Properties

Label 1728.77
Modulus $1728$
Conductor $1728$
Order $144$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1728, base_ring=CyclotomicField(144))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,135,88]))
 
pari: [g,chi] = znchar(Mod(77,1728))
 

Basic properties

Modulus: \(1728\)
Conductor: \(1728\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(144\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1728.cf

\(\chi_{1728}(5,\cdot)\) \(\chi_{1728}(29,\cdot)\) \(\chi_{1728}(77,\cdot)\) \(\chi_{1728}(101,\cdot)\) \(\chi_{1728}(149,\cdot)\) \(\chi_{1728}(173,\cdot)\) \(\chi_{1728}(221,\cdot)\) \(\chi_{1728}(245,\cdot)\) \(\chi_{1728}(293,\cdot)\) \(\chi_{1728}(317,\cdot)\) \(\chi_{1728}(365,\cdot)\) \(\chi_{1728}(389,\cdot)\) \(\chi_{1728}(437,\cdot)\) \(\chi_{1728}(461,\cdot)\) \(\chi_{1728}(509,\cdot)\) \(\chi_{1728}(533,\cdot)\) \(\chi_{1728}(581,\cdot)\) \(\chi_{1728}(605,\cdot)\) \(\chi_{1728}(653,\cdot)\) \(\chi_{1728}(677,\cdot)\) \(\chi_{1728}(725,\cdot)\) \(\chi_{1728}(749,\cdot)\) \(\chi_{1728}(797,\cdot)\) \(\chi_{1728}(821,\cdot)\) \(\chi_{1728}(869,\cdot)\) \(\chi_{1728}(893,\cdot)\) \(\chi_{1728}(941,\cdot)\) \(\chi_{1728}(965,\cdot)\) \(\chi_{1728}(1013,\cdot)\) \(\chi_{1728}(1037,\cdot)\) ...

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: $\Q(\zeta_{144})$
Fixed field: Number field defined by a degree 144 polynomial (not computed)

Values on generators

\((703,325,1217)\) → \((1,e\left(\frac{15}{16}\right),e\left(\frac{11}{18}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 1728 }(77, a) \) \(-1\)\(1\)\(e\left(\frac{143}{144}\right)\)\(e\left(\frac{11}{72}\right)\)\(e\left(\frac{91}{144}\right)\)\(e\left(\frac{137}{144}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{43}{48}\right)\)\(e\left(\frac{61}{72}\right)\)\(e\left(\frac{71}{72}\right)\)\(e\left(\frac{133}{144}\right)\)\(e\left(\frac{13}{18}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1728 }(77,a) \;\) at \(\;a = \) e.g. 2