Properties

Label 1008.2.t.k.961.9
Level $1008$
Weight $2$
Character 1008.961
Analytic conductor $8.049$
Analytic rank $0$
Dimension $22$
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: \(22\)
Relative dimension: \(11\) over \(\Q(\zeta_{3})\)
Twist minimal: no (minimal twist has level 504)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 961.9
Character \(\chi\) \(=\) 1008.961
Dual form 1008.2.t.k.193.9

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.21966 + 1.22980i) q^{3} -0.481387 q^{5} +(-2.53326 - 0.763277i) q^{7} +(-0.0248369 + 2.99990i) q^{9} +O(q^{10})\) \(q+(1.21966 + 1.22980i) q^{3} -0.481387 q^{5} +(-2.53326 - 0.763277i) q^{7} +(-0.0248369 + 2.99990i) q^{9} -3.38159 q^{11} +(-2.86067 - 4.95482i) q^{13} +(-0.587131 - 0.592012i) q^{15} +(2.75605 + 4.77362i) q^{17} +(-2.18023 + 3.77626i) q^{19} +(-2.15105 - 4.04636i) q^{21} -3.62585 q^{23} -4.76827 q^{25} +(-3.71958 + 3.62832i) q^{27} +(1.53131 - 2.65231i) q^{29} +(-4.67459 + 8.09663i) q^{31} +(-4.12441 - 4.15870i) q^{33} +(1.21948 + 0.367431i) q^{35} +(1.48552 - 2.57299i) q^{37} +(2.60441 - 9.56128i) q^{39} +(-6.29558 - 10.9043i) q^{41} +(-1.90827 + 3.30522i) q^{43} +(0.0119562 - 1.44411i) q^{45} +(-1.88282 - 3.26114i) q^{47} +(5.83482 + 3.86716i) q^{49} +(-2.50916 + 9.21161i) q^{51} +(5.57860 + 9.66242i) q^{53} +1.62786 q^{55} +(-7.30321 + 1.92452i) q^{57} +(4.21141 - 7.29438i) q^{59} +(3.64312 + 6.31007i) q^{61} +(2.35267 - 7.58056i) q^{63} +(1.37709 + 2.38519i) q^{65} +(1.28571 - 2.22692i) q^{67} +(-4.42232 - 4.45909i) q^{69} +3.94304 q^{71} +(-0.862216 - 1.49340i) q^{73} +(-5.81569 - 5.86403i) q^{75} +(8.56646 + 2.58109i) q^{77} +(-2.79980 - 4.84940i) q^{79} +(-8.99877 - 0.149016i) q^{81} +(0.119494 - 0.206970i) q^{83} +(-1.32673 - 2.29796i) q^{85} +(5.12952 - 1.35172i) q^{87} +(0.648116 - 1.12257i) q^{89} +(3.46492 + 14.7353i) q^{91} +(-15.6587 + 4.12634i) q^{93} +(1.04953 - 1.81784i) q^{95} +(-7.02669 + 12.1706i) q^{97} +(0.0839884 - 10.1444i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 22 q - 2 q^{3} - 6 q^{5} - 7 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 22 q - 2 q^{3} - 6 q^{5} - 7 q^{7} - 8 q^{9} - 6 q^{11} - 3 q^{13} + q^{15} + 7 q^{17} + q^{19} - 15 q^{21} + 4 q^{23} + 20 q^{25} + 4 q^{27} + 9 q^{29} + 4 q^{31} - 31 q^{33} - 14 q^{35} + 2 q^{37} - 8 q^{39} + 16 q^{41} + 22 q^{45} - 5 q^{47} - 15 q^{49} - 7 q^{51} + 11 q^{53} - 22 q^{55} + 7 q^{57} + 19 q^{59} - 13 q^{61} - 21 q^{63} + 13 q^{65} - 26 q^{67} - 4 q^{69} + 48 q^{71} - 35 q^{73} + 8 q^{75} - 4 q^{77} - 10 q^{79} - 8 q^{81} + 28 q^{83} - 20 q^{85} - 9 q^{87} + 6 q^{89} + 37 q^{91} - 32 q^{93} - 12 q^{95} - 29 q^{97} + 56 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.21966 + 1.22980i 0.704174 + 0.710028i
\(4\) 0 0
\(5\) −0.481387 −0.215283 −0.107641 0.994190i \(-0.534330\pi\)
−0.107641 + 0.994190i \(0.534330\pi\)
\(6\) 0 0
\(7\) −2.53326 0.763277i −0.957482 0.288491i
\(8\) 0 0
\(9\) −0.0248369 + 2.99990i −0.00827898 + 0.999966i
\(10\) 0 0
\(11\) −3.38159 −1.01959 −0.509794 0.860296i \(-0.670279\pi\)
−0.509794 + 0.860296i \(0.670279\pi\)
\(12\) 0 0
\(13\) −2.86067 4.95482i −0.793406 1.37422i −0.923846 0.382764i \(-0.874972\pi\)
0.130440 0.991456i \(-0.458361\pi\)
\(14\) 0 0
\(15\) −0.587131 0.592012i −0.151596 0.152857i
\(16\) 0 0
\(17\) 2.75605 + 4.77362i 0.668440 + 1.15777i 0.978340 + 0.207003i \(0.0663711\pi\)
−0.309900 + 0.950769i \(0.600296\pi\)
\(18\) 0 0
\(19\) −2.18023 + 3.77626i −0.500178 + 0.866334i 0.499822 + 0.866128i \(0.333399\pi\)
−1.00000 0.000205746i \(0.999935\pi\)
\(20\) 0 0
\(21\) −2.15105 4.04636i −0.469397 0.882987i
\(22\) 0 0
\(23\) −3.62585 −0.756042 −0.378021 0.925797i \(-0.623395\pi\)
−0.378021 + 0.925797i \(0.623395\pi\)
\(24\) 0 0
\(25\) −4.76827 −0.953653
\(26\) 0 0
\(27\) −3.71958 + 3.62832i −0.715833 + 0.698271i
\(28\) 0 0
\(29\) 1.53131 2.65231i 0.284358 0.492522i −0.688095 0.725620i \(-0.741553\pi\)
0.972453 + 0.233098i \(0.0748863\pi\)
\(30\) 0 0
\(31\) −4.67459 + 8.09663i −0.839581 + 1.45420i 0.0506646 + 0.998716i \(0.483866\pi\)
−0.890245 + 0.455481i \(0.849467\pi\)
\(32\) 0 0
\(33\) −4.12441 4.15870i −0.717968 0.723936i
\(34\) 0 0
\(35\) 1.21948 + 0.367431i 0.206130 + 0.0621072i
\(36\) 0 0
\(37\) 1.48552 2.57299i 0.244218 0.422997i −0.717694 0.696359i \(-0.754802\pi\)
0.961911 + 0.273361i \(0.0881355\pi\)
\(38\) 0 0
\(39\) 2.60441 9.56128i 0.417039 1.53103i
\(40\) 0 0
\(41\) −6.29558 10.9043i −0.983204 1.70296i −0.649659 0.760226i \(-0.725088\pi\)
−0.333545 0.942734i \(-0.608245\pi\)
\(42\) 0 0
\(43\) −1.90827 + 3.30522i −0.291009 + 0.504042i −0.974049 0.226339i \(-0.927324\pi\)
0.683040 + 0.730381i \(0.260658\pi\)
\(44\) 0 0
\(45\) 0.0119562 1.44411i 0.00178232 0.215275i
\(46\) 0 0
\(47\) −1.88282 3.26114i −0.274638 0.475687i 0.695406 0.718617i \(-0.255225\pi\)
−0.970044 + 0.242930i \(0.921891\pi\)
\(48\) 0 0
\(49\) 5.83482 + 3.86716i 0.833545 + 0.552451i
\(50\) 0 0
\(51\) −2.50916 + 9.21161i −0.351353 + 1.28988i
\(52\) 0 0
\(53\) 5.57860 + 9.66242i 0.766280 + 1.32724i 0.939567 + 0.342364i \(0.111228\pi\)
−0.173287 + 0.984871i \(0.555439\pi\)
\(54\) 0 0
\(55\) 1.62786 0.219500
\(56\) 0 0
\(57\) −7.30321 + 1.92452i −0.967334 + 0.254909i
\(58\) 0 0
\(59\) 4.21141 7.29438i 0.548279 0.949647i −0.450114 0.892971i \(-0.648617\pi\)
0.998393 0.0566756i \(-0.0180501\pi\)
\(60\) 0 0
\(61\) 3.64312 + 6.31007i 0.466454 + 0.807922i 0.999266 0.0383116i \(-0.0121979\pi\)
−0.532812 + 0.846234i \(0.678865\pi\)
\(62\) 0 0
\(63\) 2.35267 7.58056i 0.296409 0.955061i
