Properties

Label 867.2.e.e.616.1
Level $867$
Weight $2$
Character 867.616
Analytic conductor $6.923$
Analytic rank $0$
Dimension $4$
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] = [867,2,Mod(616,867)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(867, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("867.616");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 867 = 3 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 867.e (of order \(4\), 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: \(6.92302985525\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) 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 51)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 616.1
Root \(0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 867.616
Dual form 867.2.e.e.829.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.707107 + 0.707107i) q^{3} +2.00000 q^{4} +(2.12132 - 2.12132i) q^{5} +(-2.82843 - 2.82843i) q^{7} -1.00000i q^{9} +O(q^{10})\) \(q+(-0.707107 + 0.707107i) q^{3} +2.00000 q^{4} +(2.12132 - 2.12132i) q^{5} +(-2.82843 - 2.82843i) q^{7} -1.00000i q^{9} +(2.12132 + 2.12132i) q^{11} +(-1.41421 + 1.41421i) q^{12} +1.00000 q^{13} +3.00000i q^{15} +4.00000 q^{16} -1.00000i q^{19} +(4.24264 - 4.24264i) q^{20} +4.00000 q^{21} +(-6.36396 - 6.36396i) q^{23} -4.00000i q^{25} +(0.707107 + 0.707107i) q^{27} +(-5.65685 - 5.65685i) q^{28} +(4.24264 - 4.24264i) q^{29} +(-1.41421 + 1.41421i) q^{31} -3.00000 q^{33} -12.0000 q^{35} -2.00000i q^{36} +(2.82843 - 2.82843i) q^{37} +(-0.707107 + 0.707107i) q^{39} +(-2.12132 - 2.12132i) q^{41} +7.00000i q^{43} +(4.24264 + 4.24264i) q^{44} +(-2.12132 - 2.12132i) q^{45} +6.00000 q^{47} +(-2.82843 + 2.82843i) q^{48} +9.00000i q^{49} +2.00000 q^{52} -6.00000i q^{53} +9.00000 q^{55} +(0.707107 + 0.707107i) q^{57} -6.00000i q^{59} +6.00000i q^{60} +(5.65685 + 5.65685i) q^{61} +(-2.82843 + 2.82843i) q^{63} +8.00000 q^{64} +(2.12132 - 2.12132i) q^{65} -4.00000 q^{67} +9.00000 q^{69} +(-8.48528 + 8.48528i) q^{71} +(1.41421 - 1.41421i) q^{73} +(2.82843 + 2.82843i) q^{75} -2.00000i q^{76} -12.0000i q^{77} +(7.07107 + 7.07107i) q^{79} +(8.48528 - 8.48528i) q^{80} -1.00000 q^{81} -6.00000i q^{83} +8.00000 q^{84} +6.00000i q^{87} +(-2.82843 - 2.82843i) q^{91} +(-12.7279 - 12.7279i) q^{92} -2.00000i q^{93} +(-2.12132 - 2.12132i) q^{95} +(-11.3137 + 11.3137i) q^{97} +(2.12132 - 2.12132i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{4} + 4 q^{13} + 16 q^{16} + 16 q^{21} - 12 q^{33} - 48 q^{35} + 24 q^{47} + 8 q^{52} + 36 q^{55} + 32 q^{64} - 16 q^{67} + 36 q^{69} - 4 q^{81} + 32 q^{84}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/867\mathbb{Z}\right)^\times\).

\(n\) \(290\) \(292\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{4}\right)\)

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 1.00000 \(0\)
−1.00000 \(\pi\)
\(3\) −0.707107 + 0.707107i −0.408248 + 0.408248i
\(4\) 2.00000 1.00000
\(5\) 2.12132 2.12132i 0.948683 0.948683i −0.0500628 0.998746i \(-0.515942\pi\)
0.998746 + 0.0500628i \(0.0159421\pi\)
\(6\) 0 0
\(7\) −2.82843 2.82843i −1.06904 1.06904i −0.997433 0.0716124i \(-0.977186\pi\)
−0.0716124 0.997433i \(-0.522814\pi\)
\(8\) 0 0
\(9\) 1.00000i 0.333333i
\(10\) 0 0
\(11\) 2.12132 + 2.12132i 0.639602 + 0.639602i 0.950457 0.310855i \(-0.100615\pi\)
−0.310855 + 0.950457i \(0.600615\pi\)
\(12\) −1.41421 + 1.41421i −0.408248 + 0.408248i
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 0 0
\(15\) 3.00000i 0.774597i
\(16\) 4.00000 1.00000
\(17\) 0 0
\(18\) 0 0
\(19\) 1.00000i 0.229416i −0.993399 0.114708i \(-0.963407\pi\)
0.993399 0.114708i \(-0.0365932\pi\)
\(20\) 4.24264 4.24264i 0.948683 0.948683i
\(21\) 4.00000 0.872872
\(22\) 0 0
\(23\) −6.36396 6.36396i −1.32698 1.32698i −0.907994 0.418984i \(-0.862386\pi\)
−0.418984 0.907994i \(-0.637614\pi\)
\(24\) 0 0
\(25\) 4.00000i 0.800000i
\(26\) 0 0
\(27\) 0.707107 + 0.707107i 0.136083 + 0.136083i
\(28\) −5.65685 5.65685i −1.06904 1.06904i
\(29\) 4.24264 4.24264i 0.787839 0.787839i −0.193301 0.981140i \(-0.561919\pi\)
0.981140 + 0.193301i \(0.0619194\pi\)
\(30\) 0 0
\(31\) −1.41421 + 1.41421i −0.254000 + 0.254000i −0.822608 0.568608i \(-0.807482\pi\)
0.568608 + 0.822608i \(0.307482\pi\)
\(32\) 0 0
\(33\) −3.00000 −0.522233
\(34\) 0 0
\(35\) −12.0000 −2.02837
\(36\) 2.00000i 0.333333i
\(37\) 2.82843 2.82843i 0.464991 0.464991i −0.435297 0.900287i \(-0.643356\pi\)
0.900287 + 0.435297i \(0.143356\pi\)
\(38\) 0 0
\(39\) −0.707107 + 0.707107i −0.113228 + 0.113228i
\(40\) 0 0
\(41\) −2.12132 2.12132i −0.331295 0.331295i 0.521783 0.853078i \(-0.325267\pi\)
−0.853078 + 0.521783i \(0.825267\pi\)
\(42\) 0 0
\(43\) 7.00000i 1.06749i 0.845645 + 0.533745i \(0.179216\pi\)
−0.845645 + 0.533745i \(0.820784\pi\)
\(44\) 4.24264 + 4.24264i 0.639602 + 0.639602i
\(45\) −2.12132 2.12132i −0.316228 0.316228i
\(46\) 0 0
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) −2.82843 + 2.82843i −0.408248 + 0.408248i
\(49\) 9.00000i 1.28571i
\(50\) 0 0
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) 6.00000i 0.824163i −0.911147 0.412082i \(-0.864802\pi\)
0.911147 0.412082i \(-0.135198\pi\)
\(54\) 0 0
\(55\) 9.00000 1.21356
\(56\) 0 0
\(57\) 0.707107 + 0.707107i 0.0936586 + 0.0936586i
\(58\) 0 0
\(59\) 6.00000i 0.781133i −0.920575 0.390567i \(-0.872279\pi\)
0.920575 0.390567i \(-0.127721\pi\)
\(60\) 6.00000i 0.774597i
\(61\) 5.65685 + 5.65685i 0.724286 + 0.724286i 0.969475 0.245189i \(-0.0788501\pi\)
