Properties

Label 855.2.k.a.676.1
Level $855$
Weight $2$
Character 855.676
Analytic conductor $6.827$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [855,2,Mod(406,855)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(855, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("855.406");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 855.k (of order \(3\), degree \(2\), minimal)

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: no
Analytic conductor: \(6.82720937282\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 285)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 676.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 855.676
Dual form 855.2.k.a.406.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.00000 + 1.73205i) q^{2} +(-1.00000 - 1.73205i) q^{4} +(-0.500000 + 0.866025i) q^{5} -2.00000 q^{7} +(-1.00000 - 1.73205i) q^{10} -1.00000 q^{11} +(-1.00000 - 1.73205i) q^{13} +(2.00000 - 3.46410i) q^{14} +(2.00000 - 3.46410i) q^{16} +(1.00000 - 1.73205i) q^{17} +(0.500000 + 4.33013i) q^{19} +2.00000 q^{20} +(1.00000 - 1.73205i) q^{22} +(-2.00000 - 3.46410i) q^{23} +(-0.500000 - 0.866025i) q^{25} +4.00000 q^{26} +(2.00000 + 3.46410i) q^{28} +(-2.50000 - 4.33013i) q^{29} +9.00000 q^{31} +(4.00000 + 6.92820i) q^{32} +(2.00000 + 3.46410i) q^{34} +(1.00000 - 1.73205i) q^{35} -6.00000 q^{37} +(-8.00000 - 3.46410i) q^{38} +(3.00000 - 5.19615i) q^{41} +(5.00000 - 8.66025i) q^{43} +(1.00000 + 1.73205i) q^{44} +8.00000 q^{46} -3.00000 q^{49} +2.00000 q^{50} +(-2.00000 + 3.46410i) q^{52} +(-1.00000 - 1.73205i) q^{53} +(0.500000 - 0.866025i) q^{55} +10.0000 q^{58} +(3.50000 - 6.06218i) q^{59} +(3.50000 + 6.06218i) q^{61} +(-9.00000 + 15.5885i) q^{62} -8.00000 q^{64} +2.00000 q^{65} +(-4.00000 - 6.92820i) q^{67} -4.00000 q^{68} +(2.00000 + 3.46410i) q^{70} +(1.50000 - 2.59808i) q^{71} +(1.00000 - 1.73205i) q^{73} +(6.00000 - 10.3923i) q^{74} +(7.00000 - 5.19615i) q^{76} +2.00000 q^{77} +(5.50000 - 9.52628i) q^{79} +(2.00000 + 3.46410i) q^{80} +(6.00000 + 10.3923i) q^{82} -6.00000 q^{83} +(1.00000 + 1.73205i) q^{85} +(10.0000 + 17.3205i) q^{86} +(7.50000 + 12.9904i) q^{89} +(2.00000 + 3.46410i) q^{91} +(-4.00000 + 6.92820i) q^{92} +(-4.00000 - 1.73205i) q^{95} +(-4.00000 + 6.92820i) q^{97} +(3.00000 - 5.19615i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{4} - q^{5} - 4 q^{7} - 2 q^{10} - 2 q^{11} - 2 q^{13} + 4 q^{14} + 4 q^{16} + 2 q^{17} + q^{19} + 4 q^{20} + 2 q^{22} - 4 q^{23} - q^{25} + 8 q^{26} + 4 q^{28} - 5 q^{29} + 18 q^{31}+ \cdots + 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/855\mathbb{Z}\right)^\times\).

\(n\) \(172\) \(191\) \(496\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 + 1.73205i −0.707107 + 1.22474i 0.258819 + 0.965926i \(0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(3\) 0 0
\(4\) −1.00000 1.73205i −0.500000 0.866025i
\(5\) −0.500000 + 0.866025i −0.223607 + 0.387298i
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) −1.00000 1.73205i −0.316228 0.547723i
\(11\) −1.00000 −0.301511 −0.150756 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) 0 0
\(13\) −1.00000 1.73205i −0.277350 0.480384i 0.693375 0.720577i \(-0.256123\pi\)
−0.970725 + 0.240192i \(0.922790\pi\)
\(14\) 2.00000 3.46410i 0.534522 0.925820i
\(15\) 0 0
\(16\) 2.00000 3.46410i 0.500000 0.866025i
\(17\) 1.00000 1.73205i 0.242536 0.420084i −0.718900 0.695113i \(-0.755354\pi\)
0.961436 + 0.275029i \(0.0886875\pi\)
\(18\) 0 0
\(19\) 0.500000 + 4.33013i 0.114708 + 0.993399i
\(20\) 2.00000 0.447214
\(21\) 0 0
\(22\) 1.00000 1.73205i 0.213201 0.369274i
\(23\) −2.00000 3.46410i −0.417029 0.722315i 0.578610 0.815604i \(-0.303595\pi\)
−0.995639 + 0.0932891i \(0.970262\pi\)
\(24\) 0 0
\(25\) −0.500000 0.866025i −0.100000 0.173205i
\(26\) 4.00000 0.784465
\(27\) 0 0
\(28\) 2.00000 + 3.46410i 0.377964 + 0.654654i
\(29\) −2.50000 4.33013i −0.464238 0.804084i 0.534928 0.844897i \(-0.320339\pi\)
−0.999167 + 0.0408130i \(0.987005\pi\)
\(30\) 0 0
\(31\) 9.00000 1.61645 0.808224 0.588875i \(-0.200429\pi\)
0.808224 + 0.588875i \(0.200429\pi\)
\(32\) 4.00000 + 6.92820i 0.707107 + 1.22474i
\(33\) 0 0
\(34\) 2.00000 + 3.46410i 0.342997 + 0.594089i
\(35\) 1.00000 1.73205i 0.169031 0.292770i
\(36\) 0 0
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) −8.00000 3.46410i −1.29777 0.561951i
\(39\) 0 0
\(40\) 0 0
\(41\) 3.00000 5.19615i 0.468521 0.811503i −0.530831 0.847477i \(-0.678120\pi\)
0.999353 + 0.0359748i \(0.0114536\pi\)
\(42\) 0 0
\(43\) 5.00000 8.66025i 0.762493 1.32068i −0.179069 0.983836i \(-0.557309\pi\)
0.941562 0.336840i \(-0.109358\pi\)
\(44\) 1.00000 + 1.73205i 0.150756 + 0.261116i
\(45\) 0 0
\(46\) 8.00000 1.17954
\(47\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 2.00000 0.282843
\(51\) 0 0
\(52\) −2.00000 + 3.46410i −0.277350 + 0.480384i
\(53\) −1.00000 1.73205i −0.137361 0.237915i 0.789136 0.614218i \(-0.210529\pi\)
−0.926497 + 0.376303i \(0.877195\pi\)
\(54\) 0 0
\(55\) 0.500000 0.866025i 0.0674200 0.116775i
\(56\) 0 0
\(57\) 0 0
\(58\) 10.0000 1.31306
\(59\) 3.50000 6.06218i 0.455661 0.789228i −0.543065 0.839691i \(-0.682736\pi\)
0.998726 + 0.0504625i \(0.0160695\pi\)
\(60\) 0 0
\(61\) 3.50000 + 6.06218i 0.448129 + 0.776182i 0.998264 0.0588933i \(-0.0187572\pi\)
−0.550135 + 0.835076i \(0.685424\pi\)
\(62\) −9.00000 + 15.5885i −1.14300 + 1.97974i
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) −4.00000 6.92820i −0.488678 0.846415i 0.511237 0.859440i \(-0.329187\pi\)
−0.999915 + 0.0130248i \(0.995854\pi\)
\(68\) −4.00000 −0.485071
