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Properties

Label 1850.1621
Modulus $1850$
Conductor $925$
Order $90$
Real no
Primitive no
Minimal yes
Parity even

Related objects

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1850, base_ring=CyclotomicField(90)) M = H._module chi = DirichletCharacter(H, M([54,35]))
 
Copy content gp:[g,chi] = znchar(Mod(1621, 1850))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1850.1621");
 

Basic properties

Modulus: \(1850\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(925\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(90\)
Copy content comment:Order
 
Copy content sage:chi.multiplicative_order()
 
Copy content gp:charorder(g,chi)
 
Copy content magma:Order(chi);
 
Real: no
Copy content comment:Whether the character is real
 
Copy content sage:chi.multiplicative_order() <= 2
 
Copy content gp:charorder(g,chi) <= 2
 
Copy content magma:Order(chi) le 2;
 
Primitive: no, induced from \(\chi_{925}(696,\cdot)\)
Copy content comment:If the character is primitive
 
Copy content sage:chi.is_primitive()
 
Copy content gp:#znconreyconductor(g,chi)==1
 
Copy content magma:IsPrimitive(chi);
 
Minimal: yes
Parity: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 1850.ca

\(\chi_{1850}(21,\cdot)\) \(\chi_{1850}(41,\cdot)\) \(\chi_{1850}(141,\cdot)\) \(\chi_{1850}(321,\cdot)\) \(\chi_{1850}(361,\cdot)\) \(\chi_{1850}(391,\cdot)\) \(\chi_{1850}(411,\cdot)\) \(\chi_{1850}(511,\cdot)\) \(\chi_{1850}(521,\cdot)\) \(\chi_{1850}(691,\cdot)\) \(\chi_{1850}(731,\cdot)\) \(\chi_{1850}(761,\cdot)\) \(\chi_{1850}(781,\cdot)\) \(\chi_{1850}(881,\cdot)\) \(\chi_{1850}(891,\cdot)\) \(\chi_{1850}(1061,\cdot)\) \(\chi_{1850}(1131,\cdot)\) \(\chi_{1850}(1261,\cdot)\) \(\chi_{1850}(1431,\cdot)\) \(\chi_{1850}(1471,\cdot)\) \(\chi_{1850}(1521,\cdot)\) \(\chi_{1850}(1621,\cdot)\) \(\chi_{1850}(1631,\cdot)\) \(\chi_{1850}(1841,\cdot)\)

Copy content comment:Galois orbit
 
Copy content sage:chi.galois_orbit()
 
Copy content gp:order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 
Copy content magma:order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
 

Related number fields

Field of values: $\Q(\zeta_{45})$
Copy content comment:Field of values of chi
 
Copy content sage:CyclotomicField(chi.multiplicative_order())
 
Copy content gp:nfinit(polcyclo(charorder(g,chi)))
 
Copy content magma:CyclotomicField(Order(chi));
 
Fixed field: Number field defined by a degree 90 polynomial
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((1777,1001)\) → \((e\left(\frac{3}{5}\right),e\left(\frac{7}{18}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 1850 }(1621, a) \) \(1\)\(1\)\(e\left(\frac{14}{45}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{28}{45}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{61}{90}\right)\)\(e\left(\frac{47}{90}\right)\)\(e\left(\frac{37}{90}\right)\)\(e\left(\frac{34}{45}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{14}{15}\right)\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x) # x integer
 
Copy content gp:chareval(g,chi,x) \\ x integer, value in Q/Z
 
Copy content magma:chi(x)
 
\( \chi_{ 1850 }(1621,a) \;\) at \(\;a = \) e.g. 2