−0.245189 + 0.969475i \(0.578850\pi\)
\(62\) 0 0
\(63\) −2.82843 + 2.82843i −0.356348 + 0.356348i
\(64\) 8.00000 1.00000
\(65\) 2.12132 2.12132i 0.263117 0.263117i
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) 9.00000 1.08347
\(70\) 0 0
\(71\) −8.48528 + 8.48528i −1.00702 + 1.00702i −0.00704243 + 0.999975i \(0.502242\pi\)
−0.999975 + 0.00704243i \(0.997758\pi\)
\(72\) 0 0
\(73\) 1.41421 1.41421i 0.165521 0.165521i −0.619486 0.785007i \(-0.712659\pi\)
0.785007 + 0.619486i \(0.212659\pi\)
\(74\) 0 0
\(75\) 2.82843 + 2.82843i 0.326599 + 0.326599i
\(76\) 2.00000i 0.229416i
\(77\) 12.0000i 1.36753i
\(78\) 0 0
\(79\) 7.07107 + 7.07107i 0.795557 + 0.795557i 0.982391 0.186834i \(-0.0598227\pi\)
−0.186834 + 0.982391i \(0.559823\pi\)
\(80\) 8.48528 8.48528i 0.948683 0.948683i
\(81\) −1.00000 −0.111111
\(82\) 0 0
\(83\) 6.00000i 0.658586i −0.944228 0.329293i \(-0.893190\pi\)
0.944228 0.329293i \(-0.106810\pi\)
\(84\) 8.00000 0.872872
\(85\) 0 0
\(86\) 0 0
\(87\) 6.00000i 0.643268i
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −2.82843 2.82843i −0.296500 0.296500i
\(92\) −12.7279 12.7279i −1.32698 1.32698i
\(93\) 2.00000i 0.207390i
\(94\) 0 0
\(95\) −2.12132 2.12132i −0.217643 0.217643i
\(96\) 0 0
\(97\) −11.3137 + 11.3137i −1.14873 + 1.14873i −0.161931 + 0.986802i \(0.551772\pi\)
−0.986802 + 0.161931i \(0.948228\pi\)
\(98\) 0 0
\(99\) 2.12132 2.12132i 0.213201 0.213201i
\(100\) 8.00000i 0.800000i
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 5.00000 0.492665 0.246332 0.969185i \(-0.420775\pi\)
0.246332 + 0.969185i \(0.420775\pi\)
\(104\) 0 0
\(105\) 8.48528 8.48528i 0.828079 0.828079i
\(106\) 0 0
\(107\) 6.36396 6.36396i 0.615227 0.615227i −0.329076 0.944303i \(-0.606737\pi\)
0.944303 + 0.329076i \(0.106737\pi\)
\(108\) 1.41421 + 1.41421i 0.136083 + 0.136083i
\(109\) 14.1421 + 14.1421i 1.35457 + 1.35457i 0.880471 + 0.474100i \(0.157226\pi\)
0.474100 + 0.880471i \(0.342774\pi\)
\(110\) 0 0
\(111\) 4.00000i 0.379663i
\(112\) −11.3137 11.3137i −1.06904 1.06904i
\(113\) 6.36396 + 6.36396i 0.598671 + 0.598671i 0.939959 0.341288i \(-0.110863\pi\)
−0.341288 + 0.939959i \(0.610863\pi\)
\(114\) 0 0
\(115\) −27.0000 −2.51776
\(116\) 8.48528 8.48528i 0.787839 0.787839i
\(117\) 1.00000i 0.0924500i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 2.00000i 0.181818i
\(122\) 0 0
\(123\) 3.00000 0.270501
\(124\) −2.82843 + 2.82843i −0.254000 + 0.254000i
\(125\) 2.12132 + 2.12132i 0.189737 + 0.189737i
\(126\) 0 0
\(127\) 13.0000i 1.15356i 0.816898 + 0.576782i \(0.195692\pi\)
−0.816898 + 0.576782i \(0.804308\pi\)
\(128\) 0 0
\(129\) −4.94975 4.94975i −0.435801 0.435801i
\(130\) 0 0
\(131\) 2.12132 2.12132i 0.185341 0.185341i −0.608338 0.793678i \(-0.708163\pi\)
0.793678 + 0.608338i \(0.208163\pi\)
\(132\) −6.00000 −0.522233
\(133\) −2.82843 + 2.82843i −0.245256 + 0.245256i
\(134\) 0 0
\(135\) 3.00000 0.258199
\(136\) 0 0
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) −1.41421 + 1.41421i −0.119952 + 0.119952i −0.764535 0.644583i \(-0.777031\pi\)
0.644583 + 0.764535i \(0.277031\pi\)
\(140\) −24.0000 −2.02837
\(141\) −4.24264 + 4.24264i −0.357295 + 0.357295i
\(142\) 0 0
\(143\) 2.12132 + 2.12132i 0.177394 + 0.177394i
\(144\) 4.00000i 0.333333i
\(145\) 18.0000i 1.49482i
\(146\) 0 0
\(147\) −6.36396 6.36396i −0.524891 0.524891i
\(148\) 5.65685 5.65685i 0.464991 0.464991i
\(149\) 18.0000 1.47462 0.737309 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) 0 0
\(151\) 8.00000i 0.651031i 0.945537 + 0.325515i \(0.105538\pi\)
−0.945537 + 0.325515i \(0.894462\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 6.00000i 0.481932i
\(156\) −1.41421 + 1.41421i −0.113228 + 0.113228i
\(157\) −11.0000 −0.877896 −0.438948 0.898513i \(-0.644649\pi\)
−0.438948 + 0.898513i \(0.644649\pi\)
\(158\) 0 0
\(159\) 4.24264 + 4.24264i 0.336463 + 0.336463i
\(160\) 0 0
\(161\) 36.0000i 2.83720i
\(162\) 0 0
\(163\) 1.41421 + 1.41421i 0.110770 + 0.110770i 0.760319 0.649550i \(-0.225042\pi\)
−0.649550 + 0.760319i \(0.725042\pi\)
\(164\) −4.24264 4.24264i −0.331295 0.331295i
\(165\) −6.36396 + 6.36396i −0.495434 + 0.495434i
\(166\) 0 0
\(167\) −14.8492 + 14.8492i −1.14907 + 1.14907i −0.162333 + 0.986736i \(0.551902\pi\)
−0.986736 + 0.162333i \(0.948098\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 14.0000i 1.06749i
\(173\) −10.6066 + 10.6066i −0.806405 + 0.806405i −0.984088 0.177683i \(-0.943140\pi\)
0.177683 + 0.984088i \(0.443140\pi\)
\(174\) 0 0
\(175\) −11.3137 + 11.3137i −0.855236 + 0.855236i
\(176\) 8.48528 + 8.48528i 0.639602 + 0.639602i
\(177\) 4.24264 + 4.24264i 0.318896 + 0.318896i
\(178\) 0 0
\(179\) 6.00000i 0.448461i 0.974536 + 0.224231i \(0.0719869\pi\)
−0.974536 + 0.224231i \(0.928013\pi\)
\(180\) −4.24264 4.24264i −0.316228 0.316228i
\(181\) −9.89949 9.89949i −0.735824 0.735824i 0.235943 0.971767i \(-0.424182\pi\)
−0.971767 + 0.235943i \(0.924182\pi\)
\(182\) 0 0
\(183\) −8.00000 −0.591377
\(184\) 0 0
\(185\) 12.0000i 0.882258i
\(186\) 0 0
\(187\) 0 0
\(188\) 12.0000 0.875190
\(189\) 4.00000i 0.290957i
\(190\) 0 0
\(191\) −18.0000 −1.30243 −0.651217 0.758891i \(-0.725741\pi\)
−0.651217 + 0.758891i \(0.725741\pi\)
\(192\) −5.65685 + 5.65685i −0.408248 + 0.408248i
\(193\) 15.5563 + 15.5563i 1.11977 + 1.11977i 0.991775 + 0.127996i \(0.0408544\pi\)
0.127996 + 0.991775i \(0.459146\pi\)
\(194\) 0 0
\(195\) 3.00000i 0.214834i
\(196\) 18.0000i 1.28571i
\(197\) 2.12132 + 2.12132i 0.151138 + 0.151138i 0.778626 0.627488i \(-0.215917\pi\)