\(69\) 0 0
\(70\) 2.00000 + 3.46410i 0.239046 + 0.414039i
\(71\) 1.50000 2.59808i 0.178017 0.308335i −0.763184 0.646181i \(-0.776365\pi\)
0.941201 + 0.337846i \(0.109698\pi\)
\(72\) 0 0
\(73\) 1.00000 1.73205i 0.117041 0.202721i −0.801553 0.597924i \(-0.795992\pi\)
0.918594 + 0.395203i \(0.129326\pi\)
\(74\) 6.00000 10.3923i 0.697486 1.20808i
\(75\) 0 0
\(76\) 7.00000 5.19615i 0.802955 0.596040i
\(77\) 2.00000 0.227921
\(78\) 0 0
\(79\) 5.50000 9.52628i 0.618798 1.07179i −0.370907 0.928670i \(-0.620953\pi\)
0.989705 0.143120i \(-0.0457135\pi\)
\(80\) 2.00000 + 3.46410i 0.223607 + 0.387298i
\(81\) 0 0
\(82\) 6.00000 + 10.3923i 0.662589 + 1.14764i
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) 1.00000 + 1.73205i 0.108465 + 0.187867i
\(86\) 10.0000 + 17.3205i 1.07833 + 1.86772i
\(87\) 0 0
\(88\) 0 0
\(89\) 7.50000 + 12.9904i 0.794998 + 1.37698i 0.922840 + 0.385183i \(0.125862\pi\)
−0.127842 + 0.991795i \(0.540805\pi\)
\(90\) 0 0
\(91\) 2.00000 + 3.46410i 0.209657 + 0.363137i
\(92\) −4.00000 + 6.92820i −0.417029 + 0.722315i
\(93\) 0 0
\(94\) 0 0
\(95\) −4.00000 1.73205i −0.410391 0.177705i
\(96\) 0 0
\(97\) −4.00000 + 6.92820i −0.406138 + 0.703452i −0.994453 0.105180i \(-0.966458\pi\)
0.588315 + 0.808632i \(0.299792\pi\)
\(98\) 3.00000 5.19615i 0.303046 0.524891i
\(99\) 0 0
\(100\) −1.00000 + 1.73205i −0.100000 + 0.173205i
\(101\) −3.50000 6.06218i −0.348263 0.603209i 0.637678 0.770303i \(-0.279895\pi\)
−0.985941 + 0.167094i \(0.946562\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 4.00000 0.388514
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) −9.50000 + 16.4545i −0.909935 + 1.57605i −0.0957826 + 0.995402i \(0.530535\pi\)
−0.814152 + 0.580651i \(0.802798\pi\)
\(110\) 1.00000 + 1.73205i 0.0953463 + 0.165145i
\(111\) 0 0
\(112\) −4.00000 + 6.92820i −0.377964 + 0.654654i
\(113\) 10.0000 0.940721 0.470360 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) 0 0
\(115\) 4.00000 0.373002
\(116\) −5.00000 + 8.66025i −0.464238 + 0.804084i
\(117\) 0 0
\(118\) 7.00000 + 12.1244i 0.644402 + 1.11614i
\(119\) −2.00000 + 3.46410i −0.183340 + 0.317554i
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) −14.0000 −1.26750
\(123\) 0 0
\(124\) −9.00000 15.5885i −0.808224 1.39988i
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −2.00000 3.46410i −0.177471 0.307389i 0.763542 0.645758i \(-0.223458\pi\)
−0.941014 + 0.338368i \(0.890125\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) −2.00000 + 3.46410i −0.175412 + 0.303822i
\(131\) 8.00000 13.8564i 0.698963 1.21064i −0.269863 0.962899i \(-0.586978\pi\)
0.968826 0.247741i \(-0.0796882\pi\)
\(132\) 0 0
\(133\) −1.00000 8.66025i −0.0867110 0.750939i
\(134\) 16.0000 1.38219
\(135\) 0 0
\(136\) 0 0
\(137\) −3.00000 5.19615i −0.256307 0.443937i 0.708942 0.705266i \(-0.249173\pi\)
−0.965250 + 0.261329i \(0.915839\pi\)
\(138\) 0 0
\(139\) −10.0000 17.3205i −0.848189 1.46911i −0.882823 0.469706i \(-0.844360\pi\)
0.0346338 0.999400i \(-0.488974\pi\)
\(140\) −4.00000 −0.338062
\(141\) 0 0
\(142\) 3.00000 + 5.19615i 0.251754 + 0.436051i
\(143\) 1.00000 + 1.73205i 0.0836242 + 0.144841i
\(144\) 0 0
\(145\) 5.00000 0.415227
\(146\) 2.00000 + 3.46410i 0.165521 + 0.286691i
\(147\) 0 0
\(148\) 6.00000 + 10.3923i 0.493197 + 0.854242i
\(149\) 4.50000 7.79423i 0.368654 0.638528i −0.620701 0.784047i \(-0.713152\pi\)
0.989355 + 0.145519i \(0.0464853\pi\)
\(150\) 0 0
\(151\) −3.00000 −0.244137 −0.122068 0.992522i \(-0.538953\pi\)
−0.122068 + 0.992522i \(0.538953\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −2.00000 + 3.46410i −0.161165 + 0.279145i
\(155\) −4.50000 + 7.79423i −0.361449 + 0.626048i
\(156\) 0 0
\(157\) −6.00000 + 10.3923i −0.478852 + 0.829396i −0.999706 0.0242497i \(-0.992280\pi\)
0.520854 + 0.853646i \(0.325614\pi\)
\(158\) 11.0000 + 19.0526i 0.875113 + 1.51574i
\(159\) 0 0
\(160\) −8.00000 −0.632456
\(161\) 4.00000 + 6.92820i 0.315244 + 0.546019i
\(162\) 0 0
\(163\) −18.0000 −1.40987 −0.704934 0.709273i \(-0.749024\pi\)
−0.704934 + 0.709273i \(0.749024\pi\)
\(164\) −12.0000 −0.937043
\(165\) 0 0
\(166\) 6.00000 10.3923i 0.465690 0.806599i
\(167\) −1.00000 1.73205i −0.0773823 0.134030i 0.824737 0.565516i \(-0.191323\pi\)
−0.902120 + 0.431486i \(0.857990\pi\)
\(168\) 0 0
\(169\) 4.50000 7.79423i 0.346154 0.599556i
\(170\) −4.00000 −0.306786
\(171\) 0 0
\(172\) −20.0000 −1.52499
\(173\) 3.00000 5.19615i 0.228086 0.395056i −0.729155 0.684349i \(-0.760087\pi\)
0.957241 + 0.289292i \(0.0934200\pi\)
\(174\) 0 0
\(175\) 1.00000 + 1.73205i 0.0755929 + 0.130931i
\(176\) −2.00000 + 3.46410i −0.150756 + 0.261116i
\(177\) 0 0
\(178\) −30.0000 −2.24860
\(179\) 3.00000 0.224231 0.112115 0.993695i \(-0.464237\pi\)
0.112115 + 0.993695i \(0.464237\pi\)
\(180\) 0 0
\(181\) −5.00000 8.66025i −0.371647 0.643712i 0.618172 0.786043i \(-0.287874\pi\)
−0.989819 + 0.142331i \(0.954540\pi\)
\(182\) −8.00000 −0.592999
\(183\) 0 0
\(184\) 0 0
\(185\) 3.00000 5.19615i 0.220564 0.382029i
\(186\) 0 0
\(187\) −1.00000 + 1.73205i −0.0731272 + 0.126660i
\(188\) 0 0
\(189\) 0 0
\(190\) 7.00000 5.19615i 0.507833 0.376969i
\(191\) −15.0000 −1.08536 −0.542681 0.839939i \(-0.682591\pi\)
−0.542681 + 0.839939i \(0.682591\pi\)
\(192\) 0 0
\(193\) 13.0000 22.5167i 0.935760 1.62078i 0.162488 0.986710i \(-0.448048\pi\)
0.773272 0.634074i \(-0.218619\pi\)
\(194\) −8.00000 13.8564i −0.574367 0.994832i
\(195\) 0 0
\(196\) 3.00000 + 5.19615i 0.214286 + 0.371154i
\(197\) −10.0000 −0.712470 −0.356235 0.934396i \(-0.615940\pi\)
−0.356235 + 0.934396i \(0.615940\pi\)
\(198\) 0 0
