Properties

Label 1575.901
Modulus $1575$
Conductor $7$
Order $6$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1575, base_ring=CyclotomicField(6))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,0,5]))
 
pari: [g,chi] = znchar(Mod(901,1575))
 

Basic properties

Modulus: \(1575\)
Conductor: \(7\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(6\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{7}(5,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1575.x

\(\chi_{1575}(451,\cdot)\) \(\chi_{1575}(901,\cdot)\)

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: \(\Q(\sqrt{-3}) \)
Fixed field: \(\Q(\zeta_{7})\)

Values on generators

\((1226,127,451)\) → \((1,1,e\left(\frac{5}{6}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(8\)\(11\)\(13\)\(16\)\(17\)\(19\)\(22\)\(23\)
\( \chi_{ 1575 }(901, a) \) \(-1\)\(1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(1\)\(e\left(\frac{1}{3}\right)\)\(-1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{1}{6}\right)\)\(1\)\(e\left(\frac{2}{3}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1575 }(901,a) \;\) at \(\;a = \) e.g. 2