Properties

Label 1008.2.t.c.193.1
Level $1008$
Weight $2$
Character 1008.193
Analytic conductor $8.049$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1008,2,Mod(193,1008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1008, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.193");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.t (of order \(3\), degree \(2\), not 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: \(8.04892052375\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 193.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1008.193
Dual form 1008.2.t.c.961.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.50000 - 0.866025i) q^{3} -3.00000 q^{5} +(0.500000 + 2.59808i) q^{7} +(1.50000 - 2.59808i) q^{9} +O(q^{10})\) \(q+(1.50000 - 0.866025i) q^{3} -3.00000 q^{5} +(0.500000 + 2.59808i) q^{7} +(1.50000 - 2.59808i) q^{9} +3.00000 q^{11} +(-2.50000 + 4.33013i) q^{13} +(-4.50000 + 2.59808i) q^{15} +(-1.50000 + 2.59808i) q^{17} +(2.50000 + 4.33013i) q^{19} +(3.00000 + 3.46410i) q^{21} +3.00000 q^{23} +4.00000 q^{25} -5.19615i q^{27} +(1.50000 + 2.59808i) q^{29} +(-2.00000 - 3.46410i) q^{31} +(4.50000 - 2.59808i) q^{33} +(-1.50000 - 7.79423i) q^{35} +(3.50000 + 6.06218i) q^{37} +8.66025i q^{39} +(4.50000 - 7.79423i) q^{41} +(5.50000 + 9.52628i) q^{43} +(-4.50000 + 7.79423i) q^{45} +(-6.50000 + 2.59808i) q^{49} +5.19615i q^{51} +(1.50000 - 2.59808i) q^{53} -9.00000 q^{55} +(7.50000 + 4.33013i) q^{57} +(6.00000 + 10.3923i) q^{59} +(-1.00000 + 1.73205i) q^{61} +(7.50000 + 2.59808i) q^{63} +(7.50000 - 12.9904i) q^{65} +(-2.00000 - 3.46410i) q^{67} +(4.50000 - 2.59808i) q^{69} +(-5.50000 + 9.52628i) q^{73} +(6.00000 - 3.46410i) q^{75} +(1.50000 + 7.79423i) q^{77} +(4.00000 - 6.92820i) q^{79} +(-4.50000 - 7.79423i) q^{81} +(1.50000 + 2.59808i) q^{83} +(4.50000 - 7.79423i) q^{85} +(4.50000 + 2.59808i) q^{87} +(-7.50000 - 12.9904i) q^{89} +(-12.5000 - 4.33013i) q^{91} +(-6.00000 - 3.46410i) q^{93} +(-7.50000 - 12.9904i) q^{95} +(0.500000 + 0.866025i) q^{97} +(4.50000 - 7.79423i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{3} - 6 q^{5} + q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{3} - 6 q^{5} + q^{7} + 3 q^{9} + 6 q^{11} - 5 q^{13} - 9 q^{15} - 3 q^{17} + 5 q^{19} + 6 q^{21} + 6 q^{23} + 8 q^{25} + 3 q^{29} - 4 q^{31} + 9 q^{33} - 3 q^{35} + 7 q^{37} + 9 q^{41} + 11 q^{43} - 9 q^{45} - 13 q^{49} + 3 q^{53} - 18 q^{55} + 15 q^{57} + 12 q^{59} - 2 q^{61} + 15 q^{63} + 15 q^{65} - 4 q^{67} + 9 q^{69} - 11 q^{73} + 12 q^{75} + 3 q^{77} + 8 q^{79} - 9 q^{81} + 3 q^{83} + 9 q^{85} + 9 q^{87} - 15 q^{89} - 25 q^{91} - 12 q^{93} - 15 q^{95} + q^{97} + 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.50000 0.866025i 0.866025 0.500000i
\(4\) 0 0
\(5\) −3.00000 −1.34164 −0.670820 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) 0 0
\(7\) 0.500000 + 2.59808i 0.188982 + 0.981981i
\(8\) 0 0
\(9\) 1.50000 2.59808i 0.500000 0.866025i
\(10\) 0 0
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 0 0
\(13\) −2.50000 + 4.33013i −0.693375 + 1.20096i 0.277350 + 0.960769i \(0.410544\pi\)
−0.970725 + 0.240192i \(0.922790\pi\)
\(14\) 0 0
\(15\) −4.50000 + 2.59808i −1.16190 + 0.670820i
\(16\) 0 0
\(17\) −1.50000 + 2.59808i −0.363803 + 0.630126i −0.988583 0.150675i \(-0.951855\pi\)
0.624780 + 0.780801i \(0.285189\pi\)
\(18\) 0 0
\(19\) 2.50000 + 4.33013i 0.573539 + 0.993399i 0.996199 + 0.0871106i \(0.0277634\pi\)
−0.422659 + 0.906289i \(0.638903\pi\)
\(20\) 0 0
\(21\) 3.00000 + 3.46410i 0.654654 + 0.755929i
\(22\) 0 0
\(23\) 3.00000 0.625543 0.312772 0.949828i \(-0.398743\pi\)
0.312772 + 0.949828i \(0.398743\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) 5.19615i 1.00000i
\(28\) 0 0
\(29\) 1.50000 + 2.59808i 0.278543 + 0.482451i 0.971023 0.238987i \(-0.0768152\pi\)
−0.692480 + 0.721437i \(0.743482\pi\)
\(30\) 0 0
\(31\) −2.00000 3.46410i −0.359211 0.622171i 0.628619 0.777714i \(-0.283621\pi\)
−0.987829 + 0.155543i \(0.950287\pi\)
\(32\) 0 0
\(33\) 4.50000 2.59808i 0.783349 0.452267i
\(34\) 0 0
\(35\) −1.50000 7.79423i −0.253546 1.31747i
\(36\) 0 0
\(37\) 3.50000 + 6.06218i 0.575396 + 0.996616i 0.995998 + 0.0893706i \(0.0284856\pi\)
−0.420602 + 0.907245i \(0.638181\pi\)
\(38\) 0 0
\(39\) 8.66025i 1.38675i
\(40\) 0 0
\(41\) 4.50000 7.79423i 0.702782 1.21725i −0.264704 0.964330i \(-0.585274\pi\)
0.967486 0.252924i \(-0.0813924\pi\)
\(42\) 0 0
\(43\) 5.50000 + 9.52628i 0.838742 + 1.45274i 0.890947 + 0.454108i \(0.150042\pi\)
−0.0522047 + 0.998636i \(0.516625\pi\)
\(44\) 0 0
\(45\) −4.50000 + 7.79423i −0.670820 + 1.16190i
\(46\) 0 0
\(47\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(48\) 0 0
\(49\) −6.50000 + 2.59808i −0.928571 + 0.371154i
\(50\) 0 0
\(51\) 5.19615i 0.727607i
\(52\) 0 0
\(53\) 1.50000 2.59808i 0.206041 0.356873i −0.744423 0.667708i \(-0.767275\pi\)
0.950464 + 0.310835i \(0.100609\pi\)
\(54\) 0 0
\(55\) −9.00000 −1.21356
\(56\) 0 0
\(57\) 7.50000 + 4.33013i 0.993399 + 0.573539i
\(58\) 0 0
\(59\) 6.00000 + 10.3923i 0.781133 + 1.35296i 0.931282 + 0.364299i \(0.118692\pi\)
−0.150148 + 0.988663i \(0.547975\pi\)
\(60\) 0 0
\(61\) −1.00000 + 1.73205i −0.128037 + 0.221766i −0.922916 0.385002i \(-0.874201\pi\)
0.794879 + 0.606768i \(0.207534\pi\)
\(62\) 0 0
\(63\) 7.50000 + 2.59808i 0.944911 + 0.327327i
\(64\) 0 0
\(65\) 7.50000 12.9904i 0.930261 1.61126i
\(66\) 0 0
