Properties

Label 768.2.o.a.95.7
Level $768$
Weight $2$
Character 768.95
Analytic conductor $6.133$
Analytic rank $0$
Dimension $56$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [768,2,Mod(95,768)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(768, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([4, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("768.95");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 768.o (of order \(8\), degree \(4\), 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: \(6.13251087523\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{8})\)
Twist minimal: no (minimal twist has level 96)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label 95.7
Character \(\chi\) \(=\) 768.95
Dual form 768.2.o.a.671.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.0380372 - 1.73163i) q^{3} +(-2.18808 + 0.906333i) q^{5} +(1.93241 + 1.93241i) q^{7} +(-2.99711 + 0.131733i) q^{9} +O(q^{10})\) \(q+(-0.0380372 - 1.73163i) q^{3} +(-2.18808 + 0.906333i) q^{5} +(1.93241 + 1.93241i) q^{7} +(-2.99711 + 0.131733i) q^{9} +(-1.42447 + 0.590036i) q^{11} +(0.110405 - 0.266541i) q^{13} +(1.65266 + 3.75448i) q^{15} +6.17031 q^{17} +(7.34269 + 3.04144i) q^{19} +(3.27272 - 3.41972i) q^{21} +(1.85295 + 1.85295i) q^{23} +(0.430727 - 0.430727i) q^{25} +(0.342114 + 5.18488i) q^{27} +(2.11574 - 5.10784i) q^{29} +3.42046i q^{31} +(1.07591 + 2.44422i) q^{33} +(-5.97967 - 2.47686i) q^{35} +(-2.52377 - 6.09293i) q^{37} +(-0.465751 - 0.181042i) q^{39} +(0.753641 - 0.753641i) q^{41} +(1.57129 + 3.79343i) q^{43} +(6.43852 - 3.00462i) q^{45} -1.54798i q^{47} +0.468394i q^{49} +(-0.234701 - 10.6847i) q^{51} +(5.12700 + 12.3777i) q^{53} +(2.58210 - 2.58210i) q^{55} +(4.98737 - 12.8305i) q^{57} +(3.08775 + 7.45449i) q^{59} +(4.28571 + 1.77520i) q^{61} +(-6.04619 - 5.53707i) q^{63} +0.683277i q^{65} +(-0.531731 + 1.28371i) q^{67} +(3.13814 - 3.27910i) q^{69} +(8.72539 - 8.72539i) q^{71} +(-2.73022 - 2.73022i) q^{73} +(-0.762244 - 0.729477i) q^{75} +(-3.89285 - 1.61247i) q^{77} +2.76080 q^{79} +(8.96529 - 0.789635i) q^{81} +(-2.53133 + 6.11116i) q^{83} +(-13.5011 + 5.59235i) q^{85} +(-8.92538 - 3.46939i) q^{87} +(-4.14369 - 4.14369i) q^{89} +(0.728413 - 0.301719i) q^{91} +(5.92298 - 0.130105i) q^{93} -18.8230 q^{95} -10.3656 q^{97} +(4.19157 - 1.95605i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56 q - 4 q^{3} + 8 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 56 q - 4 q^{3} + 8 q^{7} - 4 q^{9} + 8 q^{13} + 8 q^{15} - 8 q^{19} + 4 q^{21} - 8 q^{25} - 28 q^{27} - 8 q^{33} + 8 q^{37} + 28 q^{39} - 8 q^{43} + 4 q^{45} - 16 q^{51} - 24 q^{55} - 4 q^{57} + 40 q^{61} + 56 q^{67} + 4 q^{69} - 8 q^{73} + 16 q^{75} - 16 q^{79} + 48 q^{85} - 52 q^{87} + 40 q^{91} - 8 q^{93} - 16 q^{97} + 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{3}{8}\right)\)

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
\(3\) −0.0380372 1.73163i −0.0219608 0.999759i
\(4\) 0 0
\(5\) −2.18808 + 0.906333i −0.978540 + 0.405324i −0.813884 0.581027i \(-0.802651\pi\)
−0.164655 + 0.986351i \(0.552651\pi\)
\(6\) 0 0
\(7\) 1.93241 + 1.93241i 0.730381 + 0.730381i 0.970695 0.240314i \(-0.0772504\pi\)
−0.240314 + 0.970695i \(0.577250\pi\)
\(8\) 0 0
\(9\) −2.99711 + 0.131733i −0.999035 + 0.0439110i
\(10\) 0 0
\(11\) −1.42447 + 0.590036i −0.429495 + 0.177903i −0.586949 0.809624i \(-0.699671\pi\)
0.157454 + 0.987526i \(0.449671\pi\)
\(12\) 0 0
\(13\) 0.110405 0.266541i 0.0306208 0.0739252i −0.907829 0.419340i \(-0.862262\pi\)
0.938450 + 0.345414i \(0.112262\pi\)
\(14\) 0 0
\(15\) 1.65266 + 3.75448i 0.426716 + 0.969402i
\(16\) 0 0
\(17\) 6.17031 1.49652 0.748260 0.663406i \(-0.230890\pi\)
0.748260 + 0.663406i \(0.230890\pi\)
\(18\) 0 0
\(19\) 7.34269 + 3.04144i 1.68453 + 0.697755i 0.999526 0.0307845i \(-0.00980057\pi\)
0.685004 + 0.728540i \(0.259801\pi\)
\(20\) 0 0
\(21\) 3.27272 3.41972i 0.714165 0.746245i
\(22\) 0 0
\(23\) 1.85295 + 1.85295i 0.386366 + 0.386366i 0.873389 0.487023i \(-0.161917\pi\)
−0.487023 + 0.873389i \(0.661917\pi\)
\(24\) 0 0
\(25\) 0.430727 0.430727i 0.0861453 0.0861453i
\(26\) 0 0
\(27\) 0.342114 + 5.18488i 0.0658400 + 0.997830i
\(28\) 0 0
\(29\) 2.11574 5.10784i 0.392882 0.948501i −0.596427 0.802667i \(-0.703413\pi\)
0.989309 0.145834i \(-0.0465866\pi\)
\(30\) 0 0
\(31\) 3.42046i 0.614332i 0.951656 + 0.307166i \(0.0993807\pi\)
−0.951656 + 0.307166i \(0.900619\pi\)
\(32\) 0 0
\(33\) 1.07591 + 2.44422i 0.187292 + 0.425485i
\(34\) 0 0
\(35\) −5.97967 2.47686i −1.01075 0.418666i
\(36\) 0 0
\(37\) −2.52377 6.09293i −0.414906 1.00167i −0.983801 0.179262i \(-0.942629\pi\)
0.568895 0.822410i \(-0.307371\pi\)
\(38\) 0 0
\(39\) −0.465751 0.181042i −0.0745798 0.0289900i
\(40\) 0 0
\(41\) 0.753641 0.753641i 0.117699 0.117699i −0.645804 0.763503i \(-0.723478\pi\)
0.763503 + 0.645804i \(0.223478\pi\)
\(42\) 0 0
\(43\) 1.57129 + 3.79343i 0.239619 + 0.578492i 0.997243 0.0741989i \(-0.0236400\pi\)
−0.757624 + 0.652691i \(0.773640\pi\)
\(44\) 0 0
\(45\) 6.43852 3.00462i 0.959798 0.447902i
\(46\) 0 0
\(47\) 1.54798i 0.225796i −0.993607 0.112898i \(-0.963987\pi\)
0.993607 0.112898i \(-0.0360133\pi\)
\(48\) 0 0
\(49\) 0.468394i 0.0669135i
\(50\) 0 0
\(51\) −0.234701 10.6847i −0.0328647 1.49616i
\(52\) 0 0
\(53\) 5.12700 + 12.3777i 0.704248 + 1.70020i 0.713897 + 0.700251i \(0.246928\pi\)
−0.00964932 + 0.999953i \(0.503072\pi\)
\(54\) 0 0
\(55\) 2.58210 2.58210i 0.348170 0.348170i
\(56\) 0 0
\(57\) 4.98737 12.8305i 0.660593 1.69945i
\(58\) 0 0
\(59\) 3.08775 + 7.45449i 0.401991 + 0.970492i 0.987182 + 0.159597i \(0.0510195\pi\)
