Properties

Label 441.2.e.i.361.1
Level $441$
Weight $2$
Character 441.361
Analytic conductor $3.521$
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] = [441,2,Mod(226,441)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(441, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.226");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.e (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: \(3.52140272914\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 361.1
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 441.361
Dual form 441.2.e.i.226.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.866025 - 1.50000i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-1.73205 - 3.00000i) q^{5} -1.73205 q^{8} +O(q^{10})\) \(q+(-0.866025 - 1.50000i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-1.73205 - 3.00000i) q^{5} -1.73205 q^{8} +(-3.00000 + 5.19615i) q^{10} +(-1.73205 + 3.00000i) q^{11} -2.00000 q^{13} +(2.50000 + 4.33013i) q^{16} +(1.73205 - 3.00000i) q^{17} +(-2.00000 - 3.46410i) q^{19} +3.46410 q^{20} +6.00000 q^{22} +(1.73205 + 3.00000i) q^{23} +(-3.50000 + 6.06218i) q^{25} +(1.73205 + 3.00000i) q^{26} +(-2.00000 + 3.46410i) q^{31} +(2.59808 - 4.50000i) q^{32} -6.00000 q^{34} +(-1.00000 - 1.73205i) q^{37} +(-3.46410 + 6.00000i) q^{38} +(3.00000 + 5.19615i) q^{40} -10.3923 q^{41} -4.00000 q^{43} +(-1.73205 - 3.00000i) q^{44} +(3.00000 - 5.19615i) q^{46} +(3.46410 + 6.00000i) q^{47} +12.1244 q^{50} +(1.00000 - 1.73205i) q^{52} +(3.46410 - 6.00000i) q^{53} +12.0000 q^{55} +(-3.46410 + 6.00000i) q^{59} +(-5.00000 - 8.66025i) q^{61} +6.92820 q^{62} +1.00000 q^{64} +(3.46410 + 6.00000i) q^{65} +(2.00000 - 3.46410i) q^{67} +(1.73205 + 3.00000i) q^{68} -10.3923 q^{71} +(7.00000 - 12.1244i) q^{73} +(-1.73205 + 3.00000i) q^{74} +4.00000 q^{76} +(-4.00000 - 6.92820i) q^{79} +(8.66025 - 15.0000i) q^{80} +(9.00000 + 15.5885i) q^{82} -12.0000 q^{85} +(3.46410 + 6.00000i) q^{86} +(3.00000 - 5.19615i) q^{88} +(-1.73205 - 3.00000i) q^{89} -3.46410 q^{92} +(6.00000 - 10.3923i) q^{94} +(-6.92820 + 12.0000i) q^{95} -14.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{4} - 12 q^{10} - 8 q^{13} + 10 q^{16} - 8 q^{19} + 24 q^{22} - 14 q^{25} - 8 q^{31} - 24 q^{34} - 4 q^{37} + 12 q^{40} - 16 q^{43} + 12 q^{46} + 4 q^{52} + 48 q^{55} - 20 q^{61} + 4 q^{64} + 8 q^{67} + 28 q^{73} + 16 q^{76} - 16 q^{79} + 36 q^{82} - 48 q^{85} + 12 q^{88} + 24 q^{94} - 56 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(344\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.866025 1.50000i −0.612372 1.06066i −0.990839 0.135045i \(-0.956882\pi\)
0.378467 0.925615i \(-0.376451\pi\)
\(3\) 0 0
\(4\) −0.500000 + 0.866025i −0.250000 + 0.433013i
\(5\) −1.73205 3.00000i −0.774597 1.34164i −0.935021 0.354593i \(-0.884620\pi\)
0.160424 0.987048i \(-0.448714\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −1.73205 −0.612372
\(9\) 0 0
\(10\) −3.00000 + 5.19615i −0.948683 + 1.64317i
\(11\) −1.73205 + 3.00000i −0.522233 + 0.904534i 0.477432 + 0.878668i \(0.341568\pi\)
−0.999665 + 0.0258656i \(0.991766\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 2.50000 + 4.33013i 0.625000 + 1.08253i
\(17\) 1.73205 3.00000i 0.420084 0.727607i −0.575863 0.817546i \(-0.695334\pi\)
0.995947 + 0.0899392i \(0.0286673\pi\)
\(18\) 0 0
\(19\) −2.00000 3.46410i −0.458831 0.794719i 0.540068 0.841621i \(-0.318398\pi\)
−0.998899 + 0.0469020i \(0.985065\pi\)
\(20\) 3.46410 0.774597
\(21\) 0 0
\(22\) 6.00000 1.27920
\(23\) 1.73205 + 3.00000i 0.361158 + 0.625543i 0.988152 0.153481i \(-0.0490483\pi\)
−0.626994 + 0.779024i \(0.715715\pi\)
\(24\) 0 0
\(25\) −3.50000 + 6.06218i −0.700000 + 1.21244i
\(26\) 1.73205 + 3.00000i 0.339683 + 0.588348i
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −2.00000 + 3.46410i −0.359211 + 0.622171i −0.987829 0.155543i \(-0.950287\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) 2.59808 4.50000i 0.459279 0.795495i
\(33\) 0 0
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 0 0
\(37\) −1.00000 1.73205i −0.164399 0.284747i 0.772043 0.635571i \(-0.219235\pi\)
−0.936442 + 0.350823i \(0.885902\pi\)
\(38\) −3.46410 + 6.00000i −0.561951 + 0.973329i
\(39\) 0 0
\(40\) 3.00000 + 5.19615i 0.474342 + 0.821584i
\(41\) −10.3923 −1.62301 −0.811503 0.584349i \(-0.801350\pi\)
−0.811503 + 0.584349i \(0.801350\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −1.73205 3.00000i −0.261116 0.452267i
\(45\) 0 0
\(46\) 3.00000 5.19615i 0.442326 0.766131i
\(47\) 3.46410 + 6.00000i 0.505291 + 0.875190i 0.999981 + 0.00612051i \(0.00194823\pi\)
−0.494690 + 0.869069i \(0.664718\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 12.1244 1.71464
\(51\) 0 0
\(52\) 1.00000 1.73205i 0.138675 0.240192i
\(53\) 3.46410 6.00000i 0.475831 0.824163i −0.523786 0.851850i \(-0.675481\pi\)
0.999617 + 0.0276867i \(0.00881407\pi\)
\(54\) 0 0
\(55\) 12.0000 1.61808
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −3.46410 + 6.00000i −0.450988 + 0.781133i −0.998448 0.0556984i \(-0.982261\pi\)
0.547460 + 0.836832i \(0.315595\pi\)
\(60\) 0 0
\(61\) −5.00000 8.66025i −0.640184 1.10883i −0.985391 0.170305i \(-0.945525\pi\)
0.345207 0.938527i \(-0.387809\pi\)
\(62\) 6.92820 0.879883
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 3.46410 + 6.00000i 0.429669 + 0.744208i
\(66\) 0 0
\(67\) 2.00000 3.46410i 0.244339 0.423207i −0.717607 0.696449i \(-0.754762\pi\)
0.961946 + 0.273241i \(0.0880957\pi\)