\(67\) −2.00000 3.46410i −0.244339 0.423207i 0.717607 0.696449i \(-0.245238\pi\)
−0.961946 + 0.273241i \(0.911904\pi\)
\(68\) 0 0
\(69\) 4.50000 2.59808i 0.541736 0.312772i
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −5.50000 + 9.52628i −0.643726 + 1.11497i 0.340868 + 0.940111i \(0.389279\pi\)
−0.984594 + 0.174855i \(0.944054\pi\)
\(74\) 0 0
\(75\) 6.00000 3.46410i 0.692820 0.400000i
\(76\) 0 0
\(77\) 1.50000 + 7.79423i 0.170941 + 0.888235i
\(78\) 0 0
\(79\) 4.00000 6.92820i 0.450035 0.779484i −0.548352 0.836247i \(-0.684745\pi\)
0.998388 + 0.0567635i \(0.0180781\pi\)
\(80\) 0 0
\(81\) −4.50000 7.79423i −0.500000 0.866025i
\(82\) 0 0
\(83\) 1.50000 + 2.59808i 0.164646 + 0.285176i 0.936530 0.350588i \(-0.114018\pi\)
−0.771883 + 0.635764i \(0.780685\pi\)
\(84\) 0 0
\(85\) 4.50000 7.79423i 0.488094 0.845403i
\(86\) 0 0
\(87\) 4.50000 + 2.59808i 0.482451 + 0.278543i
\(88\) 0 0
\(89\) −7.50000 12.9904i −0.794998 1.37698i −0.922840 0.385183i \(-0.874138\pi\)
0.127842 0.991795i \(-0.459195\pi\)
\(90\) 0 0
\(91\) −12.5000 4.33013i −1.31036 0.453921i
\(92\) 0 0
\(93\) −6.00000 3.46410i −0.622171 0.359211i
\(94\) 0 0
\(95\) −7.50000 12.9904i −0.769484 1.33278i
\(96\) 0 0
\(97\) 0.500000 + 0.866025i 0.0507673 + 0.0879316i 0.890292 0.455389i \(-0.150500\pi\)
−0.839525 + 0.543321i \(0.817167\pi\)
\(98\) 0 0
\(99\) 4.50000 7.79423i 0.452267 0.783349i
\(100\) 0 0
\(101\) −3.00000 −0.298511 −0.149256 0.988799i \(-0.547688\pi\)
−0.149256 + 0.988799i \(0.547688\pi\)
\(102\) 0 0
\(103\) −5.00000 −0.492665 −0.246332 0.969185i \(-0.579225\pi\)
−0.246332 + 0.969185i \(0.579225\pi\)
\(104\) 0 0
\(105\) −9.00000 10.3923i −0.878310 1.01419i
\(106\) 0 0
\(107\) −7.50000 12.9904i −0.725052 1.25583i −0.958952 0.283567i \(-0.908482\pi\)
0.233900 0.972261i \(-0.424851\pi\)
\(108\) 0 0
\(109\) 3.50000 6.06218i 0.335239 0.580651i −0.648292 0.761392i \(-0.724516\pi\)
0.983531 + 0.180741i \(0.0578495\pi\)
\(110\) 0 0
\(111\) 10.5000 + 6.06218i 0.996616 + 0.575396i
\(112\) 0 0
\(113\) −7.50000 + 12.9904i −0.705541 + 1.22203i 0.260955 + 0.965351i \(0.415962\pi\)
−0.966496 + 0.256681i \(0.917371\pi\)
\(114\) 0 0
\(115\) −9.00000 −0.839254
\(116\) 0 0
\(117\) 7.50000 + 12.9904i 0.693375 + 1.20096i
\(118\) 0 0
\(119\) −7.50000 2.59808i −0.687524 0.238165i
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) 15.5885i 1.40556i
\(124\) 0 0
\(125\) 3.00000 0.268328
\(126\) 0 0
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 0 0
\(129\) 16.5000 + 9.52628i 1.45274 + 0.838742i
\(130\) 0 0
\(131\) −3.00000 −0.262111 −0.131056 0.991375i \(-0.541837\pi\)
−0.131056 + 0.991375i \(0.541837\pi\)
\(132\) 0 0
\(133\) −10.0000 + 8.66025i −0.867110 + 0.750939i
\(134\) 0 0
\(135\) 15.5885i 1.34164i
\(136\) 0 0
\(137\) 3.00000 0.256307 0.128154 0.991754i \(-0.459095\pi\)
0.128154 + 0.991754i \(0.459095\pi\)
\(138\) 0 0
\(139\) 2.50000 4.33013i 0.212047 0.367277i −0.740308 0.672268i \(-0.765320\pi\)
0.952355 + 0.304991i \(0.0986536\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −7.50000 + 12.9904i −0.627182 + 1.08631i
\(144\) 0 0
\(145\) −4.50000 7.79423i −0.373705 0.647275i
\(146\) 0 0
\(147\) −7.50000 + 9.52628i −0.618590 + 0.785714i
\(148\) 0 0
\(149\) −3.00000 −0.245770 −0.122885 0.992421i \(-0.539215\pi\)
−0.122885 + 0.992421i \(0.539215\pi\)
\(150\) 0 0
\(151\) −11.0000 −0.895167 −0.447584 0.894242i \(-0.647715\pi\)
−0.447584 + 0.894242i \(0.647715\pi\)
\(152\) 0 0
\(153\) 4.50000 + 7.79423i 0.363803 + 0.630126i
\(154\) 0 0
\(155\) 6.00000 + 10.3923i 0.481932 + 0.834730i
\(156\) 0 0
\(157\) −7.00000 12.1244i −0.558661 0.967629i −0.997609 0.0691164i \(-0.977982\pi\)
0.438948 0.898513i \(-0.355351\pi\)
\(158\) 0 0
\(159\) 5.19615i 0.412082i
\(160\) 0 0
\(161\) 1.50000 + 7.79423i 0.118217 + 0.614271i
\(162\) 0 0
\(163\) 8.50000 + 14.7224i 0.665771 + 1.15315i 0.979076 + 0.203497i \(0.0652307\pi\)
−0.313304 + 0.949653i \(0.601436\pi\)
\(164\) 0 0
\(165\) −13.5000 + 7.79423i −1.05097 + 0.606780i
\(166\) 0 0
\(167\) 1.50000 2.59808i 0.116073 0.201045i −0.802135 0.597143i \(-0.796303\pi\)
0.918208 + 0.396098i \(0.129636\pi\)
\(168\) 0 0
\(169\) −6.00000 10.3923i −0.461538 0.799408i
\(170\) 0 0
\(171\) 15.0000 1.14708
\(172\) 0 0
\(173\) −3.00000 + 5.19615i −0.228086 + 0.395056i −0.957241 0.289292i \(-0.906580\pi\)
0.729155 + 0.684349i \(0.239913\pi\)
\(174\) 0 0
\(175\) 2.00000 + 10.3923i 0.151186 + 0.785584i
\(176\) 0 0
\(177\) 18.0000 + 10.3923i 1.35296 + 0.781133i
\(178\) 0 0
\(179\) 1.50000 2.59808i 0.112115 0.194189i −0.804508 0.593942i \(-0.797571\pi\)
0.916623 + 0.399753i \(0.130904\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) 3.46410i 0.256074i
\(184\) 0 0
\(185\) −10.5000 18.1865i −0.771975 1.33710i
\(186\) 0 0
\(187\) −4.50000 + 7.79423i −0.329073 + 0.569970i
\(188\) 0 0
\(189\) 13.5000 2.59808i 0.981981 0.188982i
\(190\) 0 0
\(191\) 6.00000 10.3923i 0.434145 0.751961i −0.563081 0.826402i \(-0.690384\pi\)
0.997225 + 0.0744412i \(0.0237173\pi\)
\(192\) 0 0
\(193\) −7.00000 12.1244i −0.503871 0.872730i −0.999990 0.00447566i \(-0.998575\pi\)
0.496119 0.868255i \(-0.334758\pi\)
\(194\) 0 0
\(195\) 25.9808i 1.86052i
\(196\) 0 0
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) −3.50000 + 6.06218i −0.248108 + 0.429736i −0.963001 0.269498i \(-0.913142\pi\)
0.714893 + 0.699234i \(0.246476\pi\)
\(200\) 0 0
\(201\) −6.00000 3.46410i −0.423207 0.244339i
