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Properties

Label 1870.1109
Modulus $1870$
Conductor $935$
Order $20$
Real no
Primitive no
Minimal yes
Parity even

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1870, base_ring=CyclotomicField(20)) M = H._module chi = DirichletCharacter(H, M([10,12,15]))
 
Copy content pari:[g,chi] = znchar(Mod(1109,1870))
 

Basic properties

Modulus: \(1870\)
Conductor: \(935\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(20\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{935}(174,\cdot)\)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 1870.cb

\(\chi_{1870}(489,\cdot)\) \(\chi_{1870}(599,\cdot)\) \(\chi_{1870}(829,\cdot)\) \(\chi_{1870}(939,\cdot)\) \(\chi_{1870}(999,\cdot)\) \(\chi_{1870}(1109,\cdot)\) \(\chi_{1870}(1169,\cdot)\) \(\chi_{1870}(1279,\cdot)\)

Copy content sage:chi.galois_orbit()
 
Copy content pari:order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{20})\)
Fixed field: Number field defined by a degree 20 polynomial

Values on generators

\((1497,1531,1431)\) → \((-1,e\left(\frac{3}{5}\right),-i)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(13\)\(19\)\(21\)\(23\)\(27\)\(29\)\(31\)
\( \chi_{ 1870 }(1109, a) \) \(1\)\(1\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{3}{10}\right)\)\(1\)\(-i\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{7}{20}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 1870 }(1109,a) \;\) at \(\;a = \) e.g. 2