\(68\) 1.73205 + 3.00000i 0.210042 + 0.363803i
\(69\) 0 0
\(70\) 0 0
\(71\) −10.3923 −1.23334 −0.616670 0.787222i \(-0.711519\pi\)
−0.616670 + 0.787222i \(0.711519\pi\)
\(72\) 0 0
\(73\) 7.00000 12.1244i 0.819288 1.41905i −0.0869195 0.996215i \(-0.527702\pi\)
0.906208 0.422833i \(-0.138964\pi\)
\(74\) −1.73205 + 3.00000i −0.201347 + 0.348743i
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) 0 0
\(79\) −4.00000 6.92820i −0.450035 0.779484i 0.548352 0.836247i \(-0.315255\pi\)
−0.998388 + 0.0567635i \(0.981922\pi\)
\(80\) 8.66025 15.0000i 0.968246 1.67705i
\(81\) 0 0
\(82\) 9.00000 + 15.5885i 0.993884 + 1.72146i
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −12.0000 −1.30158
\(86\) 3.46410 + 6.00000i 0.373544 + 0.646997i
\(87\) 0 0
\(88\) 3.00000 5.19615i 0.319801 0.553912i
\(89\) −1.73205 3.00000i −0.183597 0.317999i 0.759506 0.650500i \(-0.225441\pi\)
−0.943103 + 0.332501i \(0.892107\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −3.46410 −0.361158
\(93\) 0 0
\(94\) 6.00000 10.3923i 0.618853 1.07188i
\(95\) −6.92820 + 12.0000i −0.710819 + 1.23117i
\(96\) 0 0
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −3.50000 6.06218i −0.350000 0.606218i
\(101\) 1.73205 3.00000i 0.172345 0.298511i −0.766894 0.641774i \(-0.778199\pi\)
0.939239 + 0.343263i \(0.111532\pi\)
\(102\) 0 0
\(103\) −2.00000 3.46410i −0.197066 0.341328i 0.750510 0.660859i \(-0.229808\pi\)
−0.947576 + 0.319531i \(0.896475\pi\)
\(104\) 3.46410 0.339683
\(105\) 0 0
\(106\) −12.0000 −1.16554
\(107\) −8.66025 15.0000i −0.837218 1.45010i −0.892211 0.451618i \(-0.850847\pi\)
0.0549930 0.998487i \(-0.482486\pi\)
\(108\) 0 0
\(109\) −1.00000 + 1.73205i −0.0957826 + 0.165900i −0.909935 0.414751i \(-0.863869\pi\)
0.814152 + 0.580651i \(0.197202\pi\)
\(110\) −10.3923 18.0000i −0.990867 1.71623i
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 6.00000 10.3923i 0.559503 0.969087i
\(116\) 0 0
\(117\) 0 0
\(118\) 12.0000 1.10469
\(119\) 0 0
\(120\) 0 0
\(121\) −0.500000 0.866025i −0.0454545 0.0787296i
\(122\) −8.66025 + 15.0000i −0.784063 + 1.35804i
\(123\) 0 0
\(124\) −2.00000 3.46410i −0.179605 0.311086i
\(125\) 6.92820 0.619677
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) −6.06218 10.5000i −0.535826 0.928078i
\(129\) 0 0
\(130\) 6.00000 10.3923i 0.526235 0.911465i
\(131\) −6.92820 12.0000i −0.605320 1.04844i −0.992001 0.126231i \(-0.959712\pi\)
0.386681 0.922214i \(-0.373621\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −6.92820 −0.598506
\(135\) 0 0
\(136\) −3.00000 + 5.19615i −0.257248 + 0.445566i
\(137\) 3.46410 6.00000i 0.295958 0.512615i −0.679249 0.733908i \(-0.737694\pi\)
0.975207 + 0.221293i \(0.0710278\pi\)
\(138\) 0 0
\(139\) 16.0000 1.35710 0.678551 0.734553i \(-0.262608\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 9.00000 + 15.5885i 0.755263 + 1.30815i
\(143\) 3.46410 6.00000i 0.289683 0.501745i
\(144\) 0 0
\(145\) 0 0
\(146\) −24.2487 −2.00684
\(147\) 0 0
\(148\) 2.00000 0.164399
\(149\) −3.46410 6.00000i −0.283790 0.491539i 0.688525 0.725213i \(-0.258259\pi\)
−0.972315 + 0.233674i \(0.924925\pi\)
\(150\) 0 0
\(151\) −4.00000 + 6.92820i −0.325515 + 0.563809i −0.981617 0.190864i \(-0.938871\pi\)
0.656101 + 0.754673i \(0.272204\pi\)
\(152\) 3.46410 + 6.00000i 0.280976 + 0.486664i
\(153\) 0 0
\(154\) 0 0
\(155\) 13.8564 1.11297
\(156\) 0 0
\(157\) −5.00000 + 8.66025i −0.399043 + 0.691164i −0.993608 0.112884i \(-0.963991\pi\)
0.594565 + 0.804048i \(0.297324\pi\)
\(158\) −6.92820 + 12.0000i −0.551178 + 0.954669i
\(159\) 0 0
\(160\) −18.0000 −1.42302
\(161\) 0 0
\(162\) 0 0
\(163\) −10.0000 17.3205i −0.783260 1.35665i −0.930033 0.367477i \(-0.880222\pi\)
0.146772 0.989170i \(-0.453112\pi\)
\(164\) 5.19615 9.00000i 0.405751 0.702782i
\(165\) 0 0
\(166\) 0 0
\(167\) 20.7846 1.60836 0.804181 0.594385i \(-0.202604\pi\)
0.804181 + 0.594385i \(0.202604\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 10.3923 + 18.0000i 0.797053 + 1.38054i
\(171\) 0 0
\(172\) 2.00000 3.46410i 0.152499 0.264135i
\(173\) 8.66025 + 15.0000i 0.658427 + 1.14043i 0.981023 + 0.193892i \(0.0621112\pi\)
−0.322596 + 0.946537i \(0.604555\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −17.3205 −1.30558
\(177\) 0 0
\(178\) −3.00000 + 5.19615i −0.224860 + 0.389468i
\(179\) 8.66025 15.0000i 0.647298 1.12115i −0.336468 0.941695i \(-0.609232\pi\)
0.983766 0.179458i \(-0.0574343\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −3.00000 5.19615i −0.221163 0.383065i
\(185\) −3.46410 + 6.00000i −0.254686 + 0.441129i
\(186\) 0 0
\(187\) 6.00000 + 10.3923i 0.438763 + 0.759961i
\(188\) −6.92820 −0.505291
\(189\) 0 0
\(190\) 24.0000 1.74114
\(191\) 12.1244 + 21.0000i 0.877288 + 1.51951i 0.854306 + 0.519771i \(0.173983\pi\)
0.0229818 + 0.999736i \(0.492684\pi\)
\(192\) 0 0
\(193\) −7.00000 + 12.1244i −0.503871 + 0.872730i 0.496119 + 0.868255i \(0.334758\pi\)
−0.999990 + 0.00447566i \(0.998575\pi\)
\(194\) 12.1244 + 21.0000i 0.870478 + 1.50771i
\(195\) 0 0
\(196\) 0 0
\(197\) 20.7846 1.48084 0.740421 0.672143i \(-0.234626\pi\)
0.740421 + 0.672143i \(0.234626\pi\)
\(198\) 0 0
\(199\) −8.00000 + 13.8564i −0.567105 + 0.982255i 0.429745 + 0.902950i \(0.358603\pi\)
−0.996850 + 0.0793045i \(0.974730\pi\)
\(200\) 6.06218 10.5000i 0.428661 0.742462i
\(201\) 0 0
\(202\) −6.00000 −0.422159
\(203\) 0 0
\(204\) 0 0
\(205\) 18.0000 + 31.1769i 1.25717 + 2.17749i
\(206\) −3.46410 + 6.00000i −0.241355 + 0.418040i
\(207\) 0 0
\(208\) −5.00000 8.66025i −0.346688 0.600481i