\(202\) 0 0
\(203\) −6.00000 + 5.19615i −0.421117 + 0.364698i
\(204\) 0 0
\(205\) −13.5000 + 23.3827i −0.942881 + 1.63312i
\(206\) 0 0
\(207\) 4.50000 7.79423i 0.312772 0.541736i
\(208\) 0 0
\(209\) 7.50000 + 12.9904i 0.518786 + 0.898563i
\(210\) 0 0
\(211\) 2.50000 4.33013i 0.172107 0.298098i −0.767049 0.641588i \(-0.778276\pi\)
0.939156 + 0.343490i \(0.111609\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −16.5000 28.5788i −1.12529 1.94906i
\(216\) 0 0
\(217\) 8.00000 6.92820i 0.543075 0.470317i
\(218\) 0 0
\(219\) 19.0526i 1.28745i
\(220\) 0 0
\(221\) −7.50000 12.9904i −0.504505 0.873828i
\(222\) 0 0
\(223\) 8.50000 + 14.7224i 0.569202 + 0.985887i 0.996645 + 0.0818447i \(0.0260811\pi\)
−0.427443 + 0.904042i \(0.640586\pi\)
\(224\) 0 0
\(225\) 6.00000 10.3923i 0.400000 0.692820i
\(226\) 0 0
\(227\) −9.00000 −0.597351 −0.298675 0.954355i \(-0.596545\pi\)
−0.298675 + 0.954355i \(0.596545\pi\)
\(228\) 0 0
\(229\) 17.0000 1.12339 0.561696 0.827344i \(-0.310149\pi\)
0.561696 + 0.827344i \(0.310149\pi\)
\(230\) 0 0
\(231\) 9.00000 + 10.3923i 0.592157 + 0.683763i
\(232\) 0 0
\(233\) −13.5000 23.3827i −0.884414 1.53185i −0.846383 0.532574i \(-0.821225\pi\)
−0.0380310 0.999277i \(-0.512109\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 13.8564i 0.900070i
\(238\) 0 0
\(239\) 13.5000 23.3827i 0.873242 1.51250i 0.0146191 0.999893i \(-0.495346\pi\)
0.858623 0.512607i \(-0.171320\pi\)
\(240\) 0 0
\(241\) 23.0000 1.48156 0.740780 0.671748i \(-0.234456\pi\)
0.740780 + 0.671748i \(0.234456\pi\)
\(242\) 0 0
\(243\) −13.5000 7.79423i −0.866025 0.500000i
\(244\) 0 0
\(245\) 19.5000 7.79423i 1.24581 0.497955i
\(246\) 0 0
\(247\) −25.0000 −1.59071
\(248\) 0 0
\(249\) 4.50000 + 2.59808i 0.285176 + 0.164646i
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) 9.00000 0.565825
\(254\) 0 0
\(255\) 15.5885i 0.976187i
\(256\) 0 0
\(257\) 15.0000 0.935674 0.467837 0.883815i \(-0.345033\pi\)
0.467837 + 0.883815i \(0.345033\pi\)
\(258\) 0 0
\(259\) −14.0000 + 12.1244i −0.869918 + 0.753371i
\(260\) 0 0
\(261\) 9.00000 0.557086
\(262\) 0 0
\(263\) 9.00000 0.554964 0.277482 0.960731i \(-0.410500\pi\)
0.277482 + 0.960731i \(0.410500\pi\)
\(264\) 0 0
\(265\) −4.50000 + 7.79423i −0.276433 + 0.478796i
\(266\) 0 0
\(267\) −22.5000 12.9904i −1.37698 0.794998i
\(268\) 0 0
\(269\) −10.5000 + 18.1865i −0.640196 + 1.10885i 0.345192 + 0.938532i \(0.387814\pi\)
−0.985389 + 0.170321i \(0.945520\pi\)
\(270\) 0 0
\(271\) −6.50000 11.2583i −0.394847 0.683895i 0.598235 0.801321i \(-0.295869\pi\)
−0.993082 + 0.117426i \(0.962536\pi\)
\(272\) 0 0
\(273\) −22.5000 + 4.33013i −1.36176 + 0.262071i
\(274\) 0 0
\(275\) 12.0000 0.723627
\(276\) 0 0
\(277\) −7.00000 −0.420589 −0.210295 0.977638i \(-0.567442\pi\)
−0.210295 + 0.977638i \(0.567442\pi\)
\(278\) 0 0
\(279\) −12.0000 −0.718421
\(280\) 0 0
\(281\) −1.50000 2.59808i −0.0894825 0.154988i 0.817810 0.575488i \(-0.195188\pi\)
−0.907293 + 0.420500i \(0.861855\pi\)
\(282\) 0 0
\(283\) 4.00000 + 6.92820i 0.237775 + 0.411839i 0.960076 0.279741i \(-0.0902485\pi\)
−0.722300 + 0.691580i \(0.756915\pi\)
\(284\) 0 0
\(285\) −22.5000 12.9904i −1.33278 0.769484i
\(286\) 0 0
\(287\) 22.5000 + 7.79423i 1.32813 + 0.460079i
\(288\) 0 0
\(289\) 4.00000 + 6.92820i 0.235294 + 0.407541i
\(290\) 0 0
\(291\) 1.50000 + 0.866025i 0.0879316 + 0.0507673i
\(292\) 0 0
\(293\) 13.5000 23.3827i 0.788678 1.36603i −0.138098 0.990419i \(-0.544099\pi\)
0.926777 0.375613i \(-0.122568\pi\)
\(294\) 0 0
\(295\) −18.0000 31.1769i −1.04800 1.81519i
\(296\) 0 0
\(297\) 15.5885i 0.904534i
\(298\) 0 0
\(299\) −7.50000 + 12.9904i −0.433736 + 0.751253i
\(300\) 0 0
\(301\) −22.0000 + 19.0526i −1.26806 + 1.09817i
\(302\) 0 0
\(303\) −4.50000 + 2.59808i −0.258518 + 0.149256i
\(304\) 0 0
\(305\) 3.00000 5.19615i 0.171780 0.297531i
\(306\) 0 0
\(307\) 28.0000 1.59804 0.799022 0.601302i \(-0.205351\pi\)
0.799022 + 0.601302i \(0.205351\pi\)
\(308\) 0 0
\(309\) −7.50000 + 4.33013i −0.426660 + 0.246332i
\(310\) 0 0
\(311\) −12.0000 20.7846i −0.680458 1.17859i −0.974841 0.222900i \(-0.928448\pi\)
0.294384 0.955687i \(-0.404886\pi\)
\(312\) 0 0
\(313\) −7.00000 + 12.1244i −0.395663 + 0.685309i −0.993186 0.116543i \(-0.962819\pi\)
0.597522 + 0.801852i \(0.296152\pi\)
\(314\) 0 0
\(315\) −22.5000 7.79423i −1.26773 0.439155i
\(316\) 0 0
\(317\) −15.0000 + 25.9808i −0.842484 + 1.45922i 0.0453045 + 0.998973i \(0.485574\pi\)
−0.887788 + 0.460252i \(0.847759\pi\)
\(318\) 0 0
\(319\) 4.50000 + 7.79423i 0.251952 + 0.436393i
\(320\) 0 0
\(321\) −22.5000 12.9904i −1.25583 0.725052i
\(322\) 0 0
\(323\) −15.0000 −0.834622
\(324\) 0 0
\(325\) −10.0000 + 17.3205i −0.554700 + 0.960769i
\(326\) 0 0
\(327\) 12.1244i 0.670478i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 10.0000 17.3205i 0.549650 0.952021i −0.448649 0.893708i \(-0.648095\pi\)
0.998298 0.0583130i \(-0.0185721\pi\)
\(332\) 0 0
\(333\) 21.0000 1.15079
\(334\) 0 0
\(335\) 6.00000 + 10.3923i 0.327815 + 0.567792i
\(336\) 0 0
\(337\) 12.5000 21.6506i 0.680918 1.17939i −0.293783 0.955872i \(-0.594914\pi\)
0.974701 0.223513i \(-0.0717525\pi\)
\(338\) 0 0
\(339\) 25.9808i 1.41108i
\(340\) 0 0
\(341\) −6.00000 10.3923i −0.324918 0.562775i
\(342\) 0 0
\(343\) −10.0000 15.5885i −0.539949 0.841698i
\(344\) 0 0
\(345\) −13.5000 + 7.79423i −0.726816 + 0.419627i
\(346\) 0 0
