Properties

Label 1375.533
Modulus $1375$
Conductor $1375$
Order $100$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1375, base_ring=CyclotomicField(100))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([83,40]))
 
pari: [g,chi] = znchar(Mod(533,1375))
 

Basic properties

Modulus: \(1375\)
Conductor: \(1375\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(100\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1375.cs

\(\chi_{1375}(37,\cdot)\) \(\chi_{1375}(53,\cdot)\) \(\chi_{1375}(97,\cdot)\) \(\chi_{1375}(102,\cdot)\) \(\chi_{1375}(192,\cdot)\) \(\chi_{1375}(213,\cdot)\) \(\chi_{1375}(223,\cdot)\) \(\chi_{1375}(258,\cdot)\) \(\chi_{1375}(312,\cdot)\) \(\chi_{1375}(328,\cdot)\) \(\chi_{1375}(372,\cdot)\) \(\chi_{1375}(377,\cdot)\) \(\chi_{1375}(467,\cdot)\) \(\chi_{1375}(488,\cdot)\) \(\chi_{1375}(498,\cdot)\) \(\chi_{1375}(533,\cdot)\) \(\chi_{1375}(587,\cdot)\) \(\chi_{1375}(603,\cdot)\) \(\chi_{1375}(647,\cdot)\) \(\chi_{1375}(652,\cdot)\) \(\chi_{1375}(742,\cdot)\) \(\chi_{1375}(763,\cdot)\) \(\chi_{1375}(773,\cdot)\) \(\chi_{1375}(808,\cdot)\) \(\chi_{1375}(862,\cdot)\) \(\chi_{1375}(878,\cdot)\) \(\chi_{1375}(922,\cdot)\) \(\chi_{1375}(927,\cdot)\) \(\chi_{1375}(1017,\cdot)\) \(\chi_{1375}(1038,\cdot)\) ...

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: $\Q(\zeta_{100})$
Fixed field: Number field defined by a degree 100 polynomial

Values on generators

\((1002,376)\) → \((e\left(\frac{83}{100}\right),e\left(\frac{2}{5}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\(-1\)\(1\)\(e\left(\frac{23}{100}\right)\)\(e\left(\frac{1}{100}\right)\)\(e\left(\frac{23}{50}\right)\)\(e\left(\frac{6}{25}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{69}{100}\right)\)\(e\left(\frac{1}{50}\right)\)\(e\left(\frac{47}{100}\right)\)\(e\left(\frac{77}{100}\right)\)\(e\left(\frac{29}{50}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1375 }(533,a) \;\) at \(\;a = \) e.g. 2