\(64\) 0 0
\(65\) 1.37709 + 2.38519i 0.170807 + 0.295846i
\(66\) 0 0
\(67\) 1.28571 2.22692i 0.157075 0.272062i −0.776738 0.629824i \(-0.783127\pi\)
0.933813 + 0.357763i \(0.116460\pi\)
\(68\) 0 0
\(69\) −4.42232 4.45909i −0.532385 0.536811i
\(70\) 0 0
\(71\) 3.94304 0.467953 0.233977 0.972242i \(-0.424826\pi\)
0.233977 + 0.972242i \(0.424826\pi\)
\(72\) 0 0
\(73\) −0.862216 1.49340i −0.100915 0.174790i 0.811147 0.584842i \(-0.198844\pi\)
−0.912062 + 0.410053i \(0.865510\pi\)
\(74\) 0 0
\(75\) −5.81569 5.86403i −0.671538 0.677120i
\(76\) 0 0
\(77\) 8.56646 + 2.58109i 0.976238 + 0.294143i
\(78\) 0 0
\(79\) −2.79980 4.84940i −0.315002 0.545600i 0.664436 0.747345i \(-0.268672\pi\)
−0.979438 + 0.201746i \(0.935339\pi\)
\(80\) 0 0
\(81\) −8.99877 0.149016i −0.999863 0.0165574i
\(82\) 0 0
\(83\) 0.119494 0.206970i 0.0131162 0.0227179i −0.859393 0.511316i \(-0.829158\pi\)
0.872509 + 0.488598i \(0.162492\pi\)
\(84\) 0 0
\(85\) −1.32673 2.29796i −0.143904 0.249249i
\(86\) 0 0
\(87\) 5.12952 1.35172i 0.549942 0.144919i
\(88\) 0 0
\(89\) 0.648116 1.12257i 0.0687002 0.118992i −0.829629 0.558315i \(-0.811448\pi\)
0.898329 + 0.439323i \(0.144781\pi\)
\(90\) 0 0
\(91\) 3.46492 + 14.7353i 0.363222 + 1.54468i
\(92\) 0 0
\(93\) −15.6587 + 4.12634i −1.62373 + 0.427881i
\(94\) 0 0
\(95\) 1.04953 1.81784i 0.107680 0.186507i
\(96\) 0 0
\(97\) −7.02669 + 12.1706i −0.713452 + 1.23574i 0.250101 + 0.968220i \(0.419536\pi\)
−0.963553 + 0.267516i \(0.913797\pi\)
\(98\) 0 0
\(99\) 0.0839884 10.1444i 0.00844115 1.01955i
\(100\) 0 0
\(101\) 10.6064 1.05538 0.527690 0.849437i \(-0.323058\pi\)
0.527690 + 0.849437i \(0.323058\pi\)
\(102\) 0 0
\(103\) 0.159416 0.0157077 0.00785385 0.999969i \(-0.497500\pi\)
0.00785385 + 0.999969i \(0.497500\pi\)
\(104\) 0 0
\(105\) 1.03549 + 1.94786i 0.101053 + 0.190092i
\(106\) 0 0
\(107\) 3.99030 6.91140i 0.385757 0.668150i −0.606117 0.795375i \(-0.707274\pi\)
0.991874 + 0.127225i \(0.0406071\pi\)
\(108\) 0 0
\(109\) −6.85612 11.8751i −0.656697 1.13743i −0.981466 0.191639i \(-0.938620\pi\)
0.324769 0.945793i \(-0.394714\pi\)
\(110\) 0 0
\(111\) 4.97611 1.31129i 0.472312 0.124462i
\(112\) 0 0
\(113\) 8.98656 + 15.5652i 0.845384 + 1.46425i 0.885287 + 0.465045i \(0.153962\pi\)
−0.0399031 + 0.999204i \(0.512705\pi\)
\(114\) 0 0
\(115\) 1.74544 0.162763
\(116\) 0 0
\(117\) 14.9350 8.45865i 1.38074 0.782002i
\(118\) 0 0
\(119\) −3.33820 14.1964i −0.306012 1.30139i
\(120\) 0 0
\(121\) 0.435176 0.0395615
\(122\) 0 0
\(123\) 5.73161 21.0419i 0.516802 1.89728i
\(124\) 0 0
\(125\) 4.70232 0.420588
\(126\) 0 0
\(127\) −18.9684 −1.68317 −0.841587 0.540121i \(-0.818378\pi\)
−0.841587 + 0.540121i \(0.818378\pi\)
\(128\) 0 0
\(129\) −6.39223 + 1.68446i −0.562804 + 0.148309i
\(130\) 0 0
\(131\) 4.88232 0.426570 0.213285 0.976990i \(-0.431584\pi\)
0.213285 + 0.976990i \(0.431584\pi\)
\(132\) 0 0
\(133\) 8.40541 7.90214i 0.728842 0.685203i
\(134\) 0 0
\(135\) 1.79056 1.74663i 0.154107 0.150326i
\(136\) 0 0
\(137\) −6.47482 −0.553181 −0.276591 0.960988i \(-0.589205\pi\)
−0.276591 + 0.960988i \(0.589205\pi\)
\(138\) 0 0
\(139\) 11.3740 + 19.7003i 0.964727 + 1.67096i 0.710346 + 0.703852i \(0.248538\pi\)
0.254381 + 0.967104i \(0.418128\pi\)
\(140\) 0 0
\(141\) 1.71416 6.29300i 0.144358 0.529967i
\(142\) 0 0
\(143\) 9.67362 + 16.7552i 0.808948 + 1.40114i
\(144\) 0 0
\(145\) −0.737155 + 1.27679i −0.0612174 + 0.106032i
\(146\) 0 0
\(147\) 2.36067 + 11.8923i 0.194705 + 0.980862i
\(148\) 0 0
\(149\) −14.2046 −1.16369 −0.581843 0.813301i \(-0.697668\pi\)
−0.581843 + 0.813301i \(0.697668\pi\)
\(150\) 0 0
\(151\) −2.52259 −0.205285 −0.102643 0.994718i \(-0.532730\pi\)
−0.102643 + 0.994718i \(0.532730\pi\)
\(152\) 0 0
\(153\) −14.3888 + 8.14930i −1.16327 + 0.658832i
\(154\) 0 0
\(155\) 2.25029 3.89761i 0.180747 0.313064i
\(156\) 0 0
\(157\) −8.74064 + 15.1392i −0.697579 + 1.20824i 0.271724 + 0.962375i \(0.412406\pi\)
−0.969303 + 0.245867i \(0.920927\pi\)
\(158\) 0 0
\(159\) −5.07887 + 18.6455i −0.402780 + 1.47868i
\(160\) 0 0
\(161\) 9.18523 + 2.76753i 0.723897 + 0.218112i
\(162\) 0 0
\(163\) −0.881184 + 1.52625i −0.0690196 + 0.119546i −0.898470 0.439035i \(-0.855320\pi\)
0.829450 + 0.558580i \(0.188654\pi\)
\(164\) 0 0
\(165\) 1.98544 + 2.00194i 0.154566 + 0.155851i
\(166\) 0 0
\(167\) 3.57220 + 6.18723i 0.276425 + 0.478782i 0.970494 0.241127i \(-0.0775169\pi\)
−0.694069 + 0.719909i \(0.744184\pi\)
\(168\) 0 0
\(169\) −9.86684 + 17.0899i −0.758988 + 1.31461i
\(170\) 0 0
\(171\) −11.2742 6.63424i −0.862163 0.507333i
\(172\) 0 0
\(173\) −4.94691 8.56830i −0.376107 0.651436i 0.614385 0.789006i \(-0.289404\pi\)
−0.990492 + 0.137570i \(0.956071\pi\)
\(174\) 0 0
\(175\) 12.0793 + 3.63951i 0.913106 + 0.275121i
\(176\) 0 0
\(177\) 14.1072 3.71748i 1.06036 0.279423i
\(178\) 0 0
\(179\) −2.02967 3.51550i −0.151705 0.262761i 0.780149 0.625593i \(-0.215143\pi\)
−0.931854 + 0.362833i \(0.881810\pi\)
\(180\) 0 0
\(181\) 4.58084 0.340491 0.170246 0.985402i \(-0.445544\pi\)
0.170246 + 0.985402i \(0.445544\pi\)
\(182\) 0 0
\(183\) −3.31677 + 12.1765i −0.245183 + 0.900113i
\(184\) 0 0
\(185\) −0.715109 + 1.23861i −0.0525759 + 0.0910641i
\(186\) 0 0
\(187\) −9.31984 16.1424i −0.681534 1.18045i
\(188\) 0 0
\(189\) 12.1921 6.35242i 0.886843 0.462071i
\(190\) 0 0
\(191\) 5.59624 + 9.69298i 0.404930 + 0.701359i 0.994313 0.106494i \(-0.0339626\pi\)
−0.589383 + 0.807853i \(0.700629\pi\)
\(192\) 0 0
\(193\) −8.14679 + 14.1106i −0.586419 + 1.01571i 0.408278 + 0.912857i \(0.366129\pi\)
