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#Locate the two files
load('constructor.py');load('EHMMv3.sage')
#Define an EHMM class
HD=EHMM([1/3,2/3,1/2],[1,1,1]) #A length 3 example
print('First 8 terms of the hypergeometric series with datum', HD.alpha, HD.beta)
show(Hyp(HD.alpha,HD.beta,q,8))
p=5;
HD.coeff_p_order()
First 8 terms of the hypergeometric series with datum [1/3, 2/3, 1/2] [1, 1, 1]
\(\displaystyle 1 + \frac{1}{9} q + \frac{5}{108} q^{2} + \frac{175}{6561} q^{3} + \frac{67375}{3779136} q^{4} + \frac{49049}{3779136} q^{5} + \frac{9172163}{918330048} q^{6} + \frac{33025135}{4132485216} q^{7}\)
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In [2]:
print('parameter t first 10 q-coefficients')
print(HD.tq().O(10))
print('q-expansion using EHMM')
show(HD.q_expansion().O(40))
parameter t first 10 q-coefficients 27*q - 405*q^2 + 4617*q^3 - 45333*q^4 + 406458*q^5 - 3428487*q^6 + 27673704*q^7 - 216059157*q^8 + 1643453955*q^9 + O(q^10) q-expansion using EHMM
\(\displaystyle q - 3 q^{3} + 2 q^{7} + 9 q^{9} - 22 q^{13} + 26 q^{19} - 6 q^{21} + 25 q^{25} - 27 q^{27} - 46 q^{31} + 26 q^{37} + 66 q^{39} + O(q^{40})\)
In [3]:
HD4=EHMM([1/2,1/2,1/2,1/2],[1,1,1,1]) #Another length-4 example
print ('Datum=',HD4.alpha,HD4.beta)
### Sample functions
print ("t's q-expansion")
print (HD4.tq().O(6))
print (HD4.q_expansion().O(10))
print ("First few coefficients of $f_{8.4.a.a}$")
print (Newforms(8,4)[0])
Datum= [1/2, 1/2, 1/2, 1/2] [1, 1, 1, 1]
t's q-expansion
-64*q - 1536*q^2 - 19200*q^3 - 167936*q^4 - 1160064*q^5 + O(q^6)
q - 4*q^3 - 2*q^5 + 24*q^7 - 11*q^9 + O(q^10)
First few coefficients of $f_{8.4.a.a}$
q - 4*q^3 - 2*q^5 + O(q^6)
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### K2(j/8,1) where j=1,3,5,7###
K26K=EHMM([1/2,1/2,1/8],[1,1,1]) #Rosen's paper [arXiv:2412.07054], Class-6 K-data
print (table(K26K.conjugate()))
[1/2, 1/2, 1/8] [1, 1, 1] [1/2, 1/2, 3/8] [1, 1, 1] [1/2, 1/2, 5/8] [1, 1, 1] [1/2, 1/2, 7/8] [1, 1, 1]
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K26K.q_conjugate_exp()
Out[5]:
| \(q - 3 q^{9} - 6 q^{17} + O(q^{20})\) | \(q^{3} - q^{11} - 7 q^{19} + O(q^{20})\) | \(q^{5} + q^{13} + O(q^{20})\) | \(q^{7} + 3 q^{15} + O(q^{20})\) |
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K26K.conjugate_zigzag()
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K26K.normalized_Hp()
Out[10]:
| p | Normalized Hp |
| \(17\) | \(-6\) |
| \(41\) | \(-66\) |
| \(73\) | \(-58\) |
| \(89\) | \(102\) |
| \(97\) | \(26\) |
| \(113\) | \(66\) |
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