\(199\) 1.50000 + 2.59808i 0.106332 + 0.184173i 0.914282 0.405079i \(-0.132756\pi\)
−0.807950 + 0.589252i \(0.799423\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 14.0000 0.985037
\(203\) 5.00000 + 8.66025i 0.350931 + 0.607831i
\(204\) 0 0
\(205\) 3.00000 + 5.19615i 0.209529 + 0.362915i
\(206\) 8.00000 13.8564i 0.557386 0.965422i
\(207\) 0 0
\(208\) −8.00000 −0.554700
\(209\) −0.500000 4.33013i −0.0345857 0.299521i
\(210\) 0 0
\(211\) 13.5000 23.3827i 0.929378 1.60973i 0.145014 0.989430i \(-0.453677\pi\)
0.784364 0.620301i \(-0.212990\pi\)
\(212\) −2.00000 + 3.46410i −0.137361 + 0.237915i
\(213\) 0 0
\(214\) 0 0
\(215\) 5.00000 + 8.66025i 0.340997 + 0.590624i
\(216\) 0 0
\(217\) −18.0000 −1.22192
\(218\) −19.0000 32.9090i −1.28684 2.22888i
\(219\) 0 0
\(220\) −2.00000 −0.134840
\(221\) −4.00000 −0.269069
\(222\) 0 0
\(223\) 6.00000 10.3923i 0.401790 0.695920i −0.592152 0.805826i \(-0.701722\pi\)
0.993942 + 0.109906i \(0.0350549\pi\)
\(224\) −8.00000 13.8564i −0.534522 0.925820i
\(225\) 0 0
\(226\) −10.0000 + 17.3205i −0.665190 + 1.15214i
\(227\) −18.0000 −1.19470 −0.597351 0.801980i \(-0.703780\pi\)
−0.597351 + 0.801980i \(0.703780\pi\)
\(228\) 0 0
\(229\) −11.0000 −0.726900 −0.363450 0.931614i \(-0.618401\pi\)
−0.363450 + 0.931614i \(0.618401\pi\)
\(230\) −4.00000 + 6.92820i −0.263752 + 0.456832i
\(231\) 0 0
\(232\) 0 0
\(233\) −12.0000 + 20.7846i −0.786146 + 1.36165i 0.142166 + 0.989843i \(0.454593\pi\)
−0.928312 + 0.371802i \(0.878740\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −14.0000 −0.911322
\(237\) 0 0
\(238\) −4.00000 6.92820i −0.259281 0.449089i
\(239\) −27.0000 −1.74648 −0.873242 0.487286i \(-0.837987\pi\)
−0.873242 + 0.487286i \(0.837987\pi\)
\(240\) 0 0
\(241\) −2.50000 4.33013i −0.161039 0.278928i 0.774202 0.632938i \(-0.218151\pi\)
−0.935242 + 0.354010i \(0.884818\pi\)
\(242\) 10.0000 17.3205i 0.642824 1.11340i
\(243\) 0 0
\(244\) 7.00000 12.1244i 0.448129 0.776182i
\(245\) 1.50000 2.59808i 0.0958315 0.165985i
\(246\) 0 0
\(247\) 7.00000 5.19615i 0.445399 0.330623i
\(248\) 0 0
\(249\) 0 0
\(250\) −1.00000 + 1.73205i −0.0632456 + 0.109545i
\(251\) −8.50000 14.7224i −0.536515 0.929272i −0.999088 0.0426905i \(-0.986407\pi\)
0.462573 0.886581i \(-0.346926\pi\)
\(252\) 0 0
\(253\) 2.00000 + 3.46410i 0.125739 + 0.217786i
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) −8.00000 13.8564i −0.500000 0.866025i
\(257\) 11.0000 + 19.0526i 0.686161 + 1.18847i 0.973070 + 0.230508i \(0.0740389\pi\)
−0.286909 + 0.957958i \(0.592628\pi\)
\(258\) 0 0
\(259\) 12.0000 0.745644
\(260\) −2.00000 3.46410i −0.124035 0.214834i
\(261\) 0 0
\(262\) 16.0000 + 27.7128i 0.988483 + 1.71210i
\(263\) 10.0000 17.3205i 0.616626 1.06803i −0.373470 0.927642i \(-0.621832\pi\)
0.990097 0.140386i \(-0.0448344\pi\)
\(264\) 0 0
\(265\) 2.00000 0.122859
\(266\) 16.0000 + 6.92820i 0.981023 + 0.424795i
\(267\) 0 0
\(268\) −8.00000 + 13.8564i −0.488678 + 0.846415i
\(269\) 0.500000 0.866025i 0.0304855 0.0528025i −0.850380 0.526169i \(-0.823628\pi\)
0.880866 + 0.473366i \(0.156961\pi\)
\(270\) 0 0
\(271\) −13.5000 + 23.3827i −0.820067 + 1.42040i 0.0855654 + 0.996333i \(0.472730\pi\)
−0.905632 + 0.424064i \(0.860603\pi\)
\(272\) −4.00000 6.92820i −0.242536 0.420084i
\(273\) 0 0
\(274\) 12.0000 0.724947
\(275\) 0.500000 + 0.866025i 0.0301511 + 0.0522233i
\(276\) 0 0
\(277\) −8.00000 −0.480673 −0.240337 0.970690i \(-0.577258\pi\)
−0.240337 + 0.970690i \(0.577258\pi\)
\(278\) 40.0000 2.39904
\(279\) 0 0
\(280\) 0 0
\(281\) −13.0000 22.5167i −0.775515 1.34323i −0.934505 0.355951i \(-0.884157\pi\)
0.158990 0.987280i \(-0.449176\pi\)
\(282\) 0 0
\(283\) −1.00000 + 1.73205i −0.0594438 + 0.102960i −0.894216 0.447636i \(-0.852266\pi\)
0.834772 + 0.550596i \(0.185599\pi\)
\(284\) −6.00000 −0.356034
\(285\) 0 0
\(286\) −4.00000 −0.236525
\(287\) −6.00000 + 10.3923i −0.354169 + 0.613438i
\(288\) 0 0
\(289\) 6.50000 + 11.2583i 0.382353 + 0.662255i
\(290\) −5.00000 + 8.66025i −0.293610 + 0.508548i
\(291\) 0 0
\(292\) −4.00000 −0.234082
\(293\) 4.00000 0.233682 0.116841 0.993151i \(-0.462723\pi\)
0.116841 + 0.993151i \(0.462723\pi\)
\(294\) 0 0
\(295\) 3.50000 + 6.06218i 0.203778 + 0.352954i
\(296\) 0 0
\(297\) 0 0
\(298\) 9.00000 + 15.5885i 0.521356 + 0.903015i
\(299\) −4.00000 + 6.92820i −0.231326 + 0.400668i
\(300\) 0 0
\(301\) −10.0000 + 17.3205i −0.576390 + 0.998337i
\(302\) 3.00000 5.19615i 0.172631 0.299005i
\(303\) 0 0
\(304\) 16.0000 + 6.92820i 0.917663 + 0.397360i
\(305\) −7.00000 −0.400819
\(306\) 0 0
\(307\) 12.0000 20.7846i 0.684876 1.18624i −0.288600 0.957450i \(-0.593190\pi\)
0.973476 0.228790i \(-0.0734771\pi\)
\(308\) −2.00000 3.46410i −0.113961 0.197386i
\(309\) 0 0
\(310\) −9.00000 15.5885i −0.511166 0.885365i
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 0 0
\(313\) 10.0000 + 17.3205i 0.565233 + 0.979013i 0.997028 + 0.0770410i \(0.0245472\pi\)
−0.431795 + 0.901972i \(0.642119\pi\)
\(314\) −12.0000 20.7846i −0.677199 1.17294i
\(315\) 0 0
\(316\) −22.0000 −1.23760
\(317\) 16.0000 + 27.7128i 0.898650 + 1.55651i 0.829222 + 0.558920i \(0.188784\pi\)
0.0694277 + 0.997587i \(0.477883\pi\)
\(318\) 0 0
\(319\) 2.50000 + 4.33013i 0.139973 + 0.242441i
\(320\) 4.00000 6.92820i 0.223607 0.387298i
\(321\) 0 0
\(322\) −16.0000 −0.891645
\(323\) 8.00000 + 3.46410i 0.445132 + 0.192748i
\(324\) 0 0
\(325\) −1.00000 + 1.73205i −0.0554700 + 0.0960769i
\(326\) 18.0000 31.1769i 0.996928 1.72673i
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 8.00000 0.439720 0.219860 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) 6.00000 + 10.3923i 0.329293 + 0.570352i