−0.627488 + 0.778626i \(0.715917\pi\)
\(198\) 0 0
\(199\) −11.3137 + 11.3137i −0.802008 + 0.802008i −0.983409 0.181402i \(-0.941937\pi\)
0.181402 + 0.983409i \(0.441937\pi\)
\(200\) 0 0
\(201\) 2.82843 2.82843i 0.199502 0.199502i
\(202\) 0 0
\(203\) −24.0000 −1.68447
\(204\) 0 0
\(205\) −9.00000 −0.628587
\(206\) 0 0
\(207\) −6.36396 + 6.36396i −0.442326 + 0.442326i
\(208\) 4.00000 0.277350
\(209\) 2.12132 2.12132i 0.146735 0.146735i
\(210\) 0 0
\(211\) 1.41421 + 1.41421i 0.0973585 + 0.0973585i 0.754108 0.656750i \(-0.228069\pi\)
−0.656750 + 0.754108i \(0.728069\pi\)
\(212\) 12.0000i 0.824163i
\(213\) 12.0000i 0.822226i
\(214\) 0 0
\(215\) 14.8492 + 14.8492i 1.01271 + 1.01271i
\(216\) 0 0
\(217\) 8.00000 0.543075
\(218\) 0 0
\(219\) 2.00000i 0.135147i
\(220\) 18.0000 1.21356
\(221\) 0 0
\(222\) 0 0
\(223\) 1.00000i 0.0669650i −0.999439 0.0334825i \(-0.989340\pi\)
0.999439 0.0334825i \(-0.0106598\pi\)
\(224\) 0 0
\(225\) −4.00000 −0.266667
\(226\) 0 0
\(227\) −2.12132 2.12132i −0.140797 0.140797i 0.633195 0.773992i \(-0.281743\pi\)
−0.773992 + 0.633195i \(0.781743\pi\)
\(228\) 1.41421 + 1.41421i 0.0936586 + 0.0936586i
\(229\) 14.0000i 0.925146i −0.886581 0.462573i \(-0.846926\pi\)
0.886581 0.462573i \(-0.153074\pi\)
\(230\) 0 0
\(231\) 8.48528 + 8.48528i 0.558291 + 0.558291i
\(232\) 0 0
\(233\) 14.8492 14.8492i 0.972806 0.972806i −0.0268337 0.999640i \(-0.508542\pi\)
0.999640 + 0.0268337i \(0.00854246\pi\)
\(234\) 0 0
\(235\) 12.7279 12.7279i 0.830278 0.830278i
\(236\) 12.0000i 0.781133i
\(237\) −10.0000 −0.649570
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 12.0000i 0.774597i
\(241\) −5.65685 + 5.65685i −0.364390 + 0.364390i −0.865426 0.501036i \(-0.832952\pi\)
0.501036 + 0.865426i \(0.332952\pi\)
\(242\) 0 0
\(243\) 0.707107 0.707107i 0.0453609 0.0453609i
\(244\) 11.3137 + 11.3137i 0.724286 + 0.724286i
\(245\) 19.0919 + 19.0919i 1.21974 + 1.21974i
\(246\) 0 0
\(247\) 1.00000i 0.0636285i
\(248\) 0 0
\(249\) 4.24264 + 4.24264i 0.268866 + 0.268866i
\(250\) 0 0
\(251\) −24.0000 −1.51487 −0.757433 0.652913i \(-0.773547\pi\)
−0.757433 + 0.652913i \(0.773547\pi\)
\(252\) −5.65685 + 5.65685i −0.356348 + 0.356348i
\(253\) 27.0000i 1.69748i
\(254\) 0 0
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 12.0000i 0.748539i 0.927320 + 0.374270i \(0.122107\pi\)
−0.927320 + 0.374270i \(0.877893\pi\)
\(258\) 0 0
\(259\) −16.0000 −0.994192
\(260\) 4.24264 4.24264i 0.263117 0.263117i
\(261\) −4.24264 4.24264i −0.262613 0.262613i
\(262\) 0 0
\(263\) 12.0000i 0.739952i −0.929041 0.369976i \(-0.879366\pi\)
0.929041 0.369976i \(-0.120634\pi\)
\(264\) 0 0
\(265\) −12.7279 12.7279i −0.781870 0.781870i
\(266\) 0 0
\(267\) 0 0
\(268\) −8.00000 −0.488678
\(269\) 10.6066 10.6066i 0.646696 0.646696i −0.305497 0.952193i \(-0.598823\pi\)
0.952193 + 0.305497i \(0.0988226\pi\)
\(270\) 0 0
\(271\) 11.0000 0.668202 0.334101 0.942537i \(-0.391567\pi\)
0.334101 + 0.942537i \(0.391567\pi\)
\(272\) 0 0
\(273\) 4.00000 0.242091
\(274\) 0 0
\(275\) 8.48528 8.48528i 0.511682 0.511682i
\(276\) 18.0000 1.08347
\(277\) 1.41421 1.41421i 0.0849719 0.0849719i −0.663343 0.748315i \(-0.730863\pi\)
0.748315 + 0.663343i \(0.230863\pi\)
\(278\) 0 0
\(279\) 1.41421 + 1.41421i 0.0846668 + 0.0846668i
\(280\) 0 0
\(281\) 12.0000i 0.715860i −0.933748 0.357930i \(-0.883483\pi\)
0.933748 0.357930i \(-0.116517\pi\)
\(282\) 0 0
\(283\) 7.07107 + 7.07107i 0.420331 + 0.420331i 0.885318 0.464986i \(-0.153941\pi\)
−0.464986 + 0.885318i \(0.653941\pi\)
\(284\) −16.9706 + 16.9706i −1.00702 + 1.00702i
\(285\) 3.00000 0.177705
\(286\) 0 0
\(287\) 12.0000i 0.708338i
\(288\) 0 0
\(289\) 0 0
\(290\) 0 0
\(291\) 16.0000i 0.937937i
\(292\) 2.82843 2.82843i 0.165521 0.165521i
\(293\) 24.0000 1.40209 0.701047 0.713115i \(-0.252716\pi\)
0.701047 + 0.713115i \(0.252716\pi\)
\(294\) 0 0
\(295\) −12.7279 12.7279i −0.741048 0.741048i
\(296\) 0 0
\(297\) 3.00000i 0.174078i
\(298\) 0 0
\(299\) −6.36396 6.36396i −0.368037 0.368037i
\(300\) 5.65685 + 5.65685i 0.326599 + 0.326599i
\(301\) 19.7990 19.7990i 1.14119 1.14119i
\(302\) 0 0
\(303\) 0 0
\(304\) 4.00000i 0.229416i
\(305\) 24.0000 1.37424
\(306\) 0 0
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 24.0000i 1.36753i
\(309\) −3.53553 + 3.53553i −0.201129 + 0.201129i
\(310\) 0 0
\(311\) −16.9706 + 16.9706i −0.962312 + 0.962312i −0.999315 0.0370028i \(-0.988219\pi\)
0.0370028 + 0.999315i \(0.488219\pi\)
\(312\) 0 0
\(313\) −11.3137 11.3137i −0.639489 0.639489i 0.310941 0.950429i \(-0.399356\pi\)
−0.950429 + 0.310941i \(0.899356\pi\)
\(314\) 0 0
\(315\) 12.0000i 0.676123i
\(316\) 14.1421 + 14.1421i 0.795557 + 0.795557i
\(317\) 4.24264 + 4.24264i 0.238290 + 0.238290i 0.816142 0.577851i \(-0.196109\pi\)
−0.577851 + 0.816142i \(0.696109\pi\)
\(318\) 0 0
\(319\) 18.0000 1.00781
\(320\) 16.9706 16.9706i 0.948683 0.948683i
\(321\) 9.00000i 0.502331i
\(322\) 0 0
\(323\) 0 0
\(324\) −2.00000 −0.111111
\(325\) 4.00000i 0.221880i
\(326\) 0 0
\(327\) −20.0000 −1.10600
\(328\) 0 0
\(329\) −16.9706 16.9706i −0.935617 0.935617i
\(330\) 0 0
\(331\) 13.0000i 0.714545i 0.934000 + 0.357272i \(0.116293\pi\)
−0.934000 + 0.357272i \(0.883707\pi\)
\(332\) 12.0000i 0.658586i
\(333\) −2.82843 2.82843i −0.154997 0.154997i
\(334\) 0 0
\(335\) −8.48528 + 8.48528i −0.463600 + 0.463600i
\(336\) 16.0000 0.872872
\(337\) −9.89949 + 9.89949i −0.539260 + 0.539260i −0.923312 0.384052i \(-0.874528\pi\)
0.384052 + 0.923312i \(0.374528\pi\)
\(338\) 0 0
\(339\) −9.00000 −0.488813
\(340\) 0 0
