Properties

Label 3724.17
Modulus $3724$
Conductor $931$
Order $126$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(3724, base_ring=CyclotomicField(126))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,75,70]))
 
pari: [g,chi] = znchar(Mod(17,3724))
 

Basic properties

Modulus: \(3724\)
Conductor: \(931\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(126\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{931}(17,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3724.ep

\(\chi_{3724}(17,\cdot)\) \(\chi_{3724}(61,\cdot)\) \(\chi_{3724}(73,\cdot)\) \(\chi_{3724}(157,\cdot)\) \(\chi_{3724}(481,\cdot)\) \(\chi_{3724}(549,\cdot)\) \(\chi_{3724}(593,\cdot)\) \(\chi_{3724}(605,\cdot)\) \(\chi_{3724}(689,\cdot)\) \(\chi_{3724}(845,\cdot)\) \(\chi_{3724}(1013,\cdot)\) \(\chi_{3724}(1081,\cdot)\) \(\chi_{3724}(1125,\cdot)\) \(\chi_{3724}(1137,\cdot)\) \(\chi_{3724}(1221,\cdot)\) \(\chi_{3724}(1377,\cdot)\) \(\chi_{3724}(1545,\cdot)\) \(\chi_{3724}(1613,\cdot)\) \(\chi_{3724}(1657,\cdot)\) \(\chi_{3724}(1669,\cdot)\) \(\chi_{3724}(1753,\cdot)\) \(\chi_{3724}(1909,\cdot)\) \(\chi_{3724}(2145,\cdot)\) \(\chi_{3724}(2189,\cdot)\) \(\chi_{3724}(2201,\cdot)\) \(\chi_{3724}(2441,\cdot)\) \(\chi_{3724}(2609,\cdot)\) \(\chi_{3724}(2721,\cdot)\) \(\chi_{3724}(2733,\cdot)\) \(\chi_{3724}(2817,\cdot)\) ...

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

Related number fields

Field of values: $\Q(\zeta_{63})$
Fixed field: Number field defined by a degree 126 polynomial (not computed)

Values on generators

\((1863,3041,3137)\) → \((1,e\left(\frac{25}{42}\right),e\left(\frac{5}{9}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(23\)\(25\)\(27\)
\(-1\)\(1\)\(e\left(\frac{103}{126}\right)\)\(e\left(\frac{19}{126}\right)\)\(e\left(\frac{40}{63}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{53}{126}\right)\)\(e\left(\frac{61}{63}\right)\)\(e\left(\frac{55}{126}\right)\)\(e\left(\frac{46}{63}\right)\)\(e\left(\frac{19}{63}\right)\)\(e\left(\frac{19}{42}\right)\)
value at e.g. 2