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Properties

Label 7191.728
Modulus $7191$
Conductor $2397$
Order $368$
Real no
Primitive no
Minimal yes
Parity odd

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(7191, base_ring=CyclotomicField(368)) M = H._module chi = DirichletCharacter(H, M([184,207,40]))
 
Copy content gp:[g,chi] = znchar(Mod(728, 7191))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7191.728");
 

Basic properties

Modulus: \(7191\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2397\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(368\)
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 \(\chi_{2397}(728,\cdot)\)
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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 7191.cq

\(\chi_{7191}(44,\cdot)\) \(\chi_{7191}(62,\cdot)\) \(\chi_{7191}(80,\cdot)\) \(\chi_{7191}(107,\cdot)\) \(\chi_{7191}(116,\cdot)\) \(\chi_{7191}(125,\cdot)\) \(\chi_{7191}(233,\cdot)\) \(\chi_{7191}(278,\cdot)\) \(\chi_{7191}(368,\cdot)\) \(\chi_{7191}(386,\cdot)\) \(\chi_{7191}(449,\cdot)\) \(\chi_{7191}(503,\cdot)\) \(\chi_{7191}(530,\cdot)\) \(\chi_{7191}(539,\cdot)\) \(\chi_{7191}(575,\cdot)\) \(\chi_{7191}(584,\cdot)\) \(\chi_{7191}(602,\cdot)\) \(\chi_{7191}(656,\cdot)\) \(\chi_{7191}(728,\cdot)\) \(\chi_{7191}(809,\cdot)\) \(\chi_{7191}(872,\cdot)\) \(\chi_{7191}(881,\cdot)\) \(\chi_{7191}(890,\cdot)\) \(\chi_{7191}(908,\cdot)\) \(\chi_{7191}(962,\cdot)\) \(\chi_{7191}(980,\cdot)\) \(\chi_{7191}(998,\cdot)\) \(\chi_{7191}(1025,\cdot)\) \(\chi_{7191}(1133,\cdot)\) \(\chi_{7191}(1151,\cdot)\) ...

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: $\Q(\zeta_{368})$
Fixed field: Number field defined by a degree 368 polynomial (not computed)

Values on generators

\((3197,6769,2449)\) → \((-1,e\left(\frac{9}{16}\right),e\left(\frac{5}{46}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 7191 }(728, a) \) \(-1\)\(1\)\(e\left(\frac{61}{184}\right)\)\(e\left(\frac{61}{92}\right)\)\(e\left(\frac{155}{368}\right)\)\(e\left(\frac{245}{368}\right)\)\(e\left(\frac{183}{184}\right)\)\(e\left(\frac{277}{368}\right)\)\(e\left(\frac{73}{368}\right)\)\(e\left(\frac{41}{92}\right)\)\(e\left(\frac{367}{368}\right)\)\(e\left(\frac{15}{46}\right)\)
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)
 
\( \chi_{ 7191 }(728,a) \;\) at \(\;a = \) e.g. 2