096 0.754677i \(-0.272207\pi\)
0.656096 + 0.754677i \(0.272207\pi\)
\(594\) 0 0
\(595\) −15.0283 −0.616102
\(596\) 0 0
\(597\) 0.532698 0.0218019
\(598\) 0 0
\(599\) −1.64258 −0.0671139 −0.0335570 0.999437i \(-0.510684\pi\)
−0.0335570 + 0.999437i \(0.510684\pi\)
\(600\) 0 0
\(601\) 17.6343 0.719317 0.359658 0.933084i \(-0.382893\pi\)
0.359658 + 0.933084i \(0.382893\pi\)
\(602\) 0 0
\(603\) 3.33221 0.135698
\(604\) 0 0
\(605\) −18.1102 −0.736283
\(606\) 0 0
\(607\) 19.0275 0.772302 0.386151 0.922436i \(-0.373804\pi\)
0.386151 + 0.922436i \(0.373804\pi\)
\(608\) 0 0
\(609\) −0.0142901 −0.000579064 0
\(610\) 0 0
\(611\) −58.9310 −2.38409
\(612\) 0 0
\(613\) −42.4081 −1.71285 −0.856423 0.516275i \(-0.827318\pi\)
−0.856423 + 0.516275i \(0.827318\pi\)
\(614\) 0 0
\(615\) −0.848165 −0.0342013
\(616\) 0 0
\(617\) −4.09479 −0.164850 −0.0824250 0.996597i \(-0.526266\pi\)
−0.0824250 + 0.996597i \(0.526266\pi\)
\(618\) 0 0
\(619\) 20.0351 0.805280 0.402640 0.915358i \(-0.368093\pi\)
0.402640 + 0.915358i \(0.368093\pi\)
\(620\) 0 0
\(621\) 3.37274 0.135343
\(622\) 0 0
\(623\) 2.76899 0.110937
\(624\) 0 0
\(625\) −26.2199 −1.04880
\(626\) 0 0
\(627\) 0.902015 0.0360230
\(628\) 0 0
\(629\) −33.2135 −1.32431
\(630\) 0 0
\(631\) −19.6642 −0.782819 −0.391410 0.920217i \(-0.628012\pi\)
−0.391410 + 0.920217i \(0.628012\pi\)
\(632\) 0 0
\(633\) 1.24335 0.0494189
\(634\) 0 0
\(635\) 27.4538 1.08947
\(636\) 0 0
\(637\) 28.9160 1.14569
\(638\) 0 0
\(639\) −23.1618 −0.916265
\(640\) 0 0
\(641\) −19.9574 −0.788269 −0.394134 0.919053i \(-0.628956\pi\)
−0.394134 + 0.919053i \(0.628956\pi\)
\(642\) 0 0
\(643\) −0.498013 −0.0196397 −0.00981985 0.999952i \(-0.503126\pi\)
−0.00981985 + 0.999952i \(0.503126\pi\)
\(644\) 0 0
\(645\) 1.19930 0.0472222
\(646\) 0 0
\(647\) 5.57148 0.219037 0.109519 0.993985i \(-0.465069\pi\)
0.109519 + 0.993985i \(0.465069\pi\)
\(648\) 0 0
\(649\) −5.71980 −0.224522
\(650\) 0 0
\(651\) 0.518363 0.0203163
\(652\) 0 0
\(653\) 7.49594 0.293339 0.146669 0.989186i \(-0.453145\pi\)
0.146669 + 0.989186i \(0.453145\pi\)
\(654\) 0 0
\(655\) −5.89741 −0.230431
\(656\) 0 0
\(657\) −12.5118 −0.488132
\(658\) 0 0
\(659\) −47.1158 −1.83537 −0.917686 0.397307i \(-0.869945\pi\)
−0.917686 + 0.397307i \(0.869945\pi\)
\(660\) 0 0
\(661\) −25.9953 −1.01110 −0.505550 0.862797i \(-0.668711\pi\)
−0.505550 + 0.862797i \(0.668711\pi\)
\(662\) 0 0
\(663\) 2.85162 0.110748
\(664\) 0 0
\(665\) −12.4773 −0.483849
\(666\) 0 0
\(667\) 1.41321 0.0547199
\(668\) 0 0
\(669\) 0.586944 0.0226926