\(347\) −6.00000 10.3923i −0.322097 0.557888i 0.658824 0.752297i \(-0.271054\pi\)
−0.980921 + 0.194409i \(0.937721\pi\)
\(348\) 0 0
\(349\) −2.50000 4.33013i −0.133822 0.231786i 0.791325 0.611396i \(-0.209392\pi\)
−0.925147 + 0.379610i \(0.876058\pi\)
\(350\) 0 0
\(351\) 22.5000 + 12.9904i 1.20096 + 0.693375i
\(352\) 0 0
\(353\) −9.00000 −0.479022 −0.239511 0.970894i \(-0.576987\pi\)
−0.239511 + 0.970894i \(0.576987\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −13.5000 + 2.59808i −0.714496 + 0.137505i
\(358\) 0 0
\(359\) 7.50000 + 12.9904i 0.395835 + 0.685606i 0.993207 0.116358i \(-0.0371219\pi\)
−0.597372 + 0.801964i \(0.703789\pi\)
\(360\) 0 0
\(361\) −3.00000 + 5.19615i −0.157895 + 0.273482i
\(362\) 0 0
\(363\) −3.00000 + 1.73205i −0.157459 + 0.0909091i
\(364\) 0 0
\(365\) 16.5000 28.5788i 0.863649 1.49588i
\(366\) 0 0
\(367\) 1.00000 0.0521996 0.0260998 0.999659i \(-0.491691\pi\)
0.0260998 + 0.999659i \(0.491691\pi\)
\(368\) 0 0
\(369\) −13.5000 23.3827i −0.702782 1.21725i
\(370\) 0 0
\(371\) 7.50000 + 2.59808i 0.389381 + 0.134885i
\(372\) 0 0
\(373\) 17.0000 0.880227 0.440113 0.897942i \(-0.354938\pi\)
0.440113 + 0.897942i \(0.354938\pi\)
\(374\) 0 0
\(375\) 4.50000 2.59808i 0.232379 0.134164i
\(376\) 0 0
\(377\) −15.0000 −0.772539
\(378\) 0 0
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) 0 0
\(381\) 24.0000 13.8564i 1.22956 0.709885i
\(382\) 0 0
\(383\) 15.0000 0.766464 0.383232 0.923652i \(-0.374811\pi\)
0.383232 + 0.923652i \(0.374811\pi\)
\(384\) 0 0
\(385\) −4.50000 23.3827i −0.229341 1.19169i
\(386\) 0 0
\(387\) 33.0000 1.67748
\(388\) 0 0
\(389\) 9.00000 0.456318 0.228159 0.973624i \(-0.426729\pi\)
0.228159 + 0.973624i \(0.426729\pi\)
\(390\) 0 0
\(391\) −4.50000 + 7.79423i −0.227575 + 0.394171i
\(392\) 0 0
\(393\) −4.50000 + 2.59808i −0.226995 + 0.131056i
\(394\) 0 0
\(395\) −12.0000 + 20.7846i −0.603786 + 1.04579i
\(396\) 0 0
\(397\) −14.5000 25.1147i −0.727734 1.26047i −0.957839 0.287307i \(-0.907240\pi\)
0.230105 0.973166i \(-0.426093\pi\)
\(398\) 0 0
\(399\) −7.50000 + 21.6506i −0.375470 + 1.08389i
\(400\) 0 0
\(401\) 27.0000 1.34832 0.674158 0.738587i \(-0.264507\pi\)
0.674158 + 0.738587i \(0.264507\pi\)
\(402\) 0 0
\(403\) 20.0000 0.996271
\(404\) 0 0
\(405\) 13.5000 + 23.3827i 0.670820 + 1.16190i
\(406\) 0 0
\(407\) 10.5000 + 18.1865i 0.520466 + 0.901473i
\(408\) 0 0
\(409\) 11.0000 + 19.0526i 0.543915 + 0.942088i 0.998674 + 0.0514740i \(0.0163919\pi\)
−0.454759 + 0.890614i \(0.650275\pi\)
\(410\) 0 0
\(411\) 4.50000 2.59808i 0.221969 0.128154i
\(412\) 0 0
\(413\) −24.0000 + 20.7846i −1.18096 + 1.02274i
\(414\) 0 0
\(415\) −4.50000 7.79423i −0.220896 0.382604i
\(416\) 0 0
\(417\) 8.66025i 0.424094i
\(418\) 0 0
\(419\) −1.50000 + 2.59808i −0.0732798 + 0.126924i −0.900337 0.435194i \(-0.856680\pi\)
0.827057 + 0.562118i \(0.190013\pi\)
\(420\) 0 0
\(421\) 15.5000 + 26.8468i 0.755424 + 1.30843i 0.945163 + 0.326598i \(0.105902\pi\)
−0.189740 + 0.981834i \(0.560764\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −6.00000 + 10.3923i −0.291043 + 0.504101i
\(426\) 0 0
\(427\) −5.00000 1.73205i −0.241967 0.0838198i
\(428\) 0 0
\(429\) 25.9808i 1.25436i
\(430\) 0 0
\(431\) −1.50000 + 2.59808i −0.0722525 + 0.125145i −0.899888 0.436121i \(-0.856352\pi\)
0.827636 + 0.561266i \(0.189685\pi\)
\(432\) 0 0
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 0 0
\(435\) −13.5000 7.79423i −0.647275 0.373705i
\(436\) 0 0
\(437\) 7.50000 + 12.9904i 0.358774 + 0.621414i
\(438\) 0 0
\(439\) 4.00000 6.92820i 0.190910 0.330665i −0.754642 0.656136i \(-0.772190\pi\)
0.945552 + 0.325471i \(0.105523\pi\)
\(440\) 0 0
\(441\) −3.00000 + 20.7846i −0.142857 + 0.989743i
\(442\) 0 0
\(443\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(444\) 0 0
\(445\) 22.5000 + 38.9711i 1.06660 + 1.84741i
\(446\) 0 0
\(447\) −4.50000 + 2.59808i −0.212843 + 0.122885i
\(448\) 0 0
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 0 0
\(451\) 13.5000 23.3827i 0.635690 1.10105i
\(452\) 0 0
\(453\) −16.5000 + 9.52628i −0.775238 + 0.447584i
\(454\) 0 0
\(455\) 37.5000 + 12.9904i 1.75803 + 0.608998i
\(456\) 0 0
\(457\) 17.0000 29.4449i 0.795226 1.37737i −0.127469 0.991843i \(-0.540685\pi\)
0.922695 0.385530i \(-0.125981\pi\)
\(458\) 0 0
\(459\) 13.5000 + 7.79423i 0.630126 + 0.363803i
\(460\) 0 0
\(461\) −4.50000 7.79423i −0.209586 0.363013i 0.741998 0.670402i \(-0.233878\pi\)
−0.951584 + 0.307388i \(0.900545\pi\)
\(462\) 0 0
\(463\) 17.5000 30.3109i 0.813294 1.40867i −0.0972525 0.995260i \(-0.531005\pi\)
0.910546 0.413407i \(-0.135661\pi\)
\(464\) 0 0
\(465\) 18.0000 + 10.3923i 0.834730 + 0.481932i
\(466\) 0 0
\(467\) −1.50000 2.59808i −0.0694117 0.120225i 0.829231 0.558906i \(-0.188779\pi\)
−0.898642 + 0.438682i \(0.855446\pi\)
\(468\) 0 0
\(469\) 8.00000 6.92820i 0.369406 0.319915i
\(470\) 0 0
\(471\) −21.0000 12.1244i −0.967629 0.558661i
\(472\) 0 0
\(473\) 16.5000 + 28.5788i 0.758671 + 1.31406i
\(474\) 0 0
\(475\) 10.0000 + 17.3205i 0.458831 + 0.794719i
\(476\) 0 0
\(477\) −4.50000 7.79423i −0.206041 0.356873i
\(478\) 0 0
\(479\) 9.00000 0.411220 0.205610 0.978634i \(-0.434082\pi\)
0.205610 + 0.978634i \(0.434082\pi\)
\(480\) 0 0
\(481\) −35.0000 −1.59586
\(482\) 0 0
\(483\) 9.00000 + 10.3923i 0.409514 + 0.472866i
\(484\) 0 0
\(485\) −1.50000 2.59808i −0.0681115 0.117973i
\(486\) 0 0
\(487\) −15.5000 + 26.8468i −0.702372 + 1.21654i 0.265260 + 0.964177i \(0.414542\pi\)