−0.585191 + 0.810895i \(0.698981\pi\)
\(60\) 0 0
\(61\) 4.28571 + 1.77520i 0.548728 + 0.227291i 0.639784 0.768555i \(-0.279024\pi\)
−0.0910552 + 0.995846i \(0.529024\pi\)
\(62\) 0 0
\(63\) −6.04619 5.53707i −0.761748 0.697605i
\(64\) 0 0
\(65\) 0.683277i 0.0847501i
\(66\) 0 0
\(67\) −0.531731 + 1.28371i −0.0649613 + 0.156831i −0.953026 0.302887i \(-0.902049\pi\)
0.888065 + 0.459718i \(0.152049\pi\)
\(68\) 0 0
\(69\) 3.13814 3.27910i 0.377788 0.394758i
\(70\) 0 0
\(71\) 8.72539 8.72539i 1.03551 1.03551i 0.0361678 0.999346i \(-0.488485\pi\)
0.999346 0.0361678i \(-0.0115151\pi\)
\(72\) 0 0
\(73\) −2.73022 2.73022i −0.319548 0.319548i 0.529046 0.848593i \(-0.322550\pi\)
−0.848593 + 0.529046i \(0.822550\pi\)
\(74\) 0 0
\(75\) −0.762244 0.729477i −0.0880164 0.0842327i
\(76\) 0 0
\(77\) −3.89285 1.61247i −0.443632 0.183758i
\(78\) 0 0
\(79\) 2.76080 0.310614 0.155307 0.987866i \(-0.450363\pi\)
0.155307 + 0.987866i \(0.450363\pi\)
\(80\) 0 0
\(81\) 8.96529 0.789635i 0.996144 0.0877372i
\(82\) 0 0
\(83\) −2.53133 + 6.11116i −0.277849 + 0.670787i −0.999776 0.0211827i \(-0.993257\pi\)
0.721927 + 0.691970i \(0.243257\pi\)
\(84\) 0 0
\(85\) −13.5011 + 5.59235i −1.46440 + 0.606576i
\(86\) 0 0
\(87\) −8.92538 3.46939i −0.956901 0.371958i
\(88\) 0 0
\(89\) −4.14369 4.14369i −0.439230 0.439230i 0.452523 0.891753i \(-0.350524\pi\)
−0.891753 + 0.452523i \(0.850524\pi\)
\(90\) 0 0
\(91\) 0.728413 0.301719i 0.0763585 0.0316287i
\(92\) 0 0
\(93\) 5.92298 0.130105i 0.614184 0.0134912i
\(94\) 0 0
\(95\) −18.8230 −1.93120
\(96\) 0 0
\(97\) −10.3656 −1.05246 −0.526232 0.850341i \(-0.676396\pi\)
−0.526232 + 0.850341i \(0.676396\pi\)
\(98\) 0 0
\(99\) 4.19157 1.95605i 0.421269 0.196591i
\(100\) 0 0
\(101\) −10.5700 + 4.37825i −1.05176 + 0.435652i −0.840519 0.541783i \(-0.817750\pi\)
−0.211238 + 0.977435i \(0.567750\pi\)
\(102\) 0 0
\(103\) 10.4823 + 10.4823i 1.03285 + 1.03285i 0.999442 + 0.0334101i \(0.0106367\pi\)
0.0334101 + 0.999442i \(0.489363\pi\)
\(104\) 0 0
\(105\) −4.06156 + 10.4488i −0.396368 + 1.01970i
\(106\) 0 0
\(107\) 9.39578 3.89186i 0.908325 0.376240i 0.120910 0.992664i \(-0.461419\pi\)
0.787415 + 0.616423i \(0.211419\pi\)
\(108\) 0 0
\(109\) −3.00588 + 7.25683i −0.287911 + 0.695078i −0.999975 0.00703600i \(-0.997760\pi\)
0.712064 + 0.702114i \(0.247760\pi\)
\(110\) 0 0
\(111\) −10.4547 + 4.60201i −0.992318 + 0.436803i
\(112\) 0 0
\(113\) −13.6372 −1.28288 −0.641438 0.767175i \(-0.721662\pi\)
−0.641438 + 0.767175i \(0.721662\pi\)
\(114\) 0 0
\(115\) −5.73378 2.37501i −0.534678 0.221471i
\(116\) 0 0
\(117\) −0.295783 + 0.813396i −0.0273452 + 0.0751985i
\(118\) 0 0
\(119\) 11.9235 + 11.9235i 1.09303 + 1.09303i
\(120\) 0 0
\(121\) −6.09719 + 6.09719i −0.554290 + 0.554290i
\(122\) 0 0
\(123\) −1.33370 1.27636i −0.120255 0.115086i
\(124\) 0 0
\(125\) 3.97958 9.60756i 0.355945 0.859326i
\(126\) 0 0
\(127\) 7.75395i 0.688052i −0.938960 0.344026i \(-0.888209\pi\)
0.938960 0.344026i \(-0.111791\pi\)
\(128\) 0 0
\(129\) 6.50906 2.86519i 0.573090 0.252266i
\(130\) 0 0
\(131\) 1.16782 + 0.483728i 0.102033 + 0.0422635i 0.433116 0.901338i \(-0.357414\pi\)
−0.331083 + 0.943602i \(0.607414\pi\)
\(132\) 0 0
\(133\) 8.31177 + 20.0664i 0.720722 + 1.73998i
\(134\) 0 0
\(135\) −5.44780 11.0349i −0.468872 0.949730i
\(136\) 0 0
\(137\) 7.54494 7.54494i 0.644608 0.644608i −0.307077 0.951685i \(-0.599351\pi\)
0.951685 + 0.307077i \(0.0993509\pi\)
\(138\) 0 0
\(139\) 0.412435 + 0.995705i 0.0349822 + 0.0844546i 0.940405 0.340056i \(-0.110446\pi\)
−0.905423 + 0.424511i \(0.860446\pi\)
\(140\) 0 0
\(141\) −2.68053 + 0.0588808i −0.225742 + 0.00495866i
\(142\) 0 0
\(143\) 0.444824i 0.0371980i
\(144\) 0 0
\(145\) 13.0939i 1.08739i
\(146\) 0 0
\(147\) 0.811087 0.0178164i 0.0668973 0.00146947i
\(148\) 0 0
\(149\) −0.458938 1.10798i −0.0375977 0.0907689i 0.903965 0.427606i \(-0.140643\pi\)
−0.941563 + 0.336837i \(0.890643\pi\)
\(150\) 0 0
\(151\) −9.80869 + 9.80869i −0.798220 + 0.798220i −0.982815 0.184595i \(-0.940903\pi\)
0.184595 + 0.982815i \(0.440903\pi\)
\(152\) 0 0
\(153\) −18.4931 + 0.812833i −1.49508 + 0.0657136i
\(154\) 0 0
\(155\) −3.10007 7.48424i −0.249004 0.601148i
\(156\) 0 0
\(157\) −8.28207 3.43055i −0.660981 0.273787i 0.0268702 0.999639i \(-0.491446\pi\)
−0.687851 + 0.725852i \(0.741446\pi\)
\(158\) 0 0
\(159\) 21.2386 9.34890i 1.68433 0.741416i
\(160\) 0 0
\(161\) 7.16129i 0.564389i
\(162\) 0 0
\(163\) −1.15049 + 2.77752i −0.0901131 + 0.217552i −0.962510 0.271246i \(-0.912564\pi\)
0.872397 + 0.488798i \(0.162564\pi\)
\(164\) 0 0
\(165\) −4.56946 4.37303i −0.355732 0.340440i
\(166\) 0 0
\(167\) −1.86833 + 1.86833i −0.144576 + 0.144576i −0.775690 0.631114i \(-0.782598\pi\)
0.631114 + 0.775690i \(0.282598\pi\)
\(168\) 0 0
\(169\) 9.13353 + 9.13353i 0.702579 + 0.702579i
\(170\) 0 0
\(171\) −22.4075 8.14826i −1.71354 0.623113i
\(172\) 0 0
\(173\) −9.59196 3.97312i −0.729263 0.302071i −0.0130138 0.999915i \(-0.504143\pi\)
−0.716249 + 0.697845i \(0.754143\pi\)
\(174\) 0 0
\(175\) 1.66468 0.125838
\(176\) 0 0
\(177\) 12.7910 5.63040i 0.961430 0.423207i
\(178\) 0 0
\(179\) −0.00532113 + 0.0128464i −0.000397720 + 0.000960181i −0.924078 0.382203i \(-0.875165\pi\)
0.923681 + 0.383163i \(0.125165\pi\)
\(180\) 0 0
\(181\) 9.45181 3.91507i 0.702547 0.291005i −0.00266940 0.999996i \(-0.500850\pi\)
0.705217 + 0.708992i \(0.250850\pi\)
\(182\) 0 0
\(183\) 2.91097 7.48879i 0.215185 0.553588i
\(184\) 0 0
\(185\) 11.0444 + 11.0444i 0.812004 + 0.812004i
\(186\) 0 0
\(187\) −8.78944 + 3.64071i −0.642748 + 0.266235i
\(188\) 0 0
\(189\) −9.35819 + 10.6804i −0.680708 + 0.776885i
\(190\) 0 0
