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Properties

Label 7315.4108
Modulus $7315$
Conductor $7315$
Order $180$
Real no
Primitive yes
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(7315, base_ring=CyclotomicField(180)) M = H._module chi = DirichletCharacter(H, M([135,90,72,20]))
 
Copy content gp:[g,chi] = znchar(Mod(4108, 7315))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7315.4108");
 

Basic properties

Modulus: \(7315\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(7315\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(180\)
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: yes
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 7315.od

\(\chi_{7315}(328,\cdot)\) \(\chi_{7315}(377,\cdot)\) \(\chi_{7315}(587,\cdot)\) \(\chi_{7315}(643,\cdot)\) \(\chi_{7315}(993,\cdot)\) \(\chi_{7315}(1182,\cdot)\) \(\chi_{7315}(1203,\cdot)\) \(\chi_{7315}(1252,\cdot)\) \(\chi_{7315}(1567,\cdot)\) \(\chi_{7315}(1868,\cdot)\) \(\chi_{7315}(1917,\cdot)\) \(\chi_{7315}(1973,\cdot)\) \(\chi_{7315}(2183,\cdot)\) \(\chi_{7315}(2512,\cdot)\) \(\chi_{7315}(2533,\cdot)\) \(\chi_{7315}(2638,\cdot)\) \(\chi_{7315}(2722,\cdot)\) \(\chi_{7315}(2897,\cdot)\) \(\chi_{7315}(3177,\cdot)\) \(\chi_{7315}(3303,\cdot)\) \(\chi_{7315}(3513,\cdot)\) \(\chi_{7315}(3562,\cdot)\) \(\chi_{7315}(3842,\cdot)\) \(\chi_{7315}(4052,\cdot)\) \(\chi_{7315}(4108,\cdot)\) \(\chi_{7315}(4178,\cdot)\) \(\chi_{7315}(4227,\cdot)\) \(\chi_{7315}(4262,\cdot)\) \(\chi_{7315}(4493,\cdot)\) \(\chi_{7315}(4717,\cdot)\) ...

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: $\Q(\zeta_{180})$
Fixed field: Number field defined by a degree 180 polynomial (not computed)

Values on generators

\((2927,1046,5986,1541)\) → \((-i,-1,e\left(\frac{2}{5}\right),e\left(\frac{1}{9}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(12\)\(13\)\(16\)\(17\)
\( \chi_{ 7315 }(4108, a) \) \(1\)\(1\)\(e\left(\frac{47}{180}\right)\)\(e\left(\frac{71}{180}\right)\)\(e\left(\frac{47}{90}\right)\)\(e\left(\frac{59}{90}\right)\)\(e\left(\frac{47}{60}\right)\)\(e\left(\frac{71}{90}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{127}{180}\right)\)\(e\left(\frac{2}{45}\right)\)\(e\left(\frac{173}{180}\right)\)
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)
 
\( \chi_{ 7315 }(4108,a) \;\) at \(\;a = \) e.g. 2