\(670\) 0 0
\(671\) 9.01733 0.348110
\(672\) 0 0
\(673\) 6.98702 0.269330 0.134665 0.990891i \(-0.457004\pi\)
0.134665 + 0.990891i \(0.457004\pi\)
\(674\) 0 0
\(675\) −0.125192 −0.00481865
\(676\) 0 0
\(677\) 43.3439 1.66584 0.832922 0.553391i \(-0.186666\pi\)
0.832922 + 0.553391i \(0.186666\pi\)
\(678\) 0 0
\(679\) 2.36126 0.0906167
\(680\) 0 0
\(681\) −1.85099 −0.0709300
\(682\) 0 0
\(683\) −33.8893 −1.29674 −0.648368 0.761327i \(-0.724548\pi\)
−0.648368 + 0.761327i \(0.724548\pi\)
\(684\) 0 0
\(685\) −51.6678 −1.97413
\(686\) 0 0
\(687\) −1.18023 −0.0450287
\(688\) 0 0
\(689\) −1.44165 −0.0549226
\(690\) 0 0
\(691\) 41.2872 1.57064 0.785320 0.619090i \(-0.212499\pi\)
0.785320 + 0.619090i \(0.212499\pi\)
\(692\) 0 0
\(693\) 4.54856 0.172786
\(694\) 0 0
\(695\) −26.1455 −0.991754
\(696\) 0 0
\(697\) 34.6024 1.31066
\(698\) 0 0
\(699\) −1.10868 −0.0419343
\(700\) 0 0
\(701\) −11.8124 −0.446149 −0.223074 0.974801i \(-0.571609\pi\)
−0.223074 + 0.974801i \(0.571609\pi\)
\(702\) 0 0
\(703\) −27.5756 −1.04003
\(704\) 0 0
\(705\) 2.37386 0.0894047
\(706\) 0 0
\(707\) 12.7153 0.478207
\(708\) 0 0
\(709\) 17.4545 0.655519 0.327760 0.944761i \(-0.393706\pi\)
0.327760 + 0.944761i \(0.393706\pi\)
\(710\) 0 0
\(711\) 1.60752 0.0602868
\(712\) 0 0
\(713\) −51.2633 −1.91983
\(714\) 0 0
\(715\) 18.6654 0.698047
\(716\) 0 0
\(717\) −1.75266 −0.0654543
\(718\) 0 0
\(719\) 38.2491 1.42645 0.713225 0.700935i \(-0.247234\pi\)
0.713225 + 0.700935i \(0.247234\pi\)
\(720\) 0 0
\(721\) −2.76237 −0.102876
\(722\) 0 0
\(723\) 0.460439 0.0171239
\(724\) 0 0
\(725\) −0.0524569 −0.00194820
\(726\) 0 0
\(727\) 20.9911 0.778517 0.389259 0.921129i \(-0.372731\pi\)
0.389259 + 0.921129i \(0.372731\pi\)
\(728\) 0 0
\(729\) −26.6445 −0.986834
\(730\) 0 0
\(731\) −48.9274 −1.80964
\(732\) 0 0
\(733\) −23.9796 −0.885708 −0.442854 0.896594i \(-0.646034\pi\)
−0.442854 + 0.896594i \(0.646034\pi\)
\(734\) 0 0
\(735\) −1.16479 −0.0429641
\(736\) 0 0
\(737\) 1.96044 0.0722137
\(738\) 0 0
\(739\) 29.3841 1.08091 0.540455 0.841373i \(-0.318252\pi\)
0.540455 + 0.841373i \(0.318252\pi\)
\(740\) 0 0
\(741\) 2.36756 0.0869745
\(742\) 0 0
\(743\) 38.8340 1.42468 0.712340 0.701834i \(-0.247635\pi\)
0.712340 + 0.701834i \(0.247635\pi\)
\(744\) 0 0
\(745\) −19.6221 −0.718899
\(746\) 0 0
\(747\) 9.31887 0.340960
\(748\) 0 0
\(749\) 5.02792 0.183716
\(750\) 0 0
\(751\) −12.4911 −0.455805 −0.227903 0.973684i \(-0.573187\pi\)
−0.227903 + 0.973684i \(0.573187\pi\)
\(752\) 0 0
\(753\) −0.534374 −0.0194737
\(754\) 0 0