\(191\) 7.77941 0.562898 0.281449 0.959576i \(-0.409185\pi\)
0.281449 + 0.959576i \(0.409185\pi\)
\(192\) 0 0
\(193\) −6.23528 −0.448825 −0.224413 0.974494i \(-0.572046\pi\)
−0.224413 + 0.974494i \(0.572046\pi\)
\(194\) 0 0
\(195\) 1.18319 0.0259899i 0.0847297 0.00186118i
\(196\) 0 0
\(197\) 6.90773 2.86128i 0.492156 0.203858i −0.122781 0.992434i \(-0.539181\pi\)
0.614937 + 0.788576i \(0.289181\pi\)
\(198\) 0 0
\(199\) −12.4517 12.4517i −0.882681 0.882681i 0.111125 0.993806i \(-0.464555\pi\)
−0.993806 + 0.111125i \(0.964555\pi\)
\(200\) 0 0
\(201\) 2.24315 + 0.871935i 0.158219 + 0.0615015i
\(202\) 0 0
\(203\) 13.9589 5.78196i 0.979721 0.405814i
\(204\) 0 0
\(205\) −0.965978 + 2.33208i −0.0674668 + 0.162879i
\(206\) 0 0
\(207\) −5.79757 5.30938i −0.402959 0.369028i
\(208\) 0 0
\(209\) −12.2540 −0.847630
\(210\) 0 0
\(211\) −1.16851 0.484011i −0.0804432 0.0333207i 0.342099 0.939664i \(-0.388862\pi\)
−0.422542 + 0.906343i \(0.638862\pi\)
\(212\) 0 0
\(213\) −15.4411 14.7773i −1.05800 1.01252i
\(214\) 0 0
\(215\) −6.87622 6.87622i −0.468954 0.468954i
\(216\) 0 0
\(217\) −6.60972 + 6.60972i −0.448697 + 0.448697i
\(218\) 0 0
\(219\) −4.62388 + 4.83158i −0.312453 + 0.326488i
\(220\) 0 0
\(221\) 0.681233 1.64464i 0.0458247 0.110631i
\(222\) 0 0
\(223\) 10.5047i 0.703449i 0.936104 + 0.351724i \(0.114405\pi\)
−0.936104 + 0.351724i \(0.885595\pi\)
\(224\) 0 0
\(225\) −1.23419 + 1.34767i −0.0822795 + 0.0898450i
\(226\) 0 0
\(227\) −16.7869 6.95334i −1.11418 0.461509i −0.251806 0.967778i \(-0.581025\pi\)
−0.862376 + 0.506268i \(0.831025\pi\)
\(228\) 0 0
\(229\) −5.87646 14.1870i −0.388327 0.937505i −0.990295 0.138984i \(-0.955616\pi\)
0.601967 0.798521i \(-0.294384\pi\)
\(230\) 0 0
\(231\) −2.64414 + 6.80233i −0.173972 + 0.447560i
\(232\) 0 0
\(233\) −2.54073 + 2.54073i −0.166449 + 0.166449i −0.785416 0.618968i \(-0.787551\pi\)
0.618968 + 0.785416i \(0.287551\pi\)
\(234\) 0 0
\(235\) 1.40298 + 3.38711i 0.0915206 + 0.220950i
\(236\) 0 0
\(237\) −0.105013 4.78069i −0.00682132 0.310539i
\(238\) 0 0
\(239\) 17.6107i 1.13914i −0.821943 0.569570i \(-0.807110\pi\)
0.821943 0.569570i \(-0.192890\pi\)
\(240\) 0 0
\(241\) 6.18628i 0.398493i 0.979949 + 0.199247i \(0.0638495\pi\)
−0.979949 + 0.199247i \(0.936151\pi\)
\(242\) 0 0
\(243\) −1.70837 15.4946i −0.109592 0.993977i
\(244\) 0 0
\(245\) −0.424521 1.02489i −0.0271217 0.0654775i
\(246\) 0 0
\(247\) 1.62134 1.62134i 0.103163 0.103163i
\(248\) 0 0
\(249\) 10.6786 + 4.15088i 0.676727 + 0.263051i
\(250\) 0 0
\(251\) −9.18840 22.1828i −0.579967 1.40016i −0.892842 0.450369i \(-0.851292\pi\)
0.312876 0.949794i \(-0.398708\pi\)
\(252\) 0 0
\(253\) −3.73278 1.54617i −0.234678 0.0972067i
\(254\) 0 0
\(255\) 10.1975 + 23.1663i 0.638589 + 1.45073i
\(256\) 0 0
\(257\) 11.8836i 0.741276i −0.928777 0.370638i \(-0.879139\pi\)
0.928777 0.370638i \(-0.120861\pi\)
\(258\) 0 0
\(259\) 6.89706 16.6510i 0.428563 1.03464i
\(260\) 0 0
\(261\) −5.66821 + 15.5874i −0.350854 + 0.964838i
\(262\) 0 0
\(263\) −11.6191 + 11.6191i −0.716464 + 0.716464i −0.967879 0.251415i \(-0.919104\pi\)
0.251415 + 0.967879i \(0.419104\pi\)
\(264\) 0 0
\(265\) −22.4366 22.4366i −1.37827 1.37827i
\(266\) 0 0
\(267\) −7.01773 + 7.33296i −0.429478 + 0.448770i
\(268\) 0 0
\(269\) 13.7480 + 5.69459i 0.838228 + 0.347205i 0.760155 0.649742i \(-0.225123\pi\)
0.0780733 + 0.996948i \(0.475123\pi\)
\(270\) 0 0
\(271\) −7.07297 −0.429652 −0.214826 0.976652i \(-0.568919\pi\)
−0.214826 + 0.976652i \(0.568919\pi\)
\(272\) 0 0
\(273\) −0.550173 1.24987i −0.0332980 0.0756455i
\(274\) 0 0
\(275\) −0.359414 + 0.867703i −0.0216735 + 0.0523245i
\(276\) 0 0
\(277\) 1.13397 0.469705i 0.0681335 0.0282218i −0.348356 0.937362i \(-0.613260\pi\)
0.416490 + 0.909140i \(0.363260\pi\)
\(278\) 0 0
\(279\) −0.450587 10.2515i −0.0269759 0.613740i
\(280\) 0 0
\(281\) 12.0212 + 12.0212i 0.717122 + 0.717122i 0.968015 0.250893i \(-0.0807241\pi\)
−0.250893 + 0.968015i \(0.580724\pi\)
\(282\) 0 0
\(283\) 12.8158 5.30848i 0.761821 0.315557i 0.0322666 0.999479i \(-0.489727\pi\)
0.729555 + 0.683923i \(0.239727\pi\)
\(284\) 0 0
\(285\) 0.715973 + 32.5945i 0.0424106 + 1.93073i
\(286\) 0 0
\(287\) 2.91268 0.171930
\(288\) 0 0
\(289\) 21.0727 1.23957
\(290\) 0 0
\(291\) 0.394277 + 17.9494i 0.0231129 + 1.05221i
\(292\) 0 0
\(293\) 24.7412 10.2481i 1.44539 0.598702i 0.484295 0.874905i \(-0.339076\pi\)
0.961099 + 0.276203i \(0.0890763\pi\)
\(294\) 0 0
\(295\) −13.5125 13.5125i −0.786729 0.786729i
\(296\) 0 0
\(297\) −3.54660 7.18386i −0.205795 0.416850i
\(298\) 0 0
\(299\) 0.698461 0.289312i 0.0403930 0.0167313i
\(300\) 0 0
\(301\) −4.29408 + 10.3668i −0.247506 + 0.597533i
\(302\) 0 0
\(303\) 7.98357 + 18.1369i 0.458644 + 1.04194i
\(304\) 0 0
\(305\) −10.9864 −0.629079
\(306\) 0 0
\(307\) −0.664375 0.275193i −0.0379179 0.0157061i 0.363644 0.931538i \(-0.381532\pi\)
−0.401562 + 0.915832i \(0.631532\pi\)
\(308\) 0 0
\(309\) 17.7528 18.5502i 1.00992 1.05528i
\(310\) 0 0
\(311\) 3.60638 + 3.60638i 0.204499 + 0.204499i 0.801924 0.597425i \(-0.203810\pi\)
−0.597425 + 0.801924i \(0.703810\pi\)
\(312\) 0 0
\(313\) 23.2233 23.2233i 1.31266 1.31266i 0.393210 0.919449i \(-0.371365\pi\)
0.919449 0.393210i \(-0.128635\pi\)
\(314\) 0 0
\(315\) 18.2480 + 6.63569i 1.02816 + 0.373879i
\(316\) 0 0
\(317\) 1.42053 3.42947i 0.0797850 0.192618i −0.878954 0.476907i \(-0.841758\pi\)
0.958738 + 0.284289i \(0.0917577\pi\)
\(318\) 0 0
\(319\) 8.52434i 0.477271i
\(320\) 0 0
\(321\) −7.09666 16.1220i −0.396097 0.899843i
\(322\) 0 0
\(323\) 45.3067 + 18.7666i 2.52093 + 1.04420i
\(324\) 0 0
\(325\) −0.0672520 0.162361i −0.00373047 0.00900615i