\(341\) −6.00000 −0.324918
\(342\) 0 0
\(343\) 5.65685 5.65685i 0.305441 0.305441i
\(344\) 0 0
\(345\) 19.0919 19.0919i 1.02787 1.02787i
\(346\) 0 0
\(347\) −8.48528 8.48528i −0.455514 0.455514i 0.441666 0.897180i \(-0.354388\pi\)
−0.897180 + 0.441666i \(0.854388\pi\)
\(348\) 12.0000i 0.643268i
\(349\) 19.0000i 1.01705i 0.861048 + 0.508523i \(0.169808\pi\)
−0.861048 + 0.508523i \(0.830192\pi\)
\(350\) 0 0
\(351\) 0.707107 + 0.707107i 0.0377426 + 0.0377426i
\(352\) 0 0
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) 36.0000i 1.91068i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 18.0000 0.947368
\(362\) 0 0
\(363\) 1.41421 + 1.41421i 0.0742270 + 0.0742270i
\(364\) −5.65685 5.65685i −0.296500 0.296500i
\(365\) 6.00000i 0.314054i
\(366\) 0 0
\(367\) 5.65685 + 5.65685i 0.295285 + 0.295285i 0.839164 0.543879i \(-0.183045\pi\)
−0.543879 + 0.839164i \(0.683045\pi\)
\(368\) −25.4558 25.4558i −1.32698 1.32698i
\(369\) −2.12132 + 2.12132i −0.110432 + 0.110432i
\(370\) 0 0
\(371\) −16.9706 + 16.9706i −0.881068 + 0.881068i
\(372\) 4.00000i 0.207390i
\(373\) −22.0000 −1.13912 −0.569558 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) 0 0
\(375\) −3.00000 −0.154919
\(376\) 0 0
\(377\) 4.24264 4.24264i 0.218507 0.218507i
\(378\) 0 0
\(379\) 22.6274 22.6274i 1.16229 1.16229i 0.178320 0.983973i \(-0.442934\pi\)
0.983973 0.178320i \(-0.0570661\pi\)
\(380\) −4.24264 4.24264i −0.217643 0.217643i
\(381\) −9.19239 9.19239i −0.470940 0.470940i
\(382\) 0 0
\(383\) 12.0000i 0.613171i 0.951843 + 0.306586i \(0.0991866\pi\)
−0.951843 + 0.306586i \(0.900813\pi\)
\(384\) 0 0
\(385\) −25.4558 25.4558i −1.29735 1.29735i
\(386\) 0 0
\(387\) 7.00000 0.355830
\(388\) −22.6274 + 22.6274i −1.14873 + 1.14873i
\(389\) 36.0000i 1.82527i 0.408773 + 0.912636i \(0.365957\pi\)
−0.408773 + 0.912636i \(0.634043\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 3.00000i 0.151330i
\(394\) 0 0
\(395\) 30.0000 1.50946
\(396\) 4.24264 4.24264i 0.213201 0.213201i
\(397\) −14.1421 14.1421i −0.709773 0.709773i 0.256714 0.966487i \(-0.417360\pi\)
−0.966487 + 0.256714i \(0.917360\pi\)
\(398\) 0 0
\(399\) 4.00000i 0.200250i
\(400\) 16.0000i 0.800000i
\(401\) −10.6066 10.6066i −0.529668 0.529668i 0.390805 0.920473i \(-0.372197\pi\)
−0.920473 + 0.390805i \(0.872197\pi\)
\(402\) 0 0
\(403\) −1.41421 + 1.41421i −0.0704470 + 0.0704470i
\(404\) 0 0
\(405\) −2.12132 + 2.12132i −0.105409 + 0.105409i
\(406\) 0 0
\(407\) 12.0000 0.594818
\(408\) 0 0
\(409\) −19.0000 −0.939490 −0.469745 0.882802i \(-0.655654\pi\)
−0.469745 + 0.882802i \(0.655654\pi\)
\(410\) 0 0
\(411\) 4.24264 4.24264i 0.209274 0.209274i
\(412\) 10.0000 0.492665
\(413\) −16.9706 + 16.9706i −0.835067 + 0.835067i
\(414\) 0 0
\(415\) −12.7279 12.7279i −0.624789 0.624789i
\(416\) 0 0
\(417\) 2.00000i 0.0979404i
\(418\) 0 0
\(419\) −8.48528 8.48528i −0.414533 0.414533i 0.468781 0.883314i \(-0.344693\pi\)
−0.883314 + 0.468781i \(0.844693\pi\)
\(420\) 16.9706 16.9706i 0.828079 0.828079i
\(421\) 25.0000 1.21843 0.609213 0.793007i \(-0.291486\pi\)
0.609213 + 0.793007i \(0.291486\pi\)
\(422\) 0 0
\(423\) 6.00000i 0.291730i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 32.0000i 1.54859i
\(428\) 12.7279 12.7279i 0.615227 0.615227i
\(429\) −3.00000 −0.144841
\(430\) 0 0
\(431\) 16.9706 + 16.9706i 0.817443 + 0.817443i 0.985737 0.168294i \(-0.0538257\pi\)
−0.168294 + 0.985737i \(0.553826\pi\)
\(432\) 2.82843 + 2.82843i 0.136083 + 0.136083i
\(433\) 1.00000i 0.0480569i 0.999711 + 0.0240285i \(0.00764923\pi\)
−0.999711 + 0.0240285i \(0.992351\pi\)
\(434\) 0 0
\(435\) 12.7279 + 12.7279i 0.610257 + 0.610257i
\(436\) 28.2843 + 28.2843i 1.35457 + 1.35457i
\(437\) −6.36396 + 6.36396i −0.304430 + 0.304430i
\(438\) 0 0
\(439\) 19.7990 19.7990i 0.944954 0.944954i −0.0536078 0.998562i \(-0.517072\pi\)
0.998562 + 0.0536078i \(0.0170721\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) 0 0
\(443\) 6.00000 0.285069 0.142534 0.989790i \(-0.454475\pi\)
0.142534 + 0.989790i \(0.454475\pi\)
\(444\) 8.00000i 0.379663i
\(445\) 0 0
\(446\) 0 0
\(447\) −12.7279 + 12.7279i −0.602010 + 0.602010i
\(448\) −22.6274 22.6274i −1.06904 1.06904i
\(449\) −4.24264 4.24264i −0.200223 0.200223i 0.599873 0.800095i \(-0.295218\pi\)
−0.800095 + 0.599873i \(0.795218\pi\)
\(450\) 0 0
\(451\) 9.00000i 0.423793i
\(452\) 12.7279 + 12.7279i 0.598671 + 0.598671i
\(453\) −5.65685 5.65685i −0.265782 0.265782i
\(454\) 0 0
\(455\) −12.0000 −0.562569
\(456\) 0 0
\(457\) 19.0000i 0.888783i −0.895833 0.444391i \(-0.853420\pi\)
0.895833 0.444391i \(-0.146580\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) −54.0000 −2.51776
\(461\) 30.0000i 1.39724i 0.715493 + 0.698620i \(0.246202\pi\)
−0.715493 + 0.698620i \(0.753798\pi\)
\(462\) 0 0
\(463\) −8.00000 −0.371792 −0.185896 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) 16.9706 16.9706i 0.787839 0.787839i
\(465\) −4.24264 4.24264i −0.196748 0.196748i
\(466\) 0 0
\(467\) 42.0000i 1.94353i −0.235954 0.971764i \(-0.575822\pi\)
0.235954 0.971764i \(-0.424178\pi\)
\(468\) 2.00000i 0.0924500i
\(469\) 11.3137 + 11.3137i 0.522419 + 0.522419i
\(470\) 0 0
\(471\) 7.77817 7.77817i 0.358399 0.358399i
\(472\) 0 0
\(473\) −14.8492 + 14.8492i −0.682769 + 0.682769i
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) 0 0
\(479\) 19.0919 19.0919i 0.872330 0.872330i −0.120396 0.992726i \(-0.538416\pi\)
0.992726 + 0.120396i \(0.0384163\pi\)
\(480\) 0 0
\(481\) 2.82843 2.82843i 0.128965 0.128965i
\(482\) 0 0