−0.994697 + 0.102849i \(0.967204\pi\)
\(194\) 0 0
\(195\) −1.25373 + 4.60268i −0.0897813 + 0.329605i
\(196\) 0 0
\(197\) −3.17438 −0.226165 −0.113082 0.993586i \(-0.536072\pi\)
−0.113082 + 0.993586i \(0.536072\pi\)
\(198\) 0 0
\(199\) −1.44140 2.49658i −0.102178 0.176978i 0.810404 0.585872i \(-0.199248\pi\)
−0.912582 + 0.408894i \(0.865915\pi\)
\(200\) 0 0
\(201\) 4.30681 1.13492i 0.303779 0.0800511i
\(202\) 0 0
\(203\) −5.90367 + 5.55019i −0.414356 + 0.389547i
\(204\) 0 0
\(205\) 3.03061 + 5.24917i 0.211667 + 0.366618i
\(206\) 0 0
\(207\) 0.0900550 10.8772i 0.00625925 0.756016i
\(208\) 0 0
\(209\) 7.37264 12.7698i 0.509976 0.883305i
\(210\) 0 0
\(211\) 0.242718 + 0.420400i 0.0167094 + 0.0289415i 0.874259 0.485459i \(-0.161348\pi\)
−0.857550 + 0.514401i \(0.828014\pi\)
\(212\) 0 0
\(213\) 4.80919 + 4.84917i 0.329520 + 0.332260i
\(214\) 0 0
\(215\) 0.918617 1.59109i 0.0626492 0.108512i
\(216\) 0 0
\(217\) 18.0219 16.9429i 1.22341 1.15016i
\(218\) 0 0
\(219\) 0.784978 2.88181i 0.0530439 0.194734i
\(220\) 0 0
\(221\) 15.7683 27.3115i 1.06069 1.83717i
\(222\) 0 0
\(223\) −2.14795 + 3.72037i −0.143838 + 0.249134i −0.928939 0.370234i \(-0.879278\pi\)
0.785101 + 0.619368i \(0.212611\pi\)
\(224\) 0 0
\(225\) 0.118429 14.3043i 0.00789527 0.953621i
\(226\) 0 0
\(227\) 17.3827 1.15373 0.576866 0.816839i \(-0.304275\pi\)
0.576866 + 0.816839i \(0.304275\pi\)
\(228\) 0 0
\(229\) 7.33125 0.484463 0.242231 0.970218i \(-0.422121\pi\)
0.242231 + 0.970218i \(0.422121\pi\)
\(230\) 0 0
\(231\) 7.27397 + 13.6831i 0.478592 + 0.900284i
\(232\) 0 0
\(233\) −2.16624 + 3.75205i −0.141915 + 0.245805i −0.928218 0.372037i \(-0.878659\pi\)
0.786302 + 0.617842i \(0.211993\pi\)
\(234\) 0 0
\(235\) 0.906366 + 1.56987i 0.0591248 + 0.102407i
\(236\) 0 0
\(237\) 2.54899 9.35784i 0.165575 0.607857i
\(238\) 0 0
\(239\) −1.77960 3.08236i −0.115113 0.199381i 0.802712 0.596367i \(-0.203390\pi\)
−0.917825 + 0.396986i \(0.870056\pi\)
\(240\) 0 0
\(241\) 16.0185 1.03184 0.515921 0.856636i \(-0.327450\pi\)
0.515921 + 0.856636i \(0.327450\pi\)
\(242\) 0 0
\(243\) −10.7922 11.2485i −0.692321 0.721590i
\(244\) 0 0
\(245\) −2.80881 1.86160i −0.179448 0.118933i
\(246\) 0 0
\(247\) 24.9476 1.58738
\(248\) 0 0
\(249\) 0.400276 0.105480i 0.0253665 0.00668450i
\(250\) 0 0
\(251\) 12.8007 0.807972 0.403986 0.914765i \(-0.367625\pi\)
0.403986 + 0.914765i \(0.367625\pi\)
\(252\) 0 0
\(253\) 12.2612 0.770852
\(254\) 0 0
\(255\) 1.20788 4.43435i 0.0756402 0.277690i
\(256\) 0 0
\(257\) 16.4154 1.02396 0.511981 0.858997i \(-0.328912\pi\)
0.511981 + 0.858997i \(0.328912\pi\)
\(258\) 0 0
\(259\) −5.72711 + 5.38420i −0.355865 + 0.334558i
\(260\) 0 0
\(261\) 7.91864 + 4.65966i 0.490151 + 0.288426i
\(262\) 0 0
\(263\) −25.6528 −1.58182 −0.790910 0.611933i \(-0.790392\pi\)
−0.790910 + 0.611933i \(0.790392\pi\)
\(264\) 0 0
\(265\) −2.68547 4.65136i −0.164967 0.285731i
\(266\) 0 0
\(267\) 2.17103 0.572103i 0.132865 0.0350121i
\(268\) 0 0
\(269\) 5.35397 + 9.27335i 0.326437 + 0.565406i 0.981802 0.189906i \(-0.0608184\pi\)
−0.655365 + 0.755313i \(0.727485\pi\)
\(270\) 0 0
\(271\) 12.7513 22.0859i 0.774587 1.34162i −0.160439 0.987046i \(-0.551291\pi\)
0.935026 0.354578i \(-0.115376\pi\)
\(272\) 0 0
\(273\) −13.8955 + 22.2333i −0.840996 + 1.34562i
\(274\) 0 0
\(275\) 16.1243 0.972334
\(276\) 0 0
\(277\) −12.7825 −0.768023 −0.384012 0.923328i \(-0.625458\pi\)
−0.384012 + 0.923328i \(0.625458\pi\)
\(278\) 0 0
\(279\) −24.1729 14.2244i −1.44720 0.851591i
\(280\) 0 0
\(281\) −10.4763 + 18.1454i −0.624961 + 1.08246i 0.363587 + 0.931560i \(0.381552\pi\)
−0.988548 + 0.150904i \(0.951781\pi\)
\(282\) 0 0
\(283\) 7.53085 13.0438i 0.447663 0.775374i −0.550571 0.834788i \(-0.685590\pi\)
0.998233 + 0.0594141i \(0.0189232\pi\)
\(284\) 0 0
\(285\) 3.51567 0.926440i 0.208250 0.0548776i
\(286\) 0 0
\(287\) 7.62537 + 32.4286i 0.450112 + 1.91420i
\(288\) 0 0
\(289\) −6.69162 + 11.5902i −0.393625 + 0.681778i
\(290\) 0 0
\(291\) −23.5376 + 6.20258i −1.37980 + 0.363602i
\(292\) 0 0
\(293\) 0.134459 + 0.232890i 0.00785519 + 0.0136056i 0.869926 0.493182i \(-0.164166\pi\)
−0.862071 + 0.506787i \(0.830833\pi\)
\(294\) 0 0
\(295\) −2.02732 + 3.51142i −0.118035 + 0.204443i
\(296\) 0 0
\(297\) 12.5781 12.2695i 0.729856 0.711950i
\(298\) 0 0
\(299\) 10.3724 + 17.9654i 0.599849 + 1.03897i
\(300\) 0 0
\(301\) 7.35695 6.91645i 0.424048 0.398658i
\(302\) 0 0
\(303\) 12.9363 + 13.0439i 0.743171 + 0.749350i
\(304\) 0 0
\(305\) −1.75375 3.03759i −0.100420 0.173932i
\(306\) 0 0
\(307\) 5.03514 0.287371 0.143685 0.989623i \(-0.454105\pi\)
0.143685 + 0.989623i \(0.454105\pi\)
\(308\) 0 0
\(309\) 0.194434 + 0.196050i 0.0110609 + 0.0111529i
\(310\) 0 0
\(311\) 2.23815 3.87659i 0.126914 0.219821i −0.795566 0.605868i \(-0.792826\pi\)
0.922479 + 0.386046i \(0.126159\pi\)
\(312\) 0 0
\(313\) −5.48895 9.50715i −0.310254 0.537376i 0.668163 0.744015i \(-0.267081\pi\)
−0.978417 + 0.206639i \(0.933747\pi\)
\(314\) 0 0
\(315\) −1.13254 + 3.64918i −0.0638117 + 0.205608i
\(316\) 0 0
\(317\) −12.4826 21.6205i −0.701092 1.21433i −0.968084 0.250628i \(-0.919363\pi\)
0.266992 0.963699i \(-0.413970\pi\)
\(318\) 0 0
\(319\) −5.17828 + 8.96905i −0.289928 + 0.502170i
\(320\) 0 0
\(321\) 13.3665 3.52230i 0.746045 0.196596i
\(322\) 0 0
\(323\) −24.0352 −1.33736
\(324\) 0 0
\(325\) 13.6404 + 23.6259i 0.756635 + 1.31053i
\(326\) 0 0
\(327\) 6.24194 22.9154i 0.345180 1.26722i
\(328\) 0 0
\(329\) 2.28052 + 9.69844i 0.125729 + 0.534692i
\(330\) 0 0
\(331\) 6.01206 + 10.4132i 0.330453 + 0.572361i 0.982601 0.185731i \(-0.0594653\pi\)