\(333\) 0 0
\(334\) 4.00000 0.218870
\(335\) 8.00000 0.437087
\(336\) 0 0
\(337\) 4.00000 6.92820i 0.217894 0.377403i −0.736270 0.676688i \(-0.763415\pi\)
0.954164 + 0.299285i \(0.0967480\pi\)
\(338\) 9.00000 + 15.5885i 0.489535 + 0.847900i
\(339\) 0 0
\(340\) 2.00000 3.46410i 0.108465 0.187867i
\(341\) −9.00000 −0.487377
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 0 0
\(345\) 0 0
\(346\) 6.00000 + 10.3923i 0.322562 + 0.558694i
\(347\) −11.0000 + 19.0526i −0.590511 + 1.02279i 0.403653 + 0.914912i \(0.367740\pi\)
−0.994164 + 0.107883i \(0.965593\pi\)
\(348\) 0 0
\(349\) 6.00000 0.321173 0.160586 0.987022i \(-0.448662\pi\)
0.160586 + 0.987022i \(0.448662\pi\)
\(350\) −4.00000 −0.213809
\(351\) 0 0
\(352\) −4.00000 6.92820i −0.213201 0.369274i
\(353\) 14.0000 0.745145 0.372572 0.928003i \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) 0 0
\(355\) 1.50000 + 2.59808i 0.0796117 + 0.137892i
\(356\) 15.0000 25.9808i 0.794998 1.37698i
\(357\) 0 0
\(358\) −3.00000 + 5.19615i −0.158555 + 0.274625i
\(359\) −4.00000 + 6.92820i −0.211112 + 0.365657i −0.952063 0.305903i \(-0.901042\pi\)
0.740951 + 0.671559i \(0.234375\pi\)
\(360\) 0 0
\(361\) −18.5000 + 4.33013i −0.973684 + 0.227901i
\(362\) 20.0000 1.05118
\(363\) 0 0
\(364\) 4.00000 6.92820i 0.209657 0.363137i
\(365\) 1.00000 + 1.73205i 0.0523424 + 0.0906597i
\(366\) 0 0
\(367\) 10.0000 + 17.3205i 0.521996 + 0.904123i 0.999673 + 0.0255875i \(0.00814566\pi\)
−0.477677 + 0.878536i \(0.658521\pi\)
\(368\) −16.0000 −0.834058
\(369\) 0 0
\(370\) 6.00000 + 10.3923i 0.311925 + 0.540270i
\(371\) 2.00000 + 3.46410i 0.103835 + 0.179847i
\(372\) 0 0
\(373\) −32.0000 −1.65690 −0.828449 0.560065i \(-0.810776\pi\)
−0.828449 + 0.560065i \(0.810776\pi\)
\(374\) −2.00000 3.46410i −0.103418 0.179124i
\(375\) 0 0
\(376\) 0 0
\(377\) −5.00000 + 8.66025i −0.257513 + 0.446026i
\(378\) 0 0
\(379\) 31.0000 1.59236 0.796182 0.605058i \(-0.206850\pi\)
0.796182 + 0.605058i \(0.206850\pi\)
\(380\) 1.00000 + 8.66025i 0.0512989 + 0.444262i
\(381\) 0 0
\(382\) 15.0000 25.9808i 0.767467 1.32929i
\(383\) 13.0000 22.5167i 0.664269 1.15055i −0.315214 0.949021i \(-0.602076\pi\)
0.979483 0.201527i \(-0.0645904\pi\)
\(384\) 0 0
\(385\) −1.00000 + 1.73205i −0.0509647 + 0.0882735i
\(386\) 26.0000 + 45.0333i 1.32337 + 2.29214i
\(387\) 0 0
\(388\) 16.0000 0.812277
\(389\) 16.5000 + 28.5788i 0.836583 + 1.44900i 0.892735 + 0.450582i \(0.148784\pi\)
−0.0561516 + 0.998422i \(0.517883\pi\)
\(390\) 0 0
\(391\) −8.00000 −0.404577
\(392\) 0 0
\(393\) 0 0
\(394\) 10.0000 17.3205i 0.503793 0.872595i
\(395\) 5.50000 + 9.52628i 0.276735 + 0.479319i
\(396\) 0 0
\(397\) 10.0000 17.3205i 0.501886 0.869291i −0.498112 0.867113i \(-0.665973\pi\)
0.999998 0.00217869i \(-0.000693499\pi\)
\(398\) −6.00000 −0.300753
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) −13.5000 + 23.3827i −0.674158 + 1.16768i 0.302556 + 0.953131i \(0.402160\pi\)
−0.976714 + 0.214544i \(0.931173\pi\)
\(402\) 0 0
\(403\) −9.00000 15.5885i −0.448322 0.776516i
\(404\) −7.00000 + 12.1244i −0.348263 + 0.603209i
\(405\) 0 0
\(406\) −20.0000 −0.992583
\(407\) 6.00000 0.297409
\(408\) 0 0
\(409\) 14.5000 + 25.1147i 0.716979 + 1.24184i 0.962191 + 0.272374i \(0.0878089\pi\)
−0.245212 + 0.969469i \(0.578858\pi\)
\(410\) −12.0000 −0.592638
\(411\) 0 0
\(412\) 8.00000 + 13.8564i 0.394132 + 0.682656i
\(413\) −7.00000 + 12.1244i −0.344447 + 0.596601i
\(414\) 0 0
\(415\) 3.00000 5.19615i 0.147264 0.255069i
\(416\) 8.00000 13.8564i 0.392232 0.679366i
\(417\) 0 0
\(418\) 8.00000 + 3.46410i 0.391293 + 0.169435i
\(419\) 5.00000 0.244266 0.122133 0.992514i \(-0.461027\pi\)
0.122133 + 0.992514i \(0.461027\pi\)
\(420\) 0 0
\(421\) 3.50000 6.06218i 0.170580 0.295452i −0.768043 0.640398i \(-0.778769\pi\)
0.938623 + 0.344946i \(0.112103\pi\)
\(422\) 27.0000 + 46.7654i 1.31434 + 2.27650i
\(423\) 0 0
\(424\) 0 0
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) −7.00000 12.1244i −0.338754 0.586739i
\(428\) 0 0
\(429\) 0 0
\(430\) −20.0000 −0.964486
\(431\) 11.5000 + 19.9186i 0.553936 + 0.959444i 0.997985 + 0.0634424i \(0.0202079\pi\)
−0.444050 + 0.896002i \(0.646459\pi\)
\(432\) 0 0
\(433\) −3.00000 5.19615i −0.144171 0.249711i 0.784892 0.619632i \(-0.212718\pi\)
−0.929063 + 0.369921i \(0.879385\pi\)
\(434\) 18.0000 31.1769i 0.864028 1.49654i
\(435\) 0 0
\(436\) 38.0000 1.81987
\(437\) 14.0000 10.3923i 0.669711 0.497131i
\(438\) 0 0
\(439\) 12.5000 21.6506i 0.596592 1.03333i −0.396728 0.917936i \(-0.629854\pi\)
0.993320 0.115392i \(-0.0368124\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 4.00000 6.92820i 0.190261 0.329541i
\(443\) −4.00000 6.92820i −0.190046 0.329169i 0.755219 0.655472i \(-0.227530\pi\)
−0.945265 + 0.326303i \(0.894197\pi\)
\(444\) 0 0
\(445\) −15.0000 −0.711068
\(446\) 12.0000 + 20.7846i 0.568216 + 0.984180i
\(447\) 0 0
\(448\) 16.0000 0.755929
\(449\) 39.0000 1.84052 0.920262 0.391303i \(-0.127976\pi\)
0.920262 + 0.391303i \(0.127976\pi\)
\(450\) 0 0
\(451\) −3.00000 + 5.19615i −0.141264 + 0.244677i
\(452\) −10.0000 17.3205i −0.470360 0.814688i
\(453\) 0 0
\(454\) 18.0000 31.1769i 0.844782 1.46321i
\(455\) −4.00000 −0.187523
\(456\) 0 0
\(457\) −8.00000 −0.374224 −0.187112 0.982339i \(-0.559913\pi\)
−0.187112 + 0.982339i \(0.559913\pi\)
\(458\) 11.0000 19.0526i 0.513996 0.890268i
\(459\) 0 0
\(460\) −4.00000 6.92820i −0.186501 0.323029i
\(461\) −1.50000 + 2.59808i −0.0698620 + 0.121004i −0.898840 0.438276i \(-0.855589\pi\)
0.828978 + 0.559281i \(0.188923\pi\)