−0.967632 + 0.252367i \(0.918791\pi\)
\(488\) 0 0
\(489\) 25.5000 + 14.7224i 1.15315 + 0.665771i
\(490\) 0 0
\(491\) −19.5000 + 33.7750i −0.880023 + 1.52424i −0.0287085 + 0.999588i \(0.509139\pi\)
−0.851314 + 0.524656i \(0.824194\pi\)
\(492\) 0 0
\(493\) −9.00000 −0.405340
\(494\) 0 0
\(495\) −13.5000 + 23.3827i −0.606780 + 1.05097i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −11.0000 −0.492428 −0.246214 0.969216i \(-0.579187\pi\)
−0.246214 + 0.969216i \(0.579187\pi\)
\(500\) 0 0
\(501\) 5.19615i 0.232147i
\(502\) 0 0
\(503\) 12.0000 0.535054 0.267527 0.963550i \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) 0 0
\(505\) 9.00000 0.400495
\(506\) 0 0
\(507\) −18.0000 10.3923i −0.799408 0.461538i
\(508\) 0 0
\(509\) −27.0000 −1.19675 −0.598377 0.801215i \(-0.704187\pi\)
−0.598377 + 0.801215i \(0.704187\pi\)
\(510\) 0 0
\(511\) −27.5000 9.52628i −1.21653 0.421418i
\(512\) 0 0
\(513\) 22.5000 12.9904i 0.993399 0.573539i
\(514\) 0 0
\(515\) 15.0000 0.660979
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 10.3923i 0.456172i
\(520\) 0 0
\(521\) −1.50000 + 2.59808i −0.0657162 + 0.113824i −0.897011 0.442007i \(-0.854267\pi\)
0.831295 + 0.555831i \(0.187600\pi\)
\(522\) 0 0
\(523\) −3.50000 6.06218i −0.153044 0.265081i 0.779301 0.626650i \(-0.215574\pi\)
−0.932345 + 0.361569i \(0.882241\pi\)
\(524\) 0 0
\(525\) 12.0000 + 13.8564i 0.523723 + 0.604743i
\(526\) 0 0
\(527\) 12.0000 0.522728
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) 36.0000 1.56227
\(532\) 0 0
\(533\) 22.5000 + 38.9711i 0.974583 + 1.68803i
\(534\) 0 0
\(535\) 22.5000 + 38.9711i 0.972760 + 1.68487i
\(536\) 0 0
\(537\) 5.19615i 0.224231i
\(538\) 0 0
\(539\) −19.5000 + 7.79423i −0.839924 + 0.335721i
\(540\) 0 0
\(541\) −8.50000 14.7224i −0.365444 0.632967i 0.623404 0.781900i \(-0.285749\pi\)
−0.988847 + 0.148933i \(0.952416\pi\)
\(542\) 0 0
\(543\) −15.0000 + 8.66025i −0.643712 + 0.371647i
\(544\) 0 0
\(545\) −10.5000 + 18.1865i −0.449771 + 0.779026i
\(546\) 0 0
\(547\) 5.50000 + 9.52628i 0.235163 + 0.407314i 0.959320 0.282321i \(-0.0911043\pi\)
−0.724157 + 0.689635i \(0.757771\pi\)
\(548\) 0 0
\(549\) 3.00000 + 5.19615i 0.128037 + 0.221766i
\(550\) 0 0
\(551\) −7.50000 + 12.9904i −0.319511 + 0.553409i
\(552\) 0 0
\(553\) 20.0000 + 6.92820i 0.850487 + 0.294617i
\(554\) 0 0
\(555\) −31.5000 18.1865i −1.33710 0.771975i
\(556\) 0 0
\(557\) 1.50000 2.59808i 0.0635570 0.110084i −0.832496 0.554031i \(-0.813089\pi\)
0.896053 + 0.443947i \(0.146422\pi\)
\(558\) 0 0
\(559\) −55.0000 −2.32625
\(560\) 0 0
\(561\) 15.5885i 0.658145i
\(562\) 0 0
\(563\) −6.00000 10.3923i −0.252870 0.437983i 0.711445 0.702742i \(-0.248041\pi\)
−0.964315 + 0.264758i \(0.914708\pi\)
\(564\) 0 0
\(565\) 22.5000 38.9711i 0.946582 1.63953i
\(566\) 0 0
\(567\) 18.0000 15.5885i 0.755929 0.654654i
\(568\) 0 0
\(569\) −15.0000 + 25.9808i −0.628833 + 1.08917i 0.358954 + 0.933355i \(0.383134\pi\)
−0.987786 + 0.155815i \(0.950200\pi\)
\(570\) 0 0
\(571\) 10.0000 + 17.3205i 0.418487 + 0.724841i 0.995788 0.0916910i \(-0.0292272\pi\)
−0.577301 + 0.816532i \(0.695894\pi\)
\(572\) 0 0
\(573\) 20.7846i 0.868290i
\(574\) 0 0
\(575\) 12.0000 0.500435
\(576\) 0 0
\(577\) −5.50000 + 9.52628i −0.228968 + 0.396584i −0.957503 0.288425i \(-0.906868\pi\)
0.728535 + 0.685009i \(0.240202\pi\)
\(578\) 0 0
\(579\) −21.0000 12.1244i −0.872730 0.503871i
\(580\) 0 0
\(581\) −6.00000 + 5.19615i −0.248922 + 0.215573i
\(582\) 0 0
\(583\) 4.50000 7.79423i 0.186371 0.322804i
\(584\) 0 0
\(585\) −22.5000 38.9711i −0.930261 1.61126i
\(586\) 0 0
\(587\) −16.5000 28.5788i −0.681028 1.17957i −0.974668 0.223659i \(-0.928200\pi\)
0.293640 0.955916i \(-0.405133\pi\)
\(588\) 0 0
\(589\) 10.0000 17.3205i 0.412043 0.713679i
\(590\) 0 0
\(591\) −9.00000 + 5.19615i −0.370211 + 0.213741i
\(592\) 0 0
\(593\) 10.5000 + 18.1865i 0.431183 + 0.746831i 0.996976 0.0777165i \(-0.0247629\pi\)
−0.565792 + 0.824548i \(0.691430\pi\)
\(594\) 0 0
\(595\) 22.5000 + 7.79423i 0.922410 + 0.319532i
\(596\) 0 0
\(597\) 12.1244i 0.496217i
\(598\) 0 0
\(599\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(600\) 0 0
\(601\) 0.500000 + 0.866025i 0.0203954 + 0.0353259i 0.876043 0.482233i \(-0.160174\pi\)
−0.855648 + 0.517559i \(0.826841\pi\)
\(602\) 0 0
\(603\) −12.0000 −0.488678
\(604\) 0 0
\(605\) 6.00000 0.243935
\(606\) 0 0
\(607\) 43.0000 1.74532 0.872658 0.488332i \(-0.162394\pi\)
0.872658 + 0.488332i \(0.162394\pi\)
\(608\) 0 0
\(609\) −4.50000 + 12.9904i −0.182349 + 0.526397i
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 15.5000 26.8468i 0.626039 1.08433i −0.362300 0.932062i \(-0.618008\pi\)
0.988339 0.152270i \(-0.0486583\pi\)
\(614\) 0 0
\(615\) 46.7654i 1.88576i
\(616\) 0 0
\(617\) −1.50000 + 2.59808i −0.0603877 + 0.104595i −0.894639 0.446790i \(-0.852567\pi\)
0.834251 + 0.551385i \(0.185900\pi\)
\(618\) 0 0
\(619\) 19.0000 0.763674 0.381837 0.924230i \(-0.375291\pi\)
0.381837 + 0.924230i \(0.375291\pi\)
\(620\) 0 0
\(621\) 15.5885i 0.625543i
\(622\) 0 0
\(623\) 30.0000 25.9808i 1.20192 1.04090i
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) 22.5000 + 12.9904i 0.898563 + 0.518786i
\(628\) 0 0
\(629\) −21.0000 −0.837325
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 0 0
\(633\) 8.66025i 0.344214i
\(634\) 0 0
\(635\) −48.0000 −1.90482
\(636\) 0 0