−0.652148 + 0.758092i \(0.726132\pi\)
\(332\) 0 0
\(333\) 7.68182 + 4.52031i 0.420961 + 0.247711i
\(334\) 0 0
\(335\) −0.618925 + 1.07201i −0.0338155 + 0.0585702i
\(336\) 0 0
\(337\) −14.1286 24.4715i −0.769636 1.33305i −0.937761 0.347282i \(-0.887105\pi\)
0.168125 0.985766i \(-0.446229\pi\)
\(338\) 0 0
\(339\) −8.18153 + 30.0360i −0.444360 + 1.63133i
\(340\) 0 0
\(341\) 15.8076 27.3795i 0.856027 1.48268i
\(342\) 0 0
\(343\) −11.8294 14.2501i −0.638728 0.769433i
\(344\) 0 0
\(345\) 2.12885 + 2.14655i 0.114613 + 0.115566i
\(346\) 0 0
\(347\) 9.80293 16.9792i 0.526249 0.911490i −0.473283 0.880910i \(-0.656931\pi\)
0.999532 0.0305797i \(-0.00973535\pi\)
\(348\) 0 0
\(349\) −8.22904 + 14.2531i −0.440490 + 0.762952i −0.997726 0.0674029i \(-0.978529\pi\)
0.557236 + 0.830354i \(0.311862\pi\)
\(350\) 0 0
\(351\) 28.6182 + 8.05042i 1.52753 + 0.429700i
\(352\) 0 0
\(353\) −27.3709 −1.45680 −0.728402 0.685150i \(-0.759737\pi\)
−0.728402 + 0.685150i \(0.759737\pi\)
\(354\) 0 0
\(355\) −1.89813 −0.100742
\(356\) 0 0
\(357\) 13.3874 21.4202i 0.708535 1.13368i
\(358\) 0 0
\(359\) −7.88714 + 13.6609i −0.416267 + 0.720996i −0.995561 0.0941231i \(-0.969995\pi\)
0.579293 + 0.815119i \(0.303329\pi\)
\(360\) 0 0
\(361\) −0.00677168 0.0117289i −0.000356404 0.000617310i
\(362\) 0 0
\(363\) 0.530769 + 0.535182i 0.0278582 + 0.0280898i
\(364\) 0 0
\(365\) 0.415060 + 0.718905i 0.0217252 + 0.0376292i
\(366\) 0 0
\(367\) −18.8137 −0.982066 −0.491033 0.871141i \(-0.663381\pi\)
−0.491033 + 0.871141i \(0.663381\pi\)
\(368\) 0 0
\(369\) 32.8680 18.6153i 1.71104 0.969072i
\(370\) 0 0
\(371\) −6.75695 28.7355i −0.350803 1.49187i
\(372\) 0 0
\(373\) −17.5737 −0.909934 −0.454967 0.890508i \(-0.650349\pi\)
−0.454967 + 0.890508i \(0.650349\pi\)
\(374\) 0 0
\(375\) 5.73525 + 5.78293i 0.296167 + 0.298629i
\(376\) 0 0
\(377\) −17.5223 −0.902446
\(378\) 0 0
\(379\) −34.4618 −1.77018 −0.885091 0.465419i \(-0.845904\pi\)
−0.885091 + 0.465419i \(0.845904\pi\)
\(380\) 0 0
\(381\) −23.1351 23.3274i −1.18525 1.19510i
\(382\) 0 0
\(383\) 22.7410 1.16201 0.581005 0.813900i \(-0.302660\pi\)
0.581005 + 0.813900i \(0.302660\pi\)
\(384\) 0 0
\(385\) −4.12378 1.24250i −0.210167 0.0633239i
\(386\) 0 0
\(387\) −9.86794 5.80671i −0.501615 0.295172i
\(388\) 0 0
\(389\) 15.7638 0.799255 0.399628 0.916678i \(-0.369139\pi\)
0.399628 + 0.916678i \(0.369139\pi\)
\(390\) 0 0
\(391\) −9.99303 17.3084i −0.505369 0.875325i
\(392\) 0 0
\(393\) 5.95479 + 6.00429i 0.300379 + 0.302877i
\(394\) 0 0
\(395\) 1.34779 + 2.33444i 0.0678145 + 0.117458i
\(396\) 0 0
\(397\) −5.39875 + 9.35091i −0.270955 + 0.469308i −0.969107 0.246642i \(-0.920673\pi\)
0.698151 + 0.715950i \(0.254006\pi\)
\(398\) 0 0
\(399\) 19.9699 + 0.699052i 0.999744 + 0.0349964i
\(400\) 0 0
\(401\) 12.3295 0.615704 0.307852 0.951434i \(-0.400390\pi\)
0.307852 + 0.951434i \(0.400390\pi\)
\(402\) 0 0
\(403\) 53.4898 2.66452
\(404\) 0 0
\(405\) 4.33189 + 0.0717346i 0.215253 + 0.00356452i
\(406\) 0 0
\(407\) −5.02342 + 8.70082i −0.249002 + 0.431284i
\(408\) 0 0
\(409\) −9.31771 + 16.1387i −0.460731 + 0.798010i −0.998998 0.0447650i \(-0.985746\pi\)
0.538266 + 0.842775i \(0.319079\pi\)
\(410\) 0 0
\(411\) −7.89711 7.96276i −0.389536 0.392774i
\(412\) 0 0
\(413\) −16.2362 + 15.2641i −0.798932 + 0.751096i
\(414\) 0 0
\(415\) −0.0575230 + 0.0996328i −0.00282369 + 0.00489078i
\(416\) 0 0
\(417\) −10.3551 + 38.0155i −0.507090 + 1.86163i
\(418\) 0 0
\(419\) 5.90976 + 10.2360i 0.288711 + 0.500062i 0.973502 0.228677i \(-0.0734400\pi\)
−0.684791 + 0.728739i \(0.740107\pi\)
\(420\) 0 0
\(421\) 4.81800 8.34503i 0.234815 0.406712i −0.724404 0.689376i \(-0.757885\pi\)
0.959219 + 0.282664i \(0.0912182\pi\)
\(422\) 0 0
\(423\) 9.82986 5.56728i 0.477944 0.270690i
\(424\) 0 0
\(425\) −13.1416 22.7619i −0.637460 1.10411i
\(426\) 0 0
\(427\) −4.41265 18.7658i −0.213543 0.908139i
\(428\) 0 0
\(429\) −8.80704 + 32.3324i −0.425208 + 1.56102i
\(430\) 0 0
\(431\) 18.2913 + 31.6815i 0.881062 + 1.52604i 0.850162 + 0.526522i \(0.176504\pi\)
0.0309004 + 0.999522i \(0.490163\pi\)
\(432\) 0 0
\(433\) −7.69388 −0.369744 −0.184872 0.982763i \(-0.559187\pi\)
−0.184872 + 0.982763i \(0.559187\pi\)
\(434\) 0 0
\(435\) −2.46928 + 0.650699i −0.118393 + 0.0311986i
\(436\) 0 0
\(437\) 7.90518 13.6922i 0.378156 0.654985i
\(438\) 0 0
\(439\) −10.2717 17.7911i −0.490241 0.849122i 0.509696 0.860355i \(-0.329758\pi\)
−0.999937 + 0.0112324i \(0.996425\pi\)
\(440\) 0 0
\(441\) −11.7460 + 17.4078i −0.559333 + 0.828943i
\(442\) 0 0
\(443\) −16.8401 29.1679i −0.800098 1.38581i −0.919551 0.392970i \(-0.871448\pi\)
0.119454 0.992840i \(-0.461886\pi\)
\(444\) 0 0
\(445\) −0.311995 + 0.540391i −0.0147900 + 0.0256170i
\(446\) 0 0
\(447\) −17.3249 17.4689i −0.819438 0.826250i
\(448\) 0 0
\(449\) −22.5141 −1.06250 −0.531252 0.847214i \(-0.678278\pi\)
−0.531252 + 0.847214i \(0.678278\pi\)
\(450\) 0 0
\(451\) 21.2891 + 36.8738i 1.00246 + 1.73632i
\(452\) 0 0
\(453\) −3.07671 3.10229i −0.144556 0.145758i
\(454\) 0 0
\(455\) −1.66797 7.09340i −0.0781955 0.332544i
\(456\) 0 0
\(457\) 11.8559 + 20.5349i 0.554594 + 0.960584i 0.997935 + 0.0642314i \(0.0204596\pi\)
−0.443342 + 0.896353i \(0.646207\pi\)
\(458\) 0 0
\(459\) −27.5716 7.75601i −1.28693 0.362020i
\(460\) 0 0
\(461\) −5.57340 + 9.65342i −0.259579 + 0.449605i −0.966129 0.258059i \(-0.916917\pi\)
0.706550 + 0.707663i \(0.250251\pi\)
\(462\) 0 0
\(463\) −10.3208 17.8761i −0.479647 0.830773i 0.520080 0.854117i \(-0.325902\pi\)
−0.999727 + 0.0233441i \(0.992569\pi\)