\(462\) 0 0
\(463\) −10.0000 −0.464739 −0.232370 0.972628i \(-0.574648\pi\)
−0.232370 + 0.972628i \(0.574648\pi\)
\(464\) −20.0000 −0.928477
\(465\) 0 0
\(466\) −24.0000 41.5692i −1.11178 1.92566i
\(467\) 38.0000 1.75843 0.879215 0.476425i \(-0.158068\pi\)
0.879215 + 0.476425i \(0.158068\pi\)
\(468\) 0 0
\(469\) 8.00000 + 13.8564i 0.369406 + 0.639829i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −5.00000 + 8.66025i −0.229900 + 0.398199i
\(474\) 0 0
\(475\) 3.50000 2.59808i 0.160591 0.119208i
\(476\) 8.00000 0.366679
\(477\) 0 0
\(478\) 27.0000 46.7654i 1.23495 2.13900i
\(479\) 2.50000 + 4.33013i 0.114228 + 0.197849i 0.917471 0.397803i \(-0.130227\pi\)
−0.803243 + 0.595652i \(0.796894\pi\)
\(480\) 0 0
\(481\) 6.00000 + 10.3923i 0.273576 + 0.473848i
\(482\) 10.0000 0.455488
\(483\) 0 0
\(484\) 10.0000 + 17.3205i 0.454545 + 0.787296i
\(485\) −4.00000 6.92820i −0.181631 0.314594i
\(486\) 0 0
\(487\) 14.0000 0.634401 0.317200 0.948359i \(-0.397257\pi\)
0.317200 + 0.948359i \(0.397257\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 3.00000 + 5.19615i 0.135526 + 0.234738i
\(491\) −16.5000 + 28.5788i −0.744635 + 1.28974i 0.205731 + 0.978609i \(0.434043\pi\)
−0.950365 + 0.311136i \(0.899290\pi\)
\(492\) 0 0
\(493\) −10.0000 −0.450377
\(494\) 2.00000 + 17.3205i 0.0899843 + 0.779287i
\(495\) 0 0
\(496\) 18.0000 31.1769i 0.808224 1.39988i
\(497\) −3.00000 + 5.19615i −0.134568 + 0.233079i
\(498\) 0 0
\(499\) −10.0000 + 17.3205i −0.447661 + 0.775372i −0.998233 0.0594153i \(-0.981076\pi\)
0.550572 + 0.834788i \(0.314410\pi\)
\(500\) −1.00000 1.73205i −0.0447214 0.0774597i
\(501\) 0 0
\(502\) 34.0000 1.51749
\(503\) −22.0000 38.1051i −0.980932 1.69902i −0.658781 0.752335i \(-0.728928\pi\)
−0.322151 0.946688i \(-0.604406\pi\)
\(504\) 0 0
\(505\) 7.00000 0.311496
\(506\) −8.00000 −0.355643
\(507\) 0 0
\(508\) −4.00000 + 6.92820i −0.177471 + 0.307389i
\(509\) −15.0000 25.9808i −0.664863 1.15158i −0.979322 0.202306i \(-0.935156\pi\)
0.314459 0.949271i \(-0.398177\pi\)
\(510\) 0 0
\(511\) −2.00000 + 3.46410i −0.0884748 + 0.153243i
\(512\) 32.0000 1.41421
\(513\) 0 0
\(514\) −44.0000 −1.94076
\(515\) 4.00000 6.92820i 0.176261 0.305293i
\(516\) 0 0
\(517\) 0 0
\(518\) −12.0000 + 20.7846i −0.527250 + 0.913223i
\(519\) 0 0
\(520\) 0 0
\(521\) −21.0000 −0.920027 −0.460013 0.887912i \(-0.652155\pi\)
−0.460013 + 0.887912i \(0.652155\pi\)
\(522\) 0 0
\(523\) 11.0000 + 19.0526i 0.480996 + 0.833110i 0.999762 0.0218062i \(-0.00694167\pi\)
−0.518766 + 0.854916i \(0.673608\pi\)
\(524\) −32.0000 −1.39793
\(525\) 0 0
\(526\) 20.0000 + 34.6410i 0.872041 + 1.51042i
\(527\) 9.00000 15.5885i 0.392046 0.679044i
\(528\) 0 0
\(529\) 3.50000 6.06218i 0.152174 0.263573i
\(530\) −2.00000 + 3.46410i −0.0868744 + 0.150471i
\(531\) 0 0
\(532\) −14.0000 + 10.3923i −0.606977 + 0.450564i
\(533\) −12.0000 −0.519778
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 1.00000 + 1.73205i 0.0431131 + 0.0746740i
\(539\) 3.00000 0.129219
\(540\) 0 0
\(541\) −7.50000 12.9904i −0.322450 0.558500i 0.658543 0.752543i \(-0.271173\pi\)
−0.980993 + 0.194043i \(0.937840\pi\)
\(542\) −27.0000 46.7654i −1.15975 2.00874i
\(543\) 0 0
\(544\) 16.0000 0.685994
\(545\) −9.50000 16.4545i −0.406935 0.704833i
\(546\) 0 0
\(547\) 4.00000 + 6.92820i 0.171028 + 0.296229i 0.938779 0.344519i \(-0.111958\pi\)
−0.767752 + 0.640747i \(0.778625\pi\)
\(548\) −6.00000 + 10.3923i −0.256307 + 0.443937i
\(549\) 0 0
\(550\) −2.00000 −0.0852803
\(551\) 17.5000 12.9904i 0.745525 0.553409i
\(552\) 0 0
\(553\) −11.0000 + 19.0526i −0.467768 + 0.810197i
\(554\) 8.00000 13.8564i 0.339887 0.588702i
\(555\) 0 0
\(556\) −20.0000 + 34.6410i −0.848189 + 1.46911i
\(557\) −16.0000 27.7128i −0.677942 1.17423i −0.975600 0.219557i \(-0.929539\pi\)
0.297658 0.954673i \(-0.403795\pi\)
\(558\) 0 0
\(559\) −20.0000 −0.845910
\(560\) −4.00000 6.92820i −0.169031 0.292770i
\(561\) 0 0
\(562\) 52.0000 2.19349
\(563\) 42.0000 1.77009 0.885044 0.465506i \(-0.154128\pi\)
0.885044 + 0.465506i \(0.154128\pi\)
\(564\) 0 0
\(565\) −5.00000 + 8.66025i −0.210352 + 0.364340i
\(566\) −2.00000 3.46410i −0.0840663 0.145607i
\(567\) 0 0
\(568\) 0 0
\(569\) 39.0000 1.63497 0.817483 0.575953i \(-0.195369\pi\)
0.817483 + 0.575953i \(0.195369\pi\)
\(570\) 0 0
\(571\) −15.0000 −0.627730 −0.313865 0.949468i \(-0.601624\pi\)
−0.313865 + 0.949468i \(0.601624\pi\)
\(572\) 2.00000 3.46410i 0.0836242 0.144841i
\(573\) 0 0
\(574\) −12.0000 20.7846i −0.500870 0.867533i
\(575\) −2.00000 + 3.46410i −0.0834058 + 0.144463i
\(576\) 0 0
\(577\) 12.0000 0.499567 0.249783 0.968302i \(-0.419641\pi\)
0.249783 + 0.968302i \(0.419641\pi\)
\(578\) −26.0000 −1.08146
\(579\) 0 0
\(580\) −5.00000 8.66025i −0.207614 0.359597i
\(581\) 12.0000 0.497844
\(582\) 0 0
\(583\) 1.00000 + 1.73205i 0.0414158 + 0.0717342i
\(584\) 0 0
\(585\) 0 0
\(586\) −4.00000 + 6.92820i −0.165238 + 0.286201i
\(587\) 11.0000 19.0526i 0.454019 0.786383i −0.544613 0.838688i \(-0.683323\pi\)
0.998631 + 0.0523045i \(0.0166566\pi\)
\(588\) 0 0
\(589\) 4.50000 + 38.9711i 0.185419 + 1.60578i
\(590\) −14.0000 −0.576371
\(591\) 0 0
\(592\) −12.0000 + 20.7846i −0.493197 + 0.854242i
\(593\) 3.00000 + 5.19615i 0.123195 + 0.213380i 0.921026 0.389501i \(-0.127353\pi\)
−0.797831 + 0.602881i \(0.794019\pi\)
\(594\) 0 0
\(595\) −2.00000 3.46410i −0.0819920 0.142014i
\(596\) −18.0000 −0.737309
\(597\) 0 0
\(598\) −8.00000 13.8564i −0.327144 0.566631i
\(599\) −18.0000 31.1769i −0.735460 1.27385i −0.954521 0.298143i \(-0.903633\pi\)
0.219061 0.975711i \(-0.429701\pi\)
