Properties

Label 3724.237
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([0,45,56]))
 
pari: [g,chi] = znchar(Mod(237,3724))
 

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 \(\chi_{931}(237,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3724.eq

\(\chi_{3724}(237,\cdot)\) \(\chi_{3724}(321,\cdot)\) \(\chi_{3724}(377,\cdot)\) \(\chi_{3724}(405,\cdot)\) \(\chi_{3724}(461,\cdot)\) \(\chi_{3724}(517,\cdot)\) \(\chi_{3724}(769,\cdot)\) \(\chi_{3724}(853,\cdot)\) \(\chi_{3724}(909,\cdot)\) \(\chi_{3724}(937,\cdot)\) \(\chi_{3724}(993,\cdot)\) \(\chi_{3724}(1049,\cdot)\) \(\chi_{3724}(1301,\cdot)\) \(\chi_{3724}(1385,\cdot)\) \(\chi_{3724}(1441,\cdot)\) \(\chi_{3724}(1525,\cdot)\) \(\chi_{3724}(1581,\cdot)\) \(\chi_{3724}(1833,\cdot)\) \(\chi_{3724}(1917,\cdot)\) \(\chi_{3724}(1973,\cdot)\) \(\chi_{3724}(2001,\cdot)\) \(\chi_{3724}(2113,\cdot)\) \(\chi_{3724}(2365,\cdot)\) \(\chi_{3724}(2505,\cdot)\) \(\chi_{3724}(2533,\cdot)\) \(\chi_{3724}(2589,\cdot)\) \(\chi_{3724}(2897,\cdot)\) \(\chi_{3724}(2981,\cdot)\) \(\chi_{3724}(3065,\cdot)\) \(\chi_{3724}(3121,\cdot)\) ...

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

\((1863,3041,3137)\) → \((1,e\left(\frac{5}{14}\right),e\left(\frac{4}{9}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(23\)\(25\)\(27\)
\(-1\)\(1\)\(e\left(\frac{17}{126}\right)\)\(e\left(\frac{59}{126}\right)\)\(e\left(\frac{17}{63}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{1}{126}\right)\)\(e\left(\frac{38}{63}\right)\)\(e\left(\frac{47}{126}\right)\)\(e\left(\frac{29}{63}\right)\)\(e\left(\frac{59}{63}\right)\)\(e\left(\frac{17}{42}\right)\)
value at e.g. 2