\(464\) 0 0
\(465\) 7.53789 1.98637i 0.349561 0.0921155i
\(466\) 0 0
\(467\) 8.68477 15.0425i 0.401883 0.696082i −0.592070 0.805887i \(-0.701689\pi\)
0.993953 + 0.109804i \(0.0350224\pi\)
\(468\) 0 0
\(469\) −4.95680 + 4.66001i −0.228884 + 0.215179i
\(470\) 0 0
\(471\) −29.2789 + 7.71551i −1.34910 + 0.355512i
\(472\) 0 0
\(473\) 6.45300 11.1769i 0.296709 0.513916i
\(474\) 0 0
\(475\) 10.3959 18.0062i 0.476997 0.826182i
\(476\) 0 0
\(477\) −29.1248 + 16.4952i −1.33353 + 0.755266i
\(478\) 0 0
\(479\) 4.09033 0.186892 0.0934461 0.995624i \(-0.470212\pi\)
0.0934461 + 0.995624i \(0.470212\pi\)
\(480\) 0 0
\(481\) −16.9983 −0.775056
\(482\) 0 0
\(483\) 7.79938 + 14.6715i 0.354884 + 0.667576i
\(484\) 0 0
\(485\) 3.38256 5.85876i 0.153594 0.266033i
\(486\) 0 0
\(487\) 0.843065 + 1.46023i 0.0382029 + 0.0661694i 0.884495 0.466550i \(-0.154503\pi\)
−0.846292 + 0.532720i \(0.821170\pi\)
\(488\) 0 0
\(489\) −2.95174 + 0.777835i −0.133482 + 0.0351749i
\(490\) 0 0
\(491\) −6.85070 11.8658i −0.309168 0.535494i 0.669013 0.743251i \(-0.266717\pi\)
−0.978181 + 0.207757i \(0.933384\pi\)
\(492\) 0 0
\(493\) 16.8815 0.760305
\(494\) 0 0
\(495\) −0.0404309 + 4.88340i −0.00181723 + 0.219492i
\(496\) 0 0
\(497\) −9.98875 3.00963i −0.448057 0.135000i
\(498\) 0 0
\(499\) 6.55655 0.293511 0.146756 0.989173i \(-0.453117\pi\)
0.146756 + 0.989173i \(0.453117\pi\)
\(500\) 0 0
\(501\) −3.25220 + 11.9394i −0.145297 + 0.533415i
\(502\) 0 0
\(503\) −26.8584 −1.19756 −0.598779 0.800914i \(-0.704347\pi\)
−0.598779 + 0.800914i \(0.704347\pi\)
\(504\) 0 0
\(505\) −5.10580 −0.227205
\(506\) 0 0
\(507\) −33.0514 + 8.70962i −1.46787 + 0.386808i
\(508\) 0 0
\(509\) −39.7337 −1.76117 −0.880584 0.473891i \(-0.842849\pi\)
−0.880584 + 0.473891i \(0.842849\pi\)
\(510\) 0 0
\(511\) 1.04434 + 4.44129i 0.0461989 + 0.196471i
\(512\) 0 0
\(513\) −5.59198 21.9567i −0.246892 0.969411i
\(514\) 0 0
\(515\) −0.0767406 −0.00338160
\(516\) 0 0
\(517\) 6.36694 + 11.0279i 0.280018 + 0.485005i
\(518\) 0 0
\(519\) 4.50376 16.5342i 0.197693 0.725770i
\(520\) 0 0
\(521\) 11.7585 + 20.3663i 0.515148 + 0.892262i 0.999845 + 0.0175802i \(0.00559623\pi\)
−0.484698 + 0.874682i \(0.661070\pi\)
\(522\) 0 0
\(523\) 10.9289 18.9294i 0.477887 0.827725i −0.521791 0.853073i \(-0.674736\pi\)
0.999679 + 0.0253481i \(0.00806942\pi\)
\(524\) 0 0
\(525\) 10.2568 + 19.2941i 0.447642 + 0.842064i
\(526\) 0 0
\(527\) −51.5336 −2.24484
\(528\) 0 0
\(529\) −9.85320 −0.428400
\(530\) 0 0
\(531\) 21.7778 + 12.8150i 0.945075 + 0.556122i
\(532\) 0 0
\(533\) −36.0191 + 62.3869i −1.56016 + 2.70228i
\(534\) 0 0
\(535\) −1.92088 + 3.32706i −0.0830468 + 0.143841i
\(536\) 0 0
\(537\) 1.84785 6.78383i 0.0797407 0.292744i
\(538\) 0 0
\(539\) −19.7310 13.0772i −0.849874 0.563273i
\(540\) 0 0
\(541\) 14.0063 24.2596i 0.602178 1.04300i −0.390313 0.920682i \(-0.627633\pi\)
0.992491 0.122320i \(-0.0390334\pi\)
\(542\) 0 0
\(543\) 5.58709 + 5.63354i 0.239765 + 0.241758i
\(544\) 0 0
\(545\) 3.30045 + 5.71654i 0.141376 + 0.244870i
\(546\) 0 0
\(547\) 2.02714 3.51112i 0.0866744 0.150124i −0.819429 0.573181i \(-0.805709\pi\)
0.906103 + 0.423056i \(0.139043\pi\)
\(548\) 0 0
\(549\) −19.0201 + 10.7723i −0.811756 + 0.459749i
\(550\) 0 0
\(551\) 6.67722 + 11.5653i 0.284459 + 0.492698i
\(552\) 0 0
\(553\) 3.39119 + 14.4218i 0.144208 + 0.613278i
\(554\) 0 0
\(555\) −2.39544 + 0.631239i −0.101681 + 0.0267946i
\(556\) 0 0
\(557\) −0.926620 1.60495i −0.0392621 0.0680040i 0.845727 0.533616i \(-0.179167\pi\)
−0.884989 + 0.465612i \(0.845834\pi\)
\(558\) 0 0
\(559\) 21.8357 0.923553
\(560\) 0 0
\(561\) 8.48496 31.1499i 0.358235 1.31515i
\(562\) 0 0
\(563\) 22.2331 38.5088i 0.937013 1.62295i 0.166008 0.986124i \(-0.446912\pi\)
0.771005 0.636830i \(-0.219755\pi\)
\(564\) 0 0
\(565\) −4.32601 7.49287i −0.181997 0.315227i
\(566\) 0 0
\(567\) 22.6825 + 7.24605i 0.952575 + 0.304305i
\(568\) 0 0
\(569\) −7.63116 13.2176i −0.319915 0.554109i 0.660555 0.750778i \(-0.270321\pi\)
−0.980470 + 0.196669i \(0.936988\pi\)
\(570\) 0 0
\(571\) −12.7634 + 22.1068i −0.534130 + 0.925140i 0.465075 + 0.885271i \(0.346027\pi\)
−0.999205 + 0.0398690i \(0.987306\pi\)
\(572\) 0 0
\(573\) −5.09493 + 18.7045i −0.212844 + 0.781390i
\(574\) 0 0
\(575\) 17.2890 0.721002
\(576\) 0 0
\(577\) −3.26981 5.66348i −0.136124 0.235774i 0.789902 0.613233i \(-0.210131\pi\)
−0.926026 + 0.377459i \(0.876798\pi\)
\(578\) 0 0
\(579\) −27.2897 + 7.19130i −1.13412 + 0.298860i
\(580\) 0 0
\(581\) −0.460686 + 0.433102i −0.0191125 + 0.0179681i
\(582\) 0 0
\(583\) −18.8646 32.6744i −0.781291 1.35323i
\(584\) 0 0
\(585\) −7.18952 + 4.07188i −0.297250 + 0.168352i
\(586\) 0 0
\(587\) −15.2055 + 26.3366i −0.627597 + 1.08703i 0.360436 + 0.932784i \(0.382628\pi\)
−0.988033 + 0.154245i \(0.950705\pi\)
\(588\) 0 0
\(589\) −20.3833 35.3049i −0.839880 1.45471i
\(590\) 0 0
\(591\) −3.87167 3.90386i −0.159259 0.160583i
\(592\) 0 0
\(593\) −21.3291 + 36.9432i −0.875883 + 1.51707i −0.0200633 + 0.999799i \(0.506387\pi\)
−0.855819 + 0.517275i \(0.826947\pi\)
\(594\) 0 0
\(595\) 1.60697 + 6.83399i 0.0658792 + 0.280166i
\(596\) 0 0
\(597\) 1.31228 4.81763i 0.0537079 0.197172i
\(598\) 0 0
\(599\) −22.2787 + 38.5879i −0.910284 + 1.57666i −0.0966209 + 0.995321i \(0.530803\pi\)
−0.813663 + 0.581337i \(0.802530\pi\)
\(600\) 0 0
\(601\) 14.1961 24.5884i 0.579071 1.00298i −0.416515 0.909129i \(-0.636749\pi\)
0.995586 0.0938518i \(-0.0299180\pi\)
\(602\) 0 0
\(603\) 6.64860 + 3.91232i 0.270752 + 0.159322i
\(604\) 0 0
\(605\) −0.209488 −0.00851691
