Properties

Label 3724.109
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([0,120,49]))
 
pari: [g,chi] = znchar(Mod(109,3724))
 

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 \(\chi_{931}(109,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3724.eo

\(\chi_{3724}(109,\cdot)\) \(\chi_{3724}(193,\cdot)\) \(\chi_{3724}(205,\cdot)\) \(\chi_{3724}(249,\cdot)\) \(\chi_{3724}(317,\cdot)\) \(\chi_{3724}(485,\cdot)\) \(\chi_{3724}(641,\cdot)\) \(\chi_{3724}(725,\cdot)\) \(\chi_{3724}(737,\cdot)\) \(\chi_{3724}(781,\cdot)\) \(\chi_{3724}(849,\cdot)\) \(\chi_{3724}(1017,\cdot)\) \(\chi_{3724}(1173,\cdot)\) \(\chi_{3724}(1257,\cdot)\) \(\chi_{3724}(1269,\cdot)\) \(\chi_{3724}(1313,\cdot)\) \(\chi_{3724}(1381,\cdot)\) \(\chi_{3724}(1705,\cdot)\) \(\chi_{3724}(1789,\cdot)\) \(\chi_{3724}(1801,\cdot)\) \(\chi_{3724}(1845,\cdot)\) \(\chi_{3724}(1913,\cdot)\) \(\chi_{3724}(2081,\cdot)\) \(\chi_{3724}(2237,\cdot)\) \(\chi_{3724}(2377,\cdot)\) \(\chi_{3724}(2445,\cdot)\) \(\chi_{3724}(2613,\cdot)\) \(\chi_{3724}(2769,\cdot)\) \(\chi_{3724}(2853,\cdot)\) \(\chi_{3724}(2865,\cdot)\) ...

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

\((1863,3041,3137)\) → \((1,e\left(\frac{20}{21}\right),e\left(\frac{7}{18}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(23\)\(25\)\(27\)
\(-1\)\(1\)\(e\left(\frac{1}{126}\right)\)\(e\left(\frac{53}{63}\right)\)\(e\left(\frac{1}{63}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{47}{126}\right)\)\(e\left(\frac{107}{126}\right)\)\(e\left(\frac{44}{63}\right)\)\(e\left(\frac{61}{63}\right)\)\(e\left(\frac{43}{63}\right)\)\(e\left(\frac{1}{42}\right)\)
value at e.g. 2