\(326\) 0 0
\(327\) 12.6805 + 4.92905i 0.701233 + 0.272577i
\(328\) 0 0
\(329\) 2.99133 2.99133i 0.164917 0.164917i
\(330\) 0 0
\(331\) −3.25086 7.84826i −0.178683 0.431380i 0.809008 0.587798i \(-0.200005\pi\)
−0.987691 + 0.156419i \(0.950005\pi\)
\(332\) 0 0
\(333\) 8.36666 + 17.9287i 0.458490 + 0.982487i
\(334\) 0 0
\(335\) 3.29079i 0.179795i
\(336\) 0 0
\(337\) 24.2771i 1.32246i −0.750183 0.661230i \(-0.770035\pi\)
0.750183 0.661230i \(-0.229965\pi\)
\(338\) 0 0
\(339\) 0.518719 + 23.6146i 0.0281730 + 1.28257i
\(340\) 0 0
\(341\) −2.01819 4.87235i −0.109291 0.263853i
\(342\) 0 0
\(343\) 12.6217 12.6217i 0.681509 0.681509i
\(344\) 0 0
\(345\) −3.89455 + 10.0191i −0.209676 + 0.539413i
\(346\) 0 0
\(347\) 1.37907 + 3.32937i 0.0740325 + 0.178730i 0.956564 0.291524i \(-0.0941623\pi\)
−0.882531 + 0.470254i \(0.844162\pi\)
\(348\) 0 0
\(349\) −10.7378 4.44775i −0.574782 0.238083i 0.0763062 0.997084i \(-0.475687\pi\)
−0.651089 + 0.759002i \(0.725687\pi\)
\(350\) 0 0
\(351\) 1.41975 + 0.481249i 0.0757809 + 0.0256871i
\(352\) 0 0
\(353\) 5.62531i 0.299405i −0.988731 0.149703i \(-0.952168\pi\)
0.988731 0.149703i \(-0.0478316\pi\)
\(354\) 0 0
\(355\) −11.1838 + 27.0000i −0.593572 + 1.43301i
\(356\) 0 0
\(357\) 20.1937 21.1007i 1.06876 1.11677i
\(358\) 0 0
\(359\) −22.3781 + 22.3781i −1.18107 + 1.18107i −0.201603 + 0.979467i \(0.564615\pi\)
−0.979467 + 0.201603i \(0.935385\pi\)
\(360\) 0 0
\(361\) 31.2298 + 31.2298i 1.64367 + 1.64367i
\(362\) 0 0
\(363\) 10.7900 + 10.3262i 0.566329 + 0.541984i
\(364\) 0 0
\(365\) 8.44842 + 3.49945i 0.442211 + 0.183170i
\(366\) 0 0
\(367\) 26.0135 1.35789 0.678946 0.734188i \(-0.262437\pi\)
0.678946 + 0.734188i \(0.262437\pi\)
\(368\) 0 0
\(369\) −2.15946 + 2.35802i −0.112417 + 0.122754i
\(370\) 0 0
\(371\) −14.0113 + 33.8262i −0.727428 + 1.75617i
\(372\) 0 0
\(373\) 16.7694 6.94611i 0.868286 0.359656i 0.0963434 0.995348i \(-0.469285\pi\)
0.771942 + 0.635692i \(0.219285\pi\)
\(374\) 0 0
\(375\) −16.7881 6.52573i −0.866936 0.336987i
\(376\) 0 0
\(377\) −1.12786 1.12786i −0.0580878 0.0580878i
\(378\) 0 0
\(379\) 11.8673 4.91559i 0.609582 0.252497i −0.0564681 0.998404i \(-0.517984\pi\)
0.666050 + 0.745907i \(0.267984\pi\)
\(380\) 0 0
\(381\) −13.4270 + 0.294938i −0.687886 + 0.0151101i
\(382\) 0 0
\(383\) −5.25906 −0.268725 −0.134363 0.990932i \(-0.542899\pi\)
−0.134363 + 0.990932i \(0.542899\pi\)
\(384\) 0 0
\(385\) 9.97932 0.508593
\(386\) 0 0
\(387\) −5.20904 11.1623i −0.264790 0.567412i
\(388\) 0 0
\(389\) −25.9373 + 10.7436i −1.31507 + 0.544721i −0.926360 0.376639i \(-0.877080\pi\)
−0.388712 + 0.921359i \(0.627080\pi\)
\(390\) 0 0
\(391\) 11.4333 + 11.4333i 0.578204 + 0.578204i
\(392\) 0 0
\(393\) 0.793219 2.04064i 0.0400126 0.102937i
\(394\) 0 0
\(395\) −6.04085 + 2.50220i −0.303948 + 0.125899i
\(396\) 0 0
\(397\) 14.4580 34.9046i 0.725624 1.75181i 0.0689699 0.997619i \(-0.478029\pi\)
0.656654 0.754192i \(-0.271971\pi\)
\(398\) 0 0
\(399\) 34.4315 15.1562i 1.72373 0.758759i
\(400\) 0 0
\(401\) −31.7191 −1.58397 −0.791987 0.610537i \(-0.790954\pi\)
−0.791987 + 0.610537i \(0.790954\pi\)
\(402\) 0 0
\(403\) 0.911693 + 0.377635i 0.0454146 + 0.0188114i
\(404\) 0 0
\(405\) −18.9011 + 9.85333i −0.939204 + 0.489616i
\(406\) 0 0
\(407\) 7.19010 + 7.19010i 0.356400 + 0.356400i
\(408\) 0 0
\(409\) −17.8308 + 17.8308i −0.881677 + 0.881677i −0.993705 0.112028i \(-0.964265\pi\)
0.112028 + 0.993705i \(0.464265\pi\)
\(410\) 0 0
\(411\) −13.3521 12.7781i −0.658608 0.630296i
\(412\) 0 0
\(413\) −8.43832 + 20.3719i −0.415223 + 1.00244i
\(414\) 0 0
\(415\) 15.6659i 0.769011i
\(416\) 0 0
\(417\) 1.70851 0.752059i 0.0836660 0.0368285i
\(418\) 0 0
\(419\) 2.10652 + 0.872548i 0.102910 + 0.0426268i 0.433544 0.901132i \(-0.357263\pi\)
−0.330634 + 0.943759i \(0.607263\pi\)
\(420\) 0 0
\(421\) 0.169965 + 0.410331i 0.00828357 + 0.0199983i 0.927967 0.372662i \(-0.121555\pi\)
−0.919684 + 0.392660i \(0.871555\pi\)
\(422\) 0 0
\(423\) 0.203920 + 4.63946i 0.00991492 + 0.225578i
\(424\) 0 0
\(425\) 2.65772 2.65772i 0.128918 0.128918i
\(426\) 0 0
\(427\) 4.85132 + 11.7121i 0.234772 + 0.566790i
\(428\) 0 0
\(429\) 0.770272 0.0169198i 0.0371891 0.000816898i
\(430\) 0 0
\(431\) 2.95351i 0.142266i −0.997467 0.0711329i \(-0.977339\pi\)
0.997467 0.0711329i \(-0.0226614\pi\)
\(432\) 0 0
\(433\) 31.3472i 1.50645i −0.657764 0.753224i \(-0.728497\pi\)
0.657764 0.753224i \(-0.271503\pi\)
\(434\) 0 0
\(435\) 22.6739 0.498056i 1.08713 0.0238800i
\(436\) 0 0
\(437\) 7.96999 + 19.2413i 0.381256 + 0.920434i
\(438\) 0 0
\(439\) 7.44392 7.44392i 0.355279 0.355279i −0.506791 0.862069i \(-0.669168\pi\)
0.862069 + 0.506791i \(0.169168\pi\)
\(440\) 0 0
\(441\) −0.0617029 1.40383i −0.00293824 0.0668489i
\(442\) 0 0
\(443\) −3.50021 8.45026i −0.166300 0.401484i 0.818657 0.574283i \(-0.194719\pi\)
−0.984957 + 0.172799i \(0.944719\pi\)
\(444\) 0 0
\(445\) 12.8223 + 5.31117i 0.607835 + 0.251773i
\(446\) 0 0
\(447\) −1.90115 + 0.836857i −0.0899213 + 0.0395820i
\(448\) 0 0
\(449\) 24.0900i 1.13688i 0.822725 + 0.568439i \(0.192453\pi\)
−0.822725 + 0.568439i \(0.807547\pi\)
\(450\) 0 0
\(451\) −0.628866 + 1.51822i −0.0296121 + 0.0714901i
\(452\) 0 0
\(453\) 17.3581 + 16.6120i 0.815557 + 0.780498i
\(454\) 0 0
\(455\) −1.32037 + 1.32037i −0.0618999 + 0.0618999i
\(456\) 0 0
\(457\) −13.8807 13.8807i −0.649311 0.649311i 0.303515 0.952827i \(-0.401840\pi\)
−0.952827 + 0.303515i \(0.901840\pi\)
\(458\) 0 0
\(459\) 2.11095 + 31.9923i 0.0985308 + 1.49327i
\(460\) 0 0
\(461\) −26.4361 10.9502i −1.23125 0.510001i −0.330282 0.943882i \(-0.607144\pi\)