\(637\) 5.00000 34.6410i 0.198107 1.37253i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −45.0000 −1.77739 −0.888697 0.458496i \(-0.848388\pi\)
−0.888697 + 0.458496i \(0.848388\pi\)
\(642\) 0 0
\(643\) 14.5000 25.1147i 0.571824 0.990429i −0.424555 0.905402i \(-0.639569\pi\)
0.996379 0.0850262i \(-0.0270974\pi\)
\(644\) 0 0
\(645\) −49.5000 28.5788i −1.94906 1.12529i
\(646\) 0 0
\(647\) −1.50000 + 2.59808i −0.0589711 + 0.102141i −0.894004 0.448059i \(-0.852115\pi\)
0.835033 + 0.550200i \(0.185449\pi\)
\(648\) 0 0
\(649\) 18.0000 + 31.1769i 0.706562 + 1.22380i
\(650\) 0 0
\(651\) 6.00000 17.3205i 0.235159 0.678844i
\(652\) 0 0
\(653\) 9.00000 0.352197 0.176099 0.984373i \(-0.443652\pi\)
0.176099 + 0.984373i \(0.443652\pi\)
\(654\) 0 0
\(655\) 9.00000 0.351659
\(656\) 0 0
\(657\) 16.5000 + 28.5788i 0.643726 + 1.11497i
\(658\) 0 0
\(659\) 19.5000 + 33.7750i 0.759612 + 1.31569i 0.943049 + 0.332655i \(0.107945\pi\)
−0.183436 + 0.983032i \(0.558722\pi\)
\(660\) 0 0
\(661\) −7.00000 12.1244i −0.272268 0.471583i 0.697174 0.716902i \(-0.254441\pi\)
−0.969442 + 0.245319i \(0.921107\pi\)
\(662\) 0 0
\(663\) −22.5000 12.9904i −0.873828 0.504505i
\(664\) 0 0
\(665\) 30.0000 25.9808i 1.16335 1.00749i
\(666\) 0 0
\(667\) 4.50000 + 7.79423i 0.174241 + 0.301794i
\(668\) 0 0
\(669\) 25.5000 + 14.7224i 0.985887 + 0.569202i
\(670\) 0 0
\(671\) −3.00000 + 5.19615i −0.115814 + 0.200595i
\(672\) 0 0
\(673\) −5.50000 9.52628i −0.212009 0.367211i 0.740334 0.672239i \(-0.234667\pi\)
−0.952343 + 0.305028i \(0.901334\pi\)
\(674\) 0 0
\(675\) 20.7846i 0.800000i
\(676\) 0 0
\(677\) −3.00000 + 5.19615i −0.115299 + 0.199704i −0.917899 0.396813i \(-0.870116\pi\)
0.802600 + 0.596518i \(0.203449\pi\)
\(678\) 0 0
\(679\) −2.00000 + 1.73205i −0.0767530 + 0.0664700i
\(680\) 0 0
\(681\) −13.5000 + 7.79423i −0.517321 + 0.298675i
\(682\) 0 0
\(683\) −16.5000 + 28.5788i −0.631355 + 1.09354i 0.355920 + 0.934516i \(0.384168\pi\)
−0.987275 + 0.159022i \(0.949166\pi\)
\(684\) 0 0
\(685\) −9.00000 −0.343872
\(686\) 0 0
\(687\) 25.5000 14.7224i 0.972886 0.561696i
\(688\) 0 0
\(689\) 7.50000 + 12.9904i 0.285727 + 0.494894i
\(690\) 0 0
\(691\) 10.0000 17.3205i 0.380418 0.658903i −0.610704 0.791859i \(-0.709113\pi\)
0.991122 + 0.132956i \(0.0424468\pi\)
\(692\) 0 0
\(693\) 22.5000 + 7.79423i 0.854704 + 0.296078i
\(694\) 0 0
\(695\) −7.50000 + 12.9904i −0.284491 + 0.492753i
\(696\) 0 0
\(697\) 13.5000 + 23.3827i 0.511349 + 0.885682i
\(698\) 0 0
\(699\) −40.5000 23.3827i −1.53185 0.884414i
\(700\) 0 0
\(701\) 6.00000 0.226617 0.113308 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 0 0
\(703\) −17.5000 + 30.3109i −0.660025 + 1.14320i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −1.50000 7.79423i −0.0564133 0.293132i
\(708\) 0 0
\(709\) 5.00000 8.66025i 0.187779 0.325243i −0.756730 0.653727i \(-0.773204\pi\)
0.944509 + 0.328484i \(0.106538\pi\)
\(710\) 0 0
\(711\) −12.0000 20.7846i −0.450035 0.779484i
\(712\) 0 0
\(713\) −6.00000 10.3923i −0.224702 0.389195i
\(714\) 0 0
\(715\) 22.5000 38.9711i 0.841452 1.45744i
\(716\) 0 0
\(717\) 46.7654i 1.74648i
\(718\) 0 0
\(719\) 19.5000 + 33.7750i 0.727227 + 1.25959i 0.958051 + 0.286599i \(0.0925247\pi\)
−0.230823 + 0.972996i \(0.574142\pi\)
\(720\) 0 0
\(721\) −2.50000 12.9904i −0.0931049 0.483787i
\(722\) 0 0
\(723\) 34.5000 19.9186i 1.28307 0.740780i
\(724\) 0 0
\(725\) 6.00000 + 10.3923i 0.222834 + 0.385961i
\(726\) 0 0
\(727\) 2.50000 + 4.33013i 0.0927199 + 0.160596i 0.908655 0.417548i \(-0.137111\pi\)
−0.815935 + 0.578144i \(0.803777\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) −33.0000 −1.22055
\(732\) 0 0
\(733\) 41.0000 1.51437 0.757185 0.653201i \(-0.226574\pi\)
0.757185 + 0.653201i \(0.226574\pi\)
\(734\) 0 0
\(735\) 22.5000 28.5788i 0.829925 1.05415i
\(736\) 0 0
\(737\) −6.00000 10.3923i −0.221013 0.382805i
\(738\) 0 0
\(739\) 23.5000 40.7032i 0.864461 1.49729i −0.00311943 0.999995i \(-0.500993\pi\)
0.867581 0.497296i \(-0.165674\pi\)
\(740\) 0 0
\(741\) −37.5000 + 21.6506i −1.37760 + 0.795356i
\(742\) 0 0
\(743\) 1.50000 2.59808i 0.0550297 0.0953142i −0.837198 0.546899i \(-0.815808\pi\)
0.892228 + 0.451585i \(0.149141\pi\)
\(744\) 0 0
\(745\) 9.00000 0.329734
\(746\) 0 0
\(747\) 9.00000 0.329293
\(748\) 0 0
\(749\) 30.0000 25.9808i 1.09618 0.949316i
\(750\) 0 0
\(751\) −29.0000 −1.05823 −0.529113 0.848552i \(-0.677475\pi\)
−0.529113 + 0.848552i \(0.677475\pi\)
\(752\) 0 0
\(753\) −18.0000 + 10.3923i −0.655956 + 0.378717i
\(754\) 0 0
\(755\) 33.0000 1.20099
\(756\) 0 0
\(757\) 14.0000 0.508839 0.254419 0.967094i \(-0.418116\pi\)
0.254419 + 0.967094i \(0.418116\pi\)
\(758\) 0 0
\(759\) 13.5000 7.79423i 0.490019 0.282913i
\(760\) 0 0
\(761\) 3.00000 0.108750 0.0543750 0.998521i \(-0.482683\pi\)
0.0543750 + 0.998521i \(0.482683\pi\)
\(762\) 0 0
\(763\) 17.5000 + 6.06218i 0.633543 + 0.219466i
\(764\) 0 0
\(765\) −13.5000 23.3827i −0.488094 0.845403i
\(766\) 0 0
\(767\) −60.0000 −2.16647
\(768\) 0 0
\(769\) 0.500000 0.866025i 0.0180305 0.0312297i −0.856869 0.515534i \(-0.827594\pi\)
0.874900 + 0.484304i \(0.160927\pi\)
\(770\) 0 0
\(771\) 22.5000 12.9904i 0.810318 0.467837i
\(772\) 0 0
\(773\) −10.5000 + 18.1865i −0.377659 + 0.654124i −0.990721 0.135910i \(-0.956604\pi\)
0.613062 + 0.790034i \(0.289937\pi\)
\(774\) 0 0
\(775\) −8.00000 13.8564i −0.287368 0.497737i
\(776\) 0 0
\(777\) −10.5000 + 30.3109i −0.376685 + 1.08740i