\(600\) 0 0
\(601\) −15.0000 −0.611863 −0.305931 0.952054i \(-0.598968\pi\)
−0.305931 + 0.952054i \(0.598968\pi\)
\(602\) −20.0000 34.6410i −0.815139 1.41186i
\(603\) 0 0
\(604\) 3.00000 + 5.19615i 0.122068 + 0.211428i
\(605\) 5.00000 8.66025i 0.203279 0.352089i
\(606\) 0 0
\(607\) −10.0000 −0.405887 −0.202944 0.979190i \(-0.565051\pi\)
−0.202944 + 0.979190i \(0.565051\pi\)
\(608\) −28.0000 + 20.7846i −1.13555 + 0.842927i
\(609\) 0 0
\(610\) 7.00000 12.1244i 0.283422 0.490901i
\(611\) 0 0
\(612\) 0 0
\(613\) 3.00000 5.19615i 0.121169 0.209871i −0.799060 0.601251i \(-0.794669\pi\)
0.920229 + 0.391381i \(0.128002\pi\)
\(614\) 24.0000 + 41.5692i 0.968561 + 1.67760i
\(615\) 0 0
\(616\) 0 0
\(617\) −12.0000 20.7846i −0.483102 0.836757i 0.516710 0.856161i \(-0.327157\pi\)
−0.999812 + 0.0194037i \(0.993823\pi\)
\(618\) 0 0
\(619\) −32.0000 −1.28619 −0.643094 0.765787i \(-0.722350\pi\)
−0.643094 + 0.765787i \(0.722350\pi\)
\(620\) 18.0000 0.722897
\(621\) 0 0
\(622\) 24.0000 41.5692i 0.962312 1.66677i
\(623\) −15.0000 25.9808i −0.600962 1.04090i
\(624\) 0 0
\(625\) −0.500000 + 0.866025i −0.0200000 + 0.0346410i
\(626\) −40.0000 −1.59872
\(627\) 0 0
\(628\) 24.0000 0.957704
\(629\) −6.00000 + 10.3923i −0.239236 + 0.414368i
\(630\) 0 0
\(631\) 1.50000 + 2.59808i 0.0597141 + 0.103428i 0.894337 0.447394i \(-0.147648\pi\)
−0.834623 + 0.550822i \(0.814314\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −64.0000 −2.54176
\(635\) 4.00000 0.158735
\(636\) 0 0
\(637\) 3.00000 + 5.19615i 0.118864 + 0.205879i
\(638\) −10.0000 −0.395904
\(639\) 0 0
\(640\) 0 0
\(641\) −16.5000 + 28.5788i −0.651711 + 1.12880i 0.330997 + 0.943632i \(0.392615\pi\)
−0.982708 + 0.185164i \(0.940718\pi\)
\(642\) 0 0
\(643\) −22.0000 + 38.1051i −0.867595 + 1.50272i −0.00314839 + 0.999995i \(0.501002\pi\)
−0.864447 + 0.502724i \(0.832331\pi\)
\(644\) 8.00000 13.8564i 0.315244 0.546019i
\(645\) 0 0
\(646\) −14.0000 + 10.3923i −0.550823 + 0.408880i
\(647\) −6.00000 −0.235884 −0.117942 0.993020i \(-0.537630\pi\)
−0.117942 + 0.993020i \(0.537630\pi\)
\(648\) 0 0
\(649\) −3.50000 + 6.06218i −0.137387 + 0.237961i
\(650\) −2.00000 3.46410i −0.0784465 0.135873i
\(651\) 0 0
\(652\) 18.0000 + 31.1769i 0.704934 + 1.22098i
\(653\) 6.00000 0.234798 0.117399 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) 0 0
\(655\) 8.00000 + 13.8564i 0.312586 + 0.541415i
\(656\) −12.0000 20.7846i −0.468521 0.811503i
\(657\) 0 0
\(658\) 0 0
\(659\) 20.0000 + 34.6410i 0.779089 + 1.34942i 0.932467 + 0.361255i \(0.117652\pi\)
−0.153378 + 0.988168i \(0.549015\pi\)
\(660\) 0 0
\(661\) −4.50000 7.79423i −0.175030 0.303160i 0.765142 0.643862i \(-0.222669\pi\)
−0.940172 + 0.340701i \(0.889335\pi\)
\(662\) −8.00000 + 13.8564i −0.310929 + 0.538545i
\(663\) 0 0
\(664\) 0 0
\(665\) 8.00000 + 3.46410i 0.310227 + 0.134332i
\(666\) 0 0
\(667\) −10.0000 + 17.3205i −0.387202 + 0.670653i
\(668\) −2.00000 + 3.46410i −0.0773823 + 0.134030i
\(669\) 0 0
\(670\) −8.00000 + 13.8564i −0.309067 + 0.535320i
\(671\) −3.50000 6.06218i −0.135116 0.234028i
\(672\) 0 0
\(673\) 36.0000 1.38770 0.693849 0.720121i \(-0.255914\pi\)
0.693849 + 0.720121i \(0.255914\pi\)
\(674\) 8.00000 + 13.8564i 0.308148 + 0.533729i
\(675\) 0 0
\(676\) −18.0000 −0.692308
\(677\) −46.0000 −1.76792 −0.883962 0.467559i \(-0.845134\pi\)
−0.883962 + 0.467559i \(0.845134\pi\)
\(678\) 0 0
\(679\) 8.00000 13.8564i 0.307012 0.531760i
\(680\) 0 0
\(681\) 0 0
\(682\) 9.00000 15.5885i 0.344628 0.596913i
\(683\) 24.0000 0.918334 0.459167 0.888350i \(-0.348148\pi\)
0.459167 + 0.888350i \(0.348148\pi\)
\(684\) 0 0
\(685\) 6.00000 0.229248
\(686\) −20.0000 + 34.6410i −0.763604 + 1.32260i
\(687\) 0 0
\(688\) −20.0000 34.6410i −0.762493 1.32068i
\(689\) −2.00000 + 3.46410i −0.0761939 + 0.131972i
\(690\) 0 0
\(691\) 23.0000 0.874961 0.437481 0.899228i \(-0.355871\pi\)
0.437481 + 0.899228i \(0.355871\pi\)
\(692\) −12.0000 −0.456172
\(693\) 0 0
\(694\) −22.0000 38.1051i −0.835109 1.44645i
\(695\) 20.0000 0.758643
\(696\) 0 0
\(697\) −6.00000 10.3923i −0.227266 0.393637i
\(698\) −6.00000 + 10.3923i −0.227103 + 0.393355i
\(699\) 0 0
\(700\) 2.00000 3.46410i 0.0755929 0.130931i
\(701\) −11.0000 + 19.0526i −0.415464 + 0.719605i −0.995477 0.0950021i \(-0.969714\pi\)
0.580013 + 0.814607i \(0.303048\pi\)
\(702\) 0 0
\(703\) −3.00000 25.9808i −0.113147 0.979883i
\(704\) 8.00000 0.301511
\(705\) 0 0
\(706\) −14.0000 + 24.2487i −0.526897 + 0.912612i
\(707\) 7.00000 + 12.1244i 0.263262 + 0.455983i
\(708\) 0 0
\(709\) 13.5000 + 23.3827i 0.507003 + 0.878155i 0.999967 + 0.00810550i \(0.00258009\pi\)
−0.492964 + 0.870050i \(0.664087\pi\)
\(710\) −6.00000 −0.225176
\(711\) 0 0
\(712\) 0 0
\(713\) −18.0000 31.1769i −0.674105 1.16758i
\(714\) 0 0
\(715\) −2.00000 −0.0747958
\(716\) −3.00000 5.19615i −0.112115 0.194189i
\(717\) 0 0
\(718\) −8.00000 13.8564i −0.298557 0.517116i
\(719\) −13.5000 + 23.3827i −0.503465 + 0.872027i 0.496527 + 0.868021i \(0.334608\pi\)
−0.999992 + 0.00400572i \(0.998725\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) 11.0000 36.3731i 0.409378 1.35367i
\(723\) 0 0
\(724\) −10.0000 + 17.3205i −0.371647 + 0.643712i
\(725\) −2.50000 + 4.33013i −0.0928477 + 0.160817i
\(726\) 0 0
\(727\) 1.00000 1.73205i 0.0370879 0.0642382i −0.846886 0.531775i \(-0.821525\pi\)
0.883974 + 0.467537i \(0.154858\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −4.00000 −0.148047
\(731\) −10.0000 17.3205i −0.369863 0.640622i
\(732\) 0 0
\(733\) 10.0000 0.369358 0.184679 0.982799i \(-0.440875\pi\)
0.184679 + 0.982799i \(0.440875\pi\)