\(755\) −31.1128 −1.13231
\(756\) 0 0
\(757\) 18.4743 0.671461 0.335731 0.941958i \(-0.391017\pi\)
0.335731 + 0.941958i \(0.391017\pi\)
\(758\) 0 0
\(759\) 0.991049 0.0359728
\(760\) 0 0
\(761\) −4.60628 −0.166977 −0.0834887 0.996509i \(-0.526606\pi\)
−0.0834887 + 0.996509i \(0.526606\pi\)
\(762\) 0 0
\(763\) −2.31212 −0.0837042
\(764\) 0 0
\(765\) 52.1379 1.88505
\(766\) 0 0
\(767\) −15.0130 −0.542089
\(768\) 0 0
\(769\) −41.1301 −1.48319 −0.741595 0.670848i \(-0.765930\pi\)
−0.741595 + 0.670848i \(0.765930\pi\)
\(770\) 0 0
\(771\) 2.26051 0.0814101
\(772\) 0 0
\(773\) −35.9381 −1.29260 −0.646301 0.763082i \(-0.723685\pi\)
−0.646301 + 0.763082i \(0.723685\pi\)
\(774\) 0 0
\(775\) 1.90284 0.0683519
\(776\) 0 0
\(777\) 0.306361 0.0109906
\(778\) 0 0
\(779\) 28.7287 1.02931
\(780\) 0 0
\(781\) −13.6267 −0.487603
\(782\) 0 0
\(783\) 0.0992628 0.00354736
\(784\) 0 0
\(785\) 26.0243 0.928847
\(786\) 0 0
\(787\) 32.3172 1.15198 0.575991 0.817456i \(-0.304616\pi\)
0.575991 + 0.817456i \(0.304616\pi\)
\(788\) 0 0
\(789\) 0.665360 0.0236874
\(790\) 0 0
\(791\) −8.12283 −0.288815
\(792\) 0 0
\(793\) 23.6682 0.840483
\(794\) 0 0
\(795\) 0.0580727 0.00205963
\(796\) 0 0
\(797\) 38.1044 1.34973 0.674863 0.737943i \(-0.264203\pi\)
0.674863 + 0.737943i \(0.264203\pi\)
\(798\) 0 0
\(799\) −96.8457 −3.42615
\(800\) 0 0
\(801\) −9.60649 −0.339429
\(802\) 0 0
\(803\) −7.36106 −0.259766
\(804\) 0 0
\(805\) −13.7089 −0.483175
\(806\) 0 0
\(807\) 2.23762 0.0787681
\(808\) 0 0
\(809\) −3.51704 −0.123652 −0.0618262 0.998087i \(-0.519692\pi\)
−0.0618262 + 0.998087i \(0.519692\pi\)
\(810\) 0 0
\(811\) 14.5661 0.511484 0.255742 0.966745i \(-0.417680\pi\)
0.255742 + 0.966745i \(0.417680\pi\)
\(812\) 0 0
\(813\) −0.131824 −0.00462325
\(814\) 0 0
\(815\) 9.93317 0.347944
\(816\) 0 0
\(817\) −40.6221 −1.42119
\(818\) 0 0
\(819\) 11.9388 0.417176
\(820\) 0 0
\(821\) 7.19351 0.251055 0.125528 0.992090i \(-0.459938\pi\)
0.125528 + 0.992090i \(0.459938\pi\)
\(822\) 0 0
\(823\) 54.0141 1.88281 0.941407 0.337272i \(-0.109504\pi\)
0.941407 + 0.337272i \(0.109504\pi\)
\(824\) 0 0
\(825\) −0.0367866 −0.00128075
\(826\) 0 0
\(827\) −23.2108 −0.807117 −0.403559 0.914954i \(-0.632227\pi\)
−0.403559 + 0.914954i \(0.632227\pi\)
\(828\) 0 0
\(829\) 24.3315 0.845067 0.422533 0.906347i \(-0.361141\pi\)
0.422533 + 0.906347i \(0.361141\pi\)
\(830\) 0 0
\(831\) −0.447335 −0.0155179
\(832\) 0 0
\(833\) 47.5198 1.64646
\(834\) 0 0
\(835\) 2.29286 0.0793478
\(836\) 0 0
\(837\) −3.60069 −0.124458
\(838\) 0 0
