Properties

Label 2736.1157
Modulus $2736$
Conductor $2736$
Order $36$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2736, base_ring=CyclotomicField(36))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,9,30,20]))
 
pari: [g,chi] = znchar(Mod(1157,2736))
 

Basic properties

Modulus: \(2736\)
Conductor: \(2736\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
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 2736.he

\(\chi_{2736}(5,\cdot)\) \(\chi_{2736}(101,\cdot)\) \(\chi_{2736}(149,\cdot)\) \(\chi_{2736}(389,\cdot)\) \(\chi_{2736}(821,\cdot)\) \(\chi_{2736}(1157,\cdot)\) \(\chi_{2736}(1373,\cdot)\) \(\chi_{2736}(1469,\cdot)\) \(\chi_{2736}(1517,\cdot)\) \(\chi_{2736}(1757,\cdot)\) \(\chi_{2736}(2189,\cdot)\) \(\chi_{2736}(2525,\cdot)\)

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: \(\Q(\zeta_{36})\)
Fixed field: Number field defined by a degree 36 polynomial

Values on generators

\((1711,2053,1217,1009)\) → \((1,i,e\left(\frac{5}{6}\right),e\left(\frac{5}{9}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 2736 }(1157, a) \) \(-1\)\(1\)\(e\left(\frac{11}{36}\right)\)\(e\left(\frac{1}{6}\right)\)\(-i\)\(e\left(\frac{7}{36}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{1}{36}\right)\)\(1\)\(e\left(\frac{17}{36}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2736 }(1157,a) \;\) at \(\;a = \) e.g. 2