−0.900970 + 0.433881i \(0.857144\pi\)
\(462\) 0 0
\(463\) −35.7558 −1.66171 −0.830857 0.556485i \(-0.812150\pi\)
−0.830857 + 0.556485i \(0.812150\pi\)
\(464\) 0 0
\(465\) −12.8420 + 5.65287i −0.595535 + 0.262145i
\(466\) 0 0
\(467\) −1.84875 + 4.46327i −0.0855498 + 0.206536i −0.960865 0.277018i \(-0.910654\pi\)
0.875315 + 0.483553i \(0.160654\pi\)
\(468\) 0 0
\(469\) −3.50818 + 1.45313i −0.161993 + 0.0670995i
\(470\) 0 0
\(471\) −5.62542 + 14.4720i −0.259206 + 0.666834i
\(472\) 0 0
\(473\) −4.47652 4.47652i −0.205831 0.205831i
\(474\) 0 0
\(475\) 4.47273 1.85266i 0.205223 0.0850060i
\(476\) 0 0
\(477\) −16.9967 36.4218i −0.778226 1.66764i
\(478\) 0 0
\(479\) 15.8988 0.726433 0.363216 0.931705i \(-0.381679\pi\)
0.363216 + 0.931705i \(0.381679\pi\)
\(480\) 0 0
\(481\) −1.90265 −0.0867535
\(482\) 0 0
\(483\) 12.4007 0.272395i 0.564253 0.0123944i
\(484\) 0 0
\(485\) 22.6807 9.39466i 1.02988 0.426590i
\(486\) 0 0
\(487\) 10.4283 + 10.4283i 0.472552 + 0.472552i 0.902740 0.430187i \(-0.141552\pi\)
−0.430187 + 0.902740i \(0.641552\pi\)
\(488\) 0 0
\(489\) 4.85341 + 1.88657i 0.219479 + 0.0853138i
\(490\) 0 0
\(491\) −0.306137 + 0.126806i −0.0138158 + 0.00572268i −0.389581 0.920992i \(-0.627380\pi\)
0.375765 + 0.926715i \(0.377380\pi\)
\(492\) 0 0
\(493\) 13.0547 31.5169i 0.587956 1.41945i
\(494\) 0 0
\(495\) −7.39867 + 8.07896i −0.332545 + 0.363122i
\(496\) 0 0
\(497\) 33.7220 1.51264
\(498\) 0 0
\(499\) 5.62549 + 2.33015i 0.251831 + 0.104312i 0.505028 0.863103i \(-0.331482\pi\)
−0.253197 + 0.967415i \(0.581482\pi\)
\(500\) 0 0
\(501\) 3.30633 + 3.16419i 0.147716 + 0.141366i
\(502\) 0 0
\(503\) −10.6198 10.6198i −0.473513 0.473513i 0.429536 0.903050i \(-0.358677\pi\)
−0.903050 + 0.429536i \(0.858677\pi\)
\(504\) 0 0
\(505\) 19.1599 19.1599i 0.852605 0.852605i
\(506\) 0 0
\(507\) 15.4685 16.1633i 0.686981 0.717839i
\(508\) 0 0
\(509\) −2.01172 + 4.85673i −0.0891681 + 0.215271i −0.962172 0.272442i \(-0.912169\pi\)
0.873004 + 0.487713i \(0.162169\pi\)
\(510\) 0 0
\(511\) 10.5518i 0.466783i
\(512\) 0 0
\(513\) −13.2575 + 39.1115i −0.585332 + 1.72681i
\(514\) 0 0
\(515\) −32.4366 13.4357i −1.42933 0.592046i
\(516\) 0 0
\(517\) 0.913364 + 2.20506i 0.0401697 + 0.0969783i
\(518\) 0 0
\(519\) −6.51513 + 16.7609i −0.285983 + 0.735721i
\(520\) 0 0
\(521\) 19.9974 19.9974i 0.876102 0.876102i −0.117027 0.993129i \(-0.537336\pi\)
0.993129 + 0.117027i \(0.0373363\pi\)
\(522\) 0 0
\(523\) −15.2948 36.9248i −0.668793 1.61461i −0.783632 0.621225i \(-0.786635\pi\)
0.114839 0.993384i \(-0.463365\pi\)
\(524\) 0 0
\(525\) −0.0633197 2.88261i −0.00276350 0.125808i
\(526\) 0 0
\(527\) 21.1053i 0.919360i
\(528\) 0 0
\(529\) 16.1332i 0.701443i
\(530\) 0 0
\(531\) −10.2363 21.9352i −0.444219 0.951905i
\(532\) 0 0
\(533\) −0.117671 0.284082i −0.00509688 0.0123050i
\(534\) 0 0
\(535\) −17.0314 + 17.0314i −0.736332 + 0.736332i
\(536\) 0 0
\(537\) 0.0224476 + 0.00872561i 0.000968684 + 0.000376538i
\(538\) 0 0
\(539\) −0.276370 0.667216i −0.0119041 0.0287390i
\(540\) 0 0
\(541\) 35.6786 + 14.7786i 1.53394 + 0.635380i 0.980325 0.197391i \(-0.0632468\pi\)
0.553618 + 0.832771i \(0.313247\pi\)
\(542\) 0 0
\(543\) −7.13898 16.2181i −0.306363 0.695987i
\(544\) 0 0
\(545\) 18.6029i 0.796859i
\(546\) 0 0
\(547\) 12.4112 29.9634i 0.530666 1.28114i −0.400416 0.916333i \(-0.631135\pi\)
0.931083 0.364808i \(-0.118865\pi\)
\(548\) 0 0
\(549\) −13.0786 4.75589i −0.558180 0.202976i
\(550\) 0 0
\(551\) 31.0704 31.0704i 1.32364 1.32364i
\(552\) 0 0
\(553\) 5.33499 + 5.33499i 0.226867 + 0.226867i
\(554\) 0 0
\(555\) 18.7048 19.5450i 0.793976 0.829640i
\(556\) 0 0
\(557\) −9.30090 3.85256i −0.394092 0.163238i 0.176832 0.984241i \(-0.443415\pi\)
−0.570924 + 0.821003i \(0.693415\pi\)
\(558\) 0 0
\(559\) 1.18458 0.0501025
\(560\) 0 0
\(561\) 6.63869 + 15.0816i 0.280286 + 0.636746i
\(562\) 0 0
\(563\) 1.24166 2.99763i 0.0523297 0.126335i −0.895553 0.444955i \(-0.853220\pi\)
0.947882 + 0.318620i \(0.103220\pi\)
\(564\) 0 0
\(565\) 29.8392 12.3598i 1.25535 0.519981i
\(566\) 0 0
\(567\) 18.8505 + 15.7987i 0.791646 + 0.663483i
\(568\) 0 0
\(569\) 28.3250 + 28.3250i 1.18745 + 1.18745i 0.977771 + 0.209674i \(0.0672404\pi\)
0.209674 + 0.977771i \(0.432760\pi\)
\(570\) 0 0
\(571\) −37.5476 + 15.5527i −1.57132 + 0.650862i −0.987008 0.160669i \(-0.948635\pi\)
−0.584310 + 0.811530i \(0.698635\pi\)
\(572\) 0 0
\(573\) −0.295907 13.4711i −0.0123617 0.562763i
\(574\) 0 0
\(575\) 1.59623 0.0665673
\(576\) 0 0
\(577\) 5.37886 0.223925 0.111962 0.993712i \(-0.464286\pi\)
0.111962 + 0.993712i \(0.464286\pi\)
\(578\) 0 0
\(579\) 0.237172 + 10.7972i 0.00985655 + 0.448717i
\(580\) 0 0
\(581\) −16.7008 + 6.91770i −0.692866 + 0.286994i
\(582\) 0 0
\(583\) −14.6066 14.6066i −0.604942 0.604942i
\(584\) 0 0
\(585\) −0.0900101 2.04785i −0.00372146 0.0846684i
\(586\) 0 0
\(587\) 1.77252 0.734203i 0.0731599 0.0303038i −0.345803 0.938307i \(-0.612394\pi\)
0.418963 + 0.908003i \(0.362394\pi\)
\(588\) 0 0
\(589\) −10.4031 + 25.1154i −0.428653 + 1.03486i
\(590\) 0 0
\(591\) −5.21743 11.8528i −0.214616 0.487560i
\(592\) 0 0
\(593\) 24.1664 0.992394 0.496197 0.868210i \(-0.334729\pi\)
0.496197 + 0.868210i \(0.334729\pi\)
\(594\) 0 0
\(595\) −36.8964 15.2830i −1.51260 0.626541i
\(596\) 0 0
\(597\) −21.0882 + 22.0355i −0.863084 + 0.901853i
\(598\) 0 0
\(599\) 11.6692 + 11.6692i 0.476789 + 0.476789i 0.904103 0.427314i \(-0.140540\pi\)
−0.427314 + 0.904103i \(0.640540\pi\)
\(600\) 0 0
\(601\) −8.80143 + 8.80143i −0.359018 + 0.359018i −0.863451 0.504433i \(-0.831702\pi\)
0.504433 + 0.863451i \(0.331702\pi\)
\(602\) 0 0
