Properties

Label 3724.3109
Modulus $3724$
Conductor $931$
Order $42$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Learn more

Show commands: 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([0,24,35]))
 
pari: [g,chi] = znchar(Mod(3109,3724))
 

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 \(\chi_{931}(316,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3724.dh

\(\chi_{3724}(141,\cdot)\) \(\chi_{3724}(449,\cdot)\) \(\chi_{3724}(673,\cdot)\) \(\chi_{3724}(1205,\cdot)\) \(\chi_{3724}(1513,\cdot)\) \(\chi_{3724}(1737,\cdot)\) \(\chi_{3724}(2045,\cdot)\) \(\chi_{3724}(2269,\cdot)\) \(\chi_{3724}(2577,\cdot)\) \(\chi_{3724}(2801,\cdot)\) \(\chi_{3724}(3109,\cdot)\) \(\chi_{3724}(3641,\cdot)\)

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: 42.0.4012683575982151476651625353295463545613564177862358851055893007057333972430504464680793872531537508239499.1

Values on generators

\((1863,3041,3137)\) → \((1,e\left(\frac{4}{7}\right),e\left(\frac{5}{6}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(23\)\(25\)\(27\)
\(-1\)\(1\)\(e\left(\frac{17}{42}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{13}{42}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{3}{14}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3724 }(3109,a) \;\) at \(\;a = \) e.g. 2