\(209\) 13.8564 0.958468
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 3.46410 + 6.00000i 0.237915 + 0.412082i
\(213\) 0 0
\(214\) −15.0000 + 25.9808i −1.02538 + 1.77601i
\(215\) 6.92820 + 12.0000i 0.472500 + 0.818393i
\(216\) 0 0
\(217\) 0 0
\(218\) 3.46410 0.234619
\(219\) 0 0
\(220\) −6.00000 + 10.3923i −0.404520 + 0.700649i
\(221\) −3.46410 + 6.00000i −0.233021 + 0.403604i
\(222\) 0 0
\(223\) −8.00000 −0.535720 −0.267860 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −3.46410 + 6.00000i −0.229920 + 0.398234i −0.957784 0.287488i \(-0.907180\pi\)
0.727864 + 0.685722i \(0.240513\pi\)
\(228\) 0 0
\(229\) −11.0000 19.0526i −0.726900 1.25903i −0.958187 0.286143i \(-0.907627\pi\)
0.231287 0.972886i \(-0.425707\pi\)
\(230\) −20.7846 −1.37050
\(231\) 0 0
\(232\) 0 0
\(233\) −3.46410 6.00000i −0.226941 0.393073i 0.729959 0.683491i \(-0.239539\pi\)
−0.956900 + 0.290418i \(0.906206\pi\)
\(234\) 0 0
\(235\) 12.0000 20.7846i 0.782794 1.35584i
\(236\) −3.46410 6.00000i −0.225494 0.390567i
\(237\) 0 0
\(238\) 0 0
\(239\) 10.3923 0.672222 0.336111 0.941822i \(-0.390888\pi\)
0.336111 + 0.941822i \(0.390888\pi\)
\(240\) 0 0
\(241\) −5.00000 + 8.66025i −0.322078 + 0.557856i −0.980917 0.194429i \(-0.937715\pi\)
0.658838 + 0.752285i \(0.271048\pi\)
\(242\) −0.866025 + 1.50000i −0.0556702 + 0.0964237i
\(243\) 0 0
\(244\) 10.0000 0.640184
\(245\) 0 0
\(246\) 0 0
\(247\) 4.00000 + 6.92820i 0.254514 + 0.440831i
\(248\) 3.46410 6.00000i 0.219971 0.381000i
\(249\) 0 0
\(250\) −6.00000 10.3923i −0.379473 0.657267i
\(251\) −20.7846 −1.31191 −0.655956 0.754799i \(-0.727735\pi\)
−0.655956 + 0.754799i \(0.727735\pi\)
\(252\) 0 0
\(253\) −12.0000 −0.754434
\(254\) −6.92820 12.0000i −0.434714 0.752947i
\(255\) 0 0
\(256\) −9.50000 + 16.4545i −0.593750 + 1.02841i
\(257\) −1.73205 3.00000i −0.108042 0.187135i 0.806935 0.590641i \(-0.201125\pi\)
−0.914977 + 0.403506i \(0.867792\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −6.92820 −0.429669
\(261\) 0 0
\(262\) −12.0000 + 20.7846i −0.741362 + 1.28408i
\(263\) 8.66025 15.0000i 0.534014 0.924940i −0.465196 0.885208i \(-0.654016\pi\)
0.999210 0.0397320i \(-0.0126504\pi\)
\(264\) 0 0
\(265\) −24.0000 −1.47431
\(266\) 0 0
\(267\) 0 0
\(268\) 2.00000 + 3.46410i 0.122169 + 0.211604i
\(269\) −8.66025 + 15.0000i −0.528025 + 0.914566i 0.471441 + 0.881897i \(0.343734\pi\)
−0.999466 + 0.0326687i \(0.989599\pi\)
\(270\) 0 0
\(271\) 10.0000 + 17.3205i 0.607457 + 1.05215i 0.991658 + 0.128897i \(0.0411435\pi\)
−0.384201 + 0.923249i \(0.625523\pi\)
\(272\) 17.3205 1.05021
\(273\) 0 0
\(274\) −12.0000 −0.724947
\(275\) −12.1244 21.0000i −0.731126 1.26635i
\(276\) 0 0
\(277\) 5.00000 8.66025i 0.300421 0.520344i −0.675810 0.737075i \(-0.736206\pi\)
0.976231 + 0.216731i \(0.0695395\pi\)
\(278\) −13.8564 24.0000i −0.831052 1.43942i
\(279\) 0 0
\(280\) 0 0
\(281\) −20.7846 −1.23991 −0.619953 0.784639i \(-0.712848\pi\)
−0.619953 + 0.784639i \(0.712848\pi\)
\(282\) 0 0
\(283\) −2.00000 + 3.46410i −0.118888 + 0.205919i −0.919327 0.393494i \(-0.871266\pi\)
0.800439 + 0.599414i \(0.204600\pi\)
\(284\) 5.19615 9.00000i 0.308335 0.534052i
\(285\) 0 0
\(286\) −12.0000 −0.709575
\(287\) 0 0
\(288\) 0 0
\(289\) 2.50000 + 4.33013i 0.147059 + 0.254713i
\(290\) 0 0
\(291\) 0 0
\(292\) 7.00000 + 12.1244i 0.409644 + 0.709524i
\(293\) 10.3923 0.607125 0.303562 0.952812i \(-0.401824\pi\)
0.303562 + 0.952812i \(0.401824\pi\)
\(294\) 0 0
\(295\) 24.0000 1.39733
\(296\) 1.73205 + 3.00000i 0.100673 + 0.174371i
\(297\) 0 0
\(298\) −6.00000 + 10.3923i −0.347571 + 0.602010i
\(299\) −3.46410 6.00000i −0.200334 0.346989i
\(300\) 0 0
\(301\) 0 0
\(302\) 13.8564 0.797347
\(303\) 0 0
\(304\) 10.0000 17.3205i 0.573539 0.993399i
\(305\) −17.3205 + 30.0000i −0.991769 + 1.71780i
\(306\) 0 0
\(307\) 28.0000 1.59804 0.799022 0.601302i \(-0.205351\pi\)
0.799022 + 0.601302i \(0.205351\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −12.0000 20.7846i −0.681554 1.18049i
\(311\) 17.3205 30.0000i 0.982156 1.70114i 0.328206 0.944606i \(-0.393556\pi\)
0.653950 0.756538i \(-0.273111\pi\)
\(312\) 0 0
\(313\) 1.00000 + 1.73205i 0.0565233 + 0.0979013i 0.892903 0.450250i \(-0.148665\pi\)
−0.836379 + 0.548151i \(0.815332\pi\)
\(314\) 17.3205 0.977453
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) −3.46410 6.00000i −0.194563 0.336994i 0.752194 0.658942i \(-0.228996\pi\)
−0.946757 + 0.321948i \(0.895662\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −1.73205 3.00000i −0.0968246 0.167705i
\(321\) 0 0
\(322\) 0 0
\(323\) −13.8564 −0.770991
\(324\) 0 0
\(325\) 7.00000 12.1244i 0.388290 0.672538i
\(326\) −17.3205 + 30.0000i −0.959294 + 1.66155i
\(327\) 0 0
\(328\) 18.0000 0.993884
\(329\) 0 0
\(330\) 0 0
\(331\) −10.0000 17.3205i −0.549650 0.952021i −0.998298 0.0583130i \(-0.981428\pi\)
0.448649 0.893708i \(-0.351905\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) −18.0000 31.1769i −0.984916 1.70592i
\(335\) −13.8564 −0.757056
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 7.79423 + 13.5000i 0.423950 + 0.734303i
\(339\) 0 0
\(340\) 6.00000 10.3923i 0.325396 0.563602i
\(341\) −6.92820 12.0000i −0.375183 0.649836i
\(342\) 0 0
\(343\) 0 0
\(344\) 6.92820 0.373544
\(345\) 0 0
\(346\) 15.0000 25.9808i 0.806405 1.39673i
\(347\) 8.66025 15.0000i 0.464907 0.805242i −0.534291 0.845301i \(-0.679421\pi\)
0.999197 + 0.0400587i \(0.0127545\pi\)
\(348\) 0 0