\(734\) −40.0000 −1.47643
\(735\) 0 0
\(736\) 16.0000 27.7128i 0.589768 1.02151i
\(737\) 4.00000 + 6.92820i 0.147342 + 0.255204i
\(738\) 0 0
\(739\) 26.5000 45.8993i 0.974818 1.68843i 0.294285 0.955718i \(-0.404919\pi\)
0.680534 0.732717i \(-0.261748\pi\)
\(740\) −12.0000 −0.441129
\(741\) 0 0
\(742\) −8.00000 −0.293689
\(743\) 11.0000 19.0526i 0.403551 0.698971i −0.590601 0.806964i \(-0.701109\pi\)
0.994152 + 0.107993i \(0.0344425\pi\)
\(744\) 0 0
\(745\) 4.50000 + 7.79423i 0.164867 + 0.285558i
\(746\) 32.0000 55.4256i 1.17160 2.02928i
\(747\) 0 0
\(748\) 4.00000 0.146254
\(749\) 0 0
\(750\) 0 0
\(751\) 1.50000 + 2.59808i 0.0547358 + 0.0948051i 0.892095 0.451848i \(-0.149235\pi\)
−0.837359 + 0.546653i \(0.815902\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) −10.0000 17.3205i −0.364179 0.630776i
\(755\) 1.50000 2.59808i 0.0545906 0.0945537i
\(756\) 0 0
\(757\) 7.00000 12.1244i 0.254419 0.440667i −0.710318 0.703881i \(-0.751449\pi\)
0.964738 + 0.263213i \(0.0847823\pi\)
\(758\) −31.0000 + 53.6936i −1.12597 + 1.95024i
\(759\) 0 0
\(760\) 0 0
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) 0 0
\(763\) 19.0000 32.9090i 0.687846 1.19138i
\(764\) 15.0000 + 25.9808i 0.542681 + 0.939951i
\(765\) 0 0
\(766\) 26.0000 + 45.0333i 0.939418 + 1.62712i
\(767\) −14.0000 −0.505511
\(768\) 0 0
\(769\) 21.5000 + 37.2391i 0.775310 + 1.34288i 0.934620 + 0.355647i \(0.115740\pi\)
−0.159310 + 0.987229i \(0.550927\pi\)
\(770\) −2.00000 3.46410i −0.0720750 0.124838i
\(771\) 0 0
\(772\) −52.0000 −1.87152
\(773\) 10.0000 + 17.3205i 0.359675 + 0.622975i 0.987906 0.155051i \(-0.0495542\pi\)
−0.628231 + 0.778027i \(0.716221\pi\)
\(774\) 0 0
\(775\) −4.50000 7.79423i −0.161645 0.279977i
\(776\) 0 0
\(777\) 0 0
\(778\) −66.0000 −2.36621
\(779\) 24.0000 + 10.3923i 0.859889 + 0.372343i
\(780\) 0 0
\(781\) −1.50000 + 2.59808i −0.0536742 + 0.0929665i
\(782\) 8.00000 13.8564i 0.286079 0.495504i
\(783\) 0 0
\(784\) −6.00000 + 10.3923i −0.214286 + 0.371154i
\(785\) −6.00000 10.3923i −0.214149 0.370917i
\(786\) 0 0
\(787\) −8.00000 −0.285169 −0.142585 0.989783i \(-0.545541\pi\)
−0.142585 + 0.989783i \(0.545541\pi\)
\(788\) 10.0000 + 17.3205i 0.356235 + 0.617018i
\(789\) 0 0
\(790\) −22.0000 −0.782725
\(791\) −20.0000 −0.711118
\(792\) 0 0
\(793\) 7.00000 12.1244i 0.248577 0.430548i
\(794\) 20.0000 + 34.6410i 0.709773 + 1.22936i
\(795\) 0 0
\(796\) 3.00000 5.19615i 0.106332 0.184173i
\(797\) −10.0000 −0.354218 −0.177109 0.984191i \(-0.556675\pi\)
−0.177109 + 0.984191i \(0.556675\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 4.00000 6.92820i 0.141421 0.244949i
\(801\) 0 0
\(802\) −27.0000 46.7654i −0.953403 1.65134i
\(803\) −1.00000 + 1.73205i −0.0352892 + 0.0611227i
\(804\) 0 0
\(805\) −8.00000 −0.281963
\(806\) 36.0000 1.26805
\(807\) 0 0
\(808\) 0 0
\(809\) 1.00000 0.0351581 0.0175791 0.999845i \(-0.494404\pi\)
0.0175791 + 0.999845i \(0.494404\pi\)
\(810\) 0 0
\(811\) −24.5000 42.4352i −0.860311 1.49010i −0.871629 0.490167i \(-0.836936\pi\)
0.0113172 0.999936i \(-0.496398\pi\)
\(812\) 10.0000 17.3205i 0.350931 0.607831i
\(813\) 0 0
\(814\) −6.00000 + 10.3923i −0.210300 + 0.364250i
\(815\) 9.00000 15.5885i 0.315256 0.546040i
\(816\) 0 0
\(817\) 40.0000 + 17.3205i 1.39942 + 0.605968i
\(818\) −58.0000 −2.02792
\(819\) 0 0
\(820\) 6.00000 10.3923i 0.209529 0.362915i
\(821\) −22.5000 38.9711i −0.785255 1.36010i −0.928846 0.370465i \(-0.879198\pi\)
0.143591 0.989637i \(-0.454135\pi\)
\(822\) 0 0
\(823\) 10.0000 + 17.3205i 0.348578 + 0.603755i 0.985997 0.166762i \(-0.0533313\pi\)
−0.637419 + 0.770517i \(0.719998\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) −14.0000 24.2487i −0.487122 0.843721i
\(827\) 6.00000 + 10.3923i 0.208640 + 0.361376i 0.951286 0.308308i \(-0.0997628\pi\)
−0.742646 + 0.669684i \(0.766429\pi\)
\(828\) 0 0
\(829\) −38.0000 −1.31979 −0.659897 0.751356i \(-0.729400\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(830\) 6.00000 + 10.3923i 0.208263 + 0.360722i
\(831\) 0 0
\(832\) 8.00000 + 13.8564i 0.277350 + 0.480384i
\(833\) −3.00000 + 5.19615i −0.103944 + 0.180036i
\(834\) 0 0
\(835\) 2.00000 0.0692129
\(836\) −7.00000 + 5.19615i −0.242100 + 0.179713i
\(837\) 0 0
\(838\) −5.00000 + 8.66025i −0.172722 + 0.299164i
\(839\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(840\) 0 0
\(841\) 2.00000 3.46410i 0.0689655 0.119452i
\(842\) 7.00000 + 12.1244i 0.241236 + 0.417833i
\(843\) 0 0
\(844\) −54.0000 −1.85876
\(845\) 4.50000 + 7.79423i 0.154805 + 0.268130i
\(846\) 0 0
\(847\) 20.0000 0.687208
\(848\) −8.00000 −0.274721
\(849\) 0 0
\(850\) 2.00000 3.46410i 0.0685994 0.118818i
\(851\) 12.0000 + 20.7846i 0.411355 + 0.712487i
\(852\) 0 0
\(853\) 1.00000 1.73205i 0.0342393 0.0593043i −0.848398 0.529359i \(-0.822432\pi\)
0.882637 + 0.470055i \(0.155766\pi\)
\(854\) 28.0000 0.958140
\(855\) 0 0
\(856\) 0 0
\(857\) 5.00000 8.66025i 0.170797 0.295829i −0.767902 0.640567i \(-0.778699\pi\)
0.938699 + 0.344739i \(0.112033\pi\)
\(858\) 0 0
\(859\) −22.5000 38.9711i −0.767690 1.32968i −0.938813 0.344428i \(-0.888073\pi\)
0.171122 0.985250i \(-0.445261\pi\)
\(860\) 10.0000 17.3205i 0.340997 0.590624i
\(861\) 0 0
\(862\) −46.0000 −1.56677
\(863\) 38.0000 1.29354 0.646768 0.762687i \(-0.276120\pi\)
0.646768 + 0.762687i \(0.276120\pi\)
\(864\) 0 0
\(865\) 3.00000 + 5.19615i 0.102003 + 0.176674i
\(866\) 12.0000 0.407777
\(867\) 0 0
\(868\) 18.0000 + 31.1769i 0.610960 + 1.05821i
\(869\) −5.50000 + 9.52628i −0.186575 + 0.323157i
\(870\) 0 0
\(871\) −8.00000 + 13.8564i −0.271070 + 0.469506i
\(872\) 0 0
