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Properties

Label 9300.2807
Modulus $9300$
Conductor $1860$
Order $60$
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(9300, base_ring=CyclotomicField(60)) M = H._module chi = DirichletCharacter(H, M([30,30,15,14]))
 
Copy content gp:[g,chi] = znchar(Mod(2807, 9300))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9300.2807");
 

Basic properties

Modulus: \(9300\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1860\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(60\)
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 \(\chi_{1860}(947,\cdot)\)
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 9300.lt

\(\chi_{9300}(1943,\cdot)\) \(\chi_{9300}(2243,\cdot)\) \(\chi_{9300}(2807,\cdot)\) \(\chi_{9300}(2843,\cdot)\) \(\chi_{9300}(3143,\cdot)\) \(\chi_{9300}(4043,\cdot)\) \(\chi_{9300}(4343,\cdot)\) \(\chi_{9300}(4643,\cdot)\) \(\chi_{9300}(6407,\cdot)\) \(\chi_{9300}(6707,\cdot)\) \(\chi_{9300}(7307,\cdot)\) \(\chi_{9300}(7607,\cdot)\) \(\chi_{9300}(7643,\cdot)\) \(\chi_{9300}(8507,\cdot)\) \(\chi_{9300}(8807,\cdot)\) \(\chi_{9300}(9107,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((4651,3101,2977,1801)\) → \((-1,-1,i,e\left(\frac{7}{30}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(37\)\(41\)\(43\)
\( \chi_{ 9300 }(2807, a) \) \(1\)\(1\)\(e\left(\frac{17}{60}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{19}{60}\right)\)\(e\left(\frac{23}{60}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{41}{60}\right)\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x)
 
Copy content gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
 
Copy content magma:chi(x)
 
\( \chi_{ 9300 }(2807,a) \;\) at \(\;a = \) e.g. 2