\(349\) −14.0000 −0.749403 −0.374701 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 9.00000 + 15.5885i 0.479702 + 0.830868i
\(353\) 1.73205 3.00000i 0.0921878 0.159674i −0.816244 0.577708i \(-0.803947\pi\)
0.908431 + 0.418034i \(0.137281\pi\)
\(354\) 0 0
\(355\) 18.0000 + 31.1769i 0.955341 + 1.65470i
\(356\) 3.46410 0.183597
\(357\) 0 0
\(358\) −30.0000 −1.58555
\(359\) 12.1244 + 21.0000i 0.639899 + 1.10834i 0.985455 + 0.169939i \(0.0543572\pi\)
−0.345556 + 0.938398i \(0.612310\pi\)
\(360\) 0 0
\(361\) 1.50000 2.59808i 0.0789474 0.136741i
\(362\) 1.73205 + 3.00000i 0.0910346 + 0.157676i
\(363\) 0 0
\(364\) 0 0
\(365\) −48.4974 −2.53847
\(366\) 0 0
\(367\) −8.00000 + 13.8564i −0.417597 + 0.723299i −0.995697 0.0926670i \(-0.970461\pi\)
0.578101 + 0.815966i \(0.303794\pi\)
\(368\) −8.66025 + 15.0000i −0.451447 + 0.781929i
\(369\) 0 0
\(370\) 12.0000 0.623850
\(371\) 0 0
\(372\) 0 0
\(373\) 5.00000 + 8.66025i 0.258890 + 0.448411i 0.965945 0.258748i \(-0.0833099\pi\)
−0.707055 + 0.707159i \(0.749977\pi\)
\(374\) 10.3923 18.0000i 0.537373 0.930758i
\(375\) 0 0
\(376\) −6.00000 10.3923i −0.309426 0.535942i
\(377\) 0 0
\(378\) 0 0
\(379\) −28.0000 −1.43826 −0.719132 0.694874i \(-0.755460\pi\)
−0.719132 + 0.694874i \(0.755460\pi\)
\(380\) −6.92820 12.0000i −0.355409 0.615587i
\(381\) 0 0
\(382\) 21.0000 36.3731i 1.07445 1.86101i
\(383\) −6.92820 12.0000i −0.354015 0.613171i 0.632934 0.774206i \(-0.281850\pi\)
−0.986949 + 0.161034i \(0.948517\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 24.2487 1.23423
\(387\) 0 0
\(388\) 7.00000 12.1244i 0.355371 0.615521i
\(389\) −6.92820 + 12.0000i −0.351274 + 0.608424i −0.986473 0.163924i \(-0.947585\pi\)
0.635199 + 0.772348i \(0.280918\pi\)
\(390\) 0 0
\(391\) 12.0000 0.606866
\(392\) 0 0
\(393\) 0 0
\(394\) −18.0000 31.1769i −0.906827 1.57067i
\(395\) −13.8564 + 24.0000i −0.697191 + 1.20757i
\(396\) 0 0
\(397\) 19.0000 + 32.9090i 0.953583 + 1.65165i 0.737579 + 0.675261i \(0.235969\pi\)
0.216004 + 0.976392i \(0.430698\pi\)
\(398\) 27.7128 1.38912
\(399\) 0 0
\(400\) −35.0000 −1.75000
\(401\) −3.46410 6.00000i −0.172989 0.299626i 0.766475 0.642275i \(-0.222009\pi\)
−0.939463 + 0.342649i \(0.888676\pi\)
\(402\) 0 0
\(403\) 4.00000 6.92820i 0.199254 0.345118i
\(404\) 1.73205 + 3.00000i 0.0861727 + 0.149256i
\(405\) 0 0
\(406\) 0 0
\(407\) 6.92820 0.343418
\(408\) 0 0
\(409\) 7.00000 12.1244i 0.346128 0.599511i −0.639430 0.768849i \(-0.720830\pi\)
0.985558 + 0.169338i \(0.0541630\pi\)
\(410\) 31.1769 54.0000i 1.53972 2.66687i
\(411\) 0 0
\(412\) 4.00000 0.197066
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) −5.19615 + 9.00000i −0.254762 + 0.441261i
\(417\) 0 0
\(418\) −12.0000 20.7846i −0.586939 1.01661i
\(419\) −20.7846 −1.01539 −0.507697 0.861536i \(-0.669503\pi\)
−0.507697 + 0.861536i \(0.669503\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) −17.3205 30.0000i −0.843149 1.46038i
\(423\) 0 0
\(424\) −6.00000 + 10.3923i −0.291386 + 0.504695i
\(425\) 12.1244 + 21.0000i 0.588118 + 1.01865i
\(426\) 0 0
\(427\) 0 0
\(428\) 17.3205 0.837218
\(429\) 0 0
\(430\) 12.0000 20.7846i 0.578691 1.00232i
\(431\) −1.73205 + 3.00000i −0.0834300 + 0.144505i −0.904721 0.426004i \(-0.859921\pi\)
0.821291 + 0.570509i \(0.193254\pi\)
\(432\) 0 0
\(433\) −26.0000 −1.24948 −0.624740 0.780833i \(-0.714795\pi\)
−0.624740 + 0.780833i \(0.714795\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −1.00000 1.73205i −0.0478913 0.0829502i
\(437\) 6.92820 12.0000i 0.331421 0.574038i
\(438\) 0 0
\(439\) −8.00000 13.8564i −0.381819 0.661330i 0.609503 0.792784i \(-0.291369\pi\)
−0.991322 + 0.131453i \(0.958036\pi\)
\(440\) −20.7846 −0.990867
\(441\) 0 0
\(442\) 12.0000 0.570782
\(443\) 1.73205 + 3.00000i 0.0822922 + 0.142534i 0.904234 0.427037i \(-0.140443\pi\)
−0.821942 + 0.569571i \(0.807109\pi\)
\(444\) 0 0
\(445\) −6.00000 + 10.3923i −0.284427 + 0.492642i
\(446\) 6.92820 + 12.0000i 0.328060 + 0.568216i
\(447\) 0 0
\(448\) 0 0
\(449\) 41.5692 1.96177 0.980886 0.194581i \(-0.0623348\pi\)
0.980886 + 0.194581i \(0.0623348\pi\)
\(450\) 0 0
\(451\) 18.0000 31.1769i 0.847587 1.46806i
\(452\) 0 0
\(453\) 0 0
\(454\) 12.0000 0.563188
\(455\) 0 0
\(456\) 0 0
\(457\) 5.00000 + 8.66025i 0.233890 + 0.405110i 0.958950 0.283577i \(-0.0915211\pi\)
−0.725059 + 0.688686i \(0.758188\pi\)
\(458\) −19.0526 + 33.0000i −0.890268 + 1.54199i
\(459\) 0 0
\(460\) 6.00000 + 10.3923i 0.279751 + 0.484544i
\(461\) −31.1769 −1.45205 −0.726027 0.687666i \(-0.758635\pi\)
−0.726027 + 0.687666i \(0.758635\pi\)
\(462\) 0 0
\(463\) 32.0000 1.48717 0.743583 0.668644i \(-0.233125\pi\)
0.743583 + 0.668644i \(0.233125\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −6.00000 + 10.3923i −0.277945 + 0.481414i
\(467\) 3.46410 + 6.00000i 0.160300 + 0.277647i 0.934976 0.354711i \(-0.115421\pi\)
−0.774677 + 0.632358i \(0.782087\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −41.5692 −1.91745
\(471\) 0 0
\(472\) 6.00000 10.3923i 0.276172 0.478345i
\(473\) 6.92820 12.0000i 0.318559 0.551761i
\(474\) 0 0
\(475\) 28.0000 1.28473
\(476\) 0 0
\(477\) 0 0
\(478\) −9.00000 15.5885i −0.411650 0.712999i
\(479\) −3.46410 + 6.00000i −0.158279 + 0.274147i −0.934248 0.356624i \(-0.883928\pi\)
0.775969 + 0.630771i \(0.217261\pi\)
\(480\) 0 0
\(481\) 2.00000 + 3.46410i 0.0911922 + 0.157949i
\(482\) 17.3205 0.788928
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 24.2487 + 42.0000i 1.10108 + 1.90712i
\(486\) 0 0
\(487\) 20.0000 34.6410i 0.906287 1.56973i 0.0871056 0.996199i \(-0.472238\pi\)