\(483\) −25.4558 25.4558i −1.15828 1.15828i
\(484\) 4.00000i 0.181818i
\(485\) 48.0000i 2.17957i
\(486\) 0 0
\(487\) 15.5563 + 15.5563i 0.704925 + 0.704925i 0.965464 0.260538i \(-0.0839000\pi\)
−0.260538 + 0.965464i \(0.583900\pi\)
\(488\) 0 0
\(489\) −2.00000 −0.0904431
\(490\) 0 0
\(491\) 6.00000i 0.270776i −0.990793 0.135388i \(-0.956772\pi\)
0.990793 0.135388i \(-0.0432281\pi\)
\(492\) 6.00000 0.270501
\(493\) 0 0
\(494\) 0 0
\(495\) 9.00000i 0.404520i
\(496\) −5.65685 + 5.65685i −0.254000 + 0.254000i
\(497\) 48.0000 2.15309
\(498\) 0 0
\(499\) 15.5563 + 15.5563i 0.696398 + 0.696398i 0.963632 0.267234i \(-0.0861096\pi\)
−0.267234 + 0.963632i \(0.586110\pi\)
\(500\) 4.24264 + 4.24264i 0.189737 + 0.189737i
\(501\) 21.0000i 0.938211i
\(502\) 0 0
\(503\) 10.6066 + 10.6066i 0.472925 + 0.472925i 0.902860 0.429935i \(-0.141463\pi\)
−0.429935 + 0.902860i \(0.641463\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 8.48528 8.48528i 0.376845 0.376845i
\(508\) 26.0000i 1.15356i
\(509\) 36.0000 1.59567 0.797836 0.602875i \(-0.205978\pi\)
0.797836 + 0.602875i \(0.205978\pi\)
\(510\) 0 0
\(511\) −8.00000 −0.353899
\(512\) 0 0
\(513\) 0.707107 0.707107i 0.0312195 0.0312195i
\(514\) 0 0
\(515\) 10.6066 10.6066i 0.467383 0.467383i
\(516\) −9.89949 9.89949i −0.435801 0.435801i
\(517\) 12.7279 + 12.7279i 0.559773 + 0.559773i
\(518\) 0 0
\(519\) 15.0000i 0.658427i
\(520\) 0 0
\(521\) −14.8492 14.8492i −0.650557 0.650557i 0.302570 0.953127i \(-0.402155\pi\)
−0.953127 + 0.302570i \(0.902155\pi\)
\(522\) 0 0
\(523\) −20.0000 −0.874539 −0.437269 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 4.24264 4.24264i 0.185341 0.185341i
\(525\) 16.0000i 0.698297i
\(526\) 0 0
\(527\) 0 0
\(528\) −12.0000 −0.522233
\(529\) 58.0000i 2.52174i
\(530\) 0 0
\(531\) −6.00000 −0.260378
\(532\) −5.65685 + 5.65685i −0.245256 + 0.245256i
\(533\) −2.12132 2.12132i −0.0918846 0.0918846i
\(534\) 0 0
\(535\) 27.0000i 1.16731i
\(536\) 0 0
\(537\) −4.24264 4.24264i −0.183083 0.183083i
\(538\) 0 0
\(539\) −19.0919 + 19.0919i −0.822346 + 0.822346i
\(540\) 6.00000 0.258199
\(541\) 11.3137 11.3137i 0.486414 0.486414i −0.420758 0.907173i \(-0.638236\pi\)
0.907173 + 0.420758i \(0.138236\pi\)
\(542\) 0 0
\(543\) 14.0000 0.600798
\(544\) 0 0
\(545\) 60.0000 2.57012
\(546\) 0 0
\(547\) −5.65685 + 5.65685i −0.241870 + 0.241870i −0.817623 0.575754i \(-0.804709\pi\)
0.575754 + 0.817623i \(0.304709\pi\)
\(548\) −12.0000 −0.512615
\(549\) 5.65685 5.65685i 0.241429 0.241429i
\(550\) 0 0
\(551\) −4.24264 4.24264i −0.180743 0.180743i
\(552\) 0 0
\(553\) 40.0000i 1.70097i
\(554\) 0 0
\(555\) 8.48528 + 8.48528i 0.360180 + 0.360180i
\(556\) −2.82843 + 2.82843i −0.119952 + 0.119952i
\(557\) 30.0000 1.27114 0.635570 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) 0 0
\(559\) 7.00000i 0.296068i
\(560\) −48.0000 −2.02837
\(561\) 0 0
\(562\) 0 0
\(563\) 30.0000i 1.26435i −0.774826 0.632175i \(-0.782163\pi\)
0.774826 0.632175i \(-0.217837\pi\)
\(564\) −8.48528 + 8.48528i −0.357295 + 0.357295i
\(565\) 27.0000 1.13590
\(566\) 0 0
\(567\) 2.82843 + 2.82843i 0.118783 + 0.118783i
\(568\) 0 0
\(569\) 24.0000i 1.00613i 0.864248 + 0.503066i \(0.167795\pi\)
−0.864248 + 0.503066i \(0.832205\pi\)
\(570\) 0 0
\(571\) 5.65685 + 5.65685i 0.236732 + 0.236732i 0.815495 0.578763i \(-0.196465\pi\)
−0.578763 + 0.815495i \(0.696465\pi\)
\(572\) 4.24264 + 4.24264i 0.177394 + 0.177394i
\(573\) 12.7279 12.7279i 0.531717 0.531717i
\(574\) 0 0
\(575\) −25.4558 + 25.4558i −1.06158 + 1.06158i
\(576\) 8.00000i 0.333333i
\(577\) −7.00000 −0.291414 −0.145707 0.989328i \(-0.546546\pi\)
−0.145707 + 0.989328i \(0.546546\pi\)
\(578\) 0 0
\(579\) −22.0000 −0.914289
\(580\) 36.0000i 1.49482i
\(581\) −16.9706 + 16.9706i −0.704058 + 0.704058i
\(582\) 0 0
\(583\) 12.7279 12.7279i 0.527137 0.527137i
\(584\) 0 0
\(585\) −2.12132 2.12132i −0.0877058 0.0877058i
\(586\) 0 0
\(587\) 36.0000i 1.48588i 0.669359 + 0.742940i \(0.266569\pi\)
−0.669359 + 0.742940i \(0.733431\pi\)
\(588\) −12.7279 12.7279i −0.524891 0.524891i
\(589\) 1.41421 + 1.41421i 0.0582717 + 0.0582717i
\(590\) 0 0
\(591\) −3.00000 −0.123404
\(592\) 11.3137 11.3137i 0.464991 0.464991i
\(593\) 18.0000i 0.739171i −0.929197 0.369586i \(-0.879500\pi\)
0.929197 0.369586i \(-0.120500\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 36.0000 1.47462
\(597\) 16.0000i 0.654836i
\(598\) 0 0
\(599\) 6.00000 0.245153 0.122577 0.992459i \(-0.460884\pi\)
0.122577 + 0.992459i \(0.460884\pi\)
\(600\) 0 0
\(601\) −26.8701 26.8701i −1.09605 1.09605i −0.994868 0.101185i \(-0.967737\pi\)
−0.101185 0.994868i \(-0.532263\pi\)
\(602\) 0 0
\(603\) 4.00000i 0.162893i
\(604\) 16.0000i 0.651031i
\(605\) −4.24264 4.24264i −0.172488 0.172488i
\(606\) 0 0
\(607\) 26.8701 26.8701i 1.09062 1.09062i 0.0951600 0.995462i \(-0.469664\pi\)
0.995462 0.0951600i \(-0.0303363\pi\)
\(608\) 0 0
\(609\) 16.9706 16.9706i 0.687682 0.687682i
\(610\) 0 0
\(611\) 6.00000 0.242734
\(612\) 0 0
\(613\) 11.0000 0.444286 0.222143 0.975014i \(-0.428695\pi\)
0.222143 + 0.975014i \(0.428695\pi\)
\(614\) 0 0
\(615\) 6.36396 6.36396i 0.256620 0.256620i
\(616\) 0 0
\(617\) −4.24264 + 4.24264i −0.170802 + 0.170802i −0.787332 0.616530i \(-0.788538\pi\)
0.616530 + 0.787332i \(0.288538\pi\)
\(618\) 0 0
\(619\) −7.07107 7.07107i −0.284210 0.284210i 0.550575 0.834786i \(-0.314408\pi\)
−0.834786 + 0.550575i \(0.814408\pi\)
\(620\) 12.0000i 0.481932i
\(621\) 9.00000i 0.361158i
\(622\) 0 0
\(623\) 0 0
\(624\) −2.82843 + 2.82843i −0.113228 + 0.113228i