\(873\) 0 0
\(874\) 4.00000 + 34.6410i 0.135302 + 1.17175i
\(875\) −2.00000 −0.0676123
\(876\) 0 0
\(877\) −12.0000 + 20.7846i −0.405211 + 0.701846i −0.994346 0.106188i \(-0.966135\pi\)
0.589135 + 0.808035i \(0.299469\pi\)
\(878\) 25.0000 + 43.3013i 0.843709 + 1.46135i
\(879\) 0 0
\(880\) −2.00000 3.46410i −0.0674200 0.116775i
\(881\) 1.00000 0.0336909 0.0168454 0.999858i \(-0.494638\pi\)
0.0168454 + 0.999858i \(0.494638\pi\)
\(882\) 0 0
\(883\) 14.0000 + 24.2487i 0.471138 + 0.816034i 0.999455 0.0330128i \(-0.0105102\pi\)
−0.528317 + 0.849047i \(0.677177\pi\)
\(884\) 4.00000 + 6.92820i 0.134535 + 0.233021i
\(885\) 0 0
\(886\) 16.0000 0.537531
\(887\) −6.00000 10.3923i −0.201460 0.348939i 0.747539 0.664218i \(-0.231235\pi\)
−0.948999 + 0.315279i \(0.897902\pi\)
\(888\) 0 0
\(889\) 4.00000 + 6.92820i 0.134156 + 0.232364i
\(890\) 15.0000 25.9808i 0.502801 0.870877i
\(891\) 0 0
\(892\) −24.0000 −0.803579
\(893\) 0 0
\(894\) 0 0
\(895\) −1.50000 + 2.59808i −0.0501395 + 0.0868441i
\(896\) 0 0
\(897\) 0 0
\(898\) −39.0000 + 67.5500i −1.30145 + 2.25417i
\(899\) −22.5000 38.9711i −0.750417 1.29976i
\(900\) 0 0
\(901\) −4.00000 −0.133259
\(902\) −6.00000 10.3923i −0.199778 0.346026i
\(903\) 0 0
\(904\) 0 0
\(905\) 10.0000 0.332411
\(906\) 0 0
\(907\) −2.00000 + 3.46410i −0.0664089 + 0.115024i −0.897318 0.441384i \(-0.854488\pi\)
0.830909 + 0.556408i \(0.187821\pi\)
\(908\) 18.0000 + 31.1769i 0.597351 + 1.03464i
\(909\) 0 0
\(910\) 4.00000 6.92820i 0.132599 0.229668i
\(911\) 9.00000 0.298183 0.149092 0.988823i \(-0.452365\pi\)
0.149092 + 0.988823i \(0.452365\pi\)
\(912\) 0 0
\(913\) 6.00000 0.198571
\(914\) 8.00000 13.8564i 0.264616 0.458329i
\(915\) 0 0
\(916\) 11.0000 + 19.0526i 0.363450 + 0.629514i
\(917\) −16.0000 + 27.7128i −0.528367 + 0.915158i
\(918\) 0 0
\(919\) 32.0000 1.05558 0.527791 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −3.00000 5.19615i −0.0987997 0.171126i
\(923\) −6.00000 −0.197492
\(924\) 0 0
\(925\) 3.00000 + 5.19615i 0.0986394 + 0.170848i
\(926\) 10.0000 17.3205i 0.328620 0.569187i
\(927\) 0 0
\(928\) 20.0000 34.6410i 0.656532 1.13715i
\(929\) −5.50000 + 9.52628i −0.180449 + 0.312547i −0.942034 0.335519i \(-0.891088\pi\)
0.761584 + 0.648066i \(0.224422\pi\)
\(930\) 0 0
\(931\) −1.50000 12.9904i −0.0491605 0.425743i
\(932\) 48.0000 1.57229
\(933\) 0 0
\(934\) −38.0000 + 65.8179i −1.24340 + 2.15363i
\(935\) −1.00000 1.73205i −0.0327035 0.0566441i
\(936\) 0 0
\(937\) 9.00000 + 15.5885i 0.294017 + 0.509253i 0.974756 0.223274i \(-0.0716744\pi\)
−0.680739 + 0.732526i \(0.738341\pi\)
\(938\) −32.0000 −1.04484
\(939\) 0 0
\(940\) 0 0
\(941\) 10.5000 + 18.1865i 0.342290 + 0.592864i 0.984858 0.173365i \(-0.0554641\pi\)
−0.642567 + 0.766229i \(0.722131\pi\)
\(942\) 0 0
\(943\) −24.0000 −0.781548
\(944\) −14.0000 24.2487i −0.455661 0.789228i
\(945\) 0 0
\(946\) −10.0000 17.3205i −0.325128 0.563138i
\(947\) 10.0000 17.3205i 0.324956 0.562841i −0.656547 0.754285i \(-0.727984\pi\)
0.981504 + 0.191444i \(0.0613171\pi\)
\(948\) 0 0
\(949\) −4.00000 −0.129845
\(950\) 1.00000 + 8.66025i 0.0324443 + 0.280976i
\(951\) 0 0
\(952\) 0 0
\(953\) −4.00000 + 6.92820i −0.129573 + 0.224427i −0.923511 0.383572i \(-0.874694\pi\)
0.793938 + 0.607998i \(0.208027\pi\)
\(954\) 0 0
\(955\) 7.50000 12.9904i 0.242694 0.420359i
\(956\) 27.0000 + 46.7654i 0.873242 + 1.51250i
\(957\) 0 0
\(958\) −10.0000 −0.323085
\(959\) 6.00000 + 10.3923i 0.193750 + 0.335585i
\(960\) 0 0
\(961\) 50.0000 1.61290
\(962\) −24.0000 −0.773791
\(963\) 0 0
\(964\) −5.00000 + 8.66025i −0.161039 + 0.278928i
\(965\) 13.0000 + 22.5167i 0.418485 + 0.724837i
\(966\) 0 0
\(967\) −2.00000 + 3.46410i −0.0643157 + 0.111398i −0.896390 0.443266i \(-0.853820\pi\)
0.832075 + 0.554664i \(0.187153\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 16.0000 0.513729
\(971\) −12.0000 + 20.7846i −0.385098 + 0.667010i −0.991783 0.127933i \(-0.959166\pi\)
0.606685 + 0.794943i \(0.292499\pi\)
\(972\) 0 0
\(973\) 20.0000 + 34.6410i 0.641171 + 1.11054i
\(974\) −14.0000 + 24.2487i −0.448589 + 0.776979i
\(975\) 0 0
\(976\) 28.0000 0.896258
\(977\) −18.0000 −0.575871 −0.287936 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) 0 0
\(979\) −7.50000 12.9904i −0.239701 0.415174i
\(980\) −6.00000 −0.191663
\(981\) 0 0
\(982\) −33.0000 57.1577i −1.05307 1.82397i
\(983\) −7.00000 + 12.1244i −0.223265 + 0.386707i −0.955798 0.294025i \(-0.905005\pi\)
0.732532 + 0.680732i \(0.238338\pi\)
\(984\) 0 0
\(985\) 5.00000 8.66025i 0.159313 0.275939i
\(986\) 10.0000 17.3205i 0.318465 0.551597i
\(987\) 0 0
\(988\) −16.0000 6.92820i −0.509028 0.220416i
\(989\) −40.0000 −1.27193
\(990\) 0 0
\(991\) 20.0000 34.6410i 0.635321 1.10041i −0.351126 0.936328i \(-0.614201\pi\)
0.986447 0.164080i \(-0.0524655\pi\)
\(992\) 36.0000 + 62.3538i 1.14300 + 1.97974i
\(993\) 0 0
\(994\) −6.00000 10.3923i −0.190308 0.329624i
\(995\) −3.00000 −0.0951064
\(996\) 0 0
\(997\) −26.0000 45.0333i −0.823428 1.42622i −0.903115 0.429400i \(-0.858725\pi\)
0.0796863 0.996820i \(-0.474608\pi\)
\(998\) −20.0000 34.6410i −0.633089 1.09654i
\(999\) 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 855.2.k.a.676.1 2
3.2 odd 2 285.2.i.c.106.1 2
19.7 even 3 inner 855.2.k.a.406.1 2
57.8 even 6 5415.2.a.l.1.1 1
57.11 odd 6 5415.2.a.b.1.1 1
57.26 odd 6 285.2.i.c.121.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.i.c.106.1 2 3.2 odd 2
285.2.i.c.121.1 yes 2 57.26 odd 6
855.2.k.a.406.1 2 19.7 even 3 inner
855.2.k.a.676.1 2 1.1 even 1 trivial
5415.2.a.b.1.1 1 57.11 odd 6
5415.2.a.l.1.1 1 57.8 even 6