\(778\) 0 0
\(779\) 45.0000 1.61229
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 13.5000 7.79423i 0.482451 0.278543i
\(784\) 0 0
\(785\) 21.0000 + 36.3731i 0.749522 + 1.29821i
\(786\) 0 0
\(787\) 22.0000 + 38.1051i 0.784215 + 1.35830i 0.929467 + 0.368906i \(0.120268\pi\)
−0.145251 + 0.989395i \(0.546399\pi\)
\(788\) 0 0
\(789\) 13.5000 7.79423i 0.480613 0.277482i
\(790\) 0 0
\(791\) −37.5000 12.9904i −1.33335 0.461885i
\(792\) 0 0
\(793\) −5.00000 8.66025i −0.177555 0.307535i
\(794\) 0 0
\(795\) 15.5885i 0.552866i
\(796\) 0 0
\(797\) 13.5000 23.3827i 0.478195 0.828257i −0.521493 0.853256i \(-0.674625\pi\)
0.999687 + 0.0249984i \(0.00795805\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −45.0000 −1.59000
\(802\) 0 0
\(803\) −16.5000 + 28.5788i −0.582272 + 1.00853i
\(804\) 0 0
\(805\) −4.50000 23.3827i −0.158604 0.824131i
\(806\) 0 0
\(807\) 36.3731i 1.28039i
\(808\) 0 0
\(809\) −19.5000 + 33.7750i −0.685583 + 1.18747i 0.287670 + 0.957730i \(0.407120\pi\)
−0.973253 + 0.229736i \(0.926214\pi\)
\(810\) 0 0
\(811\) −20.0000 −0.702295 −0.351147 0.936320i \(-0.614208\pi\)
−0.351147 + 0.936320i \(0.614208\pi\)
\(812\) 0 0
\(813\) −19.5000 11.2583i −0.683895 0.394847i
\(814\) 0 0
\(815\) −25.5000 44.1673i −0.893226 1.54711i
\(816\) 0 0
\(817\) −27.5000 + 47.6314i −0.962103 + 1.66641i
\(818\) 0 0
\(819\) −30.0000 + 25.9808i −1.04828 + 0.907841i
\(820\) 0 0
\(821\) 27.0000 46.7654i 0.942306 1.63212i 0.181250 0.983437i \(-0.441986\pi\)
0.761056 0.648686i \(-0.224681\pi\)
\(822\) 0 0
\(823\) −20.0000 34.6410i −0.697156 1.20751i −0.969448 0.245295i \(-0.921115\pi\)
0.272292 0.962215i \(-0.412218\pi\)
\(824\) 0 0
\(825\) 18.0000 10.3923i 0.626680 0.361814i
\(826\) 0 0
\(827\) 24.0000 0.834562 0.417281 0.908778i \(-0.362983\pi\)
0.417281 + 0.908778i \(0.362983\pi\)
\(828\) 0 0
\(829\) −20.5000 + 35.5070i −0.711994 + 1.23321i 0.252113 + 0.967698i \(0.418875\pi\)
−0.964107 + 0.265513i \(0.914459\pi\)
\(830\) 0 0
\(831\) −10.5000 + 6.06218i −0.364241 + 0.210295i
\(832\) 0 0
\(833\) 3.00000 20.7846i 0.103944 0.720144i
\(834\) 0 0
\(835\) −4.50000 + 7.79423i −0.155729 + 0.269730i
\(836\) 0 0
\(837\) −18.0000 + 10.3923i −0.622171 + 0.359211i
\(838\) 0 0
\(839\) −19.5000 33.7750i −0.673215 1.16604i −0.976987 0.213298i \(-0.931580\pi\)
0.303773 0.952745i \(-0.401754\pi\)
\(840\) 0 0
\(841\) 10.0000 17.3205i 0.344828 0.597259i
\(842\) 0 0
\(843\) −4.50000 2.59808i −0.154988 0.0894825i
\(844\) 0 0
\(845\) 18.0000 + 31.1769i 0.619219 + 1.07252i
\(846\) 0 0
\(847\) −1.00000 5.19615i −0.0343604 0.178542i
\(848\) 0 0
\(849\) 12.0000 + 6.92820i 0.411839 + 0.237775i
\(850\) 0 0
\(851\) 10.5000 + 18.1865i 0.359935 + 0.623426i
\(852\) 0 0
\(853\) −8.50000 14.7224i −0.291034 0.504086i 0.683020 0.730400i \(-0.260666\pi\)
−0.974055 + 0.226313i \(0.927333\pi\)
\(854\) 0 0
\(855\) −45.0000 −1.53897
\(856\) 0 0
\(857\) −33.0000 −1.12726 −0.563629 0.826028i \(-0.690595\pi\)
−0.563629 + 0.826028i \(0.690595\pi\)
\(858\) 0 0
\(859\) −11.0000 −0.375315 −0.187658 0.982235i \(-0.560090\pi\)
−0.187658 + 0.982235i \(0.560090\pi\)
\(860\) 0 0
\(861\) 40.5000 7.79423i 1.38024 0.265627i
\(862\) 0 0
\(863\) 7.50000 + 12.9904i 0.255303 + 0.442198i 0.964978 0.262332i \(-0.0844915\pi\)
−0.709675 + 0.704529i \(0.751158\pi\)
\(864\) 0 0
\(865\) 9.00000 15.5885i 0.306009 0.530023i
\(866\) 0 0
\(867\) 12.0000 + 6.92820i 0.407541 + 0.235294i
\(868\) 0 0
\(869\) 12.0000 20.7846i 0.407072 0.705070i
\(870\) 0 0
\(871\) 20.0000 0.677674
\(872\) 0 0
\(873\) 3.00000 0.101535
\(874\) 0 0
\(875\) 1.50000 + 7.79423i 0.0507093 + 0.263493i
\(876\) 0 0
\(877\) −43.0000 −1.45201 −0.726003 0.687691i \(-0.758624\pi\)
−0.726003 + 0.687691i \(0.758624\pi\)
\(878\) 0 0
\(879\) 46.7654i 1.57736i
\(880\) 0 0
\(881\) 6.00000 0.202145 0.101073 0.994879i \(-0.467773\pi\)
0.101073 + 0.994879i \(0.467773\pi\)
\(882\) 0 0
\(883\) 4.00000 0.134611 0.0673054 0.997732i \(-0.478560\pi\)
0.0673054 + 0.997732i \(0.478560\pi\)
\(884\) 0 0
\(885\) −54.0000 31.1769i −1.81519 1.04800i
\(886\) 0 0
\(887\) 39.0000 1.30949 0.654746 0.755849i \(-0.272776\pi\)
0.654746 + 0.755849i \(0.272776\pi\)
\(888\) 0 0
\(889\) 8.00000 + 41.5692i 0.268311 + 1.39419i
\(890\) 0 0
\(891\) −13.5000 23.3827i −0.452267 0.783349i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −4.50000 + 7.79423i −0.150418 + 0.260532i
\(896\) 0 0
\(897\) 25.9808i 0.867472i
\(898\) 0 0
\(899\) 6.00000 10.3923i 0.200111 0.346603i
\(900\) 0 0
\(901\) 4.50000 + 7.79423i 0.149917 + 0.259663i
\(902\) 0 0
\(903\) −16.5000 + 47.6314i −0.549086 + 1.58507i
\(904\) 0 0
\(905\) 30.0000 0.997234
\(906\) 0 0
\(907\) −17.0000 −0.564476 −0.282238 0.959344i \(-0.591077\pi\)
−0.282238 + 0.959344i \(0.591077\pi\)
\(908\) 0 0
\(909\) −4.50000 + 7.79423i −0.149256 + 0.258518i
\(910\) 0 0
\(911\) 4.50000 + 7.79423i 0.149092 + 0.258234i 0.930892 0.365295i \(-0.119032\pi\)
−0.781800 + 0.623529i \(0.785698\pi\)
\(912\) 0 0
\(913\) 4.50000 + 7.79423i 0.148928 + 0.257951i
\(914\) 0 0
\(915\) 10.3923i 0.343559i
\(916\) 0 0
\(917\) −1.50000 7.79423i −0.0495344 0.257388i
\(918\) 0 0
\(919\) −0.500000 0.866025i −0.0164935 0.0285675i 0.857661 0.514216i \(-0.171917\pi\)
−0.874154 + 0.485648i \(0.838584\pi\)
\(920\) 0 0
\(921\) 42.0000 24.2487i 1.38395 0.799022i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 14.0000 + 24.2487i 0.460317 + 0.797293i
\(926\) 0 0