0.819181 0.573535i \(-0.194428\pi\)
\(488\) 8.66025 + 15.0000i 0.392031 + 0.679018i
\(489\) 0 0
\(490\) 0 0
\(491\) 10.3923 0.468998 0.234499 0.972116i \(-0.424655\pi\)
0.234499 + 0.972116i \(0.424655\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 6.92820 12.0000i 0.311715 0.539906i
\(495\) 0 0
\(496\) −20.0000 −0.898027
\(497\) 0 0
\(498\) 0 0
\(499\) 2.00000 + 3.46410i 0.0895323 + 0.155074i 0.907314 0.420455i \(-0.138129\pi\)
−0.817781 + 0.575529i \(0.804796\pi\)
\(500\) −3.46410 + 6.00000i −0.154919 + 0.268328i
\(501\) 0 0
\(502\) 18.0000 + 31.1769i 0.803379 + 1.39149i
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) −12.0000 −0.533993
\(506\) 10.3923 + 18.0000i 0.461994 + 0.800198i
\(507\) 0 0
\(508\) −4.00000 + 6.92820i −0.177471 + 0.307389i
\(509\) −1.73205 3.00000i −0.0767718 0.132973i 0.825084 0.565011i \(-0.191128\pi\)
−0.901855 + 0.432038i \(0.857795\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 8.66025 0.382733
\(513\) 0 0
\(514\) −3.00000 + 5.19615i −0.132324 + 0.229192i
\(515\) −6.92820 + 12.0000i −0.305293 + 0.528783i
\(516\) 0 0
\(517\) −24.0000 −1.05552
\(518\) 0 0
\(519\) 0 0
\(520\) −6.00000 10.3923i −0.263117 0.455733i
\(521\) 1.73205 3.00000i 0.0758825 0.131432i −0.825587 0.564275i \(-0.809156\pi\)
0.901470 + 0.432842i \(0.142489\pi\)
\(522\) 0 0
\(523\) −8.00000 13.8564i −0.349816 0.605898i 0.636401 0.771358i \(-0.280422\pi\)
−0.986216 + 0.165460i \(0.947089\pi\)
\(524\) 13.8564 0.605320
\(525\) 0 0
\(526\) −30.0000 −1.30806
\(527\) 6.92820 + 12.0000i 0.301797 + 0.522728i
\(528\) 0 0
\(529\) 5.50000 9.52628i 0.239130 0.414186i
\(530\) 20.7846 + 36.0000i 0.902826 + 1.56374i
\(531\) 0 0
\(532\) 0 0
\(533\) 20.7846 0.900281
\(534\) 0 0
\(535\) −30.0000 + 51.9615i −1.29701 + 2.24649i
\(536\) −3.46410 + 6.00000i −0.149626 + 0.259161i
\(537\) 0 0
\(538\) 30.0000 1.29339
\(539\) 0 0
\(540\) 0 0
\(541\) −7.00000 12.1244i −0.300954 0.521267i 0.675399 0.737453i \(-0.263972\pi\)
−0.976352 + 0.216186i \(0.930638\pi\)
\(542\) 17.3205 30.0000i 0.743980 1.28861i
\(543\) 0 0
\(544\) −9.00000 15.5885i −0.385872 0.668350i
\(545\) 6.92820 0.296772
\(546\) 0 0
\(547\) −4.00000 −0.171028 −0.0855138 0.996337i \(-0.527253\pi\)
−0.0855138 + 0.996337i \(0.527253\pi\)
\(548\) 3.46410 + 6.00000i 0.147979 + 0.256307i
\(549\) 0 0
\(550\) −21.0000 + 36.3731i −0.895443 + 1.55095i
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) −17.3205 −0.735878
\(555\) 0 0
\(556\) −8.00000 + 13.8564i −0.339276 + 0.587643i
\(557\) 3.46410 6.00000i 0.146779 0.254228i −0.783256 0.621699i \(-0.786443\pi\)
0.930035 + 0.367471i \(0.119776\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) 18.0000 + 31.1769i 0.759284 + 1.31512i
\(563\) 17.3205 30.0000i 0.729972 1.26435i −0.226922 0.973913i \(-0.572866\pi\)
0.956894 0.290436i \(-0.0938004\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 6.92820 0.291214
\(567\) 0 0
\(568\) 18.0000 0.755263
\(569\) −3.46410 6.00000i −0.145223 0.251533i 0.784233 0.620466i \(-0.213057\pi\)
−0.929456 + 0.368933i \(0.879723\pi\)
\(570\) 0 0
\(571\) 14.0000 24.2487i 0.585882 1.01478i −0.408883 0.912587i \(-0.634082\pi\)
0.994765 0.102190i \(-0.0325850\pi\)
\(572\) 3.46410 + 6.00000i 0.144841 + 0.250873i
\(573\) 0 0
\(574\) 0 0
\(575\) −24.2487 −1.01124
\(576\) 0 0
\(577\) 1.00000 1.73205i 0.0416305 0.0721062i −0.844459 0.535620i \(-0.820078\pi\)
0.886090 + 0.463513i \(0.153411\pi\)
\(578\) 4.33013 7.50000i 0.180110 0.311959i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 12.0000 + 20.7846i 0.496989 + 0.860811i
\(584\) −12.1244 + 21.0000i −0.501709 + 0.868986i
\(585\) 0 0
\(586\) −9.00000 15.5885i −0.371787 0.643953i
\(587\) 20.7846 0.857873 0.428936 0.903335i \(-0.358888\pi\)
0.428936 + 0.903335i \(0.358888\pi\)
\(588\) 0 0
\(589\) 16.0000 0.659269
\(590\) −20.7846 36.0000i −0.855689 1.48210i
\(591\) 0 0
\(592\) 5.00000 8.66025i 0.205499 0.355934i
\(593\) −12.1244 21.0000i −0.497888 0.862367i 0.502109 0.864804i \(-0.332557\pi\)
−0.999997 + 0.00243746i \(0.999224\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 6.92820 0.283790
\(597\) 0 0
\(598\) −6.00000 + 10.3923i −0.245358 + 0.424973i
\(599\) −22.5167 + 39.0000i −0.920006 + 1.59350i −0.120603 + 0.992701i \(0.538483\pi\)
−0.799402 + 0.600796i \(0.794850\pi\)
\(600\) 0 0
\(601\) 22.0000 0.897399 0.448699 0.893683i \(-0.351887\pi\)
0.448699 + 0.893683i \(0.351887\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −4.00000 6.92820i −0.162758 0.281905i
\(605\) −1.73205 + 3.00000i −0.0704179 + 0.121967i
\(606\) 0 0
\(607\) 4.00000 + 6.92820i 0.162355 + 0.281207i 0.935713 0.352763i \(-0.114758\pi\)
−0.773358 + 0.633970i \(0.781424\pi\)
\(608\) −20.7846 −0.842927
\(609\) 0 0
\(610\) 60.0000 2.42933
\(611\) −6.92820 12.0000i −0.280285 0.485468i
\(612\) 0 0
\(613\) −19.0000 + 32.9090i −0.767403 + 1.32918i 0.171564 + 0.985173i \(0.445118\pi\)
−0.938967 + 0.344008i \(0.888215\pi\)
\(614\) −24.2487 42.0000i −0.978598 1.69498i
\(615\) 0 0
\(616\) 0 0
\(617\) −20.7846 −0.836757 −0.418378 0.908273i \(-0.637401\pi\)
−0.418378 + 0.908273i \(0.637401\pi\)
\(618\) 0 0
\(619\) 4.00000 6.92820i 0.160774 0.278468i −0.774373 0.632730i \(-0.781934\pi\)
0.935146 + 0.354262i \(0.115268\pi\)
\(620\) −6.92820 + 12.0000i −0.278243 + 0.481932i
\(621\) 0 0
\(622\) −60.0000 −2.40578
\(623\) 0 0
\(624\) 0 0
\(625\) 5.50000 + 9.52628i 0.220000 + 0.381051i
\(626\) 1.73205 3.00000i 0.0692267 0.119904i
\(627\) 0 0