\(839\) −41.5082 −1.43302 −0.716511 0.697576i \(-0.754262\pi\)
−0.716511 + 0.697576i \(0.754262\pi\)
\(840\) 0 0
\(841\) −28.9584 −0.998566
\(842\) 0 0
\(843\) −1.48797 −0.0512485
\(844\) 0 0
\(845\) 19.1848 0.659977
\(846\) 0 0
\(847\) −6.81502 −0.234167
\(848\) 0 0
\(849\) −1.36854 −0.0469683
\(850\) 0 0
\(851\) −30.2974 −1.03858
\(852\) 0 0
\(853\) 35.3947 1.21189 0.605946 0.795506i \(-0.292795\pi\)
0.605946 + 0.795506i \(0.292795\pi\)
\(854\) 0 0
\(855\) 43.2876 1.48041
\(856\) 0 0
\(857\) −11.7570 −0.401613 −0.200806 0.979631i \(-0.564356\pi\)
−0.200806 + 0.979631i \(0.564356\pi\)
\(858\) 0 0
\(859\) −44.7215 −1.52588 −0.762939 0.646471i \(-0.776244\pi\)
−0.762939 + 0.646471i \(0.776244\pi\)
\(860\) 0 0
\(861\) −0.319172 −0.0108774
\(862\) 0 0
\(863\) −34.0891 −1.16041 −0.580204 0.814471i \(-0.697027\pi\)
−0.580204 + 0.814471i \(0.697027\pi\)
\(864\) 0 0
\(865\) −46.8990 −1.59462
\(866\) 0 0
\(867\) 3.30571 0.112268
\(868\) 0 0
\(869\) 0.945752 0.0320824
\(870\) 0 0
\(871\) 5.14566 0.174354
\(872\) 0 0
\(873\) −8.19193 −0.277255
\(874\) 0 0
\(875\) −9.38283 −0.317198
\(876\) 0 0
\(877\) 32.8745 1.11009 0.555046 0.831820i \(-0.312701\pi\)
0.555046 + 0.831820i \(0.312701\pi\)
\(878\) 0 0
\(879\) −0.857879 −0.0289355
\(880\) 0 0
\(881\) −46.0350 −1.55096 −0.775480 0.631373i \(-0.782492\pi\)
−0.775480 + 0.631373i \(0.782492\pi\)
\(882\) 0 0
\(883\) 49.6701 1.67153 0.835765 0.549087i \(-0.185024\pi\)
0.835765 + 0.549087i \(0.185024\pi\)
\(884\) 0 0
\(885\) 0.604755 0.0203286
\(886\) 0 0
\(887\) −19.3697 −0.650370 −0.325185 0.945650i \(-0.605427\pi\)
−0.325185 + 0.945650i \(0.605427\pi\)
\(888\) 0 0
\(889\) 10.3311 0.346494
\(890\) 0 0
\(891\) −15.7455 −0.527495
\(892\) 0 0
\(893\) −80.4064 −2.69070
\(894\) 0 0
\(895\) −0.552802 −0.0184781
\(896\) 0 0
\(897\) 2.60125 0.0868533
\(898\) 0 0
\(899\) −1.50873 −0.0503188
\(900\) 0 0
\(901\) −2.36918 −0.0789287
\(902\) 0 0
\(903\) 0.451306 0.0150185
\(904\) 0 0
\(905\) −21.5499 −0.716344
\(906\) 0 0
\(907\) 4.28749 0.142364 0.0711819 0.997463i \(-0.477323\pi\)
0.0711819 + 0.997463i \(0.477323\pi\)
\(908\) 0 0
\(909\) −44.1132 −1.46314
\(910\) 0 0
\(911\) 6.67992 0.221315 0.110658 0.993859i \(-0.464704\pi\)
0.110658 + 0.993859i \(0.464704\pi\)
\(912\) 0 0
\(913\) 5.48256 0.181446
\(914\) 0 0
\(915\) −0.953404 −0.0315186
\(916\) 0 0
\(917\) −2.21925 −0.0732860
\(918\) 0 0
\(919\) −6.93378 −0.228724 −0.114362 0.993439i \(-0.536482\pi\)
−0.114362 + 0.993439i \(0.536482\pi\)
\(920\) 0 0
\(921\) −2.51526 −0.0828807
\(922\) 0 0