\(603\) 1.42455 3.91747i 0.0580121 0.159532i
\(604\) 0 0
\(605\) 7.81507 18.8672i 0.317728 0.767062i
\(606\) 0 0
\(607\) 31.9215i 1.29565i 0.761788 + 0.647826i \(0.224322\pi\)
−0.761788 + 0.647826i \(0.775678\pi\)
\(608\) 0 0
\(609\) −10.5432 23.9517i −0.427231 0.970573i
\(610\) 0 0
\(611\) −0.412600 0.170905i −0.0166920 0.00691406i
\(612\) 0 0
\(613\) −7.35458 17.7555i −0.297049 0.717139i −0.999983 0.00588391i \(-0.998127\pi\)
0.702934 0.711255i \(-0.251873\pi\)
\(614\) 0 0
\(615\) 4.07504 + 1.58401i 0.164322 + 0.0638736i
\(616\) 0 0
\(617\) −13.5933 + 13.5933i −0.547246 + 0.547246i −0.925643 0.378397i \(-0.876475\pi\)
0.378397 + 0.925643i \(0.376475\pi\)
\(618\) 0 0
\(619\) −3.75210 9.05838i −0.150810 0.364087i 0.830362 0.557224i \(-0.188134\pi\)
−0.981172 + 0.193137i \(0.938134\pi\)
\(620\) 0 0
\(621\) −8.97338 + 10.2412i −0.360089 + 0.410966i
\(622\) 0 0
\(623\) 16.0146i 0.641611i
\(624\) 0 0
\(625\) 27.6746i 1.10699i
\(626\) 0 0
\(627\) 0.466109 + 21.2195i 0.0186146 + 0.847425i
\(628\) 0 0
\(629\) −15.5725 37.5953i −0.620915 1.49902i
\(630\) 0 0
\(631\) −5.39207 + 5.39207i −0.214655 + 0.214655i −0.806242 0.591587i \(-0.798502\pi\)
0.591587 + 0.806242i \(0.298502\pi\)
\(632\) 0 0
\(633\) −0.793682 + 2.04183i −0.0315460 + 0.0811556i
\(634\) 0 0
\(635\) 7.02766 + 16.9663i 0.278884 + 0.673286i
\(636\) 0 0
\(637\) 0.124846 + 0.0517131i 0.00494659 + 0.00204895i
\(638\) 0 0
\(639\) −25.0015 + 27.3003i −0.989044 + 1.07999i
\(640\) 0 0
\(641\) 1.82203i 0.0719659i −0.999352 0.0359830i \(-0.988544\pi\)
0.999352 0.0359830i \(-0.0114562\pi\)
\(642\) 0 0
\(643\) −10.4095 + 25.1308i −0.410511 + 0.991062i 0.574489 + 0.818512i \(0.305201\pi\)
−0.985001 + 0.172550i \(0.944799\pi\)
\(644\) 0 0
\(645\) −11.6455 + 12.1686i −0.458542 + 0.479140i
\(646\) 0 0
\(647\) 27.3258 27.3258i 1.07429 1.07429i 0.0772785 0.997010i \(-0.475377\pi\)
0.997010 0.0772785i \(-0.0246231\pi\)
\(648\) 0 0
\(649\) −8.79684 8.79684i −0.345306 0.345306i
\(650\) 0 0
\(651\) 11.6970 + 11.1942i 0.458442 + 0.438735i
\(652\) 0 0
\(653\) 21.5061 + 8.90811i 0.841598 + 0.348601i 0.761483 0.648184i \(-0.224471\pi\)
0.0801145 + 0.996786i \(0.474471\pi\)
\(654\) 0 0
\(655\) −2.99371 −0.116974
\(656\) 0 0
\(657\) 8.54241 + 7.82309i 0.333271 + 0.305208i
\(658\) 0 0
\(659\) 17.2778 41.7123i 0.673048 1.62488i −0.103355 0.994644i \(-0.532958\pi\)
0.776403 0.630237i \(-0.217042\pi\)
\(660\) 0 0
\(661\) −44.7407 + 18.5322i −1.74021 + 0.720819i −0.741453 + 0.671005i \(0.765863\pi\)
−0.998759 + 0.0498140i \(0.984137\pi\)
\(662\) 0 0
\(663\) −2.87383 1.11709i −0.111610 0.0433841i
\(664\) 0 0
\(665\) −36.3736 36.3736i −1.41051 1.41051i
\(666\) 0 0
\(667\) 13.3849 5.54420i 0.518265 0.214672i
\(668\) 0 0
\(669\) 18.1903 0.399570i 0.703279 0.0154483i
\(670\) 0 0
\(671\) −7.15231 −0.276112
\(672\) 0 0
\(673\) −10.8139 −0.416844 −0.208422 0.978039i \(-0.566833\pi\)
−0.208422 + 0.978039i \(0.566833\pi\)
\(674\) 0 0
\(675\) 2.38062 + 2.08591i 0.0916302 + 0.0802866i
\(676\) 0 0
\(677\) −16.7436 + 6.93541i −0.643507 + 0.266549i −0.680480 0.732767i \(-0.738229\pi\)
0.0369727 + 0.999316i \(0.488229\pi\)
\(678\) 0 0
\(679\) −20.0305 20.0305i −0.768701 0.768701i
\(680\) 0 0
\(681\) −11.4021 + 29.3332i −0.436930 + 1.12405i
\(682\) 0 0
\(683\) −43.5660 + 18.0456i −1.66701 + 0.690497i −0.998580 0.0532798i \(-0.983032\pi\)
−0.668428 + 0.743777i \(0.733032\pi\)
\(684\) 0 0
\(685\) −9.67071 + 23.3472i −0.369499 + 0.892049i
\(686\) 0 0
\(687\) −24.3432 + 10.7155i −0.928751 + 0.408822i
\(688\) 0 0
\(689\) 3.86521 0.147253
\(690\) 0 0
\(691\) −5.29763 2.19435i −0.201531 0.0834771i 0.279635 0.960106i \(-0.409787\pi\)
−0.481166 + 0.876629i \(0.659787\pi\)
\(692\) 0 0
\(693\) 11.8797 + 4.31994i 0.451273 + 0.164101i
\(694\) 0 0
\(695\) −1.80488 1.80488i −0.0684630 0.0684630i
\(696\) 0 0
\(697\) 4.65020 4.65020i 0.176139 0.176139i
\(698\) 0 0
\(699\) 4.49625 + 4.30297i 0.170064 + 0.162753i
\(700\) 0 0
\(701\) −13.7184 + 33.1190i −0.518135 + 1.25089i 0.420912 + 0.907101i \(0.361710\pi\)
−0.939047 + 0.343787i \(0.888290\pi\)
\(702\) 0 0
\(703\) 52.4144i 1.97685i
\(704\) 0 0
\(705\) 5.81186 2.55829i 0.218887 0.0963508i
\(706\) 0 0
\(707\) −28.8861 11.9650i −1.08638 0.449991i
\(708\) 0 0
\(709\) 1.39804 + 3.37516i 0.0525044 + 0.126757i 0.947955 0.318403i \(-0.103147\pi\)
−0.895451 + 0.445160i \(0.853147\pi\)
\(710\) 0 0
\(711\) −8.27440 + 0.363688i −0.310314 + 0.0136394i
\(712\) 0 0
\(713\) −6.33792 + 6.33792i −0.237357 + 0.237357i
\(714\) 0 0
\(715\) −0.403158 0.973311i −0.0150773 0.0363998i
\(716\) 0 0
\(717\) −30.4952 + 0.669861i −1.13887 + 0.0250164i
\(718\) 0 0
\(719\) 25.3851i 0.946706i 0.880873 + 0.473353i \(0.156956\pi\)
−0.880873 + 0.473353i \(0.843044\pi\)
\(720\) 0 0
\(721\) 40.5121i 1.50875i
\(722\) 0 0
\(723\) 10.7124 0.235309i 0.398397 0.00875122i
\(724\) 0 0
\(725\) −1.28878 3.11138i −0.0478640 0.115554i
\(726\) 0 0
\(727\) 32.5294 32.5294i 1.20645 1.20645i 0.234280 0.972169i \(-0.424727\pi\)
0.972169 0.234280i \(-0.0752734\pi\)
\(728\) 0 0
\(729\) −26.7659 + 3.54764i −0.991330 + 0.131394i
\(730\) 0 0
\(731\) 9.69534 + 23.4066i 0.358595 + 0.865725i
\(732\) 0 0
\(733\) −26.3419 10.9112i −0.972961 0.403014i −0.161147 0.986930i \(-0.551519\pi\)
−0.811814 + 0.583917i \(0.801519\pi\)
\(734\) 0 0
\(735\) −1.75858 + 0.774099i −0.0648661 + 0.0285531i
\(736\) 0 0
\(737\) 2.14236i 0.0789147i
\(738\) 0 0
\(739\) 13.4201 32.3991i 0.493668 1.19182i −0.459172 0.888347i \(-0.651854\pi\)
0.952840 0.303473i \(-0.0981461\pi\)
\(740\) 0 0
\(741\) −2.86924 2.74589i −0.105404 0.100873i
\(742\) 0 0
\(743\) 5.42669 5.42669i 0.199086 0.199086i −0.600522 0.799608i \(-0.705041\pi\)