\(628\) −5.00000 8.66025i −0.199522 0.345582i
\(629\) −6.92820 −0.276246
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 6.92820 + 12.0000i 0.275589 + 0.477334i
\(633\) 0 0
\(634\) −6.00000 + 10.3923i −0.238290 + 0.412731i
\(635\) −13.8564 24.0000i −0.549875 0.952411i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) −21.0000 + 36.3731i −0.830098 + 1.43777i
\(641\) 24.2487 42.0000i 0.957767 1.65890i 0.229860 0.973224i \(-0.426173\pi\)
0.727906 0.685677i \(-0.240494\pi\)
\(642\) 0 0
\(643\) −20.0000 −0.788723 −0.394362 0.918955i \(-0.629034\pi\)
−0.394362 + 0.918955i \(0.629034\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 12.0000 + 20.7846i 0.472134 + 0.817760i
\(647\) −3.46410 + 6.00000i −0.136188 + 0.235884i −0.926051 0.377399i \(-0.876818\pi\)
0.789863 + 0.613284i \(0.210152\pi\)
\(648\) 0 0
\(649\) −12.0000 20.7846i −0.471041 0.815867i
\(650\) −24.2487 −0.951113
\(651\) 0 0
\(652\) 20.0000 0.783260
\(653\) −13.8564 24.0000i −0.542243 0.939193i −0.998775 0.0494855i \(-0.984242\pi\)
0.456532 0.889707i \(-0.349091\pi\)
\(654\) 0 0
\(655\) −24.0000 + 41.5692i −0.937758 + 1.62424i
\(656\) −25.9808 45.0000i −1.01438 1.75695i
\(657\) 0 0
\(658\) 0 0
\(659\) −10.3923 −0.404827 −0.202413 0.979300i \(-0.564878\pi\)
−0.202413 + 0.979300i \(0.564878\pi\)
\(660\) 0 0
\(661\) −5.00000 + 8.66025i −0.194477 + 0.336845i −0.946729 0.322031i \(-0.895634\pi\)
0.752252 + 0.658876i \(0.228968\pi\)
\(662\) −17.3205 + 30.0000i −0.673181 + 1.16598i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −10.3923 + 18.0000i −0.402090 + 0.696441i
\(669\) 0 0
\(670\) 12.0000 + 20.7846i 0.463600 + 0.802980i
\(671\) 34.6410 1.33730
\(672\) 0 0
\(673\) −10.0000 −0.385472 −0.192736 0.981251i \(-0.561736\pi\)
−0.192736 + 0.981251i \(0.561736\pi\)
\(674\) −12.1244 21.0000i −0.467013 0.808890i
\(675\) 0 0
\(676\) 4.50000 7.79423i 0.173077 0.299778i
\(677\) −12.1244 21.0000i −0.465977 0.807096i 0.533268 0.845946i \(-0.320964\pi\)
−0.999245 + 0.0388507i \(0.987630\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 20.7846 0.797053
\(681\) 0 0
\(682\) −12.0000 + 20.7846i −0.459504 + 0.795884i
\(683\) −12.1244 + 21.0000i −0.463926 + 0.803543i −0.999152 0.0411658i \(-0.986893\pi\)
0.535227 + 0.844708i \(0.320226\pi\)
\(684\) 0 0
\(685\) −24.0000 −0.916993
\(686\) 0 0
\(687\) 0 0
\(688\) −10.0000 17.3205i −0.381246 0.660338i
\(689\) −6.92820 + 12.0000i −0.263944 + 0.457164i
\(690\) 0 0
\(691\) 4.00000 + 6.92820i 0.152167 + 0.263561i 0.932024 0.362397i \(-0.118041\pi\)
−0.779857 + 0.625958i \(0.784708\pi\)
\(692\) −17.3205 −0.658427
\(693\) 0 0
\(694\) −30.0000 −1.13878
\(695\) −27.7128 48.0000i −1.05121 1.82074i
\(696\) 0 0
\(697\) −18.0000 + 31.1769i −0.681799 + 1.18091i
\(698\) 12.1244 + 21.0000i 0.458914 + 0.794862i
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) −4.00000 + 6.92820i −0.150863 + 0.261302i
\(704\) −1.73205 + 3.00000i −0.0652791 + 0.113067i
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) 0 0
\(708\) 0 0
\(709\) −13.0000 22.5167i −0.488225 0.845631i 0.511683 0.859174i \(-0.329022\pi\)
−0.999908 + 0.0135434i \(0.995689\pi\)
\(710\) 31.1769 54.0000i 1.17005 2.02658i
\(711\) 0 0
\(712\) 3.00000 + 5.19615i 0.112430 + 0.194734i
\(713\) −13.8564 −0.518927
\(714\) 0 0
\(715\) −24.0000 −0.897549
\(716\) 8.66025 + 15.0000i 0.323649 + 0.560576i
\(717\) 0 0
\(718\) 21.0000 36.3731i 0.783713 1.35743i
\(719\) 13.8564 + 24.0000i 0.516757 + 0.895049i 0.999811 + 0.0194584i \(0.00619418\pi\)
−0.483054 + 0.875591i \(0.660472\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −5.19615 −0.193381
\(723\) 0 0
\(724\) 1.00000 1.73205i 0.0371647 0.0643712i
\(725\) 0 0
\(726\) 0 0
\(727\) 4.00000 0.148352 0.0741759 0.997245i \(-0.476367\pi\)
0.0741759 + 0.997245i \(0.476367\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 42.0000 + 72.7461i 1.55449 + 2.69246i
\(731\) −6.92820 + 12.0000i −0.256249 + 0.443836i
\(732\) 0 0
\(733\) −11.0000 19.0526i −0.406294 0.703722i 0.588177 0.808732i \(-0.299846\pi\)
−0.994471 + 0.105010i \(0.966513\pi\)
\(734\) 27.7128 1.02290
\(735\) 0 0
\(736\) 18.0000 0.663489
\(737\) 6.92820 + 12.0000i 0.255204 + 0.442026i
\(738\) 0 0
\(739\) −10.0000 + 17.3205i −0.367856 + 0.637145i −0.989230 0.146369i \(-0.953241\pi\)
0.621374 + 0.783514i \(0.286575\pi\)
\(740\) −3.46410 6.00000i −0.127343 0.220564i
\(741\) 0 0
\(742\) 0 0
\(743\) −10.3923 −0.381257 −0.190628 0.981662i \(-0.561053\pi\)
−0.190628 + 0.981662i \(0.561053\pi\)
\(744\) 0 0
\(745\) −12.0000 + 20.7846i −0.439646 + 0.761489i
\(746\) 8.66025 15.0000i 0.317074 0.549189i
\(747\) 0 0
\(748\) −12.0000 −0.438763
\(749\) 0 0
\(750\) 0 0
\(751\) −4.00000 6.92820i −0.145962 0.252814i 0.783769 0.621052i \(-0.213294\pi\)
−0.929731 + 0.368238i \(0.879961\pi\)
\(752\) −17.3205 + 30.0000i −0.631614 + 1.09399i
\(753\) 0 0
\(754\) 0 0
\(755\) 27.7128 1.00857
\(756\) 0 0
\(757\) 26.0000 0.944986 0.472493 0.881334i \(-0.343354\pi\)
0.472493 + 0.881334i \(0.343354\pi\)
\(758\) 24.2487 + 42.0000i 0.880753 + 1.52551i
\(759\) 0 0
\(760\) 12.0000 20.7846i 0.435286 0.753937i
\(761\) 19.0526 + 33.0000i 0.690655 + 1.19625i 0.971624 + 0.236532i \(0.0760109\pi\)
−0.280969 + 0.959717i \(0.590656\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −24.2487 −0.877288
\(765\) 0 0
\(766\) −12.0000 + 20.7846i −0.433578 + 0.750978i
\(767\) 6.92820 12.0000i 0.250163 0.433295i
\(768\) 0 0
\(769\) 22.0000 0.793340 0.396670 0.917961i \(-0.370166\pi\)
