Properties

Label 3724.125
Modulus $3724$
Conductor $931$
Order $42$
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(42))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,3,28]))
 
pari: [g,chi] = znchar(Mod(125,3724))
 

Basic properties

Modulus: \(3724\)
Conductor: \(931\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{931}(125,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3724.dd

\(\chi_{3724}(125,\cdot)\) \(\chi_{3724}(349,\cdot)\) \(\chi_{3724}(657,\cdot)\) \(\chi_{3724}(1189,\cdot)\) \(\chi_{3724}(1413,\cdot)\) \(\chi_{3724}(1721,\cdot)\) \(\chi_{3724}(1945,\cdot)\) \(\chi_{3724}(2477,\cdot)\) \(\chi_{3724}(2785,\cdot)\) \(\chi_{3724}(3009,\cdot)\) \(\chi_{3724}(3317,\cdot)\) \(\chi_{3724}(3541,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: 42.0.1539762816644858660747420157759729772758206099865060266674536067216088093085649093762782333896247463.1

Values on generators

\((1863,3041,3137)\) → \((1,e\left(\frac{1}{14}\right),e\left(\frac{2}{3}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(23\)\(25\)\(27\)
\(-1\)\(1\)\(e\left(\frac{31}{42}\right)\)\(e\left(\frac{31}{42}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{29}{42}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{19}{42}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{3}{14}\right)\)
value at e.g. 2