\(625\) 29.0000 1.16000
\(626\) 0 0
\(627\) 3.00000i 0.119808i
\(628\) −22.0000 −0.877896
\(629\) 0 0
\(630\) 0 0
\(631\) 37.0000i 1.47295i −0.676467 0.736473i \(-0.736490\pi\)
0.676467 0.736473i \(-0.263510\pi\)
\(632\) 0 0
\(633\) −2.00000 −0.0794929
\(634\) 0 0
\(635\) 27.5772 + 27.5772i 1.09437 + 1.09437i
\(636\) 8.48528 + 8.48528i 0.336463 + 0.336463i
\(637\) 9.00000i 0.356593i
\(638\) 0 0
\(639\) 8.48528 + 8.48528i 0.335673 + 0.335673i
\(640\) 0 0
\(641\) −23.3345 + 23.3345i −0.921658 + 0.921658i −0.997147 0.0754884i \(-0.975948\pi\)
0.0754884 + 0.997147i \(0.475948\pi\)
\(642\) 0 0
\(643\) −22.6274 + 22.6274i −0.892338 + 0.892338i −0.994743 0.102405i \(-0.967346\pi\)
0.102405 + 0.994743i \(0.467346\pi\)
\(644\) 72.0000i 2.83720i
\(645\) −21.0000 −0.826874
\(646\) 0 0
\(647\) 18.0000 0.707653 0.353827 0.935311i \(-0.384880\pi\)
0.353827 + 0.935311i \(0.384880\pi\)
\(648\) 0 0
\(649\) 12.7279 12.7279i 0.499615 0.499615i
\(650\) 0 0
\(651\) −5.65685 + 5.65685i −0.221710 + 0.221710i
\(652\) 2.82843 + 2.82843i 0.110770 + 0.110770i
\(653\) 19.0919 + 19.0919i 0.747123 + 0.747123i 0.973938 0.226815i \(-0.0728312\pi\)
−0.226815 + 0.973938i \(0.572831\pi\)
\(654\) 0 0
\(655\) 9.00000i 0.351659i
\(656\) −8.48528 8.48528i −0.331295 0.331295i
\(657\) −1.41421 1.41421i −0.0551737 0.0551737i
\(658\) 0 0
\(659\) −6.00000 −0.233727 −0.116863 0.993148i \(-0.537284\pi\)
−0.116863 + 0.993148i \(0.537284\pi\)
\(660\) −12.7279 + 12.7279i −0.495434 + 0.495434i
\(661\) 31.0000i 1.20576i −0.797832 0.602880i \(-0.794020\pi\)
0.797832 0.602880i \(-0.205980\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 12.0000i 0.465340i
\(666\) 0 0
\(667\) −54.0000 −2.09089
\(668\) −29.6985 + 29.6985i −1.14907 + 1.14907i
\(669\) 0.707107 + 0.707107i 0.0273383 + 0.0273383i
\(670\) 0 0
\(671\) 24.0000i 0.926510i
\(672\) 0 0
\(673\) −15.5563 15.5563i −0.599653 0.599653i 0.340567 0.940220i \(-0.389381\pi\)
−0.940220 + 0.340567i \(0.889381\pi\)
\(674\) 0 0
\(675\) 2.82843 2.82843i 0.108866 0.108866i
\(676\) −24.0000 −0.923077
\(677\) 19.0919 19.0919i 0.733761 0.733761i −0.237602 0.971363i \(-0.576361\pi\)
0.971363 + 0.237602i \(0.0763614\pi\)
\(678\) 0 0
\(679\) 64.0000 2.45609
\(680\) 0 0
\(681\) 3.00000 0.114960
\(682\) 0 0
\(683\) −2.12132 + 2.12132i −0.0811701 + 0.0811701i −0.746526 0.665356i \(-0.768280\pi\)
0.665356 + 0.746526i \(0.268280\pi\)
\(684\) −2.00000 −0.0764719
\(685\) −12.7279 + 12.7279i −0.486309 + 0.486309i
\(686\) 0 0
\(687\) 9.89949 + 9.89949i 0.377689 + 0.377689i
\(688\) 28.0000i 1.06749i
\(689\) 6.00000i 0.228582i
\(690\) 0 0
\(691\) −14.1421 14.1421i −0.537992 0.537992i 0.384947 0.922939i \(-0.374220\pi\)
−0.922939 + 0.384947i \(0.874220\pi\)
\(692\) −21.2132 + 21.2132i −0.806405 + 0.806405i
\(693\) −12.0000 −0.455842
\(694\) 0 0
\(695\) 6.00000i 0.227593i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 21.0000i 0.794293i
\(700\) −22.6274 + 22.6274i −0.855236 + 0.855236i
\(701\) −24.0000 −0.906467 −0.453234 0.891392i \(-0.649730\pi\)
−0.453234 + 0.891392i \(0.649730\pi\)
\(702\) 0 0
\(703\) −2.82843 2.82843i −0.106676 0.106676i
\(704\) 16.9706 + 16.9706i 0.639602 + 0.639602i
\(705\) 18.0000i 0.677919i
\(706\) 0 0
\(707\) 0 0
\(708\) 8.48528 + 8.48528i 0.318896 + 0.318896i
\(709\) 9.89949 9.89949i 0.371783 0.371783i −0.496343 0.868126i \(-0.665324\pi\)
0.868126 + 0.496343i \(0.165324\pi\)
\(710\) 0 0
\(711\) 7.07107 7.07107i 0.265186 0.265186i
\(712\) 0 0
\(713\) 18.0000 0.674105
\(714\) 0 0
\(715\) 9.00000 0.336581
\(716\) 12.0000i 0.448461i
\(717\) 8.48528 8.48528i 0.316889 0.316889i
\(718\) 0 0
\(719\) 2.12132 2.12132i 0.0791119 0.0791119i −0.666444 0.745555i \(-0.732184\pi\)
0.745555 + 0.666444i \(0.232184\pi\)
\(720\) −8.48528 8.48528i −0.316228 0.316228i
\(721\) −14.1421 14.1421i −0.526681 0.526681i
\(722\) 0 0
\(723\) 8.00000i 0.297523i
\(724\) −19.7990 19.7990i −0.735824 0.735824i
\(725\) −16.9706 16.9706i −0.630271 0.630271i
\(726\) 0 0
\(727\) −8.00000 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 0 0
\(729\) 1.00000i 0.0370370i
\(730\) 0 0
\(731\) 0 0
\(732\) −16.0000 −0.591377
\(733\) 2.00000i 0.0738717i 0.999318 + 0.0369358i \(0.0117597\pi\)
−0.999318 + 0.0369358i \(0.988240\pi\)
\(734\) 0 0
\(735\) −27.0000 −0.995910
\(736\) 0 0
\(737\) −8.48528 8.48528i −0.312559 0.312559i
\(738\) 0 0
\(739\) 1.00000i 0.0367856i 0.999831 + 0.0183928i \(0.00585494\pi\)
−0.999831 + 0.0183928i \(0.994145\pi\)
\(740\) 24.0000i 0.882258i
\(741\) 0.707107 + 0.707107i 0.0259762 + 0.0259762i
\(742\) 0 0
\(743\) 16.9706 16.9706i 0.622590 0.622590i −0.323603 0.946193i \(-0.604894\pi\)
0.946193 + 0.323603i \(0.104894\pi\)
\(744\) 0 0
\(745\) 38.1838 38.1838i 1.39894 1.39894i
\(746\) 0 0
\(747\) −6.00000 −0.219529
\(748\) 0 0
\(749\) −36.0000 −1.31541
\(750\) 0 0
\(751\) 32.5269 32.5269i 1.18692 1.18692i 0.209010 0.977913i \(-0.432976\pi\)
0.977913 0.209010i \(-0.0670243\pi\)
\(752\) 24.0000 0.875190
\(753\) 16.9706 16.9706i 0.618442 0.618442i
\(754\) 0 0
\(755\) 16.9706 + 16.9706i 0.617622 + 0.617622i
\(756\) 8.00000i 0.290957i
\(757\) 43.0000i 1.56286i 0.623992 + 0.781431i \(0.285510\pi\)
−0.623992 + 0.781431i \(0.714490\pi\)
\(758\) 0 0
\(759\) 19.0919 + 19.0919i 0.692991 + 0.692991i
\(760\) 0 0
\(761\) −30.0000 −1.08750 −0.543750 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) 0 0
\(763\) 80.0000i 2.89619i
\(764\) −36.0000 −1.30243
\(765\) 0 0
\(766\) 0 0
\(767\) 6.00000i 0.216647i
\(768\) −11.3137 + 11.3137i −0.408248 + 0.408248i