\(606\) 0 0
\(607\) −14.0278 −0.569372 −0.284686 0.958621i \(-0.591889\pi\)
−0.284686 + 0.958621i \(0.591889\pi\)
\(608\) 0 0
\(609\) −14.0261 0.490990i −0.568368 0.0198959i
\(610\) 0 0
\(611\) −10.7723 + 18.6581i −0.435799 + 0.754826i
\(612\) 0 0
\(613\) −9.97062 17.2696i −0.402709 0.697513i 0.591342 0.806421i \(-0.298598\pi\)
−0.994052 + 0.108907i \(0.965265\pi\)
\(614\) 0 0
\(615\) −2.75912 + 10.1293i −0.111259 + 0.408452i
\(616\) 0 0
\(617\) −1.51584 2.62551i −0.0610254 0.105699i 0.833899 0.551918i \(-0.186104\pi\)
−0.894924 + 0.446219i \(0.852770\pi\)
\(618\) 0 0
\(619\) −2.55431 −0.102666 −0.0513331 0.998682i \(-0.516347\pi\)
−0.0513331 + 0.998682i \(0.516347\pi\)
\(620\) 0 0
\(621\) 13.4866 13.1558i 0.541200 0.527923i
\(622\) 0 0
\(623\) −2.49868 + 2.34907i −0.100107 + 0.0941136i
\(624\) 0 0
\(625\) 21.5777 0.863108
\(626\) 0 0
\(627\) 24.6965 6.50795i 0.986282 0.259903i
\(628\) 0 0
\(629\) 16.3766 0.652980
\(630\) 0 0
\(631\) −37.1162 −1.47757 −0.738786 0.673941i \(-0.764600\pi\)
−0.738786 + 0.673941i \(0.764600\pi\)
\(632\) 0 0
\(633\) −0.220975 + 0.811243i −0.00878297 + 0.0322440i
\(634\) 0 0
\(635\) 9.13115 0.362358
\(636\) 0 0
\(637\) 2.46960 39.9731i 0.0978491 1.58379i
\(638\) 0 0
\(639\) −0.0979331 + 11.8287i −0.00387417 + 0.467937i
\(640\) 0 0
\(641\) 21.0968 0.833275 0.416638 0.909073i \(-0.363208\pi\)
0.416638 + 0.909073i \(0.363208\pi\)
\(642\) 0 0
\(643\) −9.31948 16.1418i −0.367524 0.636571i 0.621654 0.783292i \(-0.286461\pi\)
−0.989178 + 0.146722i \(0.953128\pi\)
\(644\) 0 0
\(645\) 3.07714 0.810879i 0.121162 0.0319283i
\(646\) 0 0
\(647\) −4.78509 8.28801i −0.188121 0.325835i 0.756503 0.653991i \(-0.226906\pi\)
−0.944624 + 0.328155i \(0.893573\pi\)
\(648\) 0 0
\(649\) −14.2413 + 24.6666i −0.559019 + 0.968249i
\(650\) 0 0
\(651\) 42.8171 + 1.49883i 1.67813 + 0.0587437i
\(652\) 0 0
\(653\) −32.8560 −1.28575 −0.642877 0.765970i \(-0.722259\pi\)
−0.642877 + 0.765970i \(0.722259\pi\)
\(654\) 0 0
\(655\) −2.35028 −0.0918332
\(656\) 0 0
\(657\) 4.50147 2.54947i 0.175619 0.0994642i
\(658\) 0 0
\(659\) −24.8011 + 42.9568i −0.966114 + 1.67336i −0.259521 + 0.965738i \(0.583565\pi\)
−0.706593 + 0.707620i \(0.749769\pi\)
\(660\) 0 0
\(661\) 1.65895 2.87338i 0.0645255 0.111761i −0.831958 0.554839i \(-0.812780\pi\)
0.896483 + 0.443077i \(0.146113\pi\)
\(662\) 0 0
\(663\) 52.8198 13.9189i 2.05135 0.540566i
\(664\) 0 0
\(665\) −4.04626 + 3.80399i −0.156907 + 0.147512i
\(666\) 0 0
\(667\) −5.55232 + 9.61690i −0.214987 + 0.372368i
\(668\) 0 0
\(669\) −7.19511 + 1.89604i −0.278179 + 0.0733050i
\(670\) 0 0
\(671\) −12.3196 21.3381i −0.475591 0.823748i
\(672\) 0 0
\(673\) 21.8005 37.7597i 0.840349 1.45553i −0.0492503 0.998786i \(-0.515683\pi\)
0.889600 0.456741i \(-0.150983\pi\)
\(674\) 0 0
\(675\) 17.7359 17.3008i 0.682657 0.665909i
\(676\) 0 0
\(677\) 9.14039 + 15.8316i 0.351294 + 0.608459i 0.986476 0.163903i \(-0.0524085\pi\)
−0.635183 + 0.772362i \(0.719075\pi\)
\(678\) 0 0
\(679\) 27.0900 25.4680i 1.03962 0.977371i
\(680\) 0 0
\(681\) 21.2011 + 21.3774i 0.812428 + 0.819182i
\(682\) 0 0
\(683\) −22.5380 39.0369i −0.862391 1.49371i −0.869614 0.493732i \(-0.835633\pi\)
0.00722317 0.999974i \(-0.497701\pi\)
\(684\) 0 0
\(685\) 3.11689 0.119090
\(686\) 0 0
\(687\) 8.94167 + 9.01601i 0.341146 + 0.343982i
\(688\) 0 0
\(689\) 31.9171 55.2820i 1.21594 2.10608i
\(690\) 0 0
\(691\) 20.8977 + 36.1960i 0.794988 + 1.37696i 0.922847 + 0.385167i \(0.125856\pi\)
−0.127859 + 0.991792i \(0.540811\pi\)
\(692\) 0 0
\(693\) −7.95577 + 25.6344i −0.302215 + 0.973770i
\(694\) 0 0
\(695\) −5.47528 9.48346i −0.207689 0.359728i
\(696\) 0 0
\(697\) 34.7019 60.1054i 1.31443 2.27665i
\(698\) 0 0
\(699\) −7.25637 + 1.91218i −0.274461 + 0.0723253i
\(700\) 0 0
\(701\) 19.3967 0.732604 0.366302 0.930496i \(-0.380624\pi\)
0.366302 + 0.930496i \(0.380624\pi\)
\(702\) 0 0
\(703\) 6.47753 + 11.2194i 0.244305 + 0.423148i
\(704\) 0 0
\(705\) −0.825173 + 3.02937i −0.0310778 + 0.114093i
\(706\) 0 0
\(707\) −26.8689 8.09565i −1.01051 0.304468i
\(708\) 0 0
\(709\) 8.61542 + 14.9223i 0.323559 + 0.560420i 0.981220 0.192894i \(-0.0617873\pi\)
−0.657661 + 0.753314i \(0.728454\pi\)
\(710\) 0 0
\(711\) 14.6172 8.27867i 0.548189 0.310474i
\(712\) 0 0
\(713\) 16.9494 29.3572i 0.634759 1.09943i
\(714\) 0 0
\(715\) −4.65675 8.06573i −0.174153 0.301641i
\(716\) 0 0
\(717\) 1.62018 5.94800i 0.0605068 0.222132i
\(718\) 0 0
\(719\) −5.08444 + 8.80650i −0.189617 + 0.328427i −0.945123 0.326715i \(-0.894058\pi\)
0.755505 + 0.655143i \(0.227391\pi\)
\(720\) 0 0
\(721\) −0.403841 0.121678i −0.0150398 0.00453153i
\(722\) 0 0
\(723\) 19.5372 + 19.6996i 0.726596 + 0.732637i
\(724\) 0 0
\(725\) −7.30172 + 12.6469i −0.271179 + 0.469696i
\(726\) 0 0
\(727\) 0.0914356 0.158371i 0.00339116 0.00587366i −0.864325 0.502934i \(-0.832254\pi\)
0.867716 + 0.497060i \(0.165587\pi\)
\(728\) 0 0
\(729\) 0.670536 26.9917i 0.0248347 0.999692i
\(730\) 0 0
\(731\) −21.0372 −0.778088
\(732\) 0 0
\(733\) 41.9343 1.54888 0.774440 0.632647i \(-0.218032\pi\)
0.774440 + 0.632647i \(0.218032\pi\)
\(734\) 0 0
\(735\) −1.13640 5.72481i −0.0419167 0.211163i
\(736\) 0 0
\(737\) −4.34776 + 7.53054i −0.160152 + 0.277391i
\(738\) 0 0
\(739\) 11.8013 + 20.4404i 0.434116 + 0.751911i 0.997223 0.0744729i \(-0.0237274\pi\)
−0.563107 + 0.826384i \(0.690394\pi\)
\(740\) 0 0
\(741\) 30.4277 + 30.6807i 1.11779 + 1.12708i
\(742\) 0 0
\(743\) 11.1821 + 19.3680i 0.410233 + 0.710544i 0.994915 0.100718i \(-0.0321141\pi\)
−0.584682 + 0.811263i \(0.698781\pi\)