0.396670 + 0.917961i \(0.370166\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −7.00000 12.1244i −0.251936 0.436365i
\(773\) 22.5167 39.0000i 0.809868 1.40273i −0.103087 0.994672i \(-0.532872\pi\)
0.912955 0.408060i \(-0.133795\pi\)
\(774\) 0 0
\(775\) −14.0000 24.2487i −0.502895 0.871039i
\(776\) 24.2487 0.870478
\(777\) 0 0
\(778\) 24.0000 0.860442
\(779\) 20.7846 + 36.0000i 0.744686 + 1.28983i
\(780\) 0 0
\(781\) 18.0000 31.1769i 0.644091 1.11560i
\(782\) −10.3923 18.0000i −0.371628 0.643679i
\(783\) 0 0
\(784\) 0 0
\(785\) 34.6410 1.23639
\(786\) 0 0
\(787\) 16.0000 27.7128i 0.570338 0.987855i −0.426193 0.904632i \(-0.640145\pi\)
0.996531 0.0832226i \(-0.0265213\pi\)
\(788\) −10.3923 + 18.0000i −0.370211 + 0.641223i
\(789\) 0 0
\(790\) 48.0000 1.70776
\(791\) 0 0
\(792\) 0 0
\(793\) 10.0000 + 17.3205i 0.355110 + 0.615069i
\(794\) 32.9090 57.0000i 1.16790 2.02285i
\(795\) 0 0
\(796\) −8.00000 13.8564i −0.283552 0.491127i
\(797\) −31.1769 −1.10434 −0.552171 0.833731i \(-0.686201\pi\)
−0.552171 + 0.833731i \(0.686201\pi\)
\(798\) 0 0
\(799\) 24.0000 0.849059
\(800\) 18.1865 + 31.5000i 0.642991 + 1.11369i
\(801\) 0 0
\(802\) −6.00000 + 10.3923i −0.211867 + 0.366965i
\(803\) 24.2487 + 42.0000i 0.855718 + 1.48215i
\(804\) 0 0
\(805\) 0 0
\(806\) −13.8564 −0.488071
\(807\) 0 0
\(808\) −3.00000 + 5.19615i −0.105540 + 0.182800i
\(809\) 13.8564 24.0000i 0.487165 0.843795i −0.512726 0.858552i \(-0.671364\pi\)
0.999891 + 0.0147574i \(0.00469758\pi\)
\(810\) 0 0
\(811\) 16.0000 0.561836 0.280918 0.959732i \(-0.409361\pi\)
0.280918 + 0.959732i \(0.409361\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −6.00000 10.3923i −0.210300 0.364250i
\(815\) −34.6410 + 60.0000i −1.21342 + 2.10171i
\(816\) 0 0
\(817\) 8.00000 + 13.8564i 0.279885 + 0.484774i
\(818\) −24.2487 −0.847836
\(819\) 0 0
\(820\) −36.0000 −1.25717
\(821\) −3.46410 6.00000i −0.120898 0.209401i 0.799224 0.601033i \(-0.205244\pi\)
−0.920122 + 0.391632i \(0.871911\pi\)
\(822\) 0 0
\(823\) −16.0000 + 27.7128i −0.557725 + 0.966008i 0.439961 + 0.898017i \(0.354992\pi\)
−0.997686 + 0.0679910i \(0.978341\pi\)
\(824\) 3.46410 + 6.00000i 0.120678 + 0.209020i
\(825\) 0 0
\(826\) 0 0
\(827\) −10.3923 −0.361376 −0.180688 0.983540i \(-0.557832\pi\)
−0.180688 + 0.983540i \(0.557832\pi\)
\(828\) 0 0
\(829\) 25.0000 43.3013i 0.868286 1.50392i 0.00453881 0.999990i \(-0.498555\pi\)
0.863747 0.503926i \(-0.168111\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −2.00000 −0.0693375
\(833\) 0 0
\(834\) 0 0
\(835\) −36.0000 62.3538i −1.24583 2.15784i
\(836\) −6.92820 + 12.0000i −0.239617 + 0.415029i
\(837\) 0 0
\(838\) 18.0000 + 31.1769i 0.621800 + 1.07699i
\(839\) −20.7846 −0.717564 −0.358782 0.933421i \(-0.616808\pi\)
−0.358782 + 0.933421i \(0.616808\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 8.66025 + 15.0000i 0.298452 + 0.516934i
\(843\) 0 0
\(844\) −10.0000 + 17.3205i −0.344214 + 0.596196i
\(845\) 15.5885 + 27.0000i 0.536259 + 0.928828i
\(846\) 0 0
\(847\) 0 0
\(848\) 34.6410 1.18958
\(849\) 0 0
\(850\) 21.0000 36.3731i 0.720294 1.24759i
\(851\) 3.46410 6.00000i 0.118748 0.205677i
\(852\) 0 0
\(853\) 10.0000 0.342393 0.171197 0.985237i \(-0.445237\pi\)
0.171197 + 0.985237i \(0.445237\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 15.0000 + 25.9808i 0.512689 + 0.888004i
\(857\) −8.66025 + 15.0000i −0.295829 + 0.512390i −0.975177 0.221425i \(-0.928929\pi\)
0.679349 + 0.733816i \(0.262262\pi\)
\(858\) 0 0
\(859\) −2.00000 3.46410i −0.0682391 0.118194i 0.829887 0.557931i \(-0.188405\pi\)
−0.898126 + 0.439738i \(0.855071\pi\)
\(860\) −13.8564 −0.472500
\(861\) 0 0
\(862\) 6.00000 0.204361
\(863\) −19.0526 33.0000i −0.648557 1.12333i −0.983468 0.181083i \(-0.942040\pi\)
0.334911 0.942250i \(-0.391294\pi\)
\(864\) 0 0
\(865\) 30.0000 51.9615i 1.02003 1.76674i
\(866\) 22.5167 + 39.0000i 0.765147 + 1.32527i
\(867\) 0 0
\(868\) 0 0
\(869\) 27.7128 0.940093
\(870\) 0 0
\(871\) −4.00000 + 6.92820i −0.135535 + 0.234753i
\(872\) 1.73205 3.00000i 0.0586546 0.101593i
\(873\) 0 0
\(874\) −24.0000 −0.811812
\(875\) 0 0
\(876\) 0 0
\(877\) −7.00000 12.1244i −0.236373 0.409410i 0.723298 0.690536i \(-0.242625\pi\)
−0.959671 + 0.281126i \(0.909292\pi\)
\(878\) −13.8564 + 24.0000i −0.467631 + 0.809961i
\(879\) 0 0
\(880\) 30.0000 + 51.9615i 1.01130 + 1.75162i
\(881\) 51.9615 1.75063 0.875314 0.483555i \(-0.160655\pi\)
0.875314 + 0.483555i \(0.160655\pi\)
\(882\) 0 0
\(883\) −52.0000 −1.74994 −0.874970 0.484178i \(-0.839119\pi\)
−0.874970 + 0.484178i \(0.839119\pi\)
\(884\) −3.46410 6.00000i −0.116510 0.201802i
\(885\) 0 0
\(886\) 3.00000 5.19615i 0.100787 0.174568i
\(887\) 3.46410 + 6.00000i 0.116313 + 0.201460i 0.918304 0.395876i \(-0.129559\pi\)
−0.801991 + 0.597336i \(0.796226\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 20.7846 0.696702
\(891\) 0 0
\(892\) 4.00000 6.92820i 0.133930 0.231973i
\(893\) 13.8564 24.0000i 0.463687 0.803129i
\(894\) 0 0
\(895\) −60.0000 −2.00558
\(896\) 0 0
\(897\) 0 0
\(898\) −36.0000 62.3538i −1.20134 2.08077i
\(899\) 0 0
\(900\) 0 0
\(901\) −12.0000 20.7846i −0.399778 0.692436i
\(902\) −62.3538 −2.07616
\(903\) 0 0
\(904\) 0 0
\(905\) 3.46410 + 6.00000i 0.115151 + 0.199447i
\(906\) 0 0
\(907\) 14.0000 24.2487i 0.464862 0.805165i −0.534333 0.845274i \(-0.679437\pi\)
0.999195 + 0.0401089i \(0.0127705\pi\)
\(908\) −3.46410 6.00000i −0.114960 0.199117i
\(909\) 0 0
\(910\) 0 0
\(911\) 31.1769 1.03294 0.516469 0.856306i \(-0.327246\pi\)