\(927\) −7.50000 + 12.9904i −0.246332 + 0.426660i
\(928\) 0 0
\(929\) 9.00000 15.5885i 0.295280 0.511441i −0.679770 0.733426i \(-0.737920\pi\)
0.975050 + 0.221985i \(0.0712536\pi\)
\(930\) 0 0
\(931\) −27.5000 21.6506i −0.901276 0.709571i
\(932\) 0 0
\(933\) −36.0000 20.7846i −1.17859 0.680458i
\(934\) 0 0
\(935\) 13.5000 23.3827i 0.441497 0.764696i
\(936\) 0 0
\(937\) −34.0000 −1.11073 −0.555366 0.831606i \(-0.687422\pi\)
−0.555366 + 0.831606i \(0.687422\pi\)
\(938\) 0 0
\(939\) 24.2487i 0.791327i
\(940\) 0 0
\(941\) −27.0000 46.7654i −0.880175 1.52451i −0.851146 0.524929i \(-0.824092\pi\)
−0.0290288 0.999579i \(-0.509241\pi\)
\(942\) 0 0
\(943\) 13.5000 23.3827i 0.439620 0.761445i
\(944\) 0 0
\(945\) −40.5000 + 7.79423i −1.31747 + 0.253546i
\(946\) 0 0
\(947\) −6.00000 + 10.3923i −0.194974 + 0.337705i −0.946892 0.321552i \(-0.895796\pi\)
0.751918 + 0.659256i \(0.229129\pi\)
\(948\) 0 0
\(949\) −27.5000 47.6314i −0.892688 1.54618i
\(950\) 0 0
\(951\) 51.9615i 1.68497i
\(952\) 0 0
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 0 0
\(955\) −18.0000 + 31.1769i −0.582466 + 1.00886i
\(956\) 0 0
\(957\) 13.5000 + 7.79423i 0.436393 + 0.251952i
\(958\) 0 0
\(959\) 1.50000 + 7.79423i 0.0484375 + 0.251689i
\(960\) 0 0
\(961\) 7.50000 12.9904i 0.241935 0.419045i
\(962\) 0 0
\(963\) −45.0000 −1.45010
\(964\) 0 0
\(965\) 21.0000 + 36.3731i 0.676014 + 1.17089i
\(966\) 0 0
\(967\) −24.5000 + 42.4352i −0.787867 + 1.36463i 0.139404 + 0.990236i \(0.455481\pi\)
−0.927271 + 0.374390i \(0.877852\pi\)
\(968\) 0 0
\(969\) −22.5000 + 12.9904i −0.722804 + 0.417311i
\(970\) 0 0
\(971\) −13.5000 23.3827i −0.433236 0.750386i 0.563914 0.825833i \(-0.309295\pi\)
−0.997150 + 0.0754473i \(0.975962\pi\)
\(972\) 0 0
\(973\) 12.5000 + 4.33013i 0.400732 + 0.138817i
\(974\) 0 0
\(975\) 34.6410i 1.10940i
\(976\) 0 0
\(977\) −3.00000 5.19615i −0.0959785 0.166240i 0.814038 0.580812i \(-0.197265\pi\)
−0.910017 + 0.414572i \(0.863931\pi\)
\(978\) 0 0
\(979\) −22.5000 38.9711i −0.719103 1.24552i
\(980\) 0 0
\(981\) −10.5000 18.1865i −0.335239 0.580651i
\(982\) 0 0
\(983\) 21.0000 0.669796 0.334898 0.942254i \(-0.391298\pi\)
0.334898 + 0.942254i \(0.391298\pi\)
\(984\) 0 0
\(985\) 18.0000 0.573528
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 16.5000 + 28.5788i 0.524669 + 0.908754i
\(990\) 0 0
\(991\) 14.5000 25.1147i 0.460608 0.797796i −0.538384 0.842700i \(-0.680965\pi\)
0.998991 + 0.0449040i \(0.0142982\pi\)
\(992\) 0 0
\(993\) 34.6410i 1.09930i
\(994\) 0 0
\(995\) 10.5000 18.1865i 0.332872 0.576552i
\(996\) 0 0
\(997\) 41.0000 1.29848 0.649242 0.760582i \(-0.275086\pi\)
0.649242 + 0.760582i \(0.275086\pi\)
\(998\) 0 0
\(999\) 31.5000 18.1865i 0.996616 0.575396i
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 1008.2.t.c.193.1 2
3.2 odd 2 3024.2.t.f.1873.1 2
4.3 odd 2 126.2.h.a.67.1 yes 2
7.2 even 3 1008.2.q.e.625.1 2
9.2 odd 6 3024.2.q.a.2881.1 2
9.7 even 3 1008.2.q.e.529.1 2
12.11 even 2 378.2.h.b.361.1 2
21.2 odd 6 3024.2.q.a.2305.1 2
28.3 even 6 882.2.f.a.589.1 2
28.11 odd 6 882.2.f.e.589.1 2
28.19 even 6 882.2.e.h.373.1 2
28.23 odd 6 126.2.e.b.121.1 yes 2
28.27 even 2 882.2.h.e.67.1 2
36.7 odd 6 126.2.e.b.25.1 2
36.11 even 6 378.2.e.a.235.1 2
36.23 even 6 1134.2.g.f.487.1 2
36.31 odd 6 1134.2.g.d.487.1 2
63.2 odd 6 3024.2.t.f.289.1 2
63.16 even 3 inner 1008.2.t.c.961.1 2
84.11 even 6 2646.2.f.e.1765.1 2
84.23 even 6 378.2.e.a.37.1 2
84.47 odd 6 2646.2.e.e.1549.1 2
84.59 odd 6 2646.2.f.i.1765.1 2
84.83 odd 2 2646.2.h.f.361.1 2
252.11 even 6 2646.2.f.e.883.1 2
252.23 even 6 1134.2.g.f.163.1 2
252.31 even 6 7938.2.a.bd.1.1 1
252.47 odd 6 2646.2.h.f.667.1 2
252.59 odd 6 7938.2.a.c.1.1 1
252.67 odd 6 7938.2.a.r.1.1 1
252.79 odd 6 126.2.h.a.79.1 yes 2
252.83 odd 6 2646.2.e.e.2125.1 2
252.95 even 6 7938.2.a.o.1.1 1
252.115 even 6 882.2.f.a.295.1 2
252.151 odd 6 882.2.f.e.295.1 2
252.187 even 6 882.2.h.e.79.1 2
252.191 even 6 378.2.h.b.289.1 2
252.223 even 6 882.2.e.h.655.1 2
252.227 odd 6 2646.2.f.i.883.1 2
252.247 odd 6 1134.2.g.d.163.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.2.e.b.25.1 2 36.7 odd 6
126.2.e.b.121.1 yes 2 28.23 odd 6
126.2.h.a.67.1 yes 2 4.3 odd 2
126.2.h.a.79.1 yes 2 252.79 odd 6
378.2.e.a.37.1 2 84.23 even 6
378.2.e.a.235.1 2 36.11 even 6
378.2.h.b.289.1 2 252.191 even 6
378.2.h.b.361.1 2 12.11 even 2
882.2.e.h.373.1 2 28.19 even 6
882.2.e.h.655.1 2 252.223 even 6
882.2.f.a.295.1 2 252.115 even 6
882.2.f.a.589.1 2 28.3 even 6
882.2.f.e.295.1 2 252.151 odd 6
882.2.f.e.589.1 2 28.11 odd 6
882.2.h.e.67.1 2 28.27 even 2
882.2.h.e.79.1 2 252.187 even 6
1008.2.q.e.529.1 2 9.7 even 3
1008.2.q.e.625.1 2 7.2 even 3
1008.2.t.c.193.1 2 1.1 even 1 trivial
1008.2.t.c.961.1 2 63.16 even 3 inner
1134.2.g.d.163.1 2 252.247 odd 6
1134.2.g.d.487.1 2 36.31 odd 6
1134.2.g.f.163.1 2 252.23 even 6
1134.2.g.f.487.1 2 36.23 even 6
2646.2.e.e.1549.1 2 84.47 odd 6
2646.2.e.e.2125.1 2 252.83 odd 6
2646.2.f.e.883.1 2 252.11 even 6
2646.2.f.e.1765.1 2 84.11 even 6
2646.2.f.i.883.1 2 252.227 odd 6
2646.2.f.i.1765.1 2 84.59 odd 6
2646.2.h.f.361.1 2 84.83 odd 2
2646.2.h.f.667.1 2 252.47 odd 6
3024.2.q.a.2305.1 2 21.2 odd 6
3024.2.q.a.2881.1 2 9.2 odd 6
3024.2.t.f.289.1 2 63.2 odd 6
3024.2.t.f.1873.1 2 3.2 odd 2
7938.2.a.c.1.1 1 252.59 odd 6
7938.2.a.o.1.1 1 252.95 even 6
7938.2.a.r.1.1 1 252.67 odd 6
7938.2.a.bd.1.1 1 252.31 even 6