\(744\) 0 0
\(745\) 6.83791 0.250522
\(746\) 0 0
\(747\) 0.617922 + 0.363611i 0.0226086 + 0.0133038i
\(748\) 0 0
\(749\) −15.3838 + 14.4627i −0.562111 + 0.528454i
\(750\) 0 0
\(751\) −31.0462 −1.13289 −0.566445 0.824099i \(-0.691682\pi\)
−0.566445 + 0.824099i \(0.691682\pi\)
\(752\) 0 0
\(753\) 15.6125 + 15.7423i 0.568952 + 0.573682i
\(754\) 0 0
\(755\) 1.21434 0.0441943
\(756\) 0 0
\(757\) −44.0639 −1.60153 −0.800764 0.598980i \(-0.795573\pi\)
−0.800764 + 0.598980i \(0.795573\pi\)
\(758\) 0 0
\(759\) 14.9545 + 15.0788i 0.542814 + 0.547327i
\(760\) 0 0
\(761\) 5.74941 0.208416 0.104208 0.994556i \(-0.466769\pi\)
0.104208 + 0.994556i \(0.466769\pi\)
\(762\) 0 0
\(763\) 8.30431 + 35.3159i 0.300636 + 1.27852i
\(764\) 0 0
\(765\) 6.92659 3.92297i 0.250431 0.141835i
\(766\) 0 0
\(767\) −48.1898 −1.74003
\(768\) 0 0
\(769\) −7.48401 12.9627i −0.269880 0.467446i 0.698950 0.715170i \(-0.253651\pi\)
−0.968831 + 0.247724i \(0.920317\pi\)
\(770\) 0 0
\(771\) 20.0212 + 20.1877i 0.721048 + 0.727042i
\(772\) 0 0
\(773\) −10.0605 17.4253i −0.361850 0.626743i 0.626415 0.779490i \(-0.284521\pi\)
−0.988265 + 0.152747i \(0.951188\pi\)
\(774\) 0 0
\(775\) 22.2897 38.6069i 0.800669 1.38680i
\(776\) 0 0
\(777\) −13.6067 0.476306i −0.488136 0.0170874i
\(778\) 0 0
\(779\) 54.9031 1.96711
\(780\) 0 0
\(781\) −13.3338 −0.477120
\(782\) 0 0
\(783\) 3.92761 + 15.4216i 0.140361 + 0.551123i
\(784\) 0 0
\(785\) 4.20763 7.28783i 0.150177 0.260114i
\(786\) 0 0
\(787\) −12.0572 + 20.8837i −0.429794 + 0.744425i −0.996855 0.0792508i \(-0.974747\pi\)
0.567061 + 0.823676i \(0.308081\pi\)
\(788\) 0 0
\(789\) −31.2878 31.5479i −1.11388 1.12314i
\(790\) 0 0
\(791\) −10.8848 46.2899i −0.387017 1.64588i
\(792\) 0 0
\(793\) 20.8435 36.1021i 0.740175 1.28202i
\(794\) 0 0
\(795\) 2.44490 8.97570i 0.0867117 0.318335i
\(796\) 0 0
\(797\) −1.54611 2.67794i −0.0547661 0.0948576i 0.837343 0.546678i \(-0.184108\pi\)
−0.892109 + 0.451821i \(0.850775\pi\)
\(798\) 0 0
\(799\) 10.3783 17.9758i 0.367158 0.635936i
\(800\) 0 0
\(801\) 3.35150 + 1.97216i 0.118419 + 0.0696830i
\(802\) 0 0
\(803\) 2.91567 + 5.05008i 0.102892 + 0.178213i
\(804\) 0 0
\(805\) −4.42165 1.33225i −0.155843 0.0469557i
\(806\) 0 0
\(807\) −4.87436 + 17.8947i −0.171586 + 0.629924i
\(808\) 0 0
\(809\) −3.32067 5.75157i −0.116749 0.202215i 0.801729 0.597688i \(-0.203914\pi\)
−0.918477 + 0.395473i \(0.870581\pi\)
\(810\) 0 0
\(811\) 23.7806 0.835049 0.417525 0.908666i \(-0.362898\pi\)
0.417525 + 0.908666i \(0.362898\pi\)
\(812\) 0 0
\(813\) 42.7137 11.2558i 1.49803 0.394758i
\(814\) 0 0
\(815\) 0.424190 0.734719i 0.0148587 0.0257361i
\(816\) 0 0
\(817\) −8.32093 14.4123i −0.291112 0.504222i
\(818\) 0 0
\(819\) −44.2905 + 10.0284i −1.54764 + 0.350421i
\(820\) 0 0
\(821\) 24.9151 + 43.1543i 0.869544 + 1.50609i 0.862464 + 0.506119i \(0.168920\pi\)
0.00708033 + 0.999975i \(0.497746\pi\)
\(822\) 0 0
\(823\) −4.72691 + 8.18726i −0.164770 + 0.285390i −0.936574 0.350471i \(-0.886021\pi\)
0.771804 + 0.635861i \(0.219355\pi\)
\(824\) 0 0
\(825\) 19.6663 + 19.8298i 0.684692 + 0.690384i
\(826\) 0 0
\(827\) 7.95385 0.276582 0.138291 0.990392i \(-0.455839\pi\)
0.138291 + 0.990392i \(0.455839\pi\)
\(828\) 0 0
\(829\) 6.22333 + 10.7791i 0.216145 + 0.374374i 0.953626 0.300993i \(-0.0973182\pi\)
−0.737481 + 0.675368i \(0.763985\pi\)
\(830\) 0 0
\(831\) −15.5903 15.7199i −0.540822 0.545318i
\(832\) 0 0
\(833\) −2.37928 + 38.5113i −0.0824373 + 1.33434i
\(834\) 0 0
\(835\) −1.71961 2.97845i −0.0595096 0.103074i
\(836\) 0 0
\(837\) −11.9897 47.0770i −0.414424 1.62722i
\(838\) 0 0
\(839\) 22.5163 38.9994i 0.777350 1.34641i −0.156114 0.987739i \(-0.549897\pi\)
0.933464 0.358671i \(-0.116770\pi\)
\(840\) 0 0
\(841\) 9.81015 + 16.9917i 0.338281 + 0.585920i
\(842\) 0 0
\(843\) −35.0928 + 9.24757i −1.20866 + 0.318503i
\(844\) 0 0
\(845\) 4.74977 8.22684i 0.163397 0.283012i
\(846\) 0 0
\(847\) −1.10242 0.332160i −0.0378794 0.0114131i
\(848\) 0 0
\(849\) 25.2264 6.64761i 0.865770 0.228145i
\(850\) 0 0
\(851\) −5.38627 + 9.32929i −0.184639 + 0.319804i
\(852\) 0 0
\(853\) −1.15007 + 1.99198i −0.0393777 + 0.0682042i −0.885043 0.465510i \(-0.845871\pi\)
0.845665 + 0.533714i \(0.179204\pi\)
\(854\) 0 0
\(855\) 5.42728 + 3.19364i 0.185609 + 0.109220i
\(856\) 0 0
\(857\) 12.6695 0.432780 0.216390 0.976307i \(-0.430572\pi\)
0.216390 + 0.976307i \(0.430572\pi\)
\(858\) 0 0
\(859\) −47.4748 −1.61982 −0.809910 0.586554i \(-0.800484\pi\)
−0.809910 + 0.586554i \(0.800484\pi\)
\(860\) 0 0
\(861\) −30.5804 + 48.9297i −1.04218 + 1.66752i
\(862\) 0 0
\(863\) 16.1445 27.9630i 0.549564 0.951873i −0.448740 0.893662i \(-0.648127\pi\)
0.998304 0.0582109i \(-0.0185396\pi\)
\(864\) 0 0
\(865\) 2.38138 + 4.12467i 0.0809693 + 0.140243i
\(866\) 0 0
\(867\) −22.4152 + 5.90681i −0.761262 + 0.200606i
\(868\) 0 0
\(869\) 9.46779 + 16.3987i 0.321173 + 0.556287i
\(870\) 0 0
\(871\) −14.7120 −0.498497
\(872\) 0 0
\(873\) −36.3360 21.3816i −1.22979 0.723659i
\(874\) 0 0
\(875\) −11.9122 3.58917i −0.402706 0.121336i
\(876\) 0 0
\(877\) 27.7589 0.937352 0.468676 0.883370i \(-0.344731\pi\)
0.468676 + 0.883370i \(0.344731\pi\)
\(878\) 0 0
\(879\) −0.122414 + 0.449406i −0.00412893 + 0.0151581i
\(880\) 0 0
\(881\) −22.7696 −0.767126 −0.383563 0.923515i \(-0.625303\pi\)
−0.383563 + 0.923515i \(0.625303\pi\)
\(882\) 0 0
\(883\) −4.65312 −0.156590 −0.0782950 0.996930i \(-0.524948\pi\)
−0.0782950 + 0.996930i \(0.524948\pi\)
\(884\) 0 0
\(885\) −6.79100 + 1.78955i −0.228277 + 0.0601550i
\(886\) 0 0