\(769\) −41.0000 −1.47850 −0.739249 0.673432i \(-0.764819\pi\)
−0.739249 + 0.673432i \(0.764819\pi\)
\(770\) 0 0
\(771\) −8.48528 8.48528i −0.305590 0.305590i
\(772\) 31.1127 + 31.1127i 1.11977 + 1.11977i
\(773\) 24.0000i 0.863220i −0.902060 0.431610i \(-0.857946\pi\)
0.902060 0.431610i \(-0.142054\pi\)
\(774\) 0 0
\(775\) 5.65685 + 5.65685i 0.203200 + 0.203200i
\(776\) 0 0
\(777\) 11.3137 11.3137i 0.405877 0.405877i
\(778\) 0 0
\(779\) −2.12132 + 2.12132i −0.0760042 + 0.0760042i
\(780\) 6.00000i 0.214834i
\(781\) −36.0000 −1.28818
\(782\) 0 0
\(783\) 6.00000 0.214423
\(784\) 36.0000i 1.28571i
\(785\) −23.3345 + 23.3345i −0.832845 + 0.832845i
\(786\) 0 0
\(787\) −28.2843 + 28.2843i −1.00823 + 1.00823i −0.00825949 + 0.999966i \(0.502629\pi\)
−0.999966 + 0.00825949i \(0.997371\pi\)
\(788\) 4.24264 + 4.24264i 0.151138 + 0.151138i
\(789\) 8.48528 + 8.48528i 0.302084 + 0.302084i
\(790\) 0 0
\(791\) 36.0000i 1.28001i
\(792\) 0 0
\(793\) 5.65685 + 5.65685i 0.200881 + 0.200881i
\(794\) 0 0
\(795\) 18.0000 0.638394
\(796\) −22.6274 + 22.6274i −0.802008 + 0.802008i
\(797\) 48.0000i 1.70025i −0.526583 0.850124i \(-0.676527\pi\)
0.526583 0.850124i \(-0.323473\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 6.00000 0.211735
\(804\) 5.65685 5.65685i 0.199502 0.199502i
\(805\) 76.3675 + 76.3675i 2.69160 + 2.69160i
\(806\) 0 0
\(807\) 15.0000i 0.528025i
\(808\) 0 0
\(809\) −36.0624 36.0624i −1.26789 1.26789i −0.947177 0.320711i \(-0.896078\pi\)
−0.320711 0.947177i \(-0.603922\pi\)
\(810\) 0 0
\(811\) −7.07107 + 7.07107i −0.248299 + 0.248299i −0.820272 0.571973i \(-0.806178\pi\)
0.571973 + 0.820272i \(0.306178\pi\)
\(812\) −48.0000 −1.68447
\(813\) −7.77817 + 7.77817i −0.272792 + 0.272792i
\(814\) 0 0
\(815\) 6.00000 0.210171
\(816\) 0 0
\(817\) 7.00000 0.244899
\(818\) 0 0
\(819\) −2.82843 + 2.82843i −0.0988332 + 0.0988332i
\(820\) −18.0000 −0.628587
\(821\) −14.8492 + 14.8492i −0.518242 + 0.518242i −0.917039 0.398797i \(-0.869428\pi\)
0.398797 + 0.917039i \(0.369428\pi\)
\(822\) 0 0
\(823\) −19.7990 19.7990i −0.690149 0.690149i 0.272115 0.962265i \(-0.412277\pi\)
−0.962265 + 0.272115i \(0.912277\pi\)
\(824\) 0 0
\(825\) 12.0000i 0.417786i
\(826\) 0 0
\(827\) 14.8492 + 14.8492i 0.516359 + 0.516359i 0.916468 0.400109i \(-0.131028\pi\)
−0.400109 + 0.916468i \(0.631028\pi\)
\(828\) −12.7279 + 12.7279i −0.442326 + 0.442326i
\(829\) −14.0000 −0.486240 −0.243120 0.969996i \(-0.578171\pi\)
−0.243120 + 0.969996i \(0.578171\pi\)
\(830\) 0 0
\(831\) 2.00000i 0.0693792i
\(832\) 8.00000 0.277350
\(833\) 0 0
\(834\) 0 0
\(835\) 63.0000i 2.18020i
\(836\) 4.24264 4.24264i 0.146735 0.146735i
\(837\) −2.00000 −0.0691301
\(838\) 0 0
\(839\) 40.3051 + 40.3051i 1.39149 + 1.39149i 0.822003 + 0.569483i \(0.192857\pi\)
0.569483 + 0.822003i \(0.307143\pi\)
\(840\) 0 0
\(841\) 7.00000i 0.241379i
\(842\) 0 0
\(843\) 8.48528 + 8.48528i 0.292249 + 0.292249i
\(844\) 2.82843 + 2.82843i 0.0973585 + 0.0973585i
\(845\) −25.4558 + 25.4558i −0.875708 + 0.875708i
\(846\) 0 0
\(847\) −5.65685 + 5.65685i −0.194372 + 0.194372i
\(848\) 24.0000i 0.824163i
\(849\) −10.0000 −0.343199
\(850\) 0 0
\(851\) −36.0000 −1.23406
\(852\) 24.0000i 0.822226i
\(853\) −18.3848 + 18.3848i −0.629483 + 0.629483i −0.947938 0.318455i \(-0.896836\pi\)
0.318455 + 0.947938i \(0.396836\pi\)
\(854\) 0 0
\(855\) −2.12132 + 2.12132i −0.0725476 + 0.0725476i
\(856\) 0 0
\(857\) −12.7279 12.7279i −0.434778 0.434778i 0.455472 0.890250i \(-0.349470\pi\)
−0.890250 + 0.455472i \(0.849470\pi\)
\(858\) 0 0
\(859\) 20.0000i 0.682391i −0.939992 0.341196i \(-0.889168\pi\)
0.939992 0.341196i \(-0.110832\pi\)
\(860\) 29.6985 + 29.6985i 1.01271 + 1.01271i
\(861\) −8.48528 8.48528i −0.289178 0.289178i
\(862\) 0 0
\(863\) −24.0000 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) 0 0
\(865\) 45.0000i 1.53005i
\(866\) 0 0
\(867\) 0 0
\(868\) 16.0000 0.543075
\(869\) 30.0000i 1.01768i
\(870\) 0 0
\(871\) −4.00000 −0.135535
\(872\) 0 0
\(873\) 11.3137 + 11.3137i 0.382911 + 0.382911i
\(874\) 0 0
\(875\) 12.0000i 0.405674i
\(876\) 4.00000i 0.135147i
\(877\) 9.89949 + 9.89949i 0.334282 + 0.334282i 0.854210 0.519928i \(-0.174041\pi\)
−0.519928 + 0.854210i \(0.674041\pi\)
\(878\) 0 0
\(879\) −16.9706 + 16.9706i −0.572403 + 0.572403i
\(880\) 36.0000 1.21356
\(881\) −12.7279 + 12.7279i −0.428815 + 0.428815i −0.888224 0.459410i \(-0.848061\pi\)
0.459410 + 0.888224i \(0.348061\pi\)
\(882\) 0 0
\(883\) 11.0000 0.370179 0.185090 0.982722i \(-0.440742\pi\)
0.185090 + 0.982722i \(0.440742\pi\)
\(884\) 0 0
\(885\) 18.0000 0.605063
\(886\) 0 0
\(887\) −27.5772 + 27.5772i −0.925951 + 0.925951i −0.997441 0.0714907i \(-0.977224\pi\)
0.0714907 + 0.997441i \(0.477224\pi\)
\(888\) 0 0
\(889\) 36.7696 36.7696i 1.23321 1.23321i
\(890\) 0 0
\(891\) −2.12132 2.12132i −0.0710669 0.0710669i
\(892\) 2.00000i 0.0669650i
\(893\) 6.00000i 0.200782i
\(894\) 0 0
\(895\) 12.7279 + 12.7279i 0.425448 + 0.425448i
\(896\) 0 0
\(897\) 9.00000 0.300501
\(898\) 0 0
\(899\) 12.0000i 0.400222i
\(900\) −8.00000 −0.266667
\(901\) 0 0
\(902\) 0 0
\(903\) 28.0000i 0.931782i
\(904\) 0 0
\(905\) −42.0000 −1.39613
\(906\) 0 0
\(907\) −1.41421 1.41421i −0.0469582 0.0469582i 0.683238 0.730196i \(-0.260571\pi\)
−0.730196 + 0.683238i \(0.760571\pi\)
\(908\) −4.24264 4.24264i −0.140797 0.140797i
\(909\) 0 0
\(910\) 0 0
\(911\) −19.0919 19.0919i −0.632542 0.632542i 0.316163 0.948705i \(-0.397605\pi\)
−0.948705 + 0.316163i \(0.897605\pi\)
\(912\) 2.82843 + 2.82843i 0.0936586 + 0.0936586i
