Properties

Label 2736.307
Modulus $2736$
Conductor $304$
Order $36$
Real no
Primitive no
Minimal yes
Parity even

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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([18,27,0,26]))
 
pari: [g,chi] = znchar(Mod(307,2736))
 

Basic properties

Modulus: \(2736\)
Conductor: \(304\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{304}(3,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2736.gt

\(\chi_{2736}(91,\cdot)\) \(\chi_{2736}(307,\cdot)\) \(\chi_{2736}(451,\cdot)\) \(\chi_{2736}(523,\cdot)\) \(\chi_{2736}(667,\cdot)\) \(\chi_{2736}(811,\cdot)\) \(\chi_{2736}(1459,\cdot)\) \(\chi_{2736}(1675,\cdot)\) \(\chi_{2736}(1819,\cdot)\) \(\chi_{2736}(1891,\cdot)\) \(\chi_{2736}(2035,\cdot)\) \(\chi_{2736}(2179,\cdot)\)

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: 36.36.19036714782161565107424425435655777110146017378670996611401194085493506048.1

Values on generators

\((1711,2053,1217,1009)\) → \((-1,-i,1,e\left(\frac{13}{18}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 2736 }(307, a) \) \(1\)\(1\)\(e\left(\frac{11}{36}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{31}{36}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{19}{36}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{23}{36}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2736 }(307,a) \;\) at \(\;a = \) e.g. 2