0.799608 + 0.600522i \(0.205041\pi\)
\(744\) 0 0
\(745\) 2.00839 + 2.00839i 0.0735817 + 0.0735817i
\(746\) 0 0
\(747\) 6.78161 18.6493i 0.248126 0.682341i
\(748\) 0 0
\(749\) 25.6771 + 10.6358i 0.938222 + 0.388624i
\(750\) 0 0
\(751\) −3.67683 −0.134170 −0.0670848 0.997747i \(-0.521370\pi\)
−0.0670848 + 0.997747i \(0.521370\pi\)
\(752\) 0 0
\(753\) −38.0629 + 16.7547i −1.38709 + 0.610576i
\(754\) 0 0
\(755\) 12.5723 30.3521i 0.457552 1.10463i
\(756\) 0 0
\(757\) 13.0816 5.41857i 0.475458 0.196941i −0.132068 0.991241i \(-0.542162\pi\)
0.607526 + 0.794299i \(0.292162\pi\)
\(758\) 0 0
\(759\) −2.53541 + 6.52262i −0.0920296 + 0.236756i
\(760\) 0 0
\(761\) −25.8277 25.8277i −0.936254 0.936254i 0.0618325 0.998087i \(-0.480306\pi\)
−0.998087 + 0.0618325i \(0.980306\pi\)
\(762\) 0 0
\(763\) −19.8317 + 8.21457i −0.717957 + 0.297387i
\(764\) 0 0
\(765\) 39.7276 18.5394i 1.43636 0.670294i
\(766\) 0 0
\(767\) 2.32783 0.0840532
\(768\) 0 0
\(769\) −22.8386 −0.823580 −0.411790 0.911279i \(-0.635096\pi\)
−0.411790 + 0.911279i \(0.635096\pi\)
\(770\) 0 0
\(771\) −20.5780 + 0.452017i −0.741097 + 0.0162790i
\(772\) 0 0
\(773\) −8.07057 + 3.34294i −0.290278 + 0.120237i −0.523071 0.852289i \(-0.675214\pi\)
0.232793 + 0.972526i \(0.425214\pi\)
\(774\) 0 0
\(775\) 1.47328 + 1.47328i 0.0529219 + 0.0529219i
\(776\) 0 0
\(777\) −29.0957 11.3098i −1.04380 0.405738i
\(778\) 0 0
\(779\) 7.82591 3.24160i 0.280392 0.116142i
\(780\) 0 0
\(781\) −7.28079 + 17.5774i −0.260527 + 0.628968i
\(782\) 0 0
\(783\) 27.2073 + 9.22236i 0.972311 + 0.329580i
\(784\) 0 0
\(785\) 21.2311 0.757769
\(786\) 0 0
\(787\) 23.1792 + 9.60114i 0.826249 + 0.342244i 0.755417 0.655245i \(-0.227434\pi\)
0.0708325 + 0.997488i \(0.477434\pi\)
\(788\) 0 0
\(789\) 20.5620 + 19.6781i 0.732026 + 0.700557i
\(790\) 0 0
\(791\) −26.3525 26.3525i −0.936989 0.936989i
\(792\) 0 0
\(793\) 0.946326 0.946326i 0.0336050 0.0336050i
\(794\) 0 0
\(795\) −37.9985 + 39.7054i −1.34767 + 1.40820i
\(796\) 0 0
\(797\) 9.64229 23.2785i 0.341547 0.824568i −0.656012 0.754750i \(-0.727758\pi\)
0.997560 0.0698182i \(-0.0222419\pi\)
\(798\) 0 0
\(799\) 9.55151i 0.337908i
\(800\) 0 0
\(801\) 12.9649 + 11.8732i 0.458093 + 0.419519i
\(802\) 0 0
\(803\) 5.50005 + 2.27819i 0.194092 + 0.0803957i
\(804\) 0 0
\(805\) −6.49052 15.6695i −0.228761 0.552277i
\(806\) 0 0
\(807\) 9.33801 24.0230i 0.328713 0.845651i
\(808\) 0 0
\(809\) −24.9240 + 24.9240i −0.876282 + 0.876282i −0.993148 0.116866i \(-0.962715\pi\)
0.116866 + 0.993148i \(0.462715\pi\)
\(810\) 0 0
\(811\) 6.09225 + 14.7080i 0.213928 + 0.516467i 0.994020 0.109197i \(-0.0348280\pi\)
−0.780092 + 0.625664i \(0.784828\pi\)
\(812\) 0 0
\(813\) 0.269036 + 12.2478i 0.00943550 + 0.429549i
\(814\) 0 0
\(815\) 7.12017i 0.249409i
\(816\) 0 0
\(817\) 32.6330i 1.14168i
\(818\) 0 0
\(819\) −2.14339 + 1.00024i −0.0748960 + 0.0349512i
\(820\) 0 0
\(821\) 9.49765 + 22.9294i 0.331470 + 0.800240i 0.998476 + 0.0551876i \(0.0175757\pi\)
−0.667006 + 0.745053i \(0.732424\pi\)
\(822\) 0 0
\(823\) −27.2557 + 27.2557i −0.950072 + 0.950072i −0.998812 0.0487391i \(-0.984480\pi\)
0.0487391 + 0.998812i \(0.484480\pi\)
\(824\) 0 0
\(825\) 1.51621 + 0.589369i 0.0527878 + 0.0205192i
\(826\) 0 0
\(827\) 20.6867 + 49.9420i 0.719345 + 1.73665i 0.675207 + 0.737628i \(0.264054\pi\)
0.0441379 + 0.999025i \(0.485946\pi\)
\(828\) 0 0
\(829\) 0.359915 + 0.149082i 0.0125004 + 0.00517782i 0.388925 0.921269i \(-0.372847\pi\)
−0.376424 + 0.926447i \(0.622847\pi\)
\(830\) 0 0
\(831\) −0.856489 1.94575i −0.0297113 0.0674973i
\(832\) 0 0
\(833\) 2.89014i 0.100137i
\(834\) 0 0
\(835\) 2.39473 5.78138i 0.0828730 0.200073i
\(836\) 0 0
\(837\) −17.7347 + 1.17019i −0.612999 + 0.0404476i
\(838\) 0 0
\(839\) −27.7697 + 27.7697i −0.958717 + 0.958717i −0.999181 0.0404644i \(-0.987116\pi\)
0.0404644 + 0.999181i \(0.487116\pi\)
\(840\) 0 0
\(841\) −1.10756 1.10756i −0.0381918 0.0381918i
\(842\) 0 0
\(843\) 20.3590 21.2735i 0.701201 0.732698i
\(844\) 0 0
\(845\) −28.2629 11.7069i −0.972275 0.402729i
\(846\) 0 0
\(847\) −23.5645 −0.809686
\(848\) 0 0
\(849\) −9.67982 21.9904i −0.332211 0.754707i
\(850\) 0 0
\(851\) 6.61346 15.9663i 0.226706 0.547317i
\(852\) 0 0
\(853\) −20.4516 + 8.47133i −0.700250 + 0.290053i −0.704263 0.709939i \(-0.748722\pi\)
0.00401354 + 0.999992i \(0.498722\pi\)
\(854\) 0 0
\(855\) 56.4145 2.47960i 1.92933 0.0848007i
\(856\) 0 0
\(857\) −13.1734 13.1734i −0.449996 0.449996i 0.445357 0.895353i \(-0.353077\pi\)
−0.895353 + 0.445357i \(0.853077\pi\)
\(858\) 0 0
\(859\) −7.55122 + 3.12782i −0.257644 + 0.106720i −0.507767 0.861494i \(-0.669529\pi\)
0.250123 + 0.968214i \(0.419529\pi\)
\(860\) 0 0
\(861\) −0.110790 5.04370i −0.00377572 0.171889i
\(862\) 0 0
\(863\) −19.3590 −0.658989 −0.329494 0.944158i \(-0.606878\pi\)
−0.329494 + 0.944158i \(0.606878\pi\)
\(864\) 0 0
\(865\) 24.5889 0.836049
\(866\) 0 0
\(867\) −0.801546 36.4902i −0.0272219 1.23927i
\(868\) 0 0
\(869\) −3.93268 + 1.62897i −0.133407 + 0.0552591i
\(870\) 0 0
\(871\) 0.283457 + 0.283457i 0.00960456 + 0.00960456i
\(872\) 0 0
\(873\) 31.0667 1.36549i 1.05145 0.0462147i
\(874\) 0 0
\(875\) 26.2559 10.8755i 0.887611 0.367661i
\(876\) 0 0
\(877\) 6.24223 15.0701i 0.210785 0.508880i −0.782759 0.622325i \(-0.786188\pi\)
0.993544 + 0.113444i \(0.0361884\pi\)
\(878\) 0 0
\(879\) −18.6871 42.4528i −0.630299 1.43190i
\(880\) 0 0
\(881\) 36.6844 1.23593 0.617964 0.786207i \(-0.287958\pi\)
0.617964 + 0.786207i \(0.287958\pi\)
\(882\) 0 0
\(883\) 11.4178 + 4.72942i 0.384240 + 0.159158i 0.566438 0.824104i \(-0.308321\pi\)
