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Properties

Label 1832.13
Modulus $1832$
Conductor $1832$
Order $76$
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(1832, base_ring=CyclotomicField(76)) M = H._module chi = DirichletCharacter(H, M([0,38,39]))
 
Copy content gp:[g,chi] = znchar(Mod(13, 1832))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1832.13");
 

Basic properties

Modulus: \(1832\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1832\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(76\)
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 1832.bj

\(\chi_{1832}(13,\cdot)\) \(\chi_{1832}(21,\cdot)\) \(\chi_{1832}(93,\cdot)\) \(\chi_{1832}(101,\cdot)\) \(\chi_{1832}(109,\cdot)\) \(\chi_{1832}(141,\cdot)\) \(\chi_{1832}(197,\cdot)\) \(\chi_{1832}(221,\cdot)\) \(\chi_{1832}(237,\cdot)\) \(\chi_{1832}(261,\cdot)\) \(\chi_{1832}(317,\cdot)\) \(\chi_{1832}(349,\cdot)\) \(\chi_{1832}(357,\cdot)\) \(\chi_{1832}(365,\cdot)\) \(\chi_{1832}(437,\cdot)\) \(\chi_{1832}(445,\cdot)\) \(\chi_{1832}(573,\cdot)\) \(\chi_{1832}(581,\cdot)\) \(\chi_{1832}(653,\cdot)\) \(\chi_{1832}(685,\cdot)\) \(\chi_{1832}(709,\cdot)\) \(\chi_{1832}(717,\cdot)\) \(\chi_{1832}(741,\cdot)\) \(\chi_{1832}(773,\cdot)\) \(\chi_{1832}(1061,\cdot)\) \(\chi_{1832}(1093,\cdot)\) \(\chi_{1832}(1197,\cdot)\) \(\chi_{1832}(1229,\cdot)\) \(\chi_{1832}(1517,\cdot)\) \(\chi_{1832}(1549,\cdot)\) ...

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: $\Q(\zeta_{76})$
Fixed field: Number field defined by a degree 76 polynomial

Values on generators

\((1375,917,1609)\) → \((1,-1,e\left(\frac{39}{76}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 1832 }(13, a) \) \(-1\)\(1\)\(e\left(\frac{9}{38}\right)\)\(e\left(\frac{15}{19}\right)\)\(e\left(\frac{69}{76}\right)\)\(e\left(\frac{9}{19}\right)\)\(e\left(\frac{12}{19}\right)\)\(e\left(\frac{41}{76}\right)\)\(e\left(\frac{1}{38}\right)\)\(e\left(\frac{15}{19}\right)\)\(e\left(\frac{27}{38}\right)\)\(e\left(\frac{11}{76}\right)\)
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)
 
\( \chi_{ 1832 }(13,a) \;\) at \(\;a = \) e.g. 2