Properties

Label 3969.703
Modulus $3969$
Conductor $441$
Order $42$
Real no
Primitive no
Minimal no
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(3969, base_ring=CyclotomicField(42))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([28,25]))
 
pari: [g,chi] = znchar(Mod(703,3969))
 

Basic properties

Modulus: \(3969\)
Conductor: \(441\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{441}(115,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3969.br

\(\chi_{3969}(136,\cdot)\) \(\chi_{3969}(271,\cdot)\) \(\chi_{3969}(703,\cdot)\) \(\chi_{3969}(838,\cdot)\) \(\chi_{3969}(1270,\cdot)\) \(\chi_{3969}(1405,\cdot)\) \(\chi_{3969}(1837,\cdot)\) \(\chi_{3969}(1972,\cdot)\) \(\chi_{3969}(2404,\cdot)\) \(\chi_{3969}(2539,\cdot)\) \(\chi_{3969}(3538,\cdot)\) \(\chi_{3969}(3673,\cdot)\)

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: 42.0.61849948934846323740928964041516234392013738413062346563659921389600804608476019954673203847.1

Values on generators

\((2108,3727)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{25}{42}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 3969 }(703, a) \) \(-1\)\(1\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{25}{42}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{31}{42}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{41}{42}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{37}{42}\right)\)\(e\left(\frac{5}{6}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3969 }(703,a) \;\) at \(\;a = \) e.g. 2