\(923\) −35.7668 −1.17728
\(924\) 0 0
\(925\) 1.12461 0.0369768
\(926\) 0 0
\(927\) 9.58350 0.314764
\(928\) 0 0
\(929\) 36.7537 1.20585 0.602924 0.797798i \(-0.294002\pi\)
0.602924 + 0.797798i \(0.294002\pi\)
\(930\) 0 0
\(931\) 39.4534 1.29303
\(932\) 0 0
\(933\) −0.620621 −0.0203182
\(934\) 0 0
\(935\) 30.6743 1.00316
\(936\) 0 0
\(937\) 14.9060 0.486957 0.243479 0.969906i \(-0.421711\pi\)
0.243479 + 0.969906i \(0.421711\pi\)
\(938\) 0 0
\(939\) 0.151766 0.00495268
\(940\) 0 0
\(941\) 33.0487 1.07736 0.538679 0.842511i \(-0.318924\pi\)
0.538679 + 0.842511i \(0.318924\pi\)
\(942\) 0 0
\(943\) 31.5644 1.02788
\(944\) 0 0
\(945\) −0.962900 −0.0313231
\(946\) 0 0
\(947\) −20.7285 −0.673585 −0.336792 0.941579i \(-0.609342\pi\)
−0.336792 + 0.941579i \(0.609342\pi\)
\(948\) 0 0
\(949\) −19.3209 −0.627184
\(950\) 0 0
\(951\) −1.96935 −0.0638606
\(952\) 0 0
\(953\) −4.46024 −0.144481 −0.0722407 0.997387i \(-0.523015\pi\)
−0.0722407 + 0.997387i \(0.523015\pi\)
\(954\) 0 0
\(955\) −23.7054 −0.767088
\(956\) 0 0
\(957\) 0.0291675 0.000942851 0
\(958\) 0 0
\(959\) −19.4431 −0.627849
\(960\) 0 0
\(961\) 23.7279 0.765417
\(962\) 0 0
\(963\) −17.4434 −0.562106
\(964\) 0 0
\(965\) 29.6428 0.954235
\(966\) 0 0
\(967\) −30.6707 −0.986302 −0.493151 0.869944i \(-0.664155\pi\)
−0.493151 + 0.869944i \(0.664155\pi\)
\(968\) 0 0
\(969\) 3.89079 0.124990
\(970\) 0 0
\(971\) −15.0650 −0.483458 −0.241729 0.970344i \(-0.577714\pi\)
−0.241729 + 0.970344i \(0.577714\pi\)
\(972\) 0 0
\(973\) −9.83878 −0.315417
\(974\) 0 0
\(975\) −0.0965556 −0.00309225
\(976\) 0 0
\(977\) −27.6234 −0.883752 −0.441876 0.897076i \(-0.645687\pi\)
−0.441876 + 0.897076i \(0.645687\pi\)
\(978\) 0 0
\(979\) −5.65178 −0.180632
\(980\) 0 0
\(981\) 8.02144 0.256105
\(982\) 0 0
\(983\) 25.8238 0.823650 0.411825 0.911263i \(-0.364892\pi\)
0.411825 + 0.911263i \(0.364892\pi\)
\(984\) 0 0
\(985\) −24.8359 −0.791337
\(986\) 0 0
\(987\) 0.893305 0.0284342
\(988\) 0 0
\(989\) −44.6317 −1.41921
\(990\) 0 0
\(991\) 21.6481 0.687675 0.343838 0.939029i \(-0.388273\pi\)
0.343838 + 0.939029i \(0.388273\pi\)
\(992\) 0 0
\(993\) 1.57922 0.0501151
\(994\) 0 0
\(995\) 15.0401 0.476805
\(996\) 0 0
\(997\) 11.0483 0.349902 0.174951 0.984577i \(-0.444023\pi\)
0.174951 + 0.984577i \(0.444023\pi\)
\(998\) 0 0
\(999\) −2.12806 −0.0673289
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.c.1.5 9
4.3 odd 2 2672.2.a.m.1.5 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1336.2.a.c.1.5 9 1.1 even 1 trivial
2672.2.a.m.1.5 9 4.3 odd 2