\(887\) 47.2867 1.58773 0.793866 0.608093i \(-0.208065\pi\)
0.793866 + 0.608093i \(0.208065\pi\)
\(888\) 0 0
\(889\) 48.0519 + 14.4781i 1.61161 + 0.485581i
\(890\) 0 0
\(891\) 30.4302 + 0.503913i 1.01945 + 0.0168817i
\(892\) 0 0
\(893\) 16.4199 0.549471
\(894\) 0 0
\(895\) 0.977058 + 1.69231i 0.0326595 + 0.0565678i
\(896\) 0 0
\(897\) −9.44319 + 34.6678i −0.315299 + 1.15752i
\(898\) 0 0
\(899\) 14.3165 + 24.7970i 0.477483 + 0.827025i
\(900\) 0 0
\(901\) −30.7498 + 53.2602i −1.02442 + 1.77436i
\(902\) 0 0
\(903\) 17.4789 + 0.611855i 0.581661 + 0.0203613i
\(904\) 0 0
\(905\) −2.20516 −0.0733019
\(906\) 0 0
\(907\) 6.06612 0.201422 0.100711 0.994916i \(-0.467888\pi\)
0.100711 + 0.994916i \(0.467888\pi\)
\(908\) 0 0
\(909\) −0.263431 + 31.8182i −0.00873747 + 1.05534i
\(910\) 0 0
\(911\) −11.3223 + 19.6109i −0.375126 + 0.649737i −0.990346 0.138618i \(-0.955734\pi\)
0.615220 + 0.788356i \(0.289067\pi\)
\(912\) 0 0
\(913\) −0.404081 + 0.699889i −0.0133731 + 0.0231630i
\(914\) 0 0
\(915\) 1.59665 5.86161i 0.0527836 0.193779i
\(916\) 0 0
\(917\) −12.3682 3.72656i −0.408433 0.123062i
\(918\) 0 0
\(919\) 22.3902 38.7810i 0.738585 1.27927i −0.214548 0.976713i \(-0.568828\pi\)
0.953133 0.302553i \(-0.0978388\pi\)
\(920\) 0 0
\(921\) 6.14118 + 6.19224i 0.202359 + 0.204041i
\(922\) 0 0
\(923\) −11.2797 19.5371i −0.371277 0.643071i
\(924\) 0 0
\(925\) −7.08335 + 12.2687i −0.232899 + 0.403393i
\(926\) 0 0
\(927\) −0.00395939 + 0.478231i −0.000130044 + 0.0157072i
\(928\) 0 0
\(929\) 0.552620 + 0.957166i 0.0181309 + 0.0314036i 0.874948 0.484216i \(-0.160895\pi\)
−0.856818 + 0.515620i \(0.827562\pi\)
\(930\) 0 0
\(931\) −27.3246 + 13.6025i −0.895528 + 0.445805i
\(932\) 0 0
\(933\) 7.49724 1.97565i 0.245449 0.0646800i
\(934\) 0 0
\(935\) 4.48645 + 7.77076i 0.146723 + 0.254131i
\(936\) 0 0
\(937\) 44.7012 1.46033 0.730163 0.683273i \(-0.239444\pi\)
0.730163 + 0.683273i \(0.239444\pi\)
\(938\) 0 0
\(939\) 4.99725 18.3459i 0.163079 0.598695i
\(940\) 0 0
\(941\) 0.749661 1.29845i 0.0244383 0.0423283i −0.853548 0.521015i \(-0.825554\pi\)
0.877986 + 0.478687i \(0.158887\pi\)
\(942\) 0 0
\(943\) 22.8268 + 39.5372i 0.743344 + 1.28751i
\(944\) 0 0
\(945\) −5.86911 + 3.05797i −0.190922 + 0.0994759i
\(946\) 0 0
\(947\) 4.75447 + 8.23498i 0.154499 + 0.267601i 0.932877 0.360196i \(-0.117290\pi\)
−0.778377 + 0.627797i \(0.783957\pi\)
\(948\) 0 0
\(949\) −4.93303 + 8.54426i −0.160133 + 0.277358i
\(950\) 0 0
\(951\) 11.3644 41.7209i 0.368515 1.35289i
\(952\) 0 0
\(953\) 15.2064 0.492585 0.246292 0.969196i \(-0.420788\pi\)
0.246292 + 0.969196i \(0.420788\pi\)
\(954\) 0 0
\(955\) −2.69396 4.66607i −0.0871745 0.150991i
\(956\) 0 0
\(957\) −17.3459 + 4.57096i −0.560715 + 0.147758i
\(958\) 0 0
\(959\) 16.4024 + 4.94208i 0.529661 + 0.159588i
\(960\) 0 0
\(961\) −28.2036 48.8500i −0.909792 1.57581i
\(962\) 0 0
\(963\) 20.6344 + 12.1421i 0.664933 + 0.391275i
\(964\) 0 0
\(965\) 3.92176 6.79268i 0.126246 0.218664i
\(966\) 0 0
\(967\) 17.9319 + 31.0589i 0.576649 + 0.998786i 0.995860 + 0.0908973i \(0.0289735\pi\)
−0.419211 + 0.907889i \(0.637693\pi\)
\(968\) 0 0
\(969\) −29.3149 29.5586i −0.941731 0.949561i
\(970\) 0 0
\(971\) 3.72746 6.45615i 0.119620 0.207188i −0.799997 0.600004i \(-0.795166\pi\)
0.919617 + 0.392816i \(0.128499\pi\)
\(972\) 0 0
\(973\) −13.7765 58.5874i −0.441653 1.87823i
\(974\) 0 0
\(975\) −12.4185 + 45.5907i −0.397710 + 1.46007i
\(976\) 0 0
\(977\) 19.9756 34.5988i 0.639076 1.10691i −0.346559 0.938028i \(-0.612650\pi\)
0.985636 0.168885i \(-0.0540166\pi\)
\(978\) 0 0
\(979\) −2.19167 + 3.79608i −0.0700460 + 0.121323i
\(980\) 0 0
\(981\) 35.7945 20.2727i 1.14283 0.647258i
\(982\) 0 0
\(983\) −8.38416 −0.267413 −0.133707 0.991021i \(-0.542688\pi\)
−0.133707 + 0.991021i \(0.542688\pi\)
\(984\) 0 0
\(985\) 1.52810 0.0486894
\(986\) 0 0
\(987\) −9.14571 + 14.6334i −0.291111 + 0.465788i
\(988\) 0 0
\(989\) 6.91911 11.9843i 0.220015 0.381077i
\(990\) 0 0
\(991\) 21.2345 + 36.7792i 0.674536 + 1.16833i 0.976604 + 0.215044i \(0.0689896\pi\)
−0.302068 + 0.953286i \(0.597677\pi\)
\(992\) 0 0
\(993\) −5.47349 + 20.0942i −0.173696 + 0.637672i
\(994\) 0 0
\(995\) 0.693871 + 1.20182i 0.0219972 + 0.0381003i
\(996\) 0 0
\(997\) −30.9632 −0.980613 −0.490306 0.871550i \(-0.663115\pi\)
−0.490306 + 0.871550i \(0.663115\pi\)
\(998\) 0 0
\(999\) 3.81015 + 14.9604i 0.120548 + 0.473326i
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.k.961.9 22
3.2 odd 2 3024.2.t.l.289.6 22
4.3 odd 2 504.2.t.d.457.3 yes 22
7.4 even 3 1008.2.q.k.529.7 22
9.4 even 3 1008.2.q.k.625.7 22
9.5 odd 6 3024.2.q.k.2305.6 22
12.11 even 2 1512.2.t.d.289.6 22
21.11 odd 6 3024.2.q.k.2881.6 22
28.11 odd 6 504.2.q.d.25.5 22
36.23 even 6 1512.2.q.c.793.6 22
36.31 odd 6 504.2.q.d.121.5 yes 22
63.4 even 3 inner 1008.2.t.k.193.9 22
63.32 odd 6 3024.2.t.l.1873.6 22
84.11 even 6 1512.2.q.c.1369.6 22
252.67 odd 6 504.2.t.d.193.3 yes 22
252.95 even 6 1512.2.t.d.361.6 22
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.q.d.25.5 22 28.11 odd 6
504.2.q.d.121.5 yes 22 36.31 odd 6
504.2.t.d.193.3 yes 22 252.67 odd 6
504.2.t.d.457.3 yes 22 4.3 odd 2
1008.2.q.k.529.7 22 7.4 even 3
1008.2.q.k.625.7 22 9.4 even 3
1008.2.t.k.193.9 22 63.4 even 3 inner
1008.2.t.k.961.9 22 1.1 even 1 trivial
1512.2.q.c.793.6 22 36.23 even 6
1512.2.q.c.1369.6 22 84.11 even 6
1512.2.t.d.289.6 22 12.11 even 2
1512.2.t.d.361.6 22 252.95 even 6
3024.2.q.k.2305.6 22 9.5 odd 6
3024.2.q.k.2881.6 22 21.11 odd 6
3024.2.t.l.289.6 22 3.2 odd 2
3024.2.t.l.1873.6 22 63.32 odd 6