\(913\) 12.7279 12.7279i 0.421233 0.421233i
\(914\) 0 0
\(915\) −16.9706 + 16.9706i −0.561029 + 0.561029i
\(916\) 28.0000i 0.925146i
\(917\) −12.0000 −0.396275
\(918\) 0 0
\(919\) 11.0000 0.362857 0.181428 0.983404i \(-0.441928\pi\)
0.181428 + 0.983404i \(0.441928\pi\)
\(920\) 0 0
\(921\) −14.1421 + 14.1421i −0.465999 + 0.465999i
\(922\) 0 0
\(923\) −8.48528 + 8.48528i −0.279296 + 0.279296i
\(924\) 16.9706 + 16.9706i 0.558291 + 0.558291i
\(925\) −11.3137 11.3137i −0.371992 0.371992i
\(926\) 0 0
\(927\) 5.00000i 0.164222i
\(928\) 0 0
\(929\) −10.6066 10.6066i −0.347991 0.347991i 0.511370 0.859361i \(-0.329138\pi\)
−0.859361 + 0.511370i \(0.829138\pi\)
\(930\) 0 0
\(931\) 9.00000 0.294963
\(932\) 29.6985 29.6985i 0.972806 0.972806i
\(933\) 24.0000i 0.785725i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 22.0000i 0.718709i −0.933201 0.359354i \(-0.882997\pi\)
0.933201 0.359354i \(-0.117003\pi\)
\(938\) 0 0
\(939\) 16.0000 0.522140
\(940\) 25.4558 25.4558i 0.830278 0.830278i
\(941\) −12.7279 12.7279i −0.414918 0.414918i 0.468529 0.883448i \(-0.344784\pi\)
−0.883448 + 0.468529i \(0.844784\pi\)
\(942\) 0 0
\(943\) 27.0000i 0.879241i
\(944\) 24.0000i 0.781133i
\(945\) −8.48528 8.48528i −0.276026 0.276026i
\(946\) 0 0
\(947\) 25.4558 25.4558i 0.827204 0.827204i −0.159925 0.987129i \(-0.551125\pi\)
0.987129 + 0.159925i \(0.0511253\pi\)
\(948\) −20.0000 −0.649570
\(949\) 1.41421 1.41421i 0.0459073 0.0459073i
\(950\) 0 0
\(951\) −6.00000 −0.194563
\(952\) 0 0
\(953\) 36.0000 1.16615 0.583077 0.812417i \(-0.301849\pi\)
0.583077 + 0.812417i \(0.301849\pi\)
\(954\) 0 0
\(955\) −38.1838 + 38.1838i −1.23560 + 1.23560i
\(956\) −24.0000 −0.776215
\(957\) −12.7279 + 12.7279i −0.411435 + 0.411435i
\(958\) 0 0
\(959\) 16.9706 + 16.9706i 0.548008 + 0.548008i
\(960\) 24.0000i 0.774597i
\(961\) 27.0000i 0.870968i
\(962\) 0 0
\(963\) −6.36396 6.36396i −0.205076 0.205076i
\(964\) −11.3137 + 11.3137i −0.364390 + 0.364390i
\(965\) 66.0000 2.12462
\(966\) 0 0
\(967\) 41.0000i 1.31847i 0.751936 + 0.659236i \(0.229120\pi\)
−0.751936 + 0.659236i \(0.770880\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 6.00000i 0.192549i 0.995355 + 0.0962746i \(0.0306927\pi\)
−0.995355 + 0.0962746i \(0.969307\pi\)
\(972\) 1.41421 1.41421i 0.0453609 0.0453609i
\(973\) 8.00000 0.256468
\(974\) 0 0
\(975\) 2.82843 + 2.82843i 0.0905822 + 0.0905822i
\(976\) 22.6274 + 22.6274i 0.724286 + 0.724286i
\(977\) 30.0000i 0.959785i 0.877327 + 0.479893i \(0.159324\pi\)
−0.877327 + 0.479893i \(0.840676\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 38.1838 + 38.1838i 1.21974 + 1.21974i
\(981\) 14.1421 14.1421i 0.451524 0.451524i
\(982\) 0 0
\(983\) 6.36396 6.36396i 0.202979 0.202979i −0.598296 0.801275i \(-0.704155\pi\)
0.801275 + 0.598296i \(0.204155\pi\)
\(984\) 0 0
\(985\) 9.00000 0.286764
\(986\) 0 0
\(987\) 24.0000 0.763928
\(988\) 2.00000i 0.0636285i
\(989\) 44.5477 44.5477i 1.41654 1.41654i
\(990\) 0 0
\(991\) −36.7696 + 36.7696i −1.16802 + 1.16802i −0.185351 + 0.982672i \(0.559342\pi\)
−0.982672 + 0.185351i \(0.940658\pi\)
\(992\) 0 0
\(993\) −9.19239 9.19239i −0.291712 0.291712i
\(994\) 0 0
\(995\) 48.0000i 1.52170i
\(996\) 8.48528 + 8.48528i 0.268866 + 0.268866i
\(997\) −43.8406 43.8406i −1.38845 1.38845i −0.828589 0.559857i \(-0.810856\pi\)
−0.559857 0.828589i \(-0.689144\pi\)
\(998\) 0 0
\(999\) 4.00000 0.126554
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 867.2.e.e.616.1 4
17.2 even 8 867.2.d.a.577.2 2
17.3 odd 16 867.2.h.c.733.1 8
17.4 even 4 inner 867.2.e.e.829.2 4
17.5 odd 16 867.2.h.c.712.1 8
17.6 odd 16 867.2.h.c.688.1 8
17.7 odd 16 867.2.h.c.757.2 8
17.8 even 8 51.2.a.a.1.1 1
17.9 even 8 867.2.a.c.1.1 1
17.10 odd 16 867.2.h.c.757.1 8
17.11 odd 16 867.2.h.c.688.2 8
17.12 odd 16 867.2.h.c.712.2 8
17.13 even 4 inner 867.2.e.e.829.1 4
17.14 odd 16 867.2.h.c.733.2 8
17.15 even 8 867.2.d.a.577.1 2
17.16 even 2 inner 867.2.e.e.616.2 4
51.8 odd 8 153.2.a.b.1.1 1
51.26 odd 8 2601.2.a.f.1.1 1
68.59 odd 8 816.2.a.g.1.1 1
85.8 odd 8 1275.2.b.b.1174.2 2
85.42 odd 8 1275.2.b.b.1174.1 2
85.59 even 8 1275.2.a.d.1.1 1
119.76 odd 8 2499.2.a.d.1.1 1
136.59 odd 8 3264.2.a.r.1.1 1
136.93 even 8 3264.2.a.a.1.1 1
187.76 odd 8 6171.2.a.e.1.1 1
204.59 even 8 2448.2.a.c.1.1 1
221.25 even 8 8619.2.a.g.1.1 1
255.59 odd 8 3825.2.a.i.1.1 1
357.314 even 8 7497.2.a.j.1.1 1
408.59 even 8 9792.2.a.cd.1.1 1
408.365 odd 8 9792.2.a.by.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
51.2.a.a.1.1 1 17.8 even 8
153.2.a.b.1.1 1 51.8 odd 8
816.2.a.g.1.1 1 68.59 odd 8
867.2.a.c.1.1 1 17.9 even 8
867.2.d.a.577.1 2 17.15 even 8
867.2.d.a.577.2 2 17.2 even 8
867.2.e.e.616.1 4 1.1 even 1 trivial
867.2.e.e.616.2 4 17.16 even 2 inner
867.2.e.e.829.1 4 17.13 even 4 inner
867.2.e.e.829.2 4 17.4 even 4 inner
867.2.h.c.688.1 8 17.6 odd 16
867.2.h.c.688.2 8 17.11 odd 16
867.2.h.c.712.1 8 17.5 odd 16
867.2.h.c.712.2 8 17.12 odd 16
867.2.h.c.733.1 8 17.3 odd 16
867.2.h.c.733.2 8 17.14 odd 16
867.2.h.c.757.1 8 17.10 odd 16
867.2.h.c.757.2 8 17.7 odd 16
1275.2.a.d.1.1 1 85.59 even 8
1275.2.b.b.1174.1 2 85.42 odd 8
1275.2.b.b.1174.2 2 85.8 odd 8
2448.2.a.c.1.1 1 204.59 even 8
2499.2.a.d.1.1 1 119.76 odd 8
2601.2.a.f.1.1 1 51.26 odd 8
3264.2.a.a.1.1 1 136.93 even 8
3264.2.a.r.1.1 1 136.59 odd 8
3825.2.a.i.1.1 1 255.59 odd 8
6171.2.a.e.1.1 1 187.76 odd 8
7497.2.a.j.1.1 1 357.314 even 8
8619.2.a.g.1.1 1 221.25 even 8
9792.2.a.by.1.1 1 408.365 odd 8
9792.2.a.cd.1.1 1 408.59 even 8