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Properties

Label 1152.797
Modulus $1152$
Conductor $1152$
Order $96$
Real no
Primitive yes
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(1152, base_ring=CyclotomicField(96)) M = H._module chi = DirichletCharacter(H, M([0,81,80]))
 
Copy content gp:[g,chi] = znchar(Mod(797, 1152))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1152.797");
 

Basic properties

Modulus: \(1152\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1152\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(96\)
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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 1152.bt

\(\chi_{1152}(5,\cdot)\) \(\chi_{1152}(29,\cdot)\) \(\chi_{1152}(77,\cdot)\) \(\chi_{1152}(101,\cdot)\) \(\chi_{1152}(149,\cdot)\) \(\chi_{1152}(173,\cdot)\) \(\chi_{1152}(221,\cdot)\) \(\chi_{1152}(245,\cdot)\) \(\chi_{1152}(293,\cdot)\) \(\chi_{1152}(317,\cdot)\) \(\chi_{1152}(365,\cdot)\) \(\chi_{1152}(389,\cdot)\) \(\chi_{1152}(437,\cdot)\) \(\chi_{1152}(461,\cdot)\) \(\chi_{1152}(509,\cdot)\) \(\chi_{1152}(533,\cdot)\) \(\chi_{1152}(581,\cdot)\) \(\chi_{1152}(605,\cdot)\) \(\chi_{1152}(653,\cdot)\) \(\chi_{1152}(677,\cdot)\) \(\chi_{1152}(725,\cdot)\) \(\chi_{1152}(749,\cdot)\) \(\chi_{1152}(797,\cdot)\) \(\chi_{1152}(821,\cdot)\) \(\chi_{1152}(869,\cdot)\) \(\chi_{1152}(893,\cdot)\) \(\chi_{1152}(941,\cdot)\) \(\chi_{1152}(965,\cdot)\) \(\chi_{1152}(1013,\cdot)\) \(\chi_{1152}(1037,\cdot)\) ...

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: $\Q(\zeta_{96})$
Fixed field: Number field defined by a degree 96 polynomial

Values on generators

\((127,901,641)\) → \((1,e\left(\frac{27}{32}\right),e\left(\frac{5}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 1152 }(797, a) \) \(-1\)\(1\)\(e\left(\frac{1}{96}\right)\)\(e\left(\frac{37}{48}\right)\)\(e\left(\frac{53}{96}\right)\)\(e\left(\frac{31}{96}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{13}{32}\right)\)\(e\left(\frac{47}{48}\right)\)\(e\left(\frac{1}{48}\right)\)\(e\left(\frac{59}{96}\right)\)\(e\left(\frac{5}{12}\right)\)
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)
 
\( \chi_{ 1152 }(797,a) \;\) at \(\;a = \) e.g. 2