0.516469 + 0.856306i \(0.327246\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 8.66025 15.0000i 0.286456 0.496156i
\(915\) 0 0
\(916\) 22.0000 0.726900
\(917\) 0 0
\(918\) 0 0
\(919\) 8.00000 + 13.8564i 0.263896 + 0.457081i 0.967274 0.253735i \(-0.0816592\pi\)
−0.703378 + 0.710816i \(0.748326\pi\)
\(920\) −10.3923 + 18.0000i −0.342624 + 0.593442i
\(921\) 0 0
\(922\) 27.0000 + 46.7654i 0.889198 + 1.54014i
\(923\) 20.7846 0.684134
\(924\) 0 0
\(925\) 14.0000 0.460317
\(926\) −27.7128 48.0000i −0.910700 1.57738i
\(927\) 0 0
\(928\) 0 0
\(929\) −22.5167 39.0000i −0.738748 1.27955i −0.953059 0.302783i \(-0.902084\pi\)
0.214312 0.976765i \(-0.431249\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 6.92820 0.226941
\(933\) 0 0
\(934\) 6.00000 10.3923i 0.196326 0.340047i
\(935\) 20.7846 36.0000i 0.679729 1.17733i
\(936\) 0 0
\(937\) 46.0000 1.50275 0.751377 0.659873i \(-0.229390\pi\)
0.751377 + 0.659873i \(0.229390\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 12.0000 + 20.7846i 0.391397 + 0.677919i
\(941\) −8.66025 + 15.0000i −0.282316 + 0.488986i −0.971955 0.235168i \(-0.924436\pi\)
0.689639 + 0.724154i \(0.257769\pi\)
\(942\) 0 0
\(943\) −18.0000 31.1769i −0.586161 1.01526i
\(944\) −34.6410 −1.12747
\(945\) 0 0
\(946\) −24.0000 −0.780307
\(947\) 12.1244 + 21.0000i 0.393989 + 0.682408i 0.992972 0.118354i \(-0.0377616\pi\)
−0.598983 + 0.800762i \(0.704428\pi\)
\(948\) 0 0
\(949\) −14.0000 + 24.2487i −0.454459 + 0.787146i
\(950\) −24.2487 42.0000i −0.786732 1.36266i
\(951\) 0 0
\(952\) 0 0
\(953\) −41.5692 −1.34656 −0.673280 0.739388i \(-0.735115\pi\)
−0.673280 + 0.739388i \(0.735115\pi\)
\(954\) 0 0
\(955\) 42.0000 72.7461i 1.35909 2.35401i
\(956\) −5.19615 + 9.00000i −0.168056 + 0.291081i
\(957\) 0 0
\(958\) 12.0000 0.387702
\(959\) 0 0
\(960\) 0 0
\(961\) 7.50000 + 12.9904i 0.241935 + 0.419045i
\(962\) 3.46410 6.00000i 0.111687 0.193448i
\(963\) 0 0
\(964\) −5.00000 8.66025i −0.161039 0.278928i
\(965\) 48.4974 1.56119
\(966\) 0 0
\(967\) −16.0000 −0.514525 −0.257263 0.966342i \(-0.582821\pi\)
−0.257263 + 0.966342i \(0.582821\pi\)
\(968\) 0.866025 + 1.50000i 0.0278351 + 0.0482118i
\(969\) 0 0
\(970\) 42.0000 72.7461i 1.34854 2.33574i
\(971\) 13.8564 + 24.0000i 0.444673 + 0.770197i 0.998029 0.0627481i \(-0.0199865\pi\)
−0.553356 + 0.832945i \(0.686653\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −69.2820 −2.21994
\(975\) 0 0
\(976\) 25.0000 43.3013i 0.800230 1.38604i
\(977\) −17.3205 + 30.0000i −0.554132 + 0.959785i 0.443838 + 0.896107i \(0.353616\pi\)
−0.997970 + 0.0636782i \(0.979717\pi\)
\(978\) 0 0
\(979\) 12.0000 0.383522
\(980\) 0 0
\(981\) 0 0
\(982\) −9.00000 15.5885i −0.287202 0.497448i
\(983\) 6.92820 12.0000i 0.220975 0.382741i −0.734129 0.679010i \(-0.762409\pi\)
0.955104 + 0.296269i \(0.0957426\pi\)
\(984\) 0 0
\(985\) −36.0000 62.3538i −1.14706 1.98676i
\(986\) 0 0
\(987\) 0 0
\(988\) −8.00000 −0.254514
\(989\) −6.92820 12.0000i −0.220304 0.381578i
\(990\) 0 0
\(991\) −28.0000 + 48.4974i −0.889449 + 1.54057i −0.0489218 + 0.998803i \(0.515578\pi\)
−0.840528 + 0.541769i \(0.817755\pi\)
\(992\) 10.3923 + 18.0000i 0.329956 + 0.571501i
\(993\) 0 0
\(994\) 0 0
\(995\) 55.4256 1.75711
\(996\) 0 0
\(997\) −5.00000 + 8.66025i −0.158352 + 0.274273i −0.934274 0.356555i \(-0.883951\pi\)
0.775923 + 0.630828i \(0.217285\pi\)
\(998\) 3.46410 6.00000i 0.109654 0.189927i
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 441.2.e.i.361.1 4
3.2 odd 2 inner 441.2.e.i.361.2 4
7.2 even 3 inner 441.2.e.i.226.1 4
7.3 odd 6 63.2.a.b.1.2 yes 2
7.4 even 3 441.2.a.g.1.2 2
7.5 odd 6 441.2.e.j.226.1 4
7.6 odd 2 441.2.e.j.361.1 4
21.2 odd 6 inner 441.2.e.i.226.2 4
21.5 even 6 441.2.e.j.226.2 4
21.11 odd 6 441.2.a.g.1.1 2
21.17 even 6 63.2.a.b.1.1 2
21.20 even 2 441.2.e.j.361.2 4
28.3 even 6 1008.2.a.n.1.1 2
28.11 odd 6 7056.2.a.cm.1.2 2
35.3 even 12 1575.2.d.i.1324.1 4
35.17 even 12 1575.2.d.i.1324.4 4
35.24 odd 6 1575.2.a.q.1.1 2
56.3 even 6 4032.2.a.bq.1.2 2
56.45 odd 6 4032.2.a.bt.1.2 2
63.31 odd 6 567.2.f.j.379.1 4
63.38 even 6 567.2.f.j.190.2 4
63.52 odd 6 567.2.f.j.190.1 4
63.59 even 6 567.2.f.j.379.2 4
77.10 even 6 7623.2.a.bi.1.1 2
84.11 even 6 7056.2.a.cm.1.1 2
84.59 odd 6 1008.2.a.n.1.2 2
105.17 odd 12 1575.2.d.i.1324.2 4
105.38 odd 12 1575.2.d.i.1324.3 4
105.59 even 6 1575.2.a.q.1.2 2
168.59 odd 6 4032.2.a.bq.1.1 2
168.101 even 6 4032.2.a.bt.1.1 2
231.164 odd 6 7623.2.a.bi.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.a.b.1.1 2 21.17 even 6
63.2.a.b.1.2 yes 2 7.3 odd 6
441.2.a.g.1.1 2 21.11 odd 6
441.2.a.g.1.2 2 7.4 even 3
441.2.e.i.226.1 4 7.2 even 3 inner
441.2.e.i.226.2 4 21.2 odd 6 inner
441.2.e.i.361.1 4 1.1 even 1 trivial
441.2.e.i.361.2 4 3.2 odd 2 inner
441.2.e.j.226.1 4 7.5 odd 6
441.2.e.j.226.2 4 21.5 even 6
441.2.e.j.361.1 4 7.6 odd 2
441.2.e.j.361.2 4 21.20 even 2
567.2.f.j.190.1 4 63.52 odd 6
567.2.f.j.190.2 4 63.38 even 6
567.2.f.j.379.1 4 63.31 odd 6
567.2.f.j.379.2 4 63.59 even 6
1008.2.a.n.1.1 2 28.3 even 6
1008.2.a.n.1.2 2 84.59 odd 6
1575.2.a.q.1.1 2 35.24 odd 6
1575.2.a.q.1.2 2 105.59 even 6
1575.2.d.i.1324.1 4 35.3 even 12
1575.2.d.i.1324.2 4 105.17 odd 12
1575.2.d.i.1324.3 4 105.38 odd 12
1575.2.d.i.1324.4 4 35.17 even 12
4032.2.a.bq.1.1 2 168.59 odd 6
4032.2.a.bq.1.2 2 56.3 even 6
4032.2.a.bt.1.1 2 168.101 even 6
4032.2.a.bt.1.2 2 56.45 odd 6
7056.2.a.cm.1.1 2 84.11 even 6
7056.2.a.cm.1.2 2 28.11 odd 6
7623.2.a.bi.1.1 2 77.10 even 6
7623.2.a.bi.1.2 2 231.164 odd 6