−0.182198 + 0.983262i \(0.558321\pi\)
\(884\) 0 0
\(885\) −22.8847 + 23.9127i −0.769262 + 0.803816i
\(886\) 0 0
\(887\) −34.3135 34.3135i −1.15213 1.15213i −0.986124 0.166009i \(-0.946912\pi\)
−0.166009 0.986124i \(-0.553088\pi\)
\(888\) 0 0
\(889\) 14.9838 14.9838i 0.502540 0.502540i
\(890\) 0 0
\(891\) −12.3049 + 6.41466i −0.412230 + 0.214899i
\(892\) 0 0
\(893\) 4.70809 11.3663i 0.157550 0.380360i
\(894\) 0 0
\(895\) 0.0329316i 0.00110078i
\(896\) 0 0
\(897\) −0.527550 1.19847i −0.0176144 0.0400159i
\(898\) 0 0
\(899\) 17.4711 + 7.23678i 0.582695 + 0.241360i
\(900\) 0 0
\(901\) 31.6352 + 76.3741i 1.05392 + 2.54439i
\(902\) 0 0
\(903\) 18.1149 + 7.04144i 0.602825 + 0.234324i
\(904\) 0 0
\(905\) −17.1330 + 17.1330i −0.569519 + 0.569519i
\(906\) 0 0
\(907\) 5.32554 + 12.8570i 0.176832 + 0.426910i 0.987299 0.158875i \(-0.0507866\pi\)
−0.810467 + 0.585784i \(0.800787\pi\)
\(908\) 0 0
\(909\) 31.1027 14.5145i 1.03161 0.481415i
\(910\) 0 0
\(911\) 10.5385i 0.349157i −0.984643 0.174579i \(-0.944144\pi\)
0.984643 0.174579i \(-0.0558563\pi\)
\(912\) 0 0
\(913\) 10.1988i 0.337530i
\(914\) 0 0
\(915\) 0.417891 + 19.0244i 0.0138151 + 0.628927i
\(916\) 0 0
\(917\) 1.32195 + 3.19147i 0.0436546 + 0.105392i
\(918\) 0 0
\(919\) −11.9849 + 11.9849i −0.395346 + 0.395346i −0.876588 0.481242i \(-0.840186\pi\)
0.481242 + 0.876588i \(0.340186\pi\)
\(920\) 0 0
\(921\) −0.451262 + 1.16092i −0.0148696 + 0.0382536i
\(922\) 0 0
\(923\) −1.36235 3.28900i −0.0448423 0.108259i
\(924\) 0 0
\(925\) −3.71144 1.53733i −0.122032 0.0505471i
\(926\) 0 0
\(927\) −32.7974 30.0357i −1.07721 0.986502i
\(928\) 0 0
\(929\) 0.514845i 0.0168915i −0.999964 0.00844575i \(-0.997312\pi\)
0.999964 0.00844575i \(-0.00268840\pi\)
\(930\) 0 0
\(931\) −1.42460 + 3.43928i −0.0466892 + 0.112718i
\(932\) 0 0
\(933\) 6.10775 6.38210i 0.199959 0.208941i
\(934\) 0 0
\(935\) 15.9323 15.9323i 0.521043 0.521043i
\(936\) 0 0
\(937\) −15.6989 15.6989i −0.512862 0.512862i 0.402540 0.915402i \(-0.368127\pi\)
−0.915402 + 0.402540i \(0.868127\pi\)
\(938\) 0 0
\(939\) −41.0976 39.3309i −1.34117 1.28351i
\(940\) 0 0
\(941\) 8.26022 + 3.42149i 0.269275 + 0.111538i 0.513236 0.858248i \(-0.328447\pi\)
−0.243960 + 0.969785i \(0.578447\pi\)
\(942\) 0 0
\(943\) 2.79291 0.0909497
\(944\) 0 0
\(945\) 10.7965 31.8512i 0.351210 1.03612i
\(946\) 0 0
\(947\) 1.67384 4.04101i 0.0543926 0.131315i −0.894347 0.447373i \(-0.852359\pi\)
0.948740 + 0.316058i \(0.102359\pi\)
\(948\) 0 0
\(949\) −1.02914 + 0.426286i −0.0334074 + 0.0138378i
\(950\) 0 0
\(951\) −5.99261 2.32939i −0.194324 0.0755357i
\(952\) 0 0
\(953\) −32.0579 32.0579i −1.03846 1.03846i −0.999230 0.0392279i \(-0.987510\pi\)
−0.0392279 0.999230i \(-0.512490\pi\)
\(954\) 0 0
\(955\) −17.0220 + 7.05074i −0.550819 + 0.228157i
\(956\) 0 0
\(957\) 14.7610 0.324242i 0.477156 0.0104813i
\(958\) 0 0
\(959\) 29.1598 0.941619
\(960\) 0 0
\(961\) 19.3005 0.622596
\(962\) 0 0
\(963\) −27.6475 + 12.9021i −0.890927 + 0.415763i
\(964\) 0 0
\(965\) 13.6433 5.65124i 0.439193 0.181920i
\(966\) 0 0
\(967\) 37.4258 + 37.4258i 1.20353 + 1.20353i 0.973085 + 0.230448i \(0.0740190\pi\)
0.230448 + 0.973085i \(0.425981\pi\)
\(968\) 0 0
\(969\) 30.7736 79.1684i 0.988591 2.54326i
\(970\) 0 0
\(971\) −33.0754 + 13.7003i −1.06144 + 0.439663i −0.843962 0.536402i \(-0.819783\pi\)
−0.217478 + 0.976065i \(0.569783\pi\)
\(972\) 0 0
\(973\) −1.12712 + 2.72110i −0.0361337 + 0.0872344i
\(974\) 0 0
\(975\) −0.278591 + 0.122632i −0.00892206 + 0.00392735i
\(976\) 0 0
\(977\) 39.7489 1.27168 0.635839 0.771821i \(-0.280654\pi\)
0.635839 + 0.771821i \(0.280654\pi\)
\(978\) 0 0
\(979\) 8.34750 + 3.45765i 0.266787 + 0.110507i
\(980\) 0 0
\(981\) 8.05297 22.1455i 0.257112 0.707050i
\(982\) 0 0
\(983\) 29.8565 + 29.8565i 0.952276 + 0.952276i 0.998912 0.0466360i \(-0.0148501\pi\)
−0.0466360 + 0.998912i \(0.514850\pi\)
\(984\) 0 0
\(985\) −12.5214 + 12.5214i −0.398965 + 0.398965i
\(986\) 0 0
\(987\) −5.29366 5.06610i −0.168499 0.161256i
\(988\) 0 0
\(989\) −4.11750 + 9.94053i −0.130929 + 0.316091i
\(990\) 0 0
\(991\) 56.2662i 1.78736i 0.448710 + 0.893678i \(0.351884\pi\)
−0.448710 + 0.893678i \(0.648116\pi\)
\(992\) 0 0
\(993\) −13.4667 + 5.92782i −0.427352 + 0.188114i
\(994\) 0 0
\(995\) 38.5309 + 15.9600i 1.22151 + 0.505966i
\(996\) 0 0
\(997\) −13.8797 33.5085i −0.439574 1.06122i −0.976096 0.217339i \(-0.930262\pi\)
0.536523 0.843886i \(-0.319738\pi\)
\(998\) 0 0
\(999\) 30.7277 15.1699i 0.972181 0.479956i
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 768.2.o.a.95.7 56
3.2 odd 2 inner 768.2.o.a.95.12 56
4.3 odd 2 768.2.o.b.95.8 56
8.3 odd 2 96.2.o.a.35.2 yes 56
8.5 even 2 384.2.o.a.47.8 56
12.11 even 2 768.2.o.b.95.3 56
24.5 odd 2 384.2.o.a.47.3 56
24.11 even 2 96.2.o.a.35.13 yes 56
32.5 even 8 96.2.o.a.11.13 yes 56
32.11 odd 8 inner 768.2.o.a.671.12 56
32.21 even 8 768.2.o.b.671.3 56
32.27 odd 8 384.2.o.a.335.3 56
96.5 odd 8 96.2.o.a.11.2 56
96.11 even 8 inner 768.2.o.a.671.7 56
96.53 odd 8 768.2.o.b.671.8 56
96.59 even 8 384.2.o.a.335.8 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
96.2.o.a.11.2 56 96.5 odd 8
96.2.o.a.11.13 yes 56 32.5 even 8
96.2.o.a.35.2 yes 56 8.3 odd 2
96.2.o.a.35.13 yes 56 24.11 even 2
384.2.o.a.47.3 56 24.5 odd 2
384.2.o.a.47.8 56 8.5 even 2
384.2.o.a.335.3 56 32.27 odd 8
384.2.o.a.335.8 56 96.59 even 8
768.2.o.a.95.7 56 1.1 even 1 trivial
768.2.o.a.95.12 56 3.2 odd 2 inner
768.2.o.a.671.7 56 96.11 even 8 inner
768.2.o.a.671.12 56 32.11 odd 8 inner
768.2.o.b.95.3 56 12.11 even 2
768.2.o.b.95.8 56 4.3 odd 2
768.2.o.b.671.3 56 32.21 even 8
768.2.o.b.671.8 56 96.53 odd 8